ilyas thesis

QUANTUM DOTS IN PHOTONIC CRYSTALS: FROM

QUANTUM INFORMATION PROCESSING TO SINGLE PHOTON

NONLINEAR OPTICS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Ilya Fushman

December 2008

UMI Number: 3343582

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(Stephen E. Harris)

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hi

Abstract

Photons are attractive candidates for both quantum and classical information pro

cessing where they act as essential carriers of information and can greatly reduce the

operating power, respectively. Efficient photon routing and switching devices are re

quired for both applications, and necessitate the development of optical nonlinearities

that work at single photon-single emitter levels.

This work presents experimental and theoretical efforts toward the realization of

nonlinear optical devices that operate at low photon numbers and reach single photon

levels that are suitable for quantum information processing. We show that a single

quantum dot coupled to a photonic crystal cavity can be used to realize a controlled

phase gate between photons. In addition, this work also describes attempts at im

proving the operation of all-optical switches and modulators with the use of photonic

bandgap devices where the density of photon states is modified and tight photon con

finement leads to enhanced field strengths. We show that the combination of cavities

with standard nonlinearities can be used to realize fast optoelectronic modulators and

switches. Finally we review efforts to combine this photonic technology with novel

light emitters that operate at room temperature and have the potential to realize

functional and cost effective quantum information processing devices.

The main results of this work are the development of an experimental technique for

coherent probing of a quantum dot inside a photonic crystal cavity and the realization

of a controlled phase shift interaction between photons on the semiconductor chip

[1, 2]. This interaction is enabled by the nonlinearity of a single quantum dot that is

embedded in an optical microcavity.

IV

Acknowledgements

During my time at Stanford I had the pleasure of interacting and working together

with many inspirational people who have helped me grow as a scientist and human

being. In particular, I would like to thank my advisor Jelena Vuckovic for the support

and education she has provided throughout the years. Under her guidance, I was able

to explore many different topics and learn how to succeed and fail. I would also like to

thank my close friends, co-workers and collaborators Dirk Englund, Andrei Faraon,

and Edo Waks without whom the most exciting results would not be possible.

I would like to thank my oral and reading committee members, Professors Steve

Harris, David Miller, Steve Quake and David Goldhaber-Gordon. I would also like

to thank Prof. Harris and Prof. Miller for taking time to discuss academic questions

with me throughout my time at Stanford.

My time in the Vuckovic group greatly benefitted from the members Hatice Altug,

Maria Makarova, Bryan Ellis, Yiyang Gong, Kelley Rivoire, Arka Majumdar and Jesse

Lu.

I would like to thank Prof. Vanessa Sih for her help with developing electrical

contacts and teaching me about electron spins in quantum dots.

Throughout my time at Stanford I benefitted from many discussions with Ofer

Levi and Michelle Povinelli, as well as Shanhui Fan, and members of the Yamamoto

Group: David Fattal, Thaddeus Ladd, Kai-Mei Fu, Chales Santori, David Press,

Na-Young Kim.

In my last year at Stanford, I had the opportunity to work with Tornas Sarmiento

in Prof. James Harris' group. I would like to thank both Tomas and Prof. Harris for

teaching me about semiconductor optoelectronic devices and semiconductor growth.

v

Finally I would like to thank my mother Dina and my father David, who made

everything possible in my life. This thesis is dedicated to you.

VI

Contents

Abstract iv

Acknowledgements v

1 Introduction 1

1.1 Overview 1

1.2 Quantum dots 3

1.3 Photonic crystal cavities 3

1.4 Interaction between coherent light and a quantum dot in a photonic

crystal cavity 6

1.4.1 Weak coupling regime 8

1.4.2 Strong coupling regime 9

1.4.3 Transmission of light through the cavity: weak and strong cou

pling regime 10

1.4.4 How a cavity-QD system can be used for quantum information

processing? 12

1.5 Brief introduction to quantum information processing 14

1.5.1 Encoding information in photon states 16

1.5.2 Quantum operations on photons 17

2 Photonic Crystal Cavity Design 20

2.1 Introduction 20

2.2 A simple look at photonic crystal design 21

2.3 Numerical solution methods 22

vn

2.4 Bloch modes, reciprocal space and cavities 24

2.5 Inverse Approach 26

2.5.1 Simplified relation between Q of a cavity mode and its k-space

Distribution 27

2.5.2 Optimal k-space distribution 30

2.5.3 Inverse problem approach to designing PC cavities 33

2.5.4 General trend of Q/V 33

2.5.5 Estimating Photonic Crystal Design from /c-space Field Distri

bution 35

2.6 Genetic Algorithms 41

2.6.1 Algorithm description 42

2.6.2 Algorithm implementation 42

2.6.3 Simulation results: Optimizing planar photonic cavity cavities 44

2.7 Conclusions 45

3 Optical nonlinearities in PC waveguides 48

3.1 Introduction 48

3.2 QND photon detection with Kerr nonlinearities 49

3.3 Pulse Propagation in PC Waveguides 53

3.4 Nonlinear Phase Shift 57

3.5 Conclusion 59

4 Nonlienarities for QIP 61

4.1 Introduction 61

4.2 Measurement description 62

4.3 Coherent probing of a cavity-QD system 62

4.4 Phase measurement 65

4.5 Controlling the phase 70

4.5.1 Control and signal beams at the same wavelength (Acontr.0/ =

^signal) • *-*

4.5.2 Control and signal beams at different wavelengths (\c<mtroi ¥"

•^signal) '^

Vlll

4.6 Conclusion 76

5 Towards room temperature cavity-QED 80

5.1 Introduction 80

5.2 Room temperature operation 81

5.3 Conclusion 87

6 Ultra Fast Modulation 88

6.1 Introduction 88

6.2 Free Carrier Tuning 90

6.3 Thermal Tuning 93

6.4 Conclusion 95

7 Fabrication 98

7.1 Introduction 98

7.2 PC fabrication 99

7.2.1 Sample preparation 100

7.2.2 Cleaving 101

7.2.3 Exposure 101

7.2.4 Development 101

7.2.5 Etching 101

7.2.6 Undercut 102

7.2.7 Wet Oxidation 103

7.2.8 Wet Oxidation Undercut 103

7.3 PMMA doped with colloidal QDs 103

7.3.1 PMMA 103

7.3.2 Dissolving QDs in PMMA 103

7.4 PC laser/detector and electrical contact fabrication 104

7.4.1 Wafer design 105

7.4.2 Mask design 105

7.4.3 Lithography process 107

7.4.4 Wet Etch 109

IX

7.4.5 Metal contacts 110

7.4.6 Electrical isolation by resist I l l

7.4.7 Device fabrication flow 112

8 Conclusion and Future Directions 113

8.1 Cavity QED and quantum information processing 113

8.2 Classical Information Processing 115

A Equivalence between the CNOT and CZ gate 116

B Derivation of Cavity Radiative Loss 118

C Pulses in a nonlinear PC waveguide 121

C.0.1 Derivation of the propagation equations 121

D Cavity QED Experiment and Derivations 128

D.l Experimental setup 128

D.2 Quantum dot visibility 129

D.3 Quantum dot saturation 129

D.4 Amplitude and phase of the interference signal 130

D.5 Computational model 132

Bibliography 135

x

List of Tables

2.1 Q values of structures derived with inverse-approach 1 39

3.1 values for coupling 7 for different modes of the waveguide in units of

a'2, and mode volumes for each unit cell of the waveguide (in units of

a - 3 ) . 1 and 2 refer to the first and second modes of the waveguide. . 56

4.1 Nonlinear parameters and phase modulation derived from experimental

data for the strongly (first row) and weakly (second row) coupled QDs.

A(f) is a maximum differential phase shift (A</>=<̂ >(nc)-</>(0)) which is

achieved at the intra-cavity photon number nc in the last column. . . 76

7.1 Pquest etch recipe for GaAs membranes with thickness up to 200 nm. 102

XI

List of Figures

1.1 (A) Atomic force microscope image of InAs/GaAs semiconductor quan

tum dots grown by molecular beam epitaxy. The typical dot diameter

is 20 nm. (B) Typical emission spectrum of a quantum dot. Typi

cal dot wavelengths are around 920 to 940 nm, with the distribution

determined by the dot diameter variation. (C) The quantum dot is

formed by small InAs islands that are grown by self assembly between

GaAs layers. The three dimensional confinement in the quantum dot

leads to discrete energy state of the electron and hole. (D) Optical

excitation pathways of the QD. The main QD transition is the low

est energy transition with a typical bulk lifetime of 1 ns and emission

wavelength of ~ 920 nm. The emission wavelength is determined by

the In content in InxGsLi_xAs that is used to form the dot. There are

three ways to optically excite the quantum dot. Electron-hole pairs

can be injected into the surrounding GaAs via above-band excitation

(i), higher order (n > 1) states can be driven resonantly (ii), and the

n = l transition can be driven directly (iii). For above-band excitation

the phonon relaxation time of the electron-hole pair into the n = l level

takes on average lOps. The line width of QD's in bulk is close to the

Fourier limit and on the order of 10 /xeV 4

xn

2 (a) Scanning electron micrograph of a photonic crystal cavity, (b)

Conceptual device that integrates multiple cavities with waveguides

on the same chip, (c) the cavity enhances emission from quantum

dots that are coupled to it, while suppressing emission from decoupled

quantum dots 7

3 (a) The QD is a two-level system with an excited and ground state. It

decays with a rate 7 to free space when it is embedded in bulk material,

and couples to the cavity with a rate g. The cavity field decays with

a rate K — UJC/2Q. (b) Probability of an excited dot staying in the

excited state as a function of time (in units of (2K) _ 1 ) . The bulk dot

decays slowly once it is in the excited state (black line). As the cavity-

QD coupling rate g increases, the dot decays more quickly (red line).

Once g > K /2 , the dot and cavity begin to exchange energy coherently

and the decay profile oscillates 9

4 11

5 13

xin

A two-dimensional photonic crystal and light confinement mechanisms.

In-plane confinement is given by Bragg scattering due to the refractive

index contrast at the PC holes. Out-of-plane confinement is the result

of total internal reflection at the interface between the higher index

PC and lower index surroundings. A cavity is realized by introducing

a defect into the periodic lattice of holes with period a and radius

r. The thickness of the membrane is given by d. Waveguides can be

formed by removing rows of defects, and coupling between cavities and

waveguides can be controlled by the number of holes between them and

their relative orientation. In this work, the majority of the cavities were

fabricated in GaAs with an index of 3.6 and measured in vacuum with

an index of 1. The typical membrane thickness d is 160nm and the

periodicity a is 246nm corresponding to a resonant cavity wavelength

in the range of 910-950nm, and the radius r is typically on the order

of 70nm

(a) Schematic representation of the photonic crystal lattice with lattice

vectors a,i,a,2 that map out the triangular lattice, (b) The reciprocal

lattice corresponding to the triangular lattice in (a) with reciprocal

lattice vectors 61,62, where |6i| = |62| = 4ir/\/3a. The first Brillouin

zone is indicated by the blue line and the irreducible Brillouin zone

is shaded in green. The black circle corresponds to the light line (or

cone) given by Eq. 2.1. In (c) we show the resulting band structure of

the triangular crystal with r = 0.3a, d = 0.65a and n = 3.6. The filled

circles correspond to frequencies of confined states inside the lattice at

a particular value of the wave vector as it traces out the edge of the

irreducible Brillouin zone indicated by the green line in the inset in (c).

States above the light line (solid line) correspond to lossy states (with

non-imaginary values of kz)

xiv

3 Left: Band diagram for hexagonal waveguide in TJ direction, with

r/a = 0.3, d/a = 0.65, n = 3.6. The bandgap (wedged between the

gray regions) contains three modes. Mode B00 can be pulled inside

the bandgap by additional neighbor hole tuning. Right: Bz of con

fined modes of hexagonal waveguide. The modes are indexed by the

5-field's even ("e") or odd ("o") parities in the x and y directions,

respectively. The confined cavity modes B00, Bee, and Beo required

additional structure perturbations for shifting into the bandgap. This

was done by changing the diameters of neighboring holes 26

4 Estimating the radiated power and Q± from the known near field at

the surface S 28

5 Comparison of Q factors derived from Eq. (2.13) (squares) to those

calculated with FDTD (circles). Top: cavity made by removing three

holes along the TJ direction confining the Boe mode. The Q factor is

tuned by shifting the holes closest to the defect as shown by the red

arrow. The x-axis gives the shift as a fraction of the periodicity a.

Bottom: the X dipole cavity described in [3]. The Q factor is tuned

by stretching the center line of holes in the TX direction, as shown by

the arrow. The rr-axis gives the dislocation in terms of the periodicity

6 Idealized cavity modes at the surface S above the PC slab; all with

mode volume ~ (A/n)3. (a-c) Mode with sine and Gaussian envelopes

in x and y, respectively: Hz(x,y), FT2(i/z), and K(kx,ky) inside the

light cone; (d-g) Mode Boe with Gaussian envelopes in the x and y

directions : Hz(x,y), FT2(HZ), K(kx,ky), and Q±(o-x/a) as well as

Q±/V . Qj_ was calculated using Eq. (2.13) and Ez was neglected.;

(h-i) Mode Bee with Gaussian envelopes in x and y can be confined to

radiate preferentially upward 34

xv

FDTD simulations for the derived Gaussian cavity (a-c) and the de

rived sine cavity (d-f). Gaussian: (a) Bz\ (b) \E\; (c) FT pattern of

Bz taken above the PC slab (blue) and target pattern (red). Sine: (d)

Bz; (e) \E\; (f) FT pattern of Bz taken above the PC slab (blue) and

target pattern (red) (The target FT for the sine cavity appears jagged

due to sampling, since the function was expressed with the resolution

of the simulations). The cavities were simulated with a discretiza

tion of 20 points per period a, PC slab hole radius r — 0.3a, slab

thickness of 0.6a and refractive index 3.6. Starting at the center, the

defect hole radii in units of periodicity a are: (0,0,0.025,0.05,0.075,0.1

, 0.075, 0.075,0.1,0.125,0.125,0.125,0.1,0.125,0.15,0.3,0.3) for the sine

cavity, and (0.025,0.025,0.05,0.1,0.225) for the Gaussian cavity. . . .

Top-left: Real-space mode profile after optimizing for closest-match to

a sine-envelope target mode. Top-right: k-space mode profile of the

optimized simulated mode and a sine-envelope target mode. Bottom:

Real-space and k-space mode profiles for matching against a sinc2-

envelope target mode. All: Red curves represent the real-space (k-

space) mode profiles of the optimized fields, blue curves represents the

real-space (k-space) mode profiles of the target fields

The setup for QND detection of a signal beam with average expected

photon number Ns and phase <j>3ig. The probe is split on the 50/50

beamsplitter BS into two beams with photon numbers Np and phases

<f>probe- The mirrors Mi and M2 pass the signal and reflect the probe

with negligible losses. The two beams co-propagate in one arm of

the interferometer through a PC waveguide made in a medium with a

large Kerr nonlinearity (x(3)). The signal Ns is unmodified and only

the phase is distorted to <p'sig. The probe photon number is preserved,

but the phase is modified to 4>probe — 4>probe oc Na. The probe phase

rotation 4>Wobe ~~ 4>probe is detected by a homodyne measurement on

detectors Dt, D2 and yields the photon number Na

xvi

2 (a) waveguide mode dispersions calculated by the 3D Finite Difference

Time Domain (FDTD) method. The solid (black) line is the light line

in the photonic crystal, above which modes are not confined by total

internal reflection. The insets show the Bz profiles of the even (i) and

odd (ii) modes at the k = n/a point, (b) Group velocities of the two

modes derived from the dispersion curves via numerical differentiation

(vg — g|). The group velocity is greatly reduced at the k — ir/a point. 52

3 Amplitude of E field for k = \\, | f and \ 55

4 (a)Phase shift due to a single signal photon with a lifetime of 200 ps,

after propagation through a 100 \±m AlGaAs PC waveguide with a

narrow probe and no group velocity mismatch as a function of the

group velocity vg normalized by the speed of light c. (b) The energy

required for an external pulse to obtain a SNR of 1. In (a) and (b) it

is assumed that the signal and probe are at 1500 nm and two-photon

absorption is not present. In (c) we plot the phase for the case of the

signal photon in waveguide 1 at 1550 nm and probe at 1620 nm in

waveguide 0. The required probe energy for this scheme is shown in

(d). In all plots, the blue and red curves correspond to both the signal

and the probe in waveguide modes 0 or 1. The black curve corresponds

to the probe and signal in different waveguide modes 58

1 (a) The amplitude of cavity transmitted photons with and without

the quantum dot. (b) The phase of cavity transmitted photons with

and without the quantum dot. In the presence of the QD the phase

has an abrupt modulation of up to n across the dot resonance, (c)

The phase of photons can be controlled by either saturating the dot,

where in principle a 7r/2 modulation can occur, or by shifting the dot

resonance, where a IT phase shift would result, (d) In this experiment

a "one-sided" cavity was used. The reflection coefficient is given by

r(uj) = Eout/Ein 63

xvn

2 A cross polarization setup is used to reject direct laser scatter from the

sample and collect only cavity-coupled photons. Instead of tuning the

probing laser, the cavity and QD wavelength are shifted with temper

ature by heating a "heating pad" with a 980 nm laser that does not

excite carriers in GaAs and the QD. The QD shifts « 3 — 4 times faster

than the cavity as detailed in [4]. The temperature is varied period

ically between 20K and 27K and the amplitude of the cavity-coupled

probing beam is collected on the spectrometer 64

3 66

4 Experimental setup (A). Vertically polarized control (wavelength Ac)

and signal (wavelength \s) beams are sent to the PC cavity (Inset) via

a polarizing beamsplitter (PBS). A quarter wave plate (QWP) (fast

axis 9 from vertical) changes the relative phase and amplitude (£(#))

of components polarized along and orthogonal to the cavity. Only the

reflection coefficient r(uj) for cavity-coupled light (at |—45°)) depends

on the input frequency and amplitude. The PBS transmits horizontally

polarized light to a detector D. (B) Theoretical model for the phase of

signal beam 0. The signal phase 0i changes to 02,03 when the control

and signal beams are resonant or detuned respectively, and nc = 0.3.

The nonlinear phase shift due to the increase in power is shown as A0i.

The wavelength detuned control shifts the phase 03 relative to 0i by

the AC Stark effect [5]. 03 is asymmetric because the cavity-coupled

control power depends on the cavity and QD wavelengths during the

temperature scan (C). The temperature was scanned from 20 to 27

K. (D) Measured reflectivity R for different QWP angles and fit by

theoretical model Eq. D.8. (E) Phase of the reflected beam, extracted

from model fits in (D) 68

5 Phase and amplitude of the interfering beam as a function of the QWP

angle 9 in units of 7T 69

xvni

4.6 A. Is(u) taken for several values of 9. Interference between the refer

ence beam and cavity coupled beam are clearly visible. B. Extracted

amplitude |r(u;)| and phase arctan(Im(r(uj))/Re(r{ui))) of the cavity

reflection coefficient 69

4.7 Nonlinear response of the QD-PC cavity system to single wavelength

excitation near saturation at control photon number nc=0.6 (A,B).

Each temperature scan count corresponds to a particular detuning be

tween the cavity and QD as in Fig.4.4C. At a detuning of 0.014 nm

(gr/3.5) from the dot resonance (vertical line in B), the phase changes

by 0.247T when nc increases from 0.08 to 3 (C). The phases derived

from experimental scans (points) agree with theory (solid line). The

dashed red curve is the fit to experimental results evaluated at con

trol powers of 2nc. The signal phase shift due to the doubled signal

photon number </>(nc) — </>(2nc) is maximized at nc=0.1 (arrow). (D)

The main loss mechanism due to fluorescence from the quantum dot

corresponds to ~ 1% photon loss. (E) Reflectivity power dependence.

Points correspond to experimental data for reflectivity (R) normalized

by the calculated value of reflectivity from a cavity with no QD (RQ). 71

xix

Interaction between a control and signal beam at different wavelengths.

The signal beam at Xs (A-i) is detuned by 0.027 nm (« g) from the

control beam at Ac (A-ii) and positioned to coincide with the cavity-

dot crossing-point (A-iii). For each measurement, a sequence of scans

is taken (A i-iii). The quantum dot and cavity trajectories are shown

in (A-iii). We track the amplitudes at both wavelengths in each frame

(A i-iii) to subtract fluorescence backgrounds, which are magnified 10

times in (B) and (C) (these are fluorescence backgrounds detected at

control and signal wavelengths, respectively.). The QD-induced dip is

clearly visible in (B) when only the signal (solid blue line) is on, and in

(C) when only the control (dashed line) is on. This feature disappears

when both beams are on in (D). In (D), the spectra are normalized

in order to clearly show saturation. The signal and control powers

were 100 nW and 200 nW measured before the lens, corresponding to

cavity coupled signal and control photon numbers ns «0.2 and nc ^0.3,

respectively

Nonlinear response of a weakly coupled quantum dot inside the cavity

to excitation with control and signal beam wavelengths separated by

0.027 nm (~ g). The reflectivity of a signal beam with ns=0.2 photons

per cavity lifetime is shown in (A) for three values of the control beam

photon number nc. The quantum dot saturates almost completely

when nc=1.3, which corresponds to a power of 1 fiW measured before

the objective lens. The data is fit with a full quantum model, which

allows us to extract the signal phase shown in (B). In (C) the amplitude

of the reflected signal beam when it is 0.009 nm (sa g/3) away from the

dot resonance (vertical line in A,B) is shown as a function of control

beam photon number nc. In (D), we show the difference between the

phase shift of the signal beam when the control beam is on (</>) and

when the control is off (0o) as a function of nc at the same time point

as in (C)

xx

4.10 Simulated differential phase shift 4>{2n) — 4>(n) as a function of probe

detuning from the cavity u>i — toc when the QD is on resonance with

the cavity ui^ot — wc = 0. A maximum of 0.1 n occurs when the aver

age photon number inside the cavity is close to nc « 0.1 as observed

experimentally 78

4.11 A. Differential phase shift 4>(2n) — cf)(n) as a function of probe detuning

from the cavity u>i — uic for different QD detunings Udot —uc. B. Maxima

from A vs dot detuning in units of g. Theory predicts that for large

detunings, the differential phase shift approaches n/2. Also plotted is

the average intra-cavity photon number (nc) corresponding to the par

ticular maximum differential phase shift. If the lowest photon number

operating point is desired, the detuning should be uidot — uc « g. . . 79

5.1 Influence of 7 on the transmission through a cavity with an embedded

QD. As 7 increases due to phonon scattering at elevated temperatures,

the transmission dip diminishes and the transmitted amplitude falls

due to energy losses into phonon modes. For the plots it is assumed

that g = K 82

5.2 Left: Scanning electron micrograph showing the photonic crystal cavity

(a). Middle: Simulated electric field intensity of the x (b) and y (c)

dipole modes in the asymmetric cavity. The measured Q factors are

400 and 200 respectively 83

5.3 PbS quantum dot spectra: 850 and 950 dot spectra taken on a bulk

silicon wafer 83

5.4 Cavity resonances mapped out by quantum dots in PMMA. Left: Po

larization dependence of modes confirming that they are x and y dipole

modes. Right: Ex dipole mode measured at two orthogonal polariza

tions. Angles refer to analyzer positions 85

xxi

(a): Scanning electron micrograph of the L3 type cavity fabricated in

a GaAs material with a high density of InAs quantum dots, (b): high-

Q mode electric field amplitude distribution, as predicted by FDTD

simulations, (c): FDTD simulations of frequency and Q changes as

An/n changes from ±10"3 -» ±10_ 1 . A high-Q (QHQ = 20000) and

low-L (QLQ = 2000) cavity were tuned: (qi) AQ/Q for An > 0 and

Q = QHQ, (<?2) AQ/Q for An > 0 and Q = QLQ, (ui) ALO/UJ for

An < O , ^ ) Aw/w for An > 0 for both high Q and low Q modes, (qj)

AQ/Q for An < 0 Q = QLQ, (q4) AQ/Q for An < 0 Q = QHQ. The

lines An/n for An > 0 and An < 0 are also plotted and overlap exactly

with u>2 and u\. As can be seen, the magnitude of the relative frequency

change is independent of Q, but the higher Q cavity is degraded more

strongly by the change in index. For an increase in n, the Q increases

due to stronger Total Internal Reflection confinement in the slab, as

expected

Numerical model of a free-carrier tuned cavity. In (a) the cavity is

always illuminated by a light source. Panel (b) shows the cavity reso

nance at the peak of the free carrier distribution (t=0) and 50 ps later,

as indicated by the yellow arrows in (a). The time-integrated spec

trum is shown as the asymmetric black line (labeled Sp) in (b), and

corresponds to the signal seen on the spectrometer, which is the inte

gral over the whole time window of the shifted cavity. The asymmetric

spectrum indicates shifting. In (c) and (d) the same data is plotted,

but now we consider the cavity illuminated only by QD emission with

a turn-on delay of 30 ps due to the carrier capture lifetime rc, and a

QD lifetime of 200 ps. In (d) the dashed line is the un-normalized t=0

spectrum, which now appears much smaller in magnitude. Further

more, the asymmetry of the line is even smaller in this case

xxn

Experimental result of FC cavity tuning for the L3 cavity. In (a) the

cavity is always illuminated by a light source and pulsed with a 3ps

Ti:Sapphire pulse. Panel (b) shows the cavity resonance at the peak

of the FC distribution (t=0) and 50 ps later, as indicated by the yellow

arrows in (a). In order to verify that the cavity tunes at the arrival

at the pulse, we combine the pulsed excitation with a weak CW above

band pump. The emission due to the CW source is always present, and

is in the box labeled CW in (a). This very weak emission is reproduced

in Panel (b) as the broad background with a peak at the cold cavity

resonance in (b). The time-integrated spectrum is shown as the black

line (Spectrometer) in (b). In (c) and (d) the same data is plotted,

but now we consider the cavity illuminated only by QD emission pulsed

by 10 [iW from the Ti:Sapphire source. In (d) suppression by about

.4-.35 at the cold cavity resonance can be seen. The inset shows a

strongly asymmetric spectrum of a dipole type cavity under excitation

of lOO/uH^ and the same cavity at low power after prolonged excitation.

Such strong excitation degrades the Q

Thermal tuning of the L3 cavity under CW excitation (a): Measured

AUJJUJ (left axis) and AQ/Q (right axis) as a function of pump power

for the L3 cavity, obtained from the fits to the spectra shown in (b).

The Q initially increases due to moderate gain and then degrades, while

u! shifts linearly. The straight dashed line fits Aw/w = 3 x 10 - 3 x Pin —

5 x 10~5 with 95% confidence and with root mean square deviation of

« 0.99. At very high power, the change in frequency does not follow

the same trend. The inset in (b) shows a plot of AU/{OJ/Q), which is

a measure of the number of lines that we shift the cavity by. A shift

of three line widths is obtained

xxin

1 A. Overall process flow for fabrication. B Sample GaAs wafer. C

Exposed resist (inset is poor outcome). D Etched resist (inset is poor

outcome). E final undercut PC. The light area around the structure

indicates a successful undercut 100

2 (A) A simulation of the band structure under zero and IV reverse

bias for the structure shown in the inset of (A) are shown. The bias

applied through the Schottky contact is used to control the tunneling

of electrons from the n-doped layer into the InAs QDs in the middle of

the intrinsically doped layer. Appropriate biasing aligns the QD levels

with the Fermi level of the n-doped layer for enhanced tunneling. (B)

A fabricated device in which the p and n contacts have been made

to the appropriate layers. The pattern was made by multiple aligned

writes with the electron beam tool. A wet etch is used to etch down

to the n-GaAs layer of the structure. (C) A preliminary measurement

for the sample under reverse bias. Slight crossing of lines, which is

indicative of charging is potentially observed. The overall line shift is

due to heating in the sample due to reverse breakdown of the diode.

Further measurements could not be made, as the sample shorted. . . 106

3 (A) A simulation of the band structure under zero bias for the structure

shown in (B). In (C) an optical micrograph of the final fabricated

device is shown. In (D) a scanning electron micrograph of the same

device reveals that the metal liftoff with a single layer of resist leaves

a significant edge 107

xxiv

7.4 (A) An optical micrograph of the TLM structure next to active de

vices. The structure consists of metal pads with varying spacing. The

width of the pad should be larger than the spacing. (B) A plot and

fit of resistance R versus pad spacing. R is obtained from the slope

of the ohmic IV curves measured between successive pads in (A). The

slope of the fit gives the sheet resistance Rsh and the intercept is twice

the contact resistance Rc. (C) IV measurements on the fabricated de

vice validate diode behavior and show current limitation due to series

resistance at large forward bias. The device acts as a detector in the

reverse bias configuration 108

D.l Experimental setup. Vertically polarized control (wavelength Ac) and

signal (wavelength Ag) beams are sent to the PC cavity (Inset) via

a polarizing beamsplitter (PBS). A quarter wave plate (QWP) (fast

axis 9 from vertical) changes the relative phase and amplitude (C($))

of components polarized along and orthogonal to the cavity. Only the

reflection coefficient r(uj) for cavity-coupled light (at |—45°)) depends

on the input frequency and amplitude. The PBS transmits horizontally

polarized light to a detector D. (B) Theoretical model for the phase of

signal beam 6 130

xxv

D.2 The quantum dot and cavity wavelengths (AQ£>,Acat,) are extracted from spectra

taken as a function of the temperature scan count (A). The probing laser is posi

tioned at a wavelength (Ac) that is close to the point of crossing between the QD

and cavity trajectories in (C). Thus, each point along the 'temperature scan count'

corresponds to different offsets between the quantum dot and cavity. By tracking

the amplitude of the probing laser, we can find the reflectivity signal and extract

the phase in B. The point of maximum phase contrast corresponds to the verti

cal dashed line in B (also in C) and to a particular offset between the cavity and

dot wavelengths. When the cavity and QD wavelength are fixed, and the laser is

scanned along the dashed line in B, the signal shown in D would be obtained. The

dot and cavity detunings are indicated by the two dashed lines. In the experiment,

the laser is positioned at Ac, which overlaps with \cav in D. The point of maximum

phase contrast, which is the point where we find the phase, coincides with Xcav and

Ac in this case 134

xxv i

Chapter 1

Introduction

1.1 Overview

Photonic quantum and classical information processing are long-standing and rapidly

developing fields of research that had their inception with the development of the

working laser by Maiman in 1960, and the pioneering work of atomic physics. Today,

more than ever, these fields are coming closer to each other and closer to deeply im

pacting computing, communication and the sharing of information. In the realm of

classical information processing, the integration of optical interconnects for board-to-

board, core-to-core, chip-to-chip and potentially on-chip communication is inevitable

and is being actively pursued by companies such as Hewlett-Packard, IBM, Sun Mi

crosystems, Intel, and Luxtera. These efforts are driven by the astronomical savings

in power consumption that could result from replacing copper wires with photonic

channels. Photonics has the potential to account for 90% power savings in today's

chipsets 1 and is already used to significantly reduce the power losses associated with

2% efficient copper transmission lines in data centers 2. Although quantum effects

may not be fully exploited to yield functioning quantum computers in the near future,

lMWhy optics and why now?", Greg Astfalk, Office of Strategy and Technology at Hewlett-Packard: the power spent on communication and data transfer is 33-50% for the processor, 50% in memory, and 90% in the chipset. Delivered in a presentation at the HP Labs Photonic Interconnect Forum, May 12, 2008

2At the time of this thesis, several companies have unveiled optical "wires" for use in datacenters as replacements of lossy electrical connections.

1

CHAPTER 1. INTRODUCTION 2

they have great potential for enhancing the performance of optoelectronic devices.

Enhanced light matter interactions due to quantum effects can already be used to

create extremely low threshold lasers [6], optical modulators[7], logic gates between

photons [2], and switches operating at single photon energies on a semiconductor

chip [2, 1]. This thesis explores how these effects are enabled by the extremely small

optical volumes and high quality factors of photonic crystal cavities, which are an

electromagnetic breadboard for tailoring the interaction between photons and atoms

on the chip. As the aforementioned devices are nonlinear in nature, this thesis fo

cuses on our exploration of engineered high efficiency nonlinear materials that can be

tailored for particular applications.

In this chapter, I begin by reviewing quantum dots, resonators, and field quanti

zation in small volumes. The small optical volume of photonic crystal cavities results

in large electric fields even due to a single photon. Furthermore, the resonator leads

to a modified density of states for photons, and this effect can be used to enhance

or suppress real and virtual optical absorption processes. I then discuss how photons

can be used to encode quantum bits and demonstrate that a cavity with an embedded

nonlinear medium can be used to realize quantum logic between photons on the chip.

I then review the interaction between photons and quantum dot that is embedded

in a cavity. I briefly review the concepts of classical nonlinear processes, such as the

intensity dependent refractive index, and show how these are affected by the presence

of the cavity. Although losses of off-resonant nonlinearities cannot be efficiently con

trolled, the resonant nonlinearities due to the refractive index of a single emitter can

be greatly enhanced and modified with the use of a photonic cavity, yielding interac

tions between photons at the single photon level and modulation of the transmission

through the cavity with up to 99% contrast.

In Chapter 2, I will review the basics of photonic crystal cavities and discuss our

development of theoretical and numerical methods for the modeling of photonic band

gap cavities. I will focus on the inverse design approach and genetic algorithms as

well as the design of asymmetric cavities with differing refractive indices on the top

and bottom side of the PC membrane. In Chapter 3, I will introduce our attempts

at enhancing classical nonlinearities with the use of photonic bandgap cavities in the


context of quantum non-demolition measurements, and use this as a motivation for

• the results of Chapter 4, where I will more deeply explore the interaction between a

quantum dot and an optical cavity and the use of this system as a high efficiency non

linear optical element for controlled phase interaction between photons. In Chapter 5

I will review our attempts at realizing cavity-quantum dot interactions at room tem

perature. In Chapter 6 I will present our results on ultra-fast all-optical modulation

in photonic crystal cavities.

1.2 Quantum dots

Some of the greatest advances in experimental quantum information processing have

been realized in experiments with atoms and trapped ions [8, 9, 10, 11]. In order to

develop scalable and manufacturable analogues of such experiments, it is advanta

geous to follow in the steps of the semiconductor industry and look for a solid-state

semiconductor based implementation of quantum memories and technologies. Since

photons will be used as carriers of information, an optically active semiconductor

quantum bit is required. A natural choice is a quantum dot (QD), which acts as a

single atom with discrete energy states, optically active transitions, and controllable

spin states for the development of quantum memories. The quantum dot is formed

by capping a chunk of a low band gap semiconductor with a higher bandgap sur

rounding material, which results in three-dimensional confinement for electrons and

holes, and the formation of discrete energy levels. For the purposes of this thesis,

the quantum dot acts as a two-level atom with an excited and ground state, that

is accessible to optical transitions. The main features of a quantum dot along with

excitation strategies are illustrated in Fig. 1.1.

1.3 Photonic crystal cavities

The use of optical cavities enhances the interaction between light and matter. In the

simplest case, the optical cavity recirculates photons and allows the interaction to

occur several times before the photons leak out of the cavity mirrors. The number


o i.as

Conduction Band

Discrete Energies n=2

n=l

B

JLLJ., - L 981 932 933 934

X [nm]

GaAs In As GaAs

n=l n=2

Valence Band Hole

Figure 1.1: (A) Atomic force microscope image of InAs/GaAs semiconductor quantum dots grown by molecular beam epitaxy. The typical dot diameter is 20 nm. (B) Typical emission spectrum of a quantum dot. Typical dot wavelengths are around 920 to 940 nm. with the distribution determined by the dot diameter variation. (C) The quantum dot is formed by small InAs islands that are grown by self assembly between GaAs layers. The three dimensional confinement in the quantum dot leads to discrete energy state of the electron and hole. (D) Optical excitation pathways of the QD. The main QD transition is the lowest energy transition with a typical bulk lifetime of 1 ns and emission wavelength of « 920 nm. The emission wavelength is determined by the In content in IncGai_.EAs that is used to form the dot. There are three ways to optically excite the quantum dot. Electron-hole pairs can be injected into the surrounding GaAs via above-band excitation (i), higher order (n > 1) states can be driven resonantly (ii), and the n = l transition can be driven directly (iii). For above-band excitation the plionon relaxation time of the electron-hole pair into the n=l level takes on average lOps. The line width of QD's in bulk is close to the Fourier limit and on the order of 10 //,eV.


of roundtrips is proportional to the quality factor (Q) of the cavity. However, given

a fixed Q, as the resonator size decreases, the optical energy is confined to a smaller

volume, and the field intensity increases. In a small mode volume resonator, the

frequency spacing between modes is quite large, and the density of optical states

is strongly modified. By Fermi's golden rule, this implies that optical transitions

can be both enhanced and suppressed. As will be shown in Chapter 2, photonic

crystals (PCs) are a suitable platform for such experiments and can be used as a

breadboard for photonics on the chip. Photonic crystals can be readily fabricated

in most semiconductors due to a rich history of semiconductor processing. This

allows photonic crystals to be scalable, integrable, and easily combined with current

technologies. Furthermore, the combination with active semiconductor materials such

as quantum dots and quantum wells results in a variety of novel devices and effects. In

Fig. 1.2 we show a typical photonic crystal cavity that was fabricated in GaAs, and a

conceptual device that combines several cavities and connects them with waveguides.

The influence of the PC cavity on QD emission is illustrated in Fig. 1.2c, where the

emission from QD's that are not resonant with the cavity is suppressed, while that

from quantum dots is enhanced.

The electric field E inside a resonator of volume V is given by:

E{f) = ^ { a { t ) + a\t))u{r)u (1.1)

Where e = e(f) is the material dielectric function, u{f) is the normalized spatial part

of the single photon wave function, given by the solution of Maxwell's equations, u is

the polarization of the field, u> is the photon frequency, a(t), at(t) are the annihilation

and creation operators and the cavity mode volume V, and rrvu is the dipole moment

of the emitter. Optical transitions can be formulated in terms of the dipole interaction

Hamiltonian Hi [12]:

#7 = -/7 • i? = hg(a+ + a') (a(i) + a\t)) (1.2)

Here a+ = |e) (g\ = a_ are the raising and lowering operators for the driven dipole


that is assumed to have an excited |e) and ground state \g), while g is the vacuum

Rabi frequency, which determines the dipole-field coupling rate. Retaining only the

energy conserving terms (i.e. those that excite the dipole and destroy a photon, and

vice versa):

Hi = fig (a+a(t) + a\t)a~) (1.3)

The vacuum Rabi frequency inside the cavity g is then

The mode volume V is defined as:

max{e \E\2} \nj

In the above expression A is the wavelength of light and n is the refractive index of

the medium surrounding the dipole. In typical resonators C, » 1, and we can see that

the electric field of a single photon is small. However, in a photonic crystal resonator,

C = 0(1), with a lower bound of 1/8, and the fields are significantly enhanced.

For comparison, typical atomic cavities have Q ~ 0(1000). Therefore, interaction

strengths between the cavity field and the two-level system are significantly enhanced

in a photonic crystal cavity.

1.4 Interaction between coherent light and a quan

tum dot in a photonic crystal cavity

The theory of an emitter coupled to an optical cavity can be found in Refs. [13, 14].

As described above, the novelty of photonic crystal cavities is that the optical mode

volume is very small (on the order of (A/n)3, where n is the refractive index of the

material in which the dipole is embedded, and A is the resonant wavelength of the

cavity). Therefore, even with moderate quality factors, highly nonlinear effects can

CHAPTER 1. INTRODUCTION i

Figure 1.2: (a) Scanning electron micrograph of a photonic crystal cavity, (b) Conceptual device that integrates multiple cavities with waveguides on the same chip. (c) the cavity enhances emission from quantum dots that are coupled to it, while suppressing emission from decoupled quantum dots.

occur in this system.

We consider the quantum dot to be a two level system with a long-lived ground

state, and a clipole decay rate 7/2TT = 0.2GHz. InAs quantum dot line widths at low

temperatures (w 10 —20A") are generally close to being radiatively limited, and so the

dipole dephasing rate can be approximated to be one half of the dipole decay rate.

The coupling of the quantum dot dipole to the cavity electric field is given by the

Rabi frequency given by Eq. 1.4. Values of g clearly depend on position and dipole

alignment, with the maximum dipole moment of gmax « ZtiOGHz in this system.

These parameters are illustrated in Fig. 1.3 (a).

When a quantum dot (transition frequency u;,/) is resonant with the PC cavity at

frequency ajc, the dot emission is modified. The degree of modification depends on

the parameters g. 7 and the field decay rate K = aJc/(2Q). By solving the interaction


Hamiltonian, the eigenfrequencies of the coupled cavity-dot system are found to be

[15]:

^ = —2-+z{-r)±i{ 2 ) ~g (L6)

This equation can be taken to two limits, one in which the square root is real, and

the other, in which it is imaginary. In the first case, the two eigenvalues are real and

correspond to decay rates of the cavity and quantum dot. This regime is typically

termed the weak coupling regime. In the case of an imaginary square root, the

composite quantum dot system decays at one rate, and oscillates at two frequencies

corresponding to the coherent exchange of photons between the dot and cavity.

1.4.1 Weak coupling regime

For our systems, K/2TT SS 16GHZ, The eigenfrequencies in the weak coupling regime

can be found by expanding the square root in Eq. 1.6 for the case K » (ujc—uid, g) »

7 as:

Q2

u+=ud + i^- (1.7) K

U- ~ LUC + in (1.8)

In this case, w_ corresponds to the cavity and u+ corresponds to the dot dipole.

The dot's excited state population decays with a rate 2g2//c inside the cavity. As can

be seen, the value of the dot decay is modified relative to the bulk value 7. The ratio

of the lifetime in the cavity 7C to that in the bulk 7c/7 = 2u;+/7 is dependent on

position, dipole alignment and frequency detuning according to

7c 3 Q (\^ 7 47r2 V \n

E(f) V ffl-e

(1.9)

(1.10) ,WM)J \\fl\J 1 + 4 Q 2 ( v _ 1 y

At the point of maximum spatial and frequency alignment, the ratio 7c/7 reduces to


the Purcell factor F

cavity mode is then li

4§2^ (^) • The fraction of photons emitted into the single

f+F where / « 0.2 accounts for the reduction in the available

1). As the interaction states inside the photonic band gap material (f/reespace

strength is increased, the excited state lifetime of the quantum dot shortens and the

probability of finding the clot in the excited state decays faster, as shown by the red

line relative to the uncoupled dot (black line) in Fig.1.3 (b).

(a) (b) ,

~e

CD CL,

t[{2H)-1

Figure 1.3: (a) The QD is a two-level system with an excited and ground state. It decays with a rate 7 to free space when it is embedded in bulk material, and couples to the cavity with a rate g. The cavity field decays with a rate K = LUC/2Q. (b) Probability of an excited dot staying in the excited state as a function of time (in units of (2K;) - 1 ) . The bulk dot decays slowly once it is in the excited state (black line). As the cavity-QD coupling rate g increases, the dot decays more quickly (red line). Once g > K/2, the dot and cavity begin to exchange energy coherently and the decay profile oscillates.

1.4.2 Strong coupl ing regime

In the strong coupling regime, as in the weak coupling regime, the eigenvalues of the

cavity and dot can be found from Eq. 1.6 with the condition that g > £. In this

case (in the limit of g » K,-f,u>c — u>d) the square root becomes imaginary. The


cavity and quantum dot cannot be treated separately, but exist in a time dependent

superposition state, with photons exchanging energy between the two at a rate of

« 2g. For large g, the eigenvalues of this system are

a)± = (̂ + ̂ )+i(l+2)± j W ^ + ' K - ^ y (1.U)

K(^) + . (^+ I)± 9 ( 1 1 2 )

The oscillation is illustrated by the blue and green lines in Fig. 1.3 (b). By improving

the fabrication of our PC cavities, we have been able to observe this effect in cavities

with Q's in the range of 10000 to 15000, with the best case of 25,500. Figure 1.4 shows

a scanning electron microscope (SEM) image of such a cavity with Q=10,500 and the

photoluminescence observed from the cavity as a quantum dot is tuned through the

cavity resonance via temperature. However, since the dot is strongly coupled to the

cavity, the dot line never crosses the cavity line. Instead, when the two are exactly

on resonance, we observe the characteristic splitting predicted by Equation 1.12. For

clarity, we track the quantum dot and cavity peaks at the different temperature points

and plot them in Figure 1.4 (c).

1.4.3 Transmission of light through the cavity: weak and

strong coupling regime

So far we have only considered the evolution of the cavity-QD system when the dot

starts in the excited state and decays by emitting a photon to the cavity, which

subsequently leaks it to free space. However, the power and utility of this system for

quantum information processing comes from its response to an external driving field,

where it can be used to create logic gates and nodes in a quantum network [16, 14]. In

both the strong and weak coupling regimes, the QD modifies the cavity transmission

properties. In the strong coupling regime, driving the cavity-QD system on resonance

leads to the simultaneous excitation of the two coupled modes and they interfere

destructively. It was realized by Waks and Vuckovic in [14] that strong coupling is


J \^J: • \J>:

V,-*' V—/'

Vwr* \ - /

935.1

• (b ) -

-

-

.

-

^

_vv_ TV y\ J\

_/v. f\ / \

/ \

/v A, /V A

T=36.80K

T=36.40K"

T=36.00K

T=35.60K

T=35.20K-

T=34.80K

T=34.70K

T=34.65K

T=34.60K

T=34.20K-

T=33.80K

T=33.40K"

T=33.00K

934.5 935 935.5 A,(nm)

936

Figure 1.4: (a) Scanning electron microscope (SEM) image of a PC cavity in which strong coupling is observed, (b) Photoluminescence of a quantum dot strongly coupled to a photonic crystal cavity, for different values of detuning between the cavity and dot. The dot wavelength is controlled by the temperature of the sample, which is controlled by the cryostat. The cold-cavity Q-factor is around 10500, and the mode volume is « 0.75 (A/n)3 . (c) Peak positions of photoluminescence in Panel (b) as a function of temperature. The dot and cavity line never cross, indicating that they behave as predicted for the strong coupling regime.

not necessary to obtain destructive interference in a one-dimensional system. That is,

even if the dot is in the Purcell regime, destructive interference between dot-scattered

and cavity-scattered photons occurs and a quantum node can be realized. This means

that even solid state systems in which phonon dephasing and fabrication may impede

strong coupling, quantum information processing is feasible.


The theory for the transmission function of a cavity containing a well coupled

emitter under weak excitation can be found in [13, 14], and has been extended to the

strong excitation regime in [8, 17, 18]. Transmission through a coupled cavity-QD

system under weak excitation by coherent light of frequency ui is given by:

T = r) K

i{ujc - u) + K + j i (cu d -a j )+7

(1.13)

where 7? is a factor that accounts for the coupling efficiency between the driving field

and the cavity. The cavity transmission under weak and strong excitation is shown

in Fig. 1.5. The strength of the excitation is given by the average number of photons

(nc) coupled to the resonator from the coherent driving laser field. At low excitation

levels of less than one photon per QD lifetime, photons falling within the bandwidth 2

given by the QD's modified spontaneous emission rate ^- cannot pass through the

cavity due to destructive interference between the driving field and the QD scattered

photons. As the photon number nc becomes comparable to one photon per QD

lifetime, the QD cannot follow the driving field and incomplete interference leads to

transmission of light. In the strong coupling regime, this effect can be explained by the

splitting of the cavity-QD eigenfrequencies and a lack of a photon energy eigenstate

at the care cavity or QD frequency [19]. This QD saturation is extremely nonlinear

as can be seen in Fig. 1.5. The methods of probing this regime and the realization of

a photon-photon interaction by means of this nonlinearity are presented in Chapter

4.

1.4.4 How a cavity-QD system can be used for quantum in

formation processing?

In the quantum information processing scheme proposed by Duan and Kimble [16], an

atom (or a quantum dot in our case) possesses ground, metastable, and excited states,

and the ground-to-excited state transition is strongly coupled to a cavity. When

the quantum dot is in the ground state, an incident photon that is sent at the dot

resonance cannot enter the cavity due to the vacuum Rabi splitting of the energy levels


Cavity transmission for g=K Cavity transmission for g=K/4

w-coc [g] o)-coc [g]

Figure 1.5: The transmission of a coherent field at a frequency uii through a cavity with frequency uic depends on the presence of a QD, the coupling strength g, and the strength of the driving field. In (a) the transmission of a cavity with a strongly coupled QD (g = K) is shown for various values of the average photon number inside the cavity. As the driving strength increases, the QD saturates nonlinearly an tends toward the empty cavity transmission function (dashed line). In (b) the same series is shown for a weakly coupled dot g = K/4 . AS in the case of a strongly coupled system, the QD prohibits transmission at its resonance.

and cannot pass. However, when the dot is in the decoupled (metastable) state, the

photon can be transmitted. Thus, the cavity essentially acts as a read/write interface

between a quantum memory that is realized by the quantum dot. The interference of

photons that interact with such cavities can be used to realize entangling operations on

QDS and therefore this system acts as a resource for quantum information processing.

Furthermore, such a strongly coupled system can be used to realize a quantum state

preparation device, because incident coherent light with Poissonian photon statistics

will be converted to non-Poissonian light, since the probability of transmitting higher

photon number states is suppressed at the dot resonance. In the ultimate limit of

strong coupling with high-Q cavities, a photon number state generating device can

be realized [19].

Unfortunately the realization of a robust three-level QD-cavity system is not

straightforward and may require controlled charging of a QD in a strong magnetic


field [20]. However, the nonlinear behavior of the QD response to the driving field

(shown in Fig.1.5) can be used to realize deterministic controlled logic between pho

tonic qubits even with only a two-level system. Such a scheme for quantum logic

between photons was first realized by Turchette et. al. in an atomic system [8], and

extended to the solid state in our work [2].

1.5 Brief introduction to quantum information pro

cessing

Since their inception in 1981 [21], quantum computers have been an active area of

research. Their greatest promise lies in the ability to solve intractable problems

that are of fundamental importance to fundamental science [21], drug discovery, and

information processing [22, 23]. The main concepts behind quantum information

processing will be discussed below.

In parallel, the application of quantum systems to unconditionally secure commu

nication has emerged as a novel technology [24, 25, 26, 27] and has seen the most

progress in deployment and implementation, with several systems currently operating

around the world and several companies commercializing this technology 3.

Most quantum information processing systems for computation or communication

rely on photons as carriers of information [24, 28]. Photons are ideally suited for this

task because they propagate over long distances with low loss, therefore preserving

information, and do not require large operating powers. Photons can also exist in

several convenient orthogonal logic states: horizontal and vertical polarization (or

right hand and left hand circular), propagation in one of two physical channels, and

propagation in one of two time slots.

There are two requirements in quantum information processing with photons [29].

First, one must be able to manipulate each photon individually. For logical bits stored

3Companics providing quantum cryptography systems: id Quantique sells quantum key distribution products, and was used in the 2007 Swiss national elections to transmit ballot information in Geneva MagiQ Technologies sells quantum devices for cryptography SmartQuantum provides hardware solutions for quantum and digital cryptography


in the horizontal or vertical polarization, this is easily accomplished with waveplates.

Second, a two-photon logic gate is required. Such a gate takes two photons and, if

both are vertically polarized, changes one of them to horizontal. Since photons do

not interact with each other, such a gate requires a nonlinear medium whose response

depends on the number of photons. Finding a sufficiently nonlinear medium that has

low losses has been a great challenge and one of the biggest impediments to realizing

such gates. It will be shown in Chapter 4 that such a medium can be engineered by

placing a semiconductor quantum dot inside an optical cavity in a photonic crystal.

The advantage of quantum bits for information processing stems from the number

of states that a string of N quantum bits can occupy. A classical bit can only exist in

either one of two states [classical) — |0) or [classical] — |1). A quantum bit, on the

other hand, can exist in an arbitrary superposition of these states [quantum) = a |0) +

b |1) where \a\ +[b\ = 1. For example, we can start with two physical quantum bits in

the state [quantum) = |0) |0) and rotate each bit to the state 4|(|0) + |1)). The overall

state becomes [quantum) = \ (|0) |0) + |0) |1) + |1) |0) + |1) |1)). Thus, N quantum

bits can be used to create all 2N states of length N, which are all possible classical input

bit strings. Quantum mechanical operations are linear, and therefore the "quantum

computing device" that operates on the physical bits computes all possible outputs

for all possible inputs in a single computational step. However, only one bit string

of length N can be measured at the outcome of the computation. Thus, it is not

possible to retrieve all the answers, and the quantum computer must take advantage

of the quantum encoding before the state is measured. Fortunately, two powerful

algorithms due to Shor and Grover exist, and are sufficiently interesting to motivate

research in this area. Although Shor's algorithm for prime number factorization has

been the most notorious for its promise to break encryption, Graver's algorithm for

unsorted database searches is particularly useful for problems such as drug discovery

and processing of large data-sets [22, 23].


1.5.1 Encoding information in photon states

The single photon is an elementary constituent of light and can be used to encode

information in several schemes. The fundamental difference between a single photon

and a classical photon pulse can be illustrated by the output of a beamsplitter acting

on either of the states. It is well known that a 50/50 non-polarizing beamsplitter

(NPBS) divides an incident coherent pulse \a) (photon number is given by \a\ ) into

two equal energy pulses with 1/2 of the energy of the input pulse. A single photon,

however, can only go into one of the ports. Denoting by UNPBS the beamsplitter

operator, and assigning labels 1 and 2 to the two input ports and 3 and 4 to the two

output ports, the action of the NPBS on a coherent state and a vacuum state arriving

at ports 1 and 2, respectively is:

UNPBS |a)i |0)2 = a \

7 2 / 3 +

a \

7i/4

Which means that we have two photon packets in two output ports with equal ampli

tude. However, when the beamsplitter acts on a single photon, the photon can only

go into one of two ports:

1

V2 UNPBS | l)x |0)2 = ^ = (|0)3 |1}4 + |1)3 |0}4)

This is an entangled state of the photon and the two channels and is a powerful

medium for the transmission and communication of quantum information between

quantum nodes. There are several ways to encode quantum information in photons:

Dual rail:

The above example results in a quantum correlation between the state of the photon

and the two ports of the beamsplitter. This is the "dual-rail" representation, where

the single photon can exist in one of two channels spatial channels. The above single

photon state is then the superposition state of logical qubits ^ ( | 0 ) + |1))

Single rail (time energy):

We can also divide the photons between different time bins that are determined by a

computer clock. Thus a photon can exist in superpositions of occupying several time


bins.

Polarization encoding:

The photon can exist in superpositions of polarization states such as horizontal and

vertical (\H) , \V)) or right hand and left hand circular (|±) = ^(\H) ± i \V))).

The results of our work are most directly applicable to polarization and single

rail encoding strategies. In order to take advantage of the dual rail encoding scheme,

momentum preserving resonators, such as ring resonators, may be necessary and will

be the subject of future research.

1.5.2 Quantum operations on photons

It was shown by DiVincenzo [29] that single and two-qubit gates are universal for

quantum computation. Several single qubit gates and two-bit gates can be considered.

However, a complete set is formed from arbitrary single qubit rotations and the two-

photon controlled-NOT (CNOT) gate.

In our work, qubits are encoded in the polarization state of photons, such as

horizontal and vertical (\H), \V)) or circular (\H) ±i\V)) polarization. The optical

cavity used in our experiments is a one-sided polarizing cavity that is formed from a

linearly polarized PC cavity with a distributed Bragg reflector (DBR) placed under

neath. The DBR eliminates radiative losses from the back mirror of the cavity. The

particular geometry of this PC cavity results in a fundamental mode that is linearly

polarized (see Chapter 2). Thus, only photons that are polarized along the cavity can

couple to it can interact with each other via a cavity-embedded nonlinear medium.

Single qubit gates

In the polarization encoding, an arbitrary single qubit state is given by:

\i,)=a\H) + b\V) (1.14)

with \a\ +\b\ = 1 with a, b being complex numbers. Single qubit gates manipulate the

complex coefficient a,b. An arbitrary polarization state can be generated with phase


plates. The particularly useful gate operations are the Hadamard gate, controlled

phase gate, TT/8 gate and the X,Y,Z gates [30]. The Hadamard gate is one of the

most useful, and is given by:

(1.15)

It can be easily seen that the Hadamard transformation is realized with a A/2

wave plate (HWP) set to 7r/8.

/ cos(2<9) sin(20) \ HWP(8) = V ' (1.16)

\ sin(20) - cos(26>) /

Two-qubit gates

The two-qubit state (using qubits encoded in photon polarization) can be written

as \V)S\H)C, where subscripts label the signal photon s and control photon c. The

controlled-NOT (CNOT) gate changes the state of the signal photon conditioned on

the state of the control photon:

\H)a\H)c-*\H)a\H)c (1.17)

\V)s\H)c^\V)s\H)c (1.18)

\H).\V)e^\V),\V)e (1.19)

\V)a\V)e-*\H)a\V)c (1.20)

In a photonic system, the CNOT gate can be obtained from another two-photon

interaction called the controlled phase gate and manipulation of the two photons via

Hadamard transforms (enabled by waveplates). For polarization encoded qubits, the

controlled phase gate can be easily realized by a linearly polarized cavity containing a

nonlinear optical medium whose refractive index is sensitive to the number of photons

inside of it. This means that given signal and control photon numbers ns and nc inside


the cavity, the phase shift <f> acquired by the photons inside the cavity satisfies:

|0(nc) + (f)(ns) - 4>{nc + na)\ = A > 0 (1.21)

In the case of A=7T, the interaction results in a controlled-phase (controlled-X,i.e.

CZ) gate. The details of the operation of the CZ gate, the transformation between

the CNOT and CZ gate, and the realization of such a gate in a linearly polarized

nonlinear cavity are given in Appendix A.

In Chapter 3 we will show that classical nonlinearities cannot easily satisfy these

conditions at the single photon levels, and present our preliminary results toward the

realization of the CZ gate in photonic crystal cavities in Chapter 4.

Chapter 2

Photonic Crystal Cavity Design

2.1 Introduction

Photonic crystals can be viewed as a " breadboard" for photonics on the semiconduc

tor chip. These crystals are formed by periodically changing the refractive index of

a thin semiconductor slab in two dimensions. Through clever manipulation of the

refractive index, a variety of optical elements including cavities, waveguides, focusing

and dispersive elements can be made and connected on the same chip in one mono

lithic step. However, the design problem is not simple. The space of solutions may

have many local optima and analytical techniques are seldom used. The optical res

onator is the most relevant element for quantum information processing experiments,

and in this chapter we discuss our efforts at reducing the complexity of techniques

used in finding optimal solutions for the design of cavities with high quality factors

and small optical volumes.

The main result is an analytical model that reduces the design problem from a

computationally intensive search to a simple analytical inversion of Maxwell's equa

tions when the cavity is formed along a one direction of high symmetry [31] given in

Section 2.5

Although the results of this approach are quite satisfactory, the resulting devices

are difficult to fabricate. Using the intuition developed in our analytical work, we

20

CHAPTER 2. PHOTONIC CRYSTAL CAVITY DESIGN 21

present an optimization approach based on Genetic algorithms [32] that can be con

strained to produce realistic devices in Section 2.6.

2.2 A simple look at photonic crystal design

Photonic crystals (PCs) are made by introducing periodic variations in refractive

index that lead to a bandstructure for photons in one, two and three dimensions. The

periodic nature of the refractive index leads to Bloch states of photons supported in

the structure and results in the formation of energy band gaps for photons, as a result

of the distributed Bragg reflection (DBR). Photons with energies in the energy gap

cannot propagate through such a structure and can therefore be confined and trapped

in defects in the periodic dielectric structure. In the directions without periodicity,

light is confined by total internal reflection (TIR). The operating principle of a two-

dimensional PC are shown in Fig. 2.1.

The periodicity of the dielectric constant allows us to describe the in-plane wave

vector for photons in terms of the irreducible Brillouin zone of the underlying periodic

lattice. The direct and reciprocal lattice along with the irreducible Brillouin zone and

photon energy band structure for a triangular lattice PC are shown in Fig. 2.2, where

it can be seen that the reciprocal lattice is also triangular, but rotated relative to the

original lattice by 30°.

As can be seen from Fig. 2.2 the so-called light line accounts for the boundary

between lossy and confined modes. The light line is given by the condition:

^ = ̂ T S (2.1)

where kx, ky are the in-plane wave vector components, A is the wavelength and no is

the index of refraction of the medium surrounding the PC. The problem of designing

high quality cavities with long photon storage times can be reduced to the problem

of shaping the mode in reciprocal space (k-space) to contain most of its energy below

the light line. The k-space distribution is strongly affected by the dielectric struc

ture of the PC, and the refractive index must be designed to minimize the spread


and minimize the amplitude of k vectors close to the origin of the Brillouin zone (T

point ). This concept forms the basis of our discussion in the next few sections in

which we will first formulate the out-of-plane losses in terms of the electromagnetic

energy distribution in reciprocal space within the Brillouin zone. Then, given a near-

optimal distribution we will find the refractive index profile that can accommodate

this solution. We show that this procedure is analytically solvable under symme

try constraints, and that this concept can be more generally applied to numerical

optimizations.

Figure 2.1: A two-dimensional photonic crystal and light confinement mechanisms. In-plane confinement is given by Bragg scattering due to the refractive index contrast at the PC holes. Out-of-plane confinement is the result of total internal reflection at the interface between the higher index PC and lower index surroundings. A cavity is realized by introducing a defect into the periodic lattice of holes with period a and radius r. The thickness of the membrane is given by d. Waveguides can be formed by removing rows of defects, and coupling between cavities and waveguides can be controlled by the number of holes between them and their relative orientation. In this work, the majority of the cavities were fabricated in GaAs with an index of 3.6 and measured in vacuum with an index of 1. The typical membrane thickness d is 160nm and the periodicity a is 246nm corresponding to a resonant cavity wavelength in the range of 910-950nm. and the radius r is typically on the order of 70nm.

2.3 Numerical solution methods

The evolution of electromagnetic radiation inside a dielectric material is given by the

well known Maxwell equations [33]. In source-free media with linear susceptibilities.


Figure 2.2: (a) Schematic representation of the photonic crystal lattice with lattice vectors ai,a2 that map out the triangular lattice, (b) The reciprocal lattice corresponding to the triangular lattice in (a) with reciprocal lattice vectors 61,62, where |6i| = |62| = 47r/\/3a. The first Brillouin zone is indicated by the blue line and the irreducible Brillouin zone is shaded in green. The black circle corresponds to the light line (or cone) given by Eq. 2.1. In (c) we show the resulting band structure of the triangular crystal with r = 0.3a, d = 0.65a and n = 3.6. The filled circles correspond to frequencies of confined states inside the lattice at a particular value of the wave vector as it traces out the edge of the irreducible Brillouin zone indicated by the green line in the inset in (c). States above the light line (solid line) correspond to lossy states (with non-imaginary values of kz).


the equations reduce to the following wave equation for the electric field (with a

similar equation for the magnetic field):

V x V x £ ( r > - ^ ^ T (2-2)

The wave equation is numerically solved in the time domain using the Finite Dif

ference Time Domain - FDTD algorithm, which accurately models radiative losses and

simulates boundary conditions that appropriately simulate free space; the overview

of these numerical methods can be found in Ref. [34]. In most of the work PCs are

discretized with 20 points (unit spatial increments) per period a.

2.4 Bloch modes, reciprocal space and cavities

As discussed previously, our optimization technique relies on tailoring the electro

magnetic mode distribution in reciprocal space. In this section we give the Fourier

formulation of the fundamental PC lattice and waveguide modes. The cavity is formed

by closing the ends of the PC waveguide and can therefore be expanded in waveguide

modes. This results in a formulation of the electromagnetic problem that is conve

nient for an analytical solution, as the mode is constrained to translational invariance

in one dimension.

The periodic refractive index e(f) — e(f+R) can be written as a Fourier sum over

spatial frequency components in the periodic plane:

e(r) = J2 W** (2-3) G

Here G are the reciprocal lattice vectors in the (kx, ky) plane and are defined by

G • R = 27rm for integer m. The real and reciprocal lattice vectors for the square and

hexagonal lattices with periodicity a are:


Square Lattice:

Rmj = max + jay (2.4) - 2-rrq A 2TTI A

G? / = x H y a a

Hexagonal Lattice:

- (x + yy/3) {x- yV3) Rmj = ma- + ja- (2.5)

^ = _4TT_ (-xV3 + y) 4TT (xy/3 + y)

ay/3 2 a\/3 2

where m, j , q and / are integers. The electromagnetic field corresponding to a

particular wave vector k inside such a periodic medium can be expressed as a Bloch

mode [35]:

Ej: = e^J2\G^ (2.6) G

One-dimensional Bloch modes can be formed inside of photonic crystal waveguides

that are formed by linear defects in the PC lattice. In Fig. 2.4, we plot the dispersion

of a waveguide in the YJ direction of a hexagonal lattice PC along with modes at

several points at the edge of the Brillouin zone.

Cavity modes can then be formed by closing a portion of a waveguide, i.e., by

introducing mirrors to confine a portion of the waveguide mode. In case of perfect

mirrors, the standing wave is described by H = akQHkQ + a_fc0i/_fe0. (Here we focus

on TE-like PC modes, and discuss primarily Hz, although similar relations can be

written for all other field components). Imperfect mirrors introduce a phase shift upon

reflection; moreover, the reduction of the distance between the mirrors (shortening of

the cavity) broadens the distribution of k vectors in the mode to some width Ak. The

optimization problem can then be reduced to obtaining the appropriate distribution of

holes at the "closing" points of the waveguide, that lead to reflections and scattering


Figure 2.3: Left: Band diagram for hexagonal waveguide in IV direction, with r/a = 0.3, d/a = 0.65, n = 3.6. The bandgap (wedged between the gray regions) contains three modes. Mode B00 can be pulled inside the bandgap by additional neighbor hole tuning. Right: Bz of confined modes of hexagonal waveguide. The modes are indexed by the 5-field's even ("e") or odd ("o") parities in the x and y directions, respectively. The confined cavity modes B00, Bee, and Beo required additional structure perturbations for shifting into the bandgap. This was done by changing the diameters of neighboring holes.

that lead to cancellation of far-field radiated components.

2.5 Inverse Approach

This section describes a general recipe for designing high-quality factor (Q) photonic

crystal cavities with small mode volumes. It is based on an equal contribution of

I.F., Dirk Englund and Jelena Vuckovic as described in [31]. We first derive a simple

expression for out-of-plane losses in terms of the /c-space distribution of the cavity

mode in a layer just above the PC membrane. Using this, we select a field that will

result in a high Q. We then derive an analytical relation between the cavity field

and the dielectric constant along a high symmetry direction, and use it to confine our

desired mode. By employing this inverse problem approach, we are able to design

photonic crystal cavities with Q > 4 • 106 and mode volumes V ~ (A/n)3. Our


approach completely eliminates parameter space searches in photonic crystal cavity

design, and allows rapid optimization of a large range of photonic crystal cavities.

Finally, we study the limit of the out-of-plane cavity Q and mode volume ratio.

2.5.1 Simplified relation between Q of a cavity mode and its

fc-space Distribution

In order to simplify PC cavity optimization, we first derive an analytical relation

between the near-field pattern of the cavity mode and its quality factor in this section.

Q measures how well the cavity confines light and is defined as

Q = ^ y (2-7)

where u is the angular frequency of the confined mode. The mode energy is

(U) = J \{eE2 + ^H2)dV (2.8)

The difficulty lies in calculating P, the far-field radiation intensity.

Following Vuckovic's prior work [36], we consider the in-plane and out-of-plane

mode loss mechanisms in two-dimensional photonic crystals of finite depth separately:

<P> = ^ l ) + (Px> (2-9)

or

£ = 4 + £ (210)

In-plane confinement occurs through DBR. For frequencies well inside the photonic

band gap, this confinement can be made arbitrarily high (i.e., Q\\ arbitrarily large) by

addition of PC layers. On the other hand, out-of-plane confinement, which dictates

Q±, depends on the modal k-distribution that is not confined by TIR. This distribu

tion is highly sensitive to the exact mode pattern and must be optimized by careful

tuning of the PC defect. Assuming that the cavity mode is well inside the photonic

band gap, Qj_ gives the upper limit for the total Q-factor of the cavity mode.


Given a PC cavity or waveguide, we can compute the near-field using Finite

Difference Time Domain (FDTD) analysis. As described in Reference [3], the near-

field pattern at a surface S above the PC slab contains the complete information

about the out-of-plane radiation losses of the mode, and thus about Q± (Fig. 2.4).

<x,y,z

Figure 2.4: Estimating the radiated power and Q± from the known near field at the surface S

The total time-averaged power radiated into the half-space above the surface S

is:

TT/2 2TT

f f d9d(i>sm{9)K(0,<t>), (2.11)

o o

where K(9, (j>) is the radiated power per unit solid angle. In Appendix B, we derive

a very simple form for K in terms of 2D Fourier Transforms (FTs) of Hz and Ez at

the surface S, after expressing the angles 9, 4> in terms of kx and ky:

77 A* K{kx,ky) = 2^2^.2

1 \FT2{EZ)\2 + \FT2(HZ (2.12)

Here, r\ = 4/—, A is the mode wavelength in air, k = 2ir/\, and fc|j = (kx,ky) =


k(s'm6cos(fi,sin#sin</>) and kz = kcos(6) denote the in-plane and out-of-plane k-

components, respectively. In Cartesian coordinates, the radiated power (2.11) can

thus be re-written as the integral over the light cone, k^ < k. Substituting (2.12) into

(2.11) gives

V f ^ ' / / CtKxu,Ky

2A k Jk <k &ii ~\FT2(EZ)\2 + \FT2(HZ)\2

V (2.13)

This is the simplified expression we were seeking. It gives the out-of-plane ra

diation loss as the light cone integral of the simple radiation term (2.12), evaluated

above the PC slab. Substituting Eq. (2.8) and Eq. (2.13) into Eq. (2.7) thus yields

a straightforward calculation of the Q for a given mode. In the following sections,

when considering the qualitative behavior of (2.13), we will restrict ourselves to TE-

like modes, described at the slab center by the triad (Ex, Ey,Hz), that have Hz even

in at least one dimension x or y. For such modes, the term \FT2(HZ)\2 in (2.13) just

above the slab is dominant, and \FT2(EZ)\2 can be neglected in predicting the general

trend of Q.

The figure of merit for cavity design depends on the application: for spontaneous

emission rate enhancement through the Purcell effect, one desires maximal Q/V; for

nonlinear optical effects Q2/V; while for the strong coupling regime of cavity QED,

maximizing ratios gJK ~ Q/W and g/j ~ 1/y/V is important. In these expressions,

V is the cavity mode volume: V = (/e(r)\E(r)\2d3r)/m&x(e(f)\E(r)\2), g is the

emitter-cavity field coupling, and K and 7 are the cavity field and emitter dipole

decay rates, respectively, as introduced previously [36].

Thus, for a given mode pattern, we have derived a simple set of equations that

allow easy evaluations of the relevant figures of merit. In the next section, we address

the problem of finding the field pattern that optimizes these figures of merit and later

derive the necessary PC to support it.


2.5.2 Opt imal k-space d is tr ibut ion

The magnetic field in a closed waveguide cavity can be written in terms of the forward

and backward propagating waveguide modes as:

ko+Ak/2

#.(*,*) ~ E E (\G^+ A-i&-**) ^ (2-14) G feo-Afe/2

A similar expansion of Hz can be made at the surface S directly above the PC

slab (see Fig. 2.4), which is relevant for calculation of radiation losses. The Fourier

transform of the above equation gives the fc-space distribution of the cavity mode,

with coefficients A% Q and A_^ Q. The distribution peaks are positioned at ±k0 ± G,

with widths directly proportional to Ak. The fc-space distribution is mapped to other

points in Fourier space by the reciprocal lattice vectors G. To reduce radiative losses,

the mapping of components into the light cone should be minimized [3]. Therefore, the

center of the mode distribution k0 should be positioned at the edge of the first Brillouin

zone, which is the region in &-space that cannot be mapped into the light cone by

any reciprocal lattice vector G (see Fig. 2.2). For example, this region corresponds to

k0 = ±jx^kx±jy^ky for the square lattice, where jx,jy e 0,1. Clearly, \jx\ = \jy\ = 1

is a better choice for k0, since it defines the edge point of the 1st Brillouin zone

which is farthest from the light cone. Thus, modes centered at this point and fc-space

broadened due to confinement, will radiate the least. Similarly, the optimum k0 for

the cavities resonating in the TJ direction of the hexagonal lattice is ko = ±£fcx (as

it is for the cavity from Ref. [37], and for the TX direction is k0 = ±-2y=ky (as it is

for the cavity from Refs. [36] and [3]).

Assuming that the optimum choice of k0 at the edge of the first Brillouin zone has

been made, the summation over G can be neglected in Eq. (2.14), because it only

gives additional Fourier components which are even further away from the light cone

and do not contribute to the calculation of the radiation losses. In that case, we can

express the field distribution as:

Hz(x,y)= [ fdkxdkyA(kx,ky)e^, (2.15)


where A(kx, ky) is the Fourier space envelope of the mode, which is some function cen

tered at k0 = (±kox, ±koy) with the full-width half-maximum (FWHM) determined

by Ak = (Akx,Aky) in the kx and ky directions respectively. Eq. (2.15) implies

that Hz(x,y) and A(kx,ky) are related by 2D Fourier transforms. For example, if

A(kx, ky) can be approximated by a Gaussian centered at kQ = (k0x, k0y) and with the

FWHM of (Akx, Aky), the real space field distribution Hz(x, y) is a function periodic

in the x and y directions with the spatial frequencies of kox and k0y, respectively, and

modulated by Gaussian envelope with the widths Ax ~ l/Akx and Ay ~ 1/Aky.

Therefore, the properties of the Fourier transforms imply that the extent of the mode

in the Fourier space AA; is inversely proportional to the mode extent in real space

(i.e., the cavity length), making the problem of Q maximization even more challeng

ing when V needs to be simultaneously minimized. This has already been attempted

in the past for a dipole cavity [36, 3] and a linear defect [37, 38, 39], by using extensive

parameters space search. In the following sections, we will design high Q cavities by

completely eliminating the need for parameter space searches and iterative trial and

error approaches.

There are two main applications of Eq. (2.13). First, this formulation of the

cavity Q factor allows us to investigate the theoretical limits of this parameter and

its relation to the mode volume of the cavity. Second, it allows us to quantify the

effect of our perturbation on the optimization of Q using only one or two layers of the

computational field and almost negligible computational time compared to standard

numerical methods. We applied Eq. (2.13) to cavities obtained from an iterative

parameter space search. These cavities were previously studied in [3] and [37]. The

results for Q using Eq. (2.13) at S, as well as full first-principle FDTD simulations are

shown in Fig. 2.5. A good match is observed. Therefore, our expression (2.13) is a

valid measure of the radiative properties of the cavity and can be used to theoretically

approach the design problem; we can also use this form to speed up the optimization

of the cavity parameters. The discrepancy between Eq. (2.13) and FDTD is primarily

due to discretization errors.


o o 4

O FDTD

D Estimated

0 0 0 0 0 0 0 I- ) 0 0 0 0 0 0 (

o o «-+o o I- ) 0 0 0 0 0 0 C

0 0 0 0 0 0 0

3.05 0 0.05 0.1 0.15 0.2 0.25 Shift (in units of the periodicity a)

1.4

1.2

O FDTD

• Estimated

) 0 0 0 0 ( o o o o o

0.05 0.1 0.15 Shift (in units of the periodicity a)

0.3 0.35

0.2

Figure 2.5: Comparison of Q factors derived from Eq. (2.13) (squares) to those calculated with FDTD (circles). Top: cavity made by removing three holes along the YJ direction confining the Boe mode. The Q factor is tuned by shifting the holes closest to the defect as shown by the red arrow. The x-scxis gives the shift as a fraction of the periodicity a. Bottom: the X dipole cavity described in [3]. The Q factor is tuned by stretching the center line of holes in the FX direction, as shown by the arrow. The x-axis gives the dislocation in terms of the periodicity a.


2.5.3 Inverse problem approach to designing PC cavities

In the inverse approach, we begin with a desired in-plane Fourier decomposition of

the resonant mode, FT2(H(f)), chosen again to minimize radiation losses given by

Eq. (2.13). The difficulty lies with designing a structure that supports the field which

is approximately equal to the target field, H{f).

In this section, we first estimate the general behavior of Q/V for structures of

varying mode volume. Then we present two approaches for analytically estimating

the PC structure e(f) from the desired &-space distribution FT2(H(r)). As mentioned

in Sec.2.5.1, we restrict the analysis to TE-like modes Beo,Boe, and Bee (Fig.2.4(b))

for which we can approximate the trend of the radiation (2.13) by considering only Hz

at the surface S just above the PC slab. Moreover, to make a rough estimate of the

cavity dielectric constant distribution from the desired Hz field on S, we approximate

that Hz at S is close to Hz at the slab center.

2.5.4 General trend of Q/V

The simplification described above allows us to study the general behavior of Q/V

for a cavity with varying mode volume. Here, we assume that a structure has been

found to support the desired field Hz.

We again start from the expression for radiated power, Eq. (2.13), and calculate

Q using Eq. (2.7). All that is required of the cavity field is that its FT at the surface

S above the slab be distributed around the four points kx$ = ±n/a,kyo = ±-^=-, to

minimize the components inside the light cone. As an example, we choose a field

with a Gaussian envelope. For now, let us consider mode symmetry Boe. The Fourier

Transform of the Hz field is then given by

FT2{HZ) = Y, sign(kx0)exp(-(kx - kx0)(ax/V2)2 - (kv - ky0)(ay/V2)2), (2.16)

where ox and ay denote the modal width in real space. The mode and its FT are

shown in Fig.2.6 (d-e). We use Eq.2.13 without the Ez terms to estimate the trend in

Q, as described above. As the mode volume grows, the radiation inside the light cone


shrinks exponentially. This results in an exponential increase in Q. This relationship

is shown in Fig.2.6(g) for field Boe at frequency a/A = 0.248. At the same time, the

mode volume grows linearly with ax. The growth of Q/V is therefore dominantly

exponental , and we can find the optimal Q for a particular choice of mode volume

(i.e. <jx) of the Gaussian mode cavity.

(a)Hz WFT^hy <c)K(kx,ky)

a,ky/jt t

f /v i / / /

J ' 1 / a - 2

(i) K(kx,kv)

0.5 1.5 2

Figure 2.6: Idealized cavity modes at the surface S above the PC slab; all with mode volume ~ (A/n)3. (a-c) Mode with sine and Gaussian envelopes in x and y, respectively: Hz(x,y), FT2(HZ), and K(kx,ky) inside the light cone; (d-g) Mode Boe

with Gaussian envelopes in the x and y directions : Hz(x,y), FT2{HZ), K(kx,ky), and Q±(<Jx/a) as well as Q±/V . Q± was calculated using Eq. (2.13) and Ez was neglected.; (h-i) Mode Bee with Gaussian envelopes in x and y can be confined to radiate preferentially upward.

According to Fig.2.6(g), very large Qs can be reached with large mode volumes

and there does not seem to be an upper bound on Qj_. As the mode volume of the

Gaussian cavity increases, the radiative Fourier components vanish exponentially, but

are never zero. A complete lack of Fourier components in the light cone should result

in the highest possible Q. As an example of such a field, we propose a mode with a


sine envelope in x and a Gaussian one in y. The FT of this mode in Fig.2.6(b) is

described by

FT2{HZ)= Yl eM~(K ~ ky0)2(ay/V2)2)Rect(kx - kx0,Akx), (2.17)

KxQ tfcyO

where Rect(kx, Akx) is a rectangular function of width Akx and centered at kx.

The Fourier-transform implies that the cavity mode is described by a sine function

in x whose width is inversely proportional to the width of the rectangle Rect(kx, Akx).

To our knowledge, this target field has not been previously considered in PC cavity

design. This field is shown in Fig.2.6(a-c). Though it has no out-of-plane losses, this

field drops off as - and therefore requires a larger structure than the Gaussian field

for confinement.

Over the past years, many new designs with ever-higher theoretical quality factors

have been suggested [39]. In light of our result that Q/V increases exponentially with

mode size, these large Qs are not surprising.

It is interesting to note that Eq. (2.12) also allows one to calculate the field

required to radiate with a desired radiation distribution. For example, many applica

tions require radiation with a strong vertical component; waveguide modes with even

Hz can be confined for this purpose so that K(kXlky) dominates losses at the origin

in k—space, as shown for instance in Fig. 2.6(h-i) for the confined mode pattern Bee

of Fig.2.4.

2.5.5 Estimating Photonic Crystal Design from £>space Field

Distribution

Now we introduce two analytical ways of estimating the dielectric structure e(f) that

supports a cavity field that is approximately equal to the desired field Hc. These

methods directly calculate the dielectric profile from the desired field distribution,

without any dynamic tuning of PC parameters, and are thus computationally fast.

We focus on TE-like modes, since they see a large bandgap and exhibit electric-field


maxima at the slab center. For TE-like modes, Hc = Hcz at the center of the slab,

and Hc w Hcz at the surface. First, we relate Hc to one of the allowed waveguide

fields Hw. The fields Hc and Hw at the center of the PC slab (z — 0) are solutions to

the homogeneous wave equation with the corresponding refractive indices e.c and ew,

respectively.

f)2 ff 1 - ^ o — ^ = ulimHc = V x - V x | (2.18)

dt2 ec

-^o^-~ = (JtfM>Hw = V x i v x f f f f i (2.19) Oh €w

Here UJC and LOW are the frequencies of the cavity and waveguide fields. We expand

the cavity mode into waveguide Bloch modes:

HC = Y, CkUk{f)e^-^t] (2.20)

where uk is the periodic part of the Bloch wave. Assuming that the cavity field is

composed of the waveguide modes with k ss fc0, we can approximate uk{r) ~ Uko{r),

which leads to the slowly varying envelope approximation:

Hc « Uk^rje1^-"^ J2 ckei^-^)-r-^k-Uw)t) = HwHe, (2.21)

k

where the waveguide mode Hw = uiio(f)e^ko'r~UJ,nt"1 and the cavity field envelope He — V ^ c, ei{(k-k0)-r-(ujk-ujw)t)

The cavity and waveguide fields FT-distributions are concentrated at the edge of -, |2

the Brillouin zone, where uJi ss UJ2 + a k — ko and a « l (i.e., the band is nearly

flat). Differentiating (2.21) in time twice gives:

d2H, = -u2Hc = -J2 W^eW-^ [wl + a\k- kn\

2} r ^ j

dt2

k

-Hw ^ C k e ^ - k ^ - ^ - ^ [u2w + a\k- k0\

2] = -co2wHwHe + aHwV2He (2.22)


Thus, for a <C 1 and finite V2ife, LUC « UJW, i.e., the cavity resonance is very close

to the frequency of the dominant waveguide mode. The condition a < l , also implies

that <jjk ~ UJW, i.e. He ss J2n Ckei{k~h°yif-

Estimating Photonic Crystal Design from fc-space field Distribution: Ap

proach 1

Let us express the cavity dielectric constant ec as ec = ewee, where ee is the unknown

envelope. Hc is a solution of Eq. (2.18) and each waveguide mode Uk{r)eli-k'?~lJJkf:)

satisfies Eq. (3.2). Prom previous arguments, for k within the range corresponding

to the cavity mode, UJC « u>w « ujk. Thus, a linear superposition of waveguide modes T^2kckuk{;r)el{-k'r~UJkt'> = HeHw = Hc also satisfies Eq. (3.2), i.e. the cavity mode is

also a solution of Eq. (3.2) for a slowly varying envelope. We assume that the mode

is TE-like, so that the H field only has a z component at the center of the slab.

Then inserting Hc from (2.21) into Eq.(2.18) and Eq. (3.2) and subtracting the two

equations with ec = ewee, yields a partial differential equation for ec(x,y):

dx ^ e

-l)dxHc + 9y l ( I - i )<yf c Mo(^ - UJ2C)HC « 0 (2.23)

For this approach, we consider waveguide modes with Boe symmetry in Fig.2.4(b).

These modes are even in y, so the partial derivatives in y in (2.23) vanish at y = 0.

The resulting simplified first-order differential equation in l/ee can then be solved

directly near y — 0, and the solution is —(— — l)dxHc « C , where C is an arbitrary

constant of integration. In our analysis ew corresponds to removing holes along one

line (x-axis) in the PC lattice. The cavity is created by introducing holes into this

waveguide, which means that — — 1 > 0. The solution holds when we take the

absolute value of both its sides, and for C > 0, this leads to the following result for

the cavity dielectric constant near y = 0:

C is a positive constant of integration, and Hc = HwHe, where Hw is the known


waveguide field and He is the desired field envelope. C can be chosen by fixing the

value of ec at some x, leading to a particular solution for ec. In our cavity designs we

chose C such that the value of ec is close to ew at the cavity center. To implement

this design in a practical structure, we need to approximate this continuous ec by

means of a binary function with low and high-index materials e/ and eh, respectively.

We do this by finding, in every period j , the air hole radius r, that gives the same

field-weighted averaged index on the a>axis:

fja+a/2 ja+a/2

/ (eh + (ei -eh)Rect(ja,rj)) Ec dx = ec(x) Ec dx, (2.25) Jja—a/2 Jja—a/2

where Ec is estimated from a linear superposition of waveguide modes as Ec oc V x Hc.

We assume that the holes are centered at the positions of the unperturbed hexagonal

lattice PC holes.

The radii rj thus give the required index profile along the x symmetry axis. The

exact shape of the holes in 3D is secondary - we choose cylindrical holes for conve

nience. Furthermore, we are free to preserve the original hexagonal crystal structure

of the PC far away from the cavity where the field is vanishing.

To illustrate the power of this inverse approach, we now design PC cavities that

support the Gaussian and sine-type modes of Eq. (2.16), (2.17). In each case, we start

with the waveguide field Boe of Fig.2.4(b) confined in a line-defect of a hexagonal PC.

The calculated dielectric structures and FDTD simulated fields inside them are shown

in Fig. 2.7. The FT fields on S also show a close match and very little power radiated

inside the light cone (Fig.2.7(c,f)). This results in very large Q values, estimated from

Q± to limit computational constraints. These estimates were done in two ways, using

first principles FDTD simulations [36], and direct integration of lossy components

by Eq. (2.13). The results are listed in Table 2.1 and show an improvement of

roughly three orders of magnitude over the unmodified structure of Fig. 2.5 with

as small increase in mode volume. Furthermore, a fit of the resulting field pattern

to a Gaussian envelope multiplied by a Sine, yielded a value of ax/a « 1.6, which,

according to the plot in Fig. 2.6 g., puts us at the attainable limit of Q± at this mode

volume.


In our FDTD simulations, we verified that Q± correctly estimates Q by noting

that Q± did not change appreciably as the number of PC periods in the x— and

y— directions, Nx and Ny, was increased: for the Gaussian-type (sine-type) mode,

increasing the simulation size from Nx = 13, Nv = 13 (Nx = 21, Ny = 9) PC periods

to Nx = 25, Ny = 13 (Nx = 33, Ny = 13) changed quality factors from Q\\ = 22 •

103, Q± = 1.4 • 106 (Q|| = 17 • 103, Q± = 4.2 • 106) to Q,, = 180 • 103, Q± = 1.48 • 106

(Q|j = 260 • 103, Q± = 4.0 • 106). (The number of PC periods in the x-direction in

which the holes are modulated to introduce a cavity is 9 and 29 for Gaussian and

sine cavity, respectively, while both cavities consist of only one line of defect holes

in the y-direction.) Thus, with enough periods, the quality factors would be limited

to Q±, as summarized in the table. In the calculation of Q, the vertically emitted

power (F||) was estimated from the fields a distance ~ 0.25 • A above the PC surface.

Note that the frequencies a/A closely match those of the original waveguide field Boe

(a/Aca„=0.251), validating the assumption in the derivation.

Table 2.1: Q values of structures derived with inverse-approach 1

Gaussian Sine

Unmodified 3-hole defect

^V "cav

0.248 0.247 0.251

Qcav (freq. filter) 1.4 • 10b

4.2 • 106

6.6 • 103

Qcav (Eq. (2.13)) 1.6 • 10° 4.3 • 106

6.4 • 103

' mode \ v )

0.85 1.43 0.63

Estimating Photonic Crystal Design from A;-space Field Distribution: Ap

proach 2

We now derive a closed-form expression for ec(x, y) that is valid in the whole PC

plane (instead of the center line only). Again, begin with the cavity field H(r) — zHc

consisting of the product of the waveguide field and a slowly varying envelope, Hc =

HwHe, and treat the cavity dielectric constant as: — = —^—h 7-. In the PC plane,


Figure 2.7: FDTD simulations for the derived Gaussian cavity (a-c) and the derived sine cavity (d-f). Gaussian: (a) Bz; (b) \E\; (c) FT pattern of Bz taken above the PC slab (blue) and target pattern (red). Sine: (d) Bz; (e) \E\; (f) FT pattern of Bz taken above the PC slab (blue) and target pattern (red) (The target FT for the sine cavity appears jagged due to sampling, since the function was expressed with the resolution of the simulations). The cavities were simulated with a discretization of 20 points per period a, PC slab hole radius r = 0.3a, slab thickness of 0.6a and refractive index 3.6. Starting at the center, the defect hole radii in units of periodicity a are: (0,0,0.025,0.05,0.075,0.1 ,0.075,0.075,0.1,0.125,0.125,0.125,0.1,0.125,0.15,0.3,0.3) for the sine cavity, and (0.025,0.025,0.05,0.1,0.225) for the Gaussian cavity.

Eq. (2.18, 3.2) for a TE-like mode can be rewritten as

-<4lM>Hc = V - ( - V # c ) (2.26)

-LU2wfi0Hw = V - ( - V # w ) (2.27)

Multiplying the last equation by He, subtracting from the first, and recalling that

UJC ~ tow yields

u2wIM)HeHw - uj2

cii0Hc = HoHc(u>2w - LO2

C) W 0 (2.28)

= V • ( -Vi f c ) - HeV • (—VHW) €c tw

sa V - ( — V H C ) (2.29) tpert


where the last line results after some algebra and dropping spatial derivatives of

the slowly varying envelope He. This relation is a quasilinear partial differential

equation in l/epert. With boundary conditions that can be estimated from the original

waveguide field, this equation can in principle be solved for ec (e.g., [40]).

Alternatively, one can find a formal solution for epert by assuming a vector function

£(r) chosen to satisfy the boundary conditions, so that

VHC = V x £ (2.30) £pert

or 1 V x £• VH*

71 = IV/T p C (2"31)

^pert | v ±±c\

This gives is the formal solution of the full dielectric constant ec = (e~^rt + e^1) - 1 in

the plane of the photonic crystal.

2.6 Genetic Algorithms

Although the inverse methods yield interesting results, they do not take into account

fabrication difficulties, and so may yield designs that cannot be realized. It is difficult

to introduce design constraints into the inverse approach. Brute force techniques can

easily incorporate fabrication constraints, but are computationally intensive, and so

a smart design technique is necessary. In large parameter spaces, the optimization

problem may have multiple local optima, and the typical gradient search methods

and convex optimization may not be applicable to the modeling of photonic crystal

resonators. One of the most promising numerical methods for the rapid design of

photonic structures is the approach of genetic algorithms (GAs). This section is

taken from Ref. [32], which came as a result of my mentorship of Joel Goh, who

was an undergraduate researcher in the Vuckovic group. I will first introduce and

describe the algorithm and then give an example application of the algorithm to

the optimization of the far field radiation of a one-dimensional resonator, which uses

either integration over the Fourier space to minimize power losses, or requires the

resulting electromagnetic mode to be closest to a desired field profile, as in Section


2.5. Thus, we can apply the theory used in the previous sections as a fitness measure

in our optimization algorithm.

2.6.1 Algorithm description

Genetic algorithms (also known as Evolutionary algorithms) are a class of optimiza

tion algorithms that apply principles of natural evolution to optimize a given ob

jective [41, 42, 43]. In the genetic optimization of a problem, different solutions to

the problem are picked (usually randomly), and a measure of fitness is assigned to

each solution. On a given generation of the design, a set of operations, analogous

to mutation and reproduction in natural selection, are performed on these solutions

to create a new generation of solutions, which should theoretically be "fitter" than

their parents. This process is repeated until the algorithm terminates, typically after

a pre-defined number of generations, or after a particularly "fit" solution is found, or

more generally, when a generation of solutions meets some pre-defined convergence

criterion.

2.6.2 Algorithm implementation

Genetic algorithms have already been used in PC design to find non-intuitive large-

bandgap designs [44, 45] and for designing PC fibers [46]. In this work, we implement

a general GA to optimize 1 and 2-dimensional photonic bandgap structures, and show

that it is able to robustly optimize these structures for a wide variety of objectives.

In the 1-dimensional case, we consider the design of planar photonic crystal cavities

(which are infinite in extent in the remaining two spatial dimensions) by varying the

widths of dielectric stacks; and in the 2-dimensional case, we perform the genetic

optimization by varying the sizes of circular holes in a triangular lattice. These

approaches were chosen because the search space is conveniently well-constrained in

these paradigms, and the optimized structures (for the triangular lattice) can be easily

fabricated.

In addition, we used the following parameters for the implementation of our GA:


Chromosome encoding. We used a direct-chromosome encoding, where the var

ious optimization parameters were stored in a vector. For the current sim

ulations, for simplicity, we only varied the radii of cylindrical holes in a tri

angular lattice. Our implementation can be easily modified to include other

optimization parameters as well, such as the positions of the various holes, or

the refractive index of the dielectric material.

Selection. We used fitness-proportionate selection (also known as roulette-wheel

selection), to choose parent chromosomes for mating. In this selection scheme,

a chromosome is selected with a probability Pj that is proportional to its fitness

fi, as shown in Eq. (2.32).

Pi = -j~- (2-32)

IVIating. After a pair of parent chromosomes Vparent,\ and vparentt2 were selected,

they were mated to produce a child chromosome vchud by taking a random

convex combination of the parent vectors, as in Eq. (2.33).

A ~ Z7(0,1)

V child = ^parent,! + (1 — tyvparent,2 (2.33)

Muta t ion . Mutation was used to introduce diversity in the population. We used

two types of mutation in our simulations, a random-point crossover and a gaus-

sian mutation.

1) Random-point crossover: For an original chromosome vector vorig of

length N, we select a random index, k, from 0 to N as the crossover point, and

swap the two halves of v^g to produce the mutated vector, vmut, as represented


in Eq. (2.34).

Vorig = (Vl,V2j..-,VN)T

k ~ U{0,1,2,....,N}

Vmut = (vk+1,Vk+2,...,VN,Vi,V2,...,Vk^i)T (2.34)

2) Gaussian mutation: To mutate a chromosome vector by Gaussian mu

tation, we define each element of vmut to be independent and identically dis

tributed Gaussian Random Variables with mean v^g and a standard deviation

proportional to the corresponding elements of vorig. This searches the space in

the vicinity of the original chromosome vector vorig-

v™ut ^N^f^a2), i e {0,1, 2, . . . . ,#} (2.35)

a2 is an algorithm-specific variance, and can be tuned to change the extent of

parameter-space exploration due to mutation.

Cloning. To ensure that the maximum fitness of the population does not decrease,

we copied (cloned) the top few chromosomes with the highest fitness in each

generation and inserted them into the next generation.

2.6.3 Simulation results: Optimizing planar photonic cavity

cavities

Q-factor maximization

As in Section 2.5, we wish to minimize the fc-space amplitude inside the light cone, to

minimize radiation losses. Since it was shown that the inverse design problem leads

to satisfactory results when the problem is reduced to an effectively one-dimensional

problem, we decided to use this one dimensional problem as the starting point of the

optimization. We used one-dimensional photonic crystals as an approximation to the

closed waveguide cavities, and simulated them using the standard Transfer Matrix


method for the E-field [35]. The reflectance spectrum of the each cavity was obtained

using the Transfer Matrix method, and we used a heuristic peak-finding algorithm to

automatically search the spectrum for sharp resonance peaks. The resulting resonant

modes were then evaluated according to the chosen fitness function (which differed

depending on our optimization objective), and the maximum fitness found from all

the resonant modes was assigned as the fitness for the particular cavity. The fitness

function was taken as:

fitness ex I / \fsim{x) - ftarget(x)\2dx I (2.36)

The results are shown in Fig. 2.8, where we applied the GA to finding a dielectric

structure that supports an electric field with a sine and sine2 like envelopes that

should result in a high degree of suppression of radiative components at the k\\ = 0

point. Although the matching to both envelopes is not exact, it is quite promising.

Part of the error arises due to the finite simulation structure and discretization used

in the simulations. This work demonstrates that the GA is a good candidate for

optimizing PC cavities when the calculation time per structure is short, since many

trial solutions must be made per convergence cycle.

2.7 Conclusions

We have described a simple recipe for designing two-dimensional photonic crystal

cavities. Although the approach is general, we have demonstrated its utility on the

design of cavities with very large Q > 106 and near-minimal mode volume ~ (A/n)3.

These values follow our theoretically estimated value of Q±/V for the cavity with the

Gaussian field envelope, which means that we were able to find the maximal Q for

the given mode volume V under our assumptions. Our approach is analytical, and

the results are obtained within a single computational step. We first derive a simple

expression of the modal out-of-plane radiative loss and demonstrate its utility by

the straightforward calculation of Q factors on several cavity designs. Based on this

radiation expression, the recipe begins with choosing the FT mode pattern that gives


Figure 2.8: Top-left: Real-space mode profile after optimizing for closest-match to a sine-envelope target mode. Top-right: k-space mode profile of the optimized simulated mode and a sine-envelope target mode. Bottom: Real-space and k-space mode profiles for matching against a sinc2-envelope target mode. All: Red curves represent the real-space (k-space) mode profiles of the optimized fields, blue curves represents the real-space (k-space) mode profiles of the target fields.

the desired radiation losses. For high-Q cavities with minimal radiative loss inside

the light cone, we show that the transform of the mode should be centered at the

extremes of the Brillouin Zone, as far removed from the light cone as possible. Next we

proved that for a cavity mode with a Gaussian envelope, Q/V grows exponentially

with mode volume V, while the cavity with the sine field envelope should lead to

even higher Q's by completely eliminating the Fourier components in the light cone.

Finally, we derived approximate solutions to the inverse problem of designing a cavity

that supports a desired cavity mode. This approach yields very simple design guides

that lead to very large Q/V. Since it eliminates the need for lengthy trial-and-error

optimization, our recipe enables rapid and efficient design of a wide range of PC

cavities.

We have also shown that the Genetic Algorithm can be successfully applied to


designing photonic structures when the conditions on the reciprocal space field profile

are chosen as fitness parameters. This kind of algorithm can be successfully applied

to optimization problems where the instances of the problem can be quickly solved.

Chapter 3

Optical nonlinearities in PC

waveguides

3.1 Introduction

Photonic crystal waveguides are well suited to facilitate nonlinear optical interac

tions. The typical cross-sectional dimensions of PC waveguides are on the order of

a2 « (\/n) , and result in large electric field amplitudes inside the structures. The

dispersion of the PC waveguide contains regions of extremely low group velocity.

Of particular interest to quantum optics and information processing are third order

Kerr nonlinearities characterized by a refractive index that depends on the number

of photons that are interacting with it. Such nonlinearities have been shown to be

extremely interesting for the realization of photon-photon interactions and quantum

nondemolition (QND) measurement of photons [47, 48, 49], or highly sensitive clas

sical detectors.

We show that, in principle, such experiments may be feasible with current photonic

technologies [50]. The major drawback to enhancing nonlinear interactions between

photons due to classical nonlinear processes is the fact that the losses due to absorp

tion and scattering are equally enhanced by the PC. Since the classical off-resonant

nonlinearity is unaffected by the PC, we cannot reduce radiative losses. It will be

shown in Chapter 4 that loss processes in resonant nonlinearities can be manipulated

48

CHAPTER 3. OPTICAL NONLINEARITIES IN PC WAVEGUIDES 49

with PC structures and are promising for photon-photon interactions and nonlinear

optics at single photon power levels.

3.2 QND photon detection with Kerr nonlineari-

ties

The principle of such a QND measurement is shown in Fig. 3.2, and is modeled

after Ref. [51]. We consider the case of a signal pulse from either a photon number

emitter or a coherent state. A typical single photon source is an InAs quantum dot

(QD) coupled to a PC cavity as in [52]. The radiative lifetime of such QD's coupled

to PC cavities is « 0.2 — Ins. In the QND measurement this signal with average

photon number Ns and phase <j>sig is combined with a strong coherent probe with

average photon number Np and phase </>pro(,e in one arm of a Michelson interferometer.

Due to the nonlinear Kerr medium, the probe acquires a phase shift that is directly

proportional to the signal photon number and is destructively detected, while the

signal acquires a phase shift, but is retained for further use. The main source of

losses in this scheme are first and second order material absorption. The probe phase

delay consists of two terms </>pro&e = 4>s + <f>P where <f>a is the probe phase shift due to

the signal photon number Ns and <pp is the phase shift due to self-phase modulation

of the probe with photon number Np.

The Kerr effect is typically characterized by either an intensity dependent refrac

tive index ra2 or a third-order susceptibility x^3\ which are related via 3 x ^ = cn2ri2,

where c is the speed of light and n is the intensity independent refractive index of

the material [53]. We will focus on Aluminium Gallium Arsenide (AlGaAs), which

has a high n2 ~ 1.51CT13£p- at a wavelength of 1500 nm and a high refractive index

n « 3.4 [54, 55]. This large refractive index is attractive for the fabrication of PC

devices and can be combined with our current QD sources.

An order of magnitude estimate of the phase due to the nonlinearity can be gained

by expanding the index n + n2I, where / is the light intensity. The intensity

dependent term adds a phase to the detection beam p, over the co-propagation length

CHAPTER 3. OPTICAL NONLINEARITIES IN PC WAVEGUIDES

Kerr Medium

x ( 3 )

Ns,4>sig 35*8*8*5* N8,(j)sig

Mi « • • • M2

Np,<fiprobe NpA probe

/ BS \ s

Figure 3.1: The setup for QND detection of a signal beam with average expected photon number iV, and phase 0sig. The probe is split on the 50/50 beamsplitter BS into two beams with, photon numbers Np and phases dproi,e. The mirrors il/ t

and M-2 pass the signal and reflect the probe with negligible losses. The two beams co-propagate in one arm of the interferometer through a PC waveguide made in a medium with a large Kerr nonlinearity (,\;(3)). The signal JV, is unmodified and only the phase is distorted to Qmg. The probe photon number is preserved, but the phase

is detected is modified to $ 'obe 9Pro6e oc Ns. The probe phase rotation o obe O probe

by a homodyne measurement on detectors Di, D2 and yields the photon number Ar.,.


L that is given by

AsApTsA

for a signal and probe photons with wavelength \s,\p and Ns signal photons,

which for an area of A — l//m2, length L = lOO/im, a lifetime of TS = Ins and

XPtS « 1.5/um gives A $ « 8 x 10~13 x iVs. In the case of a coherent probe with signal

in a number state, the signal to noise ratio of the detection scheme is (402JVP)_1,

where Np is the probe average photon number, and 4>s is the phase due to a single

signal photon [51]. This means that Np > 1023 probe photons (a 48k J pulse with a

nanosecond width) are needed for the bulk experiment in order to overcome the shot

noise in a typical ridge waveguide. In the PC waveguide the group velocity is reduced

to v « c/100 [56, 57], and leads to an increased propagation time and reduced pulse

width, which, along with an area reduction of approximately (250nm)2//um2 requires

on the order of 10_1 J per nanosecond pulse for an SNR of 1.

The PC interferometer would be made in a free standing membrane of AlGaAs

that is patterned by a hexagonal lattice of air holes. The waveguides are made by

removing rows of holes. Fig. 3.2 shows a PC waveguide and the dispersion relation

for two modes. The group velocity is significantly reduced at the band edge (kx = - ) ,

which makes this an attractive operating point. Numerical precision allows us to

estimate that vg < c x 10~2. Since the Kerr effect depends on the intensity overlap,

either two spectrally different points on the same PC waveguide band, or on different

bands can be chosen. In the first case, the intensity overlap is maximized, but there is

a potential for a large group velocity mismatch. In the second case the mode overlap

is sacrificed in favor of matching the group velocities. In principle, — w 10~6 for a

Ins pulse, which means that points with very close j values can be chosen, and the

proximity is limited by the ability to filter, or by the wavelength requirements for the

pulse and probe.


0.4 0.6 k(jt/a)

Figure 3.2: (a) waveguide mode dispersions calculated by the 3D Finite Difference Time Domain (FDTD) method. The solid (black) line is the light line in the photonic crystal, above which modes are not confined by total internal reflection. The insets show the Bz profiles of the even (i) and odd (ii) modes at the k = n/a point, (b) Group velocities of the two modes derived from the dispersion curves via numerical differentiation (Vg = ^). The group velocity is greatly reduced at the k = n/a point.


3.3 Pulse Propagation in PC Waveguides

First, we derive the equations of motion for the signal and probe pulse in the inter

ferometer. The eigenstates of the PC waveguide are solutions to:

V > V » l . - i ? f ,3,,

where e(f) is the relative spatially varying waveguide dielectric constant. The solu

tions are Bloch modes n™(f)el(fc2-"W;) where u™(r + az) = u™(r) for the lattice with

periodicity a, z is the direction of propagation along the waveguide, and m is the

index of the band of particular symmetry. These modes satisfy the wave equation:

V x V x (uk(r)eikz) = ^ ! e ( r > f c ( r ) e f c (3.3) cr

The Bloch state is re-normalized for convenience in the last part of the chapter. The

waveguide modes can be shown to obey the following orthogonality conditions :

/ d 3 re ( r )« ,e*( f c - f e > = 8mn8kk, (3.4)

where the integral is taken over the whole space Q. The Bloch modes un,um are

orthogonal over a unit cell of the waveguide, and in the case when the difference

k — k! is small, the integral may be taken over a unit cell. In what follows, index m is

dropped, unless it is necessary, and e = e(r). The waveguide modes can be rewritten

to solve a different Hermitian operator:

6 = -^=V x V x -^ (3.5)

The eigenstates of this operator are (r \u, m, k) = v^/T\f)el{~hz^ul{-k^ with eigenvalue

" a , and (u,m,k \u, n, I) = &m,n&k,i by Eq. 3.4. The inner product denotes integra

tion over all physical space.

A pulse propagating in the PC waveguide in the presence of the weak nonlinearity

may be written as E — 4j J dkA(k,t)\u,m,k) where A(k,t) is a time dependent


coefficient of each k component. The k-space range over which the integrand is

appreciable depends on the frequency distribution of the pulse. For pulses with ^ «

10~6, and group velocity of v = j^j, ^ sa 10~4. So the integrand in the expression

for E is dominated by a particular k about which the pulse can be expanded:

. i[{k-k0)z-(u(k)-wo)t] E= f dkA{k,t)uk(r)ei{kz-"{k)t) ^ Uk^e1^02-^ f dkA(k,t)t

(3.6)

expanding u(k) = co(k0) + ^\k=ko(k - k0) + \^\k=k(j{k - k0)2 = ui0 + vgq + ^^q2

with q = k — k0 gives:

uk0(r)e«k°*-^ [dqA(q + k0,t)eilq{'-v''t)-^^t] E

= uko(*y to'-"*') x F(z, t) = 4= \u, ko) x F(z, t) (3.7)

Here F is a slowly spatially and time varying envelope of the signal or probe, which

extends over many periods of the waveguide. In order to determine the interaction of

pulses propagating in the PC waveguide, we need to know the evolution of such an

envelope. In Appendix C, first order perturbation theory is applied to the operator

O to determine the evolution of the Fourier components of the envelope. To first

order in the nonlinear perturbation, and with negligible group velocity dispersion,

the evolution of two pulses (S and P) in the same waveguide, but possibly coupled to

different waveguide modes {s,p) is given by (see Appendix C):

S = qnus(%,s\S\2 + 27 s ,p |P|2)5 - vsS' + i\d-§±S" (3.8)

P = i |«wp(7 p j , |P |2 + 2lpJS\2)P - vpP' + i^P" (3.9)

with:

7-* = - [ d3re(f)\us\2\up\

2 (3.10) a J A

The dot in the above equation denotes differentiation in time, and the prime is


Figure 3.3: Amplitude of E field for k = |f, | f and f

a derivative in the direction of propagation (z). The overlap 7SiP, has dimensions of

m~2, due to our re-normalization of the Bloch state to fAd3r\u\2 — a. The function

e has a value of n2 in the material and is zero in air. Here, p and s label the probe

and signal modes at a particular k point, and vs and vp are the group velocities of

the signal and probe pulses respectively; K is defined as cn2 (see Appendix C). The

intensity of each pulse is enhanced by ^ - , relative to bulk, as can be shown from

Poynting's theorem [33]. When there is no nonlinear coupling, each pulse propagates

with group velocity i>s,(p), and spreads according to | Vg^p). The above equations are

derived in Appendix C and can be used to investigate self focusing, soliton formation,

and other effects in PC waveguides. In the presence of the nonlinearity, the pulses

experience self-phase modulation due to 7s(p),s(p) and cross-phase modulation due to

7SjP terms. The integral for j ^ p , which gives the coupling strength, is taken over a unit

cell of the waveguide, and is normalized by the length of the period a (see Appendix

C for further details).

The shape of us and up, and hence the values of the 7 terms, is not strongly k

dependent within numerical error for a wide range of wave vectors, as determined by

3D Finite Difference Time Domain (FDTD) simulations (Figure 3.3), and dkUk ~ 0.

Thus, the coupling strength j3tP only depends on the waveguide branch for the modes

and not the particular k point. The total effective interaction strength is k dependent,

since the group velocity determines the propagation time. The coupling strengths in

units of a - 2 , and mode volumes of the unit cell of the waveguide in units of or3,


Table 3.1: values for coupling 7 for different modes of the waveguide in units of a - 2 , and mode volumes for each unit cell of the waveguide (in units of a - 3 ) . 1 and 2 refer to the first and second modes of the waveguide.

Ui,j

1i,j

''mode

1,1 6.4 x 1(T2

3.9 x l f r 1

2,2 7.9 x 1(T2

2.8 x 1CT1

1,2 1.4 x lO"2

2.5 x 1G-1

are shown in the Table 3.1. The mode volume for each waveguide mode is defined

as VL,I(2,2) = (e\ui(2)\2)m1

ax JAd3re\ui^)\2, ana^ the m °de volume for the overlap is

Vi,2 = (eMKI taL /A^HN-Each equation can be transformed into a coordinate frame moving with the probe

and signal respectively via x — z — vst and x — z — vpt. The dispersion terms in 3.8

complicate the solution. We will assume that the length of the waveguide is small

enough so that the measurement of the induced phase and the measurement of the

phase on the probe is unaffected by the dispersion throughout the propagation. With

^ r and -^ neglected, the solution and upper bounds on the phases on the probe

after time t = — are:

P(z') = Exp[-iKujp f0{iP,P\P(z>)\2 + 2lsJS(z> + Avt')\2)dt']P(0)

<j>P = lKupf*jPtP\P(z')\2dt' w ±KUJP-/P^\P{Z')\2

<f>S = KUJp f0 1s,p\S(z' + Avt')\2dt' » KUpls^\S{z')\2

(3.11)

(3.12)

(3.13)

(3.14)

Where vp and vs are the group velocities of the signal and probe beams and L is

the interaction length. <f>s gives the phase shift of the probe due to the signal photon

number.


3.4 Nonlinear Phase Shift

The phase <f>s is the phase on the probe due to the nonlinear interaction with the

signal, and gives the signal photon number. In Appendix C we derive that the ideal

case of a negligible group velocity mismatch and a narrow probe, gives the phase shift

per signal photon of: L I .

<Ps,ideai = cn2jS)Phuisojp (3.15) VpVsTs

L is the length of the PC and r3 is the temporal width of the signal wave packet.

Both the signal and probe wave undergo material absorption and scattering due

to waveguide losses. PC waveguide losses are already as low as 14dB/cm and will

improve with time [58]. The material absorption consists of the linear absorption

coefficient a.\ and the nonlinear coefficients, of which we will only consider the two

photon absorption coefficient a2. In the case of AlGaAs at the half band gap, the

values of ot\ and a2 were found to be w .1cm -1 and ss . 2 ^ respectively [54]. Thus,

ax limits us to L— RS 10cm. For uJ and sub-//J pulses, a2 results in a smaller

attenuation length on the order of 50/um at best. Thus, experiments with the signal

and probe at the half-gap are not feasible. In order to circumvent pump depletion

due to two photon absorption, a pump at even longer wavelengths above 1550 nm

should be used [59]. In that case, a2 is close to zero, and we will assume that the

100 /im PC waveguide length is the limit. In this case, the pump will propagate

in the lower branch of the waveguide, while the signal should couple to the upper

branch. For example for a signal at 1550 nm in the upper waveguide branch, a pump

at 1620 nm should be used in order to have both beams velocity matched at the

ir/a point. We briefly mention that the GalnAsP material system has a\ « 1cm-1,

and n2 ~ 5 x \Q~l2~ and most likely similar two-photon absorption, which means

that both materials are suitable candidates for an experiment. While the nonlinearity

is enhanced closer to the band-edge of the semiconductor band gap, the absorption

increases accordingly and reduced the interaction length.

The phase due to a single photon in signal S and the energy required for an SNR

of 1 for number state detection in AlGaAs, are plotted in Fig. 3.4. We plot both the


100 200 300 400 500 v.

(b) Probe Energy [deal (7)

l . x l O 4

100 2 0 0 3 0 0 4 0 0 500v„

(d) Probe Energyt/)

l . x l O 5

•>2

100 200 300 400 500 v. 100 200 300 400 500 v„

Figure 3.4: (a)Phase shift due to a single signal photon with a lifetime of 200 ps, after propagation through a 100 jim AlGaAs PC waveguide with a narrow probe and no group velocity mismatch as a function of the group velocity vg normalized by the speed of light c. (b) The energy required for an external pulse to obtain a SNR of 1. In (a) and (b) it is assumed that the signal and probe are at 1500 nm and two-photon absorption is not present. In (c) we plot the phase for the case of the signal photon in waveguide 1 at 1550 nm and probe at 1620 nm in waveguide 0. The required probe energy for this scheme is shown in (d). In all plots, the blue and red curves correspond to both the signal and the probe in waveguide modes 0 or 1. The black curve corresponds to the probe and signal in different waveguide modes

ideal case, in which two-photon absorption is negligible, and the reality in which the

pump is at 1620nm.

There are two sources of noise in this experiment in the case of an ideal detector.

One is the phase noise due to intrinsic noise of the signal beam, and the other is the

interferometer noise due to the uncertainty of the probe photon number. Following

[51], it can be shown that in the case of a coherent signal state with mean photon

number (hs) = Ns and coherent probe with mean photon number (np) = Ns the

uncertainty in the detected signal is (Anlobserved) = {An2sintrinsic) + {-^. There are

two cases of interest: the signal in coherent and number states. For the coherent


state,

7 ^ r = ̂ + 4 * X (3.16) \L-^"'s,observed/ J v «

In the case of the signal in the number state, the intrinsic noise of the signal disap

pears,

\^ns,observed) = A A2 AT (3.17)

4(f) jNp

When the probe photon number is reasonably large, we can relax the requirement on

Np. If the tolerated error for coherent state detection is E = /3N3, then the condition

is Np = {A4>23[3Ns)~

l, and (5 < 1. Thus, detection of 1000 signal photons with an error

of 100 (/? = 0.1), would require 50 — lOOnJ. For smaller signal photon numbers, the

level of tolerated error decreases, and the requirement is more stringent than number

state detection, since ^- > 1.

3.5 Conclusion

In conclusion, we have derived the equations of motion for a probe and signal wave

interacting via the third order nonlinearity in a photonic crystal waveguide. Within

the slowly varying envelope approximation, the equations yield intuitive results, and

are essentially identical to the equations of propagation for pulses in nonlinear fibers

and materials, if the plane waves used in the mode expansion of the electromagnetic

fields are replaced by Bloch waves. However, the use of PC waveguides leads to the

necessary enhancement of the pulse intensities due to the small mode volume and

reduced group velocity of the pulses. We have shown that for the case of a very long

wavelength probe pulse, that does not suffer from two-photon absorption in the Al-

GaAs material system, the energy requirement on the probe wave is within attainable

values («» fiJ in sub ns pulses). Since the sources of such pulses are external to the PC

waveguide, the generated probe pulse can be broader than the pulse desired in the PC

waveguide, due to contraction by the group velocity. Our derivation has assumed that

coupling into the waveguides and the beamsplitter implementation in a PC waveg

uide are perfect, and the scattering loss of the waveguide can be neglected. This is of

course a gross generalization. High coupling efficiencies and low loss propagation over


10's of microns have been shown, and will only improve in time. We are in general

most strongly limited by two photon absorption in this proposal. Other material sys

tems such as GalnAsP [60], which also exhibit higher n? values that AlGaAs, could

also be considered for this implementation, but ultimately atomic resonance systems

and systems with a high phase shift and low loss are necessary [5, 48]. In principle,

an on-chip QND photon number detector could be a component of a photonic crystal

based quantum circuit, or can serve as a sensitive intensity detector and switch. The

derivation presented here, can be easily extended to other types of intensity and field

dependent nonlinearities, and can be used to analyze other nonlinear optical effects

in PC waveguides, as well as soliton formation and propagation.

Chapter 4

Nonlinearities for quantum

information processing

4.1 Introduction

As introduced in Chapter 1, the cavity-QD system is highly nonlinear. The coupling

of the QD strongly modifies the transmission function of a PC cavity and prohibits

photons from passing through at the QD resonance in both the weak and strong

coupling regimes (see Section 1.4.3). This single-photon sensitive cavity transmission

along with two stable QD ground states can be exploited for quantum information

processing as detailed in [16, 14]. Additionally, the QD response to the driving field

is itself nonlinear due to QD saturation at high driving field strengths. The QD

can only scatter one photon per Rabi flop, and as the strength of the driving field

that is incident on the cavity increases, the QD cannot follow the driving field and

photons are transmitted. This indicates that the response of the QD to two photons

that arrive simultaneously differs from that of two photons arriving separately added

together. Thus, even without the availability of stable ground states, the QD can be

exploited to interact single photons and can be used for quantum logic operations.

This kind of two-photon interaction scheme was first realized by Turchette et. al. in

an experiment with atoms [8]. To date, the largest nonlinearities have been realized

with single atoms and atomic ensembles. We showed in [2] that a single quantum

61

CHAPTER 4. NONLIENARITIES FOR QIP 62

dot (QD) coupled to a PC nanocavity can facilitate controlled phase and amplitude

modulation between two modes of light at the single photon level. We observed

phase shifts up to n/4 and amplitude modulation up to 50%. This is accomplished

by varying the photon number in the control beam at the wavelength which is the

same as the signal, or at a wavelength which is detuned by several QD linewidths from

the signal. Our results present a step towards quantum logic devices and quantum

nondemolition measurements on a chip. In this chapter I will discuss the controlled

phase shift experiment and result. The following discussion is based on References

[2, 1], where I.F., Dirk Englund and Andrei Faraon were equal contributors.

4.2 Measurement description

The concept of the controlled phase interaction from an experimental point is illus

trated in Fig. 4.1. A QD inside the PC cavity modifies the cavity transmission func

tion and with it the phase of photons that are reflected from it due to the dispersive

component of the QD susceptibility. Thus, by controlling the quantum dot, the phase

of cavity scattered photons can be altered. If this control is exerted through another

photon stream, a controlled phase shift results. In our experiments, a one-sided cavity

was used. In such a case, the reflection coefficient r(u) = E^t/Em « 1 — -A-p w 1,

where F is the Purcell factor and / is fraction of energy emitted into modes other

than the cavity mode (see Section 1.4.3). Thus in a one-sided cavity, only the phase

of cavity reflected photons is modified. As will be described further on, our experi

mental implementation results in a reflection coefficient of the one-sided cavity that

is equivalent to a transmission coefficient of a two-sided cavity.

4.3 Coherent probing of a cavity-QD system

Before a controlled phase interaction between photons inside a PC cavity can be

attempted, coherent interaction between an external laser field and the cavity must

be realized in order to be able to map out the cavity transmission function and the

phase of cavity scattered photons. A diagram of experimental setup for coherently

CHAPTER 4. NONLIENAMTIES FOR QIP 63

(d) detuning [K]

f -0.5 0 0.5 1 15 2

detuning [K]

(C)

in _̂

out =

r(uj) = \r(cj)\ el -0.5 0 0.5

detuning [K]

Figure 4.1: (a) The amplitude of cavity transmitted photons with and without the quantum dot. (b) The phase of cavity transmitted photons with and without the quantum dot. In the presence of the QD the phase has an abrupt modulation of up to 7T across the dot resonance, (c) The phase of photons can be controlled by either saturating the dot, where in principle a TT/2 modulation can occur, or by shifting the dot resonance, where a n phase shift would result, (cl) In this experiment a "one-sided" cavity was used. The reflection coefficient is given by r(uj) = Eout/Ein

probing the cavity is shown in Fig. 4.2. Instead of scanning the probing laser beam,

the probe is held fixed, and the QD and cavity are scanned through it by means of a

temperature tuning technique given in Ref. [4].

As shown in Fig. 4.3, the QD strongly modifies the cavity transmission spectrum

as expected. This measurement, given in Ref. [1] was the first demonstration of

the coherent probing of a cavity-QED system in solid state, and constitutes a major

breakthrough in our ability to access and manipulate QD qubits. The results of the

measurement are shown in Fig. 4.3(b). From the width of the splitting in (a), we

obtain a value of g/2ir = 8 GHz= K/2. As expected from the theory in Eq. 4.4, the


927.32 927.59 927.86 928.13 928.40

20K ^probe

X27K

Figure 4.2: A cross polarization setup is used to reject direct laser scatter from the sample and collect only cavity-coupled photons. Instead of tuning the probing laser, the cavity and QD wavelength are shifted with temperature by heating a "heating pad'1 with a 980 run laser that does not excite carriers in GaAs and the QD. The QD shifts ~ 3 — 4 times faster than the cavity as detailed in [4]. The temperature is varied periodically between 20K and 27K and the amplitude of the cavity-coupled probing beam is collected on the spectrometer.


cavity reflectivity is strongly modified. The observed reflectivity does not drop down

directly to zero at the dot resonance. We attribute this discrepancy to experimental

noise. In particular, we observed that the temperature of the heating laser fluctuated

by 0.7%, which resulted in a fluctuation of the quantum dot on the order of 0.005 nm.

We obtain good agreement with the theory, when we account for this fluctuation. The

coupling efficiency was found to be 2%, and indicates that without cross-polarization,

the observation of this effect would be nearly impossible without a lock-in technique.

Next, we explore the nonlinear behavior of this system, by increasing the power

of the probing laser. As shown in Figure 4.3(c), we observe saturation at power levels

which correspond to several photons in the cavity, which indicates one of the largest

optical nonlinearities available in the solid state systems. The major advantage of

this nonlinearity is that photons are conserved, because they are primarily re-emitted

into the cavity mode. While the coupling efficiency of 2% in this experiment is quite

low, this can be greatly improved when the probe laser is coupled to the cavity

via a waveguide [61], and this nonlinearity can be used for on-chip photon-photon

interactions.

4.4 Phase measurement

We measure the phase of cavity-reflected photons by interfering them with a reference

beam of known amplitude and phase (Fig. 4.4). As in the previous section, the reflec

tivity of the linearly polarized cavity is isolated from background laser scatter using

a cross-polarized setup [62, 1]. Here, the cavity field decay rate is n/2n = 16GHz,

corresponding to a quality factor Q=10000. The quantum dot has an estimated spon

taneous emission rate 7/27r = 0.2GHz. In the described experiments, we employ two

quantum dots: a strongly coupled QD with vacuum Rabi frequency <?/2-7r = 16GHz,

and a weakly coupled one with g/2n = 8GHz.

The reference beam is introduced by inserting a quarter wave plate (QWP) be

tween the beamsplitter and the cavity. The QWP converts the linearly polarized

signal into an elliptically polarized beam with components parallel and orthogonal to

the cavity polarization. In this way, both the signal and interfering beam traverse the


* H (experiment)

0 20 40 60

temperature scan count

(0

Figure 4.3: (a) A strongly coupled quantum dot, which exhibits anti-crossing with a coupling g/2w = 8GHz = K / 2 is resonantly probed. Spectra at different values of the heating laser power (i.e. at different temperatures) show how the dot traverses the cavity, (b) The probing laser (probe A) frequency is set closely to the point of anticrossing as shown in the bottom panel. The temperature tuning shifts both the dot (QD) and cavity (Cavity) lines through the probe. The temperature tuning is driven by a triangular wave. In the top panel of (b) we show the amplitude of the reflected beam, which traces out the cavity reflectivity. A fit to the theory (Eq. 4.4) with cavity and dot parameters taken from spectral da ta in (a.) matches the experimental data. When we take thermal fluctuations due to heating laser power stability into account in our theory, the fit matches the data very well. In (c) we progressively increase the power of the probe laser. We observe that the dip begins to saturate at powers, which correspond to ~ I photon inside the cavity per cavity lifetime. The powers were measured before the objective lens, while photon numbers are obtained from theoretical fits.

same path, and we do not need to account for noise due to mechanical and thermal

fluctuations in the interferometer. After reflection from the sample, these two com

ponents acquire a relative phase. The detected signal /., is an interference between

the cavity reflected component and the reference field:

926.5 9267 926.9

wavelength (nm)

(a)

00 150 200


(b)


Is(oj) = \A(9)(r(oj) + ei^)\2, (4.1)

where A{9) is a coefficient that depends on the QWP angle 6 relative to the

vertical polarization of the PBS, r{uj) is the frequency and power-dependent cavity

reflectivity, and ty(0) is the reference phase delay. A{6) and \I/(0) (Fig. 4.5) are given

by (for a derivation see Appendix D):

m$) = cos2(2fl)+isin(2fl) cos2 ( 2 0 ) - i sin (20)'

The amplitude at the detector is proportional to the modulus squared of A(9), where

A{9) = %- (cos2 {26) - i sin {29)) , (4.3)

We extract the cavity reflection coefficient r{u) from the detected signal IS{LO) in

two ways. First, we make no assumptions about the theoretical form of r(u>) and

collect Is{u) for several values of the QWP angle 9. From this, we can extract the

real and imaginary parts of r(uj) via a nonlinear fit as shown in Fig. 4.6. This agrees

with the theoretical form of r{uj) given by:

r M = f ^ fl2 - 1, (4-4)

Where the factor ^/rj accounts for the coupling efficiency into the cavity mode. We

performed fits to several values of 9 as shown in Fig. 4.4C, and extract the phase

of the cavity reflected photons in Fig. 4.4D. As expected, the phase varies abruptly

across the QD resonance.

The second method is illustrated in Fig. 4.6. There the reflectivity is taken at

several settings of the QWP Fig. 4.6A. It is assumed that the resulting amplitudes

are the result of interference between a known reference beam and an unknown cavity

signal with real and imaginary components. The amplitude and phase of the reference

are given by Eq. D.10 and Eq. D.9 respectively, and are shown in Fig. 4.5 as a

function of QWP angle. The nonlinear fit allows us to extract the real and imaginary


I l\>

• i f

I, ig>l

QD

3 ; control a c )^: j -

v l | I signal {Xs) >v\^j~

IV + CWI/*) Q ^ P IT) ^W> 1 ^AA.*-

*%/%/*" '

r(«) | \> - CW I / )

r . . . . . . . . . . . ^ 0.7 0.65 0.6 0.5S

K 0.S ^•O.AS

0.4 0.35 0.3 0 25

' . B A -<t>,

M<h 80 85 90 95 100 105 110 temperature scan count

j

PBS i4(«){r(w) + e ' * w } M S

D 100.

927.5

temperature scan count 100 120 140


-It/2

signal phase §

40 60 80 100 120 140


Figure 4.4: Experimental setup (A). Vertically polarized control (wavelength A,.) and signal (wavelength A.,) beams are sent to the PC cavity (Inset) via a polarizing beamsplitter (PBS). A quarter wave plate (QWP) (fast axis 9 from vertical) changes the relative phase and amplitude (C(#)) of components polarized along and orthogonal to the cavity. Only the reflection coefficient r(ui) for cavity-coupled light (at |— 45°}) depends on the input frequency and amplitude. The PBS transmits horizontally polarized light to a detector D. (B) Theoretical model for the phase of signal beam 0. The signal phase p t changes to 02-0^ when the control and signal beams are resonant or detuned respectively, and nc = 0.3. The nonlinear phase shift due to the increase in power is shown as A0L. The wavelength detuned control shifts the phase 03 relative to 0L by the AC Stark effect [5]. 03 is asymmetric because the cavity-coupled control power depends on the cavity and QD wavelengths during the temperature scan (C). The temperature was scanned from 20 to 27 K. (D) Measured reflectivity R for different QWP angles and fit by theoretical model Eq. D.8. (E) Phase of the reflected beam, extracted from model fits in (D).


parts of the unknown cavity signal, which are shown as Re(r(u)) and Im(r(uj)) in

Fig. 4.6B and give the phase of the cavity reflectivity as arctan(Re(r(ui))/Im(r(uj))).

Both methods are in good agreement and validate the theoretical form Eq. 4.4.

Figure 4.5: Phase and amplitude of the interfering beam as a function of the QWP angle 9 in units of n.

0 20 40 60 80 100 120 140 160 180 200 ~°% 20 40 60 80 100 120 140 160 180 20o"°''

temperature scan count scan count

Figure 4.6: A. Is(u/) taken for several values of 9. Interference between the reference beam and cavity coupled beam are clearly visible. B. Extracted amplitude \r(uj)\ and phase arctan(Im(r{uj))/Re(r(u>))) of the cavity reflection coefficient.


4.5 Controlling the phase

4.5.1 Control and signal beams at the same wavelength

control ~ ^signal)

In the controlled phase shift measurements, we first consider the case when the control

and signal have the same wavelength (they could potentially be distinguished by

polarization or incident direction, although the current experiment is performed with

one beam). When the control and signal are at the same wavelength, the nonlinear

interaction between them (Fig.4.7 A,B) arises from the saturation of the QD in the

presence of cavity coupled photons [1] see Fig.4.3. Saturation occurs when the average

photon number inside the cavity reaches approximately one photon per modified QD

lifetime given by n/g2. The cavity photon number is nc = r]Pin/(2K,hujc), given the

input power Pin, control frequency uc, and coupling efficiency r\ s=s 2 — 5% in our

experimental setup. The observed QD-induced dip does not fully reach zero at low

powers, as expected from theory[1, 14], because of QD wavelength jitter and blinking

(see Appendix D).

We observe a phase modulation of 0.24-7T (43°) when the control photon number is

increased from ric=0.08 to 3 and the wavelength is set ~ 0.014 nm (5/3.5) away from

the anti-crossing point (Fig.4.7C). The reflectivity amplitude R normalized by the

cavity reflectivity without a dot R0 is shown for the same detuning in Fig.4.7E and

changes from 50% to 100% at saturation. The excitation powers are 40nW and 1.3/iW

measured before the objective lens (corresponding to nc of 0.08 and 3 respectively),

and indicate a coupling efficiency of up to 5%. However, the coupling efficiency

fluctuated due to sample drift during the experiment. Therefore, we estimate control

powers from fits to the data, and give power levels measured before the objective lens

for reference.

In the context of quantum gates[63, 64, 8], we are interested in the signal photon's

phase change due to a single control photon. When the control and signal have the

same wavelength (AC=AS) and the same duration, the change is given by the difference

between the phase evaluated at nc and 2nc (Fig.4.7C). We measure a maximum

A


X(nm) ric

Figure 4.7: Nonlinear response of the QD-PC cavity system to single wavelength

excitation near saturation at control photon number nc—0.G (A,B)- Each temperature

scan count corresponds to a particular detuning between the cavity and QD as in

Fig.4.4C. At a detuning of 0.014 nm (g/3.5) from the dot resonance (vertical line in B),

the phase changes by 0.247T when nc increases from 0.08 to 3 (C). The phases derived

from experimental scans (points) agree with theory (solid line). The dashed red

curve is the fit to experimental results evaluated at control powers of 2nc. The signal

phase shift due to the doubled signal photon number (f)(nc) — </>(2nc) is maximized at

nc=0.1 (arrow). (D) The main loss mechanism due to fluorescence from the quantum

dot corresponds to ~ 1% photon loss. (E) Reflectivity power dependence. Points

correspond to experimental data for reflectivity (R) normalized by the calculated

value of reflectivity from a cavity with no QD (i?o).


differential phase shift of 0.077T (12°) when nc =0.1. The differential amplitude is

maximized at a higher nc=0.43, where it changes by 15% when nc is increased to 2nc

(Fig.4.7E). Theoretically we estimate a maximum of ~ 0.157T (27°) for phase and 20%

amplitude modulation with our system parameters.

Conventionally, the intensity-dependent refractive index n2 or the Kerr coefficient

X ^ describes the strength of a nonlinear medium in which the nonlinearity is propor

tional to the photon number [53]. The cavity-embedded QD is highly nonlinear and is

not well described as a pure Kerr medium. However, for weak excitations, we can still

approximate the nonlinear index and susceptibility from the relationship between the

acquired signal phase shift <f>s and n2 given by (f>s — ^x^-p^^fi, where Acav ss (A/n)

is the cavity area, and c/2nn gives the propagation length in GaAs with refractive

index n=3.5. From our experimental data at very low values of control power we infer

n2 ~ 2.7 x 10~5 cm2/W and x(3) = 2.4 x 10~10 m2/V2. This value is many orders of

magnitude larger than most fast optical nonlinearities in solid state materials.

Spontaneous emission from the QD into modes other than the cavity reduces the

performance of quantum gates due to photon losses. In Fig.4.7D we show a 1%

photon loss due to incoherent fluorescent emission from the quantum dot, which is

driven 0.014 nm away from resonance by the signal laser. Fluorescence loss is expected

to scale as Fpc/(F + Fpc) ~ 0.15%, where F — 160 is the QD Purcell factor in the

PC cavity, and FPC ~ 0.25 is the suppression of the QD radiative rate due to the PC

lattice[65]. The observed 1% is higher than the expected value for losses, but within

error, since Fpc strongly depends on the dot position and can at most be unity.

Radiation from nearby emitters cannot be excluded from this signal and therefore

fluorescence losses from the addressed QD may be lower[65].

4.5.2 Control and signal beams at different wavelengths {\Controi

^•signal)

For applications such as quantum nondemolition (QND) detection and optical control,

it is advantageous to spectrally separate the control and signal beams. Here we detune

the control beam by AA = —0.027 nm (~ g) with respect to the signal beam, which is


again aligned to the QD/cavity intersection (Fig.4.8A). The number of signal photons

per QD lifetime (ns) is fixed and the control photon number (nc) is varied. In these

measurements a weakly coupled quantum dot with g/2Tt « K/47T=8 GHZ was used.

Saturation power scales with the modified spontaneous emission rate g2/it, and so the

smaller g value permits lower control powers and reduces background noise. In Fig.4.8

we show the principle of the measurement. First, the signal and control are turned

on independently and the QD dip is visible in Fig.4.8(B,C). The dip disappears when

the two beams are turned on simultaneously and interact in Fig.4.8D. For better

visibility at high control powers, the signal power in Fig.4.8 was set to 100 nW before

the objective lens, corresponding to ns=0.2 signal photons in the cavity per cavity

lifetime.

In Fig.4.9 we show experimental results for phase shifts with control and signal

beams at different wavelengths. Here, the signal phase is affected by the saturation of

the QD and a frequency shift of the QD due to the AC Stark effect, which can be used

to realize large phase shifts [5]. The signal reflectivity and phase as functions of control

beam photon number are shown in Fig.4.9 A,B. We fit both the signal and control data

by a full quantum simulation and derive the underlying signal phase shift as a function

of control photon number[66]. The reflectivity at the signal wavelength saturates

completely when the control photon number reaches nc=1.3, which corresponds to

1/uW of power measured before the objective lens. The associated phase modulation is

0.13 7T at the signal detuning of 0.009 nm (sa g/3) from the dot resonance. The phase

behavior in Fig.4.9B is asymmetric with respect to the center of the quantum dot-

induced dip because the coupling of the control beam changes with the temperature

scan.

We fix the signal wavelength 0.009 nm (<y/3) away from the quantum dot resonance

and determine the phase and amplitude modulation for a range of values of nc. The

signal phase 4> relative to the signal phase with no control (f)0 — (j){nc = 0) is shown in

Fig.4.9D. The maximum observed phase shift is 0.167T (23.4°) when n c =l . The largest

nonlinear phase change is observed for nc = 0.05, where (f>(nc) — </>(0)=0.057r (9°).

These values give a nonlinear index of n2 « 1.8 x 10~5 cm2/W, or x ^ ~ 1-6 x 10 -10

m2/V2 for a detuning of 0.027 nm (« g) between the signal and control. This value


927.0 927.2 927.4 927.6 20 40 60 80 100 120

A,(nm) temperature scan count

Figure 4.8: Interaction between a control and signal beam at different wavelengths.

The signal beam at Xs (A-i) is detuned by 0.027 nm (« g) from the control beam

at Ac (A-ii) and positioned to coincide with the cavity-dot crossing-point (A-iii). For

each measurement, a sequence of scans is taken (A i-iii). The quantum dot and

cavity trajectories are shown in (A-iii). We track the amplitudes at both wavelengths

in each frame (A i-iii) to subtract fluorescence backgrounds, which are magnified 10

times in (B) and (C) (these are fluorescence backgrounds detected at control and

signal wavelengths, respectively.). The QD-induced dip is clearly visible in (B) when

only the signal (solid blue line) is on, and in (C) when only the control (dashed line)

is on. This feature disappears when both beams are on in (D). In (D), the spectra

are normalized in order to clearly show saturation. The signal and control powers

were 100 nW and 200 nW measured before the lens, corresponding to cavity coupled

signal and control photon numbers ns «0.2 and nc ~0.3, respectively.


120

-e-

0.7

0.6

0.5

0.4

0.3

B

•7 ° \

n = 1.3 0

— n=0.3 c

n = 0.05 c

**K

60 70 80 90

temperature scan count 100 10" 10

control power (nc)

Figure 4.9: Nonlinear response of a weakly coupled quantum dot inside the cavity

to excitation with control and signal beam wavelengths separated by 0.027 nm (as g).

The reflectivity of a signal beam with ns=0.2 photons per cavity lifetime is shown

in (A) for three values of the control beam photon number nc. The quantum dot

saturates almost completely when roc=1.3, which corresponds to a power of 1 fxW

measured before the objective lens. The data is fit with a full quantum model, which

allows us to extract the signal phase shown in (B). In (C) the amplitude of the reflected

signal beam when it is 0.009 nm (« g/3) away from the dot resonance (vertical line

in A,B) is shown as a function of control beam photon number nc. In (D), we show

the difference between the phase shift of the signal beam when the control beam is

on ((f)) and when the control is off (<̂ o) as a function of nc at the same time point as

in (C).


Table 4.1: Nonlinear parameters and phase modulation derived from experimental

data for the strongly (first row) and weakly (second row) coupled QDs. A0 is a

maximum differential phase shift (A(f>=(f)(nc)-<f)(0)) which is achieved at the intra-

cavity photon number nc in the last column.

g/2-K (GHz) 16 8

K - AQD (nm) 0.014 {g/3.5) 0.009 (g/3)

Xs - \c (nm) 0

0.027

n2 (Cm2/W) 2.7 x 10"5

1.8 x 10~5

x(3) ( m 2 / ^ )

2.4 x 10"iU

1.6 x 10-10

A<l> 0.0157T 0.057T

nc

0.01 0.05

is similar to that of the QD with larger g. Numerical simulations indicate that the

relative magnitude of nonlinearities due to these two quantum dots strongly depends

on the laser frequency. The nonlinearities for the two cases are summarized in Table

4.1.

4.6 Conclusion

We have shown that the phase and amplitude of the signal beam reflected from

the strongly coupled QD-PC cavity system strongly depends on the control photon

number. Furthermore, the magnitude and bandwidth of the Kerr nonlinearity x^

observed in this experiment are rivaled only by measurements in atomic ensembles[67,

68]. The current implementation of the QD/PC-cavity system is already promising

for low-power and quantum nondemolition photon detectors[69, 70, 71]. We have

shown that the phase and amplitude of the signal strongly depend on the control

photon number when the signal and control photons are spectrally separated.

To realize useful quantum logic gates, controlled n phase shifts are necessary[8, 30].

This will require repeated interactions. Such cascading requires coupling efficien

cies that are higher than the observed 2-5%. This technical challenge can be over

come. We have already demonstrated architecture for a QD cavity-waveguide cou

pled quantum network[72] with coupling efficiency above 50% between two nodes,

and cavity-waveguide couplers [73] with coupling efficiency reaching 90%, as well as


coherent probing of a strongly coupled QD-PC cavity system in a circuit configuration

[61]. The observed fluorescence losses are already sufficiently low to allow scalable

computation[74], and can be further improved with increases in cavity Q. The ability

to tailor photon-QD interactions by photonic crystal fabrication makes this a highly

versatile platform for a variety of quantum optics experiments with great potential

for compact scalable quantum devices.

The observed differential phase shifts are far from the expected values of 7r/2 for

full saturation and n for full Stark shifts. This is in part due to the structure of the

nonlinearity, and in part due to the parameters which were used in our experiments.

In the case of signal and control beams at distinct frequencies, the control beam

was applied close to the QD resonance, and so both saturation and Stark shift are

included in the observed measurements. The operating point was mostly dictated by

the available laser sources and experimental resolution. Analytically, however, we can

span a larger parameter space for expected values of the phase shift. In Fig. 4.6, we

show the phase shift as a function of frequency when the QD is on resonance with the

cavity. The simulation parameters for g and K are the same as used in our experiment.

The maximum phase shift in this case is PH 0.17T as in the experiment. In Fig. 4.11

we show the phase shift as the quantum dot is detuned from the cavity. Here, a near

7r/2 phase shift occurs at a high detuning and is close to what is expected from full

saturation.


Differential phase saturation at OOL^SOJ In units o l n

Figure 4.10: Simulated differential phase shift 4>(2n) — 4>(n) as a function of probe

detuning from the cavity coi — uc when the QD is on resonance with the cavity uj^ot —

UJC = 0. A maximum of 0.1 IT occurs when the average photon number inside the

cavity is close to Tip ~ 0.1 as observed experimentally.


Figure 4.11: A. Differential phase shift (p{2n) — <p{n) as a function of probe detuning

from the cavity ujt — JJC for different QD detunings jjjA)t — vjr. B. Maxima from A vs

dot detuning in units of g. Theory predicts that for large detunings, the differential

phase shift approaches ir/2. Also plotted is the average intra-cavity photon number

(nr) corresponding to the particular maximum differential phase shift. If the lowest

photon number operating point is desired, the detuning should be njdot — u>r ~ ().

Chapter 5

Towards room temperature

cavity-QED

5.1 Introduction

In Chapter 4, it was shown that single quantum dots coupled to photonic crystal

cavities can serve as building blocks of quantum information processing devices. Cur

rent state of the art experiments are performed at cryogenic temperatures, because

electron-hole pairs cannot be confined in In As quantum dots at elevated temperatures.

However, a variety of emitters operate at room temperature and can potentially serve

as a two or three level system for quantum information processing experiments. Room

temperature generation of single photons has been observed from single molecules [75],

nitrogen vacancy centers [76], and CdSe quantum dots [77], but collection efficien

cies were reduced due to lack of coupling to good cavities. These sources, operate

at visible wavelengths and are therefore difficult to combine with cavities made in

high index semiconductor slabs. In contrast, PbS quantum dots (and other colloidal

quantum dots such as PbSe) can be made to cover a very broad wavelength range.

Furthermore, these dots can be easily deposited onto passive structures in a low in

dex polymer and can be used to map out the resonances of the structures [78]. This

method is potentially easier than transmission and reflectivity type measurements on

single cavities. Lastly, PbS quantum dots are successfully used as fluorescent labels

80

CHAPTER 5. TOWARDS ROOM TEMPERATURE CAVITY-QED 81

in biological imaging applications. Cavity enhanced emission and collection efficiency

from such dots could prove to be a valuable technique for targeted signal amplification

of a target molecule bound to or in the vicinity of the resonant cavity.

5.2 Room temperature operation

Room temperature operation requires exceptionally high-Q and low-V cavities in or

der to overcome the phonon broadening of the main quantum dot transition line. This

is illustrated in Fig. 5.1 where we plot the transmission function of a cavity with an

embedded QD whose bulk decay rate 7 is allowed to increase due to a mechanism

such as phonon broadening. It can be seen that as 7 increases, the QD induced trans

mission dip saturates and the amplitude of the transmitted component diminishes,

as energy is scattered out of the system. The reduction in the transmission dip can

only be mitigated by increasing g or decreasing K since this drives the system into

stronger coupling where losses to the cavity mode dominate scattering losses. The

same challenges are faced by attempts of using cavity-coupled nitrogen vacancies in

diamond as quantum memory nodes [79].

Photonic crystals are potentially ideally suited for the realization of cavities for

coupling such room temperature emitters due to the high Q/V ratios available in

these devices and the ability to fabricate them in a variety of substrates. However,

the combination of PC cavities with external emitters had not been demonstrated

prior to our work described here [78]. Photonic crystal cavities were made in a 160

nm thick AlGaAs (33% Al) membrane on top of 500 nm of AlAs. The cavities

were defined in 3% 450 K molecular weight (KmW) PMMA with the Raith Electron

Beam Lithography system. The patterns were transferred from PMMA developed

in 3:1 Methyl Isobutyl Ketone (MIBK) to the membrane with an electron-cyclotron

resonance (ECR) plasma etch process. The Al rich substrate was then oxidized at

420° C for 10 minutes in the presence of water vapor in order to create an index

contrast of 3.4:1.8 between membrane and substrate. A scanning electron micrograph

of the sample structure is shown in Fig. 5.2. PbS quantum dots emitting at 850

nm and 950 nm were obtained from Evident Technologies. The dots were dissolved


Transmission as a function of y toi g=K

Figure 5.1: Influence of 7 on the transmission through a cavity with an embedded QD. As 7 increases due to phonon scattering at elevated temperatures, the transmission dip diminishes and the transmitted amplitude falls due to energy losses into phonon modes. For the plots it is assumed that g = K

together in 1% 75 KmW PMMA at a concentration of 0.1 mg/ml in toluene and

spun onto the structures at 2 krpm, resulting in a 20nm-100nm membrane (bulk

spectra shown in Fig. 5.3). The coating was not uniform due to the small chip size

~ (5mm2) and presence of structures on the chip surface. This concentration of

emitters corresponds to ~ 102 dots per fim2 in a 100 nm membrane. The dots were

kept under vacuum, and excited with femtosecond pulses from a Ti:Saph laser at 760

nm. Cavity spectra, collected under pulsed excitation, are shown in Fig. 5.4. The

spectra reveal two orthogonally polarized modes as expected for this type of photonic

crystal cavity. The linearly increasing background observed on the spectra can be

explained by considering the emission profile of the dots and the reflectivity of the

PMMA/AlGaAs/AlOx structure, which leads to an almost linear emission profile of

dots on the AlGaAs slab.

The cavity used is shown in Fig. 5.2. The coupling between a cavity and emitter

depends on both the spatial alignment and the orientation of the emitter dipole


Figure 5.2: Left: Scanning electron micrograph showing the photonic crystal cavity (a). Middle: Simulated electric field intensity of the x (b) and y (c) dipole modes in the asymmetric cavity. The measured Q factors are 400 and 200 respectively.

Figure 5.3: PbS quantum dot spectra: 850 and 950 dot spectra taken on a bulk silicon wafer.


moment relative to the cavity field. The chosen cavities support two dipole type

(electric field primarily polarized in x and y) cavity modes with maxima in the central

hole in (Fig. 5.2). The asymmetric cavity was modeled using Finite Difference Time

Domain (FDTD) methods [80]. The simulations predict the splitting between the two

resonant peaks to be w 43 nm, which is somewhat larger than the splitting of 37.5

nm observed in the experiment and is likely due to deviations of fabricated structures

from simulated ones. The quality factor (Q) is the figure of merit for these cavities,

and can be written as 1 - l 1

Q = ^ + O l { }

The above description separates the cavity losses into an out-of-plane loss which leads

to directly observed emission, and losses into the Bragg mirrors in the plane of the

two dimensional crystal. The cavities investigated here have experimentally observed

x-dipole Q's « 400 and y-dipole Q's « 200. These measurements correspond to the

total Q given in Eq. 5.1, which is determined by the lowest of Q\\ and Q±. Simulations

for the asymmetric cavity predict that the out of plane Q± « 8000 for the x-dipole

mode and ~ 4000 for the y-dipole. The simulated in-plane Q\\ was only ss 450 and

400 for these modes, and thus limits the total Q. The two components are calculated

by measuring the energy loss through planes above and below the PC slab (Q±), and

through the sides between the planes (Q\\). Q is unaffected in this case by the etch

depth into the AlOx substrate and the hole profile inside the substrate [81], since

the structure is back-filled with PMMA, whose refractive index is approximately that

of AlOx. The slight non-uniformity of the center hole, which stems from a lower

electron dose delivered to the PMMA due to proximity effects, is small relative to

the resonant wavelength, and therefore does not contribute greatly. The two cavity

modes are shown in Fig. 5.2 b. In a perfectly symmetric cavity, where the two holes

above and below the defect are unperturbed, the x and y modes are degenerate and

have a Q of « 400. We shift the four holes in order to increase the Q factor for the x

dipole mode, as has been discussed in [80].

The photonic crystal can both enhance and reduce spontaneous emission (SE)

rates for the quantum dots. The enhancement is desirable for photon generation and


Mnm) X(nm)

Figure 5.4: Cavity resonances mapped out by quantum dots in PMMA. Left: Polarization dependence of modes confirming that they are x and y dipole modes. Right: Ex dipole mode measured at two orthogonal polarizations. Angles refer to analyzer positions.

spectroscopy applications due to increased signal rate and strength. The figure of

merit for this modification is the Purcell factor F = jp-. Here T is the emission rate 1 o

in the cavity, and F0 is that without the cavity. The observed spontaneous emission

rate for an emitter with dipole moment /I at a point on the cavity can be derived

from Fermi's Golden rule [82], and the total rate takes the form

rjL = J-Jj—)' f +Fpc (5.2) r o ^ max{£} \jl\ j AAC

2 + 4(A - Ac)2

Here E is the electric field at the position of the emitter, Vmod,e is the cavity mode

volume, AA is the detuning from the cavity resonance wavelength Ac. For an emitter

that is on resonance with the cavity and has a dipole moment aligned with the field,

Fc = ( A i 3u— ) • The mode volume is calculated from the FDTD simulation results y 47T n Vmotie J

as Vmode = f ,L,i d3r, with e as the position dependent dielectric constant. For . • . . max{e .E } •

this cavity, the volume has a value of Vmode = 0.96-f, where n is the index in AlGaAs.

The term Fpc describes the modification of the SE rate due to the presence of the

photonic crystal lattice and modes other than the cavity mode. This modification

results in a suppression of emission [83, 65]. The cavity mode volume can be quite


accurately derived from FDTD calculations, and with the cavity Q derived from

the spectra, we calculate an Fc ~ 30. We can also derive an average value of the

suppression Fpc from the experimental data. The total intensity collected by the

spectrometer 1$ is given by the emission rate Eq. 5.2, integrated over the spatial and

spectral density of the emitters.

1+= T0JdflJd\Jd3fp(X,f,p) x (5.3)

Here (p is the value of the polarizer angle, r\c is the collection efficiency due to the

cavity radiation profile, r/o is the collection efficiency from an emitter embedded in the

PMMA layer and uncoupled to the cavity, p(X, f, p) contains the spectral and spatial

distribution of the dots and the orientations of their dipole moments. The value of

r/c can be estimated from numerically integrating the PC emission profile over the

numerical aperture (NA) of the lens, and is rjc ~ 8%. The coupling efficiency of the

bulk emitter is given by the integral over the sub-critical solid angle defined by the NA o dvcsin,(— )

of the objective lens 770 = ^ /„ d<f> J0 npMMA sm(6)d0 « 3% (Here NA = 0.5

and TIPMMA = 1-5). Since the cavity modes are primarily linearly polarized, the ratio

of the integrated intensities for polarizer angles of 0° and 90° gives:

J dp dX d3fp(\, f, jl) M \ AA^

R^^-^Fc1^ V —4- + 1 (5.4) I90o r)Q J dn d\ d3r FPC p(A, r, fi)

Using the value of Fc and the x-dipole field, the average value of Fpc over the

lattice around the cavity can thus be estimated from the spectra. Assuming random

dipole orientation, the integral over /2 gives | . We take the spectral distribution of the

dots to be Gaussian and centered at 850 and 950 nm with a FWHM of 100 nm. The

cavity line shape is Lorentzian with a FWHM of 2 nm. Only dots which are excited

by the pump are contributing to the intensity, and so the spatial density corresponds

to the excitation of a uniform dot distribution by a Gaussian pulse with a FWHM of

600nm in the x,y plane and uniform in z, since the layer is thin relative to the focal


depth. The integral over the field is done numerically with the simulated cavity field

components. The spectral data shown in Fig. 5.4 b. gives R ?s 1.07. Using the value

of Fc = 30, Fpc is estimated to be Fpc ~ 0.6, which qualitatively agrees with the

values found in [83, 65].

5.3 Conclusion

In conclusion, we have shown coupling of PbS quantum dots dissolved in PMMA to

photonic crystal cavities at room temperature. The dot emission maps out the cavity

resonances and is enhanced relative to the bulk emission by a maximal Purcell factor

of 30. Using the emission spectra with and without cavity lines, we derive a photonic

crystal lattice spontaneous emission rate modification of « 0.6. To our knowledge this

is the first demonstration of coupling of colloidal quantum dots to photonic crystal

cavities, and the first use of such dots as a broadband on-chip source for cavity

characterization. Following our work, other demonstrations of colloidal QD coupling

to PC cavities have been made [84, 85]. Since such dots are available in a variety of

emission wavelengths, this characterization technique can be applied to many different

semiconductor devices. The deposition technique presented in this work is easy to use

and allows many different types of emitters to be deposited and re-deposited onto both

passive and doped structures. Furthermore, E-beam lithography can then be used to

remove dot-doped PMMA from regions around the cavity, and the resulting emission

will only come from the emitters in the cavity. The use of colloidal quantum dots for

characterization of PC structures is similar to internal light source techniques [86],

but has a number of advantages: first, our method permits characterization of passive

PC structures, where internal light sources are not available. Moreover, our technique

allows deposition of sources on select areas on the chip, as well as re-deposition and

removal after characterization. Finally, our technique, combined with tuning of the

QD concentration in PMMA, can enable isolation of a single emitter inside the cavity,

which would be beneficial for construction of single photon sources that are cheap and

reusable, as opposed to those based on self-assembled QD's embedded in a PC cavity

[65].

Chapter 6

Ultra Fast Modulation

6.1 Introduction

Nonlinear optical switching in photonic networks is a promising approach for ultra-

fast low-power optical data processing and storage [87]. In addition, optical data

processing will be essential for optics-based quantum information processing systems.

A number of elements of an all optical network have been proposed and demonstrated

in silicon photonic crystals [88, 89]. Fast tuning of the photonic crystal bands has

also been demonstrated [90, 91]. Here, we directly observe ultrafast (« 20 GHz)

nonlinear optical tuning of photonic crystal (PC) cavities containing quantum dots

(QD). We perform the fast tuning via free carrier injection, which alters the cavity

refractive index, and observe it directly in the time domain. Three material effects

can be used to quickly alter the refractive index. The first is the index change due to

free carrier (FC) generation, which is discussed in this work, and has been explored

elsewhere [90]. The cavity resonance shifts to shorter wavelengths due to the free-

carrier effect. Switching via free-carrier generation is limited by the lifetime of free

carriers and depends strongly on the material system and geometry of the device. In

our case, the large surface area and small mode volume of the PC reduce the lifetime

of free carriers in GaAs. Free carriers can alternatively be swept out of the cavity

by applying a potential across the device [92]. The second effect that can be used

to modify the refractive index is the Kerr effect, which is promising for a variety of

88

CHAPTER 6. ULTRA FAST MODULATION 89

other applications [93, 50] and, in principle, should result in modulation rates exceed

ing THz. However, the free carrier effect is more easily achieved in the GaAs PC

considered here. The third effect is thermal tuning (TT) via optical heating of the

sample through absorption of the pump laser. This process is much slower than free

carrier and Kerr effects and shifts the cavity resonance to longer wavelengths due to

the temperature dependence of the refractive index. The time scale for this process

is on the order of microseconds. Here we consider these two processes for modulating

cavity resonances, and focus on the higher-speed FC tuning. Here we consider both

FC and TT for modulating cavity resonances.

Photonic crystal samples investigated in this study are grown by molecular beam

epitaxy on a Si n-doped GaAs (100) substrate with a 1 jim buffer layer. The sample

contains a 10 period distributed Bragg reflector (DBR) mirror consisting of alternat

ing layers of AlAs/GaAs with thicknesses of 80.2/67.6 nm respectively. A 918 nm

sacrificial layer of Al0.sGa0.2As is located above the DBR mirror. The active region

consists of a 150 nm thick GaAs region with a centered InGaAs/GaAs QD layer. QDs

self-assemble during epitaxy operating in the Stranski-Krastanov growth mode with

a density of 100 dots per /j,m2. InGaAs islands are partially covered with GaAs and

annealed before completely capping with GaAs. This procedure blue shifts the QDs

emission wavelengths [94] towards the spectral region where Si-based detectors are

more efficient.

PC cavities, such as those shown in Fig.6.1, were fabricated in GaAs membranes

using standard electron beam lithography and reactive ion etching techniques. Finite

Difference Time Domain (FDTD) simulations predict that the fundamental resonance

in the cavity has a field maximum in the high index region (Fig.6.1), and thus a

change in the value of the dielectric constant should affect these modes strongly.

We investigated the dipole cavity (Fig.6.1), the linear three-hole defect cavity [38],

and the linear two-hole defect cavity designs. The experimentally observed Q's for

all three cavities were in the range of 1000-2000 (optimized cavities can have much

higher Q's), and consequently the experimental tuning results were similar for all

three cavities.

Photonic crystal cavities were made to spectrally overlap with the QD emission,

http://Al0.sGa0.2As


and are visible above the QD emission background due to an increased emission rate

and collection efficiency of dots coupled to the cavity. Quantum dot emission was

excited with a Ti:Sapphire laser tuned to 750 nm in a pulsed or CW configuration.

In the pulsed mode, the pump produced 3ps pulses at an 80 MHz repetition rate.

Tuning was achieved by pulsing the cavity with appropriate pump power. The cavity

emission was detected on a spectrometer and on a streak camera for the time resolved

measurements.

Tuning is achieved by quickly changing the value of the dielectric constant e = n2

of the cavity with a control pulse. The magnitude of the refractive index shift An

can be estimated from

Aco l[Ae\E\2dV An ftj ' ' Ri (6 1)

LU 2 fe\E\2dV n K }

Above, ui is the resonance of the un-shifted cavity, \E\2 is the amplitude of the cavity

mode, and the integral goes over all space. In order to shift by a line width, we require

~ = 7j, which gives An = 0. Finite difference time domain (FDTD) calculations

indeed verify that for a linear cavity with Q 1000, a An ~ 10 - 2 shifts the resonance

by more than a line width, as seen in Fig. 6.1.

As described above, two tuning mechanisms were investigated in this work. The

first is temperature tuning, which is quite slow (on the time scale of microseconds).

The second is the free carrier induced refractive index change, which is found to occur

on the time scale of tens of picoseconds. Therefore, we can look at the two effects

separately in the time domain.

6.2 Free Carrier Tuning

In the case of Free Carrier (FC) tuning only,

An(t) = An / e(t) =

n n


Figure 6.1: (a): Scanning electron micrograph of the L3 type cavity fabricated in a GtiAs material with a high density of InAs quantum clots, (b): high-Q mode electric field amplitude distribution, as predicted by FDTD simulations, (c): FDTD simulations of frequency and Q changes as An/n changes from ±10~'! —> ±10" l . A high-Q (QHQ = 20000) and low-L (QLQ = 2000) cavity were tuned: (qx) AQ/Q for An > 0 and Q = QHQ- (q-i) AQ/Q for An > 0 and Q = QLQ, LJJ\) AJJ/^J for

An < 0,(^2) AUJJUJ for An > 0 for both high Q and low Q modes. (q-,$) AQ/Q for An < 0 Q = QLQ, (en) AQ/Q for An < 0 Q = QHQ- The lines An/n for An > 0 and An. < 0 are also plotted and overlap exactly with ^2

a n d u/[. As can be seen, the magnitude of the relative frequency change is independent of Q, but the higher Q cavity is degraded more strongly by the change in index. For an increase in 11, the Q increases due to stronger Total Internal Reflection confinement in the slab, as expected.


where Nfc(t) is the density of free carriers in the GaAs slab, and the value of r\ is

given in terms of fundamental constants (e0,c), DC refractive index (n0), charge (e),

effective electron mass (m*) and wavelength (A) as 77 = — 87r2e2eA

n m . [95] , and we

calculate rj sa 10~21cm3 for our system.

The FC density changes with the pump photon number density P(t), with pulse

width TP, in time t as: dNfc 1 Ar Pit) , ,

The carriers decay with — = — -\—-—|- —, where rT,Tm are the radiative and

non-radiative recombination times of free carriers, and rc is the relaxation time (or

capture time) into the QDs. While rc « 30 — 50ps « rr,Tnr, the dot capture is

not the dominant relaxation process. The dots saturate for the duration of the dot

recombination lifetime r^ « 2Q0ps — lns, and, because the dot density is much smaller

than the FC density, the effective capture time is much longer. Qualitatively, we can

describe this effect by lengthening rc by a factor 1/x as rc —> rc/x » Tr,Tnr, where

x « 1 is essentially the ratio of QD to FC densities. The FC density is then given

by:

Nfc(t) = Nfc(0)e rtc+e *•/= / e-tr.-L±^ (6.4) JO Tp

In order to shift the cavity resonance by a line width (An/n — 10~3), we need

NfC « 1018cm-3 according to Eq. 6.2. Taking into account the GaAs absorption

coefficient a « 104cm_1, reflection losses from the 160 nm GaAs membrane (R =

Cai rI" f faA3)2 ~ -3), lens losses (50%), and an approximately 5/um spot size, average

3ps pulse powers as low as 1-10 /iW should yield the desired shifts of order ^ « 10^3.

In our experiment, we monitor the cavity resonance during the tuning process

using QD emission. Thus, we need to account for the delay between the pump and

onset of emission in QDs. The QDs are excited by free carriers according to:

^ = _±^+^£ („.5) at Td TC

From Eqs. 6.5 and 6.4, the QD population (assuming no excited dots at carriers


at t=0) is given by:

t_

Nqd(t) = / e v V / e^P(t")dfdt' (6.6)

where rp is the pump pulse width, rqrf RS 200 ps is the average cavity coupled QD

lifetime, TJC « 30ps is the FC lifetime, and P(t) is the pump photon number density.

The observed spectrum is that of a Lorentzian with a time - varying central frequency

u>o(t) (for simplicity, we assume that the Q factor is time invariant), which we define

as:

5(Wl«) = (l + 4 g 2 ( l - ^ ) ) - 1 (6.7)

ur

The numerical results are shown in Fig. 6.2. We find that going beyond 10's of

[iW does not result in a larger shift, but destroys and shifts the cavity Q permanently.

The experimental data is shown in Fig. 6.3. We used moderate power («10 /iW)

to shift the cavity by one half line width. Stronger excitation results in higher shifts

as indicated by an extremely asymmetric spectrum shown on the inset in (d) of Fig.

6.3, where 100 /J,W were used. However, prolonged excitation at this power leads to

a sharp reduction in Q over time.

6.3 Thermal Tuning

In the case of thermal tuning, the refractive index is given by:

An(t)

n = (3T (6.8)

where j3 is the material dependent coefficient giving the refractive index dependence

on temperature.

Continuous wave above-band excitation of the sample results in both free carrier

generation and heating. In this case, the heating mechanism dominates, and the

cavity red-shifts. The predominant effect on the dielectric constant is the change in

the band gap with temperature due to lattice expansion and phonon population. The


Figure 6.2: Numerical model of a free-carrier timed cavity. In (a) the cavity is always illuminated by a light source. Panel (b) shows the cavity resonance at the peak of the free carrier distribution (t=0) and 50 ps later, as indicated by the yellow arrows in (a). The time-integrated spectrum is shown as the asymmetric black line (labeled Sp) in (b), and corresponds to the signal seen on the spectrometer, which is the integral over the whole time window of the shifted cavity. The asymmetric spectrum indicates shifting. In (c) and (d) the same data is plotted, but now we consider the cavity illuminated only by QD emission with a turn-on delay of 30 ps due to the carrier capture lifetime rc, and a QD lifetime of 200 ps. In (d) the dashed line is the un-normalized t=0 spectrum, which now appears much smaller in magnitude. Furthermore, the asymmetry of the line is even smaller in this case.

cavity itself could potentially expand, but since the thermal expansion coefficient of

GaAs is on the order of 10~6A'~L. this is insignificant. As the cavity red-shifts, the Q

first increases due to gain and then drops clue to absorption losses. The experimental


data for thermal tuning is shown in Fig. 6.4. From a fit to the frequency shift, we

obtain /? » 3 x 10~3.

6.4 Conclusion

In conclusion, we show that fast (20 GHz) all-optical tuning of GaAs cavities with

reasonable pump powers (10 /J.W). Under these conditions the cavity is shifted by

almost a line width, which leads to suppression of transmission at the cold-cavity

frequency by « 1/e. The suppression depends on the Q of the cavity and for cavities

with Q RS 4000, shifts by a full line width would be obtained. Thus, fast control over

photon propagation in a GaAs based PC network is easily achieved and can be used

to control the elements of an optical or quantum on-chip network. Free carrier tuning

strongly depends on the geometry of the cavity, since a larger surface area leads to a

shorter FC lifetime. Thus, our future work will focus on identifying optimal designs

for shifting and a demonstration of an active switch based on the combination of PC

cavities and waveguides.


X(nm) ?v(nm)

Figure 6.3: Experimental result of FC cavity tuning for the L3 cavity. In (a) the cavity is always illuminated by a light source and pulsed with a 3ps TkSapphire pulse. Panel (b) shows the cavity resonance at the peak of the FC distribution ( t=0) and 50 ps later, as indicated by the yellow arrows in (a ) . In order to verify that the cavity tunes at the arrival at the pulse, we combine the pulsed excitation with a weak CW above band pump. The emission due to the CW source is always present, and is in the box labeled CW in (a) . This very weak emission is reproduced in Panel (b) as the broad background with a peak at the cold cavity resonance in (b ) . The time-integrated spectrum is shown as the black line (Spectrometer) in (b ) . In (c) and (d) the same data is plotted, but now we consider the cavity illuminated only by QD emission pulsed by 10 f.i\V from the TkSapphire source. In (d) suppression by about .4-.35 at the cold cavity resonance can be seen. The inset shows a strongly asymmetric spectrum of a dipole type cavity under excitation of 100/iIF and the same cavitv at low power after prolonged excitation. Such strong excitation degrades the

Q.


2 3 4 5 Pump Power (mW)

Figure 6.4: Thermal tuning of the L3 cavity under CW excitation (a): Measured Aui/uJ (left axis) and AQ/Q (right axis) as a function of pump power for the L3 cavity, obtained from the fits to the spectra shown in (b). The Q initially increases clue to moderate gain and then degrades, while uj shifts linearly. The straight dashed line fits Au/w = 3 x 10"-3 x Pir x 10 ° with 95% confidence and with root mean square deviation of ~ 0.99. At very high power, the change in frequency does not follow the same trend. The inset in (b) shows a plot of AU)/(UJ/Q), which is a measure of the number of lines that we shift the cavity by. A shift of three line widths is obtained.

Chapter 7

Fabrication

7.1 Introduction

In this chapter I review the fabrication techniques used to make our standard devices

and review our early attempts at making electrically controlled PC structures for

realizing a three-level system in an InAs QD for quantum memory applications, as

well as an electrically driven modulator/detector/source in a PC doped with quan

tum wells. Much of the research in the field of PC crystals is driven by simulations

and modeling such as that discussed in Chapter 2. However, there is a large dis

crepancy between the predicted designs and the performance of fabricated devices.

Much of this difference is due to the fabrication process - some due to the quality of

equipment used (which cannot be rectified), and some due to the methodology used.

In Section 7.2, I will review our progress toward realizing high quality PC cavities

in the Stanford Nanofabrication Facility (SNF). While the tools available elsewhere

will dramatically modify the outcomes in other fabs, the general methodology of a

systematic fabrication parameter space and cleanliness that were vital to our success

should be applicable everywhere.

Next to the quality and repeatability of fabrication, electrical control over PC

structures is the single most important technical challenge for the field of PC enabled

devices and their success and integration into real-world technologies. Due to the

thin structure of PC membranes, a high sheet resistance and tendency of current to

98

CHAPTER 7. FABRICATION 99

recombine at the contact the problem of placing electrical contacts on PC structures

is nontrivial. In Section 7.4, I review our early attempts at realizing electrically

controlled structures for quantum information processing and classical information

processing applications. For quantum optics, robust electrical control can result in

controlled charge loading into the quantum dot and, with an application of a magnetic

field, a quantum memory. For classical devices, a variety of monolithically integrated

devices such as modulators, detectors, and light sources could be realized.

7.2 PC fabrication

The devices are fabricated in high index semiconductors such as Si, GaAs, and Al-

GaAs. Fabrication in SiN and lower index materials is also possible [96], but limits the

quality factor of these devices due to low TIR confinement. In the next sections, the

numerical and analytical techniques for optimizing devices will be described. How

ever, in practice it turns out that reaching the exact design can prove difficult due

to an uncertainty in processes and accumulation of errors during the fabrication pro

cess. At the writing of this thesis, the highest Q demonstrated in GaAs at cryogenic

temperatures in our group was « 25,000. This is on par with the best devices found

in literature, and is limited most likely by fabrication and scattering losses. It took

several years to reach this Q. Two significant steps that increased the fabrication

performace were as follows:

• The r/a parameter space was experimentally spanned in order to create the

devices that operate in the desired frequency range. An optimal value of the

periodicity a=246 nm and radius r=60 nm (written, which resulted in 140 nm

diameter fabricated holes) was found. This raised the Q from a maximum of

« 3,000 to ^8 ,000 .

• The cavities were extensively cleaned after fabrication. Cleaning of the cavities

after fabrication raised the Q to the 10,000 to 20,000 range. It was found by

Faraon that a long acetone soak cleaned the devices and reduced scattering

losses.


JJKX h—I *•• •«• ummm-nmmm

Figure 7.1: A. Overall process flow for fabrication. B Sample GaAs wafer. C Exposed resist (inset is poor outcome). D Etched resist (inset is poor outcome). E final undercut PC. The light area around the structure indicates a successful undercut.

Thus, in the fabrication of PC devices, simulations have to be used as guidelines and

the fabrication parameter space must be sampled.

The overall GaAs PC fabrication process is illustrated in Fig. 7.1.

7.2.1 Sample preparation

• Clean wafers in Acetone and Isopropyl Alcohol for 10 niin each.

• Bake wafers at 90° C for 1 niin to evaporate solvents

• Cover bottom of wafer with blue tape

• Spin 4.5% 950K MW PMMA (in Anisole) at 3500 rpm for 40 sec. Dispense

resist with a glass pipette for small pieces, or pour from a beaker for larger

wafers. Syringes tend to stick due to solvent and PMMA.


• Remove blue tape and bake for 2 min at 180-200° C

• Cleave or re-attach blue tape to keep sample in one place. Keep in N2 atmo

sphere in low light.

7.2.2 Cleaving

To cleave GaAs samples score the desired area on the opposing sides of the cleave

with a diamond scribe. The sample should cleave under light pressure applied around

the cleave marks along one of the cleave planes. Keep the sample on some cleanroom

paper to allow it to bend. To cleave Si, you have to flick the sample with the scribe,

because there are no cleave planes as in GaAs.

7.2.3 Exposure

Expose the PMMA with the Raith E-beam tool. Use the 10 fim aperture and 10 kV

for best results. Typical dose is 140 f.iC. Make sure that the step size is below 10 nm

and write speed is below 10 mm/s. Best results are obtained with a 3 nm step and

4mm/s.

7.2.4 Development

• Develop exposed PMMA in 3:1 Isopropanol to Methyl Iso Butyl Ketone for 45

s with light agitation.

• Rinse in Isorpopanol for 30 sec.

• Bake for 1-2 min at 90°.

7.2.5 Etching

• For thermal contact glue the chip to a Si wafer carrier in the carrier in the

Electron Cyclotron Resonance Reactive Ion Etcher (ECR RIE) Pquest.

• Use a glass pipette to put a small amount of PMMA on the carrier.


• Place chip on top of droplet. Make sure there is not too much resist to go over

the sides.

• Bake wafer with chip at 90° C for 1-2 min. DO NOT BAKE AT HIGH TEMP.

PMMA reflows.

• Etch in Pquest given recipe in Table 7.1.

• Strip residual PMMA by sonicating the chip in acetone.

• Soak in Isopropanol.

Table 7.1: Pquest etch recipe for GaAs membranes with thickness up to 200 nm.

Ar (seem) BC13 (seem) CI2 (seem) process Press. (mTorr) ECR power (W) RF power (W) He backside (mTorr) temperature (C) time (s)

Step 1 15 10 2.5 2 0 0 10 13 70

Step 2 (ignite) 15 10 2.5 2

400 47 10 13

until ignition

Step 3 (etch) 15 10 2.5 2

200 47 10 13

220

7.2.6 Undercut

• For 80% Al content AlGaAs use 6% HF. Dunk sample into HF and water

repeatedly (« 8 dunks with 3 s in HF each).

• Do not blow dry the sample.

• Carry it to the solvent wet bench in a small amount of water and use Isopropanol

to remove the water.


• Put wet sample into Isopropanol and then take it out and dry under low N2

flow.

• Inspect under microscope. Repeat if necessary.

7.2.7 Wet Oxidation

• set up GaAs oxidation furnace as per instructions: make sure the bubbler is

working and gas flow and water temperature are correct.

• put sample on glass boat

• drive boat very slowly into furnace

• to undercut a typical PC structure, leave there for 30 min, but times can vary

based on vapor and gas flows.

• remove sample carefully and slowly

7.2.8 Wet Oxidation Undercut

undercut oxidized sample in KOH and rinse in water similar to HF undercut.

7.3 P M M A doped with colloidal QDs

7.3.1 PMMA

Combine PMMA of a desired molecular weight with a solvent such as Toluene in a

closed container. Stir on a magnetic stirrer overnight, or until beads are dissolved.

Filter the PMMA prior to use for best results.

7.3.2 Disso lv ing Q D s in P M M A

If the QD's are in Toluene, simply combine them with the desired volume of premade

PMMA to obtain the proper concentration. If the dots are not in the same solvent, a


solvent transfer may be necessary if a high volume of dots is needed. Otherwise, the

two solvents can be mixed.

7.4 PC laser/detector and electrical contact fabri

cation

In Chapter 6 we discussed all-optical modulation of PC cavities. However, to become

truly useful in signal processing and low power optical switching and light sources,

electrical control over PC structures must be exerted. In this section I discuss our

preliminary efforts toward realizing electrical control over PC chips. These efforts were

not only motivated by applications to optical signal processing, but also the possibility

of electron loading into QDs in order to realize a three-level system and a quantum

memory. As this was the first serious effort towards electrical control of PC devices,

the results are incomplete. However, a significant amount of progress was made

in creating electrical contacts on the chip and characterizing these contacts. With

optoelectronic applications as the main focus, a PIN diode structure with one and

three quantum wells was grown by Tomas Sarmiento in Prof. James Harris' group. In

parallel, a PIN diode sample with a layer of quantum dots was grown by Prof. Pierre

Petroff's group at UCSB. In all cases, the p-doped, intrinsic and n-doped layers had a

thickness on the order of 100 nm and were grown on an insulating AlGaAs layer with

80% Al. Thus, one of the challenges was a precise chemical wet etch that targets

each layer. Targeted ohmic contacts were fabricated and measured. Preliminary

experiments with quantum dot samples showed resistive heating effects and some

indication of control over the dot carrier population in reverse bias. Unfortunately few

of the samples survived to carry out quantifiable experiments. For the quantum well

devices, ohmic contacts were successfully fabricated. However, there was insufficient

time left to continue with this project.


7.4.1 Wafer design

The sample design was modeled on existing devices such as VCSELS and p-i-n diodes

in consultation with Profs. James Harris and Pierre Petroff. The sample structure,

band simulations, a fabricated device and a preliminary measurement for the QD are

shown in Fig. 7.2. The band structure was obtained with the "ID Poisson" software

written by Prof. Gregory Snider and available from http : //www.nd.edu/ gsnider/.

A preliminary device for optoelectronic applications is shown in Fig. 7.3. The fab

rication of this device is given in the next sections. The electrical properties of the

optoelectronic device were found to strongly depend on the quality of contacts. To

characterize the contacts a transmission line measurement (TLM) structure is fabri

cated along with the active devices. Measurements on this structure give the contact

resistance and allow the quality of the contacts to be acessed. Once ohmic contacts

are made, an IV curve for the device is obtained in order to quantify the limitations

due to series resistance and reverse breakdown. The TLM structures and measure

ment as well as an IV curve under dark and light conditions for the device in Fig. 7.3

are shown in Fig. 7.4. The forward current is limited in this device due to resistance

effects. However, the device already functions as a fairly sensitive detector for room

lights.

7.4.2 Mask design

When using electron beam lithography to define the contacts, no physical mask is

necessary, but the writes must be carefully designed to allow overlay exposure of

patterns. For optical lithography a standard mask with multiple patterns can be

made if the mask can move relative to the device in the exposure system. Otherwise,

a mask for each layer must be made. Several things should be kept in mind when

designing the mask:

• The mask should not be opaque everywhere outside the pattern, so that the

chip can be properly positioned.

• The alignment tolerances are on the order of 5 fim on standard optical systems,

http://www.nd.edu/


Sb02 Sample with Schottky and - I V bias

Ec+V (eV) - Ev+V (eV) Ec (eV)

-Ev(eV)

0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03 x(A)

(B)

920 921 wavelength (nm)

922

Figure 7.2: (A) A simulation of the band structure under zero and IV reverse bias for the structure shown in the inset of (A) are shown. The bias applied through the Schottky contact is used to control the tunneling of electrons from the n-doped layer into the InAs QDs in the middle of the intrinsically doped layer. Appropriate biasing aligns the QD levels with the Fermi level of the n-doped layer for enhanced tunneling. (B) A fabricated device in which the p and n contacts have been made to the appropriate layers. The pattern was made by multiple aligned writes with the electron beam tool. A wet etch is used to etch down to the n-GaAs layer of the structure. (C) A preliminary measurement for the sample under reverse bias. Slight crossing of lines, which is indicative of charging is potentially observed. The overall line shift is due to heating in the sample clue to reverse breakdown of the diode. Further measurements could not be made, as the sample shorted.


(A) 2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

-2.5

I ' I • 1 • ] 1 • 1 ' 1

Ec(eV) n(cm-3) t \ . Ev(eV) p(cm-3)

in 7 ca

rrie

r densi

ty (

cm"3)

1

5x10" (8)

Material

p*-GaAs

p-GaAs

i-GaAs

InaaGaAsoa

I-GaAs

i-GaAs

n-GaAs

Alo.aGaAso.2

Substrate

Doping (cnv=)

Iel9

Iel8

Iel8

insulating

Thickness (A)

100/200

400

250

80x5

80x5

250

400

10000

Figure 7.3: (A) A simulation of the band structure under zero bias for the structure shown in (B). In (C) an optical micrograph of the final fabricated device is shown. In (D) a scanning electron micrograph of the same device reveals that the metal liftoff with a single layer of resist leaves a significant edge.

and so a 10 /im margin should be included in any part of the mask (i.e. a

pattern can shift by 5 [im and not short any device).

• Alignment marks on successive layers should be made bigger than those on the

previous layer, so that the previous alignment mark is visible during the align

process. A 5 fxm increase in size is good.

• An edgebead removal mask is necessary when high precision is required - oth

erwise there cannot be good contact between the mask and the sample.

7.4.3 Lithography process

Electron beam lithography has been discussed in the previous sections.

CHAPTER 7. FABRICATION

(A) (B)

mi i

i • • i

: : i

*(O

h

,7"

R = 10.981 d + 190.63 ,

R2 = 0.9953 X''

X

y* y

* Measured R Fit

-20 (1 20 40 60 80

(C)

d(um)

Bright and Dark I V response

--89000 T

V (Volts)

Figure 7.4: (A) An optical micrograph of the TLM structure next to active devices. The structure consists of metal pads with varying spacing. The width of the pad should be larger than the spacing. (B) A plot and fit of resistance R versus pad spacing. R is obtained from the slope of the ohmic IV curves measured between successive pads in (A). The slope of the fit gives the sheet resistance Rsh and the intercept is twice the contact resistance Rc. (C) IV measurements on the fabricated device validate diode behavior and show current limitation due to series resistance at large forward bias. The device acts as a detector in the reverse bias configuration.


For optical lithography using Shipley 3612 photoresist:

• Cover the backside of the chip with blue tape.

• Spin coat resist at 4000 rpm for a 1/im thick film.

• Remove blue tape and bake resist at 90° C for 60s.

• For edgebead and coarse features: expose 3 times with 8s each and 5s between

exposures. Use a large gap (e.g. 70 /im).

• For fine features: expose 1 time with 4s. Use a small gap (e.g. 30-40 //m).

• Develop in LDD26W for approximately 60s. Check under optical for cleared

resist.

7.4.4 Wet Etch

For controlled etching to a layer without an etch stop, a slow etch should be used.

If possible, the resist should be baked at 125° C for 30 minutes in order to make it

more etch resistant. The following etches work well for GaAs:

• 200:8:4 t^ChH^SO^H^C^ - 1-2 nm/s etch rate at room temperature.

• 25:1:1 H20:H3P04:H202 - 2-3 nm/s etch rate at room temperature.

All etching is followed by a deionized (DI) water rinse and Nitrogen blow dry. The

etch rate should be checked on the alphastep.

The rates of both of these etches change over time is small etchant quantities are

mixed. Since the etch works by peroxide oxidation of the semiconductor and a strip

of the oxide in the acid. The etches cannot be premixed. A larger volume of etchant

ensures that the etch composition does not change significantly, and furthermore

means that a smaller relative error is made when mixing the components.


7.4.5 Metal contacts

Metal contacts are made by depositing the appropriate metal through a mask onto

the sample. The simplest mask is a patterned photoresist. A two-layer resist can be

used for optimal contact performance, but a single layer resist works well for contacts

that are relatively thin. For contacts with a thickness of several 100 nm, the 1/im

of Shipley 3612 yields satisfactory results. Once the metal is deposited, the resist

mask is removed in a liftoff process. The best results are obtained with an overnight

soak in acetone. Sonication may aid in the liftoff process, but can also damage the

contact. When doing liftoff it is usually best not to let metal particles dry on the

sample. They will be difficult to remove.

Before contact deposition, the samples should be cleaned in 15:1 H20:NaOH for

30s to remove oxides. Once contacts to the n and p doped layers are made, large Au

contact pads can be deposited in one step to allow probe contacts and wire bonding

to the sample.

When low resistance Ohmic contacts are required, a transmission line measure

ment (TLM) structure should be made to measure the contact resistance and Ohmic-

ity of the contacts.

N-Contact

The n contact is typically an annealed contact and may need to be done first depend

ing on the thermal constraints for the sample. A description of such contacts is given

in Ref. [97]. During the anneal, Ge diffuses into GaAs and alloys with it.

The following contact recipe was used, with the first Ge layer on the n-doped

GaAs:

• Ge(108A)/Au(102A)/Ge(62A)/Au(236A)/Ni(100A)/Au(2000A)

The contacts were annealed with the following recipe:

• Ramp 0° C to 420° C for 60s

• Hold at 420° C for 60s

• Cool


P-Contact

P contacts are difficult to make for low levels of doping, and best results were obtained

for doping levels of 5el8. A description of such contacts is given in Ref. [98]. The

standard contact does not include the first layer below, but this layer was found to

decrease the contact resistance by a factor of 4.

The following contact recipe was used:

• Pt(100A)/Ti(200A)/Pt(100A)/Au(2000A)

P contacts do not need to be annealed, but do not degrade with the aforementioned

anneal recipe.

7.4.6 Electrical isolation by resist

Several materials can be used for electrical isolation including polymers and deposited

nitrides and oxides. We chose a hard-baked photoresist due to it's reflow properties

and ease of deposition and pattern definition. The resist process is as follows:

• spin AZ9260 photoresist at 3000 rpm for 60s.

• bake spun resist at 100°C for 3 minutes on a hot plate

• remove edge bead with a 100 s exposure

• develop the resist in 421K developer for 2 minutes

• expose the desired pattern for 40 s

• develop the resist in 42IK developer for 1 minute

The resist is then cured in the BLUEM Oven in N2 atmosphere with the following

recipe:

• 1.5 hr ramp to 140°C

• 2 hr hold


• 4.5 hr ramp to 325°C

• 2 hr hold

• 1 hr ramp to 25°C

7.4.7 Device fabrication flow

The overall device fabrication flow for the quantum well device is as follows:

• Lithography for wet etch.

• Wet etch.

• Lithography for n-Contact.

• n-Contact metal deposition.

• n-Contact liftoff.

• n-Contact anneal.

• Lithography for p-Contact.

• p-Contact metal deposition.

• p-Contact liftoff.

• Lithography for electrical isolation.

• Resist hard bake for electrical isolation.

• Lithography for contact pads.

• Metal deposition for contact pads.

Chapter 8

Conclusion and Future Directions

8.1 Cavity QED and quantum information pro

cessing

In our cavity-QED work we have so far demonstrated the fundamental capability of

the PC-QD system to serve as a platform for on-chip quantum information processing

and cavity-QED experiments. The two main results - experimental realization of

coherent probing of a strongly coupled cavity-QD system [1] and of controlled phase

interaction between individual photons [2] have shown that on-chip solid state systems

can be used to perform fundamental cavity-QED quantum optics experiments on the

chip.

In particular:

• We have developed, and for the first time demonstrated coherent driving of an

InAs QD inside a PC cavity on a semiconductor chip. This is a major enabling

step to exerting coherent optical control over the PC-QD system and quantum

optics experiments on the chip.

• We have shown that a single InAs quantum dot can significantly modify the

transmission properties of a PC cavity in the regimes of strong and weak cou

pling with low scattering losses. This verifies that that PC-QD system is a

viable platform for scalable quantum information processing on the chip.

113

CHAPTER 8. CONCLUSION AND FUTURE DIRECTIONS 114

• We have shown that the response of the QD is highly nonlinear and used this

effect to perform a proof of concept controlled-phase interaction between two

photon streams at the single photon level, in which we demonstrated that a

single photon is able to impart a 0.077T phase shift on another identical pho

ton. With an appropriate choice of cavity-QD detuning this phase shift can be

increased to RS 7r/2, and could be used in an all-optical quantum information

processing device.

• Although it was not discussed in this thesis, we have also demonstrated the

phenomena of photon blockade and photon induced tunneling by measuring

the photon statistics of a cavity-coupled beam in the presence of the quantum

dot [19]. We have shown that the QD can be used to convert a coherent state

of light into an anti-bunched or bunched stream of photons. With reasonable

improvements in cavity quality factors, such a system can be used to generate

on-demand single photons, and is a demonstration of the highest possible optical

nonlinearity.

• We have shown in our early work that emitters operating at room temperature

can be coupled to the PC cavity via a simple spin-on technique. Thus, with

improved light sources and cavities, such systems could potentially be used to

realize quantum information processing devices that operate at or above liquid

nitrogen temperatures.

These experiments serve as a springboard for further investigation into quantum

logic and memory [16] and quantum systems such as repeaters [99, 100], and sim

ulators and computers [30]. However, to achieve such goals several technological

challenges must be overcome.

Future work will have to focus on two main challenges: scalability and coherent

storage. The promise of the PC platform is that it allows robust and fast scaling

to multiple qubit systems. Device variability introduced in the fabrication process,

as well as the inherent size variation (sa 5%) in QDs dramatically limit scalability

in the system, due to frequency misalignment between resonators and dots on the

chip. These challenges have been in part adressed by the work of Andrei Faraon in

CHAPTER 8. CONCLUSION AND FUTURE DIRECTIONS 115

our group, who has developed techniques to tune QDs and resonators independently

on the same chip [101, 102]. Still the possibility of aligning more than two QDs

and cavities is not very large in these systems, and quantum information processing

protocols that are insensitive to frequency differences should be investigated.

In order to truly implement quantum information processing proposals such as

Ref. [16, 99, 100], a quantum memory with a long storage time is required. The QDs

used in this work were two-level systems with a ground and excited state. A third

level that is accessible through optical or electrical signals is necessary for quantum

information processing protocols. Such a level can be realized with a charged QD

inside a magnetic field [20, 103]. Though alternative protocols based on polarization

splitting of QDs can in principle be used to make a three-level system, the storage

times in such protocols are too low to realize truly functioning devices, and attempts

should be made to bring the proven approach of Ref. [103] inside a PC cavity.

8.2 Classical Information Processing

We have developed a variety of modeling and optimization tools for designing high

quality PC resonators. Due to the size and the planar and monolithic nature of

such devices, they are attractive candidates for low power optoelectronic devices and

photonics for rack-to-rack, board-to-board and chip-to-chip communication in data

centers and future computers [104].

We have shown that such devices can yield highly nonlinear elements and take

advantage of material nonlinearities. Furthermore, we have shown that the free-carrier

induced refractive index changes can greatly affect the resonances of PC cavities with

lower power inputs when compared to classical optoelectronic switches. Thus, PC

based resonators may reduce power consumption in routers and switches. However,

the main challenge that has yet to be overcome is the development of robust electrical

control over such devices.

Appendix A

Equivalence between the CNOT

and CZ gate

The matrix representation of the CNOT gate in the basis {\H)s\H)c,\H)s\V)c,

\V)a\H)e,\V)a\V)e}is:

/ 1 0 0 0 \

0 1 0 0 CNOT =

0 0 0 1

\ 0 0 1 0 /

The controlled-Z (CZ) gate, that is given by the following matrix:

/ 1 0 0 0 \

0 1 0 0

0 0 1 0

\ 0 0 0 - 1 /

(A.l)

CZ (A.2)

Suppose that the cavity realizes the CZ gate. That is, the nonlinear interaction

delays photons that are coupled to the vartically polarized cavity and phase shifts

the state |V") |V) relative to the other states by re. Further, if the second qubit is

Hadamard transformed by a waveplate on the way into and out of the cavity. The

116

APPENDIX A. EQUIVALENCE BETWEEN THE CNOT AND CZ GATE 117

following happens:

Ic ® HWP(ir/8)s \V)C \V)a = \V)C (\V)a + \H)S)

CZ\V)e(\V)a + \H),) = \V)c(-\V)a + \H)a)

Is ® HWP(ir/8)c \V)C (- \V)a + \H)a) -> - \V)C \H)a

(A.3)

(A-4)

(A.5)

Thus the overall transformation is an identity on the control qubit and two Hadamard

transforms on the signal qubit. This can be written as Hs ® Ic:

1 / 1 1 U = Hs <g> I x = - =

y/2 \ 1 -

This leads to CNOT:

1

1 0

0 1

1

72

/ 1 1 0 0 \

1 - 1 0 0

0 0 1 1

\ o o i - l )

CNOT = £/f -CZ-U =

( 1 0 0 0 \

0 1 0 0

0 0 0 1

V o o i o )

(A.6)

(A.7)

In the case of circularly polarized qubits |±) = \H) ±i\V), passing the control

photon through a A/4 (QWP) waveplate allows us to do the CNOT operation directly.

The A/4 maps |+) —>• \V) and |—) —• \H). In that case, the component i \VV) always

changes sign and switches rotation angle.

I+)I±> \V){\H)±i\V))

(X/4)cmtroi^\V)(\H)±i\V))

CZ-*\V)(\H)^i\V)) = \V)\T)

W) IT) - (A/4)c n,trol IT)

(A.8)

(A.9)

(A.10)

Appendix B

Derivation of Cavity Radiative

Loss

The radiated power per unit solid angle K(9,4>) can be expressed in terms of the

radiation vectors N and L in spherical polar coordinates (r, 9, </>) :

K(9, JL 8A2

Ne + ^ V

2

+ N$- u_ V

(B.l)

where r\ = J^- The radiation vectors in spherical polar coordinates can be expressed

from their components in Cartesian coordinates:

Ng = (Nx cos 4> + Ny sin (j>) cos 9

Nj, = —Nx sin <f> + Ny cos 4>,

(B.2)

and similarly for Lg and L<p. As described in Reference [3], the radiation vectors in

Cartesian coordinates are proportional to 2D Fourier transforms of the parallel (x

and y) field components at the surface S (Fig. 2.4):

118

APPENDIX B. DERIVATION OF CAVITY RADIATIVE LOSS 119

Nx =

Ny --

L>x -

Ly -

= -FT2{HV)

= FT2(HX)

~- FT2(Ey)

fc|

fe.

-- -FT2{EX)

k\\

kz

x v = k(—,—) = ksm9(cos<j)x + smcj)y)

= kcosO,

(B.3)

where k = 2TT/\, A is the mode wavelength in air, and k\\ = ksinl

Here the 2D Fourier Transform of the function f(x,y) is

FT2(f(x,y)) = J Jdxdyf(x,y)e^x^

= J Jdxdvf(x,y)e^+k^

(B.4)

(B.5)

Substitution of expressions (B.3) and (B.4) into (B.l) now yields an expression for

the radiated power (2.11) in terms of the FTs of the four scalars Hx, Hy, Ex, and Ey.

This expression is in general difficult to track analytically. We will now use Maxwell's

relations to express (B.l) in terms of only two scalars, Hz and Ez.

Noting that for a bounded function g, FT2(^) = —ikxFT2(g) (similarly for ^-),

we can re-write No as

N0 h\k (-kxFT2(Hv) + kyFT2(Hx)) (B.6)

i kz FTJdHy 9Hx' k\\k dx dy '

kzce0 FT2{EZ

APPENDIX B. DERIVATION OF CAVITY RADIATIVE LOSS 120

where the last step follows from Maxwell's Eq. V x H = e0Wr = iue0E. Similarly,

we find Le = ™ F T 2 ( i ^ ) . From V • H = 0 and V • E = 0 at the surface S, it also

follows that Nj, = =±FT2(^) and L0 = -f F T 2 ( ^ ) . Substituting these expressions

into Eq. (B.l) yields

K[kx, ky) — 8\2k2 T]z

kzFT2{Ez) + iFT2{ dEz

dz + a IT

kzFT2{Hz) + iFT2{--^-)

(B-7)

Furthermore, Ez, Hz oc exp(ikzz) for propagating waves inside the light cone

(which are the only ones that determine P), implying that FT2{^-) = —ikzFT2(Ez)

and similarly for Hz. This allows further simplification of the previous expression to

Tik K(kx, ky) = 2\^k2 \FT2{EZ)\2 + \FT2{HZ)[ (B.8)

Substituting this result back into the expression for total radiated power (2.11)

gives the required result (2.13):

TT/2 2TT

= I (d6dcj)sm(6)K(9,<t>)

o o

— / dkxdky—J\ykx,ky)\J{kx,k. Jk»<k

2A k 7fcn<fc fcii

(B.9)

\\FT2(EZ)\2 + \FT2(HZ)\2

Above, J(kx, ky) is the Jacobian resulting from the change of coordinates from

{B,4>) to (kx,ky).

Appendix C

Pulses in a nonlinear PC waveguide

C.0.1 Derivation of the propagation equations

The probe envelope evolves according to (with q = k — ho):

^ = | J dkA(k, i)e'[(fc-fco)2-Mfc)-^)t]

at 1 „2 dv9

(C.l)

ftjdqA{q+k,Mq{z-V3t)~w

= J dq{d-^hA _ t{Vgq + ±%Ltf)A(q + fc0, f))e'[ ,(-'*t)-* ,a3M = f rfga(A(g+fco,t)ci[g(^-f9t)-^

2^-t]

= j dq§A^L^'-v't)-^^t] - Vg§-zp + qdit&p

in the case of nonlinearity, the wave equation with c~2 = [i0e0 is:

„ „ - 1 d2(e(r)E) d2P

And we can rewrite this equation in terms of the previously introduced Hermitian

field operator as:

121

APPENDIX C. PULSES IN A NONLINEAR PC WAVEGUIDE 122

Here, in general, the polarizability P is given by

Pi = ,(3) £o 2 ^ 2 ^ xUk,iE(u'p)iE(u,')jE(vp)i< (C.4) j,k,l m,n,p

(3)

where Xij,k,i a r e *n e components of the third order nonlinearity tensor. In our case,

we consider the system to be isotropic, the response instantaneous, and only two

frequencies (UJS,UJP) to be present. To find the time evolution of the coefficients, we

use the wave equation. In the presence of the nonlinearity we expand e ~ e + 5 and

O as:

0 + AO = | V x V x | -

6 + Ad « 6 - |[f 6 + of] = 6 - ±{f, 6} £ 4=V x V x 4= + -7-V x V x - M + 0 5\2

(!) J(C5)

Let (u'\ = (u,k',n'\, \u) = \u,k,n), A = A(k,t). Then we have:

(u'\ (O + AO) J dkA \u) - - (u'\ jigiJdkA \u)

J dkA({u'\ 6 \u) + {u'\ AO) \u) = - (u'\ ± J dk(A - u2A - 2iuA) \u)

J dkA (u'\ AO \u) = ^J dkA {u') u

-\$dkA (u'\ s-0 + Of \u) = ^A{k')

-±fdkA(u'\ f \u) (*£ + £) = 2-fA{k>) '-A{k')

(C.6)

J dkA(k) (ic\ - \u) ,/i 5 2iu)

Above, we neglect second derivatives of the envelope and combine the two fre

quency terms as UJ '2 •UJ 2UJ2 + vs(k — k') ~ 2UJ2, because the (k — k') term will lead

to the derivative of the slowly varying envelope multiplied by the nonlinearity and is

very small. Since the frequency bandwidth of the envelope is small, the envelope is

slowly varying in time, and the second order time variation in the coefficients A(k)

is neglected. Furthermore, we have also assumed that the inner product (v!) u is

roughly unchanged by the nonlinearity - it remains a delta function. The perturba

tion 8s^p contains both the real and imaginary parts of the third order susceptibility.

The real part is responsible for the cross phase modulation, while the nonlinear term


gives the two photon absorption of the signal and probe. The figure of merit for the

feasibility of the experiments is the phase-shift gained per loss length (- point). The

full perturbation can be written as:

8StP = iax + 360(xl3) + «xi3 ))( |£K,P) |2 + 2\E{uPtS)\2) (C.7)

Where x ^ is the third order polarizability (which is assumed to have only one value

and to be infinitely fast), and E(ujStP) = {S, P}ukspel^ks-pZ~u'a^ is the electric field of

the two modes. The linear loss ct\ is the imaginary part of the dielectric constant.

The value of \r c a n be determined from the experimentally observed bulk in

tensity dependent refractive index defined via n — n + n2I, where I is the average

field intensity. For a bulk material I is given by I = \^ —\E\2 = ^ce0n\E\2. Thus,

3^(3) _ c n2n 2_ p o r AlGaAs at the wavelength of 1.5 fim, n2 ~ 1.5 x 1CT 1 3 ^ ;

furthermore, the index is very similar for TE and TM polarization in AlGaAs slab

waveguides [54]. Since x ^ measures the response of the local charge distribution to

the local electric field, the coefficient itself is a material property and is not modified

in the PCW, except possibly due to surface effects (e.g. reduced response or arti

ficially added birefringence). Thus, we can derive the value of the coefficient from

bulk experiments and combine it with the modified electromagnetic fields to get the

resulting effect in the PCW. The x ^ coupling term only exists in the material, and

we can replace the n2 term with a dielectric which is equal to the spatially patterned

index of AlGaAs in the PCW and is zero in the air. We set 3 x ^ = cri2e(f). And

we define K = ceon2, so that the perturbation due to real part of the nonlinearity

becomes <5SiP = Ke(f)(\E(iOS:P)\2 + 2\E(iOPtS)\2) . The loss terms are similarly deter

mined from a fit to atotai = oti + o.^!- The linear loss a\ results in an exponential

decay of the signal with a characteristic length (a i ) _ 1 . The nonlinear loss gives a

characteristic length of (a^ / ) - 1 . We will drop the losses for now, in order to derive

the equation of motion for the pulses, and will assume that the Bloch components

of the eigenstate \u) and \u') belong to the same waveguide branch n = n!. Each of

un,k, un,k' is t h e n roughly given by some central k component that is modulated by an

envelope un^eikz ~ Mnfcoe«(feo2-wo*)e»[(fc-fco)z-Mfc)-"o)«]_ \ye now insert the exact form


for (u'\ and \u) into the above equation, and only look at the cross phase modulation

component on the probe due to the signal. We take n to be branch of the pump mode

p, and m to be that of the signal s, and only treat the perturbation due to the signal

explicitly. Also, we will assume for simplicity that the signal group velocity vs at k

and k', as well as the dispersion ^ - , so that we can combine them in the expansion

of the Bloch state. Eq. C.6 then implies:

A{k') = iup J dk" j dh' A{k")Kl\up\2\us\

2\S{z')\2e^k"^

« iupn J dk"A(k") J ^ / |5(2 /)|2e l [ ( fc"- fc ' )z '- (a j ( fc" )-w(fe ' ) )^/Arf3re>s |2 |«p |2 (C.8)

« WJ„K7SIP / dk"A(k") f dz'\S{z')\2e^k"-k'^-^k"^k'W

This is now inserted back into the evolution equation for the envelope C.l, and

we re-substitute k — fc0 = q and keep the frequency term in the form uj(k) — oj(ko) for

convenience.

/dkA{k,t)ei{k-ko)z-(-'Jj{k)-U0^ =

= *wpK7a,p JdkJ dk"A{k") J '̂|5(y)|2ei[(fe"-fc)^-(^(fe")-uJ(fc))t]el[(fc-fco)^(^(fe)-^o)t]

= iujpK^StP f dz' J dk''A{k'')\S{z')\2ei[{k''z'-koz)~{bj{k'')~^)t]) J dketk^-z,)

- iupKls<p J dz'\S(z')\2elk^z'-^5(z - z') J dk"A(k")e^k"-k°>'-^k"^W

= iujpKjSiP Jdz'\S(z')\2eiko^'^6(z - z')P(z')

= iupK-fStP\S(z)\2P(z)

The y/e terms cancel the denominator of the perturbation. The term 7SjP contains

an effective area integral j S t P = ^ JAd3re|ris |2 |np |2 « J dxdye\ua\

2\up\2. Since the

Bloch states are periodic, their integral in each unit cell ("fs,p) is the same, and we

simply weigh it by the average value of the slowly varying envelope in that cell to

find the integral over the whole volume. A is the unit cell volume. S and P are only

functions of time and the propagation coordinate (z here), and are uniform in the

transverse (x,y) plane. The term K contains the strength of the nonlinearity. We now

re-normalize the Bloch state and the field:


Nshujs =11 f d3fe0e(f)\S\2\us\2 (CIO)

« [dze0\S\2- f d3fe(r)\us\2 (C.ll)

J a JA

Nshcos =fdze0\S\2 (C.12)

1 = 1 [ d3fe{f)\us\2 (C.13)

"A The above normalization means that [7] = m~2 and [\S\2] = Volt2. Thus the term

KJLJ\S\2 has units of s_1 , as desired. We now insert the form for A into C.l:

The refractive index due to cross-phase modulation is twice that of self phase

modulation [53]. The evolution of the slowly varying envelope is given by:

5 = qKius(7s>s\S\2 + 2ls,p\P\2)S - vsS' + i\%S" (C.14)

P = q>Kup(%,p\P\2 + 2%tS\S\2)P - vpP> + i\^P" (C.15)

We will now show the qualitative behavior of the two pulses, assuming a weak

interaction. Take P = p{z,t)ei^z't) and S = r]{z, t)e^z't]. Inserting P into C.15, the

real and imaginary parts satisfy:

Real:

p+(Vp+d^)p = --*-<$'p (C.16)

Imaginary:

i\>p + vv4p = -KLUP^PIPP2 + 27s,p7?2)p + -&-(p" - {c^fp) (C.17)

If we assume that in C.16 the second derivative term vanishes, then we can see

that the envelope moves along a characteristic given by vp + -^4> « vp(ko+ (/>'), which,

in the natter regions of the dispersion curve is very close to vp(ko). If 7^ (^-) < < 1,

then C.17 simplifies to give an equation for the phase along a characteristic given by


vP(k0 + *-)•

dvp -.

<P + vP(t>' + -|-(<£')2 = -«wp(7p,pP2 + 2js,Pr]2) (C18)

Since we will not generally be able to generate a pulse that is a solution to the

nonlinear system, we will assume that the pulses are Gaussian and drop the dispersion

terms for simplicity of the analysis. We are essentially assuming that the envelopes

are very slowly varying and that the interaction time and dispersion are not strong

enough to affect the phase measurement, which is dominated by the group velocity

term rather than the group velocity dispersion term. Thus, we will rewrite Eq. C.15

in a frame moving with the group velocity in terms of z' = z — vpt and t' = t, and

neglect the terms with -~^ and ^ . Since the two pulses may have different group

velocities, we have Av = (vp—vs). If the group velocity dispersion terms are neglected,

each envelope is only a function of z'.

dJ^ « ^p(lP,P\P(z')\2 + 2laj>\S(z' + Avt)\2)P(z>) (C.19)

The solution, and upper bounds on the phases after time t = — are:

P(z') = P(0)EXP[^KU;P J\iPtP\P(z')\2 + 27stP\S(z' + Avt')\2)dt'} = P ( 0 ) e ^ + t ^ 2 0 )

<J)P = \KUP I lp,p\P{z')\2dt' ~ \KUJP1P,P-\P{Z'\$.2\) l J 0 Z Vp

<t>s = «wP / ls>P\S(z' + Avt')\2dt' « KLUP1S,P-\S{Z'\$.22) JO vp

If the second arm of the interferometer is adjusted for a phase shift of 7r/2, the

difference in the intensity signal on the two detectors gives the phase, and thus

an estimate of the photon number. The total integrated signal energy is Idet =

fffd3re0e\P(z')\2\un\2sin((f>s). Starting from NBhw„ = J ^ d * ' e 0 |S(*')|2 = J^dz 'eo lS lM*' )

with | 5 | 2 = ^ g ^ (similarly, \P(z')\2 = \P\2p(z') and \P\2 = ^j£%j), the integral


for Idet is re-ordered again:

het « f f J d'reolPiz'^MzyKl2 (C.23)

« f dz'e0\P{z')\2^s{z')- f d3re\up\2 (C.24)

J a J A

= J dz'e0\P(z')\2<f>s(z') (C25)

~ KWP7S)P -^—- / dzp(z )s{z ) (C.26) Wp €0LefftS Lieff,P J

In the case when P is much narrower than the signal pulse S, we can view it as a quasi

delta function p(z') « Lefftp5(x — x0). The effective length can be approximated by

the spatial width of the pulse, which is TSVS, the product of the temporal width and

the group velocity of S. In this approximation the detected intensity is :

Idet ss KWp7piS —-Nptkjp (C.27) vp e0Tsva

2 2 LNSNP = cn2n LOpuis^PtS (C.28)

Vp TSVS

And the phase shift per photon of S is N ^ e ^ = cn2^p,shiosLVpf-~^r• In order to

make a comparison to our initial plane wave argument, we can identify 7P)S as the

effective inverse area, -£— as the effective pulse bandwidth in the waveguide, and —

as the enhanced interaction length (time).

Appendix D

Cavity QED Experiment and

Derivations

D.l Experimental setup

In our measurements a strongly coupled quantum dot is identified by the QD/cavity

anti-crossing signature in photoluminescence. Anti-crossing is observed by temperature-

tuning a QD through the cavity resonance, using the method detailed in Ref. [4].

During the temperature scan, a 980 nm diode heating laser is incident upon a metal

pad next to the cavity and modulated by a triangular wave form, whose period de

termines the heating cycle. Both the cavity and quantum dot wavelengths shift with

temperature, with the dot shifting three times faster, as shown in Fig.lC of the main

text. Thus each point of the temperature scan corresponds to a particular detuning

between the QD and cavity, as the system is probed at a fixed point by the reso

nant laser beam. This is illustrated in Fig.D.2, where we show how the quantum

dot and cavity trajectories and phase shifts are determined. A narrow-band (5 MHz

linewidth) frequency tunable external cavity diode laser is focused into the cavity and

observed in cross-polarization, as detailed in Ref. [1] and shown in Fig.lA of the main

text. The sample temperature changes from 20 to 27 K during the heating cycle.

This signal beam probes the cavity-QD system at various detunings between the QD

and cavity. The control beam is combined with the signal beam input path and also

128

APPENDIX D. CAVITY QED EXPERIMENT AND DERIVATIONS 129

directed at the cavity.

D.2 Quantum dot visibility

The visibility of the QD-induced feature is reduced due to wavelength jitter (through

thermal fluctuations [1]) as well as QD blinking between an optically active and an

optically dark state. From an analysis of second-order coherence, similar to that

presented in Ref.[105], we deduce a dark-state probability of 25%. This limits the

QD-induced features to a visibility of V=(hright - Idark)l{hright + /darfc)=75%. With

blinking and sampling taken into account, the theoretical fits agree well with experi

mental data. In more recent experiments, we have observed V=90% dip visibility for

quantum dots that do not exhibit blinking.

D.3 Quantum dot saturation

The response of the quantum dot to the driving electric field is given by its suscep

tibility x[53]. This function determines the dot's radiative properties and gives the

phase and amplitude of the field that interacts with the quantum dot. When the

control beam is resonant with the QD, \ can be expressed as [18],

X - *F P. (D.l)

where Pin is the input power, and F is the Purcell factor F=^-. x is a nonlinear

function, and is typically given by an expansion in the driving electric field E as

X = X^1' +X^ 'E + x^E2 +.... The field E is related to the input power via Pin/Acav —

2no^e0/n0 \E\ , where Acav « ( ̂ -J is the cavity area, no is the refractive index

of GaAs and e0 and /J,0 are the dielectric constant and permeability of free space.

Generally, the Kerr coefficient x^ gives the phase modulation. Another conventional

measure of nonlinearity is the intensity-dependent refractive index n2, which is related

to the Kerr coefficient via n2 j p -j^- = ^— \E\ [53]. Here the units of n2 are cm2/W,

c is the speed of light in m/s, and x*^ is in units of m2/V2. A beam propagating


through a cavity with such a nonlinear medium picks up a phase shift that is given

bv 6 = ^~i-p^-^-—. Thus. n--> and y('3' can be determined from the change in o with

changes in Pin.

D.4 Amplitude and phase of the interference sig

nal

In this section we derive the expressions for the amplitude and phase pf the interfer

ence signal observed in our experiment. The experiment setup is reproduced in Fig.

D.l for convenience.

• • i.'

r.' [ • • • • # • • # • • • • • « t, . -

I ; control (Xc) # v v " L

/ \A^J

i | | signal (ks) > w -

l\> + CWI/> ~*f\/\t

QWP

* W -

r(«) | \> - CW \A

0.7 0.65 0.6 0.55

fi 0.5 •e-0.45

0.4 0.35 0.3 0 25

B A - i f .

yf\ ~*:

v. /l/~^C~(')'

]Wi 80 85 90 95 100 105 110

• \


*\/\/»

PBS A(9) {r(u>) + e*W} ^

Figure D.l: Experimental setup. Vertically polarized control (wavelength A(.) and signal (wavelength A,5) beams are sent to the PC cavity (Inset) via a polarizing beamsplitter (PBS). A quarter wave plate (QWP) (fast axis 6 from vertical) changes the relative phase and amplitude (C(#)) of components polarized along and orthogonal to the cavity. Only the reflection coefficient r(jj) for cavity-coupled light (at (—45°)) depends on the input frequency and amplitude. The PBS transmits horizontally polarized light to a detector D. (B) Theoretical model for the phase of signal beam


The detected signal is modeled by considering the propagation through each opti

cal element. Vertically polarized light passes through the QWP to the cavity that is

tilted by 7r/4. The component of the incident beam along the cavity is reflected with

coefficient r(ui), while the orthogonal component reflects from the distributed bragg

reflector (DBR) with TUBR- The beam then passes back through the QWP and the

horizontal component is passed by the beamsplitter.

In beamsplitter polarization the vertical input and horizontal output are

* - : . " - ( M )

Rotation of the QWP and cavity relative to vertical are accounted for by applying

the rotation matrix:

,„x I cos(#) -sin(0) \ , „ , R(0) =[ K ' y ' ) (D.3)

\ sin(0) cos{6) J

The QWP (with a horizontal fast axis) rotated by 9 from vertical is given by

QWP(6) = eiv/4R(d)• ( l °)-nr1(d) (D.4) 0 i

Interaction with the cavity is given by

RC = R(ir/4) • I r ( w ) ° ) • R-\TT/4) (D.5) \ 0 rDBR J

The transfer matrix for the scattering process is then

T - QWP(9) • RC • QWP(9) (D.6)


and the field going to the detector is given by

E^t = Transposed) • QWP(9) • RC • QWP{9) • V (D.7)

= | (r(to)(cos2(29) - isin(29)) - rDBR(cos2(29) + isin(29)))

From which, in the limit rDBR —>• — 1 we obtain

Is(u>) = \E0Ut\2 = I ^ X r ^ + e^))!2 , (D.8)

with the interference term

. m = cos2 (29) +ism(29)

cos2 (20) -ism (20)' l '

and detected amplitude

A(9) = %- (cos2 (20) - i sin (20)) , (D.10)

D.5 Computational model

The Hamiltonian for the two-beam interaction with a cavity embedded quantum dot

is given by:

H = fkuQDaz + hujcava)a + hg(a+a + aV_) + ^ ( E ^ ^ + Ese~iulst)(a + af) (D.ll)

Where LOQD, ^cav, wc, LOS are the quantum dot transition frequency, cavity resonant fre

quency and the control and signal frequencies respectively. Es, Ec are the control and

signal driving fields respectively. The cavity photon number is given by ncav = (a^a).

Fits to the data were performed with the Quantum Optics Toolbox[66] which numer

ically solves this Hamiltonian with a photon number basis that ensured convergence

(up to 30 photons). In the case of frequency detuned beams, a matrix continued

fractions method was used to solve for the cavity mode at the signal frequency us un

der excitation of a driving field with two frequency components UJC,US. This method


expands operators in a Fourier series and solves for coefficients at the appropriate

frequencies. The matlab code is available upon request.


50 60 70 80 90 100 110 120 130 140 150 160 928.3 928.4 928.5 928.6 928.7 temperature scan count X(nm)

Fi gu re D.2: The quantum dot and cavity wavelengths (AQ£i,AcatJ) are extracted from spectra taken

as a function of the temperature scan count (A). The probing laser is positioned at a wavelength

(Ac) that is close to the point of crossing between the QD and cavity trajectories in (C). Thus, each

point along the 'temperature scan count' corresponds to different offsets between the quantum dot

and cavity. By tracking the amplitude of the probing laser, we can find the reflectivity signal and

extract the phase in B. The point of maximum phase contrast corresponds to the vertical dashed line

in B (also in C) and to a particular offset between the cavity and dot wavelengths. When the cavity

and QD wavelength are fixed, and the laser is scanned along the dashed line in B, the signal shown

in D would be obtained. The dot and cavity detunings are indicated by the two dashed lines. In

the experiment, the laser is positioned at Ac, which overlaps with \cav in D. The point of maximum

phase contrast, which is the point where we find the phase, coincides with Aca„ and Ac in this case.

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