COHERENT CONTROL OF BLOCH OSCILLATIONS IN SEMICONDUCTOR

SUPERLATTICES BY MEANS OF OPTICAL PULSE SHAPING

A Thesis

Submitted to the Faculty

of

Purdue University

by

Riccardo Fanciulli

In Partial Fulfillment of the

Requirements for the Degree

of

Doctor of Philosophy

August 2004

ii

Fatti non foste a viver come bruti, ma per seguir virtute e canoscenza Dante, Divina

Comedia, Inferno Canto XXVI

iii

ACKNOWLEDGMENTS

I wish first of all to acknowledge my parents, Sauro and Minnie for their unconditioned

love and support over all these years. I am always trying to follow their example.

A big thank you goes also to Melanie, my personal software (and not only) support

person.

I would also like to remember my friends Stefano and Ute, Mark, Fabio, Edie, Gio-

vanna, Maria Rita, Rajesh, and many more that gave me friendship and support in

the difficult moments. I would like to acknowledge my dear Gerardo for teaching me

the love for books as a little child and Sergio for pushing me to look beyond the big

ocean towards the land of the wide plains. I would also like to mention my beloved

Tango Club and Fred and Yermen who brought lightness and fun in otherwise hard

times.

From a scientific point of view, I would like to acknowledge my advisor, Professor

A.M. Weiner for his patience and for letting me free to try and experiment different

approaches. I owe much also to the insight and lead of Professor M.M. Dignam and

to the many useful discussions with my ”German counterpart”, Dirk Meinhold.

A thank you goes also to Dan for sharing his wide laboratory experience and to my

coworkers for all the times they had to seat through my talks on quantum mechanics

at our group meetings.

Finally I would like to thank the Physics Department of Purdue for providing the

funds necessary to subsistence and Professors D.D. Nolte and A.S. Hirsch for their

availability.

iv

TABLE OF CONTENTS

Page

table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Bloch oscillations (BOs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Semiclassical Introdution to the Bloch oscillations . . . . . . . . . . . 1

1.2 Quantum mechanical introduction to the Bloch oscillations . . . . . . 3

1.3 Many particles picture: excitonic effects . . . . . . . . . . . . . . . . . 7

1.3.1 Intrawell excitons . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Interwell excitons . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.3 Excitonic effects on the oscillator strength . . . . . . . . . . . 9

1.4 Absorption spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Density of states of a superlattice structure . . . . . . . . . . . 9

1.4.2 Excitonic absorption . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.3 Absorption with external electric field . . . . . . . . . . . . . . 12

2 Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Detection of the BOs through THz radiation detection . . . . . . . . 15

2.2.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Photo-conductive sampling (PC-S) . . . . . . . . . . . . . . . 16

2.2.3 Electro-optic sampling (EO-S) . . . . . . . . . . . . . . . . . . 17

2.2.4 Bloch oscillations and THz radiation . . . . . . . . . . . . . . 19

2.3 BO detection with Degenerate Four Wave Mixing (DFWM) . . . . . 22

2.3.1 Introduction to the theory of DFWM applied to the BOs . . . 22

v

2.3.2 DFWM: a physical picture . . . . . . . . . . . . . . . . . . . . 23

2.3.3 DFWM: a density matrix approach . . . . . . . . . . . . . . . 27

2.3.4 DFWM: beyond the finite-order approach . . . . . . . . . . . 29

3 Shaping and characterization of ultrafast laser pulses . . . . . . . . . . . . 33

3.1 Pulse shaping by means of spatial light modulators (SLM) . . . . . . 33

3.1.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 The functioning of the SLM . . . . . . . . . . . . . . . . . . . 35

3.2 Autocorrelation measurements . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Cross-correlation measurements . . . . . . . . . . . . . . . . . . . . . 40

3.4 Second Harmonic Generation - Frequency Resolved Optical Gating(SHG-FROG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Phase Retrieval by Gating in Frequency Domain with an SLM . . . . 44

4 Bloch Oscillations in superlattices: Samples growth and preparation . . . . 46

4.1 Superlattice samples for experiments in transmission . . . . . . . . . 46

5 Spectrally Resolved Degenerate Four Wave Mixing (SR-DFWM) . . . . . . 51

5.1 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Pulse Shaper: a few numbers . . . . . . . . . . . . . . . . . . 53

5.1.2 Shaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.3 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.4 CCD Camera: a few numbers . . . . . . . . . . . . . . . . . . 55

5.1.5 Monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.6 IR camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Bloch Oscillations and SR-DFWM: unshaped results . . . . . . . . . 57

5.3 Method for monitoring the electronic dynamics through SR-DFWM . 59

6 Coherent control of Bloch Oscillations in superlattices: towards a THz os-cillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Modes of the Bloch oscillations: Breathing mode and Non-breathingmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Changing mode of oscillation: a switch for the Bloch THz oscillator . 65

6.3 Control over the initial condition of the electronic oscillatory motion . 75

vi

6.4 Presence of an internal DC field . . . . . . . . . . . . . . . . . . . . . 75

6.5 Selective excitation of Wannier-Stark states . . . . . . . . . . . . . . 79

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A Sample processing: a step-by-step guide . . . . . . . . . . . . . . . . . . . . 89

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vii

LIST OF FIGURES

Figure Page

1.1 Lowest miniband of the superlattice in the first Brillouin zone. Theelectron moves along the band in k space. . . . . . . . . . . . . . . . 3

1.2 Probability of finding the electron in a part of the Brillouin zone ata time t. Lighter shades indicate higher probability. The oscillatorymotion is resulting clearly. . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Binding energy for interwell excitons calculated as a function of electricfield in a 3nm/3nm GaAs/Al0.35Ga0.65As superlattice. The dashedcurves are for p < 0. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Oscillator strength of transition p = -1 normalized by the transitionp = 0 measured versus electric field in GaAs/AlAs superlattices withdifferent barrier widths LB. The well width was always LW = 3.1 nm.The dashed curves are for p < 0. [2] . . . . . . . . . . . . . . . . . . . 9

1.5 Schematic of the behaviour of the DOS in a superlattice with no exter-nal fields applied. The DOS in the center of the minibands should beflat. The plateaus between the minibands are due to the 2D characterof the kinetic energy in the (x,y) plane. Also, the extreme 2D (dashed)and 3D (top) cases are shown. [3] . . . . . . . . . . . . . . . . . . . . 11

1.6 Absorption spectra for GaAs/AlGaAs superlattice as a function of theelectric field calculated in the tight-binding formalism. ε = −4(Eg +E1e + E1h)/(∆e + ∆h) and f = 4eFd/(∆e + ∆h) are respectively thereduced photon energy and the reduced electric field [4]. . . . . . . . 12

1.7 Schematic of the superlattice structure with an applied electric field. . 13

1.8 Derivative of the absorption spectrum as a function of the externallyapplied bias. The sample is a 67/17 Aas used for the experiments ofcoherent control reported in the following chapters. . . . . . . . . . . 13

1.9 Example of a photocurrent spectrum taken on one of our first samples.A field is applied and by varying it one could see the linear motionof the levels. The sample is a 67/17 Aas used for the experiments ofcoherent control reported in the following chapters. . . . . . . . . . . 14

2.1 Schematic of the setup for photoconductive-sampling measurement [5] 17

viii

2.2 Schematic of the setup for electrooptic sampling measurements [5] . . 18

2.3 Schematic of an electro-optic sampling experiment. . . . . . . . . . . 19

2.4 Bias dependence of the detected THz transients after excitation by op-tical pulses with photon energy of 1.54eV (centered on the excitonicWS resonances of the lowest heavy-hole transitions). Sample temper-ature: 10K [6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Transmittive electro-optic signals for different applied voltages in theWS regime of the superlattice at an excitation density of 2x109cm−2.The curves are shifted proportional to the voltage applied. [7] . . . . 21

2.6 Internal EO-S data from reference [7] . . . . . . . . . . . . . . . . . . 22

2.7 Schematic of the of the amplitude of the polarization grating (at a fixedtime) created by the interaction of the two pulses with wave vectorsk1 and k2 inside the sample. . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Schematic of the decaying of the amplitude of the polarization gratingcreated by the interaction of the two pulses with wave vectors k1 andk2 inside the sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9 Schematic of the temporal evolution of the polarization grating in thepresence of Bloch oscillations. The slower temporal modulation duethe internal electronic dynamics is now visible. . . . . . . . . . . . . . 27

2.10 The time integrated intensity of the DFWM versus delay τ is reportedfor ωlaser = ω00 − 2.27ωBO, n0=13 and a density of 6.36x109 cm−2 forthe three different directions Kn indicated. For K−3 results for n0=3(dashed line) and n0=5 (dotted line) are also plotted. [8] . . . . . . . 30

2.11 The hh−1 spectrally resolved degenerate four wave mixing peak energyversus delay τ is plotted for ωlaser = ω00− 2.27ωBO and densities givenabove each curve (in units of 109 cm−2). The curve offsets are real effectdue to intraband renormalization of the DC field. The inset shows peakoscillation amplitudes as a function of density for (ωlaser−ω00)/ωBO =-1.24 (circles), -2.27 (triangles), and -2.83 (squares).[8] . . . . . . . . . 31

3.1 Schematic of a fourier transform pulse shaper with capability of shapingamplitude and phase of the spectrum separately at the same time. Theslabs filled with diagonal lines are two parallel x polarizers, while thecentral slab represents the SLM (or two SLMs) . . . . . . . . . . . . . 34

3.2 Schematic of a spatial light modulator. ITO is the semitransparent,conductive material that is used to apply field in the z direction [9].The width of each SLM pixel in our set-up is actually 100 µm with agap of 2 µm between adjacent pixels. . . . . . . . . . . . . . . . . . . 35

ix

3.3 Schematic of the effect of the external bias over the liquid crystals inthe spatial light modulator.[9] . . . . . . . . . . . . . . . . . . . . . . 36

3.4 (top) Transmission data for a pulse centered at 800 nm going througha 45 SLM where a layer is kept at fixed voltage (zero here) and onthe other the bias is swept from the minimum to the maximum usablevoltage. (bottom) the data shown in the top plot are used togetherwith eq.3.1 to find the phase introduced by the layer to the incominglight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Schematic of a real time autocorrelator. . . . . . . . . . . . . . . . . . 39

3.6 Cross correlation data of a shaped pulse. The effect of the probe finitewidth is visible in the raised minima that otherwise are supposed tobe at sitting at zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 Spectrogram of the pump pulse when a marked third order chirp wasapplied with the pulse shaper. . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Example of chirped pulse where different frequency components arriveat different times. From top to bottom it is shown how the initial chirp(top) was taken care for. . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9 (a) Spectrum of the studied pulsed laser. Each sampled slice of thespectrum is reported separately. (b) (circles) Relative position of thelong pulses in time domain. (line) Third order polynomial fit. Aquadratic behaviour is clearly visible. . . . . . . . . . . . . . . . . . . 45

4.1 First and second electronic minibands calculated for an infinite 67/17structure at low temperature. . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Schematic of the growth of the SL samples (courtesy of Professor K.Kohler from the Max Plank Institut in Freiburg, Germany). . . . . . 47

4.3 Sample etched prepared for an experiment in transmission. [3] . . . . 48

4.4 IV characteristic curve of the sample described in fig. 4.2 . . . . . . . 49

4.5 Calculation of the absorption fanchart for the SL system. The areaof the squares is proportional to the dipole matrix of the transition.Courtesy of our collaborator Prof. Marc M. Dignam, Queen’s Univer-sity Ontario, Canada. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Schematic of the SR-DFWM set-up. AC = auto-correlator. CC =cross-correlator. OSA = optical spectrum analyzer. SLM = spatiallight modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Spectral dispersion of the pulse shaper in meV . . . . . . . . . . . . . 53

5.3 Quantum efficiency of the CCD used (VISAR) . . . . . . . . . . . . . 55

x

5.4 Spectral dispersion of the Monochromator/CCD system in meV for the1200 lines/mm grating . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Resolution measured for the TRIAX 550 with the MicroMax-800PBCCD as a function of the entrance slit of the Monochromator. Thegrating installed was the 1200 lines/mm. The curve in red is a simpleexponential fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.6 Spectrum of the DFWM signal at zero delay for a 67/17 structure withapplied bias. The hh−1 (left) and hh−2 (right) are represented. . . . . 58

5.7 (a) Spectrally integrated DFWM versus delay of the probe. (b) Spec-trally integrated DFWM for the only hh−2 peak versus delay of theprobe. (c) Position of the hh−2 transition (the max amplitude is ofabout 0.5 meV) versus delay of the probe. (d) Height of the hh−2 peakversus delay of the probe. . . . . . . . . . . . . . . . . . . . . . . . . 59

5.8 Absorption spectrum of the superlattice as a function of externallyapplied field. The color chart is in units of m−1. . . . . . . . . . . . . 60

6.1 (top in black and white) Probability of finding the electron at a givenposition in space along z(each maximum corresponds to the center ofa different well). On the right (left) column the dynamics of the non-breathing (breathing) mode is shown. (bottom in color) The samedynamics is described through a color contour plot displaying position(x) and time (y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.2 Position of the electronic center of mass versus time for the cases ofbreathing mode (large amplitude) and non-breathing mode (small am-plitude). Notice how the smaller oscillations are characterized by dou-ble frequency. This contribution at 2ωBO frequency is due to the over-lap between the wave function of the hh±1 states. This effect will beexplained in better detail in the next paragraph. . . . . . . . . . . . . 65

6.3 (a) Absorption spectrum taken at the external field Fbias= 15 kV/cm.The three transitions visible are the hh−1,hh0 and hh+1. (b) Full spec-trum of the unshaped laser and spectrum of the pump modulated toachieve the breathing mode. . . . . . . . . . . . . . . . . . . . . . . . 66

6.4 Schematic of the overlap hole/electron wave functions. The differentsigns of the Ψ−1(x) (red) and the Ψ+1(x) (blue) in the central well areevident. The electric field is pointing to the right. . . . . . . . . . . . 67

6.5 Spectrogram for the case of pure amplitude modulation of the pumpto achieve the breathing mode. . . . . . . . . . . . . . . . . . . . . . 69

xi

6.6 Shift of the transition energies for the hh+1 (blue), hh0 (black) and hh−1

(red) transitions. The breathing mode is obtained with a flat, [0,0,0],phase profile. The right scale in all the plots reports the estimatedmagnitude of the generated internal oscillatory THz field as estimatedusing the quasi-static model. . . . . . . . . . . . . . . . . . . . . . . . 70

6.7 Spectrogram for the case of amplitude and phase modulation ([0,0,π])of the pump to achieve the non-breathing mode. . . . . . . . . . . . . 71

6.8 Shift of the transition energies for the hh+1 (blue), hh0 (black) and hh−1

(red) transitions. The non-breathing mode is obtained with a phaseprofile [0,0,π]. The right scale in all the plots reports the magnitudeof the generated internal oscillatory THz field as estimated using thequasi-static model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.9 Height of the hh−1 SR-DFWM peak vs. delay of the probe for a pumpphase profile [0,0,0] (full line) and [0,0,π] (dotted line). . . . . . . . . 73

6.10 (black) Height of the hh−1 SR-DFWM peak vs. delay of the probe fora pump phase profile [0,0,0] (red) Cross correlation of the pump pulse.The heights of the first pulses of both curves are brought artificiallytogether. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.11 Effect of the coherent control. (black dots) a phase of [π,0,0] is appliedto the pump spectrum. (gray stars) a phase of [0,0,π] is applied tothe pump spectrum. The right scale reports the magnitude of thegenerated internal oscillatory THz field as estimated using the quasi-static model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.12 Spectrogram of the hh−1 and hh−2 transitions. The pump is shaped toexcite resonantly at τ = 0. . . . . . . . . . . . . . . . . . . . . . . . . 76

6.13 spectrum of the pump used for the DC experiment reported. . . . . . 77

6.14 Peak position of the hh−1 and hh−2 transitions as a function of delaytime of the probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.15 Spectra of the pump (red) and of the pump (black). The central com-ponent of the pump (that is blocked here) would sit at approximately1.58 eV. The hh−1 transition corresponds to the peak at lower energy. 81

6.16 Spectrogram of the 2-color experiment where only the hh−1 and hh+1

transitions are excited. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.17 Height of the hh0 (red) and hh−1 (black) spectral components of theDFWM as a function of the delay of the probe. The phase profile ofthe pump spectrum is [0,π]. . . . . . . . . . . . . . . . . . . . . . . . 83

xii

6.18 (bottom) Height of the hh−1 spectral component of the DFWM as afunction of the delay of the probe. (top) Position of the hh−1 transitionversus delay. The phase profile of the exciting pump spectrum is [0,π]. 84

A.1 Photograph of the superlattice sample in the cryostat. The central,smaller window has a thickness of less than 2 µm and a diameter ofabout 2-3 mm. The wires for front and back contacts are also visible.the Contacts where obtained with thickened silver paint. . . . . . . . 89

xiii

ABSTRACT

Fanciulli, Riccardo. Ph.D., Purdue University, August, 2004. Coherent Control ofBloch Oscillations in Semiconductor Superlattices by Means of Optical Pulse Shaping.Major Professor: A.M. Weiner.

The Bloch oscillations are a natural effect that may be observed when an electron

is subject to a periodic potential (period d) and to a constant electric field, F . This

effect is introduced in a semiclassical picture, first, and then in the quantum picture of

beatings among localized wavefunctions corresponding to equally-spaced levels. An

introduction to the techniques of observation of the Bloch oscillations is included,

with special focus on the degenerate four wave mixing (DFWM) technique employed

in our experiments. We did not observe directly the THz waveform from the sample,

but used an indirect observation through DFWM to estimate the internal THz field

and follow its change over time. The main focus of our work is the application of

the laser pulse shaping technique to the Bloch oscillations. Laser pulse shaping is a

new technique that allows to take a broad-band laser pulse and apply an amplitude

and/or phase modulation to its spectrum. In our work the emphasis was placed

on the creation of a tunable THz oscillator. We created, with our pulse shaper in

amplitude modulation mode, a pulse that shares the same time period as the expected

Bloch oscillations. This creates a kicked-oscillator kind of dynamics and makes the BO

observable for up to 15 oscillations. Once the prolonged internal THz field was created,

with phase modulation we achieved a switch for the oscillator that was capable to

turn on and off the internal THz field. We then showed full control over the electronic

dynamics by reversing the initial conditions of the electronic motion. This effect was

again achieved by phase modulating the spectrum of the pump pulse. We then studied

the presence of an optically created internal DC field in the superlattice structure.

xiv

The effects of this DC field on the THz oscillator are discussed (broadening of the line,

chirp). Finally, while doubling the frequency of our electronic oscillator, we observed

an interesting effect of the DFWM where a transition that was not excited by the

pump pulse (~k1), upon arrival of the wide-band probe (~k2), was nonetheless emitting

a signal in the direction 2~k2-~k1.

1

1. Bloch oscillations (BOs)

1.1 Semiclassical Introdution to the Bloch oscillations

The standard picture of electronic transport in solids with an applied static electric

field is given by drift transport and was first studied by Drude [10]. In Drude’s picture

the carriers move ballistically until they encounter a scattering center and at that

point their momentum changes. The drift velocity is determined by a balance between

energy gain from the field during the mean free time and energy changes during elastic

and inelastic scattering processes. Finally, the overall process is resumed in Ohm’s

law:

~J = σ ~E (1.1)

where the conductivity is given by:

σ =e2nτ

m. (1.2)

and τ is the mean free time, n is the carrier density and m the mass of the carriers.

A description of the electrons as ballistically moving between two subsequent scat-

tering events is justified if the carriers remain always close to the conduction band

lower edge (free-like electrons). However, when the field is high enough that the car-

riers can reach higher, non parabolic, portions of the band before being scattered,

the electrons won’t behave like free electrons anymore. If the scattering rate is low

enough, they will start to oscillate. This effect was predicted for the first time by

Felix Bloch in 1928 [11] and it is a direct consequence of the electronic band structure

in a crystalline solid. According to Bloch’s acceleration theorem the change of the

quasi-momentum vector ~k of an electron in a periodic lattice subject to an external

electric field ~F is given by:

~dk

dt= eF. (1.3)

2

Since the electronic band structure is periodic in ~k space, it is clear therefore that an

electron not subject to scattering will perform an oscillatory motion in energy [12].

This periodic motion will be characterized by a period

τBO =h

eFd(1.4)

Where d is the lattice parameter that sets the 1st Brillouin zone width to 2π/d. This

oscillatory motion was named Bloch oscillation. In the case of zero field applied the

localization length of an electron in a lattice is mainly due to a certain deviation from

the coherence in the making of the lattice itself (slight changes in the periodicity)

or to the presence of impurities and is roughly of the order of several periods. Still

remaining within the semiclassical (scattering-free) picture, we can estimate the lo-

calization length of the electron in the case of non-zero field ~F applied. The work

done by the field on the electron to move it from the center of the 1st Brillouin zone

to the edge of it is given by:

W = LmaxeF (1.5)

And therefore, equating that work to the width ∆ of the band, one can find that the

maximum displacement of the oscillation is given by:

Lmax =∆

eF(1.6)

And if we are interested in the behaviour of the speed of the oscillating electrons,

according to Erehnfest’s theorem along its motion will be given by

v =1

~dE(k)

dk(1.7)

The speed of the electron moving along the band in ~k/energy space is equivalent to

the one of a pendulum or of any other harmonic oscillating system (see Fig.1.1).

3

Figure 1.1. Lowest miniband of the superlattice in the first Brillouin zone.The electron moves along the band in k space.

Given that common periodicities in natural semiconductor lattices are of the order

of 5 A and that scattering times T2 are of about 1.5 ps, one can see how a TBO > 20 ps

makes the observation of this effect impossible. This was behind the reason for using

man-engineered lattices (superlattices (SL)) where the periodicity can be brought to

be large enough (about 20 times in our samples) to make TBO < T2.

1.2 Quantum mechanical introduction to the Bloch oscillations

The previous introduction to the Bloch Oscillations had a purely semiclassical

character. Now we would like to introduce the same concepts from a purely quantum

mechanical point of view.

When an electric field is applied to a periodic structure like a superlattice, the trans-

lational symmetry is broken and the delocalization of the eigenvectors of the Hamil-

tonian (Bloch functions) is replaced by a rather strong localization that can now be

better described by mean of the Wannier functions (hence the name Wannier-Stark

ladder (WSL)). The Wannier functions are the equivalent of a Fourier transform in

the ~k space of the Bloch functions described in the real space. Their definition in

4

terms of the Bloch functions is reported in eq. 1.16 and their main feature is their

strong localization in real space.

In the one particle approximation (no excitonic effects) the problem to be solved

is the following:

[− h2

2m∗(z)

d2

dz2+ eFz + U0(z)

]ψ(z) = Eψ(z) (1.8)

Where U0(z) is the potential describing the superlattice and F is the field ap-

plied on the superlattice along the direction of growth, z. This problem has been

approached mainly in two ways. The first method [1, 13, 14] studies a finite number

of wells embedded between constant potential to give bound solutions. The solutions

are written in terms of Airy functions (solutions of the triangular single quantum well

case) and at the interfaces the Daniel-Ben Duke conditions [15] are used. The results

are valid as long as the localization length is shorter than the structure itself.

The second approach [4, 16] is represented by a tight-binding (first-neighbor) one.

This method may also take partially into account the excitonic effect, at least within

the same well (intra-well excitons). A set of wave functions for the SL is written as

linear combination of the single well wave functions

Ψm(z) =∑

n

C(n−m)fe(z − nd) (1.9)

Here d is the SL period and n is the well number. The functions fe(z−nd) may take

into account the intra-well coulomb interaction, so the levels constituting the WSL in

the end will reflect this excitonic effect. Also, only the 1S-like states of each well are

taken into account as it was shown that the ratio in the oscillator strengths between

such states and the higher energy ones exceeds one order of magnitude [2]. For this

reason one can use the well number n as a good quantum number in this theory.

Now, considering only the first neighbor overlap and neglecting the coupling with the

other bands, the coefficients Cn are found to be:

C(n−m) = Jn−m(∆

2eFd) (1.10)

5

Where J is the Bessel function of first order, ∆ is the first miniband width and F is

the electric field applied.

To show the effect of the Wannier-Stark (WS) localization, let’s consider the case when

a strong electric field is applied (case that suits well the tight binding approach). In

this case ∆ ¿ 2eFd and one could find that:

Jn−m

(∆

2eFd

)≈ 1

(n−m)!

(∆

4eFd

)|n−m|(1.11)

which, in the complete strong field limit yields

C(n−m) → 1 for n = m

C(n−m) → 0 for n 6= m (1.12)

This approach works better when the field is strong enough to localize the electron

mostly in one well, but not enough to make Zener tunnelling between minibands a

likely effect. Comparing the length Lmax and the period d, one can find that for the

structures of interest (i.e. for miniband widths ∆ of about 30 meV) the field must

be roughly 15 kV/cm, which is indeed reasonable from an experimental point of view

for such structures where the intrinsic region is typically about 2 µm thick.

Still in this tight-binding formulation, a wave packet excited by a fast laser pulse can

be described on this basis of WS states in this way:

Ψ(z) =∑

p

Cp e(−iωBOpt)∑

n

J(n−p)fe(z − nd) (1.13)

Where the constant Cp describes how much of the electron is promoted to the p level

from valence band (see fig. 1.7) and ωBO = ~eFd

(frequency associated to the spacing

between the levels of the WSL). Along this line, it is now possible to calculate the

expectation value of the exciton position along z:

< z >= d∑

p

C2pp + cos(ωt)L

∑p

Cp−1Cp (1.14)

and also estimate the localization of the wave packet calculating the expectation value

6

of z2

< z2 >=< z >2 +L2

2+ d2

∑p

C2pp

2

+ cos(ωt)Ld∑

p

CpCp−1(2p− 1)

+ cos(2ωt)L2

2

∑p

CpCp−2 (1.15)

This method can be connected to the semiclassical picture in k-space that we used

earlier to introduce the Bloch oscillations. This is achieved by expanding the wave

function in terms of Wannier functions

φWk (z) =

1√N

∑n

e(iknd)φ(z − nd) (1.16)

Using this basis and the form for Ψ that we had already found before, the probability

amplitude for the electrons to be in states with a given k is [17]:

g(k, t) =d√

N

2πe−iθ sin(kd)

∑p

Cpe−ip(kd+ωt) (1.17)

Student Version of MATLAB

Figure 1.2. Probability of finding the electron in a part of the Brillouinzone at a time t. Lighter shades indicate higher probability. The oscilla-tory motion is resulting clearly.

7

Both with the real space and the ~k space descriptions of the single-particle problem

one can see that choosing certain configurations of the parameters Cn, the oscillations

in < z > and < z2 > can be controlled. This is a key property of this system and of

great importance for our experimental work. We will come back to this properties in

further detail in chapter 6.

This was the picture of non-interacting excitons. In the following paragraph we would

like to introduce briefly a further level of complexity that builds on top of the problem

just presented. When the exciton binding energy is comparable to the spacing of the

WS ladder, the effect of the coulomb interaction can be well observable.

1.3 Many particles picture: excitonic effects

We would like to discuss some of the effects that one can expect due to the coulomb

interaction between electrons in conduction band and holes. In the zero-field case,

in a strongly coupled SL (thin barriers and wells) the binding energy of an exciton

approaches its value in bulk (4.7 meV with a Bohr radius of about 120 A). Quite

interestingly, when the field is turned on one finds a different behavior for excitons

where the e and the h are in the same well (intrawell) or in different wells (interwell).

1.3.1 Intrawell excitons

In the case of intrawell excitons a raise in the bias, by increasing the localization

of both e and h, leads to a raise in the binding energy (which is proportional to

the spatial overlap between the two particles wave functions). As the applied field

approaches the level for which both e and h are completely localized in the same well,

the value of the binding energy approaches the one for a single well with triangularly

shaped bottom. To give an example of the dependence of the binding energy on the

applied bias it can be interesting to take a look at the numbers for a 40/40 ASL

structure made of GaAs/Al0.3Ga0.7As. The binding energy at F = 0 is 8 meV while

at F = 20 kV/cm it increases to 12 meV. This effect will be visible in chapter 6 where

the fan charts showing the transition energies versus field strength are shown. The

change in the central, so called E00 transition (see fig. 1.7) is, in fact, mostly due to

the effect described above and not to bands bending like the other transitions present

8

in the fanchart.

1.3.2 Interwell excitons

The effect of the applied field on interwell excitons has not been clearly measured

to the best of our knowledge, but qualitatively we can expect that as the field increases

and the localization becomes stronger, the interwell overlap between e and h should go

down and so should the binding energy of this kind of excitons. Lacking experimental

results, it will be interesting to take a look at the numerical results [1, 18]. The

different curves in fig. 1.3 denoted by different P s are calculated for an integer number

P of periods separating e and h.

Figure 1.3. Binding energy for interwell excitons calculated as a functionof electric field in a 3nm/3nm GaAs/Al0.35Ga0.65As superlattice. Thedashed curves are for p < 0. [1].

From this plot we can also see the general increase expected for the binding energy

in the case of the intra-well exciton. We can also notice that the binding energy

reaches a maximum for a certain value of the field. This maximum corresponds to

the field that maximizes the probability of having e and h in the same well (or, better,

this field maximizes the overlap of electrons and holes that still maintain the main

peak in two different wells) [1].

9

1.3.3 Excitonic effects on the oscillator strength

Knowing the oscillator strength of each transition as a function of the applied field

can also be useful if we want to control the oscillations.

Figure 1.4. Oscillator strength of transition p = -1 normalized by thetransition p = 0 measured versus electric field in GaAs/AlAs superlatticeswith different barrier widths LB. The well width was always LW = 3.1nm. The dashed curves are for p < 0. [2]

From the calculated data in fig. 1.4 we can see that due to asymmetry in the

electronic wave function (as in the position of main peak respect to the center of the

well) the transition with p < 0 (higher energy respect to the Ehh0) are more likely

to happen then their lower energy counterparts. This asymmetry simply originates

from the asymmetry of the SL structure that is induced by the electric field.

Summarizing what we have seen so far, the Bloch oscillations can be seen as quantum

beatings in a system made of equally spaced energy levels whose spacing depends on

the applied bias. The levels can be calculated in first approximation without taking

the coulomb interaction (e-h) into account, but if we want to better understand and

more rigorously describe the system such interaction has to be considered as well.

1.4 Absorption spectrum

1.4.1 Density of states of a superlattice structure

Dividing the energy into the component due to the SL structure along z and the

component due to the free motion in the (x,y) plane, we can write the total energy

10

in this way:

ε(n, q, k⊥, σ) =~2k2

⊥2m∗ + εn(q) (1.18)

Where n is the index of the miniband, q is the quasi-momentum in the reduced

Brillouin zone, k⊥ and m∗ are the magnitude of the quasi-momentum and the effective

mass in the (x,y) plane and σ is the spin that, in this case, doesn’t affect the energy.

Using the general expression for the density of states gives us the expression:

%(ε) =∑

n,q,k⊥,σ

δ(ε− εn,q − ~

2k2⊥

2m∗), (1.19)

that can be simplified passing to the integral form in k⊥ and playing with the prop-

erties of the delta function. The expression that one obtains is:

%(ε) =SNd

π2

m∗

~2

∑n

∫ πd

0

dqΘ[ε− εn,q

]=

∑n

%n(ε) (1.20)

Where Nd is the total length of the SL, S is the area of the layers, Θ is the step

function and the %n are defined as densities of states for each miniband (assigning

to each miniband a miniband gap as well). In general the miniband εn,q has a finite

width, so the result is that for energies below the minimum of the band the %n will

be zero (in the case of the 1st one), while for energies above the maximum of the

miniband the %n is a constant equal to SNdπ2

m∗~2 . In order to see what happens for

energies in between we can use the results from the tight-binding approach that gives

us a (first) miniband of the form:

εn,q = Sn − 2tncos(qd) (1.21)

From this equation and in view of what we found before for energies out of the

miniband, we find that the density of states has a behaviour as reported in fig. 1.5.

11

Figure 1.5. Schematic of the behaviour of the DOS in a superlattice withno external fields applied. The DOS in the center of the minibands shouldbe flat. The plateaus between the minibands are due to the 2D characterof the kinetic energy in the (x,y) plane. Also, the extreme 2D (dashed)and 3D (top) cases are shown. [3]

The plateaus between the minibands, due to the (kx, ky) plane are spaced by

SNdπ2

m∗~2 which is exactly N times the quantization of the same kind of plateaus in the

single quantum well case.

1.4.2 Excitonic absorption

Excitons, with their binding energy, introduce levels below the bottom of each

miniband. The confinement increased by the coulomb interaction (both along z and

in the plane of the layers) increases the oscillator strength by a great amount (in a

100 Angstrom QW the excitonic absorption peak is about 30 times stronger than the

QW levels absorption). The probability of the optical transition is proportional to

the overlap integral of the envelope functions in the z direction and to the overlap

of electron and hole for r⊥=0 (in the plane (x,y) of the layers). The enhanced lo-

calization due to the Coulomb interaction acts on both of these elements and that is

what makes the absorption line of the exciton strong. The exciton is also character-

ized by a kinetic energy (in our case corresponding to a motion in the (x,y) plane)

~2K⊥2

2(me+mh). This energy, together with the hydrogen-like energy spectrum coming from

the problem in the center of mass reference, would yield a density of states similar in

shape to the one of a single quantum well. Steps that get closer and closer and flat

12

density between them. What makes us see peaks in the absorption spectrum is the

selection rule (valid in dipole approximation) according to which optical transitions

can only take place between states that have zero momentum K⊥. This selection

rule basically states the conservation of the in-plane total momentum. As a last note,

since we mentioned that the transition probability is proportional to the overlap of

the electron and hole wave functions in r⊥=0, we would like to point out that this

implies that only the hydrogen-like states with zero total angular momentum (1S, 2S

etc.) will be visible through absorption. In our case, in fact, we shall always consider

only the 1S excitonic states that have been proven to give a much stronger line [2].

1.4.3 Absorption with external electric field

As we have already shown, when an electric field is applied along the z direction

(or direction of growth), the translational symmetry is broken and the minibands

split into several transitions as shown in Fig. 1.6. The number associated to each

plateau in fig. 1.6 corresponds to the transitions in a way that will hopefully result

clear from fig. 1.7.

Energy

Figure 1.6. Absorption spectra for GaAs/AlGaAs superlattice as afunction of the electric field calculated in the tight-binding formalism.ε = −4(Eg + E1e + E1h)/(∆e + ∆h) and f = 4eFd/(∆e + ∆h) are respec-tively the reduced photon energy and the reduced electric field [4].

13

p = -2 0 -1 1 p = 2

Ene

rgy

Direction of growth z

Figure 1.7. Schematic of the superlattice structure with an applied electricfield.

Figure 1.8. Derivative of the absorption spectrum as a function of the ex-ternally applied bias. The sample is a 67/17 Aas used for the experimentsof coherent control reported in the following chapters.

The creation and suppression of the absorption lines as a function of the applied

14

bias can be see in fig. 1.8 where the derivative of the linear absorption is reported as

a function of the bias. This kind of fanchart will be of key importance in our work

and we shall get back to it in the following chapters.

The kind of fanchart reported in fig. 1.8 is often measured using photocurrent data

where light at different frequencies is shined on the sample and the resulting photocur-

rent is measured. An example of a photocurrent spectrum is reported in fig. 1.9. An

electric field was applied and one can clearly see the three strongest peaks representing

the hh−1,hh0 and hh+1 transitions. Notice that in section 1.3.3, when explaining the

contribution of the excitonic effects to the WSL we have reported the behaviour (the

theoretical prediction) of the oscillator strength of the transitions described above as

a function of the external field. Finally an example of a chart reporting the position

of the transitions versus applied field is reported later on in Fig.4.5.

770 780 790 800 810

0.0

0.2

0.4

0.6

0.8

Photocurrent T = 6K hh

+1

hh-1

hh0

Cur

rent

(A

.U.)

Wavelength of incident light (nm)

Figure 1.9. Example of a photocurrent spectrum taken on one of our firstsamples. A field is applied and by varying it one could see the linearmotion of the levels. The sample is a 67/17 Aas used for the experimentsof coherent control reported in the following chapters.

15

2. Detection Methods

2.1 Introduction

There are a few different ways to observe Bloch Oscillations, but two are the

main signatures that make the oscillations observable. The more intuitive one is

represented by the field emitted by the charged oscillating dipole. Such field can be

detected and it bears clearly the signature of the oscillations (i.e. the dependence of

the period on the applied bias). Another signature of the oscillations is found in the

third order optical field emitted from the SL when a pump pulse and a probe laser

pulse are sent on the sample (Four Wave Mixing).

2.2 Detection of the BOs through THz radiation detection

2.2.1 General introduction

A clear signature of the spatial oscillation of a charged particle is, naturally, the

radiation that the charge, being subject to acceleration, emits. In our case there are

two particles in the system. The hole h that, because of its heavier effective mass

(about 5 times heavier) is well localized in its original well even for weak fields [19]

and the electron e in conduction band that, on the other hand, undergoes a motion

quite well described by eq.1.14 as an oscillatory one:

< z >= d∑

p

C2pp + cos(ωt)L

∑p

Cp−1Cp (2.1)

From eq.2.1 we can derive the acceleration of the e that yields the emitted field:

Erad =q

c2

n× (n× ~a)

R. (2.2)

Where q is the absolute value of the electronic charge, n is the direction from the

source to the observer, R is the observer’s distance from the source and ~a is the

16

acceleration of the electron. Due to real world limits (like having a finite dephasing

time of the oscillators as fast as 1.5ps or having optical phonons with absorption

energy of 36 meV) that constrain the choice of the SL structure and of the applied bias,

the frequencies of the oscillations that have been observed are all roughly between 0.1

and 30 THz (corresponding to 0.4 and 124 meV). In this range of frequencies there

are mainly two ways to detect light. Chronologically the first approach is represented

by using a dipole antenna in which the current passing through the two poles of

the dipole is modulated by the incoming THz field [20]. The second, and now often

preferred, approach makes use of the electro-optic (EO) properties of crystals like

ZnSe, ZnTe or GaAs where the EO coefficients are particularly strong [21]. In the

following paragraphs we will try to introduce the fundamental aspects of these two

techniques.

2.2.2 Photo-conductive sampling (PC-S)

A photoconductive (PC) antenna [20] consists of two electrodes deposited a few

tens of microns apart on top of a semiconductor (SC) material (GaAs or low-temperature-

grown GaAs, for example). When these antennae are used as detectors no bias is

applied to the poles of the dipole. The time gate for the measurement is opened

when a probe pulse arrives and, focusing on the gap, creates carriers in conduction

band. The temporal resolution (duration of the temporal window of a single shot

measurement) is given by the recombination time of the photo-excited electrons (this

time depends on the semiconducting material chosen and is very critical). During

this time (that is of the order of 0.3-1 ps for annealed low-temperature grown GaAs)

the electrons are accelerated by the incoming THz field and by measuring the current

one can obtain a number proportional to the magnitude of the field, or rather, of its

integral over such temporal window.

17

Figure 2.1. Schematic of the setup for photoconductive-sampling mea-surement [5]

As we said, the relaxation time of the electrons sets the temporal resolution and

therefore in order to obtain the best results, this detection technique should be used

only in those cases when the THz field to be detected is expected to vary on longer

temporal scales than the relaxation time of the SC material. This fact makes this tech-

nique suitable for detecting frequencies not higher than about 3THz (corresponding

to a relaxation time of about 0.3ps). Anything faster than that would be integrated

over and information on the shape of the THz pulse would be lost.

2.2.3 Electro-optic sampling (EO-S)

This technique [21] makes again use of two fast (100 fs or faster) laser pulses.

A Pump pulse first creates the THz radiation and a probe pulse is then used to

detect it. The probe pulse and THz fields pass collinearly through an electro-optic

crystal (ZnTe, ZnSe or GaAs, to name a few). Since the temporal feature of the

THz field is much slower than the pulse width of the laser (10-100 fs), the THz can

be considered as a quasi-dc field. Whenever a dc field is applied to an electro-optic

solid, the ellipsoid of the indexes of refraction is modified. In the case of the materials

18

reported above what happens is that, because of the presence of the quasi-DC THz

field, the polarization of the probe laser pulse is rotated by an amount proportional

to the magnitude of the THz field. By monitoring the rotation in polarization of the

probe pulse (with a polarizer beam splitter, for example) as a function of its delay

respect to the THz pulse, the magnitude of the THz field can, in principle, be studied

with a temporal resolution equal to the duration of the probe laser pulse.

Figure 2.2. Schematic of the setup for electrooptic sampling measurements[5]

As we said, this technique allows, in principle, a temporal resolution equal to the

probe pulse duration. In fact there is a problem of group velocity mismatch (GVM)

between the THz and any pulsed source in the N-IR. To overcome this problem thin

EO crystals have been used (of the order of 100 µm), although, to the detriment of

the signal and, as a consequence , of the signal-to-noise ratio.

19

Figure 2.3. Schematic of an electro-optic sampling experiment.

2.2.4 Bloch oscillations and THz radiation

The Bloch oscillations have been detected through THz detection making use of

both the techniques described before. The first detection (but not the first detection

ever of BO) of the THz radiation from BOs was attained through PC-S detection in

1994 [6]. In Fig. 2.4 are reported the results of such an experiment.

20

Figure 2.4. Bias dependence of the detected THz transients after exci-tation by optical pulses with photon energy of 1.54eV (centered on theexcitonic WS resonances of the lowest heavy-hole transitions). Sampletemperature: 10K [6])

More recently [22] the BOs have been observed through internal EO-S also called

transmissive EO-S (TEO-S). In these experiments the, usually external to the source,

electro-optic crystal is replaced by the superlattice itself. In this way the area of the

sample that acts as a source of THz radiation is the same one that, through the EO

effect acts on the probe pulse and, rotating its polarization, allows the detection. The

main difference with respect to the EO-S as introduced before is that in the previous

case the far field deriving from the oscillation of the dipole was detected, while in this

case the dipole itself is the one being detected. In Fig. 2.5 some of the results from

this kind of experiments are reported.

21

Figure 2.5. Transmittive electro-optic signals for different applied voltagesin the WS regime of the superlattice at an excitation density of 2x109cm−2.The curves are shifted proportional to the voltage applied. [7]

These TEO-S experiments have been quite criticized, though, by other groups.

Other effects, like first order pump-probe and higher order non-linear interactions are

likely to be playing a strong role in the detected signal that has to be very carefully

analyzed.

Finally, I would like to mention that the same kind of experiment with TEO-S has

been reported also at room temperature [7], but the results (shown in Fig. 2.6)

seem again somewhat in doubt and the fact that there was no follow-up to this rapid

communications article makes them a little suspicious.

22

Figure 2.6. Internal EO-S data from reference [7]

2.3 BO detection with Degenerate Four Wave Mixing (DFWM)

2.3.1 Introduction to the theory of DFWM applied to the BOs

Four wave mixing (FWM) is a third-order non-linear technique where, in general,

three waves with wave vectors ~k1,~k2 and ~k3 interact together to give in output a fourth

one in the ~k4 direction. To study the electronic dynamics of Bloch oscillations, a sim-

plified version of FWM is used, the so called degenerate four wave mixing (DFWM).

In this mode only two pulses, ~k1 and ~k2 impinge on the SL. It is exactly the same

principle, except that one of the two pulses interacts with the other one and with

itself to give again a third-order signal (third order in the incident optical fields).

DFWM on SL was actually the first technique that showed clearly the signs of the

Bloch Oscillations[23]. The theory behind this technique is certainly less straightfor-

ward with respect to the two techniques previously introduced for the THz detection.

As extensively as this technique has been used on the SL systems, a clear physical

picture of the link between the DFWM signal and the electronic motion of the Bloch

23

oscillating electrons is somewhat missing (to the best of our knowledge). In the first

paper [24] where the idea of using DFWM to observe the BO appears, the authors

dealt with delta-like laser pulses (in time domain) and gave solutions only for times

long with respect to the dephasing time T2 in the region of the so called photon-echo.

The conditions imposed are quite distant from the ones required to describe the ac-

tual experiments that would eventually be performed. More thorough theories have

been developed in recent years (of which we will see some interesting examples in the

following), but a simple graphical picture of the nature of the DFWM and its link to

the electronic dynamics is still somewhat missing.

2.3.2 DFWM: a physical picture

In the following we would like to give a physical picture of the experiments aimed

at studying the BO with DFWM. In order to do so we’ll need to introduce a few

concepts and we shall do it in an orderly manner, step by step so to build, in the end,

a physical description of the physics going on in the system.

Interband polarization

The interband polarization in general comes from the different shape of the wave

functions of holes and electrons in a solid. When an electron is excited in conduc-

tion band two wave functions characterized by opposite charge and different spatial

distribution are created. This is already enough to create an electric dipole. In our

case we also have the effect of the external bias applied to the SL. In the (x,y) plane

of each well the interband polarization or internal polarization of the exciton is on

average zero as we are exciting 1S-like excitons [17]. Along the z direction (direction

of growth of the SL) things will be different though. Let’s consider the simple case

of a single quantum well in a semiconductor. When an electron in the well is excited

from valence to conduction band both the electron and hole wave functions are spread

symmetrically with respect to the center of the wells. When an electric field is applied

along the direction of growth z, the well is tilted in such a way (”triangular well”)

that the lowest level in conduction will be characterized by a wave function shifted

towards a side (the one where the ”dip” is) and viceversa the hole wavefuntion will

24

be shifted towards the opposite side. The applied bias, therefore, has the effect of

creating a further polarization. The sum of these two polarizations is what we will

call interband polarization in our system.

Laser pulse in the 2-level system

When a 2-level system is excited in resonance by a laser pulse, the electron that is ex-

cited maintains a coherence with the exciting field in that it keeps oscillating between

the lower state and upper state at the frequency of the exciting field (which, working

at resonance, is the same as the one of the transition). This effect, which in a perfect

single and isolated system could go on indefinitely, is in reality limited to a certain

temporal window because of scattering with other particles (loss of coherence at first

in a time T2 and eventually loss of energy in a longer time T1). As we have seen in

the previous paragraph, if lower and higher states have different spatial distribution,

this oscillation between the two states will correspond to a dipole oscillating at the

frequency characteristic of the transition (in our case about 380 THz or 2.6 fs).

Ensemble of 2-level systems in an (x,y) plane

Now let’s imagine to have a set of these 2-levels oscillators uniformly distributed on

a plane (this plane, in our real system is created by the directions of the two laser

pulses pump (~k1) and probe (~k2)). When the exciting pulse, that we shall call pump,

comes in from a certain direction it, not only excites the oscillators (each single 2-

level system), but it gives them a spatial phase that reflects its own phase ei~k1·~r. We

can then visualize these (interband polarization) dipoles as oscillating in a way such

that the front of the wave will seem to be moving along the same direction k1 as the

pump pulse. This oscillation, as we have seen in the previous paragraph, will keep

going on also after the pump pulse is gone from the region. This is a key point in the

application of DFWM.

DFWM signal in time and frequency domain for an ensemble of non-

interacting 2-level systems

To explain the DFWM signal from a semiconductor sample, then, one could visualize

this system using a picture like the one reported in fig. 2.7.

25

k2

k1 k

2k1

Figure 2.7. Schematic of the of the amplitude of the polarization grating(at a fixed time) created by the interaction of the two pulses with wavevectors k1 and k2 inside the sample.

The interband polarization along the z direction of growth can be plotted in the

plane (x,y) and will display (in our case of interest) a wave vector k2-k1. If one,

then, plots one slice of the grating (in the direction perpendicular to its wave vector)

and plots it vs. time (fig. 2.8), will see an exponential decay in the strength of the

grating. This decay will only be part of the reason of the decaying of the DFWM

signal as also the dipoles that are scattered are themselves getting out of phase due to

elastic scattering that is mainly due to exciton-exciton interaction (excitation-induced

dephasing).

26

0

0.5

1

1.5

2 05

1015

2025

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Space (A.U.)

Time (A.U.)

Gra

ting

inte

nsity

(A

.U.)

Student Version of MATLAB

Figure 2.8. Schematic of the decaying of the amplitude of the polarizationgrating created by the interaction of the two pulses with wave vectors k1

and k2 inside the sample.

DFWM signal in the presence of Bloch oscillations

At this point we need to add to our picture the effect of the internal electronic

BO dynamics. This is done introducing an intraband polarization in the z direc-

tion. This polarization arises from the dipole elements between excited WS states

(< ΨWSi |z|ΨWS

j >). This intraband polarization oscillates with a period of about 400-

500 fs (2-2.5 THz) for our experimental conditions. This slower oscillating dipole has

the effect of modulating in time domain the strength of the (interband) polarization

grating making it more or less diffractive over time. This is the key to understanding

the link between the internal dynamics of quantum beating of the Bloch oscillations

and the oscillations that one observes in the DFWM in time domain. By sending the

probe in with different delays we are basically probing the efficiency of the polariza-

tion grating and monitoring the slower modulation due to the intraband polarization

dominated by the BO dynamics. A schematic of the efficiency of the polarization

grating in the presence of the BO is reported in fig. 2.9

27

0

0.5

1

1.5

2 05

1015

2025

−1

−0.5

0

0.5

1

Space (A.U.)

Time (A.U.)G

ratin

g in

tens

ity (

A.U

.)

Student Version of MATLAB

Figure 2.9. Schematic of the temporal evolution of the polarization gratingin the presence of Bloch oscillations. The slower temporal modulation duethe internal electronic dynamics is now visible.

The oscillations at ωBO frequency that we will show in the the plots of DFWM

vs. delay (chapter 5.2), then, simply reflect the increase in diffraction efficiency of

the excitonic grating created in the direction ±(~k2 − ~k1).

2.3.3 DFWM: a density matrix approach

We would like to introduce now a simplified (but not too far from the real case)

theoretical treatment of our system where the discussion of the DFWM technique and

its results can possibly be better handled and explained. A simple non-interacting

picture that we may use to describe our system is given by a reservoir of (N+1)-level

systems (a ground state and N excited ones) excited by two laser pulses (possibly of

different shapes) delayed one respect to the other by a time τ . Having to deal with a

large number of these systems we will need to use a density matrix approach. Within

this formalism, then, the problem to be solved is the following:

−ihd%i,j

dt= [Hi,n, %n,j]− (

1

T)i,j (2.3)

where H is the Hamiltonian describing both the system and its interaction with the

electric field of the pulses. The matrix ( 1T)i,j is used to introduce finite lifetimes

for both the populations (diagonal elements) and the dipoles (off-diagonal elements).

28

The Hamiltonian operator H can be written as:

Hij = Eiδij + ~µij ·(~Epump(t) + ~Eprobe(t− τ)

)(2.4)

Where Ei are the energy eigenvalues corresponding to the N+1 levels ~µij is the dipole

operator defined as:

~µij =< Ψi|e~r|Ψj > (2.5)

and the two fields within envelope function approximation are defined as such:

Epu,pr = fpu,pr(t)eiω0 te−i kpu,pr ·r (2.6)

In the approximation of non-interacting excitons one can solve this problem in a base

of excitonic WS wave functions Ψi(x) (see fig. 6.4 for an example). In view of the

previous definitions, the system of partial differential equations equations to be solved

takes the following form:

d%ij

dt= −iΩij %ij − i

~[~µ, %

]ij· ( ~Epump(t) + ~Eprobe(t− τ)

)(2.7)

where:

Ωij =(Ei − Ej

~)− ( 1

T

)ij. (2.8)

Equation 2.7 cannot be solved exactly and therefore an approximate, perturbative

approach is applied [25]. The method consists in finding solutions to the problem

depending on the nth power of the incident field and then using the result to write

down the next equation to be solved for the (n+1)st power of the electric field in this

way:

d%(n+1)ij

dt= −iΩij %

(n)ij − i

~[~µ, %(n)

] · ( ~Epump(t) + ~Eprobe(t− τ)), (2.9)

with an initial condition given by:

%0ij = δi0δ0j. (2.10)

With enough time and computing power we could solve the problem up to any pertur-

bative order. Once the third order density matrix is obtained, the internal interband

polarization of the ensemble along the 2~k2 − ~k1 direction can be calculated using:

~P(3)interb = Tr[%

(3)i0 < Ψ0|e~r|Ψj >] (2.11)

29

and this third-order polarization can be simply used in the non-linear wave equation:

∇ ~E(~r, ω)− ω2

c2ε(ω) · ~E(~r, ω) =

4πω2

c2~P

(3)interb(~r, ω) (2.12)

to calculate the exact field whose intensity is being observed in our experiments.

As we explained in more physical terms in the previous paragraph and as one could

see carrying on the calculation reported above, at the third order the modulation

of the density matrix intraband terms (the terms that mix the excited states one

another) over the interband elements (the terms that mix the excited states with the

ground state in valence band) results into a clear signature of the Bloch oscillation.

2.3.4 DFWM: beyond the finite-order approach

The standard way to approach the description of the FWM is basically the one

reported in the previous paragraph. The so called dynamics controlled truncation

(DCT) theory of Axt and Stahl is used [25] where the optical response of the sys-

tem is expanded in powers of the exciting optical fields. This theory has been very

successful and describes well both the intra-excitonic and inter-excitonic interactions

to a given order in the optical field. However, it is intrinsically a perturbative ap-

proach. One of our collaborators, Professor Marc Dignam, has developed, together

with Professor Margaret Hawton, a theory that is based on excitonic Bloch equations

valid for infinite-order in the exciting optical field [8, 26]. It is beyond the task of

this paragraph to go into the detail of this theory, but we would like to report the

hamiltonian operator used in this state-of-the-art approach. We hope in this way to

give a flavor for what interactions are thought to be important to describe our system:

H =

n0by2∑µ;n=−n0

~ωµB†µ,nBµ,n − V

∑n=±1

(Eopt∗n ·Pinter

n + H.c.) +

+ V

2n0by2∑m=−2n0

[1

2εPintra−m − ETHzδm,0] ·Pintra

m (2.13)

The first term describes the excitonic population through the creation/destruction

operators Bµ,n where µ describes the internal motion within the exciton and n refers

30

to Kn the center of mass wave vector. The second term takes care of the interaction

of the system with the incoming optical field. The third term describes the intraband

exciton-exciton interaction and is carried on until convergence at some number n0.

Finally the fourth term describes the interaction between the excitons and a THz field.

In this term will be incorporated the contribution coming from the dipole internal

field created by the displacement e−/h and, if present, the interaction with an applied

external THz field (not present in our experiments). For a 67/17 A GaAs/Al0.3Ga0.7As

SL with 15 kV/cm bias, Professors Dignam and Hawton solved this kind of problem

on the basis of the excitonic states of the biased SL. In fig.2.10 we report their results

[8, 26] for the time integrated DFWM.

Figure 2.10. The time integrated intensity of the DFWM versus delay τis reported for ωlaser = ω00 − 2.27ωBO, n0=13 and a density of 6.36x109

cm−2 for the three different directions Kn indicated. For K−3 results forn0=3 (dashed line) and n0=5 (dotted line) are also plotted. [8]

31

One can see how important the terms at higher orders in the field can be with the

3~k2 − 2~k1 being as large as about one half of the 2~k2 − 1~k1 term. These signals can

be picked up in the experiments and we now know that they can be quite important.

Also, we can see from fig.2.10 that the exciton-exciton interaction is very important as

a (already fairly complicated) simulation that takes into account interactions between

excitons with as different wave vectors as K±6 (n0=3) is way off target with intensities

that are almost double as much as the ones at convergence (n0=13).

Professors Dignam and Hawton were also able to calculate the shift in energy of the

excitonic transitions over a BO period (see chapter 5.3). This apparently could not

be obtained directly in the DCT theory. The result is reported in fig. 2.11.

Figure 2.11. The hh−1 spectrally resolved degenerate four wave mixingpeak energy versus delay τ is plotted for ωlaser = ω00 − 2.27ωBO anddensities given above each curve (in units of 109 cm−2). The curve offsetsare real effect due to intraband renormalization of the DC field. The insetshows peak oscillation amplitudes as a function of density for (ωlaser −ω00)/ωBO = -1.24 (circles), -2.27 (triangles), and -2.83 (squares).[8]

The amplitude of the peak oscillations is quite smaller compared to what was

obtained previously experimentally. This could be the effect of screening of exciton-

32

exciton interactions via incoherent carriers. In fact, already changing the static per-

mittivity in the SL from 12.5ε0 down to 8ε0 can change the amplitude of the shift by

a factor of 3 or more and can bring the calculated values in good agreement with the

experimental results.

The presence of incoherent carriers is very important in the description of FWM

experiments in biased SL [26]. These carriers are the result of the continuum of

unbound excitons that lose the coherence with the optical field very quickly due to

elastic exciton-exciton scattering (their coherence lasts only for the duration of the

optical pulse).

33

3. Shaping and characterization of ultrafast laser pulses

The key point of our work on Bloch oscillations is represented by the shaping of the

exciting pump pulse in order to control the coherent oscillations by inhibiting certain

transitions, enhancing others or delaying some excitations respect to others. We will

come back in chapter 6 on the purpose of pulse shaping in our system of interest.

Here we would like to introduce the technique itself along with the methods used to

check the shape of our pulses.

3.1 Pulse shaping by means of spatial light modulators (SLM)

In our research the capability of selecting and delaying the frequency components

within the initial pulse bandwidth plays a key role. This is obtained through the so

called ”Fourier transform pulse shaper” [9] using SLMs as computer-controlled optical

masks. The use of these SLMs makes it easy and fast to superimpose a certain shape

and/or phase profile onto the initial laser spectrum. In this way it is possible, for

example, to select two narrow (we shall see how narrow they can be) regions of the

spectrum and, at the same time, delay one of them respect to the other. We shall

come back to this example to show what the limits of our apparatus are. Now we

would like to go through the general functioning of the pulse shaper.

3.1.1 General overview

A general overview of the Fourier transform pulse shaper can be obtained from

fig. 3.1.

34

f f f f

Figure 3.1. Schematic of a fourier transform pulse shaper with capabilityof shaping amplitude and phase of the spectrum separately at the sametime. The slabs filled with diagonal lines are two parallel x polarizers,while the central slab represents the SLM (or two SLMs)

The input pulse, characterized by a certain bandwidth ∆ω (in our case ∆ω is about

46.5 rad/s (or ∆λ of about 15 nm) is sent on a diffraction grating (1800 lines/mm or

periodicity of 555 nm) that diffracts the different frequency components to different

angles. Due to the non-null diameter of the beam that hits the grating, several rays

characterized by the same frequency scatter off the grating with the same angle, but

not collinearly. By placing a lens exactly one focal length f away from the grating

all these rays at the same frequency are reunited and eventually focused one focal

length after the lens on what we will call the Fourier plane. In this plane there is a

correspondence between physical space and frequency space. By placing a mask (be

it an amplitude or a phase modulation) in that plane it is possible to manipulate the

spectrum and, therefore, also the temporal shape of the pulse. From this ”space-to-

frequency” plane on, the structure of the shaper is symmetric as all is left to do is

reconstruct the pulse by sending the different frequency components on the second

grating. This configuration is, for obvious reason, called a ”four-f configuration”.

35

3.1.2 The functioning of the SLM

The SLM that we have used in our set-up is made of two parallel polarizers and

two layers made of 128 elements of liquid crystals cells. Fig. 3.2 gives clearer view of

the device.

100 µm

2 µm gaps

Figure 3.2. Schematic of a spatial light modulator. ITO is the semitrans-parent, conductive material that is used to apply field in the z direction [9].The width of each SLM pixel in our set-up is actually 100 µm with a gapof 2 µm between adjacent pixels.

The liquid crystals used in each one of the 128 elements are elongated molecules

whose orientation can be controlled by applying an external bias in the z direction

across the element.

36

Figure 3.3. Schematic of the effect of the external bias over the liquidcrystals in the spatial light modulator.[9]

Each molecule has a main axis and a secondary axis that correspond to higher or

lower values of the refractive index along those directions. By applying the bias and

rotating the crystals (see fig. 3.3) it is possible to change the existing birefringence,

making each layer the analog of a variable wave-plate. We will see in the following

that one can achieve both amplitude and phase modulation using two separate SLMs

at the Fourier plane of the shaper.

Looking now at fig. 3.1, we can follow the path of a pulse that undergoes a shaping

operation. Using the notation for the axes reported in fig. 3.2, the light goes through

37

a polarizer with axis parallel to x. In absence of bias applied to the liquid crystals

the elements are designed to have the main axis at 45 with the x axis and at 90

with each other (±45 configuration). In case of zero bias on both the layers, we

can imagine that the effect of polarization rotation induced by the first layer will be

cancelled by the one due to the second layer and the light, polarized along x should

pass with no losses. When some bias is applied, the liquid crystals try to minimize the

energy, by adjusting to the external field and they rotate their main axes towards the

field lines, along z. What happens then is that some polarization rotation is induced

along with the introduction of a total phase. In a configuration like the one in Fig.

3.2, by going through the algebra one could find that for an input field polarized along

x the field emerging out of the SLM before the second polarizer is given by [9]:

~EOUT = EINei(∆φ1+∆φ2)

2

[x cos

(i(∆φ1 −∆φ2)

2

)+ iy sin

((∆φ1 −∆φ2)

2

)](3.1)

Where ∆φ1,2 are the phases gained (or retardations) by the light when passing through

the first and second layer. As we can see from eq. 3.1 the difference between those

two phases takes care of the rotation in the (x,y) plane and therefore of the amplitude

modulation (thanks to the second polarizer), while the sum of the phases introduces

a common phase, that is a delay. It is then clear how with these two degrees of

freedom it is possible to achieve at one time both amplitude and phase modulation

of the spectrum of our pulse. Being able to shape in this way the spectrum of the

pulse give us the freedom to change its shape in time domain as well by applying the

FFT of the desired temporal shape to the SLM.

In order to achieve a precise pulse shaping with this method it is required to know

exactly what the correspondence is between each of the 4096 voltage levels (0-10

Volts) applicable to each element of the two SLMs and the phase ∆φ1,2 induced. For

this reason the first step is to go through a calibration measurement where the light is

sent through the SLM and, keeping one of the two layers at a fixed level, the voltage

level of the second layer is changed to the other layer and the amount of transmitted

light is measured. An example of the result of such a calibration is reported in fig. 3.4.

38

0 1000 2000 3000 4000

0

1

2

3

4

5

6

0 1000 2000 3000 4000

0

1

3

4

5

6

8

9

Pow

er (

A.U

.)

Pha

se (

2 pi

rad

)

Voltage levels (0-10V)

Figure 3.4. (top) Transmission data for a pulse centered at 800 nm goingthrough a 45 SLM where a layer is kept at fixed voltage (zero here) andon the other the bias is swept from the minimum to the maximum usablevoltage. (bottom) the data shown in the top plot are used together witheq.3.1 to find the phase introduced by the layer to the incoming light.

3.2 Autocorrelation measurements

The autocorrelation technique is often used to measure the temporal shape of

unshaped pulses. The principle [27] is quite simple. As we can see in fig. 3.5, the

laser beam is split in two. After one of the beams is delayed in one of the two arms

(probe pulse), the beams are reunited, focused and spatially overlapped on a second

harmonic (SH) generating crystal.

39

PMTvibrational stage

2nd harmoniccrystal

BS

Figure 3.5. Schematic of a real time autocorrelator.

Due to its non linear nature the SH signal is much stronger when the two pulses

are temporally overlapped. In this way, by varying the mutual delay we can obtain

a SH pulse envelope from which it is then possible to extract information about the

original pulse width. On the assumption that the intensity is low enough to avoid

effects of pump depletion and that the bandwidth of the SH process is broader than

the input pulse bandwidth [27, 28], the average SH intensity (i.e. the measured power)

is given by:

< ISHG(t) >∝<| a(t) |4> (1 +2 <| a(t) |2| a(t− τ) |2>

<| a(t) |4> (3.2)

Where a(t) is the amplitude of the two input fields (assumed to be the same in this

particular case) and τ is the variable delay. A useful way of rewriting eq. 3.2 is the

following:

< PSHG(t) >∝<| a(t) |4> (1 + G2(τ)

)

G2(τ) =< I(t)I(t− τ) >

| I(t) |2 (3.3)

where I(t) ∝| a(t) |2. From these last equations it is clear that taking the Fourier

transform of G2(τ) yields the modulus square of the Fourier transform of the input

40

intensity | I(ω) |2. Under the assumption that the intensity input profile in time

domain is symmetric we can use the fact that its Fourier transform will be real and

in that case no phase retrieval is needed when looking for I(t). In that case we find

that the input intensity profile is given by

I(t)pulse ∝√F(ISHG)(ω). (3.4)

While the assumption of symmetry is allowing us to forget about the phase, it is

certainly limiting and shaping the information that we can acquire from these exper-

iments. This is an important limiting factor of this technique and only allows us the

study of unshaped pulses that are expected to be practically chirp-free and symmet-

ric. We have used a real time autocorrelator (where the intensity of the SH signal

can be read continuously with a scope) in order to minimize any chirp that could be

introduced in our set-up by spectrally diffractive components like prisms and gratings

or simply by the glass and other transparent, but dispersive, materials our pulses go

through before reaching the SL sample.

3.3 Cross-correlation measurements

The cross correlation technique allows us to study pulses that are either heavily

chirped or that have been shaped by mean of a pulse shaper (or both cases together).

The principle at work is the same as in the case of the autocorrelation, but now one

of the two pulses, the unshaped probe, acts like an optical gate for the study of the

other, shaped, pulse. In this case the assumptions of symmetric and chirp-free pulse

have to be dropped, because the shaped pulse, as we shall see later on, is likely to

be voluntarily chirped or shaped in an asymmetric fashion. The assumption that we

are going to make is rather that in time domain the gate pulse (the probe) is very

short (delta function-like) respect to the features of the pump pulse. In this way the

features of the pump won’t be washed out by the finite width of the probe. Under

this assumption the measured second harmonic power < PSHG > is given by:

< PSHG(τ) >t∝< Ipump(t)Iprobe(t− τ) >t, (3.5)

41

where the time average is taken integrating at the detector. An example of a cross

correlation trace is reported in fig. 3.6. This piece of data shows clearly the effect of

the finite duration of the probe pulse. Not having a δ-like probe pulse in time domain

limits the resolution at which the pump shape can be resolved. In fact, in principle,

given that the spectrum of the pulse shown in fig. 3.6 pulse was a train of peaks neatly

spaced, we should see in time domain peaks that are again neatly separated (with a

region of zero power between them).

-2 -1 0 1 2

0

5

10

15

20

25

30

35

40

Inte

nsity

(A

.U.)

Delay (ps)

Figure 3.6. Cross correlation data of a shaped pulse. The effect of theprobe finite width is visible in the raised minima that otherwise are sup-posed to be at sitting at zero.

Due to the finite duration of the probe, instead, we can clearly see that the minima

don’t reach down to zero power, but are raised.

42

3.4 Second Harmonic Generation - Frequency Resolved Optical Gating

(SHG-FROG)

While the first two methods provide only partial information on the intensity

and phase profile of a laser pulse, the FROG technique was demonstrated (when

using an unchirped probe) to provide full information. The base concept is that by

performing a cross correlation with a narrow pulse P2 one can sample in time domain

the pulse P1 and then study the frequency component of this sampled window. The

experimental data of a FROG set-up are 3D spectrograms that display the intensity

of each frequency component for the different delays τ of the probing pulse P2. An

example of a spectrogram with a marked 3rd order chirp is reported in fig. 3.7.

Figure 3.7. Spectrogram of the pump pulse when a marked third orderchirp was applied with the pulse shaper.

One can see how the central components arrive earlier than the side ones and

display some sort of echoes at latter times.

In principle one would like to imagine the spectrogram for the perfect (unchirped)

pulse as a box with perfectly horizontal and vertical sides that indicate the same

arrival time for all the frequencies. In reality both in frequency and time domain, the

43

spectrogram will be smoothed off due to the gaussian-like shape of the probing pulse.

Figure 3.8. Example of chirped pulse where different frequency compo-nents arrive at different times. From top to bottom it is shown how theinitial chirp (top) was taken care for.

A sort of ellipse with major axes parallel to the frequency and time axes is really

44

what we are aiming for.

In fig. 3.8 we report an example of a process of ”un-chirping” using the pulse shaper.

In this case the negative chirp that was given through the pulse shaper was a quadratic

one and more specifically (top to bottom in fig. 3.8): 0.30x10−26 Hz−2, 1.14x10−26

Hz−2 and 1.83x10−26 Hz−2.

In principle from such spectrograms one could achieve a full picture of both the

intensity and the phase profile through a recursive algorithm. For our scope we only

needed to use these spectrograms make sure that the chirp was being minimized

through the prism pair and the pulse shaper. This can be checked by simply looking

at the spectrograms and making sure that all the frequency components arrive at

the same time. A limit to this approach comes from the fact that in our case pump

and probe sometimes had different chirp and therefore the probe pulse that in frog

is supposed to be transform limited, sometimes may not have been such. For this

reason and seeking a faster approach, we looked at other possible ways to check our

pulses chirp.

3.5 Phase Retrieval by Gating in Frequency Domain with an SLM

This is a very simple technique where we took advantage of the filtering properties

of the pulse shaper [29]. The chirp of a broad-band pulse is simply the relative

delay among its frequency components. We can use the pulse shaper to sample the

frequency components of the pump. At the output of the shaper we obtain a long

pulse (about 2.5 ps in our case) in time domain. The relative position of these pulses

for the different frequency samples gives us full information on the chirp. If, for

example, we had no chirp at all, all the frequency components would arrive at the

same time and we should expect to observe that. In fig. 3.9 we report an example of

the results from this kind of technique.

45

374 376 378 380 382 384 386

4 6 8

10 12 14 16 18 20 22 24

374 376 378 380 382 384 386

-50

0

50

100

150

(b)

(a)

Pul

se s

pect

rum

(A

.U.)

Frequency (THz)

Rel

ativ

e de

lay

(

fs)

Frequency (THz)

Figure 3.9. (a) Spectrum of the studied pulsed laser. Each sampled sliceof the spectrum is reported separately. (b) (circles) Relative position ofthe long pulses in time domain. (line) Third order polynomial fit. Aquadratic behaviour is clearly visible.

The plot of the relative position in time of the gated pulses versus the central

frequency of the narrow filter is reported. After obtaining such a piece of data, one

can use the following equation,

φ(ω) = −∫ ω

−∞τ(ω′)dω′, (3.6)

where τ is the arrival time of the long pulse in output from the shaper/filter. In

our example, after taking a polynomial fit and integrating according to eq. 3.6 , we

obtain a (mainly) quadratic and cubic chirp of about −9.2 (2π rad)/THz2 and 0.016

(2π rad)/THz3 respectively.

46

4. Bloch Oscillations in superlattices: Samples growth and

preparation

4.1 Superlattice samples for experiments in transmission

The Superlattice in our system is created by alternating layers of GaAs and

Al0.3Ga0.7As. Since the latter has bandgap larger than the former, a series of wells

and barriers are created in conduction as in valence band. The periodicity can vary

among the samples used in order to have larger or smaller miniband width (see Fig.

4.1) and therefore more or less WS levels for a given bias voltage. The periodicity is

always such that the electronic coherence spans over several periods when no bias is

applied.

-4x10 6 -3x10 6 -2x10 6 -1x10 6 0 1x10 6 2x10 6 3x10 6 4x10 6

40

50

60

70

200

250

300

30 meV

Ene

rgy

(meV

)

Quasi-momentum (cm -1 )

Figure 4.1. First and second electronic minibands calculated for an infinite67/17 structure at low temperature.

47

We report here the data for the ”67/17” sample that was used to obtain the results

shown in the following chapters. The sample is an undoped GaAs/Al0.3Ga0.7As SL

embedded between an ohmic Au contact and a semi-transparent Cr/Au Schottky

contact. The schematic of the sample is reported in Fig.4.2.

Figure 4.2. Schematic of the growth of the SL samples (courtesy of Pro-fessor K. Kohler from the Max Plank Institut in Freiburg, Germany).

In the substrate, already covered with the ohmic contact, is opened a window that

goes down through the wafer all the way to the SL leaving only less than 2 µm of

material (including the Schottky contact). The bias is applied to the ohmic contact

and is passed to the ”active” optical region through an n-doped layer that is the

uppermost layer remained exposed in the back of the sample after the wet etching

has taken place. In the end, when ready for the DFWM experiment, the sample has

the structure reported in Fig. 4.3.

48

Figure 4.3. Sample etched prepared for an experiment in transmission.[3]

The sample is characterized by measuring the I-V curve and the absorption spec-

trum as a function of the applied bias. The I-V can tell us how good is the quality of

the sample from a transport point of view. The flatter the I-V is in the reverse bias re-

gion before break down, the better. A high current could in fact lead to a destruction

of the Bloch oscillations due to scattering with the free (free in the growth direction

as opposed to the localized oscillating electrons) carriers. In Fig.4.4 is reported an

example of the I-V curves taken at low T (10K) and without light being shined on

the sample.

49

Figure 4.4. IV characteristic curve of the sample described in fig. 4.2

From the absorption spectra as a function of applied bias we can obtain informa-

tion about the position of the levels as well as the oscillator strength of the different

transitions. This information proves very useful at the moment of looking for the

DFWM signal in the final level of these experiments. In fig. 4.5 a fan chart for the

sample ”67/17” is reported. From such a calculated fan-chart (courtesy of Profes-

sor Marc Dignam) one can follow the position of the transitions and the oscillator

strength of the different transitions versus the externally applied field.

50

Figure 4.5. Calculation of the absorption fanchart for the SL system. Thearea of the squares is proportional to the dipole matrix of the transition.Courtesy of our collaborator Prof. Marc M. Dignam, Queen’s UniversityOntario, Canada.

Now that the samples used and their characterization has been explained we

should like to introduce our experimental set-up for DFWM experiments.

51

5. Spectrally Resolved Degenerate Four Wave Mixing

(SR-DFWM)

5.1 Experimental Set-Up

In our SR-DFWM experiments the spectrum of the 2~k2-~k1 component of the non-

linear signal coming off the back face of the sample is recorded for different delay

times of the probe respect to the pump pulse. A schematic of the set-up is shown in

fig. 5.1.

Ti:Sa

Monochromator

CC

AC

AC

AC

CCD

IR

camera

OSA

SLM

stage

PCTV

Ti:Sa

Monochromator

CC

AC

AC

AC

AC

AC

AC

CCD

IR

camera

OSA

SLM

stage

PCTV

Figure 5.1. Schematic of the SR-DFWM set-up. AC = auto-correlator.CC = cross-correlator. OSA = optical spectrum analyzer. SLM = spatiallight modulator.

The laser pulses (Ti:Sa 40 fs-15 nm at about 790 nm) are compressed with a prism

pair to take care of any chirp that may be introduced by all the various components

in the beam path between the laser and the sample. This is especially important

for the probe pulses as in the pump path, chirp can be taken care for also through

52

the pulse shaper. After the compressors the laser light is split in two parts by

mean of a polarizer beam-splitter cube that allows us to set the ratio of the power

of pump and probe by adjusting a half-wave plate. The pump arm goes through a

pulse shaper (more about this in the following) and after that the laser can either

go directly on the sample or to a cross-correlator (together with the probe). The

probe beam goes through a variable delay line (whose stage has a resolution of 0.1

micron or about 2/3 of a fs) and then it can go directly on the sample or to the

cross-correlator. In order to make sure that in both the pump and probe pulses the

chirp is minimized, autocorrelations are taken at different points along the path

(see fig. 5.1) and compared to the expected duration for a transform-limited pulse of

the same bandwidth. A shaker attached to the retro-reflector sitting on top of the

delay stage modulates the light of the probe with a frequency of about 100 Hz. The

function of the shaker is to ensure that small vibrations in the two arms of pump and

probe don’t introduce noise by scrambling the relative phase of the carrier sinusoidal

function of pump and probe. To do that we just scramble the phase randomly and

average.

The sample is kept on the cold finger of a continuous-flow cryostat using liquid

helium that allows us to keep it easily at a temperature of about 10 K. The surface of

the sample is imaged and magnified on an IR camera and shown on a monitor in order

to ensure that both the temporal and spatial overlap of the two beams are achieved.

This imaging system after the sample is optimized to achieve max magnification and

at the same time the optimum input angle to match the monochromator acceptance

angle. We will go back to the properties and requirements of this imaging system.

The pump and probe beams, pointing respectively in the direction ~k1 and ~k2, are

blocked after the sample with an iris. The DFWM signal 2~k2 − ~k1 is let through and

goes to the monochromator. In the monochromator (a Jobin-Ivon TRIAX550) we

used both gratings of 1200 lines/mm or 600 lines/mm depending on the need. These

gratings gave us a spectral dispersion of about 0.045 meV and 0.09 meV respectively

(measured with a CW He-Ne laser). At the output of the monochromator a CCD

53

camera (Princeton Instruments, MicroMax 1000x800) is used to detect the light and

obtain the full spectrum at once. Unless stated otherwise, the excitation density used

for the pump is about 5x109e/well · cm2 · pulse and the power of the probe is taken

as roughly double as much as the one from the pump (to improve the signal/noise).

The pump and probe pulses were always collinearly polarized.

In the following we would like to introduce the key component of the set-up in a little

more detail.

5.1.1 Pulse Shaper: a few numbers

We already introduced the pulse shaping technique in detail in chapter 3.1. Here

we would like to give some experimental numbers that may result useful to the reader.

50 55 60 65 70 75 80 85 90

1.810

1.815

1.820

1.825

1.830

1.835

Y = A + B * X Parameter Value Error ---------------------------------------------- A -329.6324 20.67827 B 12.76701 0.29062

Ene

rgy

(eV

)

SLM pixels

Figure 5.2. Spectral dispersion of the pulse shaper in meV

We were using two 1800 lines/mm gratings, two achromatic lenses of focal length

8.5 cm and a two-layer spatial light modulator with 128 elements per layer. Each

element has a width of 100 µm. Given the input diameter of the pump beam imping-

54

ing on the grating and the numbers reported above, a spectral dispersion of about

0.66 meV/pxlSLM was measured by applying amplitude modulation to the laser and

observing the modulated spectrum with the monochromator (see fig. 5.2).

5.1.2 Shaker

The shaker introduces a modulation in the delay of the probe of about 2 periods

of the carrier (about 5 fs). The shaking was achieved with a piezo element and the

amplitude of delay modulation was checked by observing the interference fringes of

pump and probe at the the surface of the sample. The frequency was high enough

to ensure that we were averaging over a large number of oscillations (usually at least

more than 100).

5.1.3 Imaging system

The active surface of the sample is imaged with a first lens on a small-aperture

iris. The surface of the small iris is then imaged on the entrance slit of the (imaging

1:1) monochromator and also on an IR camera for experimental convenience. The

small-aperture iris acts as a spatial filter to further suppress the stray light from the

probe. A first iris is filtering out most of the undesired stray light, but some light that

is scattered in random directions from the rough surface of the sample, may follow

the path of the DFWM signal. It is then important to be able to magnify on the iris

the imaged active area of the sample as much as possible. In this way we can try

and look for a region with little background coming from the stray light. After this

magnification (around a factor of 8), one needs to demagnify the beam as much as

possible in order to achieve the optimal spectral resolution without losing too much

signal when narrowing the input slit of the monochromator to match the limit given

by the width of the CCD pixels of 15 µm. Another parameter to take into account

was the input acceptance angle of the monochromator. Finally, another non-trivial

constraint was represented by the limited available space between the sample and the

monochromator).

55

5.1.4 CCD Camera: a few numbers

The CCD camera used was a back-illuminated Princeton Instruments MicroMax-

800PB. Its 1000x800 pixels are 15x15 µm wide and the total active surface is 15x12

mm. The spectral range is 200-1080 nm with a plateau above 50% of quantum

efficiency between 350 and 900 nm (see fig. 5.3). This feature has allowed us to use

it also for the SHG-FROG experiments where light at 780 nm / 2 = 390 nm was

detected. In fig. 5.3 we report the plot of the quantum efficiency for the VISAR chip

used (broader bandwidth).

1

Figure 5.3. Quantum efficiency of the CCD used (VISAR)

After checking that we were in the region of linear response of the CCD, we

measured the integrated number of counts from the CCD and the impinging power

just before the CCD itself. Passing from power to energy through the integration

time, we could estimate a number of 41 photons/count at 800 nm (against a spec

value of 8 photons/count).

5.1.5 Monochromator

The monochromator used is a Jobin-Ivon Triax 550 imaging system. An imaging

monochromator is simply one whose internal mirrors are such that there is no mag-

56

nification from the plane of the entrance slit to the one of the exit slit or of the CCD.

This means that in our case, on the plane of the CCD we will have a 1:1 image of the

entrance slit for each different color. It is understandable, then, how the resolution

depends on the frequency spreading of the grating (that in returns depends also on

the focal length of the mirrors) and on the minimum between the size of the entrance

slit and the width of a CCD pixel (15 µm). A plot of the energy vs. CCD pixel

number is shown in fig. 5.4.

0 200 400 600 800 1000

1.79

1.80

1.81

1.82

1.83

1.84 SLOPE = 0.052 meV/CCD pxl

Y = A + B * X Parameter Value Error ---------------------------------------------- A 1.79302 1.94934E-4 B 5.1781E-5 3.40901E-7 ----------------------------------------------

Ene

rgy

(eV

)

CCD pixels

Figure 5.4. Spectral dispersion of the Monochromator/CCD system inmeV for the 1200 lines/mm grating

The best resolution obtained was of about 0.09 meV for the 600 lines/mm grating

and of 0.045 for the 1200 lines/mm grating (see fig. 5.5).

57

0 100 200 300 400 500

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Re

solu

tio

n (

me

V)

Slit Aperture ( m)µ

Figure 5.5. Resolution measured for the TRIAX 550 with the MicroMax-800PB CCD as a function of the entrance slit of the Monochromator.The grating installed was the 1200 lines/mm. The curve in red is a simpleexponential fit.

5.1.6 IR camera

The IR camera is used to observe the surface of the sample and the two beams

focused on it. It is an invaluable instrument to 1) check the quality of the surface

and steer clear of rough spots that will flood the SR-DFWM signal with stray light 2)

achieve spatial overlap 3) achieve and optimize the temporal overlap by observing the

interference fringes. Furthermore, when one is using a generous total optical power

(about 1 mW) to find the DFWM signal, the DFWM signal can actually be observed

directly on the IR.

5.2 Bloch Oscillations and SR-DFWM: unshaped results

We would like to show now some of the results obtained with the SR-DFWM

technique using unshaped pulses. In fig. 5.6 the spectral peak position of the SR-

58

DFWM is reported for zero delay. As we can see each transition is represented by a

peak with intensity proportional to the oscillator strength of the transition itself as

well as the power of the laser in that spectral region.

Figure 5.6. Spectrum of the DFWM signal at zero delay for a 67/17 struc-ture with applied bias. The hh−1 (left) and hh−2 (right) are represented.

By plotting the whole integrated spectrum vs. delay time we can observe the

oscillations in time domain. An example of that is reported in fig. 5.7 (a). The same

kind of information can be achieved plotting the height (d) or the integrated spectrum

(b) of a peak corresponding to a specific transition (in fig. 5.7 for the hh−2).

59

0.5

me

V

Figure 5.7. (a) Spectrally integrated DFWM versus delay of the probe.(b) Spectrally integrated DFWM for the only hh−2 peak versus delay ofthe probe. (c) Position of the hh−2 transition (the max amplitude is ofabout 0.5 meV) versus delay of the probe. (d) Height of the hh−2 peakversus delay of the probe.

From this kind of data, assuming the two pulses are short with respect to the sys-

tem dynamics one can estimate the time T2 controlling the decaying of the coherence

among all the oscillating dipoles. The main reason for measuring the spectrum of the

DFWM signal (fig. 5.7 (c)) will be discussed in the next paragraph. This piece of

data will result a key point in determining the extent of our control on the coherent

dynamics of the BO excitons.

5.3 Method for monitoring the electronic dynamics through SR-DFWM

In order to monitor the effects of our control over the electronic dynamics, we need

a technique that may enable us to follow the motion of the oscillating excitons. The

technique described in this paragraph was first introduced by [30]. When the pump

60

pulse promotes the electron (e) to the conduction band and creates a hole (h) in

valence band, the characteristics of these two particles are very different. Due to the

different curvature of valence and conduction bands, the hole, more massive (due to

the smaller radius of curvature of its band) is very localized in the quantum well where

it was created. On the other hand, the electron, promoted into a superposition of

WS states, oscillates over a few wells. The displacement of e and h causes an internal

dipole field that, when considering the whole ensemble of the oscillators, resembles

a plasmon field with a positive sheet of charge in the middle and a negative one

oscillating along z. This internal oscillating field alternatively screens and reinforces

the external field. The effect of this internal field is to bend the bands and shift the

transition energies.

Figure 5.8. Absorption spectrum of the superlattice as a function of ex-ternally applied field. The color chart is in units of m−1.

By using the fanchart reported in fig. 5.8 and the positions of the transition ener-

gies at different delay τ of the probe pulse, we can effectively estimate the amplitude

61

of the internal dipole field. Measuring separately the excitation density, one can es-

timate the relative position of the center of mass of the electrons with respect to the

position of the holes at each delay τ of the probe.

Using displacements of the order of one SL period (84 A), excitation densities of

about 5x109 e/(pulse cm2 well) and an average of 12.85εo for the dielectric constant

index of the SL, one finds an expected energy shift of about 0.4 meV (or about 0.22

kV/cm). As we’ll see in chapter 6.2, this number is indeed very close to what was

experimentally observed.

62

6. Coherent control of Bloch Oscillations in superlattices:

towards a THz oscillator

There are different reasons that have lead us to apply shaped pulses to the excita-

tion of BO [31]. One of the most applied ideas was linked to the creation of a THz

oscillator that is tunable and persistent over a few cycles. This can be achieved by

forcing the Bloch oscillator with an optical pulse, the shaped pump. When the pump

spectrum is amplitude modulated to contain only the spectral components resonant

with the 1S bound excitonic levels, its envelope in time domain in shaped to display

a train of pulses with the same periodicity of the BO. In this way, recreating the

coherence of the population (whose coherence would otherwise decay in about 1.5

ps) every period, one can create a driven oscillator kind of internal dynamics. As we

introduced already in chapter 2, the oscillation of the electron with respect to the

hole (within each exciton) produces radiation in the THz range. In this chapter we

will show how this source of radiation can be fully controlled; its frequency can be

tuned and its amplitude can be switched it on or off with the stroke of a key on our

computer.

Another reason for selecting the part of the pump resonant with the 1S bound exci-

tonic transitions is that we were trying to minimize the creation of unbound excitons

with the pump. These excitons, due to their freedom of movement, have a higher rate

of elastic scattering and lose their collective phase very quickly, forming a plasma of

incoherent carriers whose action on the dynamics of the bound (BO) excitons seems

to play a very important role [32].

In all the experiments reported in the following chapter we are using 67/17 GaAs/AlGaAs

samples exactly like the one reported in chapter 4.1. The excitation density used is

63

about 5x109e/cm2wellpulse and the applied voltage is of about 2.2 Volts that yields

an electric field of about 15 kV/cm.

6.1 Modes of the Bloch oscillations: Breathing mode and Non-breathing

mode

Depending on the symmetry (or lack of it) in the relative motion of the electron

respect to the hole, the excitonic Bloch oscillations can be subdivided into two main

modes. In a parallelism with molecular physics, the breathing mode (BM) is achieved

when the oscillating electron leaks out of its well of origin (where the hole is local-

ized) in a symmetric fashion, increasing or decreasing its presence at the same rate

in the left and right first-neighboring wells. In the BM case, the center of mass of

the electron is barely, if at all, moving and there is very little transverse dipole field

(along the z direction of growth of the SL).

On the other hand, when the excitation condition is not symmetric, the center of mass

of the electron displays a clear spatial oscillation in what is called a non-breathing

mode (NBM). In this case the displacement between hole and electron creates an

internal transverse dipole field that oscillates with the frequency of the Bloch oscilla-

tions (typically 2-2.5 THz).

A simple example of a calculated electronic wave packet displaying BM (left column)

and NBM (right column) is reported in Fig. 6.1. The wavepacket was written on

a base of wavefunctions of the Wannier-Stark levels calculated in the presence of

Coulomb interaction.

64

Figure 6.1. (top in black and white) Probability of finding the electronat a given position in space along z(each maximum corresponds to thecenter of a different well). On the right (left) column the dynamics of thenon-breathing (breathing) mode is shown. (bottom in color) The samedynamics is described through a color contour plot displaying position (x)and time (y).

If we calculate the position of the center of mass of the electron for the two different

65

modes, one can see (Fig. 6.2) how the two cases are expected to produce very different

amplitudes for the oscillatory motion of the center of mass of the electron. Since the

hole in valence band remains well localized in the well where it was created due to its

large effective mass, larger amplitudes in the electronic motion translate into larger

internal dipolar fields created by the internal electronic dynamics.

Figure 6.2. Position of the electronic center of mass versus time for thecases of breathing mode (large amplitude) and non-breathing mode (smallamplitude). Notice how the smaller oscillations are characterized by dou-ble frequency. This contribution at 2ωBO frequency is due to the overlapbetween the wave function of the hh±1 states. This effect will be explainedin better detail in the next paragraph.

6.2 Changing mode of oscillation: a switch for the Bloch THz oscillator

In the first step of our experiments of coherent control we make use of the pulse

shaper as a pure amplitude modulator. our goal is to create a BM by selectively ex-

citing the hh−1, hh0 and hh+1 transitions. If we look at either the calculated fanchart

(Fig. 4.5) or the measured one (Fig. 5.8), we can see how the oscillator strength is not

symmetric with respect to the line of the hh0 transition. The hh−1 is always stronger

than the hh+1 due to an asymmetry of the WS excitonic wave functions towards the

66

lower-energy side of the SL. In fig. 6.3 (a) one can see that for the applied external

field of choice (15 kV/cm) the hh−1 line is about three times stronger than the hh+1.

In Fig. 6.3 (b) both the spectrum of the unshaped laser and of the shaped pump are

reported. The relative intensity of the side peaks in the pump spectrum is chosen

to counterbalance the effect of the difference in the dipole element for the two tran-

sitions hh+1 and hh−1. Had we required a different frequency for the THz radiation

produced, the ratio would have been different, but the principle would have still been

the same.

1540 1560 1580 1600 1620 1640

Pow

er S

pect

rum

(A

rb.U

.)

Energy (meV)

a 0

a -1 = 3.1 * a

+1

a +1

a -1

Figure 6.3. (a) Absorption spectrum taken at the external field Fbias= 15kV/cm. The three transitions visible are the hh−1,hh0 and hh+1. (b) Fullspectrum of the unshaped laser and spectrum of the pump modulated toachieve the breathing mode.

A simple way to explain how we are going to achieve a BM is the following. By

covering the three transitions hh−1,hh0,hh+1 with the pump spectrum, the electron

67

is sent into a superposition of three WS eigenfunctions:

Ψ(x) = A−1e+iωBOtΨ−1(x) + A0Ψ0(x) + A+1e

−iωBOtΨ+1(x). (6.1)

Here, An is a complex number proportional to the product of Sn and Mn, where Sn

is the laser complex spectral amplitude resonant with the nth transition and Mn is

the interband dipole matrix element. Due to the alternating sign of the WSL wave

function on the low-energy side, M−1 and M+1 have opposite signs [17].

Figure 6.4. Schematic of the overlap hole/electron wave functions. Thedifferent signs of the Ψ−1(x) (red) and the Ψ+1(x) (blue) in the centralwell are evident. The electric field is pointing to the right.

This can be understood looking at fig. 6.4, where the spatial overlap between the

hole and the electronic wave functions is reported for the three transitions of interest.

The overlap of the hole wave function with a negative peak of the Ψ−1(x) and a

positive peak of the Ψ+1(x) is the reason of the difference in sign between M−1 and

M+1.

68

From eq. 6.1 we can calculate the expectation value for the position of the excited

single electron:

Xe(t) =(|A−1|2x−1 + |A0|2x0 + |A+1|2x+1

)+

− 2|A∗−1A0| · |x−1,0| cos(ωBOt + φ0,−1)+ (6.2)

+ 2|A∗+1A0| · |x+1,0| cos(ωBOt + φ+1,0)+

+ 2|A∗−1A+1| · |x−1,+1| cos(2ωBOt + φ−1,+1),

where φn is the phase of Sn, xn,m ≡ ∫ +∞−∞ Ψ∗

n(x)xΨm(x)dx, and xn ≡ xn,mδn,m. Within

a very good approximation x−1,0 and x+1,0 are equal [17], while the last term can be

neglected due to the small ratio |x−1,+1|/|x−1,0|.By amplitude modulating the pump spectrum, we can make the amplitudes |A−1|and |A+1| equal. By making sure that the pump spectrum has a flat phase profile, we

can also make sure that the two parameters φ−1,0 and φ+1,0 from eq. 6.2 are equal. In

this way the oscillatory behaviour of the 2nd and 3rd terms of eq. 6.2 is cancelled out

and we have only a negligible oscillation of the center of mass of the electron respect

to the hole. In fig. 6.5 we report the spectrogram of the SR-DFWM for the case

of flat phase across the pump spectrum ([0,0,0]). The excitation density was about

5x109e/pulse ·well · cm2. This is a number that is reported in literature as being low

enough to make exciton-exciton scattering events unimportant.

69

Figure 6.5. Spectrogram for the case of pure amplitude modulation of thepump to achieve the breathing mode.

A number of things can be observed from this spectrogram. First of all we can

verify the kicked-oscillator kind of behaviour where we have prolonged the window of

observation of the oscillations to at least 7 periods. We can also observe the difference

in strength between the hh−1 and the hh+1 components. This is arises from the fact

that, although we are compensating for the difference in absorption between hh±1

transitions, the DFWM signal is proportional also to the square of the probe field

that is not shaped to compensate for the different oscillator strengths. As expected,

having excited a BM, we cannot observe from fig.6.5 any sign of oscillatory modulation

70

in the position of the three spectral components. However, we usually really need to

plot a more thorough study of the SR-DFWM peaks as a function of delay to say

something about this point. Such a study is reported for this case of pure amplitude

modulation in fig.6.6.

1601

1602

1603

1586

1587

1588

-750 -500 -250 0 250 500 750 1000

1573

1574

hh +1

hh 0

Est

imat

ed I

nter

nal T

Hz

Fiel

d (k

V/c

m)

Ene

rgy

(meV

)

hh -1

Delay (fs)

-0.6

-0.3

0.0

0.3

0.6

-0.6

-0.3

0.0

0.3

0.6

-0.6

-0.3

0.0

0.3

0.6

Figure 6.6. Shift of the transition energies for the hh+1 (blue), hh0 (black)and hh−1 (red) transitions. The breathing mode is obtained with a flat,[0,0,0], phase profile. The right scale in all the plots reports the estimatedmagnitude of the generated internal oscillatory THz field as estimatedusing the quasi-static model.

Again, no sign of modulation is clearly visible in the plot of fig. 6.6.

In fig. 6.7 we report now the spectrogram for the case of amplitude and phase modu-

lation of the pump spectrum. In this case a π phase was applied to the higher energy

component of the pump spectrum (we will use this kind of notation from now on

[0,0,π]). This is the case in which, according to eq. 6.2, the amplitude of the oscil-

71

lations of the electronic center of mass is maximized. This in return maximizes the

amplitude of the internal oscillatory dipole and its effect on the peak shifts.

Figure 6.7. Spectrogram for the case of amplitude and phase modulation([0,0,π]) of the pump to achieve the non-breathing mode.

As we already mentioned, it is rather hard from this kind of plot to observe

oscillations in the peaks position, but in this case one can nonetheless observe some

oscillations in the hh−1 in fig. 6.7. The modulation in the peaks positions is reported

as a function of probe delay in fig. 6.8.

72

1601

1602

1603

-750 -500 -250 0 250 500

1573

1574

1586

1587

1588

hh +1

hh -1

Delay (fs)

hh 0

Ene

rgy

(meV

)

-0.6

-0.3

0.0

0.3

0.6

-0.6

-0.3

0.0

0.3

0.6

Est

imat

ed I

nter

nal T

Hz

Fiel

d (k

V/c

m)

-0.6

-0.3

0.0

0.3

0.6

Figure 6.8. Shift of the transition energies for the hh+1 (blue), hh0 (black)and hh−1 (red) transitions. The non-breathing mode is obtained with aphase profile [0,0,π]. The right scale in all the plots reports the magnitudeof the generated internal oscillatory THz field as estimated using the quasi-static model.

Comparing the plots of the peaks positions for the BM (fig. 6.6) and NBM ( 6.8)

cases, the difference is rather striking. We should observe a number of things from

these plots. First of all, as expected, we observe a strong modulation in the peak shifts

in the second case (NBM) as opposed to the first case (BM). The position of the hh0

transition doesn’t display a strong dependence on the phase of the pump spectrum.

This was expected as the hh0 is less responsive to the band bending. The hh−1 and

the hh+1 display opposite slopes in the fanchart (see fig. 1.8) of peak position vs.

applied field. For this reason we expected to see the two side transitions to oscillate π

out of phase. This is well visible in the plot of fig. 6.8. The difference in the amplitude

73

of the oscillations of the hh−1 and the hh+1 transitions, however, is not explained by

our simple quasi-static model, where both excitonic effects and the effect of having

the oscillations in a plasma of unbound, incoherent electrons is not being taken into

account.

In fig. 6.9 we report the height of the hh−1 peak as a function of time for both the

[0,0,0] excitation (full line) and the [0,0,π] excitation (dotted line).

-1000 -750 -500 -250 0 250 500 750 1000

0.0

0.2

0.4

0.6

0.8

1.0

FW

M h

h -1

pea

k he

ight

(A

.U.)

Delay (fs)

Figure 6.9. Height of the hh−1 SR-DFWM peak vs. delay of the probefor a pump phase profile [0,0,0] (full line) and [0,0,π] (dotted line).

The structure observed in the last part of the dotted plot is likely due to the

internal intraband dynamics of the BO electrons. In fact, the internal dipole created

by the displacement of electron and hole along z, changes the energy of the transitions

while the pump pulse is still acting on the system. Once again, this structure in the

temporal behaviour of the DFWM can be interpreted as a sign of the shortcomings

74

of the quasi-static model in which transient phenomena are not really being properly

treated. Finally we would like to show together in the same plot (see fig. 6.10) the

time-integrated DFWM signal ([0,0,0] excitation of the BM) for the hh−1 transition

and the cross correlation of the pump as a function of the delay of the probe.

-1000 -500 0 500 1000

0.0

0.5

1.0

1.5

Inte

nsity

(A

.U.)

Delay (fs)

Figure 6.10. (black) Height of the hh−1 SR-DFWM peak vs. delay of theprobe for a pump phase profile [0,0,0] (red) Cross correlation of the pumppulse. The heights of the first pulses of both curves are brought artificiallytogether.

One can observe that the DFWM follows quite closely the shape of the pump

except that the last peak of the DFWM is not present in the cross correlation. This

is a clear signature of the internal dynamics of BO and a reminder that we are dealing

with a system where the optical excitation has a duration similar to the one of the

internal excitonic dynamics. This may make the description of this system rather

complicated.

75

6.3 Control over the initial condition of the electronic oscillatory motion

Bringing, now, the control one step further, we can observe from eq. 6.2 how the

effect of changing the relative phase between Φ−1 and Φ+1 is effectively to change the

initial condition of oscillation of the center of mass of the electron. In our parallelism

between the excitonic oscillation and a classical oscillator, we can, therefore, say that

we can control the initial position and direction of the electron.

This is shown in Fig. 6.11, where the peak shift for the hh−1 transition is reported

for the two phase profiles [0,0,π] and [π,0,0].

-1000 -500 0 500 1000

-0.4

-0.2

0.0

0.2

0.4

1572.4

1572.8

1573.2

1573.6

1574.0

Intr

aban

d el

ectr

ic fi

eld

(kV

/cm

) Delay (fs)

Pea

k P

ositi

on (

eV)

Figure 6.11. Effect of the coherent control. (black dots) a phase of [π,0,0]is applied to the pump spectrum. (gray stars) a phase of [0,0,π] is appliedto the pump spectrum. The right scale reports the magnitude of thegenerated internal oscillatory THz field as estimated using the quasi-staticmodel.

6.4 Presence of an internal DC field

In this case we focus on exciting selectively only the hh−1 and hh−2 transitions.

In such a case the excitation is not symmetric anymore around the central well where

the hole resides. For this reason we may expect to see some non-oscillating (DC)

76

dipole field acting on the position of our transitions. This internal DC field could

have some important effects on the THz train of pulses and it seems important to

observe its presence, time scale and, in general, its temporal behaviour.

In fig. 6.12 we show a spectrogram for the 2-color experiment where the pump is

amplitude modulated to be resonant with the hh−1 and the hh−2 transitions. The

excitation density is about 5x109 e/(pulse cm2 well). In this case the pump was not

transform limited as some (mainly) quadratic chirp was present. The magnitude of

this chirp was estimated to about 10−27 Hz−2 (corresponding to a delay between blue

and red components at the FWHM of the pulse of about 40 fs over an oscillation

period of the pulse train of about 500 fs). Besides being a rather small effect in

absolute sense, the effect of this limited chirp over the DC shift seems to play a minor

role in the result of some of our preliminary simulations.

Figure 6.12. Spectrogram of the hh−1 and hh−2 transitions. The pump isshaped to excite resonantly at τ = 0.

77

The pump spectrum is also reported in fig. 6.13. The position of the two lines

in the pump spectrum was chosen to be resonant for zero delay between pump and

probe.

1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585 1.590

1

2

3

4

5

6

Inte

nsity

(A

.U.)

Energy (eV)

Figure 6.13. spectrum of the pump used for the DC experiment reported.

As observed also before, although we can already see the presence of a clear

DC shift in the peaks position from fig. 6.12, this effect as well as the oscillatory

modulation in the peaks positions becomes clearer when we study the position of the

transitions for each spectrum of the scan separately. The result of this study is shown

in fig. 6.14. One can clearly see a strong DC shift (about 0.5 meV/ps). Following the

simple, single-particle, approach that we used in the previous paragraph, we should

indeed observe an internal DC field.

78

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000

1.563

1.564

1.572

1.573

Ene

rgy

(eV

)

Delay (fs)

Figure 6.14. Peak position of the hh−1 and hh−2 transitions as a functionof delay time of the probe.

In fact in this case the position of the center of mass in time is described, in our

simple model, by:

Xe(t) =(|A−2|2x−2 + |A−1|2x−1

)+ |A∗

−2A−1| · 2x−2,−1 cos(ωBOt + φ−1,−2). (6.3)

Within this excitation scheme we should expect the presence of an internal DC dipole

arising from the first, non-oscillatory term of eq. 6.3.

Due to the different decaying times of the populations (of the order of 10ps or

longer [8]) and the coherences (about 1ps [33]), one can expect the building up of

an internal, non-oscillating dipolar field due to the first term of eq. 6.3. unfortu-

nately, this explanation predicts the building up of an internal field that, screening

79

the external one, would level off the slope of the electronic bands. This would increase

the energy of the hh−1 and hh−2 transitions. The observation is contradicting this

model in that the transitions are observed to be shifting to lower energies. This is

therefore a clear example of the limits of our picture of single-electron and probably

also of the quasi-static model for the internal dipolar field used so far. A possible

explanation for this effect may reside in the presence of an optical excited plasma of

unbound electrons that are always excited along with the 1S bound-exciton levels [32].

Professor M.M. Dignam is working on the simulation of our experiments as we write.

The simulations results are in fair agreement with our experimental data, but the

introduction of the continuum (unbound excitons) in the calculation seems to be

needed to obtain good agreement. Although with our shaped pulses we aimed at

the selective excitation of the bound excitons only, the excitation of the continuum

is really unavoidable as a spectral component exciting the hh+1 in the side well will

excite continuum in the wells at lower energy down the slope of the SL conduction

band. In our case, then, the persistence of the optical field of the shaped pump during

the internal BO dynamics is proving to be a rather challenging problem to describe.

The results of this theory are not available as this thesis is being written.

As a last remark on these data, we should like to point out how this DC field is not

expected and indeed is not visible in the case in which the pump spectrum is shaped

to achieve the breathing mode. Considering the implications of the DC energy shift

and particularly of the relative shift between the two transitions, one can envision a

broadening of the bandwidth of the THz oscillator that we are creating and control-

ling with our shaped optical pulses. The previous plots demonstrate that, not only is

the BM configuration the only one where we can achieve a switch for the THz oscil-

lator, but it’s also the only one (due to its symmetry) where this effect of broadening

of the oscillations is not present.

6.5 Selective excitation of Wannier-Stark states

We would now like to introduce another experiment of control that stresses some

interesting characteristics of the nature of the DFWM signal.

80

The third order (2~k2−~k1) signal of the DFWM is the result of the interference among

the excitonic dipoles created by the two pulses. If we regard these oscillators as

independent, we should expect each SR-DFWM spectral component (related to each

electronic transition covered) to be independent of the others. If, for example, we

sent on the sample two CW laser beams (~k2 and ~k1) tuned on the hh−1 transition, we

should expect a SR-DFWM signal in the 2~k2−~k1 direction at the same frequency. No

other frequency being excited, we shouldn’t expect any other frequency to be present

in the SR-DFWM in the 2~k2 − ~k1 direction. We could envision doing the same with

the hh0 transition and seeing only the frequency of the hh0 in output. This is what

we call the physical picture of independent oscillators. Our next experiment shows

how the real physical system is not so simple.

Building on our three-colors experiments we blocked the hh0-resonant component in

the pump spectrum to leave only the side bands resonant with the hh±1 (see fig. 6.15).

The spectrum of the probe is still unshaped and covers all three transitions.

81

1.54 1.56 1.58 1.60 1.62

0.0

0.2

0.4

0.6

0.8

1.0 The two spectra are not in scale

Pow

er S

pect

rum

(A

rb.U

.)

Energy (eV)

Figure 6.15. Spectra of the pump (red) and of the pump (black). Thecentral component of the pump (that is blocked here) would sit at approx-imately 1.58 eV. The hh−1 transition corresponds to the peak at lowerenergy.

In the simple independent-oscillators picture, we would expect to see a SR-DFWM

that contains no frequency corresponding to the unexcited (by the pump) hh0 tran-

sition. As we can observe in fig. 6.16, the spectrogram of these data contains a clear

signal at the hh0 energy.

82

Figure 6.16. Spectrogram of the 2-color experiment where only the hh−1

and hh+1 transitions are excited.

In the scan reported above in fig. 6.16, the intensity of the laser had been in-

creased with respect to the previous three-colors experiment by a factor of about 2.5

(this power is also now distributed on only two spectral components). Due to the

presence of some background from the stray light (that was subtracted afterwards),

the CCD was saturated in correspondence to the hh+1 transition. The optical pulse

was optimized to observe the hh−1 and hh0 signals. That resulted into two very clear

peaks. The modulation in time domain follows closely the shape of the driving pump

pulse.

In fig. 6.17 we report the detailed study of the hh0 and hh−1 peak heights versus the

delay of the probe for the case of resonant 2-color pump with a phase profile of [0,π].

83

-1500 -1000 -500 0 500 1000 1500

0

200

400

600

800

1000

1200

DF

WM

pea

k he

ight

for

hh

0 and

hh

-1 (

Arb

. U.)

Delay (fs)

Figure 6.17. Height of the hh0 (red) and hh−1 (black) spectral componentsof the DFWM as a function of the delay of the probe. The phase profileof the pump spectrum is [0,π].

The intensity of the component of the SR-DFWM resonant with the hh0 results

even stronger than the lower energy one. Noticing that the probe spectrum was

roughly the same in the region of the hh0 transition than in the one of the hh−1, the

strength of the DFWM from fig. 6.17 can be used directly to compare the efficiencies

of the two gratings and we see how at this excitation density, the diffraction power of

the non-linear-interaction grating (see below) is stronger, roughly double as efficient

as the normal DFWM (non-interacting-picture) polarization grating.

Blocking the excitation of the middle well, we also would expect not to see any

important modulation in the position of the transition of the hh−1. In fact now only

84

the interaction between the second-neighbors wave functions Ψ±1 is present. The

term due to their interaction:

X2ωBOe (t) = 2|A∗

−1A+1| · |x−1,+1| cos(2ωBOt + φ−1,+1), (6.4)

had been completely neglected in eq. 6.2 due to the little overlap between the Ψ−1(x)

and the Ψ+1(x) appearing in the equation above in the term x−1,+1. However, as

we can observe from fig. 6.18, the term above plays now an important role. We

can observe a clear modulation with amplitude only about half of the one observed

previously in our 3-color experiments.

-1000 -750 -500 -250 0 250 500 750 -5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

100

200

300

400

500

600

700

Ene

rgy

(meV

- 1

565.

5 m

eV)

Delay (fs)

Hei

ght o

f hh

-1 F

WM

pea

k (A

rb. U

nit)

Figure 6.18. (bottom) Height of the hh−1 spectral component of theDFWM as a function of the delay of the probe. (top) Position of thehh−1 transition versus delay. The phase profile of the exciting pump spec-trum is [0,π].

This seems to be due to the fact that, by increasing the excitation density by a fac-

tor of 2.5 (between only two spectral components now), we have effectively increased

85

the strength of the internal dipole field by at least the same factor. This apparently

compensates here for the smaller overlap between the wave functions.

To corroborate further our observation we should also notice how the hh−1 peak posi-

tion is now oscillating at double the BO frequency as expected from eq. 6.4. We have,

therefore, effectively doubled the frequency of the THz oscillator whose full control

was demonstrated in the previous paragraphs.

The presence of a signal from a transition that was not excited by both pulses has

been reported previously in MQW systems [34, 35] (not in SL systems to the best

of our knowledge). The dependence of the hh0 oscillators from the ones at the other

two frequencies has been attributed to a modification of the environment of the hh0

state [36] due to the presence of the surrounding, excitons. Interactions among car-

riers with very different energies have a dramatic effect on the coherent response of

semiconductors. For low intensities excitonic optical non linearities have been treated

using optical Bloch equations only (dilute system limit). However both excitation-

induced-dephasing (EID) and excitation-induced-shift (EIS) can play an important

role even for low excitation densities [37]. To describe in simple terms the linear and

nonlinear optical response of a homogeneously broadened system, one can write the

linear susceptibility χ as:

χ =∑

n

fn

(ω − ωn) + γn

(6.5)

where the index n runs over all the exciton states, ωn and γn are the exciton energies

and widths and fn are the oscillator strengths of each transition. It was shown that

any optical system will produce a nonlinear response upon the arrival of the probe if

anyone of the following parameters depends on the incident light of the pump [38, 39].

The easiest approach is to describe the semiconductor through simple optical Bloch

equations for atomic systems and include EID phenomenologically1 :

i~∂

∂t%01(ω) = V01(%00(ω)− %11(ω)) + ~ω%01(ω)− i~(γ + σ%11(ω))%01(ω). (6.6)

Here %00 and %11 are the populations density matrix elements, V is the interaction

1EIS was reported to play a lesser role at our excitation densities [39]

86

hamiltonian that contains interactions with the optical fields. The last term in eq. 6.6

−i~ · (σ%11(ω)) · %01(ω) describes the EID effect where the dephasing is proportional

to the population of the excitons created. It was shown analytically [39] that in

solving this problem, the EID term gives rise to an additional DFWM signal. This

is the explanation for the, somewhat counterintuitive, experimental result reported

here. The population grating propagating along the ±(~k2 − ~k1) direction modulates

in the same direction, through the EID, a grating in the dephasing of the interband

polarization of the hh0 transition. This grating gives rise to a rather strong signal by

diffracting the probe (that covers the hh0) in the 2~k2 − ~k1 direction.

This tells us that at the range (1010e/cm2wellpulse ) of excitation densities used in

our experiments the excitation-induced-dephasing is a rather strong effect.

LIST OF REFERENCES

87

LIST OF REFERENCES

[1] R.P. Leavitt et al. Phys. Rev. B, 42:11784, 1990.

[2] M. Dignam et al. Phys. Rev. B, 43:4097, 1991.

[3] Editor H.T. Grahn. Semiconductor superlattices. 1995.

[4] J. Bleuse et al. Phys. Rev. Lett., 60:220, 1988.

[5] Sang Gyu Park. The generation, shaping and measurement of coherent terahertzradiation. PhD. Thesis, Purdue University, 1998.

[6] H.G. Roskos et al. Jpn. J. Appl. Phys., 34:1370, 1994.

[7] T. Dekorsy et al. Phys. Rev. B, 51:RC17275, 1995.

[8] M. Hawton and M.M. Dignam. Phys. Rev. Lett., 91:267402, 2003.

[9] A.M. Weiner. Rev. of Sci. Instr., 71:1929, 2000.

[10] P. Drude. Ann. Phys. Lpz., 1:566, 1900.

[11] F. Bloch. Z. Phys., 52:555, 1928.

[12] C. Zener. Proc. R. Soc. A, 145:523, 1934.

[13] P.W.A. McIlroy. J. Appl. Phys., 59:3532, 1986.

[14] J. Barrau et al. Sol. State Commun., 74:147, 1990.

[15] G. Bastard. Wave mechanics applied to semiconductor heterostructures. HalstedPress, N.Y., 1988.

[16] P. Voisin et al. Phys. Rev. Lett., 61:1639, 1988.

[17] M. Dignam et al. Phys. Rev. B, 49:10502, 1994.

[18] J.A. Brum et al. Surf. Sci., 229:472, 1990.

[19] A.M. Bouchard et al. Phys. Rev. B, 52:5105, 1995.

[20] D.H. Auston. IEEE J. Q.E., 19:639, 1983.

[21] Q. Wu and X.-C. Zhang. Appl. Phys. Lett., 67:352, 1995.

[22] T. Dekorsy et al. Phys. Rev. B, 50:8106, 1994.

[23] K. Leo et al. Sol. State Commun., 84:943, 1992.

88

[24] G. Von Plessen et al. Phys. Rev. B, 45:9185, 1992.

[25] V.M. Axt et al. Phys. rev. b. 93:195, 1994.

[26] M.M. Dignam and M. Hawton. Phys. Rev. B, 67:035329, 2003.

[27] A.M. Weiner. Ultrafast optics: Course notes.

[28] R.W. Boyd. Nonlinear optics. 1992.

[29] J.L.A. chilla et al. Opt. lett. 16:39, 1991.

[30] V.G. Lyssenko et al. Phys. Rev. Lett., 79:301, 1997.

[31] A.M. Weiner and Y. Liu. Hot Carriers in Semiconductors, page 169, 1996.

[32] F. Loeser et al. Phys. Rev. Lett., 85:4763, 2000.

[33] H.J. Bakker et al. Phys. stat. sol. b. 206:443, 1998.

[34] S.T. Cundiff et al. Phys. Rev. Lett., 77:1107, 1996.

[35] V.G. Lyssenko et al. Appl. phys. a. 78:441, 2004.

[36] J.B. Stark et al. Optics of Semiconductor Nanostructures. Akademie Verlag,Berlin, 1993.

[37] J.M. Schacklette and S.T. Cundiff. 66:045309, 2002.

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[39] H. Wang et al. Phys. rev. lett. 49:R1551, 1994.

APPENDIX

89

A. Sample processing: a step-by-step guide

As reported in chapter 4.1 our samples were prepared to be used in transmission

geometry by Dirk Meinhold at the Technische Universitat of Dresden. An area of the

order of 2x2 mm2 was etched down to a thickness of less than 3 µm (see fig. A.1).

Figure A.1. Photograph of the superlattice sample in the cryostat. Thecentral, smaller window has a thickness of less than 2 µm and a diameterof about 2-3 mm. The wires for front and back contacts are also visible.the Contacts where obtained with thickened silver paint.

In the following we will describe the step by step recipe to prepare our samples

in close detail. It’ll be useful to review the the schematic of the sample growth (see

fig. 4.2).

• Place the sample face down on a clean glass slide.

• Clean the free side of the sample with methanol and acetone

90

• Tape the sample down using only the edges.

• Evaporate 200 nm of SnAu (Ohmic contact).

• Remove the sample from the glass.

• Heat the sample up:

– Place the sample in an oven at rough vacuum (1 mbar).

– Heat up the sample slowly (3-4 minutes) to reach a max temperature of

about 350C.

– Leave the sample at max temperature for about 1 minute.

– Switch off the heating resistor and leave the sample to cool down for about

1/2 hour.

• Clean the other side of the sample with methanol and acetone.

• Tape the sample on glass with the uncoated face up.

• Apply one drop of HF or opticleanTM for 1 minute to remove oxide.

• Paying attention to complete the following steps as quickly as possible to avoid

the formation of oxide in less than a minute or so, rinse the sample with DI water,

dry it with nitrogen, put it in the evaporator chamber and run the roughing

pump.

• Evaporate 1 nm Cu + 5 nm Au (Schottky contact).

• Contact the Schottky side with a 25 µm gold wire.

• Glue the sample to a sapphire base (shottky side down) using epoxy (warm up

the glue on the sapphire base to prevent bubbling between the surfaces).

• Wait 1 day for the glue to be dry.

• Cover the sample and part of the base with black wax.

• Etch the free part of the sample in a solution 100:15 of H2O2(30%) and NH3(25%)

(estimated speed: 10 µm/min) up to an estimated remaining thickness of about

40 µm.

91

• Change the solution to 100:3 H2O2(30%) NH3(25%).

• As soon as a little hole is visible, pass to a weaker solution 100:1 or even 100:0.5

(immerse for as short as 10s and then look at it).

• Stop when you have enough area for the experiment (2x2 mm or even less).

VITA

92

VITA

Riccardo Fanciulli was born in Viareggio, Italy on April 4th 1972.

Following a degree at the Scientific high school in Viareggio, Riccardo received a lau-

rea degree in physics at the university of Pisa (Italy) where he worked with Prof.

Fabio Beltram and Franco Bassani at the Scuola Normale Superiore. After receiving

his degree in Pisa, Riccardo worked as assistant researcher at the CHREA institute

in Sophia Antipolis (France) before starting the PhD program at Purdue Univer-

sity.