quantum state tomography of slow and stored light

for submission to the Board of Trustees for action. The Program Director may request additional information, an interview with the applicant, or a visit to the applicant’s organization. The full proposal, including staff summary and analysis, is made available to the Trustees for their consideration and decision.

The applicant is notified promptly when a decision has been reached. While some level of merit is evident in nearly every proposal received by the Trust, only a fraction of the requests reviewed can result in awards. When an application has been declined, it will not be carried over for future consideration. Under normal circumstances, re-submission of a proposal that was declined is not encouraged.

Each proposal becomes the property of the Trust and will not be returned. It will be treated as a privileged communication with the understanding, however, that it may be peer reviewed.

All letters of inquiry and completed formal applications should be mailed in hard copy to:

John Van Zytveld, Ph.D. Senior Program Director M. J. Murdock Charitable Trust P. O. Box 1618 Vancouver, WA 98668

For More Help If your questions have not been answered by this document or you need some additional information, please call us at 360.694.8415.

Mailing Address: M. J. Murdock Charitable Trust PO Box 1618 Vancouver, Washington 98668

Office Location: M. J. Murdock Executive Plaza 703 Broadway, Suite 710 Vancouver, Washington 98660

Contact:Phone WA: 360.694.8415 Phone OR: 503.285.4086 Fax: 360.694.1819 Website: www.murdock-trust.org

22

QUANTUM STATE TOMOGRAPHY OF SLOW

AND STORED LIGHTAndrew M. C. Dawes, Noah T. Holte, Hunter A. Dassonville

Pacific UniversityReed College Physics Seminar

February 27, 2013

Friday, March 29, 13

Quantum State of Light

All the “knowable” information about an optical signal.

For a plane wave:

frequency*amplitudephasepropagation directionpolarization*

{

* we’ll ignore these for todayFriday, March 29, 13

Preserving the Quantum State

Fidelity: how well does a stored light system preserve the quantum state?

Efficiency: how well does a stored light system preserve the signal amplitude?

Storing information in the quantum state is delicate


Quantum State of Slow Light

Text


and Stopped Light


Slow & Stopped Light

medium. The width of the transparencywindow, and thus vg, is a function of theatomic density and the control beam in-tensity, and is therefore under experimen-tal control. In particular, vg decreases near-ly linearly with both quantities.

A signal pulse of light entering anatomic EIT medium undergoes the fol-lowing dynamics (see Fig. 2). The atomsare pumped into one of their groundstates and coupled by the control beam.Upon entering the atomic storage cell, thefront edge of the signal pulse abruptlyslows down because of the medium’s lowgroup velocity. Since the front edge isslowed first and the rear edge last, the sig-nal pulse ends up considerably com-pressed by a factor of c/vg once it is fully inthe EIT medium. However, the peak am-plitude of the signal pulse is unchanged bythis spatial compression. Consequently,the signal pulse contains much less energywhen in the EIT medium. Since the atom-ic gas gains only negligible amounts of en-ergy (the excited state is never populated!),the excess energy must be carried awaysomehow: this job is performed by thecontrol beam. A fraction of the photonsthat make up the signal pulse are convert-ed into a “spin wave” via a two-photon

transition that maps the signal pulse into acoherent superposition of the atomicstates, |g1! and |g2! . In so doing, the excessenergy of the signal photons is expendedin the creation of new control beam pho-tons. The resulting atomic spin wave trav-els together with the signal pulse at the re-duced vg in a coupled excitation known asa “dark-state polariton.” Upon reachingthe far end of the cell, the signal pulse isfully reconstituted via the reverse process:i.e., pulse expansion and acceleration of vgback to c. The only difference between thisreleased signal pulse and a pulse which haspropagated only through vacuum is thatthe EIT medium has induced a delay in thereleased pulse of ! = (1/vg-1/c)L , where Lis the cell length. This pulse delay processis often referred to as “slow light.”

The subject of slow light attracted greatattention with the remarkable experimentconducted by Lene Hau and co-workers in1999, in which an ultracold gas of Naatoms was used to slow light pulses to agroup velocity of 17 m per second. Fur-thermore, this work made use of a largeoptical density to compress the pulse en-tirely into the medium. Soon thereafter,the groups of Marlan Scully and DmitryBudker demonstrated very slow group ve-

locities using warm atomic vapor. All theseexperiments built upon important earlierwork on EIT,2 which had observed pulsedelays considerably exceeding the pulseduration.1

Even in an ideal EIT system, however,there is an important limitation to theslow light technique. Only signal frequen-cies that lie within the narrow transparen-cy window can propagate unabsorbedthrough the medium. Since the windowwidth, as well as the group velocity, areproportional to the control intensity, themaximum pulse delay is inversely propor-tional to the pulse length to be slowed.(The proportionality constant depends onthe optical density of the EIT medium.) Itis therefore not possible to reduce thegroup velocity to zero with the techniquedescribed above. One more twist to theEIT bag of tricks is needed.

“Stopping light”To bring a light pulse to a complete stop(i.e., vg=0), “dynamic EIT” is employed.2,3

Once a light pulse is compressed into anEIT medium and propagating as a dark-state polariton, its properties can be mod-ified by simply changing the intensity ofthe control beam: if the control beam in-tensity is decreased, then the group veloci-ty is further slowed. This also implies thatthe contribution of photons to the polari-ton state is reduced. In particular, if thecontrol beam is turned off after the signalpulse has been compressed into the EITmedium, two things happen: the polaritonbecomes purely atomic, and its group ve-locity is reduced to zero. At this point, in-formation originally carried by the pho-tons (pulse shape, amplitude, length, po-larization, etc.) is fully mapped onto thelong-lived ground or spin states of theatoms, |g1! and |g2! . As long as this “light-storage” or “trapping” process is sufficient-ly smooth, the frequency spectrum of thepolariton will narrow continuously as thepulse slows, so that it always remains with-in the transparency window. The storedlight pulse can be easily retrieved by sim-ply turning the control beam back on andre-accelerating the stopped dark-state po-lariton. For this reason, there is, in princi-ple, no loss associated with the trappingprocedure.

The light-storage process is illustratedin Fig. 3, which shows the evolution of thesignal light pulse, spin coherence, and po-lariton, when the control beam is turned

THE STORY BEHIND “STOPPED LIGHT”

52 Optics & Photonics News ! May 2002

Figure 3.A dark-state polariton can be stopped and re-accelerated by ramping the control field intensi-ty as shown in (a).The broken line shows the mixing angle between photonic and spin states ".The co-herent amplitudes of the polariton #, the electric field E of the photon, and the spin coherence S, are plot-ted in (b-d).

0 50 100 1500

0.5

1

1.5

2

2.5

d

! (z,t)cb

0 40 80 120z

100

150

50

0

t

0 50 100 1500

0.5

1

1.5

2

2.5

z

b

(z,t)"

0 40 80 120

150

100

50

0

t

0 50 100 1500

0.5

1

1.5

2

2.5

c

E(z,t)

0 40 80 120z

100

50

0

150

t

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

t0

0.5

1

#/2$

0 50 100 150

a(0)%

%(t)

Control field Polariton

Spin CoherencePhoton

(a) (b)

(c) (d)


Storing Information


Temporal Optimization (Novikova et al.)

Novikova et al. “Optimal control of light pulse storage and retrieval,” PRL 98, 243602 (2007).


Next step: Spatial mode optimization?

Zeuthen et al. “Three-dimensional theory of quantum memories based on lambda-type atomic ensembles,” PRA 84, 043838 (2011).

*Compare to recent full 3D theory

Rb vapor

In Out*

StateDetector


Problems!

Rb vapor

In Out*

PD

PD

Local Oscillator

-

LO and signal aren’t mode-matched!


Problems!

Rb vapor

In Out*

PD

PD

Local Oscillator

-

LO and signal aren’t mode-matched!

A new approach needs to keep mode informationFriday, March 29, 13

Quantum Optics


Quantum Optics

u(x, t) = u0ei(k ·x�!t)

E = u⇤(x, t)a† + u(x, t)a

Mode function (plane wave)

Electric field operator

xp =1p2

�a

† + a

�

yp =ip2

�a† � a

�Quadrature operators


Quantum Optics

u(x, t) = u0ei(k ·x�!t)

E = u⇤(x, t)a† + u(x, t)a

Mode function (plane wave)

Electric field operator

xp =1p2

�a

† + a

�

yp =ip2

�a† � a

�Quadrature operators

~ cos and sin components of E-field


Optical Phase Space

Classical Optics Quantum Optics

Uncertainty

xp

yp

xp

yp

Quadratures are the axes in phase space


Wigner and Q functions

• Quasi-probability distributions• Representation of the quantum state• 3D look at phase space

(Wigner ✽ Gaussian)


Example of Wigner and Q-functions

Schrödinger cat state | i / |↵i+ |� ↵i

Wigner Q-function


Example of Wigner and Q-functions

Schrödinger cat state | i / |↵i+ |� ↵i

Wigner Q-function

Can’t repres

ent entan

glement


Measuring the Quantum State of Light

Balanced Homodyne Tomography

Balanced Array Detection

Unbalanced Array Detection

Smithey et al. PRL 70, 1244 (1993)

Beck PRL 84, 5748 (2000)

Beck et al. PRL 87, 253601 (2000)

Beck PRL 84, 5748 (2000)Friday, March 29, 13

Quantum State Tomography

Constructing the quantum state of light from measurements of the quadrature components


A New Approach


Unbalanced Array Detection of Spatial Modes

Local Oscillator

Signal

CCD Array

x

q

q ~ 5 mrad


Unbalanced Array Detection - Theory

Local Oscillator

Signal CCD Array

S(x) = |ELO(x) + ES(x) exp(ikS ·x)|2

= |ELO(x)|2 + |ES(x)|2 + [E⇤LO(x)ES(x) exp(ikS ·x) + c.c.]

x

kS

k

k


Unbalanced Array Detection - TheoryFourier Transform of detected intensity:

f(k) = eE⇤LO(�k)⌦ eES(k)

where

eS(k) = eE⇤LO(�k)⌦ eELO(k) + eE⇤

S(�k)⌦ eES(k)

+ f(k� kS) + f⇤(�k� kS)

2nd order classical LO noise

0



nj = a†j aj

aj =X

k

exp [�i2⇡jk/N ]

ˆbk

bk

=

8><

>:

b(vac)k

�N/2 k < �M,

b(lo)k

�M k M,

b(s)k

M < k < N/2.

ˆKp =

1pN

X

j

exp [i2⇡pj/N ] nj

Each detector pixel

measures all modes

(Signal + LO + vacuum)

Fourier transform CCD output to measure:

(p is the index of the measured mode)Friday, March 29, 13


Kp =MX

k=�M

⇣�⇤k b

(s)k+p + �k b

†(vac)k�p

⌘

Kp = �⇤0 b

(s)p + �0b

†(vac)�p .

b

(s)p =

1p2(xp + iyp)

Assume LO is strong (i.e. classical) field:

Assume LO is in a single plane-wave mode:

Each entry in the FFT output is Kp (for mode p)A measurement of the signal quadratures + a vacuum component


A Toy Model


Model intensity

(visibility = 0.02, exaggerated by auto-range)

x (pixel #)

y (pixel #)

Interference of two plane waves at θ = 5 mrad

Signal = LO / 100


Calculate FFT


Calculate FFTPick one kx & histogram Re and Im values


Model histogram (in phase space)

-15 -12 -9 -6 -30

-12

-10

-8

-6

-4

-2

2A) B)

C)0

400

300

200

100

X1

X2

A) Signal = LO / 100 B) Signal = LO / 1000

C) Signal = LO / 100 with 1 rad phase shift


The Experiment


Unbalanced Array Detection - Experiment

• Two resonant laser fields (Control, Probe)

• CCD needs to be low noise & high QE

• AOM beams need stable phase relationship

Requirements:


3 Tunable Laser Systems

Commercial optics mount

780 nm laser diode

Diffraction grating

Piezo stack controls grating angle

Δν ~ 6 GHz


• 98% Quantum Efficiency

• Peak detection λ ~ 780 nm

• 1340 x 400 pixels

• 2 - 3 e- per pixel per hr (LN cooled)

High quantum-efficiency CCD installed

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050

Wavelength (nm)

Qua

ntum

E!

cien

cy (%

)

BR_eXcelon B_eXcelon BR B F (for non-eXcelon cameras only)With optional UV coating

780 nm


Acousto-Optic Modulators

ν0 + 80 MHz

ν0 - 80 MHz

Sound Wave (80 MHz)

Reflected Sound Wave

Light in

ν0

Light out

ν0


AOMs installed

• 1 x 2 cm

• Optical post mount

• Deflection ~ 10 mrad


AOM Drivers


AOM Driver (finished)


80 MHz (1.4 kHz BW)


AOM Beat Signal

AOM AOM

50/50

Interference Pattern

Stable phase relationship


What’s Next?

1) Proof of principle experiment with plane waves2) Implement slow light protocol in warm Rubidium vapor3) Measure state of slow light in Rb vapor

Repeat with stopped light in warm Rb, then switch to cold (trapped) Rb vapor.


for submission to the Board of Trustees for action. The Program Director may request additional information, an interview with the applicant, or a visit to the applicant’s organization. The full proposal, including staff summary and analysis, is made available to the Trustees for their consideration and decision.

The applicant is notified promptly when a decision has been reached. While some level of merit is evident in nearly every proposal received by the Trust, only a fraction of the requests reviewed can result in awards. When an application has been declined, it will not be carried over for future consideration. Under normal circumstances, re-submission of a proposal that was declined is not encouraged.

Each proposal becomes the property of the Trust and will not be returned. It will be treated as a privileged communication with the understanding, however, that it may be peer reviewed.

All letters of inquiry and completed formal applications should be mailed in hard copy to:

John Van Zytveld, Ph.D. Senior Program Director M. J. Murdock Charitable Trust P. O. Box 1618 Vancouver, WA 98668

For More Help If your questions have not been answered by this document or you need some additional information, please call us at 360.694.8415.

Mailing Address: M. J. Murdock Charitable Trust PO Box 1618 Vancouver, Washington 98668

Office Location: M. J. Murdock Executive Plaza 703 Broadway, Suite 710 Vancouver, Washington 98660

Contact:Phone WA: 360.694.8415 Phone OR: 503.285.4086 Fax: 360.694.1819 Website: www.murdock-trust.org

22THANKS:

Noah T. HolteHunter A. Dassonville

Marcus KienlenSimone Carpenter

Jennifer NovakBryson Vivas

NSFRCSA

PRISM (Pacific U.)Murdock Foundation


quantum state tomography of slow and stored light

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