Quantum state tomography of slow and stored light

22

Trustees for action. The Program Director may All letters of inquiry and completed formal applications should be mailed in
n, an interview with the applicant, or a visit to hard copy to:
he full proposal, including staff summary and John Van Zytveld, Ph.D.

QUANTUM STATE
he Trustees for their consideration and decision. Senior Program Director
ptly when a decision has been reached. While M. J. Murdock Charitable Trust
n nearly every proposal received by the Trust, only P. O. Box 1618
wed can result in awards. When an application has Vancouver, WA 98668

TOMOGRAPHY OF SLOW
ried over for future consideration. Under normal For More Help
f a proposal that was declined is not encouraged. If your questions have not been answered by this document or you need some
perty of the Trust and will not be returned. It will additional information, please call us at 360.694.8415.
munication with the understanding, however, that

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

Ofﬁce Location:
Andrew M. C. Dawes, Noah T. Holte, Hunter A. Dassonville M. J. Murdock Executive Plaza
703 Broadway, Suite 710
Paciﬁc University Vancouver, Washington 98660

Reed College Physics Seminar 360.694.8415
Contact:
Phone WA:

February 27, 2013 Fax: 360.694.1819
Phone OR: 503.285.4086

Website: www.murdock-trust.org

Friday, March 29, 13

Quantum State of Light
All the “knowable” information about an optical signal.

frequency*

{
amplitude
For a plane wave: phase
propagation direction
polarization*

* we’ll ignore these for today

Preserving the Quantum State

Storing information in the quantum state is delicate

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

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


Quantum State of Slow Light

Text


and Stopped Light


Slow & Stopped Light
2.5

(a) Control field (b) Polariton
2

11 Ψ(z,t)
θ 1.5 150
0.8
π/2
0.6 100
0.5 Ω (t)
1
t
Ω (0)
0.4
0.5 50
0.2
a
00 0
0
0 25 50 75 100 125 150 0 50 100 150
0 50 100 150 0 40 80 120
t z
2.5
2.5
(c) Photon (d) Spin Coherence
σcb(z,t)
2
2
E(z,t)
1.5 150 1.5 150

1 100 1 100
t t
0.5 50 0.5 50

0
0
0 50 100 150
c 0
0
0 50 100 150
0 40 80 120 0 40 80 120
z z
Figure 3.A dark-state polariton can be stopped and re-accelerated by ramping the contro
ty as shown in (a).The broken line shows the mixing angle between photonic and spin stat
herent amplitudes of the polariton ␺, the electric field E of the photon, and the spin coheren
ted in (b-d).

medium. The width of the transparency transition that maps the signal
window, and thus vg , is a function of the coherent superposition of t
atomic density and the control beam in- states, |g1ʹ and |g 2ʹ . In so doing
tensity, and is therefore under experimen- energy of the signal photons i
tal control. In particular, vg decreases near- in the creation of new control
ly linearly with both quantities. tons. The resulting atomic spin

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?

In Out*
State
Rb vapor Detector

*Compare to recent full 3D theory

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

Problems!

-

PD
In Out*

Rb vapor PD

Local Oscillator

LO and signal aren’t mode-matched!


Problems!

-

PD
In Out*

Rb vapor PD

Local Oscillator

LO and signal aren’t mode-matched!

A new approach needs to keep mode information

Quantum Optics


Quantum Optics

i(k · x !t)
u(x, t) = u0 e Mode function (plane wave)

ˆ
E = u⇤ (x, t)ˆ† + u(x, t)ˆ
a a Electric ﬁeld operator

1 †
xp = p a + a
ˆ ˆ ˆ
2
Quadrature operators
i
yp = p a †
ˆ ˆ a
ˆ
2


Quantum Optics

i(k · x !t)
u(x, t) = u0 e Mode function (plane wave)

ˆ
E = u⇤ (x, t)ˆ† + u(x, t)ˆ
a a Electric ﬁeld operator

1 †
xp = p a + a of E -ﬁeld
ents
ˆ ˆ ˆ
2
compon Quadrature operators
os and sin
~c i
yp = p a †
ˆ ˆ a
ˆ
2


Optical Phase Space
Quadratures are the axes in phase space

Classical Optics Quantum Optics

Uncertainty
yp yp

xp xp


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 e nt
gl em
nt an
t e
esen
epr
’t r
C an


Measuring the Quantum State of Light
Balanced Homodyne Tomography

Balanced Array Detection

Smithey et al. PRL 70, 1244 (1993)

Unbalanced Array Detection

Beck PRL 84, 5748 (2000)

Beck et al. PRL 87, 253601 (2000)

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

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

q CCD Array

Signal

x

q ~ 5 mrad


Unbalanced Array Detection - Theory
Local Oscillator k

CCD Array
Signal k
kS

x

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


Fourier Transform of detected intensity:
0
e e⇤ e e⇤ e
S(k) = ELO ( k) ⌦ ELO (k) + ES ( k) ⌦ ES (k)
⇤
+ f (k kS ) + f ( k kS )
where 2nd order classical LO noise
eLO ( k) ⌦ ES (k)
f (k) = E ⇤ e


†
Each detector pixel nj =
ˆ aj aj
ˆ ˆ

X
measures all modes aj =
ˆ exp [ i2⇡jk/N ] ˆk
b
k

8 (vac)
>ˆk
<b N/2  k < M,
ˆk = ˆ(lo)
(Signal + LO + vacuum) b bk M  k  M,
> (s)
:ˆ
bk M < k < N/2.

Fourier transform CCD ˆ 1 X
Kp = p exp [i2⇡pj/N ] nj
ˆ
output to measure: N j
(p is the index of the measured mode)


Assume LO is strong X ⇣
M ⌘
ˆ
Kp = ⇤ˆ(s) ˆ†(vac)
(i.e. classical) ﬁeld: k bk+p + k bk p
k= M

Assume LO is in a single ˆ ⇤ˆ(s) ˆ†(vac) .
plane-wave mode: Kp = 0 bp + 0b p

ˆ(s) 1
bp = p (ˆp + iˆp )
x y
2

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

Interference of two plane waves at θ = 5 mrad

Signal = LO / 100

y (pixel #)

x (pixel #)

(visibility = 0.02, exaggerated by auto-range)

Calculate FFT


Calculate FFT
Pick one kx & histogram Re and Im values


Model histogram (in phase space)

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

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

A) B) 2
X1
400
0
-15 -12 -9 -6 -3 -2 300

-4
X2 200
-6
100
-8
C)
-10 0
-12

The Experiment


Unbalanced Array Detection - Experiment

• Two resonant laser ﬁelds (Control, Probe)

Requirements: • CCD needs to be low noise & high QE

• AOM beams need stable phase relationship


3 Tunable Laser Systems

Commercial optics
mount

780 nm laser diode

Diffraction grating

Piezo stack controls
grating angle

Δν ~ 6 GHz


High quantum-efﬁciency CCD installed

• 98% Quantum Efﬁciency

• Peak detection λ ~ 780 nm 780 nm

•
100%

1340 x 400 pixels 90%

80%

70%

• 2 - 3 per pixel per hr
e- Quantum Efficiency (%)
60%

50%

(LN cooled) 40%

30%

20%

10%

0%
250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050

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

Wavelength (nm)


Acousto-Optic Modulators
Reﬂected Sound Wave

ν0 + 80 MHz

Light in Light out

ν0 ν0

ν0 - 80 MHz

Sound Wave (80 MHz)


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

Stable phase
relationship Interference Pattern

50/50

AOM AOM


What’s Next?

1) Proof of principle experiment with plane waves
2) Implement slow light protocol in warm Rubidium vapor
3) Measure state of slow light in Rb vapor

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


Trustees for action. The Program Director may
n, an interview with the applicant, or a visit to
THANKS:
All letters of inquiry and completed formal applications should be mailed in
hard copy to:
22

he full proposal, including staff summary and John Van Zytveld, Ph.D.
he Trustees for their consideration and decision. Senior Program Director
ptly when a decision has been reached. While M. J. Murdock Charitable Trust
n nearly every proposal received by the Trust, only P. O. Box 1618

Noah T. Holte For More Help
wed can result in awards. When an application has
ried over for future consideration. Under normal
Vancouver, WA 98668

Hunter A. It will
perty of the Trust and will not be returned.
Dassonvillequestions have not beencall us at 360.694.8415. or you need some
f a proposal that was declined is not encouraged. If your
additional information, please
answered by this document

Marcus Kienlen
munication with the understanding, however, that
NSF
Simone Carpenter M. J. MurdockRCSA
Mailing Address:
Charitable Trust

Jennifer Novak PRISM (Paciﬁc U.)
PO Box 1618

Bryson Vivas Murdock Foundation
Ofﬁce Location:
M. J. Murdock Executive Plaza
703 Broadway, Suite 710

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


Quantum state tomography of slow and stored light

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Quantum state tomography of slow and stored light