Low-Light-Level Cross Phase Modulation with Cold Atoms

J.H. Shieh, W.-J. Wang and Ying-Cheng Chen

Institute of Atomic and Molecular Sciences, Academia Sinica, NTHU AMO Seminar, March 3, 2009

IAMS

Outline

• Introduction to Cross Phase Modulation (XPM)• Briefs on Electromagnetically induced transparency (EIT)• Introductions on EIT-based XPM schemes• Considerations on few-photon-level XPM• Our experimental progress towards the goal• Prospective

Cross Phase Modulation

• One of the holey grail in nonlinear optics is to observe the π radian mutual phase shift for two single photon pulses with small absorption loss.

atomsPhoton-photon has no coupling in free space, at least at low field strengthwhere QED effect is not significant.

Photon can couple to photon via the media (e.g. atoms).

Probe light

Control light

Cross phase modulation: Phase change of the probe pulse under the presence of control pulse and media. The Kerr effect.

Without control light

With control light

φ

Kerr effect: n=n0+n2I

XPM Application: Controlled-NOT gate for Quantum Computation

• CNOT and single qubit gates can be used to implement an arbitrary unitary operation on n qubits and therefore are universal for quantum computation.

• Single photon XPM can be used to implement the quantum phase gate and CNOT gate

For a good introductory article, see 陳易馨 & 余怡德 CPS Physics Bimonthly, 524, Oct. 2008

Truth table for CNOT gateTCTCTCTC

TCTCTCTC

0111;1101

1010;0000

Signal

Probe

PBS PBS

Atoms

Control qubit

Target qubit

1

0

10 or

XPM in Generation of Entangled Photon Pairs

• Single photon π phase shift XPM can be implemented to generate entangled photon pairs which are important in quantum teleportation.

• For a close look at quantum teleportation, see e.g. 2008 AMO summer school presentation file in NCTS website given by 陳岳男 .

Signal

Probe

PBS PBS

Atoms

1

0

450 linear polarization2/)10(

CC

2/)0110(TCTC

Entangled photon pairs

XPM in Quantum Nondemolition Measurement

• QND measurement of the photon number by dispersive atom field coupling. See. e.g. Scully, Quantum Optics, Sec.19-3.

Meter beam

Signal beam

Electromagnetically Induced Transparency (EIT),the Phenomena

Atomic sample

Probe laser

Coupling laser

Probe laser

Attenuated

Pass through again

|1>

|3>

p 31

31

|1>

|3>

p32

2>

c32

Attenuated

Optical density=2

Transparent!

0,5.0,0 3 cc

For a review, see Rev. Mod. Phys. 77, 633,2005

EIT, the Physical Explanation• Destructive interference between excitation pathways

• Destructive interference between two dressed states

• Dark state or resonance

probe coupling

＝ ＋ ＋ ＋…

Path i Path iiiPath ii

2totA

|1>

|2>

|3>

coupling

|2>

|3>

|1>

c

probe

probeNNN

NNN

,3sin1,2cos)(2

,3cos1,2sin)(1

c

c

2tan

01)(21)(1 pp EdNEdN

coupling

|3>

probeDiagonalize theHA+HAC+HAP

c

p

dark

tan

2sin1cos A limiting case of coherentpopulation trapping effectwhere

pc

EIT and the Slow Light

• Group velocity of the probe pulse

• The steep dispersion in addition with no loss has important effect on the probe propagation in EIT medium : the lossless slow light.

00

0 |)/(|

1|

)(/)(

pp

p

pp

p

pp

pg

pppp

ddnn

c

ddkdk

dv

nckk

Vg<17m/s, Hau et.al. Nature397,594,1999

More on Slow Light

• If γ/Γ3<<1 & δc=0,

• Higher order dispersion and finite EIT bandwidth still cause slightly pulse spreading.

• At δp=0, one obtains (N: atom density)

• Group delay for a sample with length L

• The slow light has practical applications as optical delay line.

1

)(2

4

3)Re( 2

2

31

2

3

n

ON

pc

p

231

2

2

3,

1 cg

gg cNn

n

cv

231

231

2

)2

3()

11(

cc

g

gd ODLN

c

Ln

cvL

3122 cc considering the decoherence

Line Narrowing Effect with Large OD Gas

• Has been observed in warm gas, PRL 79,2959,1997.

• Intensity transmission

• For medium and large OD,

• Delay Bandwidth product ~

313

2

3

))()(2(

)(

8

3

))Im(exp(

pp

cp

ii

iN

kLT

3

2

OD

c

EIT

OD

Slow Light : Dark-State Polariton

2

2

21

cos

0),(]cos[

;tan

),(sin),(cos),(

cv

tzz

ct

kkkn

etzNtzEtz

g

pcg

kzip

coupling

1>

|2>

|3>

coupling

1>

|2>

|3>

probe

coupling

1>

|2>

|3>

probe

Lukin&Fleischhauer, PRL 84,5094,2000

Light component

Matter component: atomic spin coherence

EIT and the Photon Storage• By adiabatically turn off the coupling light, the probe pulse can completely

transfer to atomic spin coherence and stored in the medium and can be retrieved back to light pulse later on when adiabatically turn on the coupling.

• This effect can be used as a quantum memory for photons.

• The photon storage and retrieved process has been proved to be a phase coherent process by Yu’s team.

coupling

probe

Hau et.al. Nature, 409,490,2001 Y.F. Chen et.al. PRA 72, 033812, 2005

Basic EIT-based XPM scheme: N-type system

3

21

4

couplingprobe signal

Without signal

With signal beam

• The scheme was proposed by Schmidt & Imamoğlu (Opt. Lett. 21,1936,1996).• The signal beam cause ac Stark shift on state 2 and introduce a cross phase modulation on probe. • The EIT guarantee low probe loss with the transparency window if choosing larger enough coupling field. • The scheme was firstly demonstrated by Yi-Fu Zhu’s team (PRL91,093601,2003).

Dual Slow Light Scheme for Weak Field XPM

• Use a second species atom and a second coupling pulse to allow signal pulse becoming a slow light pulse.

• Both signal and probe pulses are tuned to the same slow group velocity. This allow long interaction time between these two weak pulses to gain significant mutual phase shift.

1. Long delay time -> high OD

2. Avoid loss due to absorption by decreasing the decoherence rate

Both signal and probe are slow-light pulses

Lukin & ImamoğluPRL 84,1419,2000

Tightly focusing Long interaction time

Single Species Dual Slow Light Weak Field XPM

• The symmetric arrangement guarantee the matched group velocity for the two weak probe pulses for long interaction length.

• Each probe pulse is the signal pulse of the N-type system for the other EIT system.

• The Zeeman shift >> Natural linewidth for dispersive coupling in the N-type system.

• Two coupling fields are needed.• The orthogonal coupling and probe requires very cold

sample to minimize the residual Doppler broadening. • The two probe pulses can be used to implement the q

uantum phase gate with the qubit as polarization states.

PRA 65,033833,2002

),(, PSiq iii

signal

probe

PC

i

PC

PC

i

PC

PC

i

PC

PC

i

PC

Pnlin

Cnlin

Plin

Clin

PClin

Plin

C

PC

e

e

e

e

)(

)(

)(

)(

0

0

00

N-Tripod Scheme XPM

• Both signal and probe form EIT with single coupling beam and can be tuned with coupling intensity and detuning and population difference to match the group velocity.

• Both signal and probe can be on exact EIT resonance but still experience XPM.

• Signal beam cause a cross phase modulation on probe but also a self phase modulation.

• Utilize Zeeman shift such that signal beam is mainly dispersive coupling in the N-type system

• Similar to previous scheme, one can implement the quantum phase gate with polarization state as qubits..

1

2 3

45

c

ps

Cs

m=06S1/2,F=4

6S1/2,F=3

6P3/2,F=3m=1 m=3

m=0

m=2

PRL 97,063901,2006 & Opt. Comm. 681,2040,2008

N-TripodProbe Signal

Im(χ)

Re(χ)

Nonlinear susceptibility

Linear susceptibility

The Light-Storage XPM Scheme

• Proposed and demonstrated by Yu’s team, PRL 96,043603,2006. • Convert and store probe light into atomic spin coherence. Signal be

am is applied during the storage time to interact with atom. • A phase shift of 440 for probe pulse was obtained with a 2-μs signal

pulse of Rabi frequency 0.32Γ with 65% transmission.• The energy per unit area of signal beam, Ω2τ,affect the phase shift a

t a fixed detuning. • To increase the phase shift, one has to tightly focus the probe and si

gnal beam to increase the interaction strength and also to increase the interaction time of signal with atoms.

coupling

probe

signal

222

222

4

4

τ

Phase shift

Probe attenuation e-α

Few-Photon-Level XPM

• Large Kerr Nonlinearity

• Low loss

• Strong focusing to increase the atom-laser interaction strength

• Long atom-laser interaction time

• Possible schemes

1. Using the high-finesse cavity

2. Couple atoms and Light into the hollow core fiber

3. Using the EIT-based stationary light scheme together with transverse waveguiding effect

• All are challenging !

cold atom

Signal beam

Coupling&probe

High-finesse cavity

Experimental Progress : Dense atomic medium

• In all EIT-based applications, the large optical density together with small decoherence rate are crucial requirements.

• We have obtained optical density > 100 using 2-dimensional MOT.

• To obtain atomic sample with even larger OD, the optical dipole trapping is required and is underway.

Optics Express. Chen & Yu 16,3754(2008)

7cm

Absorption Spectrum

Optical density=105

trapping beam

trapping

Coils&cell

trapping

Atom cloud probe

Small Decoherence Rate

• To obtain small decoherence rate, good mutual coherence between coupling and probe lasers and small inhomogeneity in magnetic field are crucial.

• We obtained phase locking via the modulation with vertical-cavity surface-emitting laser (VCSEL) and injection locking.

• Obtained phase locking between coupling and VCSEL with 13dBm modulation power at ~9 GHz with sideband power ratio ~ 15%.

• Checked with beatnote by spectrum analyzer showing that the linewidth between coupling and probe to be < 10Hz, the instrument resolution limit.

• Has been applied in the EIT experiments.

Coupling ECDL

VCSEL

ProbeDL

λ/2

PBS

frequency

coupling

VCSEL

Probe

~9GHz

Bias-TeeIdc

RF

Compensation of Stray Magnetic Field• Three pairs of compensation coils with relative l

arge size (~50 cmφ) are used with 6 independent current channels applied to null the stray field and gradients.

• Stary field is minimized to < 30mG level by the linewidth of EIT spectrum.

• Non-metal support for MOT magnetic coils is used to avoid the induced eddy current when the MOT coils are turned off.

• Current through MOT coils is rapidly turned off (~30 μs) by field-effect-transitors together with large damping resistor in parallel with coils.

• Two layers of mu-metal are used to shield the ion pump nearby the MOT.

• Even with all of these, we observed that the MOT inhomogeneous magnetic field decay completely after 2 ms possibly due to the induction of nearby metal stuffs.

• We are using Faraday rotation effect to allow a fast and quantitative diagnosis of stray field situations.

~3GHz away from resonance

PRA, 59,4836,1999

BL

Typical EIT Spectrum

• Obtained EIT with ~50% transmission at 200kHz FWHM for OD~70 with coupling intensity ~5 mW/cm2.

Line Narrowing Effect with Large OD Gas

• Has been observed in warm gas, PRL 79,2959,1997.

• Intensity transmission

• For medium and large OD,

Increasing theOD of atom cloud

313

2

3

))()(2(

)(

8

3

))Im(exp(

pp

cp

ii

iN

kLT

3

2

OD

c

EIT

The Slow Light

• The typical delay time is 5-10 μs. • Delay time > 10μs is observed at the expense of higher loss.

The Light Storage

• Observed the light storage signal. However, the storage time is still short. Better compensation on the stray magnetic field is underway to decrease the decoherence rate further.

Prespectives

• Short term:

1. Study the oscillatory effect of transverse magnetic field on slow light.

2. Realize the N-Tripod XPM in the two-dimensional MOT

• Middle term:

1. Obtain even higher OD gas in the optical dipole trap.

2. Study the transverse waveguiding effect and XPM enhancement

• Long term

1. Use the optical cavity with the light-storage XPM scheme to reach the few-photon-level XPM.

2. …

Oscillatory Behavior on Slow Light with Transverse Magnetic Field

• Why the oscillating behavior in transverse magnetic field?

• What does the period corresponding to?

• Does this behavior exist in Cs?• If yes, what is the value of the

period?• Preliminary experiment on Cs

do suggest similar transverse magnetic field effect on delay time. Better calibration on compensation coils using Faraday rotation is underway.

87Rb, Yu’s team

N-Tripod XPM Scheme

• Optical pumping and/or microwave transitions to prepare the population.

• Magnetic field ~ 30 Gauss to allow dispersive coupling for N-type system (Δ~4Γ)

• Beatnote interferometer proposed by Yu’s team will be utilized to see the XPM phase shift.

• The experiment should be quite straightforward! Cs

m=06S1/2,F=4

6S1/2,F=3

6P3/2,F=3m=1 m=3

m=0

m=2

PRA 72,033812,2005