dephasing and noise in weakly-coupled bose-einstein condensates amichay vardi

Dephasing and noise in weakly-coupled Bose-Einstein condensates

Amichay Vardi

Y. Khodorkovsky, G. Kurizki, and AV

PRL 100, 220403 (2008), e-print arXiv:0805.1832

Erez Boukobza, Maya Chuchem, Doron Cohen, and AV

PRL, in press (2009), e-print arXiv:0812.1204

I. Tikhonenkov and AV

PRL, submitted, e-print arXiv:0904.2121

Matter-wave interference

QuickTime™ and a decompressor

are needed to see this picture.

Andrews et. al., Science 275, 637 (1997)

Fringe visibility is proportional to SP coherence

Freely expanding condensates

Coherent preparation

Equal populations:

Well defined relative-phase

Population difference:

Fock preparation

Population difference: N1 - N2

Undefined relative phase between the two BECs

Does the Fock preparation give interference fringes ?

Fringes in the Fock preparationFock states are superpositions of coherent states:

Any single-shot interferometric measurement constitutes a single phase-projection .

Each shot gives fringes with random phase:

While the multi-shot density averages out to:

Coherent splitting of a BECT. Schumm et al., Nature Physics 1, 57 (2005)

Coherent splitting of a BECThe mere existence of interference patterns does not indicateInitial SP coherence - need to verify reproducible fringe position.

T. Schumm et al., Nature Physics 1, 57 (2005)

Outline

• Assume a coherent preparation.• Interactions cause ‘Phase-Diffusion’. How long will

SP coherence survive ?

1. PD between weakly-coupled BECs - ‘Phase Locking’.2. Control of PD by noise.3. Sub shot-noise interferometry and PD between

atoms and molecules.

Model: a bosonic Josephson junction

Total number conservation

Hence coherence is characterized by the length of the Bloch vector restricted to be inside the sphere .

Fringe Visibility: Lx

Coherent = classical states

are needed to see this picture.QuickTime™ and a decompressor

SU(2) coherent states:QuickTime™ and a decompressor

Gross-Pitaevskii classical (mean-field) energy functional with :

Interaction regimes

Rabi regimeWeak interaction, linear

(perturbed) Lx eigenstates

QuickTime™ and a decompressorare needed to see this picture. QuickTime™ and a decompressor

QuickTime™ and a decompressorare needed to see this picture.Josephson regimeIntermediate strong interaction

Nonlinear ‘islands’ in a linear ‘sea’

Separated by ‘figure-8’ separatrix

QuickTime™ and a decompressorare needed to see this picture.Fock regimeStrong interaction, nonlinear

‘sea’ area less than the Planck cell

(perturbed) Lz eigenstates

Classical dynamicsu>2 ‘self trapping’

A. Smerzi et al., PRL 79, 4750 (1997).M. Albeiz et al., PRL 95, 010402 (2005).

Phase ‘diffusion’ in the Fock regime• Coherent state preparation: binomial superposition of Fock states

• Evolve with J = 0 , U ≠ 0, .

• For and

First phase diffusion experiment

M. Greiner, O. Mandel, T. Haensch., and I. Bloch, Nature 419, 51 (2002).

Slow phase-diffusion as a probe of number-squeezing

G.-B. Jo et Al., PRL 98, 030407 (2007)

‘Phase locking’ S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer,

Nature 449, 324 (2007)

u ≈ 5 u ≈ ∞u ≈ 300u ≈ 100

N ~ 1000QuickTime™ and a

decompressorare needed to see this picture.

Phase-diffusion between weakly coupled condensates

N=1000

E. Boukobza, M. Chuchem, D. Cohen, and AV, PRL, in press (2009).

Phase locking in the Josephson regime is phase-sensitive :

Semiclassical quantizationQuickTime™ and a

QuickTime™ and a decompressorare needed to see this picture.Planck cell:QuickTime™ and a

Low energy ‘sea’ levels

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Separatrix levels

High energy degenerate ‘island’ Um2 levels

‘sea’

‘islands’

separatrix

Semiclassical interpretation

How good is semiclassics ?QuickTime™ and a

n=1000u=1000

Correlation time of Phase-diffusion

Linearization aboutQuickTime™ and a decompressorare needed to see this picture.

Quantum Zeno control of phase-diffusion

Y. Khodorkovsky, G. Kurizki, and AV, PRL 100, 220403 (2008)

• Long correlation times: tc for phase diffusion in BEC is of order ms

• Slow down phase diffusion by frequent measurements / noise.

• Since phase diffusion is along the Lx axis, noise has to project onto onto Lx (measure odd-even population imbalance - quasimomentum).

• Hence site indiscriminate noise such as stochastic modulation of the barrier height.

QZE reminder

For t«tc=, SP coherence decays quadratically

Frequent projective measurements of Lx (g1,2(1)) at intervals:

SP dephasing slows down as t 0

L. A. Khalfin, JETP Lett. 8, 65 (1968).B. Misra and E. C. G. Sudarshan, J. Math. Phys. Sci. 18, 756 (1977).

QZE limit: Uncorrelated, Markovian noise:

Quantum kinetic master equation:

Linearization of the master equation gives the QZE result:

QZE control of phase-diffusion

Bose enhancement of QZE

As opposed to log(N) (or N1/2) decoherence-free diffusion time:

Extended phase-diffusion time, depends linearly on N:

Preparation with noise numerical (lines) vs. analytic (symbols)

Josephson:

Comparison with local noise

Initial coherent state Noiseless dynamicsMacroscopic ‘cat’ state

Site-localized noise z=0.05J Site-indiscriminate noise x=J

N = 30u = 2t = 2.4 J

Atom-molecule dephasing in a sub-shot-noise SU(1,1) matter-wave interferometer

Optical coupling

Feshbach resonance

Undepleted pump:

SU(1,1)QuickTime™ and a

are needed to see this picture.Casimir operator:

are needed to see this picture.Fock states:

k - Bargmann index m = 0,1,2,…

Two-atom coherent states QuickTime™ and a

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Atom-molecule interferometer

(b),(c)

(c) (b)

(a)(d)

Heisenberg-limited precision

Introduce interactions - dephasing

For sinh >> 1

sin(ut) ~ utsinh~ 2n

Fringe visibilityQuickTime™ and a

(b),(c)

Time domain Frequency domain

Co-Authors

Gershon KurizkiWeizmann Institute

Doron CohenBGU

Maya ChuchemMSc, BGU

Erez BoukobzaPostdoc, BGU

Amichay VardiBGU

Igor TikhonenkovResearch Fellow, BGU

Yuri KhodorkovskyMSc BGU WIS

Conclusions• Phase diffusion between weakly-coupled Bose condensates

(Josephson regime) is phase-sensitive.

• It has a long (~1-10ms) correlation time.

• Thus, it may be slowed down by frequent (projective) or continuous measurement of the primitive quasi-momentum.

• The obtained QZE is Bose stimulated due to the transition from log(N)- to N-dependent characteristic diffusion times.

• In atom-molecule systems, the inherent phase-squeezing may be use to do interferometry below the standard quantum limit.

• But, since it comes at the price of number-stretching, atom-molecule phase diffusion time is ~1/N.

dephasing and noise in weakly-coupled bose-einstein condensates amichay vardi

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