Dephasing and noise in weakly- coupled Bose-Einstein condensates Amichay Vardi Y. Khodorkovsky, G. Kurizki, and AV PRL 100, (2008), e-print arXiv: Erez Boukobza, Maya Chuchem, Doron Cohen, and AV PRL, in press (2009), e-print arXiv: I. Tikhonenkov and AV PRL, submitted, e-print arXiv:

Matter-wave interference Andrews et. al., Science 275, 637 (1997) Fringe visibility is proportional to SP coherence = N 1 + N 2

Freely expanding condensates d x z y

Coherent preparation Equal populations: Well defined relative-phase Population difference:

Fock preparation Population difference: N 1 - N 2 Undefined relative phase between the two BECs Does the Fock preparation give interference fringes ?

Fringes in the Fock preparation Fock 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 BEC T. Schumm et al., Nature Physics 1, 57 (2005)

Coherent splitting of a BEC The mere existence of interference patterns does not indicate Initial 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: LxLx LyLy LzLz 

Coherent = classical states SU(2) coherent states: Gross-Pitaevskii classical (mean-field) energy functional with  :

Interaction regimes Rabi regime Weak interaction, linear (perturbed) L x eigenstates Josephson regime Intermediate strong interaction Nonlinear ‘islands’ in a linear ‘sea’ Separated by ‘figure-8’ separatrix Fock regime Strong interaction, nonlinear ‘sea’ area less than the Planck cell (perturbed) L z eigenstates

Classical dynamics u>2 ‘ self trapping’ A. Smerzi et al., PRL 79, 4750 (1997). M. Albeiz et al., PRL 95, (2005).

Phase ‘diffusion’ in the Fock regime Coherent state preparation: binomial superposition of Fock states Evolve with J = 0, U ≠ 0, . Ut For  and

t d / t rev First phase diffusion experiment M. Greiner, O. Mandel, T. Haensch., and I. Bloch, Nature 419, 51 (2002). VBVB VAVA

Slow phase-diffusion as a probe of number-squeezing G.-B. Jo et Al., PRL 98, (2007)

‘Phase locking’ S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature 449, 324 (2007) u ≈ 5u ≈ ∞u ≈ 300u ≈ 100 N ~ 1000

Phase-diffusion between weakly coupled condensates u=10 4 u=10 3 u=10 2 u=10  N=1000 E. Boukobza, M. Chuchem, D. Cohen, and AV, PRL, in press (2009). Phase locking in the Josephson regime is phase-sensitive :

Semiclassical quantization Planck cell: Low energy ‘sea’ levels Separatrix levels High energy degenerate ‘island’ Um 2 levels ‘sea’ ‘islands’ separatrix

Semiclassical interpretation

How good is semiclassics ? n=1000 u=1000

Correlation time of Phase-diffusion Linearization about

Quantum Zeno control of phase-diffusion Y. Khodorkovsky, G. Kurizki, and AV, PRL 100, (2008) Long correlation times: t c for phase diffusion in BEC is of order ms Slow down phase diffusion by frequent measurements / noise. Since phase diffusion is along the L x axis, noise has to project onto onto L x (measure odd-even population imbalance - quasimomentum). Hence site indiscriminate noise such as stochastic modulation of the barrier height.

QZE reminder For t«t c =   , SP coherence decays quadratically Frequent projective measurements of L x ( g 1,2 (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 N 1/2 ) decoherence-free diffusion time: Extended phase-diffusion time, depends linearly on N :

 Preparation with noise numerical (lines) vs. analytic (symbols) N=100 N=200 N=400 N=100N=200N=400 N=100 N=150 N=300 N=100 Rabi: Josephson:

Comparison with local noise Initial coherent state Noiseless dynamics Macroscopic ‘cat’ state Site-localized noise  z =0.05J Site-indiscriminate noise  x =J N = 30 u = 2 t = 2.4 J

Atom-molecule dephasing in a sub-shot- noise SU(1,1) matter-wave interferometer 2E a EmEm E    Optical coupling 2E a EmEm  Feshbach resonance Undepleted pump:

SU(1,1) Casimir operator: Fock states: k - Bargmann index m = 0,1,2,…

Two-atom coherent states KxKx KyKy KzKz  }

Atom-molecule interferometer KxKx KyKy KzKz (a) (b),(c) (d) KyKy KzKz KxKx (c) (b) (a)(d)

Heisenberg-limited precision

Introduce interactions - dephasing For sinh  >> 1 sin(ut) ~ ut sinh  ~ 2n

Fringe visibility KxKx KyKy KzKz (a) (b),(c) (d) Time domainFrequency domain

Co-Authors Gershon Kurizki Weizmann Institute Doron Cohen BGU Maya Chuchem MSc, BGU Erez Boukobza Postdoc, BGU Amichay Vardi BGU Igor Tikhonenkov Research Fellow, BGU Yuri Khodorkovsky MSc 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.