Dephasing and noise in weakly-coupled Bose-Einstein condensates Amichay Vardi

1. Dephasing and noise in weakly-coupled Bose-Einstein condensatesAmichay 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

2. Andrews et. al., Science 275, 637 (1997) = N1 + N2 Fringe visibility is proportional to SP coherence Matter-wave interference

3. x d z y Freely expanding condensates

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

5. Fock preparation Population difference: N1 - N2 Undefined relative phase between the two BECs Does the Fock preparation give interference fringes ?

6. Fock states are superpositions of coherent states: Any single-shot interferometric measurement constitutes a single phase-projection . Each shot gives fringes with random phase: Fringes in the Fock preparation While the multi-shot density averages out to:

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

8. 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)

9. Outline • Assume a coherent preparation. • Interactions cause ‘Phase-Diffusion’. How long will SP coherence survive ? • PD between weakly-coupled BECs - ‘Phase Locking’. • Control of PD by noise. • Sub shot-noise interferometry and PD between atoms and molecules.

10. Model: a bosonic Josephson junction

11. Hence coherence is characterized by the length of the Bloch vector restricted to be inside the sphere . Lz  Fringe Visibility: Lx Ly Total number conservation

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

13. Interaction regimes Rabi regime Weak interaction, linear (perturbed) Lx 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) Lz eigenstates

14. Classical dynamicsu>2 ‘self trapping’A. Smerzi et al., PRL 79, 4750 (1997).M. Albeiz et al., PRL 95, 010402 (2005).

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

16. VB td / trev VA First phase diffusion experiment M. Greiner, O. Mandel, T. Haensch., and I. Bloch, Nature 419, 51 (2002).

17. Slow phase-diffusion as a probe of number-squeezingG.-B. Jo et Al., PRL 98, 030407 (2007)

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

19.   N=1000 u=104 u=103 u=102 u=10 Phase-diffusion between weakly coupled condensates E. Boukobza, M. Chuchem, D. Cohen, and AV, PRL, in press (2009). Phase locking in the Josephson regime is phase-sensitive :

20. Planck cell: Low energy ‘sea’ levels Separatrix levels High energy degenerate ‘island’ Um2levels Semiclassical quantization ‘islands’ separatrix ‘sea’

21. Semiclassical interpretation

22. How good is semiclassics ? n=1000 u=1000

23. Linearization about Correlation time of Phase-diffusion

24. Quantum Zeno control of phase-diffusion Y. Khodorkovsky, G. Kurizki, and AV, PRL 100, 220403 (2008) • Long correlation times: tcfor 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.

25. For t«tc=,SP coherence decays quadratically QZE reminder 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).

26. QZE control of phase-diffusion QZE limit: Uncorrelated, Markovian noise: Quantum kinetic master equation: Linearization of the master equation gives the QZE result:

27. Bose enhancement of QZE Extended phase-diffusion time, depends linearly on N: As opposed to log(N) (or N1/2)decoherence-free diffusion time:

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

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

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

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

32. } Kz  Kx Ky Two-atom coherent states

33. Kz Kz (d) (c) (b) (b),(c) Ky Ky (d) (a) (a) Kx Kx Atom-molecule interferometer

34. Heisenberg-limited precision

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

36. Time domain Frequency domain Fringe visibility Kz (d) (b),(c) (a) Ky Kx

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

38. 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.