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
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Andrews et. al., Science 275, 637 (1997)
Fringe visibility is proportional to SP coherence
= N1
+ N2
Freely expanding condensates
d
x
zy
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)
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Coherent splitting of a BECThe mere existence of interference patterns does not indicateInitial SP coherence - need to verify reproducible fringe position.
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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
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Total number conservation
Hence coherence is characterized by the length of the Bloch vector restricted to be inside the sphere .
Fringe Visibility: Lx
Ly
Lz
Coherent = classical states
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SU(2) coherent states:QuickTime™ and a decompressor
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Gross-Pitaevskii classical (mean-field) energy functional with :
Interaction regimes
Rabi regimeWeak interaction, linear
(perturbed) Lx eigenstates
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QuickTime™ and a decompressorare needed to see this picture.Josephson regimeIntermediate strong interaction
Nonlinear ‘islands’ in a linear ‘sea’
Separated by ‘figure-8’ separatrix
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QuickTime™ and a decompressorare needed to see this picture.Fock regimeStrong interaction, nonlinear
‘sea’ area less than the Planck cell
(perturbed) Lz eigenstates
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Classical dynamicsu>2 ‘self trapping’
A. Smerzi et al., PRL 79, 4750 (1997).M. Albeiz et al., PRL 95, 010402 (2005).
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Phase ‘diffusion’ in the Fock regime• Coherent state preparation: binomial superposition of Fock states
• Evolve with J = 0 , U ≠ 0, .
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Ut
• For and
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t d /
t rev
First phase diffusion experiment
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M. Greiner, O. Mandel, T. Haensch., and I. Bloch, Nature 419, 51 (2002).
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VB
VA
Slow phase-diffusion as a probe of number-squeezing
G.-B. Jo et Al., PRL 98, 030407 (2007)
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‘Phase locking’ S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer,
Nature 449, 324 (2007)
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u ≈ 5 u ≈ ∞u ≈ 300u ≈ 100
N ~ 1000QuickTime™ and a
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Phase-diffusion between weakly coupled condensates
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u=104
u=103
u=102
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 :
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Semiclassical quantizationQuickTime™ and a
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Low energy ‘sea’ levels
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Separatrix levels
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High energy degenerate ‘island’ Um2 levels
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‘sea’
‘islands’
separatrix
Semiclassical interpretation
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How good is semiclassics ?QuickTime™ and a
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n=1000u=1000
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Correlation time of Phase-diffusion
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Linearization aboutQuickTime™ and a decompressorare needed to see this picture.
Quantum Zeno control of phase-diffusion
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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.
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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:
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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)
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N=100
N=200
N=400
N=100
N=200
N=400
N=100
N=150
N=300
N=100
N=100
Rabi:
Josephson:
Comparison with local noise
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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
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2Ea
Em
E
Optical coupling
2EaEm
Feshbach resonance
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Undepleted pump:
SU(1,1)QuickTime™ and a
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k - Bargmann index m = 0,1,2,…
Two-atom coherent states QuickTime™ and a
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Kx
Ky
Kz
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Atom-molecule interferometer
Kx
Ky
Kz
(a)
(b),(c)
(d)
Ky
Kz
Kx
(c) (b)
(a)(d)
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Heisenberg-limited precision
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Introduce interactions - dephasing
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For sinh >> 1
sin(ut) ~ utsinh~ 2n
Fringe visibilityQuickTime™ and a
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Kx
Ky
Kz
(a)
(b),(c)
(d)
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Time domain Frequency domain
Co-Authors
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Gershon KurizkiWeizmann Institute
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Doron CohenBGU
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Maya ChuchemMSc, BGU
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Erez BoukobzaPostdoc, BGU
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Amichay VardiBGU
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Igor TikhonenkovResearch Fellow, BGU
Yuri KhodorkovskyMSc BGU WIS
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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.