Number squeezing, quantum fluctuations and oscillations in mesoscopic Bose Josephson junctions

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Number squeezing, quantum fluctuations and
oscillations in mesoscopic Bose Josephson
junctions
Frank Hekking Université Joseph
Fourier Laboratoire de Physique et Modélisation
des Milieux Condensés Maison des Magistères Jean
Perrin CNRS-Grenoble, France
Together with G. Ferrini and A. Minguzzi
Symposium on Quantum Phenomena and Devices at Low
Temperatures ULTI users meeting, Espoo, Finland,
28.-30.3.2008
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Outline

• Introduction Motivation
• Josephson effect with superconductors and cold
  atomic gases
• similarities and differences

2. Quantum theory of Bose Josephson junction-
Two mode model ground states
3. Behavior of momentum distribution function-
Bose Josephson junction as a generator of
Schroedingers cat
Conclusions
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1. Introduction MotivationJosephson effect
with superconductors and cold atomic gases
similarities and differences
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Superconducting tunnel junction
Josephson relations
Small Josephson junction two energy scales
Equation of motion
Hamiltonian
Commutator
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Superconductor vs. insulator behavior
Well-defined charge
Well-defined phase
Josephson effect
Coulomb blockade
f
Q
I
I
Ic
Zero-voltage state
Zero-current state
Vc
V
V
6
Cooper pair box
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Bose-Einstein condensates
(Anderson et al., Science 95 Davis et al. PRL
95)
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Mean-field theory Gross-Pitaevskii equation
Kinetic energy
Interaction potential
Condensate wave function
Confining potential
(Pitaevskii, Sov. Phys JETP 61 Gross, Nuovo
Cimento 63)
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Two-mode model
Confining potential double well trap
Collective variables
K
Ignore single particle excited states
Mean field Hamiltonian
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Classical behavior Josephson oscillations
self-trapping
(Smerzi et al. PRL 97)
(Albiez et al. PRL 05)
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2. Quantum theory of Bose Josephson junctionTwo
mode model ground states
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Quantum two-mode model
Two-mode Hubbard model
Number-phase operators
Quantum phase model
Mean field phase Hamiltonian for large N
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Mapping onto a spin model
Tunnelling
Tunnel current
Particle imbalance
Spin Hamiltonian
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Ground state for isolated wells
Isolated wells K 0
Relative-number Fock states jgt
Minimize energy
Fixed particle number states, no number
fluctuations
-N/2 lt n Int (n0) lt N/2
D n 0
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Ground state for noninteracting particles
Noninteracting case Us 0
Coherent state agt
Minimize energy
f 0, q p/2
D n N1/2
Fixed phase f, large number fluctuations
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Ground state for general case number fluctuations
Particle phase duality
D cos f
D n
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Energy spectrum, particle number
0.005
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Energy spectrum, particle number
0.1
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Energy spectrum, particle number
0.25
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3. Behavior of momentum distributionBose
Josephson junction as a generator
ofSchroedingers cat
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Time-of-flight expansion towards momentum
distribution
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Momentum distribution a measurable quantity
Definition
One-body density matrix
Double-well trap
F0
d
(Gati et al. PRL 06)
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Momentum distribution in the ground state
g 0.0005
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Momentum distribution in the ground state
g 0.005
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Momentum distribution in the ground state
g 0.5
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Momentum distribution in the ground state
g 50
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Time evolution of a coherent state (1)
Initial state
f0, q p/2
Sudden raise of the barrier dynamics governed by
interaction Hamiltonian
Periodic time evolution
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Time evolution of a coherent state (2)
Intermediate times
Schroedinger cat states
Examples q 2, 4
T/8
Cat state at time T/8
Phase content of cat state
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Conclusions

• Bose Josephson junctions an experimental reality

• Quantum effects number and phase fluctuations

(See also Averin et al. cond-mat 08)

• Momentum distribution function an
  experimentally
• measurable quantity

• Possibility to create Schroedinger cat states