One Dimensional Bosons in a Harmonic trap

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One Dimensional Bosons in a Harmonic trap

• Sung-po Chao
• Rutgers University
• 2008/02/20 Journal club

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Outline

• Newtons cradle (Nature 440,p900,2006)
• Integrable theory Lieb-Liniger model (PR
  130,p.1605,1963)
• Introduction of the experimental model (PRL
  81,p938 cond-mat 0606771)
• Experimental setup and result (Nature
  440,p900,2006 Nature 429,p277,2004)
• Conclusion/Questions.

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• Newtons cradle A toy made in 1967 showing
• energy and momentum conservation for two
• particle collision.
• An example of integrable classical system (If
  no energy loss in collision).
• Q What about the quantum version??

(Am. J. Phys. 49, 761 (1981))
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One dimensional bosons interact with delta
function potential
Equations satisfy
The exact solution is obtained by separating
regions and find solution within each region.
Spectrum is obtained by imposing periodic
boundary condition Bethe Ansatz
(PR 130,p1605,1963)
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Solution of 1D Bosons in ground state
Written in dimensionless parameters for ground
state energy
Zero Temp Quasi-particle distribution given by
Q Is it possible to find it experimentally?
(PR 130,p1605)
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1/2
M. Olshanni proposed a cigar shaped potential may
demonstrate 1D behavior
with
For S-wave scattering and ??gtEz, we can do
perturbation for z directional scattered wave
and the effective interaction is given by
1/2
With g 1/n a ? ??
(PRL 81,p938,1998)
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What to expect in the cigar shape potential?

• Tow limiting cases can be computed for
  distribution function (? defined as
  dimensionless interaction strength controllable
  by n and ?)
• (a) Tonks-Giradeau regime
• ??8 (Spreading in k)
• (c) Thomas-Fermi regime
• ??0 (BEC)

(PRL 86,p5413,2001)
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Experimental Setup of Optical Trap
(PRA 71, p 011602)
(Science 305,p 1125)
To confine the bosons a 3rd harmonic trap is
added in z-axis.
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Interacting bosons in a cigar shape potential
x
z
z
Nature 429,277-281 (2004)
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Quantum Newtons Cradle
(Nature 440,900-903,2006)
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Raw data taken at different time
(Nature 440,900-903,2006)
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Data after correcting heating

(Nature 440,900-903)
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Conclusion and Outlook