One Dimensional Bosons in a Harmonic trap

Sung-po Chao

Rutgers University2008/02/20 Journal club

Outline

• Newton’s 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.

(Am. J. Phys. 49, 761 (1981))

•Newton’s 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??

One dimensional bosons interact with delta function potential

Equations satisfy:

(PR 130,p1605,1963)

The exact solution is obtained by separating regions and find solution within each region. Spectrum is obtained by imposing periodic boundary condition: Bethe Ansatz

Solution of 1D Bosons in ground state

Zero Temp Quasi-particle

distribution given by:

(PR 130,p1605)

Written in dimensionless parameters

for ground state energy:

Q: Is it possible to find it experimentally?

(PRL 81,p938,1998)

M. Olshanni proposed a cigar shaped potential may demonstrate 1D behavior

with

For S-wave scattering and ω⊥ ＞ Ez, we can do perturbation for z

directional scattered wave and the effective interaction is given by:

With g ～ 1/n ; a ⊥ ～ ω⊥

1/2

1/2

What to expect in the cigar shape potential?

Tow limiting cases can be computed for distribution function: (γ defined as dimensionless interaction strength controllable by n and ω)

(a) Tonks-Giradeau regime γ→∞ (Spreading in k)(c) Thomas-Fermi regime γ→0 (BEC)

(PRL 86,p5413,2001)

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.

Interacting bosons in a cigar shape potential

Nature 429,277-281 (2004)

x

z

z

Quantum Newton’s Cradle

(Nature 440,900-903,2006)

Raw data taken at different time

(Nature 440,900-903,2006)

Data after correcting heating

(Nature 440,900-903)

Conclusion and Outlook