1. One Dimensional Bosons in a Harmonic trap Sung-po Chao Rutgers University 2008/02/20 Journal club

2. 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.

3. (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??

4. 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

5. 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?

6. (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

7. 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)

8. Experimental Setup of Optical Trap (PRA 71, p 011602) (Science 305,p 1125) To confine the bosons a 3 rd harmonic trap is added in z-axis.

9. Interacting bosons in a cigar shape potential Nature 429,277-281 (2004) x z z

10. Quantum Newton’s Cradle (Nature 440,900-903,2006)

11. Raw data taken at different time (Nature 440,900-903,2006)

12. Data after correcting heating (Nature 440,900-903)

13. Conclusion and Outlook