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One Dimensional Bosons in a Harmonic trap

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

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One Dimensional Bosons in a Harmonic trap

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

  4. 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)

  5. 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)

  6. 1/2 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: 1/2 With g ～1/n ; a ⊥ ～ω⊥ (PRL 81,p938,1998)

  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 3rd harmonic trap is added in z-axis.

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

  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

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