Download scattering Bethe Ansatz approach - Condensed Matter Journal Club

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)
• Experimental setup and result (Nature
440,p900,2006; Nature 429,p277,2004)
• Conclusion/Questions.
•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))
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)
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)
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:
With g
～1/n ; a ⊥ ～ω⊥
1/2
(PRL 81,p938,1998)
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
x
z
z
Nature 429,277-281 (2004)
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