1. Some aspects of 1D Bose gases
陈 澍 (Shu Chen)
中国科学院物理研究所
Institute of Physics
Chinese Academy of Sciences
July 02, 2007 Taiyuan

2. Outlines Introduction: 1D quantum gas Bose gas in hard-wall trap
1D spinor gas

3. Confinement of atoms by harmonic trap
3D harmonic trap
Quasi-1d: cigar-shape trap
Transverse motion frozen
7Li
6Li

4. Realization of 1D quantum gas in optical lattice
For a 2D optical lattice, the atoms are confined to an array of tightly confining 1D potential tubes.
In each tube, radial motion confined to zero point oscillations
effective 1D quantum gas
Experiments with 1D condensates:
A. Goerlitz et al., PRL (2001), F. Schreck et al. PRL (2001), M. Greiner et al. PRL (2001)
more recently:
H. Moritz et al., PRL (2003), Nature 429, 277 (2004), Science, 305, 1125 (2004)

5. One dimensional Bose gas
Requirements: 1D bosonic quantum gas, tightly confined in two dimensions and only weakly confined along the axial direction
Parameter  governs the crossover from weak to strong interacting regime
Interaction energy
Kinetic energy

6. Cartoon for the 1D Bose gas in a trap
Weakly interacting TF regime
Strongly interacting Tonks regime
Tonks gas was realized experimentally
B. Paredes et al., Nature 429, 277 (2004),
T. Kinoshita et al., Science, 305, 1125 (2004)

7. Effective 1D Hamiltonian
Olshanii PRL 81 (1998) 938
Transverse motion frozen
Projection onto transverse ground state yields

8. V(x)=0  Lieb-Liniger model
Exactly solved by Bethe ansatz
Energy density given by
For harmonic trap, no exact solution, however, one can work in the modified GP theory
LDA + Exact result
works even in strongly interacting regime!

9. 1D Gross-Pitaevskii theory
Modified G-P equation
Well describe the density profile, but overestimate the interference

10. How good is the M-GP theory?
Density distribution in Tonks limit:
Kolomeisky et. al. PRL 85, 1146, 2000
Perfect agreement with the exact result by Bose-Fermi mapping

11. Tonks gas = hard-core boson gas
strong repulsion  avoiding point-contact occupation
effectively described by the boundary condition
Bose-Fermi mapping
Tonks 1936
Girardeau 1960

12. Tonks gas
Density distribution same as the free-fermion’s
Momentum distribution

13. Comment for the modified GP theory
Describe the density profile well, but overestimate the interference
If you use GPE to study the interference, you can always get interference no matter how strong the interaction is, even in the TG limit.
How to account properly the effect of interaction?
Quantum fluctuations suppress interference

14. Density phase representation
Dynamics of (x,t) governed by
Small parameter ( and ) expansion
zero order  time dependent GP equation
first order  EOMs for  and 
with 

16. Effective Hamiltonian for quantum fluctuation operators
EOM of 
for TF and
for TG
S. Chen & Egger PRA 2003

17. Formulas for interference signal
Interference signal around meeting point (±L/2)
Our task is to evaluate W(x,x',t)
Quantum fluctuations suppress interference
-L/2
x=0
y=0
y=x±L/2
L/2
L/2

18. Interference vs interaction
Trapping and expansion: initial preparation
Interference affected by interaction
(0)=14.3, T=0
(0)=0.001
Thomas-Fermi regime
S. Chen & Egger PRA 2003
Tonks regime

19. Interference in Tonks limit
Density profile: same with free-Fermion’s
Interference signal
No interference fringes: phase difference and
cancellation of fringes from different orbits

20. Some experimental progress

22. Outlines Introduction: 1D quantum gas Bose gas in hard-wall trap
A solvable example of many-body system exhibiting crossover from BEC to Tonks gas
1D spinor gas

23. Experimental realization of 1D Bose gas in hard-wall trap
Phys. Rev. A 71, (2005).
Model: Lieb-Linger model with open boundary condition
Regime

24. Model for1D interacting Bose gas in hard-wall trap
M. Gaudin, Phys. Rev. A, 4, 386 (1971). for c >0
We will study the full physical regime:
Wave function:
According to the symmetry condition, can be obtained by permutation of
Boundary condition:

25. Exact solution of 1D Bose gas in hard-wall trap
Bethe ansatz wave function:
Bethe ansatz equations (BAE):
Eigenenergy :
GS solutions correspond to:

26. Quasi-momentum distribution (c>0)
GS density of state in k-space:
N=200 and c=0.1,1,10,100. Inset:N=1000 and c= 10.

27. Density distribution (c>0)
Continuous crossover from weakly interacting Bose gas to Tonks gas
N=4
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, (2006)

28. Density distribution in harmonic trap

30. Attractive Bose gas with c<0
GP theory: collapse of BEC (3D)!
Attractive Bose gas in hard wall trap:
what a picture can exact results tell us?

31. 1D Bose gas in hard-wall trap (c<0)
Two body problem
BAE
GS energy
(Ground state)

32. Example: solution for the 10-atom system
5 Dimers
4-string solution + three 2-string
6-string solution + two 2-string
8-string solution + one 2-string
10-string solution

33. Crossover for the N-atom system (c<0)
Weak attractive interaction regime
N/2 Dimers
Intermediate regime
(N-M)-string solution + M/2 2-string solution (1<M<N )
Strong attractive interaction regime N-string solution

34. Density distribution (c<0)
N= N=4
the density profile matches the case of c=0. Formation of a compounded particle with mass Nm
GS energy:

35. The second order correlation function
The atoms tend to cluster together more easily for the attractive interaction and the atoms bunch closer as the interaction becomes stronger. For the repulsive interactions, the atoms avoid each other and the atom-bunching
reduces and vanishes finally for increasing interactions.

36. Outlines Introduction: 1D quantum gas Bose gas in hard-wall trap
1D spinor gas

37. Spin-1 Bose gas (spinor gas with F=1)
Realized in optical trap spin is not polarized
Spinor symmetric interaction of F=1 atoms
Ferromagnetic interactions for
Antiferromagnetic interactions for

38. Spin-1 Bose gas The second quantized Hamiltonian of Spin-1 Bose gas:
In the mean field approach, the spin-dependent energy functional:
Effective in the weakly interacting regime but not able to describe the
density distribution correctly in the Tonks-Girardeau regime.

39. Modified Gross-Pitaevskii Equations (GPEs)
By using the exact BA　solution, the interaction effect is properly taken into account.
The spin-dependent term can be expressed as:
(＊) The only processes that change the spin states occur when an atom in the state
scatters with another in the state giving two atoms in the state, or vice versa.
The conservation quantity:
The particle number
Magnetization

40. Density distribution of the GS of (FM)
Thomas-Fermi regime
m= m=0.2
Black lines: + component Red lines: 0 component Green lines: - component

41. Density distribution of the ground state of
Tonks-Girardeau regime
(c), m= (d) m=0.2
Black lines: + component Red lines: 0 component Green lines: - component
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, (2006)

42. Phase separation induced by anisotropic spin-spin interaction
(FM)
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen
EPJD (2007)

43. Density distribution of the GS of (AFM)
Thomas-Fermi regime
m=0
m=0.2
Black lines: + component Red lines: 0 component Green lines: - component

44. No phase separation induced for AFM interaction

45. Summary General theory for 1D gas beyond MFT
Exact results of the 1D interacting Bose gases in hard-wall trap
Density distributions for the 1D spinor gases in harmonic trap

46. Acknowledgements References:
Collaboraters:
Dr. Yajiang Hao
Institute of Physics, Chinese Academy of Sciences
Prof. YunBo Zhang and Prof. J.-Q. Liang
ShanXi University
Prof. R Egger
Duesserdorf University
Financial support: NSF of China, Bairen program of CAS
References:
S. Chen and R. Egger, Phys. Rev. A 73, (2003)
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, (2006)
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, (2006)
Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, EPJD (2007)

47. Thank you for your attention!
谢谢大家！
Thank you for your attention!