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

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

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

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)

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

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)

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

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!

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

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

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

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

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

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 

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

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

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

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

Some experimental progress

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

Experimental realization of 1D Bose gas in hard-wall trap Phys. Rev. A 71, 041604 (2005). Model: Lieb-Linger model with open boundary condition Regime

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:

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

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.

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, 063617 (2006)

Density distribution in harmonic trap

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?

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

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

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

Density distribution (c<0) N=2 N=4 the density profile matches the case of c=0. Formation of a compounded particle with mass Nm GS energy:

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.

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

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

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.

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

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

Density distribution of the ground state of Tonks-Girardeau regime (c), m=0 (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, 053605 (2006)

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

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

No phase separation induced for AFM interaction

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

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, 053605 (2003) Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, 053605 (2006) Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, Phys. Rev. A 73, 063617 (2006) Y. Hao, Y. Zhang, J.Q. Liang and S. Chen, EPJD (2007)

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