Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Some aspects of 1D Bose gases: evolution from BEC to Tonks gas 陈 澍 (Shu Chen) 中国科学院物理研究所 Institute of Physics Chinese Academy of Sciences Aug 15, 2007 KITPC Outlines Introduction: Bose 1D quantum gas gas in hard-wall trap A solvable example of many-body system exhibiting crossover from BEC to Tonks gas Modified GP theory for 1D quantum gas Confinement of atoms by harmonic trap 3D harmonic trap 1 V m x2 x 2 y2 y 2 z2 z 2 2 Quasi-1d: cigar-shape trap y z x , kBT 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 mg 2 0 Interaction energy Kinetic energy I ~ ng 2n2 K ~ m I K 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 ^ ^ ^ x, y, z x 0 y, z Projection onto transverse ground state yields 2 ^ ^ g ^ 2 H dx x x V x, t x x x 2 2m ^ g 2 2ma1D a1D d 2 2a s as 1 1 . 46 d d 2 / m V(x)=0 Lieb-Liniger model Exactly solved by Bethe ansatz N 2 2 2c xi x j E 1i j N i 1 xi GS energy density given by g0 /2 ( 0 ) 2 2 2 0 / 6m 1 1 For harmonic trap, no exact solution, however, one can work in the modified GP theory works even in strongly interacting regime! For hard-wall trap, exact solution is available. Outlines Introduction: Bose 1D quantum gas gas in hard-wall trap A solvable example of many-body system exhibiting crossover from BEC to Tonks gas Modified GP theory for 1D quantum 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 N 2 2 2c xi x j E 1i j N i 1 xi Boundary condition: 0, x2 , xN x1 , xN L 0 Model for1D interacting Bose gas in hard-wall trap N 2 2 2c xi x j E 1i j N i 1 xi R : 0 x1 , x2 , M. Gaudin, Phys. Rev. A, 4, 386 (1971). for c >0 , xN L c We will study the full physical regime: Wave function: x1 , x2 , , xN x p1 , x p2 , P According to the symmetry condition, x p1 , x p2 , permutation of x1 , xN , x pN x p1 x p2 , x pN x pN can be obtained by Exact solution of 1D Bose gas in hard-wall trap Bethe ansatz wave function: x1 , xN AP exp i rj k p j x j P , r1 , , rN j Bethe ansatz equations (BAE): exp i 2k j L N ikl ik j c ikl ik j c l 1 j ikl ik j c ikl ik j c c c k j L n j arctan arctan k j kl k j kl l 1 j N k j L nj r j 1 . k j kl k j kl arctan arctan c c l 1 j N Eigenenergy : E j 1 k N 2 j GS solutions correspond to: n j 11 j N nj j 1 j N 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. k j k j 1 1 L 2 k j 1 k j Density distribution (c>0) dxN x, x2 , L x N dx2 0 L 0 dx1 , xN x, x2 , dxN x1 , x2 , , xN , xN 2 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 One body density matrix Momentum density distribution (c>0) n(k) for TG gas is different from that of free Fermi gas N=4 Y. Hao, Y. Zhang, and S. Chen, (2007) preprint 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 n1 n2 c arctan , 2 2 1 2 c L ln , 2 2 c L k1 i, k 2 i , k 2 GS energy E k 2 2 2 2 n1 n2 1 (Ground state) Example: solution for the 10-atom system 1.42 c 0 5 Dimers 1.72 c 1.42 4-string solution + three 2-string 1.88 c 1.72 6-string solution + two 2-string 3.02 c 1.88 8-string solution + one 2-string c 3.02 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 GS energy: N=4 the density profile matches the case of c=0. Formation of a compounded particle with mass Nm 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: Bose 1D quantum gas gas in hard-wall trap A solvable example of many-body system exhibiting crossover from BEC to Tonks gas Modified GP theory for 1D quantum gas 1D Gross-Pitaevskii theory G-P equation in the weakly interacting limit x i t 0 [ V ( x, t ) g 0 ] 0 2m 2 2 0 0 e 0 failure of MFT (G-P thoery for BEC) in the strongly interacting regime! Local density approximation (LDA) The LDA assuming that locally the system behaves like a uniform gas Modified Gross-Pitaevskii theory Modified G-P equation x2 i t 0 [ V ( x, t ) ( 0 ( x, t ))] 0 2m 2 0 0 e 0 g 0 ( 0 ) [ 0 ( 0 )] 2 2 2 0 0 / 2m Well describe the density profile, but overestimate the interference 1 1 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 Tonks gas = hard-core boson gas Tonks 1936 strong repulsion avoiding point-contact occupation effectively described by the boundary condition Bose-Fermi mapping 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 S. Chen & Egger PRA 2003 Density phase representation Density phase representation ^ with ( x)ei x i 0 x x 0 ( x) ( x)e [( x), ( x' )] i x x' Dynamics of (x,t) governed by 2 x i t [ V ( x, t ) g ] 2m 2 Small parameter ( and ) expansion zero order time dependent GP equation first order EOMs for and Effective Hamiltonian for quantum fluctuation operators Effective Hamiltonian 2 2 0 1 0 2 2 H dx x x0 x 2 0 2m 2m EOM of . b 1 ( 0 ) ( D ) D x [ 0 x ] b m 0 1 for TF and 2 S. Chen & Egger PRA 2003 . for TG D t x(b/ b) x Formulas for interference signal Interference signal around meeting point (±L/2) I ( x, t ) 0 ( x, t ) 0 ( x ', t ) 2 Re W ( x, x ', t ) W ( x, x ', t ) ( x, t ) ( x ', t ) x x ' L W ( x, x' , t ) W0 ( x, x' , t )e F ( x , x ',t ) Our task is to evaluate W(x,x',t) -L/2 x=0 L/2 y=x±L/ 2 W0 ( x, x ', t ) 0 ( x, t ) 0 ( x ', t )ei[0 ( x ,t ) 0 ( x ',t )] F ( x, x ', t ) [( x, t ) ( x ', t )]2 / 2 -L/2 y=0 Quantum fluctuations suppress interference L/2 Interference vs interaction Trapping and expansion: initial preparation V x, t 1 m x2 x 2 t 2 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 Spin-1 Bose gas (spinor gas with F=1) Realized in optical trap spin is not polarized Spinor symmetric interaction of F=1 atoms U ij xi x j U 0 P0 U 2 P2 xi x j c0 c2 Fi Fj c0 4m 2 a0 2 a2 3 , c2 4m 2 a2 a0 3 Spin-1 Bose gas The second quantized Hamiltonian of Spin-1 Bose gas: 2 ^ ^ c0 d2 H dx i Vext x i 2 2 m dx 2 2 ^ ^ ^ ^ c2 ^ ^ : i i : k i F F j l ij kl 2 In the mean field approach, the spin-dependent energy functional: 2 * c2 * * d2 E dx i Vext x i k i F F j l 2 ij kl 2 m dx 2 where i i i 2 and c0 /2. i If c2=0, the model is integrable for V(x)=0. Modified Gross-Pitaevskii Equations (GPEs) By using the exact BA solution, the interaction effect is properly taken into account. c0 /2, e 2 2 2 / 6m, 2m 2 1 1 The spin-dependent term can be expressed as: *k *i F F j l ij kl 2 2 2 0 2 0 2 2*0 2 2 02** (*) (*) The only processes that change the spin states occur when an atom in the mF 1 state scatters with another in the mF 1 state giving two atoms in the mF 0 state, or vice versa. The conservation quantity: The particle number Magnetization N N N0 N M N N Density distribution of the GS of 87 Rb (FM) Thomas-Fermi regime m=0 m=0.2 Tonks-Girardeau regime 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 87 Rb (FM) Y. Hao, Y. Zhang, J.Q. Liang and S. Chen EPJD (2007) No phase separation induced for AFM interaction 23 Na (AFM) Summary Exact results of the 1D interacting Bose gases in hard-wall trap General theory for 1D gas beyond MFT Acknowledgements 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 谢谢大家! Thank you for your attention!