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1 Dynamics, Quantum Transport and Quantum Phase Transition of Cold Atoms Wu-Ming Liu( 刘伍明 ) (Institute of Physics, Chinese Academy of Sciences) wmliu@aphy.iphy.ac.cn Phone:

2 Collaborators Prof. Jiu-Qing Liang (Shanxi Univ.) Prof. Qian Niu (Texas Univ.) Prof. Shun-Qing Shen (HongKong Univ.) Prof. Zi-Dong Zhang (IMR, CAS)

3 1. Introduction 2. Dynamics Outline 3. Quantum transport 4. Quantum phase transition 5. Summary

4 物质的第五态玻色凝聚与费米子凝聚国家自然科学基金委员会 Bose-Einstein statistics (1924) Fermi-Dirac statistics (1926) 数理学部实验物理讲习班 Bose-Einstein condensate Fermi sea E F Bose enhancement hω T = (0.83 N) 1/3 C k B Pauli exclusion hω T << T = (6 N) F k B 1/3

5 1. Introduction 1. Introduction

6 1913 H. K. Onnes 低温研究, 液氦超导 1962 Lev Davidovich Landau 液氦超流理论 1972 J. Bardeen, L. N. Cooper, J.R. Schrieffer 超导 BCS 理论 1973 Ivar Giaever 超导隧道效应, Brian D. Josephson 超导隧道结 1978 Pyotr Leonidovich Kapitsa 氦 II 超流性 1982 Kenneth G. Wilson 连续相变重整化群理论 1987 K. Alexander Muller, J. Georg Bednorz 发现高温超导 1996 D. D. Osheroff, D. M. Lee, R. C. Richardson 发现氦三超流性 2001 E.A. Cornell, W. Ketterle, C.E. Wieman 玻色 - 爱因斯坦凝聚 2003 A. A. Abrikosov, U. L. Ginzburg, A. J. Leggett 超导超流

7 玻色 - 爱因斯坦凝聚的主要方向 coherence matter wave BEC molecule atom laser Coherent matter wave condensed-matter physics BEC quantum fluids Mott insulator nonlinear physics solitons vortices

8 Science 26 篇,Nature, 31 篇

10 1. Dynamics W. Ketterle, Science 275, 637 (1997).

11 W.M. Liu, B. Wu, Q. Niu, Nonlinear effects in interference of Bose-Einstein condensates, Phys. Rev. Lett. 84, 2294 (2000). (SCI 引用 105 次 )

12 Many-body Hamiltonian The mean field theory Gross-Pitaevskii equation () () ( ) ( ) ( ) ( ) ( ) r r r r V r r drdr r V m r dr H ext Ψ Ψ Ψ Ψ + Ψ + Ψ = ' ' ' ' ( ) ( ) ( ) t r t r t r, ' ~,, Ψ + = Φ Ψ ( ) () ( ) t r m a r V m t r t i ext, 4 2, Φ Φ + + = Φ π

13 Gross-Pitaevskii equation π a ( ) 2 i Φ = + Vext r + Φ Φ t 2m m Long time solution x 2 2 α ( ) Φ = + t α ( k) = log(1 r( k) ) 2π g x x i i2 α ( ) log(4 t) 2 1 (, ) t t t x t e O( t log t)

14 Theoretical explanation Fringe position 1 k± n =± 2 En =± 2 V + n 2V 2 '' Central fringe ( ) " 12 Δ k = k k = 4 V + V

15 Experimental prediction: 国家自然科学基金委员会 1. Energy level 2. Many wave packets 数理学部实验物理讲习班 Ratio of level width to level spacing δ k n δ E n 2 g E w E Δ k n = = Δ E n ( ) 2 n n e

17 Phase separation 国家自然科学基金委员会 PRL 81, 1539, 1543 (1998). 数理学部实验物理讲习班

18 W.D. Li, X.J. Zhou, Y.Q. Wang, J.Q. Liang, W.M. Liu, Time evolution of relative phase in two-component Bose-Einstein condensates with a coupling drive, Phys. Rev. A64, (2001). (SCI 引用 21 次 )

19 Z. X. Liang, Z. D. Zhang, W. M. Liu, Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential, Phys. Rev. Lett.. 74, (2005). (SCI 引用 31 次 )

20 Many-body Hamiltonian () () () ( ) ( ) ( ) () r r r r V r r drdr r V m r dr H ext Ψ Ψ Ψ Ψ + Ψ + Ψ = ' ' ' ' ( ) ( ) ( ) t r t r t r, ' ~,, Ψ + Φ = Ψ ( ) () ( ) t r m a r V m t r t i ext, 4 2, Φ Φ + + = Φ π The mean field theory Gross-Pitaevskii equation

22 可调幅的原子激光

23 可调频的原子激光

24 Z.W. Xie,, Z.X. Cao, E.I. Kats, W.M. Liu, Nonlinear dynamics of dipolar Bose-Einstein condensate in optical lattice, Phys. Rev. A 71, (2005).

26 Z.W. Xie, W.P. Zhang, S.T. Chui, W.M. Liu, Magnetic solitons of spinor Bose-Einstein condensates in optical lattice, Phys. Rev. A69, (2004). (SCI 引用 11 次 )

27 国家自然科学基金委员会 Z.D. Li, P.B. He, L.Li, J.Q. Liang, W.M. Liu, 数理学部实验物理讲习班 Magnetic soliton and Soliton collision of spinor Bose-Einstein condensates in optical lattice, Phys. Rev. A71, (2005). (SCI 引用 10 次 )

28 L. Li, B.A. Malomed, D. Mihalache, W.M. Liu, Exact soliton-on on-plane-wave solutions for two-component Bose- Einstein condensates, Phys. Rev. E 73, (2006).

29 2. Quantum Transport W.M. Liu, W.B. Fan, W.M. Zheng, J.Q. Liang, S.T. Chui, Quantum tunneling of Bose-Einstein condensates in optical lattices under gravity, Phys. Rev. Lett. 88, (2002). (SCI 引用 77 次 )

30 Potential energy and Bloch bands Landau-Zener tunneling Barrier between lattices is low Localized level between lattices is coupling Miniband Adiabatic approximation Tunneling between delocalized states in different Bloch bands

31 Tilted bands and Wannier-Stark ladders Wannier-Stark tunneling An external field Wavefunction of miniband is localization Miniband is divided into discrete level Wannier-Stark ladder Tunneling between localized states in different individual wells Wannier-Stark localized states

32 国家自然科学基金委员会 Temperature dependence 数理学部实验物理讲习班 0 w0 k T Γ ( T) = Γ (1 e B ) e 432V w max 0 e w 0 kbt At high temperature: Arrhenius law Γ = AR ω 2 0 π e V max / k T B

33 At intermediate temperature: Thermally assisted tunneling Crossover temperature T cr = 2 hw π 0 k B T cr = 257nK U ( x, y) = 2.1E l R At low temperature: Pure quantum tunneling

35 Quantum transport through an atomic quantum dot coupled to superfluid Bose- Einstein condensate reservoirs J. M. Wang, J. Q. Liang, W. M. Liu,

36 Fig. 1 Schematic diagram of Josephson atomic quantum device, which is an atomic quantum dot coupled to two Bose-Einstein condensate reservoirs via Raman transition with Rabi frequency \Omega and detuning \delta, where the bosons in state a> in reservoir is confined in a shallow trap, the atom in state d> in quantum dot is localized in a tight trap. A large on-site interaction U_d allows only a single atom in the quantum dot.

$(a) Effect of Rabi frequency Omega: the black, red and green lines correspond to Omega= 0.02, 0.04, 0.06 khz, U_d=1000 khz, \Delta = 2 \pi \times 0.41 khz, Gamma_L= Gamma_R= Gamma/2.$

40 Fig. 2 The transmission probability T(E) versus the energy of incident boson E, where the parameters are chosen as delta_l = delta_r=0, phi=0. (a) Effect of Rabi frequency Omega: the black, red and green lines correspond to Omega= 0.02, 0.04, 0.06 khz, U_d=1000 khz, \Delta = 2 \pi \times 0.41 khz, Gamma_L= Gamma_R= Gamma/2. (b) Effect of interaction parameter \Delta : the black, red, and green lines correspond to \Delta = 2 \pi \times 0.1 khz, 2 \pi \times 0.41 khz, 2 pi times 1.0 khz, U_d=10 khz, Omega= 0.02 khz.

$Fig. 3 The Josephson current I_Jc versus the phase difference \phi between two BEC reservoirs. (a) Effect of the Rabi frequency \Omega: the black, red, and green lines correspond to \Omega= 0.$

41 Fig. 3 The Josephson current I_Jc versus the phase difference \phi between two BEC reservoirs. (a) Effect of the Rabi frequency \Omega: the black, red, and green lines correspond to \Omega= 0.02, 0.04, 0.06 khz, \delta_l= \delta_r=0. (b) Effect of the Raman detuning \delta: the black, red, green and blue dotted lines correspond to \delta_l= \delta_r= 0, 1, 2, 3 khz, \Omega= 0.02 khz.

42 Fig. 4 Schematic diagram of superfluid quantum interference devices which is two atomic quantum dots coupled to two Bose-Einstein condensate reservoirs.

43 3. Quantum phase transition

44 J.J. Liang, J.Q. Liang, W.M. Liu, Quantum phase transition of condensed bosons in optical lattices, Phys. Rev. A68, (2003). (SCI 引用 26 次 )

45 Z.W. Xie,, W.M. Liu, Superfluid Mott insulator transition of dipolar bosons in an optical lattice, Phys. Rev. A70, (2004).

47 G.P. Zheng,, J.Q. Liang, W.M. Liu, Phase diagram of two-species Bose-Einstein condensates in an optical lattice, Phys. Rev. A71, (2005)

49 Magnetic phase transition of cold atoms in optical lattices P.B. He, P. Li, S.Q. Shen, W.M. Liu,

50 Fig. 1 (a) Two selected optical potential along x direction, the potential V_x2 (solid line) has twice period of the V_x1 (dashed line). (b) Total potential in x axis, a_1, a_2, V_1, V_2 are the widths and heights of barriers of intra-dimer and inter-dimer. (c) The whole figure of potential in x-y plane. (d) Geometry of the 2D coupled spin ladder, J_1, J_2, J_3 are intradimer, inter-dimer and inter-ladder spin-spin couplings.

54 Fig. 2 The energy spectrums of different inter-dimer coupling in spin liquid phase. The solid, dashed, dotted lines correspond to coupling strength J_2/J_1= J_3/J_1 =0.1, 0.3, 0.5. The optical lattice parameters V_x1=5E_r, V_x2 =20E_r, V_y=28E_r for J_2/J_1= J_3 /J_1=0.1; V_x1=16E_r, V_x2= 10E_r,V_y=6E_r for J_2/J_1= J_3/J_1= 0.3; V_x1=20E_r, V_x2=12E_r, V_y=7E_r for J_2/J_1=J_3/J_1 =0.5. E_r= (\hbar^2k^2_x)/(2m), m is mass of 6^Li atom, a_s = 2.41nm, m= 9.99 \times 10^{-26} kg, k_x =2k_y =5.93 \mu m^{-1}, \omega_z = 6600 Hz.

55 Fig. 4 Phase diagram controlled by the spin coupling strength. In the regime of energy gap \Delta >0, the system is spin disordered or in a spin liquid phase, a magnetically ordered phase appears at \Delta =0. The phase boundary can be determined by equation: (J_2 + 2 J_3) s^2 = J_1/4 - \mu, the amplitude of condensate s and chemical potential \mu can be given by variational method. J_1, J_2, J_3 are the functions of lattice parameters, the phase diagram is described by the amplitudes of standing waves in Fig. 4.

56 Fig. 5 Phase diagram in the V_x1 - V_x2 plane for different V_y. The wave vector satisfies k_x=2k_y. V_x1, V_x2 and V_y are rescaled by the recoil energy E_r.

57 Fig. 6 Energy spectrum in magnetically ordered phase. The upper dashed line is longitudinal mode and the lower solid one is doubly degenerate transverse mode. The coupling strength J_2/J_1 =J_3/J_1 = 0.7. \lambda = 0.6. The corresponding parameters are V_x1 = 27E_r, V_x2= 16E_r, V_y = 10E_r.

58 5. Summary 5.1. Importance and applications

59 5.2. Future topics Bose-Einstein condensates become an ultralow-temperature laboratory for atom optics, collisional physics, manybody physics, superfluidity, quantized vortices, Josephson junctions and quantum phase transitions.

60 Thanks!

幻灯片 1

幻灯片 1 Quantum Optics email: ygu@pku.edu.cn Tel: 67588 354 Quantum mechanics +Optics PPT 1 1 λ ħ Planck λ 0 ħ 0 3 λ 0 ħ 0. Maxwell 4 λ 0 ħ 0 m=0) 5 Ψ Ψ = LΨ I dx matter -di=αidx α I I=I 0 e - αl 6 Maxwell D=