Dynamics of a Trapped Bose-Condensed Gas at Finite Temperatures
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1 Dynamics of a Trapped Bose-Condensed Gas at Finite Temperatures Tetsuro Nikuni, Tokyo University of Science Collaborators: Allan Griffin, University of Toronto (Canada) Eugene Zaremba, Queen s University (Canada) James E. Williams, NIST-Gaithersburg (USA) Brian Jackson, Queen s University (Canada) Overview of theoretical studies on the dynamics of trapped atomic Bose gases at finite temperatures The role of the noncondensate atoms. Dynamics in a Spin-1/2 Bose gas Recent JILA experiments.
2 Background Bose-Einstein condensation (BEC) in laser-cooled atomic gases has become an exciting new field of research in physics, since the first atomic condensate was created in BEC of 23 Na atoms (MIT) T>T c T T c T<<T c v y v x Velocity distribution
3 Trapped Atomic Bose Gas Trap potential U ext (r) = m ( 2 ω xx 2 +ω y y 2 +ω z z 2 ) Interaction V(r 1 r 2 ) gδ(r 1 r 2 ) g = 4πh2 a m a: s-wave scattering length Although the gas is very dilute (na 3 <<1), the interaction plays a crucial role in dynamic phenomena Many-body physics.
4 Bose-Condensate Wavefunction How is this new degree of freedom couples into the normal degrees of freedom? How is the normal fluid modified by the superfluid component? iθ (r,t ) Φ(r,t) = n c (r,t)e v c (r,t) = h m θ(r,t) v c = 0 Irrotational fluid Superfluid
5 Boltzmann Equation Normal Bose Gas (T>T BEC ) t + p m U(r,t) p f (r,p,t) = C 22[ f ] U(r,t) = δu ext (r,t) +U ext (r) + 2gn(r,t) small driving perturbation trap potential Hartree-Fock mean field (very small 0) Collision Integral 1+f 1+f 2 1+f 3 1+f 4 C 22 [f]=g 2 - f 3 f 4 f f 2
6 The Boltzmann equation was used to study various dynamical properties in trapped Bose gases above T BEC [see Chap. 11 of Pethick & Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge (2001)]. One can use various techniques: variational method, moment method, ergodic approximation, test-particle simulation. In the collision-dominated regime, one can use the Chapman- Enskog method to derive hydrodynamic equations for a few macroscopic variables. Griffin, Wu & Stringari, PRL 78, 1838 (1997). Nikuni & Griffin, JLTP 111, 793 (1998).
7 Microscopic Derivation of the Boltzmann Equation (T>T BEC ) Nonequilibrium Green s function G(r 1 t 1,r 2 t 2 ) = ψ ˆ + (r 1,t 1 ) ψ ˆ (r 2,t 2 ) Semiclassical Wigner distribution function f (r,p,t) = d r e ip r /h ψ ˆ + (r + r / 2,t) ψ ˆ (r r / 2,t) For details, see Kadanoff & Baym, Quantum Statistical Mechanics Benjamin, New York (1962). Zubarev, Morozov & Ropke, Statistical Mechanics of Nonequilibrium Processes, Akademie Verlag, Berlin (1997).
8 Bose-Condensed Trapped Gas (T<T BEC ) A condensate is described by a complex order parameter: Φ(r,t) iθ (r,t ) ψ ˆ (r,t) = n c (r,t)e Gross-Pitaevskii (GP) Equation for the pure condensate at T=0 (in effect, T<0.4T BEC ) ih Φ(r,t) t = h2 2 2m + U ext (r) + gn c (r,t) Φ(r,t) It gives an excellent quantitative description of the dynamic behavior, such as collective modes, vortices, solitons.
9 Condensate Dynamics (MIT) Quadrupole Oscillation Vortex lattice
10 Trapped Bose-Condensed Gas at Finite T Trap potential U ext (r) = m 2 ω 2 0 r 2 Condensate Noncondensate x The GP equation will be modified by coupling to the noncondensate atoms.
11 Generalized GP Equation at Finite T ih Φ t = h2 2 2m + U ext + gn c Φ + 2g n Φ + gm Φ * + g ψ + ψ ψ ψ (r,t) = ψ ˆ (r,t) Φ(r,t) n (r,t) = ψ + (r,t) ψ (r,t) noncondensate operator m (r,t) = ψ (r,t) ψ (r,t) We need equations for the noncondensate Green s functions: G 11 (rt, r t ) = ψ + (r,t) ψ ( r, t ), G 12 (rt, r t ) = ψ (r,t) ψ ( r, t ) The semiclassical Wigner distribution function is defined as f (r,p,t) = d r e ip r /h ψ + (r + r / 2,t) ψ (r r / 2,t)
12 Kinetic Theory for Bose-Condensed Gases Homogeneous Bose Gas Kane & Kadanoff, J. Math. Phys. 6, 1902 (1965). Kirkpatrick & Dorfman, JLTP 58, 308 (1985). Trapped Bose Gas Proukakis, Burnett & Stoof, PRA 57, 1230 (1998). Gardiner & Zoller, PRA 61, (2000), and Refs. therein. Stoof, JLTP 114, 11 (1999). Walser, Williams, Cooper & Holland, PRA 59, 3878 (1999). Zaremba, Nikuni & Griffin (ZNG), JLTP 116, 277 (1999). Imamovic-Tomasovic & Griffin, JLTP 122, 617 (2001). Many other papers All these kinetic theories have essentially the same structure, while different approximations are used in detail.
13 ZNG Model Restricted to temperatures high enough (but T<T BEC!) so that thermal excitations can be described as atoms moving in a self-consistent Hartree-Fock (HF) mean field ε p (r,t) = p2 2m + U (r) + 2g[n (r,t)+ ext c n (r,t)] = p2 2m + U(r,t) New kinetic equation: t + p m U(r,t) p f (r,p,t) = C 12[ f,φ]+ C 22 [ f ] C 12 : Collisions between atoms in condensate and noncondensate components
14 1+f C 12 Collision Integral n 1+f 1+f c f 1+f 2 C 12 [f,f]=g 2 f 2 f 3 1+f 2 1+f 3 n c f 3 f 3 n c n c 1+f 3 1+f n 3 c f n c f f 2 f f 2 condensate atom energy ε c (r,t) = h2 2 n c 2m n c +U ext + g( n c + 2n )+ mv c 2 2 condensate momentum mv c (r,t) = h θ(r,t)
15 Generalized Gross-Pitaevskii Equation ih Φ(r,t) t = h2 2 2m +U ext (r)+ gn c (r,t) + 2g n (r,t) ir(r,t) Φ(r,t) R(r,t) = hγ 12 (r,t) 2n c (r,t) Γ 12 = dp C (2πh) 3 12 [ f,φ] It is coupled to the noncondensate component through the HF mean field and C 12 collisions. Collisions between atoms in condensate and noncondensate may change number of atoms in condensate N c.
16 Collisionless v.s. Hydrodynamic Recall two domains at T 0 Collisionless Mean fields dominate T some collective mode ω=2π/t τ coll mean collision time Hydrodynamic T Local equilibrium τ coll
17 Collisionless Dynamics Collective modes (Bijlsma & Stoof, Jackson & Zaremba) Scissors mode (Jackson & Zaremba, T. Nikuni) Condensate growth (Gardiner & Zoller, Bijlsma, Zaremba & Stoof, Yamashita et al.) Vortex nucleation (Williams et al., Tsubota, et al., Gardiner et al.) Theoretical methods Moment method collective modes Ergodic approximation condensate growth N-particle simulation recently developed by Jackson & Zaremba: can solve a lot of dynamical problems!
18 Condensate Growth MIT experiment (1999) Bijlsma, Zaremba & Stoof PRA 62, (2000) Similar results by Gardiner & coworkers
19 N-Body Simulation Semiclassical kinetic equation is solved by the N-body (test particles) simulation. Collisions are treated by the Monte-Carlo sampling technique. Jackson & Zaremba, cond-mat/ GP equation is solved by the sprit-operator Fast Fourier Transform (FFT) method. f (r, p,t) N i=1 δ[r r i (t)] δ[p p i (t)] Φ(r,t)
20 Quadrupole Collective Mode JILA experiment: Jin et al., PRL 78, 764 (1997) Theory: Jackson & Zaremba, PRL 88, (2002) Oscillations of cloud widths Frequency Damping
21 Transverse Breathing Mode ENS experiment: Chevy et al., PRL 88, (2002) Theory: Jackson & Zaremba, cond-mat/ Theory v.s. Experiment A(t)
22 Two-Fluid Hydrodynamics In the collision-dominated regime, we can derive two-fluid hydrodynamics starting from the coupled ZNG equations. Following Chapman-Enskog for classical Boltzmann equation, we expand the noncondensate distribution around local equilibrium: f local (r,p,t) = exp β [p mv n(r,t)] 2 2m + U(r,t) µ (r,t) 1 1 This leads to a closed set of hydrodynamic equations for a few macroscopic quantities. Note that the condensate is always hydrodynamic n c, v c.
23 Landau-Khalatnikov (LK) Equations Our two-fluid equations can be written in the LK form. For details, see Nikuni & Griffin, PRA 63, (2001). δn t + δj = 0 Density δj µ t = δp x µ δn U + η δv nµ x µ x ν x ν ζ 1 [ ρ s (δv s δv n )]+ ζ 2 δv n x µ + δv nν x µ { } 2 3 δ µν δv n Momentum density δv s t = δµ m ζ 3 [ ρ s (δv s δv n )] ζ 4 δv n Superfluid velocity δs t + (sδv n) = (κ δt ) T Entropy
24 Local Transport Coefficients Thermal conductivity Shear viscosity κ(r) = 5 2 τ κ n 0 k 2 B T m B κ η(r) = τ η n 0 k B TB η Four second (bulk) viscosity coefficients ζ 1 = ζ 4 = gn c0 3m σ Hτ µ, ζ 2 = gn 2 c0 9 σ Hτ µ, ζ 3 = g σ m 2 H τ µ B κ, B η, σ H are the renormalization factors involving the equilibrium thermodynamic quantities. The three relaxation times τ κ,τ η,τ µ determine how fast the system reaches local equilibrium.
25 Transport Relaxation Times Nikuni & Griffin, PRA 65, (2002) Classical relaxation rate 1/t cl =nsv ~ th (v th ~(k B T/m) 1/2 ) Relaxation rate in a trapped BEC gas 1/t BE =n c sv th Relaxation times for the MIT trap t cl >> t BE (position in a trap) Collisions with the condensate enhances the relaxation rate.
26 Spin Dynamics in a Dilute Bose Gas Trapped 87 Rb Gas in JILA Experiment (T>T BEC ) F = 2 rf microwave 2,1 Bloch vector S Ω r Ω z 1,-1 F = 1 Effective 2 two-level system: 1 Ω External field S=1/2 Bose gas Ω r
27 Exciting the Spin Dynamics p/2 pulse: All the atoms are prepared in the coherent superposition. density Bloch vector S x D 2æ 1æ Inhomogeneous D(x): Transverse spin component precesses at different rates, depending on position in trap D(x) f(x) x
28 Nonlinear Dynamics: Spin Segregation Lewandowski et al. PRL 88, (2002) 1æ 2æ n 1 (x) n 2 (x) The mean field is usually negligible for the thermal gas equilibrium properties: gn<<k B T. x
29 Kinetic Theory for a Spin-1/2 Bose Gas See for example, Lhuillier & Laloë, J. Physique 43, 197 (1982). We must include internal coherence in kinetic theory: f ˆ = f 11 f 12 f 21 f 22 f ij (r,p,t) ψ ˆ + j (r,t) ψ ˆ i ( r,t) Work with center of mass & spin variables f (r, p,t) Tr[ f ˆ (r,p,t)] r r σ (r,p,t) Tr[ τ ˆ f ˆ (r,p,t)] r τ ˆ = ( τ ˆ x, τ ˆ y, τ ˆ y ): Pauli matrices Spin kinetic equation σ r t + p m σ r r U ext p σ Ω r n + g 12 h exchange interaction r S σ r = σ r t coll
30 Theoretical Results Williams, Nikuni & Clark, PRL 88, (2002) The exchange interaction induces the longitudinal spin oscillation. We compare numerical simulation result with experimental data.
31 Linear Response: Spin Waves JILA experiment: Mcguirk et al., PRL 89, (2002) Theory: Nikuni, Williams, and Clark, cond-mat/ We study transverse spin fluctuations: r σ σ r r σ = 0 r + δσ { 0,0 n } 0 =, 0 Linearized equation for the transverse spin σ + =σ x +iσ y σ + t + p m σ + U ext p σ + + i g ( h f 0δS + n 0 δσ + ) = σ + t coll We solve for dipole and quadrupole modes by moment method (analytical) and simulation (numerical).
32 Quadrupole spin wave excited by (z) z 2. θ φ total density Beautiful Experimental Data! z D=const. D 0
33 Spin Oscillations Theory Experiment
34 Frequency and Damping
35 Briefly, what happens below T c? Two-component spinor order parameter Φ(r,t) = Φ 1(r,t ) e i = n Φ 2 (r,t) c e iθ c α c 2 cos β c 2 e iα c 2 sin β c 2 Condensate spin vector r S c (r,t ) = n c (cos α c sin β c,sin α c sin β c, cos β c ) How does condensate affect the spin waves? How does thermal cloud affect the condensate spin textures (such as vortex)?
36 Summary Dynamics of a trapped Bose gas at finite T is described by a generalized GP equation for the condensate and a quantum kinetic equation for noncondensate atoms. N-particle simulation of the ZNG kinetic theory by Jackson & Zaremba successfully explains various dynamic phenomena. Two-fluid hydrodynamics in the high-density collisiondominated regime: MIT ( 23 Na), Amsterdam ( 87 Rb), ENS (He * ). Spin-1/2 kinetic theory for the recent JILA experiments.
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