The Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs

The Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs i ( ) t Φ (r, t) = 2 2 2m + V ext(r) + g Φ (r, t) 2 Φ (r, t) (Mewes et al., 1996) 26/11/2009 Stefano Carignano 1

Contents 1 Introduction 2 Derivation of the Gross-Pitaevskii Equation The Ideal Bose Gas in a Harmonic Trap Formalism / approximations The Gross-Pitaevskii Equation 3 Applications of the Gross-Pitaevskii Equation The Thomas-Fermi Approximation Perturbations to the Ground State The Free Expansion of a BEC Vortices The Collapse of a BEC 4 Summary 26/11/2009 Stefano Carignano 2

Motivation (Mewes et al., 1996) Description of the BEC ground state Experiments usually involve the evolution of the BEC BECs cannot be understood without knowing their hydrodynamic properties Experimentally realized BECs in cold atoms are inhomogeneous systems BECs usually contain a large number of atoms many-body problem 26/11/2009 Stefano Carignano 4

The Ideal Bose Gas in a Harmonic Trap Neglect all atom-atom interactions atoms don t "see" each other ( Potential of the trap: V ext (r) = m 2 ω 2 x x 2 + ωy 2y 2 + ωz 2z2) Problem described by the Schrödinger Eq. for one particle HO Ground-state solution ( 1 ϕ 0 (r) = πa 2 ho with the oscillator length a ho = ) [ exp m ( ωx x 2 + ω y y 2 + ω z z 2)] 2 mω ho, average osc. freq. ω ho = ( ) 1/3 ω x ω y ω z 26/11/2009 Stefano Carignano 6

The Ideal Bose Gas in a Harmonic Trap Axially symmetric traps: Define λ := ωz ω ( V ext (r) = m 2 ω2 r 2 + λ 2 z 2) λ < 1 cigar-shaped trap λ = 1 spherical trap λ > 1 pancake-shaped trap 26/11/2009 Stefano Carignano 7

The Ideal Bose Gas in a Harmonic Trap Axially symmetric traps: Define λ := ωz ω ( V ext (r) = m 2 ω2 r 2 + λ 2 z 2) λ < 1 cigar-shaped trap λ = 1 spherical trap λ > 1 pancake-shaped trap Density of a cold Bose gas E.g. for N = 5000, T = 0.9T c, spherical trap n 0 : condensate, n T : thermal states (Dalfovo et al., 1999) 26/11/2009 Stefano Carignano 7

The Ideal Bose Gas in a Harmonic Trap Axially symmetric traps: Define λ := ωz ω ( V ext (r) = m 2 ω2 r 2 + λ 2 z 2) λ < 1 cigar-shaped trap λ = 1 spherical trap λ > 1 pancake-shaped trap Density of a cold Bose gas E.g. for N = 5000, T = 0.9T c, spherical trap Is the ground state really like that? n 0 : condensate, n T : thermal states (Dalfovo et al., 1999) 26/11/2009 Stefano Carignano 7

The formalism Interactions between atoms: can we neglect them? To get exact solutions, we should start from The Many-Body Hamiltonian Ĥ = dr ˆΨ (r) [ 2 2m 2 + V ext (r) + 1 ] dr ˆΨ (r )V (r r ) ˆΨ(r ) ˆΨ(r) 2 where ˆΨ(r) and ˆΨ (r) are the boson field operators. Krauth, 1996: "Direct" calculation for N = 10 4, "hard-sphere" potential via Monte-Carlo method. Usually N is much larger Direct calculations are numerically too heavy for most cases 26/11/2009 Stefano Carignano 8

So we start approximating... In general: ˆΨ(r) = α ψ α(r)â α, with â α n 0,..., n α,... = n α n 0,..., n α 1,... â α n 0,..., n α,... = n α + 1 n 0,..., n α + 1,... If N 1 and (almost) all particles are in the ground state (Bogoliubov, 1947): â 0 n 0,... â 0 n 0,... N 0 n 0,... More generally: ˆΨ(r) = ψ 0 (r)â 0 + ψ α (r)â α N 0 ψ 0 (r) + ψ α (r)â α α>0 ˆΨ(r, t) = Φ(r, t) + ˆΨ (r, t) α>0 BEC now described by a scalar "condensate wave function" Φ Thermal cloud described by a perturbation ˆΨ 26/11/2009 Stefano Carignano 9

The Effective Potential If the gas is cold and dilute... Atom-atom interaction are determined by low-energy 2-particle scattering. The mean distance to neighboring particle is large Only the long range properties of the potential are relevant. Meaning that we can write V (r r ) gδ(r r ) with g = 4π 2 a, where a is the S-wave scattering length m 26/11/2009 Stefano Carignano 10

The equation of motion Heisenberg equation for the field operator ˆΨ(r, t): i ˆΨ(r, t) = [ ˆΨ(r, t), Ĥ ] t ] = [ 2 2 2m + V ext(r) + dr ˆΨ (r, t)v (r r ) ˆΨ(r, t) ˆΨ(r, t) Assume T=0 neglect perturbation-term ˆΨ (r): i [ ] t Φ(r, t) = 2 2 2m + V ext(r) + dr V (r r )Φ (r, t)φ(r, t) Φ(r, t) Insert the effective potential V (r r ) = gδ(r r ) What we get is... 26/11/2009 Stefano Carignano 11

The Gross-Pitaevskii Equation Deploying the mean-field approximation and the effective potential yields The Gross-Pitaevskii equation (GPE) i [ ] t Φ(r, t) = 2 2 2m + V ext(r) + g Φ(r, t) 2 Φ(r, t) With Φ(r, t) = Φ(r)e iµt/, we obtain the static GPE: ] µφ(r) = [ 2 2 2m + V ext(r) + g Φ(r) 2 Φ(r) derived by Eugene P. Gross and Lev P. Pitaevskii independently in 1961 26/11/2009 Stefano Carignano 12

Ground state Compare with experimental data the ground-state wave function calculated - - - neglecting interaction (ideal gas) by deploying the GPE The ideal gas description fails to describe the BEC! GPE results instead reproduce experimental data very well N Na = 80000, trap: spherical (Dalfovo et al., 1999; exp. data: Hau et al., 1998) 26/11/2009 Stefano Carignano 14

Relevant scales in the system The Gross Pitaevskii eq. is expected to work well if n a 3 1 few particles in a "scattering volume" a 3 Typical values: a 1 6nm, n 10 13 10 15 cm 3 n a 3 < 10 3 Why cannot we neglect interactions? E int E kin gn n N2 a a 3 ho N ω ho Na 2 ho = N a a ho Typical values: a /a ho 10 3 N a a ho 10 10 4 26/11/2009 Stefano Carignano 15

The ground state The interaction strongly modifies the shape of the ground state Dependence on N a > 0 repulsive interaction broadening of the wave function Φ(r) gets "flattened" BEC wave function for various N a a ho in a spherical trap (Dalfovo et al., 1999) 26/11/2009 Stefano Carignano 16

The Thomas-Fermi Approximation For large N a a ho Φ(r) flat 2 Φ(r) is negligible at most parts of the distribution In this limit the GPE gives The Thomas-Fermi approximation n(r) = { g 1 [µ V ext (r)] µ > V ext (r) 0 µ < V ext (r) Analytical results! N a a ho = 100, distance in a ho (Dalfovo et al., 1999[adapted]) 26/11/2009 Stefano Carignano 17

Dynamics: Perturbations to the Ground State Dynamics: time-dependent GP eq. Write Φ(r, t) = e iµt/ [ φ(r) + u(r)e iωt + v (r)e iωt] Linearize the GPE ωu(r) = [ H 0 µ + 2gφ 2 (r) ] u(r) + gφ 2 (r)v(r) ωv(r) = [ H 0 µ + 2gφ 2 (r) ] v(r) + gφ 2 (r)u(r) with H 0 = 2 2 2m + V ext(r) 26/11/2009 Stefano Carignano 18

Perturbations to the Ground State - Example: m L = 0 and m L = 2 oscillations 87 Rb atoms, axially sym. trap (λ = 8) (Jin et al., 1996 [adapted]) Calculate eigenfrequencies of oscillations for different N Asymptotic regime for N Hydrodynamics (Jin et al., 1996 [adapted], Calculations: Edwards et al., Esry et al., Stringari) 26/11/2009 Stefano Carignano 19

Hydrodynamics Limit N a a ho 1: hydrodynamic theory of superfluids Write Φ(r, t) = n(r, t)e is(r,t) The time-dependent Gross Pitaevskii equation gives t n + (vn) = 0 continuity equation with the velocity field v(r, t) = S(r, t) m BEC is irrotational The GPE gives also m ( t v + V ext + gn 2 2m n 2 n + mv ) 2 = 0 2 Neglecting kinetic pressure Euler equation for frictionless hydrodynamics the BEC is superfluid 26/11/2009 Stefano Carignano 20

The Free Expansion of a BEC Expanding BEC Start with Thomas-Fermi Use the hydrodynamic equations Ansatz n(r, t) = a 0 (t) 3 i=1 a i(t)r 2 i is a solution of the equations Parabolic shape is conserved for all t scaling Size of the BEC given by R i (t) = R i (0)b i (t) For axially symmetric traps: t = 1ms, 5ms, 10ms, 20ms, 30ms, 45ms (Mewes et al., 1996) d 2 dt 2 b = ω2 b 3 b ; z Solve numerically d 2 dt 2 b z = ω2 z b 2 b2 z 26/11/2009 Stefano Carignano 21

The Free Expansion of a BEC For λ << 1 simple analytical solutions for the b i (t) can be found Aspect ratio R /Z over t Settings as in experiment: (a) λ = 0.099, (b) λ = 0.065 Ideal gas (Dalfovo et al., 1999, exp. data: (a) Ernst et al., 1998; (b) Stamper-Kurn et al., 1998) 26/11/2009 Stefano Carignano 22

Vortices in a superfluid Recall that Φ = ne is, v = m S from which v = 0 irrotational flow Φ single-valued phase circulation = 2πl, l integer S ds = m v ds = 2πl v ds = 2πl m = h m l κl (Onsager & Feynman) Tangential velocity in a vortex: v ds = 2πrv(r) = κl v = κl 2πr 26/11/2009 Stefano Carignano 23

Vortices in the BCE Quantized vortex along z-axis φ(r) = n(r, z) exp (ilϕ), l integer Angular momentum along z-axis: L z = Nl Tang. velocity v T = mr l The GPE yields (Ketterle et al., MIT) [ 2 2 2m + 2 l 2 2mr 2 + m ] ( n(r ω 2 2 r 2 + ω2 z z2) + gn(r, z), z) = µ n(r, z) For l 0: n(r, z) = 0 along z-axis due to the centrifugal term 26/11/2009 Stefano Carignano 24

Vortices Numerical Calculation N Rb = 10 4, spherical trap a ho = 0.791µm ground state (l = 0) vortex with l = 1 vortex with l = 1 for an ideal gas inset: xz-plane of the l = 1 vortex (Dalfovo et al., 1999) 26/11/2009 Stefano Carignano 25

The Collapse of a BEC If a < 0: At what N cr does the BEC collapse? Simplified calculation for a spherical trap: Calculate the GP energy for a gaussian ansatz with width w Look for minimum of E(w) Find N cr for which local min. vanishes (Dalfovo et al., 1999) 26/11/2009 Stefano Carignano 26

The Collapse of a BEC - Evolution add term to the GPE that describe three-body recombination: [ i t 2 2 2m + V ext(r) + g Φ(r, t) 2 i ] 2 K 3 Φ(r, t) 4 Φ(r, t) = 0 V ext (r) = 1 2 mω2 ( r 2 + ν2 z 2) + κ 2 k 2 L 2m cos2 (k L z) Problem K 3 is not known K 3 has to be fitted to experimental data However: no choice of K 3 yields correct values for all observables at the same time N(t) 16000 12000 8000 4000 κ = 0, ν = 0.39 expt, κ = 0, ν = 0.39 κ = 4, ν = 0.39 κ = 4, ν = 1 κ = 0, ν = 1 κ = 0, ν = 5 a in = 7a 0 a col = -30a 0 0 0 10 20 30 40 50 Time (ms) 26/11/2009 Stefano Carignano 27 (Adhikari, 2008)

Summary The Gross-Pitaevskii equation (GPE) employs a mean-field approach an effective potential and is thus applicable for cold (T = 0), dilute ( n a 3 1) BECs GPE gives a good description of perturbations to the ground state hydrodynamic phenomena, e.g. free expansion and vortices... but cannot describe processes that include depletion, like the collapse of a BEC Further information: F. Dalfovo and S. Giorgini et al., Rev. Mod. Phys., 71, 3, 1999 26/11/2009 Stefano Carignano 29

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Gross Pitaevskii energy functional: E[Φ] = dr [ 2 2m Φ 2 + V ext Φ 2 + g Φ 4 ] plus constraint δe δφ = 0 dr Φ 2 = N lagrange multiplier µ 26/11/2009 Stefano Carignano 31