The Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs RHI seminar Pascal Büscher i ( t Φ (r, t) = 2 2 ) 2m + V ext(r) + g Φ (r, t) 2 Φ (r, t) 27 Nov 2008 RHI seminar Pascal Büscher 1 (Stamper-Kurn and Ketterle, 1998)
Contents 1 Introduction 2 Derivation of the Gross-Pitaevskii Equation The Ideal Bose Gas in a Harmonic Trap The Many-Body Hamiltonian The Effective Potential The Gross-Pitaevskii Equation The Thomas-Fermi Approximation 3 Applications of the Gross-Pitaevskii Equation Perturbations to the Ground State The Free Expansion of a BEC Vortices The Collapse of a BEC 4 Summary 27 Nov 2008 RHI seminar Pascal Büscher 2
Introduction (Stamper-Kurn and Ketterle, 1998) Experiments usually involve the evolution of the BEC BECs cannot be understood without understanding the hydrodynamic properties BECs usually contain a large number of atoms many-body problem Find an effective way to describe the evolution of BECs 27 Nov 2008 RHI seminar Pascal Büscher 4
The Ideal Bose Gas in a Harmonic Trap - a first approach 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 2 y 2 + ωz 2 z 2) Problem described [ by the Schrödinger ] Eq for one particle in a HO i t ϕ(r, t) = 2 2 2m + V ext(r) ϕ(r, t) Ground-state solution ( 1 ϕ 0 (r) = πaho 2 with the oscillator length a ho = ) [ exp m ( ωx x 2 + ω y y 2 + ω z z 2)] 2 The particle density is given by n(r) = N ϕ 0 (r) 2 mω ho, average osc freq ω ho = (ω x ω y ω z ) 1/3 27 Nov 2008 RHI seminar Pascal Büscher 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 Density of a cold Bose gas Eg for N = 5000, T = 09T c, spherical trap Looks ok at first glance, but fails to describe the ground state realisitcally ( later) n 0 : condensate, n T : thermal states (Dalfovo et al, 1999) 27 Nov 2008 RHI seminar Pascal Büscher 7
The Many-Body Hamiltonian However, there are interactions between atoms! for exact solutions, we have to solve The Many-Body Hamiltonian Ĥ = dr ˆΨ (r) [ 2 2m 2 + V ext (r) + 1 2 ] dr ˆΨ (r )V (r r ) ˆΨ(r ) ˆΨ(r) with ˆΨ(r) ( ˆΨ (r)) being the particle annihilation (creation) operators for the boson field Krauth, 1996: Calculation for N = 10 4, "hard-sphere" potential via Monte-Carlo method Usually N much larger Direct Calculation heavy/impracticable for most cases 27 Nov 2008 RHI seminar Pascal Büscher 8
Mean-Field approximation In general: ˆΨ(r) = α Ψ α(r)a α, with a α n 0,, n α, = n α n 0,, n α 1, a α n 0,, n α, = n α + 1 n 0,, n α + 1, one particle annihilation op one particle creation op n α given by ˆn α = a αa α If N 1 and (almost) all particles in the ground state (Bogoliubov, 1947): Substitute a 0, a 0 with N 0 ˆΨ(r, t) = Φ(r, t) + ˆΨ (r, t): BEC now described by a scalar function Φ thermal cloud described by a perturbation ˆΨ Here: 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) 2 Φ(r, t) 27 Nov 2008 RHI seminar Pascal Büscher 9
The Effective Potential Gas cold: Atom-atom interaction determined by low-energy 2-particle scattering Gas dilute: mean distance to neighboring particle large only lim r of scattering amplitude is relevant Effective potential is applicable! ( Nov 6th: M Freudenberger) dr V (r r ) Φ(r, t) 2 = dr gδ(r r ) Φ(r, t) 2 = g Φ(r, t) 2 with g = 4π 2 m a, where a is the s-wave scattering length 27 Nov 2008 RHI seminar Pascal Büscher 10
The Gross-Pitaevskii Equation Deploying the mean-field approximation and the effective potential, the time-dependent Schrödinger eq 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 i µt, we obtain the static GPE: µφ(r) = [ 2 2 ] 2m + V ext(r) + g Φ(r) 2 Φ(r) first derived by Eugene P Gross and Lev P Pitaevskii independently in 1961 27 Nov 2008 RHI seminar Pascal Büscher 11
Comparision with experimental data Compare ground-state wave function calculated - - - neglecting interaction (ideal gas) by deploying the GPE with experimental data GPE results in good accordance with experimental data, while the ideal gas fails to describe the BEC N Na = 80000, trap: spherical (Dalfovo et al, 1999; exp data: Hau et al, 1998) 27 Nov 2008 RHI seminar Pascal Büscher 12
The Diluteness of a BEC Weakness of interaction determined by n a 3 1 However, not only interaction-term, but also kinetic energy has to be considered: E int N2 a a 3 ho = N a E kin a ho Na 2 ho is a good quantity describing the diluteness of a BEC BEC wave function for = 0, 1, 10, 100 in a spherical trap N a a ho (Dalfovo et al, 1999) 27 Nov 2008 RHI seminar Pascal Büscher 13
The Thomas-Fermi Approximation Prev slide: for large Na a ho Φ(r) flat 2 Φ(r) is negligible at most parts of the distribution The GPE yields: The Thomas-Fermi approximation { g 1 [µ V n(r) = ext (r)] µ > V ext (r) 0 µ < V ext (r) Analytical calculations possible! Na a ho = 100, distance in a ho (Dalfovo et al, 1999[adapted]) 27 Nov 2008 RHI seminar Pascal Büscher 14
Perturbations to the Ground State Apply the ansatz: 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) where H 0 = 2 2 2m + V ext(r) 27 Nov 2008 RHI seminar Pascal Büscher 16
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 Result: Asymptotic for N ( Hydrodynamics) 27 Nov 2008 RHI seminar Pascal Büscher 17 (Jin et al, 1996 [adapted], Calculations: Edwards et al, Esry et al, Stringari)
Hydrodynamics In the limit of Na a ho 1: hydrodynamic theory of superfluids Use ansatz: Φ(r, t) = n(r, t)e is(r,t) the velocity field is then: v(r, t) = m S(r, t) BEC is irrotational as long as there are no singularities The GPE then reduces to m t v + ( V ext + gn t n + (vn) = 0 ) = 0 Euler equation 2 2m n 2 n + mv 2 2 continuity equation 27 Nov 2008 RHI seminar Pascal Büscher 18
The Free Expansion of a BEC Expanding BEC Use again Φ(r, t) = n(r, t)e is(r,t) with n(r, t) = a 0 (t) 3 i=1 a i(t)r 2 i Ansatz implies that the parabolic shape is conserved for all t S(r, t) determined by n(r, t) Size of the BEC given by R i (t) = R i (0)b i (t) = for axially symmetric traps: a0 (t) a i (t) 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 27 Nov 2008 RHI seminar Pascal Büscher 19
The Free Expansion of a BEC Aspect ratio R /Z over t Settings as in experiment: (a) λ = 0099, (b) λ = 0065 Limit λ 0 Calculations in very good accordance with experimental results (Dalfovo et al, 1999, exp data: (a) Ernst et al, 1998; (b) Stamper-Kurn et al, 1998) 27 Nov 2008 RHI seminar Pascal Büscher 20
Vortices For simplicity: quantized vortex along z-axis Ansatz: φ(r) = n(r, z) exp (iκϕ) with κ the (integer) quantum number of the vortex Angular momentum along z-axis: L z = Nκ The GPE yields [ 2 2 2m + 2 κ 2 2mr 2 + m ( ω 2 2 r 2 + ωz 2 z 2) ] n(r + gn(r, z), t) = µ n(r, t) For κ 0: n(r, t) = 0 along z-axis due to the centrifugal term 27 Nov 2008 RHI seminar Pascal Büscher 21
Vortices Numerical Calculation N Rb = 10 4, spherical trap a ho = 0791µm ground state (κ = 0) vortex with κ = 1 vortex with κ = 1 for an ideal gas inset: xz-plane of the κ = 1 vortex (Dalfovo et al, 1999) 27 Nov 2008 RHI seminar Pascal Büscher 22
The Collapse of a BEC If a < 0: At what N cr does the BEC collapse? Simplified calculation for a spherical trap: N cr a Calculate E with the GPE for a Gaussian ansatz with the width a ho w Look for minimum of E [Φ w ] Find N cr for which local min vanishes a ho 0671 N Exact calculation: cr a a ho = 0575 (Ruprecht et al, 1995) However: dynamics of collapse beyond GPE (Dalfovo et al, 1999) 27 Nov 2008 RHI seminar Pascal Büscher 23
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 K 3 675 10 27 cm 6 /s V ext (r) = 1 2 mω2 ( r 2 + ν2 z 2) +κ 2 k 2 L 2m cos2 (k L z) N(t) 16000 12000 8000 κ = 0, ν = 039 expt, κ = 0, ν = 039 κ = 4, ν = 039 κ = 4, ν = 1 κ = 0, ν = 1 κ = 0, ν = 5 4000 (Adhikari, 2008) a in = 7a 0 a col = -30a 0 0 0 10 20 30 40 50 Time (ms) 27 Nov 2008 RHI seminar Pascal Büscher 24
The Collapse of a BEC - Evolution 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 The GPE fails to completely describe the depletion process! 27 Nov 2008 RHI seminar Pascal Büscher 25 (Adhikari, 2008)
Summary The Gross-Pitaevskii equation (GPE) employs a mean-field approach an effective potential small) BECs with Thomas-Fermi approximation analytic description possible GPE gives a good description of perturbations to the ground state hydrodynamic phenomena, eg free expansion and vortices but cannot describe processes that include depletion, eg the collapse of a BEC and is thus applicable for cold (T = 0), weakly-interacting ( N a a ho Further information: F Dalfovo and S Giorgini et al, Rev Mod Phys, 71, 3, 1999 27 Nov 2008 RHI seminar Pascal Büscher 27
Thank you for your attention! 27 Nov 2008 RHI seminar Pascal Büscher 28