Roton Mode in Dipolar Bose-Einstein Condensates

Transcription

1 Roton Mode in Dipolar Bose-Einstein Condensates Sandeep Indian Institute of Science Department of Physics, Bangalore March 14, 2013

2 BECs vs Dipolar Bose-Einstein Condensates Although quantum gases are very dilute systems, most of their properties are governed by the interaction between particles. In the ultracold regime, only s-wave scattering between the bosons can take place. This allows one to replace the real interatomic potential by a pseudo-potential U contact (r) = 4π 2 a m δ(r) The interatomic potential is thus short range, isotropic, and characterized by a single parameter, the s-wave scattering length a. In DBECs bosonic particles interact via the long-range, anisotropic dipole-dipole interaction, in addition to the short-range and isotropic contact interaction. (QCMJC) Dipolar Quantum Gases 2 / 27

3 Dipole-Dipole interaction The energy due to the dipole-dipole interaction U dd (r) = C dd (e 1 e 2 ) r 2 3 (e 1 r) (e 2 r) 4π r 5. C dd = µ 0 µ 2 (magnetic dipoles), C dd = d 2 /ε 0 (electric dipoles) For a polarized sample where all dipoles point in the same direction z U dd (r) = C dd 1 3 cos 2 θ 4π r 3 Dipole-dipole interaction is long-range ( 1/r 3 ) and anisotropic. (QCMJC) Dipolar Quantum Gases 3 / 27

4 Dipole-Dipole interaction As θ varies between 0 and π/2, the factor 1 3 cos 2 θ varies between 2 and 1. e 1 (a) r e 2 e 1 (b) θ r e 2 DDI is repulsive for particles sitting side by side, while it is attractive for dipoles in a head-to-tail orientation. (c) (d) For a central potential falling off at large distances like 1/r n. δ l (k) k 2l+1 for l < (n 3)/2 δ l (k) k n 2 otherwise One cannot replace the true potential by a short-range, isotropic contact interaction for dipolar BECs. T. Lahaye et al., Rep. Prog. Phys. 72, (2009) (QCMJC) Dipolar Quantum Gases 4 / 27

5 Electric vs magnetic dipoles In order to realize a quantum gas with significant DDI, one can use particles having either an electric dipole moment d, or a magnetic dipole moment µ. Usually, the dipolar coupling is much higher in the electric case. For atomic systems, d q e a 0, where q e is the electron charge and a 0 the Bohr radius. while µ µ B. Hence µ 0 µ 2 d 2 /ε 0 α , where α 1/137 is the fine structure constant. By equating the characteristic energy scale of DDI C dd /r 3 with zero point kinetic energy 2 /(mr 2 ), one obtains the typical length scale for DDI a dd = C ddm 3 2 (QCMJC) Dipolar Quantum Gases 5 / 27

6 Dipolar vs contact interactions The ratio of a dd to the a gives an estimate of the relative strengths of DDI against the contact interaction. ɛ dd = a dd a ɛ dd = for 87 Rb and 0.16 for 52 Cr. = 4πC dd 3U 0 A very appealing feature of 52 Cr is the existence of several Feshbach resonances. These allow to tune the scattering length a, Close to resonance, a varies with the applied external magnetic field B as ( a = a bg 1 ) B B 0 (QCMJC) Dipolar Quantum Gases 6 / 27

7 Non-local Gross-Pitaevskii equation To include dipolar effects, we need to add an extra term to the mean-field potential g ψ 2 to account for the effect of the DDI i ψ t = 2 2m ψ + ( V ext + g ψ 2 + Φ dd ) ψ. where Φ dd Φ(r, t) = ψ(r, t) 2 U dd (r r ) d 3 r. is the dipolar contribution to the mean field interaction For numerical calculations, it is convenient to use the Fourier transform of the DDI Ũ dd (k) = U dd (r)e ik r d 3 r ( = C dd cos 2 α 1/3 ), where α is the angle between k and the polarization direction. (QCMJC) Dipolar Quantum Gases 7 / 27

8 Excitations in Homogeneous gas The properties of elementary excitations may be investigated by considering small deviations of the state of the gas from equilibrium and finding periodic solutions of the non-local GP equation. An equivalent approach is to use the hydrodynamic formulation - Substitute ψ = n exp[iφ] in non-local GP equation. m v t where v φ n + (nv) = 0, t ( mv 2 = 2 + gn + V ext + Φ dd 2 ) n, 2m n Linearisng the above eqs. and considering v and perturbation in density (δn)as small, one can obtain the travelling wave solutions [δn exp(iq.r ωt)] (QCMJC) Dipolar Quantum Gases 8 / 27

9 Excitations in Homogeneous gas ω = k [ n 0 g + C ] dd m 3 (3 cos2 α 1) + 2 k 2 4m 2, For C dd /3g > 1, implies that a dipolar uniform condensate is unstable as phonons (k 0) acquire imaginary frequencies. Homogeneous condensate with attractive contact interactions (a < 0) is unstable, as the Bogoliubov excitations have imaginary frequencies at low momentum. (QCMJC) Dipolar Quantum Gases 9 / 27

10 Trapped Dipolar Gases A prominent effect of the DDI is to elongate the condensate along the direction of polarization. The density distribution is given, in the Thomas-Fermi limit n(r) = n 0 (1 r 2 /R 2 ) To first order in ɛ dd the mean-field dipolar potential created by this distribution Φ dd (r) = ɛ ddmω 2 ( 1 3 cos 2 θ ) 5 r 2 if r < R R 5 r 3 if r > R Stability of trapped dipolar Bose-Einstein condensates depends strongly on the trapping geometry. T. Lahaye et al., Rep. Prog. Phys. 72, (2009) (QCMJC) Dipolar Quantum Gases 10 / 27

11 Effect of trap geometry on condensates stability A BEC with pure contact attractive interactions (a < 0) is unstable in the homogeneous case. In a trap, an attractive BEC is stable if N a a ho 0.58 Due to the anisotropy of the DDI, the partially attractive character of the interaction can be hidden by confining the atoms more strongly in the direction of polarization. (a) (b) T. Lahaye et al., Rep. Prog. Phys. 72, (2009) (QCMJC) Dipolar Quantum Gases 11 / 27

12 Dipolar condensates in oblate traps condensate of dipolar particles harmonically confined in the direction of the dipoles (z) and uniform in two other directions (ρ = {x, y}) i { t ψ(r, t) = 2 2m + m 2 ω2 z 2 + g ψ(r, t) 2 } +d 2 dr V d (r r ) ψ(r, t) 2 ψ(r, t), The ground state wave function (ψ 0 (z) exp ( iµt) ) is independent of the in-plane coordinate ρ. Then, integrating over dρ in the dipole-dipole term where g d = 8πd 2 /3 { 2 2m + m } 2 ω2 z 2 + (g + g d )ψ0(z) µ 2 ψ 0 (z) = 0, For µ ω, using TF approx. ψ 2 0 (z) = n 0(1 z 2 /L 2 ) where n 0 = µ/(g + g d ) and L = (2µ/mω 2 ) 1/2 (QCMJC) Dipolar Quantum Gases 12 / 27

13 Linearizing GP eq. around the ground state solution ψ 0 (z), [by adding δψ 0 (r) = u(z)e i(qρ+ωt) + v (z)e i(qρ+ωt) to ψ 0 (z)], one obtains the Bogoliubov-de Gennes (BdG) equations for the excitations where ɛf = [ 2 2m dz 2 + q2 + ψ ] 0 f + H kin f +, ψ 0 ɛf + = H kin f + H int [f ], H int [f ] = 2(g d + g)f (z)ψ 2 0(z) (3/2)g d q ψ 0 (z) dz f (z )ψ 0 (z ) exp ( q z z ). The term in blue originates from the non-local character of the dipole-dipole interaction and gives rise to the momentum dependence of an effective coupling strength. L. Santos et al., Phys. Rev. Lett. 90, (2003) (QCMJC) Dipolar Quantum Gases 13 / 27

14 Solutions of BdG equations In the limit of low in-plane momenta ql 1, the term in blue can be neglected. BdG equations become identical to those for the condensate with the contact interaction characterized by coupling constant (g d + g). Dispersion relation ɛ 0 (q) = c s q; c s = (2µ/3m) 1/2. For ql 1, the excitations become 3D and the effective coupling strength decreases. The interaction term is then reduced to H int [f ] = (2g g d )ψ 2 0(z)f (z) The eqs. are similar to the BdG equations for the excitations of a condensate with short-range interactions characterized by a coupling constant (2g g d ). L. Santos et al., Phys. Rev. Lett. 90, (2003) (QCMJC) Dipolar Quantum Gases 14 / 27

15 Roton excitation If the parameter β = g/g d > 1/2, this coupling constant is positive and one has excitation energies which are real and positive for any momentum q and condensate density n 0. For β < 1/2, the coupling constant is negative and one easily shows that at sufficiently large density the condensate becomes unstable. For ql 1, in TF approximation the dispersion relation ɛ 2 (q) = E 2 q + ɛ 2 j (q) = E 2 q + 2 ω 2 ; β = 1/2, ql 1. (2β 1)(5 + 2β) 3(1 + β)(2 + β) E qµ + 2 ω 2 ; ql 1. L. Santos et al., Phys. Rev. Lett. 90, (2003) (QCMJC) Dipolar Quantum Gases 15 / 27

16 Tunability of roton excitation For β close to 1/2, the roton minimum is located at q = (16µδ/15 ω) 1/2 1/l 0 where δ = 1/2 β, l 0 = ( /mω) 1/2. And ɛ min = [ 2 ω 2 (8µδ/15) 2 ] 1/2 For µδ/ ω = 15/8 the minimum energy reaches zero at q = 2/l 0. (QCMJC) Dipolar Quantum Gases 16 / 27

17 Tunability of DDI By using a rotating polarizing field, it is possible to tune (decrease) the DDI. For the case of magnetic dipoles, the polarizing magnetic field B(t) = Be(t) with e(t) = cos ϕe z + sin ϕ [cos(ωt)e x + sin(ωt)e y ] U dd (t) = C dd 1 3 cos 2 [ θ 3 cos 2 ] ϕ 1 4π r 3 2 Ω ϕ Ω θ r The last factor between brackets decreases from 1 to 1/2 when the tilt angle ϕ varies from 0 to π/2, and vanishes when ϕ is equal to the magic angle θ m. (QCMJC) Dipolar Quantum Gases 17 / 27

18 How does roton mode manifest itself? It has been suggested that boundaries in superfliud 4 He, including vortex cores, should give rise to radial density oscillations whose length scale is characteristic of the roton wavelength. Calculations of vortex states in a DBEC in a highly oblate trap have exhibited similar radial structures. The ground state wavefunction of the condensate with a vortex at center is of form Ψ(r, t) = ψ(ρ, z)e ikϕ The presence of the centrifugal term ( m 2 /ρ 2 ) acts as a perturbation, leading to the mixing of the ground state wavefunction with roton mode. R. M. Wilson et al., Phys. Rev. Lett. 100, (2008) (QCMJC) Dipolar Quantum Gases 18 / 27

19 Ψ f roton U Ψ e roton f roton. A perturbation produced blue-detuned laser along the trap axis, taking the form U (r) = A exp ( ρ 2 /2ρ 2 0 ) can also excite the roton mode, and hence leading to the ripples in the density profile. Ψ(z = 0) ρ/a ho R. M. Wilson et al., Phys. Rev. Lett. 100, (2008) D = D = U (r) + U (r) Trap Potential ( hωρ) (QCMJC) Dipolar Quantum Gases 19 / 27

20 Critical Superfluid Velocity in a Trapped Dipolar Gas Landau critical velocity v L below which elementary excitations in the fluid could not be excited while conserving energy and momentum The breaking of superfluidity in the simulations can be quantified by calculating the depletion of the condensate. Ψ(r, t) ψ 0 (ρ, z)e iµt + { c j (t)u j (ρ, z)e i(mϕ ω j t) j +c j (t)v j (ρ, z)e i(mϕ ω j t) } e iµt where ω j is the quasiparticle energy, m is the projection of the quasiparticle momentum onto the z-axis and µ is the chemical potential of the ground state. The quasiparticle occupations are then given by n j (t) = c j (t) 2 dr ( u j (r ) 2 + v j (r ) 2 ). (QCMJC) Dipolar Quantum Gases 20 / 27

21 Decrease in v L due to roton excitation x 10 5 ntot D = 17.5 D = 70.1 D = v (a ρ ω ρ ) D Nd 2 and a ρ = /Mω ρ ω/ωρ ω/ωρ 20 a) 15 D = c) 15 D = k ρ a ρ b) D = d) 15 D = k ρ a ρ R. M. Wilson et al., Phys. Rev. Lett. 104, (2010) (QCMJC) Dipolar Quantum Gases 21 / 27

22 Anisotropic superfluidity in a dipolar Bose gas Time-dependent GPE for the quasi-2d system, } i t ψ = { 2 2m 2 + V p + g ψ 2 + g d Φ ψ Φ (ρ, t) = 4π 3 F 1 [ñ(k, t)f( klz 2 )], by the convolution theorem. Here, F is the 2D Fourier transform operator and ñ(k, t) = F[n(ρ, t)] F(q) = cos 2 (α)f (q) + sin 2 (α)f (q) where α is the angle between ẑ and the polarization vector ˆd. F (q) = π(q 2 d/q)e q2 erfc(q) F (q) = 2 3 πqe q2 erfc(q) where q d is the wave vector along the direction of the projection of d onto the xy plane. (QCMJC) Dipolar Quantum Gases 22 / 27

23 The dispersion relation of a homogeneous q2d DBEC is given in Bogoliubov theory by ω(k) = k k 2 g ( 1 + 4π 3 βf ( klz 2 )). For α = 0 (polarization along the trap axis) this dispersion does not depend on the direction of the quasiparticle propagation. However, for α 0, or for nonzero projection of ˆd onto the x-y plane, the direction of k becomes important in describing the quasiparticles of the system. As v L is given in terms of the dispersion relation ω(k), it then also depends on the direction of k, and thus is an anisotropic quantity. U. R. Fischer, Phys. Rev. A 73, (R) (2006); C. Ticknor et al., Phys. Rev. Lett. 106, (2011) (QCMJC) Dipolar Quantum Gases 23 / 27

24 ω (µ ) vl (ξ /τ ) /4 1/2 η/π k (1/ξ ) y/ξ* b) x/ξ* Density c) x,y/ξ* C. Ticknor et al., Phys. Rev. Lett. 106, (2011) (QCMJC) Dipolar Quantum Gases 24 / 27

25 Anisotropic drag force The force at time t is given by F(t) = d 2 ρ ψ(ρ, t) 2 Vp (ρ, t) Fx (µ /ξ ) 20 x a) A p = 0.1µ, σ p = 2.0ξ v/c Maximum vortex number A p = 1.0µ, σ p = 2.0ξ 5 b) v/c T Winiecki, J. Phys. B: At. Mol. Opt. Phys. 33, 4069, (2000); C. Ticknor et al., Phys. Rev. Lett. 106, (2011) (QCMJC) Dipolar Quantum Gases 25 / 27

26 Conclusions The pancake-shaped dipolar condensates can support roton excitations, which can tuned by changing density, DDI, trapping potential, etc. Roton mode leads to the decrease in critical superfluid velocity For the dipoles with non-zero projection on xy-plane, the dispersion relation is anisotropic. This in turn leads to anisotropic superfluidity. Superfluid is more robust in the direction parallel the direction of the projection of d on xy-plane. (QCMJC) Dipolar Quantum Gases 26 / 27

27 THANK YOU (QCMJC) Dipolar Quantum Gases 27 / 27