Nonlinear BEC Dynamics by Harmonic Modulation of s-wave Scattering Length

Transcription

1 Nonlinear BEC Dynamics by Harmonic Modulation of s-wave Scattering Length I. Vidanović, A. Balaž, H. Al-Jibbouri 2, A. Pelster 3 Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia 2 Institut für Theoretische Physik, Freie Universität Berlin, Germany 3 Fachbereich Physik, Universität Duisburg-Essen, Germany Supported by Serbian Ministry of Science (ON4035, ON707, PI-BEC), DAAD - German Academic and Exchange Service (PI-BEC), and European Commission (EGI-InSPIRE, PRACE-IP and HP-SEE). Universität Duisburg-Essen December 9, 200

2 Overview Introduction Introduction Experiments with ultracold atoms BEC with modulated interaction - recent experiment Mean-field description Gaussian approximation Spherically symmetric BEC Perturbative approach GP numerics Cylindrically symmetric BEC Experimental setup

3 Experiments with ultracold atoms Experiments with ultracold atoms BEC with modulated interaction Intensive progress in the field of ultracold atoms has been recognized by Nobel prize for physics in 200 Cold alkali atoms: Rb, Na, Li, K... T nk, ρ 0 4 cm 3 Cold bosons, cold fermions Harmonic trap, optical lattice Short-range interactions, long-range dipolar interactions Tunable quantum systems concerning dimensionality, type and strength of interactions

4 BEC with modulated interaction Experiments with ultracold atoms BEC with modulated interaction Motivation - recent experiment by Randy Hulet s group at Rice University and by Vanderlai Bagnato s group at São Paulo University: PRA 8, (200) BEC of 7 Li is confined in a cylindrical trap Time-dependent modulation of atomic interactions via a Feshbach resonance Excitation of the lowestlying quadrupole mode 300 µm Interesting setup for studying nonlinear BEC dynamics

5 Mean-field description Introduction Mean-field description Gaussian approximation Gross-Pitaevskii equation assuming T = 0 (no thermal excitations) ] ψ( r, t) ı = [ 2 + V ( r) + g ψ( r, t)) 2 ψ( r, t) t 2m ψ( r, t) is a condensate wave-function V ( r) = 2 mω2 ρ(ρ 2 + λ 2 z 2 ) is a harmonic trap potential, l = /mω ρ is a characteristic harmonic oscillator length effective interaction between atoms is given by gδ( r) g = 4π 2 Na m, a is s-wave scattering length, N is number of atoms in the condensate

6 Feshbach resonance Introduction Mean-field description Gaussian approximation Scattering length depends on external magnetic field PRL 02, : 7 Li ) a(b) = a BG ( + B B a BG = 24.5 a 0, B = G, 0 = 92 G Scattering Length (a 0 ) Magnetic Field (G) Scattering length can be modulated using external magnetic field via a Feshbach resonance B(t) = B av + δb cos Ωt, a(t) a av + δa cos Ωt a av = a(b av), δa = abg δb (B av B ) 2 B av = 565 G, δb = 4 G, a av 3a 0, δa 2a 0

7 Gaussian approximation () Mean-field description Gaussian approximation To simplify calculations and to obtain analytical insight, we approximate density of atoms by a Gaussian For an axially symmetric trap [ ψ(ρ, z, t) = C(t) exp ] [ ρ 2 2 u(t) + 2 ıρ2 A u(t) exp ] z 2 2 v(t) + 2 ız2 A v(t) By extremizing corresponding action, we obtain two ordinary differential equations, PRL 77, 5320 (996) In the dimensionless form Interaction: p(t) = d 2 u(t) + u(t) dt 2 u(t) p(t) 3 u(t) 3 v(t) = 0 d 2 v(t) dt 2 + λ 2 v(t) v(t) 3 2 Na(t)/l π p(t) u(t) 2 v(t) 2 = 0

8 Gaussian approximation (2) Mean-field description Gaussian approximation Using this type of approximation and relying on the linear stability analysis, frequencies of low-lying collective modes have been analytically calculated Equilibrium widths u 0 = u 3 0 Linear stability analysis δu + δu + p, λ 2 v u 3 0 = + p 0 v0 v0 3 u 2 0 v2 0 u(t) = u 0 + δu(t), v(t) = v 0 + δv(t) ( p ) + δv p u 4 0 u 4 0 v0 u 3 0 v2 0 ( δv + δv λ p v0 4 u 2 0 v3 0 ) + δu 2p u 3 0 v2 0 = 0 = 0

9 Gaussian approximation (3) Mean-field description Gaussian approximation Previous system of equations can be decoupled by a linear transformation As a result, we have frequencies of two low-lying modes - quadrupole mode ω Q0 and breathing mode ω B0 ω B0,Q0 = [ ( ) ] ( ) 2 ( ) λ 2 p ± λ 2 + p + 8 p 4u 2 0 v3 0 4u 2 0 v3 0 4u 3 0 v2 0 Quadrupole mode ω Q0 Breathing mode ω B0 p = 5, λ = 0.02, ω Q0 = 2π 8.2 Hz, ω B0 = 2π 462 Hz

10 Gaussian approximation (4) Mean-field description Gaussian approximation Due to the nonlinear form of the underlying GP equation, we have nonlinearity induced shifts in the frequencies of low-lying modes (beyond linear response) Our aim is to describe collective modes induced by harmonic modulation of interaction p(t) p + q cos Ωt q - modulation amplitude, Ω - modulation frequency For Ω close to some BEC eigenmode we expect resonances - large amplitude oscillations and role of nonlinear terms becomes crucial

11 Perturbative approach GP analysis - () d 2 u(t) dt 2 + u(t) u(t) 3 p u(t) 4 q cos Ωt = 0 u(t) 4 p = 0.4, q = 0., u(0) = u 0, u(0) = 0 u(t) =, analytics =, numerics u(t) = 2, analytics = 2, numerics Linear stability analysis: ω 0 = Dynamics depends on Ω u(t) = 2.04, numerics t u(t) = 4, analytics = 4, numerics t t t

12 (2) Perturbative approach GP analysis Resonant behaviour for Ω ω 0 and Ω 2ω 0 00 q = 0.3 q = Amplitude Clearly, collective modes are shifted

13 () Introduction Perturbative approach GP analysis We look at the Fourier transform of u(t), p = 0.4, q = 0. and Ω = = e-05 e

14 (2) Introduction Perturbative approach GP analysis We have two basic modes Ω and ω 0 and many higher-order harmonics 00 = 2 00 = = =

15 (3) Introduction Perturbative approach GP analysis of the breathing mode is significantly shifted in the resonant region 0 = 00 =

16 Perturbative approach - method Perturbative approach GP analysis Linearization of the variational equation yields for vanishing driving q = 0 zeroth order collective mode ω = ω 0 of oscillations around the time-independent solution u 0 : ω 0 = + 3 u p, u u 5 0 p = 0 0 u 4 0 u 3 0 To calculate the collective mode to higher orders, we rescale time as s = ωt: ω 2 ü(s) + u(s) u(s) p 3 u(s) q Ωs cos 4 u(s) 4 ω = 0 We assume the following perturbative expansions in q: u(s) = u 0 + q u (s) + q 2 u 2(s) + q 3 u 3(s) +... ω = ω 0 + q ω + q 2 ω 2 + q 3 ω

17 Perturbative approach - method Perturbative approach GP analysis This leads to a hierarchical system of equations: ω 2 0 ü (s) + ω 2 0 u (s) = u 4 0 sin Ωs ω ω0 2 ü 2(s) + ω0 2 u 2(s) = 2ω 0 ω ü (s) 4 u (s) sin Ωs u 5 0 ω + α u(s)2 ω0 2 ü 3(s) + ω0 2 u 3(s) = 2ω 0 ω 2 ü (s) 2β u (s) 3 + 2α u (s)u 2(s) ω 2 ü (s) + 0 u (s) 2 sin Ωs u 6 0 ω 4 u 2(s) sin Ωs u 5 2ω0 ω ü2(s) 0 ω where α = 0p/u /u5 0 and β = 0p/u /u6 0. We determine ω and ω 2 by imposing cancellation of secular terms - Poincaré-Lindstedt method

18 Perturbative approach - method Perturbative approach GP analysis Secular term - explanation ẍ(t) + ω 2 x(t) + C cos(ωt) = 0 x(t) = A cos(ωt) + B sin(ωt) C 2ω t sin(ωt) }{{} linear in t In order to have properly behaved perturbative expansion, we impose cancellation of secular terms by appropriately adjusting ω and ω 2 Another way of reasoning u(t) = A cos ωt+a t sin ωt A cos ωt cos ωt+ A sin ωt sin ωt ω u(t) A cos[(ω ω)t], ω = A A

19 Perturbative approach - results Perturbative approach GP analysis of the breathing mode vs. driving frequency Ω Result in second order of perturbation theory ω = ω 0 + q 2 Polynomial(Ω) (Ω 2 ω 2 0 )2 (Ω 2 4ω 2 0 ) analytical numerical analytical numerical p = 0.4, q = 0. p =, q = 0.2

20 GP analysis () Introduction Perturbative approach GP analysis Comparison of the solution of time-dependent GP equation with solution obtained using Gaussian approximation, p = 0.4, q = variational, = GP numerics, = variational, =2 GP numerics, =2 Condensate width.35.3 Condensate width Time Time Good quantitative agreement even for long times

21 GP analysis (2) Introduction Perturbative approach GP analysis e+06 Comparison becomes even more evident 0000 by looking at the 000 Fourier spectrum of 00 the solution of GP equation, p = 0.4, 0 q = 0.2, Ω = 2 GP numerics GP numerics 2 GP numerics

22 Experimental setup p =, q = 0.2, λ = 0.3 Linear stability analysis: quadrupole mode ω Q0 = , monopole mode ω B0 = = 0.4, analytics = 0.4, numerics =, analytics =, numerics v(t) u(t) t = 0.4, analytics = 0.4, numerics t v(t) u(t) t =, analytics.2 =, numerics t

23 () Introduction Experimental setup We have three basic modes: ω Q, ω B, Ω and many higher-order harmonics Axial width, = 0.4 Radial width, = Axial width, = 0.4 Q B

24 (2) Introduction Experimental setup of quadrupole mode ω Q versus driving frequency Ω of breathing mode ω B versus driving frequency Ω Q0 analytics numerics Poles: ω Q0, ω B0 ω Q0, 2ω Q0, ω Q0 + ω B0, ω B B0.992 analytics numerics Poles: ω Q0, ω B0 ω Q0, ω B0, ω Q0 + ω B0, 2ω B0

25 Experimental setup Experimental setup - condensate dynamics Comparison of the solution of time-dependent GP equation with Gaussian approximation p = 5, q = 0, λ = 0.02 and Ω = 0.05 Axial condensate width variational GP numerics Time Radial condensate width variational GP numerics Time

26 Experimental setup - frequency shifts Experimental setup In the experiment: ω B >> ω Q, Ω (0, 3ω Q ), large modulation amplitude Strong excitation of quadrupole mode Excitation of breathing mode in the radial direction shifts of quadrupole mode of about 0% are present Q0 analytical Q numerical 0.03 Q

27 Experimental setup Experimental setup - experimental analysis Resonance curve from PRA 8, (200) has been obtained by a very simplified approach Experimental data were fit to the linear combination of two basic harmonics v(t) = v 0 + v Ω sin(ωt + Φ) + v Q sin(ω Q0 t + φ) Higher harmonics were neglected completely shifts were not included in the analysis More careful analysis of experimental data is necessary 0.0 Fractional Amplitude Modulation ( / 2 ) (Hz)

28 Conclusions Introduction Motivated by recent experimental results, we have studied nonlinear BEC dynamics induced by harmonically modulated interaction We have used a combination of an analytic perturbative approach, numerics based on Gaussian approximation and numerics based on full time-dependent GP equation Relevant excitation spectra have been presented and prominent nonlinear features have been found: mode coupling, higher harmonics generation and significant shifts in the frequencies of collective modes Our results are relevant for future experimental designs that will include mixtures of cold gases and their dynamical response to harmonically modulated interactions

29 Outlook Introduction Parametric stabilization of attractive BEC Confinement induced resonance We try to excite quadrupole mode only ( ) uρ (t) u(t) =, u(0) = u u z (t) eq + ɛ u Q0, u(0) = 0 Nonlinearity leads to the coupling of quadrupole and breathing mode This coupling is particulary strong for certain values of trap anisotropy λ Signicant frequency shifts in the frequencies of collective modes may appear