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Physical Review A

Collective excitations of trapped binary mixtures of Bose-Einstein condensed gases

Robert Graham, Daniel F. Walls

Published Jan 1, 1998• 81 citations
Publisher

Abstract

The linearized time-dependent coupled Gross-Pitaevskii equations describing the long-wavelength excitations of Bose-Einstein condensed binary mixtures are solved in harmonic traps for the case where only the binary phase is present. The dispersion law contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate, but with different prefactors, depending on the ratios of the three $s$-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.

DOI
10.1103/physreva.57.484

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The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.

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The linearized time-dependent coupled Gross-Pitaevskii equations describing the long-wavelength excitations of Bose-Einstein condensed binary mixtures are solved in harmonic traps for the case where only the binary phase is present. The dispersion law contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate, but with different prefactors, depending on the ratios of the three sss-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.
The linearised time dependent coupled Gross-Pitaevskii equations describing the long wavelength excitations of Bose-condensed binary mixtures are solved, in the bulk and in harmonic traps for the case where only the binary phase is present. In the former case we obtain two zero-sound branches.In the latter case the dispersion law also contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate,but with different prefactors, depending on the ratios of the three s-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.
The nonlinear coupled Gross-Pitaevskii equation governing the dynamics of the two-component Bose-Einstein condensate (TBEC) is shown to admit sinusoidal, propagating-wave solutions in quasi-one-dimensional geometry in a trap. The solutions exist for a wide parameter range, which illustrates a procedure for coherent control of these modes through temporal modulation of the parameters, like scattering length and oscillator frequency. The effects of time-dependent coupling and the trap variation on the condensate profile are explained. The TBEC has also been investigated in the presence of an optical lattice potential, where the superfluid phase is found to exist under general conditions.
We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave-function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance zzz and the phase difference $\ensuremath{\phi}$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\ensuremath{\xi}$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z, \ensuremath{\phi}$, and $\ensuremath{\xi}$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.
We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
In this Thesis we study steady state solutions and dynamical evolutions of two– component atomic Bose–Einstein Condensates. We initially investigate the equilibrium properties of condensate mixtures in harmonic trapping potentials at zero temperature. Subsequently we simulate the coupled growth of these condensates by inclusion of damping terms. Finally, we investigate the evolution of coupled Bose gases via the so-called classical–field method. A recent experiment [D. J. McCarron et al., Phys. Rev. A, 84, 011603(R) (2011)] achieved Bose–Einstein Condensation of a two–species 87Rb–133Cs phase segregated mixture in harmonic trapping potentials. Depending on relative atom numbers of the two species, three distinct regimes of density distributions were observed. For these experimental parameters, we investigate the corresponding time–independent ground state solutions through numerical simulations of the coupled Gross–Pitaevskii equations. By including experimentally relevant shifts between the traps, we observe a range of structures including ‘ball and shell’ formations and axially/radially separated states. These are found to be very sensitive to the trap shifts. For all three experimental regimes, our numerical simulations reveal good qualitative agreement. The observed experimental profiles cannot be guaranteed to be fully equilibrated. This, coupled with the rapid sympathetic cooling of the experimental system, leads to a situation where growth may play a determining factor in the density structures formed. To investigate this further, we introduce phenomenological damping to describe the associated condensate growth/decay, revealing a range of transient structures. However, such a model always predicts the predominance of one condensate species over longer evolution times. Work undertaken by collaborators with the more elaborate Stochastic Projected Gross–Pitaevskii equations, which can describe condensate formation by coupling to a heat bath, predicts the spontaneous formation of dark–bright solitons. Motivated by this, we show how the presence of solitons can affect the condensate distribution, thus highlighting the overall dynamical role in the emerging patterns. Finally, we use classical field methods to analyse the evolution of non trapped Bose gases from strongly nonequilibrium initial distributions. The contrast between miscible (overlapping) and immiscible (phase segregated) components gives rise to important distinctions for condensate fractions and the formation of domains and vortices. In addition, splitting the particles of a single component thermalised state into two components is investigated. We then study the effects of suddenly quenching the strength of the interspecies interactions. Under suitable conditions, this quench generates isotropic vortex tangles. While this tangle subsequently decays over time, we propose how a repeat sequence of quenches at regular intervals could be employed to drive the tangle, thereby potentially providing a novel route to the generation of quantum turbulence.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.