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Physical Review A

Low-temperature Bose-Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach

Yvan Castin, Ralph Dum

Published Apr 1, 1998• 372 citations
Publisher

Abstract

We present a method to calculate the dynamics of very-low-temperature Bose-Einstein condensates in time-dependent traps. We consider a system with a well-defined number of particles, rather than a system in a coherent state with a well-defined phase. This preserves the $U(1)$ symmetry of the problem. We use a systematic asymptotic expansion in the square root of the fraction of noncondensed particles. In lowest order we recover the time-dependent Gross-Pitaevskii equation for the condensate wave function. The next order gives the linear dynamics of noncondensed particles. The higher order gives corrections to the time-dependent Gross-Pitaevskii equation including the effects of noncondensed particles on the condensate. We compare this method with the Bogoliubov--de Gennes approach.

DOI
10.1103/physreva.57.3008

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We present a method to calculate the dynamics of very-low-temperature Bose-Einstein condensates in time-dependent traps. We consider a system with a well-defined number of particles, rather than a system in a coherent state with a well-defined phase. This preserves the U(1)U(1)U(1) symmetry of the problem. We use a systematic asymptotic expansion in the square root of the fraction of noncondensed particles. In lowest order we recover the time-dependent Gross-Pitaevskii equation for the condensate wave function. The next order gives the linear dynamics of noncondensed particles. The higher order gives corrections to the time-dependent Gross-Pitaevskii equation including the effects of noncondensed particles on the condensate. We compare this method with the Bogoliubov--de Gennes approach.
We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for NNN-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations, which we then use to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional non-linear solutions of the time-dependent Gross–Pitaevskii equation.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations. We then use the hydrodynamic equations to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional solutions of the time-dependent Gross-Pitaevskii equation. General formulation In the previous chapter we saw that the equilibrium structure of the condensate is described by a time-independent Schrödinger equation with a nonlinear contribution to the potential to take into account interactions between particles.
We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N\mathit{N}N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2\mathit{N}^{-1/2}N−1/2 in the limit of large N\mathit{N}N.
We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates using the numerical solution of the Gross-Pitaevskii (GP) equation with both spherical and axial symmetries. We consider time-evolution problems initiated by suddenly changing the interatomic scattering length or harmonic trapping potential in a stationary condensate. These changes introduce oscillations in the condensate which are studied in detail. We use a time iterative split-step method for the solution of the time-dependent GP equation, where all nonlinear and linear non-derivative terms are treated separately from the time propagation with the kinetic energy terms. Even for an arbitrarily strong nonlinear term this leads to extremely accurate and stable results after millions of time iterations of the original equation.