Nonstationary excitations in Bose–Einstein condensates under the action of periodically varying scattering length with time dependent frequencies

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Abstract

We investigate nonstationary excitations in 3D Bose–Einstein condensates in a spherically symmetric trap potential under the modulation of scattering length with slowly varying frequencies (adiabatic modulation). By numerically solving the Gross–Pitaevskii equation we observe a stepwise increase in the amplitude of oscillation due to successive phase locking between the driving frequency and nonlinear frequency. Such a nonstationary excitation has been shown to exist by an analytic approach using a variational procedure and perturbation theory in the action–angle variables. By using a canonical perturbation theory, we have identified the successive resonance excitations whenever the driven frequency matches the nonlinear frequency or its subharmonics.

Introduction

The successful experimental realization of trapped Bose–Einstein condensates in alkali metal atoms has triggered immense interest in understanding the various properties of ultracold matter [1], [2]. The properties of the condensate wavefunction are usually described by a mean field Gross–Pitaevskii equation [3]. For the past couple of years, there has been increased interest in studying the properties of Bose–Einstein condensates with time varying trap potentials and scattering lengths both experimentally and theoretically [4], [5], [6], [7], [8]. In particular, temporal periodic modulation of scattering length by exploiting a Feshbach resonance is given central importance in recent times. Earlier studies show that, in certain circumstances, periodically varying scattering length can stabilize the collapsing condensate [6], [7], [8]. It may be noted that this approach was directly suggested by an earlier study of stabilization in terms of nonlinear optics [9]. However, very recently it has been shown that for a sign alternating nonlinearity an increase in the frequency of oscillations accelerates collapse [10].

From another point of view, these periodic modulations lead to resonance phenomena in the condensates. Resonance is an interesting feature of an oscillation under the action of an external periodic force manifesting in a large amplitude, when the frequency of the external force equals an integral multiple of the natural frequency of oscillation. Although the phenomenon of resonance is well understood in linear systems, the nonlinearity that arises in the Bose–Einstein condensates due to the interatomic interactions leads to an important problem of nonlinear resonance of wide interest.

In general, there are two ways of driving a nonlinear oscillator by a small driving periodic force, namely, external and parametric. If the driving frequency is constant (that is, the driving force is exactly periodic with time), the initial growth of the oscillator’s amplitude with time is arrested by its nonlinearity. On the other hand, if the driving frequency is slowly varying with time (the driving force is almost periodic with time), the oscillator can stay phase locked but, on average, increase its amplitude with time, or a persistent growth in the amplitude takes place, and this phenomenon is known as autoresonance. In such systems, the nonlinear frequency is slowly varying with time. This autoresonance phenomenon is also referred to as adiabatic nonlinear phase locking and synchronization. However, in certain nonlinear systems, the frequency is independent of the amplitude of oscillation [11], [12]. In these systems, when the driving frequency is slowly varying with time, one might expect successive resonance excitations at subharmonic frequencies. We refer to this type of resonance as a kind of subharmonic autoresonance [13].

In the present work, we identify such a subharmonic autoresonance in Bose–Einstein condensates where it shows successive resonance excitations due to periodic modulation with slowly varying frequency. In Bose–Einstein condensates it is experimentally feasible to vary the scattering length by either magnetically or optically inducing a Feshbach resonance [4], [14]. Earlier works on Bose–Einstein condensation report periodic modulation with constant frequency of scattering length [5], [6], [7], [8]. Along these lines, it is of potential interest to understand the dynamics of Bose–Einstein condensates under the action of periodically varying scattering length with slowly varying frequency. Motivated by the above, in this paper, we study the effect of such an excitation on the 3D Bose–Einstein condensates in a spherically symmetric trap potential and investigate the nature of parametric resonance. In particular, we point out the stepwise increase in amplitude of oscillation due to successive phase locking between the driving frequency and the nonlinear frequency (or its subharmonics) of the system.

This paper is organized as follows. In Section 2 we discuss the properties of Bose–Einstein condensates driven by a periodic force with time varying frequency from numerical simulations using a pseudo-spectral method. In Section 3 we describe the variational procedure and derive a reduced system of ordinary differential equations (ODEs) to describe the dynamics of the condensate width. In Section 4 we analyze the width dynamics using a perturbed action–angle variable theory for the reduced system of ODEs. Then in Section 5, by applying a canonical perturbation theory, we deduce the approximate nonlinear frequencies which are responsible for successive resonance excitation in BEC under the periodic modulation with slowly varying frequency. Finally in Section 6, we give a summary and conclusions.

Section snippets

Nonlinear Gross–Pitaevskii equation

At ultralow temperatures, the time dependent Bose–Einstein condensate wavefunction Ψ(r;τ) at position r and time τ may be described by the following mean field nonlinear GP equation [3]: [ħ222m+V(r)+gˆN|Ψ(r;τ)|2iħτ]Ψ(r;τ)=0. Here m is the mass and N is the number of atoms in the condensate, gˆ=4πħ2ã/m is the strength of interatomic interaction, with ã being the periodically varying atomic scattering length. The normalization condition of the wavefunction is dr|Ψ(r;τ)|2=1. The

Variational procedure

The variational method is one of the simplest ways to analyze the dynamics of the Bose–Einstein condensate [6], [19], [20]. In this method the original GP equation (2) is reduced to a system of ODEs, with fewer variables describing the condensate wave packet, by assuming a suitable trial wavefunction [6], [19], [20]. For convenience, Eq. (2) can be rewritten as (using the definition φ(r,t)=rϕ(r,t))iϕt=2ϕr22rϕr+r24ϕ+g(t)|ϕ|2ϕ. According to the variational method we assume a Gaussian

Perturbation theory in the action–angle variables

In order to have a clearer understanding of the resonance phenomena, we analyze Eq. (16) by constructing the action–angle variables. Treating Eq. (16) as a nearly integrable system (ϵ(t)1), it is more convenient to consider the action–angle variables in which one can employ a perturbation theory [23].

Canonical perturbation theory

In order to understand the resonance phenomena one has to find the approximate nonlinear frequency of the above system which is responsible for the resonant excitations. For this purpose we adopt a canonical perturbation theory to find the nonlinear frequency of near integrable systems [23], [24], [25]. The basic idea of this method is to rewrite the perturbed system (30) in a new set of action–angle variables (J,ϕ) by a canonical transformation to a new Hamiltonian K(J), which depends on the

Summary and conclusions

In summary, we have studied the autoresonant excitations in Bose–Einstein condensates under the action of external periodic modulation with time dependent frequency. By numerically solving the corresponding Gross–Pitaevskii equation we have observed that there occurs a successive phase locking with stepwise increase in the overall amplitude of oscillation. We have employed a variational procedure using a Gaussian trial wavefunction to simplify the problem to fewer coordinates. Then the reduced

Acknowledgments

This work was supported in part by the Department of Science and Technology (DST) and National Board for Higher Mathematics (Department of Atomic Energy), Government of India.

References (26)

  • F.K. Abdullaev et al.

    Physica D

    (2003)
  • K.B. Davis et al.

    Phys. Rev. Lett.

    (1995)
  • J.R. Enshera et al.

    Phys. Rev. Lett.

    (1996)
  • F. Dalfovo et al.

    Rev. Modern Phys.

    (1999)
  • S. Inouye et al.

    Nature

    (1998)
  • K. Staliunas et al.

    Phys. Rev. Lett.

    (2002)
  • H. Saito et al.

    Phys. Rev. Lett.

    (2003)
  • F.K. Abdullaev et al.

    Phys. Rev. A

    (2003)
  • S.K. Adhikari

    Phys. Rev. A

    (2004)
  • I. Towers et al.

    J. Opt. Soc. Amer. B

    (2002)
  • V.V. Konotop et al.

    Phys. Rev. Lett.

    (2005)
  • V.K. Chandrasekar et al.

    Phys. Rev. E

    (2005)
  • V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, in: Proc. 3rd Natnl. Conf. Nonlinear Systems & Dynamics, Univ....
  • Cited by (0)

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