Three-dimensional dynamics of vortex-lattice formation in Bose-Einstein condensates

Abstract

We numerically study the dynamics of vortex lattice formation in a rotating cigar-shaped BoseEinstein condensate. The study is a three-dimensional simulation of the Gross-Pitaevskii equation with a phenomenological dissipation term. The simulations reveal previously unknown dynamical features of the vortex nucleation process, in which the condensate undergoes a strongly turbulent stage and the penetrating vortex lines vibrate rapidly. The vibrations arise from spontaneous excitation of Kelvin waves on the proto-vortices during a surface wave instability, caused by an inhomogeneity of the condensate density along the elongated axial direction.PACS numbers: 03.75. Fi, 67.40.Db Atomic-gas Bose-Einstein condensates (BECs) provide a versatile testing ground for superfluidity, particularly in systems with an externally driven rotation. Several experimental groups have observed the formation of a lattice of quantized vortices in such a rotating BEC [1,2,3,4]. The experiment that largely motivates the present study is a direct observation of nonlinear dynamical phenomena such as vortex nucleation and lattice formation of a rotating BEC in real time [5]. There are some theoretical attempts to understand the dynamical properties of vortex lattice formation in this system [6,7,8,9,10,11,12]. Among these studies, the experimental observations are well reproduced by two-dimensional (2D) numerical simulations of the timedependent Gross-Pitaevskii equation (GPE) with a phenomenological dissipation term [7,8,9,13]. This paper presents the full three-dimensional (3D) dynamics of vortex lattice formation in a rotating BEC through the numerical simulations of the time-dependent GPE with a dissipation term. Except for a few preliminary works [6,11,12] and studies of the static configurations of vortices [14,15], this problem has been analyzed using only 2D simulations. In contrast to 2D simulations, a 3D simulation gives significant improvement for describing real 3D systems. For the present study, these improvements are (i) the 3D simulation includes the axial (z) component, which increases the number of degrees of freedom of the fluctuation modes, and (ii) the 3D simulation can treat observed physical phenomena associated with "vortex lines", such as bending [14], Kelvin waves (helical displacements of the vortex core) [16], and reconnection [17]. These features make the dynamical process of vortex lattice formation richer than that of the 2D case. Recently, Lobo et al. studied the 3D dynamics of rotating atomic-gas BECs using classical field theory in which the dynamics is described by the energy-conserving timedependent GPE and both the condensate wave function and its fluctuation modes are approximated by a single c-number field [11]. The dissipative process and the 3D vortex dynamics of a rotating BEC in a sphericallysymmetric trap was discussed in Ref. [12]. In our present study, we take account of the phenomenological dissipation so as to reproduce the entire vortex formation process that was observed in the experiments ...

Quantum turbulence in a trapped Bose-Einstein condensate

“…Shortly after trapped BECs were first realized, experimental groups [19,20] reported vortex lattice structures, as well as the crystallization dynamics of these structures under rotation. These dynamics have been successfully confirmed quantitatively by the numerical simulation of the GP equation [21,22].However, in the experimental research of trapped BECs, another important phenomenon of quantized vortices, namely QT, has not been adequately studied. Noting that quantized vortices are observable and that almost all physical parameters of trapped BECs are controllable, such systems are an ideal prototype for truly controllable QT, which is not possible for superfluid helium.…”

“…At the turbulent state, n(l)∆l obeys the scaling property n(l)∆l ∝ l −α for 2πξ < l < 2πR TF . This reflects the self-similar Richardson cascade in which large vortices entering the condensate from the surface [21,22] are divided into smaller vortices, which is first confirmed in turbulence with the framework of the Gross-Pitaevskii equation. The scaling exponent α is close to unity, which is consistent with those given by Araki et al (α ≃ 1.34) [12] and Mitani et al (α ≃ 1) [28].…”

Quantum turbulence—from superfluid helium to atomic Bose–Einstein condensates

“…This observation has been reproduced by a simulation of the Gross-Pitaevskii (GP) equation for the macroscopic wavefunction Ψ(x, t) = |Ψ(x, t)|e iθ(x,t) in two-dimensional [25,26] and threedimensional [27] spaces. The corresponding GP equation in a frame rotating with frequency Ω = Ωẑ is given by…”

Quantum Turbulence in Trapped Bose-Einstein Condensates

“…Shortly after trapped BECs were first realized, experimental groups [13,14] reported vortex lattice structures, as well as the crystallization dynamics of these structures under rotation. These dynamics has been successfully confirmed quantitatively by the numerical simulation of the GP model [15,16].…”

“…This idea was applied to atomic Bose-Einstein condensates, and some groups have succeeded in making and observing the formation of vortex lattices [13,14]. Our group has made the numerical simulation of the GP equation with dissipation, thus showing the whole story of the dynamics consistent with the observations [15,16]. The whole scenario revealed by the numerical studies is as follows.…”