Computer Physics Communications 180 (2009) 854–860 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Efficiently computing vortex lattices in rapid rotating Bose–Einstein condensates ∗ Rong Zeng a, Yanzhi Zhang b, a Department of Electrical Engineering, Tsinghua University, Beijing, 100084, China b Department of Mathematics, National University of Singapore, Singapore 117543 article info abstract Article history: We propose an efficient and spectrally accurate numerical method for computing vortex lattices in rapid Received 21 March 2008 rotating Bose–Einstein condensates (BECs), especially with strong repulsive interatomic interaction. The Received in revised form 24 November 2008 key ingredient of this method is to discretize the normalized gradient flow by Fourier spectral method in Accepted 2 December 2008 space and by semi-implicit Euler method in time. Different vortex lattice structures of condensate ground Availableonline7December2008 states in two-dimensional (2D) and 3D rapid rotating BECs are reported for both harmonic and harmonic- PACS: plus-quartic potentials. In addition, vortex lattices in rotating BECs with optical lattice potentials are also 11.10.Jj presented. 11.10.St © 2008 Elsevier B.V. All rights reserved. 31.15.-p Keywords: Rotating Bose–Einstein condensate Gross–Pitaevskii equation Vortex lattices Strong repulsive interaction regime Angular momentum rotation 1. Introduction There are many different numerical methods proposed in the literature to compute the stationary state of non-rotating and ro- Gaseous Bose–Einstein condensates (BECs) offer a versatile test- tating BECs. For example, an imaginary time method was used ing ground for the study of superfluidity where quantized vor- in [10,11] forfindinggroundstatesofBECs,ahybridthreesteps tices play an important role. The first observation of a single Runge–Kutta–Crank–Nicolson scheme was proposed in [12,13] for vortex line was in weakly interacting alkali gases by using the computing S-shape or U-shape vortex lines in 3D BECs, an adap- Reman transition phase-printing method [1,2]. Multiply charged tive step-size Runge–Kutta finite difference method was applied vortices were also created by using the topological phase engineer- in [14,15] for studying the nucleation of vortex arrays in ro- ing method [3]. Recently, vortex lattices containing more than one tating anisotropic BECs, and a backward Euler finite difference hundred vortices were observed by rotating the condensate with method was proposed in [16,17] for computing ground states of a laser spoon [4–7]. It is expected that more complicated vortex non-rotating and rotating BECs. More approaches can be found clusters can be created in the future, and such states would enable in [18–24] and references therein. In all the above methods, the various opportunities, ranging from investigating the properties second- or fourth-order compact finite difference scheme is used of random polynomials [8] to using vortices in quantum memo- to discretize space derivatives. Due to the finite order accuracy ries [9]. In addition, recent experimental developments enable one (usually second-order, fourth-order or even sixth-order) of the spa- to create a confinement potential either tighter than a harmonic tial discretization, these methods have difficulties to get the accu- potential or as an optical lattice, which opens possible methods to rate results in rapid rotating BECs, especially with strong repulsive explore the nature of BECs with/without the oscillating potential. interaction. Because in this case, very complicated vortex lattice All these developments spur great interests in the study of vortex structures may appear in the condensate [25–27], and the high lattice structures of condensate ground state in rapid rotating BECs spatial resolution of the numerical method is strongly demanded. with strong repulsive interaction. In this paper, we present an efficient and spectrally accurate numerical method to compute vortex lattices of condensate ground state in rapid rotating BECs, especially when the interatomic inter- Corresponding author. Current address: Department of Scientific Computing, * action is strongly repulsive. This method discretizes the normalized Florida State University, Tallahassee, FL 32306-4120, USA. E-mail addresses: [email protected] (R. Zeng), [email protected] gradient flow, also known as the Gross–Pitaevskii equation (GPE) in (Y. Zhang). the imaginary time, by Fourier spectral method for spatial deriva- 0010-4655/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2008.12.003 R. Zeng, Y. Zhang / Computer Physics Communications 180 (2009) 854–860 855 tives and backward Euler scheme for time derivatives. The fast with γy = ωy/ωx and γz = ωz/ωx. The wave function satisfies that direct Poisson solver is applied to solve the large linear system 2 at each time step, and also the stabilization technique is used such ψ(x,t) dx = 1, t 0. (2.7) that the time step can be chosen as large as possible [28].Vor- R3 tex lattice structures of condensate ground state are reported in the 2D and 3D rapid rotating BECs with different external poten- Furthermore, if it is tightly confined in the axial direction, i.e. = tials. We obtain results in the strong interacting regime which are γz 1andγy O (1), the dynamics of a BEC can be well approx- imated by the 2D GPE under the normalization (2.7) [12,16]: not reported in the literature. In addition, we also compute vortex lattices of rotating BEC with optical lattice potentials not reported 1 = − ∇2 + + 2 − before. In fact, our preliminary aim of the paper is not to find new i∂t ψ(x,t) V 2(x) β2 ψ(x,t) Ω Lz ψ(x,t) (2.8) 2 physics, but to propose an efficient and most accurate numerical method for computing vortex lattices of condensate ground state with x = (x, y)T ∈ R2, and in rapid rotating BECs. 1 = 2 + 2 2 ≈ γz This paper is organized as follows. In Section 2 we introduce V 2(x) x γy y ,β2 β3 . the model under the investigation and discuss different trapping 2 2π potentials. In Section 3 the numerical methods are introduced in In recent experiments, other potentials are also applied to study detail. Vortex lattices of condensate ground state in 2D and 3D the behavior of rapid rotating BECs. For example, the harmonic- rotating BECs are reported in Sections 4 and 5, respectively. Finally, plus-quartic potential has the form [25,30,31] we make the conclusions in Section 6. (1 − α)r2 + κr4, d = 2, V (x) = (2.9) d − 2 + 4 + 2 2 = 2. The Gross–Pitaevskii equation (1 α)r κr γz z , d 3, where r = x2 + y2, and α, κ and γ are positive constants; the At temperatures T much smaller than the critical temperature z harmonic-plus-optical lattice potential reads [23] Tc [29], a rotating BEC trapped in an external potential can be de- 1 2 2 scribed by a macroscopic wave function ψ(x,t) which obeys the (x + y + V opt), d = 2, V (x) = 2 (2.10) Gross–Pitaevskii equation (GPE). In the rotating frame with a fre- d 1 2 2 2 2 (x + y + γ z + V opt), d = 3, quency Ω around the z-axis, the GPE reads 2 z = 2 + 2 ¯ 2 where V opt(x, y) V 0(sin (κx) sin (κ y)) is the optical lattice ∂ψ(x,t) h 2 2 ih¯ = − ∇ + V trap(x) + g|ψ| − Ω Lz ψ(x,t), t 0, potential with V 0 and κ two positive constants. ∂t 2m The ground state solution φg (x) of the d-dimensional (d = 2, 3) (2.1) GPE is defined as which minimizes the Gross–Pitaevskii energy where x = x y z T ∈ R3 is the spatial coordinate vector, h¯ is the ( , , ) 1 2 2 βd 4 2 = |∇ | + | | + | | = ¯ Eβd,Ω (φ) φ V d(x) φ φ Planck constant, m is the atomic mass, g 4πh as/m represents 2 2 the strength of the interaction between particles with a (posi- Rd s tive for repulsive interaction and negative for attractive interaction) − ∗ the s-wave scattering length, and Lz =−ih¯ (x∂y − y∂x) is the z- Ω Re(φ Lzφ) dx, (2.11) component of the angular momentum. V trap(x) is a real-valued and satisfies the normalization constraint external potential whose shape is determined by the type of sys- tem under investigations, and if the harmonic trapping potential is 2 = considered, it has the form φ(x) dx 1, (2.12) m Rd = 2 2 + 2 2 + 2 2 V trap(x) ω x ω y ω z , (2.2) ∗ 2 x y z where f and Re( f ) denote the conjugate and the real part of the function f , respectively. where ωx, ωy and ωz are the trapping frequencies in x-, y-, and z-direction, respectively. The wave function is normalized by 3. Numerical methods 2 ψ(x,t) dx = N, t 0 (2.3) In the literature (see, e.g. [17,23,28]), the minimizer of Eβd,Ω (φ) R3 was found by applying an imaginary time (i.e. t →−it) in the GPE with N the total number of atoms. and evolving a gradient flow with discrete normalization [16].In For the numerical purpose, it is convenient to rescale the spa- this section, we will first introduce the normalized gradient flow tial and temporal variables. By assuming ωx = min{ωx, ωy, ωz} and under the rotational frame and then present spectral type methods introducing to discretize it. √ t Nψ x = a x, t = ,Ω= ω Ω, ψ = (2.4) 3.1. Normalized gradient flow 0 x 3/2 ωx a0 Choose a time step t > 0 and define the time sequence tn = = ¯ with a0 h/2mωx, the GPE (2.1) can be reduced to the following n t for n = 0, 1,....Foreachtimeinterval[tn,tn+1), the gradient dimensionless form (removing for simplicity): flow (or called as the GPE in imaginary time) has the form [12,16] ∂ψ(x,t) 1 = − ∇2 + + 2 − 1 2 2 i V 3(x) β3 ψ(x,t) Ω Lz ψ(x,t), (2.5) ∂t φ(x,t) = ∇ − V d(x) − βd φ(x,t) + Ω Lz φ(x,t), (3.13) ∂t 2 2 where β3 = 4π Nas/a0 and Lz =−i(x∂y − y∂x).