Dynamical classical superfluid-insulator transition in a Bose-Einstein condensate on an optical lattice

We predict a dynamical classical superfluid-insulator transition in a Bose-Einstein condensate (BEC) trapped in a combined optical and axially-symmetric harmonic potentials initiated by a periodic modulation of the radial trapping potential. The transition is marked by a loss of phase coherence in the BEC and a subsequent destruction of the interference pattern upon free expansion. For a weak modulation of the radial potential the phase coherence is maintained. For a stronger modulation and a longer time of holding in the modulated trap, the phase coherence is destroyed signaling a classical superfluid-insulator transition. The results are illustrated by a complete numerical solution of the axially-symmetric mean-field Gross-Pitaevskii equation for a repulsive BEC. Suggestion for future experiment is made. PACS numbers: 03.75.-b, 03.75.Kk, 03.75.Lm Superfluid-Insulator Transition in a Condensate 2 The experimental loading of a cigar-shaped Bose-Einstein condensate (BEC) in both one[1,2] and three-dimensional [3] optical lattice potentials has allowed the study of quantum phase effects on a macroscopic scale such as interference of matter waves [4]. There have been several theoretical studies on different aspects of a BEC in one[5] and three-dimensional [6] optical lattice potentials. The phase coherence between different sites of a trapped BEC on an optical lattice has been established in recent experiments [1,2,7,3,8] through the formation of distinct interference pattern when the traps are removed. In a one-dimensional optical lattice potential the expanding pattern consists of a large central piece and two smaller ones moving in opposite directions on a straight line [7]. In a two-dimensional optical lattice potential the pattern consists of a large central piece and eight others on the sides of an expanding square [6]. In a three-dimensional optical lattice potential the pattern consists of a large central piece and twenty six others on the surface of an expanding cube [3, 6]. The interference pattern is a consequence of phase coherence in the BEC generated due to free quantum tunneling of atoms from one optical lattice site to another originating in the superfluid state of the system [3, 9]. Equal phase at all points or a slowly (and orderly) varying phase are the ideal examples of coherent phase. On the other hand, a rapidly (or arbitrarily) varying phase in space is usually incoherent. It has been demonstrated for a three-dimensional optical trap potential by Greiner et al. [3] that, as the strength of the optical potential traps is increased, the quantum tunneling of condensed atoms from one optical site to another is stopped resulting in a loss of superfluidity and phase coherence in the BEC. Consequently, no interference pattern is formed upon free expansion of such a BEC which is termed a Mott insulator state. This phenomenon represents a superfluid to Mott insulator quantum phase transition. The phase on an optical lattice site and the number of atoms in that site play the roles of conjugate variables obeying the Heisenberg uncertainty principle of quantum mechanics [9]. In the superfluid state the coherent phase is considered to be known and consequently the number of atoms on each site is unknown thus allowing a free movement of atoms from one site to another [3]. In the Mott insulator state the phase is entirely arbitrary across the optical lattice sites and the number of atoms at each site is fixed and their free passage from one site to another is stopped. As the strength of the optical potential traps in the Mott insulator state is reduced the superfluidity is restored in a short time via a Mott insulator to superfluid quantum phase transition. This reversible quantum phase transition may occur at absolute zero (0 K) and is driven by Heisenberg’s uncertainty principle [3] and not by thermal fluctuations involving energy as in a classical phase transition. As the temperature approaches absolute zero all thermal fluctuations die out and at 0 K classical phase transitions are necessarily excluded. Following a suggestion by Smerzi et al. [10], Cataliotti et al. [11] have demonstrated in a novel experiment the loss of phase coherence and superfluidity in a BEC trapped in a one-dimensional optical-lattice and harmonic potentials when the center of the harmonic potential is suddenly displaced along the optical lattice through a distance larger than a critical value. Then a modulational instability takes place in the BEC and Superfluid-Insulator Transition in a Condensate 3 it cannot reorganize itself quickly enough and the phase coherence and superfluidity of the BEC are lost. The loss of superfluidity is manifested in the destruction of the interference pattern upon free expansion. However, for displacements smaller than the critical distance the BEC can reorganize itself and the superfluidity is maintained [7, 11]. Distinct from the quantum phase transition observed by Greiner et al. [3], this modulational instability responsible for the superfluid-insulator transition is classical in nature [10, 11]. This process is also different from the Landau dissipation mechanism [10, 12], occurring when the fluid velocity is greater than local speed of sound. When Landau instability occurs, the system lowers energy by emitting phonons [12]. The present classical dynamical transition can be well described [13,10,12] by the mean-field Gross-Pitaevskii (GP) equation [14]. The above modulational instability is not the unique dynamical classical process leading to a superfluid-insulator transition. Many other classical processes leading to a rapid movement in the condensate can lead to such a transition [15]. The movement should be rapid enough so that the BEC cannot reorganize itself to evolve through phase coherent states. In [11] a rapid translation of the BEC through the optical lattice sites leads to the destruction of phase coherence. Here we suggest that a rapid oscillation of the BEC may also lead to a superfluid-insulator transition. The oscillation is initiated by a periodic modulation of the magnetic trapping potential ∼ ω in the radial direction via ω → ω(1 + A sin(Ωτ)) where τ is time, A an amplitude, ω is the radial trapping frequency, and Ω is the frequency of modulation. Such modulation of the trapping potential is known to generate resonant (collective) excitations in the BEC which have been studied both theoretically [16] and experimentally [17] in the absence of an optical lattice potential. The study of such excitations in the presence of an optical lattice potential has just began [18]. Similar collective excitation generated by a periodic modulation of the atomic scattering length [19] has also been shown to lead to a classical superfluid-insulator transition [20]. In the quantum phase transition [3] the Mott insulator state has a perfectly smooth probability distribution (modulus of the wave function) across the optical lattice sites whereas the phase of the wave function across the optical lattice sites remains entirely arbitrary. In the dynamical classical transition considered in this work, because of classical oscillation of the BEC, the insulator state in the joint traps is marked by a partially disturbed (nonsmooth) probability distribution across the optical lattice sites in addition to the loss of phase coherence. However, the information about the destruction of superfluidity in both quantum and classical cases is not solely contained in the initial probability distribution. Consequently, the BEC needs to be released from the joint traps and the formation of the interference pattern studied for a definite conclusion about the destruction of superfluidity. Superfluid-Insulator Transition in a Condensate 4 As the present transition is classical or mean-field-type in nature, we base the present study on the numerical solution of the time-dependent mean-field axiallysymmetric GP equation [14] in the presence of a combined harmonic and optical potential traps. The time-dependent BEC wave function Ψ(r; t) at position r and time t is described by the following mean-field nonlinear GP equation [14]