Light cone dynamics and reverse Kibble-Zurek mechanism in two-dimensional superfluids following a quantum quench L. Mathey 1 and A. Polkovnikov 2 $^{1}$Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD 20899 $^{2}$Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215 (Dated: November 15, 2018) We study the dynamics of the relative phase of a bilayer of two-dimensional superfluids after the two superfluids have been decoupled. We find that on short time scales the relative phase shows “light cone” like dynamics and creates a metastable superfluid state, which can be supercritical. We also demonstrate similar light cone dynamics for the transverse field Ising model. On longer time scales the supercritical state relaxes to a disordered state due to dynamical vortex unbinding. This scenario of dynamically suppressed vortex proliferation constitutes a reverse-Kibble-Zurek effect . We study this effect both numerically using truncated Wigner approximation and analytically within a newly suggested time dependent renormalization group approach (RG). In particular, within RG we show that there are two possible fixed points for the real time evolution corresponding to the superfluid and normal steady states. So depending on the initial conditions and the microscopic parameters of the Hamiltonian the system undergoes a non-equilibrium phase transition of the Kosterlitz-Thouless type. The time scales for the vortex unbinding near the critical point are exponentially divergent, similar to the equilibrium case. PACS numbers: I. INTRODUCTION state. We note that experiments in a similar setup in- volving separation of two 1D superfluids were reported in Ref. [22] and the corresponding theoretical analysis was done in Refs. [23–25]. Unlike the 2D case, phonon fluctuations in 1D result in the exponential decay of the correlation functions and nonlinear effects in the form of phase slips do not bring qualitative changes to the be- havior of the correlation functions at least at low initial temperatures $^{24}$. The technological advances of trapping and manip- ulating ultra-cold atom systems provide an opportu- nity to study many-body dynamics with unprecedented clarity. The realization of Bose-Einstein condensates in ultra-cold atom systems $^{1}$, the Mott insulator transi- tion $^{2}$, the BEC-BCS transition$^{3}$, the Kosterlitz-Thouless transition $^{4–6}$, demonstrated that this technology can be used as a quantum simulator of many-body phases. Here, the static state of a system in equilibrium is created and studied. Various dynamical aspects of ultra-cold atom systems have also been probed, such as dipole oscillations $^{7}$, vortex excitations$^{8}$, and soliton dynamics$^{9}$, absence of equilibration in one-dimensional bosonic sys- tems $^{10}$, spontaneous formation of vortices in spinor con- densates 11 and many others (see Ref. [12] for a recent review). In Ref. [13], vortices excitations were created via laser stirring. In these experiments, the dynamics of only a few degrees of freedom were studied, such as the center of mass motion, or the dynamical evolution of a vortex. These experimental developments stimu- lated a considerable theoretical interest in understanding non-equilibrium quantum dynamics including analysis of dynamics following sudden quenches $^{14}$, studying connec- tions between dynamics and thermodynamics $^{15}$, dynam- ics through quantum critical points $^{16}$. In equilibrium for the uncoupled layers, there are two possible phases. At low temperatures atoms in each layer (which we regard as identical) form a (quasi-)superfluid phase while at high temperatures they form a normal Bose gas. These phases can be distinguished by the long- range behavior of the single particle correlation function G ( x ) = 〈 b $^{†}$(0) b ( x ) 〉 ≈ ρ 〈 exp[ i ( φ ( x ) − φ (0))] 〉 , where b ( x ) is the single particle operator, ρ is the atom density, and φ is the phase. We note that a rotor representa- tion of bosons b ( x ) ∼ √ ρ ( x ) exp[ iφ ( x )] is possible when the healing length characterizing the characteristic length scale of density fluctuations is short compared to other length scales in the problem. Under the same conditions the density fluctuations are negligible if we are interested in long distance physics. In the superfluid phase this function shows algebraic scaling, G ( x ) ∼ | x | − τ/ $^{4}$, where the scaling exponent τ is proportional to the temperature τ ≈ T /T$_{c}$ , with T$_{c}$ being the Kosterlitz-Thouless temper- ature. At the transition point we have G ( x ) ∼ | x | − 1 / $^{4}$. Above the transition, the correlation function shows ex- ponential scaling G ( x ) ∼ exp( −| x | /ξ ), with some corre- lation length ξ which diverges near the transition tem- perature. The algebraic scaling of the superfluid phase is due the thermally excited phonon (Bogoliubov’s) modes. In two dimensions these fluctuations generate quasi-long range order, rather than true long range order. The tran- The focus of this paper is a detailed analysis of the full many body dynamics following the quench in a two- dimensional quantum rotor model. Physically we imag- ine the situation where two initially strongly coupled su- perfluids are suddenly separated and we are interested in the evolution of the relative phase between the two superfluids. In particular, we will be interested in the question of how the system relaxes to the equilibrium arXiv:1001.0098v1 [cond-mat.quant-gas] 31 Dec 2009