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arXiv:0903.3863 · Laser Physics Letters

Nonlinear dynamics of ultracold gases in double-well lattices

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

Published Oct 21, 2008• 21 citations
arXivHTMLPDF

Abstract

An ultracold gas is considered, loaded into a lattice, each site of which is formed by a double-well potential. Initial conditions, after the loading, correspond to a nonequilibrium state. The nonlinear dynamics of the system, starting with a nonequilibrium state, is analysed in the local-field approximation. The importance of taking into account attenuation, caused by particle collisions, is emphasized. The presence of this attenuation dramatically influences the system dynamics.

DOI
10.1002/lapl.200810111

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An ultracold gas is considered, loaded into a lattice, each site of which is formed by a double-well potential. Initial conditions, after the loading, correspond to a nonequilibrium state. The nonlinear dynamics of the system, starting with a nonequilibrium state, is analysed in the local-field approximation. The importance of taking into account attenuation, caused by particle collisions, is emphasized. The presence of this attenuation dramatically influences the system dynamics.
Cold atoms, loaded into an optical lattice with double-well sites, are considered. Pseudospin representation for an effective Hamiltonian is derived. The system in equilibrium displays two phases, ordered and disordered. The second-order phase transition between the phases can be driven either by temperature or by changing the system parameters. Collective pseudospin excitations have a gap disappearing at the phase-transition point. Dynamics of atoms is studied when they are loaded into the lattice in an initially nonequilibrium state. It is shown that the temporal evolution of atoms, contrary to their equilibrium thermodynamics, cannot be described in the mean-field approximation, since it results in a structurally unstable dynamical system, but a more accurate description is necessary taking into account attenuation effects.
The absence of energy dissipation leads to an intriguing out-of-equilibrium dynamics for ultracold polar gases in optical lattices, characterized by the formation of dynamically bound on-site and inter-site clusters of two or more particles, and by an effective blockade repulsion. These effects combined with the controlled preparation of initial states available in cold-gas experiments can be employed to create interesting out-of-equilibrium states. These include quasiequilibrated effectively repulsive 1D gases for attractive dipolar interactions and dynamically bound crystals. Furthermore, nonequilibrium polar lattice gases can offer a promising scenario for the study of quasi-many-body localization in the absence of quenched disorder. This fascinating out-of-equilibrium dynamics for ultracold polar gases in optical lattices may be accessible in on-going experiments.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
We consider an ultracold quantum degenerate gas in an optical lattice inside a cavity. This system represents a simple but key model for “quantum optics with quantum gases,” where a quantum description of both light and atomic motion is equally important. Due to the dynamical entanglement of atomic motion and light, the measurement of light affects the many-body atomic state as well. The conditional atomic dynamics can be described using the Quantum Monte Carlo Wave Function Simulation method. In this paper, we emphasize how this usually complicated numerical procedure can be reduced to an analytical solution after some assumptions and approximations valid for macroscopic Bose-Einstein condensates (BEC) with large atom numbers. The theory can be applied for lattices with both low filling factors (e.g. one atom per lattice site in average) and very high filling factors (e.g., a BEC in a double-well potential). The purity of the resulting multipartite entangled atomic state is analyzed.
In this thesis we investigate strongly-correlated states of ultracold bosonic atoms in rotating ring lattices and arrays of double-well potentials. In the first part of the thesis, we study the tunneling dynamics of ultracold bosons in double-well potentials. In the non-interacting limit single-particle transitions dominate, while in the interaction-dominated regime correlated tunneling of all particles prevails. At intermediate times of the many-particle flopping process correlated states occur, but the timescales of these processes increase dramatically with the number of particles. Using an array of double-well potentials, a large number of such few-particle superposition states can be produced in parallel. In the second part of the thesis, we study the effects of rotation on ultracold bosons confined to one-dimensional ring lattices. We find that at commensurate filling there exists a critical rotation frequency, at which the ground state of the weakly-interacting gas is fragmented into a macroscopic superposition of different quasi-momentum states. We demonstrate that the generation of such superposition states using slightly non-uniform ring lattices has several practical advantages. Moreover, we show that different quasi-momentum states can be distinguished in time-of-flight absorption imaging and propose to probe correlations via the many-body oscillations induced by a sudden change in the rotation frequency. Finally, we compare these macroscopic superposition states to those occurring in superconducting quantum interference devices. In the third part of the thesis, we demonstrate the creation of entangled states with ultracold bosonic atoms by dynamical manipulation of the shape of the lattice potential. To this end, we consider an optical superlattice that allows both the splitting of each site into a double-well potential and the variation of the height of the potential barrier between the sites. We show how to use this array of double-well potentials to perform entangling operations between neighboring qubits encoded on the Zeeman levels of the atoms. As one possible application, we present a method of realizing a resource state for measurement-based quantum computation via Bell-pair measurements. In the final part of the thesis, we study ultracold bosons on a two-dimensional square lattice in the presence of an effective magnetic field and point out a couple of features this system has in common with ultracold bosons in one-dimensional rotating ring lattices.
A large-scale dynamical simulation of the superfluid--Mott-insulator transition in a gas of ultracold atoms placed in an optical lattice is performed using the time-dependent Gutzwiller mean-field approach. This approximate treatment allows us to take into account most of the details of the recent experiment [Greiner et al., Nature (London) 415, 39 (2002)] where by changing the depth of the lattice potential an adiabatic transition from a superfluid to a Mott insulator state has been reported. Our simulations reveal a significant excitation of the system with a transition to insulator in restricted regions of the trap only.