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Physical Review A

Atomic transport on one-dimensional optical lattices

W. Greenwood, Paul H. Pax, Pierre Meystre

Published Sep 1, 1997• 15 citations
Publisher

Abstract

We carry out quantum and semiclassical numerical simulations of the dynamics, particularly diffusion, of atoms in an optical lattice. We consider specifically the situation of a $J=1/2\ensuremath{\leftrightarrow}J=3/2$ atomic transition, whose experimental counterpart can be realized in ${\mathrm{Mg}}^{+}$ experiments. The semiclassical results, which cover a larger range of parameters than would be practical for fully quantum calculations, confirm and extend the work of Marksteiner et al. [Phys. Rev. A 53, 3409 (1996)], characterize the transition from normal to anomalous diffusion and the atomic trajectories which give rise to it, and provide a physical explanation for the difference between these two regimes. The full quantum calculations reveal uniquely quantum features of individual atomic trajectories, and in addition, predict significantly lower diffusion, by a factor of 1.2 to 2, than their semiclassical counterparts.

DOI
10.1103/physreva.56.2109

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•Jan 1, 2009•0 citations
We carry out quantum and semiclassical numerical simulations of the dynamics, particularly diffusion, of atoms in an optical lattice. We consider specifically the situation of a J=1/2\ensuremath↔J=3/2J=1/2\ensuremath{\leftrightarrow}J=3/2J=1/2\ensuremath↔J=3/2 atomic transition, whose experimental counterpart can be realized in Mg+{\mathrm{Mg}}^{+}Mg+ experiments. The semiclassical results, which cover a larger range of parameters than would be practical for fully quantum calculations, confirm and extend the work of Marksteiner et al. [Phys. Rev. A 53, 3409 (1996)], characterize the transition from normal to anomalous diffusion and the atomic trajectories which give rise to it, and provide a physical explanation for the difference between these two regimes. The full quantum calculations reveal uniquely quantum features of individual atomic trajectories, and in addition, predict significantly lower diffusion, by a factor of 1.2 to 2, than their semiclassical counterparts.
We study dissipative transport of spontaneously emitting atoms in a one-dimensional (1D) standing-wave laser field in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wave-function method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical experiments and confirmed analytically that chaotic atomic transport can take the form either of ballistic motion or a random walking with specific statistical properties. The character of spatial and momentum diffusion in the ballistic atomic transport is shown to change abruptly in the atom-laser detuning regime where the Hamiltonian dynamics is irregular in the sense of dynamical chaos. We find a clear correlation between the behavior of the momentum diffusion coefficient and Hamiltonian chaos probability which is a manifestation of chaoticity of the fundamental atom-light interaction in the diffusivelike dissipative atomic transport. We propose to measure a linear extent of atomic clouds in a 1D optical lattice and predict that, beginning with those values of the mean cloud's momentum for which the probability of Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds should increase sharply. A sensitive dependence of statistical characteristics of dissipative transport on the values of the detuning allows us to manipulate the atomic transport by changing the laser frequency.
Motivated by experimental progress in strongly coupled atom-photon systems in optical cavities, we study theoretically the quantum dynamics of atoms coupled to a one-dimensional dynamical optical lattice. The dynamical lattice is chosen to have a period that is incommensurate with that of an underlying static lattice, leading to a dynamical version of the Aubry-Andr\'e model which can cause localization of single-particle wave functions. We show that atomic wave packets in this dynamical lattice generically spread via anomalous diffusion, which can be tuned between superdiffusive and subdiffusive regimes. This anomalous diffusion arises from an interplay between Anderson localization and quantum fluctuations of the cavity field.
In view of the compelling need for coherent atom transport as a prerequisite for a variety of emerging quantum technologies, we investigate such transport on the example of an adjustable family of two-dimensional optical lattices [L. Tarruell {\em et al.}, Nature (London) {\bf 483}, 302 (2012)] that includes square, honeycomb, dimerized, and 1D-chains lattices as its special cases; dynamical optical lattices of this type have already been utilized for the demonstration of topological pumping and the realization of two-qubit quantum gates with neutral atoms. At the outset, we propose the appropriate arrangements of acousto-optic modulators that give rise to a frequency imbalance between counterpropagating laser beams, thus leading to the dynamical-lattice effect in an arbitrary direction in the lattice plane. We subsequently obtain the dynamical-lattice trajectories that enable atom transport in the lattices under consideration using two classes of control schemes: (i) shortcuts to adiabaticity (STA) in the form of inverse engineering based on a dynamical invariant of Lewis-Riesenfeld type, and (ii) their modification, known as enhanced STA (eSTA), which is well-suited for the treatment of anharmonic trapping potentials. We then quantify the resulting atom dynamics using transport fidelities computed from the numerical solutions of the relevant time-dependent Schrödinger equations. By doing so for various choices of the system parameters and transport directions, we demonstrate that -- except in the special case of the dimerized lattice -- the eSTA method consistently outperforms its STA counterpart, both in terms of the achievable transport times and the robustness of the resulting transport against small variations of optical-lattice depths.
Recently, anomalous superdiffusion of ultracold 87Rb atoms in an optical lattice has been observed along with a fat-tailed, Lévy type, spatial distribution. The anomalous exponents were found to depend on the depth of the optical potential. We find, within the framework of the semiclassical theory of Sisyphus cooling, three distinct phases of the dynamics as the optical potential depth is lowered: normal diffusion; Lévy diffusion; and x∼t(3/2) scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Lévy walk, with strong correlations between the length and duration of the excursions. We derive a fractional diffusion equation describing the atomic cloud, and the corresponding anomalous diffusion coefficient.
Ultracold atomic gases in optical lattices provide an unique framework to study quantum phenomena in strongly correlated systems. In addition to the precise control over all relevant parameters in the experiment, these experiments can be mapped to a fundamental model from solid-state physics. For the bosonic version of this model, the so-called Bose-Hubbard model, a phase transition from a superfluid to a Mott insulator was theoretically predicted and later experimentally observed in an ultracold gas of Rb-87 atoms in three-dimensional as well as in one-dimensional optical lattices. Apart from homogeneous optical lattices one can introduce more complex lattice topologies such as two-color superlattices which give rise to a rich phase diagram including more exotic phases like the Bose-glass. We employ various powerful many-body techniques like exact diagonalization in complete and truncated Hilbert spaces and the Density-Matrix Renormalization Group (DMRG) algorithm to study the phase diagrams of the one-dimensional Bose-Hubbard and the one-dimensional Bose-Fermi-Hubbard Hamiltonian. Most theoretical studies of these systems discuss the phase diagrams with respect to the generic parameters of the Hubbard model. These Hubbard parameters, however, depend non-trivially on the control parameters used in experiments. The focus of this work is on the ab-initio calculation of the phase diagram of ultracold Rb-87 in one-dimensional optical superlattices starting directly from the experimental setup. To this end, we first employ band-structure calculations to extract the Hubbard parameters from the experimental parameters. Then, we use state-of-the-art DMRG techniques to solve the many-body problem for realistic particle numbers and lattice sizes that occur in experiments. Our results show that by using the intensities of the two laser fields forming the two-color superlattice as control parameters while keeping all other experimental parameters fixed, it is possible to access all relevant quantum phases of the system. Furthermore, we found out that the critical values of the laser intensities for the different phase transitions depend strongly on a third parameter that has to be included for a realistic description of the experiment. This third parameter is the strength of a harmonic trapping potential which accounts for the Gaussian shape of the laser fields and an additional magnetic potential used to confine the atoms in the center of the trap.