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Journal of Statistical Mechanics Theory and Experiment

Conductivity with cold atoms in optical lattices

Andrey R. Kolovsky

Published Feb 4, 2009• 6 citations
Publisher

Abstract

This paper studies a simple model of conductivity—a quantum particle (an atom) in a lattice, subject to a static field and interacting with the bath of Bose atoms. Results from direct numerical simulations of the system dynamics and statistical analysis of the eigenstates are compared with predictions from linear response theory, analytical solutions of the master equation, and the theory of quantum chaos. It is shown, in particular, that the behavior of the current–current correlation function, entering Kubo’s equation, is entirely defined by the chaotic properties of the system and that the phenomenological relaxation constant, entering the master equation, is actually encoded in this correlation function.

DOI
10.1088/1742-5468/2009/02/p02018

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This paper studies a simple model of conductivity—a quantum particle (an atom) in a lattice, subject to a static field and interacting with the bath of Bose atoms. Results from direct numerical simulations of the system dynamics and statistical analysis of the eigenstates are compared with predictions from linear response theory, analytical solutions of the master equation, and the theory of quantum chaos. It is shown, in particular, that the behavior of the current–current correlation function, entering Kubo’s equation, is entirely defined by the chaotic properties of the system and that the phenomenological relaxation constant, entering the master equation, is actually encoded in this correlation function.
Motivated by the recent advancements in experimental techniques in the cold atomic systems, we study the dependence of the conductivity on momentum in a system of strongly interacting bosons in an optical lattice. By employing the quantum rotor approach to the Bose-Hubbard model we calculate the current-current correlation function and subsequently obtain the analytic formula for the momentum-dependent longitudinal conductivity. We analyze the behavior of the conductivity for the square and cubic lattices in both, the superfluid and Mott insulator phases. As a consequence of the particle-hole symmetry, the conductivity for a uniformly filled lattice in the superfluid phase exhibits a linear dependence for a surprisingly wide range of momenta around k=0\mathbf{k}=0k=0. This allows us to predict the value of the group velocity of the particle excitations. We also consider the impact of finite temperature and discover that it leads to an additional conductivity channel, which is aligned along the direction of the probe field and located within the energy gap.
We measure the conductivity of neutral fermions in a cubic lattice. A uniform force is applied at a single frequency by periodic displacement of the trapping potential, and the current response is observed though the global displacement of atoms. In the linear response regime, the steady-state ratio between current and force gives the ac conductivity. The tensor properties of the conductivity reveal temporal and spatial symmetries of the sample. Similarly to optical conductivity commonly measured in materials with reflectivity, exact sum rules relate dynamical measurements to thermodynamic properties. We explore the effect of lattice depth, temperature, interaction strength, and atom number on conductivity. Using a relaxation-time approximation, we extract the transport time of current. Our work enables a direct comparison between cold-atom experiments and conductivity studies of materials.
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.
Quantum simulations are quickly becoming an indispensable tool for studying particle transport in correlated lattice models. One of the central topics in the study of transport is bad-metal behavior, characterized by the direct-current resistivity linear in temperature. In the fermionic Hubbard model, optical conductivity has been studied extensively, and a recent optical lattice experiment demonstrated bad-metal behavior in qualitative agreement with theory. Far less is known about transport in the bosonic Hubbard model. We investigate the conductivity in the Bose-Hubbard model and focus on the regime of strong interactions and high temperatures. We use numerically exact calculations for small lattice sizes. At weak tunneling, we find multiple peaks in the optical conductivity that stem from the Hubbard bands present in the many-body spectrum. This feature slowly washes out as the tunneling rate gets stronger. At high temperature, we identify a regime of TTT-linear resistivity, as expected. When the interactions are very strong, the leading inverse-temperature coefficient in conductivity is proportional to the tunneling amplitude. As the tunneling becomes stronger, this dependence takes quadratic form. At very strong coupling and half filling, we identify a separate linear resistivity regime at lower temperature, corresponding to the hard-core boson regime. Additionally, we unexpectedly observe that at half filling, in a big part of the phase diagram, conductivity is an increasing function of the coupling constant before it saturates at the hard-core-boson result. We explain this feature based on the analysis of the many-body energy spectrum and the contributions to conductivity of individual eigenstates of the system.
Abstract The optical conductivity is the basic defining property of materials characterizing the current response toward time-dependent electric fields. In this work, following the approach of Kubo’s response theory, we study the general properties of the nonlinear optical conductivities of quantum many-body systems both in equilibrium and non-equilibrium. We obtain an expression of the second- and the third-order optical conductivity in terms of correlation functions and present a perturbative proof of the generalized Kohn formula proposed recently. We also discuss a generalization of the f-sum rule to a non-equilibrium setting by focusing on the instantaneous response.
We discuss applications of the theory of quantum chaos to one of the paradigm models of many-body quantum physics — the Bose–Hubbard (BH) model, which describes, in particular, interacting ultracold Bose atoms in an optical lattice. After preliminary, pure quantum analysis of the system we introduce the classical counterpart of the BH model and the governing semiclassical equations of motion. We analyze these equations for the problem of Bloch oscillations (BOs) of cold atoms where a number of experimental results are available. The paper is written for nonexperts and can be viewed as an introduction to the field.