Theory of chaotic atomic transport in an optical lattice

Abstract

A semiclassical theory of chaotic atomic transport in a one-dimensional nondissipative optical lattice is developed. Using the basic equations of motion for the Bloch and translational atomic variables, we derive a stochastic map for the synchronized component of the atomic dipole moment that determines the center-of-mass motion. We find the analytical relations between the atomic and lattice parameters under which atoms typically alternate between flying through the lattice and being trapped in the wells of the optical potential. We use the stochastic map to derive formulas for the probability density functions (PDFs) for the flight and trapping events. Statistical properties of chaotic atomic transport strongly depend on the relations between the atomic and lattice parameters. We show that there is a good quantitative agreement between the analytical PDFs and those computed with the stochastic map and the basic equations of motion for different ranges of the parameters. Typical flight and trapping PDFs are shown to be broad distributions with power law "heads" with the slope −1.5 and exponential "tails". The lengths of the power law and exponential parts of the PDFs depend on the values of the parameters and can be varied continuously. We find analytical conditions, under which deterministic atomic transport has fractal properties, and explain a hierarchical structure of the dynamical fractals.

Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension

“…In physics they emerge in situations where there is no characteristic length scale, such as near phase transitions [4], in turbulent flow [5], in quantum phase diffusion [6] or when a system is out of thermal equilibrium [7]. In fact, anomalous transport properties are intimately linked to non-linear chaotic dynamics which naturally appears in many physical systems [8,9]. A simple diffusion model we have in mind is that of particles in real space, each having a velocity which fluctuates in time due to interaction with a bath.…”

Nonlinear control of chaotic walking of atoms in an optical lattice

“…In our recent paper [14] we developed a theory of atomic transport in an optical lattice (in the absence of SE) based on a specific behavior of the variable u which performs shallow and fast oscillations between the nodes of the standing laser wave and changes suddenly its value when atoms cross the nodes. The theory predicts deterministic chaotic transport at small values of the detuning |∆| ≪ 1 whose statistical properties are very well described by a stochastic map for the deterministic variable u [14]. In fact, atom moves in a rigid optical lattice just like as in a random optical potential with a complicated alternation of oscillations in potential wells and flights over many wells when it can change its direction of motion many times.…”

“…However, the length of these fragments depends strongly on the value of the detuning. At very small value ∆ = −10 −5 (when coherent atomic dynamics is practically regular [14] and atom performs a random walk due to SE), P fl ∼ T −1.5 , whereas at larger values of |∆| the power-law fragments are much shorter. In fact, both c and D depend on the value of H, therefore, Eqs.…”

“…Thus, in both the systems the condition H ≈ ±u defines the value of the energy which allows atoms to stop and turn back. Transformation of power-law fragments p-4 into exponential tails is explained in the purely Hamiltonian system [14] by a limitation of the quantity u, whereas in the system with SE it is explained by a drift of the energy H. Both of those factors prevent randomly walking quantities to go far from their critical values and decrease exponentially the probability of long atomic flights.…”

A group-theoretical approach to study atomic motion in a laser field

“…The motion in the phase space takes place on a threedimensional manifold and may be chaotic in some ranges of the control parameters, the values of the maximal Rabi and atomic recoil frequencies. A number of nonlinear dynamical effects have been analytically and numerically demonstrated with this system: chaotic Rabi oscillations [34,35], Hamiltonian chaotic atomic transport and dynamical fractals [36,37,38], Lévy flights and anomalous diffusion [39,35]. These effects are caused by local instability of the center-of-mass motion in a laser field.…”