Routes towards the experimental observation of the large fluctuations due to chaos-assisted tunneling effects with cold atoms

Abstract

In the presence of a complex classical dynamics associated with a mixed phase space, a quantum wave function can tunnel between two stable islands through the chaotic sea, an effect that has no classical counterpart. This phenomenon, referred to as chaos assisted tunneling, is characterized by large fluctuations of the tunneling rate when a parameter is varied. To date the full extent of this effect as well as the associated statistical distribution have never been observed in a quantum system. Here we analyze the possibility of characterizing these effects accurately in a cold atom experiment.Using realistic values of the parameters of an experimental setup, we examine through analytical estimates and extensive numerical simulations a specific system that can be implemented with cold atoms, the atomic modulated pendulum. We assess the efficiency of three possible routes to observe in detail chaos assisted tunneling properties. Our main conclusion is that due to the fragility of the symmetry between positive and negative momenta as a function of quasimomentum, it is very challenging to use tunneling between classical islands centered on fixed points with opposite momentum. We show that it is more promising to use islands symmetric in position space, and characterize the regime where it could be done. The proposed experiment could be realized with current state-of-the-art technology.

Chaos-assisted tunneling resonances in a synthetic Floquet superlattice

“…We engineer the phase and amplitude of the lattice via three acousto-optic modulators ( 72 ): One is dedicated to the control of the lattice laser intensity, and the others, driven by phase-locked synthesizers, control the relative phase, φ, between the two lattice beams. Introducing the dimensionless variables p = 2π P /( m ω d ), x = 2π X / d , where m is the atomic mass, d = 532 nm is the lattice spacing, and X and P are the position and momentum along the standing wave and normalizing the time t to the modulation angular frequency ω, the Hamiltonian that governs the dynamics reads ( 40 , 41 ) …”

“…To experimentally implement this Hamiltonian, we first load the BEC in a static lattice by a smooth increase of the lattice intensity ( 72 ). We then apply the protocol ( 41 ) described in Fig. 1 : The lattice is suddenly shifted at the desired position by an abrupt change of the phase φ, and its amplitude is subsequently modulated for an even number of modulation periods n × 2 T (see below).…”

“…To perform measurements on our system, we continue the modulation for an extra T /2 duration. This amounts to performing a π/2 rotation in phase space to transfer in momentum space the information encoded in position space (see Supplementary Text) ( 41 , 71 ). After a 25-ms time of flight, the interferences between the matter waves originating from the different cells of the optical lattice create a diffraction pattern (see Figs.…”

“…By increasing γ in the range [0.18,0.36] at fixed ε = 0.268, the initial central stable island splits into two, and even three, stable islands (see Fig. 2C ) ( 41 , 74 ). This process is a Hopf-type bifurcation well known in dynamical systems (see Supplementary Text) ( 75 ).…”

“…We have observed similar chaos-assisted resonances at different parameter values (see Supplementary Text). The choice of the stable islands symmetric in position was motivated by their relatively large size, a characteristic that facilitates the efficient loading of the BEC, and the robustness of the expected tunnel oscillation with respect to the quasi-momentum width ( 41 ). The period-2 nature of the resonance [also suggested in ( 76 )] leads to a situation where the symmetry is a classical property independent of quasi-momentum.…”