Chaos-assisted tunneling resonances in a synthetic Floquet superlattice

Abstract

The field of quantum simulation, which aims at using a tunable quantum system to simulate another, has been developing fast in the past years as an alternative to the all-purpose quantum computer. So far, most efforts in this domain have been directed to either fully regular or fully chaotic systems. Here, we focus on the intermediate regime, where regular orbits are surrounded by a large sea of chaotic trajectories. We observe a quantum chaos transport mechanism, called chaos-assisted tunneling, that translates in sharp resonances of the tunneling rate and provides previously unexplored possibilities for quantum simulation. More specifically, using Bose-Einstein condensates in a driven optical lattice, we experimentally demonstrate and characterize these resonances. Our work paves the way for quantum simulations with long-range transport and quantum control through complexity.

Fig. 1. Sketch of the experimental procedure.

(A) By an abrupt phase shift φ of the lattice, the atomic clouds, initially at the bottom of the static lattice wells (blue dotted line), are suddenly placed on the slope of the shifted lattice wells (blue solid line). In the phase space (x,p) representation shown in (B), it corresponds to a displacement along the x axis. (C) The amplitude modulation of the lattice depth generates a mixed phase space with regular islands (blue closed orbits) surrounded by a chaotic sea (red zone). With the appropriate phase shift before the modulation, the wave packet can be placed on a lateral regular island (e.g., the right one). Then, the wave packet tunnels back and forth between the two stable symmetric lateral islands (tunneling rate J), leading to two wave packets R and L. (D) The observation of the tunneling requires a phase space rotation (black arrows), which transfers to momentum space the information encoded in position space: The population on the right (resp. left) island gains a negative (resp. positive) momentum. (E) Stack of integrated experimental images taken every two modulation periods. The images reflect the momentum distribution after a 25-ms time of flight, its periodicity h/d is directly related to the lattice periodicity. Classically, atoms are expected to stay on the same side (initially in the right island). The first three experimental tunneling oscillations are shown.

Fig. 2. Observation of the bifurcation.

(A and B) Typical images obtained after time of flight, when the wave packet lay initially in a regular island or in the chaotic sea, respectively. In case (A), the momentum distribution remains narrow, while the wave function spreads over many orders in case (B). On both images, the white dashed line indicates the zero-momentum component. (C) Evolution of the stroboscopic phase space for increasing values of the dimensionless depth γ and fixed ε = 0.268, showing the splitting of the initial central regular island into two and then into three islands. Phase portraits were realized with 10,000 iterations of 107 initial positions equally spaced on the x and p axes. (D) Evolution of the position of the center of the regular islands, as a function of γ: experimental results obtained for ℏ_eff = 0.203 (empty marks) and ℏ_eff = 0.232 (filled marks). The analytical prediction is indicated as a dot-dashed line. The shaded area reveals the limited resolution governed by ℏ_eff.

Fig. 3. Sketch of a chaos-assisted tunneling (CAT) resonance resonance.

(A) Quasi-energy spectrum of a regular doublet (red and blue) and a chaotic (green) state. An avoided crossing occurs between the chaotic (green) state and the regular state (red) part of the doublet having the same symmetry. As ℏ_eff is varied, these two states mix and repel. (B) The resulting tunneling oscillation frequencies (proportional to differences in quasi-energies) exhibit two contributions and strong variations. (C to F) Phase space representations of quantum states involved in the crossing [Husimi function; see (84)]. Regular, antisymmetric (C) and symmetric (F) states are localized near integrable orbits; the chaotic states (D) lie in the chaotic sea, while the mixed one (E) overlaps with both structures.

Fig. 4. Experimental observation of a CAT resonance and comparison with regular dynamical tunneling.

(A and B) Experimentally measured tunneling frequencies (red dots) as a function of ${\mathrm{\hslash }}_{\text{eff}}^{-1}$ compared with numerical simulation in the case where all lattice cells are equally populated (blue lines) for the parameters (A) ε = 0.14, γ = 0.249 ± 0.002 and (B) ε = 0.24, γ = 0.375 ± 0.005. Tunneling frequencies are extracted through Fourier transform of the experimental or numerical oscillations (see Materials and Methods). Blue shaded area corresponds to the experimental uncertainty on γ, with a shade intensity giving the height of the Fourier peak. The gray shaded area of (B) reveals the zone in which a good agreement with experiments requires realistic simulations that take into account the initial finite size of the BEC (see Materials and Methods). The corresponding stroboscopic classical phase spaces are plotted in (C). (D to F) Population in each island and for different values of ${\mathrm{\hslash }}_{\text{eff}}^{-1}$ [corresponding to the dotted lines in (B)] as a function of time (blue, left island; red, right island); top: experimental data, bottom: realistic simulations with 13 cells initially equally populated. The first three oscillations of (D) are represented in Fig. 1E.