Nonlinear Dynamics in a Synthetic Momentum-State Lattice

The scope of analog simulation in atomic, molecular, and optical systems has expanded greatly over the past decades. Recently, the idea of synthetic dimensions—in which transport occurs in a space spanned by internal or motional states coupled by field-driven transitions—has played a key role in this expansion. While approaches based on synthetic dimensions have led to rapid advances in single-particle Hamiltonian engineering, strong interaction effects have been conspicuously absent from most synthetic dimensions platforms. Here, in a lattice of coupled atomic momentum states, we show that atomic interactions result in large and qualitative changes to dynamics in the synthetic dimension. We explore how the interplay of nonlinear interactions and coherent tunneling enriches the dynamics of a one-band tight-binding model giving rise to macroscopic self-trapping and phase-driven Josephson dynamics with a nonsinusoidal current-phase relationship, which can be viewed as stemming from a nonlinear band structure arising from interactions.

Figure 1

Synthetic momentum-state lattices with atom-atom interactions. (a) Two-photon Bragg transitions individually couple atomic momentum modes along a synthetic dimension. (b) The synthetic lattice features intersite tunneling terms with controlled amplitude $J_i$ and phase ${\phi }_i$ set by the Bragg laser amplitudes and phases, lattice site energies ${ϵ}_i$ set by the two-photon Bragg detuning, and naturally occurring long-ranged and mode-dependent atomic interactions: Atoms in the same site experience a repulsive pairwise interaction of strength $u$, and atoms in different sites have a repulsive interaction of strength $2u$.

Figure 2

Macroscopic self-trapping in an array of laser-coupled momentum states. (a) Atoms initialized to a single lattice site evolve in a one-dimensional lattice with uniform tunneling $J$. Self-trapping occurs when the collective interaction strength $U$ exceeds the tunneling bandwidth $4J$. (b),(c) Population dynamics for strong [$J/h=1281(5)\mathrm{Hz}$] and weak tunneling [$J/h=93.3(2)\mathrm{Hz}$]. Left: data from integrated optical density (OD) images after 18 ms time of flight averaged over five experimental realizations. Right: dynamics from numerical simulations. (d) Integrated OD images plotted vs the tunneling strength $J$ taken at times of $t=1.5\hslash /J$. (e) Standard deviation of the atomic distributions from (d) shown alongside the simulation results. The solid blue curve assumes the ideal Hamiltonian after application of the rotating wave approximation, while the dashed red curve includes residual time dependence due to off-resonant effects. The expected self-trapping transition $U/J=4$ is shown as a vertical gray line. Data for (d),(e) are averaged over 20 experimental realizations, and the error bars in (e) are the standard error of the mean. Numerical simulations in (b),(c),(e) assume a homogeneous mean-field energy of $U/h=520\mathrm{Hz}$.

Figure 3

Phase-driven dynamics in a synthetic bosonic Josephson junction array. (a) Atoms are prepared in a superposition state on two lattice sites $n=-1$ and $n=0$ with equal weight and a relative phase of $\varphi$. Starting at time $t=0$, the full lattice is turned on and the atoms evolve for a time $\mathrm{\Delta }t$. We then measure the atomic distribution, from which we determine the mean displacement $\overline{n}$ (mean position $+0.5$) and spread ${\sigma }_n$ (standard deviation of position) of the wave packet. (b) In the noninteracting limit, $\overline{n}$ and ${\sigma }_n$ can be related to the single-band energy $E(q)$ of quasimomentum states in the synthetic lattice, as the wave packets with phase offset $\varphi$ populate a spread of quasimomentum states about a mean synthetic quasimomentum $q_0=\varphi$. The plots show the band energy $E(q)$, the group velocity $v_g\propto dE/dq$, and the dispersion $d^{2}E/d{q}^2$ of the synthetic lattice. (c) Integrated density patterns (after two tunneling times) vs $\varphi$ for the regimes of strong [left, $J/h=1253(12)\mathrm{Hz}$] and weak tunneling [right, $J/h=156.7(1.3)\mathrm{Hz}$]. (d),(e) $\overline{n}$ vs $\varphi$ and ${\sigma }_n$ vs $\varphi$, respectively, for several tunneling values [$J/h=\{156.7(1.3),257(3),402(3),583(6),1253(12)\}$]. Dots are experimental data and the lines are theory. The results for different tunneling values are offset for clarity in (d). No offset is applied for (e). The theory curves in (d),(e) are based on a real-space GPE simulation, as described in the text. Data for (c)–(e) are averaged over ten experimental realizations, with error bars in (d),(e) representing the standard error of the mean. The datasets in (c)–(e) were taken over a $2\pi$ range of $\varphi$ values, uniformly shifted in phase to correct for phase shifts acquired during the wave packet initialization, and then mapped onto a larger and redundant $4\pi$ range. These redundant regions are distinguished by a gray background.