Cat states in a driven superfluid: role of signal shape and switching protocol

TL;DR

This work analyzes a 1D Bose-Hubbard chain under kinetic driving with zero average hopping, yielding an effective Hamiltonian $H_{\mathrm{eff}}$ whose interaction matrix elements are encoded in $\Gamma(y)$ with $y=\kappa g$ and $\kappa=J/\omega$. The system undergoes a Mott-to-cat-state transition, forming a macroscopic superposition of quasi-condensates at $k=\pm \pi/2$; the authors study robustness to driving waveform shapes, initial phase, external flux, and adiabatic ramp protocols. They show that time-symmetric waveforms preserve the cat-state properties, while a non-symmetric sawtooth reduces them due to the complex $\Gamma(y)$; phase offsets do not alter ground-state properties, and flux shifts the momentum distribution. An adiabatic ramp from the Mott state to the cat state can be optimized, with cosinusoidal driving offering the best fidelity for shorter ramps, providing practical guidance for realizing macroscopic superpositions in cold-atom experiments.

Abstract

We investigate the behavior of a one-dimensional Bose-Hubbard model whose kinetic energy is made to oscillate with zero time-average. The effective dynamics is governed by an atypical many-body Hamiltonian where only even-order hopping processes are allowed. At a critical value of the driving, the system passes from a Mott insulator to a superfluid formed by a cat-like superposition of two quasi-condensates with opposite non-zero momenta. We analyze the robustness of this unconventional ground state against variations of a number of system parameters. In particular we study the effect of the waveform and the switching protocol of the driving signal. Knowledge of the sensitivity of the system to these parameter variations allows us to gauge the robustness of the exotic physical behavior.

Figures (5)

• Figure 1: The matrix element defining the effective interaction between bosons, $\Gamma$, for the various signal shapes, plotted as a function of the variable $y=\kappa g(k_\ell, k_m, k_n, k_p)$. The analytical expressions for $\Gamma(y)$ are given in Section \ref{['sec:3']}. For the sawtooth case, where $\Gamma$ cannot be made real (see discussion in Sections \ref{['sec:3']} and \ref{['sec:4']}), we plot $|\Gamma(y)|$.
• Figure 2: Two particle momentum density, $\rho^{(2)}(k,k')$ as a function of $\kappa$ for each signal shape. For $\kappa=0$ the system is in the Mott state, and $\rho^{(2)}(k,k')$ is peaked along the diagonal $k = k'$. As $\kappa$ increases, isolated peaks form at $\pm(\pi/2, \pi/2)$, indicating the formation of the superfluid cat-state.
• Figure 3: Momentum density, $\rho(k)$, as a function of $\kappa$. (a) Square wave driving. The momentum density is initially flat for $\kappa=0$ when the system is in the Mott state. As $\kappa$ increases the system passes through a phase transition and $\rho(k)$ develops two peaks at $k = \pm \pi/2$, indicating the formation of the superfluid cat state. (b) Sawtooth driving. In contrast to the square wave case, the peaks in $\rho(k)$ develop more slowly, and are much less pronounced.
• Figure 4: Two-particle momentum density $\rho^{(2)}(k,k')$ for several values of $\Phi$, with $\kappa=0.6$. When $\Phi=0$ the peaks are centered on $\pm(\pi/2,\pi/2)$. For $\Phi = \pi/8$ each peak is smeared over two values of momenta, as the momentum shift produced by $\Phi$ is incommensurate with the reciprocal lattice momenta. $\Phi = \pi/4$ is commensurate with the reciprocal lattice and the peaks are shifted by one momentum spacing, $-\pi/4$, along the diagonal.
• Figure 5: Fidelity of the ramp protocol in preparing the cat state at $\kappa = 0.8$, as measured by $\chi$. (a) For a ramp-time of $T_{\mathrm{ramp}}=320 T$ the fidelity of the final state depends strongly on the phase of the driving. Cosinusoidal driving (black solid line) shows the best performance, and sinusoidal driving (blue dot-dashed line) the worst. An intermediate phase, $f(t)=\cos(\omega t - \pi/4)$ (red dashed line) lies between these results. The value of $\chi$ in the true ground state of the system is shown by the horizontal dashed line. (b) When the ramp-time is increased to to $T_{\mathrm{ramp}}=640 T$ the driving phase barely affects the result. (c) Dependence of $\chi$ on the phase of the driving for the two ramp-times. Error bars indicate the amplitude of the oscillations of the final state. Physical parameters: $U=1$, $\omega=50$.