Quantum dynamics of the small-polaron formation in a superconducting analog simulator

We propose a scheme for investigating the nonequilibrium aspects of small-polaron physics using an array of superconducting qubits and microwave resonators. This system, which can be realized with transmon or gatemon qubits, serves as an analog simulator for a lattice model describing a nonlocal coupling of a quantum particle (excitation) to dispersionless phonons. We study its dynamics following an excitation-phonon (qubit-resonator) interaction quench using a numerically exact approach based on a Chebyshev-moment expansion of the time-evolution operator of the system. We thereby glean heretofore unavailable insights into the process of the small-polaron formation resulting from strongly momentum-dependent excitation-phonon interactions, most prominently about its inherent dynamical timescale. To further characterize this complex process, we evaluate the excitation-phonon entanglement entropy and show that initially prepared bare-excitation Bloch states here dynamically evolve into small-polaron states that are close to being maximally entangled. Finally, by computing the dynamical variances of the phonon position and momentum quadratures, we demonstrate a pronounced non-Gaussian character of the latter states, with a strong antisqueezing in both quadratures.

Figure 1

Schematic diagram of the analog-simulator circuit containing SC qubits $Q_n$ (with charging and Josephson energies $E_C^s$ and $E_J^s$, respectively), resonators $R_n$, and coupler circuits $B_n$ with three Josephson junctions $\left(n=1,...,N\right)$. ${\varphi }_n^l$ and ${\varphi }_n^u$ are total fluxes threading the lower and upper loops of $B_n$, respectively. Qubit $Q_n$ interacts with its neighbors through circuits $B_{n-1}$ and $B_n$.

Figure 2

Expected phonon number after an e-ph interaction quench at $t=0$ for $k_0=\pi /2$. Inset: the probability to remain in the initial bare-excitation Bloch state.

Figure 3

SP formation time ${\tau }_{\mathrm{sp}}$ for varying initial bare-excitation quasimomenta $k_0$ within the Brillouin zone and different choices of values for ${\varphi }_{\mathrm{dc}}$ and $\delta \omega$.

Figure 4

Dependence of the SP formation time ${\tau }_{\mathrm{sp}}$ on the effective coupling strength ${\lambda }_{\mathrm{eff}}$, shown for $\delta \omega /2\pi =200\mathrm{MHz}$ and different initial quasimomenta $k_0$.

Figure 5

Typical time dependence of the dynamical variances $S_x\left(t\right)$ and $S_p\left(t\right)$. The parameter values are indicated in the plot.

Figure 6

Time dependence of the e-ph entanglement entropy for ${\varphi }_{\mathrm{dc}}=0.975\pi$ and different choices of values for $k_0$ and $\delta \omega$.

Figure 7

Short-time behavior of the e-ph entanglement entropy for ${\varphi }_{\mathrm{dc}}=0.975\pi$ and different choices of values for $k_0$ and $\delta \omega$.