Quantum-corrected Floquet dynamics: bridging fully quantum and semiclassical regimes

Abstract

Semiclassical descriptions of a few-level system coupled to an electromagnetic field mode reduce the field to a time-dependent driving term. Although such methods are widely used, the underlying quantum character of the field generates corrections to the dynamics that can become significant. Here we develop a general approach for systematically calculating quantum corrections to the time-dependent Floquet dynamics that emerges in the semiclassical limit. Taking the Rabi model of a spin-field system as an example, we show how approximate analytical expressions for short-time quantum corrections to the semiclassical dynamics can be obtained. This framework helps explain the emergence of semiclassical behavior, sheds new light on collapse and revival dynamics in the Rabi model, and provides a tool for assessing corrections to semiclassical control techniques.

Figures (3)

• Figure 1: Collapse of Rabi oscillations for different initial field states in the JCM. The excited-state probability of the spin is plotted against time. Parameter values are $\alpha = 10$, $\lambda = 0.05$, and $\Omega = \omega_0 = 1$. Numerical solutions of the JCM Hamiltonian (magenta) are compared with Eq. \ref{['fock_collapse_osc']} (black); dashed grey lines indicate the collapse envelope from Eq. \ref{['fock_collapse_osc']}. The initial state of the system is $\lvert\Phi(0)\rangle = \lvert+z\rangle \otimes \lvert\alpha, n\rangle$ with $n=$ (a) $0$, (b) $1$, (c) $2$, (d) $10$.
• Figure 2: Collapse of Rabi oscillations in the resonant Rabi model ($\Omega = \omega_0 = 1$). The initial state is $\lvert+z\rangle\otimes\lvert\alpha,1\rangle$. Numerical solutions of the full Rabi Hamiltonian (magenta) are compared with the analytical solution (black) obtained from the FBRWA together with an analytical approximation for the Floquet states from Finney1994. The parameters are (a) $\lambda = 0.02,\,\alpha = 10$, (b) $\lambda = 0.02,\,\alpha = 20$ and (c) $\lambda = 0.01,\,\alpha = 40$.
• Figure 3: Collapse of Rabi oscillations with the field initialized in the superposition state given in Eq. \ref{['initial_sup']}. The excited-state probability of the spin is plotted as a function of time. Parameter values are $\beta = 10$, $\lambda = 0.05$, and $\Omega = \omega_0 = 1$. Numerical solution of the JCM Hamiltonian (magenta) is compared with the analytical expression given in Eq. \ref{['sup_collapse_osc']} (black).