Convergence to semiclassicality in the quantum Rabi model

We investigate the emergence of semiclassical dynamics in the quantum Rabi model using a recently developed limiting procedure that formally establishes correspondence with the semiclassical Rabi Hamiltonian [E. K. Twyeffort Irish and A. D. Armour, Phys. Rev. Lett. 129, 183603 (2022)]. While the limit itself is defined at the Hamiltonian level, how it is reached depends on the choice of quantum states. Defining a set of quantitative measures that capture the differences between quantum and semiclassical dynamics, we examine convergence to the semiclassical limit when the field is prepared in a displaced number state. These states, which interpolate to Fock states for zero displacement, are more general than the set of coherent states usually employed when considering the emergence of semiclassical behavior. Numerical computations of these measures consistently demonstrate the progressive emergence of semiclassical behavior as the joint limit of vanishing coupling and infinite displacement is approached. Complementing the numerical results, analytical approximations are developed that reproduce the behavior in the vicinity of the semiclassical limit with a high degree of fidelity and allow scaling relations to be derived. Although any initial displaced number state will eventually converge to the corresponding semiclassical dynamics as the limit is taken, the rate of convergence depends on the Fock number n of the state. States with larger values of n, which behave less classically than coherent states, converge more slowly to the limit.