Quantum-classical crossover of a field mode

We explore the quantum-classical crossover in the behavior of a quantum field mode. The quantum behavior of a two-state system—a qubit—coupled to the field is used as a probe. Collapse and revival of the qubit inversion form the signature for quantum behavior of the field and continuous Rabi oscillations form the signature for classical behavior of the field. We demonstrate both limits in a single model for the full coupled system, for field states with the same average field strength, and so for qubits with the same Rabi frequency.

Figure 1

(Color online) Qubit inversion, $⟨{\sigma }_z⟩$, as a function of dimensionless time $\omega t$ for the resonant cases of Rabi oscillations (dotted; light gray/magenta) in a classical field and collapse and revival [dark gray (red) heavy line] in a quantum field. For the classical field case $\nu /\omega =8$ is used, corresponding to the same dominant Rabi frequency (see [17]) as in the quantum field case, which uses coherent state (3) with $\alpha =\sqrt{15}$ and $\lambda /\omega =1$. Also shown [in green (light gray) solid] line is the qubit entropy (in nats), which indicates the degree of entanglement with the field.

Figure 2

(Color online) Results [dark gray (red)] showing qubit inversion, $⟨{\sigma }_z⟩$, for different values of dissipation (and therefore drive) applied to the field. These illustrate collapse and revival, suppression of collapse, and an approach to complete Rabi oscillations as dissipation and drive are increased. For each individual QSD run shown, the qubit entropy in nats [light gray (green)] is superimposed.

Figure 3

(Color online) Results [dark gray (red)] showing qubit inversion, $⟨{\sigma }_z⟩$, for different values of dissipation (and therefore drive) applied to the field. These plots focus on the parameter range where the revival of Rabi oscillations begins to emerge. For each individual QSD run shown, the qubit entropy in nats [light gray (green)] is superimposed.