Deep Strong Coupling Regime of the Jaynes-Cummings Model

We study the quantum dynamics of a two-level system interacting with a quantized harmonic oscillator in the deep strong coupling regime (DSC) of the Jaynes-Cummings model, that is, when the coupling strength $g$ is comparable or larger than the oscillator frequency $\omega$ ($g/\omega \gtrsim 1$). In this case, the rotating-wave approximation cannot be applied or treated perturbatively in general. We propose an intuitive and predictive physical frame to describe the DSC regime where photon number wave packets bounce back and forth along parity chains of the Hilbert space, while producing collapse and revivals of the initial population. We exemplify our physical frame with numerical and analytical considerations in the qubit population, photon statistics, and Wigner phase space.

Figure 1

(a),(b) Round trip of a photon number wave packet and collapse revivals due to DSC dynamics with initial state $|+,0_b⟩=|g,0_a⟩$. (c) Collapse revivals with secondary peaks due to counterpropagating photon number wave packets starting in initial state $|+,2_b⟩=|g,2_a⟩$. For all cases, ${\omega }_0=0$ and $g/\omega =2$.

Figure 2

(a) Photon statistics at different times of the evolution with ${\omega }_0=0.5\omega$. (b) Comparison of probability $P_{+,0_b}(t)$ calculated for ${\omega }_0=0$ (solid line) and ${\omega }_0=0.5\omega$ (dashed line). In all simulations the initial state is $|+,0_b⟩$ and $g/\omega =2$.

Figure 3

Modulus of the Wigner function $|W(x,p)|$ and trajectory ($x$, $p$) for Eq. (4). (a) For ${\omega }_0=0$ the Wigner function is a symmetric Gaussian (a coherent state) and moves clockwise in phase space along a circle. (b)–(d) For ${\omega }_0=0.5\omega$ the Wigner function loses its symmetry and moves clockwise along a spiral trajectory as shown in the sequence corresponding to $t=0.5$, 1, 5. In both simulations the initial state is $|+,0_b⟩$ and $g/\omega =2$.

Figure 4

Collapses and revivals of $P_{+,0_b}(t)$ for $g/\omega =2$ and (a) ${\omega }_0=0.3\omega$ and (b) ${\omega }_0=0.5\omega$ We plot the exact numerical solution (area) at first order in ${\omega }_0/\omega$ (solid) and a two-mode approximation (red, dashed). (c) Distribution of probability of the different detunings ${\delta }_{n_b}=(\hslash \omega n_b-E_{n_b})/\hslash {\omega }_0$ weighted by their contribution to the wave function given in Eq. (8). We have marked the level $N_b^r$ which is used for the two-mode approximation in curves (a) and (b).

Figure 5

Parity chains for the two-qubit DSC regime.