Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians
$$ \widehat H(t)=\begin{cases} \widehat H,\quad &t_i<t<t_i+t_\mathrm{int}, \\ \omega_0\widehat N,\quad &t_i+t_\mathrm{int}<t<t_{i+1}, \end{cases} $$
describe a maser with $N$ two-level atoms coupled to a single mode of a quantized field inside the maser cavity; here, $t_i$, $i=1,2,\dots,N_a$, are discrete times, $N_a$ is large ($\sim 10^5$), $\widehat{N}$ is the number operator in the Heisenberg–Weyl (HW) algebra, and $\omega_0$ is the cavity mode frequency. The $N$ atoms form an $(N+1)$-dimensional representation of the $su(2)$ Lie algebra, the single mode forming a representation of the HW algebra. We suppose that $N$ atoms in the excited state enter the cavity at each $t_i$ and leave at $t_i+t_\mathrm{int}$. With all damping and finite-temperature effects neglected, this model for $N=1$ describes the one-atom micromaser currently in operation with $^{85}$Rb atoms making microwave transitions between two high Rydberg states. We show that $\widehat{H}$ is completely integrable in the quantum sense for any $N=1,2,\dots$ and derive a second-order nonlinear ordinary differential equation (ODE) that determines the evolution of the inversion operator $S^Z(t)$ in the $su(2)$ Lie algebra. For $N=1$ and under the nonlinear condition $[S^Z(t)]^2= (1/)4\hat{I}$, this ODE linearizes to the operator form of the harmonic oscillator equation, which we solve. For $N=1$, the motion in the extended Hilbert space $\mathcal H$ can be a limit-cycle motion combining the motion of the atom under this nonlinear condition with the tending of the photon number $n$ to $n_0$ determined by $\sqrt{n_0+1}gt_\mathrm{int}=r\pi$ $($where $r$ is an integer and $g$ is the atom-field coupling constant$)$. The motion is steady for each value of $t_i$; at each $t_i$, the atom-field state is $|e\rangle|n_0\rangle$, where $|e\rangle$ is the excited state of the two-level atom and $\widehat{N}|n_0\rangle=n_0|n_0\rangle$. Using a suitable loop algebra, we derive a Lax pair formulation of the operator equations of motion during the times $t_\mathrm{int}$ for any $N$. For $N=2$ and $N=3$, the nonlinear operator equations linearize under appropriate additional nonlinear conditions; we obtain operator solutions for $N=2$ and $N=3$. We then give the $N=2$ masing solution. Having investigated the semiclassical limits of the nonlinear operator equations of motion, we conclude that “quantum chaos” cannot be created in an $N$-atom micromaser for any value of $N$. One difficulty is the proper form of the semiclassical limits for the $N$-atom operator problems. Because these $c$-number semiclassical forms have an unstable singular point, “quantum chaos” might be created by driving the real quantum system with an additional external microwave field coupled to the maser cavity.