Quantum integrability and quantum chaos in the micromaser

Abstract

The time-dependent quantum Hamiltonians $$\hat H\left( t \right) = \left\{ {\begin{array}{*{20}c} {\hat H,t_i< t< t_i + t_{int} } \\ {\omega _0 \hat N,t_i + t_{int}< t< t_i + 1,} \\ \end{array} } \right.$$ describe a maser with N two-level atoms coupled to a single mode of a quantized field inside the maser cavity: here, ti, i=1,2,…,Na, are discrete times, Na is large (∼105), $$\hat N$$ is the number operator in the Heisenberg-Weyl (HW) algebra, and ω0 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 ti and leave at ti+t int . With all damping and finite-temperature effects neglected, this model for N=1 describes the one-atom micromaser currently in operation with85Rb atoms making microwave transitions between two high Rydberg states. We show that $$\hat H$$ is completely integrable in the quantum sense for any N-1,2,… and derive a second-order nonlinear ordinary differential equation (ODE) that determines the evolution of the inversion operator SZ(t) in the su(2) Lie algebra. For N=1 and under the nonlinear condition $$\left[ {S^Z \left( t \right)} \right]^2 = \left( {1/} \right)4\hat I$$ , 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 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 n0 determined by $$\sqrt {n_0 + 1} gt_{int} = r\pi $$ (where r is an integer and g is the atom-field coupling constant). The motion is steady for each value of ti; at each ti, the atom-field state is |e>|n0>, where |e> is the excited state of the two-level atom and $$\left. {\left. {\hat N} \right|n_0 } \right\rangle = \left. {\left. {n_0 } \right|n_0 } \right\rangle $$ . Using a suitable loop algebra, we derive a Lax pair formulation of the operator equations of motion during the times t 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.