Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians (H) over cap(t) = {omega(0)(N) over cap i, t(i) + t(int) < t(i+1), t(i) < t < t(i) + t(int), describe a maser with N two-level atoms coupled to a single mode of a quant ized field inside the maser cavity; here, t(i), i = 1,2, ..., N-a, are disc rete times, N-a is large (similar to 10(5)), (N) over cap is the number ope rator 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(int). With all damping and finite-temperature effects neglected, this model for N = 1 describes the one-atom micromaser currentl y in operation with Rb-85 atoms making microwave transitions between two hi gh Rydberg states. We show that (H) over cap is completely integrable in th e quantum sense for any N = 1, 2,... and derive a second-order nonlinear or dinary differential equation (ODE) that determines the evolution of the inv ersion operator S-Z(t) in the su(2) Lie algebra. For N = 1 and under the no nlinear condition [s(Z)(t)](2) = (1/)4 (I) over cap, this ODE linearizes to the operator form of the harmonic oscillator equation, which we solve. For N = 1, the motion in the extended Nilbert space H can be a limit-cycle mot ion combining the motion of the atom under this nonlinear condition with th e tending of the photon number n to n(0) determined by root 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 stat e is \e>\n(0)>, where \e> is the excited state of the two-level atom and (N ) over cap\n(0)> = n(0)\n(0)>. Using a suitable loop algebra, we derive a L ax pair formulation of the operator equations of motion during the times ti ,t for any N. For N = 2 and N = 3, the nonlinear operator equations lineari ze under appropriate additional nonlinear conditions; we obtain operator so lutions 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 se miclassical 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.