Linear adiabatic invariants and coherent states

Abstract

The Born-Fock adiabatic theorem is extended to all orders for some quadratic quantum systems with finitely or infinitely degenerate energy spectra. A prescription is given for obtaining adiabatic invariants to any order. For any quadratic quantum system with N degrees of freedom there are 2N linear adiabatic invariant series, which correspond to the 2N exact invariants. The exact quantum mechanical solution for any nonstationary quadratic quantum system is also constructed by making use of the coherent-state representation: The Green's function, coherent states, transition amplitudes and probabilities and their generating functions are obtained explicitly. Two particular systems, the N-dimensional time-dependent general oscillator and charged particle motion in a varying and uniform electromagnetic field, are considered in greater detail as examples.

Introduction to quantum optics

“…where the 2N-vector B(t) is the integral of motion linear in the annihilation and creation operators found in [1], [9] and [18]. This ansatz follows from the statement that the density operator of the Hamiltonian system is the integral of motion, and its matrix elements in any basis must depend on appropriate integrals of motion.…”

“…This ansatz follows from the statement that the density operator of the Hamiltonian system is the integral of motion, and its matrix elements in any basis must depend on appropriate integrals of motion. In particular, the Wigner function and Q-function depend just on linear invariants found in [1], [9] and [18].…”

“…where time-dependent arguments are linear integrals of motion of the quadratic system found in [1], [9] and [18]. The same ansatz is used for the Q-function.…”

“…For N-mode parametric oscillator, the analogous amplitude has been expressed in terms of Hermite polynomials of 2N variables, i.e. the overlap integral of two generic Hermite polynomials of N variables with a Gaussian function (Frank-Condon factor for a polyatomic molecule) has been evaluated in [9], [18]. The corresponding result uses the formula…”

“…Two time-dependent integrals of motion which are linear forms in position and momentum for the classical and quantum oscillator with time-dependent frequency were found in [4]; for a charge moving in varying in time uniform magnetic field, this was done in [5] (in the absense of the electric field), [6], [7] (nonstationary magnetic field with the "circular" gauge of the vector potential plus uniform nonstationary electric field, and [8] (for the Landau gauge of the time-dependent vector potential plus nonstationary electric field). For the multimode nonstationary oscillatory systems, such new integrals of motion, both of Ermakov's type (quadratic in positions and momenta) and linear in position and momenta, generalizing the results of [4] were constructed in [9]. Recently the discussed time-dependent invariants were obtained using Noether's theorem in [10]- [12].…”