Generalized coherent states for time-dependent and nonlinear Hamiltonian operators via complex Riccati equations

Abstract

Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfills a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

“…In the case Ω = 0, the expressions (38) coincide with the creation and annihilation operators reported by Nienhuis and Allen for the HG modes in free space [25]. Those operators have also appeared as the conserved creation and annihilation operators of the harmonic states for the free particle [37], and as generalized (invariant) ladder operators of the parametric harmonic oscillator [38].…”

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

“…is well known in the literature and finds many application in physics [22-25, 30, 57, 64, 65, 82-85, 90, 91]. It arises quite naturally in the studies of parametric oscillators [22][23][24][25]30], in the description of structured light in varying media [82][83][84][85], and in the study of non-Hermitian Hamiltonians with real spectrum [57,64,65]. The key to solve (C-1) is to consider the homogeneous linear equation q + Ω 2 (t) q = 0, (C-2) which coincides with the equation of motion for a classical parametric oscillator.…”

Quantum operations: technical or fundamental challenge?

“…Some of these phenomena were independently observed in [40,41]. An ample collection of more general squeezing operations was described by Dodonov [42]; the manipulation of complex Hamiltonians see [43]. The idea of controlling the finite dimensional qubit systems by deforming the closed dynamical processes reappears also in the recent development [44,45,46].…”