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

TL;DR

This paper extends the concept of coherent states to systems with time-dependent and nonlinear Hamiltonians using complex Riccati equations, enabling analysis of wave packets with evolving widths in various quantum systems.

Contribution

It introduces a generalized framework for coherent states applicable to time-dependent, nonlinear, and dissipative quantum systems via complex Riccati equations.

Findings

Derived explicit forms of generalized coherent states for diverse systems.

Showed the connection between Riccati equations and supersymmetric quantum mechanics.

Demonstrated the applicability to systems with dissipative environments.

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.