Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations

Abstract

The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include the time-variable as a parameter of the transformation. The new potentials are nonstationary and define Hamiltonians which are not integrals of motion for the system under study. We take the stationary oscillator of constant frequency to produce nonstationary oscillators, and also provide an invariant that serves to define uniquely the state of the system. In this sense our approach completes the program of the BSS method since the eigenfunctions of the invariant are an orthonormal basis for the space of solutions of the related Schrödinger equation. The orthonormality holds when the involved functions are evaluated at the same time. The dynamical algebra of the nonstationary oscillators is generated by properly chosen ladder operators and coincides with the Heisenberg algebra. We also construct the related coherent states and show that they form an overcomplete set that minimizes the quadratures defined by the ladder operators. These states are not invariant under time-evolution since their time-dependence relies on the basis of states and not on the complex eigenvalue that labels them. Some concrete examples are provided.

Quasienergy operators and generalized squeezed states for systems of trapped ions

“…A recent paper introduces a protocol aimed at generalizing these states, that may be employed to build the squeezed states for any kind of quantum models [86]. The squeezed states linked to the 3D harmonic oscillator can be built as eigenstates of linear contribution of ladder operators which are associated to the generalized Heisenberg algebra [87,88]. Multimode macroscopic states consisting of a superposition of spin coherent states that are generated in a trapped ion system are introduced in [89].…”

Nonstationary deformed singular oscillator: quantum invariants and the factorization method

“…Remarkably, even if the Hamiltonian is time-independent, there is a constant of motion, different from the Hamiltonian Ĥ1 . The latter is a fact that was explored for the stationary oscillator [28], and it was used in the construction of new solvable time-dependent models [35,40].…”

“…This relates a well known model with another one which is unknown and to be determined. The method has been proved to be useful in the construction of new families of exactly-solvable deformed nonstationary oscillators [32][33][34][35]. However, the method by itself does not provide information about the constants of motion of the system, which have to be computed in a different way.…”

Freezable bound states in the continuum for time-dependent quantum potentials

“…[27] and references therein), including the adiabatical approximation [28] and perturbation theory [25]. A powerful strategy to construct time-dependent solutions to the Schrödinger equation from its stationary version is through point transformations [29][30][31][32][33]. These transformations, in combination with first-order timeindependent SUSY, have allowed extending the number of solvable time-dependent examples, from the infinite potential well with a moving wall to the trigonometric Pöschl-Teller potential [26], by transforming the stationary Schrödinger equation to a time-dependent equation that in the remote past/future connects with the solutions of the free particle.…”