Quantum manipulation through finite fluctuations for a generalized parametric oscillator using a lie algebra representation

Abstract

In this paper, we focus on the effects of temporally compact fluctuations on the parameters m(t) (mass), (frequency), and (spatial dilation coefficient), for the generalized Caldirola-Kanai Hamiltonian. We propose an algorithm such that given control over the timing of repeated compact fluctuations, one can produce any time-evolution associated with complete parameter control of the generalized Caldirola-Kanai Hamiltonian. Computational simplicity is achieved in this endeavor using a faithful representation of the Hamiltonian’s associated Lie algebra. Further simplification occurs as a result of using unitarity conditions for the time-evolution operator. We apply our analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. Furthermore, we use the faithful Lie algebra representation to show that any unitary evolution of this generalized Caldirola-Kanai system is given by the squeeze operator multiplied with a phase factor. We also show that the classical evolution of a probabilistic Gaussian phase space distribution and the evolution of the Wigner function of a coherent state in the quantum Caldirola-Kanai system are identical.