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arXiv:0912.4178 · Journal of Physics B Atomic Molecular and Optical Physics

Transitionless quantum drivings for the harmonic oscillator

Juan Gonzalo Muga, Xi Chen, Sara Ibañez, Ion Lizuain, A. Ruschhaupt

Published Apr 6, 2010• 117 citations
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Abstract

Two methods to change a quantum harmonic oscillator frequency without transitions in a finite time are described and compared. The first method, a transitionless-tracking algorithm, makes use of a generalized harmonic oscillator and a non-local potential. The second method, based on engineering an invariant of motion, only modifies the harmonic frequency in time, keeping the potential local at all times.

DOI
10.1088/0953-4075/43/8/085509

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Two methods to change a quantum harmonic oscillator frequency without transitions in a finite time are described and compared. The first method, a transitionless-tracking algorithm, makes use of a generalized harmonic oscillator and a non-local potential. The second method, based on engineering an invariant of motion, only modifies the harmonic frequency in time, keeping the potential local at all times.
The open driven quantum harmonic oscillator is studied for three different possible mechanisms of driving; a generalization of the forced oscillator, driving due to a time-dependent coupling to a two-level system, and driving due to an adiabatic modulation of the frequency. The first case encompasses all possible configurations going from the standard forced harmonic oscillator to the classical pumping type of driving typically used in quantum optics. In the second case, an additional two-level system, coupled to the oscillator at a time-varying rate, is assumed and the reduced dynamics of the oscillator and the qubit is examined. In the driving due to adiabatic modulation, the dynamics of a harmonic oscillator whose frequency slowly varies in time is studied. The study done in this paper is focused on a harmonic type of driving, although the solutions are derived for an arbitrary type of driving. In all the cases, the oscillator is assumed to be in contact with a zero- or finite-temperature environment, which provides dissipation to the system. The treatment done in this paper is fully analytic and the derivation of the solutions is based on the employment of the double Fourier transform of the Wigner function.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.