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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Wigner functions in the Paul trap

G. Schrade, Vladimir I. Man’ko, Wolfgang P. Schleich, Roy J. Glauber

Published Jun 1, 1995• 73 citations
Publisher

Abstract

The authors review the theory of the harmonic oscillator with time-dependent frequency by means of an approach based on an operator constant of the motion. With the help of this operator constant we define the ground state, the excited states and a coherent state of the oscillator and discuss the time dependence of these states through their Wigner functions. The authors derive the Wigner function of an arbitrary state at time t evolving in the time-dependent harmonic potential. Moreover, they calculate the correlation coefficient between position and momentum, which appears in the Schrodinger uncertainty relation. The authors illustrate their results for the example of a charged particle in the Paul trap.

DOI
10.1088/1355-5111/7/3/009

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The authors review the theory of the harmonic oscillator with time-dependent frequency by means of an approach based on an operator constant of the motion. With the help of this operator constant we define the ground state, the excited states and a coherent state of the oscillator and discuss the time dependence of these states through their Wigner functions. The authors derive the Wigner function of an arbitrary state at time t evolving in the time-dependent harmonic potential. Moreover, they calculate the correlation coefficient between position and momentum, which appears in the Schrodinger uncertainty relation. The authors illustrate their results for the example of a charged particle in the Paul trap.
At exact resonance we derive the linear and quadratic invariants for time-dependent coupled linear oscillators in the presence of second harmonic generation. Employing these invariants we introduce an accurate definition of the Dirac operators from which the wavefunctions in both coherent and Fock (number) states representations are calculated. We also derive the W-Wigner function of an arbitrary state at time t, evolving in the system under consideration. Moreover we calculate the correlation coefficient between position and momentum and we find that it is identical in both fields whatever the state used.
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.
We solve the quantum mechanical problem of the motion of charged particle in a quadrupole field that varies arbitrarily with time. The time-dependent wave functions are shown to be in a simple one-to-one correspondence with the wave functions of an ordinary static-field harmonic oscillator. We define and generate a set of coherent states for the charged particle that provide Gaussian localization about the classical particle trajectories. When the field variation is periodic in time, these minimum uncertainty states are shown to pulsate periodically in size and in phase. Under the action of external fields, coherent states retain their coherence. We discuss the way in which the Kapitza pseudopotential approach approximates the present exact results.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.