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Journal of Physics A Mathematical and General

Schrödinger cat states in a Penning trap

Octavio Castaños, Shahen Hacyan, Ramón López–Peña, Vladimir I. Man’ko

Published Jan 30, 1998• 11 citations
Publisher

Abstract

The time-dependent Schrödinger equation of an ion moving in an asymmetric Penning trap is solved. Properties of the evolution of squeezed and Schrödinger cat states in the trap are studied. Analytic expressions for the energy expectation values, dispersions and correlations of the position and momentum operators of different macroscopic states, solutions of the time-dependent Schrödinger equation of the trap, are obtained. The probability densities in coordinate and momentum representations are also given. As an example, the behaviour of Schrödinger cat states of a proton moving in the trap is described numerically.

DOI
10.1088/0305-4470/31/4/012

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The time-dependent Schrödinger equation of an ion moving in an asymmetric Penning trap is solved. Properties of the evolution of squeezed and Schrödinger cat states in the trap are studied. Analytic expressions for the energy expectation values, dispersions and correlations of the position and momentum operators of different macroscopic states, solutions of the time-dependent Schrödinger equation of the trap, are obtained. The probability densities in coordinate and momentum representations are also given. As an example, the behaviour of Schrödinger cat states of a proton moving in the trap is described numerically.
We study the time evolution of a charged particle in a Penning trap in the framework of open quantum systems. The interaction with the environment is taken into account by imposing Lindblad operators which are linear in the canonical variables. For the special case of a proton in an asymmetric Penning trap, we compare the dynamics with earlier results obtained from the unitary time-dependent Schrödinger equation. A possibility of estimating the spatial decoherence time of the system is discussed, and approximate decoherence time scales are given for different ions.
We study the evolution of coherent and Schr\"odinger cat states in radio-frequency traps, which were used to bound Hg+{\mathrm{Hg}}^{+}Hg+ ions. The quantum confinement is determined by evaluating the expectation values and dispersions of the position and momentum operators, together with the probability densities in coordinate and momenta representations. Also the Wigner functions associated to this nonstationary quantum system, a parametric oscillator, are constructed.
This paper can serve as an illustration for a course on quantum mechanics. It gives a non-trivial example of the time evolution of a system, which includes a particle in the potential of two δ-traps: one of which is in the fixed position, and the other moves with constant velocity. The numerical solution of the Schrödinger equation represents the behaviour of the vector of state in quite a transparent way; one can see rough pictures of the processes of charge transfer and ionization. The numerical procedure is explained, and an exercise can be carried out in constructing other time-dependent systems, containing δ-potentials.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
The two-mode even and odd coherent Schrödinger cat states are introduced for the generalized stationary and non-stationary harmonic oscillator. The expected values of positions and momenta and their dispersion matrices are calculated for the Schrödinger cat states. The quadrature squeezing and correlation phenomenon are studied and the Wigner and Husimi functions are constructed explicitly for these states. The photon statistics of the two-mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function, which is expressed in terms of Hermite polynomials. The photon number, means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon.