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The manipulation problem in Quantum Mechanics

J C David Fernández

Published Jan 1, 1999• 3 citations
OpenAlex

Abstract

We explain the meaning of dynamical manipulation, and we illustrate its mechanism by using a system composed of a charged particle in a Penning trap. It is shown that by means of appropriate electric shocks (delta-like pulses) applied to the trap walls one can induce the squeezing transformation. The geometric phases associated to some cyclic evolutions, induced either by the standard fields of the Penning trap or by the superposition of these plus a rotating magnetic field, are analysed.

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We explain the meaning of dynamical manipulation, and we illustrate its mechanism by using a system composed of a charged particle in a Penning trap. It is shown that by means of appropriate electric shocks (delta-like pulses) applied to the trap walls one can induce the squeezing transformation. The geometric phases associated to some cyclic evolutions, induced either by the standard fields of the Penning trap or by the superposition of these plus a rotating magnetic field, are analysed.
We propose a scheme to engineer an effective spin Hamiltonian starting from a system of electrons confined in micro-Penning-traps. By means of appropriate sequences of electromagnetic pulses, alternated with periods of free evolution, we control the shape and strength of the spin-spin interaction. Moreover, we can modify the effective magnetic field experienced by the particle spin. This procedure enables us to reproduce notable quantum spin systems, such as Ising and XYXYXY models. Thanks to its scalability, our scheme can be applied to a fairly large number of trapped particles within the reach of near future technology.
The quantum harmonic oscillator in a nonuniformly rotating frame is studied. It is shown that squeezed states and particles are generated by inertial effects. The analytic forms of the squeezing and Bogoliubov coefficients are given explicitly in terms of the solutions of the classical equations of motion. The squeezed states have the minumum dispersion allowed by the Robertson uncertainty relations for an arbitrary number of observables (the explicit form of these relations is obtained). The problem of an ``abrupt twist'' is worked out in detail as an example. The formalism can also be applied to the problem of a charged particle in a Penning trap, where it is shown that squeezed states are produced, with a squeezing frequency of the order of the magnetron frequency. In both examples, analytic expressions for the variances of the position and momentum are obtained. \textcopyright{} 1996 The American Physical Society.
Abstract We present the quantum theory of the elliptical Penning trap, i.e. the general case where the cylindrical symmetry of the electrostatic trapping potential around the trapping magnetic field axis is broken. The theory applies to both slightly and highly elliptical traps, where it is shown that the difference between the quantum states of particles in these traps corresponds to a variation of the degree of squeezing of their motional modes in the xy-plane. In a trap with tunable ellipticity, such as the Geonium Ghip planar Penning trap, it follows that control of the ellipticity via the trapping voltages enables squeezing of the quantum states of the particle. We discuss the adiabatic preparation of such squeezed states, which follows naturally from the appearance of an avoided crossing between the diabatic levels of the coupled motional states of the particle.
In this work, we study the geometric phase and the dynamics of electric and magnetic dipoles. We present your origins, some relevant works in this area as well as your results. The appearance of a geometric phase is explained when a study of quantum dynamics of the particle is made, during a complete cycle and that is subject to cyclic adiabatic processes. The present study begins with an introduction about the geometric phase, where we present examples used to illustrate it. Next, the adiabatic theorem was presented in both classical and quantum systems and a discourse on adiabatic invariants. After this, we present its final mathematical formulation and also its obtaining for a spin 1/2 particle. Soon after, we obtained the Lagrangian and its Hamiltonian, we find the quantum phase that the particle acquired in the efect Aharonov- Bohm and Aharonov - Casher. We have also seen the quantum dynamics of both electric and magnetic dipoles, where we start from the Lagrangian and study its field configurations, we reach the Ananda phase, where in the next step we obtain phases for some of the Field configurations in the Ananda phase from phase of Aharonov - Casher where we made the electric dipole moment equal to zero, as well as the He - Mckellar - Welkens phase with magnetic dipole moment equal to zero, and the Casella efect thus obtaining a particle case of the Ananda phase.
In [1], a new way to generate an observable geometric phase on a quantum system by means of a completely incoherent phenomenon has been proposed. The basic idea is to force the ground state of the system to evolve cyclically by “adiabatically” manipulating the environment with which it interacts. The specific scheme analyzed in [1], consisting of a multilevel atom interacting with a broadband squeezed vacuum bosonic bath whose squeezing parameters are smoothly changed in time along a closed loop, is here solved in a more direct way. This new solution emphasizes how the geometric phase at the ground state of the system is indeed due to a purely incoherent dynamics.