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New Journal of Physics

Coherent states sometimes look like squeezed states and vice versa: the Paul trap

Michael Martin Nieto, D. Rodney Truax

Published Sep 12, 2000• 18 citations
Publisher

Abstract

Using the Paul trap as a model, we point out that the same wavefunctions can be variously coherent or squeezed states, depending upon the system to which they are applied.
DOI
10.1088/1367-2630/2/1/318

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We laser-cool single beryllium ions in a Paul trap to the ground (n equals 0) quantum harmonic oscillator state with greater than 90% probability. From this starting point, we can put the atom into various quantum states of motion by application of optical and rf electric fields. Some of these states resemble classical states (the coherent states), while others are intrinsically quantum, such as number states or squeezed states. We have created entangled position and spin superposition states (Schrodinger cat states), where the atom's spatial wavefunction is split into two widely separated wave packets. We have developed methods to reconstruct the density matrices and Wigner functions of arbitrary motional quantum states. These methods should make it possible to study decoherence of quantum superposition states and the transition from quantum to classical behavior. Calculations of the decoherence of superpositions of coherent states are presented.
We present a general model of a quantum parametric oscillator heated by coupled fluctuating fields. Two kinds of external fields are considered: an external fluctuating driving force, and a noise in the basic frequency of the oscillator. The energy increase and the change in the square variances of position and momentum produced in such systems are calculated. As a particular example, we study the case of a Paul trap and the evolution of coherent and squeezed states under general conditions. The analysis is also extended to the evolution of superpositions of coherent states.
Quasiclassical dynamics of trapped ions is described by applying the time dependent variational principle (TDVP) on coherent state orbits, introduced as sub-manifolds of the quantum state space. The classical Hamilton functions associated to the system are obtained as the expected values of the quantum Hamiltonian on symplectic coherent states. Such formalism can be applied to Hamilton functions which are nonlinear in the infinitesimal generators of a dynamical symmetry group. We give the explicit expression of the quantum Hamilton function in case of nonlinear electromagnetic traps. Discrete quasienergy spectra result, and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A dequantization algorithm is also suggested. The method is then applied to combined (Paul and Penning) and ideal Paul traps. We obtain the explicit expressions of the classical Hamilton functions for such particular traps, as well as the classical Hamilton equations of motion in complex coordinates for an octupole trap. In the pseudopotential approximation, the points of minimum of the Hamiton function define equilibrium configurations for trapped ions, of interest for quantum logic.