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Journal of Optics B Quantum and Semiclassical Optics

Noise-induced heating of squeezed states in atom traps

Shahen Hacyan, Rocio J uregui

Published Feb 26, 2003• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/5/2/304

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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.
By using the Lewis–Riesenfeld quantum invariant theory and properly choosing Hermitian invariant operator, a closed solution of the Schrödinger equation is derived for two forced quantum oscillators with mixing of two modes, and the quantum fluctuations in the output fields are evaluated. For the initial two-mode squeezed number or squeezed coherent state, in some particular conditions, the time evolution of the oscillators can not only preserve the initial two-mode squeezing, but also produce squeezing in the individual modes; and exhibit a periodical squeezing behaviour. For the initial two-mode number state or coherent state, there is no squeezing in the individual and mutual quadrature phases of the two-mode fields. Furthermore, regardless of which state above being initially considered, the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters. In particular, for the initial two-mode coherent state, the variances of the output fields are also independent of other parameters in the Hamiltonian, and always preserve their initial values 1/4.
Time-evolution of squeezed coherent states of a generalized Caldirola-Kanai type quantum parametric oscillator is found explicitly using the exact evolution operator obtained by the Wei-Norman algebraic approach. Properties of these states are investigated according to the parameters of the unitary squeeze operator and the time-variable parameters of the generalized quadratic Hamiltonian. As an application, we consider exactly solvable quantum models with specific frequency modification for which the corresponding classical oscillator is in underdamping case and driving forces are of sinusoidal type. For each model we explicitly provide the evolution of the squeezed coherent states and discuss their behavior.
We consider two different kinds of fluctuations in an ion trap potential: external fluctuating electrical fields, which cause statistical movement (``wobbling'') of the ion relative to the center of the trap, and fluctuations of the spring constant, which are due to fluctuations of the ac component of the potential applied in the Paul trap for ions. We write down master equations for both cases and, averaging out the noise, obtain expressions for the heating of the ion. We compare our results to previous results for far-off resonance optical traps and heating in ion traps. The effect of fluctuating external electrical fields for a quantum gate operation (controlled-NOT) is determined and the fidelity for that operation derived.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
This paper is concerned with the dynamics of coupled quantum oscillators in the presence of a classical harmonic driving force. By means of a theorem which describes the evolution of Gaussian distributions of coherent states, the solutions of the equations of motion are presented in a very compact form. Specific applications deal with forced oscillation and parametric amplification. The theoretical models include systems of damping oscillators with a continuous frequency distribution. The approach to thermal equilibrium in the absence of a driving force is discussed as a special case.