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Journal of Physics A Mathematical and General

The vacuum excitation and squeezing properties of two quantum oscillators with delta-kicked interactions

Jeong‐Young Ji, David Peak

Published Apr 10, 2001• 2 citations
Publisher

Abstract

The vacuum excitation and squeezing of two harmonic oscillators with delta-kicked interactions for four types of elementary coupling are studied. The exact quantum motion for the Heisenberg operators and the explicit form of squeezing operators are found. The variances are calculated to study the squeezing properties for position, momentum and generalized quadrature operators.

DOI
10.1088/0305-4470/34/16/311

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The vacuum excitation and squeezing of two harmonic oscillators with delta-kicked interactions for four types of elementary coupling are studied. The exact quantum motion for the Heisenberg operators and the explicit form of squeezing operators are found. The variances are calculated to study the squeezing properties for position, momentum and generalized quadrature operators.
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.
We have generalized the two-step approach to the solution of systems of NNN coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.
By virtue of the technique of integration within an ordered product of operators, we derive the normal ordering expansion of a one- and two-mode combination squeezing operator for two harmonic oscillators with coordinate-momentum coupling. It turns out that this squeezing operator just diagonalizes the Hamiltonian H = p 1 2 /2 m 1 + m 1 ω 1 2 x 1 2 /2+ p 2 2 /2 m 2 + m 2 ω 2 2 x 2 2 /2−λ x 2 p 1 , so its ground state is a one- and two-mode combination squeezed state. Quantum fluctuation in the ground state is calculated.
"Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position."
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.