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16th Congress of the International Commission for Optics: Optics as a Key to High Technology

Squeezing by frequency changes

T. Kiss

Published Jul 26, 1993• 0 citations
Publisher

Abstract

The problem of quantum harmonic oscillator with time-dependent frequency has been solved analytically only for special co(t) functions. In this paper (Sec. 1.) we give a recursive method to determine the creation and annihilation operators for any step-like 0)(0 (t) functions (i. e. functions being constant between jumps). In Sec. 2. we apply this method to jumps between two values of w. In Sec 3. we approximate more general functions with a series of small jumps, we examine the Fourier series of the function defined in Sec. 2, and show that even the first Fourier component leads to increasing squeezing.

DOI
10.1117/12.2308461

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The problem of quantum harmonic oscillator with time-dependent frequency has been solved analytically only for special co(t) functions. In this paper (Sec. 1.) we give a recursive method to determine the creation and annihilation operators for any step-like 0)(0 (t) functions (i. e. functions being constant between jumps). In Sec. 2. we apply this method to jumps between two values of w. In Sec 3. we approximate more general functions with a series of small jumps, we examine the Fourier series of the function defined in Sec. 2, and show that even the first Fourier component leads to increasing squeezing.
The time evolution of a quantum harmonic oscillator with a series of sudden jumps of the mass or the frequency is determined in the form of a recursion relation. An approximating method is developed for determining the time evolution of harmonic oscillators with arbitrary derivable functions of the frequency or the mass. The approximate solution is shown to tend to the analytical one in the limiting case. As a demonstration of the approximating method, the solution of the problem of damped oscillation in the square of the oscillator frequency is presented.
It is shown that squeezing will inevitably be generated in a harmonic oscillator with time-dependent frequency for any nonvanishing rate of frequency change, sudden or smooth. The general form of the time-dependent transition probabilities among the eigenstates of the initial Hamiltonian is obtained with an exact operator approach. Results for the case of a sudden frequency jump are explicitly given as an illustration. It is also pointed out that to generate squeezing in a harmonic oscillator, the Hamiltonian does not have to be a unitary squeeze transformation of the free Hamiltonian.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
The author examines the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency. Using the evolution operator method developed by Cheng and Fung, he investigates the evolution of a coherent state of a time-dependent harmonic oscillator as well as its squeezing and coherence property. Two specific models of frequency variation are explored; namely, one whose frequency is exponentially decreasing in time and the other with a periodic frequency. His analyses indicates that the wavefunction of the time-dependent oscillator starts as a coherent state at t=O and evolves as a squeezed state at a later time. It is also shown that squeezing cannot be generated by an adiabatic change of the oscillator frequency because the variances of q and p turn out to be adiabatic invariants, whereas a sudden change can produce squeezing.