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Conference on Lasers and Electro-Optics

Squeezing Effects in Mechanical Oscillators

Y. Ben-Aryeh

Published Jan 1, 2009• 0 citations
Publisher

Abstract

A slow increase of the amplitude and frequency of a mechanical harmonic oscillator leads to squeezing effects, which are related to the use of the adiabatic approximation. An order of magnitude for this squeezing is calculated and it is suggested to use the present theory for mechanical oscillators in traps and for resonant detectors. The other extreme case of 'impact' interaction is analyzed in which the time of interaction is small relative to the time period of the harmonic oscillator. In the latter case the time development is without any squeezing but broadening effects for the mechanical wave packets might be important.

DOI
10.1364/iqec.2009.ifd3

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A slow increase of the amplitude and frequency of a mechanical harmonic oscillator leads to squeezing effects, which are related to the use of the adiabatic approximation. An order of magnitude for this squeezing is calculated and it is suggested to use the present theory for mechanical oscillators in traps and for resonant detectors. The other extreme case of 'impact' interaction is analyzed in which the time of interaction is small relative to the time period of the harmonic oscillator. In the latter case the time development is without any squeezing but broadening effects for the mechanical wave packets might be important.
The general theory of harmonic oscillators with time-dependent frequency and 'effective mass' is related in the present work to the use of the adiabatic approximation. A slow 'pumping' of energy into the mechanical oscillator leads to a slow increase of the amplitude and frequency with squeezing effects by which the noise in one quadrature is increasing and in the other quadrature is decreasing. An order of magnitude for the squeezing effect is calculated. It is suggested to use this theory for mechanical oscillators in traps and for resonant detectors. The other extreme case of 'impulse' interaction in which the time of interaction is very small relative to the time period of the harmonic oscillator is analyzed. In the 'impact' approximation the time development is without any squeezing, but broadening effects might be important under special conditions. The 'impact' approximation might be related to the detection of gravitational waves by Michelson interferometers.
The Hilbert space of a harmonic oscillator with frequency ω can be spanned by the overcomplete set of coherent states of a second oscillator with frequency ω'. These states are squeezed states of the first oscillator. We examine the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency and explore the two limiting cases of an adiabatic and a sudden change. Only the latter results in squeezing, while in the former symmetrical uncertainities in coordinate and momentum turn out to be adiabatic invariants.
With time and space transformation, we first solve the Schrödinger equation of the time-dependent harmonic oscillator (TDHO) system. The properties of the squeezing in the case of , i.e., the Paul trap and , namely a stable frequency interfered by a single-pulsing-type disturbance, are investigated by using function series expansion. Results in the former case are compared with the numerical calculations carried out before. It is found that the condition related to the most strong squeezing is the direct result of our method, not any approximation is needed. Solutions of the latter one are valuable for the study of the ion trap, where different cases are studied in detail. Finally we made a discussion for the measurement and observation of a trapped ion by using squeezing properties.
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.
In this work, the dynamics of the deformed one-dimensional harmonic oscillator with minimal length uncertainty is examined and the analytical solutions for time evolution of position and momentum operators are presented in which the rough approximation that neglects the higher order terms in BakerHausdor lemma is avoided. Based on these analytical solutions the uncertainties for position and momentum operators are calculated in a coherent state, and an unexpected squeezing effect in both coordinate and momentum directions is found in comparison with ordinary harmonic oscillator. Obviously such a squeezing effect is induced by the minimal length uncertainty (gravitational effects). Our results are applied to the electrons trapped in strong magnetic fields to examine the degree of the existing squeezing effect in a real system, which shows the squeezing degree induced by minimal length uncertainty is very small.
Although the problem of electromagnetic radiation by a quantum harmonic oscillator is considered in textbooks on quantum mechanics, some of its aspects have remained unclear until now. By this, we mean that usually the initial quantum states of both the oscillator and the field are assumed to be characterized by a definite energy level of the oscillator and definite occupation numbers of the field modes. In connection with growing interest in squeezed states, it would be interesting to analyze the general case when the initial states of both subsystems are arbitrary superpositions of energy eigenstates. This problem was considered in other work, where the power of the spontaneous emission was calculated in the case of an arbitrary oscillator's initial state, but the field was initially in a vacuum state. In the present article, we calculate the rate of the oscillator average energy, squeezing, and correlation parameter change under the influence of an arbitrary external radiation field. Some other problems relating to the interaction between quantum particles (atoms) or oscillators where the electromagnetic radiation is an arbitrary (in particular squeezed) state were investigated.