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Physical Review A

Squeezing and frequency jump of a harmonic oscillator

Fan Hong‐Yi, Haider R. Zaidi

Published Apr 1, 1988• 79 citations
Publisher

Abstract

The theoretical basis of a mechanism for squeezing through a frequency jump of a harmonic oscillator is examined. It is found that a related squeezing transformation exists, but its interpretation in terms of a frequency change is not possible. A reinterpretation of the results in terms of a scaling transformation in coordinate or momentum variables is given. Moreover, a sudden jump is not essential, and squeezing can evolve continuously. The equations of motion for such a process are obtained and the interaction responsible for squeezing is obtained.

DOI
10.1103/physreva.37.2985

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Comment on ‘‘Squeezing and frequency jump of a harmonic oscillator’’

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The theoretical basis of a mechanism for squeezing through a frequency jump of a harmonic oscillator is examined. It is found that a related squeezing transformation exists, but its interpretation in terms of a frequency change is not possible. A reinterpretation of the results in terms of a scaling transformation in coordinate or momentum variables is given. Moreover, a sudden jump is not essential, and squeezing can evolve continuously. The equations of motion for such a process are obtained and the interaction responsible for squeezing is obtained.
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.
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.
The control of microscopic particle behavior based on a specific external field has always been a research hotspot in the field of physics. Many studies have been exploring various methods to manipulate and control the behavior of particles at a microscopic level. In this work, we investigate the phenomenon of single-particle squeezing induced by frequency jumping in a two-dimensional rotating harmonic oscillator potential. Squeezing, as a quantum mechanical phenomenon, has attracted significant attention due to its potential applications in various fields. It refers to the reduction of fluctuations in certain physical quantities, allowing for more precise measurement results. Squeezing phenomena have been extensively studied in different physical systems, including optics, atomic physics, and solid-state physics. However, there have been few reports on the quantum state squeezing phenomenon induced by frequency jumping in a rotating harmonic oscillator potential. Therefore, our study aims to fill this gap and shed light on this intriguing phenomenon. To explore the squeezing phenomenon induced by frequency jumping, we focus on the fluctuations and squeezing of the single particle’s cyclotron radius coordinate and center-guided coordinate in the two-dimensional rotating harmonic oscillator potential. Through numerical simulations and theoretical analysis, we can understand the influence of frequency jumping on the degree of squeezing and reveal the underlying physical mechanism of squeezing evolution. In this work, we first investigate the influence of frequency jumping on the squeezing evolution of the cyclotron radius mode. By carefully selecting appropriate jumping moments, we analyze the influence of frequency jumping on the degree of squeezing. Our research results show that the degree of squeezing in the cyclotron radius coordinate remains unchanged at the jumping moment. However, we observe a stronger squeezing phenomenon in the subsequent evolution process. This indicates that frequency jumping plays a crucial role in squeezing evolution of the cyclotron radius mode. Furthermore, we focus on the squeezing evolution of the center-guided mode during frequency jumping. By selecting suitable parameters, we analyze the squeezing and evolution of two squeezing modes: the divergent mode and the oscillatory mode. Interestingly, we discover the existence of a critical potential trap aspect ratio, which is determined by the rotation angular velocity of the external potential. When the aspect ratio approaches this critical value, the squeezing mode undergoes a transition, and a significant squeezing phenomenon appears in the oscillatory mode. This finding provides valuable insights into the origin and control of squeezing phenomena. Finally, we discuss the potential applications of these squeezing phenomena. Squeezing has significant implications in the fields of quantum sensing and quantum information processing. Through a deeper understanding of the squeezing evolution process caused by frequency jump, we can better control the microscopic particle behavior through external field. This knowledge opens up new possibilities for future physical research and technical applications.
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.