Canadian Journal of Physics

Squeezing caused by a change of mass in a harmonic oscillator

Fan Hong-Yi, Haider R. Zaidi

Published Feb 1, 1989• 9 citations

Abstract

It is shown that a mass change in a harmonic oscillator generates a squeezing transformation. Time-independent as well as time-dependent transformations are investigated. An expression for the interaction Hamiltonian responsible for squeezing and the equations of motion for the time evolution are derived.

DOI: 10.1139/p89-025

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Canadian Journal of Physics

Squeezing caused by a change of mass in a harmonic oscillator

Fan Hong-Yi, Haider R. Zaidi

Published Feb 1, 1989• 9 citations

Publisher

Abstract

DOI: 10.1139/p89-025

Related papers

Squeezing and frequency jump of a harmonic oscillator

Fan Hong‐Yi, Haider R. Zaidi

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.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.

Physical Review AApr 1, 198879 citations

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Xin Ma, William Rhodes

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.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.

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We show that a harmonic oscillator subject to a sudden change of mas s produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and isalso applicable for rapid, although not necessarily instantaneous, mass variations. Ke.ywords: Solutions in closed form; quantum mechanics; exact invariants. Mostramos que un oscilador armónico sujeto a un cambio repentino de masa produce estados comprimidos. Nuestro estudio está basado en una solución analítica aproximada para el oscilador armónico dependiente del tiempo. El tratamiento continuo que estudiamos difiere de estudios anteriores en los cuales se igualan las soluciones en la discontinuidad. Nuestro formalismo requiere una transformación ad hoc de la ecuación diferencial original y es aplicable también para variaciones de masa rápidas, no solo instantáneas. Descriptores: Soluciones analíticas; mecánica cuántica; invariantes exactos. PACS: 03.65.La; 03.65.Ge; 02.30.Ik 1. Introduction ing another transforrnation that produces a monotonic time-dependent parameter. The transformed equation can then be sol ved with the continuum approach, leading to squeezing of the momentum variables at certain times. An interesting ad-vantage of this method is that the change of mass need not be a step function in the strict mathematical sense. The require-ment, which is physically more plausible, is that the change in mass should take place in a time span much shorter than the characteristic period of the oscillator. The plan of the manuscript is the following: in Seco 2, we write the quadratic and linear invariants for a time-dependent mass and exhibit their relation with the Hamiltonian. In Seco 3, we discuss two transforrnations that translate the clas-sical time-dependent mass equation into the time dependent trequency problem. The analytic solution for a rapid varia-tion of the time dependent mass using the appropriate trans-formation is then presented. In Seco 4, we exhibit how a sudden change of mas s in the quantum oscillator produces squeezing. Section 5 is left for conclusions. 2. Time dependent massWe show that a harmonic oscillator subject to a sudden change of mas s produces squeezed states. Our study is based on an approximate analytic solution to the time-dependent harmonic oscillator equation with a subperiod function parameter. This continuous treatment differs from former studies that involve the matching of two time-independent solutions at the discontinuity. This formalism requires an ad hoc transformation of the original differential equation and isalso applicable for rapid, although not necessarily instantaneous, mass variations. Ke.ywords: Solutions in closed form; quantum mechanics; exact invariants. Mostramos que un oscilador armónico sujeto a un cambio repentino de masa produce estados comprimidos. Nuestro estudio está basado en una solución analítica aproximada para el oscilador armónico dependiente del tiempo. El tratamiento continuo que estudiamos difiere de estudios anteriores en los cuales se igualan las soluciones en la discontinuidad. Nuestro formalismo requiere una transformación ad hoc de la ecuación diferencial original y es aplicable también para variaciones de masa rápidas, no solo instantáneas. Descriptores: Soluciones analíticas; mecánica cuántica; invariantes exactos. PACS: 03.65.La; 03.65.Ge; 02.30.Ik 1. Introduction ing another transforrnation that produces a monotonic time-dependent parameter. The transformed equation can then be sol ved with the continuum approach, leading to squeezing of the momentum variables at certain times. An interesting ad-vantage of this method is that the change of mass need not be a step function in the strict mathematical sense. The require-ment, which is physically more plausible, is that the change in mass should take place in a time span much shorter than the characteristic period of the oscillator. The plan of the manuscript is the following: in Seco 2, we write the quadratic and linear invariants for a time-dependent mass and exhibit their relation with the Hamiltonian. In Seco 3, we discuss two transforrnations that translate the clas-sical time-dependent mass equation into the time dependent trequency problem. The analytic solution for a rapid varia-tion of the time dependent mass using the appropriate trans-formation is then presented. In Seco 4, we exhibit how a sudden change of mas s in the quantum oscillator produces squeezing. Section 5 is left for conclusions. 2. Time dependent mass

Feb 1, 20070 citations

Design of squeezing

Hong-Yi Fan, Alfred Wünsche

The design of squeezing by a time-dependent harmonic oscillator is considered. The predetermined general time-dependent squeezing transformations are taken as the starting point from which the time evolution operator U(t,0) and then the corresponding time-dependent Hamiltonian are derived. The technique of integration within an ordered product of operators and the coherent-state representation are used in the intermediate derivations, where U(t,0) is obtained as an operator image of the classical symplectic transformation in phase space by a phase-space integral over coherent states.The design of squeezing by a time-dependent harmonic oscillator is considered. The predetermined general time-dependent squeezing transformations are taken as the starting point from which the time evolution operator U(t,0) and then the corresponding time-dependent Hamiltonian are derived. The technique of integration within an ordered product of operators and the coherent-state representation are used in the intermediate derivations, where U(t,0) is obtained as an operator image of the classical symplectic transformation in phase space by a phase-space integral over coherent states.

Journal of Optics B Quantum and Semiclassical OpticsJun 21, 200024 citations

Squeezing induced in a harmonic oscillator by a sudden change in mass or frequency

M. Sebawe Abdalla, R. K. Colegrave

The Kanai-Caldirola (Bateman) Hamiltonian is used to derive the dynamics of a simple harmonic oscillator, initially in a minimum uncertainty state, under the influence of an external agency which causes the mass parameter to change from ${\mathit{M}}_{0}$ to ${\mathit{M}}_{1}$ in a short time \ensuremath{\epsilon}. Then the frequency changes from ${\mathrm{\ensuremath{\omega}}}_{0}$ to ${\mathrm{\ensuremath{\omega}}}_{1}$=( ${\mathit{M}}_{0}$ / ${\mathit{M}}_{1}$ )${\mathrm{\ensuremath{\omega}}}_{0}$+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). In the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}0, no squeezing or loss of coherence occurs. If ${\mathit{M}}_{1}$ / ${\mathit{M}}_{0}$ =1\ifmmode\pm\else\textpm\fi{}\ensuremath{\eta} (0\ensuremath{\ll}1), then a squeezing of order ${\mathrm{\ensuremath{\epsilon}}}^{2}$\ensuremath{\eta} occurs. If ${\mathit{M}}_{1}$ / ${\mathit{M}}_{0}$ is appreciably different from unity, then the quadrature variances are unequal but the state no longer has minimum uncertainty. An application could be made in quantum optics.The Kanai-Caldirola (Bateman) Hamiltonian is used to derive the dynamics of a simple harmonic oscillator, initially in a minimum uncertainty state, under the influence of an external agency which causes the mass parameter to change from ${\mathit{M}}_{0}$ to ${\mathit{M}}_{1}$ in a short time \ensuremath{\epsilon}. Then the frequency changes from ${\mathrm{\ensuremath{\omega}}}_{0}$ to ${\mathrm{\ensuremath{\omega}}}_{1}$=(${\mathit{M}}_{0}$/${\mathit{M}}_{1}$)${\mathrm{\ensuremath{\omega}}}_{0}$+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). In the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}0, no squeezing or loss of coherence occurs. If ${\mathit{M}}_{1}$/${\mathit{M}}_{0}$=1\ifmmode\pm\else\textpm\fi{}\ensuremath{\eta} (0\ensuremath{\ll}1), then a squeezing of order ${\mathrm{\ensuremath{\epsilon}}}^{2}$\ensuremath{\eta} occurs. If ${\mathit{M}}_{1}$/${\mathit{M}}_{0}$ is appreciably different from unity, then the quadrature variances are unequal but the state no longer has minimum uncertainty. An application could be made in quantum optics.

Physical Review AAug 1, 199318 citations

{\mathit{M}}_{0}

{\mathit{M}}_{1}

in a short time \ensuremath{\epsilon}. Then the frequency changes from ${\mathrm{\ensuremath{\omega}}}_{0}$ to ${\mathrm{\ensuremath{\omega}}}_{1}$=(

{\mathit{M}}_{0}

{\mathit{M}}_{1}

)${\mathrm{\ensuremath{\omega}}}_{0}$+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). In the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}0, no squeezing or loss of coherence occurs. If

{\mathit{M}}_{1}

{\mathit{M}}_{0}

=1\ifmmode\pm\else\textpm\fi{}\ensuremath{\eta} (0\ensuremath{\ll}1), then a squeezing of order ${\mathrm{\ensuremath{\epsilon}}}^{2}$\ensuremath{\eta} occurs. If

{\mathit{M}}_{1}

{\mathit{M}}_{0}

is appreciably different from unity, then the quadrature variances are unequal but the state no longer has minimum uncertainty. An application could be made in quantum optics.