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.

Canadian Journal of PhysicsFeb 1, 19899 citations

Squeezing and Frequency Changes in Harmonic Oscillations

R. Graham

Journal of Modern OpticsJun 1, 198781 citations

Journal of Yanbian University

Squeezing caused by a change of frequency in a harmonic oscillator

Zheng Li

Published Jan 1, 2001• 0 citations

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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.

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Canadian Journal of PhysicsFeb 1, 19899 citations

Squeezing and Frequency Changes in Harmonic Oscillations

R. Graham

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 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.

Journal of Modern OpticsJun 1, 198781 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.