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Communications in Theoretical Physics

Density Matrix and Squeezed State for Two Harmonic Oscillators with a Kinetic Coupling

Fan Hong‐Yi, Ying Zhang

Published Jan 30, 2000• 0 citations
Publisher

Abstract

We study two harmonic oscillators with a kinetic coupling system. By taking a unitary transformation approach, we turn the system into the Fock space of two independent harmonic oscillators and derive the density matrix for it. The corresponding unitary operator is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of the integration within an ordered product of operators, we manifestly show that the ground state of the system is a squeezed state.

DOI
10.1088/0253-6102/33/1/69

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We study two harmonic oscillators with a kinetic coupling system. By taking a unitary transformation approach, we turn the system into the Fock space of two independent harmonic oscillators and derive the density matrix for it. The corresponding unitary operator is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of the integration within an ordered product of operators, we manifestly show that the ground state of the system is a squeezed state.
We derive the density matrix ρ(x1, x2; x1', x2', β) for two coupled harmonic oscillators by taking a unitary transformation approach. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of integration within ordered product and the coordinate representation of U we manifestly show that the ground state of the coupled oscillators is a squeezed state. Based on the derived ρ(x1, x2; x1', x2', β) we calculate the average coupling energy.
By virtue of the technique of integration within an ordered product of operators, we derive the normal ordering expansion of a one- and two-mode combination squeezing operator for two harmonic oscillators with coordinate-momentum coupling. It turns out that this squeezing operator just diagonalizes the Hamiltonian H = p 1 2 /2 m 1 + m 1 ω 1 2 x 1 2 /2+ p 2 2 /2 m 2 + m 2 ω 2 2 x 2 2 /2−λ x 2 p 1 , so its ground state is a one- and two-mode combination squeezed state. Quantum fluctuation in the ground state is calculated.
It is shown that n>0 simple-harmonic-oscillator squeezed states have the same time-evolution behavior as the fundamental (n=0) squeezed state. That is, the probability density \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ retains its original shape as time progresses, except that the scales of the ordinate and abscissa oscillate in a reciprocal manner with frequency 2\ensuremath{\omega}. This allows the time-evolved squeezing operator S${\mathrm{^}}_{\mathit{H}}$(\ensuremath{\lambda},t) to be written as a product of simple factors. Moreover, if a thermal distribution with density matrix \ensuremath{\rho}^ is squeezed, then the mixed squeezed state with minimum free energy has a density matrix given by a unitary transformation \ensuremath{\rho}${\mathrm{^}}_{\mathit{s}}$=S^(\ensuremath{\lambda})\ensuremath{\rho}^S^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(\ensuremath{\lambda}), where \ensuremath{\lambda} is a suitable squeezing parameter.