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Optics Communications

The solutions of the generalized classical and quantum harmonic oscillators with time dependent mass, frequency, two-photon parameter and external force: The squeezing effects

S. K. Singh, Swapan Mandal

Published Aug 4, 2010• 15 citations
Publisher
DOI
10.1016/j.optcom.2010.07.009

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For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals O(m/ℏ)O( m / \hbar)O(m/ℏ), when ℏ/m≪1\hbar / m \ll 1ℏ/m≪1. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for nnn non-interacting quantum harmonic oscillators with masses m1,...,mnm_1, ..., m_nm1​,...,mn​ and frequencies ω1,...,ωnω_1, ..., ω_nω1​,...,ωn​ can be derived as an approximation to the Newtonian mechanics of nnn non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals O(m~/ℏ)O( \tilde{m} / \hbar)O(m~/ℏ), where m~\tilde{m}m~ is the mass of the minimum mass oscillator, when ℏ/m~≪1\hbar/\tilde{m} \ll 1ℏ/m~≪1. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, nnn non-interacting quantum harmonic oscillators are approximately nothing but nnn non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.