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Physical Review A

Induced transitions and energy of a damped oscillator

P. Croxson

Published Jan 1, 1994• 16 citations
Publisher

Abstract

Damped harmonic-oscillator pseudostationary-state wave functions are used to calculate transitions probabilities for a simple harmonic oscillator (SHO) subjected to damping from time t=0. The mean and variance of the energy operator are obtained in the state which was initially an SHO stationary state. This state is an instantaneous eigenstate of the energy at times when sin(\ensuremath{\omega}t)=0, \ensuremath{\omega} being the reduced frequency.

DOI
10.1103/physreva.49.588

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Comment on “Induced transitions and energy of a damped oscillator”

George J. Papadopoulos, Ioannis A. Hadjiagapiou

In a recent paper Croxson [Phys. Rev. A $49,$ 588 (1994)] reported dissipation-induced transition probabilities from the ground state to higher states for a quantum harmonic oscillator. A partial approximate result was given for the oscillator to remain in its ground state, while for higher transitions the situation became complicated. Our approach provides an exact simple form closed expression for any order of transition. In addition, we supply the evolution of the work done by the system on its environment.

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Physical Review A•Jun 1, 1989•40 citations
Damped harmonic-oscillator pseudostationary-state wave functions are used to calculate transitions probabilities for a simple harmonic oscillator (SHO) subjected to damping from time t=0. The mean and variance of the energy operator are obtained in the state which was initially an SHO stationary state. This state is an instantaneous eigenstate of the energy at times when sin(\ensuremath{\omega}t)=0, \ensuremath{\omega} being the reduced frequency.
A harmonic oscillator subject to the combined effects of damping and pulsating is represented by a Kanai–Caldirola Hamiltonian. The equations of motion are solved in the Heisenberg picture in the case of weak pulsation. The rotating-wave approximation (RWA) is used to obtain the motion in the neighborhood of the principal resonance. The RWA Schrödinger equation is solved exactly and pseudostationary and quasicoherent states are described. The transition probability between quasicoherent and coherent states is obtained and the gain in energy is discussed.
We investigate the dissipation-induced transition probabilities between any two eigenstates of a simple harmonic oscillator. Using the method developed by Yu and Sun [Phys. Rev. A 49, 592 (1994)], the general analytical expressions for the transition probabilities are obtained. The special cases: transition probabilities from the ground state to the first few excited states are then discussed in detail. Different from the previous studies in the literature where only the effect of damping was considered, it is found that the Brownian motion makes the transitions between states of different parity possible. The limitations of the applicability of our results are also discussed.
The time-dependent Schrdinger equation for the harmonic oscillator with time-dependent frequency ω=ω(t) and mass m=m(0)ω(0)ω(t)exp[2c∫t0ω(t)dt] can be solved by means of simple transformation and separation of variables.The wave functions are obtained for c21,c2=1 and c21,respectively.When |x|→∞,the wave function for c21 tends to zero but it is not so for c2≥1.Finally,some special cases are given for the time-dependent harmonic oscillator and one of them is similar to a damped harmonic oscillator.
We have obtained explicit analytical formulas for the mean energy and its variance (characterizing the energy fluctuations) of a quantum harmonic oscillator with time-dependent frequency in the adiabatic regimes after the frequency passes through zero. The behavior of energy turns out to be quite different in two cases: when the frequency remains real and when it becomes imaginary. In the first case, the mean energy always increases when the frequency returns to its initial value, and the increment coefficient is determined by the exponent in the power law of the frequency crossing zero. On the other hand, if the frequency becomes imaginary, the absolute value of mean energy increases exponentially, even in the adiabatic regime, unless the Hamiltonian becomes time independent. Small corrections to the leading terms of simple adiabatic approximate formulas are crucial in this case, due to the unstable nature of the motion.
In a recent paper Croxson [Phys. Rev. A 49,49,49, 588 (1994)] reported dissipation-induced transition probabilities from the ground state to higher states for a quantum harmonic oscillator. A partial approximate result was given for the oscillator to remain in its ground state, while for higher transitions the situation became complicated. Our approach provides an exact simple form closed expression for any order of transition. In addition, we supply the evolution of the work done by the system on its environment.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.