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.
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.
Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels
E1 and
E2 and obtain exact analytical expressions for the coefficients
c1 and
c2 of the final state. The expressions are general since the coefficients
α1 and
α2 of the initial superposition state are free parameters constrained only by
∣α1∣2+∣α2∣2=1. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability
Pα1,α2→2 of an initial superposition state to transition to the second eigenstate. Armed with this general expression we study some interesting special cases. With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap
E2−E1 and hence on the relative phase between the two eigenstates. For specific values of the interaction strength
β, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.
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.
We consider Ullersma's model of a damped harmonic oscillator interacting with a heat bath. We show that for large damping the oscillator does not undergo Brownian motion at any temperature. For high temperatures non-Markovian effects show up in initial slips only. For low temperatures we find, as further anomalies, power-law decay of time-dependent expectation values as well as a strong squeezing of the displacement and momentum uncertainties; again these anomalies are most strongly pronounced for strong damping.