Perturbations of the excited quantum oscillator: From number states to statistical distributions D. K. Sunko Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia B. Gumhalter a) Institute of Physics of the University, P.O. Box 304, 10001Zagreb, Croatia Received 10 February 2003; accepted 9 May 2003 We discuss the transitions that an external time-dependent perturbation can induce upon a quantum harmonic oscillator in an excited initial state. In particular, we show how to describe transitions of the oscillator from initial states characterized by statistical distributions. These results should be useful for interpretations of the properties of weakly dispersive bosonic excitations in quantum systems whose dynamics is investigated by time or energy resolved spectroscopies. © 2004 American Association of Physics Teachers. DOI: 10.1119/1.1587703 I. INTRODUCTION: THE FORCED OSCILLATOR MODEL In many areas of quantum physics the forced oscillator model provides a paradigm for demonstrating exact nonper- turbative solutions to systems subjected to strong time- dependent perturbations. Many authors have used this model as a testing ground for nearly all the many-body techniques developed during that time. 1–5 In addition, a number of real- istic problems can be fruitfully treated within this simple model 5–16 or its generalizations. 7,17–21 The present work revisits the same problem once more with a different agenda. Our primary aim is to describe how to use the forced oscillator model to calculate the excitation probabilities for an oscillator whose initial state is character- ized by a statistical distribution. In Sec. II we first introduce the forced oscillator model in second quantization notation. We begin with the usual exact solution for the evolution operator of the oscillator subject to an external time- dependent force coupled linearly to the oscillator displace- ment; the corresponding scattering operator is derived by a limiting procedure. We present in Sec. III some useful novel derivations of the expressions for state-to-state transition probabilities of the oscillator. By observing the symmetries that these probabilities should satisfy, we are able to demon- strate some interconnections between the present formulas and the ones obtained in various earlier treatments of the same problem, giving more insight into the temporal evolu- tion of the studied system. In particular, in the last part of Sec. III and Appendix A we show the equivalence of the results of two alternative operator disentanglements that ap- pear naturally in the calculation of transition matrix ele- ments. The main body of the paper is in Sec. IV, which gives the first systematic study of transitions of the forced quantum oscillator whose initial state is characterized by a statistical distribution. We derive two interesting results. The first is the generating function for the transition probabilities, obtained using the results of Sec. III. As a by-product, we also find a surprisingly simple shortcut to the spectral density of a per- turbed oscillator initially in thermal equilibrium, derived in textbooks by a rather involved time-dependent Green’s func- tion formalism. 5 Our second result is a general expression for the transition probabilities as a functional of the initial dis- tribution of oscillator states, which should enable the tack- ling of a wide class of physical problems that can be de- scribed by the forced oscillator model. Although this expression cannot be generally evaluated in a closed form, it efficiently provides a formula for the probability of a transi- tion into any particular final state. All that is required is that the generating function of the initial distribution be known. As an example, we introduce the two-pulse problem: one pulse excites the oscillator from its ground state into a known distribution of excited states. The second pulse then acts on this distribution, and the above-mentioned formula provides the excitation probability of any final state. The calculation is worked out in some detail, showing for a con- crete example that the result is symmetric with respect to the order of application of the two pulses. II. PROBLEM AND NOTATION The Hamiltonian corresponding to the forced oscillator model can be written in the simple form: H =H 0 +V  t  , 1 where H 0 =  0  a † a + 1 2  2 is the Hamiltonian of an unperturbed quantized harmonic oscillator of mass m and characteristic frequency  0 , with a † and a denoting the usual creation and annihilation opera- tors of noninteracting bosons, respectively. The eigenstates | n  of H 0 are the eigenstates of the number operator N ˆ =a † a , that is, they satisfy N ˆ | n  =n | n  , 3 and H 0 | n  =  0  n + 1 2  | n  , 4 with the eigenvalues n =0,1,2,..., which are the number of bosons in the state | n  with the total energy n   0 . The perturbation V ( t ) is taken to describe the linear cou- pling of the oscillator displacement to an external time- dependent force F ( t ): V  t  =-u 0  a +a †  F  t  , 5 231 231 Am. J. Phys. 72 2, February 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers