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American Journal of Physics

Long-Time Transition Probabilities for a Two-Level System Interacting with a Stochastic Electromagnetic Field

F.W. Cummings

Published Dec 1, 1962• 10 citations
Publisher

Abstract

The purpose of this paper is to demonstrate solutions of Schrödinger's equations for a two-level quantum system which is coupled to a radiation field. The electric (or magnetic) field, which is considered as a definite (if unknown) function of time E(t), is characterized by a power spectrum or its related correlation function. Long-time transition probabilities are obtained in the form of a contour integral under the assumption of small atom-field coupling. The contour integral can be easily evaluated for many cases of interest. In particular, it is shown how a two level system in interaction with a blackbody approaches the Maxwell-Boltzmann distribution as t → infin.

DOI
10.1119/1.1941846

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The purpose of this paper is to demonstrate solutions of Schrödinger's equations for a two-level quantum system which is coupled to a radiation field. The electric (or magnetic) field, which is considered as a definite (if unknown) function of time E(t), is characterized by a power spectrum or its related correlation function. Long-time transition probabilities are obtained in the form of a contour integral under the assumption of small atom-field coupling. The contour integral can be easily evaluated for many cases of interest. In particular, it is shown how a two level system in interaction with a blackbody approaches the Maxwell-Boltzmann distribution as t → infin.
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.
The power spectrum of the light scattered by a two-level atom driven near resonance by a monochromatic classical electric field is evaluated. The atom is assumed to relax to equilibrium with the driving field via radiation damping, which is treated by explicitly coupling the atom to the quantized electromagnetic field modes. The power spectrum of the scattered field is directly obtainable from the two-time atomic dipole moment correlation function, which is evaluated by a method based on a Markoff-type assumption analogous to that used to evaluate the time evolution of single-time atomic expectation values.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.
The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.
In this paper the physical entities such as transition probabilities and the density of states are related to appropriate electromagnetic-field correlation functions and to appropriate response functions. Such response functions have already been computed in a previous paper and therefore these can be used to obtain surface-dependent corrections. It is shown how the density of states and hence Planck's law depends on the presence of surfaces. I explicitly calculate the correction terms for the case of a small blackbody bounded by two plane conducting surfaces. An appreciable correction occurs if the linear dimensions of the blackbody are of the order of a wavelength. Next electric-dipole-type transitions in atomic systems are considered and a straightforward perturbation theory is used to obtain one-photon transition probabilities in terms of the surface-dependent response functions. As an illustration of the surface-dependent terms, the transitions in presence of a conducting surface are considered. The transition probabilities show a marked increase or decrease depending on whether the dipole transition is parallel or perpendicular to the surface. Both stationary and nonstationary fields are considered. As a special case of nonstationary fields, the transitions in a coherent field are considered in detail. It is also shown how the coherent radiation field in presence of dielectrics can be realized. It is found that if the radiation field, in arbitrary geometries, is initially in vacuum state then at later times it would be found in a coherent state if perturbed by an external (ccc-number) electromagnetic field.
It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schr\"odinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schr\"odinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.