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Physical Review A

Interaction between a Nonlinear Oscillator and a Radiation Field

I. R. Senitzky

Published Sep 1, 1972• 41 citations
Publisher

Abstract

The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy $\ensuremath{\hbar}\ensuremath{\omega}{L}_{3}$, that describes the collective effect of a number of identical two-level (or spin-\textonehalf{}) systems under given idealized conditions, a large total angular momentum quantum number ${L}_{0}$ corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay---with the NO initially excited and the radiation field in the ground state---and forced oscillation---with the NO subject to a prescribed resonant driving field---are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large ${L}_{0}$, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as ${L}_{3}(0)$ approaches ${L}_{0}$ (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to ${L}_{3}(t)$. It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of ${L}_{3}(t)$ to a constant (negative) value---or steady state---at which the power absorbed equals the power radiated; in the strong-field case, ${L}_{3}(t)$ oscillates periodically between the limits $\ifmmode\pm\else\textpm\fi{}{L}_{0}$, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for ${L}_{0}=\frac{1}{2}$, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of $〈{L}_{3}(t)〉$ and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with $〈{L}_{3}(0)〉=\ensuremath{-}{L}_{0}, 〈{L}_{3}(t)〉$ approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For ${L}_{0}>\frac{1}{2}$, the free decay is treated by a method that involves the derivation of a set of expressions for the $k\mathrm{th}$ derivative of $〈{L}_{3}(t)〉$ as an expectation value of a polynomial in ${L}_{3}$ of order $k+1$. This set, together with the eigenvalue equation for ${L}_{3}$, is shown to lead to a solution for all the moments of ${L}_{3}$. The method is used to obtain complete solutions for several low values of ${L}_{0}$, and also to obtain initial derivatives of $〈{L}_{3}(t)〉$ as polynomials in ${L}_{0}$ for $〈{L}_{3}(0)〉={L}_{0}$. For large ${L}_{0}$, a comparison of classical and quantum-mechanical equations shows that only the condition $〈{L}_{3}(0)〉\ensuremath{\approx}{L}_{0}$ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition $〈{L}_{3}(0)〉={L}_{0}$ solved quantum mechanically up to such a time ${t}_{1}$ by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ensuremath{\ge}{t}_{1}$. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of $〈{L}_{3}(t)〉$---for arbitrary ${L}_{0}$---is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of ${L}_{0}$. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and $〈{L}_{2}^{3}(t)〉$ is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since $〈{L}_{3}(t)〉$ describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.

DOI
10.1103/physreva.6.1175

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The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy ${L}_{3}$ corresponds to the pendulum potential energy, the LO energy $n$ corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\ensuremath{\gamma}K}{\ensuremath{\omega}}$ is proportional to the angular momentum $K$ of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for ${L}_{3}(t)$ is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\ensuremath{\equiv}{L}_{3}+n)$ and $K$. ${L}_{3}$ oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants $E$ and $K$ become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that $〈{L}_{3}〉$ is constant, does not exist, because of spontaneous emission; it is also found that $〈{L}_{3}〉$ oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.

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The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy \ensuremathℏ\ensuremathωL3\ensuremath{\hbar}\ensuremath{\omega}{L}_{3}\ensuremathℏ\ensuremathωL3​, that describes the collective effect of a number of identical two-level (or spin-\textonehalf{}) systems under given idealized conditions, a large total angular momentum quantum number L0{L}_{0}L0​ corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay---with the NO initially excited and the radiation field in the ground state---and forced oscillation---with the NO subject to a prescribed resonant driving field---are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large L0{L}_{0}L0​, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as L3(0){L}_{3}(0)L3​(0) approaches L0{L}_{0}L0​ (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to L3(t){L}_{3}(t)L3​(t). It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of L3(t){L}_{3}(t)L3​(t) to a constant (negative) value---or steady state---at which the power absorbed equals the power radiated; in the strong-field case, L3(t){L}_{3}(t)L3​(t) oscillates periodically between the limits \ifmmode±\else\textpm\fiL0\ifmmode\pm\else\textpm\fi{}{L}_{0}\ifmmode±\else\textpm\fiL0​, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for L0=12{L}_{0}=\frac{1}{2}L0​=21​, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of 〈L3(t)〉〈{L}_{3}(t)〉〈L3​(t)〉 and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with 〈L3(0)〉=\ensuremath−L0,〈L3(t)〉〈{L}_{3}(0)〉=\ensuremath{-}{L}_{0}, 〈{L}_{3}(t)〉〈L3​(0)〉=\ensuremath−L0​,〈L3​(t)〉 approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For L0>12{L}_{0}>\frac{1}{2}L0​>21​, the free decay is treated by a method that involves the derivation of a set of expressions for the kthk\mathrm{th}kth derivative of 〈L3(t)〉〈{L}_{3}(t)〉〈L3​(t)〉 as an expectation value of a polynomial in L3{L}_{3}L3​ of order k+1k+1k+1. This set, together with the eigenvalue equation for L3{L}_{3}L3​, is shown to lead to a solution for all the moments of L3{L}_{3}L3​. The method is used to obtain complete solutions for several low values of L0{L}_{0}L0​, and also to obtain initial derivatives of 〈L3(t)〉〈{L}_{3}(t)〉〈L3​(t)〉 as polynomials in L0{L}_{0}L0​ for 〈L3(0)〉=L0〈{L}_{3}(0)〉={L}_{0}〈L3​(0)〉=L0​. For large L0{L}_{0}L0​, a comparison of classical and quantum-mechanical equations shows that only the condition 〈L3(0)〉\ensuremath≈L0〈{L}_{3}(0)〉\ensuremath{\approx}{L}_{0}〈L3​(0)〉\ensuremath≈L0​ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition 〈L3(0)〉=L0〈{L}_{3}(0)〉={L}_{0}〈L3​(0)〉=L0​ solved quantum mechanically up to such a time t1{t}_{1}t1​ by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for t\ensuremath≥t1t\ensuremath{\ge}{t}_{1}t\ensuremath≥t1​. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of 〈L3(t)〉〈{L}_{3}(t)〉〈L3​(t)〉---for arbitrary L0{L}_{0}L0​---is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of L0{L}_{0}L0​. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and 〈L23(t)〉〈{L}_{2}^{3}(t)〉〈L23​(t)〉 is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since 〈L3(t)〉〈{L}_{3}(t)〉〈L3​(t)〉 describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.
The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy L3{L}_{3}L3​ corresponds to the pendulum potential energy, the LO energy nnn corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\ensuremath{\gamma}K}{\ensuremath{\omega}}$ is proportional to the angular momentum KKK of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for L3(t){L}_{3}(t)L3​(t) is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\ensuremath{\equiv}{L}_{3}+n)$ and KKK. L3{L}_{3}L3​ oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants EEE and KKK become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that 〈L3〉〈{L}_{3}〉〈L3​〉 is constant, does not exist, because of spontaneous emission; it is also found that 〈L3〉〈{L}_{3}〉〈L3​〉 oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.
In recent years interest has significantly quickened in investigation of nonlinear optical and spectroscopic phenomena in such unique objects as Rydberg atoms (the atoms in the energy states near the ionization limit) and van der Waals complexes (dimers). These objects are characterized, in particular, by that their energy states are strongly degenerate due to a large value of angular (rotational) momentum. Here, the problem arises of adequate theoretical description of experimental results, especially when polarization peculiarities of nonlinear radiation processes are considered. The problem is caused by the existence of a huge number of quantum magnetic sublevels of energy states, which are, in the general case, coherently coupled to each other and take part in the radiation processes. Figure 1 gives a vivid presentation of the problem arising even in such a seemingly simple case when one need describe the radiation transitions caused by a strong monochromatic field between two energy levels with large values of the angular momentum J. If the field polarization is arbitrary, the whole set of the magnetic sublevels turn out to be coupled by radiative transitions. Formally, the problem becomes essentially multilevel. In this case not only the hope fades of solving it analytically using the traditional quantum-mechanical approach, but the problems of numerical solution grow abruptly with increasing quantum number J as well. On the other hand, as is often the case, every cloud has a silver lining. With large J we approach the situation when the rotational motion of particles can be described classically. It turns out that the transition from the quantum description of orientation states of the angular momentum to the classical one simplifies the problem radically: it reduces to the problem of radiation processes in the simplest model of nondegenerate states. In this case the rotational motion manifests itself in a dependence of the density matrix elements on orientation angles of the angular momentum as on a parameter.
Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an NNN-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.
We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its n=1n=1n=1 state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.
A large number of "molecules," which are taken to be angular momentum oscillators of frequency $\ensuremath{\nu}$, are considered to interact through their dipole moment with two electromagnetic cavity modes of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$, respectively, where $\ensuremath{\nu}={\ensuremath{\omega}}_{1}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\omega}}_{2}$. The behavior of the combined system is analyzed up to fourth order in perturbation theory. Phenomena known as two-photon emission or absorption, parametric amplification, various types of Raman emission, and frequency conversion are exhibited. The analytical method used is applicable to both a quantum-mechanical and a classical description of the modes and molecules; it displays explicitly the resulting differences. Expressions for the rate of energy change of the modes and molecules indicate resonance in second and fourth order. The second-order results are the same for the classical and quantum-mechanical treatments, while the fourth-order results show certain differences, which are examined in detail. The conditions under which each of the above phenomena occurs are discussed in a self-consistent manner, both classically and quantum-mechanically. It is shown that the second-order expression and the most significant part of the fourth-order expression can be derived very simply from a resonance interaction Hamiltonian that consists of a sum of products of three variables---two referring to the two modes, and the third to the molecules. The relationship between this resonance Hamiltonian and the fundamental dipole-moment Hamiltonian is examined, and the approximations by which the resonance Hamiltonian may be derived are considered. Arguments concerning the applicability of perturbation theory to steady-state situations are presented; it is pointed out that the second-order expressions for the energy transfer between molecules and field may be applicable to such a situation, in which case the classical and quantum-mechanical treatments are equivalent.
We consider the radiation from a system of NNN identical two-level atoms coupled to a continuum of quantized em modes, and possibly, to an external driving field near resonance. The atoms can be distributed over a region large in comparison to the resonant wavelength, but smaller than the spontaneous pulse length. Radiation rates and correlation functions are expressed in terms of expectation values of time-dependent atomic operators, which are shown to satisfy coupled first-order differential equations involving similar atomic operators and the initial radiation operators. The corresponding equations for the expectation values simplify considerably if no driving field is present. Similar results are derived for a model in which each atom is replaced by a harmonic oscillator.