We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectories---the solutions of Hamiltonian equations of motion for a given nonlinear system---we construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.
The semi-classical theory of radiation-matter coupling misses local-field effects that may alter the pulse time-ordering and cascading that leads to the generation of new signals. These are then introduced macroscopically by solving Maxwell's equations. This procedure is convenient and intuitive but ad hoc. We show that both effects emerge naturally by including coupling to quantum modes of the radiation field that are initially in the vacuum state to second order. This approach is systematic and suggests a more general class of corrections that only arise in a QED framework. In the semi-classical theory, which only includes classical field modes, the susceptibility of a collection of N non-interacting molecules is additive and scales as N. Second-order coupling to a vacuum mode generates an effective retarded interaction that leads to cascading and local field effects both of which scale as N(2).
Classical optical frameworks such as the discrete dipole approximation (DDA) assume that the linear spectrum of coupled quantum emitters can be computed solely from the linear susceptibilities of individual constituents. However, recent polariton studies show that cavity linear response can encode nonlinear optical susceptibilities. Here, we demonstrate that this phenomenon is more general: emitter-emitter interactions allow nonlinearities of individual emitters to emerge in the linear response of arrays, without cavities or permutational symmetry. To illustrate this phenomenon, we show linear spectra for coupled heterodimers and linear chains, and demonstrate that Raman features of individual monomers show up as vibrational sidebands of collective resonances. Moreover, tuning Raman-type anharmonicities enables systematic control of spectral features, establishing a genuine quantum optical effect in molecular aggregates and quantum emitter arrays, which goes beyond mean-field descriptions in light-matter interactions.
We develop a theory of photon-echo phenomena in harmonic vibrational modes. Although classical harmonic oscillators cannot produce any nonlinear optical signal in the linear response limit, we demonstrate that quantum harmonic oscillators that are coupled to any physically reasonable bath can give rise to novel nonlinear optical behavior, even in the perturbative limit. We show that photon echoes in high-frequency vibrational modes are strongly affected by both population relaxation and pure dephasing. The time dependence of the echo signal is shown to be highly sensitive to the amount of inhomogeneous broadening in the vibrational line. As an example, we develop the simple model of population relaxation resulting from linear coupling to the bath and pure dephasing resulting from quadratic coupling to the bath. Counter to the classical picture, echo signal is present when the only coupling to the bath is linear, but absent when the only coupling is quadratic.
The problem of quantum state exchange between three coupled quantum harmonic oscillators (which can represent interacting modes of the electromagnetic field in nonlinear crystal or interaction between laser beams and vibrational modes of ions in traps) is considered. We review different definitions of state exchange and find conditions of its possibility for arbitrary initial states in a chain of three oscillators with bilinear resonance coupling (corresponding to the rotating wave approximation), analyzing the structure of integrals of motion and propagators of the Schr\"odinger equation, as well as the Wigner functions. The influence of different parameters (such as detunings between coupling constants and temperatures of the intermediate oscillator and ``receiver'') on the fidelity of exchange is analyzed.
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
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, 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) approaches
L0 (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to
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) to a constant (negative) value---or steady state---at which the power absorbed equals the power radiated; in the strong-field case,
L3(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
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)〉 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
L0>21, the free decay is treated by a method that involves the derivation of a set of expressions for the
kth derivative of
〈L3(t)〉 as an expectation value of a polynomial in
L3 of order
k+1. This set, together with the eigenvalue equation for
L3, is shown to lead to a solution for all the moments of
L3. The method is used to obtain complete solutions for several low values of
L0, and also to obtain initial derivatives of
〈L3(t)〉 as polynomials in
L0 for
〈L3(0)〉=L0. For large
L0, 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
〈L3(0)〉=L0 solved quantum mechanically up to such a time
t1 by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ensuremath{\ge}{t}_{1}$. 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
〈L3(t)〉---for arbitrary
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. 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)〉 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)〉 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.