Integrable Models of Quantum Optics

A list of exactly solvable many-body models of quantum optics is presented. The problem of the cascade radiation spectrum of an arbitrary multilevel system, the solution to which contains tasks considered earlier as special cases, is highlighted. Examples of spectra for a three-level system, a harmonic oscillator, the Dicke model, and an equidistant system with a constant rate of spontaneous transition are given.


INTRODUCTION
The simplest model of quantum optics is a twolevel atom that interacts resonantly with a single-mode quantum electromagnetic field (the Jaynes-Cummings model [1]). It allows us to determine the states of an atomic-photon system and describe its dynamics accurately. However, the task becomes substantially non-trivial for multimode field. Examples of a multimode field include a field propagating in a long waveguide and a field in an open three-dimensional space. A common way of describing a system of atoms interacting with a multimode quantum electromagnetic field is based on excluding the field's degrees of freedom and describing the atomic subsystem in terms of a density matrix. Information on the states of the field itself is in this case partially lost. The development of quantum optics allows us to describe both an atomic and a photon subsystem. Solving the Schrödinger equation directly for an atomic-photon system in terms of multiphoton states is generally not possible. In some cases, however, including some well-known models of quantum optics, such a description is feasible.

BRIEF OVERVIEW
A key example of this is the Weisskopf-Wigner model [2], which describes the spontaneous radiation of a two-level atom. It was this that produced the generally accepted idealization, in which the dynamics of the decay of the excited state is described by exponential law and the radiation spectrum when the observation time tends to infinity turns out to be a Lorentzian with width Γ. This idealization includes (a) the rotating-wave approximation, (b) no dependence of on photon frequency in a scale much wider than the line width, and (c) formally unlimited frequency interval The following examples refer to (a) the decay of a harmonic oscillator from the second [3,4] and the third [5] excited levels; (b) a system of two atoms with a single excitation, in both 1D (spatially separated atoms in a single-mode waveguide) [6,7] and 3D (closely spaced atoms with possible allowance for dipole-dipole interaction [8]); and finally (c) the Dicke model [9].
Recently, however, a solution was obtained for a theoretical model that summarizes the results from earlier studies. This is a model of cascade spontaneous transitions for an arbitrary multilevel system with a single restriction: one to one (Fig. 1). Below, we present the final formula, describe a numerical calculation algorithm, and several specific examples.

SPECTRUM OF SPONTANEOUS RADIATION FOR QUANTUM SYSTEMS WITH CASCADE TRANSITIONS
The general expression for the transition spectrum in the notations given in Fig. 1 has the form (1) It is assumed that the quantum system is in state and the radiation field is in the vacuum state at initial time t = 0. The problem is formulated in terms of amplitudes of states that are direct products of the (N − k)th state N k of the quantum system and the k-partial state of the radiation field with ordered set of frequencies Here, the word "spectrum" means (2) with amplitude A 0 , calculated at Note that the normalization of the spectrum corresponds to the radiation of N photons; i.e.,

RECURRENT SCHEME FOR CALCULATING
For high values of N, direct calculations using formula (1) can be time-consuming. A fast recurrence scheme should in this case be used [10]. This includes the set of auxiliary functions where and These functions are calculated in sequence: (3) They are included in the recurrent scheme for calculating the spectrum: (4) with Lorentzian upon transition 1 → 0, in accordance with the Weisskopf-Wigner idealization.
The scheme described by formulas (3) and (4) was used to calculate some of the spectra shown below.

ARBITRARY THREE-LEVEL SYSTEM
It is convenient to present the results in the form of the real and imaginary parts of the complex Lorentzian In these denotations, the expression for the radiation spectrum has the form (6) where is the difference between transition frequencies 1 → 0 and 2 → 1. It can be seen that, at , the spectrum is simply the sum of two Lorentzians at transition frequencies 1 → 0 and 2 → 1 with widths and respectively-a well known result (see, e.g., [11]). Another special case is the oscillator, where Formula (6) in this case completely reproduces the results in [4] and the classical result of Weisskopf and Wigner when [3], where the spectrum degenerates into one Lorentzian with width

HARMONIC OSCILLATOR The last result is generalized to when the initial state of the harmonic oscillator is arbitrary level
The spectrum is reduced to as soon as = const and for all M, regardless of the initial value of N. Proof of this, which is known as the harmonic oscillator paradox, was given in [10] in two ways: directly, using formula (1), and applying recurrent scheme (2)-(3). In [12], it was noted that "… the Weisskopf and Wigner approach requires consideration of the full wave function of the system (the oscillator plus the radiation field) and is therefore extremely cumbersome, especially when n 1." It was in [12] where this property was derived via the density matrix formalism using the correlation function of the creation and annihilation operators. It was confirmed by the calculations in [5] for when N = 3 that occupied six pages of formulas. However, development becomes quite simple using general formula (1).

EQUIDISTANT SYSTEM WITH = const
Using formula (6) for when and a nontrivial feature can be noted: some narrowing of the spectrum in comparison to It turns out that this tendency continues with an increase in the number of transitions, as can be seen from the spectra presented in Fig. 2. In addition, the spectrum normalized to unity tends to the delta function when N → ∞ [10]: There is a strong analogy between this example and the elastic scattering of the weak monochromatic radiation of an atom, when field frequency is far from the frequencies of atomic transitions. Scattering can then be considered as an infinite sequence of transitions between quasienergy (Floquet) states separated by an value. As is well known, the resonance fluorescence spectrum in this limiting case really does degenerate into the delta function.

THE DICKE MODEL
The Dicke model is a system of closely spaced N twolevel atoms. The radiation of a fully excited system corresponds to a cascade of transitions between equidistant (energy difference ) collective levels (Dicke states [13]). The radiation spectrum calculated using the Bethe ansatz approach [9] has the form of general formula (1) with rates In limit N 1, the spectrum has the form [10] (7) Figure 3 illustrates the transition to the limiting case with the growth of N.

CONCLUSIONS
It was shown that the exact solution to the problem of the cascade radiation spectrum of a multi-level system with a relay unbranched nature of de-excitation contains as special cases a number of solutions to the problems considered earlier.