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# Coherent and Incoherent States of the Radiation Field

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## Abstract

Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.
... Since then, this problem has been revisited recurrently in different contexts either in physics or in engineering. Examples are abundant in condensed matter, statistical mechanics and quantum field theory among others [3][4][5][6]. ...
... Each cell k of size accounts for a lattice site, where radiation is locally at thermal equilibrium with surrounding walls. The three processes (1,2,3) namely, in-cell creation and annihilation (red), bulk (green) and boundaries (purple) inter-cell exchanges are indicated. ...
... where L ⊥ accounts for process (1) and L ≡ L bulk + L boundary accounts for processes (2) and (3). Expressions of these generators are given in the Supplementary Material S IV in [20]. ...
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Non-equilibrium radiation is addressed theoretically by means of a stochastic lattice-gas model. We consider a resonating transmission line composed of a chain of radiation resonators, each at a local equilibrium, whose boundaries are in thermal contact with two blackbody reservoirs at different temperatures. In the long chain limit, the stationary state of the non-equilibrium radiation is obtained in a closed form. The corresponding spectral energy density departs from the Planck expression, yet it obeys a useful scaling form. A macroscopic fluctuating hydrodynamic limit is obtained leading to a Langevin equation whose transport parameters are calculated. In this macroscopic limit, we identify a local temperature which characterises the spectral energy density. The generality of our approach is discussed and applications for the interaction of non-equilibrium radiation with matter are suggested.
... To correctly identify QEPs, we also need to know the eigenvectors that correspond to the analyzed eigenfrequencies. The eigenvectors Y 1 ,Ȳ 1 , Y 2 , andȲ 2 given in Eq. (49), arising in the diagonalization of the dynamics matrix of the Heisenberg-Langevin equations (45), and belonging in turn to the eigenfrequencies Ω 1 , −Ω * 1 , Ω 2 , and −Ω * 2 , may be used to form the eigenvectors of the dynamics matrices of FOMs with increasing order. They directly represent the eigenvectors of the first-order FOMs dynamics matrix and, when formed into the supervector Y ≡ (Y 1 ,Ȳ 1 , Y 2 ,Ȳ 2 ), they allow to express the eigenvectors of the dynamics matrix of a pth-order FOMs via the tensor product Y ⊗ . . . ...
... We assume the 'average' reservoir twolevel atom in the ground state, i.e. σ −σ+ R = 1 and σ +σ− R = 0, and so we have γ d ′ ≡ γ d σ −σ+ R = γ d using the damping constant γ d . Using the Glauber-Sudarshan representation of statistical operator [49,50] ...
Preprint
Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine and induced) and hidden exceptional points not recognized directly in amplitude spectra are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in open quantum infinitely dimensional systems.
... The incompatibility of physical observables also limits our ability to associate to them joint probability distributions. A well-known example is represented by the formulation of the quantum mechanics of continuous variable systems in phase-space [4][5][6][7][8]. Classically, the state of a physical system can be represented by a joint probability distribution over its phase-space. ...
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... The associated Glauber first order correlation function [21,22] then reads ...
Preprint
The de Broglie-Bohm interpretation is a no-collapse interpretation, which implies that we are in principle surrounded by empty waves generated by all particles of the universe, empty waves that will never collapse. It is common to establish an analogy between these pilot-waves and 3D radio-waves, which are nearly devoided of energy but carry nevertheless information to which we may have access after an amplification process. Here we show that this analogy is limited: if we consider empty waves in configuration space, an effective collapse occurs when a detector clicks and the 3ND empty wave associated to a particle may not influence another particle (even if these two particles are identical, e.g. bosons as in the example considered here).
... where 2 (x) = (x) (x * ). [45] Now we can express Equation (A1) as p(n, m) = 1 n!m! ∫ d 2 1 d 2 2 P 1 , 2 ( 1 , 2 )( | 1 | 2 + ) n ( | 2 | 2 + ) m e −( | 1 | 2 + ) e −( | 2 | 2 + ) (A3) and integration yields immediately p(n, m) = 1 n!m! ( | 1 | 2 + ) n ( | 2 | 2 + ) m e −(n+2 ) (A4) ...
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Exceptional points are complex‐valued spectral singularities that lead to a host of intriguing features such as loss‐induced transparency—a counterintuitive process in which an increase in the system's overall loss can lead to enhanced transmission. In general, the associated enhancements scale with the order of the exceptional points. Consequently, it is of great interest to devise new strategies to implement realistic devices capable of exhibiting high‐order exceptional points. Here, it is shown that high‐order N$N$‐photon exceptional points can be generated by exciting non‐Hermitian waveguide arrangements with coherent light states. Using photon‐number resolving detectors it then becomes possible to observe N$N$‐photon enhanced loss‐induced transparency in the quantum realm. Further, it is analytically shown that the number‐resolved dynamics occurring in the same nonconservative waveguide arrays will exhibit eigenspectral ramifications having several exceptional points associated to different sets of eigenmodes and dissipation rates. Exceptional points are complex‐valued spectral singularities that lead to a host of intriguing features such as loss‐induced transparency—a counterintuitive process in which an increase in the system's overall loss can lead to enhanced transmission. It is shown that high‐order N‐photon exceptional points can be generated by exciting non‐Hermitian waveguide arrangements with coherent light states and employing photon‐number‐resolving detectors.
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The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n\mathrm{th}$ order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
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DOI:https://doi.org/10.1103/PhysRevLett.10.276
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