Abstract The modern theory of coherence is based on correlation functions. A generic example could be written 〈 V ∗ ( t 1 ) V ( t 2 ) 〉 , denoting an average of products of the values of a signal V ( t ) at two specified times. Here we infer that t is a degree of freedom that the signal depends on. Typically, physical variables depend on more than one degree of freedom, and recognition of this has prompted attention to some interesting questions for the correlation functions and the several coherences that can be attributed to the same optical field. We examine some of the questions arising from the standpoint of experimental contexts. Degree of polarizability and degree of entanglement (classical non-separability) can serve as starting points for quantitative assignments.
Abstract The modern theory of coherence is based on correlation functions. A generic example could be written 〈 V ∗ ( t 1 ) V ( t 2 ) 〉 , denoting an average of products of the values of a signal V ( t ) at two specified times. Here we infer that t is a degree of freedom that the signal depends on. Typically, physical variables depend on more than one degree of freedom, and recognition of this has prompted attention to some interesting questions for the correlation functions and the several coherences that can be attributed to the same optical field. We examine some of the questions arising from the standpoint of experimental contexts. Degree of polarizability and degree of entanglement (classical non-separability) can serve as starting points for quantitative assignments.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
More than a century ago, starting with Michelson, the field of classical coherence has developed rapidly. By studying and uncovering the coherence properties of light, many useful applications were discovered. In modern times, these applications have seen large use in fields like astronomy, where the properties of light can be used to discover stars and determine their radius, for example. Another class of correlations, namely quantum correlations, which were discovered in the beginning of the twentieth century, have gained much attention from the scientific community in the last two decades. In particular, the field of quantum information developed, promising great computational power by using quantum correlations to build computers. Currently, quantum computation is a very active field bringing together physicists, mathematicians, engineers, chemists, and computer scientists to find solutions to the problems encountered in building quantum computers.\n\nI consider some classical coherence effects of the degree of cross polarization (DCP) on the Hanbury-Brown Twiss effect, with a specific focus on Gaussian Schell-model beams. I show that the DCP is necessary, in general, to determine the correlations in intensity fluctuations of a beam at two different points. As for quantum correlations, I consider entanglement in realistic systems: one in two-qubit systems, and the other in continuous variable quantum systems. In the former case, when the temperature of the system is finite, entanglement always decays in a finite time. However, in the latter case, entanglement is long-lived, although in the long run it is not of much practical use. Finally, I unravel the relationship between quantum discord and quantum entanglement, as well as quantum discord and entropy for the most general two-qubit systems, and I identify the states that define the boundaries of these relationships.
Introduction In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points ( r 1 , t 1 ) and ( r 2 , t 2 ). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources. In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory. Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.
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 nth 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.
Abstract Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.
Abstract Any light that isn’t perfectly monochromatic (and that’s every kind of light) necessarily has some degree of randomness. In Chapter 9 we described this randomness, how it originates and how (within limits) it may be controlled. However, there were obvious inadequacies: having no quantity to represent a ‘degree of coherence’, we could not give a precise condition for coherence to disappear or for fringes to be so indistinct that they should be considered absent. Moreover, we found ourselves using awkward wordings: ‘obtaining knowledge’; ‘permits us to predict’; ‘so clumsy that interference fails to manifest itself’; ‘a wavetrain is the whole wave defined for all t’. The author of this book used words with more than usual care in Chapter 9, in an attempt at giving a correct impression and excluding common misconceptions; but the need for that care was itself an indication that something better was called for.