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arXiv:1812.03854

A sincere tribute to E.C.G Sudarshan's phenomenal contribution toward quantum theory of optical coherence

Arindam Kumar Chatterjee, Anik Rudra, Soham Chakraborty

Published Nov 21, 2018• 0 citations
arXivHTMLPDF

Abstract

The diagonal representation and optical equivalence theorem are the E. C. G. Sudarshan's mid 20th century adventures in non-classical optics. It basically deals with a quantum mechanical description of photons to explain the quantum properties of light. Inspired by Sudarshan's pioneering work we try to explain the every minute mathematical details of his paper "Equivalence of semi-classical and quantum mechanical descriptions of statistical light beams". In this article we are going to go through some of the basics in developing quantum optics, then land up in E.C.G's original work and try to present it as rigorous as possible. We show some of its important applications in various classes of physics problems.

DOI
10.48550/arxiv.1812.03854

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The diagonal representation and optical equivalence theorem are the E. C. G. Sudarshan's mid 20th century adventures in non-classical optics. It basically deals with a quantum mechanical description of photons to explain the quantum properties of light. Inspired by Sudarshan's pioneering work we try to explain the every minute mathematical details of his paper "Equivalence of semi-classical and quantum mechanical descriptions of statistical light beams". In this article we are going to go through some of the basics in developing quantum optics, then land up in E.C.G's original work and try to present it as rigorous as possible. We show some of its important applications in various classes of physics problems.
The essential of this monograph is to reproduce the results of Sudarshans paper Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams published in 1963. To go in that direction we also describe the linear harmonic oscillator and its algebraic equivalence with a single-mode quantized beam of light. Coherent states and their overcompleteness properties along with the diagonal representation of the density matrix are essential as a prerequisite to reproduce the result. Most of the textbooks meant for graduate students describe linear harmonic oscillators in great detail but they cryptically mention coherent states and their associated properties. So it is difficult for the students to understand the essence of Sudarshans paper which contains only seven equations. This monograph is written in such a lucid manner that without the help of experts one can understand it.
The comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive PPP representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without PPP representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.
Modern attempts to understand light go back to Newton who considered light to be particles, the so called corpuscular theory, and the other school of Huygens, Young and others. Huygens and Young viewpoint emphasised the wave property. This difference of opinions persisted for close to two centuries till Maxwell theory solidly established light as a wave phenomenon associated with Electromagnetism. A serious schism so to speak was introduced into the theory of light with the understanding of "Light gas", the so called Black Body radiation. Einstein's photon hypothesis and Bose's successful derivation of the Planck spectrum put the corpuscular theory back in place, while classical description continues to be used as well. One may then wonder if this being a classical limit may have to be corrected when dealing with quantum phenomena. To most people's surprise, the classical description was in fact a subset of the full quantum description, and the classical states of light could be shown to be subsumed within the fully quantum description without having to take the limit of Planck's constant h going to zero. This was the resolution provided by E. C. G. Sudarshan's Diagonal representation as the most general formalism for dealing with light.
Coherence is theorized as the normalized correlation between two superposed optical signals, usually replicated by a two-beam interferometer or by a double-slit system. Prevailing coherence theory is presented as field–field correlation, as it does not go into the technical details of how the measurable transformations are physically generated and registered. Teaching coherence theory using a semiclassical model has been very successful as is evidenced in literature from the extensive reference to the text by M. Born and E. Wolf. This has been further strengthened by the work of Sudarshan, who demonstrated the optical equivalence between Wolf’s classical theory and Glauber’s quantum coherence theories. The discussion about the classical concept of coherence remains incomplete without presenting the van Cittert–Zernike theorem on spatial coherence, because the concept of coherence evolved with historic attempts to understand spatial coherence.
Statistical and quantum optics provides a clear, up-to-date and comprehensive introduction to the recent field of modern optics dealing with coherence and fluctuations of light. The first two chapters discuss the statistical frame work needed for later part of the book followed by systematic account of coherence theory with a detailed discussed on photon statistics. Description of quantum theory of coherence is preceded by a introductory chapter on field quantization. A detailed description of non-classical light highlighted by phenomena of anti-bunching, Sub-Poissonian statistics, production and detection of squeezed light and coherent state follows. Next is the discussion on E-P-R paradox on reality, locality and completeness of quantum mechanics bringing out clearly the importance of Bell's inequalities. The concluding two chapters put forward forcefully the new fundamental concepts based on quantum superposition and entanglement leading to two photon interference, quantum cryptography and teleportation.