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ON THE QUANTUM MECHANICAL ASPECTS OF THE WAVESHAPE OF ELECTROMAGNETIC RADIATION.

Gerard Lachs

Published Sep 8, 1964• 0 citations
OpenAlex

Abstract

Previous work of this contract on the waveshape aspects of light from a classical point of view led to an interest to investigate the quantum mechanical aspects of the waveshape of light. This work contains some results of this investigation. One of the results obtained is an upper bound on the channel capacity where the only noise in the channel is due to the zero point field. The P-representation of a general state in terms of the coherent states is utilized to obtain the probability densities of the in-phase component, quadrature component, envelope, and the number of photons of the field. This is done for thermal radiation and for radiation which is a mixture of thermal radiation and coherent radiation with the result that the thermal radiation behaves as pure Gaussian noise while the coherent radiation behaves as a classical signal. The quantum mechanical formalism is utilized to obtain the rules of superposition for a large class of fields. The results show that the signal part can be treated classically and the noise part is nearly the same as would be found from the classical superposition of noise.

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Applied Sciences•Oct 15, 2024•1 citation
Previous work of this contract on the waveshape aspects of light from a classical point of view led to an interest to investigate the quantum mechanical aspects of the waveshape of light. This work contains some results of this investigation. One of the results obtained is an upper bound on the channel capacity where the only noise in the channel is due to the zero point field. The P-representation of a general state in terms of the coherent states is utilized to obtain the probability densities of the in-phase component, quadrature component, envelope, and the number of photons of the field. This is done for thermal radiation and for radiation which is a mixture of thermal radiation and coherent radiation with the result that the thermal radiation behaves as pure Gaussian noise while the coherent radiation behaves as a classical signal. The quantum mechanical formalism is utilized to obtain the rules of superposition for a large class of fields. The results show that the signal part can be treated classically and the noise part is nearly the same as would be found from the classical superposition of noise.
Some of the statistical properties of a mode of propagation for an electromagnetic field which is a super-position of thermal and coherent radiation are derived. It is found that the electric field and the magnetic field have Gaussian probability densities. The variance of the mixed field is the same as the variance of the thermal part of the field alone, while the average of the mixed field is the same as the average of the coherent field alone. It is pointed out that this result differs from the classical theory in that the zero-point field appears only once in the variance for the mixed field while it appears once in the variance of each of the constituent fields. This result is extended to a more general class of fields, and for that class it is shown that except for the zero-point field, the quantum properties of superposition are the same as the classical properties of superposition of noisy fields. The probability distribution for the number of photons in a mode of mixed radiation is also derived. The results shows that there are fluctuations in the number of photons that arise, because of interference effects.
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.
In Chapter 4 we defined different representations of the elec- tromagnetic (EM) field: the Fock, thermal and coherent state representations, and discussed in details properties of the EM field in these representations. We have seen that in the coherent state the fluctuations in the two quadratures of the EM field amplitudes are equal and minimize the uncertainty product given by the Heisenberg’s uncertainty relation. In other words, the quantum fluctuations of the field in a coherent state are equal to the zero- point fluctuations and are randomly distributed in phase. These zero-point fluctuations represent the standard quantum limit to the reduction of noise in a signal. Even an ideal laser operating in a pure coherent state would still possess quantum noise due to zero-point fluctuations.
Squeezing of radiation is a purely quantum mechanical phenomenon which has no classical counterpart. This quantum effect is expected to manifest itself in optical processes in which the nonlinear response of the system to the radiation field plays an important role. The concept of squeezing started in mid 1980s. Since then there has been an enormous upsurge of interest in this field owing to its low noise property. In the present paper various aspects of squeezing of radiation are studied and possible means of generation of squeezed light in which the quantum fluctuations are reduced below shot noise limit are investigated. The amplitude of the electric field of a mode of an electromagnetic field is always associated with quantum mechanical fluctuations. Squeezing states are characterised by reduced quantum fluctuations in one quadrature component of the field at the expense of increased fluctuations in the other noncommuting component. A fully quantum mechanical approach is followed. In the present paper we intend to review different aspects of squeezing and higher order squeezing in higher order Raman processes as well as in multiwave mixing processes. A comparative study between different processes are made which will pave a way for easier generation of squeezed light.
The problem of including quantum effects within existing methods of communication system analysis is investigated.A mathematical model is postulated for describing narrow bandwidth systems.Utilizing the Glauber P representation, the model permits analysis of both real field detection (measurement of electric and magnetic field magnitudes) and photon detection.The resulting equation which describes photon detection is shown to be consistent with previous work which has been proven experimentally.Unique aspects of the postulated model are its ability to describe both random and deterministic modulation functions in conventional terms, and its ability to demonstrate the inapplicability of the "photon channel" model for describing present-day systems.An analysis of quantum effects at the transmitter shows that modulation of the radiated field must take place by one of two distinctly different processes.Wave modulation is defined as a process which conveys information in the amplitude and phase of the electromagnetic field.Photon modulation is defined as a process which conveys information in the exact number of photons per pulse of the transmitted field (i.e., the "photon channel" of Stern, Gordon, ani othfrs).A wave modulated signal is shown to describe present-day modulation schemes and, moreover, to yield random photon fluctuations.It is concluded a wave modulated signal is not capable of describing a photon modulated signal.In contrast, a photon modulated signal is shown to possess a completely random phase fluctuation thus demonstrating the inability of the "photon channel" to describe presently used wave modulation schemes.For wave modulation the Glauber P function is shown to be identical (with a change of variable) to the classical joint probability density of the quadrature components of a narrowband sinusoid used in conventional analysis.Equations are derived which "transform" classical probability distributions into distributions which include quantum effects.The "partitioning noise" studied by Hagfors and Bowen is found to constitute a source of noise only in the "photon channel."The "partitioning effect" is shown to merely preserve the Poisson character of photon counts in wave modulated fields and therefore does not introduce a noise in present-day systems.For a wave modulated system with real field detection it is shown that photon noise can be included in conventional analysis by adding to the normal input noise, a noise density of hf c /2 Watts/Hertz where h is Planck's constant and f is the system carrier frequency.Use of photon heterodyne detection requires the addition of an input noise density of hf c Watts/Hertz.The photon noise is exactly additive Gaussian for real field detection and approximately additive Gaussian for photon heterodyne detection.Detection by counting the received photons is shown to yield neither Gaussian nor Poissonian statistics.However, with an "ideal measurement process," when the effective received noise density n c is much less than hf c , the variance of the counts are found to approach that of a Poisson distribution.Conversely, when nc is much larger than hf c , the .variance approaches that derivable from a classical analysis.A channel capacity equation for a wave modulated system is found which is equal to those derived by Lachs and Jelsma.The equation is shown to give capacities less than that derived by Gordon.
We carry out a theoretical investigation about quantum noise in coherent mixtures of states with different statistics. On the one hand, we investigate the dependence between the behavior wave-particle and associated noise. On the other hand, we are interested in studying the persistence of chaotic behavior in mixtures of states with quantum and classical behavior. For this, we perform mixtures of single photons with coherent states of different averages. We find that the noise associated with quantum chaos depends on the proportion of the quantum–classical mixture and that quantum wave states are more efficient than classical wave states in terms of their randomness properties.