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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Minimum-Uncertainty States |γ〉 and the States |n〉γ of the Electromagnetic Field

Paul Bertrand, K. Moy, Eli A. Mishkin

Published Sep 15, 1971• 9 citations
Publisher

Abstract

Two vector bases suitable for the representations of states of the electromagnetic field are introduced. The minimum-uncertainty states $|\ensuremath{\gamma}〉$ form an overcomplete nonorthogonal set. The ${|n〉}_{\ensuremath{\gamma}}$ states form a complete orthogonal set. The various relations they satisfy are derived via an oscillator frequency transformation. The corresponding similarity transformation of the non-normal operator changes the physical character of its characteristic states. The photon statistics of the states $|\ensuremath{\gamma}〉$ and ${|n〉}_{\ensuremath{\gamma}}$ are shown to display anticorrelation effects.

DOI
10.1103/physrevd.4.1909

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Two vector bases suitable for the representations of states of the electromagnetic field are introduced. The minimum-uncertainty states ∣\ensuremathγ〉|\ensuremath{\gamma}〉∣\ensuremathγ〉 form an overcomplete nonorthogonal set. The ∣n〉\ensuremathγ{|n〉}_{\ensuremath{\gamma}}∣n〉\ensuremathγ​ states form a complete orthogonal set. The various relations they satisfy are derived via an oscillator frequency transformation. The corresponding similarity transformation of the non-normal operator changes the physical character of its characteristic states. The photon statistics of the states ∣\ensuremathγ〉|\ensuremath{\gamma}〉∣\ensuremathγ〉 and ∣n〉\ensuremathγ{|n〉}_{\ensuremath{\gamma}}∣n〉\ensuremathγ​ are shown to display anticorrelation effects.
It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides ∣n〉|n〉∣n〉, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators bbb and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables xxx, ppp, and HHH, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states ∣n〉|n〉∣n〉. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator aaa is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.
The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.
Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.
For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.
The conventional Fourier transform has a well-known uncertainty relation that is defined in terms of the first and second moments of both a function and its Fourier transform. It is also well known that Gaussian functions, when translated to an arbitrary centre and supplemented by a linear phase factor, provide a complete set of minimum uncertainty states (MUSs) that exactly achieves the lower bound set by this uncertainty relation. A similarly general set of MUSs and uncertainty relations are derived here for discrete and/or periodic generalizations of the Fourier transform, namely for the discrete Fourier transform and the Fourier series. These extensions require a modified definition for the width of a periodic distribution, and they lead to more complex uncertainty relations that turn out to depend on the centroid location and mean frequency of the distribution. The derivations lead to novel generalizations of Hermite–Gaussian functions and, like Gaussians, the MUSs can play a special role in a range of Fourier applications.