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CRC Press eBooks

Wave-Packet Photons and Bi-Photons

Ole Keller

Published Aug 6, 2024• 0 citations
Publisher

Abstract

A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).

DOI
10.1201/9781003029458-8

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arXiv:1608.06479•Journal of Mathematical Physics•Aug 1, 2017•25 citations
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
The physics of the photon vacuum, one-photon wave packets, and bi-photons are the subjects of the books SECTION II. Here we present an overview of these subjects paying particular attention to fundamental aspects. Cavity elec- trodynamics is often used to study free-photon states. However, such states constitutes an idealization since photon-matter interaction always is present at least in the surface region of the cavity wall. This means that even a photon- empty state does not exist in a rigorous physical sense. The transverse gauge photon states coupled to the wall of a closed cavity is discussed in a novel man- ner on the basis of a microscopic version of the Ewald-Oseen extinction theorem. Single-photon physics is introduced with background in the mean position state introduced in Chapter 3 . It is shown that a wave-packet one-photon wave func- tion in direct space satisfies a dynamical equation of a a Hamilton-like form. Quantum states containing two photons may be entangled, and such states are here called bi-photons. Using wave-packet bases for the two photons, a brief discussion of entangled transverse gauge photon states is given both in Hilbert space and direct space.
In Sec. 15.4 we introduced polychromatic single-photon states in Hilbert space [Eq. (15.80)], |Φ〉 = L− 32 ∑ i φi|Pi〉, i = (q, s), (16.1) by linear superposition of monochromatic plane (P) one-photon states |Pi〉 ≡ |1i〉 = aˆ†i |0〉. (16.2) In the one-photon subspace the |Pi〉 states satisfy the orthonormality, 〈Pj |Pi〉 = δij , (16.3) and completeness ∑ i |Pi〉〈Pi| = 1 (16.4) relations. Instead of taking the |Pi〉’s as a basis set for the expansion of |Φ〉, one may use different sets of one-photon wave-packet states, as we now shall see. The generalization to wave-packet expansion is of importance from a fundamental point of view, and in this chapter we shall realize the usefulness of such an expansion for single-photon correlation studies.
The monochromatic Dirac and polychromatic Titulaer–Glauber quantized field theories (QFTs) of electromagnetism are derived from a photon-energy wave function in much the same way that one derives QFT for electrons, i.e., by quantization of a single-particle wave function. The photon wave function and its equation of motion are established from the Einstein energy–momentum–mass relation, assuming a local energy density. This yields a theory of photon wave mechanics (PWM). The proper Lorentz-invariant single-photon scalar product is found to be non-local in coordinate space, and is shown to correspond to orthogonalization of the Titulaer–Glauber wave-packet modes. The wave functions of PWM and mode functions of QFT are shown to be equivalent, evolving via identical equations of motion, and completely describe photonic states. We generalize PWM to two or more photons, and show how to switch between the PWM and QFT viewpoints. The second-order coherence tensors of classical coherence theory and the two-photon wave functions are shown to propagate equivalently. We give examples of beam-like states, which can be used as photon wave functions in PWM, or modes in QFT. We propose a practical mode converter based on spectral filtering to convert between wave packets and their corresponding biorthogonal dual wave packets.
A review is given of the space-time wave mechanics of single photons, a subject with an almost century long history. The Landau-Peierls photon wave function, which is related nonlocally to the electromagnetic field is first described, and thereafter the so-called energy wave function, based on the positive-frequency Riemann-Silberstein vectors, is discussed. Recent attempts to understand the birth process of a photon emerging from a single atom are summarized. The polychromatic photon concept is introduced, and it is indicated how the wave mechanics of polychromatic photons can be upgraded to wave train quantum electrodynamics. A brief description of particle (photon) position operators is given, and it is shown that photons usually are only algebraically confined in an emission process. Finally, it is demonstrated that the profile of the birth domain of a radio-frequency photon emitted in a hyperfine spin transition shows an exponential falloff with the distance from the atomic nucleus. The length parameter in this falloff is the Bohr radius.
Abstract The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.