We compare different measures for photon localization in terms of two-dimensional Gaussian wave packets of width Delta k. We find that all measures start to coalesce if the wave packet has evolved for times which are larger than a few (c Delta k)^{-1}. However, the Landau-Peierls wave function yields the largest initial offset <r>-ct > 0 while the Fourier transform of the helicity components of the k-space wave packet yields the smallest offset <r>-ct > 0 and converges towards <r>-ct = 0 fastest. We also discuss local detection of photons through a model detector consisting of ions in ion traps. The position-dependent detection probabilities are inferred from the scattering matrix. We find that the local detection probability for photons can be expressed in terms of three of the proposed localization measures, viz. the Landau-Peierls wave function, the energy wave function, and the Hawton density. Those three proposals also remain close throughout the time evolution of the single-photon wave packet. However, the energy wave function is locally related to electromagnetic fields and is therefore confirmed as the preferred tool for the description of local photon densities. The novelty of our observations concerns three items, viz. the possibility to express local photon absorption matrix elements through three of the proposed localization measures, the coalescence of all the proposed localization measures after sufficiently long evolution of a photon wave packet, and the striking differences of some of the proposed localization measures during early wave packet evolution.
We compare different measures for photon localization in terms of two-dimensional Gaussian wave packets of width Delta k. We find that all measures start to coalesce if the wave packet has evolved for times which are larger than a few (c Delta k)^{-1}. However, the Landau-Peierls wave function yields the largest initial offset <r>-ct > 0 while the Fourier transform of the helicity components of the k-space wave packet yields the smallest offset <r>-ct > 0 and converges towards <r>-ct = 0 fastest. We also discuss local detection of photons through a model detector consisting of ions in ion traps. The position-dependent detection probabilities are inferred from the scattering matrix. We find that the local detection probability for photons can be expressed in terms of three of the proposed localization measures, viz. the Landau-Peierls wave function, the energy wave function, and the Hawton density. Those three proposals also remain close throughout the time evolution of the single-photon wave packet. However, the energy wave function is locally related to electromagnetic fields and is therefore confirmed as the preferred tool for the description of local photon densities. The novelty of our observations concerns three items, viz. the possibility to express local photon absorption matrix elements through three of the proposed localization measures, the coalescence of all the proposed localization measures after sufficiently long evolution of a photon wave packet, and the striking differences of some of the proposed localization measures during early wave packet evolution.
Whilst quantum electrodynamics (QED) underwent an impressive development and reached its maturity in the middle of the last century, one of its basic concepts—the photon wave function in free space—was deprived of such fortune. Although the photon wave function in coordinate representation was introduced already in 1930 by Landau and Peierls, the concept was found to suffer from inherent difficulties that were not overcome during the century —see review (Bialynicki-Birula, 1996). The common explanation presented in textbooks, e.g., (Akhiezer & Berestetskii, 1965; Mandel & Wolf, 1995), may be summed up as follows: (i) no position operator exists for the photon, (ii) while the position wave function may be localized near a space-time point, the measurable quantities like the electromagnetic field vectors, energy, and the photodetection probability remain spread out due to their non-local relation with the position wave function.
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.
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) eld. However, Heisenberg’s uncertainty principle tells us that a state of denite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indenite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with nite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
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 expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals K(κ) and E(κ) with modulus κ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value q of the position eigenvalue, but also on a plane containing q and that they decay as inverse powers of the distance from q.
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.