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arXiv:1309.0620 · Physica Scripta

The photon detection operator and complementarity between an electric detector and a magnetic detector

Shogo Tanimura

Published Apr 1, 2014• 0 citations
arXivHTMLPDF

Abstract

It had been a long standing problem that there is no consistent definition of photon position operator nor photon number density in the context of quantum theory. In this paper we derive the photon detection operator, which defines location of photon absorption, by applying the theory of indirect measurement to quantum electrodynamics. It is shown that the photon detection probability depends on electric properties of a photon-absorbing atom, in particular, on both electric and magnetic dipole moments of the atom. An experiment is proposed, in which the complementarity of wave-particle nature of light will be tested. It is also discussed that the complementarity is related to the non-commutativity of the electric and the magnetic fields.

DOI
10.1088/0031-8949/2014/t160/014039

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It had been a long standing problem that there is no consistent definition of photon position operator nor photon number density in the context of quantum theory. In this paper we derive the photon detection operator, which defines location of photon absorption, by applying the theory of indirect measurement to quantum electrodynamics. It is shown that the photon detection probability depends on electric properties of a photon-absorbing atom, in particular, on both electric and magnetic dipole moments of the atom. An experiment is proposed, in which the complementarity of wave-particle nature of light will be tested. It is also discussed that the complementarity is related to the non-commutativity of the electric and the magnetic fields.
A quantum-mechanical theory of photon detection is presented which takes fully into account the attenuation of the field due to the detection process. The time evolution of the joint quantum state of the detector and the field is found, to all orders in perturbation theory. Formulas are derived for the probabilities of absorbing specified numbers of photons within a given time interval, and for the correlations in the positions and times at which absorptions take place. The results are free from the inconsistencies which arise in the conventional theory of photon detection, and remain valid even when the field becomes appreciably attenuated during the experiment. It is found that an initially coherent field state remains coherent during its interaction with a given detector, and that its amplitude becomes attenuated by an amount which is completely independent of the number of counts which the detector records. This independence is shown to be a simple consequence of the Poisson quantum-number distribution of coherent fields. The counting statistics for arbitrary fields are expressed in a way which shows explicitly the relationship between the attenuation of the field and the absorption of quanta by the detector. The analysis is performed by first treating the case in which the field is confined within a homogeneous detecting medium throughout the experiment, and then generalizing to the case in which the field spends a limited amount of time in a spatially localized detecting region. The detector is assumed to consist of harmonic oscillators, which are shown to represent a suitable formal model for the absorption of radiation by large numbers of conventional detecting atoms.
In biorthogonal quantum mechanics, the eigenvectors of a quasi-Hermitian operator and those of its adjoint are biorthogonal and complete and the probability for a transition from a quantum state to any one of these eigenvectors is positive definite. We apply this formalism to the long standing problem of the position observable in quantum field theory. The dual bases are positive and negative frequency one-particle states created by the field operator and its conjugate and biorthogonality is a consequence of their commutation relations. In these biorthogonal bases the position operator is covariant and the Klein-Gordon wave function is localized. We find that the invariant probability for a transition from a one-photon state to a position eigenvector is the first order Glauber correlation function, bridging the gap between photon counting and the sensitivity of light detectors to electromagnetic energy density.
It is well known that the expectation value of the electric field operator in a number (Fock) state vanishes. However, a measurement of the electric field must yield some nonzero value each time an experiment is performed. What values can be obtained, and with what probability, is perhaps less well known. In this work we determine the probability distribution for the electric-field amplitude of an optical beam prepared in a single-photon state, using some techniques of quantum field theory. For the purpose of illustration, we apply our theory to the analysis of an ideal experiment where two spatially separated detectors can measure either the amplitude of the electric field, or the intensity of a single-photon optical beam. We find that while the photon cannot be detected in two disjoint positions at the same time, a measurement of the amplitude of the electric field carried by ditto photon, would yield a nonzero value at both detectors. These results show that a quantum field is both local with respect to its discrete excitations, i.e., the particles, and nonlocal with respect to its continuous amplitude.
The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.
The main principles of the previously constructed photon quantum mechanics allowing one to calculate the probability density of detection of the photon using the its wave function in the coordinate representation both in classical experiments and in experiments with single photons are presented. The theory is supplemented by solving the Cauchy problem for the corresponding equation, to which the photon wave function satisfies. The results of modeling the photon wave function with Gaussian pulse distribution for femtosecond laser radiation are presented. It is concluded that for well-polarized radiation the probability density of photon detection is proportional to the envelope of the square of the “electric field strength”. The Young's interference experiment is explained from the point of view of photon quantum mechanics. It is claimed that the coherence of the radiation should be consistent with the “separation” of the wave function of the photon falling on the beam splitter in experiments with single photons. This statement substantially weakens the dilemma of wave-particle duality, but it sets new challenges for the further development of physics. It is emphasized that for their solution the new experiments are needed to identify the properties of the physical vacuum. The physical nature of the photon is discussed as a result of the propagation of a spin wave with Planck parameters in a physical vacuum.
Quantum theory does not have a mechanism that explains how Nature implements probabilities. Thus, the main objective of this paper is to present new directions for the photonics research for the control of photon localization from probabilistic properties, and another step towards a probability field theory. The expectation is to improve photon collection and loss mitigation. The paper delves into the physics of photon probability control to explain the basis for the 4 new proposed experiments. It is known that photons are not affected by the presence of electric or magnetic fields. Therefore, an alternative question is, can photon probabilities be controlled? Probability control means vectoring and modulation. Vectoring is the control of the direction of localization, and modulation is the control of the distance to localization. This paper proposes a control mechanism by rethinking the foundations of quantum theory, using (i) a modified Schrodinger wave function, (ii) a new structure for particle design, (iii) the existence of subspace (x, y, zand no t), (iv) that all particles consists of a disc of the modified Schrodinger or the probabilistic wave function, orthogonal to the particle’s motion vector and (v) all other particle properties (e.g. electromagnetic wave, charge, mass, etc.) are added to this structure. The shape of the new probabilistic wave function is very close to that of the Schrodinger wave function. It is proposed that probabilities can be controlled by altering the electric and magnetic field densities, as probabilities are function the electric and magnetic fields. Thus, a new formula for the Airy Pattern. Modified Airy Pattern experiments are proposed to confirm these findings. These include measuring the photon’s electric field amplitude, the electric fields in materials, using the Airy Pattern to filter photons by their phase. The new theoretical results confirm that probability is a function of the wavelength. Finally, more than 4 experiments are proposed.