The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group SU(2)⊗U(1) using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates.
The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group SU(2)⊗U(1) using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates.
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.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
Introduction: In the rapidly developing areas of photonics associated with quantum teleportation of photons, quantum cryptography, and quantum computing, ideas are becoming increasingly relevant about the certain localized states of single or several entangled photons moving in space and time. As is known, in quantum mechanics, localized states of particles can only be described using wave functions in coordinate representation – the corresponding wave packets. Aims: To explain Young's one- and two-photon experiments using coordinate 6-component quantum mechanical and 1-component quasi-classical photon wave functions. Methods: The article outlines the method developed in authors' previous work in constructing the 6-component coordinate wave function of free photons in the form of the wave packet (an integral over the entire momentum space and sum over two possible values +1 and -1 of the helicity). The basis functions are the circularly polarized monochromatic waves, corresponding to certain values of the momentum, helicity, and photon energy operators and forming a complete orthonormal set of generalized eigenfunctions of these operators. This photon wave function (PWF) is normalized to the unit probability of finding the photon anywhere in space. Both the basis functions and the coordinate PWF satisfy a Schrödinger-type equation derived from the Maxwell-Lorentz equations written in quantum mechanical form first introduced by Majorana. Then the results of modeling and consideration of the space-time evolution of the wave packet with Gaussian momentum distribution corresponding to the directed photon emission for 80 fs are briefly presented. Further, in view of the possibility of constructing the coordinate PWF, the basic formula for the waveparticle duality of photons and particles is formulated. It is emphasized that the photon is fundamentally different from “ordinary” particles, since, according to the authors, the photon is the result of the propagation of a certain complex spin wave in a physical vacuum, the details of which can only be considered simultaneously with the study of the structure of the physical vacuum at Planckian distances. Results: Young`s one- and two-photon experiments are explained using various options for modeling 6-component coordinate PWFs, including approximate functions corresponding to spherically symmetric and electric dipole photon radiation. Conclusion: This explanation is illustrated using specific examples of radiation and, at the same time, is compared with numerical simulations for spherically symmetric radiation and the use of 1-component quasi-classical PWFs.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.
Abstract A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
A closed, uncountably infinite dimensional system of “quantum object + thermostat” is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrödinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in potential determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial differential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase transitions of second kind may occur in the expressions for microscopic quantum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscillator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermodynamics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.