A gauge field treatment of a current oscillating at frequency ν of interacting neutral atoms leads to a set of matter-wave duals to Maxwell's equations for the electromagnetic field. In contrast to electromagnetics, the velocity of propagation has a lower limit rather than upper limit, and the wave impedance of otherwise free space is negative real-valued rather than 377 Ω. Quantization of the field leads to the matteron, the gauge boson dual to the photon. Unlike the photon, the matteron is bound to an atom and carries negative rather than positive energy, causing the source of the current to undergo cooling. Eigenstates of the combined matter and gauge field annihilation operator define the coherent state of the matter-wave field, which exhibits classical coherence in the limit of large excitation.
A gauge field treatment of a current oscillating at frequency ν of interacting neutral atoms leads to a set of matter-wave duals to Maxwell's equations for the electromagnetic field. In contrast to electromagnetics, the velocity of propagation has a lower limit rather than upper limit, and the wave impedance of otherwise free space is negative real-valued rather than 377 Ω. Quantization of the field leads to the matteron, the gauge boson dual to the photon. Unlike the photon, the matteron is bound to an atom and carries negative rather than positive energy, causing the source of the current to undergo cooling. Eigenstates of the combined matter and gauge field annihilation operator define the coherent state of the matter-wave field, which exhibits classical coherence in the limit of large excitation.
A self-consistent theoretical treatment of a triple-well atomtronic transistor circuit reveals the mechanism of gain, conditions of oscillation, and properties of the subsequent coherent matter waves emitted by the circuit. A Bose-condensed reservoir of atoms in a large source well provides a chemical potential that drives circuit dynamics. The theory is based on the ansatz that a condensate arises in the transistor gate well as a displaced ground state, that is, one that undergoes dipole oscillation in the well. That gate atoms remain condensed and oscillating is shown to be a consequence of the cooling induced by the emission of a matter wave into the vacuum. Key circuit parameters such as the transistor transconductance and output current are derived by transitioning to a classical equivalent circuit model. Voltage-like and current-like matter-wave circuit wave fields are introduced in analogy with microwave circuits, as well as an impedance relationship between the two. This leads to a notion of a classically coherent matter wave that is the dual of a coherent electromagnetic wave and which is distinct from a de Broglie matter wave associated with cold atoms. Subjecting the emitted atom flux to an atomic potential that will reduce the de Broglie wavelength, for example, will increase the classical matter-wave wavelength. Quantization of the classical matter-wave fields leads to the dual of the photon that is identified not as an atom but as something else, which is here dubbed a ``matteron.''
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.
Introduction . In the previous chapter we have obtained the form of the coupling between matter and the electromagnetic field. We are thus in a position to treat electrodynamics in the presence of charges and currents. We start by obtaining the Euler-Lagrange equations of the matter-electromagnetic field system. These turn out to be the Maxwell-Lorentz equations, which is a set of coupled equations of motion in which the matter field acts as a source for the electromagnetic field and vice versa. In Section 4.2 we derive the Hamiltonian of the complete system in the Coulomb gauge, and this leads to the so-called minimal coupling Hamiltonian, containing both the electromagnetic potential and the matter-field amplitude. Specializing the nonrelativistic matter field to the case of a neutral atom, consisting of the field of electrons in a static nuclear potential, in Section 4.3 we obtain the atom-photon Hamiltonian in the minimal coupling scheme. Some of the basic processes induced by the atom-photon interaction part of the Hamiltonian are also discussed in this section. This gives us the possibility of introducing at this stage a fundamental simplification of the interaction Hamiltonian, namely the electric dipole approximation. The minimal coupling scheme, however, is not the only possible atom-field coupling.
We revisit the quantum oscillator model of the electromagnetic field and conclude that, while the nonlocal positive and negative frequency ladder operators generate a photon Fock basis, the Hermitian field operators obtained by second quantization of real Maxwell fields describe photon-antiphoton pairs that couple locally to Fermionic matter and can be modeled classically. Their commutation relations define a scalar product that can be the basis of a first quantized theory of single photons. Since a one-photon state collapses to a zero-photon state when the photon is counted, the field describing it must be interpreted as a probability amplitude.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
Maxwell matter waves emerge from a perspective, complementary to de Broglie's, that matter is fundamentally a wave phenomenon whose particle aspects are revealed by quantum mechanics. Their quantum mechanical description is derived through the introduction of a matter vector potential, having frequency ω0, to Schrodinger's equation for a massive particle. Maxwell matter waves are then seen to be coherent excitations of a single-mode of the matter-wave field. In the classical regime, their mechanics is captured by a matter analog of Maxwell's equations for the electromagnetic field. As such, Maxwell matter waves enable a spectrum of systems that have useful optical analogs, such as resonant matter-wave interferometric sensors and matter-wave parametric oscillators. These waves are associated with a wavelength that is tied to the drive frequency ω0 rather than to the massive particle's energy, as is ordinarily the case with de Broglie matter waves. As a result, simple interferometric measurements lead to different outcomes for the two types of waves. While their apparent departure from de Broglie character is surprising, Maxwell matter waves are wholly consistent with quantum mechanics.