We revisit the interaction of a first-quantized atomic system (consisting of two charged quantum particles) with the quantum electromagnetic field, pointing out the subtleties related to the gauge nature of electromagnetism and the effect of multipole approximations. We connect the full minimal-coupling model with the typical effective models used in quantum optics and relativistic quantum information such as the Unruh-DeWitt (UDW) model and the dipole coupling approximation. We point out in what regimes different degrees of approximation are reasonable and in what cases effective models need to be refined to capture the features of the light-matter interaction. This is particularly important when considering the center of mass (COM) of the atom as a quantum system that can be delocalized over multiple trajectories. For example, we show that the simplest UDW approximation with a quantum COM fails to capture crucial R\"ontgen terms coupling COM and internal atomic degrees of freedom with each other and the field. Finally we show how effective dipole interaction models can be covariantly prescribed for relativistically moving atoms.

Quantum delocalization, gauge, and quantum optics: Light-matter interaction in relativistic quantum information

Maxwell's equations are formulated as a relativistic “Schrodinger-like equation” for a single photon of a given helicity. The probability density of the photon satisfies an equation of continuity. The energy eigenvalue problem gives both positive and negative energies. The Feynman concept of antiparticles is applied here to show that the negative-energy states going backward in time (t → −t) give antiphoton states, which are photon states with the opposite helicity. For a given mode, properties of a photon, such as energy, linear momentum, total angular momentum, orbital angular momentum, and spin are equivalent in both classical electromagnetic field theory and quantum theory. The single-photon Schrodinger equation is field (or “second”) quantized in a gauge-invariant way to obtain the quantum field theory of many photons. This approach treats the quantization of electromagnetic radiation in a way that parallels the quantization of material particles and, thus, provides a unified treatment.

A Relativistic Schrödinger-like Equation for a Photon and Its Second Quantization

We present three schemes to go beyond the electric-dipole approximation in X-ray absorption spectroscopy calculations within a four-component relativistic framework. The first is based on the full semi-classical light-matter interaction operator, and the two others on a truncated interaction within Coulomb gauge (velocity representation) and multipolar gauge (length representation). We generalize the derivation of multipolar gauge to an arbitrary expansion point and show that the potentials corresponding to different expansion point are related by a gauge transformation, provided the expansion is not truncated. This suggests that the observed gauge-origin dependence in multipolar gauge is more than just a finite-basis set effect. The simplicity of the relativistic formalism enables arbitrary-order implementations of the truncated interactions, with and without rotational averaging, allowing us to test their convergence behavior numerically by comparison to the full formulation. We confirm the observation that the oscillator strength of the electric-dipole allowed ligand K-edge transition of TiCl$_4$, when calculated to second order in the wave vector, become negative, but also show that inclusion of higher-order contributions allows convergence to the result obtained using the full light-matter interaction. However, at higher energies, the slow convergence of such expansions becomes dramatic and renders such approaches at best impractical. When going beyond the electric-dipole approximation, we therefore recommend the use of the full light-matter interaction.

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Beyond the electric-dipole approximation in simulations of X-ray absorption spectroscopy: Lessons from relativistic theory

We present three schemes to go beyond the electric-dipole approximation in x-ray absorption spectroscopy calculations within a four-component relativistic framework. The first is based on the full semi-classical light-matter interaction operator and the two others on a truncated interaction within the Coulomb gauge (velocity representation) and multipolar gauge (length representation). We generalize the derivation of the multipolar gauge to an arbitrary expansion point and show that the potentials corresponding to different expansion points are related by a gauge transformation, provided that the expansion is not truncated. This suggests that the observed gauge-origin dependence in the multipolar gauge is more than just a finite-basis set effect. The simplicity of the relativistic formalism enables arbitrary-order implementations of the truncated interactions, with and without rotational averaging, allowing us to test their convergence behavior numerically by comparison to the full formulation. We confirm the observation that the oscillator strength of the electric-dipole allowed ligand K-edge transition of TiCl4, when calculated to the second order in the wave vector, becomes negative but also show that inclusion of higher-order contributions allows convergence to the result obtained using the full light-matter interaction. However, at higher energies, the slow convergence of such expansions becomes dramatic and renders such approaches at best impractical. When going beyond the electric-dipole approximation, we therefore recommend the use of the full light-matter interaction.

https://hal.archives-ouvertes.fr/hal-02899747/document

Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory.

The action principle is used to derive, by an entirely algebraic approach, gauge transformations of the full vacuum-to-vacuum transition amplitude (generating functional) from the Coulomb gauge to arbitrary covariant gauges and in turn to the celebrated Fock–Schwinger (FS) gauge for the Abelian (QED) gauge theory without recourse to path integrals or to commutation rules and without making use of delta functionals. The interest in the FS gauge, in particular, is that it leads to Faddeev–Popov ghosts-free non-Abelian gauge theories. This method is expected to be applicable to non-Abelian gauge theories including supersymmetric ones.

Action Principle and Algebraic Approach to Gauge Transformations in Gauge Theories

The free electromagnetic field is canonically quantised in a gauge-invariant way by interpreting the Fourier coefficients of the magnetic induction field B as generalised coordinates, and the coefficients of the electric field E as their conjugate momenta. The usual commutation relations among the components of E and B are obtained. A canonical transformation, corresponding to a rotation in generalised phase space, is made on the Fourier coefficients. This transformation is shown to give a duality transformation on the electric and magnetic fields. The free-field Maxwell equations and the commutation relations are invariant under duality transformations. However, if interactions are introduced, the invariance under duality transformations is broken, and the original canonical theory should be used.

Gauge-invariant canonical quantisation of the electromagnetic field and duality transformations

Operator gauge transformations on the electromagnetic potentials in quantum electrodynamics are obtained by requiring form invariance of the Schrodinger equation of a single particle under arbitrary space- and time-dependent unitary transformations. These operator gauge transformations are found to have the same form as in non-Abelian gauge field theories. By considering the Lorentz force on the particle, we show that the definition of the electromagnetic-field strength operator in terms of the potentials must also have the same form as in non-Abelian gauge field theory. Since the Hamiltonian is gauge dependent, a distinction is made between the Hamiltonian and the gauge-invariant energy operator of the particle. Local conservation of energy for the particle is derived and it is shown that the particle gains (loses) energy at the same rate as expected classically. An operator gauge transformation is made to obtain potentials in the multipolar gauge and the multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian in the multipolar gauge.

D. H. Kobe

Papers

Gauge-invariant canonical quantisation of the electromagnetic field and duality transformations

Operator gauge transformations in nonrelativistic quantum electrodynamics: Application to the multipolar Hamiltonian