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arXiv:1807.08001 · Physical review. D/Physical review. D.

p^·A^ vs x^·E^ : Gauge invariance in quantum optics and quantum field theory

Nicholas Funai, Jorma Louko, Eduardo Martín-Martínez

Published Mar 26, 2019• 23 citations
arXivHTMLPDF

Abstract

We compare the predictions of the fundamentally motivated minimal coupling ($\stackrel{^}{\mathbit{p}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{A}}$) and the ubiquitous dipole coupling ($\stackrel{^}{\mathbit{x}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{E}}$) in the light-matter interaction. By studying the light-matter interaction for hydrogenlike atoms we find that the dipole approximation cannot be a priori justified to analyze the physics of vacuum excitations (a very important phenomenon in relativistic quantum information) since a dominant wavelength is absent in those problems, no matter how small (as compared to any frequency scale) the atom is. Remarkably, we show that the dipole approximation in those regimes can still be valid as long as the interaction time is longer than the light-crossing time of the atoms, which is a very reasonable assumption. We also highlight some of the subtleties that one has to be careful with when working with the explicitly gauge noninvariant nature of the minimal coupling, and we compare it with the explicitly gauge invariant dipole coupling.

DOI
10.1103/physrevd.99.065014

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We compare the predictions of the fundamentally motivated minimal coupling (\stackrel{^}{\mathbit{p}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{A}}) and the ubiquitous dipole coupling (\stackrel{^}{\mathbit{x}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{E}}) in the light-matter interaction. By studying the light-matter interaction for hydrogenlike atoms we find that the dipole approximation cannot be a priori justified to analyze the physics of vacuum excitations (a very important phenomenon in relativistic quantum information) since a dominant wavelength is absent in those problems, no matter how small (as compared to any frequency scale) the atom is. Remarkably, we show that the dipole approximation in those regimes can still be valid as long as the interaction time is longer than the light-crossing time of the atoms, which is a very reasonable assumption. We also highlight some of the subtleties that one has to be careful with when working with the explicitly gauge noninvariant nature of the minimal coupling, and we compare it with the explicitly gauge invariant dipole coupling.
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.
In this article we discuss the invariance under general changes of reference frame of all the physical predictions of particle detector models in quantum field theory in general and, in particular, of those used in quantum optics to model atoms interacting with light. We find explicitly how the light-matter interaction Hamiltonians change under general coordinate transformations, and analyze the subtleties of the Hamiltonians commonly used to describe the light-matter interaction when relativistic motion is taken into account.
We revisit the derivation of Rabi- and Dicke-type models, which are commonly used for the study of quantum light-matter interactions in cavity and circuit QED. We demonstrate that the validity of the two-level approximation, which is an essential step in this derivation, depends explicitly on the choice of gauge once the system enters the ultrastrong-coupling regime. In particular, while in the electric dipole gauge the two-level approximation can be performed as long as the Rabi frequency remains much smaller than the energies of all higher-lying levels, it can dramatically fail in the Coulomb gauge, even for systems with an extremely anharmonic spectrum. We extensively investigate this phenomenon both in the single-dipole (Rabi) and multidipole (Dicke) case, and consider the specific examples of dipoles confined by double-well and square-well potentials, and of circuit QED systems with flux qubits coupled to an LC resonator.
The loss of gauge invariance in models of light-matter interaction which arises from material and photonic space truncation can pose significant challenges to conventional quantum optical models when matter and light strongly hybridize. In structured photonic environments, necessary in practice to achieve strong light-matter coupling, a rigorous model of field quantization within the medium is also needed. Here we use the framework of macroscopic QED by quantizing the fields in an arbitrary material system, with a spatially dependent dispersive and absorptive dielectric, starting from a fundamental light-matter action. We truncate the material and mode degrees of freedom while respecting the gauge principle by imposing a partial gauge-fixing constraint during canonical quantization, which admits a large number of gauges including the Coulomb and multipolar gauges commonly used in quantum optics. We also consider gauge conditions with explicit time dependence, enabling us to unambiguously introduce additional phenomenologically time-dependent light-matter interactions in any gauge. Our results allow one to derive rigorous nonrelativistic models of ultrastrong light-matter interactions in structured photonic environments with no gauge ambiguity. Results for two-level systems and the dipole approximation are discussed, as well as how to go beyond the dipole approximation for effective single-particle models. By comparing with the limiting case of an inhomogeneous dielectric, where dispersion and absorption can be neglected and the fields can be expanded in terms of the generalized transverse eigenfunctions of the dielectric, we show how lossy systems can introduce an additional gauge ambiguity, which we resolve and predict to have fundamental implications for open quantum system models. Finally, we show how observables in mode-truncated systems can be calculated without ambiguity by using a simple gauge-invariant model of photodetection.
The Dicke model, which describes the dipolar coupling between NNN two-level atoms and a quantized electromagnetic field, seemingly violates gauge invariance in the presence of ultrastrong light-matter coupling, a regime that is now experimentally accessible in many physical systems. Specifically, it has been shown that, while the two-level approximation can work well in the dipole gauge, the Coulomb gauge fails to provide the correct spectra in the ultrastrong coupling regime. Here we show that, taking into account the nonlocality of the atomic potential induced by the two-level approximation, gauge invariance is fully restored for arbitrary interaction strengths, even in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. Finally, we express the Hopfield model, a general description based on the quantization of a linear dielectric medium, in a manifestly gauge-invariant form, and show that the Dicke model in the dilute regime can be regarded as a particular case of the more general Hopfield model.
This article deals with the fundamental problem of light-matter interaction in the quantum theory. Although it is described through the vector potential in quantum electrodynamics, it is believed by some that a hamiltonian involving only the electric and the magnetic fields is preferable. In the literature this hamiltonian is known as the Power-Zienau-Woolley hamiltonian. We question its validity and show that it is not equivalent to the minimal-coupling hamiltonian. In this article, we show that these two hamiltonians are not connected through a gauge transformation. We find that the gauge is not fixed in the Power-Zienau-Woolley hamiltonian. The interaction term is written in one gauge whereas the rest of the hamiltonian is written in another gauge. The Power-Zienau-Woolley hamiltonian and the minimal-coupling one are related through a unitary transformation that does not fulfill the gauge fixing constraints. Consequently, they predict different physical results. In this letter, we provide the correct quantum theory in the multipolar gauge with a hamiltonian involving only the physical fields.