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arXiv:2003.04899 · Scientific Reports

Avoiding gauge ambiguities in cavity quantum electrodynamics

Dominic M. Rouse, Brendon W. Lovett, Erik M. Gauger, Niclas Westerberg

Published Feb 19, 2021• 12 citations
arXivHTMLPDF

Abstract

Systems of interacting charges and fields are ubiquitous in physics. Recently, it has been shown that Hamiltonians derived using different gauges can yield different physical results when matter degrees of freedom are truncated to a few low-lying energy eigenstates. This effect is particularly prominent in the ultra-strong coupling regime. Such ambiguities arise because transformations reshuffle the partition between light and matter degrees of freedom and so level truncation is a gauge dependent approximation. To avoid this gauge ambiguity, we redefine the electromagnetic fields in terms of potentials for which the resulting canonical momenta and Hamiltonian are explicitly unchanged by the gauge choice of this theory. Instead the light/matter partition is assigned by the intuitive choice of separating an electric field between displacement and polarisation contributions. This approach is an attractive choice in typical cavity quantum electrodynamics situations.

DOI
10.1038/s41598-021-83214-z

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Systems of interacting charges and fields are ubiquitous in physics. Recently, it has been shown that Hamiltonians derived using different gauges can yield different physical results when matter degrees of freedom are truncated to a few low-lying energy eigenstates. This effect is particularly prominent in the ultra-strong coupling regime. Such ambiguities arise because transformations reshuffle the partition between light and matter degrees of freedom and so level truncation is a gauge dependent approximation. To avoid this gauge ambiguity, we redefine the electromagnetic fields in terms of potentials for which the resulting canonical momenta and Hamiltonian are explicitly unchanged by the gauge choice of this theory. Instead the light/matter partition is assigned by the intuitive choice of separating an electric field between displacement and polarisation contributions. This approach is an attractive choice in typical cavity quantum electrodynamics situations.
This work provides the fundamental theoretical framework for molecular cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the electronic state truncation. We conjecture that such ambiguity arises because not all operators are consistently constrained in the same truncated electronic subspace for both gauges. We resolve this ambiguity by constructing a unitary transformation operator that properly constrains all light-matter interaction terms in the same subspace. We further derive an equivalent and yet convenient expression for the Coulomb gauge Hamiltonian under the truncated subspace. We finally provide the analytical and numerical results of a model molecular system coupled to the cavity to demonstrate the validity of our theory.
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 interaction between quantized electromagnetic fields in cavities and natural or artificial atoms has played a crucial role in developing our understanding of light-matter interactions and quantum technologies.Recently, new regimes beyond the weak and strong light-matter coupling of cavity-QED have been explored in several settings, wherein the light-matter coupling rate becomes comparable to (ultrastrong coupling) or even exceeds (deep-strong coupling) the photon frequency.These ultrastrong coupling regimes can give rise to new physical effects and applications, and they challenge our understanding of cavity QED; fundamental issues like the proper definition of subsystems, their quantum measurements, the structure of light-matter ground states, and the analysis of time-dependent interactions are subject to gauge ambiguities that lead to even qualitatively distinct predictions.The resolution of these ambiguities is important for understanding and designing next-generation quantum devices that can operate in extreme coupling regimes.Here we discuss and provide solutions to these ambiguities by adopting an approach based on operational procedures involving measurements on the individual light and matter components of the interacting system.
This article reviews the meaning of gauge freedom in quantum electrodynamics and emphasizes its consistency with examples and analyses. It thus demonstrates that ambiguities and recent controversies are resolved by proper identification of physically relevant observables of field and matter subsystems. Since these observables differ for different gauges, light and matter subsystems can only be defined relative to the choice of gauge, with implications for the identification of effective few-dimensional models, entanglement, phase transitions, and cavity quantum electrodynamics effects beyond weak-coupling regimes.
This work provides the fundamental theoretical framework for few-mode cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the photonic mode truncation. We first propose a general framework to resolve ambiguities for an arbitrary truncation in a given gauge. Then, we specifically consider the case of mode truncation, deriving gauge invariant expressions for both the Coulomb and dipole gauge Hamiltonians that naturally reduce to the commonly used single-mode Hamiltonians when considering a single-mode truncation. We finally provide the analytical and numerical results of both atomic and molecular model systems coupled to the cavity to demonstrate the validity of our theory.
Ultrastrong-coupling between an effective two-level system and radiation offers immense potential for advancing both fundamental and applied quantum electrodynamics (QED). Such regimes cannot be treated using the rotating-wave approximation, which applied to the quantum Rabi model yields the apparently less fundamental Jaynes-Cummings model. However, the implications of gauge-freedom for ultrastrong-coupling QED are yet to be recognised. Here we show that when truncating the material system to only two levels, each gauge gives a different effective description whose predictions can vary significantly for ultrastrong-coupling. Rabi models are obtained through specific gauge choices, but so too is a Jaynes-Cummings model without needing the rotating-wave approximation. Analysing a circuit QED setup, we find that this Jaynes-Cummings model is not only valid well beyond the rotating-wave approximation, it is often more accurate than the Rabi model. Among the many implications of this finding is that the system's ground state is not necessarily highly entangled, which is usually considered a hallmark of ultrastrong-coupling light-matter physics.