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Ultrastrong time-dependent light-matter interactions are gauge-relative

Adam Stokes, Ahsan Nazir

Published Jan 19, 2021• 10 citations
OpenAlex

Abstract

Time-dependent light-matter interactions are a widespread means by which to describe controllable experimental operations. They can be viewed as an approximation in which a third system - the control system - is treated as external within the Hamiltonian. We demonstrate that this results in non-equivalence between gauges. We provide a physical example in which each different non-equivalent model coincides with a gauge-invariant description applied in a different experimental situation. The qualitative final-time predictions obtained from these models, including entanglement and photon number, depend on the gauge within which the time-dependent coupling assumption is made. This occurs whenever the interaction switching is sufficiently strong and non-adiabatic even if the coupling vanishes at the preparation and measurement stages of the protocol, at which times the subsystems are unique and experimentally addressable.

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•May 16, 2018•1 citation
Time-dependent light-matter interactions are a widespread means by which to describe controllable experimental operations. They can be viewed as an approximation in which a third system - the control system - is treated as external within the Hamiltonian. We demonstrate that this results in non-equivalence between gauges. We provide a physical example in which each different non-equivalent model coincides with a gauge-invariant description applied in a different experimental situation. The qualitative final-time predictions obtained from these models, including entanglement and photon number, depend on the gauge within which the time-dependent coupling assumption is made. This occurs whenever the interaction switching is sufficiently strong and non-adiabatic even if the coupling vanishes at the preparation and measurement stages of the protocol, at which times the subsystems are unique and experimentally addressable.
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.
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.
We describe the physical relativity of light and matter quantum subsystems, their correlations, and energy exchanges. We examine the most commonly adopted definitions of atoms and photons, noting the significant difference in their localisation properties when expressed in terms of primitive manifestly gauge-invariant and local fields. As a result, different behaviours for entanglement generation and energy exchange occur for different definitions. We explore such differences in detail using toy models of a single photonic mode interacting with one and two dipoles.
After truncating the matter Hilbert space of a single-mode light-matter model, gauge transformations leave the spectrum invariant whereas the ground-state photon number is gauge dependent, hence, its meaning remains disputed. We show that this gauge dependence obeys an exact algebraic law whenever the gauge transformation acts as a displacement of the field coupled to a matter operator GGG. Namely, the ground-state photon number is a quadratic polynomial in the gauge parameter λλλ, ⟨a†a⟩λ\langle a^{\dagger} a\rangle_λ⟨a†a⟩λ​ === ⟨a†a⟩0\langle a^{\dagger} a\rangle_{0}⟨a†a⟩0​ +++ CλCλCλ +++ Qλ2Qλ^2Qλ2. Here the coefficients are the correlator C=⟨G p⟩C=\langle G\,p\rangleC=⟨Gp⟩ of GGG and the field momentum ppp, and Q=⟨G2⟩Q=\langle G^2\rangleQ=⟨G2⟩. Its minimum over all gauges is naturally gauge-invariant, and for the quantum Rabi model it is determined by the ground-state energy alone. We show that it is strictly positive for the quantum Rabi and Dicke models with level splitting, so that a residual photon content survives in every gauge. At weak coupling the gauge of minimal photon number is the Jaynes-Cummings gauge, and for the Dicke model it remains so at all couplings in the thermodynamic limit.
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.
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.