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arXiv:2404.03205 · Chinese Physics Letters

Optimal Dynamical Gauge in the Quantum Rabi Model

Yuqi Qing, Liangsheng Li, Wen‐Long You, Yueheng Lan, Maoxin Liu

Published Jan 21, 2025• 1 citation
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Abstract

Abstract We have investigated the gauge dependence of physical observables in a quantum Rabi model under different potential fields arising from the Hilbert space truncation of the atomic degrees of freedom. In both the square-well and harmonic potentials, the optimal gauge for the ground-state energy depends on the cavity frequency, which is optimal in the low-frequency limit, whereas the Coulomb gauge is optimal in the high-frequency limit. For a dynamic quantity, such as an out-of-time-order correlator, we have demonstrated the necessity of introducing an optimal dynamic gauge. This study provides deeper insight into the intricate relationship between gauge choice and the dynamics of quantum electrodynamics systems, resulting in more accurate theoretical frameworks.

DOI
10.1088/0256-307x/42/2/020501

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Abstract We have investigated the gauge dependence of physical observables in a quantum Rabi model under different potential fields arising from the Hilbert space truncation of the atomic degrees of freedom. In both the square-well and harmonic potentials, the optimal gauge for the ground-state energy depends on the cavity frequency, which is optimal in the low-frequency limit, whereas the Coulomb gauge is optimal in the high-frequency limit. For a dynamic quantity, such as an out-of-time-order correlator, we have demonstrated the necessity of introducing an optimal dynamic gauge. This study provides deeper insight into the intricate relationship between gauge choice and the dynamics of quantum electrodynamics systems, resulting in more accurate theoretical frameworks.
Multimode cavity quantum electrodynamics describes, for example, the coupling between an atom and a multimode electromagnetic resonator. The gauge choice is important for practical calculations in truncated Hilbert spaces, because the exact gauge invariance is recovered only in the whole space. An optimal gauge can be defined as the one predicting the most accurate observables for the same number of atomic levels and modes. Different metrics quantifying the gauge performance can be introduced depending on the observable of interest. In this paper we demonstrate that the optimal choice is generally mode dependent, i.e., a different gauge is needed for each cavity mode. While the choice of gauge becomes more important for increasing light-matter interaction, we also show that the optimal gauge does not correspond to the situation where the entanglement between light and matter is the smallest.
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.
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.
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.
A quantum dipole interacting with an optical cavity is one of the key models in cavity quantum electrodynamics (cavity-QED). To treat this system theoretically, the typical approach is to truncate the dipole to two levels. However, it has been shown that in the ultrastrong-coupling regime, this truncation naively destroys gauge invariance. By truncating in a manner consistent with the gauge principle, we introduce master equations to compute gauge-invariant emission spectra and quantum correlation functions which show significant disagreement with previous results obtained using the standard quantum Rabi model, with quantitative differences already present in the strong coupling regime. Explicit examples are shown using both the dipole gauge and the Coulomb gauge.