Abstract
The quantum Rabi model is a widespread description of the coupling between a two-level system and a quantized single mode of an electromagnetic resonator. Issues about this model's gauge invariance have been raised. These issues become evident when the light-matter interaction reaches the so-called ultrastrong coupling regime. Recently, a modified quantum Rabi model able to provide gauge-invariant physical results (e.g., energy levels, expectation values of observables, quantum probabilities) in any interaction regime was introduced [Nature Physics 15, 803 (2019)]. Here we provide an alternative derivation of this result, based on the implementation in two-state systems of the gauge principle, which is the principle from which all the fundamental interactions in quantum field theory are derived. The adopted procedure can be regarded as the two-site version of the general method used to implement the gauge principle in lattice gauge theories. Applying this method, we also obtain the gauge-invariant quantum Rabi model for asymmetric two-state systems, and the multi-mode gauge-invariant quantum Rabi model beyond the dipole approximation.
Highlights
The ultrastrong and deep-strong coupling (USC and DSC) between individual or collections of effective twolevel systems (TLSs) and the electromagnetic field has been realized in a variety of settings [1, 2]
The problem of a quantum-mechanical system whose state is effectively restricted to a two-dimensional Hilbert space is ubiquitous in physics and chemistry [17]
Since this remains true beyond TLSs, and since almost every practical calculation involving field-matter interactions is carried out cutting the infinite amount of information provided by exact infinitedimensional Hilbert spaces, the unpleasant conclusion of Stokes and Nazir [4, 5] is that gauge invariance and the gauge principle do not work in most practical cases
Summary
The ultrastrong and deep-strong coupling (USC and DSC) between individual or collections of effective twolevel systems (TLSs) and the electromagnetic field has been realized in a variety of settings [1, 2]. The source of gauge violation has been identified, and a general method for the derivation of lightmatter Hamiltonians in truncated Hilbert spaces able to produce gauge-invariant physical results has been developed [6] (see related work [7,8,9]). This gauge invariance was achieved by compensating the non-localities introduced in the construction of the effective Hamiltonians. V, we provide a reply to the key points raised by Stokes and Nazir [5]
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