Gauge principle and gauge invariance in two-level systems

PHYSICAL REVIEW A 103, 053703 (2021) Gauge principle and gauge invariance in two-level systems Salvatore Savasta ,1Omar Di Stefano ,1,*Alessio Settineri,1David Zueco ,2,3Stephen Hughes ,4and Franco Nori 5,6 1Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina, Italy 2Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain 3Fundación ARAID, Campus Río Ebro, 50018 Zaragoza, Spain 4Department of Physics, Engineering Physics, and Astronomy, Queen’s University, Kingston, Ontario, Canada, K7L 3N6 5Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan 6Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Received 14 October 2020; accepted 23 March 2021; published 7 May 2021) 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 [O. Di Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nat. Phys. 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 multimode gauge-invariant quantum Rabi model beyond the dipole approximation. DOI: 10.1103/PhysRevA.103.053703 I. INTRODUCTION Recently, it was argued that truncations of the atomic Hilbert space, to obtain a two-level description of the matter system, violate the gauge principle [1–3]. Such violations become particularly relevant in the ultrastrong and deep-strong coupling (USC and DSC) regimes. These extreme regimes have been realized between individual or collections of effective two-level systems (TLSs) and the electromagnetic field in a variety of settings [4,5]. In the USC (DSC) regime of quantum light-matter interaction the coupling strength becomes comparable (larger) than the transition frequencies of the system. The authors of Ref. [1] demonstrated that, 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. However, the two-level approximation can drastically fail in the Coulomb gauge, even for systems with an extremely anharmonic spectrum. The impact of the truncation of the Hilbert space of the matter system to only two states was also studied in Ref. [2], by introducing a one-parameter (α) set of gauge transformations. The authors found that each value of the parameter produces a distinct quantum Rabi model (QRM), thus providing distinct physical predictions. Investigating a matter system with a lower anharmonicity (with respect to that considered in *Corresponding author: [email protected] Ref. [1]), they used the gauge parameter αas a fit parameter to determine the optimal QRM for a specific set of system parameters, by comparing the obtained α-dependent lowestenergy states and levels with the corresponding predictions of the nontruncated gauge-invariant model. The surprising result [2] is that, according to this procedure, in several circumstances the optimal gauge is the so-called Jaynes-Cummings (JC) gauge, a gauge where the counterrotating terms are automatically absent. Recently, the source of gauge violation was identified [6] and a general method for the derivation of light-matter Hamiltonians in truncated Hilbert spaces able to produce gauge-invariant physical results was developed [6] (see also relatedworkinRefs.[7–9]). This gauge invariance was achieved by compensating the nonlocalities introduced in the construction of the effective Hamiltonians. Consequently, the resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly from the standard one, but provides exactly the same energy levels obtained by using the dipole gauge, as it should, because physical observable quantities must be gauge invariant. A recent overview of these gauge issues in TLSs can be found in Ref. [10]. Very recently, the validity of the gauge-invariant QRM developed in Ref. [6] has been put into question [3]. Specifically, it was claimed that the truncation of the Hilbert space necessarily ruined gauge-invariance. In this paper, however, we confirm that the gauge principle applies also to TLSs, as required by any consistent description of light-matter interactions. Specifically, we 2469-9926/2021/103(5)/053703(8) 053703-1 ©2021 American Physical Society SALVATORE SAVASTA et al. PHYSICAL REVIEW A 103, 053703 (2021) formulate, in a fully consistent and physically meaningful way, the fundamental gauge principle in two-state systems. The derivation described here can also be regarded as the two-site version of the general method for lattice gauge theories [11], which represent the most advanced and commonly used tool for describing gauge theories in the presence of a truncated infinite-dimensional Hilbert space. When a gauge theory is regularized on the lattice, it is essential to maintain its invariance under gauge transformations [11]. An analogous approachwas developed as early as 1933 [12] for the description of tightly bound electrons in a crystal in the presence of a slowly varying magnetic vector potential (see, e.g., also Refs. [13–15]). Moreover, applying this method further, we also obtain the multimode gauge-invariant QRM beyond the dipole approximation. II. GAUGE PRINCIPLE In this section, we recall some fundamental concepts, which we will apply in the next sections. In quantum field theory, the coupling of particles with fields is constructed in such a way that the theory is invariant under a gauge transformation [16]. Here, we limit the theoretical model to consider U(1) invariance. For symmetry groups that are noncommutative, this approach can be generalized to nonabelian gauge theories [11,16]. Let us consider the transformation of the particle field ψ→exp(iqθ)ψ. This transformation represents a symmetry of the free action of the particle (e.g., the Dirac action) if θ is a constant, but we want to consider a generic function θ(x) (local phase transformation). However, the free Dirac action is not invariant under local phase transformations because the factor exp[iqθ(x)] does not commute with ∂μ.Atthesame time, it is known that the action of the free electromagnetic field is invariant under the following gauge transformation: Aμ→Aμ−∂μθ. (1) It is then possible to replace, in the action, the derivative ∂μ with a covariant derivative of ψas Dμψ=(∂μ+iqAμ)ψ, (2) so that Dμψ→eiqθDμψ, (3) even when θdepends on x. It is now easy to construct a Lagrangian with a local U(1) invariance. It suffices to replace all derivatives ∂μwith covariant derivatives Dμ. The same procedure, leading to the well-known minimal coupling replacement, can be applied to describe the interaction of a nonrelativistic particle with the electromagnetic field. Considering a particle of mass mwith a geometrical coordinate xand a potential V(x), the Hamiltonian of such a particle interacting with the electromagnetic field can be written as ˆ Hgi 0=1 2m[ˆp−qA(x)]2+V(x),(4) where ˆp=−id/dx is the momentum of the particle (here ¯h=1). It turns out that the expectation values ψ|ˆ Hgi 0|ψare Potential Position x (a) (b) FIG. 1. A double-well system in the two-state limit. The symbols E0and E1are the two lowest-energy levels, well separated in energy by the next higher energy level E2. Panel (a) also shows the square modulus of the two wave functions localized in the well, obtained as linear combinations of the two lowest-energy wave functions displayed in panel (b). invariant under local phase transformations, ψ(x)→eiqθ(x)ψ(x),(5) thanks to the presence of the gauge field A(x). Note that the function of a continuous degree of freedom ψ(x), lives in the infinite-dimensional space of all square-integrable functions, and the local phase transformation transforms a state vector in this space into a different vector in the same space. Finally, we observe that the total Hamiltonian, in addition to ˆ Hgi 0, includes the free Hamiltonian for the gauge field. III. DOUBLE-WELL SYSTEMS IN THE TWO-STATE LIMIT 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]. In the simplest examples, the system simply possesses a degree of freedom that can take only two values. For example, the spin projection in the case of a spin-1/2 particle or the polarization in the case of a photon. In addition to these intrinsic two-state systems, a more common situation is that the system has a continuous degree of freedom x, for example, a geometrical coordinate, and a potential energy function V(x) depending on it, with two separate minima [17](seeFig.1). Let us assume that the barrier height Vis large enough that the system dynamics can be adequately described by a two-dimensional Hilbert space spanned by the two ground states in the two wells |Land |R. 053703-2 GAUGE PRINCIPLE AND GAUGE INVARIANCE IN … PHYSICAL REVIEW A 103, 053703 (2021) Potential Position x (a) (b) FIG. 2. A symmetric double-well system in the two-state limit. The symbols E0and E1are the two lowest-energy levels, well separated in energy by the next higher-energy level E2. Panel (a) also shows the square modulus of the two wave functions localized in the well, obtained as symmetric and antisimmetric combinations of the two lowest-energy wave functions displayed in panel (b). The motion in the two-dimensional Hilbert space can be adequately described by the simple Hamiltonian ˆ H0= j=L,R Ej|jj|−t(|RL|+H.c.),(6) where the tunneling coefficient is given by t=L|ˆ H0|R, and ˆ H0=ˆp2 2m+V(x)(7) is the usual system Hamiltonian. If the potential is an even function of the geometrical coordinate, namely V(x)=V(−x) (see Fig. 2), then EL=ER, and we can fix EL=ER=0. Introducing the Pauli operator ˆρx=|LR|+H.c., we obtain ˆ H0=−tˆρx,(8) whose eigenstates, delocalized in the two wells, are the well-known symmetric and antisymmetric combinations [see Fig. 2(b)], |S= 1 √2(|R+|L), |A= 1 √2(|R−|L),(9) with eigenvalues EA,S=±t, so that =EA−ES=2t, and we assume t>0. The Hamiltonian in Eq. (6) can be written in diagonal form as ˆ H0=(/2)ˆσz,(10) where ˆσz=−ˆρx=|AA|−|SS|. Note, to distinguish between the different basis states for the operator representations, we use ˆσifor the |A−|Sbasis, and ˆρifor the |L−|R basis. Thus, for example, the diagonal ˆσzoperator becomes nondiagonal in the |L−|Rbasis. It is worth noting that this elementary analysis is not restricted to the case of a double-well potential. Analogous considerations can be carried out for systems with different potential shapes, displaying two (e.g., lowest energy) levels well separated in energy from the next higher level. The wave functions ψL(x)=x|Land ψR(x)=x|Lcan be obtained from the symmetric and antisymmetric combinations of ψS(x) and ψA(x) (see Fig. 1), which can be obtained exactly as the two lowest-energy eigenfunctions of the Schrödinger problem described by the Hamiltonian in Eq. (7). The gap =2t is obtained from the difference between the corresponding eigenvalues. This two-state tunneling model is a well-known formalism to describe many realistic systems, including the ammonia molecule, coupled quantum dots, and superconducting flux-qubits. The case of a potential of the effective particle which does not display inversion symmetry can also be easily addressed. For example, consider an asymmetric double well potential, as shown in Fig. 1. In this case, Eq. (6) can be expressed as ˆ H0= 2ˆρz− 2ˆρx.(11) The quantity is the detuning parameter, that is, the difference in the ground-state energies of the states localized in the two wells in the absence of tunneling. The Hamiltonian in Eq. (11) can be trivially diagonalized with eigenvalues ±ωq/2, where ωq=√2+2. IV. GAUGE PRINCIPLE IN TWO-LEVEL SYSTEMS The question arises if it is possible to save the gauge principle when, under the conditions described above, such a particle is adequately described by states confined in a two-dimensional complex space. If we apply an arbitrary local phase transformation to, e.g., the wave function ψA(x)= x|A:ψA(x)→ψ A(x)=eiqθ(x)ψA(x), it happens that, in general, ψ A(x)= cSψS(x)+cAψA(x), where cAand cSare complex coefficients. Thus the general local phase transformation does not guarantee that the system can still be described as a two-state system. According to this analysis, those works claiming gauge noninvariance due to material truncation in ultrastrong-coupling QED [18] (we would say at any coupling strength, except negligible), at first sight, might appear to be correct. The direct consequence of this conclusion would be that two-level models, widespread in physics and chemistry, are too simple to implement their interaction with a gauge field, according to the general principle from which the fundamental interactions in physics are obtained. Since adding to the particle system description a few additional levels does not change this point, the conclusion would be even more dramatic. Moreover, according to the authors of Refs. [2,3], this leads to several nonequivalent models of light-matter interactions providing different physical results. One might then claim the “death of the gauge principle” and of gauge invariance in truncated Hilbert spaces, namely in 053703-3 SALVATORE SAVASTA et al. PHYSICAL REVIEW A 103, 053703 (2021) almost all cases where theoreticians try to provide quantitative predictions to be compared with actual experiments. Our view is drastically different: We find that the breakdown of gauge invariance is the direct consequence of an inconsistent approach of reducing the information (Hilbert space truncation) on the effective particle, without accordingly reducing the information, by the same amount, on the phase θ(x) determining the transformation in Eq. (5). In physics, the approximations must be done with care, and they must be consistent. We start by observing that the two-state system defined in Eq. (6) still has a geometric coordinate, which, however, can assume only two values:xj(with j=L,R), that we can approximately identify with the position of the two minima of the double-well potential. More precisely, and more generally, they are xR=R|x|R,xL=L|x|L.(12) Here, parity symmetry implies xL=−xR. In the following we will use the shorthand R|x|R=a/2. Hence, the operator describing the geometric coordinate can be written as [17] X=(a/2)ˆρz, where ˆρz≡|RR|−|LL|. We observe that the terms proportional to tin the Hamiltonian in Eq. (6)orEq.(8), implies that these can be regarded as nonlocal Hamiltonians, i.e., with an effective potential depending on two distinct coordinates. Nonlocality here comes from the hopping term t=R|ˆ H0|L, which is determined by the interplay of the kinetic energy term and of the potential energy in ˆ H0. It is clear that the consistent and meaningful local gauge transformation corresponds to the following transformation: |ψ=cL|L+cR|R→|ψ=eiqθLcL|L+eiqθRcR|R, (13) where |ψis a generic state in the two-dimensional Hilbert space, and θjare arbitrary real-valued parameters. It is easy to show that the expectation values of ˆ H0are not invariant under the local transformation in Eq. (13). They are only invariant under a uniform phase change: |ψ→ exp(iqθ)|ψ. However, one can introduce in the Hamiltonian field-dependent factors, that compensate the difference in the phase transformation from one point to the other. Specifically, following the general procedure of lattice gauge theory, we can consider the parallel transporter (a unitary finitedimensional matrix), introduced by Wilson [11,19,20] Uxk+a,xk=exp iq xk+a xk dxA(x),(14) where A(x) is the gauge field. After the gauge transformation of the field, A(x)=A(x)+dθ/dx,(15) the transporter then transforms as U xk+a,xk=eiq θ(xk+a)Uxk+a,xke−iq θ(xk),(16) which is now discrete. This property can also be used to implement gauge invariant Hamiltonians in two-state systems. A. Symmetric two-state systems Properly introducing the parallel transporter in Eq. (14) into Eq. (8), we obtain a gauge-invariant two-level model ˆ Hgi 0=−t|RL|UxR,xL+H.c.(17) Gauge invariance can be directly verified: ψ||RL|U xR,xL+H.c.|φ =ψ|(|RL|UxR,xL+H.c.)|φ, where |ψand |φare two generic states in the vector space spanned by |Land |R. By neglecting the spatial variations of the field potential A(x) on the distance a=xR−xL, (dipole approximation). The Hamiltonian in Eq. (17) can be written as ˆ Hgi 0=−t|RL|eiqaA +H.c.(18) Using Eq. (9) and the Euler formula, the Hamiltonian in Eq. (18) can be expressed using the diagonal basis of ˆ H0,as ˆ Hgi 0= 2[ˆσzcos (qaA)+ˆσysin (qaA)],(19) where ˆσy=−i(|AS|−|SA|). Using Eqs. (9) and (12), then qa/2=qA|x|S.(20) This coincides precisely with the transition matrix element of the dipole moment as in Ref. [6]. Considering a quantized field ˆ A, the total light-matter Hamiltonian also contains the free-field contribution, ˆ Hph,so that ˆ H= 2[ˆσzcos (qa ˆ A)+ˆσysin (qa ˆ A)] +ˆ Hph .(21) For the simplest case of a single-mode electromagnetic resonator, the potential can be expanded in terms of the mode photon destruction and creation operators. Around x=0, ˆ A=A0(ˆa+ˆa†), where A0(assumed real) is the zero-pointfluctuation amplitude of the field in the spatial region spanned by the effective particle. We also have ˆ Hph =ωph ˆa†ˆa, where ωph is the resonance frequency of the cavity mode. It can be useful to define the normalized coupling strength parameter [6] η=q(a/2)A0,(22) so that Eq. (21) can be written as ˆ H= 2{ˆσzcos [2η(ˆa+ˆa†)] +ˆσysin [2η(ˆa+ˆa†)]}+ωph ˆa†ˆa. (23) Using the relations ˆρz≡|RR|−|LL|=|AS|+ |SA|≡ˆσx, the Hamiltonian in Eq. (17) can also be expressed as ˆ H=ˆ Uˆ H0ˆ U†,(24) where ˆ U=exp (iqa ˆ Aˆσx/2).(25) 053703-4 GAUGE PRINCIPLE AND GAUGE INVARIANCE IN … PHYSICAL REVIEW A 103, 053703 (2021) Equations (24) and (25) coincide with Eqs. (8) and (9)of Ref. [6], which represents one of our main results. It is also interesting to rewrite the coordinate-dependent phase transformation in Eq. (13) as the application of a unitary operator on the system states. Defining φ=(θR+θL)/2 and θ=(θR−θL)/2, Eq. (13) can be written as |ψ→|ψ=eiqφeiqθˆσx|ψ.(26) This shows that the coordinate-dependent phase change of a generic state of a TLS is equivalent to a global phase change, which produces no effect, plus a rotation in the Bloch sphere, which can be compensated for by introducing a gauge field as in Eq. (24). Notice also that Eq. (26) coincides with the result presented in the first section of the Supplementary Information of Ref. [6], obtained with a different, but equivalent approach. In summary, the method presented here can be regarded as the two-site version (with the additional dipole approximation) of the general method for lattice gauge theories [11], which represents the most advanced and sophisticated tool for describing gauge theories in the presence of truncation of infinite-dimensional Hilbert spaces. These results eliminate any concern about the validity of the results presented in Ref. [6], raised by Ref. [3]. We conclude this subsection by noting that Eq. (17) can be also used, without applying the dipole approximation, to obtain the (multimode) gauge-invariant QRM beyond the dipole approximation. Specifically, without applying the dipole approximation to Eq. (17), after the same steps to obtain Eq. (23), we obtain ˆ H= 2ˆσzcosqxR xL dx ˆ A(x) +ˆσysinqxR xL dx ˆ A(x)+ˆ Hph .(27) One interesting consequence of this result is that it introduces a natural cutoff for the interaction of high-energy modes of the electromagnetic field with a TLS. In particular, owing to cancellation effects in the integrals in Eq. (27), the resulting coupling strength between the TLS and the mode goes rapidly to zero when the mode wavelength becomes shorter than a/2=A|x|S. This finding can stimulate further investigations beyond the dipole approximation, without having to introduce a cutoff frequency by hand. It is worth noticing that this derivation of the gaugeinvariant QRM does not require the introduction of an externally controlled two-site lattice spacing, in contrast to general lattice gauge theories. In the present case, the effective spacing abetween the two sites is only determined by the transition matrix element of the position operator between the two lowest-energy states of the effective particle a=2A|x|S, which in turn determines the dipole moment of the transition qa/2. B. Asymmetric two-state systems The results in this section can be directly generalized to also address the case of a potential of the effective particle which does not display inversion symmetry. It was shown that the interaction (in the USC and DSC limit) of these TLSs (without inversion symmetry), with photons in resonators, can lead to a number of interesting phenomena [21–27]. In this case, Eq. (11) provides the bare TLS Hamiltonian. Note that the first term in Eq. (11) is not affected by the two-state local phase transformation in Eq. (13), hence the gauge-invariant version of Eq. (11) can be written as ˆ Hgi 0= 2ˆρz− 2(|RL|UxR,xL+H.c.),(28) which, in the dipole approximation, reads ˆ Hgi 0= 2ˆρz− 2|RL|eiqaA +H.c..(29) This can be expressed as ˆ Hgi 0= 2ˆρz− 2[ˆρxcos (qaA)−ˆρysin (qaA)],(30) which can also be written in the more compact form ˆ Hgi 0=ˆ Uˆ H0ˆ U†,(31) where ˆ U=exp [iqaAˆρz/2].(32) Equations (31) and (32) represent the minimal coupling replacement for TLSs, derived directly from the fundamental gauge principle. We observe that the operator ˆ X=aˆρz/2 represents the geometrical-coordinate operator for the two-state system, with eigenvalues ±a/2. The Hamiltonian in Eq. (30) can be directly generalized beyond the dipole approximation with the following replacement: aA →a/2 −a/2 dxA(x).(33) Considering a single-mode electromagnetic resonator, the total Hamiltonian becomes ˆ H=ωph ˆa†ˆa+ 2ˆρz− 2{ˆρxcos [2η(ˆa+ˆa†)] −ˆρysin [2η(ˆa+ˆa†)]}.(34) Since the operator ˆ Xis the position operator in the twostate space, the unitary operator ˆ U†=ˆ Talso corresponds to the operator which implements the PZW unitary transformation [28], leading to the dipole-gauge representation ˆ Hd=ˆ U†ˆ Hˆ U=ωph ˆa†ˆa+ 2ˆρz− 2ˆρx −iηωph(ˆa−ˆa†)ˆρz+η2ˆ I,(35) where we used ˆρ2 z=ˆ I, with Ithe identity operator for the two-state system. Note that ˆ Hdcoincides with the Hamiltonian describing a flux qubit interacting with an LC oscillator [25]. Since the Hamiltonians in Eqs. (34) and (35) are related by a gauge (unitary) transformation, their eigenvalues Ejcoincide. Figure 3displays their energy spectra, defined as (Ej− E0)/ωph as a function of the normalized coupling strength, where E0is the ground-state energy. The spectra were obtained at zero detuning: ωq=ωph. In particular, Fig. 3(a) displays the energy spectrum in the absence of symmetry breaking (=0), namely that of the standard QRM [6]. 053703-5 SALVATORE SAVASTA et al. PHYSICAL REVIEW A 103, 053703 (2021) 0123 0 1 2 3 4 5 0123 0 1 2 3 4 5 Normalized light–matter coupling Normalized Energy Levels (a) (b) FIG. 3. Normalized energy spectra of the QRMs for the (a) symmetric and (b) asymmetric TLS. Energy differences (Ej−E0)/ωph (Ejare energy eigenvalues) as a function of the normalized coupling strength η, calculated at zero detuning: ωq=ωph. Panel (a) displays the spectrum for the standard QRM (symmetric TLS), while the spectrum for the QRM for the asymmetric TLS is shown in (b). Figure 3(b) is obtained using =0.2ωph. Such a symmetry breaking gives rise to a number of interesting features. In particular, we observe that the level crossings present in Fig. 3(a) convert into avoided-level crossings. The appearance of these splittings is a signature of the hybridization between states with different parity. Note that, in the Jaynes Cummings model (the QRM after the rotating wave approximation) the number of excitations is conserved. In the QRM, owing to the counterrotating terms, such a number is no longer conserved. However, its parity remains a good quantum number [4]. For = 0, also this symmetry is removed. A peculiar feature of the QRM consists of energy levels Ej−E0which tend to become flat and “two-fold degenerate” in the extreme coupling limit. Figure 3(b) shows that this degeneracy is removed and in the limit η→∞, it is converted into a gap exactly equal to . V. DISCUSSION AND CONCLUSIONS This work discussed the connection between the QRM, an essential and widespread model in quantum optics, and lattice gauge theory, and shows that the results in Ref. [6], obtained with a completely different approach, fit well in the great tradition of lattice gauge theories opened by Wilson [11]. Lattice-gauge theories constitute a powerful reference example, where it is possible and also vital to maintain the gauge invariance of a theory after reducing the infinite amount of information associated to a continuous coordinate [11]. The gauge principle is based on the concept of local phase change of the system wave function. The approach in Ref. [6], using the energy eigenstates as a basis, does not provide, in a direct way, such a locality concept. However, this work shows how the results in Ref. [6] can be derived using the key concept of local phase change of the system wave function. Of course, measurements (as, e.g., experimental clicks or transmission amplitudes) yield data that do not care about gauge representations. Therefore, if approximations are applied consistently, as theorists, we should provide numbers which are not affected by gauge transformations. Theorists can use different representations, but all of them must be consistent. This paper, as well as Ref. [6], shows that this is the case even under extreme conditions, e.g., in the USC and DSC regimes of light-matter interaction. To demonstrate the versatility of the prescription used here, we also presented the gauge-invariant formulation in the case of asymmetric two-state systems interacting with the electromagnetic field, extending the results in Ref. [6] to the case of asymmetric two-state systems interacting with the electromagnetic field. The corresponding energy spectrum, for a single-mode field, as a function of the normalized coupling strength, shows the impact of breaking parity symmetry in the USC regime. In addition, the method used here allowed us to obtain the gauge-invariant QRM beyond the dipole approximation. We highlight that the double-well potential, considered in this work, is not essential for the derivation of the QRMs derived here. This potential has been used just as an example, for two main reasons: (i) it allows us to visualize in a clear way the concept of a two-valued geometric coordinate and the relationship of the present approach with lattice gauge theory; (ii) because the double-well potential can display very high anharmonicity, which is essential to keep valid the two-level approximation in the presence of deep ultrastrong coupling. The general approach for potentials different from the doublewell consists of (i) writing the position operator in the two energy eigenstates basis and diagonalazing it; (ii) writing the bare atomic Hamiltonian in the basis of the eigenstates of the position operator, so that Eq. (11) is obtained; and (iii) applying the parallel transporter in Eq. (14) to the nondiagonal elements of the Hamiltonian in Eq. (11). The QRMs developed here and in Ref. [6] are gaugeinvariant and satisfy the gauge principle for truncated models. Naturally, they are able to reproduce the energy levels of the full (nottruncated) model [6] only when the two-level approximation is a valid option. In quantum optics, it is known that the two-level approximation can be applied only when the higher energy levels of the system provides a negligible contribution to the resonant dynamics of the light-matter system [29]. As discussed in Ref. [6], it is useful to define an anharmoncity coefficient of the matter system: μ= (ω2,1−ω1,0)/ω1,0, where ωi,j=ωi−ωj, and ωiindicates the eigenfrequency of the ith energy level of the matter system. The two-level approximation works fine if the normalized coupling ηis significantly smaller than μ. In other words, additional transitions beyond ω1,0in the system can be neglected only if |ωi,j−ωc||gi,j|∼η|ωi,j|, where gi,jis the coupling rate of the transition at ωi,jwith cavity photons. Of course, using the QRM for systems when η∼μcan provide 053703-6 GAUGE PRINCIPLE AND GAUGE INVARIANCE IN … PHYSICAL REVIEW A 103, 053703 (2021) wrong or unphysical results [2]. In this case we may expect that, taking into account a few additional levels and applying the gauge-invariant framework [6], can restore the agreement with the full model, at least for the lowest-energy levels of the system. Our results on the connection between the QRM and lattice-gauge theory can hopefully stimulate the development of lattice gauge models for the study of USC cavity QED in oneand two-dimensional systems, as well as of interacting electron systems [30–33]. It would also be interesting to apply lattice-gauge theory to investigate cavity QED systems beyond the dipole approximation [34]. ACKNOWLEDGMENTS F.N. is supported in part by: Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP) program, the Moonshot R&D Grant No. JPMJMS2061, and the Centers of Research Excellence in Science and Technology (CREST) Grant No. JPMJCR1676], the Japan Society for the Promotion of Science (JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134 and the JSPS – RFBR Grant No. JPJSBP120194828], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. S.H. acknowledges funding from the Canadian Foundation for Innovation and the Natural Sciences and Engineering Research Council of Canada. S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF1910065). [1] D. 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