Abstract
Ultrastrongcoupling between twolevel systems and radiation is important for both fundamental and applied quantum electrodynamics (QED). Such regimes are identified by the breakdown of the rotatingwave approximation, which applied to the quantum Rabi model (QRM) yields the apparently less fundamental JaynesCummings model (JCM). We show that when truncating the material system to two levels, each gauge gives a different description whose predictions vary significantly for ultrastrongcoupling. QRMs are obtained through specific gauge choices, but so too is a JCM without needing the rotatingwave approximation. Analysing a circuit QED setup, we find that this JCM provides more accurate predictions than the QRM for the ground state, and often for the first excited state as well. Thus, JaynesCummings physics is not restricted to lightmatter coupling below the ultrastrong limit. Among the many implications is that the system’s ground state is not necessarily highly entangled, which is usually considered a hallmark of ultrastrongcoupling.
Introduction
Progress in experimental cavity and circuit quantum electrodynamics has granted unprecedented access to the strong, ultrastrong and deep–strong light–matter coupling regimes^{1,2,3,4,5,6,7,8,9,10,11,12,13,14}. Recently, circuit QED experiments involving a single LCoscillator mode with frequency ω coupled to a flux qubit with transition frequency ω_{m} have realised coupling g as large as g/ω ranging from 0.72 to 1.34, with \(g/{\omega} _{\mathrm{m}} \gg 1\)^{11}. Such regimes offer a new testing ground for the foundations of quantum theory, and offer opportunities for the development of quantum technologies.
Our interest is in material systems that possess anharmonic spectra, and which are commonly truncated to two levels (qubits). In this case, conventional forms of light–matter interaction Hamiltonian yield the socalled quantum Rabi model (QRM) that consists of a linear interaction between the radiation mode and the qubit. Performing the rotatingwave approximation (RWA) then yields the celebrated Jaynes–Cummings model (JCM), which owing to its simple exact solution, has provided deep physical understanding in a wide range of contexts^{15,16,17,18}. In the ultrastrongcoupling regime where 0.1 < g/ω < 1, the RWA is no longer valid^{3,5,11} and it is therefore widely believed that the JCM breaks down. For this reason, the QRM is considered indispensible and has found myriad applications in condensed matter, quantum optics and quantum information theory^{19,20,21,22,23,24}. A disadvantage of the QRM when compared with the JCM is the lack of any simple solution, which makes its physical interpretation more difficult^{25}. Despite this difficulty, the QRM is known to possess some markedly different physical features compared with the JCM. For example, the JCM predicts that there is no atom–photon entanglement within the ground state, while the ground state of the QRM is highly entangled within the ultrastrongcoupling regime^{26}.
It was noted some time ago in the context of scattering theory that retaining only a subset of states raises the prospect of gauge noninvariance^{27,28,29,30,31,32,33,34,35}. Yet, when the coupling is weak, it is possible to elicit gauge invariance through systematically accounting for the effects of the truncation^{36}, and the choice of gauge has no practical implications for the qualitative physical conclusions. It has also been shown in the traditional setting of a single atom weakly coupled to a (multimode) radiation reservior, that numberconserving (JCMtype) light–matter interaction Hamiltonians can be obtained without recourse to the RWA^{37,38,39}.
Very recently, the validity of twolevel truncations performed in the Coulomb and multipolar gauges has been assessed^{40,41}. The multipolar gauge was found to offer a more accurate QRM than the Coulomb gauge for the particular systems and regimes considered there. This was directly attributed to differences in the corresponding forms of coupling. Specifically, contributions of material levels above the first two were found to be suppressed for dipolemoment matrix elements that feature in the multipolargauge coupling, but not for canonical momentum matrix elements that feature in the Coulombgauge coupling.
While Refs. ^{40,41} provide valuable comparisons of the Coulomb and multipolar gauges, we employ a more general approach whereby gauge freedom is encoded into the value of a single real parameter. Our methods are applicable to arbitrary systems in QED, including both cavity and circuit QED implementations. We show that corresponding to a given unique light–matter Hamiltonian, there is a continuous infinity of nonequivalent twolevel models, each of which corresponds to a different choice of gauge. We thereby obtain the most general possible Hermitian interaction operator that is bilinear in qubit and oscillator raising and lowering operators, and which is therefore more general than the JCM or QRM forms. We show that a specific choice of gauge, which we call the JC gauge, yields a JCM without any need for the RWA. There are also two gauges that yield distinct QRMs. To understand the implications of our approach within the ultrastrongcoupling regime, we consider in detail a fluxoniumLCoscillator circuit QED system. We show that the breakdown of the RWA in strong and ultrastrongcoupling regimes does not imply a breakdown of the JCM.
Results
Our key findings are as follows:

(i)
A finitelevel truncation of the matter system ruins the gauge invariance of the theory. In the ultrastrongcoupling regime, the predictions relating to the same physical observable are generally significantly different within any two distinct twolevel models. However, it remains meaningful to ask which truncation produces the best approximation of the unique physics. We are able to determine the accuracy of approximate twolevel models by benchmarking against the unique predictions of the nontruncated (exact and gaugeinvariant) theory.

(ii)
Each twolevel model admits a RWA, which yields a corresponding JCM. The only exception to this occurs in the case of the twolevel model associated with the JC gauge, wherein the counterrotating terms are automatically absent. This JCM is valid far beyond the regime of validity of the RWA as applied to the QRM. It follows that Jaynes–Cummings physics is not necessarily restricted to the weakcoupling regime. In particular, independent of the coupling strength, the ground state is not entangled in the JCgauge twolevel model.

(iii)
When focusing on predictions that involve the lowestlying energy eigenstates of the composite system, the JCgauge twolevel model nearly always outperforms the available QRMs within the regimes of interest. Thus, the JCM can and should be used in various situations previously thought to require use of the QRM.
Light–matter Hamiltonian
We first present our approach within the context of cavity QED. We consider a material system with charge e and mass m described by position and velocity variables r and \(\dot{\mathbf{r}}\), respectively, and with potential energy V(r). The material system interacts with an electromagnetic field described by the gaugeinvariant transverse vector potential A and the associated transverse electric field \( \,{\dot{\mathbf A}} = {\mathbf{E}}_{\mathrm{T}}\). The total vector potential is given by A_{tot} = A + A_{L}, where the longitudinal part \({\bf A}_{\mathrm{L}}\) determines the gauge. In the Coulomb gauge, A_{L} = 0 so A_{tot} = A. The scalar potential A_{0} that then accompanies A is, upto a factor of e, the Coulomb potential. As is wellknown, the Maxwell–Lorentz equations are invariant under a gauge transformation taking the form A_{0} → A_{0} − ∂χ⧸∂t, A → A + ∇χ, where A_{L} = ∇χ and χ is an arbitrary function. Here, we employ a formulation in which this gauge freedom is contained within a single real parameter α, which determines the gauge through the function χ_{α}. This function in turn defines a Lagrangian L_{α} (see Methods). The value α = 0 specifies the Coulomb gauge, while the Poincaré (multipolar) gauge also commonly used in atomic physics is obtained by choosing α = 1.
Moving to the Hamiltonian description canonical momenta are defined in the usual way as \({\mathbf{p}}_{\alpha} = \partial L_{\alpha} /\partial \dot{\mathbf{r}}\) and \({{{\Pi}}}_{\alpha} = \delta L_{\alpha} /\delta \dot{{\mathbf{A}}}\). Quantisation of the system is carried out using Dirac’s method^{42}, full details of which are given in Supplementary Note 1. As in conventional derivations of the QRM and JCM, we restrict our attention to a singlecavity mode. Recently, it was shown that the singlemode approximation can break down in the ultrastrongcoupling regime, and in particular that it eliminates the requisite spatiotemporal structure necessary to elicit causal signal propagation^{43}. However, the singlemode approximation does not result in a breakdown of gauge invariance because gauge transformations remain unitary in the singlemode theory. The generalisation to the multimode case is straightforward^{36,39}, but is not necessary for understanding the implications of gauge freedom within twolevel models. Following conventional derivations, we also make the electric dipole approximation, which similarly does not affect the gauge invariance of the theory.
With these simplifications, the αgauge canonical momenta p_{α}, Π_{α} are related to manifestly gaugeinvariant observables by
where \(\widehat {\mathbf{d}} =  e{\mathbf{r}}\) is the material dipole moment, v denotes the cavity volume, ω denotes the cavity frequency and ε is a cavity unit polarisation vector. The Hamiltonian is the sum of material and cavity energies
where \(E_{{\mathrm{matter}}} = m{\dot{\mathbf{r}}}^2/2 + V(\mathbf{r})\) and \(E_{{\mathrm{cavity}}} = v({\mathbf{E}}_{\mathrm{T}}^2 + \omega ^2{\mathbf{A}}^2)/2\). The Hamiltonian is expressible in terms of the αgauge canonical operators using Eqs. (1) and (2), with the wellknown Coulombgauge (α = 0) and Poincarégauge (α = 1) forms obtained as specific examples.
The energy is a particular example of a gaugeinvariant observable, which in Eq. (3) has been expressed as a function of the elementary gaugeinvariant observables \({\mathbf{x}} = \{ {\mathbf{r}}, {\dot{\mathbf{r}}} ,{\mathbf{A}},{\mathbf{E}}_{\mathrm{T}}\}\). More generally, when written in terms of x, any observable O possesses a unique functional form O ≡ O(x). The theory is gaugeinvariant in that the predictions concerning any gaugeinvariant observable can be calculated using any gauge, and these predictions are unique. The canonical momenta {p_{α}, Π_{α}} are, however, manifestly gaugedependent in that for each different α, they constitute different functions of the gaugeinvariant observables x. When written in terms of canonical operators \({\bf y}_{\alpha}\) = {r, p_{α}, A, Π_{α}}, an observable O generally possesses an αdependent functional form O = o^{α}(\({\bf y}_{\alpha}\)). The canonical operators belonging to fixed gauges α and α′ are related using the unitary gaugefixing transformation \(R_{\alpha \alpha{\prime} } = {\mathrm{exp}}[{\mathrm{i}}(\alpha  \alpha{\prime} )\widehat {\mathbf{d}} \cdot {\mathbf{A}}]\). This implies that distinct functional forms o^{α} and \(o^{\alpha{\prime}}\) of the observable O are related according to
This equation expresses the uniqueness of physical observables independent of the chosen gauge.
The unitarity of the gauge transformation R_{αα′} also ensures that in all gauges, the canonical operators satisfy the canonical commutation relations [r_{i}, p_{α,j}] = iδ_{ij}, \([A_i, {\Pi}_{\alpha ,j}] = {\mathrm{i}}{\mathbf{\varepsilon} }_i {\mathbf{\varepsilon}}_j/v\) with all the remaining commutators between canonical operators being zero. These relations allow us to decompose the state space \({\cal H}\) of the light–matter system into αdependent matter and cavity state spaces \({\cal H}_{\mathrm{m}}^{\alpha}\) and \({\cal H}_{\mathrm{c}}^{\alpha}\) such that \({\cal H} = {\cal H}_{\mathrm{m}}^{\alpha} \otimes {\cal H}_{\mathrm{c}}^{\alpha}\). The eigenstates of the canonical operators r, p_{α} provide a basis for the material space \({\cal H}_{\mathrm{m}}^{\alpha}\) while the eigenstates of the canonical operators A, Π_{α} provide a basis for the cavity space \({\cal H}_{\mathrm{c}}^{\alpha}\). It is not possible to define gaugeinvariant (αindependent) light and matter quantum subsystem state spaces directly in terms of the gaugeinvariant observables x, because Eqs. (1) and (2) along with the canonical commutation relations imply that \([m{\dot{r}}_{i},E_{{\mathrm{T}},j}]\) = −ie ε_{i}ε_{j}/v ≠ 0.
The present theory yields unique physical predictions despite the αdependence of the quantum subsystems. This is because the representation of an observable by operators is unique, as expressed by Eq. (4), which implies that the average of an observable O in the state \(\left\psi\right\rangle\) is unambiguously \(\left\langle \psi \right O \left\psi\right\rangle\). The αdependence of the quantum subsystems is, however, an important feature of the theory, which is made transparent within our formulation. An approximation performed on one of the quantum subsystems will constitute a different approximation in each gauge, and may ruin the gauge invariance of the theory.
Nonequivalent twolevel models
In conventional approaches, a gauge is chosen at the outset and the Hamiltonian is partitioned into matter and cavity bare energies plus an interaction part. Here, we follow this same procedure, but with the important exception that the gauge is left open rather than fixed. This is achieved through substitution of Eqs. (1) and (2) into Eq. (3), which casts the total Hamiltonian in the form \(H = H_{\mathrm{m}}^{\alpha} ({\mathbf{r}},{\mathbf{p}}_{\alpha} ) \otimes I_{\mathrm{c}}^{\alpha} + I_{\mathrm{m}}^{\alpha} \otimes H_{\mathrm{c}}^{\alpha} ({\mathbf{A}},{{{\Pi}}}_{\alpha} ) + V^{\alpha} ({\mathbf{y}}_{\alpha} )\). Here, \(I_{\mathrm{m}}^{\alpha}\) and \(I_{\mathrm{c}}^{\alpha}\) are the identity operators in \({\cal H}_{\mathrm{m}}^{\alpha}\) and \({\cal H}_{\mathrm{c}}^{\alpha}\), respectively, \(H_{\mathrm{m}}^{\alpha}\) and \(H_{\mathrm{c}}^{\alpha}\) are material and cavity bare energies in \({\cal H}_{\mathrm{m}}^{\alpha}\) and \({\cal H}_{\mathrm{c}}^{\alpha}\), respectively, and V^{α} denotes the interaction Hamiltonian. The explicit forms of \(H_{\mathrm{m}}^{\alpha} , H_{\mathrm{c}}^{\alpha}\) and V^{α} are given in Eqs. (9)–(11) in Methods.
One of the most useful and widespread approximations in light–matter theory is a twolevel truncation of the material system, whereby only the first two eigenstates \(\left\epsilon _0^{\alpha} \right\rangle ,\, \left \epsilon _1^{\alpha} \right\rangle\) of the material bare energy \(H_{\mathrm{m}}^{\alpha}\) are retained. Our approach reveals that this procedure ruins the uniqueness of physical predictions that results from Eq. (4). Using the projection \(P^{\alpha} = \left\epsilon _0^{\alpha} \right\rangle \left\langle \epsilon _0^{\alpha} \right + \left\epsilon _1^{\alpha} \right\rangle \left\langle \epsilon _1^{\alpha} \right\), we obtain the αgauge twolevel model Hamiltonian
where \(u_{\alpha} ^ \pm = \pm ({\mathbf{d}} \cdot {\mathbf{\varepsilon}} )[\omega _{\alpha} \alpha \mp \omega _{\mathrm{m}}(1  \alpha )]/\sqrt {2\omega _{\alpha} v}\) and Δ_{α} = \(\epsilon_0\) + α^{2}(d ⋅ ε)^{2}/2v is an αdependent zeropoint shift. The transition dipole moment \({\mathbf{d}} = \left\langle \epsilon _1^{\alpha} \right  e{\mathbf{r}}\left\epsilon _0^{\alpha} \right\rangle\), which is assumed to be real, is αindependent, because r commutes with R_{αα′}. The material Hamiltonian’s eigenvalues \({\epsilon}_0\) and \({\epsilon}_1\) = ω_{m} + \(\epsilon_0\) corresponding to material states \(\left {\epsilon _0^{\alpha} } \right\rangle\) and \(\left {\epsilon_1^{\alpha} } \right\rangle\), respectively, are also αindependent because \(H_{\mathrm{m}}^{\alpha} = R_{\alpha \alpha{\prime} }H_{\mathrm{m}}^{\alpha{\prime} }R_{\alpha \alpha{\prime} }^{  1}\). The complete derivation of Eq. (5) is given in Methods.
An important topic relating to twolevel models and the choice of gauge in light–matter physics concerns the occurrence or otherwise of a superradiant phase transition in the Dicke model at strong coupling^{40,44,45,46,47,48}. A precursor already occurs in the QRM, whereby beyond a critical coupling point, an exponential closure of the first transition energy occurs^{49,50,51}. We note that in Eq. (5), counterrotating and numberconserving interactions generally have different coupling strengths, and a strict bound cannot be given for either coupling independent of the material potential, except if α = 0. It follows that the standard “nogo theorem” concerning the groundstate instability of a single dipole, holds in general only in the Coulomb gauge^{41,44,45,46,47}. An arbitrarygauge analysis of this topic is important, but lies beyond the scope of this article and will be discussed elsewhere.
We are concerned with the αdependence of predictions obtained when using the Hamiltonian in Eq. (5). This Hamiltonian has neither JC nor Rabi form, because \(u_{\alpha} ^ + \, \ne\, u_{\alpha} ^  \) and \(u_{\alpha} ^ +\, \ne\, 0\) except when particular values of α are chosen. Specifically, two distinct QRMs are obtained for the choices α = 0 and α = 1, which are nothing but the Coulomb and Poincarégauge QRMs frequently encountered in quantum optics. On the other hand, by choosing α = α_{JC}, which solves the coupled equations α_{JC}(ω_{m} + ω_{JC}) = ω_{m} and \(\omega _{{\mathrm{JC}}}^2 = \omega ^2 + e^2(1  \alpha _{{\mathrm{JC}}})^2/mv\), we obtain \(u_{{\mathrm{JC}}}^ + \equiv 0\) and \(u_{{\mathrm{JC}}}^  =  2({\mathbf{d}} \cdot {\mathbf{\varepsilon}} )\omega _{\mathrm{m}}\sqrt {\omega _{{\mathrm{JC}}}} /[\sqrt {2v} (\omega _{{\mathrm{JC}}} + \omega _{\mathrm{m}})]\). This choice therefore yields a JC Hamiltonian without any need for the RWA. The JCM derived in this way possesses the same advantage of exact solvability as conventional JCMs obtained as RWAs of the Coulomb and Poincarégauge QRMs. However, the states \(\left\epsilon_{\mathrm {JC}}\right\rangle\), operators \(\sigma _{{\mathrm{JC}}}^ {\pm}\) and parameters \(u_{{\mathrm{JC}}}^ \), \(\omega_{\mathrm {JC}}\) are different from their counterparts within conventional JCMs. In particular, the renormalised cavity frequency \({\omega}_{\mathrm {JC}}\) together with the zeropoint shift Δ_{JC} yields a groundstate energy that is a nonconstant function of the Coulombgauge and multipolargauge QRM coupling parameters.
Having derived an expression for the energy, most properties of practical interest can now be calculated using the twolevel model associated with any gauge. This includes atomic populations and coherences, as well as various cavity properties, such as photon number. It is, however, possible to go further by defining the twolevel representation of any additional observable of interest O as \(O_2^{\alpha} = P^{\alpha} OP^{\alpha}\). Restricting the state space \({\cal H}_{\mathrm{m}}^{\alpha}\) to the twodimensional subspace spanned by the eigenstates \(\left {\epsilon _0^{\alpha} } \right\rangle , \left {\epsilon _1^{\alpha} } \right\rangle\) then completes the construction of the twolevel model.
Twolevel models corresponding to distinct gauges α and α′ must be distinguished, because when α ≠ α′, the projection P^{α} involves all eigenstates of \(H_{\mathrm{m}}^{\alpha{\prime}}\), and similarly P^{α′} involves all eigenstates of \(H_{\mathrm{m}}^{\alpha}\). This is because the gauge transformation does not have product form; R_{αα′} ≠ R_{m} ⊗ R_{c}. A pictorial representation of the relationship between different gauges and their associated twolevel models is given in Fig. 1. After a twolevel truncation, the uniqueness of the representation of observables expressed by Eq. (4) no longer holds, that is, \(O_2^{\alpha}\, \ne\, O_2^{\alpha{\prime} }\) when α ≠ α′. Distinct twolevel models will therefore give different predictions for the same physical quantity. An observable of particular importance is the energy represented by the Hamiltonian, which we focus on hereafter. There is generally no simple relation between distinct twolevel model Hamiltonians \(H_2^{\alpha}\) and \(H_2^{\alpha{\prime} }\), when α ≠ α′. In fact, it was noted some time ago that twolevel models associated with different gauges can give different results even in the weakcoupling regime^{52}. However, provided that the twolevel modification of the operator algebra is accounted for, it can be shown that certain twolevel model predictions are gaugeinvariant upto order d^{2 }^{36}. This is discussed in more detail in Supplementary Note 2. Regardless, one expects predictions of twolevel models corresponding to different gauges to be significantly different when the coupling is sufficiently strong. We show how a comparison of the predictions of different twolevel models can be achieved for an arbitrary observable in Methods. We show further that if the material system is a harmonic oscillator, then it is possible to derive a JCM that is necessarily more accurate than any derivable QRM for finding groundstate averages.
Application to ultrastrong coupling in circuit QED
When considering less artificial systems than a material oscillator, the relative accuracies of twolevel models is more difficult to determine. We now consider an experimentally relevant circuit QED setup consisting of a fluxonium atom coupled to an LC oscillator. The fluxonium is described by the flux variables ϕ, \(\dot \phi\) and the external flux ϕ_{ext}, along with three energy parameters E_{c}, E_{J} and E_{l} that are the capacitive energy, tunnelling Josephson energy and inductive energy, respectively. The external flux ϕ_{ext} = π⧸2e specifies maximum frustration of the atom. The LC oscillator is described by analogous flux variables \(\theta , {\dot \theta}\), with inductance L and capacitance C defining the oscillator frequency \(\omega = 1/\sqrt {LC}\).
In terms of \({\mathbf{x}} = \{ \phi ,{\theta} ,{\dot \phi} ,{\dot \theta \}}\), the functional form of an observable O is unique \(O \equiv O(\mathbf{x})\). On the other hand, different canonical operators \({\mathbf{y}}_{\alpha} = \{ \phi ,\xi _{\alpha} ,\theta _{\alpha} ,\zeta \}\) are related by \(\theta _{\alpha} = R_{0\alpha }^{  1}\theta _0R_{0\alpha }\) and \(\xi _{\alpha} = R_{0\alpha }^{  1}\xi _0R_{0\alpha }\), where \(R_{0\alpha } = {\mathrm{e}}^{{\mathrm{i}}\alpha \zeta \phi }\) is a unitary gauge transformation with α real and dimensionless. Here, ξ_{α} and ζ are canonical momenta conjugate to ϕ and θ_{α}, respectively. The gauge choices α = 0 and α = 1 are called the charge gauge and flux gauge, respectively^{53}. The Hamiltonian H describing the system is derived in Supplementary Note 3 and is given in Methods.
In exactly the same way as for the cavity QED Hamiltonian, the projection P_{α} onto the first two eigenstates \(\left {\epsilon _0^{\alpha} } \right\rangle , \left {\epsilon _1^{\alpha} } \right\rangle\) of the material bare energy \(H_{\mathrm{m}}^{\alpha}\) can be used to obtain an αdependent twolevel model Hamiltonian, which at maximal frustration reads
where \(u_{\alpha} ^ \pm = \varphi [\alpha \omega _{\alpha} \mp (1  \alpha )\omega _{\mathrm{m}}]/\sqrt {2\omega _{\alpha} L}\) and Δ_{α} = \(\epsilon_0\) + α^{2}φ^{2}/2L, in which \(\varphi = \left\langle {\epsilon _1^{\alpha} } \right\phi \left {\epsilon _0^{\alpha} } \right\rangle = \varphi ^ \ast\) and \(\epsilon_0\) denotes the ground energy of \(H_{\mathrm{m}}^{\alpha}\). The twolevel system parameters ω_{m}, φ and \(\epsilon_0\) depend implicitly on E_{c}, E_{J}, E_{l} and ϕ_{ext}. The renormalised cavity frequency is \(\omega _{\alpha} = \omega \sqrt {1 + 2E_{\mathrm{c}}(1  \alpha )^2C/e^2}\). Away from the maximal frustration point, the flux ϕ possesses diagonal matrix elements in the basis \(\left\{ {\left {\epsilon _0^{\alpha} } \right\rangle ,\,\left {\epsilon _1^{\alpha} } \right\rangle } \right\}\), such that \(\sigma _{\alpha} ^ + \sigma _{\alpha} ^ \) and \(\sigma _{\alpha} ^  \sigma _{\alpha} ^ +\) are also linearly coupled to the mode operators \(c_{\alpha} , c_{\alpha} ^\dagger\). In analogy to the cavity QED case, the charge and flux gauges yield distinct Rabi Hamiltonians, but there also exists a value α = α_{JC} = ω_{m}/(ω_{m} + ω_{JC}) such that \(u_{\alpha} ^ + \equiv 0\), which casts the Hamiltonian in JC form.
The ratio δ = ω/ω_{m} in which ω_{m} is taken as the qubit transition at maximal frustration ϕ_{ext} = π⧸2e, specifies the relative qubit–oscillator detuning. To quantify the relative coupling strength, we use the ratio η = g/ω where \(g = \varphi \sqrt {\omega /2L}\). The parameters g and ω are the coupling strength and cavity frequency of the fluxgauge QRM, but we note that the corresponding parameters associated with any other twolevel model could also be used. For different α, the αdependent twolevel truncation yields different predicted behaviour of physical observables as functions of the model parameters δ, η and ϕ_{ext}. In contrast, the exact predictions resulting from the nontruncated model are αindependent (gaugeinvariant).
We begin by determining how the ground energy G and first excited energy E vary with the detuning δ at maximal frustration ϕ_{ext} = π⧸2e and fixed coupling η = 1 (Fig. 2). Regimes with large δ are presently more experimentally relevant^{11,13,14}, yet, unless δ is relatively small (δ < 1), we find that all twolevel models become inaccurate in predicting eigenvalues \(E_{n} \, > \, E\) of the nontruncated Hamiltonian. This can be traced to the occurence of resonances in energy shifts, which occur for large δ (see Supplementary Note 4). Indeed, deviations from the predictions of the QRM have been observed experimentally for such E_{n} within the ultrastrongcoupling regime^{14}.
We focus primarily on the experimentally relevant large δ regime by choosing δ = 5. Other detunings may also be considered and various results for the cases δ = 1 (resonance) and δ = 1/5 are presented in Supplementary Note 5. In Fig. 3a, b, we compare the ground and first excited energies found using various twolevel models with the corresponding gaugeinvariant energies of the exact theory. The ground and excitedlevel shifts are obtained by subtracting the corresponding (bare) eigenenergies of the noninteracting system. At maximal frustration, the shift of the ground state can be identified as the Bloch–Siegert shift^{4}. The first transition shift is the difference between the ground and excited shifts, and is commonly termed the Lamb shift by analogy with atomic hydrogen^{14}. In the RWA, the couplingdependent zeropoint contribution ω_{α}/2 + Δ_{α} in Eq. (6) gives the ground energy. For α ≠ \({\alpha}_{\mathrm {JC}}\), this results in an incorrect expression for the Lamb shift even for weak coupling^{36,54} (see also Supplementary Note 2). It is therefore unsurprising that the flux and chargegauge JCMs are inaccurate in predicting the associated dressed energies within the ultrastrongcoupling regime, as illustrated in Fig. 3a, b. In contrast, for the twolevel model of the JC gauge (α = \({\alpha}_{\mathrm{JC}}\)), the RWA is no longer an approximation. The ground energy \(\omega_{\mathrm{JC}}\)/2 + \(\Delta_{\mathrm{JC}}\), is different from the results of the RWA applied in the α = 0 and α = 1 gauges, and it does lead to the expected expression for the Lamb shift within the weakcoupling regime^{36} (see Supplementary Note 2). Thus, even though the Hamiltonian has Jaynes–Cummings form, it is not evident that like the charge and fluxgauge JCMs, the JCgauge twolevel model will necessarily be inaccurate in predicting dressed energies within the ultrastrongcoupling regime. Indeed, Fig. 3a, b shows that the JCgauge twolevel model is not only more accurate than the flux and chargegauge JCMs, it is also more accurate than the flux and chargegauge QRMs.
To determine which twolevel model yields the most accurate lowest energy eigenstates, we compute the ground and first excitedstate fidelities \(F_G^{\alpha} = \left {\left\langle {G_2^{\alpha} {\big}G} \right\rangle } \right^2\) and \(F_E^{\alpha} = \left {\left\langle {E_2^{\alpha} {\big}E} \right\rangle } \right^2\), where \(G \rangle \,{\mathrm{and}}\, E \rangle\) are the exact ground and first excited eigenstates of the nontruncated Hamiltonian H, while \(\left {G_2^{\alpha} } \right\rangle\) and \(\left {E_2^{\alpha} } \right\rangle\) are the corresponding eigenstates of \(H_2^{\alpha}\). Figure 3c, d shows that the JCgauge model is more accurate than both QRMs, and much more accurate than conventional JCMs, especially in the case of the ground state. Since the JCgauge twolevel model tends to produce a more accurate representation of the lowest two energy states of the system, it is natural to suppose that it will generally be more accurate than the QRM in predicting observable averages in these states. This is verified for the cases of groundstate photon number averages in Supplementary Note 6.
To link with recent experiments in which circuit properties are measured for varying external flux ϕ_{ext}, Fig. 4 shows the behaviour with ϕ_{ext} of the lowest dressed energies when η = 1/2. The JCgauge again yields the most accurate twolevel model (Fig. 4a, b) despite the clear breakdown of the RWA (Fig. 4c, d). It follows that Jaynes–Cummings physics is not synonymous with the RWA, and that a departure from Jaynes–Cummings physics is not implied within the ultrastrongcoupling regime. For larger η, twolevel models become increasingly inaccurate, though the JC gauge continues to give the best agreement with exact energies even within the deep–strong coupling regime (see Supplementary Note 5).
Discussion
The behaviour shown in Figs. 2–4 can be understood by deriving an effective Hamiltonian valid sufficiently far from resonance (dispersive regime)^{55}, details of which are given in Supplementary Note 4. In this context, let us first consider the flux gauge, wherein the light and matter systems are coupled through the material position operator ϕ. The matrix elements of this operator between material states \(\left {\epsilon _n^1} \right\rangle\) are largest between adjacent levels n, n ± 1^{41} (see Supplementary Note 4). Thus, provided higher material levels are sufficiently separated from the lowest two, the coupling to them can be neglected, unless the light–matter coupling η is very large, or δ is large enough that several material energies lie within the first oscillator band ω. For such large δ, contributions of energy denominators in the effective Hamiltonian become large due to the occurence of resonances \(\epsilon _{ni}\sim \omega\), \(\epsilon_{ni} = \epsilon_{n}\, −\, \epsilon_{i}\), i = 0, 1, n > 1 (see Supplementary Note 4). The fluxgauge QRM is therefore qualitatively accurate if δ and η are sufficiently small. This includes accurately predicting higher system energy levels E_{n} > E as well as the first two levels G and E^{41} (see Supplementary Note 5).
In the charge gauge, the light–matter coupling occurs via the material canonical momentum ξ_{0}, for which matrix elements involving higher levels are not suppressed (see Supplementary Note 4). Independent of δ, when the coupling is sufficiently large, they cannot generally be neglected even for highly anharmonic material spectra, so the chargegauge QRM generally breaks down^{41}. However, the ratio of the fluxgauge QRMcoupling strength g and the coupling strength \(\tilde g_0\) of the chargegauge QRM, increases as δ increases (see Supplementary Note 4). For large enough δ, the chargegauge coupling is significantly weaker than that of the flux gauge to the extent that for sufficiently large δ and provided η does not become too large, the chargegauge QRM is qualitatively accurate for the ground level G, and occasionally for the first level E (Figs. 2–4).
In the general αgauge, all fluxgauge coupling terms are weighted by α and all chargegauge coupling terms by 1 − α. By tuning α, the αgauge twolevel model smoothly interpolates between the two available QRMs. In particular, the \({\alpha}_{\mathrm{JC}}\)gauge JCM is defined such that the counterrotating terms that give the dominant contribution to deviations between the exact and twolevel model ground states are eliminated (see Supplementary Note 4). This allows us to understand why the \({\alpha}_{\mathrm{JC}}\)gauge JCM accurately represents the ground state across all parameter regimes. As δ and η increase, the \({\alpha}_{\mathrm{JC}}\) gauge becomes predominantly chargelike (see Supplementary Note 4) and like the chargegauge QRM becomes inaccurate for predicting levels E_{n} > E.
Quite generally, twolevel models remain most accurate in predicting the first two system levels G and E. For the lowest such levels of certain circuit QED systems, spectroscopic experimental data have been matched to the predictions of the QRM defined by the Hamiltonian \(h =  ({\mathrm{\Delta }}\sigma ^z + \epsilon \sigma ^x)/2 + \omega a^\dagger a + g\prime \sigma ^x(a + a^\dagger )\), where Δ and \(\epsilon\) are tunnelling and bias parameters, respectively, and g′ denotes the coupling strength^{11,13,14}. In Ref. ^{13} for example, the parameters Δ, g′ and ω are treated as constant fitting parameters, while \(\epsilon\) is externally variable. It is important to note, however, that fitting transitions between eigenenergies of h to experimental data does not preclude the possibility of fitting other models to experimental data.
It is possible to rotate the fluxgauge QRM \(H_2^1\) into the form of h, but upon doing so, each of Δ, \(\epsilon\) and g′ are found to be nontrivial functions of ϕ_{ext}. In particular, for the fluxoniumLC system we consider, g′ and Δ do not remain constant while varying \(\epsilon\) by varying ϕ_{ext}. Moreover, the αgauge twolevel model cannot be uniquely specified in terms of the parameters of h. Whenever ϕ_{ext} ≠ π/2e, these properties obstruct meaningful comparison between our results and experimental results of the kind found for example in Ref. ^{13}.
More relevant experimental results for the system we consider are given in Ref. ^{2}, where spectroscopic data were found to agree well with the nontruncated fluxoniumLC Hamiltonian H of Eq. (13). There, the fluxonium energies E_{c}, E_{l} and E_{J} were treated as fitting parameters. Our results show that using such a fitting procedure, the JCgauge twolevel model would offer better agreement with experimental data than the QRM, at least for the lowest two levels G and E. This occurs over the full range of δ shown in Fig. 2 with only a few exceptions in the case of the excited state E when δ is small (see Supplementary Note 5).
The results presented here open up multiple avenues for further investigation. For example, our more general form of twolevel model in which the gauge is left open is capable (albeit fortuitously) of exactly predicting a given energy value, but it remains to be understood in more detail. A comprehensive comparison of different methods for deriving twolevel model descriptions is also yet to be performed.
An investigation of the implications of the arbitrarygauge formalism for the occurence of phase transitions in multidipole systems constitutes further important work. The dependence on arbitrarygauge parameters of weaker truncations such as threelevel atomic models remains to be investigated as does the generalisation to multimode situations for structured photonic environments. We note that issues with the singlemode approximation have been recognised and discussed elsewhere^{5,43}, but that this approximation does not result in a breakdown of gauge invariance and does not therefore affect the results reported here. Within exact (nontruncated) models determining the dependence on the gauge parameter of light–matter entanglement, as well as averages of local light and matter observables such as photon number, is of experimental relevance and is important for applications. This too will be investigated in further work.
Methods
Lagrangians in different gauges
The Coulombgauge Lagrangian is denoted L_{0} and is given in Supplementary Note 1. More generally, the αgauge Lagrangian yielding the same correct equations of motion as L_{0} is L_{α} = L_{0} − dχ_{α}/dt, where the function χ_{α} is defined as
Here, P_{mult} denotes the usual multipolar transverse polarisation field. Latin indices denote spatial components and repeated indices are summed. This χ_{α} is the generator of the unitary Power–Zienau–Woolley transformation, multiplied by α. The αdependence of the Lagrangian can be understood as the underlying cause of the αdependence of the canonical momenta \({\mathbf{p}}_{\alpha} = \partial L_{\alpha} /\partial \dot{\mathbf{r}}\) and \({{{\Pi}}}_{\alpha} = \delta L_{\alpha} /\delta \dot{\mathbf{A}}\).
Derivation of cavity QED twolevel model Hamiltonian
Substituting Eqs. (1) and (2) into Eq. (3) yields the Hamiltonian written in terms of canonical operators \({\bf y}_{\alpha}\) as \(H = H_{\mathrm{m}}^{\alpha} + H_{\mathrm{c}}^{\alpha} + V^{\alpha}\) where
The Hamiltonian has a hybrid form between the Coulomb and multipolar gauges. Coulombgauge coupling terms are weighted by 1 − α while multipolargauge coupling terms are weighted by α. The interaction includes the quadratic “A^{2}” and “\(\widehat {\mathbf{d}}^2\)” selfenergy terms in addition to the linear coupling terms “p_{α}⋅A” and “\(\widehat {\mathbf{d}} \cdot {{{\Pi}}}_{\alpha}\)”. This approach is easily adapted to describe multimode fields and more than one dipole^{36}.
The first two eigenstates of the material bare energy \(H_{\mathrm{m}}^{\alpha}\) are denoted \(\left {\epsilon _0^{\alpha} } \right\rangle\) and \(\left {\epsilon _1^{\alpha} } \right\rangle\), and the projection onto this subspace is \(P^{\alpha} = \left {\epsilon _0^{\alpha} } \right\rangle \left\langle {\epsilon _0^{\alpha} } \right + \left {\epsilon _1^{\alpha} } \right\rangle \left\langle {\epsilon _1^{\alpha} } \right\). The operator \(H_{\mathrm{m}}^{\alpha}\) admits the twolevel truncation \(H_{{\mathrm{m}},2}^{\alpha} = P^{\alpha} H_{\mathrm{m}}^{\alpha} P^{\alpha} = \epsilon _0 + \omega _{\mathrm{m}}^{\alpha} \sigma _{\alpha} ^ + \sigma _{\alpha} ^ \), where ω_{m} = \(\epsilon_1\) − \(\epsilon_0\), \(\sigma _{\alpha} ^ + = \left {\epsilon _1^{\alpha} } \right\rangle \left\langle {\epsilon _0^{\alpha} } \right\) and \(\sigma _{\alpha} ^  = \left {\epsilon _0^{\alpha} } \right\rangle \left\langle {\epsilon _1^{\alpha} } \right\). The eigenvalues \(\epsilon_0\) and \(\epsilon_1\) = ω_{m} + \(\epsilon_0\) corresponding to \(\left \epsilon _0^{\alpha}\right\rangle\) and \(\left \epsilon _1^{\alpha} \right\rangle\), respectively, are αindependent because \(H_{\mathrm{m}}^{\alpha} = R_{\alpha \alpha\prime}H_{\mathrm{m}}^{\alpha\prime}R_{\alpha \alpha\prime}^{  1}\). In practice, twolevel model Hamiltonians are found by first defining the interaction Hamiltonian as \(V_2^{\alpha} = V^{\alpha} (P^{\alpha} {\mathbf{y}}_{\alpha} P^{\alpha} )\) and then combining this interaction with the bare energies to obtain the total Hamiltonian
If the interaction Hamiltonian V^{α} is linear in r and \({\bf p}_{\alpha}\) then the twolevel model Hamiltonian can also be written \(H_2^{\alpha} = P^{\alpha} HP^{\alpha}\). This is not the case for H in Eq. (11) due to the “\(\widehat {\mathbf{d}}^2\)” term, which demonstrates the availability of different methods for deriving truncated models. Here, we adopt the approach most frequently encountered in the literature, and outline other methods in Supplementary Note 2.
We can now define an arbitrarygauge twolevel model associated with the Hamiltonian H in Eq. (11) by using the definition (12). The projection P^{α} does not alter the “A^{2}” and \(H_{\mathrm{c}}^{\alpha}\) terms of Eq. (11), because these terms depend on the cavity canonical operators only. Combining them gives the renormalised cavity energy \(H_{\mathrm{c}}^{\alpha} + e^2/2m(1  \alpha )^2{\mathbf{A}}^2 = \omega _{\alpha} (c_{\alpha} ^\dagger c_{\alpha} + 1/2)\) with renormalised cavity frequency \(\omega _{\alpha} = \omega \sqrt {1 + e^2(1  \alpha )^2/mv\omega ^2}\). The \(c_{\alpha} , c_{\alpha} ^\dagger\) are cavity ladder operators of the renormalised energy satisfying \([c_{\alpha} ,c_{\alpha} ^\dagger ] = 1\). In terms of these operators, the Hamiltonian \(H_2^{\alpha}\) defined by Eq. (12) is given by Eq. (5).
Method for comparing twolevel model predictions
A comparison of the predictions that different twolevel models yield for an arbitrary observable requires that we determine how a given physical state is represented within each twolevel model. To this end, consider an observable A with the property that both the exact representation A and the twolevel model representation \(A_2^{\alpha}\) possess nondegenerate discrete spectra. The eigenvalues a_{n} of A and \(a_{2,n}^{\alpha}\) of \(A_2^{\alpha}\) are in onetoone correspondence such that the eigenstates \(\leftA_n\right\rangle\) and \(\left {A_{2,n}^{\alpha} } \right\rangle\) can be assumed to represent the same physical state. An arbitrary physical state can then be constructed via linear combination; the physical state \(\left \psi \right\rangle = \sum\nolimits_n \psi _n\left {A_n} \right\rangle , \mathop {\sum}\nolimits_n \left\psi _n\right^2 = 1\) within the exact theory, is represented within the αgauge twolevel model by \(\left {\psi _2^{\alpha} } \right\rangle = \mathop {\sum}\nolimits_n \psi _n\left {A_{2,n}^{\alpha} } \right\rangle\). A natural choice of observable A for the purpose of representing states is the energy A = H, which we consider in Results.
The most accurate twolevel model for the purpose of predicting the average \(\left\langle \psi \right O \left\psi\right\rangle\) of an arbitrary observable O, which may or may not equal A, is found by selecting the gauge α for which the difference between the exact and twolevel model prediction, \(z^{\alpha} (O,\psi ) = \left {\left\langle \psi \rightO\left \psi \right\rangle  \left\langle {\psi _2^{\alpha} } \rightO_2^{\alpha} \left {\psi _2^{\alpha} } \right\rangle } \right\), is minimised. Since twolevel models are indispensable practical tools within cavity and circuit QED, it is important to ascertain which twolevel models yield the best approximations of physical averages that are of interest in applications. In Results, the energy is considered, both to represent states (A = H) and as the observable of interest (O = H). The averages \(\left\langle A_n\right O \leftA_n\right\rangle\) are then nothing but the eigenvalues E_{n} of H.
As an example illustrating how the relative accuracies of twolevel models can be determined, let us consider the quantities z^{α}(O, G) where G denotes the ground state of a composite cavitycharge system. The charge is assumed to be confined in all directions except the direction ε of the cavity mode polarisation. In this direction, it oscillates harmonically with bare frequency ω_{m}. In the gauge specified by choosing α = ω_{m}/(ω_{m} + ω), the matter oscillator can be described by ladder operators for which the interaction Hamiltonian takes numberconserving form^{37}. The exact ground state G is then the vacuum state of these modes, and the projection \(P^{\mathrm{JC}}\) onto the first two material levels in this gauge defines a twolevel JCM with ground state \(\left {G_2^{{\mathrm{JC}}}} \right\rangle = P^{{\mathrm{JC}}}\left G \right\rangle = \left G \right\rangle\). It follows that z^{α}(O, G) = 0 for all O with \(O_2^{\alpha} = P^{\alpha} OP^{\alpha}\). Thus, if the material system is a harmonic oscillator, it is possible to derive a JCM that is necessarily more accurate than any derivable QRM for finding groundstate averages.
FluxoniumLC twolevel model Hamiltonian
The derivation in Supplementary Note 3 yields the αgauge fluxoniumLC Hamiltonian
The fluxonium bare energy is defined as
The projection onto the first two eigenstates of this operator is used along with H in Eq. (13) to define a twolevel model Hamiltonian in precisely the same way as in the cavity QED case. The final result is given in Eq. (6).
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by the UK Engineering and Physical Sciences Research Council, grant no. EP/N008154/1. We thank Zach BlundenCodd for useful discussions.
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Correspondence to Adam Stokes or Ahsan Nazir.
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