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 $G$. Namely, the ground-state photon number is a quadratic polynomial in the gauge parameter $λ$, $\langle a^{\dagger} a\rangle_λ$ $=$ $\langle a^{\dagger} a\rangle_{0}$ $+$ $Cλ$ $+$ $Qλ^2$. Here the coefficients are the correlator $C=\langle G\,p\rangle$ of $G$ and the field momentum $p$, and $Q=\langle G^2\rangle$. 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.
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The paper asks what gauge-independent information can be extracted from the gauge-dependent ground-state photon number in truncated single-mode light–matter models. For gauge transformations $U_\lambda=\exp[i\lambda G(a+a^\dagger)]$, it derives the exact identity $n_\lambda=n_0+C\lambda+Q\lambda^2$, where $C=\langle Gp\rangle$, $p=i(a^\dagger-a)$, and $Q=\langle G^2\rangle$. Minimization gives the gauge-orbit invariant $n_{\min}=n_0-C^2/(4Q)$; Cauchy–Schwarz establishes nonnegativity.
For the gauge-corrected quantum Rabi model, $Q=1$. Mapping to the standard Rabi Hamiltonian and applying Hellmann–Feynman relations yields $n_{\min}=\partial_{\omega_c}E_0-\tfrac14(\partial_gE_0)^2$. A coherent-state argument proves strict positivity for nonzero coupling and positive level splitting. The minimizing gauge approaches the Jaynes–Cummings gauge at weak coupling and the Coulomb gauge at strong coupling. Exact-solution calculations and numerical diagonalization evaluate these quantities.
For the gauge-consistent Dicke model, the curvature is the collective dipole second moment. Including dipole self-energy suppresses its growth and prevents superradiance. Holstein–Primakoff analysis gives, at fixed collective coupling in the thermodynamic limit, $\alpha_{\min}=\omega_x/(\omega_c+\omega_x)$ at all couplings and $n_{\min}=(R-\Sigma)^2/(4R\Sigma)$, with $\Sigma=\omega_c+\omega_x$ and $R=\sqrt{\Sigma^2+4\omega_c\omega_xg^2}$.
Operationally, the Rabi minimum equals the mean conditional photon number after dipole measurement and subtraction of conditional field amplitudes; for Dicke systems this quantity is only a lower bound. Such measurements require fast protocols, and ordinary photodetection does not count ground-state virtual photons. The general identity assumes displacement-form gauge transformations; multimode extensions require commuting matter generators. The minimizing gauge’s location between Coulomb and dipole gauges is partly numerical.
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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 G. Namely, the ground-state photon number is a quadratic polynomial in the gauge parameter λ, ⟨a†a⟩λ=⟨a†a⟩0+Cλ+Qλ2. Here the coefficients are the correlator C=⟨Gp⟩ of G and the field momentum p, and Q=⟨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 paper asks what gauge-independent information can be extracted from the gauge-dependent ground-state photon number in truncated single-mode light–matter models. For gauge transformations Uλ=exp[iλG(a+a†)], it derives the exact identity nλ=n0+Cλ+Qλ2, where C=⟨Gp⟩, p=i(a†−a), and Q=⟨G2⟩. Minimization gives the gauge-orbit invariant nmin=n0−C2/(4Q); Cauchy–Schwarz establishes nonnegativity.
For the gauge-corrected quantum Rabi model, Q=1. Mapping to the standard Rabi Hamiltonian and applying Hellmann–Feynman relations yields nmin=∂ωcE0−41(∂gE0)2. A coherent-state argument proves strict positivity for nonzero coupling and positive level splitting. The minimizing gauge approaches the Jaynes–Cummings gauge at weak coupling and the Coulomb gauge at strong coupling. Exact-solution calculations and numerical diagonalization evaluate these quantities.
For the gauge-consistent Dicke model, the curvature is the collective dipole second moment. Including dipole self-energy suppresses its growth and prevents superradiance. Holstein–Primakoff analysis gives, at fixed collective coupling in the thermodynamic limit, αmin=ωx/(ωc+ωx) at all couplings and nmin=(R−Σ)2/(4RΣ), with Σ=ωc+ωx and R=Σ2+4ωcωxg2.
Operationally, the Rabi minimum equals the mean conditional photon number after dipole measurement and subtraction of conditional field amplitudes; for Dicke systems this quantity is only a lower bound. Such measurements require fast protocols, and ordinary photodetection does not count ground-state virtual photons. The general identity assumes displacement-form gauge transformations; multimode extensions require commuting matter generators. The minimizing gauge’s location between Coulomb and dipole gauges is partly numerical.
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.
Abstract We investigate the ground state of two physically motivated modifications of the Dicke model. The first modification corresponds to particles whose phase space contains only two states, for example, artificially created qubits. The second modification describes two-level systems that arise as a result of truncating the full Hilbert space of atoms to two levels that are in resonance with the electromagnetic field and are described by the gauge-invariant Dicke model. We demonstrate that the behavior of these systems is qualitatively distinct in both cases. In particular, in the first scenario, a superradiant phase transition (SPT) into a state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter η = 〈 a ˆ 〉 which would indicate condensation into a coherent state remains zero, meaning that the SPT is forbidden. It resolves the long-standing discussion about the SPT in the Dicke model. At the same time, the average number of photons n ¯ = 〈 a ˆ † a ˆ 〉 ≠ 0 , and the collective excitation in the system manifests as a non-classical ‘squeezed’ state of the field. We analyze the observable characteristics of both systems in a wide range of variation of their parameters.
Ultrastrong-coupling between two-level systems and radiation is important for both fundamental and applied quantum electrodynamics (QED). Such regimes are identified by the breakdown of the rotating-wave approximation, which applied to the quantum Rabi model (QRM) yields the apparently less fundamental Jaynes-Cummings model (JCM). We show that when truncating the material system to two levels, each gauge gives a different description whose predictions vary significantly for ultrastrong-coupling. QRMs are obtained through specific gauge choices, but so too is a JCM without needing the rotating-wave 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, Jaynes-Cummings physics is not restricted to light-matter 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 ultrastrong-coupling.
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.
The prototypical system constituted by a two-level atom interacting with a quantized single-mode electromagnetic field is described by the quantum Rabi model (QRM). The QRM is potentially valid at any light-matter interaction regime, ranging from the weak (where the decay rates exceeds the coupling rate) to the deep strong coupling (where the interaction rate exceeds the bare transition frequencies of the subsystems). However, when reaching the ultrastrong coupling regime, several theoretical issues may prevent the correct description of the observable dynamics of such a system: (i) the standard quantum optics master equation fails to correctly describe the interaction of this system with the reservoirs; (ii) the correct output photon rate is no longer proportional to the intracavity photon number; and (iii) they appear to violate gauge invariance. Here, we study the photon flux emission rate of this system under the incoherent excitation of the two-level atom for any light-matter interaction strength and consider different effective temperatures. The dependence of the emission spectra on the coupling strength is the result of the interplay between energy levels, matrix elements of the observables, and the density of states of the reservoirs. Within this approach, we also study the occurrence of light-matter decoupling in the deep strong-coupling regime and show how all of the obtained results are gauge invariant.
Self-adjoint extensions of k-photon light-matter Hamiltonians
Multiphoton light-matter interactions, in which a bosonic mode exchanges k excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators H=Hmat⊗I+I⊗ωa∗a+Σ⊗(a∗)k+Σ∗⊗ak on H⊗L2(R), coupling a single bosonic mode to an arbitrary matter system through a bounded operator Σ. When Σ is normal and nonzero, we prove that H is self-adjoint if and only if k≤2; for k≥3 we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of Σ. The normality of Σ is optimal: a k-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every k. We illustrate our results on the k-photon Rabi and Dicke models.