Abstract The quantum Rabi model, which describes the interaction between a simplified atom and a mode of the electromagnetic field, is a cornerstone of modern quantum optics. One of the key assumptions of the model is that the ‘atom’ is a perfect two-level system. We explore what happens when one generalises the atom to a multilevel system, with m ground and n excited states coupled to the same field. We focus on the case where the excited and ground states form distinct, well-separated, manifolds of near-degenerate levels (so that the spacing between the excited states on a given manifold is much less than the average spacing between the manifolds) and consider either uniform or random couplings between the individual ground and excited levels. We find that the system reduces approximately to a direct sum of Rabi models with a range of different couplings. Importantly, the strongest coupling is enhanced in a way that depends on the number of levels, for the simple case where n = m, the coupling scales with n for uniform couplings and 2 n for random couplings (in the limit of large n). Our work thus suggests that multilevel Rabi systems could provide an attractive alternative route to accessing regimes of very strong coupling in light-matter systems.
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.
Within the Rabi quantum model, it has been shown theoretically that a two-level atom ultrastrongly coupled to an electromagnetic field generates or absorbs photons. Photons can be generated if the field is initially in the vacuum state. Under certain initial states of the atom + field system, the absorption of photons is possible in the field mode in the ultrastrong atom–field coupling regime. If the atom is initially in the ground (unexcited) state and the field is in the vacuum state, photons can be generated under resonance condition ωa ≈ ωf or ξ ≡ ωa/ωf ≈ 1, where ωa and ωf are the atomic transition and field frequencies, respectively, in the ultrastrong coupling regime. At the negative detuning when ξ
≪ 1 or ωa
≪ ωf, Rabi oscillations of the average number of photons
⟨n^⟩t with
0⩽⟨n^⟩t⩽nmax≫1 are observed in the case of the ultrastrong coupling with the atom–field coupling constant
g~≡|g|/ωf∼1 ; in this case, the population of the excited atomic state is Pe(t) ≈ 0.5. At a large positive detuning when ξ
≫ 1, photons are not generated, i.e.,
⟨n^⟩≈0 , and the atom remains in the initial state, i.e., Pe(t) ≈ 0. The statistics of photons in the generation regime is nearly chaotic: the variance of photons is much larger than the level of the coherent field state (i.e., is super-Poisson). Field photons are absorbed without the excitation of the atom in the ultrastrong coupling regime in the case of the coherent initial state of the field
(⟨n^(t=0⟩>0) for certain positive detuning values. In this case, the field becomes sub-Poisson.
In quantum electrodynamics, an artificial circuit can couple to a quantized microwave radiation much stronger than its real atomic counterpart. The celebrated quantum Rabi model describes the simplest interaction of a two-level system with a single-mode boson field. When the coupling is arbitrary large, the bare multilevel structure of a realistic atom cannot be ignored even if the is strongly anharmonic. We explored this situation theoretically for flux (fluxonium) and charge (Cooper pair box) type multi-level atoms at maximal frustration and identified which spectral features of the quantum Rabi model survive and which are renormalized for arbitrary large coupling. We provide a quantitative comparison with the ideal quantum Rabi model by inspecting not only the energy level spectrum, but also the entanglement spectrum. Despite significant renormalization of the low-energy energy spectrum in the fluxonium case, the key quantum Rabi feature -- nearly-degenerate vacuum consisting of an atomic state entangled with a multi-photon field -- appears in both circuits when the coupling is sufficiently large. Like in the quantum Rabi model, for very large couplings the entanglement spectrum is dominated by only two, nearly equal eigenvalues, in spite of the fact that a large number of bare atomic states are actually involved in the ground state. We interpret the emergence of the vacuum degeneracy in both circuits as an environmental suppression of flux/charge tunneling due to their dressing by virtual low-/high-impedance photons in the resonator. For flux tunneling, the dressing is nothing else than the shunting of a Josephson atom with a large capacitance of the resonator. Suppression of charge tunneling appears to have the same origin as the dynamical Coulomb blockade of transport in tunnel junctions connected to resistive leads.
Abstract In the present paper we show that the Hamiltonian describing the resonant interaction of N two-level systems with a single-mode electromagnetic quantum field in the Coulomb gauge can be diagonalized with a high degree of accuracy using a simple basis set of states. This allows one to find an analytical approximation for the eigenvectors and eigenvalues of the system, which interpolates the numerical solution in a broad range of the coupling constant values. For the first time, we show that the energy spectrum of Dicke model (DM) as a function of N (with N ≫ 1 ) satisfies the simple scaling relation. In addition, the introduced basis states provide a regular way of calculating the corrections and estimating the convergence to the exact numerical solution. Analytical approximations and numerical results describing the energy spectrum are valid for both quantum Rabi model ( N = 1) and DM for N ⩾ 2 atoms.
Abstract We investigate a hybrid quantum system consisting of two coupled optical cavities, each containing an ensemble of coherently pumped artificial or real two-level atoms. The system is modelled within the dressed-state formalism, accounting for coherent driving and dissipative interactions with independent electromagnetic reservoirs. Focusing on the regime where the Rabi frequency greatly exceeds the collective spontaneous emission and cavity decay rates, respectively, we derive the equations of motion characterizing the system and analyse its steady-state behaviours. Special attention is given to the scenario in which only one cavity contains a pumped multi-atom ensemble, and as a consequence, coherent excitation transfer to the second cavity occurs via cavity–cavity coupling. In this case, the mean photon number in the second cavity mode scales proportional to the squared number of atoms, i.e., N2 from the first cavity. The proposed model offers insight into light–matter interaction in composite systems and suggests viable mechanisms for engineering collective interactions and non-local excitation transport in quantum optical platforms.
We present a perturbative analysis of a Rabi model where the coupling between the quantized single-mode electromagnetic field and the two-level atom depends on the field intensity. Upon modeling the matter–radiation coupling through the Holstein–Primakoff realization of algebra su(1,1), we evaluate first- and second-order eigenenergies and eigenstates both in the weak-coupling regime (atom transition frequency smaller than the coupling strength) and in the strong-coupling regime. In the first case, among various effects, we observe a quadratic dependence on the photon number of energy eigenvalues and the possible formation of level doublets. In the strong-coupling case, the perturbative analysis becomes considerably complex due to the su(1,1)-valued form of the unperturbed Hamiltonian. The critical condition for the transition to an almost continuous spectrum is found in terms of the model parameters.