The quantum Rabi model (QRM) is a cornerstone in the study of light-matter interactions within cavity and circuit quantum electrodynamics (QED). It effectively captures the dynamics of a two-level system coupled to a single-mode resonator, serving as a foundation for understanding quantum optical phenomena in a great variety of systems. However, this model may produce inaccurate results for large coupling strengths, even in systems with high anharmonicity. Moreover, issues of gauge invariance further undermine its reliability. In this work, we introduce a renormalized QRM that incorporates the effective influence of higher atomic energy levels, providing a significantly more accurate representation of the system while still maintaining a two-level description. To demonstrate the versatility of this approach, we present two different examples: an atom in a double-well potential and a superconducting artificial atom (fluxonium qubit). This procedure opens new possibilities for precisely engineering and understanding cavity and circuit QED systems, which are highly sought-after, especially for quantum information processing.
The quantum Rabi model (QRM) is a cornerstone in the study of light-matter interactions within cavity and circuit quantum electrodynamics (QED). It effectively captures the dynamics of a two-level system coupled to a single-mode resonator, serving as a foundation for understanding quantum optical phenomena in a great variety of systems. However, this model may produce inaccurate results for large coupling strengths, even in systems with high anharmonicity. Moreover, issues of gauge invariance further undermine its reliability. In this work, we introduce a renormalized QRM that incorporates the effective influence of higher atomic energy levels, providing a significantly more accurate representation of the system while still maintaining a two-level description. To demonstrate the versatility of this approach, we present two different examples: an atom in a double-well potential and a superconducting artificial atom (fluxonium qubit). This procedure opens new possibilities for precisely engineering and understanding cavity and circuit QED systems, which are highly sought-after, especially for quantum information processing.
The quantum Rabi model (QRM) is a cornerstone in the study of light-matter interactions within cavity and circuit quantum electrodynamics (QED). It effectively captures the dynamics of a two-level system coupled to a single-mode resonator, serving as a foundation for understanding quantum optical phenomena in a great variety of systems. However, this model may produce inaccurate results for large coupling strengths, even in systems with high anharmonicity. Moreover, issues of gauge invariance further undermine its reliability. In this work, we introduce a renormalized QRM that incorporates the effective influence of higher atomic energy levels, providing a significantly more accurate representation of the system while still maintaining a two-level description. To demonstrate the versatility of this approach, we present two different examples: an atom in a double-well potential and a superconducting artificial atom (fluxonium qubit). This procedure opens new possibilities for precisely engineering and understanding cavity and circuit QED systems, which are highly sought-after, especially for quantum information processing. The quantum Rabi model is widely used to describe how light interacts with atoms or artificial atoms in cavities and circuits, but it often becomes inaccurate in modern strong-coupling experiments. The authors introduce a renormalized version that accounts for higher atomic levels while keeping a simple two-level picture, greatly improving accuracy.
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.
We revisit the derivation of Rabi- and Dicke-type models, which are commonly used for the study of quantum light-matter interactions in cavity and circuit QED. We demonstrate that the validity of the two-level approximation, which is an essential step in this derivation, depends explicitly on the choice of gauge once the system enters the ultrastrong-coupling regime. In particular, while 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, it can dramatically fail in the Coulomb gauge, even for systems with an extremely anharmonic spectrum. We extensively investigate this phenomenon both in the single-dipole (Rabi) and multidipole (Dicke) case, and consider the specific examples of dipoles confined by double-well and square-well potentials, and of circuit QED systems with flux qubits coupled to an LC resonator.
In circuit quantum electrodynamics (circuit QED), an artificial 'circuit atom' 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 large enough, the bare multilevel structure of a realistic circuit atom cannot be ignored even if the circuit is strongly anharmonic. We explored this situation theoretically for flux (fluxonium) and charge (Cooper pair box) type multi-level circuits tuned to their respective flux/charge degeneracy points. We identified which spectral features of the quantum Rabi model survive and which are renormalized for large coupling. Despite significant renormalization of the low-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 types of 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 atom-resonator ground state. We interpret the emergence of the two-fold degeneracy of the vacuum of 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 is a manifestation of the dynamical Coulomb blockade of transport in tunnel junctions connected to resistive leads.
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.
The quantum Rabi model (QRM), describing a two-level system coupled to a quantum harmonic oscillator, is one of the simplest and most ubiquitous interaction models in quantum mechanics. Despite its simplicity, the QRM has found applications in various fields of physics. The most exciting realization of the QRM nowadays is arguably the circuit quantum electrodynamics (QED) system, where an on-chip artificial atom (or a qubit) is coupled to a transmission line or a waveguide. The qubits in circuit QED systems contain a bias term, which breaks the Z_2 symmetry in the QRM. Therefore, the precise model realized in circuit QED systems is referred to as the asymmetric quantum Rabi model (AQRM). This thesis explores theoretical methods to treat the AQRM and develops a complete approach to describe its energy landscape. The hidden symmetry of the AQRM is investigated in detail, and seen to be a rather general phenomenon in light-matter interaction models. The equivalence between the generalised Poschl-Teller potential and the spectrum of the AQRM is also established. Chapter 1 is dedicated to reviewing the analytic solutions and conventional approximations to the QRM/AQRM. Regarding the analytic solutions, the analytic results in terms of the transcendental functions determining the eigenspectrum of the AQRM are collected. Special emphasis is placed on the isolated closed-form exceptional solutions to the AQRM. Due to the lack of simple closed-form expressions for the general eigenvalues, some well-known approximations are considered. In Chapter 2, a generalised adiabatic approximation (GAA) is developed to calculate the eigenvalues of the excited states of the QRM and apply it to the AQRM. The GAA is based on the similarity between the exact exceptional solutions and the Laguerre polynomials. By construction, the GAA always has exact exceptional energy and approximates the regular spectrum with rather good agreement in a large parameter regime. Furthermore, the GAA is applied to the AQRM and shown to correctly recover the conical intersections in the energy landscape. The geometric phases around these conical intersections are calculated analytically as an illustrative example. In Chapter 3, a physically motivated variational wave function is presented for the ground state of the AQRM. The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states and relies only on three variational parameters in the regimes of interest. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. The results show that the variational expansion is a significant improvement over the existing approximations for the AQRM. In Chapter 4, the hidden symmetry and tunnelling dynamics in the AQRM and several related models are analysed in detail. The AQRM has a broken Z_2 symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases of the related models, stable level crossings typical of the Z_2-symmetric quantum Rabi model are recovered. It is thus shown that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models. In Chapter 5, starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the AQRM, a spectral equivalence is established with QES hyperbolic Schrodinger potentials. This leads to particular QES Poschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Poschl-Teller potentials. Conclusions and future directions are presented in the final Chapter 6.