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Is the quantum Rabi model adequate in circuit QED for any atom-resonator coupling?

Vladimir Manucharyan, Alexandre Baksic, Cristiano Ciuti

Published Dec 9, 2016• 0 citations
OpenAlex

Abstract

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.

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Resilience of the quantum Rabi model in circuit QED

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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.
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.
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.
In cavity quantum electrodynamics (QED), the key physical systems include Fabry-Pérot cavity resonators and multilevel cold atoms. Since ultra-strong electromagnetic interaction between photons and quantized cavity modes occurs in cavity QED, many novel quantum optical effects such as optical Rabi oscillation, quantum collapse/revival of atomic-level population and atom-field quantum entanglement can be realized in such quantum microcavities. As is known in quantum optics, these schemes can be generalized to circuit QED, which is a circuit analog of cavity QED. Fundamental devices in circuit QED are superconducting coplanar waveguide microwave resonators and mesoscopic Cooper-pair boxes. Here, the coplanar waveguide resonator with photons stored can be identified as a resonant cavity and the Cooper-pair boxes with large electric dipoles play the role of multilevel cold atoms. Thus, all the quantum optical effects in conventional cavity QED can also be exhibited in the present superconducting circuit QED system. We consider the theoretical working mechanism of superconducting charge qubits based on tunable Josephson junction coupling. It can be found that an analog of optical Rabi oscillation in Cooper-pair boxes can be achieved by tuning an applied external magnetic field. Time evolution of energy levels in a Cooper-pair box, which is manipulated by a time-dependent magnetic flux through its Josephson-junction ring, is studied within the framework of time-dependent quantum mechanics.
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.
Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number nnn scales as n\sqrt{n}n​ for nnn up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.
In circuit QED, protocols for quantum gates and readout of superconducting qubits often rely on the dispersive regime, reached when the qubit-photon detuning $\ensuremath{\Delta}$ is large compared to the mutual coupling strength. For qubits including the Cooper-pair box and transmon, selection rules dramatically restrict the contributions to dispersive level shifts $\ensuremath{\chi}$. By contrast, in the absence of selection rules many virtual transitions contribute to $\ensuremath{\chi}$ and can produce sizable dispersive shifts even at large detuning. We present theory for a generic multilevel qudit capacitively coupled to one or multiple harmonic modes and give general expressions for the effective Hamiltonian in second- and fourth-order perturbation theory. Applying our results to the fluxonium system, we show that the absence of strong selection rules explains the surprisingly large dispersive shifts observed in experiments and leads to the prediction of a two-photon vacuum Rabi splitting. Quantitative predictions from our theory are in good agreement with experimental data over a wide range of magnetic flux and reveal that fourth-order resonances are important for the phase modulation observed in fluxonium spectroscopy.