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arXiv:2307.15437 · Nature Communications

Spectral properties of two superconducting artificial atoms coupled to a resonator in the ultrastrong coupling regime

Akiyoshi Tomonaga, Roberto Stassi, Hiroto Mukai, Franco Nori, F. Yoshihara, J. S. Tsai

Published Jun 17, 2025• 13 citations
arXivHTMLPDF

Abstract

Abstract We experimentally investigate a superconducting circuit composed of two flux qubits ultrastrongly coupled to a common LC resonator. Owing to the large anharmonicity of the flux qubits, the system can be correctly described by a generalized Dicke Hamiltonian containing spin-spin interaction terms. In the experimentally measured spectrum, an avoided level crossing provides evidence of the exotic interaction that allows the simultaneous excitation of two artificial atoms by absorbing one photon from the resonator. This multi-atom ultrastrongly coupled system opens the door to studying nonlinear optics where the number of excitations is not conserved. This enables novel processes for quantum-information processing tasks on a chip.

DOI
10.1038/s41467-025-60589-5

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Abstract We experimentally investigate a superconducting circuit composed of two flux qubits ultrastrongly coupled to a common LC resonator. Owing to the large anharmonicity of the flux qubits, the system can be correctly described by a generalized Dicke Hamiltonian containing spin-spin interaction terms. In the experimentally measured spectrum, an avoided level crossing provides evidence of the exotic interaction that allows the simultaneous excitation of two artificial atoms by absorbing one photon from the resonator. This multi-atom ultrastrongly coupled system opens the door to studying nonlinear optics where the number of excitations is not conserved. This enables novel processes for quantum-information processing tasks on a chip.
The realization of strong nonlinear coupling between single photons has been a long-standing goal in quantum optics and quantum information science, promising wide impact applications, such as all-optical deterministic quantum logic and single-photon frequency conversion. Here, we report an experimental observation of the strong coupling between a single-photon and a two-photon Fock state in an ultrastrongly-coupled circuit-QED system. This strong nonlinear interaction is realized by introducing a detuned flux qubit working as an effective coupler between two modes of a superconducting coplanar waveguide resonator. The ultrastrong light–matter interaction breaks the excitation number conservation, and an external flux bias breaks the parity conservation. The combined effect of the two enables the strong one–two-photon coupling. Quantum Rabi-like avoided crossing is resolved when tuning the two-photon resonance frequency of the first mode across the single-photon resonance frequency of the second mode. Within this new photonic regime, we observe the thresholdless second harmonic generation for a mean photon number below one. Our results represent a key step towards a new regime of quantum nonlinear optics, where individual photons can deterministically and coherently interact with each other in the absence of any stimulating fields. Systems with strong photon-photon interactions enable advanced quantum optical applications as well as the study of highly correlated light-matter states. Here the authors report strong coupling between single- and two-photon states in a superconducting circuit, enabling a new regime of nonlinear quantum optics.
We propose a promising hybrid quantum system, where a highly excited atom strongly interacts with a superconducting LCLCLC oscillator via the electric field of capacitor. An external electrostatic field is applied to tune the energy spectrum of the atom. The atomic qubit is implemented by two eigenstates near an avoided-level crossing in the dc Stark map of a Rydberg atom. Varying the electrostatic field brings the atomic-qubit transition on or off resonance with respect to the microwave resonator, leading to a strong atom-resonator coupling with an extremely large cooperativity. Like the nonlinearity induced by Josephson junctions in superconducting circuits, the large atom-resonator interface disturbs the harmonic potential of the resonator, resulting in an artificial two-level particle. Different universal two-qubit logic gates can also be performed on our hybrid system within the space where an atomic qubit couples to a single photon with an interaction strength much larger than any relaxation rates, opening the door to the cavity-mediated state transmission.
This thesis consists of two parts: the main part is a theoretical investigation of the ultrastrong coupling regime for atom-light coupling in superconducting circuits, and the second part is concerned with the development of a new high coherence flux qubit design. \n \nThe spin-boson model describes the interaction between a quantum two-level system and a continuum of bosonic modes. When the interaction strength becomes comparable to the system frequency, the system enters the ultrastrong coupling (USC) regime, where the rotating wave approximation breaks and the system dynamics need to be described nonperturbatively. Recently, the ultrastrong coupling has been achieved on a device consisting of a superconducting flux qubit coupled to an electromagnetic continuum, with the coupling strength being verified in experiments by the standard transmission method. The first project in this thesis aims to measure the dynamics of the spin-boson model in a direct and controllable way when the coupling strength is in the USC regime. We propose three experiments to measure the coherence of the two-level system including the relaxation and dephasing, and the the renormalized tunneling frequencies in the ultrastrong coupling regime. The controllable measurements are realized with a fast-switchable ultrastrong coupling system consisting of a two-loop flux qubit galvanically coupled to an open transmission line, flux bias and driving lines, and readout circuits. The design and model of the full device are presented. We demonstrate that the three proposed experiments can be well implemented based on the simulations of qubit properties and coupling strengths. \n \nThe second part of the thesis presents work on the design of a new type of capacitively shunted flux qubit. This work is motivated by improving qubit coherence time and anharmonicity, which is essential for speeding up qubit gates and enhancing the capability of quantum computing. It was demonstrated that adding shunting pads to flux qubits can drastically improve the coherence time and the reliability of qubit fabrication, but the CSFQ was shown with moderate anharmonicities. In this project, we present the a new design of CSFQ and its circuit model, which contains three Josephson junctions and three shunting pads. In experiments, the qubit spectroscopy matches well with the circuit model, which takes into account all the capacitances between the qubit and other circuits. The qubit is found to have both large anharmonicity A = ω₁₂ - ω₀₁ = 2π ⨯ 3.69 GHz, and high coherence with T₁ = 40 ± 5 µs. We also present experiments on the multilevel quantum control and multilevel relaxation measurement. We perform qutrit state tomography to reconstruct the full density matrix with the tomography fidelity reaching 99.2%. We are able to extract the time-dependent state populations for the relaxation process of a qutrit, and extract the exact transition rates between the three levels.
Recent measurement on an LC resonator magnetically coupled to a superconducting qubit [ Phys. Rev. Lett. 105 (2010) 237001 ] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes—Cummings model exactly without using the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch—Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch—Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.
Cavity quantum electrodynamics (QED) investigates the interaction between resonator-confined radiation and natural atoms. Similar phenomena can be explored using superconducting artificial atoms coupled to microwave resonators. Here, we examine incoherent and coherent spectra in a circuit QED system consisting of a flux qubit coupled to an LC resonator, considering different coupling configurations to the output transmission line. Although such systems can be effectively described by the quantum Rabi model, we demonstrate that the choice of the coupling between the resonator and the input-output channel can significantly affect the emission properties, particularly in the ultrastrong coupling regime. Specifically, we show that when the resonator is capacitively coupled with the input-output transmission line, the spectral features at zero flux offset can be mapped onto those of the cavity QED model. However, this correspondence breaks down for the mutual inductive coupling case. Extending our analysis to the parity-symmetry broken regime of the flux qubit, we further highlight how capacitive and inductive coupling mechanisms lead to distinct spectral features, emphasizing the necessity of a careful input-output treatment for an accurate description of the emission processes.
We study effective light-matter interactions in a circuit QED system consisting of a single LCLCLC resonator, which is coupled symmetrically to multiple superconducting qubits. Starting from a minimal circuit model, we demonstrate that, in addition to the usual collective qubit-photon coupling, the resulting Hamiltonian contains direct qubit-qubit interactions, which have a drastic effect on the ground- and excited-state properties of such circuits in the ultrastrong-coupling regime. In contrast to the superradiant phase transition expected from the standard Dicke model, we find an opposite mechanism, which at very strong interactions completely decouples the photon mode and projects the qubits into a highly entangled ground state. These findings resolve previous controversies over the existence of superradiant phases in circuit QED, but they more generally show that the physics of two- or multiatom cavity QED settings can differ significantly from what is commonly assumed.