Depending on the relative rates of coupling and dissipation, a light-matter coupled system is either in the weak- or strong-coupling regime. Here, we present a unique system where the coupling rate continuously increases with an externally applied magnetic field while the dissipation rate remains constant, allowing us to monitor a weak-to-strong coupling transition as a function of magnetic field. We observed a Rabi splitting of a terahertz magnon mode in yttrium orthoferrite above a threshold magnetic field of ~14 T. Based on a microscopic theoretical model, we show that with increasing magnetic field the magnons transition into magnon polaritons through an exceptional point, which will open up new opportunities for in situ control of non-Hermitian systems.

Andrey Baydin, Kenji Hayashida, Takuma Makihara, Fuyang Tay, Xiaoxuan Ma, Wei Ren, Guohong Ma, G. Timothy Noe II, Ikufumi Katayama, Jun Takeda, Hiroyuki Nojiri, Shixun Cao, Motoaki Bamba, Junichiro Kono

*Magnetically Tuned Continuous Transition from Weak to Strong Coupling in Terahertz Magnon Polaritons*

We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian, which describes a two-level system coupled to a harmonic oscillator. We show that there is a simple, intuitive correspondence between the circuit Hamiltonian and the quantum Rabi Hamiltonian. While there is an overall shift of the entire spectrum, the energy level structure of the circuit Hamiltonian up to the seventh excited states can still be fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength is larger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit is in the deep–strong-coupling regime. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the variant of the quantum Rabi Hamiltonian that is obtained using an analogous procedure for mapping the circuit variables onto Pauli and harmonic oscillator operators, even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.

Fumiki Yoshihara, Sahel Ashhab, Tomoko Fuse, Motoaki Bamba, Kouichi Semba

*Hamiltonian of a flux qubit-LC oscillator circuit in the deep–strong-coupling regime **Scientific Reports *12, 6764 (2022)

We excite the spin precession in rare-earth orthoferrite YFeO3 by the magnetic field of intense terahertz pulse and probe its dynamics by transient absorption change in the near infrared. The observed waveforms contain quasi-ferromagnetic-mode magnon oscillation and its second harmonics with a comparably strong amplitude. The result can be explained by dielectric function derived from magnetorefractive Hamiltonian. We reveal that the strong second harmonic signal microscopically originates from novel dynamics of the quasi-ferromagnetic mode magnon at nonlinear regime, wherein spin canting angle periodically oscillates.

Takayuki Kurihara, Motoaki Bamba, Hiroshi Watanabe, Makoto Nakajima, Tohru Suemoto

*Spin canting in nonlinear terahertz magnon dynamics revealed by magnetorefractive probing in orthoferrite*

In the superradiant phase transition (SRPT), coherent light and matter fields are expected to appear spontaneously in a coupled light–matter system in thermal equilibrium. However, such an equilibrium SRPT is forbidden in the case of charge-based light–matter coupling, known as no-go theorems. Here, we show that the low-temperature phase transition of ErFeO3 at a critical temperature of approximately 4 K is an equilibrium SRPT achieved through coupling between Fe3+ magnons and Er3+ spins. By verifying the efficacy of our spin model using realistic parameters evaluated via terahertz magnetospectroscopy and magnetization experiments, we demonstrate that the cooperative, ultrastrong magnon–spin coupling causes the phase transition. In contrast to prior studies on laser-driven non-equilibrium SRPTs in atomic systems, the magnonic SRPT in ErFeO3 occurs in thermal equilibrium in accordance with the originally envisioned SRPT, thereby yielding a unique ground state of a hybrid system in the ultrastrong coupling regime.

Motoaki Bamba, Xinwei Li, Nicolas Marquez Peraca, Junichiro Kono

*Magnonic superradiant phase transition **Communications Physics *5, 3 (2022)

We investigate theoretically how the ground state of a qubit–resonator (Q–R) system in the deep-strong coupling (DSC) regime is affected by the coupling to an environment. We employ as a variational ansatz for the ground state of the Q–R–environment system a superposition of coherent states displaced in qubit-state-dependent directions. We show that the reduced density matrix of the Q–R system strongly depends on how the system is coupled to the environment, i.e. capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator–waveguide (R–W) coupling. In contrast, when the two couplings are of the same type (for instance, both are inductive), by increasing the R–W coupling strength, the average number of virtual photons increases and the quantum superposition realized in the Q–R entangled ground state is partially degraded. Since the superposition becomes more fragile with increasing the Q–R coupling, there exists an optimal coupling strength to maximize the nonclassicality of the Q–R system.

Tomohiro Shitara, Motoaki Bamba, Fumiki Yoshihara, Tomoko Fuse, Sahel Ashhab, Kouichi Semba, Kazuki Koshino

*Nonclassicality of open circuit QED systems in the deep-strong coupling regime *

*New Journal of Physics *23 103009 (2021)

A system of N two-level atoms cooperatively interacting with a photonic field can be described as a single giant atom coupled to the field with interaction strength ~N^0.5. This enhancement, known as Dicke cooperativity in quantum optics, has recently become an indispensable element in quantum information technology based on strong light-matter coupling. Here, we extend the coupling

beyond the standard light-matter interaction paradigm, emulating Dicke cooperativity in a terahertz metasurface with N meta-atoms. Cooperative enhancement manifested in the form of matter-matter coupling, through the hybridization of localized surface plasmon resonance in individual meta-atoms and surface lattice resonance due to the periodic array of the meta-atoms. By

varying the lattice constant of the array, we observe a clear anticrossing behavior, a signature of strong coupling. Furthermore, through engineering of the capacitive split-gap in the meta-atoms, the coupling rate was cooperatively enhanced into the ultrastrong coupling regime by a factor of N^0.5. This room-temperature technology serves as a convenient quantum emulator of the dynamics of a qubit with a giant dipole moment coherently driven by a single bosonic field.

Riad Yahiaoui, Zizwe A. Chase, Chan Kyaw, Fuyang Tay, Andrey Baydin, G. Timothy Noe II, Junyeob Song, Junichiro Kono, Amit Agrawal, Motoaki Bamba, Thomas A. Searles

*Tunable Plasmonic Ultrastrong Coupling: Emulating Dicke Physics at Room Temperature *

arXiv:2108.02494** [physics.optics]**

Exotic quantum vacuum phenomena are predicted in cavity quantum electrodynamics (QED) systems with ultrastrong light-matter interactions. Their ground states are predicted to be vacuum squeezed states with suppressed quantum fluctuations. The source of such phenomena are antiresonant terms in the Hamiltonian, yet antiresonant interactions are typically negligible compared to resonant interactions in light-matter systems. We report an unusual coupled matter-matter system of magnons that can simulate a unique cavity QED Hamiltonian with coupling strengths that are easily tunable into the ultrastrong coupling regime and with dominant antiresonant terms. We found a novel regime where vacuum Bloch-Siegert shifts, the hallmark of antiresonant interactions, greatly exceed analogous frequency shifts from resonant interactions. Further, we theoretically explored the system’s ground state and calculated up to 5.9 dB of quantum fluctuation suppression. These observations demonstrate that magnonic systems provide an ideal platform for simulating exotic quantum vacuum phenomena predicted in ultrastrongly coupled light-matter systems.

Takuma Makihara, Kenji Hayashida, G. Timothy Noe II, Xinwei Li, Nicolas Marquez Peraca, Xiaoxuan Ma, Zuanming Jin, Wei Ren, Guohong Ma, Ikufumi Katayama, Jun Takeda, Hiroyuki Nojiri, Dmitry Turchinovich, Shixun Cao, Motoaki Bamba, and Junichiro Kono

*Ultrastrong magnon–magnon coupling dominated by antiresonant interactions **Nature Communications*** 12** (3115), published online (2021)

The ground state of the photon-matter coupled system described by the Dicke model is found to be perfectly squeezed at the quantum critical point of the superradiant phase transition (SRPT). In the presence of the counter-rotating photon-atom coupling, the ground state is analytically expressed as a two-mode squeezed vacuum in the basis of photons and atomic collective excitations. The variance of a quantum fluctuation in the two-mode basis vanishes at the SRPT critical point, with its conjugate fluctuation diverging, ideally satisfying the Heisenberg uncertainty principle.

Kenji Hayashida, Takuma Makihara, Nicolas Marquez Peraca, Diego Fallas Padilla, Han Pu, Junichiro Kono, Motoaki Bamba

*Perfect Intrinsic Squeezing at the Superradiant Phase Transition Critical Point *

One of the earliest and most intensively studied problems in quantum optics is the interaction of a two-level system (an atom) with a single photon. This simple system provides a rich platform for exploring exotic light-matter interactions and the emergence of more complex phenomena such as superradiance, which is a cooperative effect that emerges when the density of atoms is increased and coupling between them is enhanced. Going beyond the light-matter system, Li *et al.* observed analogous cooperative effects for coupled magnetic systems. The results suggest that ideas in quantum optics could be carried over and used to control and predict exotic phases in condensed matter systems.

The interaction of N two-level atoms with a single-mode light field is an extensively studied many-body problem in quantum optics, first analyzed by Dicke in the context of superradiance. A characteristic of such systems is the cooperative enhancement of the coupling strength by a factor of N^{1/2}. In this study, we extended this cooperatively enhanced coupling to a solid-state system, demonstrating that it also occurs in a magnetic solid in the form of matter-matter interaction. Specifically, the exchange interaction of N paramagnetic erbium(III) (Er^{3+}) spins with an iron(III) (Fe^{3+}) magnon field in erbium orthoferrite (ErFeO_{3}) exhibits a vacuum Rabi splitting whose magnitude is proportional to N^{1/2}. Our results provide a route for understanding, controlling, and predicting novel phases of condensed matter using concepts and tools available in quantum optics.

X. Li, M. Bamba, N. Yuan, Q. Zhang, Y. Zhao, M. Xiang, K. Xu, Z. Jin, W. Ren, G. Ma, S. Cao, D. Turchinovich, and J. Kono*Observation of Dicke Cooperativity in Magnetic Interactions*

Science **361**(6404), 794-797 (2018)

Non-perturbative coupling of photons and excitons produces hybrid particles, exciton–polaritons, which have exhibited a variety of many-body phenomena in various microcavity systems. However, the vacuum Rabi splitting (VRS), which defines the strength of photon–exciton coupling, is usually a single constant for a given system. Here, we have developed a unique architecture in which excitons in an aligned single-chirality carbon nanotube film interact with cavity photons in polarization-dependent manners. The system reveals ultrastrong coupling (VRS up to 329 meV or a coupling-strength-to-transition-energy ratio of 13.3%) for polarization parallel to the nanotube axis, whereas VRS is absent for perpendicular polarization. Between these two extremes, VRS is continuously tunable through polarization rotation with exceptional points separating crossing and anticrossing. The points between exceptional points form equienergy arcs onto which the upper and lower polaritons coalesce. The demonstrated on-demand ultrastrong coupling provides ways to explore topological properties of polaritons and quantum technology applications.

Weilu Gao, Xinwei Li, Motoaki Bamba, and Junichiro Kono

Continuous transition between weak and ultra-strong coupling through exceptional points in carbon nanotube micro-cavity exciton polaritons

Nature Photonics, published online (2018).