Down-conversion of a single photon as a probe of many-body localization.

Decay of a particle into more particles is a ubiquitous phenomenon to interacting quantum systems, taking place in colliders, nuclear reactors or solids. In a nonlinear medium, even a single photon would decay by down-converting (splitting) into lower-frequency photons with the same total energy1, at a rate given by Fermi's golden rule. However, the energy-conservation condition cannot be matched precisely if the medium is finite and only supports quantized modes. In this case, the fate of the photon becomes the long-standing question of many-body localization, originally formulated as a gedanken experiment for the lifetime of a single Fermi-liquid quasiparticle confined to a quantum dot2. Here we implement such an experiment using a superconducting multimode cavity, the nonlinearity of which was tailored to strongly violate the photon-number conservation. The resulting interaction attempts to convert a single photon excitation into a shower of low-energy photons but fails owing to the many-body localization mechanism, which manifests as a striking spectral fine structure of multiparticle resonances at the standing-wave-mode frequencies of the cavity. Each resonance was identified as a many-body state of radiation composed of photons from a broad frequency range and not obeying Fermi's golden rule theory. Our result introduces a new platform to explore the fundamentals of many-body localization without having to control many atoms or qubits3-9.

Remote gate control of topological transitions in moiré superlattices via cavity vacuum fields.

Placed in cavity resonators with three-dimensionally confined electromagnetic wave, the interaction between quasiparticles in solids can be induced by exchanging virtual cavity photons, which can have a nonlocal characteristic. Here, we investigate the possibility of utilizing this nonlocality to realize the remote control of the topological transition in mesoscopic moiré superlattices at full filling (one electron/hole per supercell) embedded in a split-ring terahertz electromagnetic resonator. We show that gate tuning one moiré superlattice can remotely drive a topological band inversion in another moiré superlattice not in contact but embedded in the same cavity. Our study of remote on/off switching of a topological transition provides a paradigm for the control of material properties via cavity vacuum fields.

Electron-Photon Chern Number in Cavity-Embedded 2D Moiré Materials.

We explore theoretically how the topological properties of 2D materials can be manipulated by cavity quantum electromagnetic fields for both resonant and off-resonant electron-photon coupling, with a focus on van der Waals moiré superlattices. We investigate an electron-photon topological Chern number for the cavity-dressed energy minibands that is well defined for any degree of hybridization and entanglement of the electron and photon states. While an off-resonant cavity mode can renormalize electronic topological phases that exist without cavity coupling, we show that when the cavity mode is resonant to electronic miniband transitions, new and higher electron-photon Chern numbers can emerge.

Dissipative Phase Transition with Driving-Controlled Spatial Dimension and Diffusive Boundary Conditions.

We investigate theoretically and experimentally a first-order dissipative phase transition, with diffusive boundary conditions and the ability to tune the spatial dimension of the system. The considered physical system is a planar semiconductor microcavity in the strong light-matter coupling regime, where polariton excitations are injected by a quasiresonant optical driving field. The spatial dimension of the system from 1D to 2D is tuned by designing the intensity profile of the driving field. We investigate the emergence of criticality by increasing the spatial size of the driven region. The system is nonlinear due to polariton-polariton interactions and the boundary conditions are diffusive because the polaritons can freely diffuse out of the driven region. We show that no phase transition occurs using a 1D driving geometry, while for a 2D geometry we do observe both in theory and experiments the emergence of a first-order phase transition. The demonstrated technique allows all-optical and in situ control of the system geometry, providing a versatile platform for exploring the many-body physics of photons.

Non-Hermitian bath model for arrays of coupled nanoresonators.

Nanophotonics systems have recently been studied under the perspective of non-Hermitian physics. Given their potential for wavefront control, nonlinear optics and quantum optics, it is crucial to develop predictive tools to assist their design. We present here a simple model relying on the coupling to an effective bath consisting of a continuum of modes to describe systems of coupled resonators, and test it on dielectric nanocylinder chains accessible to experiments. The effective coupling constants, which depend non-trivially on the distance between resonators, are extracted from numerical simulations in the case of just two coupled elements. The model predicts successfully the dispersive and reactive nature of modes for configurations with multiple resonators, as validated by numerical solutions. It can be applied to larger systems, which are hardly solvable with finite-element approaches.

Permanent Directional Heat Currents in Lattices of Optomechanical Resonators.

We study the phonon dynamics in lattices of optomechanical resonators where the mutually coupled photonic modes are coherently driven and the mechanical resonators are uncoupled and connected to independent thermal baths. We present a general procedure to obtain the effective Lindblad dynamics of the phononic modes for an arbitrary lattice geometry, where the light modes play the role of an effective reservoir that mediates the phonon nonequilibrium dynamics. We show how to stabilize stationary states exhibiting directional heat currents over arbitrary distance, despite the absence of thermal gradient and of direct coupling between the mechanical resonators.

Quantum Critical Regime in a Quadratically Driven Nonlinear Photonic Lattice.

We study an array of coupled optical cavities in the presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one and two dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.

Variational Neural-Network Ansatz for Steady States in Open Quantum Systems.

We present a general variational approach to determine the steady state of open quantum lattice systems via a neural-network approach. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof of principle, we apply the method to the dissipative quantum transverse Ising model.

Ancillary qubit spectroscopy of vacua in cavity and circuit quantum electrodynamics.

We investigate theoretically how the spectroscopy of an ancillary qubit can probe cavity (circuit) QED ground states containing photons. We consider three classes of systems (Dicke, Tavis-Cummings, and Hopfield-like models), where nontrivial vacua are the result of ultrastrong coupling between N two-level systems and a single-mode bosonic field. An ancillary qubit detuned with respect to the boson frequency is shown to reveal distinct spectral signatures depending on the type of vacua. In particular, the Lamb shift of the ancilla is sensitive to both ground state photon population and correlations. Backaction of the ancilla on the cavity ground state is investigated, taking into account the dissipation via a consistent master equation for the ultrastrong coupling regime. The conditions for high-fidelity measurements are determined.

Controlling discrete and continuous symmetries in "superradiant" phase transitions with circuit QED systems.

We explore theoretically the physics of a collection of two-level systems coupled to a single-mode bosonic field in the nonstandard configuration where each (artificial) atom is coupled to both field quadratures of the boson mode. We show that such an unusual coupling scheme can be implemented in circuit QED systems, where artificial Josephson atoms are coupled both capacitively and inductively to a superconducting resonator. We demonstrate that it is possible to pass from a discrete, paritylike Z(2) symmetry to a continuous U(1) with the appearance of photonic Goldstone and amplitude modes above a critical point even in the ultrastrong coupling regime (where the rotating wave approximation for the interaction between field and two-level systems is no longer applicable). We determine the rich phase diagram showing "superradiant" phases with different symmetries and phase boundaries of both first and second order.

Single-polariton optomechanics.

This Letter investigates a hybrid quantum system combining cavity quantum electrodynamics and optomechanics. The Hamiltonian problem of a photon mode coupled to a two-level atom via a Jaynes-Cummings coupling and to a mechanical mode via radiation pressure coupling is solved analytically. The atom-cavity polariton number operator commutes with the total Hamiltonian leading to an exact description in terms of tripartite atom-cavity-mechanics polarons. We demonstrate the possibility to obtain cooling of mechanical motion at the single-polariton level and describe the peculiar quantum statistics of phonons in such an unconventional regime.

Emergent quantum phase transition of a Josephson junction coupled to a high-impedance multimode resonator.

The physics of a single Josephson junction coupled to a resistive environment is a long-standing fundamental problem at the center of an intense debate, strongly revived by the advent of superconducting platforms with high-impedance multimode resonators. Here we investigate the emergent criticality of a junction coupled to a multimode resonator when the number of modes is increased. We demonstrate how the multimode environment renormalizes the Josephson and capacitive energies of the junction so that in the thermodynamic limit the charging energy dominates when the impedance is larger than the resistance quantum and is negligible otherwise, independently from the bare ratio between the two energy scales and the compact or extended nature of the phase of the junction. Via exact diagonalization, we find that the transition surprisingly stems from a level anticrossing involving not the ground state, but the first excited state, whose energy gap vanishes in the thermodynamic limit. We clarify the nature of the two phases by pointing at a different behavior of the ground and excited states and we show that at the transition point the spectrum displays universality not only at low frequencies. In agreement with recent experiments, we reveal striking spectral signatures of the phase transition.

Quantum phases of a multimode bosonic field coupled to flat electronic bands.

We investigate the quantum phases of systems in which a multimode bosonic field is coupled to the transitions between two flat electronic bands. In the literature, such systems are usually modeled using a single or multimode Dicke model, leading to the prediction of superradiant quantum phase transitions for large enough couplings. We show that the physics of these systems is remarkably richer than previously expected, with the system continuously interpolating between a Dicke model exhibiting a superradiant quantum phase transition and a quantum Rabi model with no phase transition.

Steady-state phases and tunneling-induced instabilities in the driven dissipative Bose-Hubbard model.

We determine the steady-state phases of a driven-dissipative Bose-Hubbard model, describing, e.g., an array of coherently pumped nonlinear cavities with a finite photon lifetime. Within a mean-field master equation approach using exact quantum solutions for the one-site problem, we show that the system exhibits a tunneling-induced transition between monostable and bistable phases. We characterize the corresponding quantum correlations, highlighting the essential differences with respect to the equilibrium case. We also find collective excitations with a flat energy-momentum dispersion over the entire Brillouin zone that trigger modulational instabilities at specific wave vectors.

Cavity QED of the graphene cyclotron transition.

We investigate theoretically the cavity quantum electrodynamics of the cyclotron transition for Dirac fermions in graphene. We show that the ultrastrong coupling regime characterized by a vacuum Rabi frequency comparable or even larger than the transition frequency can be obtained for high enough filling factors of the graphene Landau levels. Important qualitative differences occur with respect to the corresponding physics of massive electrons in a semiconductor quantum well. In particular, an instability for the ground state analogous to the one occurring in the Dicke model is predicted for an increasing value of the electron density.

Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect.

The prospect of controlling the electronic properties of materials via the vacuum fields of cavity electromagnetic resonators is emerging as one of the frontiers of condensed matter physics. We found that the enhancement of vacuum field fluctuations in subwavelength split-ring resonators strongly affects one of the most paradigmatic quantum protectorates, the quantum Hall electron transport in high-mobility two-dimensional electron gases. The observed breakdown of the topological protection of the integer quantum Hall effect is interpreted in terms of a long-range cavity-mediated electron hopping where the anti-resonant terms of the light-matter coupling Hamiltonian develop into a finite resistivity induced by the vacuum fluctuations. Our experimental platform can be used for any two-dimensional material and provides a route to manipulate electron phases in matter by means of vacuum-field engineering.

Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED.

We investigate theoretically the dynamical behavior of a qubit obtained with the two ground eigenstates of an ultrastrong coupling circuit-QED system consisting of a finite number of Josephson fluxonium atoms inductively coupled to a transmission line resonator. We show a universal set of quantum gates by using multiple transmission line resonators (each resonator represents a single qubit). We discuss the intrinsic "anisotropic" nature of noise sources for fluxonium artificial atoms. Through a master equation treatment with colored noise and many-level dynamics, we prove that, for a general class of anisotropic noise sources, the coherence time of the qubit and the fidelity of the quantum operations can be dramatically improved in an optimal regime of ultrastrong coupling, where the ground state is an entangled photonic "cat" state.

Manipulating matter by strong coupling to vacuum fields.

Over the past decade, there has been a surge of interest in the ability of hybrid light-matter states to control the properties of matter and chemical reactivity. Such hybrid states can be generated by simply placing a material in the spatially confined electromagnetic field of an optical resonator, such as that provided by two parallel mirrors. This occurs even in the dark because it is electromagnetic fluctuations of the cavity (the vacuum field) that strongly couple with the material. Experimental and theoretical studies have shown that the mere presence of these hybrid states can enhance properties such as transport, magnetism, and superconductivity and modify (bio)chemical reactivity. This emerging field is highly multidisciplinary, and much of its potential has yet to be explored.

Vacuum degeneracy of a circuit QED system in the ultrastrong coupling regime.

We investigate theoretically the quantum vacuum properties of a chain of N superconducting Josephson atoms inductively coupled to a transmission line resonator. We derive the quantum field Hamiltonian for such a circuit QED system, showing that, due to the type and strength of the interaction, a quantum phase transition can occur with a twice degenerate quantum vacuum above a critical coupling. In the finite-size case, the degeneracy is lifted, with an energy splitting decreasing exponentially with increasing values of g{2}N{2}, where g is the dimensionless vacuum Rabi coupling per artificial atom. We determine analytically the ultrastrong coupling asymptotic expression of the two degenerate vacua for an arbitrary number of artificial atoms and of resonator modes. In the ultrastrong coupling regime the degeneracy is protected with respect to random fluctuations of the transition energies of the Josephson elements.

Quantum squeezing generation versus photon localization in a disordered planar microcavity.

We investigate theoretically the nonlinear dynamics induced by an intense pump field in a disordered planar microcavity. Through a self-consistent theory, we show how the generation of quantum optical noise squeezing is affected by the breaking of the in-plane translational invariance and the occurrence of photon localization. We find that the generation of single-mode Kerr squeezing for the ideal planar case can be prevented by disorder as a result of multimode nonlinear coupling, even when the other modes are in the vacuum state. However, the excess noise is a nonmonotonic function of the disorder amplitude. In the strong localization limit, we show that the system becomes protected with respect to this fundamental coupling mechanism and that the ideal quadrature squeezing generation can be obtained.