We explore the quantum criticality of a two-site model combining quantum Rabi models with hopping interaction. Through a combination of analytical and numerical approaches, we find that the model allows the appearance of a superradiant quantum phase transition (QPT) even in the presence of strong $\mathbf{A}^2$ terms, preventing single-site superradiance. In the two-site model the effect of $\mathbf{A}^2$ terms can be surmounted by the photon delocalization from hopping, and a reversed superradiant QPT occurs as a consequence of the competition between $\mathbf{A}^2$ terms and the hopping interaction. We characterize the phase diagram and scaling functions, and extract the critical exponents in the vicinity of the critical point, thus establishing the universal behavior of the second-order phase transition. Remarkably the effective hopping strength will be enhanced if more cavities are cascaded. We also prove that the multi-qubit counterpart of the quantum Rabi dimer, i.e., the Dicke dimer, has the same properties in beating the $\mathbf{A}^2$ effect. Our work provides a way to the study of phase transitions in presence of the $\mathbf{A}^2$ terms and offers the prospect of investigating quantum-criticality physics and quantum devices in many-body systems.
We investigate the universally non-equilibrium dynamics of superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we obtain the ground states and their excitation gaps for both normal and superradiant phases via perturbation theory. We analytically extract the critical exponents from the excitation gap and the diverging length scale near the critical point, and find that the critical exponents are independent upon the anisotropy ratio. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract t…
The superradiant phase transition (SPT) controlled by the interacting strength between the two-level atom and the photons has been a hot topic in the Rabi model and the Rabi-dimer model. The latter describes two Rabi cavities coupled with an inter-cavity hopping parameter. Moreover, the SPT in the Rabi-dimer model is found to be the same universal class that in the Rabi model by investigating the correlation-length critical exponent. In this paper, we are concerned about whether the inter-cavity hopping parameter between two Rabi cavities (i.e., the Rabi-dimer model) will induce the SPT and to…
We theoretically study the superradiant phase transition (SPT) in an indirect Rabi model, where both a two-level system and a single mode bosonic field couple to an auxiliary bosonic field. We find that the indirect spin-field coupling induced by the virtual excitation of the auxiliary field can allow the occurrence of a SPT at a critical point, and the influence of the so-called $A^2$ term in the normal Rabi model is naturally avoided. In the large detuning regime, we present the analytical expression of quantum critical point in terms of the original system parameters. The critical atom-fiel…
Parity describing the symmetry of quantum mechanics wavefunction under space inversion transformation not only plays an essential role in solving quantum systems but also can be used to manipulate and measure the motional quantum states of such hybrid quantum systems as quantum Rabi model(QRM) and/or its variants through parity measurements. Here we address an exotic parity behavior of the QRM in its superradiant phase by numerical exact diagonalization, namely, the parities of eigenstates of the QRM behave irregular in the strong coupling regime but the sum of parities for each pair of eigens…
We map a quantum Rabi ring, consisting of $N$ cavities arranged in a ring geometry, into an effective magnetic model containing the XY exchange and the Dzyaloshinskii Moriya (DM) interactions. The analogue of the latter is induced by an artificial magnetic field, which modulates photon hopping between nearest-neighbor cavities with a phase. The mean-field behavior of both systems is almost identical, facilitating the description of the different phases in the quantum optical model through simple arguments of competing magnetic interactions. For the square geometry ($N=4$) the rich phase diagra…
The ground state of quantum Rabi model (QRM) exhibits rich nonclassical states including squeezed state, cat state, and entangled state in different parameter regimes. In this paper, we firstly use the polaron picture to figure out the process of superradiant phase transition (SPT) in view of the nonclassical states. According to the polaron wave functions, the squeezed state extends to whole parameter regimes, and a small but non-zero weighted antipolaron is necessary to form novel semi-cat states. Moreover, the behavior of superradiance in the QRM can be viewed as the same displacement of th…
We demonstrate that the quantum phase transition (QPT) of the Rabi model and critical dynamics near the QPT can be probed in the setup of a single trapped ion. We first show that there exists equilibrium and non-equilibrium universal functions of the Rabi model by finding a proper rescaling of the system parameters and observables. We then propose a scheme that can faithfully realize the Rabi model in the limit of a large ratio of the effective atomic transition frequency to the oscillator frequency using a single trapped-ion and therefore the QPT. It is demonstrated that the predicted univers…
Recent progress on qubit manipulation allows application of periodic driving signals on qubits. In this study, a harmonic driving field is added to a Rabi dimer to engineer photon and qubit dynamics in a circuit quantum electrodynamics device. To model environmental effects, qubits in the Rabi dimer are coupled to a phonon bath with a sub-Ohmic spectral density. A non-perturbative treatment, the Dirac-Frenkel time-dependent variational principle together with the multiple Davydov D$_2$ {\it Ansatz} is employed to explore the dynamical behavior of the tunable Rabi dimer. In the absence of the p…
We explore an extended quantum Rabi model describing the interaction between a two-mode bosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom model is found when the ratio $η$ of the atom energy scale to the bosonic field frequency approaches infinity. An analytical solution is provided when the two lowest-energy levels are degenerate. According to it, we recognize that the phase diagram of the model consists of three regions: one normal phase and two superradiant phases. The quantum phase transitions between the normal phase and the two superradiant pha…
We in this paper derive the analytical expressions of ground-state energy, average photon-number, and the atomic population by means of the spin-coherent-state variational method for arbitrary number of atoms in an optomechanical cavity. It is found that the existence of mechanical oscil- lator does not affect the phase boundary between the normal and superradiant phases. However, the superradiant phase collapses by the resonant damping of the oscillator when the atom-field coupling increases to a so-called turning point. As a consequence the system undergoes at this point an additional phase …
A Rabi dimer is used to model a recently reported circuit quantum electrodynamics system composed of two coupled transmission-line resonators with each coupled to one qubit. In this study, a phonon bath is adopted to mimic the multimode micromechanical resonators and is coupled to the qubits in the Rabi dimer. The dynamical behavior of the composite system is studied by the Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D$_{2}$ ansätze. Initially all the photons are pumped into the left resonator, and the two qubits are in the down state coupled with the …
The demonstration of enhanced spontaneous emission of nanoscaled optical emitters near metallic nanoparticles and the recent realization of a nanolaser based on surface plasmon amplification by stimulated emission of radiation (spaser) encourage the search for strong coupling regime at the nanoscale. Here we propose the concept of nanopolaritons. We demonstrate with accurate scattering calculations that the strong coupling regime of a single quantum emitter (a semiconductor quantum dot) placed in the gap between two metallic nanoparticles can be achieved. The largest dimension of the investiga…
We achieve the strong coupling regime between an ensemble of phosphorus donor spins in a highly enriched $^{28}$Si crystal and a 3D dielectric resonator. Spins were polarized beyond Boltzmann equilibrium using spin selective optical excitation of the no-phonon bound exciton transition resulting in $N$ = $3.6\cdot10^{13}$ unpaired spins in the ensemble. We observed a normal mode splitting of the spin ensemble-cavity polariton resonances of 2$g\sqrt{N}$ = 580 kHz (where each spin is coupled with strength $g$) in a cavity with a quality factor of 75,000 ($γ\ll κ\approx$ 60 kHz where $γ$ and $κ$ a…
The analysis of single-mode photon fluctuations and their counting statistics at the superradiant phase transition is presented. The study concerns the equilibrium Dicke model in a regime where the Rabi frequency, related to a coupling of the photon mode with a finite-number qubit environment, plays a role of the transition's control parameter. We use the effective Matsubara action formalism based on the representation of Pauli operators as bilinear forms with complex and Majorana fermions. Then, we address fluctuations of superradiant order parameter and quasiparticles. The average photon num…
We investigate signatures of chaos and equilibration in the quantum Rabi model, which exhibits a quantum phase transition when the ratio of the atomic level-splitting to bosonic frequency grows to infinity. We show that out-of-time-order correlator derived from the Loschmidt echo signal quickly saturates in the normal phase and reveals exponential growth in the superradiant phase which is associated with the onset of quantum chaos. Furthermore, we show that the effective time-averaged dimension of the quantum Rabi system can be large compared to the spin system size which leads to suppression …
We investigate a one-dimensional Rabi-Hubbard type of model, arranged such that a qdot is sandwiched between every cavity. The role of the qdot is to transmit photons between neighboring cavities, while simultaneously acting as a photon non-linearity. We consider three-level qdots in the $Λ$ configuration, where the left and right leg couples exclusively to the left or right cavity. This non-commuting interaction leads to two highly entangled incompressible phases, separated by a second order quantum phase transition: the qdot degrees-of-freedom act as a dynamical lattice for the photons and a…
We study the ground state phase diagrams of two-photon Dicke, the one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models using mean field, perturbation, quantum Monte Carlo (QMC), and density matrix renormalization group (DMRG) methods. We first compare mean field predictions for the phase diagram of the Dicke model with exact QMC results and find excellent agreement. The phase diagram of the JCH model is then shown to exhibit a single Mott insulator lobe with two excitons per site, a superfluid (SF, superradiant) phase and a large region of instability where the Hamiltoni…
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M. -J. Hwang et al, Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which shows a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states leads to singular…
In this paper we analyze properties of the phase transition that appears in a set of quantum optical models; Dicke, Tavis-Cummings, quantum Rabi, and finally the Jaynes-Cummings model. As the light-matter coupling is increased into the deep strong coupling regime, the ground state turns from vacuum to become a superradiant state characterized by both atomic and photonic excitations. It is pointed out that all four transitions are of the mean-field type, that quantum fluctuations are negligible, and hence these fluctuations cannot be responsible for the corresponding vacuum instability. In this…
We consider a spinless ultracold Fermi gas tightly trapped along the axis of an optical resonator and transversely illuminated by a laser closely tuned to a resonator mode. At a certain threshold pump intensity the homogeneous gas density breaks a $\mathbf{Z}_2$ symmetry towards a spatially periodic order, which collectively scatters pump photons into the cavity. We show that this known self-ordering transition also occurs for low field seeking fermionic particles when the laser light is blue-detuned to an atomic transition. The emergent superradiant optical lattice in this case is homopolar a…
We demonstrate the emergence of a non-equilibrium superradiant phase in the dissipative Rabi-Dicke model. This phase is characterized by a photonic steady state that oscillates with a frequency close to the cavity frequency, in contrast to the constant photonic steady state of the equilibrium superradiant phase in the Dicke model. We relate this superradiant phase to the population inversion of Floquet states by introducing a Schwinger representation of the driven two-level systems in the cavity. This inversion is depleted near Floquet energies that are resonant with the cavity frequency to su…
Two coupled-cavity systems, or "photonic dimers", are efficient test-beds for both fundamental optics -the realization of quantum correlated states, Josephson physics, and so forth-, and applications such as optical flip-flop memories. In this work we report on the first observation of nonlinear mode interaction in a photonic dimer formed by two semiconductor photonic crystal coupled nanolasers. For this, we investigate energy transfer between hybrid modes, which manifests as a switching from the blue-detuned (bonding) to the red-detuned (anti-bonding) modes. An mean-field model allows us to e…
In cavity quantum electrodynamics (QED), the interaction between an atomic transition and the cavity field is measured by the vacuum Rabi frequency $Ω_0$. The analogous term "circuit QED" has been introduced for Josephson junctions, because superconducting circuits behave as artificial atoms coupled to the bosonic field of a resonator. In the regime with $Ω_0$ comparable to the two-level transition frequency, "superradiant" quantum phase transitions for the cavity vacuum have been predicted, e.g. within the Dicke model. Here, we prove that if the time-independent light-matter Hamiltonian is co…
The out-of-time-order correlators (OTOCs) is used to study the quantum phase transitions (QPTs) between the normal phase and the superradiant phase in the Rabi and few-body Dicke models with large frequency ratio of theatomic level splitting to the single-mode electromagnetic radiation field frequency. The focus is on the OTOC thermally averaged with infinite temperature, which is an experimentally feasible quantity. It is shown that thecritical points can be identified by long-time averaging of the OTOC via observing its local minimum behavior. More importantly, the scaling laws of the OTOC f…
The Dicke model and Rabi model can undergo phase transitions from the normal phase to the superradiant phase at the same boundary, which can be accurately determined using some approximated approaches. The underlying mechanism for this coincidence is still unclear and the universality class of these two models is elusive. Here we prove this phase transition exactly using the path-integral approach based on the faithful Schwinger fermion representation, and give a unified phase boundary condition for these models. We demonstrate that at the phase boundary, the fluctuation of the bosonic field i…
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.
Dynamics of a system exhibits non-adiabaticity even for slow quenches near critical points. We analyze the response to disorder in quenches on a non-adiabaticity quantifier for the quantum Rabi model, which possesses a phase transition between normal and superradiant phases. We consider a disordered version of a quench in the Rabi model, in which the system residing in the ground state of an initial Hamiltonian of the normal phase is quenched to the final Hamiltonian corresponding to the critical point. The disorder is inserted either in the total time the quench or in the quench parameter its…
We investigate multiple photon-assisted Landau-Zener (LZ) transitions in a hybrid circuit quantum electrodynamics device in which each of two interacting transmission-line resonators is coupled to a qubit, and the qubits are driven by periodic driving fields and also coupled to a common phonon mode. The quantum state of the entire composite system is modeled using the multi-$\rm D_2$ Ansatz in combination with the time-dependent Dirac-Frenkel variational principle. Applying a sinusoidal driving field to one of the qubits, this device is an ideal platform to study the photon-assisted LZ transit…
