Superradiant Phase Research Articles

Chiral quantum phases and tricriticality in a Dicke triangle

The existence of quantum tricriticality and exotic phases are found in a tricritical Dicke triangle (TDT) where three cavities, each one containing an ensemble of three-level atoms, are connected to each other through the action of an artificial magnetic field. The conventional superradiant phase (SR) is connected to the normal phase through first- and second-order boundaries, with tricritical points located at the intersection of such boundaries. Apart from the SR phase, a chiral superradiant (CSR) phase is found by tuning the artificial magnetic field. This phase is characterized by a nonzero photon current and its boundary presents chiral tricritical points (CTCPs). Through the study of different critical exponents, we are able to differentiate the universality class of the CTCP and TCP from that of second-order critical points, as well as find distinctive critical behavior among the two different superradiant phases. The TDT can be implemented in various systems, including atoms in optical cavities as well as the circuit QED system, allowing the exploration of a great variety of critical manifolds.

Theory of magnon polaritons in quantum Ising materials

We present a theory of magnon-polaritons in quantum Ising materials, and develop a formalism describing the coupling between light and matter as an Ising system is tuned through its quantum critical point. The theory is applied to Ising materials having multilevel single-site Hamiltonians, in which multiple magnon modes are present, such as the insulating Ising magnet LiHoF$_4$ . We find that the magnon-photon coupling strengths may be tuned by the applied transverse field, with the coupling between the soft mode present in the quantum Ising material and a photonic resonator mode diverging at the quantum critical point of the material. A fixed system of spins will not exhibit the diamagnetic response expected when light is coupled to mobile spins or atoms. Without the diamagnetic response, one expects a divergent magnon-photon coupling strength to lead to a superradiant quantum phase transition. However, this neglects the effects of damping and decoherence present in any real system. We show that damping and decoherence may block the superradiant quantum phase transition, and lead to weak coupling between the soft magnon mode and the resonator mode. The results of the theory are applied to experimental data on the model system LiHoF$_4$ in a microwave resonator.

Understanding the Quantum Rabi Ring Using Analogies to Quantum Magnetism.

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 analog of the latter is induced by an artificial magnetic field, which modulates photon hopping between nearest-neighbor cavities with a phase. This mapping facilitates the description and understanding 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 diagram exhibits three superradiant phases denoted as ferro-superradiant, antiferro-superradiant, and chiral superradiant. In particular, the DM interaction is responsible for the chiral phase in which the energetically degenerate configurations of the order parameters are similar to the in-plane magnetizations of skyrmions with different helicities. The antiferro-superradiant phase is suppressed in the triangle geometry (N=3) as geometric frustration contributes to stabilize the chiral phase even for small values of the DM interaction. The chiral phases for odd and even N show a different scaling behavior close to the phase transition. The equivalent behavior on both systems opens the possibility of simulating chiral magnetism in a few-body quantum optical platform, as well as understanding one system using the insights gained from the other.

Quantum phase transition of a two-dimensional Rydberg atom array in an optical cavity

We study the two-dimensional Rydberg atom array in an optical cavity with help of the meanfield theory and the large-scale quantum Monte Carlo simulations. The strong dipole-dipole interactions between Rydberg atoms can make the system exhibit the crystal structure, and the coupling between two-level atom and cavity photon mode can result in the formation of the polariton. The interplay between them provides a rich quantum phase diagram including the Mott, solid-1/2, superradiant and superradiant solid phases. As the two-order co-existed phase, the superradiant solid phase breaks both translational and U(1) symmetries. Based on both numerical and analytic results, we found the region of superradiant solid is much larger than one dimensional case, so that it can be more easily observed in the experiment. Finally, we discuss how the energy gap of the Rydberg atom can affect the type of the quantum phase transition and the number of triple points.

Squeezed-light-induced quantum phase transition in the Jaynes-Cummings model

Quantum phase transition and quench dynamics in the Jaynes-Cummings model with squeezed light are investigated. We find a special unitary transformation that removes the nonintegrable squeezing interaction when the qubit frequency far outweighs the field frequency. The eigenenergies and the eigenstates of the normal and superradiant phases are derived analytically. We demonstrate that the system exhibits a second-order quantum phase transition at a phase-boundary-induced nonlinearly by squeezed light. This phase boundary requires neither an ultrastrong coupling regime nor multiqubits, which lessens the difficulty of the experiment. The entanglement entropy is very close to its maximum value as the squeezing strength increases in the superradiant phase. In quench dynamics, the nonlinear relation between the residual energy and the squeezing strength is obtained analytically.

Quantum fluctuations and correlations in open quantum Dicke models

In the vicinity of ground-state phase transitions quantum correlations can display non-analytic behavior and critical scaling. This signature of emergent collective effects has been widely investigated within a broad range of equilibrium settings. However, under nonequilibrium conditions, as found in open quantum many-body systems, characterizing quantum correlations near phase transitions is challenging. Moreover, the impact of local and collective dissipative processes on quantum correlations is not broadly understood. This is, however, indispensable for the exploitation of quantum effects in technological applications, such as sensing and metrology. Here we consider as a paradigmatic setting the superradiant phase transition of the open quantum Dicke model and characterize quantum and classical correlations across the phase diagram. We develop an approach to quantum fluctuations which allows us to show that local dissipation, which cannot be treated within the commonly employed Holstein-Primakoff approximation, rather unexpectedly leads to an enhancement of collective quantum correlations, and to the emergence of a nonequilibrium superradiant phase in which the bosonic and spin degrees of freedom of the Dicke model are entangled.

Macroscopic quantum states in Dicke model of [formula omitted]-symmetric non-Hermitian Hamiltonian and superradiant phase with imaginary atomic population

We in this paper study the macroscopic quantum states of a PT-symmetric Non-Hermitian Hamiltonian for N quantum-dots of two imaginary-levels in an optical cavity. The eigenstate energies are obtained in terms of spin-coherent-state variational method, in which PT-symmetric but non-unitary operator is used to diagonalize the effective non-Hermitian Hamiltonian. Biorthogonal sets of the trial wave functions necessarily appear as a consequence of non-Hermitian Hamiltonian. In the normal phase of zero photon-number average energies are imaginary called the PT-symmetry broken state, while the atomic population is real showing the unstable nature of the system. The energy of superradiant phase (SP) is real called unbroken symmetry state, however the atomic population is pure imaginary. The electrons in SP oscillate between upper and lower levels stimulated by the cavity field. The quantum phase transition from broken to unbroken symmetry states occurs at critical values of the atom-field coupling and nonlinear atom-photon interaction.

Quantum criticality driven by the cavity coupling in the Rabi-dimer model

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 which the universal class of the phase transition belongs. We analytically derive the phase boundary of the SPT and investigate the ground-state properties of the system. We uncover that the inter-cavity induced SPT can be apparently understood from the ground-state energy and the ground-state photon population, as well as the ground-state expectation value of the squared anti-symmetric mode. From the scaling analysis of the fidelity susceptibility, we numerically verify that the SPT driven by the cavity coupling belongs to the same universal class as the one driven by the atom–cavity interaction. Our work enriches the studies on the SPT and its critical behaviors in the Rabi-dimer model.

Enhanced photon squeezing in two-photon Dicke model

We explore the phenomena of quadrature squeezing of photons in the two-photon Dicke model under the mean-field approximation in the thermodynamic limit. An effective squeezing of the photonic quadratures beyond the standard quantum limit is observed in the superradiant phase. The magnitude of squeezing is enhanced, and it persists longer as one moves away from the phase boundary and approaches the boundary between the superradiant phase and the unbounded region. This unbounded region is encountered where the coupling strength is of the same order of magnitude as one of the fundamental frequencies of the system. We find that the squeezing is prolonged near the said boundary for varying transition energy of the qubits. Furthermore, we explore the critical behavior of photon squeezing near the unbounded region.

High temperature superradiant phase transition in quantum structures with a complex network interface.

In the present work we propose a novel, to the best of our knowledge, quantum material concept, which enables superstrong and/or ultrastrong interaction of two-level systems with the photonic field in a complex network. Within the mean field approximation we examine phase transition to superradiance that results in two excitation (polariton) branches and is accompanied by the appearance of non-zero macroscopic polarization of two-level systems. We characterize the statistical properties of networks by the first, 〈k〉, and second normalized, ζ ≡ 〈k2〉/〈k〉, moments for node degree distribution. We have shown that the Rabi frequency is essentially enhanced due to the topology of the network within the anomalous domain where 〈k〉 and ζ sufficiently grow. The multichannel (multimode) structure of matter-field interaction leads superstrong coupling that provides primary behavior of the high temperature phase transition. The results obtained pave the way for the design of new photonic and polaritonic circuits, quantum networks for efficient processing quantum information at high (room) temperatures.

Superradiant quantum phase transition for Landau polaritons with Rashba and Zeeman couplings

We develop a theory of cavity quantum electrodynamics for a two-dimensional electron gas in the presence of Rashba spin-orbit and Zeeman couplings and perpendicular magnetic field, coupled to a spatially nonuniform quantum photon field. We show that the superradiant quantum phase transition (SQPT), also known as photon condensation, can in principle occur through a pure in-plane Zeeman coupling, but it requires extremely small (unrealistic) quantum well widths or extremely fine tuning of the effective Land\'e factor which makes two Landau levels coincide. Landau level crossings can also be induced by the Rashba spin-orbit coupling and they promote the SQPT which can be obtained for certain values of the effective Land\'e factor and filling factors.

Quantum Floquet engineering with an exactly solvable tight-binding chain in a cavity

Recent experimental advances enable the manipulation of quantum matter by exploiting the quantum nature of light. However, paradigmatic exactly solvable models, such as the Dicke, Rabi or Jaynes-Cummings models for quantum-optical systems, are scarce in the corresponding solid-state, quantum materials context. Focusing on the long-wavelength limit for the light, here, we provide such an exactly solvable model given by a tight-binding chain coupled to a single cavity mode via a quantized version of the Peierls substitution. We show that perturbative expansions in the light-matter coupling have to be taken with care and can easily lead to a false superradiant phase. Furthermore, we provide an analytical expression for the groundstate in the thermodynamic limit, in which the cavity photons are squeezed by the light-matter coupling. In addition, we derive analytical expressions for the electronic single-particle spectral function and optical conductivity. We unveil quantum Floquet engineering signatures in these dynamical response functions, such as analogs to dynamical localization and replica side bands, complementing paradigmatic classical Floquet engineering results. Strikingly, the Drude weight in the optical conductivity of the electrons is partially suppressed by the presence of a single cavity mode through an induced electron-electron interaction.

Dissipative quantum dynamics, phase transitions, and non-Hermitian random matrices

We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.

Energy cat states induced by a parity-breaking excited-state quantum phase transition

We show that excited-state quantum phase transitions (ESQPTs) in a system in which the parity symmetry is broken can be used to engineer an energy-cat state -- a Schr\"odinger cat state involving a quantum superposition of both different positions and energies. By means of a generalization of the Rabi model, we show that adding a parity-breaking term annihilates the ground-state quantum phase transition between normal and superradiant phases, and induces the formation of three excited-state phases, all of them identified by means of an observable with two eigenvalues. In one of these phases, level crossings are observed in the thermodynamic limit. These allow us to separate a wavefunction in two parts: one, with lower energy, trapped within one region of the spectrum, and a second one, with higher energy, trapped within another. Finally, we show that a generalized microcanonical ensemble, including two different average energies, is required to properly describe equilibrium states in this situation. Our results illustrate yet another physical consequence of ESQPTs.

Lasing and counter-lasing phase transitions in a cavity-QED system

We study the effect of spontaneous emission and incoherent atomic pumping on the nonlinear semiclassical dynamics of the unbalanced Dicke model---a generalization of the Dicke model that features independent coupling strengths for the co- and counter-rotating interaction terms. As well as the ubiquitous superradiant behavior the Dicke model is well-known for, the addition of spontaneous emission combined with the presence of strong counter-rotating terms creates a laserlike behavior termed counter-lasing. These states appear in the semiclassical model as stable periodic orbits. We perform a comprehensive dynamical analysis of the appearance of counter-lasing in the unbalanced Dicke model subject to strong cavity dissipation, such that the cavity field can be adiabatically eliminated to yield an effective Lipkin-Meshkov-Glick (LMG) model. If the coupling strength of the corotating interactions is small, then the counter-lasing phase appears via a Hopf bifurcation of the deexcited state. We find that if the rate of spontaneous emission is small, this can lead to resurgent superradiant pulses. However, if the corotating coupling is larger, then the counter-lasing phase must emerge via the steady-state superradiant phase. Such a transition is the result of the competition of the coherent and incoherent processes that drive superradiance and counter-lasing, respectively. We observe a surprisingly complex transition between the two, associated with the formation of a chaotic attractor over a thin transitional parameter region.

Characteristics of superradiant optical phases occurring in the system of nondegenerate Λ atoms and radiation that are interacting inside a nonlinear quantum cavity

In the present article, we investigate the behavior of a large collection of three-level $\mathrm{\ensuremath{\Lambda}}$ atoms interacting with two quantized electromagnetic fields inside a second-order nonlinear quantum cavity. The nonlinearity of the cavity is supposed to be activated by the application of two classical pump fields. The total Hamiltonian of the combination, with due attention to the nonlinearity effects, is diagonalized. The corresponding ground-state energy then follows from minimizing the total Hamiltonian. The structure of the ground state indicates that four distinct optical phases can occur in such a system. These possible phases turn out to be trivial, dark, left-arm and right-arm superradiant ones. Conditions under which any of the four optical phases can actually occur are also analyzed and discussed. The analysis of the conditions, accompanied by several figures, then reveals that with a suitable choice of the pump field amplitudes and/or geometrical phases, one can intensify the two superradiant phases drastically. Moreover, we demonstrate that the dark optical phase cannot occur at all, while the trivial and superradiant ones can, in fact, coexist. Our calculations also show that transition from the trivial phase to left-arm (right-arm) superradiant one is continuous (discrete) and second (first) order in nature. Another important result of our investigation is that by adjusting the pump field strength, one can switch from the left-arm superradiant phase to the right-arm one and vice versa. This point, in turn, provides an alternative mechanism for the development of quantum optical switching devices.

Critical behavior of quantum Fisher information in finite-size open Dicke model

We explore the steady-state critical behavior of the finite-size open Dicke model—a model that incorporates spontaneous emission decay of the collective atomic spin states and decay of the cavity field. From the perspective of quantum information theory, we can often better characterize the quantum phase transition. In this paper, we characterize the super-radiant phase transition of the steady state of the open Dicke model by numerically calculating the quantum Fisher information (QFI). We calculate the QFI for the atomic state and the cavity field state, as well as their derivatives. We find that the QFI of the cavity field state is more sensitive to atomic decay, and is suppressed more severely in the presence of atomic decay. In contrast, the QFI of the atomic state is less sensitive to the photon loss of the cavity field.

Understanding and Improving Critical Metrology. Quenching Superradiant Light-Matter Systems Beyond the Critical Point

We carefully examine critical metrology and present an improved critical quantum metrology protocol which relies on quenching a system exhibiting a superradiant quantum phase transition beyond its critical point. We show that this approach can lead to an exponential increase of the quantum Fisher information in time with respect to existing critical quantum metrology protocols relying on quenching close to the critical point and observing power law behaviour. We demonstrate that the Cramér-Rao bound can be saturated in our protocol through the standard homodyne detection scheme. We explicitly show its advantage using the archetypal setting of the Dicke model and explore a quantum gas coupled to a single-mode cavity field as a potential platform. In this case an additional exponential enhancement of the quantum Fisher information can in practice be observed with the number of atoms N in the cavity, even in the absence of N-body coupling terms.

Frustrated Superradiant Phase Transition.

Frustration occurs when a system cannot find a lowest-energy configuration due to conflicting constraints. We show that a frustrated superradiant phase transition occurs when the ground-state superradiance of cavity fields due to local light-matter interactions cannot simultaneously minimize the positive photon hopping energies. We solve the Dicke trimer model on a triangle motif with both negative and positive hopping energies and show that the latter results in a sixfold degenerate ground-state manifold in which the translational symmetry is spontaneously broken. In the frustrated superradiant phase, we find that two sets of diverging time and fluctuation scales coexist, one governed by the mean-field critical exponent and another by a novel critical exponent. The latter is associated with the fluctuation in the difference of local order parameters and gives rise to site-dependent photon number critical exponents, which may serve as an experimental probe for the frustrated superradiant phase. We provide a qualitative explanation for the emergence of unconventional critical scalings and demonstrate that they are generic properties of the frustrated superradiant phase at the hand of a one-dimensional Dicke lattice with an odd number of sites. The mechanism for the frustrated superradiant phase transition discovered here applies to any lattice geometries where the antiferromagnetic ordering of neighboring sites are incompatible and therefore our work paves the way toward the exploration of frustrated phases of coupled light and matter.

Polaritonic excitations and Bose-Einstein condensation in the Rabi lattice model

The unitary transformation method is used to study the properties of the polaritonic states and Bose-Einstein condensation in the Rabi lattice model, where the on-site two-level systems (TLSs) are coupled with the intersite hopping photons. It is shown that the counter-rotating coupling (CRC) between TLS and photon, which breaks down the conservation of the excitation numbers, may induce a long-range Ising-like interaction among intersite TLSs. We have shown that the coupled TLSs and photons are hybridized to form the polaritonic states, and the corresponding ground state and polaritonic excitation spectra are calculated by diagonalizing our transformed Hamiltonian. When the photon hopping $J$ is weak, the excitation spectrum is gapped. But the gap decreases with increasing $J$, and at critical value $J={J}_{c}$ the gap becomes zero at the $\mathrm{\ensuremath{\Gamma}}$ point and the ground state becomes instable. Thus, ${J}_{c}$ is the phase transition point where the polaritons are condensed at the $\mathrm{\ensuremath{\Gamma}}$ point to form the delocalized superradiant phase. The results show that the larger detuning between TLS and photon favors the disorder insulator phase, which requires the stronger ${J}_{c}$ to get the long-range order phase. The phase diagram of the delocalized superradiant phase transition has been obtained where there are no Mott lobes since the CRC breaks down the number conservation. The ground state and the excitation spectra of the delocalized superradiant phase are also calculated. It is shown that in the delocalized superradiant state the order parameter is the nonzero ground-state average of the photon annihilation operator. The polaritonic excitation has a finite excitation gap because the CRC breaks down the number conservation but leaves only a discrete symmetry, the parity conservation.