We consider an important generalization of the Dicke model in which multi-level atoms, instead of two-level atoms as in conventional Dicke model, interact with a single photonic mode. We explore the phase diagram of a broad class of atom-photon coupling schemes and show that, under this generalization, the Dicke model can become multicritical. For a subclass of experimentally realizable schemes, multicritical conditions of arbitrary order can be expressed analytically in compact forms. We also calculate the atom-photon entanglement entropy for both critical and non-critical cases. We find that the order of the criticality strongly affects the critical entanglement entropy: higher order yields stronger entanglement. Our work provides deep insight into quantum phase transitions and multicriticality.
We experimentally study the influence of dissipation on the driven Dicke quantum phase transition, realized by coupling external degrees of freedom of a Bose-Einstein condensate to the light field of a high-finesse optical cavity. The cavity provides a natural dissipation channel, which gives rise to vacuum-induced fluctuations and allows us to observe density fluctuations of the gas in real-time. We monitor the divergence of these fluctuations over two orders of magnitude while approaching the phase transition and observe a behavior which significantly deviates from that expected for a closed…
We establish a new theoretical framework, based on a time-dependent mean field approach, to address the dynamics of the driven Dicke model. The joint evolution of both mean fields and quantum fluctuations gives rise to a rich and generally non-linear dynamics, featuring a normal (stable) regime and an unstable, super-radiant one. Various dynamical phenomena emerge, such as the spontaneous amplification of vacuum fluctuations, or the appearance of special points around which the mean-field amplitudes rotate during driven time evolution, signalling a dynamical symmetry breaking. We also provide …
We discuss the unitary quantum dynamics of the Dicke model (spin and oscillator coupled). A suitable quasiprobabilty representing the quantum state turns out to obey a Fokker-Planck equation, with drift terms representing the underlying classical Hamiltonian flow and diffusion terms describing quantum fluctuations. We show (by projecting the dynamics onto a co-moving Poincaré section) how the interplay of deterministic drift and quantum diffusion generates equilibration to the microcanonical density, under conditions of global classical chaos. The pertinent photon statistics reveals macroscopi…
In a view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) non-stationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model, we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and number…
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 i…
We derive analytical results for various quantities related to the excited-state quantum phase transitions in a class of Dicke superradiance models in the semiclassical limit. Based on a calculation of a partition sum restricted to Dicke states, we discuss the singular behavior of the derivative of the density of states and find observables like the mean (atomic) inversion and the boson (photon) number and its fluctuations at arbitrary energies. Criticality depends on energy and a parameter that quantifies the relative weight of rotating versus counter-rotating terms, and we find a close analo…
We provide an exact solution for the interplay between Bose-Einstein condensation and the Dicke-Hepp-Lieb self-organization transition of an ideal Bose gas trapped inside a single-mode optical cavity and subject to a transverse laser drive. Based on an effective action approach, we determine the full phase diagram at arbitrary temperature, which features a bi-critical point where the transitions cross. We calculate the dynamically generated band structure of the atoms and the associated supression of the critical temperature for Bose-Einstein condensation in the phase with a spontaneous period…
We consider a driven single mode Dicke-Hamiltonian coupled to a dissipative zero-temperature bath. We derive the cumulant generating function for emitted photons of this quantum-critical system by using a $P$-representation of the master equation in the thermodynamic limit. This cumulant generating function is shown to consist of two parts: a macroscopic component, which is Poissonian in nature with characteristic rate proportional to the order parameter of the system; and a part describing fluctuations which is non-trivial in form and divergent around the quantum phase transition.
We demonstrate quantum signatures of deterministic nonlinear dynamics in the transition to superradiance of a generalized open Dicke model with different coupling strengths for the co- and counter-rotating light-matter interaction terms. A first-order phase transition to coexisting normal and superradiant phases is observed, corresponding with the emergence of switching dynamics between these two phases, driven by quantum fluctuations. We show that this phase coexistence gives rise to a hysteresis loop also for the quantum mechanical system. Additionally, a transition to a superradiant oscilla…
We propose a novel technique for the creation of maximally entangled symmetric Dicke states in an ion trap using adiabatic passage, which requires only a pair of chirped pulses from a single laser and is applicable to any number of ions and excitations. By utilising a particular factorisation of the Hilbert space for multi-level ladders we show that the problem can be reduced to `bow-tie' configuration energy-level crossings. This technique is naturally robust against fluctuations in the laser intensity and the chirp rate. Even when realistic heating rates are considered, we estimate that the …
We discuss the onset of quantum chaos and thermalization in the two-mode Dicke model, which describes the dipolar interaction between an ensemble of spins and two bosonic modes. The two-mode Dicke model exhibits normal to superradiant quantum phase transition with spontaneous breaking either of a discrete or continuous symmetry. We study the behaviour of the fidelity out-of-time-order correlator derived from the Loschmidt echo signal in the quantum phases of the model and show that its exponential growth cannot be related to a classical unstable point in the general case. Moreover, we find tha…
We consider a collection of two level systems, such as qubits, embedded into a microwave cavity as a promising candidate for the realization of high power quantum batteries. In this perspective, the possibility to design devices where the conventional single-photon coupling is suppressed and the dominant interaction is mediated by two-photon processes is investigated, opening the way to an even further enhancement of the charging performance. By solving a Dicke model with both single- and two-photon coupling we determine the range of parameters where the latter unconventional interaction domin…
The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two level atoms inside a perfectly reflecting cavity which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits which behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and of atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase fr…
We investigate the near-critical dynamics of atomic density fluctuations in the non-equilibrium self-organization transition of an optically driven quantum gas coupled to a single mode of a cavity. In this system cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of an excitation mode recently observed in experiments. This phenomenon has previously been studied within a two-mode approximation for the collective motional degrees of freedom of the atomic condensate which results in an effective open-system Dicke model. Here, including the full…
We investigate the charging process of quantum battery (QB) systems in an extended Dicke model with both atomic interactions and an external driving field. We focus on the effects of the atomic interaction and the external driving field on the charging performance of QB and find that the maximum stored energy of QB has a critical phenomenon. We analyze the critical behavior and obtain the analytical expression of the critical atomic interaction. The dependence of the maximum stored energy, the energy quantum fluctuations and the maximum charging power on the number $N$ of the two-level systems…
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…
The controllability of current quantum technologies allows to implement spin-boson models where two-photon couplings are the dominating terms of light-matter interaction. In this case, when the coupling strength becomes comparable with the characteristic frequencies, a spectral collapse can take place, i.e. the discrete system spectrum can collapse into a continuous band. Here, we analyze the thermodynamic limit of the two-photon Dicke model, which describes the interaction of an ensemble of qubits with a single bosonic mode. We find that there exists a parameter regime where two-photon intera…
We calculate the transmission spectrum of a superconducting circuit realization of the Dicke model and identify spectroscopic features that can serve as signatures of the superradiant phase. In particular, we calculate the resonance frequencies of the system as functions of the bias term, which is usually absent in studies on the Dicke model but is commonly present in superconducting qubit circuits. To avoid over-complicating the proposed circuit, we assume a fixed coupling strength. This situation precludes the possibility of observing signatures of the phase transition by varying the couplin…
We study the quantum phase transition of the Dicke model in the classical oscillator limit, where it occurs already for finite spin length. In contrast to the classical spin limit, for which spin-oscillator entanglement diverges at the transition, entanglement in the classical oscillator limit remains small. We derive the quantum phase transition with identical critical behavior in the two classical limits and explain the differences with respect to quantum fluctuations around the mean-field ground state through an effective model for the oscillator degrees of freedom. With numerical data for …
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 six-fold degenerate ground-state manifold in which the translational symmetry is spontaneously broken. In the frustrated superradi…
The collective behavior of ensembles of atoms has been studied in-depth since the seminal paper of Dicke [R. H. Dicke, Phys. Rev. 93, 99 (1954)], where he demonstrated that a group of emitters in collective states is able to radiate with increased intensity and modified decay rates in particular directions, a phenomenon which he called superradiance. Here, we show that the fundamental setup of two atoms coupled to a single-mode cavity can be distinctly exceeding the free-space superradiant behavior, a phenomenon which we call hyperradiance. The effect is accompanied by strong quantum fluctuati…
Conventional criticality-based quantum metrological schemes work only at zero or very low temperature because the quantum uncertainty around the quantum phase-transition point is generally erased by thermal fluctuations with the increase of temperature. Such an ultralow-temperature requirement severely restricts the development of quantum critical metrology. In this paper, we propose a thermodynamic-criticality-enhanced quantum sensing scenario at finite temperature. In our scheme, a qubit is employed as a quantum sensor to estimate parameters of interest in the Dicke model which experiences a…
Recently, the authors of the commented PRL presented the $ N=\infty $ solution of the $ U(1)/Z_2 $ Dicke model studied by us previously. Here we point out that (1) The authors missed an important transformation relating the two parameter regimes, so their separate discussions on the two regimes is redundant. (2) Both $ N=\infty $ classical limit and $ 1/N $ quantum fluctuations have been achieved in two of our previously published papers. It is the $ 1/N $ quantum fluctuations which lead to the non-trivial new quantum phenomena. In view of only a few $ N=2\sim 9 $ qubits inside a circuit QED m…
We present simulations of the superradiant dynamics of ensembles of atoms in the presence of collective and individual atomic decay processes. We unravel the density matrix with Monte-Carlo wave-functions and identify the quantum jumps in a reduced Dicke state basis, which reflects the permutation symmetry of the identical atoms. While the number of density matrix elements in the Dicke representation increases polynomially with atom number, the quantum jump dynamics populates only a single Dicke state at the time and thus efficient simulations can be carried out for tens of thousands of atoms.…
The mean-field steady states of a generalized model of $N$ two-state systems interacting with one mode of the radiation field in the presence of external driving and dissipation are surveyed as a function of three control parameters: one governs the interaction strength relative to the resonance frequency, thus accessing the Dicke quantum phase transition, a second the relative strength of counter-rotating to rotating-wave interactions, and a third the amplitude of an external field driving the cavity mode. We unify the dissipative extension of the Dicke quantum phase transition with the recen…
In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities. One model which has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance. We consider a generalization of this model that features independently controllable strengths of the co- and counter-rotating terms of the interaction Hamiltonian. We study this system in the semiclassical (me…
We use quantum-electrodynamical approach to study the initial stage of Dicke superradiance from a system of two-level atoms. Applying the zeroth-order Magnus approximation, we obtain the expression for the mean number of quanta emitted in the presence of thermal fluctuations of the electromagnetic field.
The out-of-equilibrium dynamics of quantum systems is one of the most fascinating problems in physics, with outstanding open questions on issues such as relaxation to equilibrium. An area of particular interest concerns few-body systems, where quantum and thermal fluctuations are expected to be especially relevant. In this contribution, we present numerical results demonstrating the impact of conserved quantities (or 'charges') in the outcomes of out-of-equilibrium measurements starting from realistic equilibrium states on a few-body system implementing the Dicke model.
Coherent states provide a natural connection of quantum systems to their classical limit and are employed in various fields of physics. Here we derive general systematic expansions, with respect to quantum parameters, of expectation values of products of arbitrary operators within both oscillator coherent states and SU(2) coherent states. In particular, we generally prove that the energy fluctuations of an arbitrary Hamiltonian are in leading order entirely due to the time dependence of the classical variables. These results add to the list of wellknown properties of coherent states and are ap…
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…
The single-mode Dicke model is well-known to undergo a quantum phase transition from the so-called normal phase to the supperradiant phase (hereinafter called the "superradiant quantum phase transition"). Normally, quantum phase transitions are closely related to the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study quantum Fisher information (QFI) of the field mode and that of the atoms in the ground state of the Dicke Hamiltonian. For finite and large enough number of atoms, our numerical results show that near the critical atom…
Resonating valence bond (RVB) states are a class of entangled quantum many body wavefunctions with great significance in condensed matter physics. We propose a scheme to synthesize a family of RVB states using a cavity QED setup with two-level atoms (with states $\vert 0 \rangle$ and $\vert 1 \rangle$) coupled to a common photon mode. In the lossy cavity limit, starting with an initial state of $M$ atoms excited and $N$ atoms in the ground state, we show that this setup can be configured as a Stern Gerlach experiment. A measurement of photon emission collapses the wavefunction of atoms onto an…
We study the finite time entanglement dynamics between two dark soliton qubits due to quantum fluctuations in a quasi one dimensional Bose-Einstein condensates. Recently, dark solitons are proved to be an appealing platform for qubits due to their appreciably long life time. We explore the entanglement decay for an entangled state of two phonon coherences and the qubits to be in the diagonal basis of so called Dicke states. We observe the collapse and revival of the entanglement, depending critically on the collective damping term but independent of the qubit-qubit interaction for both the ini…
Quantum phase transitions are often embodied by the critical behavior of purely quantum quantities such as entanglement or quantum fluctuations. In critical regions, we underline a general scaling relation between the entanglement entropy and one of the most fundamental and simplest measure of the quantum fluctuations, the Heisenberg uncertainty principle. Then, we show that the latter represents a sensitive probe of superradiant quantum phase transitions in standard models of photons such as the Dicke Hamiltonian, which embodies an ensemble of two-level systems interacting with one quadrature…
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.
We present a peculiar transition triggered by infinitesimal dissipation in the interpolating Dicke-Tavis-Cummings model. The model describes a ubiquitous light-matter setting using a collection of two-level systems interacting with quantum light trapped in an optical cavity. In a previous work [Phys. Rev. Lett. 120, 183603 (2018)], dissipation was shown to extend a normal phase (dark state) into new regions of the model's parameter space. Harnessing Keldysh's action formalism to compute the response function of the light, we show that the normal phase does not merely spread but encompasses a t…
The non-integrable Dicke model and its integrable approximation, the Tavis-Cummings (TC) model, are studied as functions of both the coupling constant and the excitation energy. The present contribution extends the analysis presented in the previous paper by focusing on the statistical properties of the quantum fluctuations in the energy spectrum and their relation with the excited state quantum phase transitions (ESQPT). These properties are compared with the dynamics observed in the semi-classical versions of the models. The presence of chaos for different energies and coupling constants is …
We shift the paradigm of feedback control from the control of quantum states to the control of phase transitions in quantum systems. We show that feedback allows tuning the universality class of phase transitions via modifying its critical exponent. We expand our previous treatment [D. A. Ivanov, T. Yu. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. Lett. 124, 010603 (2020)] of Dicke model and go beyond the approximation of adiabatically eliminated light field. Both linearized and nonlinear models of spin ensembles are considered. The tunability of quantum fluctuations near the…
We derive an effective Dicke model in momentum space to describe collective effects in the quantum regime of a free-electron laser (FEL). The resulting exponential gain from a single passage of electrons allows the operation of a Quantum FEL in the high-gain mode and avoids the experimental challenges of an X-ray FEL oscillator. Moreover, we study the intensity fluctuations of the emitted radiation which turn out to be super-Poissonian.
Fluctuation theorems for autonomous work in the quantum regime
School of Physics, Peking University（北京大学物理学院）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Frontiers Science Center for Nano-Optoelectronics, Peking University（北京大学纳米光电前沿科学中心）
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究中心国际理论科学中心）；Science, Mathematics and Technology Cluster, Singapore University of Technology and Design（新加坡科技设计大学科学技术集群）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室）
We calculate the ground state and simulate the dynamics of a finite chain of spins with Ising nearest-neighbor interactions and a Dicke collective spin interaction with a single mode cavity field. We recover the signatures of first and second order phase transitions predicted by mean field theory, while for small chains, we find significant and non-trivial finite size effects. Below the first order phase transition, even quite large spin chains of 30-40 spins give rise to a mean photon number and number fluctuations significantly above the mean field vacuum result. Near the second order phase …
We study the non-integrable Dicke model, and its integrable approximation, the Tavis-Cummings model, as functions of both the coupling constant and the excitation energy. Excited-state quantum phase transitions (ESQPT) are found analyzing the density of states in the semi-classical limit and comparing it with numerical results for the quantum case in large Hilbert spaces, taking advantage of efficient methods recently developed. Two different ESQPTs are identified in both models, which are signaled as singularities in the semi-classical density of states, one {\em static} ESQPT occurs for any …
We study the late-time relaxation following a quench in a driven-dissipative quantum many-body system. We consider the open Dicke model, describing the infinite-range interactions between $N$ atoms and a single, lossy electromagnetic mode. We show that the dynamical phase transition at a critical atom-light coupling is characterized by the interplay between reservoir-driven and intrinsic relaxation processes. Above the critical coupling, small fluctuations in the occupation of the dominant quasiparticle-mode start to grow in time while the quasiparticle lifetime remains finite due to losses. N…
We study spatial spin and density self-ordering of a two-component Bose-Einstein condensate via collective Raman scattering into a linear cavity mode. The onset of the Dicke superradiance phase transition is marked by a simultaneous appearance of a crystalline density order and a spin-wave order. The latter spontaneously breaks the discrete $\mathbf{Z}_2$ symmetry between even and odd sites of the cavity optical potential. Moreover, in the superradiant state the continuous $U(1)$ symmetry of the relative phase of the two condensate wavefunctions is explicitly broken by the cavity-induced posit…
We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appea…
