We study a driven-dissipative model of spins one-half (qubits) on a lattice with nearest-neighbor interactions. Focusing on the role of spatially extended spin-spin correlations in determining the phases of the system, we characterize the spatial structure of the correlations in the steady state, as well as their temporal dynamics. In dimension one we use essentially exact matrix-product-operator simulations on large systems, and pushing these calculations to dimension two, we obtain accurate results on small cylinders. We also employ an approximation scheme based on solving the dynamics of the mean field dressed by the feedback of quantum fluctuations at leading order. This approach allows us to study the effect of correlations in large lattices with over one hundred thousand spins, as the spatial dimension is increased up to five. In dimension two and higher we find two new states that are stabilized by quantum correlations and do not exist in the mean-field limit of the model. One of these is a steady state with mean magnetization values that lie between the two bistable mean-field values, and whose correlation functions have properties reminiscent of both. The correlation length of the new phase diverges at a critical point, beyond which we find emerging a new limit cycle state with the magnetization and correlators oscillating periodically in time.
We investigate the steady-state photon-number squeezing and quantum entanglement in a network of nondegenerate optical parametric oscillators (NOPOs). We treat each NOPO with Shen's Raman laser model, whose lasing mode provides a photon-number-squeezed state. Two dissipatively coupled NOPOs satisfy Hillery-Zubairy's $HZ1$ entanglement criterion if they are pumped far above the threshold and the dissipative coupling is sufficiently larger than the NOPO cavity loss.
In this letter we propose a scheme for the preparation of steady entanglements in bosonic dissipative networks. We describe its implementation in a system of coupled cavities interacting with an engineered reservoir built up of three-level atoms. Emblematic bipartite ($Bell$ and $NOON$) and multipartite $W$-class states can be produced with high fidelity and purity.
We investigate how to enhance entanglement in the steady state of interacting two-level systems. The steady state is reached by spontaneous decay of the individual systems. When we additionally couple them to a dissipative two-level ancilla with variable eigenfrequency and coupling strength, we observe a considerable enhancement effect in the entanglement of this steady state. Moreover, we see that the increased entanglement is directly connected to the selection of certain excited states via the environment disturbing the ancilla. This effect could be used in dissipative state preparation sch…
We investigate steady-state properties of a two-dimensional incoherent-pumped dissipative Bose-Hubbard model, which describes a photon square lattice. This incoherent pumping exhibits an important environment-induced higher-order fluctuation effect, which induces a strong competition between the driven-dissipative channel, the photon-photon interaction, and the photon hopping in multi-photon processes. This new competition gives rise to a spontaneous breaking of the U(1) symmetry of system. As a result, we predict a many-body steady-state localized-delocalized phase transition and an anti-bloc…
Quantum squeezing of mechanical resonator is important for studying the macroscopic quantum effects and the precision metrology of weak forces. Here we give a theoretical study of a hybrid atom-optomechanical system in which the steady-state squeezing of the mechanical resonator can be generated via the mechanical nonlinearity and cavity cooling process. The validity of the scheme is assessed by simulating the steady-state variance of the mechanical displacement quadrature numerically. The scheme is robust against dissipation of the optical cavity, and the steady-state squeezing can be effecti…
It has been recently realized that dissipative processes can be harnessed and exploited to the end of coherent quantum control and information processing. In this spirit we consider strongly dissipative quantum systems admitting a non-trivial manifold of steady states. We show how one can enact adiabatic coherent unitary manipulations e.g., quantum logical gates, inside this {steady-state} manifold by adding a weak, time-rescaled, Hamiltonian term into the system's Liouvillian. The effective long-time dynamics is governed by a projected Hamiltonian which results from the interplay between the …
Scheme to prepare three-dimensional entangled state between a pair of Rydberg atoms is proposed via dissipative dynamics and Electromagnetic Induced Transparency (EIT) associated with the single-atom dark state. The prepared entangled state is the dark state of the whole system. The schemes are feasible no matter the system initially in arbitrary purity or mixed states and do not have accurate requirements on evolution time. In contrast to most of the former Rydberg-atombased dissipative schemes, the Rydberg-Rydberg interaction (RRI) strength do not need to satisfy a certain relation with lase…
This Article investigates dissipative preparation of entangled non-equilibrium steady states (NESS). We construct a collision model where the open system consists of two qubits which are coupled to heat reservoirs with different temperatures. The baths are modeled by sequences of qubits interacting with the open system. The model can be studied in different dynamical regimes: with and without environmental memory effects. We report that only a certain bath temperature range allows for entangled NESS. Furthermore, we obtain minimal and maximal critical values for the heat current through the sy…
We systematically study the influence of amplitude modulation on the steady-state bosonic squeezing and entanglement in a dissipative three-mode optomechanical system, where a vibrational mode of the membrane is coupled to the left and right cavity modes via the radiation pressure. Numerical simulation results show that the steady-state bosonic squeezing and entanglement can be significantly enhanced by periodically modulated external laser driving either or both ends of the cavity. Remarkably, the fact that as long as one periodically modulated external laser driving either end of the cavitie…
Inspired by a recent work [Reiter, Reeb, and Sørensen, Phys. Rev. Lett. {\bf117}, 040501 (2016)], we present a simplified proposal for dissipatively preparing a Greenberger-Horne-Zeilinger (GHZ) state of three Rydberg atoms in a cavity. The $Z$ pumping is implemented under the action of the spontaneous emission of $Λ$-type atoms and the quantum Zeno dynamics induced by strong continuous coupling. In the meantime, a dissipative Rydberg pumping breaks up the stability of the state $|{\rm GHZ}_+\rangle$ in the process of $Z$ pumping, making $|{\rm GHZ}_-\rangle$ be the unique steady state of syst…
The Geometric Phase as a Diagnostic for Driven-Dissipative Oscillators
National University of Singapore（新加坡国立大学）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit（中法联合实验室（CNRS-UNS-NUS-NTU））；Nanyang Technological University（南洋理工大学）
We present a short overview of quantum entanglement generation and preservation in a steady state. In addition to the focus on quantum entanglement stabilization, we briefly discuss the same objective for steady-state quantum coherence. The overview classifies the approaches into two main categories: hybrid drive and dissipation methods and purely dissipative schemes. Furthermore, purely dissipative schemes are discussed under two subclasses of equilibrium and nonequilibrium environments. The significance of the dissipative route to sustained quantum entanglement and challenges against it are …
We present a scheme for the dissipative preparation of an entangled steady state of two superconducting qubits in a circuit QED setup. Combining resonator photon loss, a dissipative process already present in the setup, with an effective two-photon microwave drive, we engineer an effective decay mechanism which prepares a maximally entangled state of the two qubits. This state is then maintained as the steady state of the driven, dissipative evolution. The performance of the dissipative state preparation protocol is studied analytically and verified numerically. In view of the experimental imp…
In this work, we explore the critical behaviors of fidelity susceptibility and trace distance susceptibility associated to the steady states of dissipative systems at continuous phase transitions. We investigate on two typical models, one is the dissipative spin-1/2 XYZ model on two-dimensional square lattice and the other is a driven-dissipative Kerr oscillator. We find that the susceptibilities of fidelity and trace distance exhabit singular behaviors near the critical points of phase transitions in both models. The critical points, in thermodynamic limit, extracted from the scalings of the …
We study abrupt changes in the dynamics and/or steady state of fermionic dissipative systems produced by small changes of the system parameters. Specifically, we consider open fermionic systems whose dynamics is described by master equations that are quadratic (and, under certain conditions, quartic) in creation and annihilation operators. We analyze both phase transitions in steady state, as well as "dynamical transitions". The latter are characterized by abrupt changes in the rate at which the system asymptotically approaches the steady state. We illustrate our general findings with relevant…
We investigate the steady-state phases of the dissipative spin-1/2 $J_1$-$J_2$ XYZ model on a two-dimensional square lattice. We show the next-nearest-neighboring interaction plays a crucial role in determining the steady-state properties. By means of the Gutzwiller mean-field factorization, we find the emergence of antiferromag-netic steady-state phases. The existence of such antiferromagnetic steady-state phases in thermodynamic limit is confirmed by the cluster mean-field analysis. Moreover, we find the evidence of the limit cycle phase through the largest quantum Lyapunov exponent in small…
We describe a scheme with analytic result that allows to generate steady-state entanglement for two atoms over a dissipative bosonic medium. The resonant coupling between the mediating bosonic mode and cavity modes produces three collective atomic decay channels. This dissipative dynamics, together with the unitary process induced by classical microwave fields, drives the two atoms to the symmetric or asymmetric entangled steady state conditional upon the choice of the phases of the microwave fields. The effects on the steady-state entanglement of off-resonance mediating bosonic modes are anal…
We experimentally study a driven-dissipative Josephson junction array, realized with a weakly interacting Bose Einstein condensate residing in a one-dimensional optical lattice. Engineered losses on one site act as a local dissipative process, while tunneling from the neighboring sites constitutes the driving force. We characterize the emerging steady-states of this atomtronic device. With increasing dissipation strength $γ$ the system crosses from a superfluid state, characterized by a coherent Josephson current into the lossy site to a resistive state, characterized by an incoherent hopping …
We propose a quantum optical implementation of a class of dissipative spin systems, including the XXZ and Ising model, with ultra-cold atoms in optical lattices. Employing the motional degree of freedom of the atoms and detuned Raman transitions we show how to obtain engineerable dissipation and a tunable transversal magnetic field, enabling the study of the dynamics and steady-states of dissipative spin models. As an example of effects made accessible this way, we consider small spin chains and weak dissipation and show by numerical simulation that steady-state expectation values display pron…
We have investigated the effect of counter-rotating terms on the dynamics of entanglement and quantum discord between two identical atoms interacting with a lossy single mode cavity field for a system initially in a vacuum state. The counter-rotating terms are found to lead to steady states in the long time limit which can have high quantum discord, but have no entanglement. The effect of cavity decay rate on this steady state quantum discord has been also investigated, surprisingly, the increase in cavity decay rate is found to both enhance and maximize the steady quantum discord for separabl…
We propose a dissipative method for the preparation of many-body steady entangled states in spin and fermionic chains. The scheme is accomplished by means of an engineered set of Lindbladians acting over the eigenmodes of the system, whose spectrum is assumed to be resolvable. We apply this idea to prepare a particular entangled state of a spin chain described by the XY model, emphasizing its generality and experimental feasibility. Our results show that our proposal is capable of achieving high fidelities and purities for a given target state even when dephasing and thermal dissipative proces…
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…
Steady-state spectral kissing and dissipative phase transitions
University of Connecticut（康涅狄格大学）；Cinvestav（墨西哥国家科学研究中心）；Yale University（耶鲁大学）；Universidad de Huelva（韦尔瓦大学）；CSIC（西班牙国家研究委员会）；Universidad de Granada（格拉纳达大学）
Dissipative preparation of a pure steady state usually involves a commutative action of a coherent and a dissipative dynamics on the target state. Namely, the target pure state is an eigenstate of both the coherent and dissipative parts of the dynamics. We show that working in the Zeno regime, i.e. for infinitely large dissipative coupling, one can generate a pure state by a non commutative action, in the above sense, of the coherent and dissipative dynamics. A corresponding Zeno regime pureness criterion is derived. We illustrate the approach, looking at both its theoretical and applicative a…
Directly in the thermodynamic limit, we show how to combine imaginary and real time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESS) of one-dimensional dissipative quantum lattices governed by the Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly correlated subspace of the NESS, after which real time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the d…
We present a new variational method, based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show t…
Entangled states are a key resource in fundamental quantum physics, quantum cryp-tography, and quantum computation [1].To date, controlled unitary interactions applied to a quantum system, so-called "quantum gates", have been the most widely used method to deterministically create entanglement [2]. These processes require high-fidelity state preparation as well as minimizing the decoherence that inevitably arises from coupling between the system and the environment and imperfect control of the system parameters. Here, on the contrary, we combine unitary processes with engineered dissipation to…
Driven-dissipative quantum many-body systems have attracted increasing interest in recent years as they lead to novel classes of quantum many-body phenomena. In particular, mean-field calculations predict limit cycle phases, slow oscillations instead of stationary states, in the long-time limit for a number of driven-dissipative quantum many-body systems. Using a cluster mean-field and a self-consistent Mori projector approach, we explore the persistence of such limit cycles as short range quantum correlations are taken into account in a driven-dissipative Heisenberg model.
We present a novel generic framework to approximate the non-equilibrium steady states of dissipative quantum many-body systems. It is based on the variational minimization of a suitable norm of the quantum master equation describing the dynamics. We show how to apply this approach to different classes of variational quantum states and demonstrate its successful application to a dissipative extension of the Ising model, which is of importance to ongoing experiments on ultracold Rydberg atoms. Finally, we identify several advantages of the variational approach over previously employed mean-field…
For the driven-dissipative system of two coupled bosonic modes in a nonlinear cavity resonator, we demonstrate a sequence of phase transitions from a trivial steady state to two distinct dissipative time crystalline phases. These effects are already anticipated at the level of the semiclassical analysis of the Lindblad equation using the theory of bifurcations and are further supported by the full quantum (numerical) treatment. The system is predicted to exhibit different dynamical phases characterized by an oscillating non-equilibrium steady state with non-trivial periodicity, which is a hall…
The dynamics of entanglement and quantum discord between two identical qubits strongly interacting with a common single mode leaky cavity field have been investigated beyond the rotating wave approximation (RWA) by using recently derived Lindblad type quantum optical master equation [F. Beaudoin, J.M. Gambetta, A. Blais, Phys. Rev. A {\bf 84}, 043832 (2011)] that can describe the losses of the cavity field in a strong atom-field coupling regime. Contrary to previous investigations of the same model in the dissipative regime by using the standard Lindblad quantum optical master equation in a st…
In this article, we theoretically investigate the first- and second-order quantum dissipative phase transitions of a three-mode cavity with a Hubbard interaction. In both types, there is a mean-field limit cycle phase where the local U(1)symmetry and the time-translational symmetry (TTS) of the Liouvillian super-operator are spontaneously broken (SSB). This SSB manifests itself through the appearance of an unconditionally and fully squeezed state at the cavity output, connected to the well-known Goldstone mode. By employing the Wigner function formalism hence, properly including the quantum no…
We study the effects of the quantum geometric tensor, i.e., the Berry curvature and the Fubini-Study metric, on the steady state of driven-dissipative bosonic lattices. We show that the quantum-Hall-type response of the steady-state wave function in the presence of an external potential gradient depends on all the components of the quantum geometric tensor. Looking at this steady-state Hall response, one can map out the full quantum geometric tensor of a sufficiently flat band in momentum space using a driving field localized in momentum space. We use the two-dimensional Lieb lattice as an exa…
Inverse design of a property that depends on the steady-state of an open quantum system is commonly done by grid-search type of methods. In this paper we present a new methodology that allows us to compute the gradient of the steady-state of an open quantum system with respect to any parameter of the Hamiltonian using the implicit differentiation theorem. As an example, we present a simulation of a spin-boson model where the steady-state solution is obtained using Redfield theory.
We present and analyze a new approach for the generation of atomic spin squeezed states. Our method involves the collective coupling of an atomic ensemble to a decaying mode of an open optical cavity. We demonstrate the existence of a collective atomic dark-state, decoupled from the radiation field. By explicitly constructing this state we find that it can feature spin squeezing bounded only by the Heisenberg limit. We show that such dark states can be deterministically prepared via dissipative means, thus turning dissipation into a resource for entanglement. The scaling of the phase sensitivi…
We show that short-range correlations have a dramatic impact on the steady-state phase diagram of quantum driven-dissipative systems. This effect, never observed in equilibrium, follows from the fact that ordering in the steady state is of dynamical origin, and is established only at very long times, whereas in thermodynamic equilibrium it arises from the properties of the (free) energy. To this end, by combining the cluster methods extensively used in equilibrium phase transitions to quantum trajectories and tensor-network techniques, we extend them to nonequilibrium phase transitions in diss…
We investigate the steady-state phase diagram of the dissipative spin-1/2 XYZ model on a two-dimensional triangular lattice, in which each site is coupled to a local environment. By means of cluster mean-field approximation, we find that the steady-state phases of the system are rather rich, in particular there exist various types of nonuniform antiferromagnetic phases due to the geometrical frustration. As the short-range correlations included in the analysis, the numerical results show that the oscillatory phase disappears while the triantiferromagnetic and biantiferromagnetic phases remain …
We study the transverse-field Ising model with infinite-range coupling and spontaneous emission on every site. We find that there is spin squeezing in steady state due to the presence of the transverse field. This means that there is still entanglement, despite the decoherence from spontaneous emission. We analytically calculate fluctuations beyond mean-field theory using a phase-space approach, which involves converting the master equation into a Fokker-Planck equation for the Wigner function. Our calculations are relevant to current experiments with trapped ions.
The properties of a quantum dissipative scalar field is analyzed by Caldeira-Leggett model in strong-coupling regime. The Lagrangian of the total system is canonically quantized and the full Hamiltonian is diagonalized using Fano technique. A mode-dependent probability density is introduced. The steady state energy and correlation functions at finite temperature are calculated in terms of the probability density.
We propose an efficient scheme for generating spin-squeezed states at steady state in a spin-mechanical hybrid system, where an ensemble of SiV centers are coupled to a strongly damped nanomechanical resonator. We show that,there exists a collective steady state in the system, which is exactly formed by the collective spin states plus the zero excitation state of the mechanical mode. The generation of the steady spin-squeezed state is based on a dissipative quantum dynamical process in which the mechanical dissipation plays a positive role but without destroying the target state. We demonstrat…
We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems, based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method b…
In dissipative quantum systems, strong symmetries can lead to the existence of conservation laws and multiple steady states. In this work we investigate a strong symmetry for bosonic atoms coupled to an optical cavity, an experimentally relevant system, generalizing the adiabatic elimination techniques and using numerically exact matrix product state methods. We show that for ideal bosons coupled to the cavity multiple steady states exist and in each symmetry sector a dissipative phase transition occurs at a different critical point. This implies that phases of very different nature can coexis…
The average time required for an open quantum system to reach a steady state (the steady-state time) is generally determined through a competition of coherent and incoherent (dissipative) dynamics. Here, we study this competition for a ubiquitous central-spin system, corresponding to a `central' spin-1/2 coherently coupled to ancilla spins and undergoing dissipative spin relaxation. The ancilla system can describe $N$ spins-1/2 or, equivalently, a single large spin of length $I=N/2$. We find exact analytical expressions for the steady-state time in terms of the dissipation rate, resulting in a…
We present exact results for the steady-state density matrix of a general class of driven-dissipative systems consisting of a nonlinear Kerr resonator in the presence of both coherent (one-photon) and parametric (two-photon) driving and dissipation. Thanks to the analytical solution, obtained via the complex P-representation formalism, we are able to explore any regime, including photon blockade, multiphoton resonant effects, and a mesoscopic regime with large photon density and quantum correlations. We show how the interplay between one- and two-photon driving provides a way to control the mu…
We propose various schemes for the dissipative preparation of a maximally entangled steady state of two atoms in an optical cavity. Harnessing the natural decay processes of cavity photon loss and spontaneous emission, we use an effective operator formalism to identify and engineer effective decay processes, which reach an entangled steady state of two atoms as the unique fixed point of the dissipative time evolution. We investigate various aspects that are crucial for the experimental implementation of our schemes in present-day and future cavity quantum electrodynamics systems and analytical…
A longstanding challenge in nonequilibrium thermodynamics is to predict the emergence of self-organized behaviours and functionalities typical of living matter. Despite the progress with classical complex systems, it remains far from obvious how to extrapolate these results down to the quantum scale. Here, we employ the paradigmatic master equation framework to establish that some lifelike behaviours and functionalities can indeed emerge in elementary dissipative quantum systems driven out of equilibrium. Specifically, we find both energy-avoiding (low steady dissipation) and energy-seeking be…
