We derive dynamical equations for a driven, dissipative quantum system in which the environment- induced relaxation rate is comparable to the Rabi frequency, avoiding assumptions on the frequency dependence of the environmental coupling. When the environmental coupling varies significantly on the scale of the Rabi frequency, secular or rotating wave approximations break down. Our approach avoids these approximations, yielding dynamical, periodic steady-states. This is important for the qualitative and quantitative description of the interaction between driven quantum dots and their phonon environment. The theory agrees well with recent experiments, describing the transition from asymmetric unsaturated resonances at weak driving to population inversion at strong driving.
In this work we report on a new mechanism to generate dissipative steady state entanglement in two coupled qubits driven by strong periodic ac fields. We show that steady entanglement can be generated at one side of a multiphoton resonance between a non-entangled ground state and an entangled excited state. The degree of entanglement can be tuned as a function of the amplitude of the periodic drive. A rich dynamic behavior with creation, death and revival of entanglement can be observed for certain parameter regimes, accessible in current experimental devices.
We show that an open quantum system in a non-Markovian environment can reach steady states that it cannot reach in a Markovian environment. As these steady states are unique for the non-Markovian regime, they could offer a simple way of detecting non-Markovianity, as no information about the system's transient dynamics is necessary. In particular, we study a driven two-level system (TLS) in a semi-infinite waveguide. Once the waveguide has been traced out, the TLS sees an environment with a distinct memory time. The memory time enters the equations as a time delay that can be varied to compare…
We investigate nonequilibrium energy transfer in a single-site Bose-Hubbard model coupled to two thermal baths. By including a quantum kinetic equation combined with full counting statistics, we investigate the steady state energy flux and noise power. The influence of the nonlinear Bose-Hubbard interaction on the transfer behaviors is analyzed, and the nonmonotonic features are clearly exhibited. Particularly, in the strong on-site repulsion limit, the results become identical with the nonequilibrium spin-boson model. We also extend the quantum kinetic equation to study the geometric-phase-in…
For short-range interacting systems, no Schrödinger cat state can be stable when their environment is in thermal equilibrium. We show, by studying a chain of two-level systems with nearest-neighbour Ising interactions, that this is possible when the surroundings consists of two heat reservoirs at different temperatures, or of a heat reservoir and a monochromatic field. The asymptotic state of the considered system can be a pure superposition of mesoscopically distinct states, the all-spin-up and all-spin-down states, at low temperatures. The main feature of our model leading to this result is …
We apply a full-polaron master equation and a weak-coupling non-Markovian master equation to describe the steady-state time-averaged properties of a driven two-level system, an electron coherently tunneling between double quantum dots (DQDs), interacting with a bosonic phonon bath. Comparing the results obtained using these two master equations with those from a recent DQD experiment and its corresponding weak-coupling theoretical method, we find that the original parameter set used in the experiment and theoretical method is not in the weak-coupling parameter regime. By using the full-polaron…
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…
We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinear couplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally, correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage, and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated…
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…
We present analytical solutions for the mean-field master equation of the driven-dissipative Bose-Hubbard model for cavity photons, in the limit of both weak pumping and weak dissipation. Instead of pure Mott insulator states, we find statistical mixtures with the same second-order coherence as a Fock state with n photons, but a mean photon number of n/2. These mixed states occur when n pump photons have the same energy as n interacting photons inside the nonlinear cavity and survive up to a critical tunneling coupling strength, above which a crossover to classical coherent state takes place. …
Periodically driven dynamics of open quantum systems is very interesting because typically non-equilibrium steady state is reached, which is characterized by a non-vanishing current. In this work, we study time discrete and periodically driven dynamics experimentally for a single photon that its coupled to its environment. We develop a comprehensive theory which explains the experimental observations and offers an analytical characterization of the non-equilibrium steady states of the system. We demonstrate that the periodic driving and the properties of the environment can be engineered in su…
Recent experimental advances enable the fabrication of photonic lattices in which the light propagates with non-trivial energy dispersions. When interfaced with quantum emitters, such systems yield strong collective spontaneous emission phenomena, such as perfect sub-radiance, in which the decay into the bath is completely suppressed, forming bound-states-in-the-continuum. Since such photonic lattices are generally lossy, an alternative way of probing them consists in coherently driving them to an steady-state from which photoluminescence can be extracted. Here, we formalize connections betwee…
A chiral-coupled atomic chain of two-level quantum emitters allows strong resonant dipole-dipole interactions, which enables significant collective couplings between every other emitters. We numerically obtain the steady-state phase diagram of such system under weak excitations, where interaction-driven states of crystalline orders, edge or hole excitations, and dichotomy of chiral flow are identified. We distinguish these phases by participation ratios and structure factors, and find two critical points which relate to decoherence-free subradiant sectors of the system. We further investigate …
We consider two two-level atoms fixed at different positions, driven by a monochromatic laser field, and interacting collectively with the vacuum electromagnetic field. A Born-Markov-secular master equation is used to describe the dynamics of the two atoms and their steady-state is obtained analytically for two configurations, one in which the atoms are in equivalent positions and another in which they are not. The steady-state populations of the energy levels of the free atoms, entanglement, quantum discord and degree of mixed-ness are calculated analytically as a function of the laser field …
We study periodically driven bosonic scalar field theories in the infinite N limit. It is well-known that the free theory can undergo parametric resonance under monochromatic modulation of the mass term and thereby absorb energy indefinitely. Interactions in the infinite N limit terminate this increase for any choice of the UV cutoff and driving frequency. The steady state has non-trivial correlations and is synchronized with the drive. The O(N) model at infinite N provides the first example of a clean interacting quantum system that does not heat to infinite temperature at any drive frequency…
We study an ensemble of two-level quantum systems (qubits) interacting with a common electromagnetic field in proximity of a dielectric slab whose temperature is held different from that of some far surrounding walls. We show that the dissipative dynamics of the qubits driven by this stationary and out of thermal equilibrium (OTE) field, allows the production of steady many-body entangled states, differently from the case at thermal equilibrium where steady states are always non-entangled. By studying up to ten qubits, we point out the role of symmetry in the entanglement production, which is …
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…
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 investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs at zero temperature. The EE at equilibrium obeys the logarithmic law. However, in NESSs far from equilibrium the EE grows anomalously fast, obeying the `quasi volume law,' although the conductor is driven by the zero-temperature reservoirs. This anomalous behavior arises from both the far from equilibrium condit…
We present a class of tractable non-equilibrium dynamical quantum systems which includes combinations of injection, detection and extraction of particles interspersed by unitary evolution. We show how such operations generate a hierarchy of equations tying lower correlation functions with higher order ones. The hierarchy closes for particular choices of measurements and leads to a rich class of evolutions whose long time behavior can be simulated efficiently. In particular, we use the method to describe the dynamics of current generation through a generalized quantum exclusion process, and exh…
We report on a mechanism to optimize the generation of steady-state entanglement in a system of coupled qubits driven by microwave fields. Due to the interplay between Landau-Zener-Stückerlberg pumping involving three levels and a subsequent fast relaxation channel, which is activated by tuning the qubits-reservoir couplings, a maximally entangled state can be populated. This mechanism does not require from the fine-tuning of multiphoton-resonances but depends on the sign of the qubit-qubit coupling. In particular, we find that by a proper design of the system parameters and the driving protoc…
We address the problem of analyzing the radius of convergence of perturbative expansion of non-equilibrium steady states of Lindblad driven spin chains. A simple formal approach is developed for systematically computing the perturbative expansion of small driven systems. We consider the paradigmatic model of an open $XXZ$ spin 1/2 chain with boundary supported ultralocal Lindblad dissipators and treat two different perturbative cases: (i) expansion in system-bath coupling parameter and (ii) expansion in driving (bias) parameter. In the first case (i) we find that the radius of convergence quic…
We present an explicit construction of a family of steady state density matrices for an open integrable spin-1 chain with bilinear and biquadratic interactions, also known as the Lai-Sutherland model, driven far from equilibrium by means of two oppositely polarizing Markovian dissipation channels localized at the boundary. The steady state solution exhibits n+1 fold degeneracy, for a chain of length n, due to existence of (strong) Liouvillian U(1) symmetry. The latter can be exploited to introduce a chemical potential and define a grand canonical nonequilibrium steady state ensemble. The matri…
The steady state of a driven dense ensemble of two-level atoms is determined from the competition of coherent laser excitation and decay that acts in a correlated way on several atoms simultaneously. We show that the presence of this non-local dissipation lifts the direct link between the density of excited atoms and the photon emission rate which is typically present when atoms decay independently. The non-locality disconnects these static and dynamic observables so that a dynamical transition in one does not necessarily imply a transition in the other. Furthermore, the collective nature of t…
We discuss anomalous decoherence effects at zero and finite temperatures in driven coupled quantum spin systems. By numerical simulations of the quantum master equation, it is found that the entanglement of two coupled spin qubits exhibits a non-monotonic behaviour as a function of the noise strength. The effects of noise strength, the detuning and finite temperature of independent environments on the steady state entanglement are addressed in detail. Pumped by an external field drive, non-trivial steady states can be found, the steady state entanglement increases monotonically up to a maximum…
An exact and explicit ladder-tensor-network ansatz is presented for the non-equilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is - apart from a normalization constant - a polynomial of degree 2n-2 in the coupling constant. Efficient computation of physical observables is faciliated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cos…
In this short note we provide two extensions on the recent explicit results on the matrix-product ansatz for the non-equilibrium steady state of a markovianly boundary-driven anisotropic Heisenberg XXZ spin 1/2 chain. We write a perturbative solution for the steady state density matrix in the system-batyh coupling for an arbitrary (asymmetric) set of four spin-flip rates at the two chain ends, generalizing the symmetric-driving ansatz of [Phys. Rev. Lett. 106, 217206 (2011)]. Furthermore, we generalize the exact (non-perturbative) form of the steady state for just two Lindblad channels (spin-u…
We will review some known exact solutions for the steady state of the open quantum Heisenberg $XXZ$ spin chain coupled to a pair of baths [Phys. Rev. Lett. 107, 137201 (2011).]. The dynamics is modelled by the Lindblad master equation. We also review how to calculate some relevant physical observables and provide the statistics of spin current assuming the spin chain is weakly coupled to the baths [Phys. Rev. Lett. 112, 067201 (2014).].
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, dep…
We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time-periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dip…
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 scheme for generating robust and persistent entanglement between qubits that do not interact and that are separated by a long and lossy transmission channel, using Markovian reservoir engineering. The proposal uses only the correlated decay into the common channel of remotely separated, driven single-photon qubit transitions. This simple scheme is generic and applicable to various experimental implementations, including circuit and cavity QED, with little experimental overhead compared with methods requiring dynamic control, initialization, measurement, or feedback. In addition to…
We study resonant optical excitations of atoms in a one-dimensional lattice to the Rydberg states interacting via the van der Waals potential which suppresses simultaneous excitation of neighboring atoms. Considering two- and three-level excitation schemes, we analyze the dynamics and stationary state of the continuously-driven, dissipative many-body system employing time-dependent density-matrix renormalization group (t-DMRG) simulations. We show that two-level atoms can exhibit only nearest neighbor correlations, while three-level atoms under dark-state resonant driving can develop finite-ra…
We show how to exploit algebraic relations of operators (or matrices) which constitute the non-equilibrium matrix product steady state of a boundary driven quantum spin chain to find partial differential equations determining all the $m$-point correlation functions in the continuum limit. These partial differential equations, the order of which is determined by scaling of the non-equilibrium partition function, are readily solved if we also know the boundary conditions. In this way we can avoid resorting to representation theory of the matrix product algebra. We apply our methods to study the …
The cooperation between non-Hermiticity and interaction brings about a lot of counterintuitive behaviors, which are impossible to exist in the framework of the Hermitian system. We study the effect of a non-Hermitian impurity on the Hubbard model in the context of $η$ symmetry. We show that the non-Hermitian Hubbard Hamiltonian can respect a full real spectrum even if a local non-Hermitian impurity is applied to. The balance between dissipation of single fermion and on-site pair fluctuation results in a highest-order coalescing state with off-diagonal long-range order (ODLRO). Based on the cha…
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 th…
Under the action of coherent periodic driving a generic quantum system will undergo Floquet heating and continously absorb energy until it reaches a featureless thermal state. The phase-space constraints induced by certain symmetries can, however, prevent this and allow the system to dynamically form robust steady states with off-diagonal long-range order. In this work, we take the Hubbard model on an arbitrary lattice with arbitrary filling and, by simultaneously diagonalising the two possible SU(2) symmetries of the system, we analytically construct the correlated steady states for different…
We study an integrable system that is reducible to free fermions by a Jordan-Wigner transformation which is subjected to a Fibonacci driving protocol based on two non-commuting Hamiltonians. In the high frequency limit $ω\to \infty$, we show that the system reaches a non-equilibrium steady state, up to some small fluctuations which can be quantified. For each momentum $k$, the trajectory of the stroboscopically observed state lies between two concentric circles on the Bloch sphere; the circles represent the boundaries of the small fluctuations. The residual energy is found to oscillate in a qu…
Laser technology has developed and accelerated photo-induced nonequilibrium physics from both scientific and engineering viewpoints. The Floquet engineering, i.e., controlling material properties and functionalities by time-periodic drives, is a forefront of quantum physics of light-matter interaction, but limited to ideal dissipationless systems. For the Floquet engineering extended to a variety of materials, it is vital to understand the quantum states emerging in a balance of the periodic drive and energy dissipation. Here we derive the general description for nonequilibrium steady states (…
We simulate the nonequilibrium steady state \textit{cis-trans} photoisomerization of retinal chromophore in rhodopsin based on a two-state-two-mode model coupled to a thermal environment. By analyzing the systematic trends within an inhomogeneously broadened ensemble of systems, we find that the steady state reaction quantum yield (QY) correlates strongly with the excess energy above the crossing point of the system, in agreement with the prediction of the short time dynamical wavepacket picture. However, the nontrivial dependence of the QY on the system-environment interaction indicates that …
The question of how quantum coherence facilitates energy transfer has been intensively debated in the scientific community. Since natural and artificial light-harvesting units operate under the stationary condition, we address this question via a non-equilibrium steady-state analysis of a molecular dimer irradiated by incoherent sunlight and then generalize the key predictions to arbitrarily-complex exciton networks. The central result of the steady-state analysis is the coherence-flux-efficiency relation:$η=c\sum_{i\neq j}F_{ij}κ_j=2c\sum_{i\neq j}J_{ij}{\rm Im}[ρ_{ij}]κ_j$ with $c$ the norma…
We present a systematic approach for constructing steady state density operators of Markovian dissipative evolution for open quantum chain models with integrable bulk interaction and boundary incoherent driving. The construction is based on fundamental solutions of the quantum Yang-Baxter equation pertaining to quantum algebra symmetries and their quantizations (q-deformations). In particular, we facilitate a matrix-product state description, by resorting to generic spin-s infinite-dimensional solutions associated with non-compact spins, serving as ancillary degrees of freedom. After formally …
We explore the properties of steady-state Fano coherences generated in a three-level V-system continuously pumped by polarized incoherent light in the absence of coherent driving. The ratio of the stationary coherences to excited-state populations $\mathcal{C} = (1+\frac{Δ^2}{γ(r+γ)} )^{-1}$ is maximized when the excited-state splitting $Δ$ is small compared to either the spontaneous decay rate $γ$ or the incoherent pumping rate $r$. We demonstrate that an intriguing regime exists where the $\mathcal{C}$ ratio displays a maximum as a function of the dephasing rate $γ_d$. We attribute the surpr…
In this work we analyze the validity of Lifshitz's theory for the case of non-equilibrium scenarios from a full quantum dynamical approach. We show that Lifshitz's framework for the study of the Casimir pressure is the result of considering the long-time regime (or steady state) of a well-defined fully quantized problem, subjected to initial conditions for the electromagnetic field interacting with real materials. For this, we implement the closed time path formalism developed in previous works to study the case of two half spaces (modeled as composite environments, consisting in quantum degre…
The implementation of a combination of continuous weak measurement and classical feedback provides a powerful tool for controlling the evolution of quantum systems. In this work, we investigate the potential of this approach from three perspectives. First, we consider a double-well system in the classical large-atom-number limit, deriving the exact equations of motion in the presence of feedback. Second, we consider the same system in the limit of small atom number, revealing the effect that quantum fluctuations have on the feedback scheme. Finally, we explore the behavior of modest sized Hubb…
We study a driven-dissipative Bose-Hubbard model in presence of two-particle losses and an incoherent single-particle drive on each lattice site, leading to a finite-density stationary state. Using dynamical mean-field theory (DMFT) and an impurity solver based on exact diagonalization of the associated Lindbladian, we investigate the regime of strong two-particle losses. Here, a stationary-state quantum Zeno effect emerges, as can be seen in the on-site occupation and spectral function. We show that DMFT captures this effect through its self-consistent bath. We show that, in the deep Zeno reg…
