Lindblad Master Equation Research Articles

Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect

Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.

Generalized Lindblad master equation for measurement-induced phase transition

The measurement-induced phase transition (MIPT) occurs when the system is evolving under unitary evolution together with local measurements followed by post-selection. We propose a generalized version of the Lindblad master equation as a continuous equation, to describe the dynamics of the second Rényi entropy in the MIPT. This generalized Lindblad equation explicitly takes into account the post-selection in the MIPT, which is realized as the Einstein-Podolsky-Rosen (EPR) state projection in the equation. Also, this generalized Lindblad equation preserves the Hermitian, unit trace, and positive definiteness of the density matrix. We further use the hard-core Bose-Hubbard model as a concrete example to numerically confirm that our generalized Lindblad equation is applicable to describing the MIPT.

Anderson Localization in Dissipative Lattices

AbstractAnderson localization predicts that wave spreading in disordered lattices can come to a complete halt, providing a universal mechanism for dynamical localization. In the one‐dimensional Hermitian Anderson model with uncorrelated diagonal disorder, there is a one‐to‐one correspondence between dynamical localization and spectral localization, that is, the exponential localization of all the Hamiltonian eigenfunctions. This correspondence can be broken when dealing with disordered dissipative lattices. When the system exchanges particles with the surrounding environment and random fluctuations of the dissipation are introduced, spectral localization is observed but without dynamical localization. Previous studies consider lattices with mixed conservative (Hamiltonian) and dissipative dynamics and are restricted to a semiclassical analysis. However, Anderson localization in purely dissipative lattices, displaying an entirely Lindbladian dynamics, remains largely unexplored. Here the purely‐dissipative Anderson model in the framework of a Lindblad master equation is considered, and it is shown that, akin to the semiclassical models with conservative hopping and random dissipation, one observes dynamical delocalization in spite of strong spectral localization of the Liouvillian superoperator. This result is very distinct from delocalization observed in the Anderson model with dephasing, where dynamical delocalization arises from the delocalization of the stationary state of the Liouvillian.

Non-Hermitian topological quantum states in a reservoir-engineered transmon chain

Dissipation in open systems enriches the possible symmetries of the Hamiltonians beyond the Hermitian framework, allowing the possibility of novel non-Hermitian topological phases which exhibit long-living end states that are protected against disorder. So far, non-Hermitian topology has been explored in settings where probing genuine quantum effects has been challenging. We theoretically show that a non-Hermitian topological quantum phase can be realized in a reservoir-engineered transmon chain. The spatial modulation of dissipation is obtained by coupling each transmon to a quantum circuit refrigerator, allowing in situ tuning of dissipation strength in a wide range. By solving the many-body Lindblad master equation using a combination of the density matrix renormalization group and Prosen-Seligman third quantization approaches, we show that the topological end modes and the associated phase transition are visible in simple reflection measurements with experimentally realistic parameters. Finally, we demonstrate that genuine quantum effects are observable in this system via robust and slowly decaying long-range quantum entanglement of the topological end modes, which can be generated passively starting from a locally excited transmon.

Cooperation between Coherent Control and Noises in Quantum Metrology

AbstractIn this article, the cooperation between coherent control and noises in open spin systems is studied, aiming to demonstrate that such cooperation can provide new possibilities for parametrization in quantum metrology. The cooperative scheme proposed here outperforms the standard scheme, with higher precision achieved. More specifically, the effect of cooperative interaction between coherent control and noises in conventional single‐spin systems described by Lindblad master equations is illustrated, with the magnitude of a magnetic field taken as the parameter to be estimated and encoded in the noisy dynamics besides the Hamiltonian. The scenarios of both spontaneous emission and dephasing noise have been analyzed, where the associated quantum Fisher information has been given. Furthermore, it has been demonstrated that in the realm where quantum metrology is mostly applied in practice, the precision limit under the cooperative scheme in the presence of noises can surpass the ultimate precision limit under the unitary dynamics. On the other hand, multiple‐spin systems have also been considered. It is shown that the coupling between different spins can help realize non‐local parametrization under the cooperative scheme, with the ground state made entangled. This thus leads to the improvement of precision limit, which has been proved and visualized in this article.

Noise-specific beating in the higher-level Ramsey curves of a transmon qubit

In the higher levels of superconducting transmon devices and more generally charge sensitive devices, T2* measurements made in the presence of low-frequency time-correlated 1/f charge noise and quasiparticle-induced parity flips can give an underestimation of the total dephasing time. The charge variations manifest as beating patterns observed in the overlay of several Ramsey fringe curves and are reproduced with a phenomenological Ramsey curve model, which accounts for the charge variations. T2* dephasing times, which more accurately represent the total dephasing time, are obtained. The phenomenological model is compared with a Lindblad master equation model. Both models are found to be in agreement with one another and the experimental data. Finally, the phenomenological formulation enables a simple method in which the power spectral density for the low-frequency noise can be inferred from the overlay of several Ramsey curves.

Wetting critical behavior in the quantum Ising model within the framework of Lindblad dissipative dynamics

We investigate the critical behavior, both in space and time, of the wetting interface within the coexistence region around the first-order phase transition of a fully-connected quantum Ising model in slab geometry. For that, we employ the Lindblad master equation formalism in which temperature is inherited by the coupling to a dissipative bath, rather than being a functional parameter as in the conventional Cahn's free energy. Lindblad's approach gives not only access to the dissipative dynamics and steady-state configuration of the quantum wetting interface throughout the whole phase diagram but also shows that the wetting critical behavior can be successfully exploited to characterize the phase diagram as an alternative to the direct evaluation of the free energies of the competing phases.

MuSpinSim: spin dynamics calculations for muon science

MuSpinSim is a Python software to simulate muon (µSR) experiments. In particular, it simulates the spin dynamics of a system of a muon plus other spins such as electrons and atomic nuclei. MuSpinSim can simulate various common experimental setups used in µSR, such as zero, transverse and longitudinal field experiments; and it can simulate µSR experiments that are resolved in time, field, or temperature. Furthermore, MuSpinSim can account for the effects of hyperfine, dipolar, quadrupolar and Zeeman couplings, as well as simulate quantum systems exchanging energy with the environment with the Lindblad master equation. Finally, MuSpinSim can be used to fit experimental µSR data with simulations that use all of the capabilities described above. In this work, we present the Python package MuSpinSim with all the utilities it provides to facilitate simulations of µSR experiments, and we demonstrate the effectiveness of the method with some chosen example systems.

Quantum Radiating Arrays and Their Directivity: A Dynamical Open System Approach

We investigate the directive properties of radiating quantum antennas comprised of coupled two-level (qubit) systems interacting with bosonic modes and excited by classical fields. A directivity formula inspired by classical antenna theory is proposed. The density operator of a coupled two-level quantum dot (qubit) array, excited by classical external signals with variable amplitude and phase, is evolved in time using the Gorini–Kossakowski–Sudarshan–Lindblad master equation. We illustrate the method in a numerical example where it is shown that manipulating the phase excitations of individual quantum dots may significantly enhance the directive radiation properties of the quantum dot antenna system.

Measurement-Induced Boolean Dynamics for Open Quantum Networks

In this article, we study the recursion corresponding to the measurement outcomes for open quantum networks under sequential measurements. Open quantum networks are networked quantum subsystems (e.g., qubits) with the state evolutions described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo probabilistic jumps and the corresponding measurement outcomes can be described by a vector of probabilistic Boolean variables. The induced recursion of the Boolean vectors forms a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean network can be explicitly represented through a real version of the master equation. Next, when the open quantum dynamics are relaxing in the sense that they possess a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of this relaxing property. Particularly, we show that generically, relaxing quantum dynamics leads to irreducible and aperiodic chains for the measurement outcomes. Finally, we show that for quantum consensus networks, which are a type of nonrelaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph

Lossy Micromaser Battery: Almost Pure States in the Jaynes-Cummings Regime.

We consider a micromaser model of a quantum battery, where the battery is a single mode of the electromagnetic field in a cavity, charged via repeated interactions with a stream of qubits, all prepared in the same non-equilibrium state, either incoherent or coherent, with the matter-field interaction modeled by the Jaynes-Cummings model. We show that the coherent protocol is superior to the incoherent one, in that an effective pure steady state is achieved for generic values of the model parameters. Finally, we supplement the above collision model with cavity losses, described by a Lindblad master equation. We show that battery performances, in terms of stored energy, charging power, and steady-state purity, are slightly degraded up to moderated dissipation rate. Our results show that micromasers are robust and reliable quantum batteries, thus making them a promising model for experimental implementations.

Evidencing non-Bloch dynamics in temporal topolectrical circuits

One of the core concepts from the non-Hermitian skin effect is the extended complex wavevectors (CW) in the generalized Brillouin zone (GBZ), while the origin of CW remains elusive, and further experimental demonstration of GBZ is still lacking. We show that the bulk states of an open quantum system dynamically governed by the Lindblad master equation exhibit non-Bloch evolution which results in CW. Experimentally, we present temporal topolectrical circuits to serve as simulators for the dynamics of an open system. By reconstructing the correspondence between the bulk states of an open system and circuit voltage modes through gauge scale potentials in the circuit, the non-Bloch evolution is demonstrated. Facilitated by the simulators and proper approach to characterize the non-Bloch band proposed here, the GBZ is confirmed. Our work may advance the investigation of the dissipative topological modes and provide a versatile platform for exploring the unique evolution and topology for both closed and open systems.

Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search

In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates γk(t) (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs. iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems.

Understanding Traffic Jams Using Lindblad Superoperators

We propose a model to simulate different traffic-flow conditions in terms of quantum graphs hosting an (N+1)-level dot at each site. Our model allows us to keep track of the type and of the destination of each vehicle. The traffic flow inside the system is encoded in a proper set of Lindbladian local dissipators that describe the time evolution of the system density matrix. Taking advantage of the invariance of the Lindblad master equation under inhomogeneous transformations we derive the quantum Hamiltonian for the bulk dynamics in a proper experimental setup.

Qubit-oscillator relationships in the open quantum Rabi model: the role of dissipation

Using a dissipative quantum Rabi model, we study the dynamics of a slow qubit coupled to a fast quantum harmonic oscillator interacting with a bosonic bath from weak to strong and ultra-strong coupling regimes. Solving the quantum Heisenberg equations of motion, perturbative in the internal coupling between qubit and oscillator, we derive functional relationships directly linking the qubit coordinates in the Bloch sphere to oscillator observables. We then perform accurate time-dependent Matrix Product State simulations and compare our results both with the analytical solutions of the Heisenberg equations of motion, and with numerical solutions of a Lindblad master equation, perturbative in the external coupling between oscillator and environment. Indeed, we show that, up to the strong coupling regime, the qubit state accurately fulfils the derived functional relationships. We analyse in detail the case of a qubit starting with generic coordinates on the Bloch sphere of which we evaluate the three components of the Bloch vector through the averages of oscillator observables. Interestingly, a weak to intermediate oscillator coupling to the bath is able to simplify the Bloch vector evaluation since qubit-oscillator relationships are more immediate. Moreover, by monitoring the qubit fidelity with respect to free limit, we find the parameter regime where the combined effect of internal and external couplings is able to hinder the reliable evaluation of the qubit Bloch vector. Finally, in the ultra-strong coupling regime, non-Markovian effects become robust and the dynamics of qubit and oscillator are inextricably entangled making the qubit Bloch vector evaluation difficult.Graphical

Steady state quantum statistics of a hybrid optomechanical-ferromagnet system: photon and magnon blockade

Magnon and photon blockade implementation and manipulation have significant practical applications in quantum information processing and quantum metrology due to their tight relations to single-photon and -magnon source devices. In this paper, we propose an experimentally feasible hybrid scheme for the dynamical description of the tripartite interacting system consisting of magnon and phonon modes with photons in an optomechanical system, from which we aim to explore the quantum statistics, as well as the magnon and photon blockade phenomenon. To achieve the purpose, the dissipative solution of the system is obtained with the help of the Lindblad master equation. Via employing the equal-time second-order correlation function and using the steady state solution of the system, the statistics and blockade effects of magnon and photon are analyzed and also their dependence on the parameters involved in the system are discussed. Utilizing feasible parameters, our simulations illustrate that, sub-Poissonian behavior and therefore, blockade of magnon and photon are simultaneously achieved. More importantly, the mentioned blockade effects can be obtained in a range of parameters (and not with specific) which makes our proposal easy to access, experimentally. Considering the above realizations, the introduced scheme opens up a pathway to design single-magnon and -photon generators, which are of crucial importance in advanced quantum science and technologies.

Multiple steady states and dark states induced by nonlocal dissipation in a double quantum dot

Both multiple steady states and dark states have potential applications in the quantum information processing in the presence of dissipation. Here, we propose to implement multiple steady states and dark states in a double quantum dot (DQD) system using quantum reservoir engineering. By coupling the DQD to shared reservoirs, multiple steady states of both four- and two-fold degeneracy can be achieved for specific parameters. It is proved that the occurrence of such multiple steady states is attributed to the strong symmetry of the Lindblad master equation. The multiple steady states can be well revealed by the occupation number of the DQD, which exhibits a discontinuity at the strong-symmetric points and changes drastically in the vicinities of these points. In the regime of unique steady state, the system can be stabilized to a pure state, dubbed dark state, with intact coherence in spite of the dissipation. Our work shows that a variety of novel steady states can be implemented in the DQD system, which paves the way for engineering multiple steady states and dark states in the DQD system.

Quantum Nonreciprocal Interactions via Dissipative Gauge Symmetry

One-way nonreciprocal interactions between two quantum systems are typically described by a cascaded quantum master equation, and rely on an effective breaking of time-reversal symmetry as well as the balancing of coherent and dissipative interactions. Here, we present a new approach for obtaining nonreciprocal quantum interactions that is completely distinct from cascaded quantum systems, and that does not in general require broken TRS. Our method relies on a local gauge symmetry present in any Markovian Lindblad master equation. This new kind of quantum nonreciprocity has many implications, including a new mechanism for performing dissipatively-stabilized gate operations on a target quantum system. We also introduce a new, extremely general quantum-information based metric for quantifying quantum nonreciprocity.

Lindblad master equation approach to the topological phase transition in the disordered Su-Schrieffer-Heeger model

We use the Lindblad equation method to investigate the onset of a mobility edge and the topological phase transition in the disordered SSH chain connected to two external baths in the large bias limit. From the scaling properties of the nonequilibrium stationary current flowing across the system, we recover the localization/delocalization in the disordered chain. To probe the topological phase transition in the presence of disorder, we use the even-odd differential occupancy as a means to discriminate topologically trivial from topologically nontrival phases in the out-of-equilibrium system. Eventually, we argue how to generalize our method to other systems undergoing a topological phase transition in the presence of disorder.

Stability and decoherence rates of a GKP qubit protected by dissipation⋆

We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes.