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Showing papers on "Master equation published in 2020"


Journal ArticleDOI
TL;DR: The Lindblad master equation as discussed by the authors is the most general generator of Markovian dynamics in quantum systems, and its derivation and methods of resolution can be found in this paper.
Abstract: The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.

284 citations


Journal ArticleDOI
TL;DR: The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories.
Abstract: An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

223 citations


Journal ArticleDOI
TL;DR: The hierarchical equations of motion (HEOM) theory as discussed by the authors can describe numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system.
Abstract: An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system--bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article, presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

139 citations


Journal ArticleDOI
TL;DR: A topological classification applicable to open fermionic systems governed by a general class of Lindblad master equations is uncovered, highlighting the sensitivity of topological properties to details of the system-environment coupling.
Abstract: We uncover a topological classification applicable to open fermionic systems governed by a general class of Lindblad master equations. These ``quadratic Lindbladians'' can be captured by a non-Hermitian single-particle matrix which describes internal dynamics as well as system-environment coupling. We show that this matrix must belong to one of ten non-Hermitian Bernard-LeClair symmetry classes which reduce to the Altland-Zirnbauer classes in the closed limit. The Lindblad spectrum admits a topological classification, which we show results in gapless edge excitations with finite lifetimes. Unlike previous studies of purely Hamiltonian or purely dissipative evolution, these topological edge modes are unconnected to the form of the steady state. We provide one-dimensional examples where the addition of dissipators can either preserve or destroy the closed classification of a model, highlighting the sensitivity of topological properties to details of the system-environment coupling.

123 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the transition from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach and found the signatures of the transitions as peak structures in the mutual information as a function of measurement strength.
Abstract: We investigate entanglement phase transitions from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach. We find the signatures of the transitions as peak structures in the mutual information as a function of measurement strength, as previously reported for random unitary circuits with projective measurements. At the transition points, the entanglement entropy scales logarithmically and various physical quantities scale algebraically, implying emergent conformal criticality, for both integrable and nonintegrable one-dimensional interacting Hamiltonians; however, such transitions have been argued to be absent in noninteracting regimes in some previous studies. With the aid of $U(1)$ symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.

119 citations


Journal ArticleDOI
TL;DR: A Markovian master equation in the Lindblad form is developed that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods and ensures preservation of the positivity of the density matrix.
Abstract: We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.

89 citations


Journal ArticleDOI
TL;DR: This version of KinBot tackles C, H, O and S atom containing species and unimolecular reactions, and automatically characterizes kinetically important stationary points on reactive potential energy surfaces and arranges the results into a form that lends itself easily to master equation calculations.

75 citations


Journal ArticleDOI
01 Jul 2020
TL;DR: In this paper, a broad overview of the recent progress in the field of quantum electrodynamics in the ultrastrong-coupling regime is presented, ranging from analytical estimate of ground-state properties to proper derivations of master equations and computation of photodetection signals.
Abstract: This article reviews theoretical methods developed in the last decade to understand cavity quantum electrodynamics in the ultrastrong-coupling regime, where the strength of the light-matter interaction becomes comparable to the photon frequency. Along with profound modifications of fundamental quantum optical effects giving rise to a rich phenomenology, this regime introduces significant theoretical challenges. One of the most important is the break-down of the rotating-wave approximation which neglects all non-resonant terms in light-matter interaction Hamiltonians. Consequently, a large part of the quantum optical theoretical framework has to be revisited in order to accurately account for all interaction terms in this regime. We give in this article a broad overview of the recent progress, ranging from analytical estimate of ground-state properties to proper derivations of master equations and computation of photodetection signals. For each aspect of the theory, the basic principles of the methods are illustrated on paradigmatic models such as quantum Rabi and spin-boson models. In this spirit, most of the article is devoted to effective models, relevant for the various experimental platforms in which the ultrastrong coupling has been reached, such as semiconductor microcavities and superconducting circuits. The validity of these models is discussed in the last part of the article, where we address recent debates on fundamental issues related to gauge invariance in the ultrastrong-coupling regime.

72 citations


Journal ArticleDOI
TL;DR: In this article, Naghiloo et al. generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations.
Abstract: Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti et al., Phys. Rev. A 100, 062131 (2019)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo et al., Nat. Phys. 15, 1232 (2019)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.

71 citations


Journal ArticleDOI
TL;DR: A near quantum limited detector is used to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit, and the first law of thermodynamics for an open quantum system is verified.
Abstract: We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit.

57 citations


Journal ArticleDOI
TL;DR: In this article, the authors focus on the accuracy of the reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian and provide a thorough assessment for a selection of methods (Redfield equation, quantum optical master equation, coarse-grained master equations, a related dynamical map approach, and a partial-secular approximation).
Abstract: The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time-local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straightforward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield equation, quantum optical master equation, coarse-grained master equation, a related dynamical map approach, and a partial-secular approximation). We consider secondary, here, whether or not an approach guarantees positivity. We use two qubits coupled to a Lorentzian environment in a spin-boson-like fashion modeling a generic situation with various system and bath time scales. We see that, independent of the initial state, the simple Redfield equation with time-dependent coefficients is significantly more accurate than all other methods under consideration. We emphasize that positivity violation in the Redfield equation formalism becomes relevant only in a regime where any of the perturbative master equations considered here are rendered invalid anyway. This implies that the loss of positivity should in fact be welcomed as an important feature: it indicates the breakdown of the weak-coupling assumption. In addition we present the various approaches in a self-contained way and use the behavior of their errors to provide further insight into the range of validity of each method.

Journal ArticleDOI
06 Feb 2020
TL;DR: A recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving.
Abstract: Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.

Journal ArticleDOI
TL;DR: Andersson et al. as mentioned in this paper proposed a first step in a joint work in addressing the linear stability of slowly rotating Kerr metrics, and proved an energy and Morawetz estimate for spin ��\pm 2$$ components.
Abstract: This second part of the series treats spin $$\pm 2$$ components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin $$\pm 2$$ components is proved by treating this system. This is a first step in a joint work (Andersson et al. in Stability for linearized gravity on the Kerr spacetime, arXiv:1903.03859 , 2019) in addressing the linear stability of slowly rotating Kerr metrics.

Journal ArticleDOI
TL;DR: In this article, the authors studied the real-time evolution of heavy quarkonium in the quark-gluon plasma (QGP) on the basis of the open quantum systems approach.
Abstract: In this paper we study the real-time evolution of heavy quarkonium in the quark-gluon plasma (QGP) on the basis of the open quantum systems approach. In particular, we shed light on how quantum dissipation affects the dynamics of the relative motion of the quarkonium state over time. To this end we present a novel nonequilibrium master equation for the relative motion of quarkonium in a medium, starting from Lindblad operators derived systematically from quantum field theory. In order to implement the corresponding dynamics, we deploy the well established quantum state diffusion method. In turn we reveal how the full quantum evolution can be cast in the form of a stochastic nonlinear Schr\"odinger equation. This for the first time provides a direct link from quantum chromodynamics to phenomenological models based on nonlinear Schr\"odinger equations. Proof of principle simulations in one dimension show that dissipative effects indeed allow the relative motion of the constituent quarks in a quarkonium at rest to thermalize. Dissipation turns out to be relevant already at early times well within the QGP lifetime in relativistic heavy ion collisions.

Journal ArticleDOI
24 Sep 2020-Entropy
TL;DR: A pedagogical introduction to the notion of thermodynamic length is introduced, and a geometric lower bound on entropy production in finite-time is presented, which represents a quantum generalisation of the original classical bound.
Abstract: Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.

Journal ArticleDOI
TL;DR: In this article, it was shown that the Markov approximation is valid for fixed small coupling strength and for all times, under the additional constraint that time t must not exceed an upper bound, λ 2 t ≤ constant.

Journal ArticleDOI
TL;DR: In this paper, the authors derive a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes, including thermal damping, and show that when this master equation breaks down, so do all time independent Markovian equations, including the B-R equation.
Abstract: Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the steady state properties arising from the open system dynamics described by a memoryless (Markovian) quantum collision model, corresponding to a master equation in the ultra-strong coupling regime.

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed the nonextensive thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow black hole and discussed the Tsallis' formulation of this logarithmically corrected Barrow entropy to construct the equipartition law.

Journal ArticleDOI
TL;DR: In this article, a pedagogical introduction to the notion of thermodynamic length is given, and a geometric lower bound on entropy production in finitetime is presented, which represents a quantum generalisation of the original classical bound.
Abstract: Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finitetime is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.

Journal ArticleDOI
TL;DR: In this article, the authors constructed Lorentz invariant and gauge invariant 1PI effective actions for closed and open superstrings and demonstrated that they satisfy the classical BV master equation.
Abstract: We construct Lorentz invariant and gauge invariant 1PI effective action for closed and open superstrings and demonstrate that it satisfies the classical BV master equation. We also construct the quantum master action for this theory satisfying the quantum BV master equation and generalize the construction to unoriented theories. The extra free field needed for the construction of closed superstring field theory plays a crucial role in coupling the closed strings to D-branes and orientifold planes.

Posted Content
TL;DR: In this article, all the existing master equations for quarkonium are systematically rederived as Lindblad equations in a uniform framework and the quantum Brownian motion of heavy quark pair in the QGP is studied in detail.
Abstract: Dissociation of quarkonium in quark-gluon plasma (QGP) is a long standing topic in relativistic heavy-ion collisions because it signals one of the fundamental natures of the QGP -- Debye screening due to the liberation of color degrees of freedom. Among recent new theoretical developments is the application of open quantum system framework to quarkonium in the QGP. Open system approach enables us to describe how dynamical as well as static properties of QGP influences the time evolution of quarkonium. Currently, there are several master equations for quarkonium corresponding to various scale assumptions, each derived in different theoretical frameworks. In this review, all of the existing master equations are systematically rederived as Lindblad equations in a uniform framework. Also, as one of the most relevant descriptions in relativistic heavy-ion collisions, quantum Brownian motion of heavy quark pair in the QGP is studied in detail. The quantum Brownian motion is parametrized by a few fundamental quantities of QGP such as real and imaginary parts of heavy quark potential (complex potential), heavy quark momentum diffusion constant, and thermal dipole self-energy constant. This indicates that the yields of quarkonia such as $J/\psi$ and $\Upsilon$ in the relativistic heavy-ion collisions have the potential to determine these fundamental quantities.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a protocol to realize atomic nonadiabatic holonomic quantum computation (NHQC) with two computational atoms and an auxiliary atom in the regime of Rydberg blockade, and robust laser pulses were designed via reverse engineering, so that quantum gates can be easily realized with high fidelity.
Abstract: We propose a protocol to realize atomic nonadiabatic holonomic quantum computation (NHQC) with two computational atoms and an auxiliary atom. Dynamics of the system is analyzed in the regime of Rydberg blockade, and robust laser pulses are designed via reverse engineering, so that quantum gates can be easily realized with hig h fidelities. In addition, we also study the evolution suffering from dissipation with a master equation. The result indicates that decays of atoms can be heralded by measuring the state of the auxiliary atom, and nearly perfect unitary evolution can be obtained if the auxiliary atom remains in its Rydberg state. Therefore, the protocol may be helpful to realize NHQC in a dissipative environment.

Posted Content
TL;DR: In this paper, the authors construct global in time solutions to both the scalar and vectorial master equations in potential mean field games, when the underlying space is the whole space and so, it is not compact.
Abstract: This manuscript constructs global in time solutions to the $master\ equations$ for potential Mean Field Games. The study concerns a class of Lagrangians and initial data functions, which are $displacement\ convex$ and so, it may be in dichotomy with the class of so--called $monotone$ functions, widely considered in the literature. We construct solutions to both the scalar and vectorial master equations in potential Mean Field Games, when the underlying space is the whole space $\mathbb{R}^d$ and so, it is not compact.

Journal ArticleDOI
TL;DR: In this paper, it was shown that in a self-consistent description of the dissipative dynamics in a one-band lattice, based on the stochastic Schr\"odinger equation or Lindblad master equation with a collective jump operator, the skin effect and its dynamical features are washed out.
Abstract: The non-Hermitian skin effect, i.e., eigenstate condensation at the edges in lattices with open boundaries, is an exotic manifestation of non-Hermitian systems. In Bloch theory, an effective non-Hermitian Hamiltonian is generally used to describe dissipation, which, however, is not norm preserving and neglects quantum jumps. Here it is shown that in a self-consistent description of the dissipative dynamics in a one-band lattice, based on the stochastic Schr\"odinger equation or Lindblad master equation with a collective jump operator, the skin effect and its dynamical features are washed out. Nevertheless, both short- and long-time relaxation dynamics provide a hidden signature of the skin effect found in the semiclassical limit. In particular, relaxation toward a maximally mixed state with the largest von Neumann entropy in a lattice with open boundaries is a manifestation of the semiclassical skin effect.

Journal ArticleDOI
TL;DR: Shammah et al. as mentioned in this paper studied the all-to-all connected (Anisotropic-Heisenberg) spin model with local and collective dissipations, comparing the results of mean-field (MF) theory with the solution of the Lindblad master equation.
Abstract: We study the all-to-all connected $\mathit{XYZ}$ (anisotropic-Heisenberg) spin model with local and collective dissipations, comparing the results of mean-field (MF) theory with the solution of the Lindblad master equation. Exploiting the weak $\mathcal{PT}$ symmetry of the model (referred to as Liouvillian $\mathbb{PT}$ symmetry), we efficiently calculate the Liouvillian gap, introducing the idea of an antigap, and we demonstrate the presence of a paramagnetic-to-ferromagnetic phase transition. Leveraging the permutational symmetry of the model [N. Shammah et al., Phys Rev. A 98, 063815 (2018)], we characterize criticality, finding exactly (up to numerical precision) the steady state for $N$ up to $N=95$ spins. We demonstrate that the MF theory correctly predicts the results in the thermodynamic limit in all regimes of parameters, and quantitatively describes the finite-size behavior in the small anisotropy regime. However, for an intermediate number of spins and for large anisotropy, we find a significant discrepancy between the results of the MF theory and those of the full quantum simulation. We also study other more experimentally accessible witnesses of the transition, which can be used for finite-size studies, namely, the bimodality coefficient and the angular-averaged susceptibility. In contrast to the bimodality coefficient, the angular-averaged susceptibility fails to capture the onset of the transition, in striking difference with respect to lower-dimensional studies. We also analyze the competition between local dissipative processes (which disentangle the spin system) and collective dissipative ones (generating entanglement). The nature of the phase transition is almost unaffected by the presence of these terms. Our results mark a stark difference with the common intuition that an all-to-all connected system should fall onto the mean-field solution also for intermediate number of spins.

Journal ArticleDOI
TL;DR: In this paper, a two-parameter perturbative expansion in qubit anharmonicity and the drive amplitude through a unitary transformation technique was introduced. But the authors did not consider the effect of number-nonconserving terms in the Josephson potential.
Abstract: Recent experiments in superconducting qubit systems have shown an unexpectedly strong dependence of the qubit relaxation rate on the readout drive power. This phenomenon limits the maximum measurement strength and thus the achievable readout speed and fidelity. We address this problem here and provide a plausible mechanism for drive-power dependence of relaxation rates. To this end we introduce a two-parameter perturbative expansion in qubit anharmonicity and the drive amplitude through a unitary transformation technique introduced in Part I. This approach naturally reveals number-nonconserving terms in the Josephson potential as a fundamental mechanism through which applied microwave drives can activate additional relaxation mechanisms. We present our results in terms of an effective master equation with renormalized state- and drive-dependent transition frequency and relaxation rates. Comparison of numerical results from this effective master equation to those obtained from a Lindblad master equation which only includes number-conserving terms (i.e., Kerr interactions) shows that number-nonconserving terms can lead to significant drive-power dependence of the relaxation rates. The systematic expansion technique introduced here is of general applicability to obtaining effective master equations for driven-dissipative quantum systems that contain weakly nonlinear degrees of freedom.

Book ChapterDOI
01 Jan 2020
TL;DR: In this article, the authors introduce the Mean Field Game (MFG) theory, which models differential games involving infinitely many interacting players, and focus on the Partial Differential Equations (PDEs) approach to MFGs.
Abstract: These notes are an introduction to Mean Field Game (MFG) theory, which models differential games involving infinitely many interacting players. We focus here on the Partial Differential Equations (PDEs) approach to MFGs. The two main parts of the text correspond to the two emblematic equations in MFG theory: the first part is dedicated to the MFG system, while the second part is devoted to the master equation.

Journal ArticleDOI
TL;DR: In this article, the authors present a general master equation formalism for the interaction between traveling pulses of quantum radiation and localized quantum systems, which is applicable to the transformation and interaction of pulses of radiation by their coupling to a wide class of material quantum systems.
Abstract: This paper presents a general master equation formalism for the interaction between traveling pulses of quantum radiation and localized quantum systems. Traveling fields populate a continuum of free-space radiation modes and the Jaynes-Cummings model, valid for a discrete eigenmode of a cavity, does not apply. We develop a complete input-output theory to describe the driving of quantum systems by arbitrary incident pulses of radiation and the quantum state of the field emitted into any desired outgoing temporal mode. Our theory is applicable to the transformation and interaction of pulses of radiation by their coupling to a wide class of material quantum systems. We discuss the most essential differences between quantum interactions with pulses and with discrete radiative eigenmodes and present examples relevant to quantum information protocols with optical, microwave, and acoustic waves.

Journal ArticleDOI
TL;DR: A coherent and easy-to-understand way of doing quantum mechanics in any finite-dimensional Liouville space, based on the use of Kronecker product and what is termed the `bra-flipper' operator is articulate.
Abstract: The purpose of this paper is to articulate a coherent and easy-to-understand way of doing quantum mechanics in any finite-dimensional Liouville space, based on the use of Kronecker product and what we have termed the `bra-flipper' operator. One of the greater strengths of the formalism expatiated on here is the striking similarities it bears with Dirac's bra-ket notation. For the purpose of illustrating how the formalism can be effectively employed, we use it to solve a quantum optical master equation for a two-level quantum system and find its Kraus operator sum representation. The paper is addressed to students and researchers with some basic knowledge of linear algebra who want to acquire a deeper understanding of the Liouville space formalism. The concepts are conveyed so as to make the application of the formalism to more complex problems in quantum physics straightforward and unencumbered.