Dynamics Of Open Quantum Systems Research Articles

Collisional open quantum dynamics with a generally correlated environment: Exact solvability in tensor networks

Quantum collision models are receiving increasing attention as they describe many nontrivial phenomena in the dynamics of open quantum systems. In a general scenario of both fundamental and practical interest, a quantum system repeatedly interacts with individual particles or modes, forming a correlated and structured reservoir; however, classical and quantum environment correlations greatly complicate the calculation and interpretation of the system dynamics. Here, we propose an exact solution to this problem based on the tensor network formalism. We find a natural Markovian embedding for the system dynamics, where the role of an auxiliary system is played by virtual indices of the network. The constructed embedding is amenable to an analytical treatment for a number of timely problems such as the system interaction with two-photon wave packets, structured photonic states, and one-dimensional spin chains. We also derive a time-convolution master equation and relate its memory kernel with the environment correlation function, thus revealing a clear physical picture of memory effects in the dynamics. The results advance tensor-network methods in the fields of quantum optics and quantum transport.

Non-Markovian Quantum Dynamics in Strongly Coupled Multimode Cavities Conditioned on Continuous Measurement

A new theory is introduced that describes the conditioned state of a system with continuous measurement undergoing non-Markovian open quantum systems dynamics, opening new perspectives both on a fundamental and practical level.

Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains

The unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics, which can be radically different from closed-system scenarios. Such open quantum system dynamics is generally described by Lindblad master equations, whose dynamical and steady-state properties are challenging to obtain, especially in the many-particle regime. Here, we introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function with a real-space decimation technique. Compared to other methods, such technique enables obtaining compact analytical expressions for the dynamics and steady-state properties, such as asymptotic decays or correlation lengths. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity, including the Hatano-Nelson model. The latter is especially illustrative because its surface and bulk dissipative behavior are linked due to its non-trivial topology, which manifests in directional amplification.

Open quantum systems coupled to finite baths: A hierarchy of master equations.

An open quantum system in contact with an infinite bath approaches equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium but it induces a dynamical evolution of the bath state. In this paper, we study the dynamics of open quantum systems in contact with finite baths. We obtain a hierarchy of master equationsthat improve their accuracy by including more dynamical information of the bath. For instance, as the least accurate but simplest description in the hierarchy, we obtain the conventional Born-Markov-secular master equation. Remarkably, our framework works even if the measurements of the bath energy are imperfect, which not only is more realistic but also unifies the theoretical description. Also, we discuss this formalism in detail for a particular noninteracting environment where the Boltzmann temperature and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our hierarchy of master equationsto study the central spin model.

Hamiltonian open quantum system toolkit

We present an open-source software package called “Hamiltonian Open Quantum System Toolkit" (HOQST), a collection of tools for the investigation of open quantum system dynamics in Hamiltonian quantum computing, including both quantum annealing and the gate-model of quantum computing. It features the key master equations (MEs) used in the field, suitable for describing the reduced system dynamics of an arbitrary time-dependent Hamiltonian with either weak or strong coupling to infinite-dimensional quantum baths. We present an overview of the theories behind the various MEs and provide examples to illustrate typical workflows in HOQST. We present an example that shows that HOQST can provide order of magnitude speedups compared to “Quantum Toolbox in Python" (QuTiP), for problems with time-dependent Hamiltonians. The package is ready to be deployed on high performance computing (HPC) clusters and is aimed at providing reliable open-system analysis tools for noisy intermediate-scale quantum (NISQ) devices.

From integrability to chaos in quantum Liouvillians

The dynamics of open quantum systems can be described by a Liouvillian, which in the Markovian approximation fulfills the Lindblad master equation. We present a family of integrable many-body Liouvillians based on Richardson-Gaudin models with a complex structure of the jump operators. Making use of this new region of integrability, we study the transition to chaos in terms of a two-parameter Liouvillian. The transition is characterized by the spectral statistics of the complex eigenvalues of the Liouvillian operators using the nearest neighbor spacing distribution and by the ratios between eigenvalue distances.

Geometric quantum speed limits for Markovian dynamics in open quantum systems

We study theoretically the geometric quantum speed limits (QSLs) of open quantum systems under Markovian dynamical evolution. Three types of QSL time bounds are introduced based on the geometric inequality associated with the dynamical evolution from an initial state to a final state. By illustrating three types of QSL bounds at the cases of presence or absence of system driving, we demonstrate that the unitary part, dominated by system Hamiltonian, supplies the internal motivation for a Markovian evolution which deviates from its geodesic. Specifically, in the case of unsaturated QSL bounds, the parameters of the system Hamiltonian serve as the eigen-frequency of the oscillations of geodesic distance in the time domain and, on the other hand, drive a further evolution of an open quantum system in a given time period due to its significant contribution in dynamical speedup. We present physical pictures of both saturated and unsaturated QSLs of Markovian dynamics by means of the dynamical evolution trajectories in the Bloch sphere which demonstrates the significant role of system Hamiltonian even in the case of initial mixed states. It is further indicated that whether the QSL bound is saturated is ruled by the commutator between the Hamiltonian and reduced density matrix of the quantum system. Our study provides a detailed description of QSL times and reveals the effects of system Hamiltonian on the unsaturation of QSL bounds under Markovian evolution.

Markov Approximations of the Evolution of Quantum Systems

The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.

Symmetry-resolved dynamical purification in synthetic quantum matter

When a quantum system initialized in a product state is subjected to either coherent or incoherent dynamics, the entropy of any of its connected partitions generically increases as a function of time, signalling the inevitable spreading of (quantum) information throughout the system. Here, we show that, in the presence of continuous symmetries and under ubiquitous experimental conditions, symmetry-resolved information spreading is inhibited due to the competition of coherent and incoherent dynamics: in given quantum number sectors, entropy decreases as a function of time, signalling dynamical purification. Such dynamical purification bridges between two distinct short and intermediate time regimes, characterized by a log-volume and log-area entropy law, respectively. It is generic to symmetric quantum evolution, and as such occurs for different partition geometry and topology, and classes of (local) Liouville dynamics. We then develop a protocol to measure symmetry-resolved entropies and negativities in synthetic quantum systems based on the random unitary toolbox, and demonstrate the generality of dynamical purification using experimental data from trapped ion experiments [Brydges et al., Science 364, 260 (2019)]. Our work shows that symmetry plays a key role as a magnifying glass to characterize many-body dynamics in open quantum systems, and, in particular, in noisy-intermediate scale quantum devices.

Simulation of open quantum systems by automated compression of arbitrary environments

The central challenge for describing the dynamics in open quantum systems is that the Hilbert space of typical environments is too large to be treated exactly. In some cases, such as when the environment has a short memory time or only interacts weakly with the system, approximate descriptions of the system are possible. Beyond these, numerically exact methods exist, but these are typically restricted to baths with Gaussian correlations, such as non-interacting bosons. Here we present a numerically exact method for simulating open quantum systems with arbitrary environments which consist of a set of independent degrees of freedom. Our approach automatically reduces the large number of environmental degrees of freedom to those which are most relevant. Specifically, we show how the process tensor -- which describes the effect of the environment -- can be iteratively constructed and compressed using matrix product state techniques. We demonstrate the power of this method by applying it to problems with bosonic, fermionic, and spin environments: electron transport, phonon effects and radiative decay in quantum dots, central spin dynamics, anharmonic environments, dispersive coupling to time-dependent lossy cavity modes, and superradiance. The versatility and efficiency of our automated compression of environments (ACE) method provides a practical general-purpose tool for open quantum systems.

Decoherence-Induced Exceptional Points in a Dissipative Superconducting Qubit.

Open quantum systems interacting with an environment exhibit dynamics described by the combination of dissipation and coherent Hamiltonian evolution. Taken together, these effects are captured by a Liouvillian superoperator. The degeneracies of the (generically non-Hermitian) Liouvillian are exceptional points, which are associated with critical dynamics as the system approaches steady state. We use a superconducting transmon circuit coupled to an engineered environment to observe two different types of Liouvillian exceptional points that arise either from the interplay of energy loss and decoherence or purely due to decoherence. By dynamically tuning the Liouvillian superoperators in real time we observe a non-Hermiticity-induced chiral state transfer. Our study motivates a new look at open quantum system dynamics from the vantage of Liouvillian exceptional points, enabling applications of non-Hermitian dynamics in the understanding and control of open quantum systems.

Improving the efficiency of open-quantum-system simulations using matrix product states in the interaction picture

Modeling open quantum systems---quantum systems coupled to a bath---is of value in condensed-matter theory, cavity quantum electrodynamics, nanosciences, and biophysics. The real-time simulation of open quantum systems was advanced significantly by the recent development of chain mapping techniques and the use of matrix product states that exploit the intrinsic entanglement structure in open quantum systems. The computational cost of simulating open quantum systems, however, remains high when the bath is excited to high-lying quantum states. We develop an approach to reduce the computational costs in such cases. The interaction representation for the open quantum system is used to distribute excitations among the bath degrees of freedom so that the occupation of each bath oscillator is ensured to be low. The interaction picture also causes the matrix dimensions to be much smaller in a matrix product state of a chain-mapped open quantum system than in the Schr\"odinger picture. Using the interaction representation accelerates the calculations by one to two orders of magnitude over the existing matrix-product-state method. In the regime of strong system-bath coupling and high temperatures, the speedup can be as large as three orders of magnitude. The approach developed here is especially promising to simulate the dynamics of open quantum systems in high-temperature and strong-coupling regimes.

Open quantum system dynamics and the mean force Gibbs state

The dynamical convergence of a system to the thermal distribution, or Gibbs state, is a standard assumption across all of the physical sciences. The Gibbs state is determined just by temperature and energies of the system. However, at decreasing system sizes, i.e., for nanoscale and quantum systems, the interaction with their environments is not negligible. The question then arises: Is the system's steady state still the Gibbs state? If not, how may the steady state depend on the interaction details? Here, we provide an overview of recent progress on answering these questions. We expand on the state of the art along two general avenues: First, we take the static point-of-view, which postulates the so-called mean force Gibbs state. This view is commonly adopted in the field of strong coupling thermodynamics, where modified laws of thermodynamics and nonequilibrium fluctuation relations are established on the basis of this modified state. Second, we take the dynamical point of view, originating from the field of open quantum systems, which examines the time-asymptotic steady state within two paradigms. We describe the mathematical paradigm, which proves return to equilibrium, i.e., convergence to the mean force Gibbs state, and then discuss a number of microscopic physical methods, particularly master equations. We conclude with a summary of established links between statics and equilibration dynamics and provide an extensive list of open problems. This comprehensive overview will be of interest to researchers in the wider fields of quantum thermodynamics, open quantum systems, mesoscopic physics, statistical physics, and quantum optics and will find applications whenever energy is exchanged on the nanoscale, from quantum chemistry and biology to magnetism and nanoscale heat management.

Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation.

The theory of open quantum systems lays the foundation for a substantial part of modern research in quantum science and engineering. Rooted in the dimensionality of their extended Hilbert spaces, the high computational complexity of simulating open quantum systems calls for the development of strategies to approximate their dynamics. In this Letter, we present an approach for tackling open quantum system dynamics. Using an exact probabilistic formulation of quantum physics based on positive operator-valued measure, we compactly represent quantum states with autoregressive neural networks; such networks bring significant algorithmic flexibility due to efficient exact sampling and tractable density. We further introduce the concept of string states to partially restore the symmetry of the autoregressive neural network and improve the description of local correlations. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator using a forward-backward trapezoid method and find the steady state via a variational formulation. Our approach is benchmarked on prototypical one-dimensional and two-dimensional systems, finding results which closely track the exact solution and achieve higher accuracy than alternative approaches based on using Markov chain MonteCarlo method to sample restricted Boltzmann machines. Our Letter provides general methods for understanding quantum dynamics in various contexts, as well as techniques for solving high-dimensional probabilistic differential equations in classical setups.

Revealing higher-order light and matter energy exchanges using quantum trajectories in ultrastrong coupling

The dynamics of open quantum systems is often modelled using master equations, which describe the expected outcome of an experiment (i.e., the average over many realizations of the same dynamics). Quantum trajectories, instead, model the outcome of ideal single experiments -- the ``clicks'' of a perfect detector due to, e.g., spontaneous emission. The correct description of quantum jumps, which are related to random events characterizing a sudden change in the wave function of an open quantum system, is pivotal to the definition of quantum trajectories. In this article, we extend the formalism of quantum trajectories to open quantum systems with ultrastrong coupling (USC) between light and matter by properly defining jump operators in this regime. In such systems, exotic higher-order quantum-state- and energy-transfer can take place without conserving the total number of excitations in the system. The emitted field of such USC systems bears signatures of these higher-order processes, and significantly differs from similar processes at lower coupling strengths. Notably, the emission statistics must be taken at a single quantum trajectory level, since the signatures of these processes are washed out by the ``averaging'' of a master equation. We analyze the impact of the chosen unravelling (i.e., how one collects the output field of the system) for the quantum trajectories and show that these effects of the higher-order USC processes can be revealed in experiments by constructing histograms of detected quantum jumps. We illustrate these ideas by analyzing the excitation of two atoms by a single photon~[Garziano et al., Phys. Rev. Lett.117, 043601 (2016)]. For example, quantum trajectories reveal that keeping track of the quantum jumps from the atoms allow to reconstruct both the oscillations between one photon and two atoms, as well as emerging Rabi oscillations between the two atoms.

Open quantum dynamics for plant motions

Stochastic Schrödinger equations that govern the dynamics of open quantum systems are given by the equations for signal processing. In particular, the Brownian motion that drives the wave function of the system does not represent noise, but provides purely the arrival of new information. Thus the wave function is guided by the optimal signal detection about the conditions of the environments under noisy observations. This behaviour is similar to biological systems that detect environmental cues, process this information, and adapt to them optimally by minimising uncertainties about the conditions of their environments. It is postulated that information-processing capability is a fundamental law of nature, and hence that models describing open quantum systems can equally be applied to biological systems to model their dynamics. For illustration, simple stochastic models are considered to capture heliotropic and gravitropic motions of plants. The advantage of such dynamical models is that they allow for the quantification of information processed by the plants. By considering the consequence of information erasure, it is argued that biological systems can process environmental signals relatively close to the Landauer limit of computation, and that loss of information must lie at the heart of ageing in biological systems.

Quantum correlations of localized atomic excitations in a disordered atomic chain

The atom-waveguide interface mediates significant and long-range light-matter interactions through guided modes. In this one-dimensional system, we theoretically investigate the excitation localization of multiple atomic excitations under strong position disorder. Deep in the localization side, we obtain the time evolutions of quantum correlations via Kubo cumulant expansions, which arise initially and become finite and leveled afterward, overtaking those without disorder. This indicates two distinct regimes in time: before the onset of excitation localization, the disorder engage the disturbance of quantum correlations, which is followed by disorder-assisted buildup of quantum correlations that maintain at a later stage owing to the absence of excitations diffusion. The crossing of distinct regimes is pushed further in time for longer-range correlations, which indicates a characteristic timescale needed for disorder to sustain them. We also explore the effect of directionality of couplings and resonant dipole-dipole interactions, which can drive the system toward the delocalized side when it is under chiral couplings or large dipole-dipole interaction strengths. The time-evolved quantum correlations can give insights into the studies of few-body localization phenomenon and nonequilibrium dynamics in open quantum systems.

Time-periodic interaction between a spin-pair: A quantum master equation approach

Quantum master equations describe the dynamics of open quantum systems. Such systems might be subjected to external drives and internal interactions such as direct dipolar couplings. Both the external drives and the internal interactions could be oscillatory functions of time. In particular, a spin pair undergoing magic angle sample spinning experiences such oscillatory interactions in time. In the present work, we analyze a spin pair with a time-periodic interaction by using our recently-proposed fluctuation-regulated quantum master equation [PRA, 97, 063837 (2018)] (FRQME). FRQME helps calculate the first as well as the second-order effects of all time-dependent interactions Hamiltonians. Using FRQME, we find the usual sideband pattern and also have shown the linewidths of the sidebands as a function of the coupling strength, the period of oscillation, and an environmental parameter. We show that at high spinning speed, the sidebands would have a Lorentzian shape. We predict that the dipolar contributions to the sidebands’ linewidths are smaller for a shorter correlation time of the environmental fluctuations.

Many-Body Quantum State Diffusion for Non-Markovian Dynamics in Strongly Interacting Systems.

Capturing non-Markovian dynamics of open quantum systems is generally a challenging problem, especially for strongly interacting many-body systems. In this Letter, we combine recently developed non-Markovian quantum state diffusion techniques with tensor network methods to address this challenge. As a first example, we explore a Hubbard-Holstein model with dissipative phonon modes, where this new approach allows us to quantitatively assess how correlations spread in the presence of non-Markovian dissipation in a 1D many-body system. We find regimes where correlation growth can be enhanced by these effects, offering new routes for dissipatively enhancing transport and correlation spreading, relevant for both solid state and cold atom experiments.

Digital Quantum Simulation of Open Quantum Systems Using Quantum Imaginary–Time Evolution

Quantum simulation on emerging quantum hardware is a topic of intense interest. While many studies focus on computing ground state properties or simulating unitary dynamics of closed systems, open quantum systems are an interesting target of study owing to their ubiquity and rich physical behavior. However, their non-unitary dynamics are also not natural to simulate on digital quantum devices. Here, we report algorithms for the digital quantum simulation of the dynamics of open quantum systems governed by a Lindblad equation using adaptations of the quantum imaginary time evolution (QITE) algorithm. We demonstrate the algorithms on IBM Quantum's hardware with simulations of the spontaneous emission of a two level system and the dissipative transverse field Ising model. Our work advances efforts to simulate the dynamics of open quantum systems on quantum hardware.