Dynamics Of Open Quantum Systems Research Articles

Dissipatons as generalized Brownian particles for open quantum systems: Dissipaton-embedded quantum master equation.

Dissipaton theory had been proposed as an exact, nonperturbative approach to deal with open quantum system dynamics, where the influence of the Gaussian environment is characterized by statistical quasi-particles, named dissipatons. In this work, we revisit the dissipaton equation of motion theory and establish an equivalent dissipaton-embedded quantum master equation (DQME) that gives rise to dissipatons as generalized Brownian particles. As explained in this work, the DQME supplies a direct approach to investigate the statistical characteristics of dissipatons and, thus, the physically supporting hybrid bath modes. Numerical demonstrations are carried out on the electron transfer model, exhibiting the transient statistical properties of the solvation coordinate.

Tree tensor network state approach for solving hierarchical equations of motion.

The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.

Optimality and Complexity in Measured Quantum-State Stochastic Processes

If an experimentalist observes a sequence of emitted quantum states via either projective or positive-operator-valued measurements, the outcomes form a time series. Individual time series are realizations of a stochastic process over the measurements’ classical outcomes. We recently showed that, in general, the resulting stochastic process is highly complex in two specific senses: (i) it is inherently unpredictable to varying degrees that depend on measurement choice and (ii) optimal prediction requires using an infinite number of temporal features. Here, we identify the mechanism underlying this complicatedness as generator nonunifilarity—the degeneracy between sequences of generator states and sequences of measurement outcomes. This makes it possible to quantitatively explore the influence that measurement choice has on a quantum process’ degrees of randomness and structural complexity using recently introduced methods from ergodic theory. Progress in this, though, requires quantitative measures of structure and memory in observed time series. And, success requires accurate and efficient estimation algorithms that overcome the requirement to explicitly represent an infinite set of predictive features. We provide these metrics and associated algorithms, using them to design informationally-optimal measurements of open quantum dynamical systems.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations.

We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.

QuantumDynamics.jl: A modular approach to simulations of dynamics of open quantum systems.

A simulation of the non-adiabatic dynamics of a quantum system coupled to dissipative environments poses significant challenges. New sophisticated methods are regularly being developed with an eye toward moving to larger systems and more complicated descriptions of solvents. Many of these methods, however, are quite difficult to implement and debug. Furthermore, trying to make the individual algorithms work together through a modular application programming interface can be quite difficult as well. We present a new, open-source software framework, QuantumDynamics.jl, designed to address these challenges. It provides implementations of a variety of perturbative and non-perturbative methods for simulating the dynamics of these systems. Most prominently, QuantumDynamics.jl supports hierarchical equations of motion and methods based on path integrals. An effort has been made to ensure maximum compatibility of the interface between the various methods. Additionally, QuantumDynamics.jl, being built on a high-level programming language, brings a host of modern features to explorations of systems, such as the usage of Jupyter notebooks and high level plotting, the possibility of leveraging high-performance machine learning libraries for further development. Thus, while the built-in methods can be used as end-points in themselves, the package provides an integrated platform for experimentation, exploration, and method development.

Efficient low-temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model

The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.

Full counting statistics and coherences: Fluctuation symmetry in heat transport with the unified quantum master equation.

Recently, a "unified" quantum master equationwas derived and shown to be of the Gorini-Kossakowski-Lindblad-Sudarshan form. This equationdescribes the dynamics of open quantum systems in a manner that forgoes the full secular approximation and retains the impact of coherences between eigenstates close in energy. We implement full counting statistics with the unified quantum master equationto investigate the statistics of energy currents through open quantum systems with nearly degenerate levels. We show that, in general, this equationgives rise to dynamics that satisfy fluctuation symmetry, a sufficient condition for the Second Law of Thermodynamics at the level of average fluxes. For systems with nearly degenerate energy levels, such that coherences build up, the unified equation is simultaneously thermodynamically consistent and more accurate than the fully secular master equation. We exemplify our results for a "V" system facilitating energy transport between two thermal baths at different temperatures. We compare the statistics of steady-state heat currents through this system as predicted by the unified equationto those given by the Redfield equation, which is less approximate but, in general, not thermodynamically consistent. We also compare results to the secular equation, where coherences are entirely abandoned. We find that maintaining coherences between nearly degenerate levels is essential to properly capture the current and its cumulants. On the other hand, the relative fluctuations of the heat current, which embody the thermodynamic uncertainty relation, display inconsequential dependence on quantum coherences.

Open quantum system dynamics of X-states: Entanglement sudden death and sudden birth

In this paper, the origin of disentanglement for two specific sub-classes of [Formula: see text]-states namely maximally nonlocal mixed states (MNMSs) and maximally entangled mixed states (MEMSs) is investigated analytically for a physical system consisting of two spatially separated qubits interacting with a common vacuum bath. The phenomena of entanglement sudden death (ESD) and the entanglement sudden birth (ESB) are observed, but the characteristics of ESD and ESB are found to be different for the case of two-photon coherence and single photon coherence states. The role played by initial coherence for the underlying entanglement dynamics is investigated. Further, the entanglement dynamics of MNMSs and MEMSs under different environmental noises namely phase damping, amplitude damping and random telegraph noise (RTN) noise with respect to the decay and revival of entanglement is analyzed. It is observed that the single photon coherence states are more robust against the sudden death of entanglement, indicating the usability of such states in the development of technologies for the practical implementation of quantum information processing tasks.

Interplay between Non-Markovianity of Noise and Dynamics in Quantum Systems.

The non-Markovianity of open quantum system dynamics is often associated with the bidirectional interchange of information between the system and its environment, and it is thought to be a resource for various quantum information tasks. We have investigated the non-Markovianity of the dynamics of a two-state system driven by continuous time random walk-type noise, which can be Markovian or non-Markovian depending on its residence time distribution parameters. Exact analytical expressions for the distinguishability as well as the trace distance and entropy-based non-Markovianity measures are obtained and used to investigate the interplay between the non-Markovianity of the noise and that of dynamics. Our results show that, in many cases, the dynamics are also non-Markovian when the noise is non-Markovian. However, it is possible for Markovian noise to cause non-Markovian dynamics and for non-Markovian noise to cause Markovian dynamics but only for certain parameter values.

Towards the Resolution of a Quantized Chaotic Phase-Space: The Interplay of Dynamics with Noise.

We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol-type oscillators.

Pontryagin-optimal control of a non-Hermitian qubit

Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a workaround for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity even in the case of inefficient detection.

Succinct Description and Efficient Simulation of Non-Markovian Open Quantum Systems

Non-Markovian open quantum systems represent the most general dynamics when the quantum system is coupled with a bath environment. The quantum dynamics arising from many important applications are non-Markovian. Although for special cases, such as Hamiltonian evolution and Lindblad evolution, quantum simulation algorithms have been extensively studied, efficient quantum simulations for the dynamics of non-Markovian open quantum systems remain underexplored. The most immediate obstacle for studying such systems is the lack of a universal succinct description of their dynamics. In this work, we fulfill the gap of studying such dynamics by (1) providing a succinct representation of the dynamics of non-Markovian open quantum systems with quantifiable error, and (2) developing an efficient quantum algorithm for simulating such dynamics with cost $${\mathcal {O}}(t\, \textrm{polylog}(t/\epsilon ))$$ for evolution time t and precision $$\epsilon $$ . Our derivation of the succinct representation is based on stochastic Schrödinger equations, which could lead to new alternatives to deal with open quantum systems as well.

Wave Function Realization of a Thermal Collision Model.

An efficient algorithm to simulate dynamics of open quantum system is presented. The method describes the dynamics by unraveling stochastic wave functions converging to a density operator description. The stochastic techniques are based on the quantum collision model. Modeling systems dynamics with wave functions and modeling the interaction with the environment with a collision sequence reduces the scale of the complexity significantly. The algorithm developed can be implemented on quantum computers. We introduce stochastic methods that exploit statistical characteristics of the model such as Markovianity, Brownian motion, and binary distribution. The central limit theorem is employed to study the convergence of distributions of stochastic dynamics of pure quantum states represented by wave vectors. By averaging a sample of functions in the distribution we prove and demonstrate the convergence of the dynamics to the mixed quantum state described by a density operator.

Information theoretical limits for quantum optimal control solutions: error scaling of noisy control channels.

Accurate manipulations of an open quantum system require a deep knowledge of its controllability properties and the information content of the implemented control fields. By using tools of information and quantum optimal control theory, we provide analytical bounds (information-time bounds) to characterize our capability to control the system when subject to arbitrary sources of noise. Moreover, since the presence of an external noise field induces open quantum system dynamics, we also show that the results provided by the information-time bounds are in very good agreement with the Kofman-Kurizki universal formula describing decoherence processes. Finally, we numerically test the scaling of the control accuracy as a function of the noise parameters, by means of the dressed chopped random basis (dCRAB) algorithm for quantum optimal control.

Initial correlations in open quantum systems: constructing linear dynamical maps and master equations

We investigate the dynamics of open quantum systems which are initially correlated with their environment. The strategy of our approach is to analyze how given, fixed initial correlations modify the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior with the same fixed initial environmental state, described by a completely positive dynamical map. We show that, for any predetermined initial correlations, one can introduce a linear dynamical map on the space of operators of the open system which acts like the proper dynamical map on the set of physical states and represents its unique linear extension. Furthermore, we demonstrate that this construction leads to a linear, time-local quantum master equation with generalized Lindblad structure involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian dynamics of an open quantum system can be described by means of a time-local master equation even in the case of arbitrary, fixed initial system–environment correlations. We present some illustrative examples and explain the relation of our approach to several other approaches proposed in the literature.

Witnessing non-Markovianity by quantum quasi-probability distributions

We employ frames consisting of rank-one projectors (i.e. pure quantum states) and their induced informationally complete quantum measurements to represent generally mixed quantum states by quasi-probability distributions. In the case of discrete frames on finite dimensional systems this results in a vector like representation by quasi-probability vectors, while for the continuous frame of coherent states in continuous variable (CV) systems the approach directly leads to the celebrated representation by Glauber–Sudarshan P-functions and Husimi Q-functions. We explain that the Kolmogorov distances between these quasi-probability distributions lead to upper and lower bounds of the trace distance which measures the distinguishability of quantum states. We apply these results to the dynamics of open quantum systems and construct a non-Markovianity witness based on the Kolmogorov distance of the P- and Q-functions. By means of several examples we discuss the performance of this witness and demonstrate that it is useful in the regime of high entropy states for which a direct evaluation of the trace distance is typically very demanding. For Gaussian dynamics in CV systems we even find a suitable non-Markovianity measure based on the Kolmogorov distance between the P-functions which can alternatively be used as a witness for non-Gaussianity.

Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems.

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

Semi-Markov processes in open quantum systems: Connections and applications in counting statistics.

Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.

Breakup of the proton halo nucleus 8B near barrier energies

The dynamics of a nuclear open quantum system could be revealed in the correlations between the breakup fragments of halo nuclei. The breakup mechanism of a proton halo nuclear system is of particular interest as the Coulomb polarization may play an important role, which, however, remains an open question. Here we use a highly efficient silicon detector array and measure the correlations between the breakup fragments of 8B incident on 120Sn at near-barrier energies. The energy and angular correlations can be explained by a fully quantum mechanical method based on the state-of-the-art continuum discretized coupled channel calculations. The results indicate that, compared to the neutron halo nucleus 6He, 8B presents distinctive reaction dynamics: the dominance of the elastic breakup. This breakup occurs mainly via the short-lived continuum states, almost exhausts the 7Be yield, indicating the effect of Coulomb polarization on the proton halo state. The correlation information reveals that the prompt breakup mechanism dominates, occurring predominantly on the outgoing trajectory. We also show that, as a large environment, the continuum of 8B breakup may not significantly influence elastic scattering and complete fusion.

Quantum Speed Limit for a Moving Qubit inside a Leaky Cavity

The quantum speed limit (QSL) is a theoretical lower bound of the time required for a quantum system to evolve from an arbitrary initial state to its orthogonal counterpart. This figure can be used to characterize the dynamics of open quantum systems, including non-Markovian maps. In this paper, we investigate the QSL time for a model that consists of a single qubit moving inside a leaky cavity. Notably, we show that for both weak and strong coupling regimes, the QSL time increases while we boost the velocity of the qubit inside the leaky cavity. Moreover, it is observed that by increasing the qubit velocity, the speed of the evolution tends to a constant value, and the system becomes more stable. The results provide a better understanding of the dynamics of atom-photon couplings and can be used to enhance the controllability of quantum systems.