The time-convolutionless (TCL) projection operator technique allows a systematic analysis of the non-Markovian quantum dynamics of open systems. We introduce a class of projection superoperators which project the states of the total system onto certain correlated system-environment states. It is shown that the application of the TCL technique to this class of correlated superoperators enables the non-perturbative treatment of the dynamics of system-environment models for which the standard approach fails in any finite order of the coupling strength. We demonstrate further that the correlated superoperators correspond to the idea of a best guess of conditional quantum expectations which is determined by a suitable Hilbert space average. The general approach is illustrated by means of the model of a spin which interacts through randomly distributed couplings with a finite reservoir consisting of two energy bands. Extensive numerical simulations of the full Schroedinger equation of the model reveal the power and efficiency of the method.
We prove a generalization of the quantum Markovian equation for observables. In this generalized equation, we use superoperators that are fractional powers of completely dissipative superoperators. We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups and describe the properties of this semigroup. We solve the proposed fractional quantum Markovian equation for the harmonic oscillator with linear friction. A fractional power of the Markovian superoperator can be considered a parameter describing a measure of "screening" of the environment of t…
The correlated projection superoperator techniques provide a better understanding about how correlations lead to strong non-Markovian effects in open quantum systems. Their superoperators are independent of initial state, which may not be suitable for some cases. To improve this, we develop a new approach, that is extending the composite system before use the correlated projection superoperator techniques. Such approach allows choosing different superoperators for different initial states. We apply these techniques to a simple model to illustrate the general approach. The numerical simulations…
We derive the explicit commutation relations for the generators of quantum dynamical semigroup - Markovian superoperator evolution, allowing the extension of Baker-Campbell-Hausdorff-type relations to general Lindblad-type evolutions. This provides a novel tool for exploring the interaction of time-dependent coherent and Markovian evolutions - a surprisingly rich set of behaviors which include deformation by coherent driving of Markovian terms, generation of new Lindblad terms from time-dependent noise and even a coherent driving term generated by the interaction of time-separated Markovian no…
Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the v…
We present a canonical derivation of an influence superoperator which generates the reduced dynamics of a Fermionic quantum system linearly coupled to a Fermionic environment initially at thermal equilibrium. We use this formalism to derive a generalized-Lindblad master equation (in the Markovian limit) and a generalized version of the hierarchical equations of motion valid in arbitrary parity-symmetry conditions, important for the correct evaluation of system correlation functions and spectra.
Contrary to conventional quantum mechanics, which treats measurement as instantaneous, here we explore a model for finite-time measurement. The main two-level system interacts with the measurement apparatus in a Markovian way described by the Lindblad equation, and with an environment, which does not include the measuring apparatus. To analyse the environmental effects on the final density operator, we use the Redfield approach, allowing us to consider a non-Markovian noise. In the present hybrid theory, to trace out the environmental degrees of freedom, we use a previously-developed analytic …
By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong non-exponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics…
By modeling the interaction of a system with an environment through a renewal approach, we demonstrate that completely positive non-Markovian dynamics may develop some unexplored non-standard statistical properties. The renewal approach is defined by a set of disruptive events, consisting in the action of a completely positive superoperator over the system density matrix. The random time intervals between events are described by an arbitrary waiting-time distribution. We show that, in contrast to the Markovian case, if one performs a system-preparation (measurement) at an arbitrary time, the s…
The Leggett-Garg inequalities impose restrictions on the values taken by some combinations of the two-time correlation functions of observables in order to be explainable by a noninvasive-realist classical model. While in the unitary dynamics, it is straightforward to compute these correlation functions, open system effects bring in subtleties. Specifically, for non-Markovian dynamics, which involves setting up of system-bath correlations, the Leggett-Garg measurements disrupt these correlations, making a full system-bath Hamiltonian approach natural. However, here we point out that the proble…
Perturbation theory (PT) is a powerful and commonly used tool in the investigation of closed quantum systems. In the context of open quantum systems, PT based on the Markovian quantum master equation is much less developed. The investigation of open systems mostly relies on exact diagonalization of the Liouville superoperator or quantum trajectories. In this approach, the system size is rather limited by current computational capabilities. Analogous to closed-system PT, we develop a PT suitable for open quantum systems. This proposed method is useful in the analytical understanding of open sys…
A gauge-invariant theory of small Markovian errors in quantum gate sets
Quantum Performance Laboratory, Sandia National Laboratories（桑迪亚国家实验室量子性能实验室）；Department of Physics and Astronomy, Quantum New Mexico Institute, University of New Mexico（新墨西哥大学量子新墨西哥学院物理与天文学系）；IonQ
We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level…
We investigate the coherence and non-Markovianity of a quantum tunneling system whose barrier is fluctuated by a telegraph noise, and its energy gap is modulated by Gaussian noise. With the help of averaging method, the system dynamics are analytically derived, and the analytical expression for coherence measure and non-Markovianity for the very limited parameter regimes for both initially coherent and non-coherent states are obtained. We observe non-Markovian dynamics in a situation where the Kubo number is high. It is also found that there is no strong relation between the coherence of the s…
We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems, but …
We consider the problem of the measurement of a system occurring during a finite time interval, while environmentally-induced noise decreases the system-state coherence. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analyti…
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to…
There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time-dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the EM field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open-systems: using a finite bath chosen to …
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 by wave functions and modeling the interaction with the environment with a collision sequence reduces the complexity scale significantly. The algorithm developed, can be implemented on quantum computers. We introduce stochastic methods that exploit statistical characters of the model, as Markovianity, Br…
We employ the theoretical framework of positive operator valued measures, to study Markovian open quantum systems. In particular, we discuss how a quantum system influences its environment. Using the theory of indirect measurements, we then draw conclusions about the information we could hypothetically obtain about the system by observing the environment. Although the environment is not actually observed, we can use these results to describe the change of the quantum system due to its interaction with the environment. We apply this technique to two different problems. In the first part, we stu…
Scalable simulation of non-Markovian quantum transport by stochastic-phase bath reduction
University of California, Berkeley（加州大学伯克利分校）；Flatiron Institute（平顿研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；California Institute of Technology（加州理工学院）
For complex molecules, nuclear degrees of freedom can act as an environment for the electronic `system' variables, allowing the theory and concepts of open quantum systems to be applied. However, when molecular system-environment interactions are non-perturbative and non-Markovian, numerical simulations of the complete system-environment wave function become necessary. These many body dynamics can be very expensive to simulate, and extracting finite-temperature results - which require running and averaging over many such simulations - becomes especially challenging. Here, we present numerical …
We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. The latter emerge in the unraveling of Markovian quantum master equations and/or in continuous quantum measurements. Ensemble-averaging quantum trajectories at the occurrence of quantum jumps, i.e., the jump times, gives rise to a discrete, deterministic evolution which is highly sensitive to the presence of dark states. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hami…
