Abstract We provide a quantitative evaluation of non-Markovianity (NM) for an XX chain of interacting qubits with one end coupled to a reservoir. The NM of several non-Markovian spectral densities is assessed in terms of various quantum state distance (QSD) measures. Our approach is based on the construction of the density matrix of the open chain, without the necessity of a master equation. For the quantification of NM we calculate the dynamics of the QSD measures between the Markovian-damped and various types of non-Markovian-damped cases. Since in the literature several QSD measures, appear in forms that imply trace preserving density matrices, we introduced appropriate modifications so as to render them applicable to the case of decaying traces. The results produce remarkable consistency between the various QSD measures. They also reveal a subtle and potentially useful interplay between qubit–qubit interaction and non-Markovian damping. Our calculations have also uncovered a surprisingly dramatic slowing-down of dissipation by the squared Lorentzian reservoir.
Abstract We provide a quantitative evaluation of non-Markovianity (NM) for an XX chain of interacting qubits with one end coupled to a reservoir. The NM of several non-Markovian spectral densities is assessed in terms of various quantum state distance (QSD) measures. Our approach is based on the construction of the density matrix of the open chain, without the necessity of a master equation. For the quantification of NM we calculate the dynamics of the QSD measures between the Markovian-damped and various types of non-Markovian-damped cases. Since in the literature several QSD measures, appear in forms that imply trace preserving density matrices, we introduced appropriate modifications so as to render them applicable to the case of decaying traces. The results produce remarkable consistency between the various QSD measures. They also reveal a subtle and potentially useful interplay between qubit–qubit interaction and non-Markovian damping. Our calculations have also uncovered a surprisingly dramatic slowing-down of dissipation by the squared Lorentzian reservoir.
We examine different nonlinear open quantum systems and calculate the non-Markovianity based on the distinguishability between two density matrices. We show that for a single spin (qubit) coupled to a bosonic field the non-Markovianity depends on the spectral density function and can take on any number N with 0 ≤ N ≤ 1, meaning the dynamics can be Markovian or highly non-Markovian. For the main result we consider a system of Ntot identical spins coupled to an environment in the mean-field way. Each spin is coupled to a local and the common reservoir. There are only indirect interactions between the spins through the common reservoir. In limit Ntot → 1 the subsystem consisting of a fixed set of n particles reduces to a factorized state in which each factor is a single spin evolving according to a nonlinear Hartree-Lindblad equation which is exactly solvable. For non-stationary initial spin states the non-Markovianity diverges. In this instance we can never approximate the dynamics of the quantum system by a Markovian master equation and this implies that in the mean field models, memory effects are significant.
We analyze and compare different measures for the degree of non-Markovianity in the dynamics of open quantum systems. These measures are based on the distinguishability of quantum states, which is quantified, on the one hand, by the trace distance or, more generally, by the trace norm of the Helstrom matrix and, on the other hand, by entropic quantifiers: the Jensen-Shannon divergence and the Holevo or quantum skew divergence. We explicitly construct a qubit dynamics for which the trace-norm-based non-Markovianity measure is nonzero, while all the entropic measures turn out to be zero. This leads to the surprising conclusion that the non-Markovianity measure which employs the trace norm of the Helstrom matrix is strictly stronger than all entropic non-Markovianity measures.
We analyze the dynamics of quantum correlations between two qubits coupled to a linear chain of oscillators. The chain mediates interactions between the qubits and acts as a non-Markovian reservoir. The model is amenable to an analytical solution when the initial state of the chain is Gaussian. We study the dynamics of the qubits' concurrence starting from a separable state and assuming that the chain spectrum is gapped and the chain is initially in a thermal state. We identify three relevant regimes that depend on the strength of the qubit-chain coupling in relation to the spectral gap. These are (i) the weak-coupling regime, where the qubits are entangled at the asymptotics, (ii) the strong-coupling regime, where the concurrence can exhibit collapses followed by revivals with exponentially attenuated amplitude, and (iii) the thermal damping regime, where the concurrence rapidly vanishes due to the chain's thermal excitations. In all cases, if entanglement is generated, this occurs after a finite time has elapsed. This time scale depends exponentially on the qubits' distance and is determined by the spectral properties of the chain. Entanglement irreversible decay, on the other hand, is due to the dissipative effect induced by the coupling with the chain and is controlled by the coupling strength between the chain and qubits. This study unravels the basic mechanisms leading to entanglement in a non-Markovian bath and allows one to identify the key resources for realizing quantum coherent dynamics of open systems.
In this review we discuss recent progress in the theory of open quantum systems based on non-MarkovianNon-Markovian quantum state diffusionQuantum state diffusion equation and master equationsMaster equation. In particular, we show that an exact master equation for an open quantum system consisting of a few qubits can be explicitly constructed by using the corresponding non-Markovian quantum state diffusion equation. The exact master equation arises naturally from the quantum decoherence dynamics of qubit systems collectively interacting with a colored noise. We illustrate our general theoretical formalism by the explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This exact qubit master equation accurately characterizes the time evolution of the qubit system in various parameter domains, and paves the way for investigation of the memory effect of an open quantum system in a non-Markovian regime without any approximation.
We investigate signatures of non-Markovianity in the dynamics of a periodically driven qubit coupled to a dissipative bosonic environment. We propagate the dynamics of the reduced density matrix of the qubit by integrating the numerically exact hierarchical equations of motion. Non-Markovian features are quantified by comparing on an equal footing the predictions from diverse and complementary approaches to quantum dissipation. In particular, we analyze the distinguishability of quantum states, the decay of the volume accessible to the qubit on the Hilbert space, the negativity of the canonical rates in the generalized Lindblad equation, and the relaxation of the memory kernels in the Nakajima-Zwanzig generalized quantum master equation. We study the effects of controlled driving on the coherent dynamics of the system. We show that a suitable external field can offset the ergodic relaxation of time-correlation functions, increase distinguishability over time, and strengthen non-Markovian effects witnessed by the canonical dissipation channels. We furthermore observe the phenomenon of eternal non-Markovianity for sufficiently small system-bath coupling and we discuss how this can be enhanced by modulating the frequency of the external drive. This work provides a broad theoretical analysis of quantum dissipation in the framework of open quantum dynamics and quantum information.
To study various quantum dynamics, it is important to develop effective methods to detect and distinguish different quantum dynamics. A common non-demolition approach is to couple an auxiliary system (ancilla) to the target system, and to measure the ancilla only. By doing so, the target system becomes an environment for the ancilla. Thus, different quantum dynamics of target systems will correspond to different environment properties. Here, we analytically study XX spin chains presenting different kinds of quantum dynamics, namely localized, delocalized, and dephasing dynamics, and build connections between Markovianity and non-Markovianity-the two most common properties of an environment. For a qubit coupled to the XX chain, we derived the reduced density matrix of the qubit through the projection method. Furthermore, when dephasing noise was introduced to the XX chain, we generalized the projection method by introducing an open-system interaction picture - a modification of the Dirac interaction picture. By calculating the reduced density matrix for the qubit analytically and numerically, we found that the delocalized (localized) chain corresponds to the Markovian (non-Markovian) bath when boundary effects are not considered, and the feature of the chain with dephasing noise as a bath is dependent on the dephasing strength. The three kinds of quantum dynamics can be distinguished by measuring the qubit only.