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arXiv:1901.03181 · Journal of Physics A Mathematical and Theoretical

Qubit entanglement generation by Gaussian non-Markovian dynamics

F Benatti, L Ferialdi, Stefano Marcantoni

Published Oct 22, 2018• 3 citations
arXivHTMLPDF

Abstract

Abstract We consider two qubits interacting with a common bosonic bath, but not directly between themselves. We derive the (bipartite) entanglement generation conditions for Gaussian non-Markovian dynamical maps and show that they are similar as in the Markovian regime. However, they depend on different physical coefficients and hold on different time scales. Indeed, for small times, in the non-Markovian regime entanglement is possibly generated on a shorter time scale ( ) than in the Markovian regime ( ). Moreover, although the singular coupling limit of non-Markovian dynamics yields Markovian ones, we show that the same limit does not lead from non-Markovian entanglement generation conditions to Markovian ones. Also, entanglement generation conditions do not depend on the initial time for non-Markovian open dynamics resulting from couplings to bosonic Gaussian baths, while they may depend on time for open dynamics originated by couplings to classical, stochastic Gaussian environments.

DOI
10.1088/1751-8121/aaea13

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Abstract We consider two qubits interacting with a common bosonic bath, but not directly between themselves. We derive the (bipartite) entanglement generation conditions for Gaussian non-Markovian dynamical maps and show that they are similar as in the Markovian regime. However, they depend on different physical coefficients and hold on different time scales. Indeed, for small times, in the non-Markovian regime entanglement is possibly generated on a shorter time scale ( ) than in the Markovian regime ( ). Moreover, although the singular coupling limit of non-Markovian dynamics yields Markovian ones, we show that the same limit does not lead from non-Markovian entanglement generation conditions to Markovian ones. Also, entanglement generation conditions do not depend on the initial time for non-Markovian open dynamics resulting from couplings to bosonic Gaussian baths, while they may depend on time for open dynamics originated by couplings to classical, stochastic Gaussian environments.
Non-Markovian dynamics is studied for two interacting qubits strongly coupled to a dissipative bosonic environment. We derive a non-Markovian quantum-state-diffusion (QSD) equation for the coupled two-qubit system without any approximations, and in particular, without the Markov approximation. As an application and illustration of our derived time-local QSD equation, we investigate the temporal behavior of quantum coherence dynamics. In particular, we find a strongly non-Markovian regime where entanglement generation is significantly modulated by the environmental memory. Additionally, we study residual entanglement in the steady state by analyzing the steady-state solution of the QSD equation. Finally, we discuss an approximate QSD equation.
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.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
The interplay between non-Markovian dynamics and driving fields in the survival of entanglement between two non-degenerate oscillators is considered here. Based on exact analytical results for the non-Markovian dynamics of two parametrically coupled non-degenerate oscillators in contact with non-identical independent thermal baths, the out-of-equilibrium quantum limit derived in [ Phys. Rev. Lett. 105 180501 (2010)] is generalised to the non-Markovian regime. Specifically, it is shown that non-Markovian dynamics, when compared to the Markovian case, allow for the survival of stationary entanglement at higher temperatures, with larger coupling strength to the baths and at smaller driving rates. The effect of the asymmetry of the (i) coupled oscillators, (ii) coupling strength to the baths at equal temperature, and (iii) temperature at equal coupling strength is discussed. In particular, it is shown that the non-Markovian character of the dynamics is capable of beating the resonant condition that states that the driving frequency equals the sum of the natural frequencies for the maximum rate of squeezing generation; hence, squeezing generation is more robust under non-Markovian dynamics.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the `spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
Non-Markovian dynamics of two interacting two-level qubits coupled to a bosonic bath was previously studied using the quantum-state-diffusion (QSD) equation, where a stochastic state is used to describe the system. In this study, we provide another perspective on this system by deriving the analytic form of the master equation, which describes the system with a reduced density matrix. Then, we validate the master equation by examining entanglement generation and state purity. In addition, with the master equation, we observe the effects from first-order noise and notice that a good asymptotic approximation to the master equation can be made by neglecting the first-order noise.