Problem of coherent control in non-Markovian open quantum systems

Abstract

We critically evaluate the most widespread assumption in the theoretical description of coherent control strategies for open quantum systems. We show that, for non-Markovian open systems dynamics, this fixed-dissipator assumption leads to a serious pitfall generally causing difficulties in the effective modeling of the controlled system. We show that at present, to avoid these problems, a full microscopic description of the controlled system in the presence of noise may often be necessary. We illustrate our findings with a paradigmatic example. [17]. Generally, the theoretical description of these techniques in the presence of noise is a daunting task, therefore they are typically studied under a number of assumptions concerning the type of environments the system is interacting with as well as the typical time-scales. Specifically, optimal control techniques have been so far studied, almost exclusively, in the so-called Markovian limit, that is whenever the system-environment interaction is weak and the correlations short living. In this case the master equations describing the open system dynamics are found phenomenologically or derived with microscopic approaches using numerous approximations [6] [30].In this article, we expose the difficulties in employing coherent control to compensate for environment-induced decoherence effects in non-Markovian systems. We consider the widespread assumption (fixed dissipator assumption) that the part of the master equation describing dissipation and dephasing does not change when we add a Hamiltonian control term in the unitary dynamics part. This assumption does not change the physicality of the solutions of the master equation in the Markovian case. We show, however, that this is generally not the case for non-Markovian dynamics. Hence the typical theoretical approaches to quantum control theory cannot be used in the framework of non-Markovian open quantum systems, and only a full microscopic derivation leads to physically meaningful results.For the sake of concreteness we focus on a novel concept utilizing Hamiltonian control recently introduced to counteract the detrimental effect of decoherence. In Ref.[6], the goal is to seek the control Hamiltonian that, on asymptotic time scales, optimally upholds a given target property (e.g. coherence, entanglement or fidelity with respect to a target state). The space of Hamiltonians cannot be efficiently parametrized, hence the problem is approached from a different perspective. The key idea is to optimize some target property in the set of stabilizable cycles, comprising all closed periodic trajectories ρ(t) = ρ(t + T ) for which a periodic control Hamiltonian exists such that ρ(t) solves the master equation ( = 1)with a fixed dissipator

Quantum metrology with full and fast quantum control

“…In contrast, noise processes described by timeinhomogeneous master equations have been recently considered within the context of quantum metrology in [62][63][64][65], where time-inhomogeneity has been shown to be beneficial at short time-scales-the so-called Zeno regime-where the bath can no longer be assumed to be uncorrelated from the system. Let us stress that such a regime, however, does not allow for the general control operations considered here to be unambiguously applied without explicitly modelling the environment [35,66].…”

Revealing memory effects in phase-covariant quantum master equations

“…We apply this finding to several recently introduced non-Markovianity indicators and we use this approach to make a comprehensive comparison of their sensitivity. Since our approach does not depend on the specification of the analytic form of the time-dependent decay rates, it can be applied in a variety of physical situations, such as those considered in [21][22][23][24][25][26][27].…”

Stabilizable Gaussian states

“…We note that, for conceptual clarity, in the stabilizability-related considerations the environmentally-induced dissipative part of the system dynamics is assumed to be independent of the chosen Hamiltonian; whereas in microscopic derivations the Hamiltonian in general depends on the interaction with the environment. While such assumption is unproblematic in the case of Markovian master equations, non-Markovian cases must be treated with care [14]. In this article, we only discuss Markovian master equations.…”