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arXiv:1501.07382 · New Journal of Physics

Efficiency of quantum controlled non-Markovian thermalization

V Mukherjee, Vittorio Giovannetti, Rosario Fazio, S F Huelga, T Calarco, S Montangero

Published Jun 23, 2015• 34 citations
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Abstract

We study optimal control strategies to optimize the relaxation rate towards the fixed point of a quantum system in the presence of a non-Markovian (NM) dissipative bath. Contrary to naive expectations that suggest that memory effects might be exploited to improve optimal control effectiveness, NM effects influence the optimal strategy in a non trivial way: we present a necessary condition to be satisfied so that the effectiveness of optimal control is enhanced by NM subject to suitable unitary controls. For illustration, we specialize our findings for the case of the dynamics of single qubit amplitude damping channels. The optimal control strategy presented here can be used to implement optimal cooling processes in quantum technologies and may have implications in quantum thermodynamics when assessing the efficiency of thermal micro-machines.

DOI
10.1088/1367-2630/17/6/063031

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We study optimal control strategies to optimize the relaxation rate towards the fixed point of a quantum system in the presence of a non-Markovian (NM) dissipative bath. Contrary to naive expectations that suggest that memory effects might be exploited to improve optimal control effectiveness, NM effects influence the optimal strategy in a non trivial way: we present a necessary condition to be satisfied so that the effectiveness of optimal control is enhanced by NM subject to suitable unitary controls. For illustration, we specialize our findings for the case of the dynamics of single qubit amplitude damping channels. The optimal control strategy presented here can be used to implement optimal cooling processes in quantum technologies and may have implications in quantum thermodynamics when assessing the efficiency of thermal micro-machines.
Abstract We study optimal control strategies to optimize the relaxation rate towards the fixed point of a quantum system in the presence of a non-Markovian (NM) dissipative bath. Contrary to naive expectations that suggest that memory effects might be exploited to improve optimal control effectiveness, NM effects influence the optimal strategy in a non trivial way: we present a necessary condition to be satisfied so that the effectiveness of optimal control is enhanced by NM subject to suitable unitary controls. For illustration, we specialize our findings for the case of the dynamics of single qubit amplitude damping channels. The optimal control strategy presented here can be used to implement optimal cooling processes in quantum technologies and may have implications in quantum thermodynamics when assessing the efficiency of thermal micro-machines.
We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As by-products of our analysis, we find that (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control can not speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control, it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.
An efficient optimal-control theory based on the Krotov method is introduced for a non-Markovian open quantum system with a time-nonlocal master equation in which the control parameter and the bath correlation function are correlated. This optimal-control method is developed via a quantum dissipation formulation that transforms the time-nonlocal master equation to a set of coupled linear time-local equations of motion in an extended auxiliary Liouville space. As an illustration, the optimal-control method is applied to find the control sequences for high-fidelity ZZZ gates and identity gates of a qubit embedded in a non-Markovian bath. ZZZ gates and identity gates with errors less than ${10}^{\ensuremath{-}5}$ for a wide range of bath decoherence parameters can be achieved for the non-Markovian open qubit system with control over only the ${\ensuremath{\sigma}}_{z}$ term. The control-dissipation correlation and the memory effect of the bath are crucial in achieving the high-fidelity gates.
The problem of optimal control of non-Markovian open quantum system via weak measurement is presented. Based on the non-Markovian master equation, we evaluate exactly the non-Markovian effect on the dynamics of the system of interest interacting with a dissipative reservoir. We find that the non-Markovian reservoir has dual effects on the system: dissipation and backaction. The dissipation exhausts the coherence of the quantum system, whereas the backaction revives it. Moreover, we design the control Hamiltonian with the control laws attained by the stochastic optimal control and the corresponding optimal principle. At last, we considered the exact decoherence dynamics of a qubit in a dissipative reservoir composed of harmonic oscillators, and demonstrated the effectiveness of our optimal control strategy. Simulation results showed that the coherence will completely lost in the absence of control neither in non-Markovian nor Markovian system. However, the optimal feedback control steers it to a stationary stochastic process which fluctuates around the target. In this case the decoherence can be controlled effectively, which indicates that the engineered artificial reservoirs with optimal feedback control may be designed to protect the quantum coherence in quantum information and quantum computation.
Heat-Bath Algorithmic Cooling is a set of techniques for producing highly pure quantum systems by utilizing a surrounding heat-bath and unitary interactions. These techniques originally used the thermal environment only to fully thermalize ancillas at the environment temperature. Here we extend HBAC protocols by optimizing over the thermalization strategy. We find, for any d -dimensional system in an arbitrary initial state, provably optimal cooling protocols with surprisingly simple structure and exponential convergence to the ground state. Compared to the standard ones, these schemes can use fewer or no ancillas and exploit memory effects to enhance cooling. We verify that the optimal protocols are robusts to various deviations from the ideal scenario. For a single target qubit, the optimal protocol can be well approximated with a Jaynes-Cummings interaction between the system and a single thermal bosonic mode for a wide range of environmental temperatures. This admits an experimental implementation close to the setup of a micromaser, with a performance competitive with leading proposals in the literature. The proposed protocol provides an experimental setup that illustrates how non-Markovianity can be harnessed to improve cooling. On the technical side we 1. introduce a new class of states called maximally active states and discuss their thermodynamic significance in terms of optimal unitary control, 2. introduce a new set of thermodynamic processes, called β-permutations , whose access is sufficient to simulate a generic thermalization process, 3. show how to use abstract toolbox developed within the resource theory approach to thermodynamics to perform challenging optimizations, while combining it with open quantum system dynamics tools to approximate optimal solutions within physically realistic setups.
This paper compares the cooling performance of Markovian and non-Markovian quantum collisional models for nanoscale thermodynamic devices. A minimal setup is considered: a single qubit system repeatedly collides with qubit ancillas acting as a heat bath, while a local cooling “demon” operates on the system after each collision. Two regimes are analyzed: (i) a Markovian protocol where the system interacts with a fresh thermal ancilla at every step and the ancilla is subsequently rethermalized, and (ii) a non-Markovian protocol where the system repeatedly interacts with the same initially thermal environment qubit with no reset, allowing correlations (memory) to build up. A central contribution is an efficiency definition that explicitly includes all external energetic resources, counting both the energy removed by the demon and the otherwise hidden work cost of rethermalizing the ancillas in the Markovian implementation. The demon is modeled as an amplitude damping channel; via an explicit Stinespring dilation, the extracted energy is interpreted as heat dumped into an auxiliary cold reservoir, and the paper discusses how additional demon memory/erasure (Landauer) costs would further reduce operational efficiency. Numerical simulations show that the non-Markovian protocol can cool the system to much lower final energies and achieve substantially higher efficiency than the Markovian protocol across a wide range of collision (coupling) and damping strengths. The enhancement is traced to the finite environment’s initial energy and its capacity to store correlations as usable thermodynamic resources. The results emphasize that bounds and benchmarks derived under Markovian assumptions may significantly underestimate performance in finite, memoryful environments, and that reported efficiencies should be interpreted as upper bounds when demon implementation costs are not fully modeled.