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State transfer and maintenance for non-Markovian open quantum systems in a hybrid environment via Lyapunov control method

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Abstract

This paper concerns state transfer in a non-Markovian two-level open quantum system coupled to a hybrid bath composed of bosons and fermions based on the Lyapunov control method. First, the differential equation of the decay rate factor in the hybrid environment has been achieved and after that, the effect of bath parameters on the system characteristics is investigated to not only disclose the behavior of system dynamics but also reveal that how the bath parameters affect the coherence and purity. Next, a control field has been introduced based on the Lyapunov theory where a trace distance-based exponential form has been suggested as a Lyapunov function. Moreover, a limitation is defined on control laws in order to have control fields with upper and lower bound that is suitable from the practical point of view. Finally, considering bounded and unbounded control fields, numerical simulations have been illustrated for different situations including the state transfer between two eigenstates, between two superposition states, and from a mixed state to a superposition state. The results have been compared with a control field which has been designed by considering the conventional trace distance as a Lyapunov function. The simulation outcomes demonstrate that the proposed control field governs the system state to the desired state with more accuracy and faster convergence rate rather than the conventional method.

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Abbreviations

\(\gamma_{1,2}\) :

Memory effect coefficients for bosonic and fermionic baths

\(\alpha_{1,2}\) :

Correlation strength coefficients for bosonic and fermionic baths

\(\Omega_{1,2}\) :

Central frequencies for bosonic and fermionic baths

\(K_{{\text{b}}}\) :

Correlation function of bosonic bath

\(K_{{\text{f}}}\) :

Correlation function of fermionic bath

\(\omega\) :

System frequency

\(g_{{\text{j}}} ; j = x,y\) :

Control coefficients

\(\theta_{{\text{j}}} ; j = x,y\) :

Disturbances

\(\beta\) :

Control constant for exponential control fields

\(\mu\) :

Threshold parameter in control fields equations

\(\tau\) :

Threshold for the amplitudes of control fields

\(\rho\) :

Density matrix

\(\rho_{{\text{f}}}\) :

Desired density matrix

\(f_{{\text{j}}} ; j = x,y\) :

External control field in the j-direction

\(t_{{\text{f}}}\) :

Final time of simulation

\(d\) :

Dimension of open quantum system

\({\mathcal{L}}_{{\text{D}}}\) :

Hybrid environment

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Authors and Affiliations

  1. Faulty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, 47148-71167, Babol, Iran

    Safa Khari & Zahra Rahmani

  2. Department of Electrical Engineering, Faculty of Engineering, Arak University, 38156-8-8349, Arak, Iran

    Abolghasem Daeichian

  3. Department of Physics, Babol Noshirvani University of Technology, 47148-71167, Babol, Iran

    Hossein Mehri-Dehnavi

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  1. Safa Khari
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  2. Zahra Rahmani
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Correspondence to Zahra Rahmani.

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Khari, S., Rahmani, Z., Daeichian, A. et al. State transfer and maintenance for non-Markovian open quantum systems in a hybrid environment via Lyapunov control method. Eur. Phys. J. Plus 137, 533 (2022). https://doi.org/10.1140/epjp/s13360-022-02713-8

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