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International Journal of Quantum Information

INSTANTANEOUS DARK STATE AND TRAJECTORY TRACKING

X. Y. Zhang, K. Farooq, X. X. Yi

Published Aug 1, 2013• 1 citation
Publisher

Abstract

State steering and tracking have attracted a lot of interests due to their wide applications. In this paper, we present two concrete examples of trajectory tracking of an open quantum system with time-dependent Lindblad operator which has an eigenstate |ψs(t)〉 with eigenvalue 0. We work out the first case analytically while numerically the second case. From both the cases, we find that, if changing the Lindblad operator is sufficiently slow, then the system can follow the trajectory of |ψs(t)〉 very well.

DOI
10.1142/s0219749913500469

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State steering and tracking have attracted a lot of interests due to their wide applications. In this paper, we present two concrete examples of trajectory tracking of an open quantum system with time-dependent Lindblad operator which has an eigenstate |ψs(t)〉 with eigenvalue 0. We work out the first case analytically while numerically the second case. From both the cases, we find that, if changing the Lindblad operator is sufficiently slow, then the system can follow the trajectory of |ψs(t)〉 very well.
Keeping a quantum system in a given instantaneous eigenstate is a control problem with numerous applications, e.g., in quantum information processing. The problem is even more challenging in the setting of open quantum systems, where environment-mediated transitions introduce additional decoherence channels. Adiabatic passage is a well-established solution but requires a sufficiently slow evolution time that is dictated by the adiabatic theorem. Here we develop a systematic projection theory formulation for the transitionless evolution of general open quantum systems described by time-local master equations. We derive a time-convolutionless dynamical equation for the target instantaneous eigenstate of a given time-dependent Hamiltonian. A transitionless dynamics then arises in terms of a competition between the average Hamiltonian gap and the decoherence rate, which implies optimal adiabaticity timescales. We show how eigenstate tracking can be accomplished via control pulses, without explicitly incorporating counter-diabatic driving, thus offering an alternative route to accelerate adiabaticity. We examine rectangular pulses, chaotic signals, and white noise, and find that, remarkably, the effectiveness of eigenstate tracking hardly depends on the details of the control functions. In all cases the control protocol keeps the system in the desired instantaneous eigenstate throughout the entire evolution, along an accelerated adiabatic path.
We present a control strategy to speed up the steady-state tracking for open quantum systems governed by a time-dependent Markovian master equation. With this fast trajectory tracking strategy, the quantum state strictly follows the instantaneous steady state of open quantum systems and evolves into the target state without any fidelity loss. A compact expression for the counterdiabatic term is derived in terms of a superoperator formalism in Liouville space. To be specific, we utilize a universal method to transform the counterdiabatic superoperator into the ordinary operator form. It can be seen from the operator form that in addition to the coherent control field, an incoherent control is required for the fast trajectory tracking. The control strategy is successfully applied in two examples: One is a two-level quantum system with a time-dependent Hamiltonian coupling to a heat bath, the other is a two-level system following an isothermal process.
The orbit tracking problem of a free-evolutionary target system in closed quantum systems is solved by changing it into the state transferring problem with the help of unitary transformation.The control law designed by the Lyapunov stability theorem employs a carefully constructed virtual mechanical quantity P to ensure the system convergence.The virtual mechanical quantity P is chosen by two approaches according to the forms of limit set,where P = —pf is suitable for regular limit set and a new different P is constructed for irregular one.The proposed tracking control theory is demonstrated on a four-level quantum system by means of numerical simulation experiments.
For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.
A finite-dimensional Markovian open quantum system will undergo quantum jumps between pure states, if we can monitor the bath to which it is coupled with sufficient precision. In general these jumps, plus the between-jump evolution, create a trajectory which passes through infinitely many different pure states, even for ergodic systems. However, as shown recently by us [Phys. Rev. Lett. 106, 020406 (2011)], it is possible to construct adaptive monitorings which restrict the system to jumping between a finite number of states. That is, it is possible to track the system using a finite state machine as the apparatus. In this paper we consider the question of the stability of these monitoring schemes. Restricting to cyclic jumps for a qubit, we give a strong analytical argument that these schemes are always stable and supporting analytical and numerical evidence for the example of resonance fluorescence. This example also enables us to explore a range of behaviors in the evolution of individual trajectories, for several different monitoring schemes.
A control method for trajectory tracking in non-Markovian open quantum systems is proposed in this paper. By inducing the error between the controlled state and the target state as the error state, the trajectory tracking problem of the quantum system is divided into two steps, which the regulation control of the error state and the keeping control of the error state. With the help of Lyapunov stability theorem, the tracking control law is designed to enable the non-Markovian open quantum system to track the evolutionary target quantum system over time. The control parameters are adjusted by simulation experiments, and the numerical simulation experiments are carried out. The tracking control of the quantum system when the initial states of the controlled system and the initial states of the target system are superposition states is implemented in simulation experiments. The experimental results demonstrate the feasibility of the proposed control method and the analysis is given.