On optimization of coherent and incoherent controls for two-level quantum systems

Abstract

This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic $N$-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives

“…In the analysis below, Equation ( 12) is used to take into account such bounds on variations of the incoherent control. To shorten the incoherent control time scale, the first stage of the incoherent control scheme proposed in [22] was further modified for a two-level system in [27], significantly reducing the control time scale. Such an incoherent control can be technically implemented, e.g., as it was done for controlling multi-species atomic and molecular systems with Gd 2 O 2 S : Er 3+ (6%) samples [71].…”

“…However, in some cases, one can use the environment as a useful control resource, such as, for example, in the incoherent control approach [21,22], where the spectral, generally time-dependent and non-equilibrium density of incoherent photons is used as a control function jointly with the coherent control via lasers to manipulate such a quantum system dynamics. Following this approach, various types and aspects of optimal control problems for one-and two-qubit systems were analyzed [23][24][25][26][27][28].…”

“…253-259), one-and two-step gradient projection methods (GPM-1, GPM-2) [23,24,28], etc. ; • For finite-dimensional optimization under various classes of parameterized controls, e.g., gradient ascent pulse engineering (GRAPE)-type methods (e.g., [25][26][27], Section 3, [62]) (GRAPE-type methods operate with piecewise-constant controls, matrix exponentials, and gradients), CRAB ansatz [46,63] (coherent control is considered in terms of sine, cosine, etc. ), genetic algorithm (GA) [21,64,65], dual annealing [24], etc.…”