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arXiv:2307.07450 · Quantum Reports

Control Landscape of Measurement-Assisted Transition Probability for a Three-Level Quantum System with Dynamical Symmetry

Maria A. Elovenkova, Alexander Nikolaevich Pechen

Published Jul 13, 2023• 4 citations

Abstract

Quantum systems with dynamical symmetries have conserved quantities that are preserved under coherent control. Therefore, such systems cannot be completely controlled by means of only coherent control. In particular, for such systems, the maximum transition probability between some pairs of states over all coherent controls can be less than one. However, incoherent control can break this dynamical symmetry and increase the maximum attainable transition probability. The simplest example of such a situation occurs in a three-level quantum system with dynamical symmetry, for which the maximum probability of transition between the ground and intermediate states using only coherent control is 1/2, whereas it is about 0.687 using coherent control assisted by incoherent control implemented through the non-selective measurement of the ground state, as was previously analytically computed. In this work, we study and completely characterize all critical points of the kinematic quantum control landscape for this measurement-assisted transition probability, which is considered as a function of the kinematic control parameters (Euler angles). The measurement-driven control used in this work is different from both quantum feedback and Zeno-type control. We show that all critical points are global maxima, global minima, saddle points or second-order traps. For comparison, we study the transition probability between the ground and highest excited states, as well as the case when both these transition probabilities are assisted by incoherent control implemented through the measurement of the intermediate state.

DOI: 10.3390/quantum5030035

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Quantum measurements are considered for optimal control of quantum dynamics with instantaneous and continuous observations utilized to manipulate population transfer. With an optimal set of measurements, the highest yield in a two-level system can be obtained. An analytical solution is given for the problem of population transfer by measurement-assisted coherent control in a three-level system with dynamical symmetry. The anti-Zeno effect is recovered in the controlled processes. Demonstrations show that suitable observations can be powerful tools in the manipulation of quantum dynamics.

We study quantum system dynamics driven by continuous laser fields under the measurement process. In order to take into account the system transition due to the measurement, we define the superoperator which eliminates the coherence relevant to the measured quantum states. We clarify that the dynamics of the measured states is frozen in the frequent measurement limit, while the space spanned by unmeasured states is isolated from the original system. We also derive the effective Liouvillian which governs incoherent population dynamics under the condition, in which measurements are frequently applied. We apply the formulation to two-level and Lambda-type three-level systems and clarify how the quantum measurements hinder the coherent population dynamics driven by the continuous laser fields in practical examples. Analysis on the laser field amplitude dependency of the final distribution in the t-->infinity limit suggests the possibility of the measurement-assisted quantum control.

The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology and structure provide a foundation to understand why optimal quantum control can be readily achieved.

Manipulation of a quantum system can be viewed in the framework of a control landscape defined as the physical objective as a functional of the control. Control landscape analyses have thus far considered linear quantum dynamics. This paper extends the analysis of control landscape topology to nonlinear quantum dynamics with the objective of steering a finite-level quantum system from an initial state to a final target state. The analysis rests on the assumptions that (i) the final state is reachable from the initial state, (ii) the differential mapping from the control to the state is surjective, and (iii) the control resources are unconstrained. Under these assumptions, landscape critical points (i.e., where the slope vanishes) for nonlinear quantum dynamics only appear as the global maximum and minimum; thus, the landscape is free of traps. Moreover, the landscape Hessian (i.e., the second derivative with respect to the control) at the global maximum has finite rank, indicating the presence of a large level set of optimal controls that preserve the value of the maximum. Extensive numerical simulations on finite-level models of the Gross-Pitaevskii equation confirm the trap-free nature of the landscape as well as the Hessian rank analysis, using either an applied electric field or a tunable condensate two-body interaction strength as the control. In addition, the control mechanisms arising in the numerical simulations are qualitatively assessed. These results are a generalization of previous findings for the linear Schr\"odinger equation, and show promise for successful control in a wide range of nonlinear quantum dynamics applications.

This paper explores the state-to-state transition-probability-control landscape for

n

-level open quantum systems governed by the Lindblad equation. For generic two-level systems, we show analytically that the control landscape does not possess critical points in the space of square-integrable control fields. Numerical simulations show that for a given target state the transition probability reaches its highest value at a particular finite time, and the corresponding control contains temporal subpulses, similar to that for the time optimal control of analogous closed quantum systems with unbounded controls.