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Physical Review A

Optimal control of multilevel quantum systems in the field-interaction representation

Praveen Kumar, Svetlana Malinovskaya, Vladimir S. Malinovsky

Published Sep 29, 2014• 9 citations
Publisher

Abstract

A control strategy incorporating a fixed carrier frequency constraint on the optimal field is presented. As an illustrative example we consider the creation of maximum coherence in a six-level $\ensuremath{\Lambda}$ system by solving the Schr\"odinger equation without the rotating-wave approximation in the field-interaction representation. We demonstrate that application of the optimal control theory optimization reformulated in the field-interaction representation allows one to keep the carrier frequency of the control field constant and successfully optimize off-resonant processes in multilevel quantum systems.

DOI
10.1103/physreva.90.033427

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A control strategy incorporating a fixed carrier frequency constraint on the optimal field is presented. As an illustrative example we consider the creation of maximum coherence in a six-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system by solving the Schr"odinger equation without the rotating-wave approximation in the field-interaction representation. We demonstrate that application of the optimal control theory optimization reformulated in the field-interaction representation allows one to keep the carrier frequency of the control field constant and successfully optimize off-resonant processes in multilevel quantum systems.
We consider the creation of the maximum Raman coherence in the six-level Λ system using optimal control theory. Optimal fields are designed for different initial conditions, resonant, and off-resonant, using the Krotov method including a reference field into the cost functional. Suppression of the population transfer to the intermediate level is achieved via an additional functional constraint which depends on the system dynamics. We demonstrate that the spectrum of the optimised fields has major contribution from the corresponding resonant frequencies independently of the choice of carrier frequency of the initial guess field. We also indicate that the pulse train emerges as a solution of the control problem of coherence optimisation in multi-level quantum systems.
We consider the creation of the maximum Raman coherence in the six-levelsystem using optimal control theory. Optimal fields are designed for different initial conditions, resonant, and off-resonant, using the Krotov method including a reference field into the cost functional. Suppression of the population transfer to the intermediate level is achieved via an additional functional constraint which depends on the system dynamics. We demonstrate that the spectrum of the optimised fields has major contribution from the corresponding resonant frequencies independently of the choice of carrier frequency of the initial guess field. We also indicate that the pulse train emerges as a solution of the control problem of coherence optimisation in multi-level quantum systems.
We consider a current-biased dc SQUID in the presence of an applied time-dependent bias current or magnetic flux. The phase dynamics of such a Josephson device is equivalent to that of a quantum particle trapped in a 1D anharmonic potential, subject to external time-dependent control fields, i.e. a driven multilevel quantum system. The problem of finding the required time-dependent control field that will steer the system from a given initial state to a desired final state at a specified final time is formulated in the framework of optimal-control theory. Using the spectral filter technique, we show that the selected optimal field which induces a coherent population transfer between quantum states is represented by a carrier signal having a constant frequency but which is time-varied both in amplitude and phase. The sensitivity of the optimal solution to parameter perturbations is also addressed.
We study the interplay between rotating-wave approximation and optimal control. In particular, we show that for a wide class of optimal control problems one can choose the control field such that the Hamiltonian becomes time independent under the rotating-wave approximation. Thus, we show how to recast the functional minimization defined by the optimal control problem into a simpler multivariable function minimization. We provide the analytic solution to the state-to-state transfer of the paradigmatic two-level system and to the more general star configuration of an NNN-level system. We demonstrate numerically the usefulness of this approach in the more general class of connected acyclic NNN-level systems with random spectra. Finally, we use it to design a protocol to entangle Rydberg via constant laser pulse atoms in an experimentally relevant range of parameters.
In this work, we address the problem of maximizing fidelity in a quantum state transformation process controlled in such a way as to keep decoherence within given bounds. We consider a three-level Λ-type atom subjected to Markovian decoherence characterized by non-unital decoherence channels. We introduce fidelity as the performance index for the quantum state transformation process since the goal is to maximize the similarity of the final state density operator with the one of the desired target state. We formulate the quantum optimal control problem with state constraints where the latter reflect the fact that the decoherence level remains within a pre-defined bound. Optimality conditions of Pontryagin’s Maximum Principle in Gamkrelidze’s form are given. These provide a complete set of relations enabling the computation of the optimal control strategy. This is a novel approach in the context of quantum systems.