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arXiv:1811.12785 · Physical Review A

Optimal control of coupled quantum systems based on the first-order Magnus expansion: Application to multiple dipole-dipole-coupled molecular rotors

Andrew Ma, Alicia B Magann, Tak‐San Ho, HERSCHEL A. RABITZ

Published Jul 30, 2020• 8 citations
arXivHTMLPDF

Abstract

This paper presents a method for performing approximate optimal control simulations for quantum systems with multiple coupled degrees of freedom. The time evolution is simulated using the first-order Magnus expansion in the interaction picture, where the couplings between different degrees of freedom are treated as the perturbation. A numerical implementation procedure is presented that leverages upon pairwise couplings and the separability of the zeroth-order time evolution operator to achieve a reduced computational cost, which is analyzed with respect to the number of degrees of freedom. The formulation is compatible with gradient-free methods to optimize the control field, and a stochastic hill climbing algorithm is adopted for this purpose. As illustrations, optimal control simulations are performed for systems of two and three dipole-dipole-coupled molecular rotors under the influence of a control field. For the two-rotor system, the field is optimized to achieve either orientation or entanglement objectives. For the three-rotor system, the field is optimized either to orient all three rotors in the same direction or to orient one rotor in a particular direction while the other two rotors point in the opposite direction.

DOI
10.1103/physreva.102.013115

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This paper presents a method for performing approximate optimal control simulations for quantum systems with multiple coupled degrees of freedom. The time evolution is simulated using the first-order Magnus expansion in the interaction picture, where the couplings between different degrees of freedom are treated as the perturbation. A numerical implementation procedure is presented that leverages upon pairwise couplings and the separability of the zeroth-order time evolution operator to achieve a reduced computational cost, which is analyzed with respect to the number of degrees of freedom. The formulation is compatible with gradient-free methods to optimize the control field, and a stochastic hill climbing algorithm is adopted for this purpose. As illustrations, optimal control simulations are performed for systems of two and three dipole-dipole-coupled molecular rotors under the influence of a control field. For the two-rotor system, the field is optimized to achieve either orientation or entanglement objectives. For the three-rotor system, the field is optimized either to orient all three rotors in the same direction or to orient one rotor in a particular direction while the other two rotors point in the opposite direction.
We perform quantum optimal control simulations, based on the Time-Dependent Hartree (TDH) approximation, for systems of three to five dipole-dipole coupled OCS rotors. A control electric field is used to steer all of the individual rotors, arranged in chains and regular polygons in a plane, toward either identical or unique objectives. The goal is to explore the utility of the TDH approximation to model the field-induced dynamics of multiple interacting rotors in the weak dipole-dipole coupling regime. A stochastic hill climbing approach is employed to seek an optimal control field that achieves the desired objectives at a specified target time. We first show that multiple rotors in chain and polygon geometries can be identically oriented in the same direction; these cases do not significantly depend on the presence of the dipole-dipole interaction. Additionally, in particular geometrical arrangements, we demonstrate that individual rotors can be uniquely manipulated toward different objectives with the same field. Specifically, it is shown that for a three rotor chain, the two end rotors can be identically oriented in a specific direction while keeping the middle rotor in its ground state, and for an equilateral triangle, two rotors can be identically oriented in a specific direction while the third rotor is oriented in the opposite direction. These multirotor unique objective cases exploit the shape of the field in coordination with dipole-dipole coupling between the rotors. Comparisons to numerically exact calculations, utilizing the TDH-determined fields, are given for all optimal control studies involving systems of three rotors.
The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.
Optimal control simulations are performed for orientation and entanglement of two dipole-dipole coupled identical quantum rotors. The rotors at various fixed separations lie on a model non-interacting plane with an applied control field. It is shown that optimal control of orientation or entanglement represents two contrasting control scenarios. In particular, the maximally oriented state (MOS) of the two rotors has a zero entanglement entropy and is readily attainable at all rotor separations. Whereas, the contrasting maximally entangled state (MES) has a zero orientation expectation value and is most conveniently attainable at small separations where the dipole-dipole coupling is strong. It is demonstrated that the peak orientation expectation value attained by the MOS at large separations exhibits a long time revival pattern due to the small energy splittings arising form the extremely weak dipole-dipole coupling between the degenerate product states of the two free rotors. Moreover, it is found that the peak entanglement entropy value attained by the MES remains largely unchanged as the two rotors are transported to large separations after turning off the control field. Finally, optimal control simulations of transition dynamics between the MOS and the MES reveal the intricate interplay between orientation and entanglement.
We present a time-parallelization method that enables one to accelerate the computation of quantum optimal control algorithms. We show that this approach is approximately fully efficient when based on a gradient method as optimization solver: the computational time is approximately divided by the number of available processors. The control of spin systems, molecular orientation, and Bose-Einstein condensates are used as illustrative examples to highlight the wide range of applications of this numerical scheme.
Quantum optimal control has numerous important applications ranging from pulse shaping in magnetic-resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.
The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.