The general objective of quantum control is the manipulation of atomic scale physical and chemical phenomena through the application of external control fields. These tailored fields, or photonic reagents, exhibit systematic properties analogous to those of ordinary laboratory reagents. This analogous behavior is explored further here by considering the controlled response of a family of homologous quantum systems to a single common photonic reagent. A level set of dynamically homologous quantum systems is defined as the family that produces the same value(s) for a target physical observable(s) when controlled by a common photonic reagent. This paper investigates the scope of homologous quantum system control using the level set exploration technique (L-SET). L-SET enables the identification of continuous families of dynamically homologous quantum systems. Each quantum system is specified by a point in a hypercube whose edges are labeled by Hamiltonian matrix elements. Numerical examples are presented with simple finite level systems to illustrate the L-SET concepts. Both connected and disconnected families of dynamically homologous systems are shown to exist.
The general objective of quantum control is the manipulation of atomic scale physical and chemical phenomena through the application of external control fields. These tailored fields, or photonic reagents, exhibit systematic properties analogous to those of ordinary laboratory reagents. This analogous behavior is explored further here by considering the controlled response of a family of homologous quantum systems to a single common photonic reagent. A level set of dynamically homologous quantum systems is defined as the family that produces the same value(s) for a target physical observable(s) when controlled by a common photonic reagent. This paper investigates the scope of homologous quantum system control using the level set exploration technique (L-SET). L-SET enables the identification of continuous families of dynamically homologous quantum systems. Each quantum system is specified by a point in a hypercube whose edges are labeled by Hamiltonian matrix elements. Numerical examples are presented with simple finite level systems to illustrate the L-SET concepts. Both connected and disconnected families of dynamically homologous systems are shown to exist.
In controlled quantum dynamics, a level set is defined as the collection of control fields that produce a specific value for a particular observable. This paper explores the relationship between individual solutions to a control problem, and presents the first experimentally observed quantum control level sets, which are found to be continuous submanifolds. Level sets are observed for two photon transitions where the control is the spectral phase function, which is expressed as a fourth-order polynomial. For the systems studied here, the level sets are shown to be closed surfaces in the spectral phase control space. A perturbation analysis provides insight into the observed topology of the level set, which is shown to be preserved by the low-order polynomial phase representation. Each of the multiple control fields forming a level set preserves the observable value by its own distinct manipulation of constructive and destructive quantum interferences. Thus, the richness of quantum control fields meeting a particular observable value is accompanied by an equally diverse family of control mechanisms.
Quantum control aims at the manipulation of atomic- and molecular-scale dynamics phenomena. An important objective in this regard is the understanding of dynamical control within a family of related quantum systems. To explore this issue, diffeomorphic changes in the system Hamiltonian H(s,t) are introduced by scanning over a homotopy parameter s and then monitoring the control field response needed to maintain the value of a specified target observable. This operation is implemented through a procedure referred to as diffeomorphic modulation under observable-response-preserving homotopy (D-MORPH). The governing D-MORPH differential equation determining the control laser field E(s,t) is shown to explicitly allow for innumerable solutions, with each characterized by the choice of an arbitrary function f(s,t) of s and time t. The presence of f(s,t) in the D-MORPH differential equation makes clear the origin of multiple control fields that produce the same observable objective. A stable algorithm is presented for practical execution of D-MORPH with the only criterion that the Hamiltonian H(s,t) permit reaching the objective over the full domain of s being sampled. Both analytic and numerical examples are presented to illustrate the D-MORPH concept.
Advances in quantum control theory and optimization techniques have made possible a new phenomena of quantum engineering. Chemical reactions can now be dynamically manipulated using first-principles quantum chemical models, in which a laser field, rather than a catalyst, serve as photonic reagents to selectively break a chemical bond and yield products via intramolecular rearrangement. A field pulse may also be designed to perform ultrasensitive spectroscopy, where compounds with very similar chemical properties are differentiated through their unique interaction with resonant and off-resonant photons. Despite significant progress in theoretical and laboratory quantum control, however, quantum engineering remains principally challenging due to manifestation of noise and uncertainties associated with the input and system parameters. Specifically, a small magnitude of variation in the field and Hamiltonian parameters would cause an otherwise optimal field to deviate from controlling desired quantum state transitions and reaching a particular objective. Thus, an accurate analysis of robustness of controlled quantum dynamics followed by a robust control framework is essential in achieving quantum control in a practical setting. In this work, theoretical foundations for quantum control robustness analysis are presented from both a distributional perspective - in terms of moments of the transition amplitude, interferences, and transition probability - and a worst-case perspective. In addition, an evolutionary control strategy for achieving robust control via open- and closed-loop approach are described with its mechanism of convergence elucidated using the robustness analysis method.
A quantum control landscape is defined by the value of a physical observable as a functional of the time-dependent control field E(t) for a given quantum-mechanical system. Level sets through this landscape are prescribed by a particular value of the target observable at the final dynamical time T, regardless of the intervening dynamics. We present a technique for exploring a landscape level set, where a scalar variable s is introduced to characterize trajectories along these level sets. The control fields E(s,t) accomplishing this exploration (i.e., that produce the same value of the target observable for a given system) are determined by solving a differential equation over s in conjunction with the time-dependent Schr\"odinger equation. There is full freedom to traverse a level set, and a particular trajectory is realized by making an a priori choice for a continuous function f(s,t) that appears in the differential equation for the control field. The continuous function f(s,t) can assume an arbitrary form, and thus a level set generally contains a family of controls, where each control takes the quantum system to the same final target value, but produces a distinct control mechanism. In addition, although the observable value remains invariant over the level set, other dynamical properties (e.g., the degree of robustness to control noise) are not specifically preserved and can vary greatly. Examples are presented to illustrate the continuous nature of level-set controls and their associated induced dynamical features, including continuously morphing mechanisms for population control in model quantum systems.
A traditional quantum optimal control experiment begins with a specific physical system and seeks an optimal time-dependent field to steer the evolution towards a target observable value. In a more general framework, the Hamiltonian structure may also be manipulated when the material or molecular 'stockroom' is accessible as a part of the controls. The current work takes a step in this direction by considering the converse of the normal perspective to now start with a specific fixed field and employ the system's time-independent Hamiltonian structure as the control to identify an optimal form. The Hamiltonian structure control variables are taken as the system energies and transition dipole matrix elements. An analysis is presented of the Hamiltonian structure control landscape, defined by the observable as a function of the Hamiltonian structure. A proof of system controllability is provided, showing the existence of a Hamiltonian structure that yields an arbitrary unitary transformation when working with virtually any field. The landscape analysis shows that there are no suboptimal traps (i.e., local extrema) for controllable quantum systems when unconstrained structural controls are utilized to optimize a state-to-state transition probability. This analysis is corroborated by numerical simulations on model multilevel systems. The search effort to reach the top of the Hamiltonian structure landscape is found to be nearly invariant to system dimension. A control mechanism analysis is performed, showing a wide variety of behavior for different systems at the top of the Hamiltonian structure landscape. It is also shown that reducing the number of available Hamiltonian structure controls, thus constraining the system, does not always prevent reaching the landscape top. The results from this work lay a foundation for considering the laboratory implementation of optimal Hamiltonian structure manipulation for seeking the best control performance, especially with limited electromagnetic resources.
While closed-loop control of quantum dynamics in the laboratory is proving to be broadly successful, the control mechanisms induced by the fields are often left obscure. Hamiltonian encoding (HE) was originally introduced as a method for understanding mechanisms in quantum dynamics in the context of computational simulations, based on access to the system wavefunction. As a step towards laboratory implementation of HE, this paper addresses the issues raised by the use of observables rather than the wavefunction in HE. The goal of laboratory based HE is to obtain an understanding of control mechanism through a sequence of systematic control experiments, whose collective information can identify the underlying control mechanism defined as the set of significant amplitudes connecting the initial and final states. Mechanism is determined by means of observing the dynamics of special sequences of system Hamiltonians encoded through the control field. The proposed algorithm can handle complex systems, operates with no recourse to dynamical simulations, and functions with limited understanding of the system Hamiltonian. As with the closed-loop control experiments, the HE control mechanism identification algorithm performs a new experiment each time the dynamical outcome from an encoded Hamiltonian is called for. This paper presents the basic HE algorithm in the context of physical systems described by a finite dimensional Hilbert space. The method is simulated with simple models, and the extension to more complex systems is discussed.