The dynamics induced while controlling quantum systems by optimally shaped laser pulses have often been difficult to understand in detail. A method is presented for quantifying the importance of specific sequences of quantum transitions involved in the control process. The method is based on a ``beable'' formulation of quantum mechanics due to John Bell that rigorously maps the quantum evolution onto an ensemble of stochastic trajectories over a classical state space. Detailed mechanism identification is illustrated with a model seven-level system. A general procedure is presented to extract mechanism information directly from closed-loop control experiments. Application to simulated experimental data for the model system proves robust with up to 25% noise.
The dynamics induced while controlling quantum systems by optimally shaped laser pulses have often been difficult to understand in detail. A method is presented for quantifying the importance of specific sequences of quantum transitions involved in the control process. The method is based on a ``beable'' formulation of quantum mechanics due to John Bell that rigorously maps the quantum evolution onto an ensemble of stochastic trajectories over a classical state space. Detailed mechanism identification is illustrated with a model seven-level system. A general procedure is presented to extract mechanism information directly from closed-loop control experiments. Application to simulated experimental data for the model system proves robust with up to 25% noise.
A variety of means are now available to design control fields for manipulating the evolution of quantum systems. However, the underlying physical mechanisms often remain obscure, especially in the cases of strong fields and high quantum state congestion. This paper proposes a method to quantitatively determine the various pathways taken by a quantum system in going from the initial state to the final target. The mechanism is revealed by encoding a signal in the system Hamiltonian and decoding the resultant nonlinear distortion of the signal in the system time-evolution operator. The relevant interfering pathways determined by this analysis give insight into the physical mechanisms operative during the evolution of the quantum system. A hierarchy of mechanism identification algorithms with increasing ability to extract more detailed pathway information is presented. The mechanism identification concept is presented in the context of analyzing computer simulations of controlled dynamics. As illustrations of the concept, mechanisms are identified in the control of several simple, discrete-state quantum systems. The mechanism analysis tools reveal the roles of multiple interacting quantum pathways to maximally take advantage of constructive and destructive interference. Similar procedures may be applied directly in the laboratory to identify control mechanisms without resort to computer modeling, although this extension is not addressed in this paper.
There are increasing numbers of successful numerical simulations as well as laboratory tests of controlled quantum phenomena. However, little is understood about the control mechanisms, especially in strongly driven systems. In this paper, mechanism is defined in terms of the significant quantum pathways induced by the control field, connecting the initial state to the final target state. The present work focuses on understanding mechanism in the coordinate representation, which is natural for many dynamical systems. The quantum amplitudes defining the mechanistic pathways are determined by a special procedure called coding Hamiltonians to access mechanistic pathways (CHAMP). Numerical procedures are presented for performing CHAMP calculations in a stable fashion. Mechanism determination is illustrated for excitation of a model linear triatomic molecule. Mechanisms are revealed for optimal control fields as well as for suboptimally driven systems including snapshots in time of the evolving mechanism and the effect of field noise on the mechanism. The high order multiphoton processes involved in the example produce many interfering pathways to create the final state, yet the dynamics and the mechanism are robust to reasonable levels of noise.
A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems.
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.
In this paper, employing the It\^o stochastic Schr\"odinger equation, we extend Bell's beable interpretation of quantum mechanics to encompass dissipation, decoherence, and the quantum-to-classical transition through quantum trajectories. For a particular choice of the source of stochasticity, the one leading to a dissipative Lindblad-type correction to the Hamiltonian dynamics, we find that the diffusive terms in Nelsons stochastic trajectories are naturally incorporated into Bohm's causal dynamics, yielding a unified Bohm-Nelson theory. In particular, by analyzing the interference between quantum trajectories, we clearly identify the decoherence time, as estimated from the quantum formalism. We also observe the quantum-to-classical transition in the convergence of the infinite ensemble of quantum trajectories to their classical counterparts. Finally, we show that our extended beables circumvent the problems in Bohm's causal dynamics regarding stationary states in quantum mechanics.
The control of quantum dynamics is generally accomplished by seeking a tailored electromagnetic field to meet a posed objective. A particular shaped field can be thought of as specifying a point on a quantum control landscape, which is the objective as a functional of the controls. Optimizing the pulse shape corresponds to climbing the landscape, and previous work showed that the paths taken up the landscapes, guided by a gradient algorithm, are surprisingly straight when projected into the space of control fields. The direct nature of these control trajectories can be quantified by the metric $R\ensuremath{\ge}1$, defined as the ratio of the length of the control trajectory to the Euclidean distance between its end points. The prior observation of often finding low values of R implies that the landscapes are structurally simple. In this work, we investigate whether there is a relationship between the intricacy of the control mechanism and the complexity of the trajectory taken through the control space reflected in the value of R. We use the Hamiltonian encoding procedure to identify the mechanism, and we examine control of the state-to-state transition probability. No significant correlation is found between the landscape structure, reflected in the value of R, and the control mechanism. This result has algorithmic implications, opening up the prospect of seeking fields producing particular mechanisms at little penalty in the search effort due to encountering complex landscape structure.