The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The presence of significant field fluctuations is shown to break down the evolution into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations. This process takes advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness bodes well for the future success of closed loop quantum optimal control experiments.
A relationship is shown to exist between optimally controlled quantum dynamics and field fluctuations, wherein the process of seeking optimal performance assures an inherent degree of system robustness to such fluctuations. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations by taking advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness are discussed in light of recent experiments and numerical simulations on optimal control of quantum phenomena.
Abstract In recent work, we introduced the asymptotic theory of quantum robust control, which enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we extend this theory of quantum robust control to encompass two of the most important generalizations: robust control of arbitrary quantum observables and robust control of quantum systems sustaining environmental decoherence. In addition, we present deterministic Pareto optimization algorithms that can be applied in conjunction with either asymptotic or leading order measures of robustness. This enables robust control of any observable in quantum systems with any initial density matrix, and for which the entropy can change arbitrarily during the time evolution. Methods for robust optimal control of open quantum systems are presented that maximize the expected value of a quantum control objective while minimizing the expected environmentally induced decoherence.
This work presents a general framework for assessing the quality and robustness of control over quantum dynamics induced by an optical field
E(t). The control process is expressed in terms of a cost functional, including the physical objectives, penalties, and constraints. The first variations of such cost functionals have traditionally been utilized to create designs for the controlling electric fields. Here, the second variation of the cost functional is analyzed to explore (i) whether such solutions are locally optimal, and (ii) their degree of robustness. Both issues may be assessed from the eigenvalues of the stability operator S whose kernel $K(t,\ensuremath{\tau})$ is related to $\ensuremath{\delta}\mathcal{E}(t)/\ensuremath{\delta}\mathcal{E}(\ensuremath{\tau}){|}_{c}$ for
0<t, $\ensuremath{\tau}<~T,$ where
T is the target control time. Here
c denotes the constraint that the field satisfies the optimal control dynamical equations. The eigenvalues \ensuremath{\sigma} of S satisfying $\ensuremath{\sigma}<1$ assure local optimality of the control solution, with $\ensuremath{\sigma}=1$ being the critical value separating optimal solutions from false solutions (i.e., those with negative second variational curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also correspond to $\ensuremath{\sigma}=1.$ Thus, sufficiently high sensitivity of the field at one time
t to the field at another time \ensuremath{\tau} (i.e., $\ensuremath{\sigma}>1$) will lead to a loss of local optimality. An expression is obtained for a bound on the stability operator, and this result is employed to qualitatively analyze control behavior. From this bound, the inclusion of an auxiliary operator (i.e., other than the target operator) is shown to act as a stabilizer of the control process. It is also shown that robust solutions are expected to exist in both the strong- and weak-field regimes.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.
Closed-loop optimal quantum control experiments operate in the inherent presence of laser noise. In many applications, attaining high quality results [i.e., a high signal-to-noise
(S∕N) ratio for the optimized objective] is as important as producing a high control yield. Enhancement of the
S∕N ratio will typically be in competition with the mean signal, however, the latter competition can be balanced by biasing the optimization experiments towards higher mean yields while retaining a good
S∕N ratio. Other strategies can also direct the optimization to reduce the standard deviation of the statistical signal distribution. The ability to enhance the
S∕N ratio through an optimized choice of the control is demonstrated for two condensed phase model systems: second harmonic generation in a nonlinear optical crystal and stimulated emission pumping in a dye solution.