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arXiv:1602.06250

Quantum Control Landscapes Beyond the Dipole Approximation

Benjamin Russell, HERSCHEL A. RABITZ, Re-Bing Wu

Published Feb 19, 2016• 2 citations
arXivHTMLPDF

Abstract

We investigate the control landscapes of closed, finite level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. Theoretical analysis is presented for the $n$ level case and formulas for singular controls, which are candidates for landscape traps, are compared to their analogues in the dipole approximation. A numerical analysis of the existence of traps in control landscapes beyond the dipole approximation is made in the four level case. A numerical exploration of these control landscapes is achieved by generating many random Hamiltonians which include a term quadratic in a single control field. The landscapes of such systems are found numerically to be trap free in general. This extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the magnitude of the controls resulting from optimization. It is shown numerically that including a polarizability term in an otherwise uncontrollable system removes traps from the landscapes of a specific family of systems by restoring controllability. We numerically assess the effect of a random polarizability term on the know example of a three level system with a second order trap in its control landscape. It is found that the addition of polarizability removes the trap from the landscape. The implications for laboratory control are discussed.

DOI
10.48550/arxiv.1602.06250

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We investigate the control landscapes of closed, finite level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. Theoretical analysis is presented for the nnn level case and formulas for singular controls, which are candidates for landscape traps, are compared to their analogues in the dipole approximation. A numerical analysis of the existence of traps in control landscapes beyond the dipole approximation is made in the four level case. A numerical exploration of these control landscapes is achieved by generating many random Hamiltonians which include a term quadratic in a single control field. The landscapes of such systems are found numerically to be trap free in general. This extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the magnitude of the controls resulting from optimization. It is shown numerically that including a polarizability term in an otherwise uncontrollable system removes traps from the landscapes of a specific family of systems by restoring controllability. We numerically assess the effect of a random polarizability term on the know example of a three level system with a second order trap in its control landscape. It is found that the addition of polarizability removes the trap from the landscape. The implications for laboratory control are discussed.
We investigate the control landscapes of closed n-level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. The latter term is quadratic in the control field. Theoretical analysis of singular controls is presented, which are candidates for producing landscape traps. The results for considering the presence of singular controls are compared to their counterparts in the dipole approximation (i.e., without polarizability). A numerical analysis of the existence of traps in control landscapes for generating unitary transformations beyond the dipole approximation is made upon including the polarizability term. An extensive exploration of these control landscapes is achieved by creating many random Hamiltonians which include terms linear and quadratic in a single control field. The discovered singular controls are all found not to be local optima. This result extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the fluence of the control resulting from optimization. It is also shown that including a polarizability term in an otherwise uncontrollable dipole coupled system removes traps from the corresponding control landscape by restoring controllability. We numerically assess the effect of a polarizability term on a known example of a particular three-level Λ-system with a second order trap in its control landscape. It is found that the addition of the polarizability removes the trap from the landscape. The general practical control implications of these simulations are discussed.
A proof that almost all quantum systems have trap free (that is, free from local optima) landscapes is presented for a large and physically general class of quantum system. This result offers an explanation for why gradient methods succeed so frequently in quantum control in both theory and practice. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator of closed finite dimension systems. This type of control field has been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause a control optimization progress to halt and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity axiom of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This novel condition is shown to be sufficient for a quantum system's landscape to be trap free. The control landscape for a quantum system is shown to be trap free for all but a null set of Hamiltonians using a novel geometric technique based on the parametric transversality theorem. Numerical evidence confirming this is also presented. This result is the analogue of the work of Altifini, wherein it is shown that controllability holds for all but a null set of quantum systems in the dipole approximation. The presented results indicate that by-and-large limited control resources are the most physically relevant source of landscape traps.
Abstract A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.
Manipulation of a quantum system can be viewed in the framework of a control landscape defined as the physical objective as a functional of the control. Control landscape analyses have thus far considered linear quantum dynamics. This paper extends the analysis of control landscape topology to nonlinear quantum dynamics with the objective of steering a finite-level quantum system from an initial state to a final target state. The analysis rests on the assumptions that (i) the final state is reachable from the initial state, (ii) the differential mapping from the control to the state is surjective, and (iii) the control resources are unconstrained. Under these assumptions, landscape critical points (i.e., where the slope vanishes) for nonlinear quantum dynamics only appear as the global maximum and minimum; thus, the landscape is free of traps. Moreover, the landscape Hessian (i.e., the second derivative with respect to the control) at the global maximum has finite rank, indicating the presence of a large level set of optimal controls that preserve the value of the maximum. Extensive numerical simulations on finite-level models of the Gross-Pitaevskii equation confirm the trap-free nature of the landscape as well as the Hessian rank analysis, using either an applied electric field or a tunable condensate two-body interaction strength as the control. In addition, the control mechanisms arising in the numerical simulations are qualitatively assessed. These results are a generalization of previous findings for the linear Schr\"odinger equation, and show promise for successful control in a wide range of nonlinear quantum dynamics applications.
Quantum optimal control has enjoyed wide success for a variety of theoretical and experimental objectives. These favorable results have been attributed to advantageous properties of the corresponding control landscapes, which are free from local optima if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is full rank, and (3) the control field is not constrained. This paper explores how gradient searches for globally optimal control fields are affected by deviations from assumption (2). In some quantum control problems, so-called singular critical points, at which the Jacobian is rank deficient, may exist on the landscape. Using optimal control simulations, we show that search failure is only observed when a singular critical point is also a second-order trap, which occurs if the control problem meets additional conditions involving the system Hamiltonian and/or the control objective. All known second-order traps occur at constant control fields, and we also show that they only affect searches that originate very close to them. As a result, even when such traps exist on the control landscape, they are unlikely to affect well-designed gradient optimizations under realistic searching conditions.
We present a comprehensive analysis of the landscape for full quantum-quantum control associated with the expectation value of an arbitrary observable of one quantum system controlled by another quantum system. It is shown that such full quantum-quantum control landscapes are convex, and hence devoid of local suboptima and saddle points that may exist in landscapes for quantum systems controlled by time-dependent classical fields. There is no controllability requirement for the full quantum-quantum landscape to be trap-free, although the forms of Hamiltonians, the flexibility in choosing initial state of the controller, as well as the control duration, can infulence the reachable optimal value on the landscape. All level sets of the full quantum-quantum landscape are connected convex sets. Finally, we show that the optimal solution of the full quantum-quantum control landscape can be readily determined numerically, which is demonstrated using the Jaynes-Cummings model depicting a two-level atom interacting with a quantized radiation field.