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IEEE Conference on Decision and Control (CDC)

Approximate controllability of the Schrödinger equation with a polarizability term

Nabile Boussaïd, Marco Caponigro, Thomas Chambrion

Published Dec 1, 2012• 9 citations
Publisher

Abstract

This paper is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, has to be corrected by a so-called polarizability term, involving the field induced dipole moment. Sufficient conditions for controllability between eigenstates of the free Hamiltonian are derived and control laws are explicitly given. As an illustration, the results are applied to the planar rotation of the HCN molecule.

DOI
10.1109/cdc.2012.6426619

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This paper is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, has to be corrected by a so-called polarizability term, involving the field induced dipole moment. Sufficient conditions for controllability between eigenstates of the free Hamiltonian are derived and control laws are explicitly given. As an illustration, the results are applied to the planar rotation of the HCN molecule.
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.
In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system.
This chapter describes a versatile technique to control molecular rotation and translation. The technique is based on the combined effect of external electric fields that act concurrently on the molecular permanent and induced dipole moments. This synergistic effect arises for any polar molecule, as only an anisotropic polarizability, along with a permanent dipole moment, is required. This is always available in polar molecules. The effect of the combined fields can be achieved either by superimposing an electrostatic and a far-off resonant optical field (i.e., one that supplies electric field strength but does not cause transitions) that act, respectively, on the molecule's permanent and induced electric dipole, or by making use of a unipolar electromagnetic pulse that seizes the permanent and induced dipole moments simultaneously. If the combined electric fields are homogeneous, only molecular rotation is affected, whereas inhomogeneous fields affect both molecular rotation and translation. As noted in the chapter, the ability to manipulate molecular rotation and translation has wide-ranging applications – in research areas as diverse as reaction dynamics, spectroscopy, higher harmonic generation and molecular orbital imaging, the focusing and trapping of molecules, as well as quantum simulation and computing.
We explore the possibility of forming a oriented polar molecule directly from a pair of colliding atoms. The process comprises the photoassociation and vibrational stabilization along with the molecular orientation. These processes are driven by a single time-dependent, linearly polarized control field and proceeds entirely within the electronic ground state, leveraging the presence of a permanent dipole moment. The control field is found by means of an optimal quantum control algorithm with a single target observable given by the restriction of the orientation operator on a subset of bound levels. We consider a rovibrational model system for the collision of O + H atoms and solve directly the time-dependent Schrodinger equation. We show that the optimized field is capable of enhancing the molecular orientation already induced by the photoassociation and vibrational stabilization thus yielding oriented polar molecules that can be useful for many applications.