^{1,a)}, Jason Dominy

^{2,b)}, Tak-San Ho

^{3,c)}and Herschel Rabitz

^{3,d)}

### Abstract

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 authors acknowledge support from the NSF and DARPA.

I. INTRODUCTION

II. QUANTUM CONTROL LANDSCAPES AND LEVEL SETS

III. LEVEL SET EXPLORATION TECHNIQUE

A. L-SET equations for exploration of subspaces of

B. Choice of the free functions

IV. NUMERICAL ILLUSTRATIONS

A. Deterministic free functions

B. Stochastic free functions

C. Homologous systems with varying energy differences and dipole elements: Multiple observables

V. CONCLUSION

### Key Topics

- Photonics
- 34.0
- Subspaces
- 12.0
- Differential equations
- 11.0
- Control systems
- 6.0
- Jacobians
- 6.0

## Figures

A depiction of Hamiltonian space with axes labeled by three characteristic matrix elements (out of possibly many). A particular quantum system is specified by the collection of matrix element parameters (i.e., controls); (energy level differences), (permanent dipole differences), and (transition dipole elements). Here, and represent two distinct quantum systems lying on two continuous but disconnected components of the same level set, defined as sharing an invariant value of an observable. It is implicitly understood that each quantum system is time dependent through a common photonic reagent. A level set of dynamically homologous systems is the collection of quantum systems that produce the same target observable value when driven by the common photonic reagent.

A depiction of Hamiltonian space with axes labeled by three characteristic matrix elements (out of possibly many). A particular quantum system is specified by the collection of matrix element parameters (i.e., controls); (energy level differences), (permanent dipole differences), and (transition dipole elements). Here, and represent two distinct quantum systems lying on two continuous but disconnected components of the same level set, defined as sharing an invariant value of an observable. It is implicitly understood that each quantum system is time dependent through a common photonic reagent. A level set of dynamically homologous systems is the collection of quantum systems that produce the same target observable value when driven by the common photonic reagent.

Level sets for in the subspace described by the matrix element parameters , , and . The three state quantum system has fixed energy level differences and and fixed permanent dipoles . (a) A portion of the level set for characterized by a large surface throughout the space. Each point on the level set corresponds to a member of the dynamically homologous class driven by the fixed field. (b) A portion of the level set for characterized by a large number of isolated components. Regardless of the structure of the level set (i.e., connected or disconnected), each identified quantum system yields the same value for the observable [e.g., for (b)] and is therefore a member of the same dynamically homologous class. (c) Cumulative volume fraction of quantum systems in achieving control anywhere over the interval within the domain explored in (a) and (b).

Level sets for in the subspace described by the matrix element parameters , , and . The three state quantum system has fixed energy level differences and and fixed permanent dipoles . (a) A portion of the level set for characterized by a large surface throughout the space. Each point on the level set corresponds to a member of the dynamically homologous class driven by the fixed field. (b) A portion of the level set for characterized by a large number of isolated components. Regardless of the structure of the level set (i.e., connected or disconnected), each identified quantum system yields the same value for the observable [e.g., for (b)] and is therefore a member of the same dynamically homologous class. (c) Cumulative volume fraction of quantum systems in achieving control anywhere over the interval within the domain explored in (a) and (b).

Quantum control landscapes and level set structure in , specified by and , for three state systems. The quantum control landscape (a) corresponds to the fixed photonic reagent shown in (b). The quantum control level sets are defined by the value as a function of the controling Hamiltonian variables and . The level set values range from 0.0 (black) to 1.0 (white). The quantum control landscape (c) corresponds to (not shown). The power spectra (not shown) of both fields have main peaks at and 9.0. The resonant localization of large values in (a) around and and the near independence with respect to is mainly a result of the low intensity of the field . In landscape (c), the high intensity of the field dynamically broadens the eigenstates of the system to produce a near energy continuum allowing transitions to occur throughout the domain of and . The highly fractured structure of these level sets reflects the presence of complex multipathway constructive and destructive quantum interferences. For both landscapes (a) and (c), the dipole matrix elements are fixed: , , , and .

Quantum control landscapes and level set structure in , specified by and , for three state systems. The quantum control landscape (a) corresponds to the fixed photonic reagent shown in (b). The quantum control level sets are defined by the value as a function of the controling Hamiltonian variables and . The level set values range from 0.0 (black) to 1.0 (white). The quantum control landscape (c) corresponds to (not shown). The power spectra (not shown) of both fields have main peaks at and 9.0. The resonant localization of large values in (a) around and and the near independence with respect to is mainly a result of the low intensity of the field . In landscape (c), the high intensity of the field dynamically broadens the eigenstates of the system to produce a near energy continuum allowing transitions to occur throughout the domain of and . The highly fractured structure of these level sets reflects the presence of complex multipathway constructive and destructive quantum interferences. For both landscapes (a) and (c), the dipole matrix elements are fixed: , , , and .

Two solutions to the energy difference L-SET equations in [c.f., Eq. (28)] along with the component of the level set (shown as dashed) on which both trajectories traverse. Both trajectories begin at the same initial Hamiltonian indicated by the solid dot which is given by , , and , and fixed dipole elements , , , , , and . The observable level set is specified by the invariant population transfer . (a) : The energy level difference trajectory with free functions from Ref. 19. (b) : The energy level difference trajectory with free functions from Ref. 20. The trajectories shown in (a) and (b) give an appearance of having cusps at certain locations, but this behavior arises from the perspective as the trajectories are smooth and continuous. Since both trajectories begin on the same component of the level set, the trajectories will remain on this component of the level set indefinitely. There are also other isolated components of the level set corresponding to (not shown).

Two solutions to the energy difference L-SET equations in [c.f., Eq. (28)] along with the component of the level set (shown as dashed) on which both trajectories traverse. Both trajectories begin at the same initial Hamiltonian indicated by the solid dot which is given by , , and , and fixed dipole elements , , , , , and . The observable level set is specified by the invariant population transfer . (a) : The energy level difference trajectory with free functions from Ref. 19. (b) : The energy level difference trajectory with free functions from Ref. 20. The trajectories shown in (a) and (b) give an appearance of having cusps at certain locations, but this behavior arises from the perspective as the trajectories are smooth and continuous. Since both trajectories begin on the same component of the level set, the trajectories will remain on this component of the level set indefinitely. There are also other isolated components of the level set corresponding to (not shown).

(Color) A component of the level set in shown in uniform blue for a three level system with . The energy level differences are fixed (, ). Five stochastic trajectories are shown in different colors with each originating at the same Hamiltonian (black dot) and traversing the component level set guided by the associated random free vectors. The five trajectories explore a large region of this component of the level set to infer its general shape.

(Color) A component of the level set in shown in uniform blue for a three level system with . The energy level differences are fixed (, ). Five stochastic trajectories are shown in different colors with each originating at the same Hamiltonian (black dot) and traversing the component level set guided by the associated random free vectors. The five trajectories explore a large region of this component of the level set to infer its general shape.

Energy level difference and dipole element solutions to the full L-SET equations [see Eq. (18)] for a four level system along with the final populations in each state at for each identified homologous system. The simultaneous population control goals are and and the free vector of functions is given in Ref. 21. The initial system is given in the text. (a) Energy level differences for dynamically homologous systems. The black dots indicate partially degenerate systems that are dynamically homologous to the original nondegenerate quantum system. (b) Dipole elements for the identified homologous systems. Some of the homologous systems have forbidden transitions (i.e., ) while the original system has all allowed transitions (i.e., ). These forbidden transitions imply the existence of different mechanisms for the identified homologous systems. (c) Final populations in each state at for the homologous systems identified by L-SET. The variations found in the final populations for the unconstrained states (i.e., ∣1⟩ and ∣3⟩) are indicative of the disparate mechanisms identified by L-SET over the homologous class. As expected, the final populations of the constrained target states ∣2⟩ and ∣4⟩ are constant for all homologous systems.

Energy level difference and dipole element solutions to the full L-SET equations [see Eq. (18)] for a four level system along with the final populations in each state at for each identified homologous system. The simultaneous population control goals are and and the free vector of functions is given in Ref. 21. The initial system is given in the text. (a) Energy level differences for dynamically homologous systems. The black dots indicate partially degenerate systems that are dynamically homologous to the original nondegenerate quantum system. (b) Dipole elements for the identified homologous systems. Some of the homologous systems have forbidden transitions (i.e., ) while the original system has all allowed transitions (i.e., ). These forbidden transitions imply the existence of different mechanisms for the identified homologous systems. (c) Final populations in each state at for the homologous systems identified by L-SET. The variations found in the final populations for the unconstrained states (i.e., ∣1⟩ and ∣3⟩) are indicative of the disparate mechanisms identified by L-SET over the homologous class. As expected, the final populations of the constrained target states ∣2⟩ and ∣4⟩ are constant for all homologous systems.

## Tables

Level sets with ancillary energy extremization. Asymptotic values of the elements of corresponding to the quantum systems and which are associated with and , respectively. is the cost function defined by the norm of the energy levels in Eq. (33). L-SET finds homologous systems defined by while locally extremizing . corresponds to choosing the free vector of functions , while corresponds to choosing the free vector . is the Hamiltonian with the initial energy levels indicated in the table and the fixed dipole matrix elements: , , , , , and .

Level sets with ancillary energy extremization. Asymptotic values of the elements of corresponding to the quantum systems and which are associated with and , respectively. is the cost function defined by the norm of the energy levels in Eq. (33). L-SET finds homologous systems defined by while locally extremizing . corresponds to choosing the free vector of functions , while corresponds to choosing the free vector . is the Hamiltonian with the initial energy levels indicated in the table and the fixed dipole matrix elements: , , , , , and .

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