Lobachevskii Journal of Mathematics

On Controllability of a Highly Degenerate Four-Level Quantum System with a ‘‘Chained’’ Coupling Hamiltonian

Sergey Kuznetsov, Alexander Nikolaevich Pechen

Published Jul 1, 2022• 4 citations

Abstract

Abstract Last decades have faced considerable interest in various problems of quantum control. One of them is to determine which quantum systems are controllable, meaning that all possible states of such systems can be reached from any initial one under available controls. This paper focuses on controllability exploration of a specific four-level quantum system with three levels having the same energy and driven by an electromagnetic control field which allows transitions only between adjacent basis states. Such systems appear in the analysis of higher order traps in quantum control landscapes. This paper explores a few properties of the system, including a proof it is completely controllable for any complex values of the non-zero matrix elements of the interaction Hamiltonian.

DOI: 10.1134/s1995080222100225

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Lobachevskii Journal of Mathematics

On Controllability of a Highly Degenerate Four-Level Quantum System with a ‘‘Chained’’ Coupling Hamiltonian

Sergey Kuznetsov, Alexander Nikolaevich Pechen

Published Jul 1, 2022• 4 citations

Publisher

Abstract

DOI: 10.1134/s1995080222100225

Related papers

On Controllability of $\boldsymbol{\Lambda}$ - and $\mathbf{V}$ -Atoms and Other Three-Level Systems with Two Allowed TransitionsOn Controllability of $$\boldsymbol{\Lambda}$$- and $$\mathbf{V}$$-Atoms and Other Three-Level Systems with Two Allowed Transitions

Sergei A. Kuznetsov, Alexander Nikolaevich Pechen

A three-level $\Lambda$ -atom (resp., $V$ -atom) is a three-level quantum system with a forbidden direct transition between the two lowest (resp., highest) energy states. These systems are particular examples of three-level systems with only two allowed transitions and are important in quantum optics, e.g., for modeling electromagnetically induced transparency, etc. For generality, we consider not only $\Lambda$ - and $V$ -atoms, but all other types of three-level systems with only two allowed transitions between basis states. For the most general case of arbitrary non-zero complex values of the interaction Hamiltonian, we combine in a unified way the description of the structure of the dynamical Lie algebra as well as controllability properties for such systems. It turns out that for non-degenerate transition frequencies the system is completely controllable for any non-zero interaction Hamiltonian, while for the degenerate case the system is either uncontrollable or controllable, depending on the exact values of matrix elements of the free and interaction Hamiltonians.A three-level $$\Lambda$$ -atom (resp., $$V$$ -atom) is a three-level quantum system with a forbidden direct transition between the two lowest (resp., highest) energy states. These systems are particular examples of three-level systems with only two allowed transitions and are important in quantum optics, e.g., for modeling electromagnetically induced transparency, etc. For generality, we consider not only $$\Lambda$$ - and $$V$$ -atoms, but all other types of three-level systems with only two allowed transitions between basis states. For the most general case of arbitrary non-zero complex values of the interaction Hamiltonian, we combine in a unified way the description of the structure of the dynamical Lie algebra as well as controllability properties for such systems. It turns out that for non-degenerate transition frequencies the system is completely controllable for any non-zero interaction Hamiltonian, while for the degenerate case the system is either uncontrollable or controllable, depending on the exact values of matrix elements of the free and interaction Hamiltonians.

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Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.

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Alexander Pechen, Boris Volkov

In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a $V-V$ system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a $V-V$ system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.

quant-pharXiv:2607.01217Physica ScriptaJul 2, 20260 citations

On Controllability of

\boldsymbol{\Lambda}

- and

\mathbf{V}

-Atoms and Other Three-Level Systems with Two Allowed Transitions

A three-level

\Lambda

-atom (resp.,

V

-atom) is a three-level quantum system with a forbidden direct transition between the two lowest (resp., highest) energy states. These systems are particular examples of three-level systems with only two allowed transitions and are important in quantum optics, e.g., for modeling electromagnetically induced transparency, etc. For generality, we consider not only

\Lambda

- and

V

-atoms, but all other types of three-level systems with only two allowed transitions between basis states. For the most general case of arbitrary non-zero complex values of the interaction Hamiltonian, we combine in a unified way the description of the structure of the dynamical Lie algebra as well as controllability properties for such systems. It turns out that for non-degenerate transition frequencies the system is completely controllable for any non-zero interaction Hamiltonian, while for the degenerate case the system is either uncontrollable or controllable, depending on the exact values of matrix elements of the free and interaction Hamiltonians.

\mathfrak{su}(N)

and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.

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.

In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a

V-V

system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.