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Lobachevskii Journal of Mathematics

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

Sergei A. Kuznetsov, Alexander Nikolaevich Pechen

Published Jun 1, 2023• 7 citations
Publisher

Abstract

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.

DOI
10.1134/s199508022306029x

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On Controllability of Λ\boldsymbol{\Lambda}Λ- and V\mathbf{V}V-Atoms and Other Three-Level Systems with Two Allowed Transitions
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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.
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In this paper we study controllability of a ddd-level atom interacting with the electromagnetic field in a cavity. The system is modelled by an ordered graph ΓΓΓ. The vertices of ΓΓΓ describe the energy levels and the edges allowed transitions. To each edge of ΓΓΓ we associate a harmonic oscillator representing one mode of the electromagnetic field. The dynamics of the system (drift) is given by a natural generalization of the Jaynes-Cummings Hamiltonian. If we add in addition sufficient control over the atom, the overall system (atom and em-field) becomes strongly controllable, i.e. each unitary on the system Hilbert space can be approximated with arbitrary precision in the strong topology by control unitaries. A key role in the proof is played by a topological *-algebra which is (roughly speaking) a representation of the path algebra of ΓΓΓ. It contains crucial structural information about the control problem, and is therefore an important tool for the implementation of control tasks like preparing a particular state from the ground state. This is demonstrated by a detailed discussion of different versions of three-level systems.
Mathematical modeling of the population dynamics is performed for states of a three-level system (atom) with a V-type configuration transforming a light pulse. It is assumed that the excited eigenstates of the atom are degenerate and coupled by coherent interaction, one of the states being radiating (radiative), while the other state is nonradiating (“dark”). The population dynamics of atomic states is described on the basis of numerical solutions of equations for the matrix elements of the density operator. The dependence of the efficiency of population of the atomic dark state from the values of the parameters of an irradiation pulse and from the ratio of the period of population oscillations of excited atomic states (caused by their coherent interaction) to the lifetime of the atomic radiating state is determined. Typical examples of the time dependence of the population of states of the atom considered are presented for the cases of irradiation by a short (as compared to the lifetime of the radiating state) sinusoidal light pulse and by a long rectangular light pulse with the resonance carrier frequency.