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arXiv:0802.3114

Open-Loop Control in Quantum Optics: Two-Level Atom in Modulated Optical Field

Saifullah, Sergey Borisenok

Published Feb 21, 2008• 0 citations
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Abstract

The methods of mathematical control theory are widely used in the modern physics, but still they are less popular in quantum science. We will discuss the aspects of control theory, which are the most useful in applications to the real problems of quantum optics. We apply this technique to control the behavior of the two-level quantum particles (atoms) in the modulated external optical field in the frame of the so called "semi classical model", where quantum two-level atomic system (all other levels are neglected) interacts with classical electromagnetic field. In this paper we propose a simple model of feedforward (open-loop) control for the quantum particle system, which is a basement for further investigation of two-level quantum particle in the external one-dimensional optical field.

DOI
10.48550/arxiv.0802.3114

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The methods of mathematical control theory are widely used in the modern physics, but still they are less popular in quantum science. We will discuss the aspects of control theory, which are the most useful in applications to the real problems of quantum optics. We apply this technique to control the behavior of the two-level quantum particles (atoms) in the modulated external optical field in the frame of the so called "semi classical model", where quantum two-level atomic system (all other levels are neglected) interacts with classical electromagnetic field. In this paper we propose a simple model of feedforward (open-loop) control for the quantum particle system, which is a basement for further investigation of two-level quantum particle in the external one-dimensional optical field.
Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.
Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially. In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom. Dynamical variables for a two-level atom We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.