A result (see, e.g., [3][complement BIII, page 217]) of quantum electrodynamics (based on Glauber theorem) says that classical currents and sources only generate classical light (quasi-classical states of the field). This paper provides a control theoretic interpretation of this result when the classical currents and sources are considered as control inputs: the dynamics of the quantified electrodynamics field is not controllable; the controllable part is contained into the classical dynamics. This result can be seen as the infinite dimensional analogue of the following fact (see [8]): the controllable part of the quantum harmonic oscillator corresponds to the classical dynamics of the average=-+u (u is the control input). Thus controllability can only be achieved when the field dynamics is coupled to some localized quantum dynamics. We describe here two typical models of such controlled systems: the Jaynes-Cummings model [6] (one field mode coupled to a two-level atom) and trapped ions vibrational model. Both systems correspond to experiments conducted by physicists. We recall their main approximations and validity domains. We also provide their associated formulation in terms of PDE with control.
A result (see, e.g., [3][complement BIII, page 217]) of quantum electrodynamics (based on Glauber theorem) says that classical currents and sources only generate classical light (quasi-classical states of the field). This paper provides a control theoretic interpretation of this result when the classical currents and sources are considered as control inputs: the dynamics of the quantified electrodynamics field is not controllable; the controllable part is contained into the classical dynamics. This result can be seen as the infinite dimensional analogue of the following fact (see [8]): the controllable part of the quantum harmonic oscillator corresponds to the classical dynamics of the average=-+u (u is the control input). Thus controllability can only be achieved when the field dynamics is coupled to some localized quantum dynamics. We describe here two typical models of such controlled systems: the Jaynes-Cummings model [6] (one field mode coupled to a two-level atom) and trapped ions vibrational model. Both systems correspond to experiments conducted by physicists. We recall their main approximations and validity domains. We also provide their associated formulation in terms of PDE with control.
We investigate the controllability of quantum systems in two different settings: the standard "closed" setting, in which a quantum system is seen as isolated and the control problem is formulated on the Schrödinger equation; the open setting that describes a quantum system in interaction with a larger one, of which just qualitative parameters are known, by means of the Lindblad equation on states. In the context of closed systems we focus our attention to an interesting class of models, namely the spin-boson models. The latter describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. We discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (C-QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S. Haroche. We investigate the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding -in the application to cavity QED- to an external electric and magnetic field, respectively. We review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. This result is based on a spectral analysis exploiting the non-resonances of the spectrum. As far as the relation between the Rabi and the Jaynes-Cummings Hamiltonians concerns, we treat the so called rotating wave approximation in a rigorous framework. We formulate the problem as an adiabatic limit in which the detuning frequency and the interaction strength parameter goes to zero, known as the weak-coupling regime. We prove that, under certain hypothesis on the ratio between the detuning and the coupling, the Jaynes-Cumming and the Rabi dynamics exhibit the same behaviour, more precisely the evolution operators they generate are close in norm. In the framework of open quantum systems we investigate the controllability of the Lindblad equation. We consider a control acting adiabatically on the internal part of the system, which we see as a degree of freedom that can be used to contrast the action of the environment. The adiabatic action of the control is chosen to produce a robust transition. We prove, in the prototype case of a two-level system, that the system approach a set of equilibrium points determined by the environment, i.e. the parameters that specify the Lindblad operator. On that set the system can be adiabatically steered choosing a suitable control. The analysis is based on the application of geometrical singular perturbation methods.
We investigate the controllability of the Jaynes-Cummings dynamics in the resonant and nearly resonant regime. We analyze two different types of control operators acting on the bosonic part, corresponding—in the application to cavity Quantum Electro Dynamics—to an external electric and magnetic field, respectively. For these models, we prove approximate controllability for all values of the coupling constant g∈R, except those in a countable set S* which is explicitly characterized in the statement. The proof relies on a spectral analysis which yields the non-resonance of the spectrum for every g∈R\S*.
We treat control of several two-level atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinite-dimensional system algebras. Hence we address problems arising with infinite-dimensional Lie algebras and those of unbounded operators. For the models considered, these problems can be solved by splitting the set of control Hamiltonians into two subsets: the first obeys an Abelian symmetry and can be treated in terms of infinite-dimensional Lie algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry, and its discussion introduces new arguments. Yet, full controllability can be achieved in a strong sense: e.g., in a time dependent Jaynes–Cummings model we show that, by tuning coupling constants appropriately, every unitary of the coupled system (atoms and cavity) can be approximated with arbitrarily small error.
Summary form only given, as follows. The ability to create and completely characterize a variety of fundamental quantum states has long been sought after in the laboratory since it brings to the forefront issues involving the relationship between quantum and classical physics. The interaction of a trapped atom with classical laser light constitutes a simple and well-isolated quantum system that makes experiments along these lines possible. The motion of an ion trapped in a rf-trap can be described as a quantum harmonic oscillator. If the ion is driven by a classical light field, several interactions similar to and beyond the simple Jaynes-Cummings coupling can be realized. These couplings allow for precisely controlled engineering and measurement of quantum states. We create states having both classical and nonclassical character including thermal, number, coherent, squeezed, and Schrodinger cat states. The motional quantum state is fully reconstructed with use of two novel schemes that determine the density matrix in the number state basis and the Wigner function. Our techniques allow well-controlled experiments on decoherence and related phenomena on the quantum-classical borderline.
It is proven in a previous paper that any modal approximation of the one-dimensional quantum harmonic oscillator is controllable. We prove here that, contrary to such finite-dimensional approximations, the original infinite-dimensional system is not controllable: Its controllable part is of dimension 2 and corresponds to the dynamics of the average position. More generally, we prove that, for the quantum harmonic oscillator of any dimension, similar lacks of controllability occur whatever the number of control is: the controllable part still corresponds to the average position dynamics. We show, with the quantum particle in a moving quadratic potential, that some physically interesting motion planning questions can be however solved.
In quantum control theory, the fundamental issue of controllability covers the questions whether and under which conditions a system can be steered from one pure state into another by suitably tuned time evolution operators. Even though Lie theoretic methods to analyze these aspects are well-established for finite dimensional systems, they fail to apply to those with an infinite number of levels. The Jaynes-Cummings-Hubbard model - describing two-level systems in coupled cavities - is such an infinite dimensional system. In this contribution we study its controllability. In the two cavity case we exploit symmetry arguments; we show that one part of the control Hamiltonians can be studied in terms of infinite dimensional block diagonal Lie algebras while the other part breaks this symmetry to achieve controllability. An induction on the number of cavities extends this result to the general case. Individual control of the qubit and collective control of the hopping between cavities is sufficient for both pure state and strong operator controllability. We additionally establish new criteria for the controllability of infinite dimensional quantum systems admitting symmetries.