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.
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.
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.
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.
The property of controllability of quantum systems is revisited. In the case of a system having this property, couplings to external agents are available whose adjustment can guide the state to any chosen target on a suitably defined manifold (or arbitrarily close to any such target) at any chosen time. With due consideration to unbounded operators corresponding to physical observables possessing continuous spectra, sufficient conditions for controllability based on Lie-algebraic arguments are obtained. The results are not limited to finite-dimensional systems, nor to infinite-dimensional systems with discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated for the case of the one-dimensional Poschl-Teller potential. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for controllability.
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains 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.
The property of controllability of quantum systems is explored, and conditions for controllability based on Lie-algebraic properties are enunciated. Due consideration is given to the unbounded character of some of the operators that arise and to the generally noncompact nature of the operator algebras associated with the system. Results for both finite and infinite-dimensional systems are provided; in the latter case the results are not limited to cases having purely discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated through examples, namely the Poschl-Teller potential and the repulsive radial oscillator, which reveal some of the subtleties of the controllability issue. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for its controllability.
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.