Michael Keyl’s research while affiliated with Freie Universität Berlin and other places

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski–Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term H 0 , finitely many bounded control Hamiltonians H j allowing for (at least) piecewise constant control amplitudes [Formula: see text] plus a bang-bang (i.e., on-off) switchable noise term Г V in Kossakowski–Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one as up to now it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians H j and for noise terms Г V with compact normal V.

Cavity optomechanical systems are one of the leading experimental platforms for controlling mechanical motion in the quantum regime. We exemplify that the control over cavity optomechanical systems greatly increases by coupling the cavity also to a two-level system, thereby creating a hybrid optomechanical system. If the two-level system can be driven largely independently of the cavity, we show that the nonlinearity thus introduced enables us to steer the extended system to non-classical target states of the mechanical oscillator with Wigner functions exhibiting significant negative regions. We illustrate how to use optimal control techniques beyond the linear regime to drive the hybrid system from the near ground state into a Fock target state of the mechanical oscillator. We base our numerical optimization on realistic experimental parameters for exemplifying how optimal control enables the preparation of decidedly non-classical target states, where naive control schemes fail. Our results thus pave the way for applying the toolbox of optimal control in hybrid optomechanical systems for generating non-classical mechanical states.

This chapter illustrates a unified frame of quantum systems theory in view of control engineering. In the chapter, controllability criteria is shifted from the well‐known Lie‐algebra rank condition to symmetry conditions that are easy to check yet rigorously rooted in the branching diagrams for simple subalgebras of Su(N). The chapter applies quantum systems theory for bilinear control systems of closed and open systems. It then shows how the emerging quantum systems theory links to many applications in quantum simulation and control without sacrificing mathematical rigor. The chapter points out how gradient flows form the missing link to numerical optimal control algorithms for explicit steerings (control amplitudes) for manipulating closed and open (Markovian and non‐Markovian) systems in finite dimensions. Finally, it gives an outlook on a Lie picture of systems and control theory in infinite dimensions and its application to Jaynes‐Cummings systems, for example, like atoms in a cavity.

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite dimensional open quantum dynamical systems $\Sigma$ following a unital Kossakowski-Lindblad master equation extended by controls. They shall be governed by an inevitable drift Hamiltonian $H_0$ and allowing for (at least) piecewise constant control amplitudes $u_j(t)\in{\mathbb R}$ modulating the control Hamiltonians $H_j$ plus a bang-bang (i.e.\ on-off) switchable noise term $\mathbf{\Gamma}_V$ in the GKS-Lindblad form. Generalising standard majorisation results from finite to infinite dimensions, we show that such bilinear control systems $\Sigma$ allow to approximately {\em reach any target state majorised by the initial one} -- as up to now only has been known in finite dimensional analogues. The proof of the result is currently limited to Hamiltonians $H_0, H_j$ being bounded and GKS-Lindblad noise terms $\mathbf{\Gamma}_V$ with compact V.

In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian and bounded control Hamiltonians. The drift is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control system in question. Its power is demonstrated by looking at some examples. We recover in particular previous genericity results with a much easier proof. Finally several possible generalizations are outlined.

Cavity optomechanical systems are one of the leading experimental platforms for controlling mechanical motion in the quantum regime. We exemplify that the control over cavity optomechanical systems greatly increases by coupling the cavity additionally to a two-level system, thereby creating a hybrid optomechanical system. If the two-level system can be driven largely independently of the cavity, we show that the non-linearity thus introduced enables us to steer the extended system to non-classical target states of the mechanical oscillator with Wigner functions exhibiting significant negative regions. We illustrate how to use optimal control techniques beyond the linear regime to drive the hybrid system from the near ground state into a Fock target state of the mechanical oscillator. We base our numerical optimization on realistic experimental parameters for illustrating how optimal control enables the preparation of decidedly non-classical target states, where naive control fails. Our results pave the way for applying the toolbox of optimal control in hybrid optomechanical systems for generating non-classical mechanical states.

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 this paper we study controllability of a d-level atom interacting with the electromagnetic field in a cavity. The system is modelled by an ordered graph $\Gamma$. The vertices of $\Gamma$ describe the energy levels and the edges allowed transitions. To each edge of $\Gamma$ 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 $\Gamma$. 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.

To “process” the quantum information of a typical quantum system, one has to perform many different processing steps such as free time evolution, controlled time evolution, preparations and measurements. This chapter aims at providing a unified framework for describing all these different operations. The basic idea is to interpret each processing step as a channel, which transforms the system's initial state into the output state the system attains, after completion of the processing. Each channel can be described by a positive and trace‐preserving linear map. The chapter also provides some typical examples of channels. It then discusses a relation between completely positive (cp) maps and states of bipartite systems first discovered by M. D. Choi and A. Jamiolkowski, which is very useful in establishing several fundamental properties of cp‐maps. The Stinespring dilation theorem is the central structure theorem about cp‐maps.