S. G. Schirmer

Complete controllability of quantum systems

Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.

Degrees of controllability for quantum systems and application to atomic systems

Precise definitions for different degrees of controllability for quantum systems are given, and necessary and sufficient conditions for each type of controllability are discussed. The results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies.

Complete controllability of finite-level quantum systems

Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems.

Constructive control of quantum systems using factorization of unitary operators

We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.

Criteria for reachability of quantum states

We address the question of which quantum states can be inter-converted under the action of a time-dependent Hamiltonian. In particular, we consider the problem as applied to mixed states, and investigate the difference between pure- and mixed-state controllabilities introduced in previous work. We provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group. We also address the problem of which states can be prepared if the dynamical Lie group is not sufficiently large to allow the system to be controllable.

LIMITATIONS ON QUANTUM CONTROL

In this note we give an introduction to the topic of Quantum Control, explaining what its objectives are, and describing some of its limitations.

Identification of dynamical Lie algebras for finite-level quantum control systems

The problem of identifying the dynamical Lie algebras of finite-level quantum systems subject to external control is considered, with special emphasis on systems that are not completely controllable. In particular, it is shown that the dynamical Lie algebra for an N-level system with symmetrically coupled transitions, such as a system with equally spaced energy levels and uniform transition dipole moments, is a subalgebra of so(N) if N = 2? + 1, and a subalgebra of sp(?) if N = 2?. General criteria for obtaining either so(2? + 1) or sp(?) are established.

Limits of control for quantum systems: Kinematical bounds on the optimization of observables and the question of dynamical realizability

In this paper we investigate the limits of control for mixed-state quantum systems. The constraint of unitary evolution for nondissipative quantum systems imposes kinematical bounds on the optimization of arbitrary observables. We summarize our previous results on kinematical bounds and show that these bounds are dynamically realizable for completely controllable systems. Moreover, we establish improved bounds for certain partially controllable systems. Finally, the question of dynamical realizability of the bounds for arbitrary partially controllable systems is shown to depend on the accessible sets of the associated control system on the unitary group

Robust quantum memory via quantum control

U(N)

Quantum control of population inversion in the presence of spontaneous emission

and the results of a few control computations are discussed briefly.