Physical realization of coupled Hilbert-space mirrors for quantum-state engineering

Abstract

Manipulation of superpositions of discrete quantum states has a mathematical counterpart in the motion of a unit-length statevector in an N-dimensional Hilbert space. Any such statevector motion can be regarded as a succession of two-dimensional rotations. But the desired statevector change can also be treated as a succession of reflections, the generalization of Householder transformations. In multidimensional Hilbert space such reflection sequences offer more efficient procedures for statevector manipulation than do sequences of rotations. We here show how such reflections can be designed for a system with two degenerate levels—a generalization of the traditional two-state atom—that allows the construction of propagators for angular momentum states. We use the Morris–Shore transformation to express the propagator in terms of Morris–Shore basis states and Cayley–Klein parameters, which allows us to connect properties of laser pulses to Hilbert-space motion. Under suitable conditions on the couplings and the common detuning, the propagators within each set of degenerate states represent products of generalized Householder reflections, with orthogonal vectors. We propose physical realizations of this novel geometrical object with resonant, near-resonant and far-off-resonant laser pulses. We give several examples of implementations in real atoms or molecules.

Acknowledgements

Quantum information processing, and the needed manipulation of statevectors in Hilbert space, has long interested Sir Peter Knight, in whose honour the present issue of Journal of Modern Optics has been assembled. We are pleased to offer the present article in his honour.

This work has been supported by the EU ToK project CAMEL (Grant No. MTKD-CT-2004-014427), the EU RTN project EMALI (Grant No. MRTN-CT-2006-035369), and the Alexander von Humboldt Foundation.