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

TL;DR

This paper introduces a method for quantum-state engineering using coupled Hilbert-space mirrors, which are realized through laser pulses to manipulate superpositions in degenerate quantum systems more efficiently than traditional rotation methods.

Contribution

It presents a novel approach to quantum-state manipulation employing generalized Householder reflections in degenerate systems, with practical laser pulse implementations.

Findings

Reflections enable more efficient state manipulation than rotations.

Propagators can be expressed as products of Householder reflections.

Physical realizations demonstrated in atomic and molecular systems.

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.