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.
Highlights
Manipulation of discrete quantum states has long held interest, most recently for application to quantum information processing [1]
= .), cos(An/2), where the pulse area is the phase φn is equal to π; we obtain a physical realization for the standard quantum Householder reflections (QHR) (27)
We have here extended the earlier work on QHR in Hilbert space to allow more general linkage patterns between the quantum states, with particular attention to degenerate sublevels that occur with angular momentum states
Summary
Manipulation of discrete quantum states has long held interest, most recently for application to quantum information processing [1]. The QHR implementation proposed earlier [5, 6] requires a particular multistate linkage pattern, in which a set of N low-lying degenerate states all link, via radiative interaction, with a single upper state – a generalization of the tripod linkage termed an N-pod. The expression for the resulting propagator has a clear geometric interpretation as the effect of a succession of reflections, i.e. coupled mirrors The key to this extension of the QHR is a transformation of the underlying basis states, the so-called Morris-Shore (MS) transformation [8, 9, 10]. To illustrate the procedure we develop a useful explicit analytic formalism for two upper states and present some examples
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