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.
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.
We propose a simple physical implementation of the quantum Householder reflection (QHR) $\mathbf{M}(v)=\mathbf{I}\ensuremath{-}2\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$ in a quantum system of N degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by N resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR $\mathbf{M}(v;\ensuremath{\varphi})=\mathbf{I}+({e}^{i\ensuremath{\varphi}}\ensuremath{-}1)\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$, which can be produced in the same N-pod system when the fields are appropriately detuned from resonance with the excited state. We use these two operators as building blocks in constructing arbitrary preselected unitary transformations. We show that the most general U(N) transformation can be factorized (and thereby produced) by either $N\ensuremath{-}1$ standard QHRs and an N-dimensional phase gate, or $N\ensuremath{-}1$ generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the U(N) group. As an example, we propose a recipe for constructing the quantum Fourier transform (QFT) by at most N interaction steps. For example, the QFT requires a single QHR for N=2, and only two QHRs for N=3 and 4.
Coherent manipulations of atoms using laser light The internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.
Descriptions of coherent excitation of multi-state quantum systems model changes through coupled ordinary differential equations that become more cumbersome as the number of involved quantum states increases. It is always useful to find simplifications of the equations, ideally involving exact analytic solutions. The ultimate simplification is to a set of independent pairs of quantum states. Such sets of independent two-state linkages occur naturally when we describe the excitation between two sets of degenerate Zeeman sublevels by a field that is linearly or circularly polarized. The Morris–Shore (MS) transformation, a Hilbert-space coordinate change, generalizes this procedure to replace the description of an elaborate linkage pattern of an -state Hamiltonian in the rotating-wave approximation by a set of independent two-state systems – a basis of paired bright states and excited states supplemented with dark states and spectator states. The three-state lambda-linkage found in stimulated Raman transitions is a simple example. Both bright and dark states have found application in various procedures for manipulating quantum states, either to achieve population transfer between states or to create coherent superpositions; the dark and spectator states coincide with particular dressed (or adiabatic) states. The present review describes the MS transformation, noting its historical background, and discusses examples of its use.
The Morris-Shore (MS) transformation is a powerful tool for decomposition of the dynamics of multistate quantum systems to a set of two-state systems and uncoupled single states. It assumes two sets of states wherein any state in the first set can be coupled to any state in the second set, but the states within each set are not coupled between themselves. Another important condition is the degeneracy of the states in each set, although all couplings between the states from different sets can be detuned from resonance by the same detuning. The degeneracy condition limits the application of the MS transformation in various physically interesting situations, e.g., in the presence of electric and/or magnetic fields or light shifts, which lift the degeneracy in each set of states, e.g., when these sets comprise the magnetic sublevels of levels with nonzero angular momentum. This paper extends the MS transformation to such situations, in which the states in each of the two sets are nondegenerate. To this end, we develop an alternative way for the derivation of Morris-Shore transformation, which can be applied to nondegenerate sets of states. We present an approximated eigenvalue approach, by which, in the limit of small detunings from degeneracy, we are able to generate an effective Hamiltonian that is dynamically equivalent to the nondegenerate Hamiltonian. The effective Hamiltonian can be mapped to the Morris-Shore basis with a two-step similarity transformation. After the derivation of the general framework, we demonstrate the application of this technique to the popular $\mathrm{\ensuremath{\Lambda}}$ three-state system, and the four-state tripod, double-$\mathrm{\ensuremath{\Lambda}}$, and diamond systems. In all of these systems, our formalism allows us to reduce their quantum dynamics to simpler two-state systems even in the presence of various detunings, e.g., generated by external fields of frequency drifts.
We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled W-type state or to transfer one entangled state into another. An arbitrary N-dimensional unitary transformation can be constructed by at most $N\ensuremath{-}1$ HRs, i.e., significantly faster than O(N2) steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.