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arXiv:1010.1235 · Physical Review A

Extension of the Morris-Shore transformation to multilevel ladders

Andon A. Rangelov, Nikolay V. Vitanov, Bruce W. Shore

Published Nov 3, 2006• 66 citations
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Abstract

We describe situations in which chains of $N$ degenerate quantum energy levels, coupled by time-dependent external fields, can be replaced by independent sets of chains of length $N$, $N\ensuremath{-}1,\dots{},2$ and sets of uncoupled single states. The transformation is a generalization of the two-level Morris-Shore transformation [J.R. Morris and B.W. Shore, Phys. Rev. A 27, 906 (1983)]. We illustrate the procedure with examples of three-level chains.

DOI
10.1103/physreva.74.053402

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We describe situations in which chains of NNN degenerate quantum energy levels, coupled by time-dependent external fields, can be replaced by independent sets of chains of length NNN, N\ensuremath−1,…,2N\ensuremath{-}1,\dots{},2N\ensuremath−1,…,2 and sets of uncoupled single states. The transformation is a generalization of the two-level Morris-Shore transformation [J.R. Morris and B.W. Shore, Phys. Rev. A 27, 906 (1983)]. We illustrate the procedure with examples of three-level chains.
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.
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.
Abstract The treatment of time-dependent dynamics of quantum systems involving multiple states poses considerable technical challenges. One of the most efficient approaches in treating such systems is the Morris–Shore (MS) transformation which decomposes the multistate dynamics to a set of independent systems of simpler interaction pattern and uncoupled spectator states. The standard MS transformation imposes restrictions on the time dependence of the external fields addressing the states, as it requires that both Rabi frequencies have the same time profile. In this work we treat the case of the time-dependent MS transformation, which opens prospects for a variety of physically interesting processes wherein the fields may have different time dependences. We explore the adiabatic and the double-adiabatic limit, in which we demonstrate population transfer between the MS states that results in population transfer from one set of states onto another. We demonstrate the generation of superposition states between the MS states by the techniques of half adiabatic passage and fractional stimulated Raman adiabatic passage, which translate to superpositions of all the states of the involved levels.
Examining the dynamics of a degenerate two-level system excited by coherent resonant light, we exhibit a transformation of basis states which reduces the problem to a set of independent nondegenerate one- and two-state systems. More generally, reduction of multilevel Hamiltonians requires only that the states can be collected into two sets; within each set the states are degenerate, unlinked by Hamiltonian matrix elements. The remaining Hamiltonian linkages must share the same time dependence, but are otherwise arbitrary. We apply this method to reduction of the angular momentum degeneracy of a two-level atom excited by elliptically polarized light and to multiple-laser resonant excitation of multiple levels into a common level.
We investigate a multilevel system coupled to a quantum-field mode, neglecting the damping. The model is shown to be exactly solvable in the case of equal couplings and detunings. The eigenvalues and eigensolutions are calculated. We compute the time evolution of the level populations. The behavior is compared with that of the corresponding exactly solvable semiclassical model, in which the quantum mode is replaced with an external classical field, and with that of the Maxwell–Bloch model, in which the ladder and field variables are decorrelated in the lowest nontrivial order. The semiclassical limit of the quantum system is also studied.
We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, V(t)V(t)V(t). Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses V(t)V(t)V(t).