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arXiv:0802.4254 · Physical Review A

Coherent pulsed excitation of degenerate multistate systems: Exact analytic solutions

Elica S. Kyoseva, Nikolay V. Vitanov

Published Feb 21, 2006• 82 citations
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Abstract

We show that the solution of a multistate system composed of $N$ degenerate lower (ground) states and one upper (excited) state can be reduced by using the Morris-Shore transformation to the solution of a two-state system involving only the excited state and a (bright) superposition of ground states. In addition, there are $N\ensuremath{-}1$ dark states composed of ground states. We use this decomposition to derive analytical solutions for degenerate extensions of the most popular exactly soluble models: the resonance solution, the Rabi, Landau-Zener, Rosen-Zener, Allen-Eberly, and Demkov-Kunike models. We suggest various applications of the multistate solutions, for example, as tools for creating multistate coherent superpositions by generalized resonant $\ensuremath{\pi}$ pulses. We show that such generalized $\ensuremath{\pi}$ pulses can occur even when the upper state is far off resonance, at specific detunings, which makes it possible to operate in the degenerate ground-state manifold without populating the (possibly lossy) upper state, even transiently.

DOI
10.1103/physreva.73.023420

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We show that the solution of a multistate system composed of NNN degenerate lower (ground) states and one upper (excited) state can be reduced by using the Morris-Shore transformation to the solution of a two-state system involving only the excited state and a (bright) superposition of ground states. In addition, there are N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 dark states composed of ground states. We use this decomposition to derive analytical solutions for degenerate extensions of the most popular exactly soluble models: the resonance solution, the Rabi, Landau-Zener, Rosen-Zener, Allen-Eberly, and Demkov-Kunike models. We suggest various applications of the multistate solutions, for example, as tools for creating multistate coherent superpositions by generalized resonant \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses. We show that such generalized \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses can occur even when the upper state is far off resonance, at specific detunings, which makes it possible to operate in the degenerate ground-state manifold without populating the (possibly lossy) upper state, even transiently.
We study a train of coincident pulses technique in an N-pod system driven by N pulsed fields, when all of the pulses have the same shape and are on exact resonance. We show that the solution of an N-pod system composed of N degenerate ground states and one upper excited state can be reduced using the Morris–Shore transformation to the solution of a three-state system involving an initial superposition of M ground states, the excited state and superposition of remaining ground states. We use reduced three-state propagator to create any preselected coherent superposition of states by a train of coincident pulses. A proper pulse train is used to connect any two preselected superposition states of an N-pod system. Other advantages of this method are exact analytic solution and enhancement to the robustness of system against deviation from exact pulse areas with increasing the number of coincident pulses.
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.
This paper presents an exact analytic solution for creation of any preselected coherent superposition of states in multi-lambda quantum systems by a train of coincident pulses. We show that the solution of a multi-lambda system composed of degenerate ground states, excited states, and a single initial state can be reduced by using Morris–Shore transformation to the solution of a three-state system. In this method, all of the pump and Stokes pulses have the same time dependence and the ratio of the pump pulses can be used to control the probability amplitudes of the ground states in the coherent superposition of states. Effect of spontaneous emission is efficiently suppressed by employing a train of coincident pulses. This technique resembles the technique of generalized -pulses with minimal pulse areas which is easy to reach experimentally.
A theoretical nonperturbative study of the coherent excitation of a two-state system by N consecutive equally spaced identical pulses is presented. General relations between the evolution matrix elements in the cases of one and N pulses are obtained in a closed form. For pulse envelopes allowing analytical solutions, these relations enable analytical treatment of pulse-train excitation; for pulses that can only be treated numerically, they shorten the computations by a factor of N. The relations show that the multiple-pulse excitation of a two-state system can be considered as a quantum analog of the diffraction grating. The interaction dynamics substantially depends on the way the train is produced because the phase shift that is accumulated by the probability amplitudes due to the free evolution of the system during and between the pulses differs. In the limit of weak excitation, the results recover earlier conclusions based on perturbative treatments. Simple formulas are derived for the conditions for complete population inversion (CPI) and complete population return (CPR), which differ from the single-pulse ones. These general results are applied to four particular cases that allow analytical solutions: resonant, rectangular, Rosen-Zener, and Allen-Eberly pulses. A common feature in all these cases is that the number and the amplitude of oscillations in the state populations increase with the number of pulses, while their width decreases. For rectangular pulses, CPI is possible below a given value of the ratio between the detuning and the pulse area; this value increases nearly linearly with the number of pulses.For the Rosen-Zener model, CPI is found to be possible for two and more pulses, while it is impossible for a single non-resonant pulse. It is shown that for a given number of pulses there is an upper limit on the detuning for which CPI can be observed; this limit increases logarithmically with the number of pulses. For the Allen-Eberly model, CPR is established to be possible for more than one pulse, while it is impossible for a single pulse. The cases of even and odd number of pulses are shown to lead to substantially different results. The limitations imposed on the detuning by the conditions for CPR and CPI are derived and discussed.
Abstract We present an analytically exactly solvable N -pod quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The solution is derived by using the Morris–Shore transformation to reduce the N -pod system to a -like one and then to a two state system and a set of decoupled states in two steps. In this work, we use the propagator for a reduced two state system and the inverse transformation to achieve the reduced three state propagator. It is shown that any preselected coherent superposition of all N ground states can be obtained from an initial superposition of M ground states in the presence of detuning. We use a train of tanh coincident pulses and we show that the effect of spontaneous emission is efficiently suppressed by increasing the number of pulses. Besides, we explore that a coherent superposition of ground states from the initial excited state can be implemented using tanh pulses.
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.