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Physical Review A

Controllability of population transfer to degenerate states: Analytical and numerical results for a four-level system

Suhail P. Shah, David J. Tannor, Stuart A. Rice

Published Sep 27, 2002• 19 citations
Publisher

Abstract

We examine, both numerically and analytically, the limits of controllability and the dynamics of population transfer in systems with degenerate target states embedded in a finite manifold of states. An analysis of the dressed eigenstates of coupled field-matter systems is combined with the theory of the controllability of invariant systems on Lie groups to show the existence of a control scheme using a pair of shaped laser pulses acting upon a four-state system with intermediate degenerate states and a diamond-shaped array of transition moments $({\ensuremath{\mu}}_{12},{\ensuremath{\mu}}_{24},{\ensuremath{\mu}}_{13},{\ensuremath{\mu}}_{34}\ensuremath{\ne}0,{\ensuremath{\mu}}_{14},{\ensuremath{\mu}}_{23}=0).$ The limits of the controllability of these systems are linked to the symmetry of the transition dipole moments connecting the states. A broken symmetry of the transition dipole moments $({\ensuremath{\mu}}_{12}/{\ensuremath{\mu}}_{24}\ensuremath{\ne}{\ensuremath{\mu}}_{13}/{\ensuremath{\mu}}_{34})$ permits complete population transfer to one of the pair of degenerate states. Numerical simulations of the optimized population transfer to one of a pair of degenerate states in the system agree quantitatively with the analytically predicted limits of population transfer in different regimes in the joint material-field characteristics of the system. The applicability of the four-state model to selective population transfer in vibrationally mediated photodissociation experiments is discussed.

DOI
10.1103/physreva.66.033405

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We examine, both numerically and analytically, the limits of controllability and the dynamics of population transfer in systems with degenerate target states embedded in a finite manifold of states. An analysis of the dressed eigenstates of coupled field-matter systems is combined with the theory of the controllability of invariant systems on Lie groups to show the existence of a control scheme using a pair of shaped laser pulses acting upon a four-state system with intermediate degenerate states and a diamond-shaped array of transition moments (\ensuremathμ12,\ensuremathμ24,\ensuremathμ13,\ensuremathμ34\ensuremath≠0,\ensuremathμ14,\ensuremathμ23=0).({\ensuremath{\mu}}_{12},{\ensuremath{\mu}}_{24},{\ensuremath{\mu}}_{13},{\ensuremath{\mu}}_{34}\ensuremath{\ne}0,{\ensuremath{\mu}}_{14},{\ensuremath{\mu}}_{23}=0).(\ensuremathμ12​,\ensuremathμ24​,\ensuremathμ13​,\ensuremathμ34​\ensuremath=0,\ensuremathμ14​,\ensuremathμ23​=0). The limits of the controllability of these systems are linked to the symmetry of the transition dipole moments connecting the states. A broken symmetry of the transition dipole moments (\ensuremathμ12/\ensuremathμ24\ensuremath≠\ensuremathμ13/\ensuremathμ34)({\ensuremath{\mu}}_{12}/{\ensuremath{\mu}}_{24}\ensuremath{\ne}{\ensuremath{\mu}}_{13}/{\ensuremath{\mu}}_{34})(\ensuremathμ12​/\ensuremathμ24​\ensuremath=\ensuremathμ13​/\ensuremathμ34​) permits complete population transfer to one of the pair of degenerate states. Numerical simulations of the optimized population transfer to one of a pair of degenerate states in the system agree quantitatively with the analytically predicted limits of population transfer in different regimes in the joint material-field characteristics of the system. The applicability of the four-state model to selective population transfer in vibrationally mediated photodissociation experiments is discussed.
Two simple schemes are presented for the selective population transfer in systems where the target state is degenerate with a second level. Both states are coupled with the ground state via a one photon resonance. The schemes use a combination of two laser pulses. The first pulse breaks the degeneracy of the levels by inducing a Stark shift. The second pulse is chirped across the new levels. The idea is illustrated by numerical simulations of the time-dependent Schr\"odinger equation for a four-level system. The efficient population transfer is explained using a dressed state representation. In conclusion, applications for molecules are suggested.
The manipulation of population transfer and the creation of an arbitrary superposition between two atomic states in the framework of a closed inverse Y-type four-level system driven by a series of pulse pairs and an external control field is reported. It is shown that when the laser fields are tuned to resonance with respective transitions, the non-adiabatic coupling between two degenerate adiabatic states can be reduced tremendously and complete population transfer from the initial state to the target state can be achieved in a piecewise manner. Moreover, an arbitrary superposition between the initial state and target state can be created by suitably tuning the Rabi frequency of the control laser field, inter-pair time delay, pulse number or intra-pair time delay between the pump and Stokes pulses. This scheme combines stimulated Raman adiabatic passage and temporal coherent control, and may be useful for quantum information processing and control of chemical reactions.
We analyze N-level systems for population transfer by counterintuitive pulse delay, in which $N\ensuremath{-}2$ intermediate states are strongly coupled by dressing laser fields. We have carried out specific analysis for the four- and five-level systems using a dressed-state picture, as representatives for even-N- and odd-N-level systems, respectively. We have found that the dynamics of the systems are essentially different for even-N- and odd-N-level systems, and that complete population transfer assisted by dressing fields is more restrictive for even-N-level systems than for odd-N-level systems.
We present explicit analytic expressions for adiabatic dressed states which, in suitable limits, produce complete population transfer between initial and final states of a four-state excitation chain without ever placing appreciable population into intermediate states. The population-transfer scheme is a generalization of the three-state stimulated Raman adiabatic passage that makes use of counterintuitive pulse sequences. We demonstrate analytically and numerically that the four-state system differs in a very significant qualitative way from previously studied N-state systems with odd N. In particular, we find that the presence of modest detuning from single-photon resonance is critical to the success of population transfer.
Even-N-level ladder system has been analyzed for population transfer by counterintuitive frequency-chirped laser pulse,in which N-2 intermediate states are strongly coupled by dressing laser field.The specific analysis for the four-level ladder system has been carried out by using a dressed-state picture,as representative for even-N-system.It is found that a well-controlled pulse area is required for complete population transfer from |1〉|4〉 if the dressing field is sufficiently strong and on resonance.And it is essential to detune the pump and Stokes laser so that one of the dressed intermediate states is on resonance,leading to the reduction of the whole system to the three-level system.
We report studies of state-to-state vibrational energy transfer in an isolated polyatomic molecule driven by combined stimulated Raman adiabatic passage (STIRAP) and counter-diabatic fields (CDF), using as vehicles selective population of one of a pair of near degenerate states in SCCl2_22​ and enhancing the yield of HNC in the HCN/HNC isomerization reaction. The efficiency of the population transfer within a subset of states embedded in a dense manifold of states is examined for the cases that the transition dipole moments between the initial and target states are much smaller than and comparable to the transition dipole moments between background states, and for the case that the subset states have large transition dipole moments with the background states. We show that, in a subset of states that is coupled to background states, a combination of STIRAP fields and CDFs that do not individually generate processes that are competitive with the desired population transfer can generate greater population transfer efficiency than can ordinary STIRAP with field strength and/or pulse duration similarly restricted. We also show that the exact CDF for an isolated three level system is a useful approximation to the exact CDF for three and five state systems embedded in background states, and we report a study of the stability of the STIRAP + approximate CDF control protocol to variation of the CDF.