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Journal of Modern Optics

Optimal control approach for a degenerate STIRAP

Z. Kis, Stig Stenholm

Published Jan 1, 2002• 33 citations
Publisher

Abstract

We have looked at the laser-induced excitation of a level manifold coupled through a single excited level. This generalizes the ordinary Λ configuration to many levels. It is the original situation where stimulated Raman adiabatic passage (STIRAP) has been considered. We then apply optimal control theory to determine the pulse sequence which effects a predetermined superposition state with a minimal involvement of the excited state. The program is free to find its optimal pulse sequence, and we find that a generalization of the ordinary counterintuitive situation emerges. In addition to verifying the efficiency of the counterintuitive approach, the work justifies the separation into dark and bright subspaces, where time evolution in the former has to include the dynamical effects of adiabatic corrections.

DOI
10.1080/09500340110074736

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We have looked at the laser-induced excitation of a level manifold coupled through a single excited level. This generalizes the ordinary Λ configuration to many levels. It is the original situation where stimulated Raman adiabatic passage (STIRAP) has been considered. We then apply optimal control theory to determine the pulse sequence which effects a predetermined superposition state with a minimal involvement of the excited state. The program is free to find its optimal pulse sequence, and we find that a generalization of the ordinary counterintuitive situation emerges. In addition to verifying the efficiency of the counterintuitive approach, the work justifies the separation into dark and bright subspaces, where time evolution in the former has to include the dynamical effects of adiabatic corrections.
In this paper we consider a generalization of the three-level stimulated Raman adiabatic-passage process (STIRAP). In our scheme, there are $N\ensuremath{-}1$ ground levels and one excited level. The ground levels are coupled with laser pulses through the excited level at multiphoton resonance. The ultimate aim of this scheme is to create coherent-superposition states on the ground levels. In a previous paper [J. Mod. Opt. (to be published), special issue on quantum interference] we have considered this problem from the optimal-control point of view. Here we apply a different approach: We reconsider the adiabatic approximation, which is commonly utilized to describing the STIRAP process. It is shown that in our case, in the adiabatic limit, the dark and bright subspaces of the Hamiltonian are decoupled; however, the nonadiabatic corrections influence significantly the dynamics in the dark subspace. An analytic solution is presented for the case of a five-level system. Moreover, we consider some special examples for the pulse sequences that effect prescribed final superposition states. The robustness of the scheme is studied, and the extension of our scheme is also considered.
Quantum control of lossy systems is known to be achieved by adiabatic passage via an approximate dark state relatively immune to loss, such as the emblematic example of stimulated Raman adiabatic passage (STIRAP) featuring a lossy excited state. By systematic optimal control study, via the Pontryagin maximum principle, we design alternative more efficient routes that, for a given admissible loss, feature an optimal transfer with respect to the cost defined as (i) the pulse energy (energy minimization) or (ii) the pulse duration (time minimization). The optimal controls feature remarkably simple sequences in the respective cases: (i) operating far from a dark state, of π-pulse type in the limit of low admissible loss, or (ii) close to the dark state with a counterintuitive pulse configuration sandwiched by sharp intuitive sequences, referred to as the intuitive/counterintuitive/intuitive (ICI) sequence. In the case of time optimization, the resulting stimulated Raman exact passage (STIREP) outperforms STIRAP in term of speed, accuracy, and robustness for low admissible loss.
We consider the problem of the implementation of stimulated Raman adiabatic passage (STIRAP) processes in degenerate systems, with a view to be able to steer the system wave function from an arbitrary initial superposition to an arbitrary target superposition. We examine the case of an NNN-level atomic system consisting of $N\ensuremath{-}1$ ground states coupled to a common excited state by laser pulses. We analyze the general case of initial and final superpositions belonging to the same manifold of states, and we cover also the case in which they are nonorthogonal. We demonstrate that for a given initial and target superposition, it is always possible to choose the laser pulses so that in a transformed basis the system is reduced to an effective three-level $\ensuremath{\Lambda}$ system, and standard STIRAP processes can be implemented. Our treatment leads to a simple strategy, with minimal computational complexity, which allows us to determine the laser-pulse shape required for the wanted adiabatic steering.
We demonstrate that Optimal Control Theory (OCT) with a state-dependent constraint which depends on the state of the system at each instant can reproduce the famous counterintuitive mechanism of Stimulated Raman adiabatic passage (STIRAP). We examine this behavior in a Λ-type three-level system and we show that could be applied for sequentially coupled many-level systems. We study the robustness of the two methods with respect to pulse fluctuations and the decays. We show that new OCT formulation appears to be more robust than STIRAP when a perturbation is introduced in the pulses. Such method is of great use for systems involving coherence loss such as molecular systems with dissociation or ionization limits. It also may find potential applications in the control of chemical reactions, quantum optics, and quantum information processing.
STIRAP (stimulated Raman adiabatic passage) has proven to be an efficient and robust technique for transferring population in a three-level system without populating the intermediate state. Here we show that the counterintuitive pulse sequence in STIRAP, in which the Stokes pulse precedes the pump, emerges automatically from a variant of optimal control theory we have previously called ``local'' optimization. Since local optimization is a well-defined, automated computational procedure, this opens the door to automated computation of generalized STIRAP schemes in arbitrarily complicated NNN-level coupling situations. If the coupling is sequential, a simple qualitative extension of STIRAP emerges: the Stokes pulse precedes the pump as in the three-level system. But, in addition, spanning both the Stokes and pump pulses are pulses corresponding to the transitions between the $N\ensuremath{-}2$ intermediate states with intensities about an order of magnitude greater than those of the Stokes and pump pulses. This scheme is amazingly robust, leading to almost 100% population transfer with significantly less population transfer to the $N\ensuremath{-}2$ intermediate states than in previously proposed extensions of STIRAP.
In this paper we extend the existing theory of stimulated Raman adiabatic passage (StiRAP) in three-level systems to examine the nature of selective photoexcitation using counterintuitively ordered pulses. Our goal is to develop a version of StiRAP that permits control of product selectivity in a chemical reaction. We analyze the case of selective excitation to one state in a pair of degenerate target states in a four-level system, and find that one cannot control the ratio of the populations in the two states in a resonant two-photon process. However, the extension of the system to include a fifth state and a third laser field makes selective excitation to one of the degenerate states possible. It is found that, subject to reasonable restrictions, one may accomplish complete population transfer to a single target state of a degenerate pair of states in a five-level system.