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arXiv:1201.1029 · Physical Review A

Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses

Andon A. Rangelov, Nikolay V. Vitanov

Published Apr 9, 2012• 41 citations
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Abstract

A technique for complete population transfer between the two end states $|1\ensuremath{\rangle}$ and $|3\ensuremath{\rangle}$ of a three-state quantum system with a train of $N$ pairs of resonant and coincident pump and Stokes pulses is introduced. A simple analytic formula is derived for the ratios of the pulse amplitudes in each pair for which the maximum transient population ${P}_{2}(t)$ of the middle state $|2\ensuremath{\rangle}$ is minimized, ${P}_{2}^{\mathrm{max}}={\mathrm{sin}}^{2}(\ensuremath{\pi}/4N)$. It is remarkable that, even though the pulses are on exact resonance, ${P}_{2}(t)$ is damped to negligibly small values even for a small number of pulse pairs. The population dynamics resembles generalized $\ensuremath{\pi}$ pulses for small $N$ and stimulated Raman adiabatic passage for large $N$ and therefore this technique can be viewed as a bridge between these well-known techniques.

DOI
10.1103/physreva.85.043407

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A technique for complete population transfer between the two end states ∣1\ensuremath⟩|1\ensuremath{\rangle}∣1\ensuremath⟩ and ∣3\ensuremath⟩|3\ensuremath{\rangle}∣3\ensuremath⟩ of a three-state quantum system with a train of NNN pairs of resonant and coincident pump and Stokes pulses is introduced. A simple analytic formula is derived for the ratios of the pulse amplitudes in each pair for which the maximum transient population P2(t){P}_{2}(t)P2​(t) of the middle state ∣2\ensuremath⟩|2\ensuremath{\rangle}∣2\ensuremath⟩ is minimized, P2max=sin2(\ensuremathπ/4N){P}_{2}^{\mathrm{max}}={\mathrm{sin}}^{2}(\ensuremath{\pi}/4N)P2max​=sin2(\ensuremathπ/4N). It is remarkable that, even though the pulses are on exact resonance, P2(t){P}_{2}(t)P2​(t) is damped to negligibly small values even for a small number of pulse pairs. The population dynamics resembles generalized \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses for small NNN and stimulated Raman adiabatic passage for large NNN and therefore this technique can be viewed as a bridge between these well-known techniques.
Abstract We consider coherent population transfer in multilevel quantum systems driven by time-delayed pulse trains. We show how to choose the parameters of the trains so that the population dynamics tracks the behavior of a Λ-type three-level system under stimulated Raman adiabatic passage (STIRAP). Efficient population transfer can be achieved regardless of the order and with or without overlap of the pump and the Stokes sub-pulses of the trains. Generalized STIRAP schemes of population transfer in N -level system with sequential couplings are also examined. The mechanism of the population transfer and the robustness of the proposed schemes are discussed.
We investigate ultrafast coherent population transfer in a Λ-type three-level system driven by a train of weak phase-locked pump-Stokes femtosecond pulse pairs, where the pump and Stokes pulses in each pair are applied in counterintuitive order, similar to that in the stimulated Raman adiabatic passage technique. It is shown that due to temporal constructive quantum interference and the subsequent coherent accumulation, complete population transfer can be achieved by a few pulse pairs with suitable interpair and intrapair time delays even if a single pair of pulses in the train is so weak that nearly no population transfer can take place. The method is efficient and robust to moderate variations in the interpair and intrapair time delays, number of pulse pairs, and laser intensity, which is particularly favorable to coherent population transfer in quantum systems with weak oscillator strengths.
This paper discusses a generalization of stimulated Raman adiabatic passage (STIRAP) in which the single intermediate state is replaced by N intermediate states. Each of these states is connected to the initial state ${\ensuremath{\psi}}_{i}$ with a coupling proportional to the pump pulse and to the final state ${\ensuremath{\psi}}_{f}$ with a coupling proportional to the Stokes pulse, thus forming a parallel multi-\ensuremath{\Lambda} system. It is shown that the dark (trapped) state exists only when the ratio between each pump coupling and the respective Stokes coupling is the same for all intermediate states. We derive the conditions for existence of a more general adiabatic-transfer state that includes transient contributions from the intermediate states but still transfers the population from state ${\ensuremath{\psi}}_{i}$ to state ${\ensuremath{\psi}}_{f}$ in the adiabatic limit. We present various numerical examples for success and failure of multi-\ensuremath{\Lambda} STIRAP, which illustrate the analytic predictions. Our results suggest that in the general case of arbitrary couplings, it is most appropriate to tune the pump and Stokes lasers either just below or just above all intermediate states.
The technique of stimulated Raman adiabatic passage (STIRAP) is a highly efficient tool for coherent population transfer in a chainwise-connected three-state quantum system. It is widely accepted that the two-photon resonance between the two end states of the chain is a crucial condition for STIRAP. This assumption is indeed correct when the two driving fields, pump and Stokes, produce nearly equal couplings for their respective transitions. We show analytically, supported by numeric examples, that when the pump and Stokes couplings differ significantly, the two-photon population transfer profile is distorted and its center is shifted away from the two-photon resonance; hence, the optimal operation of STIRAP demands a two-photon detuning. We derive simple analytic estimates for the center and the width of the two-photon profile. The results are of potential importance in situations when the pump and Stokes fields are of different physical natures (e.g., laser and cavity fields) or when two-photon resonance is unwanted (e.g., in the presence of residual light).
The technique of stimulated Raman adiabatic passage (STIRAP) is an efficient and robust technique for population transfer in three-state chain-wise connected quantum systems, 1-2-3. Here we analyze the transition time in STIRAP, that is the time it takes for the population to be transferred from state 1 to state 3. We derive a very simple formula for this transition time, which shows that it is inversely proportional to the nonadiabatic coupling $\mathrm{\ensuremath{\vartheta}\ifmmode \dot{}\else \.{}\fi{}}({t}_{c})$, estimated at the time tc{t}_{c}tc​ when the pump and Stokes couplings have the same value. Explicit formulas for several pulse shapes of the pump and Stokes fields are derived. Contrary to the naive expectation that the transition time might be proportional to the delay between the pump and Stokes pulses, or to the width of their overlap region, we find that the transition time is inversely proportional to the delay, i.e., larger delays produce faster transitions. We also calculate the transition time in two important variations of the STIRAP technique---bright-STIRAP and STIRAP in multistate systems---where similar features are found.
We show that the technique of Stark-chirped rapid adiabatic passage (SCRAP), hitherto used for complete population transfer between two quantum states, offers a simple and robust method for complete population transfer amongst three states in atoms and molecules. In this case SCRAP uses three laser pulses: a strong far-off-resonant pulse modifies the transition frequencies by inducing dynamic Stark shifts and thereby creating time-dependent level crossings amongst the three diabatic states, while near-resonant and moderately strong pump and Stokes pulses, appropriately offset in time, drive the population between the initial and final states via adiabatic passage. The population transfer efficiency is robust to variations in the intensities of the lasers, as long as these intensities are sufficiently large to enforce adiabatic evolution. With suitable pulse timings the population in the (possibly decaying) intermediate state can be minimized, as with stimulated Raman adiabatic passage (STIRAP). This technique applies to one-photon as well as multiphoton transitions and it is also applicable to media exhibiting inhomogeneous broadening; these features represent clear advantages over STIRAP by overcoming the inevitable dynamical Stark shifts that accompany multiphoton transitions as well as unwanted detunings, e.g., induced by Doppler shifts.