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arXiv:quant-ph/9903096 · Physical Review A

Adiabatic population transfer via multiple intermediate states

Nikolay V. Vitanov, Stig Stenholm

Published Nov 1, 1999• 65 citations
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Abstract

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.

DOI
10.1103/physreva.60.3820

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Physical Review A•Oct 8, 2010•14 citations
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 \ensuremathψi{\ensuremath{\psi}}_{i}\ensuremathψi​ with a coupling proportional to the pump pulse and to the final state \ensuremathψf{\ensuremath{\psi}}_{f}\ensuremathψf​ 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 \ensuremathψi{\ensuremath{\psi}}_{i}\ensuremathψi​ to state \ensuremathψf{\ensuremath{\psi}}_{f}\ensuremathψf​ 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.
This paper analyzes the extension of the three-state process of stimulated Raman adiabatic passage to chainwise-connected multistate systems. A necessary condition for such a process is the existence of an adiabatic-transfer state, which connects adiabatically the initial state of the chain to its final state. Various counterintuitive pulse sequences are examined, in all of which the pulse that drives the last transition of the chain precedes the pulse driving the first transition, while the pulses driving the intermediate transitions may have different timings. The paper demonstrates some important qualitative differences and similarities between the systems with odd and even number of states. In the on-resonance case, an adiabatic-transfer state always exists for an odd number of states while it never exists for an even number of states. In the off-resonance case, however, the two types of systems behave in a very similar manner and the condition for existence of an adiabatic-transfer state is essentially the same. This condition, which imposes certain limitations on the laser parameters, is derived in a simple and compact form. It is also shown analytically that, besides by large detunings, the populations of the intermediate states (which are generally nonzero during the transfer) can be damped by large intermediate couplings. It is concluded that for an odd number of states, the optimal case is the on-resonance one, with equal and large intermediate couplings. For an even number of states, the optimal (off-resonance) case is when the intermediate couplings and the detunings have similar values. Various numerical examples of success and failure of multistate population transfer confirm the analytic conclusions.
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.
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).
We examine the conditions needed to accomplish stimulated Raman adiabatic passage (STIRAP) when the three levels (ground ggg, excited eee, and final fff) are degenerate, with arbitrary couplings contributing to the pump-pulse interaction $(g\text{\ensuremath{-}}e)$ and to the Stokes-pulse interaction $(e\text{\ensuremath{-}}f)$. We show that in general a sufficient condition for complete population removal from the ggg set of degenerate states for arbitrary, pure or mixed, initial state is that the degeneracies should not decrease along the sequence ggg, eee, and fff. Using a sequence of unitary transformations we demonstrate that when this condition holds it is possible to achieve the degenerate counterpart of conventional STIRAP, whereby adiabatic passage produces complete population transfer. Indeed, the system is equivalent to a set of independent three-state systems, in each of which a STIRAP procedure can be implemented. We also examine the cases when this degeneracy constraint does not hold and show what can be accomplished in those cases. For example, for angular momentum states when the degeneracy of the ggg and fff levels is less than that of the eee level we show how a special choice for the pulse polarizations and phases can produce complete removal of population from the ggg set. Our scheme can be a powerful tool for coherent control in degenerate systems, because of its robustness, when selective addressing of the states is not required or impossible. We illustrate the analysis with several analytically solvable examples, in which the degeneracies originate from angular momentum orientation, as expressed by magnetic sublevels.
This paper begins with the premise that stimulated Raman adiabatic passage (STIRAP) from one state to another by coupling to an intermediate state necessarily requires some population of that intermediate state. The usual view of the process depends on having no population in the intermediate state, but a simple argument from the time-dependent Schr\"odinger equation shows that such adiabatic transfer cannot occur. From another perspective, removing the time dependence of the Hamiltonian by making the rotating frame transformation cancels the optical frequency oscillations but does not account for the envelopes of the pulses commonly used for STIRAP. By assuming a small but finite intermediate state amplitude ${\ensuremath{\varepsilon}}_{1}$ in the wave function, the propagator for the evolution between initial and final states is derived. Applying the propagator to the initial state produces a single compact formula for the final intermediate-state population that is readily evaluated for an arbitrary choice of laser pulse parameters. Several plots for various experimental conditions show that under the conditions normally used for STIRAP experiments the effects are typically a few percent or less, but in the end this effect puts an ultimate limit on total efficiency.
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.