The use of DensMat to model stimulated Raman adiabatic passage processes in atoms

Ž .Spectrochimica Acta Part B 56 2001 1787�1795

The use of DensMat to model stimulated Ramanadiabatic passage processes in atoms �

Denis Boudreaua,�, Daniel Martela, Peter Ljungbergb,1, Ove Axner b

a ( )Department of Chemistry, La�al Uni�ersity, Quebec City PQ , Canada G1K 7P4´bDepartment of Experimental Physics, Umea Uni�ersity, SE-901 87 Umea, Sweden˚ ˚

Received 23 February 2001; accepted 10 May 2001

Abstract

Ž .Stimulated Raman Adiabatic Passage STIRAP is a technique that is capable of transferring an almost entireatomic or molecular population from one quantum level to another by a coherent two-step excitation process. Themost remarkable property of this technique is that it is applicable to situations in which the intermediate atomic ormolecular state is exposed to a large loss rate. However, the STIRAP process has not yet been used for laserspectrochemical analysis. As a part of the ongoing development of new laser-based spectrometric techniques, thisarticle demonstrates how a previously published simulation program, DensMat, can be used to model STIRAPprocesses in atoms under various collisional and laser bandwidth conditions. � 2001 Elsevier Science B.V. All rightsreserved.

Ž .Keywords: Stimulated Raman Adiabatic Passage STIRAP ; DensMat

� This article is published in a special honor issue dedicated to Walter Slavin, in recognition of his outstanding contributions toanalytical atomic spectroscopy, in appreciation of all the time and energy spent in editing Spectrochimica Acta Part B.

� Corresponding author. Tel.: �1-418-656-3287; fax: �1-418-656-7916.Ž . Ž .E-mail addresses: [email protected] D. Boudreau , [email protected] O. Axner .

1Present address: SAAB Dynamics, Goteborg, Sweden.¨

0584-8547�01�$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 5 8 4 - 8 5 4 7 0 1 0 0 2 3 9 - 7

( )D. Boudreau et al. � Spectrochimica Acta Part B: Atomic Spectroscopy 56 2001 1787�17951788

1. Introduction

Several laser-based spectroscopic techniqueshave been investigated for their applicability tochemical analysis throughout the years. Those

Žbased on fluorescence or ionization e.g. laserinduced fluorescence, LIF, also referred to aslaser excited atomic fluorescence spectrometry,LEAFS, or resonance ionization spectrometry,

.RIS have been found to be especially versatiledue to their high sensitivity and selectivity. Acommon denominator of these techniques is thatthey aim to excite as large a fraction as possibleof the laser illuminated atoms to a specific excited

Žlevel e.g. so as to maximize the fluorescence or.ionization from that particular level .

It is well known that the phenomenon of opti-cal saturation limits the fraction of excited atomsto the upper level of a two-level atom under

Ž . Žsteady state conditions to g � g �g where g2 1 2 1and g are the degeneracy factors for the lower2

.and upper laser-connected levels, respectively� � 21�3 . Similarly, the maximum fraction of atomstransferred to the uppermost laser-connectedlevel of a three level system exposed to a two-step

Ž .excitation is g � g �g �g . This implies that,3 1 2 3at any given time and under steady state condi-tions, about half of all species in the lower laser-connected state, at most, can contribute to the

Ždetected signal 55 and 20% for ns�np�nd and.ns�np�ns atomic transitions, respectively . Obvi-

ously, although the actual signal strength dependson the fraction of atoms transferred to the upper-most laser-connected level, it can also be affectedby other processes; for example, the total fractionof atoms contributing to the signal will be differ-

Žent from the estimates above both larger and. � �smaller if optical pumping takes place 2,3 .

In addition to the limitations imposed by opti-cal saturation, if the intermediate level in a two-

2 While it is possible to temporarily reach population frac-tions above those given above by the use of ‘�-pulses’, thismode of excitation has little practical relevance for analyticalapplications since it is very sensitive to the exact energy and

Žfluency values of the laser pulses i.e. any fluctuation of eitherlaser pulse energy or duration will cause a large fluctuation of

.the detected signal .

step excitation scheme experiences a large lossŽrate to other low lying states because of radiative

.or collisional relaxation , the atomic populationaccessible to laser excitation can be drained away

� �through that intermediate level 4 . The fractionof atoms transferred to the uppermost laser-con-nected level by a step-wise laser excitation tech-nique will therefore often be, in practice, signifi-

Ž .cantly below g � g �g �g . This is particu-3 1 2 3larly the case when atoms or molecules to bedetected have a large number of lower lying states,

Žor even a few low lying metastable states e.g. Cu,.Ag and Au , that can act as trapping states during

the interaction.A novel laser-based excitation technique offer-

ing efficient and precise control of populationtransfer has been developed during recent years� �5,6 . This technique, referred to as Stimulated

Ž .Raman Adiabatic Passage STIRAP , has beenshown to be capable of achieving an almost com-

Ž .plete transfer of population close to 100% fromone quantum state to another, using two coherent

Žlight pulses in the case of two-step excitation of a.three level system . Most importantly, this effi-

cient transfer can be obtained even if the inter-mediate state is strongly coupled to other non-laser-connected states. The technique is robust inthe sense that small fluctuations in pulse inten-sity, duration, or temporal relation do not signifi-cantly affect the transfer efficiency.

The STIRAP process relies on the establish-ment of coherence between all states involved,followed by an adiabatic population transfer fromthe initial to the final state. This phenomenon ismost easily explained using the ‘dressed statepicture’ in which the combined atom light systemis described as linear combinations of the atomic

� �states and of the coherent radiation fields 7,8 . Adressed state is an eigenfunction to the combinedcoherent atom light system whereupon the systemwill reside in the same dressed state throughoutthe entire interaction.

If the temporal delay between the excitationpulses is set properly, the atoms will be forced tostart out in a particular dressed state representedby a linear combination of only the initial andfinal laser-connected states, which is thus free ofany losses through the ‘leaky’ intermediate atomic

( )D. Boudreau et al. � Spectrochimica Acta Part B: Atomic Spectroscopy 56 2001 1787�1795 1789

state. Given a smooth change in the irradiance ofthe laser pulses, the atoms will be directly trans-ferred from the lowermost to the uppermostlaser-connected atomic states. Hence, atoms canbe transferred from one particular quantum stateto another by a two-step excitation process with-out being influenced by the large loss rate fromthe intermediate state.

The first STIRAP experiments were done withcontinuous lasers about 10 years ago by Bergmann

� �et al. 5 , followed a few years later by the first� �demonstrations using pulsed lasers 9,10 . A num-

ber of experiments on atoms and molecules, aswell as simulations and modeling primarily basedupon the dressed state picture, have been carriedout since then for a variety of purposes, e.g. for:

� �atomic mirrors and beam splitters 11 ; synthesisof arbitrary quantum states in a micro-resonator� � � �12 ; atom interferometry 13 ; studies of propaga-tion properties of laser pulses in optically thick

� �media 14 ; and high resolution spectroscopy of� �molecules 15 .

STIRAP has so far been performed only usingpure substances introduced into collision-less me-

Ž .dia supersonic jets, notably . It is obvious, how-ever, that this highly efficient excitation techniquecould bring significant benefits to the field oflaser spectrochemical analysis, if the neededcoherence between quantum states can be es-tablished also in collisional atom reservoirs, suchas glow discharges, flames or plasmas. In order toinvestigate the applicability of STIRAP to laserspectrochemical analysis, we demonstrate in thispaper how a previously developed computer pro-

� �gram, DensMat 16 , can be used to model STI-RAP processes under various experimental condi-tions, with particular attention to coherence-de-stroying collisions.

2. Theory

2.1. Dressed states

Ž .Denoting the three atomic states bare states� : � : � :1 , 2 and 3 , respectively, it can be shown thatthe three dressed states of the combined atomic-

� �: � 0: � �:light system, a , a and a , can be writtenas:

� �: Ž . � : � :a �sin� t sin� 1 �cos� 2

Ž . � : Ž .�cos� t sin� 3 1

� 0: Ž . � : Ž . � : Ž .a �cos� t 1 �sin� t 3 2

� �: Ž . � : � :a �sin� t cos� 1 �sin� 2

Ž . � : Ž .�cos� t cos� 3 3

Ž .where � t is referred to as the mixing angle,given by:

Ž .� tPŽ . Ž .tan� t � 4Ž .� tS

Ž . Ž .where, in turn, � t and � t are the RabiP Sfrequencies of the first and second excitationsteps, respectively; � is a known function of theRabi frequencies and detunings that is given in

� � Žthe literature 7 and whose exact value is of.minor importance here . The dressed states are

the eigenstates of the combined atomic and laserlight system. Hence, for a fully coherent atomicsystem, once having entered a certain dressedstate, the atoms will reside in that particulardressed state throughout the entire interaction.

2.2. The STIRAP process

As is illustrated in Fig. 1, by applying twoŽ .smoothly varying e.g. Gaussian laser pulses in a

counter intuitive sequence, i.e. with the secondŽ .step excitation e.g. the Stokes pulse preceding

Ž .the first step excitation the pump pulse , theŽ .mixing angle � t will smoothly shift from 0 to

��2 during the laser interaction period. Thisimplies that the atoms, which are originally in the

� :lowest bare state 1 , start out in the dressed state� 0: � Ž . Ž .�a see Eqs. 1 � 4 . Since the dressed statesare the eigenstates of the combined atomic andlaser light system, the atoms will reside in the� 0:a state throughout the entire interaction. Fur-thermore, since the mixing angle slowly variesfrom 0 to ��2, the atoms will be squeezed into


Fig. 1. Illustration of the STIRAP process in a three levelatom. Panel a: temporal profile and relation between the

Ž . Ž .Stoke second step excitation and pump first step excitationlasers; Panel b: temporal evolution of the mixing angle; Panelc: temporal evolution of the populations of the three atomiclevels.

� :the uppermost laser-connected bare state 3 bythis adiabatic process, without ever physically be-

Ž � 0:ing in the intermediate state since the a state� :.does not have any contribution from state 2 .

This explains why the STIRAP process can pro-vide efficient transfer of atomic populations evenin the presence of high loss rates from the inter-mediate level.

3. Simulations

The DensMat program was developed as amodeling tool for the laser excitation of atomsand molecules in collisional media, and was builtupon a previously published fully time-dependenttreatment based on the density matrix formalism

� �17 . Since the STIRAP process can also be de-� �scribed in terms of this formalism 6 , it is there-

fore possible to use DensMat to model the STI-RAP process. The only fundamental differencebetween the theoretical basis of STIRAP and theformalism underlying the program is the applica-tion of a slow adiabatic turning on and off of the

Žtwo laser light fields which can be achieved inDensMat by importing an externally generatedGaussian intensity profile, using the command

.Import External Profile . Furthermore, since thisfully time-dependent formalism underlying theprogram includes the effects of elastic and inelas-

Ž .tic quenching collisions, as well as of laser band-width, it can be used to evaluate the feasibility ofestablishing a STIRAP process under pressureconditions typical of atom reservoirs used forchemical analysis.

3.1. The atomic system

In order to model a simple system that allowsus to compare the simulations with previously

� �calculated transfer efficiencies 7 , and at the sametime a system in which the STIRAP process canplay a significant role, a three level system in a

Žladder configuration, with unity degeneracy g�. 3 Ž .1 of each level, will be considered see Fig. 2 .

Ž .The intermediate level i.e. level 2 is stronglycoupled to an unspecified manifold of trapping

Ž . Žlevel s i.e. leading to an ensemble of low lyingnon-laser-connected levels or to the ionization

.continuum ; this manifold is set in DensMat aslevel 4, with a degeneracy g �100. All other4values used for the calculations are given in Table1. Note that the loss rate from the intermediatelevel into the trapping manifold is more than twoorders of magnitude larger than the spontaneous

Žemission rates out of that level 2.74 and 0.01.GHz, respectively , so as to model the case of an

intermediate level being very effectively drained

3The actual choice of degeneracy factors is of minor impor-tance for this work since simulations have shown that it hasvirtually no impact upon whether STIRAP processes will takeplace or not.


Table 1Parameter values used in the examples

Parameter Fig. 3 Fig. 4a Fig. 4b Fig. 5a Fig. 5b

Resonance wavelength, 500 nmboth excitation steps

7 �1Spontaneous emission 1�10 srate A and A21 32

Rabi frequency �, 2.74 2.74 2.74 8.7 8.7both excitation steps

9 �1Ž .�10 rad �sIrradiance, both lasers 3 3 3 30 30

2Ž .kW�cm10 7 7Spectral bandwidth, 1�10 1 1 4�10 4�10

�1Ž .FWHM, both lasers sPulse duration, FWHM, 40 4 40 4 40

Ž .both lasers nsPulse temporal profile R G R G RŽR: rectangular,

.G: GaussianStokes-to-pump laser 0 3 0 6.5 0

Ž .delay ns7 7Inelastic collision rate k 0 0 0 1�10 1�1021

�1Ž . Ž .2�1 s7 7Inelastic collision rate k 0 0 0 1�10 1�1023

�1Ž . Ž .2�3 sInelastic collision rate 2.74�109

Ž .trapping rate k24�1Ž . Ž .2�4 s

�1 7 7Ž .Elastic collision rate s 0 0 0 1�10 1�10

by non-laser-connected levels during the length ofthe interaction.

3.2. The laser fields

Ž .Three different situations cases are modeledusing laser fields that have either a rectangular

ŽFig. 2. The model atomic system see text for explanation of.symbols used .

Ži.e. constant irradiance during the interaction.time or a Gaussian temporal profile.

In case 1, the laser fields are assumed to have aŽlarge bandwidth i.e. larger than the frequencies

.of the Rabi flopping oscillations , so as to illus-trate a non-coherent system for which the densitymatrix equations revert to the conventional popu-lation equations given by the Rate Equation for-malism.

Ž .The second set of simulations case 2 demon-strates the concepts of dressed states and of theadiabatic passage process in an idealized system,in which the bandwidth of the lasers and allelastic�inelastic collisions have been neglected.The parameters for these simulations have beenchosen to allow the comparison between thesesimulations and analytical descriptions from pre-vious treatments of the concept of coherence and

� �dresses states in three level systems 7 .Ž .The third set of simulations case 3 shows the


importance of coherence and the possibility ofachieving STIRAP in the presence of collisionswhen using laser pulses having Gaussian temporalprofiles and Fourier limited bandwidths.

The Gaussian laser pulses have a Full Width atŽ .Half Maximum FWHM duration of 4 ns while

Žthe rectangular pulses have a longer duration soas to allow all populations to reach their steady-

.state values . In addition, the temporal relationbetween the two laser pulses for the STIRAP

Žcases is such that the Stokes laser between levels. Ž2 and 3 precedes the pump laser between levels.1 and 2 by a few nanoseconds, as is schematically

depicted in Fig. 1.

4. Results and discussion

4.1. Case 1 � population transfer by broadbandexcitation

Fig. 3 shows the time evolution of the variousŽ .populations n , n , n and n in the case of1 2 3 trapŽ .broadband 10 GHz excitation by two simultane-

ous rectangular laser pulses. The maximum trans-fer efficiency that can be obtained under broad-band excitation, in otherwise ideal conditions, is

Žexpected to be 33% obtained with optically satu-rating pulses, whose duration must be signifi-cantly shorter than the reciprocal of the radiative

.relaxation rate from the intermediate level . Un-der the influence of the leaky intermediate level,the third level population will reach a maximum

Žof approximately 10% after a couple of nanosec-.onds , although all population will eventually be

Žlost to the trapping manifold as long as the two.lasers are applied . In this case, the large laser

bandwidth effectively destroys any coherence inthe system, and the population developmentagrees well with that predicted by simple rateequations.

4.2. Case 2 � population transfer in a fully coherentsystem

� �As has been shown by Bergmann et al. 6 , theuse of laser pulses with a smoothly changingirradiance can lead to the establishment of a

Fig. 3. Temporal evolution of level populations in the case ofbroadband excitation by rectangular laser pulses.

STIRAP process that can in turn increase thepopulation transfer efficiency. Fig. 4a shows thetemporal evolution of level populations of a fullycoherent and collision-less three level system,when the atoms are exposed to two Gaussian-shaped laser pulses of negligible bandwidth, theStokes pulse preceding the pump pulse in thisparticular case by 3 ns. The population of thethird level reaches in this case a maximum valueof 86%, with only about 16% of the atoms beinglost to the trapping level at the end of the irradia-

Ž .tion period 20 ns . Such a high population trans-fer to the final laser-connected state is both farabove that predicted by the ordinary rate equa-tions, and a characteristic feature of the STIRAPprocess. The excitation efficiency to the final levelis limited by the spontaneous radiative relaxationfrom the third to the second level and by thelarge loss rate from the second level to the trap-ping manifold.

Fig. 4b shows the population development inthe same collision-less system, this time exposedto two simultaneous rectangular pulses, again ofnegligible bandwidth. One finds in this case thesomewhat unexpected result that only about 50%of the population will end up in the trappinglevel, while the remaining population will be splitevenly between levels 1 and 3. The reason for thisis that the simultaneous onset of two rectangularpulses will cause the mixing angle in the dressed

Žstate picture to take the value of ��4 assuming


Fig. 4. Temporal evolution of level populations in a fullyŽ .coherent and collision-less three level system exposed to: a

Ž .Gaussian-shaped; and b rectangular laser pulses, having inboth cases a negligible spectral bandwidth.

that the laser pulses have the same Rabi frequen-.cies . This implies that 50% of the initial popula-

� 0:tion of the initial state will start out in the aŽ 2Ž . .state since cos ��4 �1�2 , while the other

50% will be split into the two other dressed� �: � �: � Ž . Ž .�states, a , and a see Eqs. 1 � 4 . Further-

more, since the dressed states are the eigenstatesof the combined atomic and light field system inwhich the system will remain as long as the laser

� �: � �:interaction is ‘on’, and since the a and aŽstates include the intermediate level from which

.there is a large loss rate , the population of thesetwo dressed states will rapidly decay. The popula-

� 0:tion of the a state, on the other hand, does notinclude the intermediate state, and its population

will therefore remain unaffected by the large lossrate from level 2. This implies that, in a fullycoherent three level system submitted to a largeloss rate from the intermediate level, no morethan 50% of the population will be lost, whereas25% of the population will remain in the upper-most level, again assuming identical Rabi fre-quencies for the two transitions. This behavioragrees well with a previously published analytical

� �description 7 .

4.3. Case 3 � population transfer in the presence ofcollisions

The final case examines the temporal evolutionof the same system, now exposed both to coher-ence-destroying collisions and to more realisticlaser bandwidths. Inelastic collision rates betweenlevels 3-2, 2-1 and 3-1, as well as an elasticcollision rate, have been added and set to 107 s�1,which are typical of the collisional rates encoun-tered in a low pressure atomizer such as a glowdischarge. The laser bandwidth has been set asthe Fourier-limited value of the Gaussian-shapedpulse, i.e. 40 MHz. Finally, the irradiance of thelasers has been increased by a factor of 10 toguarantee optical saturation of the transitions,which results in a Rabi frequency of 8.7 GHz.

The temporal evolution of the atomic popula-tion, when exposed to two suitably delayed Gauss-ian pulses, is shown in Fig. 5a. It can be seen that

Ža significant part of the atomic population about.70% is still transferred into level 3, despite being

subjected to significant collisional activity and tothe influence of the leaky intermediate level, aswell as to the dephasing effect of the finite laserbandwidth. However, when the atomic system isexposed to two simultaneous rectangular excita-

Ž .tion pulses Fig. 5b , the coherence destroyingprocesses knock the atoms out of their dressedstates and drive them to be lost to a large degreeduring the excitation sequence, in stark contrastto the relative robustness of the STIRAP processas demonstrated in Fig. 5a. Under the combinedinfluence of the larger laser bandwidth and ofincreased collisional activity, the system no longerexperiences the necessary coherence needed forthe persistence of these states, and the excitation


Fig. 5. Temporal evolution of level populations in a threeŽ .level system exposed to collisions and to: a Gaussian-shaped;

Ž .and b rectangular laser pulses, having in both cases a 40-MHzspectral bandwidth.

efficiency to the final level is thus much less thanthat obtained previously.

Calculation of the excited level population as afunction of elastic and inelastic collision ratesreveals that the coherence established betweenquantum states by Gaussian-shaped pulses is pre-served until the frequency of dephasing collisionsexceeds 108 s�1. This suggests that, whereas thiscoherence can probably not be maintained inanalytical flames and plasmas operated at atmo-

Ž 9spheric pressure with collision rates in the 10�1 .s range , it should be possible to benefit from

the increased detection sensitivity provided by theSTIRAP process in moderately collisional atomreservoirs such as glow discharges and reducedpressure microwave-induced plasmas.

5. Conclusions

It is known that the STIRAP process can beused to transfer a significant portion of an atomicor molecular population from one quantum stateto another by the use of a two-step excitationmechanism in which the intensity envelope of thelasers is gradually varied in time, using a Stokes-before-pump pulse sequence. This efficient exci-tation strategy is even applicable when the inter-mediate laser-connected state is strongly coupledto other non-laser-connected states that can actas trapping states during the interaction. How-ever, STIRAP has only been demonstrated invery low pressure conditions that are not easilyamenable to atom reservoirs such as flames andplasmas; the aim of the present work was thus toestablish whether the density matrix formalismpreviously developed to model the laser excitationof atoms and molecules in collisional media couldbe used to predict the applicability of STIRAPunder pressure conditions more amenable tospectrochemical analysis. The results obtainedshow that this formalism, as implemented in theDensMat program, is a useful modeling tool forthe development of STIRAP processes as appliedto the laser excitation of target species bynanosecond laser pulses in intermediate pressuresources such as glow discharges. This implies thatthe STIRAP technique could provide substantialincreases in sensitivity and selectivity in the con-text of laser-based spectrochemical analysis in thefuture.

Acknowledgements

O.A. acknowledges support from the SwedishŽ .National Science Foundation NFR under con-

tract F 5102-915. D.B. is grateful to the NaturalScience and Engineering Research Council of

Ž .Canada NSERC , to the ‘Fonds pour la Forma-tion des Chercheurs et l’Aide a la Recherche’`Ž .FCAR of the Province of Quebec, and to the´


Association of Universities and Colleges ofŽ .Canada AUCC for financial support.

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The use of DensMat to model stimulated Raman adiabatic passage processes in atoms

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