Optics Communications 244 (2005) 423–430

www.elsevier.com/locate/optcom

Towards a FAST CARS anthrax detector: coherencepreparation using simultaneous femtosecond laser pulses

G. Beadie a, Z.E. Sariyanni b,c, Y.V. Rostovtsev b,c,d,*,T. Opatrny b, J. Reintjes a, M.O. Scully b,c,d

a US Naval Research Laboratory, Code 5614, 4555 Overlook Ave., Washington, DC 20375, USAb Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA

c Max-Planck-Institut fur Quantenoptik, Garching, D-85748, Germanyd Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

Received 16 August 2004; received in revised form 14 September 2004; accepted 16 September 2004

Abstract

Maximizing the molecular response to coherent anti-Stokes Raman spectroscopy (CARS) requires optimizing the

preparation of a vibrational quantum coherence. We simulate the amount of laser-induced coherence prepared in a

three-level system modeled after dipicolinic acid (DPA), a marker molecule for bacterial spores. The level spacings

and decoherence times were chosen to agree with experimental data observed from DPA. Nearly-maximal vibrational

coherences can be induced for 150 fs optical pulses, despite the very fast dephasing rates of the electronic transitions. It

is also shown that pulse propagation effects play an important role in the development of coherence throughout an

extended sample, due to nonlinear index effects of the molecule at the laser frequencies.

� 2004 Published by Elsevier B.V.

PACS: 42.50.Gy; 42.65.Dr; 82.53.KpKeywords: Quantum coherence; CARS; Spectroscopy

1. Introduction

Several laser spectroscopic techniques have

been applied for the detection of chemical and bio-

0030-4018/$ - see front matter � 2004 Published by Elsevier B.V.

doi:10.1016/j.optcom.2004.09.050

* Corresponding author. Tel.: +1 979 845 1722; fax: +1 979

845 2590.

E-mail address: rost@qo.tamu.edu (Y.V. Rostovtsev).

logical unknowns such as bacterial spores and

bioaerosols. These include methods such as fluo-

rescence, Raman spectroscopy, infrared absorp-

tion, laser induced breakdown, as well as mass

spectrometry for detection and analysis of gath-

ered samples. Recently we proposed a techniquebased on Raman spectroscopy which aims to en-

hance the Raman signal by orders of magnitude

Fig. 1. A three-level system. D is the detuning from one-photon

resonance (we consider three cases: resonance, D = 0; near

resonance, D = 0.1x02; off-resonance D = 0.471 x02), two-pho-

ton detuning is zero. Depending on the selection rules we can

have two cases: (a) pump field, XP, consists of field with one

frequency, xP; XS, consists of field with one frequency, xS, and

(b) each of XP and XS consist of fields with both, xP and xS

frequencies.

424 G. Beadie et al. / Optics Communications 244 (2005) 423–430

[1]. Using femtosecond adaptive spectroscopic

techniques for coherent anti-Stokes Raman spectr-

oscopy (FAST CARS), the key to the enhance-

ment is the preparation of an optimized quantum

coherence between vibrational states: the nonlin-ear signal due to molecular vibrations will be far

greater if the molecules vibrate in phase with one

another.

In this paper, our goal is to determine what la-

ser parameters are important for producing opti-

mal coherence in a molecule of interest to

biological agent detection. Our approach is to take

existing theory on simple systems and expand it tocover the effects observed in dipicolinic acid (DPA)

which is among the main constituents of bacterial

spores. We show that several standard approxima-

tions must be lifted in order to get experimentally

relevant information. The numerical work pre-

sented here is carried out in conjunction with

experiments on DPA and other molecules, re-

ported elsewhere [2,3]. The overall aim of the effortis to develop a robust technique for molecular

identification, rooted in the principles of quantum

coherence and selective molecular vibration.

Here we outline a numerical method of simulat-

ing the dynamics of laser-driven DPA, examining

the amount of vibrational coherence achieved with

different laser pulses. The laser pulses were chosen

to maximize the parameter q0,1, defined below,which describes the coherence between the vibra-

tional and ground states. We describe first the man-

ner in which the DPA parameters were chosen. In

particular, we take experimental data and use them

to motivate a useful worst-case scenario for coher-

ence preparation. Then we present the optical–

molecular interaction model, including equations

for pulse propagation, and how it was simulated.We then present the conditions under which an

optimized coherence was found. Finally, we investi-

gate coherence preparation throughout an extended

sample, taking into account the coupling between

pulse propagation and the molecular dynamics.

2. Model

We model DPA as a three level atom, as in

Fig. 1. We choose a notation such that the ground

state is level zero, and the other two are numbered

in order of increasing energy. To make the model

consistent with the observed behavior of DPA, we

first assign characteristics to the uppermost level

which match the recorded, aqueous DPA singlet–

singlet absorption band [4]: central wavelength

k02 = 272.63 nm, dipole matrix element l0,2 =0.371 A e�, and decoherence rate c0,2 = (3.03 fs)�1.The results of this fit are illustrated in Fig. 2(a),

which plots the model absorption curve on top of

the data extracted from [4]. This rapid decoherence

rate is required to match the broad extent of the

absorption band, which has a 26 nmFWHM. There

is clearly more structure to the band, but approxi-

mating the lineshape as a single, broad level repre-

sents a useful worst-case scenario. Let us note thatfor complex molecules in liquid or solid environ-

ments such relaxation times are not exceptional,

as shown for example in [5]. As discussed in Section

3, modeling the band with multiple levels only in-

creases the final coherence. Finally, we attribute to

the Raman level an energy splitting and coherence

decay rate to match DPA CARS measurements

[2]: k01 = 1/(1400 cm�1), l0,1 = 0 A e�, and c0,1 =(900 fs)�1. The dipole matrix element is chosen to

be zero based largely on symmetry arguments; a

state which is two-photon resonant is not expected

to have a one-photon matrix element. In complex

molecules such as DPA this may not be strictly true,

but we adopt this value as another aspect of the

1.00 1.10 1.200.90Normalized Optical Frequency

0.00

0.02

0.04

0.06O

ptic

al D

ensi

ty(a) (b)

1.00 1.10 1.200.90 1.00 1.10 1.200.90Normalized Optical Frequency

0.00

0.02

0.04

0.06O

ptic

al D

ensi

ty

0.00

0.02

0.04

0.06

0.00

0.02

0.04

0.06O

ptic

al D

ensi

ty

Normalized Optical Frequency

0.00

0.02

0.04

0.06

Opt

ical

Den

sity

1.00 1.10 1.200.90Normalized Optical Frequency

0.00

0.02

0.04

0.06

Opt

ical

Den

sity

0.00

0.02

0.04

0.06

0.00

0.02

0.04

0.06

Opt

ical

Den

sity

1.00 1.10 1.200.90 1.00 1.10 1.200.90

Fig. 2. Fits to absorption data of aqueous DPA, from which molecular level parameters are extracted. The horizontal axis is optical

frequency, normalized to c/(279.8 nm) and the vertical axis is optical density through 1 cm of a 10 lM solution of DPA. (a) Best fit to

approximating the absorption band as a single level. (b) Best fit to approximating the absorption band as the convolution of three

levels. The contributions of each level are shown as dashed curves.

G. Beadie et al. / Optics Communications 244 (2005) 423–430 425

worst-case scenario. Preliminary calculations using

a nonzero matrix element only increase the peak

coherence.

The model used to describe the optical interac-

tion with DPA is based on a density-matrix formu-

lation. It uses the semiclassical dipoleapproximation to describe the coupling of light

to the molecule. Evolution of the density matrix

is given by

d

dtq ¼ � i

�h½H ; q� þLðqÞ; ð1Þ

where H is the Hamiltonian and L contains the

relaxation and decoherence information. In the di-

pole approximation

H ¼ H 0 � EðtÞ � l; ð2Þwhere

H 0 ¼Xk

jwkiEkhwkj ð3Þ

is the Hamiltonian of the unperturbed molecule

with energy levels Ek, E(t) is the time-dependent

electric field of the incident laser pulses, and

l ¼Xn6¼m

jwniln;mhwmj ð4Þ

is the dipole moment operator. In principle, E and

l are vector quantities. We will assume that all

fields are linearly polarized in the same direction

and dipole moments along that direction, suitably

averaged over molecular orientation, are used in

the following equations. LðqÞ contains, in general,

terms which dictate population relaxation as well

as the decay rates cn,m of the off-diagonal coher-

ence terms qn,m. Because we consider coherence de-cay rates and laser pulses which are faster than

typical relaxation rates, we consider only the cn,mcontributions.

In the semiclassical approximation, the applied

electric field is treated as a real, time-dependent

scalar. In the simulations which follow we express

the field as the sum of two Gaussian pulses

EðtÞ �Xj

EjðtÞ

¼Xj

Aj exp�ðt � tjÞ2

s2j

" #cosðxjt þ /jÞ; ð5Þ

where the subscript j indicates a specific pulse de-

scribed by a real, peak field amplitude Aj, arrival

time tj, temporal 1/e2 intensity half-width sj, angu-lar frequency xj, and phase /j. The two pulses will

be referred to as pump and Stokes pulses, where

the pump pulse has a frequency xP, detuned below

the 0–2 transition by an amount D such that

xP ¼ ðE2 � E0Þ=�h� D, and the Stokes pulse a fre-

quency xS ¼ xP � ðE1 � E0Þ=�h. The simulations

treat the fields as plane waves. To connect with

laboratory experiments we relate the field ampli-tudes to pulse energies by assuming a Gaussian

beam with 1/e2 intensity radius rj

Energyj ¼pr2j2

ffiffiffiffiffiffiffips2j2

s1

2nje0A

2j ; ð6Þ

426 G. Beadie et al. / Optics Communications 244 (2005) 423–430

where nj is the index of refraction for frequency xj.

Shaped and chirped laser pulses, appropriate for

fractional stimulated rapid adiabatic passage or

chirped adiabatic passage, were also explored.

The rapid cn,m rates in our model system, however,rendered these techniques unable to improve the

coherence over simple Gaussian pulses so they

are not discussed further.

Numerical solutions are found not for the den-

sity matrix elements qn,m(t) themselves, but for

variables

~qn;mðtÞ � qn;mðtÞ expðiunmtÞ ð7Þ

assumed to vary slowly with time, where unm are

generalized frequencies (not necessarily equal to

xn � xm) and unm = �umn. Using these variables

and treating the fields and dipole moments as sca-

lars (as discussed after Eq. (4)) Eq. (1) becomes

d

dt~qn;m � �cn;m � i

En � Em

�h� unm

� �� �~qn;m � � �

� i

�hEðtÞ

Xk

ln;k~qk;m exp i unm � ukmð Þt½ ��

� ~qn;klk;m exp i unm � unkð Þt½ ��: ð8Þ

An illustration of how the new variables can be cho-sen to vary slowly with time, consider Eq. (6) for

E(t) = 0. In this case the new variables all exhibit a

smooth, non-oscillatory decay when each unm is

set equal to ðEn � EmÞ=�h. That is because the unper-turbed evolution of each qm,n evolves with the com-

plex phase exp½�itðEn � EmÞ=�h�. In anticipation of

utilizing the rotating wave approximation (RWA)

we choose u01 ¼ ðE0 � E1Þ=�h; u12 ¼ ðE1 � E2Þ=�h;and u02 ¼ �xP ¼ �½ðE2 � E0Þ=�h� D�. This choiceof variables is motivated by the idea that under

Raman-resonant laser excitation the lower two lev-

els evolve, primarily, at their natural frequencies

while the upper level will be driven to oscillate at

the pump laser frequency.

Up to this point, there are no approximations

made other than those inherent in the formulationof Eq. (1). Solutions of Eq. (8) can be obtained di-

rectly using the methods described below, to deter-

mine the laser-driven response of a small

molecular ensemble. To calculate the propagation

of fields within the molecular medium, however, it

is convenient to invoke the rotating wave approx-

imation. In this approximation, only those terms

of E(t) which cancel out the oscillating phase terms

in the summation of Eq. (7) are kept. Anything

else is assumed to vary too rapidly to impact the

dynamics. Within this approximation, the full setof equations becomes:

_~q1;0 ¼ �c1;0~q1;0 � i~q1;2XP þ i~q2;0X�S; ð9Þ

_~q2;0 ¼ �ðc2;0 þ iDÞ~q2;0 � ið~q2;2 � ~q0;0ÞXP

þ i~q1;0XS; ð10Þ

_~q1;2 ¼ �ðc1;2 � iDÞ~q1;2 � ið~q1;1 � ~q2;2ÞX�S � i~q1;0X

�P;

ð11Þ

_~q0;0 ¼ i~q2;0X�P � i~q0;2XP; ð12Þ

_~q1;1 ¼ i~q2;1X�S � i~q1;2XS; ð13Þ

~q0;0 þ ~q1;1 þ ~q2;2 ¼ 1; ð14Þ

where:

XPðtÞ �APl2;0

2�hexp �ðt � tPÞ2

s2P

" #; ð15Þ

XSðtÞ �ASl2;1

2�hexp �ðt � tSÞ2

s2S

" #: ð16Þ

Propagation along z at time t is determined

from two additional equations governing the prop-

agation of the fields in the RWA, as derived fromMaxwell�s equations under the slowly-varying

envelope approximation:

nPc

o

otþ o

oz

� �XPðz; tÞ ¼ �i

xPjl2;0j2

2e0nP�hcN~q2;0ðz; tÞ;

ð17Þ

nSc

o

otþ o

oz

� �XSðz; tÞ ¼ �i

xSjl2;1j2

2e0nS�hcN~q1;2ðz; tÞ;

ð18Þ

where N is the density of molecules in the sample

region and nP (nS) is the refractive index of the

molecular host material for the pump (Stokes)frequency. The right-hand side of the equations

is the polarization term for the field propagation.

G. Beadie et al. / Optics Communications 244 (2005) 423–430 427

If the density matrix elements were zero, the opti-

cal pulses would propagate unchanged. Light-in-

duced changes to the density matrix elements, as

a result of Eqs. (9)–(16), result in the nonlinear

optical response of the system through Eqs. (17)and (18).

Because of the numerical complexity of the full

propagation equations, laser parameters were cho-

sen first by optimizing the coherence generated in

the molecular model alone, Eq. (8). As these are

just a system of first-order ordinary differential

equations (ODEs) in one dimension, their solution

is relatively straightforward. In the course of thisstudy, we were able to examine the effects of differ-

ent pulse shapes, the effect of applying the RWA,

and the effect of using different models for the

DPA molecule. Pulses suggested by that effort

were then used as inputs to the set of coupled

Eqs. (9)–(18), to examine the effects of pulse prop-

agation in the excitation of an extended sample.

This required the solution of partial differentialequations over two dimensions.

For cases concerned only with Eq. (8), we

implemented an adaptive differential equation sol-

ver based on the Bulirsch–Stoer method presented

in [6]. Time steps were small fractions of an optical

period. Numerical solutions were checked: (1) for

convergence by decreasing a tolerance parameter;

(2) against ODE solvers based on Runge–Kuttaalgorithms; (3) against a wholly-different numeri-

cal algorithm, rooted in the unitary evolution of

time-dependent operators. Using the ODE solver

as a core function, we also utilized a nonlinear

optimization routine which could loop through

solutions while varying input laser parameters,

iteratively adjusting the laser pulses to optimize

the vibrational coherence. The optimization rou-tine is derived from a commercial package [7]. It

is based on a successive quadratic programming

algorithm, incorporating linear constraints on the

input variables.

The solution of Eqs. (9)–(18) is achieved by two

methods. The first method works in a retarded

time frame that allows one to integrate the equa-

tions over time at given space point, and then tomake a step in the spatial direction using a predic-

tor–corrector method. The second is the method of

lines: using a mesh of spatial coordinates to

approximate spatial derivatives by finite differ-

ences allows one to convert the partial differential

equations into a system of ODEs with respect to

time. The resulting ODEs were integrated by a

fourth order Runge–Kutta formula in the modifi-cation due to Gill [8]. The two methods provided

nearly identical results. The second technique,

the method of lines, can simulate propagation

without the restriction of the RWA. Using this

method we can explore the effect of lifting the

RWA in future work.

The propagation equations were used to find

the evolution of the fields and molecules througha sample contained between z = 0 and z = L. We

normalize the distance z by the pump beam

small-signal absorption coefficient

aP ¼ NxPjl1;2j

2

�hce0nPc0;2; ð19Þ

where eP is the dielectric constant of the molecular

host medium at xP. For example, in a 0.003 M

solution of DPA in water, aP = 35.9 cm�1 at

xP = 2pc/(272.6 nm) and nP = 1.33. Because our

peak absorption coefficient is large, we can obtain

a high optical density over a very short distance.The short path length allows us to ignore potential

dispersive effects due to a solvant, but the disper-

sive effects due to molecular medium are taken

into account automatically.

3. Numerical results

First, we present the maximum coherence

achieved in the absence of propagation. Eq. (8)

was solved using the full electric field expression,

and pulse parameters were found which optimized

the peak coherence q0,1 using the nonlinear search

algorithm. The theoretical maximum that q0,1 canachieve is 1/2. Parameters that were fixed during

the optimization included /P = /S = p/2, sP = sS =150 fs, and tP = tS = 400 fs.Optimumpulse energies,

referenced to 1/e2 intensity radii rP = rS = 275 lm,

along with their respective maximum values of

jq0,1j are presented in Table 1 for three pump config-

urations: D ¼ 0 ðresonantÞ; D ¼ 0:1ðE2 � E0Þ=�hðnear-resonantÞ; and D ¼ 0:471ðE2 � E0Þ=�h (off

Table 1

Detuning D/x20 Pump (lJ) XP (fs�1) Stokes (lJ) XS (fs�1) Max coherence q0,1

0 13.72 0.0332 22.24 0.059 0.059

0.1 70.10 0.0750 127.2 0.176 0.176

0.471 440.8 0.1881 493.2 0.349 0.349

428 G. Beadie et al. / Optics Communications 244 (2005) 423–430

resonant, matching a pump wavelength of 515 nm).

The two obvious trends in Table 1 are: (1) less laserenergy is required as the pump is tuned into reso-

nance; (2) less coherence is obtained as the pump

is tuned into resonance. While the first trend is

straightforward, the second trend is not. It is due

to the fact that a greater real population is trans-

ferred to the upper state when the lasers are reso-

nant, and this population rapidly becomes

incoherently decoupled from the excitation by thefast c0,2.

A representative plot of the coherence evolution

is presented in Fig. 3. The data from Fig. 3 were

calculated for the D = 0.1 case. Plotted are the

pump pulse envelope (equal to the Stokes enve-

lope), the coherence parameter jq0,1j, and the

upper-state population. The upper-state popula-

tion curve demonstrates the departure of the sys-tem from the ideal case: perfect coherence

between the lower two states requires that the

upper state population be zero.

Though the pulse energies of Table 1 were

optimized while holding the other parameters

fixed, the effects of other variations were also stud-

0 200 400 600 800

Time (fs)

0.0

2

4

6

0.8

1.0

e

0 4000

2

4

1.

e

1.

0.

0.

0.

1.

Val

ue

Fig. 3. Coherence preparation dynamics for excitation with

xP = 0.9x02, as given by Eq. (8). The dashed curve is the

envelope of each excitation pulse, the dotted curve is the upper

level population, and the solid curve is jq0,1j.

ied. Varying the relative arrival times of the pulses

had little impact on the peak coherence. Up to apoint, loss of beam overlap was compensated by

greater pulse energies to achieve the same peak

coherence. This is in contrast to the schemes of

adiabatic passage, where a significant improve-

ment in coherent preparation is achieved by turn-

ing on the Stokes pulse first. The difference lies

in the fact that our rapid decoherence rates violate

the conditions of adiabaticity, preventing use ofthose techniques to prepare coherences in this

model. Varying the pulsewidth was also studied.

When the two pulsewidths were kept equal to

one another and the pulse energies were kept fixed

to the values in Table 1, the peak coherence varied

by less than ±20% of the tabulated value for pulse-

widths in the range 50–500 fs. This has a practical

consequence: it suggests that laboratory pulse-widths need not be a critical factor for preparing

optimal coherence.

We also investigated the effect of the RWA on

the results. By comparing simulations calculated

under the RWA to those containing the full elec-

tric field expression we found the qualitative

behavior of the dynamics to be the same. The time

dependence of the coherence peaked at the sametime as for both models, and it had the same decay

dependence. However, the quantitative values did

change. The maximum values of jq0,1j were

0.336, 0.242, and 0.096 for D/x02 = 0.471, 0.1,

and 0.0, respectively. For the off-resonant case,

this value differs very little from the full model.

Near resonance, however, these values differ con-

siderably from those given in Table 1. Furthercomparisons between the two models will be

explored in future work.

Finally, the behavior of this three-level model

was compared to a similar five-level model. The

nature of the three-level model is to approximate

the excited-state manifold of DPA as one discrete

level. In fact, the upper state manifold of [4] is a

G. Beadie et al. / Optics Communications 244 (2005) 423–430 429

convolution of many ro-vibrational states, all

strongly perturbed by interactions with the water

solvent. From a nonlinear-optical perspective the

distinction between one broad line and a convo-

lution of narrower lines is important. For a fixedabsorption spectrum, the nonlinear response is

typically stronger when the absorption is due to

a sum of narrower lines. This is because narrower

lines indicate a slower rate of decoherence, which

generally enhances the overall nonlinear re-

sponse. For example, see [9]. To verify this for

the case of coherence preparation we extracted

best-fit coefficients from the absorption data toan upper-state manifold consisting of three levels.

The results of the fit are illustrated in Fig. 2(b).

We found that, indeed, the maximum coherence

obtained for the five-level model is better than

that of the three-level model. The improvement

in the peak jq0,1j over the three-level model was

10%, 57%, and 126% for the off-resonant, near-

resonant, and resonant pumping conditions,respectively, where resonance was measured with

respect to the lowest-lying upper state. This is

evidence for our three-level model serving as a

useful worst-case scenario for coherence prepara-

tion in DPA.

Fig. 4 shows some results of a full propagation

solution through a sample with a depth of aPz = 2.

Fig. 4. (a) Spatial and temporal behavior of coherence q0,1. (b) Tempo

(dashed lines) at different position inside the cell. 1 and 1 0 correspond

fields correspondingly, D = aPz/2 = 0; 2, 2 0 – position inside the cel

aPz/2 = 0.75; 5, 5 0 – D = aPz/2 = 1.0.

Propagation effects of resonant femtosecond

pulses in DPA have been explicitly studied in

[10]. For the present calculation we used a reso-

nant pump pulse with initial Rabi frequencies of

0.05 fs�1 per pulse: slightly higher than the opti-mum energies given in Table 1. In Fig. 4(a) we plot

the coherence parameter ~q0;1 as a function of both

time and position, while in Fig. 4(b) we show the

plot of the field envelopes of the laser pulses as a

function of retarded time, for several different

locations along z (simulations are made for

L = 1.4 mm).

Each pulse experiences amplitude and refractiveindex shifts upon traversing the sample (Fig. 4). As

expected, the resonant pump pulse is absorbed as

it propagates through the sample, due to direct

absorption by the molecules. It also exhibits a rel-

ative delay. As illustrated in Fig. 4(b) the pulse

center shifts slightly to later times deeper into the

medium, which is an indication that the molecular

system imposes a positive nonlinear refractive in-dex on the pulse. By contrast, the Stokes pulse is

both amplified and sped up by the molecules.

The amplification results from a Raman popula-

tion inversion between molecules in the ground

state and the vibrational state: (N0 > N1). The

increased group velocity results from a negative

nonlinear refractive index.

ral behavior of the pump, XP, (solid lines) and Stokes fields, XS,

to the position at the entrance of the cell for pump and Stokes

l where D = aPz/2 = 0.25; 3, 3 0 – D = aPz/2 = 0.5; 4, 4 0 – D =

430 G. Beadie et al. / Optics Communications 244 (2005) 423–430

These results are a clear demonstration that

coherence preparation, which requires a specific

set of laser parameters, must take propagation

effects into account when dealing with an optically

dense sample [10]. Note that, although in the pre-sent case the position where the coherence peaks

occurs near the point expected from just linear

absorption, if the initial pulse energies were higher

that would not be the case. In our simulation, the

optimum Rabi frequency for the pump pulse is

0.033 fs�1 (Table 1) and the maximum of the

coherence occurs near [a]Pz = 0.5 (Fig. 4a) at

which point the Rabi frequency due to linearabsorption only would have been 0.039 fs�1, in

good agreement with the optimum value. Never-

theless, a just linear absorption calculation would

be inadequate even at [a]Pz = 1.0 where it predicts

XP of 0.03 fs�1 instead of the 0.02 fs�1 value (curve

3, Fig. 4b) of our more accurate calculation which

takes into account not only non-linear effects but

pulse reshaping as well.

4. Summary

From our overall results, we can conclude sev-

eral things. First, even though the three-level

model can be seen as a worst-case scenario for

coherence preparation in DPA we are able to sim-ulate coherences near the maximum value of 1/2.

Second, it appears that the optimum pump fre-

quency for an experimental attempt at coherence

preparation would be slightly below the peak

absorption frequency. The optimum frequency

would be chosen by balancing the improvement

in coherence via detuning the laser against the pen-

alty of higher pulse energies. Third, the simula-tions suggest that there is no need for the special

pulse shaping required in adiabatic passage

schemes. Combined with the insensitivity of the

results to pulsewidth over a large pulsewidth

range, coherence preparation could be imple-

mented with simple pulses and remain robust

against standard laser pulse variations. Fourth,

the propagation simulations highlight the intraca-

cies of nonlinear pulse propagation phenomena.

Even in the absence of nonlinear optical effects

within the molecular host medium, the simulationsshow that the molecules alone exhibit enough non-

linear dielectric response to significantly alter the

pulse propagation.

Acknowledgements

We thank K. Lehmann, R. Lucht, P. Hemmer,V. Sautenkov and M.S. Zubairy for useful discus-

sions and gratefully acknowledge the support from

the Office of Naval Research, the Air Force Re-

search Laboratory (Rome, NY), Defense Ad-

vanced Research Projects Agency-QuIST, Texas

A&M University Telecommunication and Infor-

mation Task Force (TITF) Initiative, and the Ro-

bert A. Welch Foundation (#A2161).

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