Page 1

Electromagnetically induced transparency:

the thickness of the vapor column is of the order

of a light wavelength

Y. Pashayan-Leroy,1C. Leroy,1A. Sargsyan,2A. Papoyan,2and D. Sarkisyan2,*

1Institut Carnot de Bourgogne, UMR 5209 CNRS-Universit de Bourgogne, 9 Av. A Savary, BP47 870,

F–21078 Dijon Cedex, France

2Institute for Physical Research, National Academy of Sciences of Armenia, Ashtarak, 0203 Armenia

*Corresponding author: david@ipr.sc.am

Received November 28, 2006; revised March 12, 2007; accepted April 19, 2007;

posted April 30, 2007 (Doc. ID 77263); published July 19, 2007

Electromagnetically induced transparency (EIT) effect has been studied using an extremely thin cell (ETC)

with the thickness of an Rb vapor column of the order of light wavelength ? ?780 nm? and varying in the range

of 0.5?–2.5?. ?-systems on the D2line of85Rb and87Rb have been studied experimentally. Along with EIT

resonance, we study the peculiarities of velocity-selective optical pumping/saturation (VSOP) resonances,

which accompany the EIT resonance and, as a rule, are spectrally broader. It is demonstrated that size-

conditioned strongly anisotropic contribution of atoms with different velocities in an ETC causes several dra-

matic differences of the EIT and VSOP resonances formation in the ETC as compared with an ordinary

1–10 cm long cell. Particularly, in the case of the ETC, the EIT linewidth and contrast dramatically depend on

the coupling laser detuning from the exact atomic transition. A theoretical model taking into account the pe-

culiarities of transmission spectra when L=n? and L=?2n+1??/2 (n is an integer) has been developed. The

experimental transmission spectra are well described by the theoretical model developed. The possibility of

EIT resonance formation when atomic column thickness is of the order of L=0.5? and less is theoretically

predicted. © 2007 Optical Society of America

OCIS codes: 020.1670, 020.3690, 020.2930, 300.6360.

1. INTRODUCTION

There has been considerable interest in recent years

about the fascinating properties of coherent population

trapping (CPT) and the related electromagnetically in-

duced transparency (EIT). The EIT and CPT resonances

can occur in a ?-system with two long-lived states and

one excited state coupled by two laser fields [1–4]. Note

that alkali-metal vapor is very convenient for EIT and

CPT studies, and these effects have been successfully

demonstrated in atomic media prepared in a cell of ordi-

nary length ?1–100 mm?. Also of considerable interest is

the study of the possibility for miniaturization of alkali

cells for application in the CPT (EIT) experiments with-

out compromising the CPT resonance parameters. How-

ever, the reduction of the vapor cell size results in an in-

crease of the dephasing rate of coherence between the

ground states: ?21=?2?t?−1, where t=L/u, with L being

the distance between cell windows and u being the most

probable thermal velocity. The latter leads to broadening

of CPT and EIT resonances, as well as to worsening of the

EIT resonance contrast (defined as the ratio of the EIT

depth to the height of the shoulders of the EIT window).

To prevent atom-window collisions for the case of submil-

limeter thin cells, either buffer gas or paraffin-coated

walls are used [5–7]. Nevertheless, in Ref. [8], it was theo-

retically predicted for the cell with the thickness of L

?10 ?m that even in the case of pure vapor, narrow EIT

resonance can be observed. Very recently it has been ex-

perimentally demonstrated that it is possible to observe

EIT resonance using the so-called extremely thin cell

(ETC) with an atomic vapor layer smoothly controllable in

the range of L?780–1600 nm comparable with the laser

radiation wavelength resonant with the Rb D2line ??

=780 nm? [9]. The ETC was filled with pure Rb, neither

buffer gas nor paraffin-coated walls were used. Note, that

the EIT resonance linewidth in the ETC is narrower than

that of the velocity-selective optical pumping or satura-

tion (VSOP) resonance accompanying the EIT resonance.

The explanation is that the contribution of atoms with a

small velocity projection in the laser radiation direction

(i.e., atoms flying nearly parallel to the cell windows) is

enhanced thanks to their longer interaction time with la-

ser field. Due to this atomic velocity-selectivity, the ob-

served linewidth of the EIT resonance is more than by an

order narrower than that expected from the inverse of the

window-to-window flight time of the atoms. It should be

noted that in accordance with its physical nature, the

VSOP resonance cannot have a width narrower than the

natural width of the atomic transition. As for EIT reso-

nance, its width can be considerably narrower than the

natural width [2–4]. However, the increase of the interac-

tion time with a laser can lead to the increase of the

VSOP resonance amplitude and the reduction of the EIT

linewidth (see Fig. 9).

In [9] we compared our experimental results with the

theoretical model of [8] as there was no other relevant

model. However, this theoretical model considers a very

simplified three-level system for a 10 ?m thin atomic va-

Pashayan-Leroy et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. B1829

0740-3224/07/081829-10/$15.00© 2007 Optical Society of America

Page 2

por column and does not take into account the following

points: (i) the influence of the reflected beam by the sec-

ond wall of the ETC (which is important in the case when

the thickness is L??), (ii) as shown [4] in a real atom, the

ground-state coherence can be influenced by the hyperfine

structure of the excited state. Thus, the hyperfine struc-

ture of 5P3/2should be included, (iii) the influence of the

Dicke-type coherent narrowing effect, which depends on

the ratio L/? should be included; (iv) the three-level

model of [8] does not allow one to study the peculiarities

of the supplementary VSOP resonances (which appear be-

cause of the additional upper levels); (v) such important

experimental parameters, as linewidth of the coupling

and probe lasers, were not included in the model; (vi) the

model was applied to a 10 ?m thin atomic vapor column,

meanwhile for the ETC [10], the thickness of the atomic

vapor is less by an order. In this work, a theoretical model

including the practically all-important experimental pa-

rameters determining the EIT process as well as VSOP

resonance formation in the ETC has been developed, and

a comparison with the experiment is provided.

2. EXPERIMENT

The experimental setup is similar to that presented in [9].

The radiation beams ??=3 mm? of two single-frequency

diode lasers with ?=780 nm and the spectral width of ?L

=5 MHz are well superposed and focused by an F=35 cm

lens into the ETC at nearly normal incidence. The wedge-

shaped (vertical) thickness L of the gap allows one to

smoothly vary the thickness of the Rb atomic vapor col-

umn in the range from 350 to 2800 nm. This ETC oper-

ates with an oven made from nonmagnetic materials and

was placed inside the three pairs of mutually perpendicu-

lar Helmholtz coils providing the possibility to cancel the

laboratory magnetic field as well as to apply the homoge-

neous magnetic field. The laboratory magnetic field was

compensated with the accuracy of a few microtesla. It is

easy to show that an additional broadening of the EIT

resonance by the residual magnetic field is less than

0.1 MHz. The frequency reference spectra formation has

been realized with an auxiliary ETC (mentioned further

as ETC1) with the fixed thickness L=? [9]. Our ?-type

system is formed on the atomic D2line of87Rb (or85Rb).

The ground-state hyperfine levels Fg=1 and Fg=2, spaced

by 6835 MHz (or Fg=2 and Fg=3 spaced by 3036 MHz),

serve as the two lower states, and the excited state 5P3/2

serves as the common upper level (see Fig. 1). The cou-

pling laser frequency is fixed and is resonant either with

the

with the85Rb Fg=3→Fe=3 and Fg=3→Fe=2 transitions

(in some cases it is blue- or redshifted from these transi-

tions). The coupling laser frequency stabilization has

been realized by using a separate ETC with the thickness

L=?/2; this technique will be published elsewhere [11].

The probe laser frequency is scanned across87Rb Fg=1

→Fe=0,1,2 (85Rb, Fg=2→Fe=1,2,3) transitions. The

power of the coupling and probe lasers is ?1 mW and

?0.3 mW, respectively. The temperature at the window

Twand at the reservoir TR(the reservoir temperature de-

termines the atomic density inside the ETC) are ?170°C

and 150°C, respectively. These values for the reference

ETC1 are 120°C and 100°C, respectively.

As demonstrated [12–15] transmission spectra of the

probe laser in the case of ETC depend on the parameter

L/?. For low pump intensity ??1 mW/cm2?, the hyperfine

absorption linewidth exhibits oscillating behavior, with

minimum value for L=?2n+1??/2, and maximum value

for L=n?. This is the manifestation of so-called collapse

and revival of the Dicke-type coherent narrowing effect.

For high pump intensity ??1 mW/cm2?, the VSOP reso-

nances of reduced absorption, caused by VSOP effects and

centered on the hyperfine transitions occur preferentially

when L=n?, while for L=?2n+1??/2, the revival of peaks

of increased absorption is still observable with a lower

contrast.As the most interesting features are observed for

n?/2 thicknesses, L=?, 1.5?, 2?, and 2.5? values have

been exploited in the experiments as well as for the nu-

merical calculations. These thicknesses could be easily

determined in the experiment by minimal reflection from

the ETC’s gap [12]. The ETC had the area of uniform

thickness, a little less than 1 mm2, smoothly varying in

the vertical direction in the range of ?0.5?–3?. That is

why by slightly focusing the beam up to the diameter

?1 mm, we have a uniform thickness inside the beam.

When using L=?,2? under the condition of exact reso-

nance of the coupling laser with the corresponding atomic

transition, as a rule, the transmission spectrum of the

probe contains the EIT resonance, together with the

VSOP resonance of reduced absorption peak, which are

superimposed on the frequency scale and contribute in

the same way leading to absorption reduction. Meanwhile

when using L=1.5?,2.5? under the condition of exact

resonance, the transmission spectrum of the probe con-

tains the EIT resonance, together with the revival peaks

of the increased absorption, which are superimposed on

the frequency scale and contribute with opposite sign.

These features are well seen on the experimental as well

as on the calculated curves.

87Rb Fg=2→Fe=2 and Fg=2→Fe=1 transitions, or

3. OUTLINE OF THE MODEL

Modeling of EIT is achieved by considering three-level

atomic systems. In fact, most real atomic manifolds are

made up of a multiple of levels. For example, in the case

of D2transition of Rb, there is a set of four closely spaced

upper levels. To interpret our experimental results, we

consider the six-level hyperfine structure of Rb D2transi-

tion as shown in Fig. 1. The energy levels are denoted as

Fig. 1.Energy level diagram of the six-level model.

1830J. Opt. Soc. Am. B/Vol. 24, No. 8/August 2007Pashayan-Leroy et al.

Page 3

i=1,2,3,4,5,6 and ?1, ?2, ?3are the hyperfine splittings

of the upper state. A vapor of the six-level atoms confined

in an ETC of thickness L is excited under normal inci-

dence by two copropagating electromagnetic fields. Level

?2? is coupled to levels ?4?, ?5?, and ?6? by the intense laser

Ec?z,t? with frequency ?c(the coupling field). The weak

laser Ep?z,t? with frequency ?p(the probe field) couples

level ?1? to levels ?3?, ?4?, and ?5?. In a Doppler-broadened

system, the frequency detunings from the resonance of

the probe and coupling fields are defined as

?p

1,2,3= ?i1− ?p± kpvz

?i = 3,4,5?,

?1?

?c

4,5,6= ?i2− ?c± kcvz

?i = 4,5,6?,

?2?

where ?i,kis the energy difference between the levels i

and k. Here kp,cvzis the Doppler shift contribution for the

detuning of the laser field corresponding to the velocity

component vzalong the propagation vector kp,c. The plus

sign refers to the atoms with the velocity v in the positive

direction of the cell axis, and the minus sign refers to the

atoms with the velocity v in the negative direction of the

cell axis.

Our theoretical calculations are based on a standard

density-matrix approach. The dynamical behavior of the

density matrix ? is given by the Liouville equation of mo-

tion:

d?

dt= −

i

??H,?? + ??.

?3?

Here H is the Hamiltonian of the six-level atom including

the dipole interaction with the bichromatic laser radia-

tion, and ? is the relaxation matrix. All spontaneous

emission rates of the excited states are assumed to be the

same, and all ground level population relaxation rates are

ignored under the assumption that the ground states are

stable. We assume that the atomic density N is low

enough so that only atom-surface collisions are to be con-

sidered. Consistent with this condition, we ignore colli-

sional broadening ?colin our theoretical model, but we

consider a relaxation rate ?21for the coherence between

the ground-state levels, which is dominated by the finite

time of flight of atoms between the cell windows. We also

assume that the atoms lose optical excitation and all

memory about the previous state when colliding with the

cell windows, and that the incident laser beam diameter

largely exceeds the cell thickness. With the assumptions

made, the relaxation effect is taken into account exactly

by solving the temporal density-matrix equations with

proper boundary conditions for each atom separately. We

neglect the magnetic substructure of each level.

The relaxation ?21is also sensitive to amplitude or

phase fluctuations of the laser fields. We use the Wiener–

Levy phase diffusion model (see, e.g., [16–18]) to describe

the effects of the finite bandwidth of the laser field to the

coherence of the interaction. We assume that both lasers

have a Lorentzian spectrum with a full frequency width

at half maximum (FWHM) ?L(as obtained from the model

adopted). The laser bandwidths are incorporated into the

calculations by introducing relaxation terms for the non-

diagonal density-matrix elements using the procedure de-

veloped in [19,20].

Polarization of the medium on the frequency of the

probe field is determined by

P?z? = N?

j=3

5

?j1??j1

++ ?j1

−?,

?4?

where nondiagonal matrix elements ?j1

?j1

locity in the positive and negatives directions of the cell

axis, respectively. Here ?j1?j=3,4,5? is the electric dipole

moment of the optical transition ?j?→?1?.

We obtain absorption spectra by calculating numeri-

cally the set of density-matrix equations (see Appendix A

for the complete equations of motion). Details for the the-

oretical modeling appear in [13,21]. The absorption of the

probe beam is determined by the sum of imaginary parts

of three atomic coherences, i.e., Im ?31, Im ?41, and Im ?51.

The differential equations of these coherences read

+??j1?z=vzt? and

−??j1?z=L−vzt? relate to the atoms flying with the ve-

? ˙31= i?1??11− ?33? − i?2?34− i?3?35

− ?i?p

1+ ?31??31,

?5?

? ˙41= i?2??11− ?44? + i?4?21− i?1?43− i?3?45

− ?i?p

2+ ?41??41,

?6?

?51

˙= i?3??11− ?55? + i?5?21− i?1?51− i?2?54

− ?i?p

3+ ?51??51.

?7?

?i?i=1,2,3? and ?j?j=4,5,6? are the Rabi frequencies of

the probe and couple fields, respectively, and ?ijare the

decoherence rates (for the definition see Appendix A).

To include the Doppler-broadening effect, we average

the density-matrix element obtained for a single atom

over the Maxwellian velocity distribution of the atoms un-

der consideration. This is given by W?v?=?N/u???exp?

−v2/u2?, where v is the atomic velocity, N is the atomic

density, and u is the most probable velocity given by u

=?2kBT/M with T being the temperature in Kelvin and

M being the atomic mass. The Doppler broadened absorp-

tion profile is given by

Ep?F?2?

? Im??

1

Ep

2?A? = −

4??pNt2

cu?u

2t1

1

0

?

e−vz

2/u2vzdvz?

0

L/v

dt

i=3

5

?i1??i1

+?t,?+,Ep0?vzt???1 − r1e2ikpvzt?

−?t,?−,Ep0?L − vzt???1 − r1e2ikp?L−vzt????.

Here t1and t2are the transmission coefficients of the cell

windows; r1and r2are the reflections coefficients of the

windows. All the spectra presented in Section 4 are calcu-

lated for r1=r2=0.3. The factor F=1−r1r2exp?2ikL? takes

into account the Fabry–Perot nature of the cell caused by

two highly parallel cell windows. The influence of the

Fabry–Perot effect on the absorption line shape and mag-

nitude in an ETC has been studied theoretically in [13].

The probe field inside the empty cell Ep0?z? is defined in a

way:

+ ?i1

?8?

Pashayan-Leroy et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. B 1831

Page 4

Ep0?z? =

Ept1

F

?1 − r2e2ik?L−z??,

?9?

with Epbeing an external probe field. We numerically in-

tegrate Eq. (8) and obtain a velocity-averaged absorption

coefficient as a function of Raman detuning ?R=?p−?c

+?21.

4. RESULTS AND DISCUSSIONS

In Fig. 2(a), the experimental transmission spectra of the

probe laser are presented for the case where the coupling

laser is resonant with the87Rb Fg=2→Fe=2 transition,

while the probe laser is scanned across Fg=1→Fe

=0,1,2 transitions (see the upper-left inset). This mea-

surement was done for the ETC thickness L=? ?780 nm?.

The EIT resonance is seen on the upper curve together

with VSOP, and the upper inset presents the results of fit-

ting by Lorentzian profiles. FWHM of the EIT and VSOP

resonances are ?12 MHz and ?30 MHz, correspondingly

(the accuracy of the fitting is 10% to 15%). As seen, the

absorption on the Fg=1→Fe=1 transition is enhanced

due to the optical “repumping” caused by the couple field,

and the VSOP peak amplitude on the upper curve is

smaller as compared with that on the lower curve (here

the couple is blocked), indicating reduction of absorption.

In Fig. 2(b), the theoretical curve is presented for the

same parameters as in Fig. 2(a), with the Rabi frequen-

cies of couple and probe lasers being ?c=1.5? and ?p

=0.35?, correspondingly. Here ?=6 MHz is the lost rate of

the excited states. The inset presents the results of the

fitting: FWHM of the EIT and VSOP resonances are

?12 MHz and ?20 MHz, correspondingly. In both figures

(a) and (b), the lower curve is vertically shifted for conve-

nience. One can see that there is good agreement between

the experiment and the theoretical model.

An interesting feature of EIT resonance formation in

the ETC is the possibility of shifting the EIT resonance

and VSOP peak with respect to each other on the fre-

quency scale (this is not possible to realize in an ordinary

Fig. 2.

87Rb Fg=1→Fe=0,1,2 transitions (the relevant energy levels of

87Rb are presented in the upper-left inset), when the coupling la-

ser is resonant with Fg=2→Fe=2 transition. The cell thickness

L=? ?780 nm?, ?L=5 MHz. (a) Experimental results. The upper

inset presents results of the fitting by Lorentzian profiles.

FWHM of the EIT and VSOP resonances are ?12 and ?30 MHz,

correspondingly. (b) Numerical simulation, all the parameters

are the same as in (a), and ?c=1.5?, ?p=0.35?. FWHM of the

EITand VSOPresonances

correspondingly.

Transmission spectra of the probe laser scanned across

are

?12and

?20 MHz,

Fig. 3.

87Rb Fg=1→Fe=0,1,2 transitions when the coupling laser is red

detuned by 48 MHz from the Fg=2→Fe=2 transition. The cell

thickness is L=2? ?1560 nm?. (a) Experiment, ?L=5 MHz. The

middle curve is the case when the coupling laser is blocked (the

curve is vertically shifted for convenience). The lower curve (dot-

ted) presents the transmission spectra of the reference ETC1, L

=?. (b) Theoretical curve, all the parameters are the same as in

(a). The Rabi frequencies of the couple and the probe lasers are

?c=2? and ?p=0.3?, correspondingly. The arrow shows the EIT

resonance accompanied by the VSOP resonances. The lower

curve is the case when the couple is blocked.

Transmission spectra of the probe laser scanned across

1832J. Opt. Soc. Am. B/Vol. 24, No. 8/August 2007 Pashayan-Leroy et al.

Page 5

cell). Moreover, when the detuning is large, it is possible

to completely resolve the EIT resonance (however, as

shown below, this causes an additional broadening of the

EIT resonance). In Fig. 3(a), the transmission spectra of

the probe laser are shown for the case where the couple is

red detuned by 48 MHz from the87Rb Fg=2→Fe=2 tran-

sition and the probe laser is scanned across Fg=1→Fe

=0,1,2transitions. The

=2? ?1560 nm?. The pointing arrow in the upper curve

shows the EIT resonance (linewidth ?25 MHz), which is

completely resolved from the accompanying VSOP reso-

nances. The middle curve corresponds to the case where

the coupling laser is blocked (the curve is vertically

shifted for convenience). The lower curve (dotted) repre-

sents the transmission spectrum of the reference ETC1.

In Fig. 3(b), the theoretically calculated curve is pre-

sented for the Rabi frequencies of the couple and probe la-

sers ?c=2? and ?p=0.3?, correspondingly. The pointing

arrow in the upper curve shows the EIT resonance, which

is also here completely resolved from the VSOP reso-

nances. The lower curve is the case where the coupling la-

ser is blocked. Note that the repumping of VSOP reso-

nances is negligible when the detuning of the coupling

laser is large (see below).

As seen from Fig. 2, the coupling laser resonant with

one atomic transition increases the absorption of the

probe field on the other transitions due to repumping,

which consequently leads to the decrease of VSOP peak

amplitude. Moreover, depending on the couple field inten-

sity, the VSOP resonances may disappear completely and

with further increase of the intensity, appear reversed (in-

creased absorption). Particularly, a sign reversal of the

VSOP resonance is well seen for the transition Fg=1

→Fe=0 presented in Fig. 8 for the case of zero couple de-

tuning.

In Fig. 4, the (a) experimental and (b) theoretical probe

laser transmission spectra are presented for the case

where the couple laser is resonant with85Rb Fg=3→Fe

=3 transition, with the probe laser being scanned across

Fg=2→Fe=1,2,3 transitions (the relevant energy levels

are presented in the inset). On the upper curves of the fig-

ure [in both (a) and (b)], the VSOP resonances on Fg=2

→Fe=1,2 transitions (on the left from the EIT resonance

pointed by the arrow) are practically absent due to re-

pumping induced by the coupling field. The two lower

curves in both (a) and (b) show the case where the cou-

pling laser is blocked (the curves are vertically shifted for

convenience). To let the reader see the contrast, the over-

all absorption (transmission) of the probe laser is pre-

sented in Fig. 4(c) under the same conditions as in the

case of Fig. 4(a).

As mentioned above, the profiles of EIT resonances at

the thicknesses L=1.5? and 2.5? must be different from

those at L=? and L=2?. Figure 5 presents experimen-

tally observed and calculated transmission spectra of the

probe field for the cell thickness L equal to 1.5? and 2.5?.

The pointing arrows show the EIT resonances. The two

upper curves, (1) and (2), correspond to the case where

the couple laser is blue detuned by 15 MHz from

Fg=3→Fe=3 transition and the probe laser is scanned

across Fg=2→Fe=1,2,3 transitions; L=2.5?. Shown are

(1) experiment (EIT resonance linewidth ?10 MHz) and

cellthickness is

L

85Rb

(2) calculated spectra with ?c=1?, ?p=0.2?, ?L=5 MHz.

Curves (3) and (4) correspond to the case where the cou-

pling laser is resonant with85Rb Fg=3→Fe=3 transition;

L=1.5?: (3) experiment and (4) calculated spectra with

?c=1?, ?p=0.1?, ?L=5 MHz. Curve (5) is the transmis-

sion spectrum of the reference ETC1, L=?. As can be

seen, there are no VSOP resonances of reduced absorption

in Fig. 5 that are present in Figs. 2 and 3. Note that in the

case of L=? and 2?, the EIT peak (when the coupling la-

ser is in resonance with the atomic transition) is superim-

posed on the VSOP peak, which is also a peak of reduced

absorption. However, in the case of L=1.5? and L=2.5?,

the EIT peak is superimposed on the VSOP peak, which is

now the peak of increased absorption (due to the coherent

Dicke effect [12–15]). That is why, in the second case, the

Fig. 4.

85Rb Fg=2→Fe=1,2,3 transitions; the couple is resonant with

Fg=3→Fe=3 transition (the relevant energy levels are presented

in the inset). The cell thickness L=? ?780 nm?, ?L=5 MHz. (a)

Experiment. Only the EIT resonance (pointed with the arrow on

the upper curve) is present, while the VSOP resonances are ab-

sent due to a strong repumping. The lower curve presents the

case when the couple is blocked (the curve is vertically shifted for

convenience). (b) Numerical calculations with all the parameters

are the same as in (a), ?c=1? and ?p=0.26?, correspondingly. (c)

Transmission spectra of the probe laser under the same condi-

tions as in (a).

Transmission spectra of the probe laser scanned across

Pashayan-Leroy et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. B 1833

Page 6

EIT peak demonstrates a small increase in the absorption

on the wings, and further, a decrease of the absorption

[see curves (3) and (4) in Fig. 5]. Another distinction in

the behavior of the VSOP resonances accompanying the

EIT resonance is seen when comparing the “sign” of the

VSOP resonances in Figs. 2, 3, and 5.

Figure 6 shows the calculated EIT resonance linewidth

as a function of the laser radiation spectral width. The de-

pendence is obtained by fitting the theoretical probe

transmission curves when the couple is resonant with

Fg=2→Fe=2 transition (for other parameters see the fig-

ure caption). It is seen that the EIT width increases lin-

early with the laser spectral linewidth. From the figure it

is seen that even in the case of zero laser linewidths, the

residual linewidth of the EIT is 9 MHz. This linewidth is

composed of ?21and the additional broadening due to cou-

pling and probe laser intensities.

Figure 7 presents the calculated transmission spec-

trumof theprobefield

=?/2 ?390 nm? for the case where the coupling laser is

resonant with85Rb Fg=3→Fe=3 transition (for other pa-

rameters see the figure caption). Dotted curve (1) corre-

atthecellthickness

L

sponds to the case where the coupling laser is blocked.

Note, that in both cases, (1) and (2), there is an increased

absorption on VSOP resonances, opposite from the case of

L=? and 2?. The pointing arrow on curve (2) shows the

EIT resonance peak. Hence, it is possible to observe EIT

resonance at the cell thickness L=?/2 choosing correct ex-

perimental parameters and using coherently coupled la-

sers [2–4] with a small spectral linewidth of radiation.

This is a remarkable result since previous experimental

studies have shown that using only the probe laser, even

at high radiation intensity ?1 W/cm2it is not possible to

obtain a VSOP resonance at the cell thickness L=?/2

[12,15]. Impossibility to obtain a VSOP resonance of re-

duced absorption at L=?/2 was also confirmed by theoret-

ical considerations (in contrast, ?1 mW/cm2is sufficient

to form VSOP dips at L=? [14]). This is because of a

strong Dicke-type coherent narrowing regime at L=?/2,

meanwhile as we can see from Fig. 7, the EIT effect al-

lows one to form a narrow dip of reduced absorption for

L=?/2 at a lower laser intensity. Moreover, the theoreti-

cal model presented shows that, under certain conditions,

it is possible to form the EIT resonance for L?100 nm as

well. In [22] atom-surface van der Waals interaction when

L?100 nm is successfully studied exploits transmission

and selective reflection spectra. The implementation of

the EIT effect could give an additional possibility for

studying atom-surface interaction via the frequency shift

of the EIT resonance.

In Fig. 8, the calculated probe field transmission spec-

tra are shown for different values of the blue detuning of

the couple field with respect to the

transition frequency for L=? (for other parameters see

the figure caption). We have chosen the blue detuning in

order to see a pure dependence from the detuning, mean-

while in the case of red detuning, the transition Fg=1

→Fe=1 will have an influence on the EIT resonance. As

can be seen, the narrowest EIT resonance is observed

87Rb, Fg=2→Fe=2

Fig. 5.

85Rb Fg=2→Fe=1,2,3 transitions. (1) and (2) refer to the cou-

pling laser blue detuned by 15 MHz from Fg=3→Fe=3 transi-

tion; the cell thickness is L=2.5?. The pointing arrows show the

EIT resonances: (1) experiment and (2) numerical calculations.

(3) and (4) refer to the case where the couple is resonant with

Fg=3→Fe=3 transition, the cell thickness L=1.5?: (3) experi-

ment and (4) numerical calculations. (5) The transmission spec-

trum of the reference ETC1, L=?.

Transmission spectra of the probe laser scanned across

Fig. 6.

of the laser radiation spectral width (couple and probe are as-

sumed to have the same linewidth). The coupling laser is reso-

nant with

=? ?780 nm?, ?c=1.5?, ?p=0.35?.

Calculated linewidth of the EIT resonance as a function

87Rb Fg=2→Fe=2 transition, cell thickness L

Fig. 7.

cell thickness L=?/2 ?390 nm? when the couple is resonant85Rb

Fg=3→Fe=3 transition (the relevant energy levels are presented

in the inset); ?L=1 MHz, ?c=1?, ?p=0.3?. (1) the couple beam is

blocked, (2) the couple is on. EIT resonance on curve (2) is

marked by an arrow. A possibility to observe EIT resonance at

the cell thickness L=?/2 is a remarkable result (see the text).

Also on curve (2), the increase of absorption on VSOP resonances

on Fg=2→Fe=1,2 transitions is observable [compare with curve

(1)].

Calculated transmission spectra of the probe field for the

1834J. Opt. Soc. Am. B/Vol. 24, No. 8/August 2007 Pashayan-Leroy et al.

Page 7

when the coupling laser is resonant with the Fg=2→Fe

=2 transition, whereas the increase of the coupling field

detuning leads to a rapid broadening of the EIT reso-

nance. With the increase of the blue detuning from 0? to

10?, the width of EIT resonance ?EITincreases from 12 to

36 MHz. It is important to note that in cells of usual

length ?1–10 cm?, the width and amplitude of EIT reso-

nance have a weak dependence on coupling field detuning

(for a physical explanation, see below). It is also interest-

ing to note that the widths of VSOP resonances remain

practically unchanged with the increase of the coupling

field detuning (for the four cases presented in the figure,

the width of VSOP resonance?VSOPis equal to ?18 MHz).

This is clearly seen for the VSOP peak on the Fg=1→Fe

=1 transition.

To provide comparison of the EIT process and VSOP

formation with the cell of an ordinary length (COL), in

Fig. 9, the transmission spectrum of the probe laser (up-

per curve) is shown when the coupling laser ??0.5 mW? is

resonant with the85Rb Fg=3→Fe=2 transition, while the

probe laser ??0.3 mW? is scanned across Fg=2→Fe

=1,2,3 transitions (both beams are parallel, and the COL

is inserted inside Helmholtz coils). The cell length is

?1 cm, and the temperature is ?40°C. In the upper

curve, the EIT resonance (mentioned by the pointing ar-

row) is seen together with seven sub-Doppler VSOP reso-

nances. The EIT resonance has the linewidth of 2–3 MHz

(i.e., approximately two times less than the natural line-

width). The lower curve is the transmission spectrum of

the ETC1 for L=? (this spectrum serves as a reference for

the calibration of the frequency scale).

The following significant distinctions for EIT and

VSOP formations are observed when using the ETC with

an atomic vapor column of L?? as compared with the

COL:

1. In the case of exact collinear propagation of the two

laser beams in the COL, the number of sub-Doppler sat-

ellites is equal to 7 (see Fig. 9), while under the well-

defined geometry of the interaction the number of VSOP

peaks can be reduced to 5 [23]. Note that the EIT reso-

nance is always superimposed on one of the VSOP reso-

nances (the VSOP number 2 in Fig. 9). When using ETC,

the maximal number of VSOP resonances is equal to 3;

the corresponding resonance frequencies are fixed and do

not depend on the coupling laser frequency [9]. The EIT

resonance can be superimposed on one of these VSOP

resonances depending on the coupling laser frequency

only if the coupling laser frequency is in exact resonance

with the corresponding atomic transition. However, when

the coupling laser is detuned from the resonance with the

transition, the EIT peak can be clearly resolved on the

frequency scale (see Fig. 3).

2. In the case of the COL, the frequency position of the

VSOP resonance is determined by the coupling laser fre-

quency and by the values of the hyperfine splittings of the

ground and excited states. Note that the frequencies of

VSOP resonances become larger with the increase of cou-

pling laser frequency and smaller when this frequency is

reduced [24]. In contrast, in the case of the ETC, all three

VSOP resonances are positioned on the atomic transi-

tions. This is explained by the fact that the optical pump-

ing is more effective for the atoms with small longitudinal

velocity, and that the main contribution to the resonant

signal comes from the atoms flying parallel to the cell

windows, perpendicularly to the laser beam and, conse-

quently, having negligible Doppler shift. Thus, indepen-

dently of the ratio of the coupling and probe laser inten-

sities, the positions of the VSOP resonances on the

frequency scale (in the case of the COL) and EIT reso-

nance (for both the COL and the ETC) depend only on the

frequency of the coupling laser.

3. In the case of the COL, when the coupling laser is

detuned from the exact resonance by a value ?, only the

atoms with certain longitudinal velocity will participate

in the formation of the EIT resonance. For the particular

case of87Rb when the coupling laser is tuned from Fg=2,

with the probe field being scanned across the set of hyper-

fine transitions Fg=1→Fe=0,1,2 this velocity is equal to

Fig. 8.

tra as a function of the blue detuning of the couple field fre-

quency from the87Rb, Fg=2→Fe=2 transition. The cell thickness

L=? ?780 nm?, ?L=1 MHz, ?c=1.5?, ?p=0.35?, ?=6 MHz. (1)

dashed curve, coupling laser is blocked, (2) coupling laser is reso-

nant with Fg=2→Fe=2 transition, (3) couple laser is blue de-

tuned by 3?, (4) by 6?, (5) by 10?. ?EIT?12, 16, 25, and 36 MHz

for (2), (3), (4), and (5), correspondingly; ?VSOP?18 MHz for all

cases (2)–(5).

(Color online) Calculated probe field transmission spec-

Fig. 9.

when the coupling laser is resonant with the85Rb Fg=3→Fe=2

transition, while the probe laser is scanned across Fg=2→Fe

=1,2,3 transitions. The cell length is ?1 cm, and the tempera-

ture is ?40°C. In the upper curve, the EIT resonance (with the

linewidth of 2–3 MHz and mentioned by the pointing arrow) is

seen together with seven VSOP resonances. The lower curve is

the transmission spectrum of the ETC1 for the L=? and serves

as a reference for the calibration of the frequency scale.

Transmission spectrum of the probe laser (upper curve),

Pashayan-Leroy et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. B1835

Page 8

v1,2=2???c−?2i?/k, where i=1,2 with k being the wave-

number of the irradiating field. For simplicity, we con-

sider that the wavenumbers are the same for the probe

and coupling lasers. Since the atoms obey the Maxwellian

distribution, the number of atoms with the corresponding

velocity projections v1,2decreases with the increase of the

couple laser detuning ?, thus resulting in weakening of

the EIT resonance. Note, that if v1,2?vT, where vTis the

most probable thermal velocity, the weakening of the EIT

resonance is rather slow. As for the EIT width, it practi-

cally does not change (moreover, as shown in [25], the for-

mation of EIT resonance on the wings of the Doppler pro-

file of the absorption line can lead to its narrowing).

For the opposite, in the case of the ETC, the most nar-

row and the most contrasting EIT resonance is formed

when the coupling laser is resonant with the correspond-

ing atomic transition, meanwhile the increase of the cou-

pling laser detuning leads to the broadening and worsen-

ing of the contrast of the EIT resonance (see Fig. 8). The

physical explanation of this fact is that when the couple

laser is detuned from the corresponding transition by a

value of ?, only the atoms having the velocity projections

v1,2=2??/k participate in the formation of EIT, and for

these atoms the time of flight between the cell windows is

less than that when vz=0. This leads to the increase of ?21

and, consequently, to the broadening and contrast reduc-

tion of the EIT resonance.

4. In a COL, the EIT resonance width and contrast are

weakly influenced by the cell length, while in the case of

ETC, the EIT resonance line shape depends on whether

the cell length is equal to L=n? or to L=?2n+1??/2.As for

the VSOP resonances, in a COL, they demonstrate the in-

creasing of the absorption (see Fig. 9) and are weakly in-

fluenced by the cell length, while in the case of ETC, their

sign is different for L=n? and for L=?2n+1??/2 (Figs. 2

and 3; compare with Fig. 5).

5. Note that in the ETC, there is a difference in the be-

havior of the VSOP and EIT resonances in the case when

the coupling laser is detuned from the atomic transitions.

In particular, from Figs. 3 and 8, it can be seen that the

“repumping” caused by the detuning from the transition

center coupling laser is almost vanishing, while at similar

conditions, the EIT resonance is still well pronounced. On

the contrary, for the case of the COL (see Fig. 9), the EIT

resonance is superimposed on VSOP resonance (number

2) of increased absorption (due to the repumping effect) at

any coupling frequency.

In [9], it was shown that by using the ETC, one can

measure the magnitude of the magnetic field by the split-

ting of EIT resonances. Use of the COL yields much

higher B-field sensitivity [4] due to narrower EIT reso-

nances. Nevertheless, the ETC provides micrometer-

spatial resolution and is advantageous for the measure-

ment of highly nonhomogeneous magnetic fields. The

latter is important for atomic magneto-optical cooling,

nuclear magnetic resonance spectroscopy, and magnetic

resonance tomography, synchrotron radiation sources,

heavy mineral separation, and other applications.

5. CONCLUSIONS

We have studied the peculiarities of the EIT effect with

the help of the ETC with the thickness of the Rb vapor

column of the order of light wavelength ? ?780 nm?, vary-

ing in the range of 0.5?–2.5?. ?-systems on the D2line of

85Rb and87Rb have been studied experimentally. Along

with EIT resonance, we have studied the VSOP reso-

nance, which are the satellites of EIT resonance. As a

rule, the VSOP resonances are spectrally broader than

the EIT resonances. A number of dramatic differences of

the EIT and VSOP resonance formation in the ETC in

comparison with ordinary (centimeter-size) cells is re-

vealed. Particularly, in the case of the ETC, the EIT line-

width and contrast dramatically depend on the coupling

laser detuning from the exact atomic transition. The de-

veloped theoretical model takes into account the pecu-

liarities of the transmission spectra when L=n? and L

=?2n+1??/2 and describes the experiment well. The de-

pendence of EIT resonance on the probe and coupling la-

ser intensities, couple detuning from atomic resonance,

and laser spectral linewidth are also studied. The possi-

bility of the EIT resonance formation when the vapor col-

umn thickness is L=0.5? or even less is theoretically pre-

dicted. Presented peculiarities of the EIT effect when the

thickness of the atomic column vapor L?? (in our case

L=780 nmfor the Rb D2line) are of a general nature, and

this means that similar results are expected for Li, Na, K,

and Cs atomic vapors, but for the other values, the thick-

ness corresponding to the resonant wavelength, for ex-

ample, for the Na D2line, is L?590 nm.

APPENDIX A

The density-matrix equations of motion for the system

shown in Fig. 1 in the rotating frame are explicitly writ-

ten as

? ˙11= − 2 Im??1

*?31? − 2 Im??2

*?41? − 2 Im??3

*?51? + ?31?33

+ ?41?44+ ?51?55,

? ˙22= − 2 Im??4

*?42? − 2 Im??5

*?52? − 2 Im??6

*?62? + ?42?44

+ ?52?55+ ?62?66,

? ˙33= 2 Im??1

*?31? − ?33?33,

? ˙44= 2 Im??2

*?41? + 2 Im??4

*?42? − ?44?44,

? ˙55= 2 Im??3

*?51? + 2 Im??5

*?52? − ?55?55,

? ˙66= 2 Im??6

*?62? − ?66?66,

? ˙62= i?6??22− ?66? − i?4?64− i?5?65− ?i?6

c+ ?62??62,

? ˙52= i?5??22− ?55? + i?3?12− i?4?54− i?6?56

− ?i?5

c+ ?52??52,

? ˙42= i?4??22− ?44? + i?2?12− i?5?45− i?6?46

− ?i?4

c+ ?42??42,

1836J. Opt. Soc. Am. B/Vol. 24, No. 8/August 2007Pashayan-Leroy et al.

Page 9

? ˙32= i?1?12− i?4?34− i?5?35− i?6?36− ?i??5

c− ?1− ?2?

+ ?32??32,

? ˙61= i?6?21− i?2?64− i?3?65− i?1?63− ?i??6

c+ ?R?

+ ?61??61,

? ˙51= i?3??11− ?55? + i?5?21− i?1?53− i?2?54− ?i?3

p

+ ?51??51,

? ˙41= i?2??11− ?44? + i?4?21− i?1?43− i?3?45− ?i?2

p

+ ?41??41,

? ˙31= i?1??11− ?33? − i?2?34− i?3?35− ?i?1

p+ ?31??31,

? ˙65= − i?3

*?61+ i?6?25− i?5

*?62− ?i?3+ ?65??65,

? ˙64= − i?2

*?61+ i?6?24− i?4

*?62− ?i??2+ ?3? + ?64??64,

? ˙63= − i?1

*?61+ i?6?23− ?i??1+ ?2+ ?3? + ?63??63,

? ˙54= i?3?14+ i?2

*?51+ i?5?24− i?4

*?52− ?i?2+ ?54??54,

? ˙53= i?3?13− i?1

*?51+ i?5?23− ?i??1+ ?2? + ?53??53,

? ˙43= i?2?31

*− i?1

*?41+ ?i?1+ ?43??43,

? ˙21= − i?1?32

*+ i?4

*?41− i?2?42

*+ i?5

*?51− i?3?52

*+ i?6

*?61

− ?i?R+ ?21??21,

and ?ij=?ji

ation rates from state ?i? to state ?1?, and ?j2?j=4,5,6? are

the population relaxation rates from state ?j? to state ?2?;

?ii=?

i?j

*. Here ?i1?i=3,4,5? are the population relax-

?ij?i = 3,4,5,6; j = 1,2?

are the total decay rates out of state ?i?. Off-diagonal co-

efficients

2??

k

?ij=

1

?ik+?

l

?jl?

are decoherence decay rates, where k denotes all the lev-

els into which the population from level i may decay and l

denotes all the levels into which the population from level

j may decay; ?1=?13Ep/2?, ?2=?14Ep/2?, ?3=?15Ep/2?

are the Rabi frequencies of the probe field; ?4=?24Ec/2?,

?5=?25Ec/2?, ?6=?26Ec/2? are the Rabi frequencies of

the couple field. ?1, ?2, ?3are the hyperfine splittings of

the upper levels, and ?Ris the two-photon Raman detun-

ing.

ACKNOWLEDGMENTS

The authors thank A. Sarkisyan for his valuable partici-

pation in the fabrication of the ETC as well as to Yu.

Malakyan for discussions and A. Nersisyan for the curves

fitting. This work was supported, in part, by the Arme-

nian National Science and Education Fund (ANSEF)

grant 05-PS-opt-0813-233 and Scientific Co-operation be-

tween Eastern Europe and Switzerland (SCOPES) grant

IB7320-110684/1 and the International Association for

Cooperation with Scientists from the former Soviet Union

(INTAS) South-Caucasus grant 06-1000017-9001.

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