Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qilab.phys.msu.ru/papers/SPIE-4749-147-156(2002).pdf
Äàòà èçìåíåíèÿ: Mon Feb 4 18:39:35 2008
Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:01:50 2012
Êîäèðîâêà:
Spectroscopy of Coherent Dark Resonances in Samarium
V. Vladimirova,a B. A. Grishanin,a V. N. Zadkov,a N. N. Ko1achevsky,A. V. Akimov,b N. A. Kise1ev, V. N. Sorokinb and S. I. Kanorskib

J.

aFaculty of Physics and International Laser Center M. V. Lomonosov Moscow State University,Moscow 119899, Russia bp N. Lebedev Physics Institute,Russian Academy of Sciences Leninski ave. 53, Moscow 117924, Russia ABSTRACT
A theoreticalmodel of the coherent population trapping (CPT) in multilevel samarium atom and its comparison with experimental spectroscopic data are presented. Theoretical model describes a degenerated A-system in Sm atom formed of the transitions4f66s2(7F0) 4f6(7F)6s6p(3P0)9F? --Â 4f66s2(7F1) and includes also a fourth level 4f66s2(7F2),which complements the model making it an open system. An open character of the system reduces the contrast of the resonance curves in the CPT-spectra, but does not change the width of the OPT resonance. Numerical modeling of the CPT resonances in Sm atom was carried out for the case of applied longitudinal and transverse magnetic fields in 7- and 12-levelmodels, as well. Keywords: Coherent population trapping, dark resonance, samarium

1. INTRODUCTION
Coherent population trapping (CPT) phenomenon is currentlywidely used in different applications such as magnetometry, metrology, and others15 and much attention in the literaturehas been paid so far to the alkali atoms.6'7 In frequency standard applications (for cesium standard, see, for example Ref. [8]), rear-Earth atoms however have some advantages. By contrast with alkali atoms, their hyperfine structure energy levels are deeply shielded and, therefore, ground sublevels have much bigger energy splitting,on the order of 10--100 THz. Thus, direct coherent coupling of the dipole-allowedoptical transitions in a A-system formed, for instance, of samarium atom hyperfine tructurelevels with rf transition between ground levels resultsin ultra-narrow, high-finessresonances in the optical range, which is an advantage for developing new frequency standards. Th basics of OPT phenomenon arewell understood in the frame of three-level analytical model.6 For the case of multilescel systems, however, thesimple model has to be significantly complicated and analytical results in most cases becam'e impossible. Enriched energetic structureofmultilevel atomsresults also in essential modificationofthe resonance depÕdencieson driving fields parameters. In this paper, we present a theoretical model for spectroscopy of coherent dark resonances in multilevel Sm atom and compare our theoretical calculations with experimental spectroscopic data for samarium.10 While taking into account complete Zeeman structure of the involved levels requires 12-levelmodel, we show that a much simpler 4-level system gives already a reasonable agreement with the experimental observations.10 This four-levelmodel is formed of the degenerated A-system in Sm atom,

a

4f66s2(7F0)

4f6(7F)6s6p(3P0)9F?

4f66s2(7F1),

(1)

and the fourthlevel, 4f66s2(7F2),which complementsthe model making it an open system. An open character ofthe system reduces the contrastof the CPT-resonances, but does not change their widths. An extended 7- or 12-level model has alsobeenused for simulation of the coherent dark resonances spectra in Sm in longitudinal and transverse
magnetic fields, respectively. Sendcorrespondence to .V.V.: vyulia©comsim1.phys.msu.su

J

ICONO 2001: Novel Trends in Nonlinear Laser Spectroscopy and Optical Diagnostics and Lasers in Chemistry, Biophysics, and Biomedicine, A. Yu. Chikishev, et al., Eds., SPIE Vol. 4749 (2002) © 2002 SPIE · 0277-786X/02/$15.00

147


2. THREE-LEVEL MODEL
We will start with considering a three-level system in A-configuration,which is the simplest model for coherent dark resonances. Definitions of this Section will be used throughout the rest ofthe paper. In a A-configuration, two bottom closely spaced energy levels 1) and 2), split at hz, are coupled with the upper lying energy level by two driving laser fields (Fig. 1). If the transition 1) --* 2) is forbidden in dipole and two pumpingfields E1 exp(--ic4'L1t -- ipx), P22exp(--iwL2t -- i2) are in the resonance with the approximation driving transitions,one can register a sharpCPT-resonance, which reveals in absorptionspectraas a sharpminimum whenone ofthe driving fields,for instance with WL1 , is scanned throughthe resonance point at the Raman detuning

)

8R

L1

--WL2

1

0.

J=
J=o

2)
1

Figure 1. Energy diagram of a three-level atom in a A-configurationdrivenby two laser fields, WL1 and wL2. 13 and 1123 are the respective Rabi frequencies, 'yi , 'Y32, and y21 are the populationdecay rates, w12 is the pumping rate, F13, F23, and F12 are the pure dephasing rates, 8L and 5R are the laserand Raman detuning, respectively. In theory, the CPT phenomenon could be understood in terms of two coherent linear combinations 1d ) of the ground states 1+) = (f1I1) + 1lR2I2))/1eff, I--) = (11R211) 1R1I2))/1eff,

_

wherefRk
d32,

--d3kEk/h(k = 1,2) are the Rabi frequenciesdetermined by the corresponding dipole moments d31, and d3k --e(3frIk),and 1eff \/[R1I2 + IR2I2 In the rotating wave approximation (RWA), the matrixelement of the dipole interactionoperatorfor transition I--) f3) is given by

(3IVdjpI--) = hclR1clR2e_i(ol+wl)t_il (1 -- eiÆRt+io), 21eff
where K Vd1

(2)

the difference of the field phases 1,2· At the Raman resonance condition, this yields 0. As a result, such stateabsorbs and emits no photons. For the latter reason it is called the coherent The process ofoptical pumpingin the presenceofincoherent relaxation processesleads to the trapping of theentirepopulationofthesystemin thedark stateand reveals as the CPT resonance in absorption or transparency. The coherent nature of the CPT determines, in accordance with Eq. (2), its dependence on the light phases 1,2 Therefore, phasefluctuations of the driving laser fields could decrease or even destroythe CPT-condition and stabilizationof the relevant phases of the laser fields could be required. Other decoherence processes and Doppler broadening could be destructive factors for the CPT, as well.
1

L I state.) dark

= =

2

1

5

148

Proc. SPIE Vol. 4749


3. EXPERIMENTALSETUP
In experiment, two external-cavity diode lasers (ECDL) were tuned in resonance with thetransition wavelengthsin samarium: 672 nm and 686 nm. In Ref. 9 spectraofsamarium transitions havebeenstudied in detail using methods ofsub-Doppler saturationspectroscopy. The relative isotopic shifts and hyperfine-structure splittingwere determined with an accuracy of 1--2 MHz. It has alsobeen shown that spectral lines of 154Sm isotope (natural abundance of 22.75 %) are shifted with respect to the spectral lines of other isotopes by more than 1 GHz to the red, allowing us to lock reliably the lasers to the transitionsof this isotope. Experimental setup we used for the dark resonances spectroscopyis shown in Fig. 2. The Sm vapor fills a 50 cm-long cell madeof stainless steel with glasswindowsat the cell ends. The cell could be evacuated or filled with a buffer gas as required. A few grams ofsamarium were placed in thecenterofthe cell. The 15-cm central partofthe cell was heatedby a coaxial dc-powered ('5OO W) heatingcoil. The residual magnetic field inside the cell was a few fractions of an Oersted. The cell was heatedto about 1000K in order to obtain a noticeable absorption. The laser at 672 nm was locked to the corresponding transition in Sm . In Ref. 9the frequency ofthistransition was measured tobe 14863.7305 0.0015cm1. The laserat 686 nmwas slowly scanned near the second transitionofthe A system, crossing the point where the Ramandetuningis zero. Variation in thelaseroutput frequency was monitored with a one-meter-longconfocal interferometer, whose free spectralrange was 75 0.1 MHz. The single frequency operation of both lasers was controlled with a spectrum analyzer with a finess of 50 and a free spectral range of 8 GHz. To prevent a feedback, the interferometer was optically isolated from the lasers. The linearly polarized beamsof both lasers were combined in a polarizing cubewith an accuracy of rad (Fig. 2). This beam illuminated the cell with samarium vapors. The beams transmitted through thecell
were separatedby a 2400 lines/mmholographic diffraction grating and entered a two-channel detection system. The laser beam at 686 nm was modulated in front of the cell at the frequency 600 Hz using a liquid-crystal modulator, and then separated the modulatedcomponent from the detectedsignalof the other beam. In this way, we determined the change ofthe absorption of one of the beams introduced by the presence ofthe second beam.

io

A (686 nm) B (672nm)
5

gas (He,Ar)

frequency

marks

Figure 2. Experimental setup: 1--ECDL 672 nm 2--ECDL 686 nm, 3--cavity (1 m), 4--modulator
5--diffractiongrating.

600 Hz,

Energy results and g-factors of the lowest meta-stablelevels with = 0, 1,2 andthe upper populations of the metastable levels at T = 6000 C (see Ref. 9).

4. FOUR-LEVEL MODEL OF SAMARIUM ATOM levels diagram for Sm atom used in our calculations to model experimental dark resonances spectroscopy is shown in Fig. 3a. It corresponds to the transitions (1). Tables 1 and 2 give oscillator strengths, energies,

J

J

=

1 level, including

the relative

Proc. SPIE Vol. 4749

149


(a)

4f6s6p

(b)

672

J=2
J=2
J,::=1

13)

4f6s2

J=o

11)

Figure 3. Energy levels diagram for samarium atom (a) and parametersused in our calculations (b). WL1 and wL2 are the laser frequencies, l14 are 24 are the corresponding Rabi frequencies, 8L and 5R are the detuning at the 1) Â-- transition and the Ramandetuning, 741, 'Y42, and 'y43 are the radiationdecay rates from the excited state to levels 1 ), 2), and 3); F14 and "24 arethe pure dephasing rates for the transitions 1 ) --* 4) and 2) -+ 4); 'yi, 'Y32 and w12, w23 are the decay and pumping rates of the level 1) via the level 2) and 2) via the level 3); F12 and F23 are the pure dephasing rate of the 1 ) 2) and 2) 3) transition.

I)

For comparison with experimental datawe needto specify some experimental parameters to be plugged into our model. In the experiments on coherent dark resonancesspectroscopy of samarium we used two semiconductor lasers tuned at 672 nm and 686 nm, respectively.10 Both lasers operate in a single-frequencymode, at the average power of WL1 2.5 mW and WL2 12 mW, respectively. At the cell, input power densities are WL1 0.1 mW/mm2 and WL2 0.2 mW/mm2. Field strengthsare given by formula /2W/cso, where W is the laser power density, that yields ELi 274 V/m and EL2 388 V/m. The magnetic field strength used in our experiments was about 20 Oe: 15 Oe for the longitudinal field and 19 Oe for the transverse one. For the longitudinal magnetic field, the corresponding Zeeman splittingL = egH/2mc is I_&, = 1.98 x 108 --1 for the level 6s6p and LV' = 4.09 x 108 for the level 652. For the transverse magnetic field, L' = 2.50 x 108 --1 for thelevel 6s6p and Li" 5.17 x 108 _1 for thelevel 62. The dipole moment d, the Rabi frequencies 1, and decay rates 'y are calculated as:

=

=

1
+

Idjj't2

(3he2/2m)(2J

1)fjji/jj,
Oscillator strength, 8.5 x 9.5 x

Table 1. Transition oscillator strengthsfor samarium.
1ansition (J 0) -- 6s6p 652 (J 1) -- 6s6p
652

(J = (J =

Wavelength, nm
1) 1)

672.5875 686.0927

i0

f

i0

Table 2. Energy levels in samarium, which contribute to the absorption spectrum.

--

J
0
1 2

Even levels4f66s2(7F)

Energy,

cm

Odd level 4f6(7F)6s6p(3P0)YF?

g

0.00 292.58 811.92

--
1.50

Relative population at_T = 6OO'_C
1.0

J
1

Energy,

cm

g

1.50

0.6 0.24

14863.85

3.10

150

Proc. SPIE Vol. 4749


cJJ,

= djjiE/h, tjjt =

44,w3/3hc3,

wherem and e are the electron mass and charge, respectively, c is the speed of light, and Wjj' is the transition frequency. Table 3 summarizes these values for samarium. Non-zero values of parameters x21, y32 are due to the time-of-flight broadening. The pumping rates Wi2, w23 are related to the rates 'y21, 'Y32 via the temperature equilibrium condition.

Table 3. Parametersof theA-system for our mode) of samarium.
Rabifrequency,
s--i

Radiation decay Incoherent pumping

rate,

s1

rate, s--
106
W12 W23

Decay rate of the ground state,
'Y21 'Y32

s1

14
1124

1.75 x iO 2.65 x i07 --

'Y41

1.25
1.34

Y42

x x

106

y43=1.25x1O6

2.4 1.6 --

x i04 x iO

4.0 x i04
4.0 --

x

i04

5. CALCULATION TECHNIQUE Temporal dynamics of a multilevel system driven by laser fields can be writtenwith the Liouville equation

=L:15,

(3)

where L is the system Liouvillian, which takes into account both reversible and relaxation dynamics. For an Nlevel atom in the presence of laser fields it is written as a sum of contributions ¸ = + Le + ¸c + where ¸5, and i:: are the contributions due to the radiation decay, pure dephasing, detuning of the laser fields, and laser--field interaction, respectively. As it is shown in Ref. [11], it is possible to combine the advantages of the symbolic and matrixrepresentations with the possibilities of analyticcomputation and then effectivelycalculate the N2 x N2-matrix representation for ¸, Lr, Le, and L2 evenfor largeN, making useof thefollowingformula

4, 4,

4

¸,

L,

Lmm

where is a properly chosen operator basis of the matrix representation.11 The spectrum calculation problem is significantlysimplified for a stationarycase, when it is only necessary to calculate the vector representation (01 of the stationary density matrix ,ÒS as a solution of the null-space matrix problem (O(L = 0. With this solution, it is possible to calculate any stationary average value of interest,i.e., absorption of the probe field. For numerical calculationsofthe atomic populations and absorption/dispersion aspecially designed Fortran-based program, which employsthe shortly described above calculation technique, havebeendeveloped. Key advantage of this program (and algorithm) is that it requires only minimum number of active parametersthat is significantly less thanthetotal number N4ofthe matrixelements in the LiouvilleEq. (3). Another advantage ofthe program is that it can be used (and has beentested) for any model of an atom with the number ofenergy levels N 20. For larger dimensionality,additionalefforts in testing the programare required.

e

6. COMPUTER SIMULATION RESULTS
The absorption
coefficient in our four-level system modeling Sm atom is shown in Fig. 4b. For comparison, the absorption coefficientin the three-level model is shown in Fig. 4a, as well. Analysis ofthe dependenciesof Fig. 4a,b shows that introducing of the fourth level = 2 into the three-level model leaves the resonance width almost unchanged, whereas the full absorption of the four-level system is about an order of magnitude smaller than the absorption in the three-level model. This is due to the population trapping at the = 2 level via the corresponding radiationdecay channel.

J

J

Proc. SPIE Vol. 4749

151


(a)
0.3·
·-------'::

--r=o ·

006

(b)

r'2=o.oi

I

b0207
5,MHz

F12,F2, 2

75

DO

2

0

25

50

75

8,MHz
at 5L

Figure 4. Absorption coefficient in thethree-level (a) and four-level (b) modelsversus Ramandetuning 5R and different dephasing rates (s1). Corresponding models are shown in Figs. 1, 3.

=

0

6.1. Modificationof the dark resonances in applied magnetic field In the presence of a magnetic field, the considered above model of Sm atom transforms into a seven-levelmodel due to the fact that levels with J = 1 split into three components each. As a result, splitting of the level 3) results in three allowed by the selection rules transitionsto the level 1 ), the probability of each being 1/3 of the total transitionprobability 3) --* 1). Similarly, splittingof the level 2) results in 6 allowed transitionsto the levels 2) and 13) with the probability 1/6 ofthe total I) 2) transition probability. For the followingconsiderationwe shoulddistinguish between two configurationsofthe applied magnetic field--
longitudinal and transverse configurations.

6.1.1. Longitudinalconfiguration

ofthe applied magnetic

field

Energy levels diagram for Sm atom,corresponding to the applied longitudinal magnetic field configuration,is shown in Fig. 5. For crossed linear polarizations of the driving systemlaser fields the selection rules allow six transitions becauseE1 and E2±H (Lsm2 = 1) 1) and 3) (Lm1 = 7) form two three-levelA-systems. Transitions 2) +-Â 6) and 4) Â- 6) are also allowedby the selection rules, but they are not activein the excitation ofthe mentioned A-systems. Depolarization due to the collisions could be an additional decay channel in a multilevel model, by contrastto thethree-levelmodel. A mechanismofthis depolarization is thecollision-inducedtransitionsbetween thestates with different values ofthe atom'smoment onto afixed direction. In the applied magnetic field,collisionsinduce transitions between the Zeeman sublevels with different values of magnetic moment for each multiplet. In our case, the CPT the resonance occurs at the coherent superposition of the levels 1), 3) (Fig. 5). For momentum changing collisions, coherence between levels 1) and (3) is destroyed because the population of the level 3) changes. The results ofthe absorption coefficient calculations for the complete 7-level model (Fig. 5) taking into account the magnetic sublevels depolarization, Gmagn , are shownin Fig. 6. As it can be seen from Fig. 6, the depolarization process leads to the increase of the induced absorption and, additionally, in monotonous decrease of the CPT-resonance contrast. The depolarization does not change the width of the CPT-resonance. Let us now find the conditions of the CPT-resonance for considered models. For both driving laser fields only transitionswith /.m = arepossible. Laser field with the frequency WL1 is in resonance with the atomsvelocity ofwhich on the k-vector of the laserbeamsatisfythe followingequation: projections

v=1

WL1

(w13

z")(1 +

v'1/c).

(4)

The CPT condition is satisfied only if the atoms are in the resonance with the second laser frequency,wL2:
WL2

(w23

z')(1 +

v'/c).

(5)

152

Proc. SPIE Vol. 4749


17)

6)
5)

14) ,

13)

2)

J=o
Figure 5. Energy levels diagram for Sm atom in thelongitudinal magnetic field for the crossed linear polarization and Lm2 = vectors ofthe driving laserfields. Theselection rules for the laserfields with WL1 andWL2 are Lm1 = L' and is" are the Zeeman splittings of the lower and upper levels with J 0, correspondingly. respectively. I \ --G 0 --- G =0.1 G=1 G=IO
0.15

Ii)

a)

G1

--::=

(b)

"

0.14

.
-1250 -625

c

.

0.12

0.10 0.08
0.06 0.04

0 SR,MHZ

625

1250

-1250

.625

0

625

1250

ÃMHZ
8L

Figure 6. Absorption coefficientof the seven-level system (Fig. 5) versus Raman detuning 6R at depolarization constant of the magnetic sublevels ranging from 0 to 10 (a) and from 1 to 80 (b).

=

0 and

that the CPT-resonance is achieved at the two frequencies of the probe laser, wL2 = which means that the CPT-resonance line is split at the frequency 2LV(W12/w13). Additional W23 Wl2(I."/Wl3), correspond to the simultaneous excitation of the absorptionlines at the frequencies z."), and two transitions that do not form the A-system. For example, transition at the frequencyw23 + 2L" corresponds to the resonance excitation of the transitions 1) Â- 6) and I) 7). The comparison of calculated absorptionspectrum in the 7-level model with the experimental spectrum of Sm atoms vapor is shown in Fig. 7. Shown spectra are the probe laser field (672 nm) absorption spectra, received by scanning the second driving laser field frequency. Fig. 7 show good correspondence between the experimental and theoretical spectra. Fitting oftheoretical and experimental data shows the best results at Gmagn 0, which reveals a negligibly small interplay of the depolarization at the OPT in Sm atoms. Typical width of the experimentally observed CPTresonances is on the order of 5--6 MHz, which corresponds well with our theoretical calculations.

It

follows from Eqs. (4)--(5)

6.1.2. 'Iansverse configuration ofthe applied magnetic field Energy levels diagram for Sm atom, corresponding to the applied transverse magnetic field configuration, is shown

in Fig.

8.

Proc. SPIE Vol. 4749

153


0.8 0.6

(a)
.. 0
400

0.06

(b)

C

0.4 0.2 0.0 200
R12

0.03 0.02 0.01

0

200

400

-1250-1000 -750 -500 -250

0

,

Mhz

6R'

lv'

250 500 750 1000 1250

7. a) Experimental absorption spectrum of Sm vapor versus Raman detuning 8R in the presence of a longitudinal magnetic filed. The magnetic field strength is 19 Oe and the buffer gas (Ar) pressure--O.2 Torr. b) Calculated absorption spectrum of Sm atom in the seven-levelmodel (Fig. 5) versus Raman detuning 5R at = 0 and the magnetic sublevels depolarization constant Gmagn = 0. 7)

Figure

L

16)

J=

1

15) Am2

-1

Am1=

Ii)

J=0

Figure 8. Energy levels diagram for Sm atom in thetransverse magnetic field for the crossed linear polarization vectors ofthe driving laserfields. The selection rules for the laserfields with WL1 and WL2 are Lmi = 0 and Lm2 = respectively. L' and is" are the Zeeman splittings of the lowerand upper levels with J 0, correspondingly. In a transverse magnetic field H1, a linear polarization of the laser field with WL1 lies in the polarization plane) mayinduceonly the transitionswith Lm = 0, which are ir-components. At the same time, the laserfield with which are cr-components. WL2 with the polarization planeorthogonal to H1 induces the transitions with iim = In this case, two A-systems, corresponding to the transitions 1 ) --+ 6), 2) --+ 6) and Ii ) 6), are 6), excited, whereas the transitions 3) IS ) and 3) 7) are not active. Theinfluence of thedepolarization ofthe magnetic sublevels reveals in the absorption coefficientand the parameters of the CPT-resonance similarly to the caseof longitudinal magnetic field (Sec. 6.1.1). The calculation results for different depolarization constantvalues are shown in Fig. 9. The maximum of the resonance contrastis achieved at Gmagn 0. It can also be easily seen that increasing Gmagn reduces the CPT-resonance contrast, whereas its width remainsalmost the same. Similarly to the case of longitudinal magnetic field, we can calculate positions of the CPT lines in theDoppler broaden spectrumfor a transverse magnetic field. This spectrumyields alsoa splitted CPT-resonance line and the splitting value coincides with the Zeeman splitting of the sublevels 2) and 4) of the lower level = 0: Lw 2L'. The splitting ratio of the CPT-resonances for longitudinal and transverse magnetic

(1

I)

J

=

154

Proc. SPIE Vol. 4749


0.14

--G --
"G ---G
,

=0
=0.05

0.12 0.10

-- --G

=1

=5
=20

G

0

0.08 0.06

0.04
0.02

-1250-1000-750 -500 -250

0

250 500 750 1000 1250

MHz Figure 9. Absorption coefficient of the seven-level system (Fig. 8) versus Raman detuning 5R depolarization constant of the magnetic sublevelsranging from 0 to 80.
at 5L

=0

and

'
.

0.8

(a)

0.05

J1

(b)

0.6

e

Â0.4

.
-

d

0.04

0.03

,0.2 0.0 400

0.02 .
200

.
0

-

.

-

.
400

001
-1250 -1000 -750 -500 -250 0 250 500

200

750 1000 1250

J2 ,MHz

6R'

MHz

10. Figure a) Experimental absorption spectrum of Sm vapor versus Raman detuning 8R in the presence of a transverse magnetic filed. The magnetic field strength is 19 Oe and the buffer gas (Ar ) pressure--O.2 Torr. b) Calculated absorption spectrumof Sm atom in the seven-levelmodel (Fig. 5) versus Raman detuningC5R at 5L = 0 and the magnetic sublevels depolarization constant Gmagn 0. fields reads

II (Y
-- --

7. CONCLUSIONS In conclusion, we present a theoretical model for the coherent dark resonances spectroscopy for a multilevel atom, which takes into account all the coherences and decay rates in the system, as well as possible pumping rates and

)

(w12

wi3)

-- --

H1
H11

driving system laser fields. Such a model allows to fit any real multilevel atom. The model was adjusted for the samarium atomand the comparison with our experimental spectroscopicdatais presented. It isshown that multilevel energy structure of Sm atom without external magnetic field can be well fit in a simple four-levelmodel, whereas 7and 12-levelmodels work well for the cases of applied longitudinal and transverse magnetic fields, respectively.

ACKNOWLEDGMENTS
Authors from M. V. Lomonosov Moscow State University acknowledgepartial support from the RFBR grant no. 01--02--16311, the State science-technicalprograms of the RussianFederation "Fundamental Metrology" and "Nano-

Proc. SPIE Vol. 4749

155


technology", and INTAS grant no. support from the RFBR grantsno.
1/73647.

00--479,

O1--02--174--42, 01--02--174--39,

and authors from P. N. Lebedev Physics Institute acknowledgepartial and 00--15--96--586 and Volkswagen-Stiftunggrant

REFERENCES
1. C. 2. H. 3. G. 4. A. 5. A. Aizetta, A. Gozzini,L. Moi, G. Orriols, Nuovo Cimento. B. 36, 5 (1976). R. Gray, R. M. Whitly, and C. R. Stroud (Jr), Opt. Lett. 3, 218 (1978). Aizetta, L. Moi, G. Orriols, Nuovo Cimento. B. 52, 209 (1979); Opt. Commun. 42, 335 (1982). Aspect, E. Arimondo, R. Kaiser,N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1996). Kasapi, Phys. Rev. Lett. 77, 3908 (1997). E. Arimondo, "Coherent population trappingin laserspectroscopy", In: Progress in Optics, 35, 257 (1996), B. Wolf, Ed. (Elsevier, Amsterdam). R. Wynands, A. Nagel, Appi. Phys. B 68, 1 (1999). R. Holtzwarth, Th. Udem, and T. W. Haensh, Phys. Rev. Lett. 85, 2264 (2000). N.N. Kolachevskii, A.V. Akimov, N.A. Kiselev, A.A. Papchenko, V.N. Sorokin, SI. Kanorski, Optics and Spectroscopy,90(2), 164 (2001). N.N. Kolachevski, A.V. Akimov, N.A. Kiselev, A.A. Papchenko, V.N. Sorokin, SI. Kanorski, QuantumElectronics, 31(1), 61(2001). B. A. Grishanin, Quantum Stochastic Processes, located at http://comsiml.phys.msu.su/index.html (in Russian).

6.
7.

8. 9.
10.
11.

156

Proc. SPIE Vol. 4749