Steep dispersion in coherent population trapping with losses

15 May 2000

Optics Communications 178 Ž2000. 345–353 www.elsevier.comrlocateroptcom

Steep dispersion in coherent population trapping with losses F. Renzoni a,) , E. Arimondo b b

a Institut fur ¨ Laser-Physik, UniÕersitat ¨ Hamburg, Jungiusstrasse 9, D-20355 Hamburg, Germany Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica, UniÕersita` di Pisa, I-56126 Pisa, Italy

Received 27 November 1999; received in revised form 22 February 2000; accepted 1 April 2000

Abstract The dispersive properties of coherent population trapping with losses are investigated. We consider a cold sample of open three-level atoms interacting with two laser fields in the L configuration. We show that despite the population losses the slope of the dispersion at Raman resonance reaches at large interaction time a constant nonzero value. A numerical study of the propagation of a pulse in the medium in the presence of a strong coupling field shows that very small group velocities for the pulse are obtained also in the presence of population losses. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.Gy; 32.80.-t

1. Introduction Electromagnetic-induced transparency, laser without inversion, pulse matching, reduction in group velocity are quantum optics phenomena based on the coherences created by the laser radiation within a multilevel atomic Žor molecular. system. All those quantum optics phenomena have been studied extensively in the last decade w1,2x. The simplest multilevel configuration where these phenomena take place is a three level system, with a central level connected by electric dipole transitions to the two remaining ones. One transition is driven by a strong laser, denoted as coupling laser, and the second transition is driven by a weak laser, the probe one. The modifications in populations and coherences produced by the coupling laser modify the absorption and dispersion of the probe laser.

)

Corresponding author. Fax: 49 40 42838-6571.

Strong reductions of the atomic absorptions for a probe laser as result of quantum interference have been observed already in the late sixties w3x. For a three-level atomic system interacting with two laser fields in the L configuration, the absorption reduction is produced by the coherent trapping of the population into the ground state Žcoherent population trapping – CPT w4x.. More recently the possibility of realizing large modification in the light group velocity through a very steep dispersion associated to coherent population trapping narrow resonances has attracted a large experimental attention w5–9x and remarkable results for the reduction of the group velocity have been very recently reported w10–12x. To obtain very narrow atomic resonances and steep dispersion response, some limiting factors must be overcome. If the two ground states composing the dark coherent superposition have different energy, as for different hyperfine levels, the dark resonance is not completely Doppler free and for a thermal vapor a residual Doppler broadening is introduced. More-

0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 7 6 - 3

346

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

over owing to the finite interaction time of the atoms with the laser fields, a time-of-flight broadening of the dark resonance is introduced. Different experimental configurations have been used to reduce the above mentioned limiting factors. The use of a buffer gas allows the reduction of the residual Doppler broadening through the Dicke effect and leads simultaneously to an increase of the interaction time w5–7x. Very long interactions times have been realized also with cells containing antirelaxation paraffin coating in which the coherences survive tens of thousand of collisions with the walls of the cell w12x. Alternatively, in a cold atomic sample there is no residual Doppler broadening and long interaction times can be easily achieved. By following these procedures, the response of the atomic sample is reduced to the single-atom one. Often the real atomic transitions of quantum optics phenomena do not correspond to the simple theoretical models, and few authors have analyzed the experimental results taking into account the complexity of the atomic structure. Several multilevel atomic structures can be described as open systems, i.e., systems where some population is lost because of decay into levels not excited by the lasers. For instance in a L system, the presence of a low-energy sink state connected by electric dipole transitions to the excited state implies an accumulation of the population in that sink state and a decrease in the atomic coherence formation. The sink state may originate from the Zeeman structure, with Zeeman sublevels not excited by the polarized laser radiation, or by the hyperfine structure of the ground state. In the experimental investigations the open feature of a multilevel system may be eliminated through the application of a repumping radiation to empty the sink state, as investigated in w13x. Since the majority of transition schemes considered in laser spectroscopy are open ones, a number of investigations has been devoted to the study of different features of coherent population trapping with losses w13–19x. In particular it has been shown that at long interaction times the evolution of a closed and open system differ substantially w14–16x. For small detunings of the laser fields, a closed three-level L system reaches the steady state in a time scale at least equal to the excited state lifetime, and shorter increasing the Rabi frequencies of the

optical transitions in the three-level system. For an open system the time scales are completely different. At large Rabi frequencies and for small Raman detuning d R the atomic population leaks out of the system at a rate proportional to d R2 , so that the time scale gets very large at small Raman detuning. At large interaction time the quadratic dependence of the effective rate of population loss on the Raman detuning results in a law for the width of the dark resonance inversely proportional to the square root of the interaction time t, at variance with the law for closed systems where the width of the dark resonance approaches its power broadening limit with a 1rt dependence. However previous work did not examine whether in an open three-level L system the population losses modify the dispersive properties of a medium under coherent population trapping condition: that aim is the goal of the present work. Section 2 introduces the optical Bloch equations ŽOBE. of a cold sample of open three-level atoms and reports results for their numerical solution. For a closed three-level system the effect of a finite interaction time may be taken into account either directly by considering the time-dependent solution of the optical Bloch equations or by including in the steady-state solution a relaxation of the ground state coherence inversely proportional to the interaction time w20,14x. The latter approach is usually preferred. On the other hand, for an open system such an approach cannot be used and the time-dependent solution of the OBE represents the only possible strategy. Therefore we do not include in the OBE any time-of-flight relaxation term. We consider the time-dependent solution of the OBE instead. Of course, if other sources of relaxation are present Žcollisions or cross-fluctuations of the laser fields. appropriate relaxation terms have to be included. In the following, an analytic expression for the optical coherences is derived in order to show explicitly that the slope of the dispersion at Raman resonance remains finite at large interaction time despite the population losses. Thus in Section 3 an adiabatic elimination of the excited state variables is applied and the solution of the OBE is formally expressed through the eigenvalues and eigenvectors of a real matrix. In Section 4 we consider the dispersive properties of the medium at long interaction times. In Section 5 we examine the propagation of a laser

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

pulse through a cold sample of open three-level atoms in the presence of another laser field preparing the atomic system. A conclusion completes our work.

The dynamics of the atom, i.e. the evolution of the atomic density matrix r , is described by the optical Bloch equations: d dt

2. Optical Bloch equations We consider a cold sample of open three-level atoms interacting with two laser fields in the L configuration. The interaction scheme is shown in Fig. 1. The ground Žor metastable. states
rs

De a Ea "

da s va y

i"

w H0 q VAL , r x q

VAL s y

" Vc 2

ž / Et

,

Ž 1.

" Vp 2

eyi v p t
Ž 2.

as

Gout Gc q Gp

,

Ž 7.

the fractional loss rate towards other states. As explained in the Introduction we do not include in the OBE any time-of-flight relaxation term. For atoms at rest Žcold atomic sample., and neglecting the photon recoil, the OBE read:

r˙cc s Gc ree y Im V c)r˜ec ,

Ž 8a .

r˙pp s Gp ree y Im V p)r˜ep ,

Ž 8b .

r˙ee s yGexc ree q Im V c)r˜ec q Im V p)r˜ep ,

Ž 8c .

r˙˜cp s yi d R r˜cp q i r˜˙ec s y Fig. 1. Open three-level atomic system interacting with two laser fields in the L configuration.

Ž 6.

with

Ee y Ea

, Ž 3. " where De a is the atomic dipole moment, supposed to be real for the sake of simplicity.

Ž 5.

We consider here only the relaxation due to the spontaneous emission ŽS.E.. and we assume spontaneous rates Gc and Gp for the decays from the excited state
Gexc s Gc q Gp q Gout s Ž Gc q Gp . Ž 1 q a . , ,

Ž 4.

S .E .

eyi v c t
=²c < y

of frequencies vc , v p , electric field amplitudes Ec , Ep , and wavectors k a s varc, with Ž a s c,p. and c being light speed in vacuum. As standard in laser without inversion and similar processes w1,2x, the laser fields are indicated as coupling Žc. and probe Žp. lasers. The laser Rabi frequencies and detunings are indicated with Va and da respectively:

Va s

Er

1

where H0 is the Hamiltonian of the unperturbed atom and VAL describes the interaction atom-light. In the rotating wave approximation ŽRWA. the interaction Hamiltonian for the system under consideration is:

E Ž z ,t . s 12 Ec e c eyi Ž v c tyk c z . q 12 Ep e p eyiŽ v p tyk p z . q c.c.

347

qi

Gexc 2

Vp 2

V c) 2

r˜ep y i

y i dc r˜ec y i

r˜cp) ,

Vp

Vc 2

2

r˜ec) ,

Ž 8d .

Ž ree y rcc . Ž 8e .

348

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

Gexc

r˙˜ep s y

2 Vc

y i dp r˜ep y i

Vp 2

Ž ree y rpp .

r˜ . Ž 8f . 2 cp In these equations d R is the Raman detuning from the two-photon resonance: qi

d R s dc y d p .

Ž 9.

Moreover to eliminate the oscillation at the laser frequencies in the evolution equations, we have introduced the density matrix elements

r˜e a s re a e i va t

Ž a s c,p . ,

yi Ž v cyv p .t

r˜cp s rcp e

.

Ž 10 . Ž 11 .

We numerically solved the density-matrix equations, Eqs. Ž8. for an open system. Results for the real and imaginary part of the optical coherence r˜ep at different interaction times u are shown in Fig. 2. At short interaction times, as in Fig. 2Ža. and Žb., a transparency window builds up in the absorption spectrum and correspondingly the dispersion develops a steepness around Raman resonance. This behavior is quite similar to what valid for a closed system. On the other hand, while a closed system reaches a steady-state in a few excited state lifetimes, the time scale for the dynamics of an open system is much longer, as clearly illustrated in Fig. 2Žc. and Žd.. From the results of those figures it appears that the time-dependent absorption decreases because the population is pumped out of the three-level system. Being the effective loss rate an increasing function of the Raman detuning, the transparency window narrows for increasing interaction time. However the slope of the dispersion at the origin does not change significantly for increasing interaction time, even if the interval of sharp variation of the dispersion decreases with u . This behavior arises because around d R s 0 the population is trapped in the nonabsorbing state up to an interaction time dependent on d R w14x. 3. Adiabatic elimination of the excited state Fig. 2. Real parts, in Ža. and Žc., and imaginary parts, in Žb. and Žd., of the optical coherence r˜ep as a function of dp at different interaction time u . The plotted curves represent solutions of the OBE for a open three-level L system with a s1, i.e., Gc s Gp s Gexc r4. The interaction times are marked. Parameters of the calculations are: V c s Gc , V p s 0.2 Gp , dc s 0.

In the limit of weak laser fields Ž< V c <, < V p < < < Gexc y i da <, a s c,p. the excited state can be adiabatically eliminated. In this and in the following Section we will assume, without loss of generality, the Rabi frequencies to be real. Adiabatically eliminating in

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

Eqs. Ž8. the optical coherences and neglecting the terms Va ree we obtain i Vc

r˜ec Ž t . s

Gexc y 2i dc i Vp

r˜ep Ž t . s

Gexc y 2i dp

rcc q

i Vp

Gexc y 2i dc i Vc

rpp q

Gexc y 2i dp

r˜cp) ,

Ž 12 .

ree s

Dc q

rcc q

V p2

r˜cp .

Dp

1

rpp q V c V p

2 Vc Vp

dc

Ž 13 .

Gexc

Dc

1 q

Dc

Dp

Im r˜cp .

Dp

Ž 15 .

After the adiabatic eliminations, the equations for the ground state populations and coherence are

r˙cc s y

Dc

y

rcc q

Gexc y Gc

q Vc Vp y

q

Dc

Dp

Gc

Ž Gexc y Gc . dc

Gexc

Dc

rpp

Dc

rcc y

q Vc Vp

q

Gc dp Dp

q

Ž Gexc y Gp . V p2

Gp

Dp y

Im r˜cp ,

Dp

2 V c V p dc Gp

Gexc

rpp

Ž Gexc y Gp .

Dc

Dc

q

Dc y dR y

Gexc 2

ž

2 Dp

V c2

q

Dc

Dp

dc V p2

q

dp V c2

Dc

rcc y

Dp

V c V p dp

dc V p2

Dp q

q

Dc

Dp

V c2 Dp

/

Re rcp

Ž Gexc y Gp . dp Dp

/

Im r˜cp ,

Ž 16c.

rpp

dp V c2

Dc

V p2

rpp

Re r˜cp

Re r˜cp

Im r˜cp .

Ž 16d.

For a closed system a s Gout s 0 Ž Gc q Gp s G . the steady-state solution of Eqs. Ž16. can be obtained, and previous results of other authors recovered. For instance substituting the steady state of Eqs. Ž16. in Eq. Ž13. we find at the first order in V p y2 d R V p rep , , Ž 17 . G 4d R d p q i q V c2 2

ž

/

r s Ž rcc , rpp ,Re r˜cp ,Im r˜cp . ,

Ž 16a. r˙pp s

V c V p dc

V c V p Gexc

rcc y

which coincides with the result of Refs. w8,2x Žapart differences linked to the definition of the Rabi frequencies.. In the general case, introducing the vector

Re r˜cp

Dp

2 Vc Vp

V c2Gp

Im r˜˙cp s

Ž 14 .

2 Da s Gexc q 4da2 .

Gc V p2

Gexc V p2

ž

Re r˜cp

Here we shortened the notations by introducing the quantities Ž a s c,p.

Ž Gexc y Gc . V c2

2 Dc

q dR y

y

dp

y

y

V c V p Gexc

2

Adiabatically eliminating the excited state population and substituting the previous two expression in the corresponding equation Ž8c., we get

V c2

Re r˙˜cp s y

349

Eqs. Ž16. can be rewritten as

r˙ s M r

Ž 19 .

and the problem is reduced to the calculation of the eigenvalues and eigenvectors of the real Bloch matrix M. Being the matrix M nonsymmetric, for each eigenvalue l j the corresponding left Ž wj . and right Ž zj . eigenvectors are in general different. The solution of Eqs. Ž16. is written as

r Ž t . s Ý c j zj exp Ž l j t . Im r˜cp ,

Ž 16b.

Ž 18 .

Ž 20 .

with the coefficients c j determined by the density matrix at t s 0 cj s

wj P r Ž 0 . wj P z j

.

Ž 21 .

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

350

Let us consider the case of Raman resonance: dc s dp , assuming for the sake of simplicity not only two-photon resonance, but also one-photon resonance: dc s dp s 0. In this case the eigenvalues of the matrix M are

lŽ0. 1 s0 , lŽ0. 2,3 s y

lŽ0. 4 sy

Ž 22a. V c2 q V p2 2 Gexc

,

Ž 22b.

Ž Gexc y Gc . V c2 q Ž Gexc y Gp . V p2 2 Gexc

.

Ž 22c.

In the same way, exact expressions for the eigenvectors can be easily found. Their expressions will not, however, be reported here, because not particularly illuminating. Only the eigenvector corresponding to the zero eigenvalue z1Ž0. s Ž V p2 , V c2 ,y V c V p ,0 .

1

Ž V p ei v t < c: y Vc ei v t < p:. . c

(

V c2 q V p2

lŽ2. 1

p

Ž 24 .

4. Dispersion at long interaction time

™`

t

r Ž t . , c l z l exp Ž ll t . ,

Ž 25 .

where ll is the lowest eigenvalue, i.e. that closest to zero, and z l and c l the corresponding eigenvector and coefficient. As presented in w15x, for an open system the occupation of the ground states
2

Ž V c2 q V p2 . w Ž Gexc y Gc . V c2 q Ž Gexc y Gp . V p2 x 'yCdp2 .

dp2

Ž 26 . lŽ2. l ,

Substituting in Eqs. Ž25. and Ž21. at the second order in dp , and the corresponding eigenvector z1Ž1. at the first order in dp , we find that for an atomic sample initially at the thermal equilibrium Ž rcp Ž0. s 0. the long-interaction-time real part of the optical coherence r˜ep is t

™`

2 dp V c2V p

Re r˜ep ,

2

Ž V c2 q V p2 . =  V c2 Gp rcc Ž 0 . q Ž Gexc y Gc . rpp Ž 0 . qV p2 Ž Gexc y Gp . rcc Ž 0 . q Gc rpp Ž 0 . 4

½

Ž Gexc y Gc . V c2 q Ž Gexc y Gp . V p2

=exp Ž yCdp2 t . .

5 Ž 27 .

Let us now specify the region of validity of the expression Ž27.. The expression Ž25., and correspondingly Ž27., is a good approximation of the general solution Ž20. for interaction times t so large that the modes corresponding to l2 , l 3 and l4 are damped, i.e. for < l j < P t 4 1 Ž j s 2,3,4.. For small detunings, we simply substitute for l j Ž j s 2 y 4. their values at Raman resonance, and the region of validity of the expression Ž27. is given by

V c2 In the long interaction time limit, the solution Ž20. is well approximated by the single term

4V c2V p2 Gout Gexc

sy

Ž 23 .

has a immediate physical interpretation: it corresponds to the stability of the noncoupled state at Raman resonance
vector. The lowest nonzero order of the eigenvalue is the second one

Gexc

Pt41 .

Ž 28 .

Inspection of expression Ž27. allows us to verify that for the long interaction times, defined by Eq. Ž28., the slope of Re r˜ep at the origin reaches a constant nonzero value, but the frequency interval of sharp variation of the dispersion narrows without limit. 5. Pulse propagation For the propagation of laser pulses through a cold sample of open three-level atoms, we consider in Eq. Ž1. the amplitudes Ep Ž z,t . and Ec Ž z,t . dependent on the time t and the coordinate z along the propaga-

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

™

tion direction. To simplify, we take the coupling field resonant with the
™

1 E c Et

E V p Ž z ,t . q

Ez

V p Ž z ,t . s i k p r˜ep Ž z ,t . ,

Ž 29 .

where the parameter k p is given by

kp s

v p N Dep2

Gexc kp

has been derived from the numerical solution. Moreover, the pulse delay td Ž z . at the spatial position z defined as q`

Hy` t < V Ž z ,t . < p

td Ž z . s

q`

Hy`

2

dt

Ž 35 .

< V p Ž z ,t . < 2 dt

has been calculated. The average centrovelocity Õc , introduced in Ref. w22x and reducing to the group velocity for a quasi-monochromatic pulse, as function of the pulse delay is given by c Õg s . Ž 36 . ctd Ž z . 1q z

Ž 30 .

ce 0 "

with N the atomic density e 0 the vacuum susceptibility. The weak probe Beer’s penetration depth Zp is Zp s

351

.

Numerical results for the evolution of the pulse in the medium of open three levels atoms are show in Fig. 3Ža.. We used the 87 Rb D 2 line parameters

Ž 31 .

In the local coordinate Ž z ,t . s Ž z,t y zrc . the propagation equation is

E Ez

V p Ž z ,t . s i k p r˜ep Ž z ,t . .

Ž 32 .

For a slowly-varying probe amplitude the equations for the density matrix elements depending on space and time through the variables Ž z ,t . represent an obvious generalization of Eqs. Ž8.. These equations and Eq. Ž32. have been numerically solved for an initially Gaussian probe pulse

V p Ž z s 0,t . s

V0 TGexc'2p

yt 2

exp

ž / 2T 2

.

Ž 33 .

with the atoms being initially prepared in the ground
< V p Ž z ,t . < 2 < V p Ž 0,0 . < 2

Ž 34 .

Fig. 3. Normalized probe intensity Ip Ž z ,t . plotted as a function of the retarded time t for different penetration distances: z rZp s 0,1,2,3,4. In Ža. results for an open system Ž Gout s Gexc r2, Gc s Gp s Gexc r4. are shown; in Žb. results for a closed system Ž Gout s 0, Gc s Gp s Gexc r2. are reported for comparison. In both case the pulse peak height decreases with the penetration distance. For the initial Gaussian pulse V 0 s Gp r10 and T s150r Gexc . The pump Rabi frequency is V c s Gc r2.

352

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

Ž Gexc s 2p P 5.9 MHz, l s 780.1 nm. and a cold atoms density of 4 = 10 10 atomsrcm3. In Fig. 3Žb. results for a closed three-level system are reported for comparison. Notice that in both cases the pulse peak height decreases with the penetration distance, and a reduction of the probe-light centrovelocity is produced. We calculated from the data of Fig. 3Ža. for the open system that the average centrovelocity at the maximum penetration distance considered is about 240 mrs. In order to compare these results quantitatiÕely with the known behavior associated with closed system, the atomic parameters for the closed system should be chosen carefully. We may introduce a correspondence between closed and open systems by means of a repumper laser, if we neglect variation in the relaxation rates of the optical coherences. For a given open system the application of a repumper laser eliminates the decay channel out of the system, a s 0 in Eq. Ž7. w13x, with the decay rate from the excited state diminished accordingly. Then for given Gc and Gp an open system characterized by the relaxation rates Gexc s Ž Gc q Gp .Ž1 q a ., corX responds to a closed system characterized by Gexc s Gc q Gp . Data for that corresponding closed system are reported in Fig. 3Žb.. The comparison of Fig. 3Ža. with Fig. 3Žb. shows clearly that the probe-light centrovelocity is of comparable magnitude for the two cases and in fact we calculated an average centrovelocity associated to the numerical results of Fig. 3Žb. of about 220 mrs. We verified that the reduction of the probe-light centrovelocity does not depend significantly on the

Fig. 4. Light-probe centrovelocity for a gaussian pulse of width T at the penetration distance z rZp s 4 as a function of T. A medium of open three-level system is considered Ž Gout s Gexc r2, Gc s Gp s Gexc r4.. Parameter of the calculation are: V 0 s Gp r10, V c s Gc r2.

pulse length. Results of our numerical calculations are shown in Fig. 4. The weak dependence of the probe-light centrovelocity on the pulse length is a confirmation that the slope of the dispersion reaches a constant nonzero value at large interaction time. 6. Conclusions We have examined the dispersive properties of coherent population trapping with losses. We calculated the optical coherence for the probed transition and shown that despite the population losses the slope of the dispersion at Raman resonance reaches at long interaction times a constant nonzero value. We have also examined the propagation of a pulse through a medium composed of open three-level atoms in the presence of a strong coupling laser. Our numerical results show that a very effective reduction of the probe-light centrovelocity is obtained also in presence of population losses. In fact the comparison with a closed system created by applying a repumper laser to the open system, shows that the centrovelocity of the pulse propagating through the medium depends very weakly on the presence of a population loss. However for the parameters of our calculation the losses towards the external state decrease the amplitude of the transmitted probe pulse more than in the corresponding closed system. In future work it remains to be verified if by choosing different parameters for the atom-laser interaction, gigantic reductions in the group velocities, similar to those realized in Refs. w10–12x, can be achieved also in open systems. These results are promising for the practical implementations of optical devices based on the dispersive properties of coherent population trapping, similar to the magnetometer of Ref. w23x. In fact such implementations are not restricted to closed atomic systems or to simple transition schemes for which the population losses can be compensated by a repumper, and the use of highly degenerate systems, as molecular or semiconductor media, seems possible. Acknowledgements F.R. is grateful to A. Lindner for continuous support and encouragement, and to S. Stenholm for discussions and for hospitality. F.R. thanks the

F. Renzoni, E. Arimondor Optics Communications 178 (2000) 345–353

Deutsche Forschungsgemeinschaft for financial support Žproject Li 417r4-1. and the foundation C.M. Lerici for a grant to visit the Royal Institut of Technology of Stockholm.

References w1x For a review S.E. Harris, Physics Today 50 Ž1997. 36. w2x See also M.O. Scully, M.S. Zubairy, Quantum Optics, University Press, Cambridge, 1997. w3x T. Hansch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, ¨ Z. Physik 226 Ž1969. 293; T. Hansch, P. Toschek, Z. Physik ¨ 236 Ž1970. 213. w4x For a review see E. Arimondo, in: E. Wolf ŽEd.., Progress in Optics, vol. 35, Elsevier, Amsterdam, 1996, p. 257. w5x S. Brandt, A. Nagel, R. Wynands, D. Meschede, Phys. Rev. A 56 Ž1997. R1063. w6x S. Brattke, U. Kallmann, W.-D. Hartmann, Eur. Phys. J. D 3 Ž1998. 159. w7x A. Godone, F. Levi, J. Vanier, Phys. Rev. A 59 Ž1999. R12. w8x Y.-Q. Li, M. Xiao, Phys. Rev. A 51 Ž1995. R2703. w9x O. Schmidt, R. Wynands, Z. Hussein, D. Meschede, Phys. Rev. A 53 Ž1996. R27.

353

w10x L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature 397 Ž1999. 594. w11x M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett. 82 Ž1999. 5229. w12x D. Budker, D.F. Kimball, S.M. Rochester, V.V. Yashchuk, Phys. Rev. Lett. 83 Ž1999. 1767. w13x F. Renzoni, W. Maichen, L. Windholz, E. Arimondo, Phys. Rev. A 55 Ž1997. 3710. w14x F. Renzoni, E. Arimondo, Phys. Rev. A 58 Ž1998. 4717. w15x F. Renzoni, A. Lindner, E. Arimondo, Phys. Rev. A 60 Ž1999. 450. w16x F. Renzoni, E. Arimondo, Europhys. Lett. 46 Ž1999. 716. w17x V. Milner, B.M. Chernobrood, Y. Prior, Europhys. Lett. 34 Ž1996. 557. w18x V. Milner, Y. Prior, Phys. Rev. Lett. 80 Ž1998. 940. w19x Y. Zhu, S. Wang, N.M. Mulchan, Phys. Rev. A 59 Ž1999. 4005. w20x E. Arimondo, Phys. Rev. A 54 Ž1996. 2216. w21x M. Sargent III, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley, Reading, 1974; J.H. Eberly, M.L. Pons, H.R. Haq, Phys. Rev. Lett. 72 Ž1994. 56; R. Grobe, F.T. Hioe, J.H. Eberly, Phys. Rev. Lett. 73 Ž1994. 3183. w22x R.L. Smith, Am. J. Phys. 38 Ž1976. 978. w23x A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, R. Wynands, Europh. Lett. 44 Ž1998. 31.