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Effect of four-photon interactions on coherent population trapping in
B. A. Grishanin, V. N. Zadkov
International Laser Center, M. V. Lomonosov Moscow State University, 119899 Moscow, Russia

-systems

D. Meschede
Ё Ё Institut fur Angewandte Physik der Universitat Bonn, Wegelerstr. 8, D-53115 Bonn, Germany

Submitted 26 June 1997 ґ Zh. Eksp. Teor. Fiz. 113, 144 ­ 167 January 1998 The resonance fluorescence spectrum of a -system excited by two resonant light fields is calculated using a Markov analysis. Analytical formulas are derived in the strong-field limit within and beyond the rotating wave approximation. It is shown that the resonance fluorescence of the system does not vanish during coherent population trapping. Its spectrum consists of two multiplets which are similar to a triplet in the resonance fluorescence spectrum of a twolevel atom and lie at the electronic transition frequencies, together with two triplets located at the frequencies of four-photon processes involving the optical excitation fields. The latter are fundamental in character and impose limits on the lower bound of the dephasing rate for the Raman resonance owing to the effect of radiative decay of the dipole transitions on the dynamics of the ground state. The effect of four-photon dephasing on the absorption spectrum of a -system is analyzed and found to lead to a substantial reduction in the depth of a dip in the absorption spectrum which vanishes as the laser field strength is increased. © 1998 American Institute of Physics. S1063-7761 98 01001-4

1. INTRODUCTION

The interaction of electromagnetic fields with atoms is one of the most fundamental problems in quantum optics. It is known that a much wider range of effects occur in multilevel atoms than in two-level atoms owing to field-induced coherence between the atomic states and quantum interference. The three-level systems realized in -, -, and Vconfigurations play an important role in research on these effects, as they are of intermediate complexity between twolevel and multilevel atoms. A whole series of new effects have been observed in them; coherent population trapping is one of the most intriguing and has been studied intensely, both experimentally and theoretically. See the reviews by Agap'ev et al.1 and Arimondo2 and the references cited there. Coherent populating trapping shows up most clearly in a three-level system with two close long-lived levels and a third level which lies far from them - or V-systems that have been excited by two cw laser fields, so that the distant level is optically coupled to the two others. Tuning the driver fields to resonance with its dipole transitions leads to trapping of the populations of the system in a coherent superposition of the two close levels. In Raman absorption spectra this effect shows up as a very narrow dip against the background of an absorption line and in resonance fluorescence spectra it is observed as the absence of emission, which has led to its being referred to as a ``dark'' or ``coherent population trapping'' resonance. In this article we study the question of how four-photon interactions affect the coherent population trapping effect in a -system excited by two cw laser fields, in particular, how ``dark'' the coherent population trapping resonance in the
79 JETP 86 (1), January 1998

resonance fluorescence spectrum is and how deep the dip in the Raman absorption spectrum is. A rough estimate of the intensity of resonance fluorescence in a -system during coherent population trapping has been made in the rotating wave approximation3 which yields zero fluorescence intensity for two-level atoms. The same result can be seen in Fig. 5c of Narducci et al.,4 which shows a calculated fluorescence spectrum for a -system. This indicates that during coherent population trapping a -system does not radiate and the dark line is entirely absent in its resonance fluorescence spectrum. Our calculations, presented in this paper in the asymptotic limit of a strong field, show, however, that the resonance fluorescence of a system does not vanish during coherent population trapping. Its spectrum consists of two multiplets, similar to the triplet in the resonance spectrum of two-level atoms and located at the electronic transition frequencies, together with two triplets located at the frequencies of four-photon processes involving the pump light fields. The latter are fundamental in character and impose limits on the lower bound of the dephasing rate of the Raman resonance owing to the contribution to the dynamics of the ground state from radiative decay of dipole transitions. The effect of the four-photon dephasing mechanism on the absorption spectrum of a -system is analyzed and found to lead to a substantial reduction in the depth of a dip in the absorption spectrum that vanishes as the laser field intensity is increased. This article is organized as follows: Section 2 is devoted to a description of the complete Liouvillian of an atom interacting with a laser radiation field. The specific features of the excitation of two-level atoms and a -system are analyzed. In Sec. 3 the resonance fluorescence spectrum of a -system
© 1998 American Institute of Physics 79

1063-7761/98/010079-14$15.00


is calculated in the rotating wave approximation, as well as outside the range of validity of this approximation. The effect of four-photon interactions involving the driver light fields on coherent population trapping and their role in the formation of the absorption resonance and in the dispersion of the -system are analyzed in Secs. 4 and 5, respectively. Most of the voluminous mathematical calculations are carried out in the Appendix. In the Conclusion we discuss an experiment for detecting the calculated structure of the resonance fluorescence spectrum of a -system.

Thus, we can treat it as a universally small perturbation relative to the resonant excitation contribution for the amplitude of driver waves smaller than the amplitude of the intraatomic field. In a first-order approximation with respect to this parameter, the evolution superoperator has the form
t L0t

S 0,t

S 0,t

RWA

1
0

L

p

d

e

.

6

2. LIOUVILLIAN OF AN ATOM BEYOND THE RANGE OF VALIDITY OF THE ROTATING WAVE APPROXIMATION; DYNAMICAL TRANSFORMATIONS USED TO CALCULATE THE FLUORESCENCE SPECTRUM

Note that this approximation is valid if the value of the integral is less than of order unity. We now consider the specifics features of the excitation of a two-level atom and a -system.

The complete Liouvillian of an atom, which describes changes in the atomic variables according to the equation ^ ^ dA / dt L ( t ) A in Markov theory, has the form Lt L
0

2.1. Excitation of a two-level atom

For a two-level atom excited by a laser field E L cos( Lt), Eq. 5 takes the form i
0

L

L

r

LL t .

1 L
p

^ Here L 0 is the unperturbed Liouvillian ( i / ) H 0 , , including the free precession of the atom at the laser frequencies according to Eq. A4 of Appendix A. The symbol denotes a place for substituting a transformed operator. L r and L determine the dynamics of the atom owing, respectively, to relaxation and nonzero detuning of the frequencies of the driver laser fields from the resonance transition frequencies in the atom resonance excitation is described by L 0 , while L L ( t ) describes the laser excitation. In terms of the interaction representation the transformation S (0,t ) corresponding to the Liouvillian 1 takes the form S 0,t S
RWA

2 ^

^

exp
L

2i t,

L

t 7

exp 2 i

,

0,t ~0 0,t e S

L0t

,

2

where 0 is the Rabi frequency and ^ are the standard Pauli matrices. Applying Eq. 7 to the complex polarization amplitude ^ and using Eq. 6 , we find that S ( t ) ^ determines the structure of a triplet because of the presence of the term S ( t ) RWA in the resonance fluorescence spectrum of a two-level atom at a frequency of 3 L , 5 which is analogous to the known triplet at the laser excitation frequency.6,7 Integrating with respect to in Eq. 6 , we can easily show that the ratio of the corresponding amplitudes of the spectral components at the frequencies of the third harmonic and the laser light is proportional to the small quantity 0 /2 L .

where the superoperators
2.2. Excitation of a -system

S

RWA

0,t

exp L

RWA

t,

L

RWA

L

L

r

L

p

3

determine the system dynamics in the rotating wave approximation RWA , ~ 0,t S0
t

T exp
0

L

p

d

4

is the evolution superoperator for the dynamics of the system owing to the nonresonant excitation component, and Lp t e
L0t

LL t e

L0t

L

p

5

is the deviation from the average value L p of the Liouvillian for the laser excitation.5 The symbol T used in Eq. 4 denotes the time ordering of the superoperator taken in quantum mechanics. The deviation L p ( ) in Eq. 5 oscillates at the frequencies of the laser drive fields and their combinations.
80 JETP 86 (1), January 1998

Let us consider a -system consisting of three electronic levels with transition frequencies among them of 12 13 , 23 Fig. 1 . Two coherent fields E cos( Lt) and E cos( Lt) act, respectively, on the transitions 1 3 and 2 3. These fields interact with the complete dipole moment of the system determined by the operator d 13 ^ 13 d 23 ^ 23 , where ^ 13,23 are the Pauli matrices ^ 1 for the corresponding atomic transitions. As a result, the induced dipole moment of the system oscillates at frequencies L and L. As opposed to the case of two-level atoms, where both driver fields interact with one and the same atomic transition, during excitation of a -system each field interacts with two transitions. Thus, the Liouvillian corresponding to biharmonic laser excitation with a frequency detuning L takes according to Eq. B3 the form L L ig 2 ^e
it

p

^e

it

,

,

8
80

Grishanin et al.


FIG. 1. A -system a and a typical arrangement for the experimental measurement of resonance fluorescence induced by two monochromatic laser fields with frequencies L and L b . , , and 12 are the relaxation rates of the populations in the upper levels; 13 , 23 , and 12 are the dephasing rates; and, w is the rate of pumping to level 2. The fluorescence spectrum of the atoms is analyzed using a Fabry-Perot interferometer FP and a photodiode PD in a direction perpendicular to the directions of the laser Lb and atom Ab beams.

where g g 2 g 2 , g d 13E , g tor ^ is defined as ^ g
1

d 23E , and the opera-

g^

13

g^

23

.

9

In deriving Eq. 8 we have neglected terms containing sums with higher frequencies. Equation 8 obviously determines additional spectral and L , of which components at frequencies L only the components at L and L are new. The correspond to four-photon processes and should show up for a symmetric -system as a mirror reflection of the virtual levels of the subsystem of lower levels Fig. 2 owing to modulation of the 1 3 and 2 3 transitions by the intrinsic oscillatory frequency 12 of the lower level subsystem. It is known that four-photon frequency mixing leads to generation of a coherent signal at the Stokes and anti-Stokes frequencies.8 Our later calculations show, however, that these nonlinear resonances are also accompanied by sidebands because of incoherent scattering processes. The above analysis shows that the important difference between exciting a two-level atom see Eq. 7 and a system Eq. 8 is that excitation in the case of the -system is mainly determined by the biharmonic frequency detuning . In experiments this detuning is usually much smaller than the frequencies of the laser systems that are exciting the system. This means that the intensity of the additional components in the fluorescence spectrum fine structure , which is determined by the exponential factors in Eqs. 7 and 8 , is substantially higher for a -system than for a two-level atom.

where ^ ( t ) are the Heisenberg positive negative frequency operators. These operators have a time dependence only in the form of high-frequency oscillations at optical frequencies. The superoperators S (0,t ) and S ( t , t ) describe the relaxation and interaction of the atoms with the exciting laser fields during the time intervals ( 0,t ) and (t,t ) , and ^ 0 S (0,t ) is the density matrix ^ ( t ) at time t . It follows from Eq. 3 that the superoperators S (0,t ) and ) are simple exponentials of the form exp L RWA t , S(t,t according to the rotating wave approximation. The term ^ ( t ) S ( t , t ) ^ (t ) in Eq. 10 is sim^ ( t ) and ^ ( t ply the product of the two operators ) ). averaged over the fluctuations in the time interval ( t , t This averaging is carried out with the aid of the transforma) , which determines the conditional atomic tion S ( t , t quantum mechanical probability distribution function at time relative to time t . The emission spectrum of the atom t can then be calculated as the Fourier transform of the correlation function 10 . In the stationary case the density matrix in the vector representation is simply the zero vector 0 of the matrix of the evolution superoperator L RWA . Then we can obtain the stationary correlation function from Eq. 10 by averaging it over the temporal oscillations. This averaging leads to the replacement of the bilinear combination of the complete operators ^ ( t ) by two combinations:

3. CALCULATING THE FLUORESCENCE SPECTRUM OF A -SYSTEM

The spectral density of the emission from an excited atom resonance fluorescence spectrum is determined by the normally ordered two-time correlation function of the light emitted by the atom.9,10 Assuming that the atomic fluctuations are Markovian, i.e., they are independent of one an, we can write down the correlation other at times t and t function for the atomic fluorescence in the form K
81

^ 0 S 0,t ^

t S t,t

^

t

,

10

FIG. 2. A -resonance and additional resonances which determine the fine structure of the resonance fluorescence spectrum. Grishanin et al. 81

JETP 86 (1), January 1998


^

t

^

t

^

13

^

13



^

23

^

23

.

Similarly, we have K K
1

K

2

.

When we calculate the resonance fluorescence spectrum of the atom in the rotating wave approximation the correlation functions K 1,2( ) correspond to the frequencies 13 L and 23 L and when we calculate the fine structure of the spectrum beyond the range of validity of this approximation they correspond, respectively, to the frequencies 2L 2L L L . See Sec. 2.2. L and L On describing the -system with the aid of the Liouvillian in the rotating wave approximation and expanding it in terms of the eigen-projectors, we obtain the following relatively simple expression:
8

FIG. 3. The formation of Rabi nutations in a two-level atom a and in a -system b . Only a minimum set of transitions between the quasienergy states, corresponding to the set of all possible lines in the fluorescence spectrum, is shown,

K
k0

0 ^ 13· k

k^ k^
23

13

exp
k

k

i
L

L

0 ^ 23· k

exp

i

},

11

where the symbol ``·'' means that the operators are multiplied in accordance with the multiplication rules for operators and the result is presented in the form of a ket-vector; k , k , and k are the eigenvalues of the matrix and the eigenvectors of the Liouvillian L RWA . Using Eq. 11 together with Appendix B we can obtain the following expression for the correlation function that describes the structure of the resonance fluorescence spectrum of the atom outside the range of validity of the rotating wave approximation: g K 4
2 8

are proportional to the coefficient in front of the corresponding exponent in Eq. 11 , while their width and frequency shift are determined by the real and imaginary parts of the eigenvalues k . In general, the fluorescence spectrum can be calculated numerically. In an asymptotically strong field, however, as we shall show below and has been demonstrated previously4 for a special case an analytic solution can also be obtained. For simplicity let us consider a -system excited by two high-power laser fields of equal intensity. In this case we can average the relaxation of the system over the Rabi nutations, while the Hamiltonian corresponding to the laser-induced ^ precession operator L p ( i / ) H p , takes the form

2 12 k 0

0 ^ 12· k k ^
L

12

exp
k

i

L

0 exp , 12 ^ H
p

1 0 0

1 0. 0

exp

i

g ^ 2

g 2&

1 1

where the ^ 12 are the complex conjugate amplitude of the subsystem of lower levels which modulate the dipole moment of the transition. This modulation gives rise to new spectral components in the fluorescence spectrum. Recall that Eq. 12 describes only the basic structure of the fluorescence spectrum, which is determined by the parameter g / 12 , which, in turn, we assume to be small. Here we have neglected the higher order contribution which makes a nonzero contribution to the coherent component of the response in the rotating wave approximation, which component equals zero when this correction is neglected in a strong field in a first-order approximation with respect to the parameter / g for detunings , R 0 by analogy with the two-level atom7 .
3.1. Fluorescence spectrum in the rotating field approximation

The fluorescence spectrum determined by Eq. 11 is the sum of Lorentz spectrum lines whose total spectral powers
82 JETP 86 (1), January 1998

This Hamiltonian corresponds to quasi-energy states with energies that are shifted with respect to the eigenvalues of the ^ Hamiltonian H p , which equal 0, g /2 . 11 For a twolevel atom the eigenvalues of the Hamiltonian are equal to g /2 . The temporal dynamics of these mixed quasienergy states cause oscillations in the expected values of the physical variables at two different frequencies g and g /2 . The physical significance of these nutations in terms of the quasienergy levels is illustrated in Fig. 3. The Rabi nutations between the quasi-energy levels are described by a Liouvillian which, in the operator basis ^^^ n3 ,n1 ,n2 , ^ c , ^ s , ^ c , ^ s , ^ c , ^ s the indices c and s 12 12 13 13 23 23 denote the cosine and sine components, respectively , takes the form
Grishanin et al. 82


0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 g 2&

g 2 g 2 0 g 2&

0 0 0 0 g 2&

g 2 0 g 2 g 2& 0

L

p

Ё
¦

0

0

0

0

0 g 2&

0

0 g 2 0 g 2

0 g 2 0

0

0 g 2&

0

0

0

0

0

0 g 2&

0

0

0

0

0 g 2

0 g 2&

0

0

0

0

0

0

0

0 ig /2, 1/2 0 0 0 i /2 i /2 0 0 0

0

0

©
i /2

.

13

Its eigenvalues k ( k 0, ..., 8) are equal to 0, 0, 0, of eigenvectors is defined as 0 0 1/2 0 0 0 0 1/2 1/2 2 0 1/2
3/2

ig /2, ig /2, ig /2, 0 0 0 1/2 0 0 1/2 i /2 i /2 1/2 0 0 0 i /2 i /2 0 0 0 0 0 0 1/2 0 0 1/2 i /2

ig , ig , while the corresponding set

0 1/2 2
3/2

0 1/2 1/2 0 0 0 0 2 2
3/2 3/2

0 0 0 0 1/2 1/2 0 0 0

i /2 0 0 i /2 1/4 1/4

i /2 0 0 i /2 1/4 1/4

k

§

.

Let us now discuss the physical significance of the dynamical variables corresponding to the eigenvectors k . The eigenvector 0 describes the stationary excitation of a system by two laser fields of equal intensity acting on the 1 3 and 2 3 transitions, respectively. The eigenvectors 1 and 2 describe a two-dimensional stationary excitation space, a combination of the polarization of the ground state and the populations of all three levels. The eigenvectors 3 and 6 describe excitation which involves a combination of the populations of the subsystems of the lower levels and an independent combination of the polarizations 1 3 2 3; they oscillate at half the Rabi frequency, g /2. The eigenvectors 4 and 5 describe excitations which are a combination of three polarizations 1 2 and an independent combination 1 3 2 3 which also oscillate at a frequency of
83 JETP 86 (1), January 1998

g /2. The last two eigenvectors, 7 and 8 , describe the excitation of the bound state together with the populations of all three levels and the polarization of the ground state, all oscillating at a frequency of g . Therefore, the Rabi nutations of the eigenexcitations of the system for combinations of the unbound levels take place at a frequency of g /2, while the bound states oscillate at a frequency of g . See Eq. A5 . Using Eq. 13 for the nutation operator L P , we can average the Liouvillian L L r in Eq. 1 over the nutations and write it in the form of a sum of 3 3 matrices, two 2 2, and two 1 1. The last two are diagonal elements. As a result, we can obtain a simplified expression for the last three terms in Eq. 1 , which describe the overall dynamics of the system in the interaction representation, of the form
Grishanin et al. 83


13

23

/2

0
12

0 /2 &
23/4 12

L

RWA

0 0

12

/2& /4 i s /2

12/2

12

w /2

13/4

ig /2
12/2

i s /2
12

13/4

23/4

ig /2



/2

13

/4

23/4

ig /2
12

i s /2 w /2 ig ig ,
13/4 23/4

i s /2 3 /8 3 3 /8 3 /8 /8
12/8 12/8 13/4 13/4 23/4 23/4

ig /2

where the total detuning is s 2 R L 23 . The corresponding eigenvalues are given by 0 3
13 1 2 23 12

L

13

/4

The preceding analysis shows that in a strong field only the total detuning s 2 R appears among the eigenfrequencies of the system, and not the Raman detuning R . Here for 0, relaxation in the system of lower levels does not contribute to the oscillations at half the Rabi frequency:
1,2 13/4 23/4

/2 ,

ig /2.

3 3 where 1 4
13

3 3

12 12

* 1 * 2 2 2

13 13

2 2

23 23

8 ig 8 ig

/8 /8 14

1,2

12

w
23

12

i4 .

2 s

12

w

12

2

2 ig

Here 12 and w are the rates of relaxation and pumping of the lower level system, 12 is the dephasing rate in this sysare the rates of relaxation from the excited tem, and states, and 13 and 23 are the corresponding dephasing rates. Let us now discuss the eigenvalues k in detail. Note that because the relaxation operator is not selfadjoint, each eigenvalue corresponds to two eigenvectors, one of which describes the operators acting on the physical variables, while the other describes the density matrix. The eigenvalue 0 0 corresponds to the stationary state ^ st ^ 0 and the operator I 0 , which has no dynamical significance. This eigenvalue determines the coherent line in the fluorescence spectrum. The eigenvalues 1,2 describe the nonoscillatory dynamics of the system and determine the Rayleigh scattering of the fields which excite the system. The four eigenvalues 3,4,5,6 determine oscillations at half the Rabi frequency g /2 g / & and describe the contribution of field-induced resonances to the fluorescence spectrum. The last two eigenvalues 7,8 determine oscillations at the Rabi frequency g and describe the ordinary 1 3 and 2 3 nutations in the weak-field limit.
84 JETP 86 (1), January 1998

This effect can, in principle, be used to study the contribution of the lower level system to the fluorescence spectrum in an experiment where the spectra are measured as a function of the detuning for different intensities of the laser line. The measured width of the spectral components located at half the Rabi frequency is then determined directly by the relaxation rate in the lower level system. For the case of an exact resonance ( , R 0 ) , we can obtain an analytic expression for the fluorescence spectrum in the rotating wave approximation. The major difference compared to the spectra from two independent two-level systems, however, is that in the case of the -system the general coefficient in Eq. 11 , which determines the intensity of the spectral components, differs from the corresponding coefficient for the case of a two-level atom, which is simply proportional to . In coherent population trapping, this coefficient for a -system and, therefore, the intensity of the spectral components decrease by roughly a factor of 12 / , which is a small parameter. For the cesium atom,12 as an example, it is 1.6 10 3 , while for sodium2 it can be estimated to be 4.9 10 3 using published parameters.13

3.2. Fine structure of the fluorescence spectrum

For simplicity let us again consider the case of an exact resonance. Using the equations from Sec. 3.1 together with Eq. 12 , we obtain the following expressions for the coefficients c k in front of the exponential factors:
2

c

/

2 12 12 3

0

3 2/ 0, c

, 1

c

1

9 8 /
12 12

1 2/ 3 2/ ,

12 12 3

,

c

2,3,5

4,6

1 4

3 2/

Grishanin et al.

84


Isolating the contribution to ^ 13 and ^ 23 owing to fourphoton interactions outside the range of applicability of the rotating wave approximation, we can write Eq. 16 in the form ^ H ^ H
4 ph

^ H

0

^ H ^

4 ph

, ^

2

1

S0 ^

13

2

S0 ^

23

H.c.,

FIG. 4. The resonance fluorescence spectrum of a -system the 2 S 1/2 2 P 3/2 transition in the cesium atom excited by two intense laser fields into a coherent population trapping state.

c

7,8

1 16

1 3 2/
12

,

15

^ ^ where H 0 determines the standard interaction, H 4 ph describes the additional contribution owing to four-photon interactions, and S 0 is the four-photon contribution to the dynamic transformation of the -system. Using the final formulas of Appendix B for the transformed operators ^ 13 and ^ 23 , we obtain the following formula for the four-photon contribution to the Hamiltonian: ^ H
4 ph

These coefficients multiplied by the common factor g 2 /4 2 determine the intensity of the fine structure com12 ponents of the fluorescence spectrum 12 . The complete fluorescence spectrum of a -system, including the structure in the rotating wave approximation as well as the fine structure calculated above, is shown in Fig. 4. Equation 15 implies that for the typically large values of the ratio / 12 only the coefficients c 0 and c 4 c 6 are proportional to the large values of order / 12. As a result, only three lines show up in each of the two fine structure features of the fluorescence spectrum Fig. 4 . One of them is coherent i.e., has zero width with an intensity proportional to c 0 , while the other two are broadened lines with an intensity proportional to c 4 shifted to the left and right of the coherent line at half the Rabi frequency g /2.

g 2 ^

^ 2
2

1

^, ^
23

13

exp i

i
L

L

t t H.c. 17

^ ,^

exp

By calculating the commutators in Eq. 17 , we can write down the four-photon contribution with the aid of the transition operators for the low level subsystem as ^ H
4 ph

4 g^

g^
2

1

t exp i

i
L

L

t^ t^
12

12

t exp

H.c.

18

4. EFFECT OF FOUR-PHOTON INTERACTIONS ON COHERENT POPULATION TRAPPING

Using Appendix B we can easily calculate the contribution to the relaxation of the ground state from the relaxation contributions of the dipole transitions. The nature of the relaxation processes involves an interaction of the dipole moments of the 1 3 and 2 3 transitions with the vacuum fluctuations of the electromagnetic field. These interactions are described by the Hamiltonian ^ H where ^ are ^ E0 ( the (k
85

2

^^ 1

13

^^ 2

23

H.c.,

16

This implies that the vacuum electromagnetic field interacts with the lower level subsystem through four-photon processes. The efficiency of this interaction depends on the interaction constant of the laser fields with the dipole transitions of the -system. The distinctive feature of this interaction is that the emission of a vacuum photon by the 1 3 transition is accompanied by the absorption of a 1 2 transition photon, while the emission of a vacuum photon by the 2 3 transition is accompanied by the emission of a 1 2 transition photon. Conservation of energy in these processes is ensured by the four-photon interaction of the vacuum field with the laser fields, and this is reflected in the exponential terms in Eq. 18 . Following Ref. 10, we can write the relaxation operator for the low level subsystem, which corresponds to Eq. 18 , in its customary form in terms of the operator basis ^^ n1 ,n2 , ^ 1 , ^ 2 : L
12 12 12

1
1

d

13

^ r E0 r ,

^

1
2

d

23

^ r E0 r

0 0
12

0 0 w
12

the components of the vacuum electromagnetic field r) with negative frequency amplitudes integrated over spatial distribution dk 3 ( r) of the dipole moments 1,2) .
JETP 86 (1), January 1998

w

12

w 0 0

12

0 0

/2
12

0 w
12

, /2
85

0

Grishanin et al.


where 12 is the relaxation rate of the lower level subsystem. Given that the term g ^ 1 ^ 12 in Eq. 18 describes a relaxation transition from level 1 to level 2 and the term g ^ 2 ^ 12 describes a transition in the opposite direction, we obtain the following expression for the relaxation owing to the contribution of four-photon processes: c L
4 ph 12 2 13 23

c

2 2

13 23

0 0 c
2 13

0 0 c
2 23

g 4

2 2

c

2

c

0 0

0 0

/2 c
2 13

0 c
2 23

, /2

19

0

where c g / g , c g / g , and c 2 c 2 1. Equation 19 implies that the contribution of fourphoton processes to the relaxation rate constant of the lower level subsystem, 12 is given by
4 ph 12

g 8

2 2

c

2

13

c

2

g
23

2 2 13

4

/2.

This contribution leads to a fundamental lower bound on 12 . As an example, for the Cs atom and laser field intensities of 1 W/cm2, we have 4 ph 10 3 13/2 104 s 1. 12
5. THE ROLE OF FOUR-PHOTON INTERACTIONS IN THE FORMATION OF AN ABSORPTION RESONANCE AND DISPERSION

When the contribution of four-photon processes to resonance dephasing is taken into account, the relaxation operator in the expression for L RWA must include the field-dependent correction 19 . Then the results of averaging in Eqs. 20 and 21 with a natural choice of basis for the vector representation of the density matrix see Appendix A1 are described simply by the corresponding components of the vector 0 . Performing the corresponding analytic calculations and expressing the concentration of active atoms in terms of the pressure p , we obtain ~ ~ g
2 2 k

n

k

1

0.0289p

3

n3 ~

~ g

~ 2 ~12~ ~ 2 ~R g2 g ~ ~2 4 ~ 2 , ~ 12 4 12 g R 22
2 1 2

The simplest experimental possibility for observing the dark resonance is to measure the transmission and/or dispersion in atomic vapor of exciting laser waves, whose independent detection is made easier by the relatively large difference of the corresponding frequencies L and L in the neighborhood of the resonance: . The real and imaginary parts of the corresponding refractive indices are expressed in an obvious way in terms of the operators for the corresponding dipole transitions assuming that macroscopic volume averaging is valid as cg k N 0 Im k 3 , Ik

and ~ ~ g
2 2 k

n

k

0.0289p

3

n3 .

23

Here n 3 describes the population of the excited state calculated according to the formula 2 ~~ g ~ g
2

n n and cg k N 0 Re k 3 , Ik
k

3

3

2~ 2 1g 2

1~ ~ g
2 2 2 2

2

~ 2

20

~4 ~ g

12

~6 /4 g

~ g

~ 2~ ~ g 1g

2 2~ ~ 2 2~ 2 g 12 1 2 ~2 ~ ~ 2 /4 12g R 12

~ g

2 2

~ 2 ~R~ g1

1

,

n

k

1

21

where g k and I k are the corresponding Rabi frequencies and intensities of the fields, with k 1 corresponding to frequency L and k 2 to L here g 1 g and g 2 g . To calculate Eqs. 20 and 21 in the stationary case it is necessary to find the stationary density matrix in the rotating wave approximation; this matrix is represented by the corresponding zero eigenvector 0 determined from the equation 0L
86
RWA

0.

where the tilde means that the corresponding variables are normalized to . The arguments and R depend on the velocity of the atom owing to the single-photon and residual Doppler effect, while the damping 12 in the lower level system is determined by the reciprocal time of flight of the atom for a cuvette with pure vapor . Thus, in order to obtain computational data which model the experimental situation, Eqs. 22 and 23 must be averaged over a Maxwellian velocity distribution, which is done numerically. Figure 5 shows calculated resonance absorption curves for 2 S 1/2 2 P 3/2 transitions in cesium and potassium for intense pump and weak probe fields. Although four-photon dephasing is not very important for cesium, in the case of
Grishanin et al. 86

JETP 86 (1), January 1998


FIG. 5. Absorption resonances in cesium a and potassium b vapor including smooth curve and neglecting dotted line fourphoton dephasing. The dot-dashed curve in Fig. a corresponds to a calculation taking four-photon dephasing into account where the separation between the lower levels was specially reduced by a factor of 10. The field intensities in the calculations were I 1 0.01 mW/cm2 and I 2 10 mW/cm2.

potassium, for which the splitting of the ground state for the isotope 41K is only 0.25 GHz, i.e., almost two orders of magnitude smaller than the splitting in cesium, with a pump field intensity of 10 mW/cm2 the resonance in the absence of a magnetic field is essentially unobservable. Thus, the fourphoton mechanism for dephasing of the -resonance establishes a fundamental limit of ``observability'' for the absorption resonance in strong fields, by imposing a limit either on the pump field intensity or on the magnitude of the splitting in the ground state.
6. CONCLUSION

V.N.Z. is indebted for support. Thus work was Stiftung Grant No. damental Research
APPENDIX A

to the Humboldt Foundation Germany supported in part by the VolkswagenI/72944 and the Russian Fund for FunGrant No. 96-03-032867 .

Dynamic superoperator of a

-system

Four-photon interactions, therefore, play a fundamental role in the formation of fluorescence spectra, as well as of the absorption spectra and/or dispersion of a resonance -system during coherent population trapping. A typical arrangement of a possible experiment for detecting a resonance fluorescence spectrum employing an atomic beam is shown in Fig. 1b. Experiments of this sort using an atomic beam and an atom trap have been described in detail by Gauthier et al.14 and Stalgies et al.,15 respectively. The directions of the atomic and laser beams are chosen to be mutually perpendicular so as to avoid the ordinary Doppler effect. The fluorescence spectrum is analyzed with a Fabry-Perot interferometer. Calculations for the fluorescence spectrum of two-level atoms show that for an atomic beam with 109 atoms/s·mm2, using a 5-millimeter Fabry-Perot cavity with Q 104 one can expect more than 105 photons/s from a volume of diameter 100 m. For a -system, fourphoton interactions, on one hand, reduce the fluorescence intensity relative to that of a two-level atoms by a factor of ( g /2 12) 2 , and, on the other, increase it by a factor of ( / 12) 1/2. As an example, for the Cs atom a saturation intensity of 1.1 mW/cm and the corresponding parameter 102 are already achieved for a laser power of 30 mW g at a wavelength of 852 nm focussed into a spot with a diameter of about 1 mm. For these experimentally easily realized parameters, we may expect, as the calculations show, a reduction in the scattering intensity of the -system compared to two-level atoms by a factor of 2 103 and, therefore, to detect fewer than 100 photons/s, which is not a problem for modern detection systems. The authors thank D. N. Klyshko, A. Schentzle, and R. Wynands for fruitful discussions. One of the authors
87 JETP 86 (1), January 1998

Let us consider a -configuration of the quantum mechanical levels of an atom Fig. 1 acted on by two laser fields with frequencies close to a Raman resonance which is described by a Hamiltonian of the form ^ H where ^ H
a 12

^ H

a

^ HI ,

A1

22

13

33

is the intrinsic Hamiltonian of the atom the energy of level 1 is taken to be zero, so that the projection operator 1 1 is absent in the Hamiltonian and ^ H
I

g cos cos
L

L

t

13 23

31 32

g

t

is the Hamiltonian of the interaction of the atomic system with two light fields having frequencies L and L , including the dependence of the excitation on the phase of the field. The interaction constants, i.e., the Rabi frequencies, depend on the amplitudes A L and A of the external field an on the L dipole matrix elements d 13 and d 23 : 1 1

g

d 13A L ,

g

d 23A .
L

A2

Only the case when a single-photon resonance is present is of interest, i.e., when L and L are close, respectively, to ^ 13 and 23 . We can rewrite H a in the form ^ H
a

^ H

0

^ H,
Grishanin et al.

A3
87


where the ``unperturbed'' Hamiltonian ^ H 33 22, A4

2

g

1

ge

i

c

ge

i

n ),

0

L

which leads to the result g
2 2

including

biharmonic frequency detuning , describes free precession with the two L 12 L ^ laser frequencies. The ``perturbing'' Hamiltonian H b can be written in the form ^ H where , 33 2 2,

the

^ H

R

g gg
R 2

cc
i

R

g
2

2

g cn

nn

g

[e

H.c.]

R

g (c 3 2 In the basis form 3,c,n

H.c.) . the corresponding matrix has the

L

13

R

L

L

12

describe the single-photon detuning for the 1 3 transition and the two-photon Raman detuning, respectively. Both detunings can be zero with a suitable choice of laser frequencies. The dynamics of an atomic system with the Hamiltonian A3 can be characterized as a combination of fast owing to ^ ^ H 0 and slow owing to H precessions, so it is appropriate to shift to a representation of the interaction with the unperturbed unitary transformation i^ H0t .

0 ^ H g /2 0 0
R

g /2 0

0 0

0 g gg e
2

0 gg e
i 2

g

2

0 0

i

g

U0 t

exp

In the rotating wave approximation6 we can neglect the rapidly oscillating terms, so that the Hamiltonian A1 takes the form ^ H ^ H ^ H 33 3 H.c. ] 22 g /2 c A5

and, for R 0, can quickly be expanded in terms of the 2 2 matrix of a two-level system ``dressed'' with the atomic field and the 1 1 matrix of a single unbound state, i.e., the excited and bound states form an effective two-level system e c . For simplicity we redefine 1 as exp( i ) 1 and 2 as exp( i ) 2 , so that we can rewrite Eq. A6 in the form c g
1

p

R

g1

g 2 ),

n

g

1

g1

g 2 ),

and is the effective Hamiltonian in this approximation. Here we have introduced the bound state ( c ) and the unbound state ( n ) orthogonal to it: c g
1

which does not contain the phase factors explicitly. With this representation of the Hamiltonian in the rotating wave approximation, the corresponding dynamic part of the Liouvillian has the form i L

ge
1

i

1
i

ge

i

2 ),
i

^ H,

.

A7

n

g

ge

1

ge

2 ).

A6

The complete Liouvillian L RWA also contain a relaxation operator which is specified phenomenologically here.

The state c is associated with excitation of the level e with an effective coupling constant of g g 2 g 2 . For zero Raman detuning ( R 0 ) , it is easy to see that the Hamiltonian A5 describes a two-level system. This can be demonstrated most clearly by substituting the expression 2 2 corresponding to the inverse transformation A6 in Eq. A5 : 1
88

A1. Transformation of the Liouvillian in the rotating wave approximation

g

1

ge

i

c

ge

i

n ),

The initial representation of the Liouvillian in the rotating wave approximation is a matrix in the nonhermitian basis ^ ^ ek P , , where k ( , ) and , 1,2,3, which can be represented by the following complex matrix elements using Eqs. A7 :
Grishanin et al. 88

JETP 86 (1), January 1998


0 0 0 0 i

0 0 0 0 i

ig 2 ig 2 0 ig 2
12

ig 2 ig 2 0 0 ig 2
13

ig 2 0 ig 2 0 ig 2 0 0 i 0

ig 2 0 ig 2 ig 2 0 0 0 0 i
c1

12

12

L

RWA

Ё
1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
12 c n

0 0 0 ig 2 ig 2 ig 2 ig 2

w 0 0 ig 2 ig 2 0 0

w 0 0 0 0 ig 2 ig 2

R

12

0 ig 2 0 0 ig 2

R

0 i 0 0 0

0 ig 2 ig 2 0

0 i 0 0

13

23

23

©

.

For converting to the more convenient Hermitian bases, we can introduce two transformations V c and V 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1/& i/& 0 0 0 0 0 0 0 0 0 g/g 0 g /g 0 0 0 0 1/& i/& 0 0 0 0 0 0 0 0 0 0 g/g 0 g /g 0 0 0 0 0 1/& i/& 0 0 0 0 0 0 0 1/& i/& 0 0 0 0 0 0 0 g /g 0 g/g 0 0 0 0 0 0 0 0 1/& i/& 0 0 0 0 0 0 g /g 0 g/g 0 0 0 0 0 0 0 1/& i/& . ,

of the form

V

c

V

c1

A8

^^ ^^ The transformation V c introduces two pairs of polarization operators P c , P c and P n , P n for transitions to the excited level from the bound and unbound states, while V c 1 introduces the Hermitian cosine-sine operators analogs of the coordinates and momenta or the Pauli matrices ^ 1 , ^ 2 in a two-level system : ^ q ^ q ^ q
g c n

^ P ^ P ^ P

^ P ^ P ^ P
c

21

/&,

^ p ^ p ^ p
c n

g

^ iP ^ iP
c n

12

^ P ^ P
c n

21

/&,

/&, /&,

/&, /&. A9

n

^ iP

^ P

Here the subscripts g , c , and n correspond to the ground ( 1 2 , bound ( c 3 ) , and unbound ( n 3 ) subsystems.
89 JETP 86 (1), January 1998 Grishanin et al. 89


After applying the transformations A8 to the Liouvillian L operator L RWA V c 1 V c L RWA V c 1 V c 11 we obtain 0 0

RWA

in the rotating wave approximation, for the transformed g &
2

0

0 g

0 g &

0

12

12

0

0

0

0

&

L

RWA

Ё
13 13

2

0 0 0 0 g & 0 0

w 0 0 0
2

w 0 0 0
2

0

0

0 0 0
c

g

0 0 g 2
2

g &
2

&
12 R

g 0

g 2

R

12

0 0

0 g 0
2 2

0 0 g 2 g 2 0 0

g

g

c

0

& 0 g &

& 0 g & 0
n

n

©

,

A10

where we have used the following notation g/g ,
c 2 2

g /g ,
23 23

, .

n

2

13

2

23

,

The block structure of the transformed dynamic superoperator L RWA indicated by the continuous lines in Eq. A10 is discussed in more detail in Appendix A2. As opposed to the initial complex representation, the transformed operator L RWA has real matrix elements, since it ^ corresponds to the Hermitian basis e k . For a symmetric -system, with 13 0 and , Eq. A10 takes the form 23 , g g , and, therefore, 0 0 0 0 L
RWA 12 12

0 0 0
12 R 12

0 0 0 0 0

g /& g /2& g /2 g /2 0

0 0 0 0 g /2 0 0

0 g /2& g /2& 0 0 0 0 .

0 0

w 0 0 0 g /2& 0 g /2&

w 0 0 0 g /2 0 g /2&

0 0 g /& 0 0

R

0 g /2 0 0

0 0 g /2 0 0 0 0 0

A2. The block structure of the dynamic superoperator in the rotating wave approximation

The physical significance of the superoperator L RWA defined by Eq. A10 becomes most transparent on examining its block structure. It is convenient to break the matrix A10 up into blocks in accordance with a definite set of polariza90 JETP 86 (1), January 1998

tion variables A9 . By analogy, the matrix blocks can be numbered with a subscript p corresponding to the populations and by the subscripts g , c , and n corresponding to the polarizations of ground ( 1 2 ) , bound ( 1 3 ) , and unbound ( 1 2 ) subsystems. In this notation, the matrix A10 appears as
Grishanin et al. 90


L L
RWA

pp

0 L L L
gg cg ng

L L L L

0 L L
cp np

T cp T cg

L L L L

cc nc

T np T ng T nc

where the perturbation Liouvillian has the form i . A11 L
p

^ H

,

.

B4 and using Eq. B4

nn

It consists of nine nonzero independent blocks. The diagonal block L pp describes the dynamics of the populations n 3 , n c , and n n , and the blocks L gg , L cc , and L nn , the polarization dynamics, respectively, of the ground state and of the bound and unbound subsystems. The five nonzero nondiagonal matrices describe the dynamics of the -system owing to coupling among the above basis variables. The antisymmetry of these five blocks is a consequence of the purely oscillatory character of the dynamics resulting from the interaction with the external field, while the inner dynamics includes relaxation so it is also represented by the matrix elements which yield nonzero real components in the eigenvalues of the matrix L RWA . It is easy to see from the block structure of A11 that there is no connection between the populations and polarizations of the ground state, since L gp 0. This reflects the fact that the exciting field acts directly only on transitions into the excited state, while single-photon excitation of the ground state is absent. The block representation given here for the dynamic superoperator in the rotating wave approximation is convenient for qualitative discussions of the effect of the parameters of the -system on its dynamics, since it reduces to changes in only the inner structure of the blocks in the representation A11 .
APPENDIX B Superoperator calculation of the general formula for the fine structure of the spectrum

Integrating S 0 ( t ) with respect to together with Eq. B3 , we obtain S0 t g 2 ^e
it

^e

it

,

S0 t ,

B5

which describes oscillations at a frequency . The superoperator S 0 ( t ) in Eq. B5 describes the unperturbed dynamics represented by the Hamiltonian A4 , which accounts for the free precession of all the -system transitions. The latter is represented in the form S0 t S 12e
S 23
it

S 12e i t S 13

e

i Lt

S 13e

i Lt

e

i Lt

S 23e

i Lt

P0 ,

B6

where the matrices S kl for the corresponding superoperators are the one-dimensional eigen-projectors on the corresponding intrinsic precession of the variables and P 0 is the projector on the three-dimensional subspace of the nonoscillatory variables, i.e., the populations. After substituting Eq. B6 in Eq. B5 , we obtain S0 t g 2 ^, ^, ^, ^, S S S S
13

exp i

i
L L L

L

t t t t. B7

23 13

exp exp i

23

exp i

Let us calculate the two-time correlation function 10 that determines the atomic fluorescence spectrum: K ^ 0 S 0,t ^ t S t,t ^ t . B1 has the B2

Then substituting Eq. B7 into Eq. B2 and using Eq. B1 together with the relation ^ 0 S (0,t ) 0 for t because of the damping of all the eigen-oscillations corresponding to the nonzero eigenvalues, we can finally write the correlation function in the form K t,t 0 S 0 t ^ · exp L
RWA

Here the total evolution superoperator see Eq. 2 form S 0,t S
RWA

S0 t

^

,

t ~0 t , S

where S RWA ( t ) is the superoperator in the rotating wave approximation and ~ 0 ( t ) is the superoperator for the perturbed S evolution owing to the nonresonant interaction. The superoperator ~ 0 ( t ) describes the transformation of an initial system S ^ Hamiltonian H ( t ) of the form ^ ^ H t H
0

B8 where the symbol ``· '' denotes the product of transformed operators and 13 23 is the sum of the complex amplitudes oscillating at the optical frequencies. If we then apply S 0 to ^ and recall that ^ 13 and ^ 23 are eigenvectors for the eigen-projectors S 13 and S 23 , we obtain S0 t ^ g 2 ^, ^ and S0 t ^ g 2 ^, ^ ^ ,^
13 23

^ ,^

13

exp i
L

L

t B9

^ Ht

^ H

0

g 2

^e

it

^e

it

, B3

exp i

t,

where the operator ^ is defined by Eq. 9 . In first-order perturbation theory we can introduce a superoperator ~ 0 ( t ), S corresponding to the Hamiltonian B3 , in the form ~t S0
91
t

exp i

i
L

L

t t , B10

S0 t

S0 t

S0 t

L
0

p

d S0 t ,

23

exp

JETP 86 (1), January 1998

Grishanin et al.

91


where the commutators of the complex amplitudes of the 1 3 and 2 3 transitions with ^ and ^ are given by ,^ ^
13 12

2 ^
12

,

,^

^
23

12

2 ^
12

,

Let us consider the important special case in which the oscillations in Eq. C.2 are fast compared to the rates of all the relaxation processes, so that it is possible to average over these oscillations. Then the resulting effective ``reduced'' relaxation operator has the form L
re

,^

13

2

,

,^

23

2

.

L
,

,

After substitution of Eqs. B9 and B10 in Eq. B8 and leaving out terms which oscillate relative to t , with the dynamical representation in the rotating wave approximation in terms of the eigenvectors and corresponding eigenvalues Eq. B8 finally takes the form of Eq. 12 .
APPENDIX C Transformation of the time evolution superoperator in the rotating wave approximation

L

,

,

C3

For a -system with a time-independent Hamiltonian ^ , the time evolution superoperator is unitary and is given H by an exponential S ( t ) exp(L H t ) with a purely dynamic Liouvillian of the type A7 and can be written in the form Lt Ut t U
1

where it is assumed that all the frequencies correspond) are different. The ing to the atomic transitions ( first term in Eq. C.3 describes the relaxation of the populations owing to transitions from other levels ( ) and radiative decay ( ) . The second term describes the relaxation of the polarization variables. The corresponding matrix is n n , where n 3 is the number of levels in a -system. The superoperator C3 commutes with the dynamical Liouvillian, since they have an eigenbasis in common. Given this circumstance, the relaxation of the atomic oscillations is simply described by the corresponding damping rates Re L
,

t
,

exp ,

i C1

.

where the and are the Bohr eigenfrequencies and the corresponding eigenvectors of the Hamiltonian, while the unitary transformation U( t ) is specified by the relation Ut exp i^ Ht .

If these quantities are all nonzero, then the stationary zerovector ^ st has nonzero components only in population relaxation space and is actually described by the n -component zero-vector st of the n n submatrix L , .
1

2 3

Subsequently, we can use the interaction representation for the superoperator L H L r , which differs from the purely dynamical Liouvillian corresponding to S ( t ) in that it includes the relaxation superoperator L r , and treat L H as the Liouvillian for the unperturbed time evolution S ( t ). In the interaction representation the time dependent relaxation superoperator has the form Lt
,;, I r

4

5 6

7 8 9

10

exp i tL
,

11

,

C2

12

where
13 14

L

,

kl
k,l;m,n

L

kl , mn

m

n
15

denotes the matrix elements of the relaxation superoperator ^ in terms of the eigen-basis of the Hamiltonian H .

B. D. Agap'ev, M. B. Gorny, B. G. Matisov, and Yu. V. Rozhdestvenski, Usp. Fiz. Nauk 163, 1 1993 . E. Arimondo, Progress in Optics 35, 257 1996 . M. R. Ferguson, Z. Ficek, and B. J. Dalton, Phys. Rev. A 54, 2379 1996 . L. M. Narducci, M. O. Scully, G.-L. Oppo, R. Pu, and J. R. Tredicce, Phys. Rev. A 42, 1630 1990 . B. A. Grishanin, Kvant. Elektrodinamika 6, 1649 1979 . L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press 1995 . B. R. Mollow, Phys. Rev. 188, 1969 1969 . I. R. Shen, The Principles of Nonlinear Optics, Wiley, New York 1984 . R. Glauber, in Quantum Optics and Electronics, Gordon and Breach, N.Y. 1965 , p. 63. B. A. Grishanin, Quantum Electrodynamics for Radio Physicists in Russian , Izd-vo. Mosk. Un-ta., Moscow 1981 . Van M. Lu, G. Mainfray, C. Manus, and I. Tugov, Phys. Rev. A 7, 91 1973 . O. Schmidt, K.-M. Knaak, R. Wynands, and D. Meschede, Phys. Rev. A 54, R27 1996 ; S. Brandt, A. Nagel, R. Wynands, and D. Meschede, Phys. Rev. A 56, R1063 1997 . E. Arimondo, Phys. Rev. A 54, 2216 1996 . D. J. Gauthier, Y. Zhu, and T. W. Mossberg, Phys. Rev. Lett. 66, 2460 1991 . Y. Stalgies, I. Siemers, B. Appasamy, and P. E. Toschek, Europhys. Lett. 35, 259 1996 .

Translated by D. H. McNeill

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92