- Email: [email protected]
Superradiance from slightly detuned sources
Volume 74A, number S
PHYSICS LETTERS
26 November 1979
SUPERRADIANCE FROM SLIGHTLY DETUNED SOURCES C. LEONARDI and A. VAGLICA Istituto di Fisica dell’Università and Gruppo di Fisica Teorica del GNSM, Consiglio Nazionale delle Ricerche, Palermo, Italy Received 11 July 1979
We present a new quantum theory of superradiant emission by an assembly of sources having slightly different resonance T’D <(Ti’ )2 is satisfied, inhomogeneous broadening has negligible effrequencies. It shows that, if the new condition TR fects on superradiance.
As is well known, a system ofN sources initially pumped to their excited levels relaxes its energy in an interval of time TR much shorter than the single-source relaxation time. The emission process takes place at a time TD after the pumping, and TD is generally longer than TR. During this superradiant decay the assembly of sources, according to Dicke’s formulation [1] ,goes via collective states, characterized by having the maximum “cooperation number” r, which remains constant during the whole process. However, when the frequencies of the sources are not all the same, the system in its evolution can occupy states with smaller r and the decay profile can be largely modified. These modifications have been investigated in the past both by using the semiclassical approach [2—5] and fully quantum theories [6—8] and particular attention has been devoted to the formulation of the condition that should be satisfied if one wants that inhomogeneous broadening does not influence the formation of the superradiant pulse [5,8]. However, some disagreement still exists on this subject which in particular leads to different interpretations of some experimental results [7,9] as well as to different choices in the adoption of one of the following two inequalities: ,
T~> T~ or
T~> TD,
(1)
as the condition for the inhomogeneous dephasing time T~to have negligible effects on superradiance [10,11]. These difficulties are in our opinion indicative of some intriguing aspects of the superradiant behaviour of not exactly tuned sources. As a simple case we consider two sources in their excited state at t = 0 and interacting with a common radiation field which is at the beginning in its vacuum state. The hamiltonian we use is ~—
~
+
, 1~1)÷nw2~2)÷~{e(k,
l)+ê(k, 2)},
(2)
where S~(i= 1, 2) is the z-component of the spin which describes the two levels of the ith source and 11w1 is the spacing in energy between these two levels. The free radiation field, which is assumed to be one dimensional, is described by the operators ~ and âk which create and destroy one quantum of energy1iw~ in the k-mode, respectively. The last term in the hamiltonian represents the interaction of the two sources with the field, e(k, i) (ekâk~’~+ h.c.), where
~k
is the matrix element of the transition described by the operator âkS~.The positions of the two sources 309
Volume 74A, number 5
PHYSICS LETTERS
26 November 1979
do not appear in eq. (2) since we wish to study here the case in which the distance between the sources is shorter than the wavelength of the modes involved in the process. It is possible to show that the dynamical behaviour of this system can be described by a set of second-order differential equations involving the average values of the operators p3(k) = â~ãk,ê(k, 1), ê(k, 2), ~ ~ and S(1, 2) = ~ + h.c.). These equations are 2d2p(tkl)/dt2=1l(wk—wo)e+(Ikl)+lThwe_(Ikl)+416k12(S(t)+l/2)+2Iekl2S(1,2), (3a) h
2p(tkI)/dt2 —fl2~w2e+(IkI)—112&w(wk w 0)d 0)e_(lkI) 2d2e_(Ikl)/dt2~r_2h2(wk_wo)~we+(lkl)_112[(wk_wo)2+~w2Je(IkI)
h2d2e+(Ikl)/dt2
=
—J13(wk
—
w
—
,
(3b)
11
_4hIekI2~w(S(t)+1/2)_61ThwIckI2S(l,2)+8IekI2S(t)~e_(Iknj),
(3c)
and finally 112d2S(1, 2)/dt2
=
8h~wS(t) k’>O e_(Ik’I)
—
2h2d2S2(t)/dt2
—
4h26w2S(l, 2).
(3d)
We have used the following notation for the average values: p(IkI) = ~(p(k) +p(—k)),
S(t) = }(S~w+
e+(IkI)~(e+(k)+e+(—k)), e÷(k)e(k,1)+e(k,2), e_(Ikl) = ~(e_(k)
+ e(—k)),
e_(k) = e(k, 1)
-
e(k, 2).
Moreover =
1
1
~(w 2 + w1),
~w
=
~(w2
—
w1)
The equation for S(t) can be derived immediately from eq. (3a) since the number of excitations is constant and in our case S(t)=l/2— Ep(IkI).
(4)
k>O
No terms containing the difference s~’~ appear in eqs. (3). In fact, it can be explicitly shown that if both ek and the density of modes do not vary over the range of1~ frequencies involved in the process and if the initial = s~2~ during the whole decay. Furthermore, in eqs. conditions are those specified previously, one obtains S~ (3c) and (3d) use has been made of the assumption that the average values (S$~ê(k,/))(i zj) can be decoupled and replaced by the product (S —
2c~)<ê(k,f)). The set of equations we-have introduced actually consists of a large number of equations since, as k varies, new
equations appear for the unknown functionsp(~kI),e+(Ikl), e_(IkI). However, the complete system can be solved, at least in principle, exactly and in particular the behaviour time of the can bewhen 6w. We present here theinsolutions for population small valuesofofthe ~wlevels, and inS(t), particular found as a function of the detuning (5) where T 1~ (11w is the single-source relaxation time in the one-dimensional radiation field. It is given by TjQ’ 2PQ 0), where p2 (11w0) is the density of modes at 11w0.
X
We use the following expansions: 310
(2irm)
Volume 74A, number 5
PHYSICS LETTERS
x2 ~2~p((k~)+
p(Jkt) = (O)p(~k)+
e+(Ikp)~r(°)e+(Jkl)+X2~2~e+(IkI)+... 5(1,2)
=
26 November 1979
~°~S(i,2) +
x2
~2~S(l,2) +
,
e_(IkI)X~1~e(IkI)+X3 ~3~e_(IkI)+..., S(t)
...,
=
(6)
~°~S(t)+ X2 ~2~S(t) +
where (°)p(Ikl),~°~e+(IkI),etc. are the solutions when X
=
6wT
12 = 0, and (tñp(~k~), (~)e÷(~k~), etc. multiplied by ~ are the corrections to these solutions which are of order q in X. We note that e_(tkI) is the only quantity which changes sign when the two sources are exchanged and therefore it is an odd function of X. Since we are interested in the corrections to ~°~S(t)of lowest order in X, we can substitute the expansions (6) into eqs. (3) and neglect all terms of order higher than 2~p()kI), 2 in X. In~2~e+(jk!), this way a “~e()kI), new set of differential equations similar is When ob~2~S(l,2), and ~2~S~(t) definedtoineqs. eqs.(3) (6). tained in the unknown functions ~ the equations for (2)p(~kj), ~2~e+()kt), ~e(lk~) are formally solved and their solutions are summed over all the modes, the new set reduces to the following three equations: d ~~G(x)/dx
=
2 ~°~S(x) WG(x)
—
d2 ~2~S(l, 2)/dx2 = —4 ~°~S(I,2)
+
~°~S(l, 2),
(7a)
8 ~°~S(x)~‘~G(x) 4d2(~°~S(x) ~2~S(x))/dx2 —
(7b)
,
}
d ~2~S(x)/dx = —~2~S(x) (2)~g(~ 2), —
where x
=
(7c)
t/T
12 and the function G(x) = (T12/h) ~k>Oe_(Ik~) has been introduced. By solving eqs. (7a) and (7b) and substituting their formal solutions into eq. (7c) we finally obtain
+f dx~(x —x’)[2~°~S(l,2;x’)—4~°~S(x’)(~)G(xt)]
(8)
f ~°~S(x”)dx”) ~°~S(i, 2; x’).
(9)
2~S(x)/dx —(1—2 ~0~S(x))~2~5(x) d ~
,
and we have ~1~G(x) exp (_2f (0)s(x’)dxt)
f
~x’ exp (_2
The solution of eq. (8) multiplied by X2 gives the correction to the decay law of the two sources when they are slightly detuned. We wish to report here in some detail the corrections one obtains when the sources are N in number, with N very large, and are divided in two equally populated groups according to their resonance frequency which is assumed to be w 1 for the sources in one group and w2 for those in the other group. The analysis of this new system follows a procedure very similar to the one exposed above, with some obvious adjustment of notation. In fact, eq. (8) is replaced in this case by 2~S(y)/dy
=
—(2/N)[l —2(N— 1) (O)5(y)](2)5~)+
d~
f dy’ (y —y’)[2 ~°~S(l, 2;y’)—4 (O)S(yl) WG(y’)J ,(lO)
while the expression for ~1~Gis formally identical to eq. (9). Eq. (10) is easily integrable and its solution, when N ~ 1, is given by (2)S(y)
=
2 1/2 cosh (YD
—~)
f dy’ cosh2(yD —y’){artg[sinhy~}
—
artg[sinh(y~ —y’)j}2,
(11)
0
311
Volume 74A, number 5
PHYSICS LETTERS
26 November 1979
where the zeroth order solutions ~0~S(y),~°~S(l, 2) have been explicitly determined as ~-th(yD—y), ~°~S(1,2)= (1/2 —2 andy t/TR,yD TD/TR, TR = 2T 2)S(y) multiplied by A2 (6w TR)2 12/N,According TD ~ TRtoIneq. N. (11), The quantity gives the desired correction to the decay. (2)S(y) is( very small wheny ~YD and rapidly increases in an interval of a few units aroundyD, where it reaches the value YDI2~roughly. Then, asy becomes larger, ~2~S(y)decreases and approaches its asymptotic value (2)~(e~)= ~.2/4~ Therefore we can conclude that the correction to the decay law at the time TD of maximum emission intensity is given by A2(2)S(t=TD)~~6w2TRTD
(12)
,
and that the decay profile, when 6w2 TRTD ~ 1, is shifted and delayed in time by a few times TR. Moreover, the effects of the detuning should be negligible when 6w2TRTD < 1.
(13)
The theory developed here can be extended, up to second order in the detuning, to sources distributed over different resonance frequencies, provided that the distribution is symmetrical with respect to a central frequency w~. The correction to the decay law (0)S(y) previously defined is given in this case by the quantity 6w2T~(2)S(y), where 6w2 is the mean square value of the deviation with respect to w 0 ofusual the frequency the g(6w) sources, and T~ 2)S(y) is the function reported in eq. (11). For a distribution having the Iorentzian of shape = 7r1 (X [1 + 6w2(T~)2] 1, the correction to the decay is equal to (TR/T~)2(2)S(y). Condition (13) for negligible effects of the detuning becomes in this case TRTD/(T~)2< 1
(14)
which is intermediate between the two inequalities (I) and is as much different from both of them as TD is longer than TR. This condition has the same form also when the length L of the active region that includes the N sources is longer than the wavelength of the radiation modes involved in the process, provided that L is shorter than the coherence length. We note that the inequality (14) is satisfied in both experiments reported in ref. [9] and therefore our calculations suggest, differently from ref. [7] that the effects of inhomogeneous broadening should be negligible in both these cases. In fact the less restrictive character of our condition, in comparison with the one expressed by the inequality T~> TD, agrees with some considerations developed in ref. [51and at the end of ref. [91 on the basis of semiclassical results. On the other hand, condition (14) does not seem to be consistent, in the case of small detuning considered here, with some of the conclusions reached in refs. [2,6—8] In our opinion, these differences are due to the presence in these theories of too restrictive assumptions on the symmetry properties of the system in its time evolution. These assumptions should be valid in the limit of large detuning [4]. We leave to a forthcoming paper a more detailed discussion on this point. ,
,
.
One of the authors (C.L.) wishes to acknowledge a discussion with Professor C. Cohen-Tannoudji on some aspects of the subject of this paper. This work has been partially supported by the CRRN and SM, Palermo, Italy. [1]
R.H. Dicke, Phys. Rev. 93 (1954) 99. [2] J.H. Eberly, Nuovo Cimento Lett. 1(1971)182; Acta Phys. Pol. A39 (1971) 633. [3] R. Friedberg and S.R. Hartmann, Phys. Lett. 38A (1972) 227. [41 R. Jodoin and L. Mandel, Phys. Rev. A9 (1974) 873. [51J.C. MacGillivray and M.S. Feld, Phys. Rev. A14 (1976) 1169. [6] G.S. Agarval, Phys. Rev. A4 (1971) 1791. [71 E. Ressayre and A. Tallet, Phys. Rev. Lett. 30 (1973) 1239.
312
[8] R. Bonifacio and L.A. Lugiato, Phys. Rev. All (1975)
1507. [9] N. Skribanowitz, l.P. Herman, J.C. MacGilllvray and M.S. Feld, Phys. Rev. Lett. 30 (1973) 309. [10] M. Gross, C. Fabre, P. Pillet and S. Haroche, Phys. Rev. Lett. 36 (1976) 1035. [11] H.M. Gibbs, Q.H.F. Vrehen and H.M.J. Hikspoors, Phys. Rev. Lett. 39 (1977) 547; Q.I-LF. Vrehen, H.M.J. Hikspoors and H.M. Gibbs in: Coherence and quantum optics IV, eds. L. Mandel and E. Wolf (Plenum, New York, 1978).
PHYSICS LETTERS
26 November 1979
SUPERRADIANCE FROM SLIGHTLY DETUNED SOURCES C. LEONARDI and A. VAGLICA Istituto di Fisica dell’Università and Gruppo di Fisica Teorica del GNSM, Consiglio Nazionale delle Ricerche, Palermo, Italy Received 11 July 1979
We present a new quantum theory of superradiant emission by an assembly of sources having slightly different resonance T’D <(Ti’ )2 is satisfied, inhomogeneous broadening has negligible effrequencies. It shows that, if the new condition TR fects on superradiance.
As is well known, a system ofN sources initially pumped to their excited levels relaxes its energy in an interval of time TR much shorter than the single-source relaxation time. The emission process takes place at a time TD after the pumping, and TD is generally longer than TR. During this superradiant decay the assembly of sources, according to Dicke’s formulation [1] ,goes via collective states, characterized by having the maximum “cooperation number” r, which remains constant during the whole process. However, when the frequencies of the sources are not all the same, the system in its evolution can occupy states with smaller r and the decay profile can be largely modified. These modifications have been investigated in the past both by using the semiclassical approach [2—5] and fully quantum theories [6—8] and particular attention has been devoted to the formulation of the condition that should be satisfied if one wants that inhomogeneous broadening does not influence the formation of the superradiant pulse [5,8]. However, some disagreement still exists on this subject which in particular leads to different interpretations of some experimental results [7,9] as well as to different choices in the adoption of one of the following two inequalities: ,
T~> T~ or
T~> TD,
(1)
as the condition for the inhomogeneous dephasing time T~to have negligible effects on superradiance [10,11]. These difficulties are in our opinion indicative of some intriguing aspects of the superradiant behaviour of not exactly tuned sources. As a simple case we consider two sources in their excited state at t = 0 and interacting with a common radiation field which is at the beginning in its vacuum state. The hamiltonian we use is ~—
~
+
, 1~1)÷nw2~2)÷~{e(k,
l)+ê(k, 2)},
(2)
where S~(i= 1, 2) is the z-component of the spin which describes the two levels of the ith source and 11w1 is the spacing in energy between these two levels. The free radiation field, which is assumed to be one dimensional, is described by the operators ~ and âk which create and destroy one quantum of energy1iw~ in the k-mode, respectively. The last term in the hamiltonian represents the interaction of the two sources with the field, e(k, i) (ekâk~’~+ h.c.), where
~k
is the matrix element of the transition described by the operator âkS~.The positions of the two sources 309
Volume 74A, number 5
PHYSICS LETTERS
26 November 1979
do not appear in eq. (2) since we wish to study here the case in which the distance between the sources is shorter than the wavelength of the modes involved in the process. It is possible to show that the dynamical behaviour of this system can be described by a set of second-order differential equations involving the average values of the operators p3(k) = â~ãk,ê(k, 1), ê(k, 2), ~ ~ and S(1, 2) = ~ + h.c.). These equations are 2d2p(tkl)/dt2=1l(wk—wo)e+(Ikl)+lThwe_(Ikl)+416k12(S(t)+l/2)+2Iekl2S(1,2), (3a) h
2p(tkI)/dt2 —fl2~w2e+(IkI)—112&w(wk w 0)d 0)e_(lkI) 2d2e_(Ikl)/dt2~r_2h2(wk_wo)~we+(lkl)_112[(wk_wo)2+~w2Je(IkI)
h2d2e+(Ikl)/dt2
=
—J13(wk
—
w
—
,
(3b)
11
_4hIekI2~w(S(t)+1/2)_61ThwIckI2S(l,2)+8IekI2S(t)~e_(Iknj),
(3c)
and finally 112d2S(1, 2)/dt2
=
8h~wS(t) k’>O e_(Ik’I)
—
2h2d2S2(t)/dt2
—
4h26w2S(l, 2).
(3d)
We have used the following notation for the average values: p(IkI) = ~(p(k) +p(—k)),
S(t) = }(S~w+
e+(IkI)~(e+(k)+e+(—k)), e÷(k)e(k,1)+e(k,2), e_(Ikl) = ~(e_(k)
+ e(—k)),
e_(k) = e(k, 1)
-
e(k, 2).
Moreover =
1
1
~(w 2 + w1),
~w
=
~(w2
—
w1)
The equation for S(t) can be derived immediately from eq. (3a) since the number of excitations is constant and in our case S(t)=l/2— Ep(IkI).
(4)
k>O
No terms containing the difference s~’~ appear in eqs. (3). In fact, it can be explicitly shown that if both ek and the density of modes do not vary over the range of1~ frequencies involved in the process and if the initial = s~2~ during the whole decay. Furthermore, in eqs. conditions are those specified previously, one obtains S~ (3c) and (3d) use has been made of the assumption that the average values (S$~ê(k,/))(i zj) can be decoupled and replaced by the product (S —
2c~)<ê(k,f)). The set of equations we-have introduced actually consists of a large number of equations since, as k varies, new
equations appear for the unknown functionsp(~kI),e+(Ikl), e_(IkI). However, the complete system can be solved, at least in principle, exactly and in particular the behaviour time of the can bewhen 6w. We present here theinsolutions for population small valuesofofthe ~wlevels, and inS(t), particular found as a function of the detuning (5) where T 1~ (11w is the single-source relaxation time in the one-dimensional radiation field. It is given by TjQ’ 2PQ 0), where p2 (11w0) is the density of modes at 11w0.
X
We use the following expansions: 310
(2irm)
Volume 74A, number 5
PHYSICS LETTERS
x2 ~2~p((k~)+
p(Jkt) = (O)p(~k)+
e+(Ikp)~r(°)e+(Jkl)+X2~2~e+(IkI)+... 5(1,2)
=
26 November 1979
~°~S(i,2) +
x2
~2~S(l,2) +
,
e_(IkI)X~1~e(IkI)+X3 ~3~e_(IkI)+..., S(t)
...,
=
(6)
~°~S(t)+ X2 ~2~S(t) +
where (°)p(Ikl),~°~e+(IkI),etc. are the solutions when X
=
6wT
12 = 0, and (tñp(~k~), (~)e÷(~k~), etc. multiplied by ~ are the corrections to these solutions which are of order q in X. We note that e_(tkI) is the only quantity which changes sign when the two sources are exchanged and therefore it is an odd function of X. Since we are interested in the corrections to ~°~S(t)of lowest order in X, we can substitute the expansions (6) into eqs. (3) and neglect all terms of order higher than 2~p()kI), 2 in X. In~2~e+(jk!), this way a “~e()kI), new set of differential equations similar is When ob~2~S(l,2), and ~2~S~(t) definedtoineqs. eqs.(3) (6). tained in the unknown functions ~ the equations for (2)p(~kj), ~2~e+()kt), ~e(lk~) are formally solved and their solutions are summed over all the modes, the new set reduces to the following three equations: d ~~G(x)/dx
=
2 ~°~S(x) WG(x)
—
d2 ~2~S(l, 2)/dx2 = —4 ~°~S(I,2)
+
~°~S(l, 2),
(7a)
8 ~°~S(x)~‘~G(x) 4d2(~°~S(x) ~2~S(x))/dx2 —
(7b)
,
}
d ~2~S(x)/dx = —~2~S(x) (2)~g(~ 2), —
where x
=
(7c)
t/T
12 and the function G(x) = (T12/h) ~k>Oe_(Ik~) has been introduced. By solving eqs. (7a) and (7b) and substituting their formal solutions into eq. (7c) we finally obtain
+f dx~(x —x’)[2~°~S(l,2;x’)—4~°~S(x’)(~)G(xt)]
(8)
f ~°~S(x”)dx”) ~°~S(i, 2; x’).
(9)
2~S(x)/dx —(1—2 ~0~S(x))~2~5(x) d ~
,
and we have ~1~G(x) exp (_2f (0)s(x’)dxt)
f
~x’ exp (_2
The solution of eq. (8) multiplied by X2 gives the correction to the decay law of the two sources when they are slightly detuned. We wish to report here in some detail the corrections one obtains when the sources are N in number, with N very large, and are divided in two equally populated groups according to their resonance frequency which is assumed to be w 1 for the sources in one group and w2 for those in the other group. The analysis of this new system follows a procedure very similar to the one exposed above, with some obvious adjustment of notation. In fact, eq. (8) is replaced in this case by 2~S(y)/dy
=
—(2/N)[l —2(N— 1) (O)5(y)](2)5~)+
d~
f dy’ (y —y’)[2 ~°~S(l, 2;y’)—4 (O)S(yl) WG(y’)J ,(lO)
while the expression for ~1~Gis formally identical to eq. (9). Eq. (10) is easily integrable and its solution, when N ~ 1, is given by (2)S(y)
=
2 1/2 cosh (YD
—~)
f dy’ cosh2(yD —y’){artg[sinhy~}
—
artg[sinh(y~ —y’)j}2,
(11)
0
311
Volume 74A, number 5
PHYSICS LETTERS
26 November 1979
where the zeroth order solutions ~0~S(y),~°~S(l, 2) have been explicitly determined as ~-th(yD—y), ~°~S(1,2)= (1/2 —2 andy t/TR,yD TD/TR, TR = 2T 2)S(y) multiplied by A2 (6w TR)2 12/N,According TD ~ TRtoIneq. N. (11), The quantity gives the desired correction to the decay. (2)S(y) is( very small wheny ~YD and rapidly increases in an interval of a few units aroundyD, where it reaches the value YDI2~roughly. Then, asy becomes larger, ~2~S(y)decreases and approaches its asymptotic value (2)~(e~)= ~.2/4~ Therefore we can conclude that the correction to the decay law at the time TD of maximum emission intensity is given by A2(2)S(t=TD)~~6w2TRTD
(12)
,
and that the decay profile, when 6w2 TRTD ~ 1, is shifted and delayed in time by a few times TR. Moreover, the effects of the detuning should be negligible when 6w2TRTD < 1.
(13)
The theory developed here can be extended, up to second order in the detuning, to sources distributed over different resonance frequencies, provided that the distribution is symmetrical with respect to a central frequency w~. The correction to the decay law (0)S(y) previously defined is given in this case by the quantity 6w2T~(2)S(y), where 6w2 is the mean square value of the deviation with respect to w 0 ofusual the frequency the g(6w) sources, and T~ 2)S(y) is the function reported in eq. (11). For a distribution having the Iorentzian of shape = 7r1 (X [1 + 6w2(T~)2] 1, the correction to the decay is equal to (TR/T~)2(2)S(y). Condition (13) for negligible effects of the detuning becomes in this case TRTD/(T~)2< 1
(14)
which is intermediate between the two inequalities (I) and is as much different from both of them as TD is longer than TR. This condition has the same form also when the length L of the active region that includes the N sources is longer than the wavelength of the radiation modes involved in the process, provided that L is shorter than the coherence length. We note that the inequality (14) is satisfied in both experiments reported in ref. [9] and therefore our calculations suggest, differently from ref. [7] that the effects of inhomogeneous broadening should be negligible in both these cases. In fact the less restrictive character of our condition, in comparison with the one expressed by the inequality T~> TD, agrees with some considerations developed in ref. [51and at the end of ref. [91 on the basis of semiclassical results. On the other hand, condition (14) does not seem to be consistent, in the case of small detuning considered here, with some of the conclusions reached in refs. [2,6—8] In our opinion, these differences are due to the presence in these theories of too restrictive assumptions on the symmetry properties of the system in its time evolution. These assumptions should be valid in the limit of large detuning [4]. We leave to a forthcoming paper a more detailed discussion on this point. ,
,
.
One of the authors (C.L.) wishes to acknowledge a discussion with Professor C. Cohen-Tannoudji on some aspects of the subject of this paper. This work has been partially supported by the CRRN and SM, Palermo, Italy. [1]
R.H. Dicke, Phys. Rev. 93 (1954) 99. [2] J.H. Eberly, Nuovo Cimento Lett. 1(1971)182; Acta Phys. Pol. A39 (1971) 633. [3] R. Friedberg and S.R. Hartmann, Phys. Lett. 38A (1972) 227. [41 R. Jodoin and L. Mandel, Phys. Rev. A9 (1974) 873. [51J.C. MacGillivray and M.S. Feld, Phys. Rev. A14 (1976) 1169. [6] G.S. Agarval, Phys. Rev. A4 (1971) 1791. [71 E. Ressayre and A. Tallet, Phys. Rev. Lett. 30 (1973) 1239.
312
[8] R. Bonifacio and L.A. Lugiato, Phys. Rev. All (1975)
1507. [9] N. Skribanowitz, l.P. Herman, J.C. MacGilllvray and M.S. Feld, Phys. Rev. Lett. 30 (1973) 309. [10] M. Gross, C. Fabre, P. Pillet and S. Haroche, Phys. Rev. Lett. 36 (1976) 1035. [11] H.M. Gibbs, Q.H.F. Vrehen and H.M.J. Hikspoors, Phys. Rev. Lett. 39 (1977) 547; Q.I-LF. Vrehen, H.M.J. Hikspoors and H.M. Gibbs in: Coherence and quantum optics IV, eds. L. Mandel and E. Wolf (Plenum, New York, 1978).