- Email: [email protected]
Spectrum of squeezing in resonance fluorescence
Volume
52, number
OPTICS COMMUNICATIONS
2
15 November
1984
SPECTRUMOFSQUEEZINGINRESONANCEFLUORESCENCE M.J. COLLETT Department
ofphysics,
University of Waikato, Hamilton, New Zealand
D.F. WALLS’ Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106,
USA
and P. ZOLLER Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Received
2 August
Austria
1984
The spectral squeezing in resonance fluorescence from a two-level atom is calculated. It is shown that for resonance citation optimum squeezing is obtained for near resonant frequencies of the fluorescent light. An operational definition the spectrum of squeezing is given based on a discussion of possible experimental configurations.
Resonance fluorescence from a two-level atom has revealed a number of interesting quantum features of the electromagnetic field [ 11. Theoretical calculations of the fluorescence [2] and absorption spectra [3] have been verified by experimental observations [4-61. A calculation of the second order correlation function predicted the phenomenon of photon antibunching [7] which has been experimentally observed [S] together with sub-poissonian photon statistics [9]. Further calculations of the phase-dependent correlation functions showed that squeezing [lo] was present in the total fluorescent light [ 111. The phenomenon of squeezing in resonance fluorescence has been addressed in a number of recent papers [ 12-161. In this paper we wish to give a spectral analysis of the squeezing in resonance fluorescence. This may be accomplished via a study of a phase sensitive correlation function as follows. We consider a two-level atom driven by coherent light of frequency wL. In the radiation zone the positive frequency part of the electric field has the form E(+)(x,t)=EQ&,t) + $(x) u_ (t - T/c) exp [-iw, 1 On leave from the Department Waikato,
Hamilton,
(t - r/c)],
of Physics, New Zealand.
University
(1)
exof
where u_ is the slowly varying atomic lowering operator at the retarded time t - r/c. For simplicity we have suppressed the vector character of the electric field. $(x) is a geometrical factor. We define a slowly varying electric field operator with phase 8 as E, (x, t) =i
[E(+)(x, t) exp(-iwlt
+ EC-)(x, =E,(x,
+ ie)
t) exp(iwL t - ie)]
t) cos 0 + E2 (x, t) sin 8,
(2)
which for 0 = 0 and 0 = 7r/2 coincides with the m-phase and out-of-phase component of the electric field, respectively. Taking the Fourier transfom-r with respect to time, the frequency components of the quadrature phases are &(x,w)
=+ [5+)(x,
tE(-)(x,wL
oL t w) eie
-W)e-ie].
(3)
Following ref. [ 171, a generalized spectral density of the electric field E, (x, t) can be defined as the nomrally ordered variance
of
0 0304018/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
145
Volume 52, number 2 (:
OPTICS COMMUNICATIONS
.zo (x. w), .E*(x, w’) : 1 =a[(2(+)(x, w), 3+)(x, WL
t
+ W)(x,
WL
-
w),
+ (W(x,
WL
-
w’),
+ (iTp)(x,wL
-
w),
WL
Eg”‘(x,WL 9+)(x, WL E( ~‘(x,wL
0’)) e2io
+
t
0’))
f
0))
-~~
w’))
15 Nuvember 1984
to the total outgoing flux [ 121, y is the radiative decay rate of the two-level system. Substituting the expressions (7) and (8) for the spectral correlation functions into eq. (4), we tind (: EO(w).EO(wi):
)-:
S,(w):
6(w t w’)
em ‘if?I,
(4) with (A, B) = ((A - (A)) (B - (B))). The angular brackets in eq. (4) denote averaging with respect to the density operator of the system. In the case of resonance fluorescence from a two-level system, the initial state is a coherent state for the laser mode, a vacuum for the other states of the radiation field. and the atoms in the lower level. This allows the normally ordered variances of the electric field to be expressed in terms of atomic correlation functions. In particular. we find in the stationary limit
(F+)(x, t), EC-)(x,t’>) (0)) exp[-iwI(t-
X le ‘@(u
(r), u- (0)) + e2iu(u+(0).
+ (u+(4. o--(O))+ (u+(O),U_.(T))].
u+(r)) (0 1
where we identify: S(wO ): with the phase sensitive nonnally ordered spectrum of the operator6;(x. t). Obviously : Se(w) : =: So(-w): which is a consequence of the fact that it correlates frequencies from different sides of wL, The atomic correlation functions in eq. (9) can be easily calculated from the atomic matrix equations of resonance fluorescence. Confining ourselves to the case of resonant excitation we have
t’)]. (5)
and @(+I@ >t) 2E(+)(x, t’), = Q(x)
((5_ (It-
0). u (O))exp[-iwI(t-t’)]. (6)
Delayed time contributions eq. (6). The corresponding tions are (g(+)(wL
t w), $f)(OI
X s d7 (eiwT + eeiw’)
+ w’), = $(w
Co-(r),
+ w’)
u_(O)).
(7)
0
and (E((-)(WI
(10)
[ 181 have been ignored in frequency correlation func.
where y is the radiative decay rate, !L!the Rabi frequency and (u,(t)) the population inversion of the atom. By application of the quantum fluctuation regression theorem, the atomic correlation functions appearing in eq. (9) are (Ui it), u_. (0)) = (u+(O). u,(t))
~ w), z(+)(WI
+ w’),
=-1 =yS(w-w’)
~dre~i”iiu+(r).u_(0)). (8) -cc respectively. For simplicity we have dropped the argument x (the position of the detector) in eqs. (7) and (8). In addition, we have followed the usual convention and have renormalized the correlation functions 146
2522
4y”t2@
[
exp( ~- +yt)
y lo&?-y2 +-4K y2 +2R2 ?2R2
-y=
y2 + 3_R2
exp(-$yt)
exp(--5
sinh Kr
yt) cash Kt , I
(II)
Volume 52. number 2
OPTICS COMMUNICATIONS
15 November 1984
with K2 = y2 - Cl2 for t 2 0. Correlation functions of the type (a+(t), o_(O)) have been derived previously in the discussion of the spectrum of resonance fluorescence. An expression for (o_(t), o_(O)) has recently been given by Loudon [ 141. The in-phase and out-ofphase normally ordered spectra are then : Sl(w):
= -yJ d7 (eiw7 t eeiw7) 0
x 3 ((o+(7), a_(O)) + Q_(7), u_ (0)))
-
WI2
2@
(12)
y2 + a2 (y/2)2 + ,2’
Fig.
and : So:
1.
1/2,(b)
= ys
: Sz(w) : as a function of w/y for values of n2/r2 1/4,(c)
1/6,(d)
(a)
l/12:
d7 (eiw7 t e-iwT)
0 x 3 &J+(T),
o_w
-(O_(T),
o_(W)
4G - y2 - ,2 2G =_y2 4 y2 + 2s12 [(r/2)2 + sl2 -6-J]2 + (3~/2) 2 0 2’ (13) respectively. Integration the total variances (:E,,EI:)=2
Y
over all frequencies gives for
2sP y2 t2Q2
(14)
-0.21
,
,
0
,
,
,
{
3
4
Q.:,
and
(:E,,J,:)=-;.
,
I
(r2-2’2)‘2 (72 + 2sP)2
Fig.
2.
: Sl(w
=
0): and : Sz{w = 0): as a function
of n/r.
(15)
thus squeezing is present in the E2 quadrature of the total field for 2Q2 < y2 in agreement with ref. [ 111. The spectrum : S2(w): is plotted as a function of o/r for different values of the Rabi frequency in fig. 1. Maximum squeezing is seen to arise in a narrow band around w = 0 for 52 = (3 - a) y/4. In fig. 2 we plot : S2(w = 0): as a function of a/y. Squeezing is present only for fi2 < f 72. With the onset of saturation the maximum squeezing shifts from the frequency component at w = 0 to the wings of the spectrum. The maximum squeezing attainable in resonance fluorescence : S2(0): = -0.07 is considerably less than that possible ’ in ideal parametric oscillators operating near the oscillation threshold, where values of: S2(0): = -0.25 are theoretically possible [ 19,iO].
Fig. 3. : Sl(w): and : Sz(w) : for a2 = 4~~.
147
Volume
52, number
2
OPTICS
COMMUNICATIONS
In fig. 3 we have plotted: S,(w): and : S2(w): fog fL* = 4y2. It is of interest to note that for values of 12 above saturation : Sl(o): gives rise to the central conponent and : Sz(o) : to the sidebands of the Mallow fluorescent triplet [I]. Finally, let us discuss possible experimental configuratior.s to detect spectral squeezing. As has been shown .n refs. [ 12-2 11, squeezing in the total field may be detected by homodyning the signal field with a local oscillator with amplitude e and frequency w,_ and observing sub-poissonian statistics in the corresponding photon count distribution, i.e..
(An)’
--(~)=~(Y*T*IEI’(:E~(x,~).E,(x,~):).
(16)
with (Ya constant characterizing the detector efficiency. T a sufficiently short counting interval, andE@(x, t) the field at the position of the detector. If before homodyning the scattered radiation is frequency filtered by a Fabry-Perot. we have at the detector [ 181 +CC E;;‘(t) = J” J(t - t’ ) _m X expl-i(wL
+ uf) (t - t’)] Ii’
Here, af is the filter frequency detuning laser reference wL, while [ 181 J(7) = e(7) il/emmr7 .
dt’.
(17)
from our
(18)
with r the filter bandwidth (the inverse cavity filling time);ili is a normalization factor. In this way we find for the phase component E,(x, t) at the detector +Ca a: =3 dw eLiwr _m
s
x [E’+)(w, tw>3
t Ecp)(wL-w) J”(w+ wj)
eiO J.
(19)
whereJ(w) =?(--w) =N/(- iw + r) is the Fourier transform ofJ(T). Note that frequency filtering of the scattered radiation will generally introduce a frequencydependent phase shift. As we noted in the context of cq. (3), the spectrum of squeezing arises from a combination of frequency components from different sides of the laser reference frequency wL. We. therefore, generalize the above considered configuration by introducing two Fabry-Perots tuned to frequencies + wf and --wf symmetrically below and above the laser fre148
quency. This allows the phase-sensitive the detector to be written in the form E:(t)
= Jdw
emiwr 1
-
(apart from an irrelevant normalization gives a detection signal
+ ?(Wf
w)lZ
.
component
at
-*
factor) and
iLl
1
with So(w) the spectrum of squeezing calculated before. Alternatively, the fluorescent light may be direclly homodyned with a local oscillator at the laser frequency wL and the spectral components resolved bl an electronic band pass filter. This work was supported in part by the National Science Foundation under Grant No. PHY77-27084 (supplemented by funds from the National Aeronautic and Space Administration), the United States Office of Naval Research under Contract No. NOOO14-81 -I<0809. the United States Army through its Europeall Research Office and by the hsterreichische Forschungsgemeinschaft under Project No. 06/0249.
References
C. Cohen Tannoudji. in: I:ronticrs of laser spectroscop! 4s. R. Balian. S. Haroche and S. Liberman (Norttlllolland. 1977) 1’. 3; J.D. Cresser, J. Hlger. G. Leuchs. M. Rateike and Il. Walther. in: Dissipative systems in quantum optics, cd. R. Bonifacio (Springer. 1982) p. 21. [2) B.R. Mollow. Phys. Rev. 188 (1969) 1969. 131 B.R. Mallow, Phys. Rev. A5 (1972) 2217. [4] R.E. Grove, F.Y. Wu and S. Ezekiel, Phys. Rev. Lctt, 35 (1975) 1426. 151W. Hartig, W. Rasmussen, K. Schieder and H. Walther, %. Phys. A278 (1976) 205. [61 F.Y. Wu. S. Ezekiel, M. Ducloy and B.R. Mallow. Phys. Rev. Lett. 38 (1977) 1077. and D.F. Walls. J. Phys. B9 (1976) 1199. [71 H.J. Carmichael 181 H.J. Kimble, M. Dagenais and L. Mandel. Phys. Rev. Al8 (1978) 201. [91 R. Short and L. Mandel, Phys. Rev. Lctt. 51 (1983) 384. (101 For a review set D.F. Walls. Nature 306 (1983) 141.
Volume
52. number
2
OPTICS COMMUNICATIONS
[ 111 D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (1981)
709.
[ 121 L. Mandel, Phys. Rev. Lett. 49 (1982) 136. [ 131 H.F. Arnoldus and G. Nienhuis, Optica Acta 30 (1983) 1573. R. Loudon, Optics Comm. 49 (1984) 24. P.A. Lakshmi and G.S. Aganval, Phys. Rev. A29 (1984) 2260. [ 161 Z. Ficek, R. Tanas and S. Kielich, Phys. Rev. A29 (1984) 2004. [14] [15]
15 November
1984
[ 171 C.M. Savage and C.W. Gardiner, Optics Comm. 50 (1984) 173. [18] J.D. Cresser, Phys. Rep. 94 (1983) 47. [19] M.J. Collett and C.W. Gardiner, Phys. Rev. A30(1984) 1386. [20] M.J. Collett and D.F. Walls, to be published. [21] H.P. Yuen and J.M. Shapiro, I.E.E.E. Trans. Inf. Theory 24 (1978) 657; 26 (1980) 78.
149
52, number
OPTICS COMMUNICATIONS
2
15 November
1984
SPECTRUMOFSQUEEZINGINRESONANCEFLUORESCENCE M.J. COLLETT Department
ofphysics,
University of Waikato, Hamilton, New Zealand
D.F. WALLS’ Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106,
USA
and P. ZOLLER Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Received
2 August
Austria
1984
The spectral squeezing in resonance fluorescence from a two-level atom is calculated. It is shown that for resonance citation optimum squeezing is obtained for near resonant frequencies of the fluorescent light. An operational definition the spectrum of squeezing is given based on a discussion of possible experimental configurations.
Resonance fluorescence from a two-level atom has revealed a number of interesting quantum features of the electromagnetic field [ 11. Theoretical calculations of the fluorescence [2] and absorption spectra [3] have been verified by experimental observations [4-61. A calculation of the second order correlation function predicted the phenomenon of photon antibunching [7] which has been experimentally observed [S] together with sub-poissonian photon statistics [9]. Further calculations of the phase-dependent correlation functions showed that squeezing [lo] was present in the total fluorescent light [ 111. The phenomenon of squeezing in resonance fluorescence has been addressed in a number of recent papers [ 12-161. In this paper we wish to give a spectral analysis of the squeezing in resonance fluorescence. This may be accomplished via a study of a phase sensitive correlation function as follows. We consider a two-level atom driven by coherent light of frequency wL. In the radiation zone the positive frequency part of the electric field has the form E(+)(x,t)=EQ&,t) + $(x) u_ (t - T/c) exp [-iw, 1 On leave from the Department Waikato,
Hamilton,
(t - r/c)],
of Physics, New Zealand.
University
(1)
exof
where u_ is the slowly varying atomic lowering operator at the retarded time t - r/c. For simplicity we have suppressed the vector character of the electric field. $(x) is a geometrical factor. We define a slowly varying electric field operator with phase 8 as E, (x, t) =i
[E(+)(x, t) exp(-iwlt
+ EC-)(x, =E,(x,
+ ie)
t) exp(iwL t - ie)]
t) cos 0 + E2 (x, t) sin 8,
(2)
which for 0 = 0 and 0 = 7r/2 coincides with the m-phase and out-of-phase component of the electric field, respectively. Taking the Fourier transfom-r with respect to time, the frequency components of the quadrature phases are &(x,w)
=+ [5+)(x,
tE(-)(x,wL
oL t w) eie
-W)e-ie].
(3)
Following ref. [ 171, a generalized spectral density of the electric field E, (x, t) can be defined as the nomrally ordered variance
of
0 0304018/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
145
Volume 52, number 2 (:
OPTICS COMMUNICATIONS
.zo (x. w), .E*(x, w’) : 1 =a[(2(+)(x, w), 3+)(x, WL
t
+ W)(x,
WL
-
w),
+ (W(x,
WL
-
w’),
+ (iTp)(x,wL
-
w),
WL
Eg”‘(x,WL 9+)(x, WL E( ~‘(x,wL
0’)) e2io
+
t
0’))
f
0))
-~~
w’))
15 Nuvember 1984
to the total outgoing flux [ 121, y is the radiative decay rate of the two-level system. Substituting the expressions (7) and (8) for the spectral correlation functions into eq. (4), we tind (: EO(w).EO(wi):
)-:
S,(w):
6(w t w’)
em ‘if?I,
(4) with (A, B) = ((A - (A)) (B - (B))). The angular brackets in eq. (4) denote averaging with respect to the density operator of the system. In the case of resonance fluorescence from a two-level system, the initial state is a coherent state for the laser mode, a vacuum for the other states of the radiation field. and the atoms in the lower level. This allows the normally ordered variances of the electric field to be expressed in terms of atomic correlation functions. In particular. we find in the stationary limit
(F+)(x, t), EC-)(x,t’>) (0)) exp[-iwI(t-
X le ‘@(u
(r), u- (0)) + e2iu(u+(0).
+ (u+(4. o--(O))+ (u+(O),U_.(T))].
u+(r)) (0 1
where we identify: S(wO ): with the phase sensitive nonnally ordered spectrum of the operator6;(x. t). Obviously : Se(w) : =: So(-w): which is a consequence of the fact that it correlates frequencies from different sides of wL, The atomic correlation functions in eq. (9) can be easily calculated from the atomic matrix equations of resonance fluorescence. Confining ourselves to the case of resonant excitation we have
t’)]. (5)
and @(+I@ >t) 2E(+)(x, t’), = Q(x)
((5_ (It-
0). u (O))exp[-iwI(t-t’)]. (6)
Delayed time contributions eq. (6). The corresponding tions are (g(+)(wL
t w), $f)(OI
X s d7 (eiwT + eeiw’)
+ w’), = $(w
Co-(r),
+ w’)
u_(O)).
(7)
0
and (E((-)(WI
(10)
[ 181 have been ignored in frequency correlation func.
where y is the radiative decay rate, !L!the Rabi frequency and (u,(t)) the population inversion of the atom. By application of the quantum fluctuation regression theorem, the atomic correlation functions appearing in eq. (9) are (Ui it), u_. (0)) = (u+(O). u,(t))
~ w), z(+)(WI
+ w’),
=-1 =yS(w-w’)
~dre~i”iiu+(r).u_(0)). (8) -cc respectively. For simplicity we have dropped the argument x (the position of the detector) in eqs. (7) and (8). In addition, we have followed the usual convention and have renormalized the correlation functions 146
2522
4y”t2@
[
exp( ~- +yt)
y lo&?-y2 +-4K y2 +2R2 ?2R2
-y=
y2 + 3_R2
exp(-$yt)
exp(--5
sinh Kr
yt) cash Kt , I
(II)
Volume 52. number 2
OPTICS COMMUNICATIONS
15 November 1984
with K2 = y2 - Cl2 for t 2 0. Correlation functions of the type (a+(t), o_(O)) have been derived previously in the discussion of the spectrum of resonance fluorescence. An expression for (o_(t), o_(O)) has recently been given by Loudon [ 141. The in-phase and out-ofphase normally ordered spectra are then : Sl(w):
= -yJ d7 (eiw7 t eeiw7) 0
x 3 ((o+(7), a_(O)) + Q_(7), u_ (0)))
-
WI2
2@
(12)
y2 + a2 (y/2)2 + ,2’
Fig.
and : So:
1.
1/2,(b)
= ys
: Sz(w) : as a function of w/y for values of n2/r2 1/4,(c)
1/6,(d)
(a)
l/12:
d7 (eiw7 t e-iwT)
0 x 3 &J+(T),
o_w
-(O_(T),
o_(W)
4G - y2 - ,2 2G =_y2 4 y2 + 2s12 [(r/2)2 + sl2 -6-J]2 + (3~/2) 2 0 2’ (13) respectively. Integration the total variances (:E,,EI:)=2
Y
over all frequencies gives for
2sP y2 t2Q2
(14)
-0.21
,
,
0
,
,
,
{
3
4
Q.:,
and
(:E,,J,:)=-;.
,
I
(r2-2’2)‘2 (72 + 2sP)2
Fig.
2.
: Sl(w
=
0): and : Sz{w = 0): as a function
of n/r.
(15)
thus squeezing is present in the E2 quadrature of the total field for 2Q2 < y2 in agreement with ref. [ 111. The spectrum : S2(w): is plotted as a function of o/r for different values of the Rabi frequency in fig. 1. Maximum squeezing is seen to arise in a narrow band around w = 0 for 52 = (3 - a) y/4. In fig. 2 we plot : S2(w = 0): as a function of a/y. Squeezing is present only for fi2 < f 72. With the onset of saturation the maximum squeezing shifts from the frequency component at w = 0 to the wings of the spectrum. The maximum squeezing attainable in resonance fluorescence : S2(0): = -0.07 is considerably less than that possible ’ in ideal parametric oscillators operating near the oscillation threshold, where values of: S2(0): = -0.25 are theoretically possible [ 19,iO].
Fig. 3. : Sl(w): and : Sz(w) : for a2 = 4~~.
147
Volume
52, number
2
OPTICS
COMMUNICATIONS
In fig. 3 we have plotted: S,(w): and : S2(w): fog fL* = 4y2. It is of interest to note that for values of 12 above saturation : Sl(o): gives rise to the central conponent and : Sz(o) : to the sidebands of the Mallow fluorescent triplet [I]. Finally, let us discuss possible experimental configuratior.s to detect spectral squeezing. As has been shown .n refs. [ 12-2 11, squeezing in the total field may be detected by homodyning the signal field with a local oscillator with amplitude e and frequency w,_ and observing sub-poissonian statistics in the corresponding photon count distribution, i.e..
(An)’
--(~)=~(Y*T*IEI’(:E~(x,~).E,(x,~):).
(16)
with (Ya constant characterizing the detector efficiency. T a sufficiently short counting interval, andE@(x, t) the field at the position of the detector. If before homodyning the scattered radiation is frequency filtered by a Fabry-Perot. we have at the detector [ 181 +CC E;;‘(t) = J” J(t - t’ ) _m X expl-i(wL
+ uf) (t - t’)] Ii’
Here, af is the filter frequency detuning laser reference wL, while [ 181 J(7) = e(7) il/emmr7 .
dt’.
(17)
from our
(18)
with r the filter bandwidth (the inverse cavity filling time);ili is a normalization factor. In this way we find for the phase component E,(x, t) at the detector +Ca a: =3 dw eLiwr _m
s
x [E’+)(w, tw>3
t Ecp)(wL-w) J”(w+ wj)
eiO J.
(19)
whereJ(w) =?(--w) =N/(- iw + r) is the Fourier transform ofJ(T). Note that frequency filtering of the scattered radiation will generally introduce a frequencydependent phase shift. As we noted in the context of cq. (3), the spectrum of squeezing arises from a combination of frequency components from different sides of the laser reference frequency wL. We. therefore, generalize the above considered configuration by introducing two Fabry-Perots tuned to frequencies + wf and --wf symmetrically below and above the laser fre148
quency. This allows the phase-sensitive the detector to be written in the form E:(t)
= Jdw
emiwr 1
-
(apart from an irrelevant normalization gives a detection signal
+ ?(Wf
w)lZ
.
component
at
-*
factor) and
iLl
1
with So(w) the spectrum of squeezing calculated before. Alternatively, the fluorescent light may be direclly homodyned with a local oscillator at the laser frequency wL and the spectral components resolved bl an electronic band pass filter. This work was supported in part by the National Science Foundation under Grant No. PHY77-27084 (supplemented by funds from the National Aeronautic and Space Administration), the United States Office of Naval Research under Contract No. NOOO14-81 -I<0809. the United States Army through its Europeall Research Office and by the hsterreichische Forschungsgemeinschaft under Project No. 06/0249.
References
C. Cohen Tannoudji. in: I:ronticrs of laser spectroscop! 4s. R. Balian. S. Haroche and S. Liberman (Norttlllolland. 1977) 1’. 3; J.D. Cresser, J. Hlger. G. Leuchs. M. Rateike and Il. Walther. in: Dissipative systems in quantum optics, cd. R. Bonifacio (Springer. 1982) p. 21. [2) B.R. Mollow. Phys. Rev. 188 (1969) 1969. 131 B.R. Mallow, Phys. Rev. A5 (1972) 2217. [4] R.E. Grove, F.Y. Wu and S. Ezekiel, Phys. Rev. Lctt, 35 (1975) 1426. 151W. Hartig, W. Rasmussen, K. Schieder and H. Walther, %. Phys. A278 (1976) 205. [61 F.Y. Wu. S. Ezekiel, M. Ducloy and B.R. Mallow. Phys. Rev. Lett. 38 (1977) 1077. and D.F. Walls. J. Phys. B9 (1976) 1199. [71 H.J. Carmichael 181 H.J. Kimble, M. Dagenais and L. Mandel. Phys. Rev. Al8 (1978) 201. [91 R. Short and L. Mandel, Phys. Rev. Lctt. 51 (1983) 384. (101 For a review set D.F. Walls. Nature 306 (1983) 141.
Volume
52. number
2
OPTICS COMMUNICATIONS
[ 111 D.F. Walls and P. Zoller, Phys. Rev. Lett. 47 (1981)
709.
[ 121 L. Mandel, Phys. Rev. Lett. 49 (1982) 136. [ 131 H.F. Arnoldus and G. Nienhuis, Optica Acta 30 (1983) 1573. R. Loudon, Optics Comm. 49 (1984) 24. P.A. Lakshmi and G.S. Aganval, Phys. Rev. A29 (1984) 2260. [ 161 Z. Ficek, R. Tanas and S. Kielich, Phys. Rev. A29 (1984) 2004. [14] [15]
15 November
1984
[ 171 C.M. Savage and C.W. Gardiner, Optics Comm. 50 (1984) 173. [18] J.D. Cresser, Phys. Rep. 94 (1983) 47. [19] M.J. Collett and C.W. Gardiner, Phys. Rev. A30(1984) 1386. [20] M.J. Collett and D.F. Walls, to be published. [21] H.P. Yuen and J.M. Shapiro, I.E.E.E. Trans. Inf. Theory 24 (1978) 657; 26 (1980) 78.
149