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Intermodal correlations in the generation of higher harmonics
Volume 64. number
OPTIC’S C‘OMMIINI~‘.~l‘ION\
2
INTERMODAL
Received
CORRELATIONS
27 April 1987: re~lsed manuscript
IN THE GENERATION
rccclved
74 June
OF HIGHER
HARMONICS
:‘Y
I YX7
Correlations between the photons of’the fundamental and harmonic modes are considered. In the short-path the classical Cauchy-Schwarr inequaltiy is shown to be vIolated. Moreover. photon anticorrelation takes place.
approximation.
1. Introduction The problem of the production of photon antibunching. subpoissonian photon statistics and squeezing [ 1.21. intimately related with the quantum nature of light. has attracted much attention during the last decade. Analytical solutions of subpoissonian photon statistics [ 31 and squeezing [ 41 in second-harmonic generation have been found using the short-path approximation. Later, numerical calculations [ 51 have confirmed the analytical solution for subpoissonian photon statistics (SPS). It is worth noting that, already from the earliei numerical results given in ref. [ 61, SPS was evident for the second-harmonic mode. although the authors themselves did not pay attention to this fact. Wagner et al. [ 71 have performed a simulation experiment, the results of which pointed to SPS in socondharmoning generation. Squeezing in the course of second-harmonic generation within a cavity has been revealed quite recently by Kimble and Hall [ 81. Chmela [ 91 has shown that, by generating second harmonics in cascade in thm plates. littering out the generated harmonic behind each plate, SPS in the fundamental beam increases considerably. Analytical solutions for SPS [ 10.1 1 ] and squeezing [ 121 in the generation of an arbitrary kth (A-==?._3...) harmonic have been found as well. Recently, Hong and Mandel [ 131 have introduced the concept of higher-order squeezing. The processes under consideration also seem promising in this respect [ 141. Harmonics generation, as two-beam phenomena, offer moreover other possibilities of testing the quantum theory of light against the classical theory. Namely. we can study intermodal correlations and the validity of various inequalities derivable from the classical theory for the non-delayed two-beam and single-beam correlation functions. In particular, the classical Cauchy-Schwarz inequality for the second-order two-beam correlation function Cl;’ and the single-beam correlation functions Cl,’ ’ and Gj,‘) reads [ 15 ] (Gl”)‘<(;“lG’2’ I, !i
(1)
il
where the subscripts i and j refer to two different light beams. Violation of the inequality (1) has already been observed by Clauser ir This work was supported
186
by Research
[ 161 in two-photon
cascade emission
Project CPBP 0 I .07
0 030-4018/87/$03.jO 0 Elscvier Science Publishers (North-Holland Physics Publishing Division)
R.V.
Volume 64, number
OPTICS
2
15 October
COMMUNICATIONS
1987
and theoretically studied by Loudon [ 151. In the case of the two-photon laser [ 17,181 and parametric amplifier [ 19,201 the above inequality is violated too. Violation of various inequalities classically derivable for second-order correlation functions has been reviewed recently by Reid and Walls [ 201. In this paper we show violation of the inequality (1) in the quantum description of higher-harmonic generation processes and moreover, that anticorrelations between the fundamental and harmonic photons take place.
2. Equations of motion In the electric dipole approximation and for perfect phase matching, the generation the absence of damping is described in terms of the following hamiltonian: HG =fwafaf+fikwa~aA
of the kth harmonic
+cAgh(aXa~+a$“ak) ,
in
(2)
where the suffices f and k refer to the fundamental and kth harmonic mode, respectively, and g, is the mode coupling constant depending on the order of the process and involving appropriate nonlinear characteristics of the medium. The symbol c denotes the light velocity, equal for both beams with respect to phase matching. af and a, (i=f, k) are the photon creation and annihilatioin operators fulfilling the boson commutation rule [a,,
a:1 =d,, .
In the Heisenberg
(3) picture,
the time evolution
of any operator
A is given by the equation
k=(ilA)[H,A].
(4)
For A equal to a,, the commutator with the free-field part of the hamiltonian (2) gives the “rapid” oscillations of the operator a,. In turn, the commutator with the interaction part of the hamiltonian (2) leads to the “slow” part of the evolution of the operator a,, resulting from nonlinear coupling between the beams. Generally, we can write af(t)=af,(t)
exp( -iwt),
ak(t) =ak,(t)
exp( -ikwt)
.
(5)
The slowly varying parts ars( t) of the photon operators satisfy the same commutation rule (3) as the operators a,(t) and af(t) (ij=f, k). Since the problem under consideration is that of propagation of the beams and not that of fields in a cavity, we perform the interchange z= - ct, where z is the direction of propagation of the beams. As a result, the quantum equations of motion for the slowly varying parts of the annihilation operators a,,(z) have the form da,(z)ldz=ikg,
a::-‘(z)
aks(z),
dak,(z)ldz=igk
ats(z) .
(6)
The above equations cannot be solved strictly. We must have recourse to the short-path approximation procedure used by us earlier [ 31. It consists in expanding the operators (or the correlation functions) in a Taylor series above z=O. For relative simplicity of the final results, it is more convenient to calculate the successive derivatives of a,,(z) not with eq. (6), but rather immediately differentiating the preceding derivative. Hence, on bringing the photon operators to normal order within their products, we find
(7)
187
Volume 64, number 7
+(i/6)kg:z’
OPTIC‘S C’OMMIJNI(‘,.lTIONS
:z:,,Q
(71
i”r’)[
where the symbol 0 at the suffices f and k denotes that a,~,= a,,( 0) = a,( 0 1. Moreover, n,,, = a fr,u,~ is the photon-number ial coefficients.
the operators arc taken at ~~0. for instance operator. The symbols ( :“) are Newton binom-
3. Correlation functions and results It is easily checked that the sum of the photon-number
operators
n,(z)+kn,(z)=const
(9)
is an integral of motion. This is because the generation takes place in a nonabsorbing medium. Of course. the expectation value of eq. (9) represents the conservation law for the mean photon numbers. i.e., for the firstorder correlation functions G{,! ‘(z) and G::l( -_). By definition Gj,“(z)=(a:\(z)
U,,(Z))
,
(10)
where the brackets ( > stand for the quantum expectation value calculated with a density operator of the initial fields. The harmonic field is in vacuum state at the input to the medium (==O). so that the following initial conditions are fulfilled: (11)
~Ol~~,,IO~=~Ol~,,,IO~=O. The fundamental field may be initially in an arbitrary state. Hence, eq. (9) leads to the following conservation law Gh!‘(z)+kGl:‘(z)=Gk:‘=(n,,).
(I’)
where (n,,,) is the mean number of the fundamental photons incident on the medium. The nondelayed double-beam second-order correlation function (;I”( z) reads G;;‘(z)=
(a;,(z)
and, in the single-beam Gi;‘(z)=(a:\‘(=)
u:\(z)
a,,(z) u,,(z))
case, transforms
(13) into
u;,(z)).
(14)
On insertion of (8) into the definition (10) and having recourse to the commutation rule (3) and the initial conditions (11). we find the evolution of the mean number of harmonic photons in the form ’ ‘1 +... .
I 8X
(15)
Volume 64, number
OPTICS
2
COMMUNICATIONS
15 October
1987
with G$$=(ah:a&). The first-order correlation function G#)(z) is readily obtained from the conservation law (12). Although the correlation function G,&‘(z) involves the fourth-order term in z, its calculation requires only the third-order approximation in z in the photon operators. Moreover, as we shall see, the second-order correlation function CL:)(z) is calculated within accuracy up to z6. All higher approximations in (8) contribute nothing to the term z4 in Cl:‘(z) or to the terms z4 and z6 in GLzk’(z) as a result of the normal ordering of the harmonic-photon operators in these functions and the condition (11). From (13) at (7) and (8) one gets G&?(z)=g:z’Gtgk+‘)
+z4(
3Gj,fk’ +
Iz;f. r! (;) (“,‘) [GfOZk--'I+(k-
1 -I)G~~-])
+... .
(16)
In order to obtain this function we apply the photon operators a#,( z) and aa( z) calculated only within accuracy to z2. However, calculation of the function Gj$(z), which is identical with the function (16), requires the third-order approximation in the photon operators. The only classical term in the square brackets in eq. (16) is Ggk), obtainable from the function Ggk-r) at r=O. The magnitude of the cross correlations v&z) between the two modesfand k is V,(z)=G;;)(z)-CR)(z)
G&(z).
(17)
Peiina [ 111 has shown for second-harmonic generation that both beams are uncorrelated ( v#,( z) = 0) up to the second-order approximation in z if the incident fundamental field is in a coherent state. For coherent input radiation (then G& ( nfD) k), within accuracy to z4, one gets from (16)) (15) and (12)
(18) The negative value of vrk(z) arises from the purely quantum photons of the fundamental and harmonic beams. In turn, from (14) at (7) and (8) we find
terms and reflects anticorrelation
GW’(z)=G&-k&z2[2Gg+‘)+(k-l)G,$f)]+...,
G@(z)
=&z4Gtik)
‘;;;1
-
between
the
(19)
$8~~ lz;IOr! (;) (“T‘) p! (,“) (k-;-r)
Ggk-‘-“-HI
+... .
(20)
The above forms of the correlation functions have not been presented hitherto by us [ lo]. For an initially coherent fundamental beam, by (19), (20) and (16)) we arrive at
189
Volume 64. number
2
OPTICS
(‘OMMCJNICATIONS
I5 October
I ‘)87
This result, in fact, means violation of the classical Cauchy-Schwarz inequality ( I ) in the process of an arbitrary kth harmonic generation due to the quantum nature of light. Thus for coherent input radiation both the fundamental and harmonic modes after transversing the nonlinear medium exhibit not only subpoissonian photon statistics [ lo] and squeezing [ 4.121 but moreover their correlation properties are such that photon anticorrelations and violation of the Cauchy-Schwarz inequality take place. No well-behaved Glauber-Sudarashan P functions exist for these fields [ 20,2 11. In particular, for second-harmonic generation (A-= 2) the general formulas ( 18) and (2 1 ) reduce to very simple ones, namely V,z(-‘)=-eg~24(n,~,)“.
&(z)=2,!&?‘(n,,,)“.
(22)
Recently, Mandel and coworkers [ 22,231 have observed subpoissonian statistics in the process of spontaneous parametric down conversion; two- and one-photon states have been produced. This allows us to conclude that the nonclassical properties of the processes under consideration can be experimentally verified.
References [ I] D.F. Walls. Nature 280 ( 1979) 45 I; L. Mandel. Physica Scripta T12 (1986) 34. [2] D.F. Walls. Nature 306 (1983) 141. [ 31 M. Korierowski and R. Tanas. Optics Comm. 2 I ( I977 ) 229. [ 41 L. Mandel, Optics Comm. 42 (I 982) 437. [ 51J. Mostowski and K. Rzaiewski. Phys. Lett. A 66 (1978) 275. [ 61 D.F. Walls and C.T. Tindle. J. Phys. A5 (1972) 534. [ 71 J. Wagner, P. Kurowski and W. Martienssen, Zs. Phys. 33 ( 1979) 39 I. [ 81 H.J. Kimble and J.L. Hall, J. Opt. Sot. Am. 83 (I 986) P86. [9] P. Chmela, Optics Comm. 42 (1982) 201. [ IO] S. Kielich. M. Kozierowski and R. Tanas. in: Coherence and quantum optics. IV. eds. L. Mandel and E. Wolf ( Plenum. New \rork. 1978) p. 51 I. [ I 1 ] J. Peiina, Quantum statistics of hnear and nonlinear opttcal phenomena (Reidel Publishing C‘ompany. Dordrecht. Boston. Lancaster. 1984) p. 230. [ 121 M. Kozierowski and S. Kielich, Phys. Lett. A 94 (1983) 2 13. [ 131 C.K. Hong and L. Mandel, Phys. Rev. A32 (1985) 974; Phys. Rev. Lett. 54 (1985) 323. [ 141 M. Kozierowski. Phys. Rev. A34 (I 986) 3474. [ 151 R. Loudon, Rep. Prog. Phys. 43 (I 980) 9 13. [ 161 J.F. Clauser, Phys. Rev. D9 (1974) 853. [ 171 M.S. Zubairy. Phys. Lett. A 87 (1982) 162. [ 181 J. Peiinova, Optica Acta 30 (1983) 955. [ 191 K.J. McNeil and C.W. Gardiner, Phys. Rev. A28 (1983) 1560. [ 201 M.D. Reid and D.F. Walls, Phys. Rev. A34 (1986) 1260. [21] M. Hillery, Phys. Rev.A31 (1985) 338. [22] S. Friberg, C.K. Hong and L. Mandel, Phys. Rev. Lett. 54 (1985) 201 I: Optics Comm. 54 (1985) 31 I [ 231 C.K. Hong and L. mandel. Phys. Rev. Lett. 56 (1986) 58.
190
OPTIC’S C‘OMMIINI~‘.~l‘ION\
2
INTERMODAL
Received
CORRELATIONS
27 April 1987: re~lsed manuscript
IN THE GENERATION
rccclved
74 June
OF HIGHER
HARMONICS
:‘Y
I YX7
Correlations between the photons of’the fundamental and harmonic modes are considered. In the short-path the classical Cauchy-Schwarr inequaltiy is shown to be vIolated. Moreover. photon anticorrelation takes place.
approximation.
1. Introduction The problem of the production of photon antibunching. subpoissonian photon statistics and squeezing [ 1.21. intimately related with the quantum nature of light. has attracted much attention during the last decade. Analytical solutions of subpoissonian photon statistics [ 31 and squeezing [ 41 in second-harmonic generation have been found using the short-path approximation. Later, numerical calculations [ 51 have confirmed the analytical solution for subpoissonian photon statistics (SPS). It is worth noting that, already from the earliei numerical results given in ref. [ 61, SPS was evident for the second-harmonic mode. although the authors themselves did not pay attention to this fact. Wagner et al. [ 71 have performed a simulation experiment, the results of which pointed to SPS in socondharmoning generation. Squeezing in the course of second-harmonic generation within a cavity has been revealed quite recently by Kimble and Hall [ 81. Chmela [ 91 has shown that, by generating second harmonics in cascade in thm plates. littering out the generated harmonic behind each plate, SPS in the fundamental beam increases considerably. Analytical solutions for SPS [ 10.1 1 ] and squeezing [ 121 in the generation of an arbitrary kth (A-==?._3...) harmonic have been found as well. Recently, Hong and Mandel [ 131 have introduced the concept of higher-order squeezing. The processes under consideration also seem promising in this respect [ 141. Harmonics generation, as two-beam phenomena, offer moreover other possibilities of testing the quantum theory of light against the classical theory. Namely. we can study intermodal correlations and the validity of various inequalities derivable from the classical theory for the non-delayed two-beam and single-beam correlation functions. In particular, the classical Cauchy-Schwarz inequality for the second-order two-beam correlation function Cl;’ and the single-beam correlation functions Cl,’ ’ and Gj,‘) reads [ 15 ] (Gl”)‘<(;“lG’2’ I, !i
(1)
il
where the subscripts i and j refer to two different light beams. Violation of the inequality (1) has already been observed by Clauser ir This work was supported
186
by Research
[ 161 in two-photon
cascade emission
Project CPBP 0 I .07
0 030-4018/87/$03.jO 0 Elscvier Science Publishers (North-Holland Physics Publishing Division)
R.V.
Volume 64, number
OPTICS
2
15 October
COMMUNICATIONS
1987
and theoretically studied by Loudon [ 151. In the case of the two-photon laser [ 17,181 and parametric amplifier [ 19,201 the above inequality is violated too. Violation of various inequalities classically derivable for second-order correlation functions has been reviewed recently by Reid and Walls [ 201. In this paper we show violation of the inequality (1) in the quantum description of higher-harmonic generation processes and moreover, that anticorrelations between the fundamental and harmonic photons take place.
2. Equations of motion In the electric dipole approximation and for perfect phase matching, the generation the absence of damping is described in terms of the following hamiltonian: HG =fwafaf+fikwa~aA
of the kth harmonic
+cAgh(aXa~+a$“ak) ,
in
(2)
where the suffices f and k refer to the fundamental and kth harmonic mode, respectively, and g, is the mode coupling constant depending on the order of the process and involving appropriate nonlinear characteristics of the medium. The symbol c denotes the light velocity, equal for both beams with respect to phase matching. af and a, (i=f, k) are the photon creation and annihilatioin operators fulfilling the boson commutation rule [a,,
a:1 =d,, .
In the Heisenberg
(3) picture,
the time evolution
of any operator
A is given by the equation
k=(ilA)[H,A].
(4)
For A equal to a,, the commutator with the free-field part of the hamiltonian (2) gives the “rapid” oscillations of the operator a,. In turn, the commutator with the interaction part of the hamiltonian (2) leads to the “slow” part of the evolution of the operator a,, resulting from nonlinear coupling between the beams. Generally, we can write af(t)=af,(t)
exp( -iwt),
ak(t) =ak,(t)
exp( -ikwt)
.
(5)
The slowly varying parts ars( t) of the photon operators satisfy the same commutation rule (3) as the operators a,(t) and af(t) (ij=f, k). Since the problem under consideration is that of propagation of the beams and not that of fields in a cavity, we perform the interchange z= - ct, where z is the direction of propagation of the beams. As a result, the quantum equations of motion for the slowly varying parts of the annihilation operators a,,(z) have the form da,(z)ldz=ikg,
a::-‘(z)
aks(z),
dak,(z)ldz=igk
ats(z) .
(6)
The above equations cannot be solved strictly. We must have recourse to the short-path approximation procedure used by us earlier [ 31. It consists in expanding the operators (or the correlation functions) in a Taylor series above z=O. For relative simplicity of the final results, it is more convenient to calculate the successive derivatives of a,,(z) not with eq. (6), but rather immediately differentiating the preceding derivative. Hence, on bringing the photon operators to normal order within their products, we find
(7)
187
Volume 64, number 7
+(i/6)kg:z’
OPTIC‘S C’OMMIJNI(‘,.lTIONS
:z:,,Q
(71
i”r’)[
where the symbol 0 at the suffices f and k denotes that a,~,= a,,( 0) = a,( 0 1. Moreover, n,,, = a fr,u,~ is the photon-number ial coefficients.
the operators arc taken at ~~0. for instance operator. The symbols ( :“) are Newton binom-
3. Correlation functions and results It is easily checked that the sum of the photon-number
operators
n,(z)+kn,(z)=const
(9)
is an integral of motion. This is because the generation takes place in a nonabsorbing medium. Of course. the expectation value of eq. (9) represents the conservation law for the mean photon numbers. i.e., for the firstorder correlation functions G{,! ‘(z) and G::l( -_). By definition Gj,“(z)=(a:\(z)
U,,(Z))
,
(10)
where the brackets ( > stand for the quantum expectation value calculated with a density operator of the initial fields. The harmonic field is in vacuum state at the input to the medium (==O). so that the following initial conditions are fulfilled: (11)
~Ol~~,,IO~=~Ol~,,,IO~=O. The fundamental field may be initially in an arbitrary state. Hence, eq. (9) leads to the following conservation law Gh!‘(z)+kGl:‘(z)=Gk:‘=(n,,).
(I’)
where (n,,,) is the mean number of the fundamental photons incident on the medium. The nondelayed double-beam second-order correlation function (;I”( z) reads G;;‘(z)=
(a;,(z)
and, in the single-beam Gi;‘(z)=(a:\‘(=)
u:\(z)
a,,(z) u,,(z))
case, transforms
(13) into
u;,(z)).
(14)
On insertion of (8) into the definition (10) and having recourse to the commutation rule (3) and the initial conditions (11). we find the evolution of the mean number of harmonic photons in the form ’ ‘1 +... .
I 8X
(15)
Volume 64, number
OPTICS
2
COMMUNICATIONS
15 October
1987
with G$$=(ah:a&). The first-order correlation function G#)(z) is readily obtained from the conservation law (12). Although the correlation function G,&‘(z) involves the fourth-order term in z, its calculation requires only the third-order approximation in z in the photon operators. Moreover, as we shall see, the second-order correlation function CL:)(z) is calculated within accuracy up to z6. All higher approximations in (8) contribute nothing to the term z4 in Cl:‘(z) or to the terms z4 and z6 in GLzk’(z) as a result of the normal ordering of the harmonic-photon operators in these functions and the condition (11). From (13) at (7) and (8) one gets G&?(z)=g:z’Gtgk+‘)
+z4(
3Gj,fk’ +
Iz;f. r! (;) (“,‘) [GfOZk--'I+(k-
1 -I)G~~-])
+... .
(16)
In order to obtain this function we apply the photon operators a#,( z) and aa( z) calculated only within accuracy to z2. However, calculation of the function Gj$(z), which is identical with the function (16), requires the third-order approximation in the photon operators. The only classical term in the square brackets in eq. (16) is Ggk), obtainable from the function Ggk-r) at r=O. The magnitude of the cross correlations v&z) between the two modesfand k is V,(z)=G;;)(z)-CR)(z)
G&(z).
(17)
Peiina [ 111 has shown for second-harmonic generation that both beams are uncorrelated ( v#,( z) = 0) up to the second-order approximation in z if the incident fundamental field is in a coherent state. For coherent input radiation (then G& ( nfD) k), within accuracy to z4, one gets from (16)) (15) and (12)
(18) The negative value of vrk(z) arises from the purely quantum photons of the fundamental and harmonic beams. In turn, from (14) at (7) and (8) we find
terms and reflects anticorrelation
GW’(z)=G&-k&z2[2Gg+‘)+(k-l)G,$f)]+...,
G@(z)
=&z4Gtik)
‘;;;1
-
between
the
(19)
$8~~ lz;IOr! (;) (“T‘) p! (,“) (k-;-r)
Ggk-‘-“-HI
+... .
(20)
The above forms of the correlation functions have not been presented hitherto by us [ lo]. For an initially coherent fundamental beam, by (19), (20) and (16)) we arrive at
189
Volume 64. number
2
OPTICS
(‘OMMCJNICATIONS
I5 October
I ‘)87
This result, in fact, means violation of the classical Cauchy-Schwarz inequality ( I ) in the process of an arbitrary kth harmonic generation due to the quantum nature of light. Thus for coherent input radiation both the fundamental and harmonic modes after transversing the nonlinear medium exhibit not only subpoissonian photon statistics [ lo] and squeezing [ 4.121 but moreover their correlation properties are such that photon anticorrelations and violation of the Cauchy-Schwarz inequality take place. No well-behaved Glauber-Sudarashan P functions exist for these fields [ 20,2 11. In particular, for second-harmonic generation (A-= 2) the general formulas ( 18) and (2 1 ) reduce to very simple ones, namely V,z(-‘)=-eg~24(n,~,)“.
&(z)=2,!&?‘(n,,,)“.
(22)
Recently, Mandel and coworkers [ 22,231 have observed subpoissonian statistics in the process of spontaneous parametric down conversion; two- and one-photon states have been produced. This allows us to conclude that the nonclassical properties of the processes under consideration can be experimentally verified.
References [ I] D.F. Walls. Nature 280 ( 1979) 45 I; L. Mandel. Physica Scripta T12 (1986) 34. [2] D.F. Walls. Nature 306 (1983) 141. [ 31 M. Korierowski and R. Tanas. Optics Comm. 2 I ( I977 ) 229. [ 41 L. Mandel, Optics Comm. 42 (I 982) 437. [ 51J. Mostowski and K. Rzaiewski. Phys. Lett. A 66 (1978) 275. [ 61 D.F. Walls and C.T. Tindle. J. Phys. A5 (1972) 534. [ 71 J. Wagner, P. Kurowski and W. Martienssen, Zs. Phys. 33 ( 1979) 39 I. [ 81 H.J. Kimble and J.L. Hall, J. Opt. Sot. Am. 83 (I 986) P86. [9] P. Chmela, Optics Comm. 42 (1982) 201. [ IO] S. Kielich. M. Kozierowski and R. Tanas. in: Coherence and quantum optics. IV. eds. L. Mandel and E. Wolf ( Plenum. New \rork. 1978) p. 51 I. [ I 1 ] J. Peiina, Quantum statistics of hnear and nonlinear opttcal phenomena (Reidel Publishing C‘ompany. Dordrecht. Boston. Lancaster. 1984) p. 230. [ 121 M. Kozierowski and S. Kielich, Phys. Lett. A 94 (1983) 2 13. [ 131 C.K. Hong and L. Mandel, Phys. Rev. A32 (1985) 974; Phys. Rev. Lett. 54 (1985) 323. [ 141 M. Kozierowski. Phys. Rev. A34 (I 986) 3474. [ 151 R. Loudon, Rep. Prog. Phys. 43 (I 980) 9 13. [ 161 J.F. Clauser, Phys. Rev. D9 (1974) 853. [ 171 M.S. Zubairy. Phys. Lett. A 87 (1982) 162. [ 181 J. Peiinova, Optica Acta 30 (1983) 955. [ 191 K.J. McNeil and C.W. Gardiner, Phys. Rev. A28 (1983) 1560. [ 201 M.D. Reid and D.F. Walls, Phys. Rev. A34 (1986) 1260. [21] M. Hillery, Phys. Rev.A31 (1985) 338. [22] S. Friberg, C.K. Hong and L. Mandel, Phys. Rev. Lett. 54 (1985) 201 I: Optics Comm. 54 (1985) 31 I [ 231 C.K. Hong and L. mandel. Phys. Rev. Lett. 56 (1986) 58.
190