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Second-harmonic generation as a source of correlated coherent states
Volume 69, number 1
OPTICS COMMUNICATIONS
1 December 1988
S E C O N D - H A R M O N I C G E N E R A T I O N AS A SOURCE OF CORRELATED C O H E R E N T STATES ~ M. KOZIEROWSKI ~ and V.I. M A N ' K O P.N. Lebedev Institute of Physics, Academy of Sciences USSR, 117-924 Moscow, Leninsky Prospect 53, USSR
Received 18 July 1988
The field of the fundamental mode in the course of second-harmonic generation is shown to be in a correlated coherent state. Optimum squeezing occurs for the uncorrelated field quadrature components. Only then the field of the mode under consideration is in a minimum Heisenberg'suncertainty squeezed state.
1. Introduction
The last decade witnessed an intense search for photon antibunching, sub-poissonian photon statistics and squeezing which are a direct manifestation of the quantum nature of light. Harmonic-generation processes yield light which simultaneously exhibits sub-poissonian photon statistics [ 1,2 ], squeezing [ 3,4 ] and higher-order squeezing [ 5-7 ]. Moreover, both the fundamental and harmonic modes after traversing the nonlinear medium reveal correlation features such that the Cauchy-Schwarz inequality is violated and anticorrelation between the photons of the modes takes place [ 8 ]. It was also shown that by generating harmonics in cascade in thin plates and filtering out the generated harmonic behind every plate, sub-poissonian photon statistics [9] and squeezing [10] in the fundamental mode increase considerably. Recently, squeezing in the course of the secondharmonic generation within a cavity has been observed [ 11 ]. A new class of the states of the electromagnetic field corresponding to the minimum of the Sehr6dingerRobertson uncertainty principle [ 12,13 ] and termed correlated coherent states has recently been introThis work was sponsored in part by the Polish Central Project for Fundamental Research CPBP 01.06. Permanent address: Institute of Physics, A. Mickiewicz University, 60-780 Poznafi, Poland.
duced and investigated [ 14]. It is our aim to show that in the process of second-harmonic generation the field of the fundamental mode is in such a correlated coherent state. Simultaneously, one of the field quadrature components may be squeezed. Optimum squeezing occurs for the uncorrelated quadrature components. Then the field of the fundamental mode is in a particular correlated coherent state corresponding to the minimum of the Heisenberg uncertainty principle. Such states are commonly termed squeezed states [ 15-17 ]. In fact, such states had already been considered by Glauber [ 18 ] and termed minimum uncertainty states. Although squeezing in the harmonic-generation processes has been intensively examined [3,4,10] no paper has touched the problem of the uncertainty principles.
2. Definitions
We discuss here a single mode of the radiation field with frequency to and annihilation and creation operators a and a t. It is useful to introduce the slowly varying variables as and a t as = a exp(itot),
a*~ = a t exp( - i t o t ) ,
(1)
which obey the same boson commutation relation as the operators a and a*, [as, a~ ] = 1, but do not oscillate at optical frequencies. Phase-sensitive detectors respond to the envelope of the optical field only.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
71
Volume 69, number I
OPTICS COMMUNICATIONS
The canonical quantities
Q=a~+a*~, P = - i ( a s - a * ~ ) ,
(2)
correspond to two slowly varying quadrature components of the field. Usually, the quantum fluctuations associated with this pair of hermitian operators are considered as limited by the Heisenberg uncertainty principle
1 December 1988
tainly principle for the field quadrature components reads [ 14 ] ¢reao( 1 - 7 2) >i 1 ,
(9)
where
7 = ( ½ ( P Q + Q P ) - ( P ) ( Q ) )/(apao) ~/2 ,
(10)
where ap= ( ( A P ) 2) and g o = ( ( A Q ) 2) denote the variances of P and Q. The minimum uncertainty states satisfying one of the following inequalities
is the correlation degree of the quadrature components and I y I e[0, 1]. The states that satisfy the equality in eq. (9) are termed correlated coherent states [ 14 ]. It was shown [14] that squeezed states are a particular class of correlated coherent states with y=0. With respect to (6), from (9) we get
a~,
A(A+2) -B2~> 72/( 1 - y2).
at, a~ ~> 1 ,
(3)
orao
(4)
are termed squeezed states [ 15-18 ]. If one of the above inequalities is satisfied but the product of these variances is greater than the minimum of the uncertainty relation ( 3 ) we talk about squeezing in one of the field quadratures. The quantum fluctuations in one of the field quadratures are then smaller than those associated with a coherent field or with the vacuum. Obviously, the uncertainty principle requires enhanced fluctuations in the canonically conjugate quadrature component. The variances of Q and P may be presented in the form
aQ=I+A+B,
ap=l+A-B,
(5)
The above relation requires for 7# 0 enhanced fluctuations, at least in one of the quadrature components, in comparison with the relation (7). The ratio A~ IBI is simply implied to be greater than that allowed by (7). This means directly that correlations of the quadrature components degrade squeezing.
3. E q u a t i o n s
of motion
In the absence of damping and at perfect phase matching, the generation of the second-harmonic is described by the following hamiltonian (h = 1 )
H=toa~taf+2oJa*hah +cg(a*ha2f+a[2ah).
where
( 11 )
(12)
The quantity A is non-negative. With respect to the inequalities (4) one of the quadrature components is squeezed if
The subscripts f and h refer to the fundamental and harmonic mode, respectively. The factor g is the mode coupling coefficient defined by the nonlinear susceptibility tensor of the medium. The symbol c denotes the light velocity, equal for both beams with respect to phase matching. 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 = - c t , were z is the direction of propagation of the beams. Then the Heisenberg equations of motion for the slowly varying operators afs and ahs take the form
A/IBI
d a J d z = 2iga~ahs,
A=2((n)-(a*~)(a~)), B = ((Aa~*) 2) + ((Aas) 2 ) .
(6)
Here n = a*~as is the photon-number operator. From (3) and (5) it is evident that the following general condition must be satisfied, A(A+2)-B2>~0.
(7)
(8)
Then Q is squeezed for B < 0 while P is squeezed for B > 0. Squeezing increases as the ratio (8) decreases. The generalized Schr6dinger-Robertson uncer72
dahs/dz=iga2s.
(13)
In order to solve the above set of equations we must have recourse to the short-path approximation procedure [ 1,3 ]. It consists in expanding the operators
Volume 69, number 1
OPTICS COMMUNICATIONS
in a Taylor series about z = 0 , where z is the path traversed by the beams in the nonlinear medium. For paths greater than z following the interaction, we must assume that both field modes evolve as free, non-interacting modes. It was shown that for the coherent input radiation the fundamental mode reveals sub-poissonian photon statistics [1] and squeezing [3] already in the second-order approximation in z. However, in order to examine the uncertainty principles (3) and (9) we must solve afs(z) within accuracy up to z 4. On bringing the photon operators to normal order within their products, the power-series solution of the operator ars, sufficient for our further calculations, is aft(z) =afo + 2igza~oaho + g2z2 ( 2nho - nfo )afo + ~ig3z3(2nhoahoa~o - 3ahoa~nho + 2a~oa 3 --ahoa~o) I 4 4 t2 2 + gg z (4ahoahoafo -- 28nhonfoa~ -- 20nhoafo
+ 8aZoa~ + 5 a ~ a 3 +nfoafo) + ....
(14)
The subscript 0 denotes that the operators are taken at z = 0 and, for instance, aio = a~(0) = a~ (0). As always, operators associated with the two different modes commute at the same z, in particular at z = 0 . The harmonic field is in the vacuum state at the input to the medium, so that the following initial conditions are fulfilled: ahol0 ) = nho[ 0) = 0. The fundamental field may be initially in an arbitrary state. In what follows, we assume that the fundamental field is initially in a coherent state l a ) , afola)=a[a), and a = ( n f o ) ' / Z e x p ( i O ) , where (nfo) is the mean number of photons incident on the medium and 0 is the initial phase of a.
It is clear from (16) that, depending on the phase angle 0, one or the other quadrature component may be squeezed. Optimum squeezing, that is the smallest values of tref and tref occur for 0= 0, rt and 0 = it/ 2, 3rc/2, respectively. It is obvious from the forms of the variances (16) that their product is less than unity. This might suggest a violation of the uncertainty principles (3) and (9). However, the deviation from unity is proportional to z 4. In the solution (16) the terms proportional to z 4 are omitted. They ought to be taken into account if one wishes to obtain the correct form of the uncertainty principle in this order of approximation. The solutions (16) permit to examine the uncertainty relation within accuracy to the secondorder approximation in z only. Then trpftrQf= 1, and the field of the fundamental mode, accurate to z 2, might seem to be in a minimum Heisenberg's uncertainty state, i.e., in a squeezed state. In fact, however, the correlation degree ~'sis in general non-zero already in the second-order approximation in z. This suggests that the field of the fundamental mode is in a correlated state rather than in a minimum Heisenberg's uncertainty state. Within accuracy to the fourth-order approximation (14) we get Af = 2g424 ( nf0 ) 2,
Br= - 2 g Z z Z ( n f o ) cos 20 × [1-~g2z2(6(nfo)+l)]
At the above initial conditions and within accuracy to z : it was shown by Mandel [ 3 ] that
trQf = 1 -- 2g2z 2 ( nfo ) COS20
q-2g4z 4 ('nfo)2 ( 1 +cos 20) (
nfo ) cos 20,
+2g4z 4 (nfo)2 ( 1 - c o s 20) - ]g4z4 < nfo > cos 20.
(18)
(15)
After simple algebra, retaining the terms only of order z 4, we find that the product of the above variances is for the majority of phases 0 greater than unity and amounts to
(16)
treftrQf= 1 + 4g4z 4 (nro) 2 sin220.
Hence o'Qf= 1 -- 2g2z 2 ( nfo ) cos 20, ap~= 1 + 292z2 ( nfo ) cos 20.
(17)
tree = 1 + 2g2z 2 ( nro ) cos 20
4. Results
Bf=-2gez 2 (nro) cos20.
.
Hence from (5) we arrive at
"at"I g 4 z 4
At=0,
1 December 1988
(19) 73
Volume 69, number 1 In t u r n , f r o m (10, ( 1 4 ) a n d
OPTICS COMMUNICATIONS ( 1 8 ) w e get
7f = - 2g 2z2 ( nro ) sin 20 × ( 1 -l- 4g4z 4 ( n f o ) 2 sin 2 20) - 1 / 2 .
(20)
Since yf a p p e a r s in eq. ( 9 ) in the s e c o n d p o w e r it is sufficient to r e t a i n in the n u m e r a t o r the t e r m s o n l y o f o r d e r z 2. Thus, w i t h i n accuracy to z 4, f r o m ( 9 ) , ( 1 9 ) a n d ( 2 0 ) we find
ap,,aQ,( 1 - 72 ) = 1 .
(21)
T h i s m e a n s that the f u n d a m e n t a l m o d e is in a correlated c o h e r e n t state with the c o r r e l a t i o n degree given by eq. ( 2 0 ) . T h e o p t i m u m choice o f 0 for c o r r e l a t i o n is 0 = g / 4 + k g / 2 ( k = 0 , 1, 2, 3). T h e n , however, the field is n o t squeezed, n a m e l y
ap,.=aQf= 1 -{-2gaz 4 ( n ~ o ) 2 .
(22)
In turn, in the case o f the earlier discussed o p t i m u m squeezing, 7r=0. O n l y t h e n the field o f the f u n d a m e n t a l m o d e is in a s q u e e z e d state c o r r e s p o n d i n g to the m i n i m u m o f the H e i s e n b e r g u n c e r t a i n t y relation. F o r 0 # 0 , n / 2 , ~z a n d 3rt/2 the field o f this m o d e r e m a i n s in a correlated c o h e r e n t state c o r r e s p o n d i n g to the m i n i m u m o f the S c h r 6 d i n g e r - R o b e r t s o n u n c e r t a i n t y relation. O n e o f the q u a d r a t u r e s is still squeezed if a d d i t i o n a l l y 0 # r c / 4 + k g / 2 ( k = 0 , l, 2, 3) b u t t h e n s q u e e z i n g is n o t m a x i m a l . To c o n c l u d e briefly, by control over the phase 0 we
74
1 December 1988
can get the field o f the f u n d a m e n t a l m o d e in a s q u e e z e d or u n s q u e e z e d correlated c o h e r e n t state, or in a m i n i m u m H e i s e n b e r g ' s u n c e r t a i n t y squeezed state.
References [ 1 ] M. Kozierowski and R. Tanag, Optics Comm. 21 I977) 229. [2] J. Mostowski and K. Rza,2ewski, Phys. Lett. A 66 1978) 275. [ 3 ] L. Mandel, Optics Comm. 42 ( 1982 ) 437. [4] M. Kozierowski and S. Kielich, Phys. Left. A 94 1983) 213. [5] C.K. Hong and L. Mandel, Phys. Rev. A32 (1985) 974: Phys. Rev. Lett. 54 (1985) 323. [6] M. Kozierowski, Phys. Rev. A34 (1986) 3464. [ 7 ] M. Hillery, Optics Comm. 62 ( 1987 ) 135. [8]M. Kozierowski. Phys. Rev. A36 (1987) 2973: Optics Comm. 64 (1987) 186. [ 9 ] P. Chmela, R. Hor~ikand J. Pehna, Optica Acta 28 ( 1981 ) 1209; P. Chmela, Optics Comm. 42 (1982) 201. [ 10] P. Chmela, M. Kozierowski and S. Kielich, Czech. J. Phys. B37 (1987) 846. [ 11 ] H.J. Kimble and J.L. Hall, J. Opt. Soc. Am. 83 (1983) 86. [ 12] E. Schr6dinger, Berliner Berichte (1930) p. 296. [ 13 ] H.P. Robertson, Phys. Rev. 35 (1930) 667A. [ 14] V.V. Dodonov, E.V. Kurmyshev and V.I. Man'ko, Phys. Lett. A79 (1980) 150, Trudy Fian 176 ( 1986 ) 57: V.V. Dodonov and V.I. Man'ko. Trudy Fian 183 ( 1987 ) 5. [15] D. Stoler, Phys. Rev. DI (1970) 3217. [ 16] H.P. Yuen, Phys. Rev. AI3 (1976) 2226. [ 17] D.F. Walls, Nature 306 (1983) 141. [ 18] R.J. Glauber, Phys. Rev. 131 (1963) 2766.
OPTICS COMMUNICATIONS
1 December 1988
S E C O N D - H A R M O N I C G E N E R A T I O N AS A SOURCE OF CORRELATED C O H E R E N T STATES ~ M. KOZIEROWSKI ~ and V.I. M A N ' K O P.N. Lebedev Institute of Physics, Academy of Sciences USSR, 117-924 Moscow, Leninsky Prospect 53, USSR
Received 18 July 1988
The field of the fundamental mode in the course of second-harmonic generation is shown to be in a correlated coherent state. Optimum squeezing occurs for the uncorrelated field quadrature components. Only then the field of the mode under consideration is in a minimum Heisenberg'suncertainty squeezed state.
1. Introduction
The last decade witnessed an intense search for photon antibunching, sub-poissonian photon statistics and squeezing which are a direct manifestation of the quantum nature of light. Harmonic-generation processes yield light which simultaneously exhibits sub-poissonian photon statistics [ 1,2 ], squeezing [ 3,4 ] and higher-order squeezing [ 5-7 ]. Moreover, both the fundamental and harmonic modes after traversing the nonlinear medium reveal correlation features such that the Cauchy-Schwarz inequality is violated and anticorrelation between the photons of the modes takes place [ 8 ]. It was also shown that by generating harmonics in cascade in thin plates and filtering out the generated harmonic behind every plate, sub-poissonian photon statistics [9] and squeezing [10] in the fundamental mode increase considerably. Recently, squeezing in the course of the secondharmonic generation within a cavity has been observed [ 11 ]. A new class of the states of the electromagnetic field corresponding to the minimum of the Sehr6dingerRobertson uncertainty principle [ 12,13 ] and termed correlated coherent states has recently been introThis work was sponsored in part by the Polish Central Project for Fundamental Research CPBP 01.06. Permanent address: Institute of Physics, A. Mickiewicz University, 60-780 Poznafi, Poland.
duced and investigated [ 14]. It is our aim to show that in the process of second-harmonic generation the field of the fundamental mode is in such a correlated coherent state. Simultaneously, one of the field quadrature components may be squeezed. Optimum squeezing occurs for the uncorrelated quadrature components. Then the field of the fundamental mode is in a particular correlated coherent state corresponding to the minimum of the Heisenberg uncertainty principle. Such states are commonly termed squeezed states [ 15-17 ]. In fact, such states had already been considered by Glauber [ 18 ] and termed minimum uncertainty states. Although squeezing in the harmonic-generation processes has been intensively examined [3,4,10] no paper has touched the problem of the uncertainty principles.
2. Definitions
We discuss here a single mode of the radiation field with frequency to and annihilation and creation operators a and a t. It is useful to introduce the slowly varying variables as and a t as = a exp(itot),
a*~ = a t exp( - i t o t ) ,
(1)
which obey the same boson commutation relation as the operators a and a*, [as, a~ ] = 1, but do not oscillate at optical frequencies. Phase-sensitive detectors respond to the envelope of the optical field only.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
71
Volume 69, number I
OPTICS COMMUNICATIONS
The canonical quantities
Q=a~+a*~, P = - i ( a s - a * ~ ) ,
(2)
correspond to two slowly varying quadrature components of the field. Usually, the quantum fluctuations associated with this pair of hermitian operators are considered as limited by the Heisenberg uncertainty principle
1 December 1988
tainly principle for the field quadrature components reads [ 14 ] ¢reao( 1 - 7 2) >i 1 ,
(9)
where
7 = ( ½ ( P Q + Q P ) - ( P ) ( Q ) )/(apao) ~/2 ,
(10)
where ap= ( ( A P ) 2) and g o = ( ( A Q ) 2) denote the variances of P and Q. The minimum uncertainty states satisfying one of the following inequalities
is the correlation degree of the quadrature components and I y I e[0, 1]. The states that satisfy the equality in eq. (9) are termed correlated coherent states [ 14 ]. It was shown [14] that squeezed states are a particular class of correlated coherent states with y=0. With respect to (6), from (9) we get
a~,
A(A+2) -B2~> 72/( 1 - y2).
at, a~ ~> 1 ,
(3)
orao
(4)
are termed squeezed states [ 15-18 ]. If one of the above inequalities is satisfied but the product of these variances is greater than the minimum of the uncertainty relation ( 3 ) we talk about squeezing in one of the field quadratures. The quantum fluctuations in one of the field quadratures are then smaller than those associated with a coherent field or with the vacuum. Obviously, the uncertainty principle requires enhanced fluctuations in the canonically conjugate quadrature component. The variances of Q and P may be presented in the form
aQ=I+A+B,
ap=l+A-B,
(5)
The above relation requires for 7# 0 enhanced fluctuations, at least in one of the quadrature components, in comparison with the relation (7). The ratio A~ IBI is simply implied to be greater than that allowed by (7). This means directly that correlations of the quadrature components degrade squeezing.
3. E q u a t i o n s
of motion
In the absence of damping and at perfect phase matching, the generation of the second-harmonic is described by the following hamiltonian (h = 1 )
H=toa~taf+2oJa*hah +cg(a*ha2f+a[2ah).
where
( 11 )
(12)
The quantity A is non-negative. With respect to the inequalities (4) one of the quadrature components is squeezed if
The subscripts f and h refer to the fundamental and harmonic mode, respectively. The factor g is the mode coupling coefficient defined by the nonlinear susceptibility tensor of the medium. The symbol c denotes the light velocity, equal for both beams with respect to phase matching. 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 = - c t , were z is the direction of propagation of the beams. Then the Heisenberg equations of motion for the slowly varying operators afs and ahs take the form
A/IBI
d a J d z = 2iga~ahs,
A=2((n)-(a*~)(a~)), B = ((Aa~*) 2) + ((Aas) 2 ) .
(6)
Here n = a*~as is the photon-number operator. From (3) and (5) it is evident that the following general condition must be satisfied, A(A+2)-B2>~0.
(7)
(8)
Then Q is squeezed for B < 0 while P is squeezed for B > 0. Squeezing increases as the ratio (8) decreases. The generalized Schr6dinger-Robertson uncer72
dahs/dz=iga2s.
(13)
In order to solve the above set of equations we must have recourse to the short-path approximation procedure [ 1,3 ]. It consists in expanding the operators
Volume 69, number 1
OPTICS COMMUNICATIONS
in a Taylor series about z = 0 , where z is the path traversed by the beams in the nonlinear medium. For paths greater than z following the interaction, we must assume that both field modes evolve as free, non-interacting modes. It was shown that for the coherent input radiation the fundamental mode reveals sub-poissonian photon statistics [1] and squeezing [3] already in the second-order approximation in z. However, in order to examine the uncertainty principles (3) and (9) we must solve afs(z) within accuracy up to z 4. On bringing the photon operators to normal order within their products, the power-series solution of the operator ars, sufficient for our further calculations, is aft(z) =afo + 2igza~oaho + g2z2 ( 2nho - nfo )afo + ~ig3z3(2nhoahoa~o - 3ahoa~nho + 2a~oa 3 --ahoa~o) I 4 4 t2 2 + gg z (4ahoahoafo -- 28nhonfoa~ -- 20nhoafo
+ 8aZoa~ + 5 a ~ a 3 +nfoafo) + ....
(14)
The subscript 0 denotes that the operators are taken at z = 0 and, for instance, aio = a~(0) = a~ (0). As always, operators associated with the two different modes commute at the same z, in particular at z = 0 . The harmonic field is in the vacuum state at the input to the medium, so that the following initial conditions are fulfilled: ahol0 ) = nho[ 0) = 0. The fundamental field may be initially in an arbitrary state. In what follows, we assume that the fundamental field is initially in a coherent state l a ) , afola)=a[a), and a = ( n f o ) ' / Z e x p ( i O ) , where (nfo) is the mean number of photons incident on the medium and 0 is the initial phase of a.
It is clear from (16) that, depending on the phase angle 0, one or the other quadrature component may be squeezed. Optimum squeezing, that is the smallest values of tref and tref occur for 0= 0, rt and 0 = it/ 2, 3rc/2, respectively. It is obvious from the forms of the variances (16) that their product is less than unity. This might suggest a violation of the uncertainty principles (3) and (9). However, the deviation from unity is proportional to z 4. In the solution (16) the terms proportional to z 4 are omitted. They ought to be taken into account if one wishes to obtain the correct form of the uncertainty principle in this order of approximation. The solutions (16) permit to examine the uncertainty relation within accuracy to the secondorder approximation in z only. Then trpftrQf= 1, and the field of the fundamental mode, accurate to z 2, might seem to be in a minimum Heisenberg's uncertainty state, i.e., in a squeezed state. In fact, however, the correlation degree ~'sis in general non-zero already in the second-order approximation in z. This suggests that the field of the fundamental mode is in a correlated state rather than in a minimum Heisenberg's uncertainty state. Within accuracy to the fourth-order approximation (14) we get Af = 2g424 ( nf0 ) 2,
Br= - 2 g Z z Z ( n f o ) cos 20 × [1-~g2z2(6(nfo)+l)]
At the above initial conditions and within accuracy to z : it was shown by Mandel [ 3 ] that
trQf = 1 -- 2g2z 2 ( nfo ) COS20
q-2g4z 4 ('nfo)2 ( 1 +cos 20) (
nfo ) cos 20,
+2g4z 4 (nfo)2 ( 1 - c o s 20) - ]g4z4 < nfo > cos 20.
(18)
(15)
After simple algebra, retaining the terms only of order z 4, we find that the product of the above variances is for the majority of phases 0 greater than unity and amounts to
(16)
treftrQf= 1 + 4g4z 4 (nro) 2 sin220.
Hence o'Qf= 1 -- 2g2z 2 ( nfo ) cos 20, ap~= 1 + 292z2 ( nfo ) cos 20.
(17)
tree = 1 + 2g2z 2 ( nro ) cos 20
4. Results
Bf=-2gez 2 (nro) cos20.
.
Hence from (5) we arrive at
"at"I g 4 z 4
At=0,
1 December 1988
(19) 73
Volume 69, number 1 In t u r n , f r o m (10, ( 1 4 ) a n d
OPTICS COMMUNICATIONS ( 1 8 ) w e get
7f = - 2g 2z2 ( nro ) sin 20 × ( 1 -l- 4g4z 4 ( n f o ) 2 sin 2 20) - 1 / 2 .
(20)
Since yf a p p e a r s in eq. ( 9 ) in the s e c o n d p o w e r it is sufficient to r e t a i n in the n u m e r a t o r the t e r m s o n l y o f o r d e r z 2. Thus, w i t h i n accuracy to z 4, f r o m ( 9 ) , ( 1 9 ) a n d ( 2 0 ) we find
ap,,aQ,( 1 - 72 ) = 1 .
(21)
T h i s m e a n s that the f u n d a m e n t a l m o d e is in a correlated c o h e r e n t state with the c o r r e l a t i o n degree given by eq. ( 2 0 ) . T h e o p t i m u m choice o f 0 for c o r r e l a t i o n is 0 = g / 4 + k g / 2 ( k = 0 , 1, 2, 3). T h e n , however, the field is n o t squeezed, n a m e l y
ap,.=aQf= 1 -{-2gaz 4 ( n ~ o ) 2 .
(22)
In turn, in the case o f the earlier discussed o p t i m u m squeezing, 7r=0. O n l y t h e n the field o f the f u n d a m e n t a l m o d e is in a s q u e e z e d state c o r r e s p o n d i n g to the m i n i m u m o f the H e i s e n b e r g u n c e r t a i n t y relation. F o r 0 # 0 , n / 2 , ~z a n d 3rt/2 the field o f this m o d e r e m a i n s in a correlated c o h e r e n t state c o r r e s p o n d i n g to the m i n i m u m o f the S c h r 6 d i n g e r - R o b e r t s o n u n c e r t a i n t y relation. O n e o f the q u a d r a t u r e s is still squeezed if a d d i t i o n a l l y 0 # r c / 4 + k g / 2 ( k = 0 , l, 2, 3) b u t t h e n s q u e e z i n g is n o t m a x i m a l . To c o n c l u d e briefly, by control over the phase 0 we
74
1 December 1988
can get the field o f the f u n d a m e n t a l m o d e in a s q u e e z e d or u n s q u e e z e d correlated c o h e r e n t state, or in a m i n i m u m H e i s e n b e r g ' s u n c e r t a i n t y squeezed state.
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