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A comparison between degenerate paramagnetic oscillators and two-photon correlated-spontaneous-emission lasers
Volume 80, number 5,6
OPTICS COMMUNICATIONS
15 January 1991
A comparison between degenerate paramagnetic oscillators and two-photon correlated-spontaneous-emission lasers Shi Yao Zhu
Centerfor AdvancedStudies, and Departmentof Physicsand Astronomy, Universityof New Mexico, Albuquerque,NM 87131, USA Received 2 March 1990; revised manuscript received 17 September 1990
The degenerate parametric oscillator (DPO) is analyzed with the nonlinear material treated quantum mechanically. The source of squeezing in the DPO is the same as in the two-photon correlated-spontaneous-emission laser (CEL) - the level coherence. In this sense, they are similar. The amount of squeezing in DPO is the same as in the two-photon CEL at two-photon resonance, but the gain inDPO is usually much smaller than that in the two-photon CEL at two-photon resonance.
There are m a n y similarities between a degenerate p a r a m e t r i c oscillator ( D P O ) [1,2] a n d a two-photon correlated-spontaneous-emission laser ( C E L ) [ 3,4]. Both have the capability to p r o d u c e squeezed state light. A m a x i m u m 50% intracavity squeezing can be o b t a i n e d at threshold, a n d the squeezing decreases above threshold for both cases. The similarities m o t i v a t e us to investigate the connection between them. In D P O , a strong laser at frequency 20) p u m p s a nonlinear material in a cavity to p r o d u c e an output field at frequency o). Both the p u m p i n g field a n d the o u t p u t field are far o f f r e s o n a n t from the levels o f the nonlinear material, so that the spontaneous emission is negligible. The o u t p u t field p r o d u c e d by D P O could be in a squeezed state. The source o f squeezing is usually considered as a t w o - p h o t o n transition. In the t w o - p h o t o n CEL, a t o m s are p r e p a r e d in a coherent superposition o f upper and lower levels, which correlates the spontaneous emission, a n d i f the level coherence is large enough, the laser field can be in a squeezed state. The D P O can be considered to be composed o f two processes: ( i ) the p u m p i n g laser prepares the nonlinear material to a certain state; (ii) the material produces a laser field at frequency o). Is the first process a p r e p a r a t i o n o f the level coherence a n d the seco n d the correlated-spontaneous-emission lasing process? In o r d e r to answer this question, we investigate
the D P O in a different way, where the material and the output signal field are treated q u a n t u m mechanically, while the p u m p i n g laser field is treated classically by assuming that the p u m p i n g light is in an intense coherent state a n d not depleted significantly
[5]. In DPO, the nonlinear materials are far off resonant, i.e., all detunings are much greater than decay rates. The D P O is a kind o f k t2) effect. I f the frequency difference o)~c between the first excited level l a ) and the ground level I c ) is near the frequency o f the p u m p i n g field 2o), I o)ac -- 2o)l <
(1)
which is a p p r o x i m a t e l y true for c o m m o n l y used nonlinear materials in D P O , such as LiNbO3 [6]. The k t2) coefficient can be written as [7] k~,) (o) = 2 o ) - o))
=_N~2
[
(ri)¢b(rk)b~(rj)a¢ (o)ac - 2o)) (o)b~-- W)
(ri)b~(rk )~b(rj)a~ -1
(2)
+ (O)ac - 2o)) ( o ) - O~ab)J ' where the s u m m a t i o n Yb is over all levels, i.e. the level Ib ) may be above, below [a ) and even the same as the levels [a ) and [c ) . It is very clear that the field o f 2o) is associated with the transition between [c)
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
393
Volume 80, number 5,6
OPTICS COMMUNICATIONS
and l a ) , the field of 09 associated with the transitions between Ic) and Ib ) and between Ia ) and Ib ) . For the commonly used nonlinear materials in DPO, their band gaps are larger than 2hoJ (green light), i.e., there is no level between la) and Ic). The only possibility of the two photon transition from l a ) to Ic ) is the following: one photon transition from la) to a level Ib), which is above la) ( o r l a ) itself), and another one photon transition from [b) to Ic ). For simplicity, we consider only one level Ib ) above level Ia). The one photon transition from Ia ) to Ic ) at frequency 09 has no contribution to the squeezing and has been neglected. Here we simplify the ground band and first excited bands as levels. It is a good approximation if we assume that the band widths are much smaller than the band gap. Therefore, we simplify our model to a three level system, as shown in fig. 1. The system is similar to that used in ref. [5 ]. However, we consider Ib ) above Ia ) ( Ic) ground state), and far off two-photon transition. The eigenfrequencies of Ia ) , Ib ) and Ic ) are co~, Ogb and o9¢, respectively. For the present simplified model, we assume that the level l a ) decays to Ic) with a rate Yl, and the level Ib) decays to [a) and Ic) with rates Y2and 73, respectively. The decay rates of off-diagonal elements are 7pa, (fl~: fl' ). Because of eq. (1), we can make rotating wave approximation on the pumping field, but not on the signal field. The hamiltonian for such a field-atom system is
n=h
~
oJplfl)(fll+hl2a*a+h[ala)(cl
15 January 1991
pumping field, which is assumed in a coherent state all the time; and £2 is the bare cavity frequency. In the following discussion, we keep the pumping field to all order, and the signal field to first order, which is enough for the discussion on phase locking, phase noise, and linear gain of the signal field. By using the standard techniques of the laser quantum theory [8], the master equation for the reduced density matrix of the signal field is obtained, under the condition IA~I= Itoab-~ol and IAzi = Io~,c-oglAol = IoJac2o91, to be
/~= ½A[Ll aa~(paa*--a*pa ) + L2acc(a *ap-apa* ) + L~ a~¢(patat-a*pa*) + Lzaac(a*a*p-a*pa*) ] - ½F(pata,apa*) -i(O-og)atap+h.c.,
(4)
where A = 2 N g 2 / ~ with N the atom number in the cavity, LI = ( k + i~l ) - 1, L2 = k( 1 + ikfi2) - l, ~i = d,/~b~b~, k= X/~bTbc, and F is the cavity loss rate. Here apa, (the zeroth order solution) represent the level populations and coherence, which are produced by the interaction between the pumping field and the nonlinear material. They are (2Y~¢/71) lal 2 cr~ = 1 -~rc~ = 7~+d~+(4y~/y~)lot[2, a~=
2
-ia(Ta¢ -iAo) 2
7~c + A o + ( 4 7 ~ c / 7 1 ) l a l 2 "
(5a) (5b)
The zeroth order solution (all order in pumping field or) can be considered as the results of the process of level preparation. From eqs. (5), we find
r = a,b,c
+g~ (a+a t) la) (bl +g2(a+a ~) Ib) (cl + h.c.], (3) where a t (a) is the creation (annihilation) operator for the signal field; ot is the Rabi frequency of the
Iaac [2= tTcctraa-- tra2a .
(6a)
In DPO experiments, we have IAol >> Iotl to guarantee no significant population in any excited level. Under the approximation 2 o
I+Az+B
= I + A Ao
(A, B constants), eq. (6a) become l a > ~ /
laac I2= trcc~Taa,
IC>
which means an approximately full coherence between Ia ) and Ic). Therefore, the pumping of a strong laser at off-resonance is really a preparation of the level coherence. The corresponding Fokker-Planck equation for the
t
,
Fig. 1. 394
(6b)
Volume 80, number 5,6
OPTICS COMMUNICATIONS
Glauber P-representation in polar coordinates (al 8> = 81 ~>, 8 = r e i#) can be obtained. The drift and phase-diffusion coefficients are [. 0, O'aa
d'=(~2-v)-~ALk2+6~
k202 O'cc]
i +---~J
-½AxIGc Is i n ( 2 O - G ~ - 0o), A [ kaaa dr = -~ L k ~ Z
(7a)
kGc
1+ k262 (7b)
+ x l G c Icos ( 2 0 - G~ - 0 o ) ] ,
A r kaaa D° = -4r~ Lk2+012
k2[ trac [ x(1 +k202)
( 1 l+0,02~] k-~6-~- ] J ' (7c)
where x exp ( - i00) = L j - L2, O'ac = I O'ac [ exp ( - iGc). From the phase drive coefficient, we find the phase of the signal field is locked at 2~o-Gc-0o =0.
(8)
In order to have positive gain, from the amplitude drive coefficient eq. (Tb), we need laacl > k o.cc - x(l_t_k2022)
kaaUacc x(k2+# 2) •
(9)
We notice that the phase squeezing could occur as long as the second term in eq. (4c) is larger than the first one. According to the Fokker-Planck equation and by expanding d o around 0o to the first order [ 3 ], we can obtain the steady state phase variance with positive gain
1 {1 ((OO)z) >- ~r 2 X[I+~,02_02
-
k2
+ (l_k2)+kz(Ol--02) k2 +
2
x/~cc
(k2+&~)x/a~-~a:~cd) '
(10)
where the first term 1/4r 2 is the shot noise. Here eqs. (9) and (6) have been used. Since Idl I, Id2l >> Idol, we have 01 = -02. For 1011, ]0zl >> 1, k, eq. (10) reduces to
((t~0) 2) >- 1/8r 2 ,
15 January 1991 (ll)
where the same approximation to get eq. (6b) has been used. Eq. (11) means that a maximum 50% squeezing can be achieved at threshold, the well known result [ 1,2]. Recalling eq. (7c), we see that the phase diffusion coefficient would never be negative and there would be no squeezing, if there were no level coherence ( G c = 0 ) . The total noise is the sum of vacuum fluctuation (shot noise) and spontaneous fluctuation. The negligible upper level population merely leads to the noise quenching, i.e., no spontaneous fluctuation but vacuum fluctuation still exists. Therefore, it is the level coherence that leads to the noise squeezing. The squeezing in the DPO is also from the level coherence, which correlates the spontaneous emission. The two-photon emission process of the nonlinear material is indeed a two-photon correlated-spontaneous emission lasing one. From this point of view (spontaneous emission correlated by level coherence), we would conclude that the DPO is similar to the two-photon CEL. Next, we turn to discuss the difference between the DPO and the original two-photon CEL [3]. The DPO operates at far off resonance, while the twophoton CEL can operate at resonance. Consequently, the gain in two-photon CEL (operating at two-photon resonance) is much greater than that in DPO provided other conditions remain the same. The coherence between the upper and lower levels in the two-photon CEL can be created by some means such as resonant Raman process [9,10]. Another difference between them is that the phase sensitive terms in the master equation (eq. ( 4 ) ) are proportional to Ll or L2, while these terms in the original two-photon CEL (eq. ( I ) in ref. [ 3 ] ) are proportional to L,Lo o r L2L o. The key difference is the factor Lo. At two-photon resonance, Lo = l, these terms in the two master equations are the same. However, these terms in the DPO are much greater than that in the original two-photon CEL at far off two-photon resonance, because of ILol << I. This is the reason why the DPO can create large squeezing while the original two-photon CEL can't at far off two-photon resonance. In the above, we choose the level l b ) above level Ia ) . For the case where Ib ) is below Ia ) , or are Ia ) 395
Volume 80, number 5,6
OPTICS COMMUNICATIONS
and [ c ) themselves, we can o b t a i n the s a m e result for the phase variance, if Iot12/IAo12<< 1 a n d IA~I, IA2I >> IZtol, which are satisfied in the D P O experiments [ 1 1,12 ]. T h e case with one level for Ib ) can be easily extended to the case with a s u m m a t i o n over l b ) levels where A, Lt a n d L2 will be replaced by their effective ones which c o m e after the s u m m a tion. We will give the detail calculation in a regular paper. F r o m the above analysis, we s u m m a r i z e our resuits as follows. First, the source o f squeezing in the D P O and in the t w o - p h o t o n CEL is the same - level coherence. Second, because the phase sensitive terms in both cases are identical, the m a g n i t u d e o f the squeezing in the D P O is the same as that in the original two-photon C E L at t w o - p h o t o n resonance. Third, the gain in the D P O is usually m u c h smaller than that in the original t w o - p h o t o n CEL at twop h o t o n resonance, because the D P © operates at far off two-photon resonance. This work is s u p p o r t e d by the U.S. Office o f N a v a l Research.
396
15 January 1991
References [ 1] G. Milburn and D.F. Walls, Optics Comm. 36 ( 1981 ) 401. [ 2 ] L.A. Lugiato and G. Strini, Optics Comm. 41 ( t 982 ) 67. [ 3 ] M.O. Scully et al., Phys. Rev. Lett. 60 ( 1988 ) 1832. [4] N. Lu and S.Y. Zhu, Phys. Rev. A 40 (1989) 5735. [ 5 ] N.A. Ansazi, J. Gea-Banaclocheand M.S. Zubary, Phys. Rev. A41 (1990) 5179. [6 ] Properties of litium niobate (INSPEC, the Institution of Electrical Engineers, London and New York, 1989, EMIS Datareviews Series, No. 5 ). [ 7 ] Y.R. Shen, The principles of nonlinear optics (John Wiley and Sons, Inc. 1984). [ 8 ] M. Sargent 11I,M.O. Scullyand W.E+Lamb Jr., Laser Physics (Addison-Wesley, Reading, MA, 1974). [9] G. Alzetta, L. Moi and G. Orriols, Nuovo Cimento 52 B (1979) 209. [ 10] G. Orriols, Nuovo Cimento 53 B (1979) 1. [ 11 ] L.A. Wu, H.J. Kimble, J.H. Hall and H. Wu, Phys. Rev. Lett+ 57 (1986) 2520. [ 12] S. Reynaund, C. Fabre and E. Giacobino, J. Opt. Soc. Am. B4 (1987) 1520.
OPTICS COMMUNICATIONS
15 January 1991
A comparison between degenerate paramagnetic oscillators and two-photon correlated-spontaneous-emission lasers Shi Yao Zhu
Centerfor AdvancedStudies, and Departmentof Physicsand Astronomy, Universityof New Mexico, Albuquerque,NM 87131, USA Received 2 March 1990; revised manuscript received 17 September 1990
The degenerate parametric oscillator (DPO) is analyzed with the nonlinear material treated quantum mechanically. The source of squeezing in the DPO is the same as in the two-photon correlated-spontaneous-emission laser (CEL) - the level coherence. In this sense, they are similar. The amount of squeezing in DPO is the same as in the two-photon CEL at two-photon resonance, but the gain inDPO is usually much smaller than that in the two-photon CEL at two-photon resonance.
There are m a n y similarities between a degenerate p a r a m e t r i c oscillator ( D P O ) [1,2] a n d a two-photon correlated-spontaneous-emission laser ( C E L ) [ 3,4]. Both have the capability to p r o d u c e squeezed state light. A m a x i m u m 50% intracavity squeezing can be o b t a i n e d at threshold, a n d the squeezing decreases above threshold for both cases. The similarities m o t i v a t e us to investigate the connection between them. In D P O , a strong laser at frequency 20) p u m p s a nonlinear material in a cavity to p r o d u c e an output field at frequency o). Both the p u m p i n g field a n d the o u t p u t field are far o f f r e s o n a n t from the levels o f the nonlinear material, so that the spontaneous emission is negligible. The o u t p u t field p r o d u c e d by D P O could be in a squeezed state. The source o f squeezing is usually considered as a t w o - p h o t o n transition. In the t w o - p h o t o n CEL, a t o m s are p r e p a r e d in a coherent superposition o f upper and lower levels, which correlates the spontaneous emission, a n d i f the level coherence is large enough, the laser field can be in a squeezed state. The D P O can be considered to be composed o f two processes: ( i ) the p u m p i n g laser prepares the nonlinear material to a certain state; (ii) the material produces a laser field at frequency o). Is the first process a p r e p a r a t i o n o f the level coherence a n d the seco n d the correlated-spontaneous-emission lasing process? In o r d e r to answer this question, we investigate
the D P O in a different way, where the material and the output signal field are treated q u a n t u m mechanically, while the p u m p i n g laser field is treated classically by assuming that the p u m p i n g light is in an intense coherent state a n d not depleted significantly
[5]. In DPO, the nonlinear materials are far off resonant, i.e., all detunings are much greater than decay rates. The D P O is a kind o f k t2) effect. I f the frequency difference o)~c between the first excited level l a ) and the ground level I c ) is near the frequency o f the p u m p i n g field 2o), I o)ac -- 2o)l <
(1)
which is a p p r o x i m a t e l y true for c o m m o n l y used nonlinear materials in D P O , such as LiNbO3 [6]. The k t2) coefficient can be written as [7] k~,) (o) = 2 o ) - o))
=_N~2
[
(ri)¢b(rk)b~(rj)a¢ (o)ac - 2o)) (o)b~-- W)
(ri)b~(rk )~b(rj)a~ -1
(2)
+ (O)ac - 2o)) ( o ) - O~ab)J ' where the s u m m a t i o n Yb is over all levels, i.e. the level Ib ) may be above, below [a ) and even the same as the levels [a ) and [c ) . It is very clear that the field o f 2o) is associated with the transition between [c)
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
393
Volume 80, number 5,6
OPTICS COMMUNICATIONS
and l a ) , the field of 09 associated with the transitions between Ic) and Ib ) and between Ia ) and Ib ) . For the commonly used nonlinear materials in DPO, their band gaps are larger than 2hoJ (green light), i.e., there is no level between la) and Ic). The only possibility of the two photon transition from l a ) to Ic ) is the following: one photon transition from la) to a level Ib), which is above la) ( o r l a ) itself), and another one photon transition from [b) to Ic ). For simplicity, we consider only one level Ib ) above level Ia). The one photon transition from Ia ) to Ic ) at frequency 09 has no contribution to the squeezing and has been neglected. Here we simplify the ground band and first excited bands as levels. It is a good approximation if we assume that the band widths are much smaller than the band gap. Therefore, we simplify our model to a three level system, as shown in fig. 1. The system is similar to that used in ref. [5 ]. However, we consider Ib ) above Ia ) ( Ic) ground state), and far off two-photon transition. The eigenfrequencies of Ia ) , Ib ) and Ic ) are co~, Ogb and o9¢, respectively. For the present simplified model, we assume that the level l a ) decays to Ic) with a rate Yl, and the level Ib) decays to [a) and Ic) with rates Y2and 73, respectively. The decay rates of off-diagonal elements are 7pa, (fl~: fl' ). Because of eq. (1), we can make rotating wave approximation on the pumping field, but not on the signal field. The hamiltonian for such a field-atom system is
n=h
~
oJplfl)(fll+hl2a*a+h[ala)(cl
15 January 1991
pumping field, which is assumed in a coherent state all the time; and £2 is the bare cavity frequency. In the following discussion, we keep the pumping field to all order, and the signal field to first order, which is enough for the discussion on phase locking, phase noise, and linear gain of the signal field. By using the standard techniques of the laser quantum theory [8], the master equation for the reduced density matrix of the signal field is obtained, under the condition IA~I= Itoab-~ol and IAzi = Io~,c-oglAol = IoJac2o91, to be
/~= ½A[Ll aa~(paa*--a*pa ) + L2acc(a *ap-apa* ) + L~ a~¢(patat-a*pa*) + Lzaac(a*a*p-a*pa*) ] - ½F(pata,apa*) -i(O-og)atap+h.c.,
(4)
where A = 2 N g 2 / ~ with N the atom number in the cavity, LI = ( k + i~l ) - 1, L2 = k( 1 + ikfi2) - l, ~i = d,/~b~b~, k= X/~bTbc, and F is the cavity loss rate. Here apa, (the zeroth order solution) represent the level populations and coherence, which are produced by the interaction between the pumping field and the nonlinear material. They are (2Y~¢/71) lal 2 cr~ = 1 -~rc~ = 7~+d~+(4y~/y~)lot[2, a~=
2
-ia(Ta¢ -iAo) 2
7~c + A o + ( 4 7 ~ c / 7 1 ) l a l 2 "
(5a) (5b)
The zeroth order solution (all order in pumping field or) can be considered as the results of the process of level preparation. From eqs. (5), we find
r = a,b,c
+g~ (a+a t) la) (bl +g2(a+a ~) Ib) (cl + h.c.], (3) where a t (a) is the creation (annihilation) operator for the signal field; ot is the Rabi frequency of the
Iaac [2= tTcctraa-- tra2a .
(6a)
In DPO experiments, we have IAol >> Iotl to guarantee no significant population in any excited level. Under the approximation 2 o
I+Az+B
= I + A Ao
(A, B constants), eq. (6a) become l a > ~ /
laac I2= trcc~Taa,
IC>
which means an approximately full coherence between Ia ) and Ic). Therefore, the pumping of a strong laser at off-resonance is really a preparation of the level coherence. The corresponding Fokker-Planck equation for the
t
,
Fig. 1. 394
(6b)
Volume 80, number 5,6
OPTICS COMMUNICATIONS
Glauber P-representation in polar coordinates (al 8> = 81 ~>, 8 = r e i#) can be obtained. The drift and phase-diffusion coefficients are [. 0, O'aa
d'=(~2-v)-~ALk2+6~
k202 O'cc]
i +---~J
-½AxIGc Is i n ( 2 O - G ~ - 0o), A [ kaaa dr = -~ L k ~ Z
(7a)
kGc
1+ k262 (7b)
+ x l G c Icos ( 2 0 - G~ - 0 o ) ] ,
A r kaaa D° = -4r~ Lk2+012
k2[ trac [ x(1 +k202)
( 1 l+0,02~] k-~6-~- ] J ' (7c)
where x exp ( - i00) = L j - L2, O'ac = I O'ac [ exp ( - iGc). From the phase drive coefficient, we find the phase of the signal field is locked at 2~o-Gc-0o =0.
(8)
In order to have positive gain, from the amplitude drive coefficient eq. (Tb), we need laacl > k o.cc - x(l_t_k2022)
kaaUacc x(k2+# 2) •
(9)
We notice that the phase squeezing could occur as long as the second term in eq. (4c) is larger than the first one. According to the Fokker-Planck equation and by expanding d o around 0o to the first order [ 3 ], we can obtain the steady state phase variance with positive gain
1 {1 ((OO)z) >- ~r 2 X[I+~,02_02
-
k2
+ (l_k2)+kz(Ol--02) k2 +
2
x/~cc
(k2+&~)x/a~-~a:~cd) '
(10)
where the first term 1/4r 2 is the shot noise. Here eqs. (9) and (6) have been used. Since Idl I, Id2l >> Idol, we have 01 = -02. For 1011, ]0zl >> 1, k, eq. (10) reduces to
((t~0) 2) >- 1/8r 2 ,
15 January 1991 (ll)
where the same approximation to get eq. (6b) has been used. Eq. (11) means that a maximum 50% squeezing can be achieved at threshold, the well known result [ 1,2]. Recalling eq. (7c), we see that the phase diffusion coefficient would never be negative and there would be no squeezing, if there were no level coherence ( G c = 0 ) . The total noise is the sum of vacuum fluctuation (shot noise) and spontaneous fluctuation. The negligible upper level population merely leads to the noise quenching, i.e., no spontaneous fluctuation but vacuum fluctuation still exists. Therefore, it is the level coherence that leads to the noise squeezing. The squeezing in the DPO is also from the level coherence, which correlates the spontaneous emission. The two-photon emission process of the nonlinear material is indeed a two-photon correlated-spontaneous emission lasing one. From this point of view (spontaneous emission correlated by level coherence), we would conclude that the DPO is similar to the two-photon CEL. Next, we turn to discuss the difference between the DPO and the original two-photon CEL [3]. The DPO operates at far off resonance, while the twophoton CEL can operate at resonance. Consequently, the gain in two-photon CEL (operating at two-photon resonance) is much greater than that in DPO provided other conditions remain the same. The coherence between the upper and lower levels in the two-photon CEL can be created by some means such as resonant Raman process [9,10]. Another difference between them is that the phase sensitive terms in the master equation (eq. ( 4 ) ) are proportional to Ll or L2, while these terms in the original two-photon CEL (eq. ( I ) in ref. [ 3 ] ) are proportional to L,Lo o r L2L o. The key difference is the factor Lo. At two-photon resonance, Lo = l, these terms in the two master equations are the same. However, these terms in the DPO are much greater than that in the original two-photon CEL at far off two-photon resonance, because of ILol << I. This is the reason why the DPO can create large squeezing while the original two-photon CEL can't at far off two-photon resonance. In the above, we choose the level l b ) above level Ia ) . For the case where Ib ) is below Ia ) , or are Ia ) 395
Volume 80, number 5,6
OPTICS COMMUNICATIONS
and [ c ) themselves, we can o b t a i n the s a m e result for the phase variance, if Iot12/IAo12<< 1 a n d IA~I, IA2I >> IZtol, which are satisfied in the D P O experiments [ 1 1,12 ]. T h e case with one level for Ib ) can be easily extended to the case with a s u m m a t i o n over l b ) levels where A, Lt a n d L2 will be replaced by their effective ones which c o m e after the s u m m a tion. We will give the detail calculation in a regular paper. F r o m the above analysis, we s u m m a r i z e our resuits as follows. First, the source o f squeezing in the D P O and in the t w o - p h o t o n CEL is the same - level coherence. Second, because the phase sensitive terms in both cases are identical, the m a g n i t u d e o f the squeezing in the D P O is the same as that in the original two-photon C E L at t w o - p h o t o n resonance. Third, the gain in the D P O is usually m u c h smaller than that in the original t w o - p h o t o n CEL at twop h o t o n resonance, because the D P © operates at far off two-photon resonance. This work is s u p p o r t e d by the U.S. Office o f N a v a l Research.
396
15 January 1991
References [ 1] G. Milburn and D.F. Walls, Optics Comm. 36 ( 1981 ) 401. [ 2 ] L.A. Lugiato and G. Strini, Optics Comm. 41 ( t 982 ) 67. [ 3 ] M.O. Scully et al., Phys. Rev. Lett. 60 ( 1988 ) 1832. [4] N. Lu and S.Y. Zhu, Phys. Rev. A 40 (1989) 5735. [ 5 ] N.A. Ansazi, J. Gea-Banaclocheand M.S. Zubary, Phys. Rev. A41 (1990) 5179. [6 ] Properties of litium niobate (INSPEC, the Institution of Electrical Engineers, London and New York, 1989, EMIS Datareviews Series, No. 5 ). [ 7 ] Y.R. Shen, The principles of nonlinear optics (John Wiley and Sons, Inc. 1984). [ 8 ] M. Sargent 11I,M.O. Scullyand W.E+Lamb Jr., Laser Physics (Addison-Wesley, Reading, MA, 1974). [9] G. Alzetta, L. Moi and G. Orriols, Nuovo Cimento 52 B (1979) 209. [ 10] G. Orriols, Nuovo Cimento 53 B (1979) 1. [ 11 ] L.A. Wu, H.J. Kimble, J.H. Hall and H. Wu, Phys. Rev. Lett+ 57 (1986) 2520. [ 12] S. Reynaund, C. Fabre and E. Giacobino, J. Opt. Soc. Am. B4 (1987) 1520.