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Evolution of the squeezing character of an initially squeezed EM field in the Dicke model
Volume 84, number 5,6
OPTICS COMMUNICATIONS
1 August 1991
Evolution of the squeezing character of an initially squeezed EM field in the Dicke model Yi-dong Zhou and Shi-qun Li Department of Applied Physics, Tsinghua University, Belting 100084, China
and Tian-jie Chen Department of Physics, Peking University, Beijing 100871, China Received 8 January 1991; revised manuscript received 19 March 1991
The evolution of an initially squeezed electromagnetic (EM) field was studied by using the Dicke model and the perturbation approach. The study covers both one-photon and two-photon interaction between the squeezed field and the cooperative atomic system. The results show that during the early evolution stage in the one-photon Dicke model thenonclassical statistical property (squeezing) gradually decays. But under certain conditions in the two-photon model the squeezing will be enhanced. The physical meaning of these phenomena is discussed.
1. Introduction The evolution o f a quantized electromagnetic ( E M ) field when it is interacting with atoms has attracted c o m m o n interest for decades. The J a y n e s - C u m m i n g s model [ l ], in which the interaction between a singlemode EM field and a two-level atom is considered, has been widely used to discuss the evolution o f the field with different initial states. But a system consisting o f m a n y two-level atoms is much closer to the real experimental situation. For such a system, the most widely used model is the cooperative Dicke model, which was first proposed by Dicke in 1954 [2]. Butler and D r u m m o n d [3] used the Dickc model to study the onephoton interaction with an EM field which is initially in a single-mode coherent state. His results show, in the short-time approximation [4], that, if the atoms are initially fully de-excited, then at an early stage the field will be squeezed and stay in a minimum-uncertainty state. But if the atoms are initially fully excited, the field will never be squeezed. Gerry and Togeas [ 5 ] used the same model to study the same problem but two-photon interaction was considered. The results are almost the same as in ref. [ 3 ]; only the squeezing may be stronger than in the one-photon case. Here we report a study o f the nonclassical statistical property o f an EM field which initially is already in a squeezed state [ ix, ( ) , when it is interacting with a group o f atoms in the Dicke model. The interesting point of our results is that, if the atoms are initially fully de-excited, the field is weakly squeezed and strong enough, and the two-photon interaction is involved, then the squeezing o f the field will be enhanced as time elapses. As for the one-photon interaction case the result shows that, no matter whether the atoms are initially in fully excited or de-excited, the squeezing will gradually decay; and the larger the squeezing parameter (, the higher ¢r Supported by the China National Natural Science Foundation. 0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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the rate of deviating from the minimum-uncertainty state. When ~= 0, our result is consistent with refs. [ 3 ] and [5].
2. One-photon Dicke model The hamiltonian of a system in which a single-mode EM field is interacting with N two-level atoms and only one-photon processes are taken into account, is H = htoa + a + ½htooJo + hg( J + a + J - a + ) .
( 1)
The notations are the same as in ref. [3]. Precisely at resonance (to=tao), the interaction term of the hamiltonian of the system in the interaction picture is (2)
H~=hg(J+ a+ J-a + ) .
Therefore at arbitrary time t, the wavefunction of the system can be expressed as I~, (t) > = e x p ( - i H i t / h ) [g/~(0) > = [ 1 - ( i H ~ t / h ) + ( 1/2! ) (ill, t / h ) 2 + ... ] [ g/(0) > = [1 - i z ( J + a + J - a + ) - ( z 2 / 2 ) ( J + a + J - a + ) 2 +
...] Ig/(0) >
= I~/,(0)>+ I ¥ ~ ' ) ( t ) > + ...,
(3)
where ,z=gt, and the initial state is I~h(0)>=l(0)>=la,
ff>lj, m > ,
(4)
where Iot, ~> is an initially squeezed field, and IJ, m> is a cooperative state of N atoms in the Dicke model. Using the short-time approximation, and only keeping the terms up to z z, we have I ~, (t) > = [ 1 - ½z2B2aa + - ½z2A 2a + a ] I a, ~ > IJ, m > - izAa I a, ~> lJ, m + 1 > - izBa + I a, ~> IJ, m - 1 > --½r2Ea2[ Or, ~> lJ, m + 2 > -- ½T2Da+2la, (> lJ, m - 2 > ,
(5)
where A=[(J-m)(j+m+l)]
~/2,
E=[(J-m)(j+m+l)(j-m-l)(j+m+2)]
1/2,
B=[(j+rn)(j-m+l)]
1/2,
V=[(j+m)(j-m+l)(j+m-1)(j-m+2)]
~/2.
(6)
The two quadrature components of the slowly varying operator b [ = a exp (itat) ] of the EM field are X=(b+b+)/2,
(7)
Y=(b-b+)/2i.
Since [X, Y] = i / 2 , these two components satisfy the minimum-uncertainty inequality
(8)
< (AX)2 > < (Ay)2> >i 1/16. The variances of these two components are )2] ,
(9)
< (AY)2> = !4 [ 1 - 2 Re +2 - 4 ( I m )2].
(10)
<(AX)2>=4l [ l + 2 R e < b + 2 > + 2 < b + b > - 4 ( R e
Using (5) we can calculate, , : < ~/i(t) [Kb+2K + [ ~h(t) > = [ 1 + T2(B2-A 2) ] (tx 2 - s i n h s cosh s) , < Vx(t) I K b + b K + [ ~l(t) > = [ 1 + T 2 ( B 2 - A 2) ] (or2 + sinh2s) + z 2 a 2 ,
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(Wi(t) IKb + K + l ¥ 1 ( t ) ) = [ 1 + z 2 ( B 2 - A 2 ) / 2 ] o t ,
1 August 1991 ( 11 )
where K = e x p ( i c o a + at) ,
K + =exp(-icoa
+at) .
For the sake of simplicity, in calculating the expectation values we assume that a is a real number, and the squeezing angle 0 = 0, i.e., the squeezing parameter ( = s exp (i0) = s (0 ~
(12)
( ( A Y ) 2) = ~{ 1 + 2z2B2+ 2 [ 1 + z 2 ( B 2 - A 2 ) ] (sinh2s+ sinh s cosh s ) } .
(13)
2.1. A t o m s are initially f u l l y excited
Since the atoms are fully excited initially we have m = j in eq. (4), and the variances of the two quadrature components can be calculated, ( ( A X ) 2) = ¼[e-2S+4jz2+2j'c2(e-2S-- 1 ) ] ,
(14)
( ( A y ) 2 ) = 1 [e2S+4jT2+2jr2(e2S 1) ] .
(15)
( l ) These formulae tell us that at t = 0 ( ( A X ) E ) = ¼ e -2s,
( ( A y ) 2 ) = 4 l e 2..
(16)
This is just the evidence that the initial field is a pure squeezed field. (2) Formula (14) shows that as the time elapses the squeezing of the field will gradually decay. And the squeezing will only exist for times z satisfying r2< (e 2s- 1 ) / [ 2j(e2~+ 1 ) ] .
(17)
In ref. [ 6 ] it was pointed out that the short-time approximation will be valid only (¢,cl)(t) IFlc/C')(t) ) / ( ¢ / ( 0 ) I F I V ( 0 ) ) << 1,
(18)
where F=Kb+2K +,
Kb+bK + ,
or
Kb+K + .
For the case that the field is strong enough and weakly squeezed it is required that r2<< l l (2jlotl 2) .
(19)
Therefore if the field should remain squeezed during the period in which the short-time approximation is valid, the squeezing parameter of the field should satisfy (e2~-l)/(eES+l)>~l/Iotl
2.
. (20)
As to the Y component, eq. (15) tells us that it never shows the squeezing property. The term 4jr 2 exists in eqs. (14) and (15), and also in ref. [3]. This term plays an important role in the squeezing properties of the field and is independent of a and (. Even in the case of the vacuum state ( a = 0 and ( = 0 ) this term still exists and plays an important role in the evolution of the field. Therefore it must be the contribution of the spontaneous radiation, and it naturally exists in the fully excited case. For rather small z, such that the influence of the spontaneous radiation has not yet been essential, ( ( A X ) 2) < 1/4 still holds, and the X component still shows a real squeezing character. As time elapses, the influence of the spontaneous 361
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emission becomes essential and the field starts deviating from the minimum-uncertainty state and the squeezing character disappears gradually. In fact, if only the terms up to z 2 are kept, the product of the variances will be ( ( A S ) 2 ) ( ( A Y ) 2) = ~ [ 1 +2jz2(e-2*+e2"+2) ] .
(21)
From this formula we see that, as time elapses, the interaction makes the EM field deviate from the minimumuncertainty state, and the rate of deviation will be higher if the squeezing parameter is larger. (3) When s = 0 , one is back in the case that the field is initially in a pure coherent state. And the results are consistent with those presented in ref. [ 3 ].
2.2. Atoms are initially fully de-excited When the atoms are initially fully de-excited, m = - j in formula (4). The variances o f two quadrature components are ( ( A X ) 2 ) = 1 [e-2~_ 2jr2 (e-2S_ 1 )] ,
(22)
( ( A Y ) 2) = ~ [e2S-2jz2(e 2s- 1 ) ] .
(23)
( l ) For t = 0 , ( ( A X ) 2) ( ( A Y ) 2 ) = l / 1 6 ; this means the initial state is a pure squeezed state. (2) Eq. (22) tells us the squeezing will decay and it will still be squeezed only as long as rE< l / 2 j .
(24)
Since generally we have Ia I2 > 1, combining ( 19 ) and (24) we see that, in the region in which this approach is valid, the X component will remain squeezed. And from (23) and (19) we see that the V component will never be squeezed. As the terms up to z 2 are kept, the product of the variances is ((~)2)
( ( A y ) 2 ) = ~ [ 1 +2jzE(e-2s+eEs--2) ] .
(25)
Again the field will not keep being in a minimum-uncertainty state due to the interaction with the atoms. Summarizing the above discussion, the following results were obtained: In the one-photon interaction Dicke model, if the initial field is a squeezed one, then ( 1 ) the interaction will make the field not being in a minimum-uncertainty state anymore; (2) the larger s, the faster the rate of deviation from the minimum-uncertainty state; (3) both the decay rate of the squeezing and the rate of deviation from the minimum-uncertainty state are higher for the initially fully-excited case than for the initially fully de-excited case due to the existence of the spontaneous emission.
3. Two-photon Dicke model The hamiltonian of a system in which a single-mode EM field is interacting with N two-level atoms and only two-photon processes are taken into account, is
H=ho9a +a+ ½ho9oJo+ hg(J+ a2 + j - a +2) .
(26)
Precisely at resonance (2o9 = 09o), the interaction term of the hamiltonian of the system in the interaction picture is
H~ =hg(J+ a 2 + j - a +z) . 362
(27)
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Therefore at arbitrary time t, the wavefunction of the system can be expressed as I g/t ( t ) ) = [1 -- ½"c2B2a2a + 2 - ½z2A2a+2a 2 ] [a, ( ) [j, m ) -- irAa2lot, ( ) [j, m + 1 ) --irBa+2lot, ( ) lJ, m - 1 ) -
-
½z2Ea4lot, ( ) l J, m + 2 ) -½zDa+4lot, ( ) l J, m - 2 )
,
(28)
where A, B, E, D are the same as in eq. (6). In the following discussion we use the short-time approximation again. In the two-photon Dicke model the condition of validity of this approximation can be derived from (18), and is Z2<< 1/ (2jla[ 4) .
(29)
3.1. Atoms are initially fully excited Since the atoms are fully excited initially, so in eq. (28) we have m = j , and the variances for the two slowly varying quadrature components are calculated to be ( ( A X ) 2) = l{e-2S+j'c2[ ( 12e-2S+ 8)o~2 + (3e-4S+2e2S+6e-2S+ 5) ]} ,
(30)
( ( A Y ) 2 ) = l{e2S+jr2[ (4e2S+ 8)0t2+ (3e4S+ 2e-2S+ 6e2S+ 5) ]} .
(31)
The product of these two variances is (terms up to z 2 are kept) ( ( A X ) 2) ( ( A Y ) 2) = ~{1 + 2jr2 [ (8 + 4e-2"+ 4e2S)ot2+ (e-4S+e4S+4e-2S+4e2S+6) ]}.
(32)
The conclusions derived from the above three formulae are similar to the case of one-photon interaction. Only one thing has to be pointed out: for the case s = 0 (initially a coherent state rather than a squeezed state) the result is ( ( A X ) 2 ) = ~ + 4jr2+ 5jz2ot 2 ,
(33)
< (Ay)2> = ~ +4jz2+3jr2ot 2 .
(34)
The 4jr 2 term exists even if a = 0, so it is attributed to the contribution of the spontaneous emission. It did not show up in the formulae in ref. [ 5 ], although it should.
3.2. Atoms are initially fully de-excited In this casewe should have m = - j in eq. (28), and the variances are ( ( A X ) 2 ) = ]{e-2S+jz2[ ( 2 e 2 S + 6 e - 2 S - 3 e - 4 S - 5 ) - ( 12e-2S-- 8 ) a 2 ] } ,
(35)
( (A Y)2 ) = ¼{e2, + j r 2 [ (6e2~+ 2e-2s_ 3eas_ 5 ) - (4e 2~- 8 )ct 2 ] }.
( 36 )
The product of these two variances is (terms up to z 2 are kept)
((AX)2)((Ay)2)=~{I+2jz2[(4e
-
2s
+ 4 e -2 s8 ) a
2
+(e-
4s
+ e -4 4s e -
2S
-4e
2s
+6)]}.
(37)
Only when s < ½1n(3/2) ~0.203
(38)
and a 2 > 6e-2S+ 2e2~-- 3e - 4 s - 5 ) / ( 12e-2~-- 8)
(39)
can the coefficient of the z 2 term in (35) be negative. This means that, when the initial field is weakly squeezed, and the intensity is high enough, then the squeezing of the field may be enhanced. And as the initial field is 363
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intenser, the growth rate of the squeezing will be higher. It seems to be of interest that, when a field is initially weakly squeezed (s << 0.203) and strong enough, then it might be squeezed more, even though s would remain in the range s < 0.203. And in this kind of squeezing the field will not be able to keep itself in a pure squeezed state since it will not remain in a minimum-uncertainty state anymore. But the rate of deviation is lower than in the initially fully excited case. As (38) and (39) are not satisfied, the squeezing will decay again, similarly to the previous results for the one-photon process.
4. Conclusion
The short-time approximation and the one-photon and two-photon Dicke model were used to study the evolution of a field which is initially squeezed. The following conclusions were reached: ( 1 ) Due to the interaction with the atoms the field will not be able to remain in a minimum-uncertainty state, at least at the early stage. (2) The larger s, the higher will be the rate of deviation. (3) Due to the spontaneous emission, the rate of deviation for the fully excited case is higher than for the fully de-excited case. (4) In the one-photon case, the squeezing of the already squeezed component will decay due to the interaction, but it can keep being squeezed for a short period. And the other component will never show squeezing character. (5) In the two-photon case, there are some interesting results. When the intensity of the field is high enough, and the atoms are in the de-excited state, the weakly squeezed component might be further squeezed (but s remains in the range s < 0.203), at least during the early stage. (6) The rate of deviation from the minimum-uncertainty state for the two-photon case is higher than that for the one-photon case.
References
[ 1] E.T. Jaynes and C.W. Cummings, Proc. IEEE 51 (1963) 89. [2] R.H. Dicke, Phys. Rev. 93 (1954) 99. [ 3 ] M. Butler and P.D. Drummond, Optica Acta 33 (1986) 1. [4] M. Schubert and B. Wicheimi, Nonlinear optics and quantum electronics (Wiley, New York, 1986). [ 5 ] C.C. Gerry and J.B. Togeas,Optics Comm. 69 (1989) 263. [6 ] Li Shiqun et al., China Laser, to be published.
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Evolution of the squeezing character of an initially squeezed EM field in the Dicke model Yi-dong Zhou and Shi-qun Li Department of Applied Physics, Tsinghua University, Belting 100084, China
and Tian-jie Chen Department of Physics, Peking University, Beijing 100871, China Received 8 January 1991; revised manuscript received 19 March 1991
The evolution of an initially squeezed electromagnetic (EM) field was studied by using the Dicke model and the perturbation approach. The study covers both one-photon and two-photon interaction between the squeezed field and the cooperative atomic system. The results show that during the early evolution stage in the one-photon Dicke model thenonclassical statistical property (squeezing) gradually decays. But under certain conditions in the two-photon model the squeezing will be enhanced. The physical meaning of these phenomena is discussed.
1. Introduction The evolution o f a quantized electromagnetic ( E M ) field when it is interacting with atoms has attracted c o m m o n interest for decades. The J a y n e s - C u m m i n g s model [ l ], in which the interaction between a singlemode EM field and a two-level atom is considered, has been widely used to discuss the evolution o f the field with different initial states. But a system consisting o f m a n y two-level atoms is much closer to the real experimental situation. For such a system, the most widely used model is the cooperative Dicke model, which was first proposed by Dicke in 1954 [2]. Butler and D r u m m o n d [3] used the Dickc model to study the onephoton interaction with an EM field which is initially in a single-mode coherent state. His results show, in the short-time approximation [4], that, if the atoms are initially fully de-excited, then at an early stage the field will be squeezed and stay in a minimum-uncertainty state. But if the atoms are initially fully excited, the field will never be squeezed. Gerry and Togeas [ 5 ] used the same model to study the same problem but two-photon interaction was considered. The results are almost the same as in ref. [ 3 ]; only the squeezing may be stronger than in the one-photon case. Here we report a study o f the nonclassical statistical property o f an EM field which initially is already in a squeezed state [ ix, ( ) , when it is interacting with a group o f atoms in the Dicke model. The interesting point of our results is that, if the atoms are initially fully de-excited, the field is weakly squeezed and strong enough, and the two-photon interaction is involved, then the squeezing o f the field will be enhanced as time elapses. As for the one-photon interaction case the result shows that, no matter whether the atoms are initially in fully excited or de-excited, the squeezing will gradually decay; and the larger the squeezing parameter (, the higher ¢r Supported by the China National Natural Science Foundation. 0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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the rate of deviating from the minimum-uncertainty state. When ~= 0, our result is consistent with refs. [ 3 ] and [5].
2. One-photon Dicke model The hamiltonian of a system in which a single-mode EM field is interacting with N two-level atoms and only one-photon processes are taken into account, is H = htoa + a + ½htooJo + hg( J + a + J - a + ) .
( 1)
The notations are the same as in ref. [3]. Precisely at resonance (to=tao), the interaction term of the hamiltonian of the system in the interaction picture is (2)
H~=hg(J+ a+ J-a + ) .
Therefore at arbitrary time t, the wavefunction of the system can be expressed as I~, (t) > = e x p ( - i H i t / h ) [g/~(0) > = [ 1 - ( i H ~ t / h ) + ( 1/2! ) (ill, t / h ) 2 + ... ] [ g/(0) > = [1 - i z ( J + a + J - a + ) - ( z 2 / 2 ) ( J + a + J - a + ) 2 +
...] Ig/(0) >
= I~/,(0)>+ I ¥ ~ ' ) ( t ) > + ...,
(3)
where ,z=gt, and the initial state is I~h(0)>=l(0)>=la,
ff>lj, m > ,
(4)
where Iot, ~> is an initially squeezed field, and IJ, m> is a cooperative state of N atoms in the Dicke model. Using the short-time approximation, and only keeping the terms up to z z, we have I ~, (t) > = [ 1 - ½z2B2aa + - ½z2A 2a + a ] I a, ~ > IJ, m > - izAa I a, ~> lJ, m + 1 > - izBa + I a, ~> IJ, m - 1 > --½r2Ea2[ Or, ~> lJ, m + 2 > -- ½T2Da+2la, (> lJ, m - 2 > ,
(5)
where A=[(J-m)(j+m+l)]
~/2,
E=[(J-m)(j+m+l)(j-m-l)(j+m+2)]
1/2,
B=[(j+rn)(j-m+l)]
1/2,
V=[(j+m)(j-m+l)(j+m-1)(j-m+2)]
~/2.
(6)
The two quadrature components of the slowly varying operator b [ = a exp (itat) ] of the EM field are X=(b+b+)/2,
(7)
Y=(b-b+)/2i.
Since [X, Y] = i / 2 , these two components satisfy the minimum-uncertainty inequality
(8)
< (AX)2 > < (Ay)2> >i 1/16. The variances of these two components are )2] ,
(9)
< (AY)2> = !4 [ 1 - 2 Re
(10)
<(AX)2>=4l [ l + 2 R e < b + 2 > + 2 < b + b > - 4 ( R e
Using (5) we can calculate
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(Wi(t) IKb + K + l ¥ 1 ( t ) ) = [ 1 + z 2 ( B 2 - A 2 ) / 2 ] o t ,
1 August 1991 ( 11 )
where K = e x p ( i c o a + at) ,
K + =exp(-icoa
+at) .
For the sake of simplicity, in calculating the expectation values we assume that a is a real number, and the squeezing angle 0 = 0, i.e., the squeezing parameter ( = s exp (i0) = s (0 ~
(12)
( ( A Y ) 2) = ~{ 1 + 2z2B2+ 2 [ 1 + z 2 ( B 2 - A 2 ) ] (sinh2s+ sinh s cosh s ) } .
(13)
2.1. A t o m s are initially f u l l y excited
Since the atoms are fully excited initially we have m = j in eq. (4), and the variances of the two quadrature components can be calculated, ( ( A X ) 2) = ¼[e-2S+4jz2+2j'c2(e-2S-- 1 ) ] ,
(14)
( ( A y ) 2 ) = 1 [e2S+4jT2+2jr2(e2S 1) ] .
(15)
( l ) These formulae tell us that at t = 0 ( ( A X ) E ) = ¼ e -2s,
( ( A y ) 2 ) = 4 l e 2..
(16)
This is just the evidence that the initial field is a pure squeezed field. (2) Formula (14) shows that as the time elapses the squeezing of the field will gradually decay. And the squeezing will only exist for times z satisfying r2< (e 2s- 1 ) / [ 2j(e2~+ 1 ) ] .
(17)
In ref. [ 6 ] it was pointed out that the short-time approximation will be valid only (¢,cl)(t) IFlc/C')(t) ) / ( ¢ / ( 0 ) I F I V ( 0 ) ) << 1,
(18)
where F=Kb+2K +,
Kb+bK + ,
or
Kb+K + .
For the case that the field is strong enough and weakly squeezed it is required that r2<< l l (2jlotl 2) .
(19)
Therefore if the field should remain squeezed during the period in which the short-time approximation is valid, the squeezing parameter of the field should satisfy (e2~-l)/(eES+l)>~l/Iotl
2.
. (20)
As to the Y component, eq. (15) tells us that it never shows the squeezing property. The term 4jr 2 exists in eqs. (14) and (15), and also in ref. [3]. This term plays an important role in the squeezing properties of the field and is independent of a and (. Even in the case of the vacuum state ( a = 0 and ( = 0 ) this term still exists and plays an important role in the evolution of the field. Therefore it must be the contribution of the spontaneous radiation, and it naturally exists in the fully excited case. For rather small z, such that the influence of the spontaneous radiation has not yet been essential, ( ( A X ) 2) < 1/4 still holds, and the X component still shows a real squeezing character. As time elapses, the influence of the spontaneous 361
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emission becomes essential and the field starts deviating from the minimum-uncertainty state and the squeezing character disappears gradually. In fact, if only the terms up to z 2 are kept, the product of the variances will be ( ( A S ) 2 ) ( ( A Y ) 2) = ~ [ 1 +2jz2(e-2*+e2"+2) ] .
(21)
From this formula we see that, as time elapses, the interaction makes the EM field deviate from the minimumuncertainty state, and the rate of deviation will be higher if the squeezing parameter is larger. (3) When s = 0 , one is back in the case that the field is initially in a pure coherent state. And the results are consistent with those presented in ref. [ 3 ].
2.2. Atoms are initially fully de-excited When the atoms are initially fully de-excited, m = - j in formula (4). The variances o f two quadrature components are ( ( A X ) 2 ) = 1 [e-2~_ 2jr2 (e-2S_ 1 )] ,
(22)
( ( A Y ) 2) = ~ [e2S-2jz2(e 2s- 1 ) ] .
(23)
( l ) For t = 0 , ( ( A X ) 2) ( ( A Y ) 2 ) = l / 1 6 ; this means the initial state is a pure squeezed state. (2) Eq. (22) tells us the squeezing will decay and it will still be squeezed only as long as rE< l / 2 j .
(24)
Since generally we have Ia I2 > 1, combining ( 19 ) and (24) we see that, in the region in which this approach is valid, the X component will remain squeezed. And from (23) and (19) we see that the V component will never be squeezed. As the terms up to z 2 are kept, the product of the variances is ((~)2)
( ( A y ) 2 ) = ~ [ 1 +2jzE(e-2s+eEs--2) ] .
(25)
Again the field will not keep being in a minimum-uncertainty state due to the interaction with the atoms. Summarizing the above discussion, the following results were obtained: In the one-photon interaction Dicke model, if the initial field is a squeezed one, then ( 1 ) the interaction will make the field not being in a minimum-uncertainty state anymore; (2) the larger s, the faster the rate of deviation from the minimum-uncertainty state; (3) both the decay rate of the squeezing and the rate of deviation from the minimum-uncertainty state are higher for the initially fully-excited case than for the initially fully de-excited case due to the existence of the spontaneous emission.
3. Two-photon Dicke model The hamiltonian of a system in which a single-mode EM field is interacting with N two-level atoms and only two-photon processes are taken into account, is
H=ho9a +a+ ½ho9oJo+ hg(J+ a2 + j - a +2) .
(26)
Precisely at resonance (2o9 = 09o), the interaction term of the hamiltonian of the system in the interaction picture is
H~ =hg(J+ a 2 + j - a +z) . 362
(27)
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Therefore at arbitrary time t, the wavefunction of the system can be expressed as I g/t ( t ) ) = [1 -- ½"c2B2a2a + 2 - ½z2A2a+2a 2 ] [a, ( ) [j, m ) -- irAa2lot, ( ) [j, m + 1 ) --irBa+2lot, ( ) lJ, m - 1 ) -
-
½z2Ea4lot, ( ) l J, m + 2 ) -½zDa+4lot, ( ) l J, m - 2 )
,
(28)
where A, B, E, D are the same as in eq. (6). In the following discussion we use the short-time approximation again. In the two-photon Dicke model the condition of validity of this approximation can be derived from (18), and is Z2<< 1/ (2jla[ 4) .
(29)
3.1. Atoms are initially fully excited Since the atoms are fully excited initially, so in eq. (28) we have m = j , and the variances for the two slowly varying quadrature components are calculated to be ( ( A X ) 2) = l{e-2S+j'c2[ ( 12e-2S+ 8)o~2 + (3e-4S+2e2S+6e-2S+ 5) ]} ,
(30)
( ( A Y ) 2 ) = l{e2S+jr2[ (4e2S+ 8)0t2+ (3e4S+ 2e-2S+ 6e2S+ 5) ]} .
(31)
The product of these two variances is (terms up to z 2 are kept) ( ( A X ) 2) ( ( A Y ) 2) = ~{1 + 2jr2 [ (8 + 4e-2"+ 4e2S)ot2+ (e-4S+e4S+4e-2S+4e2S+6) ]}.
(32)
The conclusions derived from the above three formulae are similar to the case of one-photon interaction. Only one thing has to be pointed out: for the case s = 0 (initially a coherent state rather than a squeezed state) the result is ( ( A X ) 2 ) = ~ + 4jr2+ 5jz2ot 2 ,
(33)
< (Ay)2> = ~ +4jz2+3jr2ot 2 .
(34)
The 4jr 2 term exists even if a = 0, so it is attributed to the contribution of the spontaneous emission. It did not show up in the formulae in ref. [ 5 ], although it should.
3.2. Atoms are initially fully de-excited In this casewe should have m = - j in eq. (28), and the variances are ( ( A X ) 2 ) = ]{e-2S+jz2[ ( 2 e 2 S + 6 e - 2 S - 3 e - 4 S - 5 ) - ( 12e-2S-- 8 ) a 2 ] } ,
(35)
( (A Y)2 ) = ¼{e2, + j r 2 [ (6e2~+ 2e-2s_ 3eas_ 5 ) - (4e 2~- 8 )ct 2 ] }.
( 36 )
The product of these two variances is (terms up to z 2 are kept)
((AX)2)((Ay)2)=~{I+2jz2[(4e
-
2s
+ 4 e -2 s8 ) a
2
+(e-
4s
+ e -4 4s e -
2S
-4e
2s
+6)]}.
(37)
Only when s < ½1n(3/2) ~0.203
(38)
and a 2 > 6e-2S+ 2e2~-- 3e - 4 s - 5 ) / ( 12e-2~-- 8)
(39)
can the coefficient of the z 2 term in (35) be negative. This means that, when the initial field is weakly squeezed, and the intensity is high enough, then the squeezing of the field may be enhanced. And as the initial field is 363
Volume 84, number 5,6
OPTICS COMMUNICATIONS
1 August 1991
intenser, the growth rate of the squeezing will be higher. It seems to be of interest that, when a field is initially weakly squeezed (s << 0.203) and strong enough, then it might be squeezed more, even though s would remain in the range s < 0.203. And in this kind of squeezing the field will not be able to keep itself in a pure squeezed state since it will not remain in a minimum-uncertainty state anymore. But the rate of deviation is lower than in the initially fully excited case. As (38) and (39) are not satisfied, the squeezing will decay again, similarly to the previous results for the one-photon process.
4. Conclusion
The short-time approximation and the one-photon and two-photon Dicke model were used to study the evolution of a field which is initially squeezed. The following conclusions were reached: ( 1 ) Due to the interaction with the atoms the field will not be able to remain in a minimum-uncertainty state, at least at the early stage. (2) The larger s, the higher will be the rate of deviation. (3) Due to the spontaneous emission, the rate of deviation for the fully excited case is higher than for the fully de-excited case. (4) In the one-photon case, the squeezing of the already squeezed component will decay due to the interaction, but it can keep being squeezed for a short period. And the other component will never show squeezing character. (5) In the two-photon case, there are some interesting results. When the intensity of the field is high enough, and the atoms are in the de-excited state, the weakly squeezed component might be further squeezed (but s remains in the range s < 0.203), at least during the early stage. (6) The rate of deviation from the minimum-uncertainty state for the two-photon case is higher than that for the one-photon case.
References
[ 1] E.T. Jaynes and C.W. Cummings, Proc. IEEE 51 (1963) 89. [2] R.H. Dicke, Phys. Rev. 93 (1954) 99. [ 3 ] M. Butler and P.D. Drummond, Optica Acta 33 (1986) 1. [4] M. Schubert and B. Wicheimi, Nonlinear optics and quantum electronics (Wiley, New York, 1986). [ 5 ] C.C. Gerry and J.B. Togeas,Optics Comm. 69 (1989) 263. [6 ] Li Shiqun et al., China Laser, to be published.
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