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Periodic revivals of squeezing in an anharmonic-oscillator model with coherent light
Volume
136. number
4.5
PH’IYCS
PERIODIC REVIVALS OF SQUEEZING WITH COHERENT LIGHT Vladimir
BUPEK
LETTERS
3 Apnl 1989
4
IN AN ANHARMONIC-OSCILLATOR
MODEL
’
ICTP. P.0. B. 586. 34 I 00 Trie.ste, Iralj, Received I4 October 1988: accepted Communicated by J.P. Vigier
for publication
27 January
1989
We ha\e carried out a detailed investigation of squeezing in the anharmonic-oscillator model with coherent light. The Hamlllonian is of the form H=fwK,+iiX+h~_. where k;,. bI are the generators of the Lie algebra of the SU( 1. I ) group. We have shown that for a particular realization of the generators K the squeezing performs periodic revivals. We find analytically the maximum value of the squeezing
1. Introduction In recent years a remarkable amount of interest has been devoted to the study of the exactly solvable models which exhibit significant squeezing of the electromagnetic field (for a review on squeezed light seeref. [I]). Among others Tanas [ 2 ] has studied the interaction of coherent light with a nonabsorbing nonlinear medium modelled as an anharmonic oscillator with the Hamiltonian H=kwa’a+~M(af)‘a’.
(1)
He has found that if a large number of photons ( - 10’) is present initially in the system. then the light becomes squeezed significantly over the appropriate time scale of At _ 1Oph ( i.e. the variance in one quadrature is sufficiently less than l/4 ). Moreover. in the framework of the model described by the Hamiltonian ( 1 ). it can be observed that at longer times the variance under consideration tends to oscillate irregularly and neL,er’becomes squeezed again. So. after the first reduction of the variance. the original squeezing becomes definitely revoked. Here we can mention that the mode1 proposed by Tanas has been generalized recently by Gerry [ 3 1, ’ Permanent
address: Institute of Physics. EPRC SAS. Dtibravski cesta 9. CS-832 78 Bratislava, Czechoslovakia.
188
who has studied the k-photon anharmonic oscillator with the nonlinear term proportional to (frA/li) x (a+ )‘u’. Gerry has shown that such a model Hamiltonian interacting with the coherent light can give rise to enhanced squeezing for some values of the number of photons. Nevertheless, his results do not differ principally from those of Tanas. In the present Letter we will study a model of the anharmonic oscillator very similar to that considered by Tanas [I], the typical features of which are the periodic revivals of the squeezing of the variances of the quadrature operators. We will analyze in detail the behaviour of the system in the short-time scale (At-z 1). when squeezing occurs in one quadrature if the number of photons is sufficiently large. It will be also shown that significant squeezing in the second quadrature can be obtained for weak intensities (small number of photons) at a long-time scale (it- Jr).
2. The anharmonic-oscillator
model
First of all we want to call attention to the fact that the Hamiltonian ( 1 ) can be rewritten (up to constant factors) in terms of the generators K,,, K+ and K_ of the Lie algebra of the group SU ( I, 1 ) [ 41: H=hwK,,
+hX+
It_ .
(2)
03759601/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 136. number
4.5
PHYSICS
where the generators mutation relations
K,,, K,, which satisfy the com-
[k;,,K,]=kK,,
[K_,K+]=2Ko.
are expressed
(3)
in terms of the boson operators
u and
LETTERS
3
A
‘4, =(a)
exp[i(or-y,)] f P,, exp[ -iA(2n+ n=o
=JZ
=,/‘nexp[ xexp[
K,~=~(a+a+au+). K, =;(a+)‘,
K_=fa’.
K,
=,:u+uu+ ,
K_ =a&%.
(5)
Since we suppose the interaction of the anharmanic oscillator with ordinary coherent light, the initial state vector of the system can be written as
1 -cos
.42- (u2)
exp[2i(wt-a,)]
=n
P,, exp[ -4il(n+
f
1 )t]
21t)] ,
1)[I
PI=c,
=Eexp[-_(I-cos4&)] xexp[
Ko=;(u+u+uu+),
-fi(
-i(At+fisin2it)]
(4)
In our analysis we will also consider the Hamiltonian in the form (2). but with a different realization of the SU ( 1, 1) Lie algebra. Namely [ 51. let
.April I989
-i(4At+fisin
4At)] .
where P,, is the distribution
(9)
of the coherent
light:
P,,=~Q,,~‘=e-“5.
3. Light squeezing
(6) where cy= ,/’ fi e” and fi is the dimensionless intensity of the coherent field (i.e. the average number of photons fi= (cy (a ‘a (a) and rp is the phase of this field ). The state vector II,v( t)) of the system at t>O is given by the time-dependent Schrodinger equation: ifi$
Iv/(t))=Wyl(t)).
and can be obtained
Iv(t)>=
(7) in the following
,,~oQ,,exp[-i(nw+lnl)tlIn).
f P,,n=H. ,r=ll
a,(t)=f(ae”‘“‘-“‘+u+e-“‘“‘-“‘) u,(tj=
$
,
.
(uelt’,~‘-~)_a+e-il’,J/-“‘)
(10)
where S is an arbitrary phase chosen to be equal to 9. Since the squeezed states are defined as the states with a smaller uncertainty (variance) in one quadrature of the field than that associated with the coherent field, it is convenient to define two functions
161:
form:
(8)
For the purpose of the following calculations we will write down the mean values of the photon number operator (a ‘a). photon amplitude (a) and the squared photon amplitude (a’): d40=(a+u)=
To analyze the squeezing properties of the radiation we introduce two Hermitian time-dependent quadrature operators
S,(r)=
b:(r)-
fori=l,2.
C”“(t)
c-y”(t)
=4v(t)_
I
’
’ (11)
where the variances 6’!(t) are defined as usual, V!=(a’)-(a,)’ and Vyh(f)=i. The squeezing condition now looks very simply $(f)
(12)
So, with the function S,(t) ( i= 1, 2) going to - 1 the squeezing becomes larger and larger. 189
Volume
136, number
4.5
PHYSICS
The variances L;(t) can be expressed mean values of the photon operators functions S,(t) can be written as S, =2&+2
ReA,--4(Re.4,)‘.
SZ=2.4,,--2
ReA,-4(Im.4,)‘.
LETTERS
.A
3 .Apnl 1989
through the (9). so the
0
0.2 (13) 04
-2H(
Xcos’(At+tisin
Ut)
-2fi(
Xsin’(At+ti
.
.S,( I) versus the average pho-
(14a) 0.6 1
l-cos4&)]
1 -cos
0.4 2&)]
sin 2At) .
(14b)
fork=l.2
,...
S, = -4?ie-I”
.
_’
-04
1
ITAt Fig. 2. Time evolution
ofthe
functions
S,( 1) for A=0.25.
W)
This means that if at some moment one quadrature becomes squeezed. then this squeezing will reappear periodically at later times. It is easy to find that for L= T /2 the functions S, ( t ) are (16)
From here it follows that there is no squeezing (at t= T/2) in the first quadrature, but the second one can be squeezed maximally (up to z 37% for fi= $ ). In fig. 1 the minimum value of Sz ( t ) (corresponding to the maximum squeezing in the second quadrature) is plotted versus the mean photon number ti. From here it follows that for long-time scales and weak intensities the squeezing occurs in the second quadrature. In fig. 2 the time evolution of the functions S,(C) is plotted. It is seen from this picture that the variances are periodical and that the variance in the first 190
of the function
ii= 0.25
Further we will analyze these functions in two time scales. First in the long-time scale (At- 7~) and second in the short-time scale (it< 1). (i) Long-finze scale. The first remarkable feature of the functions S,( t ) is their exact periodicity with period T= x//1: S,(t)=S,(t+kT)
I
I -cos 2At)]
(4At + H sin 4At)
-4Hexp[
s, =4/i.
I
4&)
S,(f)=?E-2fiexp[-fi( X cos
VI
Fig. 1. The minima ton number ti.
S,(t)=2fi+?Aexp[-~(l-cos4~t)]
-4rfexp[
1.5
52
Using the explicit expressions for ‘4, (9). we obtain for the functions S,( t ) the following expressions:
Xcos(4At+Hsin
1.0 ii
05
0.0
quadrature is squeezed in its evolution, too. The maximum squeezing in this quadrature appears at times closer to the initial moment (we will show later that with increasing the average photon number the moments when the maximum squeezing in the first quadrature is achieved become closer and closer to the initial moment; of course this squeezing will reappear periodically at later times). Finally we mention that with increasing the average photon number ( tin 1 ) the squeezing in the second quadrature vanishes definitely. (ii) Short-time scale. First we will analyze the behaviour of the functions S,(t) close to the initial moment. To do so we will calculate their first and second derivatives at t=O. for which we find $S,\,=,=O.
fori=1.2.
(17)
Volume
136, number
4.5
PHYSICS
LETTERS
.+I
3 April 1989
and $s,
I,=rJ=-8A’fi(4n+3)<0,
(18a)
&
,r=o=811’n(*n+3)~0.
(18b)
From the above it follows that the first quadrature tends to be squeezed at first moments of the evolution. Moreover, the more photons in the initial state, the more rapidly the squeezing grows. One can also see that there is no tendency for squeezing in the second quadrature at initial moments. Now the question is, at what moment the first minimum of the function S, (t) can be observed. From the above considerations it is clear that it has sense to find the first minimum of this function for fi> 1. So, we have to find the first derivative of S, (I) for t> 0. Using the conditions At < 1 and fi>> 1, we find EL -4,G[8fiAtsin’(2ri/lt)-sin(4fiAt)] at The right-hand t,=
.
0
20
40
Ti 60
Fig. 3. The minima of the function S, at the moment versus the average photon number A.
ir= 10 -’
(19)
side of ( 19 ) is equal to zero for times
55 4&i * Fig. 4. The minima ton number fi.
where CX=1.3066.
(20b)
From here it follows that the moment when the maximum squeezing in the first quadrature occurs for the first time is proportional to the inverse number of photons. This can be seen also from fig. 3 obtained by numerical calculations where the minima of the function S, achieved at At= lo-’ are plotted versus the intensity a. From this picture one can see that the maximum squeezing is achieved for fiz 3 1 which is in excellent agreement with our formula (20) (n~32.6). Now we will look for the absolute minimum of the function S, (t) for Ate 1 and fi> 1. We can find it by substituting the value oft, from eq. (20) into S, and searching for the minimum of S, (t, ) as a function of ti. To do so we have to solve the equation
(21)
of the function
S,
(t) versus the average pho-
The derivative at the left-hand side of (2 1) can be easily calculated up to the second order of the expansion in the small parameter a/~:
(22) It is seen now that the maximum squeezing is reached for fi-tco. The last thing we have to do is to calculate the limit of the function s,(t) at t, for E-+co. For this limit we find limS,(t,)=+cu’(l-coscr)--sina!. ri-1Zc
(23)
Substituting the value of the parameter o (20b) into the last expression we find that the absolute minimum of the function S, is S, (t, ),,, = -0.6306
.
(24)
191
\‘olumc
136.
number
4.5
PHYSICS
This result is again in very good correspondence with computer calculations. It is seen from fig. 4. where the minimal values of the function S, are plotted l’ersus the intensity IT, that our analysis works vet well already for ii> 15. It should be stressed here that the analysis presented above is extremely useful for very large vi. tvhen the computer calculations may become difficult.
LETTERS
.A
3 .April I989
These periodic revivals of the squeezing in the first quadrature are seen also for weak intensities (see fig. 7). 4s was discussed earlier, for weak intensities one can obtain significant squeezing in the second quadrature. The masimum squeezing in this quadrature occurs periodically at times t; = ({+ k)T.
Acknowledgement
4. Conclusions We can conclude that the time evolution of the function S, (t) which describes the squeezing in the first quadrature is as follows: .L\t first moments S, tends to achieve its minimal value. Then at t= t,(20) the minimum is achieved and after this moment the squeezing starts to vanish - the more photons in the initial state, the more rapidly the maximum squeezing is achieved and then more rapidly it is revoked. After that the maximum squeezing periodically reappears at later times ri =~TF t,.where k= 1. 2. ... .
192
I am grateful to 5. Olejnik for helpful comments and to V. corn? for performing some computer calculations.
References [ I ] R. Loudon and P.L. Knight, J. Mod. Opt. 34 ( 1987 1709. [2] R. Tanas. in: Coherence and quantum optics. Vol. 5, eds. L. Mandel and E. Wolf ( Plenum. New \I’ork. 198-I) p. 643. [3] C.C. Gerry. Phys. Lett. .A 174 (1987) 237. [a] C.C. Gem, Phys. Rev. .-t 35 (1987) 2146. [5] B. Buck and C.V. Sukumar, Phys. Lett. .A 81 ( 1981 ) 132. [6] 4.S. Shumovsky. Fam Le Kien and E.I. .Aliskenderov, Phys. Lett. 4 124 (1987) 351.
136. number
4.5
PH’IYCS
PERIODIC REVIVALS OF SQUEEZING WITH COHERENT LIGHT Vladimir
BUPEK
LETTERS
3 Apnl 1989
4
IN AN ANHARMONIC-OSCILLATOR
MODEL
’
ICTP. P.0. B. 586. 34 I 00 Trie.ste, Iralj, Received I4 October 1988: accepted Communicated by J.P. Vigier
for publication
27 January
1989
We ha\e carried out a detailed investigation of squeezing in the anharmonic-oscillator model with coherent light. The Hamlllonian is of the form H=fwK,+iiX+h~_. where k;,. bI are the generators of the Lie algebra of the SU( 1. I ) group. We have shown that for a particular realization of the generators K the squeezing performs periodic revivals. We find analytically the maximum value of the squeezing
1. Introduction In recent years a remarkable amount of interest has been devoted to the study of the exactly solvable models which exhibit significant squeezing of the electromagnetic field (for a review on squeezed light seeref. [I]). Among others Tanas [ 2 ] has studied the interaction of coherent light with a nonabsorbing nonlinear medium modelled as an anharmonic oscillator with the Hamiltonian H=kwa’a+~M(af)‘a’.
(1)
He has found that if a large number of photons ( - 10’) is present initially in the system. then the light becomes squeezed significantly over the appropriate time scale of At _ 1Oph ( i.e. the variance in one quadrature is sufficiently less than l/4 ). Moreover. in the framework of the model described by the Hamiltonian ( 1 ). it can be observed that at longer times the variance under consideration tends to oscillate irregularly and neL,er’becomes squeezed again. So. after the first reduction of the variance. the original squeezing becomes definitely revoked. Here we can mention that the mode1 proposed by Tanas has been generalized recently by Gerry [ 3 1, ’ Permanent
address: Institute of Physics. EPRC SAS. Dtibravski cesta 9. CS-832 78 Bratislava, Czechoslovakia.
188
who has studied the k-photon anharmonic oscillator with the nonlinear term proportional to (frA/li) x (a+ )‘u’. Gerry has shown that such a model Hamiltonian interacting with the coherent light can give rise to enhanced squeezing for some values of the number of photons. Nevertheless, his results do not differ principally from those of Tanas. In the present Letter we will study a model of the anharmonic oscillator very similar to that considered by Tanas [I], the typical features of which are the periodic revivals of the squeezing of the variances of the quadrature operators. We will analyze in detail the behaviour of the system in the short-time scale (At-z 1). when squeezing occurs in one quadrature if the number of photons is sufficiently large. It will be also shown that significant squeezing in the second quadrature can be obtained for weak intensities (small number of photons) at a long-time scale (it- Jr).
2. The anharmonic-oscillator
model
First of all we want to call attention to the fact that the Hamiltonian ( 1 ) can be rewritten (up to constant factors) in terms of the generators K,,, K+ and K_ of the Lie algebra of the group SU ( I, 1 ) [ 41: H=hwK,,
+hX+
It_ .
(2)
03759601/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 136. number
4.5
PHYSICS
where the generators mutation relations
K,,, K,, which satisfy the com-
[k;,,K,]=kK,,
[K_,K+]=2Ko.
are expressed
(3)
in terms of the boson operators
u and
LETTERS
3
A
‘4, =(a)
exp[i(or-y,)] f P,, exp[ -iA(2n+ n=o
=JZ
=,/‘nexp[ xexp[
K,~=~(a+a+au+). K, =;(a+)‘,
K_=fa’.
K,
=,:u+uu+ ,
K_ =a&%.
(5)
Since we suppose the interaction of the anharmanic oscillator with ordinary coherent light, the initial state vector of the system can be written as
1 -cos
.42- (u2)
exp[2i(wt-a,)]
=n
P,, exp[ -4il(n+
f
1 )t]
21t)] ,
1)[I
PI=c,
=Eexp[-_(I-cos4&)] xexp[
Ko=;(u+u+uu+),
-fi(
-i(At+fisin2it)]
(4)
In our analysis we will also consider the Hamiltonian in the form (2). but with a different realization of the SU ( 1, 1) Lie algebra. Namely [ 51. let
.April I989
-i(4At+fisin
4At)] .
where P,, is the distribution
(9)
of the coherent
light:
P,,=~Q,,~‘=e-“5.
3. Light squeezing
(6) where cy= ,/’ fi e” and fi is the dimensionless intensity of the coherent field (i.e. the average number of photons fi= (cy (a ‘a (a) and rp is the phase of this field ). The state vector II,v( t)) of the system at t>O is given by the time-dependent Schrodinger equation: ifi$
Iv/(t))=Wyl(t)).
and can be obtained
Iv(t)>=
(7) in the following
,,~oQ,,exp[-i(nw+lnl)tlIn).
f P,,n=H. ,r=ll
a,(t)=f(ae”‘“‘-“‘+u+e-“‘“‘-“‘) u,(tj=
$
,
.
(uelt’,~‘-~)_a+e-il’,J/-“‘)
(10)
where S is an arbitrary phase chosen to be equal to 9. Since the squeezed states are defined as the states with a smaller uncertainty (variance) in one quadrature of the field than that associated with the coherent field, it is convenient to define two functions
161:
form:
(8)
For the purpose of the following calculations we will write down the mean values of the photon number operator (a ‘a). photon amplitude (a) and the squared photon amplitude (a’): d40=(a+u)=
To analyze the squeezing properties of the radiation we introduce two Hermitian time-dependent quadrature operators
S,(r)=
b:(r)-
fori=l,2.
C”“(t)
c-y”(t)
=4v(t)_
I
’
’ (11)
where the variances 6’!(t) are defined as usual, V!=(a’)-(a,)’ and Vyh(f)=i. The squeezing condition now looks very simply $(f)
(12)
So, with the function S,(t) ( i= 1, 2) going to - 1 the squeezing becomes larger and larger. 189
Volume
136, number
4.5
PHYSICS
The variances L;(t) can be expressed mean values of the photon operators functions S,(t) can be written as S, =2&+2
ReA,--4(Re.4,)‘.
SZ=2.4,,--2
ReA,-4(Im.4,)‘.
LETTERS
.A
3 .Apnl 1989
through the (9). so the
0
0.2 (13) 04
-2H(
Xcos’(At+tisin
Ut)
-2fi(
Xsin’(At+ti
.
.S,( I) versus the average pho-
(14a) 0.6 1
l-cos4&)]
1 -cos
0.4 2&)]
sin 2At) .
(14b)
fork=l.2
,...
S, = -4?ie-I”
.
_’
-04
1
ITAt Fig. 2. Time evolution
ofthe
functions
S,( 1) for A=0.25.
W)
This means that if at some moment one quadrature becomes squeezed. then this squeezing will reappear periodically at later times. It is easy to find that for L= T /2 the functions S, ( t ) are (16)
From here it follows that there is no squeezing (at t= T/2) in the first quadrature, but the second one can be squeezed maximally (up to z 37% for fi= $ ). In fig. 1 the minimum value of Sz ( t ) (corresponding to the maximum squeezing in the second quadrature) is plotted versus the mean photon number ti. From here it follows that for long-time scales and weak intensities the squeezing occurs in the second quadrature. In fig. 2 the time evolution of the functions S,(C) is plotted. It is seen from this picture that the variances are periodical and that the variance in the first 190
of the function
ii= 0.25
Further we will analyze these functions in two time scales. First in the long-time scale (At- 7~) and second in the short-time scale (it< 1). (i) Long-finze scale. The first remarkable feature of the functions S,( t ) is their exact periodicity with period T= x//1: S,(t)=S,(t+kT)
I
I -cos 2At)]
(4At + H sin 4At)
-4Hexp[
s, =4/i.
I
4&)
S,(f)=?E-2fiexp[-fi( X cos
VI
Fig. 1. The minima ton number ti.
S,(t)=2fi+?Aexp[-~(l-cos4~t)]
-4rfexp[
1.5
52
Using the explicit expressions for ‘4, (9). we obtain for the functions S,( t ) the following expressions:
Xcos(4At+Hsin
1.0 ii
05
0.0
quadrature is squeezed in its evolution, too. The maximum squeezing in this quadrature appears at times closer to the initial moment (we will show later that with increasing the average photon number the moments when the maximum squeezing in the first quadrature is achieved become closer and closer to the initial moment; of course this squeezing will reappear periodically at later times). Finally we mention that with increasing the average photon number ( tin 1 ) the squeezing in the second quadrature vanishes definitely. (ii) Short-time scale. First we will analyze the behaviour of the functions S,(t) close to the initial moment. To do so we will calculate their first and second derivatives at t=O. for which we find $S,\,=,=O.
fori=1.2.
(17)
Volume
136, number
4.5
PHYSICS
LETTERS
.+I
3 April 1989
and $s,
I,=rJ=-8A’fi(4n+3)<0,
(18a)
&
,r=o=811’n(*n+3)~0.
(18b)
From the above it follows that the first quadrature tends to be squeezed at first moments of the evolution. Moreover, the more photons in the initial state, the more rapidly the squeezing grows. One can also see that there is no tendency for squeezing in the second quadrature at initial moments. Now the question is, at what moment the first minimum of the function S, (t) can be observed. From the above considerations it is clear that it has sense to find the first minimum of this function for fi> 1. So, we have to find the first derivative of S, (I) for t> 0. Using the conditions At < 1 and fi>> 1, we find EL -4,G[8fiAtsin’(2ri/lt)-sin(4fiAt)] at The right-hand t,=
.
0
20
40
Ti 60
Fig. 3. The minima of the function S, at the moment versus the average photon number A.
ir= 10 -’
(19)
side of ( 19 ) is equal to zero for times
55 4&i * Fig. 4. The minima ton number fi.
where CX=1.3066.
(20b)
From here it follows that the moment when the maximum squeezing in the first quadrature occurs for the first time is proportional to the inverse number of photons. This can be seen also from fig. 3 obtained by numerical calculations where the minima of the function S, achieved at At= lo-’ are plotted versus the intensity a. From this picture one can see that the maximum squeezing is achieved for fiz 3 1 which is in excellent agreement with our formula (20) (n~32.6). Now we will look for the absolute minimum of the function S, (t) for Ate 1 and fi> 1. We can find it by substituting the value oft, from eq. (20) into S, and searching for the minimum of S, (t, ) as a function of ti. To do so we have to solve the equation
(21)
of the function
S,
(t) versus the average pho-
The derivative at the left-hand side of (2 1) can be easily calculated up to the second order of the expansion in the small parameter a/~:
(22) It is seen now that the maximum squeezing is reached for fi-tco. The last thing we have to do is to calculate the limit of the function s,(t) at t, for E-+co. For this limit we find limS,(t,)=+cu’(l-coscr)--sina!. ri-1Zc
(23)
Substituting the value of the parameter o (20b) into the last expression we find that the absolute minimum of the function S, is S, (t, ),,, = -0.6306
.
(24)
191
\‘olumc
136.
number
4.5
PHYSICS
This result is again in very good correspondence with computer calculations. It is seen from fig. 4. where the minimal values of the function S, are plotted l’ersus the intensity IT, that our analysis works vet well already for ii> 15. It should be stressed here that the analysis presented above is extremely useful for very large vi. tvhen the computer calculations may become difficult.
LETTERS
.A
3 .April I989
These periodic revivals of the squeezing in the first quadrature are seen also for weak intensities (see fig. 7). 4s was discussed earlier, for weak intensities one can obtain significant squeezing in the second quadrature. The masimum squeezing in this quadrature occurs periodically at times t; = ({+ k)T.
Acknowledgement
4. Conclusions We can conclude that the time evolution of the function S, (t) which describes the squeezing in the first quadrature is as follows: .L\t first moments S, tends to achieve its minimal value. Then at t= t,(20) the minimum is achieved and after this moment the squeezing starts to vanish - the more photons in the initial state, the more rapidly the maximum squeezing is achieved and then more rapidly it is revoked. After that the maximum squeezing periodically reappears at later times ri =~TF t,.where k= 1. 2. ... .
192
I am grateful to 5. Olejnik for helpful comments and to V. corn? for performing some computer calculations.
References [ I ] R. Loudon and P.L. Knight, J. Mod. Opt. 34 ( 1987 1709. [2] R. Tanas. in: Coherence and quantum optics. Vol. 5, eds. L. Mandel and E. Wolf ( Plenum. New \I’ork. 198-I) p. 643. [3] C.C. Gerry. Phys. Lett. .A 174 (1987) 237. [a] C.C. Gem, Phys. Rev. .-t 35 (1987) 2146. [5] B. Buck and C.V. Sukumar, Phys. Lett. .A 81 ( 1981 ) 132. [6] 4.S. Shumovsky. Fam Le Kien and E.I. .Aliskenderov, Phys. Lett. 4 124 (1987) 351.