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Phase fluctuations of coherent light in an anharmonic oscillator using the hermitian phase operator
Volume 75, number 2
OPTICS COMMUNICATIONS
15 February 1990
PHASE F L U C T U A T I O N S O F C O H E R E N T L I G H T IN AN A N H A R M O N I C O S C I L L A T O R U S I N G T H E H E R M I T I A N PHASE OPERATOR Christopher C. GERRY Department of Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA
Received 17 July 1989; revised manuscript received 10 October 1989
We study the phase fluctuations of coherent light interacting with a non-absorbing nonlinear medium, modelled as an anharmonic oscillator, by using the Pegg-Barnen formalism for hermitian phase operators.
In a recent paper [ 1 ], we studied the phase fluctuations resulting from coherent light interacting with a non-absorbing nonlinear medium modelled as an anharmonic oscillator. Such a system has been shown to give rise to squeezed light [2,3] as well as various forms of higher order squeezing [4-6 ]. In our study of the phase fluctuations, we used the phase operator formalism first introduced by Susskind and Glogower [ 7 ] (see ref. [ 8 ] for a review of this formalism). We found that, generally speaking, enhanced fluctuations in the phase (the photon counting statistics remain poissonian) appeared in the emerging light and that the number-phase uncertainty product no longer approached the m i n i m u m as for a coherent field at large photon numbers. In fact, the greater the photon number, the greater were the fluctuations in the phase. It is well known however, that the Susskind-Glogower (SG) phase operator formalism suffers from the fact that hermitian and unitary phase operators cannot be constructed. Nevertheless, very recently, Pegg and Barnett [9,10] have shown a way out of this difficulty. Instead of utilizing the whole infinite dimensional Hilbert space of the single mode field, they use a finite but arbitrarily large subspace whose dimension is allowed to become infinite after the calculation of observable quantities. This new formalism predicts that the phase properties for weak fields are quite different than those predicted by the SG operators. This has been shown to be the case for special physical states of the field such as the number 168
states and the coherent states [ 11 ] and the ideal twophoton coherent states [ 12 ] of Yuen [ 13 ]. In this paper we re-examine the phase properties of the coherent states interacting with the anharmonic oscillator using the new phase operator formalism. As in the previous calculations with this formalism, we obtain essentially the same results as with the SG phase operators in the limit of high field excitation but with deviations in the case of a low field intensity. We will not extensively review the new phase operator formalism here but refer the reader to the cited literature. The hamiltonian for the system under consideration has the form [2] /~= h~od*~ + ½/ch~*2~2 ,
( 1)
where the symbol ^ means an operator in Hilbert space and k is the anharmonicity parameter related to the medium. To remove the rapid oscillations at frequency ~o we work in the interaction picture where HI
= ~ k'hd~'2d 2 .
(2)
For an initial coherent state given as Io~)=exp(-½1o~l 2) n=o ~ (/7!) an 1/2 I n ) ,
(3)
containing average photon number iT= lal 2, the state of the system at a later time is
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 75, number 2
OPTICS COMMUNICATIONS
la, t> = e x p ( - i ½ k t a t 2 a 2) lot) Ot 2
=exp(-I°zl2/2) n=O ~ (n!) 1/2 e x p ( - i f 2 n r )
In> ,
(4) where ~2,=½n(n- l ) and r=kt. Now according to Pegg and Barnett [9,10], the phase states l0m) are given in terms of the number states I n) as 10.,)=lim ( s + l ) -1/2 Z exp(inOm)Jn), s~oo
(5)
15 February 1990
state, where the energy is finite, the expectation value of the unitary phase operator exp(im6) (m is integer) is given as [12]
~
(13)
Using the state pa, t) given in eq. (4) we find ( e x p ( i d ) ) = ( e x p ( - i d ) ) * = a exp( - [al 2)
x .=o ~,
la12"
[ n ! ( n ~ 1-)!] l/i e x p ( - i n r ) ,
(14)
and ( e x p ( 2 i ~ ) ) = (exp ( - 2i~))* = a2 exp ( - [al 2)
X z~
lal2n e x p [ - i ( 2 n + 1)r] .=o [n!(n+ 2 )!] 1/2
(COS ~> =/.~1/2
exp(-ri) t~n
×n~=O[n!(n+ l )!],/2cos(~-nr)
IOm>(O.,In>
(15)
Thus, with a = r i 1/2 exp(i~), we have
oo
m=0
.
n 0
n=0
where Om=Oo+2~m/(s+l), m = 0 , l, 2 ..... and 0o is chosen anywhere along the real line. In what follows we shall drop the l i m s ~ with the understanding that all calculations are to be done with s finite subsequently to be followed by the limit. The states 10,,) are orthonormai and complete. The number states may be expressed in terms of the phase states according to In>=
In)(n+m[
(exp(imd))p=
,
(16)
(sin 0) =~,/2 exp( - r D = ( s + l ) -'/2 ~ exp(-inO.,)lO.,).
(6)
m=O
oo
an
X ~.,o=[n!(n+l)!]t/2sin(~-nr
),
(17)
The hermitian phase operator is given as
,~o=mY= 0 O,.lOm)(O,.,I,
(7)
which, as expected [ 12 ], are in agreement with the results of ref. [ 1 ]. On the other hand, we have
where the 0., are the eigenvalues of ~o. There also exist a unitary phase operator exp(i~) with the eigenvalues exp (i0,.). The hermitian conjugate is
(cos20)=½+riexp(-r]) ~ rt~ .=o [n!(n+ 2 )!] 1/2
[exp(i~) ] t = e x p ( - i 0 ) ,
(sin20F=½-t~exp(-rI) ~ tin .=o [n!(n+ 2 )!] '/:
(8)
which has eigenvalues exp(-iOm). Cosine and sine operators are defined as cos ~= ½[exp(i¢~) +exp( - i ~ ) ] , sin ~= (1/2i) [ e x p ( i ~ ) - e x p ( - i 0 ) ] ,
(9) (10)
Xcos[2~-(2n+ l)r] ,
Xcos[2~-(2n+l)r]
,
(19)
which do differ from the result using the SG operators [ 1 ] by the amount [ 12 ] l ( ( 1 0 > (01)>
while for the squares of these operators we have
which in this case is just
cos2¢~= ¼[exp(2i¢) + exp( - 2i~) + 2 ] ,
(11)
) ( a , tlO) (Ola, t ) = ~ e x p ( - a ) ,
sin2~= - ~ [exp(2i~) +exp( - 2 i 0 ) - 2 ] .
(12)
just as for a coherent state. Obviously we have
It can also be shown that for a physically realizable
(18)
(COS20) + (sin20) = 1
(20)
(21)
(22) 169
Volume 75, number 2
OPTICS COMMUNICATIONS
which agrees with the SG result only in the limit n----~ (~O.
We now consider the distribution in phase for very high intensity fields where ~>> 1. For a realistic situation where squeezing is produced one may typically have r~~ l06 and r ~ 10 -6, see ref. [2]. The phase probability amplitude is (0,7 la, t) = e x p ( - r i / 2 )
15 February 1990
where (s+ 1 )/2n is the density of states. Using this distribution we calculate the expectation values of ~ and q~2, assuming ~ is not near 0o, to obtain (d)
= f OP(O) d O = ~ + ½ r + n r
and ( ~ 2 ) = (~)2..~_ 1/4jq+ r~r2.
×~o= (n!)~/~ exp[i(n~--Qnr) ] (OreIn>
Thus the fluctuations in the phase are
= (s+ 1 )-~/2 e x p ( - ~ / 2 )
(A~)2= ( ~ 2 ) = ( ~ ) 2
rT" × n=o ~ ~exp[i(n~-f2~r-nOm)].
(23)
As in the case of the coherent state the photon distribution is Poisson with the probability of there being n photons in the field given as exp(
p(n)=
=exp(-r~)
r~/2 . 2 - r~/2) ( ~ . ) 155 exp [1 ( n ~ - £2nr) ]
~q"/n!,
(24)
which for high ri is approximated as [ 11 ]
p(n)~-(2nff)-~/aexp[-(~q-n)2/2a] ,
(25)
which is normalized according to
f p(n)
dn= 1 .
(26)
Thus for the phase amplitude we have, using eqs. (23) and (25), (0,,, la, t) ~ ( s + 1 )-1/2 (27~h)-1/4 rC [ (¢-- Om+ ½r+~q'r) 2] × 1/4r~+ir/2 exp ])-r2-~--~r j
(27)
1/4n+r~r2.
(31)
(32)
This equation and eq. (30) are not valid for arbitrary r since the phase is bounded by 2n. We have approximated the exact discrete probability distribution with a continuous distribution, thus removing the periodicity in phase. This gives results valid in a window between 0o and 0o+2~, therefore limiting the allowed size of r. Since the photon number distribution is still poissonian we have (AN)2= rl SO that the number-phase uncertainty product is (AN)2(A0) 2= 1 + n2r2.
(33)
We see that at large rL the phase fluctuations are enhanced, in agreement with ref. [ 1 ]. We should also point out that the results of eqs. (32) and (33) have previously been obtained by Kitagawa and Yamamoto [ 14] using the SG operators in a study of the optical processing of a coherent state in a MachZehnder interferometer containing a Kerr medium which is described by the hamiltonian of eq. ( 1 ). Finally, we consider the effects of a very low intensity field. The phase probability distribution, using eq. (23), expanded to order r~ is
P(O) =
and the probability density in the phase is
I (0l~x, t ) 1 2 ~ (s+ 1 )-~
X [1 + 2 ( 4 ) 1/2 c o s ( C - 0 )
P ( 0 ) = I (01a, t)]2
+x/2r~ c o s ( 2 ~ - 2 0 - r) ] ,
[ (2r~/g) (1 +4r2r~2) ] I/2
xexp(-2r~ (¢-0+ ~ i ~ ~r+,~)2)j,
(30)
(28)
(34)
where we notice that the effect of the nonlinear interaction first appears in the term of order rL This distribution is normalized according to Oo + 2 ; t
which is normalized according to
fP(O)\
2~r j d O = l ,
f (29)
P(O) s~+- dl O =
1.
Oo
Using this distribution we obtain 170
(35)
Volume 75, number 2
OPTICS COMMUNICATIONS
15 February 1990
(~o> =Oo + 7r--2(n) l/2 sin(~-- 0o)
0.3
(36)
-- ½x/2a sin(2{--20o - - r ) ,
0.2
and 0.:1
( ~ o2 ) = 0 2 + 2z~Oo+ ~ z 2 - 4 ( f f ) , / 2 cos ( ~ - 0o) -4(2¢-0o-7r
) ( r ~ ) ' n s i n ( ~ - 0o)
+ (r~/x/2) c o s ( 2 ¢ - 2 0 o - r ) -0. 1
+ i f [ ( 1 / 8 , f 2 x ) ( 2 ¢ - 20o - r) 2
-0.2
-x/2(¢-r/2) (a)
-0.3
.3
I
I
I
I
I
-0.2
-0.1
0
0.1
0.2
0.3
0.3
-x/2x]sin(2~-2Oo-r)
,
(37)
from which we may determine the fluctuations (A¢~) 2. Note that, as for the usual two-photon coherent states, the results depend on 0o. F r o m eq. ( 3 4 ) it is apparent that for low photon n u m b e r fi, the dependence o f on z is weak i f r ~ 10 -6, for which the d i s t r i b u t i o n is indistinguishable from the case o f a weak coherent state. The polar form o f the distribution is essentially circular with
P(O)
0.2 O.J, 0 -0. ~. -0.2
(b) -0.3 -0.3
i -0.2
0.3
I
I
I
I
-0.~.
0
0.1
0.2
[
I
I
I
,3
0.2 0.:1
the center o f the circle offset from the origin along the line 0 = {. On the other hand, for an optical fiber, r could become quite large. For r>> 10 -6, the circle becomes distorted, somewhat in the shape o f a kidney, indicating the enhancement o f phase fluctuations. The effect is illustrated in fig. 1 for ~= ~ / 4 with r = 0 , 2, and 4. In summary, we have used the Pegg-Barnett [9,10] phase o p e r a t o r formalism to reconsider the phase fluctuations resulting from the interaction o f coherent light with an a n h a r m o n i c oscillator. The p r o b l e m has previously been studied [ 1 ] using the SusskindGlogower [ 7 ] operators. In the present paper we find identical results for very strong fields, as expected. At low fields, the effects differ from coherent states only for long interaction times, as might occur in an optical fiber.
References
-0. J. -0.2
(c) -0.3
[
.3
-0.2
I
-0.t
I
I
I
0
0.~.
0.2
0.3
Fig. 1. (s+ 1) P(0), plotted in polar form for ¢=n/4, with (a) r=0, (b) r=2.0, (c) r=4.0.
[ 1] C.C. Gerry, Optics Comm. 63 ( 1987 ) 278. [2] R. Tana~, in: Coherence and quantum optics, V, eds. L. Mandel and E. Wolf (New York, Plenum, 1984) p. 643. [3] G.J. Milburn, Phys. Rev. A 33 (1986) 674. [4] C.C. Gerry and S. Rodrigues, Phys. Rev. A 36 (1987) 5444. [5] R. TanaL Phys. Rev. A 38 (1988) 1091. [ 6 ] C.C. Gerry and E.R. Vrscay, Phys. Rev. A 37 ( 1988 ) 1779. [ 7 ] L Susskind and J. Glogower, Physics 1 (1964) 49. 171
Volume 75, number 2
OPTICS COMMUNICATIONS
[8]See, R. Loudon, Quantum theory of light (Oxford, Clarendon Press, 1983 ) p. 141. [ 9 ] D.T. Pegg and S.M. Barnett, Europhys. Lett. 6 ( 1988 ) 483. [ 10] D.T. Pegg and S.M. Barnett, Phys. Rev. A 39 (1989) 1665. [ 11 ] S.M. Barnett and D.T. Pegg, J. Mod. Optics 36 (1989) 7.
172
15 February 1990
[ 12 ] J.A. Vaccaro and D.T. Pegg, Optics Comm. 70 ( 1989 ) 529. [ 13 ] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 14] M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34 (1986) 3974.
OPTICS COMMUNICATIONS
15 February 1990
PHASE F L U C T U A T I O N S O F C O H E R E N T L I G H T IN AN A N H A R M O N I C O S C I L L A T O R U S I N G T H E H E R M I T I A N PHASE OPERATOR Christopher C. GERRY Department of Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA
Received 17 July 1989; revised manuscript received 10 October 1989
We study the phase fluctuations of coherent light interacting with a non-absorbing nonlinear medium, modelled as an anharmonic oscillator, by using the Pegg-Barnen formalism for hermitian phase operators.
In a recent paper [ 1 ], we studied the phase fluctuations resulting from coherent light interacting with a non-absorbing nonlinear medium modelled as an anharmonic oscillator. Such a system has been shown to give rise to squeezed light [2,3] as well as various forms of higher order squeezing [4-6 ]. In our study of the phase fluctuations, we used the phase operator formalism first introduced by Susskind and Glogower [ 7 ] (see ref. [ 8 ] for a review of this formalism). We found that, generally speaking, enhanced fluctuations in the phase (the photon counting statistics remain poissonian) appeared in the emerging light and that the number-phase uncertainty product no longer approached the m i n i m u m as for a coherent field at large photon numbers. In fact, the greater the photon number, the greater were the fluctuations in the phase. It is well known however, that the Susskind-Glogower (SG) phase operator formalism suffers from the fact that hermitian and unitary phase operators cannot be constructed. Nevertheless, very recently, Pegg and Barnett [9,10] have shown a way out of this difficulty. Instead of utilizing the whole infinite dimensional Hilbert space of the single mode field, they use a finite but arbitrarily large subspace whose dimension is allowed to become infinite after the calculation of observable quantities. This new formalism predicts that the phase properties for weak fields are quite different than those predicted by the SG operators. This has been shown to be the case for special physical states of the field such as the number 168
states and the coherent states [ 11 ] and the ideal twophoton coherent states [ 12 ] of Yuen [ 13 ]. In this paper we re-examine the phase properties of the coherent states interacting with the anharmonic oscillator using the new phase operator formalism. As in the previous calculations with this formalism, we obtain essentially the same results as with the SG phase operators in the limit of high field excitation but with deviations in the case of a low field intensity. We will not extensively review the new phase operator formalism here but refer the reader to the cited literature. The hamiltonian for the system under consideration has the form [2] /~= h~od*~ + ½/ch~*2~2 ,
( 1)
where the symbol ^ means an operator in Hilbert space and k is the anharmonicity parameter related to the medium. To remove the rapid oscillations at frequency ~o we work in the interaction picture where HI
= ~ k'hd~'2d 2 .
(2)
For an initial coherent state given as Io~)=exp(-½1o~l 2) n=o ~ (/7!) an 1/2 I n ) ,
(3)
containing average photon number iT= lal 2, the state of the system at a later time is
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 75, number 2
OPTICS COMMUNICATIONS
la, t> = e x p ( - i ½ k t a t 2 a 2) lot) Ot 2
=exp(-I°zl2/2) n=O ~ (n!) 1/2 e x p ( - i f 2 n r )
In> ,
(4) where ~2,=½n(n- l ) and r=kt. Now according to Pegg and Barnett [9,10], the phase states l0m) are given in terms of the number states I n) as 10.,)=lim ( s + l ) -1/2 Z exp(inOm)Jn), s~oo
(5)
15 February 1990
state, where the energy is finite, the expectation value of the unitary phase operator exp(im6) (m is integer) is given as [12]
~
(13)
Using the state pa, t) given in eq. (4) we find ( e x p ( i d ) ) = ( e x p ( - i d ) ) * = a exp( - [al 2)
x .=o ~,
la12"
[ n ! ( n ~ 1-)!] l/i e x p ( - i n r ) ,
(14)
and ( e x p ( 2 i ~ ) ) = (exp ( - 2i~))* = a2 exp ( - [al 2)
X z~
lal2n e x p [ - i ( 2 n + 1)r] .=o [n!(n+ 2 )!] 1/2
(COS ~> =/.~1/2
exp(-ri) t~n
×n~=O[n!(n+ l )!],/2cos(~-nr)
IOm>(O.,In>
(15)
Thus, with a = r i 1/2 exp(i~), we have
oo
m=0
.
n 0
n=0
where Om=Oo+2~m/(s+l), m = 0 , l, 2 ..... and 0o is chosen anywhere along the real line. In what follows we shall drop the l i m s ~ with the understanding that all calculations are to be done with s finite subsequently to be followed by the limit. The states 10,,) are orthonormai and complete. The number states may be expressed in terms of the phase states according to In>=
In)(n+m[
(exp(imd))p=
,
(16)
(sin 0) =~,/2 exp( - r D = ( s + l ) -'/2 ~ exp(-inO.,)lO.,).
(6)
m=O
oo
an
X ~.,o=[n!(n+l)!]t/2sin(~-nr
),
(17)
The hermitian phase operator is given as
,~o=mY= 0 O,.lOm)(O,.,I,
(7)
which, as expected [ 12 ], are in agreement with the results of ref. [ 1 ]. On the other hand, we have
where the 0., are the eigenvalues of ~o. There also exist a unitary phase operator exp(i~) with the eigenvalues exp (i0,.). The hermitian conjugate is
(cos20)=½+riexp(-r]) ~ rt~ .=o [n!(n+ 2 )!] 1/2
[exp(i~) ] t = e x p ( - i 0 ) ,
(sin20F=½-t~exp(-rI) ~ tin .=o [n!(n+ 2 )!] '/:
(8)
which has eigenvalues exp(-iOm). Cosine and sine operators are defined as cos ~= ½[exp(i¢~) +exp( - i ~ ) ] , sin ~= (1/2i) [ e x p ( i ~ ) - e x p ( - i 0 ) ] ,
(9) (10)
Xcos[2~-(2n+ l)r] ,
Xcos[2~-(2n+l)r]
,
(19)
which do differ from the result using the SG operators [ 1 ] by the amount [ 12 ] l ( ( 1 0 > (01)>
while for the squares of these operators we have
which in this case is just
cos2¢~= ¼[exp(2i¢) + exp( - 2i~) + 2 ] ,
(11)
) ( a , tlO) (Ola, t ) = ~ e x p ( - a ) ,
sin2~= - ~ [exp(2i~) +exp( - 2 i 0 ) - 2 ] .
(12)
just as for a coherent state. Obviously we have
It can also be shown that for a physically realizable
(18)
(COS20) + (sin20) = 1
(20)
(21)
(22) 169
Volume 75, number 2
OPTICS COMMUNICATIONS
which agrees with the SG result only in the limit n----~ (~O.
We now consider the distribution in phase for very high intensity fields where ~>> 1. For a realistic situation where squeezing is produced one may typically have r~~ l06 and r ~ 10 -6, see ref. [2]. The phase probability amplitude is (0,7 la, t) = e x p ( - r i / 2 )
15 February 1990
where (s+ 1 )/2n is the density of states. Using this distribution we calculate the expectation values of ~ and q~2, assuming ~ is not near 0o, to obtain (d)
= f OP(O) d O = ~ + ½ r + n r
and ( ~ 2 ) = (~)2..~_ 1/4jq+ r~r2.
×~o= (n!)~/~ exp[i(n~--Qnr) ] (OreIn>
Thus the fluctuations in the phase are
= (s+ 1 )-~/2 e x p ( - ~ / 2 )
(A~)2= ( ~ 2 ) = ( ~ ) 2
rT" × n=o ~ ~exp[i(n~-f2~r-nOm)].
(23)
As in the case of the coherent state the photon distribution is Poisson with the probability of there being n photons in the field given as exp(
p(n)=
=exp(-r~)
r~/2 . 2 - r~/2) ( ~ . ) 155 exp [1 ( n ~ - £2nr) ]
~q"/n!,
(24)
which for high ri is approximated as [ 11 ]
p(n)~-(2nff)-~/aexp[-(~q-n)2/2a] ,
(25)
which is normalized according to
f p(n)
dn= 1 .
(26)
Thus for the phase amplitude we have, using eqs. (23) and (25), (0,,, la, t) ~ ( s + 1 )-1/2 (27~h)-1/4 rC [ (¢-- Om+ ½r+~q'r) 2] × 1/4r~+ir/2 exp ])-r2-~--~r j
(27)
1/4n+r~r2.
(31)
(32)
This equation and eq. (30) are not valid for arbitrary r since the phase is bounded by 2n. We have approximated the exact discrete probability distribution with a continuous distribution, thus removing the periodicity in phase. This gives results valid in a window between 0o and 0o+2~, therefore limiting the allowed size of r. Since the photon number distribution is still poissonian we have (AN)2= rl SO that the number-phase uncertainty product is (AN)2(A0) 2= 1 + n2r2.
(33)
We see that at large rL the phase fluctuations are enhanced, in agreement with ref. [ 1 ]. We should also point out that the results of eqs. (32) and (33) have previously been obtained by Kitagawa and Yamamoto [ 14] using the SG operators in a study of the optical processing of a coherent state in a MachZehnder interferometer containing a Kerr medium which is described by the hamiltonian of eq. ( 1 ). Finally, we consider the effects of a very low intensity field. The phase probability distribution, using eq. (23), expanded to order r~ is
P(O) =
and the probability density in the phase is
I (0l~x, t ) 1 2 ~ (s+ 1 )-~
X [1 + 2 ( 4 ) 1/2 c o s ( C - 0 )
P ( 0 ) = I (01a, t)]2
+x/2r~ c o s ( 2 ~ - 2 0 - r) ] ,
[ (2r~/g) (1 +4r2r~2) ] I/2
xexp(-2r~ (¢-0+ ~ i ~ ~r+,~)2)j,
(30)
(28)
(34)
where we notice that the effect of the nonlinear interaction first appears in the term of order rL This distribution is normalized according to Oo + 2 ; t
which is normalized according to
fP(O)\
2~r j d O = l ,
f (29)
P(O) s~+- dl O =
1.
Oo
Using this distribution we obtain 170
(35)
Volume 75, number 2
OPTICS COMMUNICATIONS
15 February 1990
(~o> =Oo + 7r--2(n) l/2 sin(~-- 0o)
0.3
(36)
-- ½x/2a sin(2{--20o - - r ) ,
0.2
and 0.:1
( ~ o2 ) = 0 2 + 2z~Oo+ ~ z 2 - 4 ( f f ) , / 2 cos ( ~ - 0o) -4(2¢-0o-7r
) ( r ~ ) ' n s i n ( ~ - 0o)
+ (r~/x/2) c o s ( 2 ¢ - 2 0 o - r ) -0. 1
+ i f [ ( 1 / 8 , f 2 x ) ( 2 ¢ - 20o - r) 2
-0.2
-x/2(¢-r/2) (a)
-0.3
.3
I
I
I
I
I
-0.2
-0.1
0
0.1
0.2
0.3
0.3
-x/2x]sin(2~-2Oo-r)
,
(37)
from which we may determine the fluctuations (A¢~) 2. Note that, as for the usual two-photon coherent states, the results depend on 0o. F r o m eq. ( 3 4 ) it is apparent that for low photon n u m b e r fi, the dependence o f on z is weak i f r ~ 10 -6, for which the d i s t r i b u t i o n is indistinguishable from the case o f a weak coherent state. The polar form o f the distribution is essentially circular with
P(O)
0.2 O.J, 0 -0. ~. -0.2
(b) -0.3 -0.3
i -0.2
0.3
I
I
I
I
-0.~.
0
0.1
0.2
[
I
I
I
,3
0.2 0.:1
the center o f the circle offset from the origin along the line 0 = {. On the other hand, for an optical fiber, r could become quite large. For r>> 10 -6, the circle becomes distorted, somewhat in the shape o f a kidney, indicating the enhancement o f phase fluctuations. The effect is illustrated in fig. 1 for ~= ~ / 4 with r = 0 , 2, and 4. In summary, we have used the Pegg-Barnett [9,10] phase o p e r a t o r formalism to reconsider the phase fluctuations resulting from the interaction o f coherent light with an a n h a r m o n i c oscillator. The p r o b l e m has previously been studied [ 1 ] using the SusskindGlogower [ 7 ] operators. In the present paper we find identical results for very strong fields, as expected. At low fields, the effects differ from coherent states only for long interaction times, as might occur in an optical fiber.
References
-0. J. -0.2
(c) -0.3
[
.3
-0.2
I
-0.t
I
I
I
0
0.~.
0.2
0.3
Fig. 1. (s+ 1) P(0), plotted in polar form for ¢=n/4, with (a) r=0, (b) r=2.0, (c) r=4.0.
[ 1] C.C. Gerry, Optics Comm. 63 ( 1987 ) 278. [2] R. Tana~, in: Coherence and quantum optics, V, eds. L. Mandel and E. Wolf (New York, Plenum, 1984) p. 643. [3] G.J. Milburn, Phys. Rev. A 33 (1986) 674. [4] C.C. Gerry and S. Rodrigues, Phys. Rev. A 36 (1987) 5444. [5] R. TanaL Phys. Rev. A 38 (1988) 1091. [ 6 ] C.C. Gerry and E.R. Vrscay, Phys. Rev. A 37 ( 1988 ) 1779. [ 7 ] L Susskind and J. Glogower, Physics 1 (1964) 49. 171
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OPTICS COMMUNICATIONS
[8]See, R. Loudon, Quantum theory of light (Oxford, Clarendon Press, 1983 ) p. 141. [ 9 ] D.T. Pegg and S.M. Barnett, Europhys. Lett. 6 ( 1988 ) 483. [ 10] D.T. Pegg and S.M. Barnett, Phys. Rev. A 39 (1989) 1665. [ 11 ] S.M. Barnett and D.T. Pegg, J. Mod. Optics 36 (1989) 7.
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[ 12 ] J.A. Vaccaro and D.T. Pegg, Optics Comm. 70 ( 1989 ) 529. [ 13 ] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 14] M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34 (1986) 3974.