- Email: [email protected]
On the phase fluctuations of coherent light interacting with an anharmonic oscillator
OPTICS COMMUNICATIONS
Volume 63, number 4
15 August 1987
ON THE PHASE FLUCTUATIONS OF COHERENT LIGHT INTERACTING WITH AN ANHARMONIC OSCILLATOR Christopher C. GERRY Department ofPhysics, St. Bonaventure University, St. Bonaventure, NY 14778, USA
Received 28 January 1987
We show that ordinary coherent light interacting with a non-absorbing nonlinear medium modelled as an anharmonic oscillator gives rise to enhanced and reduced phase fluctuations as well as squeezing.
Recently there has been some discussion of the phase operators and phase fluctuations associated with squeezed states of the electromagnetic field [ 1,2]. In refs. [ 1 ] and [ 21 the squeezed states discussed are of the two photon type first considered by Yuen [ 31. In this paper we consider ordinary coherent states interacting with a nonlinear non-absorbing medium modeled as an anharmonic oscillator. This system has previously been shown to give rise to squeezed light [4] and also to higher order squeezing [ 51 in the sense of Hong and Mandel [ 61. In this paper we show that while the photon number fluctuations are identical to those of coherent states (i.e. the statistics remain poissonian) the fluctuations of the phase variables are generally enhanced compared to the coherent state values as discussed by Carruthers and Nieto [ 71. In some occasions however, the fluctuations may actually be reduced. The hamiltonian for the sytem is of the form [4] A=hwB+ci+ tMid+?
,
(1)
where the h symbol means an operator in Hilbert space and k is the anharmonicity parameter related to the third order susceptibility of the medium. This anharmonic term is known to lead to optical bistability [ 81. Heisenberg’s equation for CEreads d=-(i/fi)[ci,fi]=-i(w+kri+B)B.
(2)
It is trivial to show that fi=ri+d is a constant of the motion reflecting the conservation of photon numbers. Thus the solution of eq. (2) is simply 278
ci(t>=exp{-it[w+kZQO)]}d(O).
(3)
We introduce the quadrature operators of the field 2, =ci+b+ ) ,t2 = -i(b-d+)
,
(4)
such that [fI,z21=2i,
(5)
leading to the uncertainty relation (dX,)2 (dX,)2>1
.
(6)
We consider the 3, quadrature only which becomes squeezed whenever (AX, )’ < 1. Using eq. (3), drop ping the free evolution term, and assuming the initial state to be a coherent state 1a) with the phase of cr chosen so that (Yis real, we obtain the second order variance [ 41
+2NRe{exp( -ir)
exp[l\r(exp( -2ir) - l)]
-exp[2N(exp(-ir)-l)]},
(7)
where N= ((Y 1fi/ a )“and 7 = kt. We now consider the phase operators. We define the phase operators f?=g(i(p) in the usual way
[7,%101 B+ =f$+
&(fi+l)-“‘S,
.
(8)
Now we define “sine” and “cosine” operators S= (1/2i)(fi-JZ+)
,
e=t(l?+iff)
)
(9)
0 030-4018/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
OPTICS COMMUNICATIONS
Volume 63, number 4
15 August 1987
and in the usual way define the uncertainty relations (dN)2 (dS)%~(C)2, _tlve_N~ (dzQ2 (K)22$(s>2,
(10)
and the symmetrical form u=(LtN)2 [(LlS)2+(Llc)~]l[(S)2+(c)2]. (11)
We shall be following Carruthers and Nieto [7] by calculating the quantities, (8 being the phase angle) s(e,N)=(LfN)2
(LtS)2,
(12)
and
Q(R N) =S(e, WC6
2,
(13)
which from eq. (10) must satisfy Q( 6) L l/4. Now using eq. (3)) again dropping the free evolution term we obtain B(t)=(Z?+1)-L’2
exp(-iizfl)
B+(t)=&+(O) exp(irfi)
h(O),
(N+l)-1’2.
(14)
In terms of an initial coherent state Ia) we have (al&t)
Ia)=exp(-
la12)
m cr*“a”+‘exp( -irn) [n! (n+l)!]“2
XC n=O
(15)
*
Then it follows from eq. (9) that, with N= I a I 2,
N” c0s[2e-(2n+l)7~ n=O n! [(n+l) (n+2)]“2
.
(20)
Note that in the no interaction case, r = 0, we obtain the standard coherent state results [ 71. Also we note that even with T# 0, from eq. (3), it is obvious that N= I (YI 2 so that (LUV)~=N. Now in previous work on the squeezing obtained from this system, it was assumed that the medium was small enough (a few centimeters) so that a typical value of 7 might be 7 - 1O- 6. Then if the number of photons is N- 1O6a high degree of squeezing may be obtained [ 41. However for a medium such as an optical fiber, T may be much longer. In the present calculations we shall- take r=O. 1. This allows us to illustrate the altered phase fluctuations in a manner which is computationally feasible in view of the fact that eqs. (16), (17), (19) and (20) cannot be expressed as closed forms. In fig. 1 we show the squeezing obtained for the X, quadrature as a function of the average photon number N with r fixed at 0.1. Changing the phase of (Y merely shifts the curve along the N-axis. We choose to fix r and vary N so as to compare the phase uncertainty calculations with those of the ordinary coherent states in ref. [ 71. In fig. 2 we give the uncertainty product U which turns out to be independent of the phase angle 8. It
where we have set (Y= I a I exp( i6). Then using
(al~2(t)la>=exp(-Ia12) 00 &At”
&7+2
exp[ -i(2n+ [n! (n+2)!]“Z
XC
n=O
l)r] >
(18)
we obtain (LYl~2J~)=j-f +fNe_N
f II=0
81-0.0 2.0
e-N N” c0s[28-(2n+1)?1, [n! (n+2)!]“2
4.0
6.0
80
10.0
N (19)
Fig. 1. Squeezing of X, quadrature as a function of N for r = 0.1. 279
Volume 63, number 4
OPTICS COMMUNICATIONS
I
15 August 1987
is evident in this picture that uncertainty product U is amplified over the coherent state values. For computational practicality we have considered N only up to N= 10. In fig. 3 we show the product S( 8, N) for 6 = n/4 which like U, also appears enhanced. In fig. 4 we have Q( 6, N) which also appears to show enhanced fluctuations for high N but that for N less than about 1.7 we notice that Q is reduced compared to the coherent state value. In this paper we have investigated the phase properties of coherent light interacting with an anharmanic oscillator modeling a nonlinear non-absorbing medium. We have shown that while the photon statistics are unaltered, squeezing may be obtained as well as enhanced and reduced phase fluctuations. Apparently these states do not approach the semiclassical number phase uncertainty product for high intensity incident fields. A similar behavior was noted for the two-photon coherent states by Sanders et al.
[21. References
Fig. 3. Product S( 6, N) versus N for 6 = n/4.
Fig. 4. Uncertainty product Q( 0, N) versus N for 0 = n/4.
280
[ 1 ] R. Lynch, J. Opt. Sot. Am. B 3 (1986) 1006. [2] B.C. Sanders, SM. Bamett and P.L. Knight, Optics Comm. 58 (1986) 290. [ 41 H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 41 R. Tanas, in: Coherence and quantum optics V, eds. L. Mandel and E. Wolf (New York, Plenum, 1984) p. 643. [ 5 ] C. Gerry and S. Rodriques, to be published. [6] C.K. Hong and L. Mandel, Phys. Rev. Lett. 54 (1985) 323. [7] P. Carruthers and MM. Nieto, Phys. Rev. Lett. 14 (1965) 387. [8] P.D. Drumm0ndandD.F. Walls, J. Phys. Al3 (1980) 725. [ 91 L. Susskind and J. Glowgower, Physics I ( 1964) 49. [ lo] R. Loudon, The quantum theory of light (2nd Edition, Clarendon Press, Oxford, 1983).
Volume 63, number 4
15 August 1987
ON THE PHASE FLUCTUATIONS OF COHERENT LIGHT INTERACTING WITH AN ANHARMONIC OSCILLATOR Christopher C. GERRY Department ofPhysics, St. Bonaventure University, St. Bonaventure, NY 14778, USA
Received 28 January 1987
We show that ordinary coherent light interacting with a non-absorbing nonlinear medium modelled as an anharmonic oscillator gives rise to enhanced and reduced phase fluctuations as well as squeezing.
Recently there has been some discussion of the phase operators and phase fluctuations associated with squeezed states of the electromagnetic field [ 1,2]. In refs. [ 1 ] and [ 21 the squeezed states discussed are of the two photon type first considered by Yuen [ 31. In this paper we consider ordinary coherent states interacting with a nonlinear non-absorbing medium modeled as an anharmonic oscillator. This system has previously been shown to give rise to squeezed light [4] and also to higher order squeezing [ 51 in the sense of Hong and Mandel [ 61. In this paper we show that while the photon number fluctuations are identical to those of coherent states (i.e. the statistics remain poissonian) the fluctuations of the phase variables are generally enhanced compared to the coherent state values as discussed by Carruthers and Nieto [ 71. In some occasions however, the fluctuations may actually be reduced. The hamiltonian for the sytem is of the form [4] A=hwB+ci+ tMid+?
,
(1)
where the h symbol means an operator in Hilbert space and k is the anharmonicity parameter related to the third order susceptibility of the medium. This anharmonic term is known to lead to optical bistability [ 81. Heisenberg’s equation for CEreads d=-(i/fi)[ci,fi]=-i(w+kri+B)B.
(2)
It is trivial to show that fi=ri+d is a constant of the motion reflecting the conservation of photon numbers. Thus the solution of eq. (2) is simply 278
ci(t>=exp{-it[w+kZQO)]}d(O).
(3)
We introduce the quadrature operators of the field 2, =ci+b+ ) ,t2 = -i(b-d+)
,
(4)
such that [fI,z21=2i,
(5)
leading to the uncertainty relation (dX,)2 (dX,)2>1
.
(6)
We consider the 3, quadrature only which becomes squeezed whenever (AX, )’ < 1. Using eq. (3), drop ping the free evolution term, and assuming the initial state to be a coherent state 1a) with the phase of cr chosen so that (Yis real, we obtain the second order variance [ 41
+2NRe{exp( -ir)
exp[l\r(exp( -2ir) - l)]
-exp[2N(exp(-ir)-l)]},
(7)
where N= ((Y 1fi/ a )“and 7 = kt. We now consider the phase operators. We define the phase operators f?=g(i(p) in the usual way
[7,%101 B+ =f$+
&(fi+l)-“‘S,
.
(8)
Now we define “sine” and “cosine” operators S= (1/2i)(fi-JZ+)
,
e=t(l?+iff)
)
(9)
0 030-4018/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
OPTICS COMMUNICATIONS
Volume 63, number 4
15 August 1987
and in the usual way define the uncertainty relations (dN)2 (dS)%~(C)2, _tlve_N~ (dzQ2 (K)22$(s>2,
(10)
and the symmetrical form u=(LtN)2 [(LlS)2+(Llc)~]l[(S)2+(c)2]. (11)
We shall be following Carruthers and Nieto [7] by calculating the quantities, (8 being the phase angle) s(e,N)=(LfN)2
(LtS)2,
(12)
and
Q(R N) =S(e, WC6
2,
(13)
which from eq. (10) must satisfy Q( 6) L l/4. Now using eq. (3)) again dropping the free evolution term we obtain B(t)=(Z?+1)-L’2
exp(-iizfl)
B+(t)=&+(O) exp(irfi)
h(O),
(N+l)-1’2.
(14)
In terms of an initial coherent state Ia) we have (al&t)
Ia)=exp(-
la12)
m cr*“a”+‘exp( -irn) [n! (n+l)!]“2
XC n=O
(15)
*
Then it follows from eq. (9) that, with N= I a I 2,
N” c0s[2e-(2n+l)7~ n=O n! [(n+l) (n+2)]“2
.
(20)
Note that in the no interaction case, r = 0, we obtain the standard coherent state results [ 71. Also we note that even with T# 0, from eq. (3), it is obvious that N= I (YI 2 so that (LUV)~=N. Now in previous work on the squeezing obtained from this system, it was assumed that the medium was small enough (a few centimeters) so that a typical value of 7 might be 7 - 1O- 6. Then if the number of photons is N- 1O6a high degree of squeezing may be obtained [ 41. However for a medium such as an optical fiber, T may be much longer. In the present calculations we shall- take r=O. 1. This allows us to illustrate the altered phase fluctuations in a manner which is computationally feasible in view of the fact that eqs. (16), (17), (19) and (20) cannot be expressed as closed forms. In fig. 1 we show the squeezing obtained for the X, quadrature as a function of the average photon number N with r fixed at 0.1. Changing the phase of (Y merely shifts the curve along the N-axis. We choose to fix r and vary N so as to compare the phase uncertainty calculations with those of the ordinary coherent states in ref. [ 71. In fig. 2 we give the uncertainty product U which turns out to be independent of the phase angle 8. It
where we have set (Y= I a I exp( i6). Then using
(al~2(t)la>=exp(-Ia12) 00 &At”
&7+2
exp[ -i(2n+ [n! (n+2)!]“Z
XC
n=O
l)r] >
(18)
we obtain (LYl~2J~)=j-f +fNe_N
f II=0
81-0.0 2.0
e-N N” c0s[28-(2n+1)?1, [n! (n+2)!]“2
4.0
6.0
80
10.0
N (19)
Fig. 1. Squeezing of X, quadrature as a function of N for r = 0.1. 279
Volume 63, number 4
OPTICS COMMUNICATIONS
I
15 August 1987
is evident in this picture that uncertainty product U is amplified over the coherent state values. For computational practicality we have considered N only up to N= 10. In fig. 3 we show the product S( 8, N) for 6 = n/4 which like U, also appears enhanced. In fig. 4 we have Q( 6, N) which also appears to show enhanced fluctuations for high N but that for N less than about 1.7 we notice that Q is reduced compared to the coherent state value. In this paper we have investigated the phase properties of coherent light interacting with an anharmanic oscillator modeling a nonlinear non-absorbing medium. We have shown that while the photon statistics are unaltered, squeezing may be obtained as well as enhanced and reduced phase fluctuations. Apparently these states do not approach the semiclassical number phase uncertainty product for high intensity incident fields. A similar behavior was noted for the two-photon coherent states by Sanders et al.
[21. References
Fig. 3. Product S( 6, N) versus N for 6 = n/4.
Fig. 4. Uncertainty product Q( 0, N) versus N for 0 = n/4.
280
[ 1 ] R. Lynch, J. Opt. Sot. Am. B 3 (1986) 1006. [2] B.C. Sanders, SM. Bamett and P.L. Knight, Optics Comm. 58 (1986) 290. [ 41 H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [ 41 R. Tanas, in: Coherence and quantum optics V, eds. L. Mandel and E. Wolf (New York, Plenum, 1984) p. 643. [ 5 ] C. Gerry and S. Rodriques, to be published. [6] C.K. Hong and L. Mandel, Phys. Rev. Lett. 54 (1985) 323. [7] P. Carruthers and MM. Nieto, Phys. Rev. Lett. 14 (1965) 387. [8] P.D. Drumm0ndandD.F. Walls, J. Phys. Al3 (1980) 725. [ 91 L. Susskind and J. Glowgower, Physics I ( 1964) 49. [ lo] R. Loudon, The quantum theory of light (2nd Edition, Clarendon Press, Oxford, 1983).