- Email: [email protected]
The photon distribution of the spontaneous two-photon emission
Volume 57A, number 4
PHYSICS LETTERS
28 June 1976
THE PHOTON DISTRIBUTION OF THE SPONTANEOUS TWO-PHOTON EMISSION 0. HIROTA and S. IKEHARA Graduate School of Tokyo Electrical Engineering College, Tokyo, Japan
Received 17 March 1976 Revised manuscript received 17 May 1976 Some properties of a generalized coherent state (G.C.S.) generated by the ideal stimulated two-photon laser are discussed and the photon distribution of Gaussian mixed state of G.C.S. is derived.
A coherent state [6] has received much attention as the basic state for representation of the optical sources. But recently a generalized coherent state (G.C.S.) was derived by Stoler [1], and the general properties of the G.C.S. were discussed by Yuen [2,3]. The G.C.S. is generated by the ideal stimulated two-photon laser and it cannot be represented as a positive superposition of coherent states [2]. The second order coherence function g(2) of G.C.S. has many interesting properties. The g(2) is defined by the second factorial moment of the photon distribution: g(2)= Trp(a~2a2)(n(n—1)) ~Trp(a~a)}2 (n)2 A G.C.S. is defined as the eigenstate of generalized photon annihilation operator b:
(2)
bjia+va~,
(2)
bI13~p,v)=I3jl3:p,v),
where a, a+ are an ordinary photon annihilation and creation operator, respectively, and 11 and v are complex numbers which satisfy 1/112 1v12 = 1,so that [b, b~]= 1. Ifp and vare real, G.C.S. becomes a minimum uncertainty state with respect to the canonical quantities. The photon distribution of the G.C.S. is from the number representation of G.C.S. [1—3]as follows: —
~
~
2
n
Hn(~j~)
I~I21÷I~I2 exp(_ 21/112
)‘
(3)
where j3 = /1*13 j43*, and Hn(z) is the Hermite polynomial. We call eq. (3) Hermite distribution. The generating function of the Hermite distribution and its g(2) have been given by Yuen [3] as below: —
Q
2
+ [1 —(l—X)2r]
(4)
1(X)~~j~(l —xrP(n) =~~exp [(1—X)r—1)]1i31 wherer
[l~I2—(1—X)2lvI2]’
(l1~I2+Ivl2)2 The variance of the distribution is expressed in general cases as follows: (~n2)(n)(1+(g(2)1)(n))
(6)
The g(2) of the Hermite distribution enables one to obtain enhanced photon bunchingg(2)> 2 [4] and anti-photon bunching g(2) < 1 [51, but we can show easily that g(2) cannot attain the theoretical minimum value 1 11(n), corresponding to the photon number state. —
317
Volume 57A, number 4
PHYSICS LETTERS 1
10
28 June 1976
P(n)
—
(1)
P~(n),
(2)
B-E,
H.~
~n7=
—
(1) -2.
10
—
(2)
—5
fl
10 I
0
1
I
3
2
Fig. 1. Photon distribution of Gaussian mixed state of G.C.S.
The below-threshold radiation of the two-photon laser may be considered as the spontaneous two-photon emission which is represented by mixed states of G.C.S. Since G.C.S. cannot be represented by coherent state, and the mixed state of G.C.S. cannot be represented either, we give here the density operator based upon a G.C.S. pg=ffP(~*)/1,p,vld2~’
(7)
213 = dReI3dImP, and P(13, 3*) is a positive true probability density function. Using ‘og’ we get the photon where d distribution of the mixed state of G.C.S. Pg(fl)
=ffP(~,~*)1
(~-~f( fl~2 (H~
~
(8)
The generating function of the Pg(n) is as follows:
Q
2~.
(9)
2(X) = ffQ1(X)P(O, ~*) d We shall derive the photon distribution of the Gaussian mixed state based upon the G.C.S. as the minimum uncertainty state. We take P(j3, 13*) as Gaussian, so that we get the photon distribution: rg(fl)=~~,~
Vi~(~)L’Ktt’~1,
(10)
where E[(tt’)12] is the nth joint moment of bivariate (t, t’) Gaussian distribution with the variances o~,u~and correlation coefficient -y, and where
/
2_i A—B\ 1U2T~l+~j,
318
A+B 72AB+A—B
Volume 57A, number 4
A=/1v (1_~+(/1;)2),
PHYSICS LETTERS
B=~iv(i+~+(P;)2),
28 June 1976
N=ffI~I2P(13,~*)d2~.
If we take ~t = 1, r’ = 0 in the eq. (10), the Pg(fl) becomes the Bose—Einstein distribution (B E). Then the Q2(X) of the photon distribution eq. (10) becomes —
(11) 2r]. The first order moment is
where D = [(1—X)r—1],E = [1—(1 —X) (n)=Trpga+a=_~~N+IPl2,
(12)
and the second order coherence g(2) is g(2)
a2Q
2I~~~
2)x
2 +~—. 0(fl)2 (13) From results eqs. (6) and (13), the variance of the photon distribution as Gaussian mixed state of G.C.S. is now proved always larger than that of the B—E. These results will enable one to represent the statistical property of two-photon laser. 2
(fl) above the
References [1] [2] [3] [4]
D. Stoler, Phys. Rev. Dl (1970) 3217. H.P. Yuen, Phys. Lett. 51A (1975) 1. H.P. Yuen, Generalized coherent state of the radiation field (not yet published). K.J. McNeil and D.F. Walls, Phys. Lett. 51A (1975) 233. [51 D. Stoler, Phys. Rev. Lett. 33(1974)1397. [6] R.J. Glauber, Phys. Rev. 131 (1963) 2766.
319
PHYSICS LETTERS
28 June 1976
THE PHOTON DISTRIBUTION OF THE SPONTANEOUS TWO-PHOTON EMISSION 0. HIROTA and S. IKEHARA Graduate School of Tokyo Electrical Engineering College, Tokyo, Japan
Received 17 March 1976 Revised manuscript received 17 May 1976 Some properties of a generalized coherent state (G.C.S.) generated by the ideal stimulated two-photon laser are discussed and the photon distribution of Gaussian mixed state of G.C.S. is derived.
A coherent state [6] has received much attention as the basic state for representation of the optical sources. But recently a generalized coherent state (G.C.S.) was derived by Stoler [1], and the general properties of the G.C.S. were discussed by Yuen [2,3]. The G.C.S. is generated by the ideal stimulated two-photon laser and it cannot be represented as a positive superposition of coherent states [2]. The second order coherence function g(2) of G.C.S. has many interesting properties. The g(2) is defined by the second factorial moment of the photon distribution: g(2)= Trp(a~2a2)(n(n—1)) ~Trp(a~a)}2 (n)2 A G.C.S. is defined as the eigenstate of generalized photon annihilation operator b:
(2)
bjia+va~,
(2)
bI13~p,v)=I3jl3:p,v),
where a, a+ are an ordinary photon annihilation and creation operator, respectively, and 11 and v are complex numbers which satisfy 1/112 1v12 = 1,so that [b, b~]= 1. Ifp and vare real, G.C.S. becomes a minimum uncertainty state with respect to the canonical quantities. The photon distribution of the G.C.S. is from the number representation of G.C.S. [1—3]as follows: —
~
~
2
n
Hn(~j~)
I~I21÷I~I2 exp(_ 21/112
)‘
(3)
where j3 = /1*13 j43*, and Hn(z) is the Hermite polynomial. We call eq. (3) Hermite distribution. The generating function of the Hermite distribution and its g(2) have been given by Yuen [3] as below: —
Q
2
+ [1 —(l—X)2r]
(4)
1(X)~~j~(l —xrP(n) =~~exp [(1—X)r—1)]1i31 wherer
[l~I2—(1—X)2lvI2]’
(l1~I2+Ivl2)2 The variance of the distribution is expressed in general cases as follows: (~n2)(n)(1+(g(2)1)(n))
(6)
The g(2) of the Hermite distribution enables one to obtain enhanced photon bunchingg(2)> 2 [4] and anti-photon bunching g(2) < 1 [51, but we can show easily that g(2) cannot attain the theoretical minimum value 1 11(n), corresponding to the photon number state. —
317
Volume 57A, number 4
PHYSICS LETTERS 1
10
28 June 1976
P(n)
—
(1)
P~(n),
(2)
B-E,
H.~
~n7=
—
(1) -2.
10
—
(2)
—5
fl
10 I
0
1
I
3
2
Fig. 1. Photon distribution of Gaussian mixed state of G.C.S.
The below-threshold radiation of the two-photon laser may be considered as the spontaneous two-photon emission which is represented by mixed states of G.C.S. Since G.C.S. cannot be represented by coherent state, and the mixed state of G.C.S. cannot be represented either, we give here the density operator based upon a G.C.S. pg=ffP(~*)/1,p,vld2~’
(7)
213 = dReI3dImP, and P(13, 3*) is a positive true probability density function. Using ‘og’ we get the photon where d distribution of the mixed state of G.C.S. Pg(fl)
=ffP(~,~*)1
(~-~f( fl~2 (H~
~
(8)
The generating function of the Pg(n) is as follows:
Q
2~.
(9)
2(X) = ffQ1(X)P(O, ~*) d We shall derive the photon distribution of the Gaussian mixed state based upon the G.C.S. as the minimum uncertainty state. We take P(j3, 13*) as Gaussian, so that we get the photon distribution: rg(fl)=~~,~
Vi~(~)L’Ktt’~1,
(10)
where E[(tt’)12] is the nth joint moment of bivariate (t, t’) Gaussian distribution with the variances o~,u~and correlation coefficient -y, and where
/
2_i A—B\ 1U2T~l+~j,
318
A+B 72AB+A—B
Volume 57A, number 4
A=/1v (1_~+(/1;)2),
PHYSICS LETTERS
B=~iv(i+~+(P;)2),
28 June 1976
N=ffI~I2P(13,~*)d2~.
If we take ~t = 1, r’ = 0 in the eq. (10), the Pg(fl) becomes the Bose—Einstein distribution (B E). Then the Q2(X) of the photon distribution eq. (10) becomes —
(11) 2r]. The first order moment is
where D = [(1—X)r—1],E = [1—(1 —X) (n)=Trpga+a=_~~N+IPl2,
(12)
and the second order coherence g(2) is g(2)
a2Q
2I~~~
2)x
2 +~—. 0(fl)2 (13) From results eqs. (6) and (13), the variance of the photon distribution as Gaussian mixed state of G.C.S. is now proved always larger than that of the B—E. These results will enable one to represent the statistical property of two-photon laser. 2
(fl) above the
References [1] [2] [3] [4]
D. Stoler, Phys. Rev. Dl (1970) 3217. H.P. Yuen, Phys. Lett. 51A (1975) 1. H.P. Yuen, Generalized coherent state of the radiation field (not yet published). K.J. McNeil and D.F. Walls, Phys. Lett. 51A (1975) 233. [51 D. Stoler, Phys. Rev. Lett. 33(1974)1397. [6] R.J. Glauber, Phys. Rev. 131 (1963) 2766.
319