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Field fluctuations of two-photon coherent states
Volume 68A, number 3,4
PHYSICS LETTERS
16 October 1978
FIELD FLUCTUATIONS OF TWO-PHOTON COHERENT STATES M. SCHUBERT and W. VOGEL Friedrich-Schiller-Universiteit, Sektion Physik, GDR-69 Jena, GDR Received 28 June 1978
Two-photon coherent states described by Yuen are studied with respect to their spatial behavior. The fluctuations of the field strength and the minimum uncertainty state character are analyzed.
Yuen [1,2] has described the concept of the socalled two-photon coherent states (TC-states) which are the radiation states of ideal two-photon lasers far above the threshold. A TC-state ly> is a right-hand eigenstate of the operator b = pa + va~where a and +
.
.
.
(
c
=
_________
2
211w,
~~ =
iJ.L
2i
+
I) e
__________
e1~
(3)
.
+
LUW
.
2ikr
2ikr
a are the annthilation and creation operators of a certain radiation mode with frequency w: bly)yIy) jIy)~ly;p,~), (1)
p i.’ e The normalization condition fdE(r)Iy)J2 2 (c 2+c~) ~ (c
p and v 2are numbers which fulfil the condi= complex 1 + 1v12. The quantum fluctuation behavtion 1p1 ior of the TC-states essentially differs from that of ordinary coherent states a), which are the right-hand( eigenstates of the operator a: = I ‘2~ aa aa “
Ic1 I exp 4(c2 + c~) Vis the normalization volume, k = w/c is the wave number. From eq~(3)one can calculate the probability density w = I(E(r)Iy)I2 for the occurrence of the measurable value E at the point r. It can be easily seen from eq. (3) that the function w is a &aussian dis-
The difference between coherent and TC-states was illustrated by considering the mean-square fluctuations of the hermitian operators a 1 = (a + a+)/2 and a2 = (a a~)/2i. he operators a1 and a2ofare to a constant factorTequal to the operators theupvector potential A(r) and of the electric field strength E(r) in the Schroedinger picture at a fixed point r 0. Therefore it seems to be useful for a better insight to extend the consideration to the whole spatial reglon. On the basis of the procedure applied in ref. [3] for deriving the eigenstates IE(r)) of the operator of the electric field strength one m~ycalculate the matrix element (E(r)Iy):
tribution with regard to the field strength E. Thus, the distribution can be uniquely characterized by the
(E(r)Jy) = c
2(r) 1 exp[c2E
+
c 3E(r)],
=
I yields
L
—
expectation value (yIE(r)Iy) 2iy). and the mean-square deviation (yI(~E(r))value takes the form of a sine The expectation wave concerning the dependence on r: (yIE(r)Iy)=Csin(kr+p =
—2gIpI lyl
)
c sin (kr + cOy —
—2glvijyIsm(kr
~ +~). Y
~
4)
The amplitude C and the phase ~ depend in a rather complicated way on the complex numbersy, p, ~ and the real numberg. Detailed information on this dependence can be obtained from the second line of eq. 1!2 is equal to the (4). The quantityg = (11w/2e0 V) 321
Volume 68A, number 3,4
PHYSICS LETTERS
16 October 1978
values only for a coherent state, whereby this minof this result, TC-states theis spatially averaged fluctuations are for imum value g2. In generalization
IvI-
greater than g2, namely
v/-I v/-U
(yI(~E(r))2Iy)=g2(1 + 21p12). (7) The operators of the vector potentialA(r) and of the electric field strength E(r) may be regarded as fundamental quantities of the quantized radiation field. They fulfil the commutation relation [E(r),A(r)] = i(h/e 0 V)I, which can be considered as a generalization of the commutation relation for the quadrature components a1 and a2. The commutation relation for E(r) and A(r) yields the uncertainty relation (8) 2)((~E)2)~g4/w2.
0 Fig. 1. Mean-square field fluctuations in dependence on 2 the spatial coordinate.
A) For TC-states
root mean-square deviation I(0IE2(r)I0))~2of the vacuum state. In the special case p = 1, v = 0 the TCstate y) becomes an ordinary coherent state a) and the first summand alone represents the expectation
~I(~A(r))2Iy)(yI(~E(r))2Iy)
value.
The first summand is equal to the uncertainty product of the coherent states. The uncertainty product
The mean-square fluctuation is given by (yIC~E(r))2Iy)=g2 +g2 [21p12 +
21v1
l~Icos(2kr + P~ cO,)].
(5)
This expression is independent of y. In the special case of coherent states it attains the value g2 = (OlE2(r)I0) of the vacuum state fluctuations and does not depend on r. However, in the general case of TCstates the right-hand side of eq. (5) depends on r. In 2 + Iz.’I ui cos (2kr + spatial where 1p1 <0intervals, holds, the mean-square fluctions is less than (OiE2(r)iO) (see fig. 1). If r fulfils the condition 2 kr + p~ = it + 2nir the mean-square deviation takes its minimum value, namely mm {~yI(~E(r))2iy)} = g2( !~l ivD2. (6) With increasing i~l the minimum value tends to zero, whereas the r-intervals with fluctuations less thang2 become smaller. Fol1owi~gj4J~he spatially averaged field fluctuations IE(r))2) take their minimum
322
X [(1v12+
the uncertainty product is
1v14) (1
=
w cos2(2kr +
+
4
p~
p~))].
(9)
of TC-states depends on r. In general it takes a greater value than that of coherent states. Therefore in the sense of relation (9) TC-states cannot be regarded as minimum uncertainty states. Only at special points where the condition 2kr + = nit holds, TC. states take the minimum value of the uncertainty product. However, this condition is more general than the condition for the minimum uncertainty states with respect to a 1 and a2 it has been shown [2]; the TC-states minimize thisas uncertainty productinonly when p = 6v for a real number ~ (that means if —
References [11 H.P.
Yuen, Phys. Lett. 51A (1975) 1. [2] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [31M. Schubert and W. Vogel, Wiss. Z. Univ. Jena, Math.Nat. Reihe 27 (1978) 179.
[41H.
Paul, Fortschr. Phys. 14 (1966) 141.
PHYSICS LETTERS
16 October 1978
FIELD FLUCTUATIONS OF TWO-PHOTON COHERENT STATES M. SCHUBERT and W. VOGEL Friedrich-Schiller-Universiteit, Sektion Physik, GDR-69 Jena, GDR Received 28 June 1978
Two-photon coherent states described by Yuen are studied with respect to their spatial behavior. The fluctuations of the field strength and the minimum uncertainty state character are analyzed.
Yuen [1,2] has described the concept of the socalled two-photon coherent states (TC-states) which are the radiation states of ideal two-photon lasers far above the threshold. A TC-state ly> is a right-hand eigenstate of the operator b = pa + va~where a and +
.
.
.
(
c
=
_________
2
211w,
~~ =
iJ.L
2i
+
I) e
__________
e1~
(3)
.
+
LUW
.
2ikr
2ikr
a are the annthilation and creation operators of a certain radiation mode with frequency w: bly)yIy) jIy)~ly;p,~), (1)
p i.’ e The normalization condition fdE(r)Iy)J2 2 (c 2+c~) ~ (c
p and v 2are numbers which fulfil the condi= complex 1 + 1v12. The quantum fluctuation behavtion 1p1 ior of the TC-states essentially differs from that of ordinary coherent states a), which are the right-hand( eigenstates of the operator a: = I ‘2~ aa aa “
Ic1 I exp 4(c2 + c~) Vis the normalization volume, k = w/c is the wave number. From eq~(3)one can calculate the probability density w = I(E(r)Iy)I2 for the occurrence of the measurable value E at the point r. It can be easily seen from eq. (3) that the function w is a &aussian dis-
The difference between coherent and TC-states was illustrated by considering the mean-square fluctuations of the hermitian operators a 1 = (a + a+)/2 and a2 = (a a~)/2i. he operators a1 and a2ofare to a constant factorTequal to the operators theupvector potential A(r) and of the electric field strength E(r) in the Schroedinger picture at a fixed point r 0. Therefore it seems to be useful for a better insight to extend the consideration to the whole spatial reglon. On the basis of the procedure applied in ref. [3] for deriving the eigenstates IE(r)) of the operator of the electric field strength one m~ycalculate the matrix element (E(r)Iy):
tribution with regard to the field strength E. Thus, the distribution can be uniquely characterized by the
(E(r)Jy) = c
2(r) 1 exp[c2E
+
c 3E(r)],
=
I yields
L
—
expectation value (yIE(r)Iy) 2iy). and the mean-square deviation (yI(~E(r))value takes the form of a sine The expectation wave concerning the dependence on r: (yIE(r)Iy)=Csin(kr+p =
—2gIpI lyl
)
c sin (kr + cOy —
—2glvijyIsm(kr
~ +~). Y
~
4)
The amplitude C and the phase ~ depend in a rather complicated way on the complex numbersy, p, ~ and the real numberg. Detailed information on this dependence can be obtained from the second line of eq. 1!2 is equal to the (4). The quantityg = (11w/2e0 V) 321
Volume 68A, number 3,4
PHYSICS LETTERS
16 October 1978
values only for a coherent state, whereby this minof this result, TC-states theis spatially averaged fluctuations are for imum value g2. In generalization
IvI-
greater than g2, namely
v/-I v/-U
(yI(~E(r))2Iy)=g2(1 + 21p12). (7) The operators of the vector potentialA(r) and of the electric field strength E(r) may be regarded as fundamental quantities of the quantized radiation field. They fulfil the commutation relation [E(r),A(r)] = i(h/e 0 V)I, which can be considered as a generalization of the commutation relation for the quadrature components a1 and a2. The commutation relation for E(r) and A(r) yields the uncertainty relation (8) 2)((~E)2)~g4/w2.
0 Fig. 1. Mean-square field fluctuations in dependence on 2 the spatial coordinate.
A) For TC-states
root mean-square deviation I(0IE2(r)I0))~2of the vacuum state. In the special case p = 1, v = 0 the TCstate y) becomes an ordinary coherent state a) and the first summand alone represents the expectation
~I(~A(r))2Iy)(yI(~E(r))2Iy)
value.
The first summand is equal to the uncertainty product of the coherent states. The uncertainty product
The mean-square fluctuation is given by (yIC~E(r))2Iy)=g2 +g2 [21p12 +
21v1
l~Icos(2kr + P~ cO,)].
(5)
This expression is independent of y. In the special case of coherent states it attains the value g2 = (OlE2(r)I0) of the vacuum state fluctuations and does not depend on r. However, in the general case of TCstates the right-hand side of eq. (5) depends on r. In 2 + Iz.’I ui cos (2kr + spatial where 1p1 <0intervals, holds, the mean-square fluctions is less than (OiE2(r)iO) (see fig. 1). If r fulfils the condition 2 kr + p~ = it + 2nir the mean-square deviation takes its minimum value, namely mm {~yI(~E(r))2iy)} = g2( !~l ivD2. (6) With increasing i~l the minimum value tends to zero, whereas the r-intervals with fluctuations less thang2 become smaller. Fol1owi~gj4J~he spatially averaged field fluctuations IE(r))2) take their minimum
322
X [(1v12+
the uncertainty product is
1v14) (1
=
w cos2(2kr +
+
4
p~
p~))].
(9)
of TC-states depends on r. In general it takes a greater value than that of coherent states. Therefore in the sense of relation (9) TC-states cannot be regarded as minimum uncertainty states. Only at special points where the condition 2kr + = nit holds, TC. states take the minimum value of the uncertainty product. However, this condition is more general than the condition for the minimum uncertainty states with respect to a 1 and a2 it has been shown [2]; the TC-states minimize thisas uncertainty productinonly when p = 6v for a real number ~ (that means if —
References [11 H.P.
Yuen, Phys. Lett. 51A (1975) 1. [2] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [31M. Schubert and W. Vogel, Wiss. Z. Univ. Jena, Math.Nat. Reihe 27 (1978) 179.
[41H.
Paul, Fortschr. Phys. 14 (1966) 141.