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Maximum photon antibunching in generalized coherent states
Volume
28, number
3
OPTICS COMMUNICATIONS
March 1979
MAXIMUMPHOTONANTIBUNCHINGINGENERALIZEDCOHERENTSTATES Carl W. HELSTROM * Department of Applied Physics and Information Science, Universityof California,San Diego, L.uJolla, CA 92093, USA Received 2 October Revised manuscript
1978 received
29 November
1978
If when light falls on a photoelectron emitter the random number of photoelectrons counted in a time interval is m, the quantity [Var m - LJm)l /[L?(?n)]* is called the bunching ratio. It is zero for light coherent to all orders and positive for thermal light, but may be negative for other kinds of light. Here it is shown that when the incident light is in a generalized coherent state, the most negative value of this ratio is of the order of -NT’ where NT is the total number of incident photons.
Plane light waves are falling on the surface of a photoelectric detector, and the number m of electrons emitted in a time interval r is counted. If the time r is smaller than the coherence time rc of the light, the photoelectrons are ejected through the interaction of a single mode of the incident electromagnetic field with the atoms of the detector. Let p be the quantummechanical density operator of that mode, and let II and a+ be the associated photon annihilation and creation operators. The mean number of photoelectrons counted during the time r is g(m)
= cyTr pa+a ,
(1)
where cr = vr/hf, 77being the quantum efficiency of the detector and hfthe energy of a single photon. The variance of the number of counts is Var m = g(m) + a2 [Tr pa+2a2 -(Tr pa+a)2 ] ,
(2)
as shown by Mandel and Wolf [ 11. If the mode of the light field is in a pure coherent state, as when it has been generated by a laser operating far above threshold, the second term on the right in (2) vanishes and Var m = E(m). The number m of counts is then a Poisson random variable. When, on the other hand, the light comes from a thermal source, that second term * This research was supported dation
under grant
by the National ENC 77-04500.
in (2) is positive, and Var m > l?(m). This phenomenon, attributed to the photons in the light field, is called bunching [ 11. We define the bunching excess as A = [Var m - g(m)] /a2 = Tr pa+2a2 - (n j2
(3)
with (n) = Tr pa+a, the mean number of photons in the mode of the field. As discussed in [ I] , bunching may also manifest itself in an increased probability that a second photoelectron will be counted within a time rc after a single photoelectron is counted, as well as in a positive correlation of the numbers of photoelectrons counted simultaneously at nearby photodetectors within intervals short compared with the coherence time rc. For some states of the light field, as when p = 1MXMl and the mode contains a definite number M of photons, the bunching ratio R may be negative, a phenomenon known as anti-bunching. For such states the variance of the number m of photoelectron counts will be less than its expected value. The generalized coherent states, introduced by Stoler [2] and Lu [3], have been reported as exhibiting photon anti-bunching [4,5]. A state IP;~,u) of this kind is the right-eigenstate of the operator pa + va+ with eigenvalue 0, (pa + va’) IB;p,v) = P lP;/.~,v) ,
Science
(4)
Foun-
where 1~1~ - lv12 = 1. Yuen [6] has described how 363
Volume
28, number
3
OPTICS COMMUNICATIONS
March
1979
light of this nature might be generated by a monochromatic two-photon laser operating far above threshold. Here we wish to show that the minimum (most negative) attainable value of the bunching ratio A/(Pz)~ for these states rises toward zero roughly in proportion to -(n)~ 1 of the mean number (n) of photons in the mode is increased. From [6] the “bunching excess” is A = 21f112(4\v12•t 3)jvj2 ~ 2 Re(pvfl*2)(1
t 41~1~) t [vi2 t 21~1~ ,
(5)
and we seek its most negative value for generalized coherent states bearing a fixed average number of photons
1
0 001
(?I) = ,pr2 -I-(V/2 .
(6)
Here B = Tr(pa) = y/3 ~ @* represents the coherent “core” of the state. The bunching excess A in (5) will be most negative when /~/3*~ is real, and we lose no generality by taking ~,v,@, and fi as all real. We can then simplify (5) to A = 2f12 v[4v2(v ~ p) + 3u - ~1 t v2 t 2~4 , and by using the relations (P-v)-‘=/_l+v,
fi=(p-v)&
and (6) we can reduce this expression to A = v2(l + 2/~v) ~ 2v(/~ - v)(n).
(7)
Setting the derivative of A with respect to v equal to zero and using dp/dv = v/p lead after some algebra to the expression (77)= (p t up
v(4v3
+ 3v +/_I).
(8)
By substituting a succession of values of u into (7) and (8) we produced the curve shown in fig. 1 for the maximum anti-bunching ratio R max = -Aminl(n)2
= -[Var
m -bJm)]
/E(m)]
2
as a function of the average number (n) of photons. When (n)9 1 and v% 1, we find(n)= 16v6, whereupon R ma\ = h-'
.
(9)
In this limit, furthermore, the generalized coherent state ]fl;p,v) with parameters maximizing anti-bunching differs only slightly from a pure coherent state in the sense that the ratio 364
IO
_L&uALLLLLLL-__ IO
Fig. 1. Maximum anti-bunching number (II) of photons.
lfil2/(rz) = 1 ~ 2-- 4 ,-4
100
ratio R,,,
l&l 1000
versus the average
x ] _ 2-4/3(,)-213
is nearly equal to 1. This analysis has assumed that photoelectrons are counted over an interval r less than the coherence time rc. If on the other hand r > rC, an effective number (r/re) of independent field tnodes are interacting with the detector. Because g(:(171) and Var m will be proportional to (r/r& the maximum anti-bunching ratio must be multiplied by (r/~,)-~. Fig. 1 shows that for 7 < rc, R,,, rC, therefore, I?,,,,, 2 (n)kl (~/7,)-l
= N; 1 ,
where N, is the total number of photons incident on the detector during the counting interval T. Whether T < rc or T % rC, therefore, the maximum anti-bunching ratio attainable for light of this kind will be less than or at best of the order of the reciprocal of the total number of incident photons.
References [l] [2] [3] [4] [S] [6]
L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) D. Stoler, Phys. Rev. Dl (1970)3217. E.Y.C. Lu,Nuovo Cim. Lett. 2 (1971) 1241. E.Y.C. Lu, Nuovo Cim. Lett. 3 (1972) 585. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. H. Yuen, Phys. Rev. Al3 (1976) 2226.
231
28, number
3
OPTICS COMMUNICATIONS
March 1979
MAXIMUMPHOTONANTIBUNCHINGINGENERALIZEDCOHERENTSTATES Carl W. HELSTROM * Department of Applied Physics and Information Science, Universityof California,San Diego, L.uJolla, CA 92093, USA Received 2 October Revised manuscript
1978 received
29 November
1978
If when light falls on a photoelectron emitter the random number of photoelectrons counted in a time interval is m, the quantity [Var m - LJm)l /[L?(?n)]* is called the bunching ratio. It is zero for light coherent to all orders and positive for thermal light, but may be negative for other kinds of light. Here it is shown that when the incident light is in a generalized coherent state, the most negative value of this ratio is of the order of -NT’ where NT is the total number of incident photons.
Plane light waves are falling on the surface of a photoelectric detector, and the number m of electrons emitted in a time interval r is counted. If the time r is smaller than the coherence time rc of the light, the photoelectrons are ejected through the interaction of a single mode of the incident electromagnetic field with the atoms of the detector. Let p be the quantummechanical density operator of that mode, and let II and a+ be the associated photon annihilation and creation operators. The mean number of photoelectrons counted during the time r is g(m)
= cyTr pa+a ,
(1)
where cr = vr/hf, 77being the quantum efficiency of the detector and hfthe energy of a single photon. The variance of the number of counts is Var m = g(m) + a2 [Tr pa+2a2 -(Tr pa+a)2 ] ,
(2)
as shown by Mandel and Wolf [ 11. If the mode of the light field is in a pure coherent state, as when it has been generated by a laser operating far above threshold, the second term on the right in (2) vanishes and Var m = E(m). The number m of counts is then a Poisson random variable. When, on the other hand, the light comes from a thermal source, that second term * This research was supported dation
under grant
by the National ENC 77-04500.
in (2) is positive, and Var m > l?(m). This phenomenon, attributed to the photons in the light field, is called bunching [ 11. We define the bunching excess as A = [Var m - g(m)] /a2 = Tr pa+2a2 - (n j2
(3)
with (n) = Tr pa+a, the mean number of photons in the mode of the field. As discussed in [ I] , bunching may also manifest itself in an increased probability that a second photoelectron will be counted within a time rc after a single photoelectron is counted, as well as in a positive correlation of the numbers of photoelectrons counted simultaneously at nearby photodetectors within intervals short compared with the coherence time rc. For some states of the light field, as when p = 1MXMl and the mode contains a definite number M of photons, the bunching ratio R may be negative, a phenomenon known as anti-bunching. For such states the variance of the number m of photoelectron counts will be less than its expected value. The generalized coherent states, introduced by Stoler [2] and Lu [3], have been reported as exhibiting photon anti-bunching [4,5]. A state IP;~,u) of this kind is the right-eigenstate of the operator pa + va+ with eigenvalue 0, (pa + va’) IB;p,v) = P lP;/.~,v) ,
Science
(4)
Foun-
where 1~1~ - lv12 = 1. Yuen [6] has described how 363
Volume
28, number
3
OPTICS COMMUNICATIONS
March
1979
light of this nature might be generated by a monochromatic two-photon laser operating far above threshold. Here we wish to show that the minimum (most negative) attainable value of the bunching ratio A/(Pz)~ for these states rises toward zero roughly in proportion to -(n)~ 1 of the mean number (n) of photons in the mode is increased. From [6] the “bunching excess” is A = 21f112(4\v12•t 3)jvj2 ~ 2 Re(pvfl*2)(1
t 41~1~) t [vi2 t 21~1~ ,
(5)
and we seek its most negative value for generalized coherent states bearing a fixed average number of photons
1
0 001
(?I) = ,pr2 -I-(V/2 .
(6)
Here B = Tr(pa) = y/3 ~ @* represents the coherent “core” of the state. The bunching excess A in (5) will be most negative when /~/3*~ is real, and we lose no generality by taking ~,v,@, and fi as all real. We can then simplify (5) to A = 2f12 v[4v2(v ~ p) + 3u - ~1 t v2 t 2~4 , and by using the relations (P-v)-‘=/_l+v,
fi=(p-v)&
and (6) we can reduce this expression to A = v2(l + 2/~v) ~ 2v(/~ - v)(n).
(7)
Setting the derivative of A with respect to v equal to zero and using dp/dv = v/p lead after some algebra to the expression (77)= (p t up
v(4v3
+ 3v +/_I).
(8)
By substituting a succession of values of u into (7) and (8) we produced the curve shown in fig. 1 for the maximum anti-bunching ratio R max = -Aminl(n)2
= -[Var
m -bJm)]
/E(m)]
2
as a function of the average number (n) of photons. When (n)9 1 and v% 1, we find(n)= 16v6, whereupon R ma\ = h-'
.
(9)
In this limit, furthermore, the generalized coherent state ]fl;p,v) with parameters maximizing anti-bunching differs only slightly from a pure coherent state in the sense that the ratio 364
IO
_L&uALLLLLLL-__ IO
Fig. 1. Maximum anti-bunching number (II) of photons.
lfil2/(rz) = 1 ~ 2-- 4 ,-4
100
ratio R,,,
l&l 1000
versus the average
x ] _ 2-4/3(,)-213
is nearly equal to 1. This analysis has assumed that photoelectrons are counted over an interval r less than the coherence time rc. If on the other hand r > rC, an effective number (r/re) of independent field tnodes are interacting with the detector. Because g(:(171) and Var m will be proportional to (r/r& the maximum anti-bunching ratio must be multiplied by (r/~,)-~. Fig. 1 shows that for 7 < rc, R,,,
= N; 1 ,
where N, is the total number of photons incident on the detector during the counting interval T. Whether T < rc or T % rC, therefore, the maximum anti-bunching ratio attainable for light of this kind will be less than or at best of the order of the reciprocal of the total number of incident photons.
References [l] [2] [3] [4] [S] [6]
L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) D. Stoler, Phys. Rev. Dl (1970)3217. E.Y.C. Lu,Nuovo Cim. Lett. 2 (1971) 1241. E.Y.C. Lu, Nuovo Cim. Lett. 3 (1972) 585. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. H. Yuen, Phys. Rev. Al3 (1976) 2226.
231