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Negative hypergeometric states of the quantized radiation field
21 December 1998
PHYSICS LETTERS A
Physics Letters A 250 (1998) 88-92
Negative hypergeometric states of the quantized radiation field Hong-yi Fan a*b,Nai-le Liu b ’ CCAST (WorM Laboratory), PO Box 8730, Beijing 10W80, China h Department
of Material Science and Engineering,
China Universiry of Science and Technology, Hefei, Anhui 230026, China ’
Received 17 September 1998; accepted for publication 13 October 1998 Communicated by C.R. Doering
Abstract We introduce the negative hypergeometric states of the quantized radiation field. The importance of these states lies in the fact that they interpolate between the binomial states and negative binomial states, and tend to them in two different limits. In some other limiting cases, they degenerate to the number, coherent and Susskind-Glogower phase states, respectively. It is shown that some nonclassical properties of these states, such as antibunching effect and sub-Poissonian statistics, also display intermediary behavior. @ 1998 Published by Elsevier Science B.V.
1. Introduction
In recent years, the binomial states (BS) [ 1 ] and negative binomial states (NBS) [ 21 have attracted considerable attention of physicists in the field of quantum optics. They separately correspond to two opposite statistical distributions in classical probability theory. The BS interpolate between the coherent states (the most classical) and the number (Fock) states (the most nonclassical) [ 31, while the NBS interpolate between the coherent states and Susskind-Glogower (SG) phase states [ 41. A question thus naturally arises: are there any quantum-mechanical states which interpolate between the BS and NBS? The answer is affirmative. In Section 2 we shall introduce a new kind of intermediate states, the negative hypergeometric states (NHGS), so named because their photon statistics distribution is the negative hypergeometric distribution in probability theory. In Section 3 we show that the NHGS respectively approach the BS and NBS in two different limits. Furthermore, in some other limiting cases, they separately tend to the coherent states, the number states and the SG phase states. In this sense, the NHGS play the role of unifying these fundamental states of the quantized optical fields. In Section 4 we demonstrate that the NHGS display intermediary behavior even in their nonclassical properties, such as the antibunching effect and sub-Poissonian statistics, as compared with the properties of the BS and NBS. ’ Mailing address. 0375-9601/98/$ - see front matter @ 1998 Published by Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00818-4
H. Fan, N. LidPhysics LettersA 250 (1998) 88-92
89
2. Negative hypergeometic states We introduce the negative hypergeometric states as
(2) In} are the number states, p is a real number and s a non-negative integer, satisfying
w
M
s-c l-P+-.1-P
(3)
IIere the definition of the generalized binomial coefficient is ff 0n
=
ar(CY-I)...(ff-Tz+l) n.I
7
(4)
cy need not be an integer. The name “negative hypergeometric states” originates from the fact that their photon statistics dis~ibution ~{~~~,~,~)~* = [@:(P,s)l * is a negative hy~rgeome~~ dis~bution in probability theory [51 (seealsothe Appendix). The normalization of the NI-IGS can be seen from the combinatorial relation $(n;x)(Mi--:y)
= (M+x;y+l),
(5)
namely,
~~JWWXs) =g
(“;“)(Ml(l-~_~-s-1)(M/(~-8))-‘=l.
(6)
Eq. (5) can be derived by making use of the formula (1 + u)-fz+‘) = Cs (“;t’)( --B)~ and equating the power series expansion coefficients of u on both sides of ( 1 + u) -“( 1 + U) -Y = ( 1 + u) -(*+JJ). 3. Asymptotic behavior of the NHGS The different limits to which the three parameters M, .r, and p in Eq. ( 1) go correspond to different states in the quantized optical field. Remark I. In the limit p + 1, M and s finite, the NHGS degenerate to the vacuum state IO). In fact, noticing that M/C 1 - #3>+ oo we have [M,‘(l-~)]“-” (M-n)!
M!
[M/(1 -P)]M
-+sn*o’
(7)
90
H. Fan, N. Liu/Physics Letters A 250 (1998) 88-92
R~~rk
2. When /3 -+ s/(M + s),
where we noticed that (“it) = Sn,a, and hence the NHGS reduce to the number states IM}. Remarks 1 and 2 imply that the NHGS can be continuously ‘“tuned” from the vacuum state 10) to the number state /M) by changing p from p = s/( M + s) to p = 1. Remark 3. When p + 1, s --+ 00, but keeping ( 1 - p) ( 1 -t s) = (Y*constant ( a2 < M) (BS limit) we have M!
t@,Mwd)l*= X
[M-
(1
---, M 0n
(M-n>!
(n+s)(n+s-l)...(s+l)(l-p)” n!
(1 -P)(n+s+2>1 M[M--(l-P)][M-2(1-p)]...[M-(M-1)(1-/3)]
-P>(n+s+l)l[M-
(cr2)“(M - a2)‘+” MM
= ~~)(~)n(l
. ..[M-
(1 -P)(M-t-s)]
- $7
which tells us that the NHGS approach the ordinary BS, i.e., IM,&s)
+ fJ(:)
(g)n(l
- %)“-n]“2/n)
= 1~4,;)
(note CY*< M).
Remark 4. In the limit M -+ co, while p and s are finite (NBS limit), M!
(M-n)! --+ nts (
n )
M”[M&‘(l-/W’+s [M/( 1 - p)]‘+s+”
~~P/~~-~~~~~P/~~-~~-~~...[MP/(~-~)-s] [M/U-p)][M/(l-L+ll...[M/(l-p)-n-s] =
p’+s( 1 - p}“,
(9)
which is none other than the negative binomial distribution. Thus in this case the NHGS tend to the NBS, which are defined as ]p, s) = Cz* [(“~“)p’*“( 1 - p)“] “2 In), Since BS degenerate to the coherent and number states in two different limits [3] and the NBS to coherent and SG phase states in two different limits [ 41, the NHGS also take these fundamen~l states as their (further) limiting forms. In other words, the NHGS provide a unified approach to these irn~~nt states in the quantize radiation field.
4. Some properties of the NHGS Since the NHGS take the BS and NBS as their different limiting cases, we naturahy expect that they exhibit nonclassical properties which are intermediary between those of the BS and NBS. As summarized in Refs. [ 1,3], the ordinary BS can exhibit antibunching and sub-Poissonian behaviors; while from Ref. [4] we know that the NBS do not exhibit these effects for any p and s. Thus the NHGS ought to possess these nonclassicai features for a certain range of parameters. To verify this, let us first calculate the Mandel’s Q parameter [ 61, Q
=
(AN’>- W) (N)
’
(10)
H. Fan, N. Liu/Physics
Letters A 250 (1998) 88-92
91
which measures the deviation of the photon statistics distribution from the Poisson distribution (Q = 0). If Q < 0, the field is called sub-Poissonian; when Q > 0, the field is super-Poissonian. For the NHGS we evaluate the averages (N), (N*) and the fluctuation (AN*), @/) = (s+ l)(l -P)M 1-/3+/M
’
(1 -P+/M4)(2-2/3+@4) (AN2)= (Sfl)(l-_)M+ l-P+PM
’
(s+l)(s+2)(1-p)*M(M-l) (1 -P+PM)(2-2P+BW
_ (s+l)V-P)W (1 -p+pW
’
(11)
from which we find Q=(l_~,(~-~)(~-p+P~)-(s+l)(l-p+M)
(1 -P+PM)(2-2P+PM)
*
(12)
It then follows for Q < 0, P(M-
s>
V+p-2 1-p+kf
(13)
*
Combining the condition in Eq. (3) with Eq. ( 13), we arrive at the conclusion that the sufficient and necessary condition for the NHGS to be sub-Poissonian (for given /I and M) is P(M- V+p-2 l-/3+hf
MP
(14)
For fixed /3 and s, when M becomes large enough, it is obvious that the above condition is destroyed. The extreme case occurs when M + 00 (the NBS limit), in this case from Eq. ( 12) we see
->o .
Q+p
(15)
Thus the NBS are definitely super-Poissonian. On the other hand, in the BS limit, Q + -a2/M that the BS are definitely sub-Poissonian. A field is antibunched if its second-order correlation function g(*)(O) < 1 [7], namely, p
(0)
=
P2)- V) <
1
w*
< 0, indicating
.
Inserting Eq. ( 11) into Eq. ( 16), for the NHGS we have g’2’(o) = (s+2)(1 --P+PW(M(s+ 1)(2-2P+@kf)M
1) ’
(17)
from which we find that the NHGS will be antibunched if S>
P(M- V+p-2 l-P+M
’
(18)
which is the same as the condition of Eq. (13). In fact, the antibunching and sub-Poissonian are always coincident for single-mode and time-independent fields. In the NBS limit, g(*) (0) + (s + 2) /(s + 1) > 1; while in the BS limit, g(*) (0) + 1 - 1/M < 1, so the NBS are definitely bunching; while the BS are definitely antibunching.
92 5.
H. Fan, N. Liu/Physics Letters A 250 (1998) 88-92
Conclusion
In this Letter we have constructed the negative hypergeometric states of the quantized radiation field. Their photon statistics is the negative hypergeometric distribution in classical probability theory. These states interpolate between the binomial and negative binomial states. The intermediary nature is also embodied in their nonclassical properties, such as the antibunching effect and su~Poissonian statistics. Appendix. Negative hypergeometric distribution Assume that there are N balls, K of which are black and (N - K) are white. Suppose we choose balls at random, one by one and without replacement. Let rr be the number of our choices needed in order to exactly obtain r black balls. Then the probability of rr being m (m = r, r + 1,. . . , N - K + r) is given by the negative hypergeome~ic distribution
By a change of variables n=m-r,
r=s-t-
1,
M=N-K=(l-P)N,
we can rewrite the probability as
P{T~ = n + s + I} =
n+s (
iz I(
M/(1-p)-n-s-1 M_n
)~‘(;-P))-l>
which is just [O~(/~,S)]~, References [ 11 D. Staler, B.E.A. Saleh, M.C. Teich, Opt. Acta 32 (1985) 345. 121 A. Joshi, S.V. Lawande, Opt. Commun. 70 (1989) 21; J. Mod. Opt. 38 (1991) 2009; G.S. Agarwal, Phys. Rev. A 45 (1992) 1787. [3] A. Vidiella-Barraaco. J.A. Roversi, Phys. Rev. A 50 ( 1994) 5233. [41 K. Matsuo, Phys. Rev. A 41 ( 1990) 519. [Sl J.G. Skellam, J. R. Stat. Sot. B IO (1948) 257; L.R. Shenton, Biometrika 37 ( 1950) I1 1. 161 L. Maadel, Opt. Lett. 4 (1979) 205. 171 D.F. Walls, G.C. Miibum, Quantum Optics (Springer, Berlin, 1994).
PHYSICS LETTERS A
Physics Letters A 250 (1998) 88-92
Negative hypergeometric states of the quantized radiation field Hong-yi Fan a*b,Nai-le Liu b ’ CCAST (WorM Laboratory), PO Box 8730, Beijing 10W80, China h Department
of Material Science and Engineering,
China Universiry of Science and Technology, Hefei, Anhui 230026, China ’
Received 17 September 1998; accepted for publication 13 October 1998 Communicated by C.R. Doering
Abstract We introduce the negative hypergeometric states of the quantized radiation field. The importance of these states lies in the fact that they interpolate between the binomial states and negative binomial states, and tend to them in two different limits. In some other limiting cases, they degenerate to the number, coherent and Susskind-Glogower phase states, respectively. It is shown that some nonclassical properties of these states, such as antibunching effect and sub-Poissonian statistics, also display intermediary behavior. @ 1998 Published by Elsevier Science B.V.
1. Introduction
In recent years, the binomial states (BS) [ 1 ] and negative binomial states (NBS) [ 21 have attracted considerable attention of physicists in the field of quantum optics. They separately correspond to two opposite statistical distributions in classical probability theory. The BS interpolate between the coherent states (the most classical) and the number (Fock) states (the most nonclassical) [ 31, while the NBS interpolate between the coherent states and Susskind-Glogower (SG) phase states [ 41. A question thus naturally arises: are there any quantum-mechanical states which interpolate between the BS and NBS? The answer is affirmative. In Section 2 we shall introduce a new kind of intermediate states, the negative hypergeometric states (NHGS), so named because their photon statistics distribution is the negative hypergeometric distribution in probability theory. In Section 3 we show that the NHGS respectively approach the BS and NBS in two different limits. Furthermore, in some other limiting cases, they separately tend to the coherent states, the number states and the SG phase states. In this sense, the NHGS play the role of unifying these fundamental states of the quantized optical fields. In Section 4 we demonstrate that the NHGS display intermediary behavior even in their nonclassical properties, such as the antibunching effect and sub-Poissonian statistics, as compared with the properties of the BS and NBS. ’ Mailing address. 0375-9601/98/$ - see front matter @ 1998 Published by Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00818-4
H. Fan, N. LidPhysics LettersA 250 (1998) 88-92
89
2. Negative hypergeometic states We introduce the negative hypergeometric states as
(2) In} are the number states, p is a real number and s a non-negative integer, satisfying
w
M
s-c l-P+-.1-P
(3)
IIere the definition of the generalized binomial coefficient is ff 0n
=
ar(CY-I)...(ff-Tz+l) n.I
7
(4)
cy need not be an integer. The name “negative hypergeometric states” originates from the fact that their photon statistics dis~ibution ~{~~~,~,~)~* = [@:(P,s)l * is a negative hy~rgeome~~ dis~bution in probability theory [51 (seealsothe Appendix). The normalization of the NI-IGS can be seen from the combinatorial relation $(n;x)(Mi--:y)
= (M+x;y+l),
(5)
namely,
~~JWWXs) =g
(“;“)(Ml(l-~_~-s-1)(M/(~-8))-‘=l.
(6)
Eq. (5) can be derived by making use of the formula (1 + u)-fz+‘) = Cs (“;t’)( --B)~ and equating the power series expansion coefficients of u on both sides of ( 1 + u) -“( 1 + U) -Y = ( 1 + u) -(*+JJ). 3. Asymptotic behavior of the NHGS The different limits to which the three parameters M, .r, and p in Eq. ( 1) go correspond to different states in the quantized optical field. Remark I. In the limit p + 1, M and s finite, the NHGS degenerate to the vacuum state IO). In fact, noticing that M/C 1 - #3>+ oo we have [M,‘(l-~)]“-” (M-n)!
M!
[M/(1 -P)]M
-+sn*o’
(7)
90
H. Fan, N. Liu/Physics Letters A 250 (1998) 88-92
R~~rk
2. When /3 -+ s/(M + s),
where we noticed that (“it) = Sn,a, and hence the NHGS reduce to the number states IM}. Remarks 1 and 2 imply that the NHGS can be continuously ‘“tuned” from the vacuum state 10) to the number state /M) by changing p from p = s/( M + s) to p = 1. Remark 3. When p + 1, s --+ 00, but keeping ( 1 - p) ( 1 -t s) = (Y*constant ( a2 < M) (BS limit) we have M!
t@,Mwd)l*= X
[M-
(1
---, M 0n
(M-n>!
(n+s)(n+s-l)...(s+l)(l-p)” n!
(1 -P)(n+s+2>1 M[M--(l-P)][M-2(1-p)]...[M-(M-1)(1-/3)]
-P>(n+s+l)l[M-
(cr2)“(M - a2)‘+” MM
= ~~)(~)n(l
. ..[M-
(1 -P)(M-t-s)]
- $7
which tells us that the NHGS approach the ordinary BS, i.e., IM,&s)
+ fJ(:)
(g)n(l
- %)“-n]“2/n)
= 1~4,;)
(note CY*< M).
Remark 4. In the limit M -+ co, while p and s are finite (NBS limit), M!
(M-n)! --+ nts (
n )
M”[M&‘(l-/W’+s [M/( 1 - p)]‘+s+”
~~P/~~-~~~~~P/~~-~~-~~...[MP/(~-~)-s] [M/U-p)][M/(l-L+ll...[M/(l-p)-n-s] =
p’+s( 1 - p}“,
(9)
which is none other than the negative binomial distribution. Thus in this case the NHGS tend to the NBS, which are defined as ]p, s) = Cz* [(“~“)p’*“( 1 - p)“] “2 In), Since BS degenerate to the coherent and number states in two different limits [3] and the NBS to coherent and SG phase states in two different limits [ 41, the NHGS also take these fundamen~l states as their (further) limiting forms. In other words, the NHGS provide a unified approach to these irn~~nt states in the quantize radiation field.
4. Some properties of the NHGS Since the NHGS take the BS and NBS as their different limiting cases, we naturahy expect that they exhibit nonclassical properties which are intermediary between those of the BS and NBS. As summarized in Refs. [ 1,3], the ordinary BS can exhibit antibunching and sub-Poissonian behaviors; while from Ref. [4] we know that the NBS do not exhibit these effects for any p and s. Thus the NHGS ought to possess these nonclassicai features for a certain range of parameters. To verify this, let us first calculate the Mandel’s Q parameter [ 61, Q
=
(AN’>- W) (N)
’
(10)
H. Fan, N. Liu/Physics
Letters A 250 (1998) 88-92
91
which measures the deviation of the photon statistics distribution from the Poisson distribution (Q = 0). If Q < 0, the field is called sub-Poissonian; when Q > 0, the field is super-Poissonian. For the NHGS we evaluate the averages (N), (N*) and the fluctuation (AN*), @/) = (s+ l)(l -P)M 1-/3+/M
’
(1 -P+/M4)(2-2/3+@4) (AN2)= (Sfl)(l-_)M+ l-P+PM
’
(s+l)(s+2)(1-p)*M(M-l) (1 -P+PM)(2-2P+BW
_ (s+l)V-P)W (1 -p+pW
’
(11)
from which we find Q=(l_~,(~-~)(~-p+P~)-(s+l)(l-p+M)
(1 -P+PM)(2-2P+PM)
*
(12)
It then follows for Q < 0, P(M-
s>
V+p-2 1-p+kf
(13)
*
Combining the condition in Eq. (3) with Eq. ( 13), we arrive at the conclusion that the sufficient and necessary condition for the NHGS to be sub-Poissonian (for given /I and M) is P(M- V+p-2 l-/3+hf
MP
(14)
For fixed /3 and s, when M becomes large enough, it is obvious that the above condition is destroyed. The extreme case occurs when M + 00 (the NBS limit), in this case from Eq. ( 12) we see
->o .
Q+p
(15)
Thus the NBS are definitely super-Poissonian. On the other hand, in the BS limit, Q + -a2/M that the BS are definitely sub-Poissonian. A field is antibunched if its second-order correlation function g(*)(O) < 1 [7], namely, p
(0)
=
P2)- V) <
1
w*
< 0, indicating
.
Inserting Eq. ( 11) into Eq. ( 16), for the NHGS we have g’2’(o) = (s+2)(1 --P+PW(M(s+ 1)(2-2P+@kf)M
1) ’
(17)
from which we find that the NHGS will be antibunched if S>
P(M- V+p-2 l-P+M
’
(18)
which is the same as the condition of Eq. (13). In fact, the antibunching and sub-Poissonian are always coincident for single-mode and time-independent fields. In the NBS limit, g(*) (0) + (s + 2) /(s + 1) > 1; while in the BS limit, g(*) (0) + 1 - 1/M < 1, so the NBS are definitely bunching; while the BS are definitely antibunching.
92 5.
H. Fan, N. Liu/Physics Letters A 250 (1998) 88-92
Conclusion
In this Letter we have constructed the negative hypergeometric states of the quantized radiation field. Their photon statistics is the negative hypergeometric distribution in classical probability theory. These states interpolate between the binomial and negative binomial states. The intermediary nature is also embodied in their nonclassical properties, such as the antibunching effect and su~Poissonian statistics. Appendix. Negative hypergeometric distribution Assume that there are N balls, K of which are black and (N - K) are white. Suppose we choose balls at random, one by one and without replacement. Let rr be the number of our choices needed in order to exactly obtain r black balls. Then the probability of rr being m (m = r, r + 1,. . . , N - K + r) is given by the negative hypergeome~ic distribution
By a change of variables n=m-r,
r=s-t-
1,
M=N-K=(l-P)N,
we can rewrite the probability as
P{T~ = n + s + I} =
n+s (
iz I(
M/(1-p)-n-s-1 M_n
)~‘(;-P))-l>
which is just [O~(/~,S)]~, References [ 11 D. Staler, B.E.A. Saleh, M.C. Teich, Opt. Acta 32 (1985) 345. 121 A. Joshi, S.V. Lawande, Opt. Commun. 70 (1989) 21; J. Mod. Opt. 38 (1991) 2009; G.S. Agarwal, Phys. Rev. A 45 (1992) 1787. [3] A. Vidiella-Barraaco. J.A. Roversi, Phys. Rev. A 50 ( 1994) 5233. [41 K. Matsuo, Phys. Rev. A 41 ( 1990) 519. [Sl J.G. Skellam, J. R. Stat. Sot. B IO (1948) 257; L.R. Shenton, Biometrika 37 ( 1950) I1 1. 161 L. Maadel, Opt. Lett. 4 (1979) 205. 171 D.F. Walls, G.C. Miibum, Quantum Optics (Springer, Berlin, 1994).