Time evolution of negative binomial optical field in diffusion channel , China

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1 Chinese Physics B arxiv: v1 [quant-ph] 17 Apr 2015 Time evolution of negative binomial optical field in diffusion channel Liu Tang-Kun a, Wu Pan-Pan a, Shan Chuan-Jia a, Liu Ji-Bing a, and Fan Hong-Yi b a College of Physics and Electronic Science, Hubei Normal University, Huangshi , China b Department of Material Science and Engineering, University of Science and Technology of China, Hefei , China We find time evolution law of negative binomial optical field in diffusion channel. We reveal that by adjusting the diffusion parameter, photon number can controlled. Therefore, the diffusion process can be considered a quantum controlling scheme through photon addition. Keywords: negative binomial optical field, time evolution, diffusion channel, integration within an ordered product IWOP of operators PACS: w, p, p 1. Introduction In a recent paper [1] we have pointed out that an initial number state l l undergoing through a diffusion channel, described by the master equation [2-3] d dt ρ = κ a aρ+ρaa aρa a ρa, 1 ProjectsupportedbytheNationalBasicResearchProgramofChinaGrantNo. 2012CB922103,theNational Natural Science Foundation of China Grant Nos , and , and the Natural Science Foundation of Hubei Province of China Grant No. 2011CDA021. Corresponding author. tkliuhs@163.com

2 would become a new photon optical field, named Laguerre-polynomial-weighted chaotic state, whose density operator is ρ λ1 λ l :L l λ 2 a a 1 λ e λa a :,λ = 1 Here :: denotes normal ordering symbol, l = a n 0 / n!, L l is the Laguerre polynomial. Experimentally, this new mixed state may be implemented, i.e., when a number state enters into the diffusion channel. Remarkablely, this state is characteristic of possessing photon number Tr a aρ = l + κt at time t, so we can control photon number by adjusting the diffusion parameter κ, this mechanism may provide application in quantum controlling. To go a step further, our aim in this paper is to derive evolution law of a negative binomial state NBS in diffusion channel. Physically, when an atom absorbs some photons from a thermo light beam, then the corresponding photon field will be in a negative binomial state. We are thus challenged by the question: how an initial NBS evolves in a diffusion channel, what a final state it will be, and what is the photon number distribution in the final state. To our knowledge, such questions has not been touched in the literature before. Our paper is arranged as follows. In Sec. 2, we convert the density operator of NBS into normally ordered form. In Sec. 3 based on the Kraus-operator-solution corresponding to the diffusion channel we find the evolution law of NBS in diffusion channel. Then in Sec. 4 we calculate the photon number distribution in the final state Normally ordered form of the density operator of negative binomial state Corresponding to the negative-binomial formula m+n x m = 1+x n 1 3 m=0 m there exists negative-binomial state of quantum optical field [4] ρ 0 = n+s! s+1 1 n n n,0 < < 1, 4 n!s! where n = a n 0 / n! is the Fock state, a is the photon creation operator, 0 is the vacuum state in Fock space. NBS is intermediate between a pure thermal state and a pure coherent

3 state, and its nonclassical properties and algebraic characteristic have already been studied in detail in Refs. [11, 12]. The photon number average in this state is Tr ρ 0 a a = s+11 5 Using [ a,a ] = 1, a s n = n! n s, one can reform Eq. 4 as n s! ρ 0 = = s+1 s!1 s a s 1 a s ρ s!n s c a s c 1 n n n a s 6 where ρ c denotes a chaotic field ρ c = 1 n 1 n n n = n! = e a aln1, : a n e a a a n := : e a a : 7 Eq. 6 tells us that when some photons are detected for a chaotic state, e.g. after detecting several photons, the chaotic light field exhibits negative-binomial distribution. One can further show Tr c = 1, and tr ρ c a a = 1 1 = n c 8 is the mean number of photons of chaotic light field, according to Bose-Einstein distribution, n c = 1 e βω h 1, here β = 1 k B T, k B is the Boltzmann constant, ω is the frequency of chaotic light field. We can derive the normally ordered form of the density operator of NBS, let ln1 = f, then n c = e f / 1 e f and by introducing the coherent state representation π 2 z z = 1, z = e z 2 e za 0, and employing the technique of integration within an ordered product IWOP of operators [5-6] we reform Eq. 6 as 1 ρ 0 = a s ρ s!n s c a s = 1 ef a s e fa a a s c s!n s c = 1 ef d 2 z s!n s c π as e fa a z z a s = 1 ef d 2 z e z 2 s!n s 2 a s e fa a e za e fa a 0 z z s c π = 1 ef d 2 z e z 2 s!n s 2 a s e za e f 0 z z s c π = 1 ef s!n s c π ze f s z s :e z 2 +za e f +z a a a :

4 = 1 ef e fs : s!n s c l=0 e e f 1 a an! 2 a ae f n l l![n l!] 2 : = 1 e f s+1 :e ef 1 a a Ls a ae f : = s+1 :e a al s [ 1a a ] : 9 where we have used 0 0 =: e a a :, and the definition of Laguerre polynomials s x l n! L s x = l! 2 n l! that 9 is quite different from 1, so they represent different optical field. l= Evolution law of the negative binomial state in diffusion channel Recall in Ref. [7] by using the entangled state representation and IWOP technique we have derived the infinite sum form of ρt ρt = 1 m!n! where = m, κt m+n κt+1 m+n+1a m a a a m 1 1 a a 1 a n ρ 0 a n κt m+n m!n! κt+1 m+n+1 M m,n ρ 0 M m,n 11 m, M m,n = 1 κt m+n 1 a a m!n! κt+1 m+n+1a m a n 12 satisfying m, M m,nm m,n = 1, which is trace conservative. Now we examine time evolution of negative binomial optical field in diffusion channel. Substituting ρ 0 = 1 s!n s ca s ρ c a s into 11 we have ρt = s+1 s!1 s m, 1 κt m+n m!n! κt+1 m+n+1 1 a a m a 1 a n+s e a aln1 a s+n in which we first consider the summation over n, using a a a m 13 1 a a = e a aln,e fa a ae fa a = ae f, 14

5 we have 1 κt n n! κt+1 n e a aln a n+s e a aln1 a s+n e a aln = 2s κt n n a n+s e a aln[1 / 2 ] a s+n n! Note from Eq. 9 we have 15 a s e fa a a s = s!e fs :e ef 1a a L s a ae f : 16 it follows a n+s e a a[ln1 2ln] a s+n = n+s!e n+sln[1 /2 ] :e [1 / 2 1]a a L n+s Substituting 17 into 15 and multiplying s+1 2s 1 s κt+1 n κt n n+s! s!n! = s+1 n+s!κt n 1 n s!n!κt+1 n :L n+s s+1 1 s s! we see a a 1 2 : 17 e n+sln[1 /2 ] :e [1 / 2 1]a a L n+s a a 1 : 2 a a 1 e [1 / 2 1]a a : 2 Then we use the new generating function formula about the Laguerre polynomials [8]. we obtain 18 = = 18 n+s! λ n L n+s z = 1+λ s 1 e λz z 1+λ Ls. 19 n!s! 1+λ [ ] s+1 κt+1 : L s a 1 a [ ] s+1 κt+1 : L s a 1 a e a a κt1 2 2 e [1 /2 1]a a 20 : e [ 1 tκ+1tκ+1 1]a a : For ρt in Eq. 13 It remains to perform summation over m, using the summation technique within normal ordering we have [ ] s+1 κt+1 κt m ρt = m=0 m!κt+1 m+1 21 : a m e [ 1 tκ+1tκ+1 1]a a L s a 1 a a m : : e Ea a L s a af : where E = κt +1,F = 1 1,F +E = 22

6 C = 1 [ ] s+1 κt+1 23 Comparing ρt in 21 with ρ 0 in 9 we can see the big difference. Now we must check if Trρt = 1, in fact, using the coherent state s completeness relation 1 = π we do have Trρt Tr 0 [ z z and e bx L l xdx = b 1 l b l : e Ea a L s a af : π e tκ+1 z 2 L s 1 1 d 2 ] z π z z z dr e r 1 L s r 25 κt+1 1 s [ ] s+1 1 = 1 4. Photon number average in the final state have Now we evalute photon number average in the final state, using 21, 10 and we Tr [ ρta a ] π z a a : e Ea a L s a af : z 26 π z z {[ a,: e Ea a L s a af : ] + : e Ea a L s a af : a } z d 2 { z π z z a : eea a L s a af : + z 2 e E z 2 L s z 2 F } z s l z 2l F l s! d 2 z π ee z 2 l=0 l! 2 +C s l! π 1+E z 2 e E z 2 L s z 2 F s F l [ s! d 2 z d l=0 l! 2 l z 2l 2 ] z +1+E s l! π ee z 2 π z 2l+1 e E z 2 s F l [ ] s! l! l+1! l=0 l! 2 l l+1 +1+E s l! E E l+2 = tk + s+11 Comparing with Eq. 5 we see that after passing through a diffusion channel, the photon average of a NBS varies from s+11 to tk + s+11, Tr [ ρta a ] = tk + s+11 = tk +Tr ρ 0 a a 27

7 This result is encourageous, since by adjusting the diffusion parameter κ, we can control photon number, when κ is small, it slightly increases by an amount κt. Therefore, this diffusion process for NBS can be considered a quantum controlling scheme through photon addition. References [1] Fan HongYi, Lou SenYue and Pan XiaoYin 2014 SCIENCE CHINA-PHYSICS MECHAN- ICS & ASTRONOMY [2] Carmichael H J 1999 Statistical Methods in Ouantum Optics I, Master Equation and Fokker-Planck equations Berlin: Springer-Verlag [3] Orszag M 2000 Quantum Optics Berlin:Springer-Verlag [4] G. S. Agarwal 1992 Phys. Rev. A [5] Fan H Y, H. L. Lu and Y. Fan 2006 Ann. Phys., [6] Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt., 5 R147-R163 [7] Liu Tang-Kun, Shan Chuan-Jia, Liu Ji-Bing, and Fan Hong-Yi 2014 Chin. Phys. B [8] Fan Hong-yi, Lou Sen-yue, Pan Xiao-yin and Da Chen 2013 Acta Phys. Sin in Chinese