Recursive Construction of the Bosonic Bogoliubov Vacuum State

Transcription

1 Quant. Phy. Lett. 2, No. 1, Quantum Phyc Letter An Internatonal Journal Recurve Contructon of the Boonc Bogolubov Vacuum State Drceu Porte Jr 1, Marco O. Pnho 1 and Hláro Rodrgue 1 1 Departamento de Enno Superor, Centro Federal de Educação Tecnológca do Ro de Janero, Av. Maracanã, Brazl Receved: 7 Jun. 2012, Reved: 21 Sep. 2012, Accepted: 23 Sep Publhed onlne: 1 Apr Abtract: In th work we derve a novel procedure for obtanng the boonc Bogolubov vacuum tate by ung a recurve cheme. The vacuum tate for the new creaton and annhlaton operator explctly contructed n term of the number tate of the old operator, whch are connected by a Bogolubov tranformaton. The coeffcent of the ground tate n Fock ba are thu obtaned a excluve functon of the parameter of the Bogolubov tranformaton. Keyword: quantum phyc, boonc quantum ytem, boonc vacuum tate, Bogolubov tranformaton 1 Introducton The Bogolubov tranformaton ha been ued a a powerful tool n tudyng the properte of varou quantum ytem [1,2]. One of the advantage n ung th method that lnear canoncal tranformaton provde the exact dagonalzaton of quadratc multdmenonal Hamltonan [3, 4]. It well known that the problem of fndng the Hamltonan egenvalue and egenvector for a et of coupled harmonc ocllator can be olved n analogy wth the clacal cae, when a Bogolubov tranformaton performed. In order to obtan the complete oluton, one need the quantum tate tranformaton from the bae tate of the old operator to thoe of the new one. Ultmately, th equvalent to obtan the relaton between lnear canoncal tranformaton and the related untary operator n Hlbert pace, whch were extenvely tuded for boonc and fermonc operator by many author [5,6,7]. The untary operator correpondng to the Bogolubov tranformaton ha no clacal equvalent and t tudy contrbute to undertandng remarkable quantum apect. Partcularly, t theoretcally relevant the boonc Bogolubov vacuum, or boonc ground tate. For example, n Boe-Enten condenate tate [8, 9], and n the tudy of vacuum tructure of de Stter pace [10]. In the coherent tate repreentaton, the ground tate gven by a Gauan functon [13]. However, n ome tuaton t turn to be convenent to have the ground tate n the Fock pace, gven n term of the number tate of the old creaton and annhlaton operator. e., before the Bogolubov tranformaton carred out. In prncple, t poble to contruct the Bogolubov vacuum tate n the occupaton number repreentaton by mply changng the ba. However th a very cumberome tak whch demand a huge work. In th work we gve explct formula for the coeffcent c n1,n 2,... n 1,n 2,...,0 b, appearng n the expanon of the Bogolubov tranformed vacuum wth repect to the orgnal number of partcle ba. The formula we propoe allow a recurve computaton of the coeffcent c n1,n 2,... baed only on the parameter characterzng the Bogolubov tranformaton. The propoed new formula are an alternatve calculaton of tandard formula baed on coherent tate and Gauan ntegraton. The work organzed a follow. In Secton 2, we brefly dcu the properte of the Bogolubov tranformaton of creaton and annhlaton boonc operator. In Secton 3, we preent a recurve method of calculaton of the coeffcent of the Bogolubov tranformaton. In Secton 4 we preent the concluon. 2 The Bogolubov tranformaton Let u conder a boonc Fock pace, wth creaton and annhlaton operator, a j and a j, atfyng canoncal Correpondng author e-mal: harg@cefet-rj.br Natural Scence Publhng Cor.

2 12 D. Porte Jr, M. O. Pnho, H. Rodrgue: Boonc Bogolubov Vacuum State commutaton relaton. An orthonormal ba for the Fock pace gven by vector n 1,n 2,..., where the nteger n j pecfy how many partcle occupy the j-th one-partcle tate. The lnear canoncal tranformaton actng on creaton and annhlaton boonc operator wa frt ntroduced by N. Bogolubov n 1947 [11, 12]. In uch tranformaton the new creaton b and annhlaton b operator are related to the correpondng old operator a and a through, b b j1 j1 µ j a j + ν j a j, 1 ν j a j + µ j a j, 2 where 1,...,, beng the number of ocllator. Ung matrx notaton, we may wrte b µa + va, 3 b νa+ µa, 4 where we have denoted the matrce µ [µ jk ], ν [ν jk ], a [a ] 1, etc. The commutaton relaton atfed by the boonc operator [b j,b k ] δ jk, [b j,b k ] 0, 5 yeld the followng two matrx relaton: and µµ νν 1, 6 µν T νµ T 0. 7 The number tate n the Fock ba can be wrtten n term of the old vacuum tate, namely n 1...n a j1 1 n j! a j n j 0,...,0 a, 8 where { n 1,...,n a } repreent the egenvector of the old number operator a a. In the ame way, the contructon of the number tate n the Fock ba for the new number operator b b j can be expreed n term of the vacuum tate 0 b 0,...,0 b. Notce that 0 b are not necearly the ground tate. By defnton, the mnmum energy tate correpond to the vacuum tate aocated wth a tranformaton that dagonalze the ytem Hamltonan. In fact, any quadratc multdmenonal Hamltonan can be wrtten a H E b + b, 9 from an approprate Bogolubov tranformaton. In vew of the dcuon n the prevou ecton, we can ak how to obtan the vacuum tate 0 b n term of the old number tate n 1,...,n a gven n Eq. 8. From the formal pont of vew, th may be wrtten n the form 0 b n 1...n c n1...n n 1,...,n a. 10 The explct form of 0 b obtaned n the coherent repreentaton gven by [13], a α 1,...,α 0 b [ det µ µ ] 1/4 exp 1 2 ᾱ α σ j ᾱ ᾱ j, 11 j where α a repreent the coherent tate aocated wth the operator a a α α α, where the matrx σ defned a σ µ 1 ν. 12 Notce that the matrx µ nvertble, nce from Eq. 6 follow that detµ 0. It not dffcult to obtan 0 a, f we aume µ 1 and ν 0 wth b a n Eq. 11. Thu, we get a α 1,...,α 0 a exp 1 2 ᾱ α 13 In term of the number tate n 1,...,n a relatve to the old creaton and annhlaton operator, the change of ba of Eq. 13 yeld c n1,...,n [ det µ µ ] 1/4 d 2 α ᾱ n Π π n! [ exp ᾱα 1 2 σ j ᾱ ᾱ j ], 14 j where the ntegral n the rght hand de of the lat equaton carred out over the whole complex plane. So, we have a cloe relatonhp for the coeffcent. However, calculatng the ntegral on the rght de of the lat equaton, ung analytcal or numercal method, t uually a dffcult tak to be performed. So the determnaton of value for c n1,...,n. In partcular, for n 0, 1,...,, the lat equaton provde c 0,...,0 [ det µ µ ] 1/4. 15 It ueful to obtan the vacuum 0 b ndependent repreentaton, whch gven by 0 b [ det µ µ ] 1/4 1 exp 2 σ j a + a + j 0 a. 16 j In order to verfy Eq. 16, we jut multply t by a α 1,...,α and then ue Eq. 13. It alo poble to obtan the coeffcent {c n1,...,n } by mean of a expanon n power ere of Eq. 16. For example, for 1, the drect calculaton lead to c n n 0 b 1 ν n/2 n! µ 2µ n/2!. 17 Natural Scence Publhng Cor.

3 Quant. Phy. Lett. 2, No. 1, / 13 Neverthele, for > 1 the calculaton of the coeffcent {c n1,...,n } by expanon n power ere of Eq. 16 not eay to carry out, and th jutfe the earch for an alternatve method for calculatng the coeffcent. 3 Decrpton of the method We want to how that the vacuum tate 0 b gven n the Fock ba can be contructed avodng the ue of Eq. 14. We then take the expreon j b 0 b j n 1,...,n a But from Eq. 1 and 2, th equaton can be put n the form b 0 µ j a j + ν ja j n 1,...,n a 0, 19 from whch we get j µ j n j + 1c n1,...,n j +1,...,n + ν j n j c n1,...,n j 1,...,n The lat equaton can alo be wrtten n the matrx form c n1 +1,...,n n1 + 1 c n1 1,...,n n1. σ., c n1,...,n +1 n + 1 c n1,...,n 1 n 21 From Eq. 7 follow and ν µν T µ T 1, 22 µ 1 ν µ 1 ν T. 23 So, from Eq. 12 together wth the two lat equaton, one how that σ a ymmetrc matrx. Thu, Eq. 21 provde equaton n the form of n j c n1,...,n +1,...,n σ j j1 whch yeld the recurve equaton c n1,...,n σ n 1 + j n n + 1 c n 1,...,n j 1,...,n c n1,...,n 2,...,n 1,...,, 24 σ j n j n c n1,...,n 1,...,n j 1,...,n. 25 Eq. 24 the natural way to obtan the vacuum 0 b n Fock ba. Our goal n th work to obtan, from Eq. 24, an explct expreon for any of the coeffcent c n1,...,n gven a a functon of the parameter of the Bogolubov tranformaton. Intally, we wll dtrbute the coeffcent nto et C N {c n1,...,n n n N}, where N a nonnegatve nteger. We pont out that the coeffcent c n1...n C N are generated from the coeffcent c n 1...n C N 2. So we mut conclude that we have two ndependent group of coeffcent: c n1...n C N, for N an even nteger, and c n1...n C N, for N an odd nteger. It very mportant to note that none of the member of a group relate to the element of another group. The tartng value of the odd group are coeffcent c 0,...,1,...,0 C 1, whch can be determned from Eq. 20: µ c 1,0...,0. c 0,...,0, Snce the matrx µ alway nvertble, we have c 0,...,n 1,..., for every. Therefore, for n n an odd nteger number one ha c n1,...,n When conderng the even group, we ee that the coeffcent c 0,...,0 the only element of the et C 0, and o t taken a the ntal value. However, t can not be determned by Eq. 20, and t value yet determned by Eq. 15. Thu, t poble obtanng c n1...n C N, for every even N from Eq. 25 ung the tartng value c 0,...,0 [ det µ µ ] 1/4. In order to derve the general expreon, we need frt defne a permutaton group partcularly ueful n olvng the propoed problem. So, uppoe the product σ k σ k σ k, 29 contng of N/2 factor, where k j k j k j are nonnegatve nteger atfyng the condton 2k + j k j n. 30 Impong the condton gven by the lat equaton equvalent to requrng that each ndex mut appear n tme n the product 29. Let u call,...,n the et of all dfferent confguraton of k j atfyng Eq. 30. Formally, we can wrte,...,n {k 11,k 12,...,k 2k + j k j n, k j N andk j k j }. 31 The et,...,n equvalent to get all dfferent product n 29 obtaned by ndex permutaton, preervng j n σ j. For example, for 3, wth n 1 4, n 2 1, and n 3 3, we have the poblte Natural Scence Publhng Cor.

4 14 D. Porte Jr, M. O. Pnho, H. Rodrgue: Boonc Bogolubov Vacuum State {σ 2 11 σ 23σ 33,σ 11 σ 12 σ 13 σ 33,σ 11 σ 2 13 σ 23,σ 3 13 σ 12}, whch equvalent to wrte K 4,1,3 {k 11 2, k 23 1, k 33 1, k j 0, nallothercae,...}. 32 From,...,n we can obtan drectly,...,n 2,...,n, not conderng element wth k 0 and, for k > 0, replace k k 1. Smlarly,,...,n 1,...,n j 1,...n obtaned by deletng the element k j 0 and, for k j > 0, replace k j k j 1. Wth,...,n determned, we can now try to how that the recurve relaton Eq. 25, together wth Eq. 15, reult n the followng expreon for the coeffcent c n1,...,n : n1!...n! c n1,...,n [detµ µ] 1/4,...,n 1 σ k 2 k k! l>k σ k j j, 33 k j! where the ummaton carred out over all element of,...,n. We have thu n1!...n! c n1,...,n 2,...,n [detµ µ] 1/4 n n 1 2k σ k 1 σ k σ k 2 k k!,...,n p 2 k k. 34! q>p k! Notce that Kn1 and,...,n Kn1 repreent,...,n l 2,...,n dfferent ummaton, but multplcaton by k wll exclude any addtonal term n Kn1. Thu, we arrve,...,n at n 1 n1!...n! σ c n1,...,n 2,...,n n [detµ µ] 1/4 2k σ k σ k n,...,n p1 2 k k. 35! q>p k! Analogouly, we have σ j n j n c n1,...,n 1,...,n j 1,...,n k j n,...,n p1 σ k 2 k k! q>p n1!...n! [detµ µ] 1/4 σ k k! whch lead drectly to Eq. 25, nce 1 2k + n concludng the proof. j, 36 k j 1, 37 4 Concluon In th work we derve a new alternatve procedure where the vacuum tate for the new creaton and annhlaton operator contructed n term of the number tate of the old operator. The new and old creaton and annhlaton operator are connected by a Bogolubov tranformaton, gven by Eq. 1 and 2. Formally, the new vacuum tate can be wrtten a n Eq. 10, whch requre knowng the complete et of coeffcent {c n1,...,n }. We have hown that uch coeffcent can be obtaned recurvely from Eq. 33. It worth to menton that the coeffcent of the ground tate n Fock ba are wrtten a excluve functon of the parameter of the Bogolubov tranformaton. Fnally, t hould be mentoned that formula 34 can be undertood a a method to calculate the ntegral defned n the rght-hand de of equaton 14. Acknowledgement H. Rodrgue thank the Brazlan foundaton CNPq for the fnancal upport. D. Porte Jr thank the crtcm and uggeton of Dr. Paulo Borge. Reference [1] R. Balan, E. Brezn, Nuovo Cmento B 64, [2] V. V. Dodonov, Bogolubov tranformaton for ferm boe ytem and queezed tate generaton n cavte wth ocllatng wall, Concept of Phyc, Vol. IV, No [3] C. Tall, Dagonalzaton method for the general blnear Hamltonan of an aembly of boon, J. Math. Phy. 19, [4] O. Maldonaldo, On the Bogolubov tranformaton for quadratc boon obervable, J. Math. Phy. 34, [5] M. Mohnky and C. Quene, Lnear canoncal tranformaton and ther untary repreentaton, J. Math. Phy. 12, [6] K. Hara, S. Iwaak, Nucl. Phy. A 332, [7] Kazuo Takayanag, Nuclear Phyc A [8] Bartłomej Oleś and Krzyztof Sacha, N-conervng Bogolubov vacuum of a two-component Boe Enten condenate: denty fluctuaton cloe to a phae-eparaton condton, J. Phy. A: Math. Theor ; Jacek Dzarmaga and Krzyztof Sacha, Bogolubov theory of a Boe-Enten condenate n the partcle repreentaton, Phy. Rev. A 67, [9] U. V. Poulen, T. Meyer, and M. Lewenten, Entanglement n the Bogolubov vacuum, Phy. Rev. A 71, [10] Jmmy A. Hutaot, Vacuum ambguty n de Stter pace at trong couplng, Journal of Hgh Energy Phyc 2010, 2, 26, [11] N. Bogolubov, On the theory of uperfludty, Izv. Akad. Nauk USSR, er. fz. 11, [J. Phy. USSR 11, ; reprnted n: D. Pne, The Many-Body Problem W.A. Benjamn, New York, 1961, p ]. Natural Scence Publhng Cor.

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