SPIN COHERENT STATES DEFINED IN THE BARUT-GIRARDELLO MANNER

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1 THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eries A, OF THE ROMANIAN ACADEMY Volue 7, Nuber 4/6, pp PIN COHERENT TATE DEFINED IN THE BARUT-GIRARDELLO MANNER Duša POPOV Politehica Uiversity of Tiisoara, Departet of Physical Foudatios of Egieerig, B-dul Vasile Pârva No., 33 Tiisoara, Roaia E-ail: Abstract. I this paper we costruct, for the first tie to the best of our kowledge, the Barut- Girardello coheret states for the spi systes, ad exaie soe of their properties. We apply the previously deduced diagoal orderig operatio techique (DOOT to the spi coheret states. I this aer, iplicitly we have showed that the DOOT ca be applied ot oly to the coheret states of systes with ifiite uber of eergy boud states but also for systes with fiite boud states, aog the beig the spi systes. Key words: spi systes, coheret states, operator orderig, theral states.. INTRODUCTION I codesed atter physics ad ot oly, the ore atural basis to study the agetic probles is provided by the use of spi coheret states (Cs >, where z z exp(iϕ is the coplex variable labelig these states ad is the spi quatu uber. These states so far have bee defied usig the Klauder-Pereloov s procedure, i.e. by applyig the spi displaceet operator to the groud state ; >. I the preset paper we will show that the spi Cs ay also be defied i the Barut-Girardello aer, i.e. as the eigevalues of the lowerig operator. Moreover, i differet calculatios ivolvig Cs it is ecessary to achieve coutatio of operators or their orderig, accordig to certai rules. For the haroic oe-diesioal Cs it is used the Itegratio Withi a Ordered Рroduct of Operators (IWOP techique forulated by Fa et al. (see, e.g. [] ad refereces therei. I a previous paper [] we have proposed a ew, ore geeral, techique for orderig the product operators applicable oly to the orally ordered product of a pair of geeralized raisig ad lowerig operators A ad A. These operators defie the geeralized hypergeoetric coheret states (GH-Cs of Barut-Girardello (BG kid. As ay kids of Barut-Girardello coheret states (BG-Cs, e. g. the Cs of haroic oscillator (HO-D, the pseudoharoic oscillator ad the Morse oscillator, the Cs are also the particular cases of GH-BG-Cs. The ew elaborated techique i [] we have called the diagoal orderig operatio techique (DOOT ad we have deoted the correspodig operatio by the sybol. Because the Cs of HO-D are a particular case of GH-BG-Cs, the IWOP techique [] ca be regarded also as a particular case of the DOOT [].. PIN OPERATOR AND DOOT The Fock vectors >,,,...,, of a sigle quatu particle with total spi for a coplete orthooral set, so that >< I, (

2 Barut - Girardello spi coheret states 39 where I is the idetity operator i the Hilbert space of the sigle-ode syste. The spi operators ± x ± i y, 3 z act o a fiite ( -diesioal Hilbert space spaed by Fock vectors > as follows > ( ( ; >, > ( ; >, ( z > ( >. (3 Moreover, the practical use of spi operators frequetly requires the oral orderig, i.e. i a product, the raisig (creatio operator is foud o the left ad the lowerig (aihilatio oe o the right,. The oral ordered product is a diagoal operator i the Fock vectors basis > : > ( > L ( > (4 or, usig the copleteess relatio: ( >< L ( ><, (5 L x where, for shortess, we have used the followig otatio ( ( x x. I short, the ewly itroduced DOOT for GH-BG-Cs is based o the followig rules []: a Iside the sybol the order of the operators A ad A (geeratig the GH-BG-Cs ca be peruted like coutable operators, but so that fially will result a operator fuctio that depeds oly o the powers of orally ordered product A A, i.e. ( A ( A ( A ( A ( A A. b A sybol iside aother sybol ca be deleted. c If the itegratio is coverget, expressios cotaiig the orally ordered product of operators ca be itegrated or differetiated, with respect to c-ubers, accordig to the usual rules. I additio, the c-ubers ca be take out fro the sybol. d The projector ; λ >< ; λ of the oralized vacuu (or uperturbed groud state ;λ >, i the frae of DOOT, has the followig oral ordered for: ; λ >< ; λ, q F ( a ; b ; A A p { } { } p q i j where λ is a real paraeter whose physical eaig differs fro oe kid of quatu exaied syste to aother. Geerally, the projector ; λ >< ; λ is just the iverse of the oralizatio fuctio of the GH- BG-Cs, but havig the product operator AA as variable. Let us apply this geeral techique to the spi operators by the followig idetificatios: A ad A. I this case the real paraeter λ is just the particle total spi ad the iteger paraeters of GH-BG-Cs are p ad q as we will see below. The groud or vacuu state ; > is defied as usual, i.e. ; > ; >, while the repeatable actio of the raisig operator o the vacuu state is Γ ( ( ; λ >! > (!( >, (7 Γ ( where Γ(x is Euler Gaa fuctio ad ( a Γ( a / Γ( a is the Pochhaer sybol, particularly ( a ( Γ( a / Γ( a. Next we will use the followig short otatio: Γ ( ρ(! (!(. (8 Γ ( I this aer, the Fock vectors ket ad bra ca be expressed as (6

3 33 Duša Popov 3 ; ; ρ( < ; < ;. (9 ρ( > ( > ( Usig the DOOT rules, the copleteess relatio is successively writte as >< ( ; > < ; ( ρ( ( ; >< ; ; >< ; F( ; ;, (! fro which we obtai the expressio of the vacuu projector: ; >< ;. ( F(; ; We see that the vacuu projector is i fact the iverse of the trucated operator cofluet hypergeoetric fuctio F (; ; because the su stops at, i.e. i fact this is the iverse of a cofluet hypergeoetric operator polyoial of degree, with operatorial variable defied as ( Γ ( ( F(; ;. (! Γ (! ( ( Cosequetly, the projector o the Fock state > becoes ( ; >< ;. F(; ; (! ubstitutig this relatio i Eq. (5 ad usig the DOOT rules, we obtai ( F(; ; F(; ; L F(; ;. F(; ; Now we ca ake a fudaetal observatio: by actig oly o the trucated cofluet hypergeoetric polyoial F (; ;, the orally ordered product operator becoes equivalet to the operator i the square brackets, with respect to the DOOT rules, i.e. F( ; ; L F( ; ;. (5 Reciprocally, it follows that wheever the operator L ( / acts o the cofluet hypergeoetric polyoial F (; ; this ca be substituted by the orally ordered product operator actig o the (; ; F. iilarly, for a iteger power of the ordered product we obtai ( L F(; ;. F(; ; (3 (4 (6

4 4 Barut - Girardello spi coheret states 33 f Cosequetly, for a fuctio of the ordered product operator ( c (, ad usig the DOOT rules, we obtai, after power series expasio, f ( f L F( ; ;. (7 F(; ; Particularly, ay iteger power,,... of the particle uber operator N, ca be writte as N > < (; ; (; ; F (8 F This expressio will be useful i the ext ectio i order to calculate the Madel paraeter. 3. PIN COHERENT TATE Let us cosider a spi operator with the agitude. The spi coheret states (Cs so far have bee defied i the Klauder-Pereloov (KP aer, i.e. by applyig the spi displaceet operator to the groud state ; > [3, 4]. Now, we ca defie the Cs also i the Barut-Girardello (BG aer [5], as eigevalues of the lowerig spi operator : > z >. (9 o, the Barut - Girardello spi coheret states (BG-Cs > are defied o the etire coplex plae of variable z z exp(iϕ, < z, ϕ π, with real paraeter λ. Usig the oralizatio coditio < z ; z ; >, the expasio of BG-Cs i Fock basis is z z ; > ; >, ( F z ρ( (; ; where F (; ; z is the trucated cofluet hypergeoetric polyoial i variable z. The BG-Cs defied above accoplish all coditios required for a coheret state, as stated i [6]: Cotiuity i the coplex label, i.e. if z' z, the z '; > z ; > : li z'; > > li < z'; > < z'; >. ( z' z [ ( ] z' z The BG-Cs fulfill the resolutio of uity operator (or satisfy the copleteess relatio: d μ,( z ; z ; >< z ;, ( where the positive defied itegratio easure is structured as d z d d ; ; ϕ d ;, μ;( z h;( z ( z h;( z π π ad where the positive weight fuctio h ( z ; (3 ; ust be deteried. Here we have used the idexes ad i order to ephasize that the BG-Cs are i fact oe of particular cases of the Barut-Girardello geeralized hypergeoetric coheret states (GH-BG-Cs with the idexes p ad q []. After perforig the agular itegratio we ust solve the followig relatio >< d( z ( z h;( z ;. Γ (! F(; ; z (4 Γ (

5 33 Duša Popov 5 Chagig the expoet s leads to the tieltjes oet proble [7]: ;( The solutio of this proble i.e. h ~ ( z ; h ~ ; z ; h; z ; F (; ; z ad the fuctio ( ( [ ] ( d( z ( z h Γ z ; Γ (. (5 Γ ( J ( z [7] ad fially, the itegratio easure becoes: ϕ ( ( ; is expressed through the Bessel fuctio of the first kid d d μ ; z ; Γ ( d z F( ; ; z J ( z. (6 π z Geerally, the covergece radius R ~ of the radial itegral is deteried by calculatig the liit ~ R li ρ( [8]. Because, if, it follows also that, ad for the BG-Cs the covergece radius is ifiite [9] (Eq o, the BG-Cs are defied o the whole coplex z-plae. 3 The BG-Cs are oralized but ot orthogoal ad this is evidet fro the overlap, i.e. F (; ; z z' < z'; >. (7 F (; ; z F (; ; z' O the other had, usig the DOOT rules, the BG-Cs ca be writte i a operatorial aer as z ; > F (; ; z ; >, (8 F (; ; z ad siilarly for their dual state <. Cosequetly, the BG-Cs projector is * F (; ; z F (; ; z z ; >< z ;. (9 F (; ; z F (; ; For z we obtai the projector o the vacuu state i accordace with the rule d of DOOT. The ea value of a operator A i the BG-Cs represetatio, i.e. < A is < A > z < A > F (; ; z > ' ( z z < '; > ; ρ( '; ρ( A. (3 I the preset paper we are iterested oly i diagoal operators i the Fock vectors basis, with the eigeequatio A ; > A( > so this relatio becoes ( z A( A z F (; ; < A > z ; z (3 F (; ; z ρ( F (; ; z z o, the expectatio value of the product operator A i the BG-Cs represetatio regarded as the eigevalue of the operator A( z / z F (; ;. Particularly, for z A the result is < A > ca be ad with the correspodig eigefuctio < > z ; < L L z F (; ; z > (3 F (; ; z z

6 6 Barut - Girardello spi coheret states 333 Moreover, for a DOOT ordered fuctio like A f ( c ( f ( < > z ; f (; ; L z F z F (; ; z z, we obtai also. (33 By coparig Eqs. (7 ad (33 we ca see that it exists a correspodece betwee the orally ordered product operator ad the square of coplex variable z, i.e. z, respectively F (; ; F (; ; z. This eas that if we have to copute, i the BG-Cs represetatio, the average of soe fuctios that deped oly o the ordered product, it is sufficiet to siply replace this product of operators with z ad perfor the correspodig algebraic operatios. O the other had, the average of the iteger power,,... of the particle uber operator, or the average values of the oets of weightig distributio probability i the BG-Cs represetatio is ( z z F (; ; < N > z (34 F (; ; z ρ( F (; ; z z This result ca be verified by usig the direct forula for average, i.e. Eq. (3. Usig this forula we ca exaie the ature of weightig distributio of BG-Cs by coputig the Madel paraeter [], [], which is used as a coveiet easure of the statistics of coheret states: ( N > < N > < Q z < N >. (35 For BG-Cs this expressio fially ca be writte like i the aer preseted i [8]: Q z ( ( F (; ; x F (; ; x d d x x log log F (; ; x, (36 ( F (; ; x F (; ; x dx dx ( d where x z ad where we have used the otatio F (; ; x F (; ; x. dx The behavior of the BG-Cs, is sub-poissoia (if Q <, Poissoia (if Q or super- z z Poissoia (if Q >, ad this ca be exaied by calculatig the Madel paraeter Q with respect to z z the variable x z. Thus, the statistical properties of the BG-Cs are depedet o the aalytical properties of the expressios ivolvig fuctio F(; ; x ad their derivatives. I other words, because the expectatio value ( < N ( ΔN z ; < N > > < N is less, equal respectively higher tha the variace >, the the statistics are sub-poissoia, Poissoia, or super-poissoia. The probability to occupy the -th Fock state i the BG-Cs z ; > is P ( ( z z ; < z ; >. (37 (; ; ( F z Γ! Γ( It is obvious that for (uphysical liit z : ( P ( z with the shape paraeter ( P ; we recover the stadard Poisso distributio

7 334 Duša Popov 7 z ( P li P ( P ( exp z. (38! Γ ( x a a log x I order to calculate this relatio we have used the liit [9] (Eq : li e. x Γ( x For fiite (physical values of, the probability P ( ca be either arrower or broader tha Poisso distributio ad this situatio depeds o the values chose for ad for z. At the ed, we will say that this costructio of the BG-Cs by eas of DOOT ca be used also i the case of ixed (theral states. We preset here, oly soe cosideratios. As a exaple, we cosider a quatu syste of N tot o-iteractig particles with spi placed i a exteral costat agetic field B i the z-directio, described by the Zeea Hailtoia Hit γ B γ B z, ad eergy eigevalues E γ B (, where γ is the gyroagetic ratio. If the spi syste is i theral cotact with a reservoir (or theral bath, the correspodig state is ixed, described by the caoical equilibriu oralized desity operator ρ exp( β Hit exp( Θ z, (39 Z Z where Z is the partitio fuctio of oe spi, Θ β γ B, ad β / k B T, as usual. We idicate here just that i the Fock-vector basis, by usig the DOOT rules, the desity operator is expressed as ρ Θ Θ Θ F (; ; e ( e >< e Θ e Z Z F (; ;. (4 The exhaustive exaiatio of the ixed (theral states of spi systes, usig the DOOT, will be the subject atter of a forthcoig paper. 4. CONCLUDING REMARK I this paper we have costructed, for the first tie to the best of our kowledge, the coheret states for the spi systes usig the Barut-Girardello aer (BG-Cs, ad have exaied soe of their properties, by applyig the previously deduced diagoal orderig operatio techique (DOOT. We showed that the ewly costructed coheret states i the Barut-Girardello aer for spi systes satisfy all Klauder s iial prescriptios for a coheret state: cotiuity i the coplex label, oralizatio, o orthogoality, ad the uity operator resolutio, with positive weight fuctio of the itegratio easure. Usig the DOOT we have foud a iterestig situatio: by actig oly o the cofluet hypergeoetric polyoial F (; ;, the orally ordered product operator is equivalet to the operator L ( /, with respect to the DOOT rules. At the sae tie, the expectatio value of the product operator i the BG-Cs represetatio < > ca be regarded as the eigevalue of the operator ( z / z L actig o the correspodig eigefuctio F (; ; z. This eas that if we have to copute, i the BG-Cs represetatio, the average of soe fuctios that deped oly o the ordered product, it is sufficiet to siply replace this product of operators with z ad perfor the correspodig algebraic operatios. This way of approachig (i.e. the DOOT is ot surprisig sice

8 8 Barut - Girardello spi coheret states 335 ay kid of coheret states are actually particular cases of ore geeral coheret states the GH-BG-Cs [,, 3]. For pure states, the DOOT calculatios are the sae eve if the syste has fiite or ifiite uber of boud eergy states ad ay be successfully used to avoid ay relatively coplicated algebraic calculatios that appear whe usig the coheret state foralis. REFERENCE. H.-Y. FAN, Operator orderig i quatu optics theory ad the developet of Dirac's sybolic ethod, J. Opt. B: Quatu eiclass. Opt., 5, 4, pp. R47 R53, 3.. D. POPOV, M. POPOV, oe operatorial properties of the geeralized hypergeoetric coheret states, Phys. cr., 9, 3, 35, J. M. RADCLIFFE, oe properties of coheret spi states, J. Phys. A: Ge. Phys., 4, 3, pp , A.M. PERELOMOV, Coheret states for arbitrary Lie group, Cou. Math. Phys., 6, 3, pp. 36, 97; arxiv: athph/3. 5. A.O. BARUT, L. GIRARDELLO, New Coheret tates associated with o-copact groups, Co. Math. Phys.,,, pp. 4 55, J.R. KLAUDER, K.A. PENON, J.M. IXDENIER, Costructig coheret states through solutios of tieltjes ad Hausdorff oet probles, Phys. Rev. A, 64,, 387,. 7. A.M. MATHAI, R.K. AXENA, Geeralized Hypergeoetric Fuctios with Applicatios i tatistics ad Physical cieces, Lect. Notes Math. 348, priger-verlag, Berli, J.P. ANTOINE, J.P. GAZEAU, P. MONCEAU, J.R. KLAUDER, K.A. PENON, Teporally stable coheret states for ifiite well ad Pöschl-Teller potetials, J. Math. Phys., 4, 6, pp ,. 9. I.. GRADHTEYN, I.M. RYHIK, Table of Itegrals, eries ad Products, eveth ed., Acadeic Press, Asterda, 7.. D. POPOV,.H. DONG, N. POP, V. AJFERT,. IMON, Costructio of the Barut Girardello quasi coheret states for the Morse potetial, A. Phys., 339, pp. 34, 3.. D. POPOV, Photo added Barut Girardello coheret states of the pseudoharoic oscillator, J. Phys. A: Math. Ge., 35, pp ,.. T. APPL, D.H. CHILLER, Geeralized hypergeoetric coheret states, J. Phys. A: Math. Ge., 37, 7, pp , D. POPOV,.H. DONG, M. POPOV, Diagoal orderig operatio techique applied to Morse oscillator, A. Phys., 36, pp , 5. Received March 9, 6