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provided by Elsevier - Publisher Connector Journal of the Egyptian Mathematical Society (2011) 19,39–44
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REVIEW PAPER Some quantum optical states as realizations of Lie groups
Abdel-Shafy Fahmy Obada
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
Available online 9 November 2011
KEYWORDS Abstract We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the Lie algebras; SU(2) and SU(1,1) algebras and their coherent states. From there we go on describing the binomial Quantum states state and its extensions as realizations of the SU(2) group. This is followed by considering the neg- ative binomial states, and some squeezed states as realizations of the SU(1,1) group. Generation schemes based on physical systems are mentioned for some of these states. ª 2011 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. Open access under CC BY-NC-ND license.
1. Introduction by some preliminaries about the annihilation and creation operators and the number operators which constitute the cor- With the advances in the field of quantum optics which began ner stones of the Hesenbeg-Weyl algebra, then their eigenstates with the 60s, group theory started to infiltrate in this branch. and their coherent states are defined. The familiar algebras of Groups involving simple Lie algebras such as SU(2) and the SU(2) and SU(1,1) are introduced. Then some quantum SU(1,1) and their simple generalizations have been used to states which are realizations of the SU(2) are reviewed in Sec- study different aspects in quantum optics. However, the use tion 3. Section 4 is devoted to states as realizations of SU(1,1) of the theory of groups in quantum mechanics started with group. Some comments are given about the generations of the early days of that theory. Weyl’s book that was first pub- some of these states through physical processes. lished in German in 1928 [1] is a standing witness on this. Wider dimensions in various branches of physics benefited 2. Preliminaries greatly from the use of the group theory. Some states used in the field of quantum optics as realiza- 2.1. The harmonic oscillator tions of the SU(2) or SU(1,1) groups are reviewed. We start In the study of the harmonic oscillator, the following operators E-mail address: [email protected] are introduced: the annihilation operator a^ the creation oper- y y 1110-256X ª 2011 Egyptian Mathematical Society. Production and ator a^ and the number operator n^ ¼ a^ a^. They satisfy the com- hosting by Elsevier B.V. Open access under CC BY-NC-ND license. mutation relations
y y y Peer review under responsibility of Egyptian Mathematical Society. ½a; a ¼I; ½n; a ¼a ; ½n; a ¼ a: ð1Þ doi:10.1016/j.joems.2011.09.013 The eigen-states Œnæ of the number operator n^ are called Fock states or number states. They satisfy Production and hosting by Elsevier n^jni¼njni: ð2Þ 40 Abdel-Shafy Fahmy Obada
The non-negative integer n can be looked upon as the num- The operators Ja are the generators of the group SU(2). The ber of particles in the state. When n = 0 we call Œ0æ the vacuum angular momentum coherent state is defined by the action of state with no particles present. the rotation operator The operations of a and a on n are given by Œ æ 1 ÀÁ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Rbðh; /Þ¼exp h e i/J ei/J ; ð11Þ ajni¼ njn 1i; ayjni¼ n þ 1jn þ 1i: ð3Þ 2 þ The states {Œnæ} form a complete set and resolve the unity on the state Œj, jæ. X The angular momentum coherent state Œh,/æ is given by n n I: 4 j ih j¼ ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2j Xj jþm n 1 2j 1 jh;/i¼Rbðh;/Þjj; ji¼ cos h tan he i/ jj;mi: j þ m 2 m¼ j 2 2.1.1. Coherent states ð12Þ The coherent state Œaæ can be looked upon as an eigenstate of the operator a such that They resolve the identity operator on the space with total angular momentum j as follows Z Z ajai¼ajai: ð5Þ 2j þ 1 2p p sin h dhd/jh; /ihh; /j¼I: ð13Þ Also, it can be produced by applying the Glauber displace- 4p 0 0 ment operator which is a unitary operator on the vacuum state Œ0æ [2,3]. 2.3. The SU(1,1) group jai¼DðaÞj0i¼expðaay a aÞj0i: The notion of coherent states can be extended to any set of This is the coherent state of the Heisenberg–Weyl group operators obeying a Lie algebra. The SU(1,1) is the simplest [4,5]. non-abelian noncompact Lie group with a simple Lie algebra This state, which is a superposition of infinite series of the (For a comprehensive review we may refer to [6] and the recent Fock states with their distribution being Poissonian. It is given review book [7]). by its expansion in the number state as The SU(1,1) algebra is spanned by the three operators K1, X1 n K2, K3 which satisfy the commutation meatiness 1jaj2 a jai¼ Cnjni; Cn ¼ e 2 pffiffiffiffi : ð6Þ ½K1; K2 ¼ iK3; ½K2; K3 ¼iK1; ½K3; K1 ¼iK2: n¼0 n! This state describes to a great deal the laser field where the By using the operators K± = K1 ± iK2, hence phase is fixed while the numberR is not. The sates {Œaæ} are over- ½K3; K ¼ K and ½Kþ; K ¼ 2K3: ð14Þ d2a complete and they satisfy jaihaj p ¼ I. 2 2 2 2 The Casimir operator C ¼ K3 K1 K2 has the value C2 = k(k 1)I for any irreducible representation. Thus, repre- 2.2. The angular momentum (SU(2) group) sentation is determined by the parameter k which is called the Bargmann number. The corresponding Hilbert space is The angular momentum defined as r^ p^ as well as the spin, are spanned by the complete orthonormal basis {Œk,næ} which 2 described by the three operators Jx, Jy, and Jz which satisfy the are the eigenstates of C and K3, such that commutation relations (we take ⁄ =1) X1
½Jx; Jy ¼iJz; ½Jy; Jz ¼iJx; ½Jz; Jx ¼iJy; ð7Þ hk; njk; mi¼dnm and jk; nihk; nj¼I: n¼0 with The operations of the operators K± and K3 on Œk,næ are gi- 2 2 2 2 J ¼ Jx þ Jy þ Jz ; ven by [5] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K jk; ni¼ ðn þ 1Þð2k þ nÞ jk; n þ 1i which commutes with each component. Raising and lowering þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 15 operators are introduced through the relations K jk; ni¼ nð2k þ n 1Þ jk; n 1i ð Þ
K3jk; ni¼ðk þ nÞjk; ni J ¼ Jx iJy:
The ground state Œk,0æ satisfies K Œk,0æ = 0 while Hence the commutation relations (7) become sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½Jz; J ¼ J and ½Jþ; J ¼2Jz: ð8Þ Cð2kÞ m jk; mi¼ Kþjk; 0i: 2 n!Cð2k þ mÞ The simultaneous eignestates of the operators Jz and J de- noted by Œj,mæ are given from [5,6] There are two sets of coherent states related to the SU(1,1)
2 group namely: J jj; mi¼jðj þ 1Þjj; mi and Jzjj; mi¼mjj; mi; ð9Þ with ŒmŒ 6 j, j half integers. (i) The Perelomov coherent states.By applying the unitary The operations of J+ and J on Œj,mæ are given by operator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DPerðnÞ¼exp ðÞnKþ n K ; Jþjj; mi¼ ðj mÞðj þ m þ 1Þ jj; m þ 1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ on the ground state Œk,0æ to get [2] J jj; mi¼ ðj þ mÞðj m þ 1Þ jj; m 1i: Some quantum optical states as realizations of Lie groups 41 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X1 Then Œw(t)æ = UŒj, jæ is the coherent state (12) with 2 k Cð2k þ nÞ n ja;ki ¼ D ðnÞjk;0i¼ð1 jaj Þ a jk;ni; ð16Þ p Per Per n!C 2k h =2Bt and / ¼ . n¼0 ð Þ 2 ih ih with n = ŒnŒe , a = (tan hŒnŒ)e .(95) 3.2. Finite dimensional pair coherent state (ii) The Barut-Girardello coherent states It is defined as the eigenstate [7] It may be termed as the two-mode binomial stateŒn,qæ. It can qþ1 y q y n ðab Þ K ja; kiBG ¼ aja; kiBG; be defined as the eigen state of the operators a b þ ðq!Þ2 which can be expressed as and (a a + b b) where a,b are annihilation operators for the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X1 two modes. The states satisfy the eigen value equations [9]. a2kþ1 an ! ja; ki ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jk; ni: ð17Þ qþ1 y q BG 2 n ðab Þ I2k 1ð2jaj Þ n¼0 n!Cðn þ 2kÞ aybþ jn;qi¼njn;qi; ðaya þ bybÞjn;qi¼qjn;qi; ð20Þ ðq!Þ2 Im(x) is the modified Bessel function of the 1st kind. and takes the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi After this very quick review of these preliminaries we look Xq Xq ðq nÞ! ðq nÞ! at some states which are realization of the SU(2) and SU(1,1) jn;qi¼N nn jq n;ni; N 2 ¼ jnj2n : ð21Þ q q!n! q q!n! groups. n¼0 n¼0
This is a type of the entangled states where we find (q n) 3. SU(2) realizations particles in 1st mode and (n) particles in the 2nd mode. When we define We look at some states which can be looked upon as realiza- ayb þ aby ayb aby aya byb tions of SU(2) group. J ¼ ; J ¼ ; J ¼ ; ð22Þ x 2 y 2i z 2 which are the generators of the SU(2) group. Hence the 3.1. The single mode Binomial state raising and lowering operators are J+ = Jx + iJy = a b, J = Jx iJy = ab . These states are of the form [8] Thus is the state (21) are the coherent states of SU(2) when sffiffiffiffiffiffiffiffiffiffiffiffiffiffi we label q =2j and identify the states {Œj,m jæ} as the states XM M 2 M n jM gi¼ gnð1 jgj Þ 2 jni; ð18Þ {Œq n,næ}. This state can be generated as demonstrated in n¼0 n Ref. [9,10]. M 2 Zþ, g 2 C, Œg Œ2 6 1. 3.3. Nonlinear two-mode binomial state They have the photon-number distribution (probability of finding n photons) as An extension to the earlier state is performed by introducing M the nonlinear finite dimensional pair coherent state as the eigen PðnÞ¼ jgj2nð1 jgj2ÞM n; state satisfying n 2 q3 a 1 1 by which is the binomial distribution. 4 y qþ1 f1ðnaÞ f2ðnbÞ 5 f1ðnaÞa bf2ðnbÞþn jn;qif ¼ njn;qif; ð23Þ They are the eigen states of the operator [7] ðq!Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p B ¼ gaya þ 1 jgj MI ayaa; and with the eigen-value gM, i.e. ðna þ nbÞjn; qi¼qjn; qi; BjM; gi¼gMjM; gi: ð19Þ with the usual notation. It is expanded in the Fock states for the two modes as [10] We may note the vacuum, Fock and coherent state as lim- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xq iting cases of this state [7,8]. ðq nÞ! f ðq nÞ! jn; qi ¼ N nn 1 jq n; ni; ð24Þ As it is mentioned in Section 2.2 when we looked at the f q n!q! f q !f n ! n¼0 1ð Þ 2ð Þ SU(2) representations, the angular momentum operators J sat- isfy the relations (7); and the SU(2) coherent states which is de- with f(n)! = f(0) Æ f(1) ...f(n) and f(0) = 1. fined as the action of the rotation operator (11) on the ground In order to relate these states to SU(2) group realization, we state. Hence Eq. (12) is the SU(2) coherent state. Thus when introduce the operators h i/ f ðn Þayf ðn Þbþaf 1ðn Þbyf 1ðn Þ we take g ¼ sin 2 e and take n = j + m and M =2j, the 1 a n b 1 a 2 b h i/ Jx ¼ 2 ; binomial state j2j; sin 2 e i is the coherent state of the SU(2) ð25Þ f ðn Þayf ðn Þb af 1ðn Þbyf 1ðn Þ group. 1 a n b 1 a 2 b n^a n^b Jy ¼ 2i ; Jz ¼ 2 ; This state can be generated through the following scheme. An atom under a classical magnetic field B has the interaction which satisfy the relations [Jx,Jy]=iJz,[Jy,Jz]=iJx,[Jz,Jx]= iJ . Note that neither J nor J is hermitian, hence we define Hamiltonian H = J Æ B with the field along the direction x. y x y
Under this Hamiltonian the state Œj, jæ evolves to the binomial y 1 y 1 Jþ ¼ f1ðnaÞa f2ðnbÞb; J ¼ a b ; state. The evolution operator U(t) is given by f1ðnaÞ f2ðnbÞ
U ¼ exp itH ¼ exp itBðJþ þ J Þ: consequently [Jz,J±]=±J±,[J+,J ]=2Jz. 42 Abdel-Shafy Fahmy Obada
These operators can be thought of as generators of an ex- The state (28) is obtained by applying D(g) of Section 4.1 tended SU(2) group. When we take 2j = q and the states but with the operators given by (29), on the vacuum state Œ0æ {Œj,n jæ} to correspond to {Œq n,næ}, then we get an ex- which is the SU(1,1) realization for the nonlinear negative tended SU(2) coherent state similar to Eq. (24). Some proper- binomial state (28). ties of these states may be found in Ref. [10] and also a The case of the non-unitary f is also considered in Ref. [12]. generation scheme. 4.3. Single mode squeezed vacuum and 1st excited states 4. SU(1,1) Realizations The squeezed vacuum state is defined as the eigenstate of the oper- There are a large number of states that can be termed as real- ator b = la + ma with eigenvalue zero and Œl Œ2 Œm Œ2 =1 izations of the SU(1,1) group reviewed in Section 2.3. Here we [13,14]. It has the expansion pffiffiffiffiffiffiffi mention some of these states. X1 2n! n 2 1 jni ¼ n ð1 jnj Þ4j2ni; ð30Þ 0 2nn! 4.1. The negative binomial states n¼0 where n = tanh rei/, l = cosh r, m = sinh rei/. This state is defined as the Fock state expansion [11] While the squeezed 1st excited state is obtained as the eigen rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X1 state of the operator b with egien value 0. It has the form ðn þ MÞ! 2 Mþ1 jM; ni ¼ nnð1 jnj Þ 2 jni: ð26Þ N n!M! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n¼0 X1 ð2n þ 1Þ! 2 3 jni ¼ nnð1 jnj Þ4j2n þ 1i: ð31Þ This state follows the negative binomial distribution for the 1 2nn! n¼0 photon number distribution This can be cast as a realization of the SU(1,1) group by ðM þ nÞ! PðnÞ¼ jnj2nð1 jnj2ÞMþ1: taking M!n! ay2 a2 1 1 The special case of M = 0 is the Pascal distribution or the thermal K ¼ ; K ¼ ; K ¼ aya þ : ð32Þ þ 2 2 3 2 2 distribution. The state (26) interpolates between the pure thermal 2 2 state and the coherent state (n fi 0, M fi 1, MŒnŒ fiŒaŒ ), The Casamir operator C2 in this case hence it is termed as an intermediate state. 3 The SU(1,1) realization can be achieved by introducing the C ¼ kðk 1ÞI ¼ I: 2 16 operators The state space associated with k ¼ 1 is the even Fock sub- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 y M 3 Kþ ¼ a MI þ n^; K ¼ MI þ n^ a; Kz ¼ I þ n^; ð27Þ space with {Œ2næ} and that associated with k ¼ 4 is the odd 2 Fock subspace with {Œ2n +1æ}. The unitary operator (the which are the raising, lowering and generators of the SU(1,1) squeeze operator) is: group. * 1 y2 1 2 Thus the unitary evolution operator D(g) = exp(gK+ g SðzÞ¼expðzK z K Þ¼exp za z a : þ 2 2 K ) can be applied on the vacuum state to have the state Mþ1 The SU(1,1) coherent states are the single-mode squeezed 2 2 DðgÞj0i¼expðnKþÞ½1 jnj expð nK Þj0i 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sates. For k ¼ 4 we have squeezed vacuum X1 2 Mþ1 n ðM þ nÞ! 2 1 z ¼½1 jnj n jni¼jM; niN; M!n! n; ¼ SðzÞj0i¼jni0 of ð30Þ n ¼ tanh jzj; n¼0 4 jzj g 3 where n ¼ jgj tanh j g j. for k ¼ we have the squeezed one photon state 4
3 4.2. The non-linear negative binomial state n; ¼ SðnÞj1i¼jni of ð31Þ: 4 1 The nonlinear extension to the above state has been introduced [12]. It amounts to deform the operator a to A = af(n) where 4.4. Nonlinear squeezed states f(n) is an operator valued function. Hence the state is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X1 The use of the following operators Mþ1 ðM þ nÞ! n 2 2 y jM:niNf ¼ Nf n ð1 jnj Þ ðf ðnÞ!Þjni: ð28Þ M!n! 1 y y 2 1 2 n¼0 K ¼ ðf ðnÞa Þ ; K ¼ ðafðnÞÞ : ð33Þ þ 2 2 The commutation relation For the unitary operator function f (n)=f 1(n), we have ½A; Ay ¼½afðnÞ; fyðnÞay ¼ðn þ 1Þjfðn þ 1Þj2 njfðnÞj2: 1 y 1 1 K ¼ a a þ : It becomes [A,A ] = 1 for f (n)=f (n) i.e. unitary operator. 3 2 2 For the operator f(n) being unitary, the following SU(1,1) generators are defined Under these operators, we have the nonlinear squeezing operator pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y M 1 y2 y 2 Kþ ¼ a f ðnÞ MI þ n; Kþ ¼ MI þ n fðnÞa; Kz ¼ Iþ n: ð29Þ S ðzÞ¼exp ðzA z A Þ; where A ¼ afðnÞ: 2 f 2 Some quantum optical states as realizations of Lie groups 43
Consequently ja; n; mi¼SðnÞDðaÞjmi¼Dða0ÞSðnÞjmi; pffiffiffiffiffiffiffiffiffiffi X1 * i/ * i/ 1 2 1 ð2nÞ! with a0 = acoshr + a sinhre = la + ma , n = re . jn; i ¼ð1 jnj Þ4 ðfð2nÞ!Þnnj2ni 4 f 2nn! Its expansion in the Fock state space is given there [18]. n¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X1 Here S(n) is the unitary operator that could be expressed in 3 2 3 ð2n þ 1Þ! jn; i ¼ð1 jnj Þ4 ðfð2n þ 1Þ!Þnnj2n þ1i: ð34Þ terms of the generators of the SU(1,1) group as defined in the 4 f 2nn! n¼1 Eq. (32) . The case of the non-unitary f has been considered and dis- cussed in [15]. 4.8. Two-mode squeezed vacuum states
4.5. Single mode squeezed coherent state These states are obtained by applying the non-degenerate two- mode operator These states are the solutions of eigenvalue problem [16] y y bjbi¼la þ mayjbi¼bjbi; S2ðnÞ¼expð na b þ n abÞ; i/ i/ with l = cosh r, m = sinh re . on the vacuum state ––01,02æ with n = re . These states are ex- m i/ If we write n ¼ l ¼ e tanh r, then the state pressed in terms of the two-mode Fock states in the form [6,7] X jbi¼jb; ni¼ja; ri 1 i/ n jni2 ¼ S2ðnÞj01; 02i¼ ðtanh re Þ jn; ni: ð36Þ 2 1 cosh r ¼ð1 jnj Þ4 m These states are considered as a class of entangled states where X1 1 1 jnj ð nÞ2 exp jbj2 ðb2ei/ þ b 2e i/Þ p2ffiffiffiffiffi the numbers of quanta in both modes are equal in each 2 2 m! component. 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 m¼0 2 Such state can be considered as realization of the SU(1,1) as @b 1 jnj A coherent state of this group. This is accomplished by defining the Hm pffiffiffiffiffiffiffiffiffi jmi; ð35Þ 2n generators as follows
* 1 with b = la + ma . K ¼ ab; K ¼ ayby; K ¼ ðn þ n þ 1Þ: ð37Þ þ 3 2 a b When we use the representation (32) for the operators K±, K3, the state (35) can be cast as the operation of the operator The Casimir operator 1 y2 2 SðnÞ¼expðnKþ n K Þ¼exp ðna n a Þ; 2 2 1 1 2 C2 ¼ K3 ðKþK þ K KþÞ¼ ½ðna nbÞ 1 : on the state Œbæ of the form (6). Therefore we have 2 4 qþ1 jn; bi¼SðnÞjbi¼SðnÞDðbÞj0i: Therefore the irreducible representation with k ¼ 2 , q = 0,1,2,... is the eigenvalue of (n n ) hence the state Œm,kæ After using the disentanglement of the squeezing operator and a b ) Œn + q,næ with n =0,1,2,... applying it to the state ––bæ we get the expression (35). We may sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi use the relation X1 1 þ q 2 qþ1 ðn þ qÞ! n n; k ¼ ¼ ð1 jnj Þ 2 n jn þ q; ni: ð38Þ 2 n!q! SðzÞDðbÞ¼DðaÞSðzÞ with a ¼ lb þ mb ; n¼0 Hence we get which reduces to (36) when q =0. jn; bi¼SðnÞDðbÞj0i¼DðaÞSðnÞj0i There is another coherent state of SU(1,1) (Barut-Girardil- lo) which is the eigenfunction of the operator K namely: i.e. we displace the squeezed vacuum, or squeeze the coherent state [6]. K ja; ki ¼ aja; ki BG sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBG 2kþ1 X1 jaj an 4.6. Nonlinear squeezed coherent state ) ja; ki ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jn; ki: BG I 2 a 2k 1ð j jÞ n¼0 n!Cðn þ 2kÞ The nonlinear operator A is defined as A = af(n) and A = f (n)a where the operator valued function f(n) is a unitary opera- 4.9. Nonlinear two mode squeezed vacuum state tor i.e. f = f 1. In this case we find [A,A ]=I. The operators K ,K of the SU(1,1) are defined as is (33). ± 3 As before we have A = af (n ), B = bf (n ) and their hermi- the nonlinear realization in this case is given in [17]. With the 1 a 2 b tian conjugates, consequently appearance of the function f(m) denotes the effect of the nonlinearity. ½A; Ay ¼ðn þ 1Þf ðn þ 1Þfyðn þ 1Þ n fy ðn Þf ðn Þ; ð39Þ Also, the case for non-unitary nonlinear function has been a 1 a a a 1 a 1 a discussed there. with a similar formula for [B,B ]. y 1 ) y For the unitary case fi ¼ fi , fifi ¼ I, the nonlinear squeez- 4.7. Squeezed displaced Fock states ing operator will be a unitary operator and we have
These states are defined as application of the squeezed opera- y y y jni2f ¼ S2fðnÞj0; 0i¼expð nA B þ n ABÞj0; 0i: ð40Þ tors and the displacement operators on the Fock state Œmæ in the following form [18]. The non-unitary case has been discussed also in [17]. 44 Abdel-Shafy Fahmy Obada
4.10. SU (1,1) Intelligent states gle mode binomial states, the finite dimensional pair coherent states, and their nonlinear variants. Then came the negative For the two self-adjoint operator A, B, one obtains the uncer- binomial states, single mode squeezed vacuum, squeezed tainty relation coherent, squeezed displaced Fock states and their nonlinear 1 variants. The two mode squeezed vacuum states and their non- hðDAÞ2ihðDBÞ2i P jh½A; B ij2: linear counterparts are discussed. Finally the intelligent states 4 are mentioned. An extended version of this article with appear A state is called an intelligent state (IS) if it satisfies the elsewhere. strict equality. Such states must satisfy the eigen value equation 6. A tribute ðA ikBÞjwi¼gjwi; ð41Þ k is a positive real parameter, g a complex number. When This is a small tribute to the late Prof. Gamal M. Abd AlKader [A,B]=cI with constant c, the minimum uncertainty states [1963–2009] with whom I have had a very fruitful and most (MUS) coincide with the IS. interesting collaboration for almost two decades. Whose For the SU(1,1) the IS ––wæ are solutions of the eigenvalue friendship and amicable personality, I as well as many of his problem colleagues and students really miss. May ALLAH accept him in His Mercy. ðK1 ikK2Þjwi¼gjwi; or References ða1K þ b1KþÞjwi¼gjwi; a1 ¼ 1 þ k; b1 ¼ 1 k: ð42Þ In the basis Œn,kæ of the SU(1,1) of Eq. (15) then Œwæ is given [1] H. Weyl, The Theory of Groups and Quantum Mechanics, by Dover, New York, 1950. [2] A.M. Perelomov, Generalized Coherent States and Their X1 Applications, Springer Verlag, Berlin, 1986. jwi¼ cnjn; ki: [3] P.J. Kral, J. Mod. Opt. 37 (1990) 889; n¼0 P.J. Kral, Phys. Rev. A 42 (1990) 4177. The coefficients cn can be calculated to be related to the [4] A. Wunsche, J. Opt. B Quant. Semiclassical 2 (2000) 73. Pollaczek polynomial [19,20] [5] W.H. Louisell, Quantum Statistical Properties of Radiation, ! John Wiley, New York, 1973. n b1 pffiffiffiffiffiffiffiffiffig [6] G.J. Milburn, D.F. Walls, Quantum Optics, Springer, Berlin, cn ¼ Pn ; k 1991. a1 a1b 1 ![7] V.V. Dodonov, V.I. Man’ko, Theory of nonclassical states of nn 2 n k 2 i light, Taylor & Francis, London, 2003. n b1 Cð þ 2 Þ pffiffiffiffiffiffiffiffiffig ¼ i 2F1 m; k þ ; 2k; 2 : [8] D. Stoler, B.E.H. Saleh, M.C. Teich, Opt. Acta 32 (1985) 345. a1 n!Cð2kÞ a b 1 1 [9] A.-S.F. Obada, E.M. Khalil, Opt. Comm. 260 (2006) 19. ð43Þ [10] E.M. Khalil, J. Phys A 39 (2006) 11053. [11] A. Joshi, S.V. Lawande, Opt. Comm. 70 (1989) 21. Some special cases are discussed in [20]. For example: [12] M.S. Abdalla, A.S.F. Obada, M. Darwish, Opt. Comm. 274 (2007) 372. (i) The one mode realization include as special cases: the [13] C.C. Gerry, Phys. Rev. A 35 (1987) 2146; Barut-Girardillo state, the Perelomov C.S., and the non- G.S. Agarwal, J. Opt. Soc. Am. B 5 (1988) 1540. linear squeezed coherent states. [14] M.M. Nieto, D.R. Truax, Phys. Rev. Lett. 71 (1993) 2843. (ii) The two-mode realizations include as special cases; the [15] A.-S. F. Obada, M. Darwish, J. Opt. B 7 (2005) 57; pair coherent state as the correlated SU(1,1) CS, and A.-S. F. Obada, G.M. Abd-Al-Kader, Eur. Phys. J. D 44 (2007) nonlinear realizations. 189. [16] H.P. Yuen, Phys. Rev. A 13 (1976) 2226; B. Schnmaker, Phys. Rep. 135 (1986) 317. 5. Conclusion [17] G.M. Abd-Al-Kader, A.-S. F. Obada, Turk J. Phys. 32 (2008) 115. [18] M.V. Satyanarayana, Phys. Rev. D 32 (1985) 400; In this article we have tried to review some quantum states and P. Kral, Phys. Rev. A 42 (1990) 4177. their relations to some algebraic groups. Some of the Lie alge- [19] C. Aragone, G. Guerri, S. Salumo, J.L. Tani, J. Phys. A 7 (1974) bras and some of the relations of their operators and represen- L 149. tations especially some of their coherent states are mentioned. [20] G.M. Abd-Al-Kader, A.-S.F. Obada, Phys. Scip. 78 (2008) As realizations of these groups the discussion included the sin- 0345401.