View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of the Egyptian Mathematical Society (2011) 19,39–44 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems 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; a¼a: 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 eÀi/J À ei/J ; ð11Þ ajni¼ njn À 1i; ayjni¼ n þ 1jn þ 1i: ð3Þ 2 þ À The states {Œnæ} form a complete set and resolve the unity on the state Œj,Àjæ. 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;Àji¼ cos h tan heÀi/ jj;mi: j þ m 2 m¼j 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 À aÃaÞj0i: 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 À1jaj2 a jai¼ Cnjni; Cn ¼ e 2 pffiffiffiffi : ð6Þ ½K1; K2¼iK3; ½K2; K3¼iK1; ½K3; K1¼iK2: 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; KƼKÆ and ½Kþ; KÀ¼2K3: ð14Þ d2a complete and they satisfy jaihaj p ¼ I. 2 2 2 2 The Casimir operator C ¼ K3 À K1 À K2 has the value C2 = k(k À 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; Jy¼iJz; ½Jy; Jz¼iJx; ½Jz; Jx¼iJy; 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 KÀjk; ni¼ nð2k þ n À 1Þ jk; n À 1i ð Þ K3jk; ni¼ðk þ nÞjk; ni JÆ ¼ Jx Æ iJy: The ground state Œk,0æ satisfies KÀŒk,0æ = 0 while Hence the commutation relations (7) become sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Jz; JƼÆJÆ and ½Jþ; JÀ¼2Jz: 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 JÀ on Œj,mæ are given by operator pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi à DPerðnÞ¼exp ðÞnKþ À n KÀ ; Jþjj; mi¼ ðj À mÞðj þ m þ 1Þ jj; m þ 1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ on the ground state Œk,0æ to get [2] JÀjj; mi¼ ðj þ mÞðj À m þ 1Þ jj; m À 1i: Some quantum optical states as realizations of Lie groups 41 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X1 Then Œw(t)æ = UŒj,Àjæ 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 KÀja; 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 I2kÀ1ð2jaj 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 ðqÀ nÞ! ðq À nÞ! at some states which are realization of the SU(2) and SU(1,1) jn;qi¼N nn jq À n;ni; NÀ2 ¼ jnj2n : ð21Þ q q!n! q q!n! groups.