Coherent states for SU„3… - ePrints@IISc

Coherent states for SU „ 3 …

Manu Mathur^a)

S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Calcutta 700091, India

Diptiman Sen^b)

Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India

We define coherent states for SU共3兲 using six bosonic creation and annihilation operators. These coherent states are explicitly characterized by six complex num- bers with constraints. For the completely symmetric representations (n,0) and (0,m), only three of the bosonic operators are required. For mixed representations (n,m), all six operators are required. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties. We introduce an explicit parametrization of the group SU共3兲 and the corresponding integration measure. Finally, we discuss the path integral formalism for a problem in which the Hamiltonian is a function of SU共3兲operators at each site.

I. INTRODUCTION

Coherent states have been used for a long time in different areas of physics.^1,2In the area of quantum optics, coherent states based on the Heisenberg–Weyl group共which are described later in this work兲have been extensively used to study the interaction of a single mode of electromagnetic radiation with a two-level atomic system共for instance, the Jaynes–Cummings model兲.³Coherent states based on the noncompact Lie group SU共1,1兲have also been used to study certain problems in quantum optics.⁴ In condensed matter physics, coherent states for the Lie group SU共2兲 have been very useful for studying Heisenberg spin systems using the path integral formalism.^{5– 8}These studies have been generalized to systems with SU(N) symmetry, although such studies have usually been restricted to the completely symmetric representations.^6,9However, there is a recent discussion of coherent states for arbitrary irreducible representations of SU共3兲 in Ref. 10. The purpose of our work is to discuss a coherent state formalism which is valid for all representations of SU共3兲, and to give an explicit characterization of them in terms of complex numbers and the states of some harmonic oscillators.共Our work differs in this respect from Ref. 10 which does not use harmonic oscillator operators to define the basis states.兲As we will see, this way of charac- terization is very similar to those used for the Heisenberg–Weyl and SU共2兲coherent states. But, there are certain additional features共such as tracelessness兲which are redundant in the simpler case of SU共2兲.

One can imagine various possible applications of coherent states for SU共3兲. In quantum optics, coherent states for SU共3兲 may turn out to be useful for studying the interaction of a three-level atomic system with three modes of electromagnetic radiation 共corresponding to the three possible energy differences of the atom兲. We should also mention that there have been many other studies of SU共3兲in the recent mathematical physics literature, including the geometric phase for three-level systems¹¹and the study of Clebsch–Gordon coefficients and the outer multiplicity problem.¹²These studies do not use coherent states; however, our work is likely to shed new light on some of these studies. For instance, we will use two triplets of complex numbers z and w which

a兲Electronic mail: manu@boson.bose.res.in

b兲Electronic mail: diptiman@cts.iisc.ernet.in

are similar to the ones used in Ref. 12, except that we will normalize the triplets to unity.

Similarly, it is well known that the geometric phases in the different representations of SU共2兲may be obtained by integrating around a closed loop the overlap of two coherent states which differ infinitesimally from each other.^7,8In the same way, it should be possible to derive the geometric phases for SU共3兲representations from the coherent states discussed below.

The organization of the article is as follows. Section II will motivate our ideas and techniques using two examples which are simpler than the SU共3兲 group. We start with the standard group theoretical definitions of the coherent states of the Heisenberg–Weyl and SU共2兲groups. We then discuss another way of defining SU共2兲coherent states using the Schwinger or Holstein–Primakoff representation of the Lie algebra of SU共2兲¹³in terms of harmonic oscillator creation and annihi- lation operators. This definition is discussed in some detail as it can be extended to the SU共3兲 group. We then establish its equivalence with the standard group theoretical coherent state definition.²In Sec. III, we generalize the SU共2兲Lie algebra in terms of harmonic oscillators to the SU共3兲group, and construct the irreducible representations of SU共3兲. We describe the structure of SU共3兲matrices in an explicit way, and provide an integration measure for this eight-dimensional manifold. In Sec. IV, we use this group structure to construct a set of SU共3兲coherent states which are explicitly characterized by a set of complex numbers which are equivalent to eight real variables. We prove various identities expected for coherent states such as the resolution of identity and a transformation from a particular coherent state to the general coherent state. In Sec.

V, we provide an alternative set of coherent states for SU共3兲which require only five real variables;

although these share some of the features of the coherent states defined in Sec. IV, they have a few limitations arising from the smaller number of variables used. In Sec. VI, we discuss how coherent states can be used to develop a path integral formalism for problems involving SU共3兲variables.

II. HEISENBERG–WEYL AND SU„2…COHERENT STATES

There are many definitions of coherent states used in the literature. However, the most essen- tial ingredients common in all these definitions are the continuity and completeness properties.¹ 共1兲 These are states in a Hilbert spaceHassociated which are characterized by a set of continuous

variables兵^zជ其, and the coherent states兩zជ典 are strongly continuous functions of the labels兵^zជ其^. 共2兲 There exists a positive measure d␮^(zជ) such that the unit operatorIadmits the resolution of

identity

I⫽

冕

^{d␮共zជ兲兩zជ典具zជ兩. 共1兲}

Given a group G, the coherent states in a given representation R are functions of q parameters denoted by兵^z1,z2,...,zq其, and are defined as

兩zជ典^⬅TR共g共zជ_兲兲兩0典^{R. 共2兲}

Here T_R(g(zជ)) is a group element in the representation R, and兩0典R is a fixed vector belonging to R. In the simplest example of the Heisenberg–Weyl group, the Lie algebra contains three genera- tors. It is defined in terms of creation annihilation operators (a,a^†) satisfying

关a,a^†兴⫽I, 关a,I兴⫽0, 关a^†,I兴⫽0. 共3兲

This algebra has only one infinite dimensional irreducible representation which can be character- ized by occupation number states 兩n典⬅(a^†)ⁿ/

冑

^n!兩0典 ^{with n}⫽0,1,2, . . . . A generic group ele- ment in 共2兲 can be characterized by T(g)⫽exp(i␣I⫹za^†⫺¯a) with an anglez ␣ and a complex parameter z. Therefore,

兩␣^,z典⬁⫽exp共i␣兲兩z典^, 兩z典⫽exp共za^†⫺¯az 兲兩0典⫽_n

兺

_⫽^⬁₀ ^{Fn共z兲兩n典, 共4兲}

where the sum runs over all the basis vectors of the infinite dimensional representation, and

F_n共z兲⫽ zⁿ

冑

^{n!exp共⫺兩z兩2/2兲 共5兲}

are the coherent state expansion coefficients. This feature, i.e., an expansion of the coherent states in terms of basis vectors of a given representation with analytic functions of complex variables (F_n(z)) as coefficients, will also be present in the case of SU共2兲and SU共3兲groups. It is easy to see that Eq.共4兲provides a resolution of identity as in 共1兲with the measure d␮^(z)⫽dzdz¯.

We now briefly review the next simplest example, i.e., the coherent states associated with the SU共2兲group. The SU共2兲Lie algebra is given by a set of three angular momentum operators 兵^Jជ其

⬅兵^J1,J₂,J₃其 or equivalently by兵^J⫹,J_⫺,J₃其^{, (J}⫾⬅J₁⫾iJ₂) satisfying

关J₃,J_⫾兴⫽⫾J_⫾, 关J_⫹,J_⫺兴⫽2J₃. 共6兲 The SU共2兲 group has a Casimir operator given by Jជ_•Jជ, and the different irreducible representa- tions are characterized by its eigenvalues j ( j⫹1), where j is an integer or half-odd-integer. A given basis vector in representation j is labeled by the eigenvalue m of J₃ as兩j,m典. We charac- terize the SU共2兲 group elements U by the Euler angles, i.e., U(␪^,␾^,␺⁾

⬅exp (i␾^J3) exp(i␪^J2) exp(i␺^J3). The standard group theoretical definition共2兲takes兩0典^{j in共2兲to} be the highest weight state兩j , j典 and is of the form

兩nˆ共␪^,␾兲典j⫽U共␪^,␾^,␺兲兩j , j典^,

⫽_m

兺

_⫽⫺^⫹j _j^{Cm共␪,␾兲兩j,m典, 共7兲}

In 共7兲, the coefficients C_m(␪^,␾) are given by

C_m共␪^,␾兲⫽e^im␾

冑

共j⫹m^共兲^{2 j}!共^兲j^!⫺m兲!

冋

^sin␪2

册

^j⫺m

冋

^cos␪2

册

^{j⫹m, 共8兲}

where we have ignored possible phase factors.

The algebra in Eq.共6兲can be realized in terms of a doublet of harmonic oscillator creation and annihilation operators aជ_⬅(a₁,a₂) and aជ^†_⬅(a₁^†,a₂^†), respectively.¹³ They satisfy the simpler bosonic commutation relation关a_i,a^†_j兴⫽␦i j with i, j⫽1,2. The vacuum state is兩0,0典. In terms of these operators,

J^a⬅¹2a_i^†共␴^a兲i ja_j, 共9兲 where␴^a denote the Pauli matrices.共We will generally use the convention that repeated indices are summed over兲. It is easy to check that the operators in共9兲satisfy the SU共2兲Lie algebra with the Casimir Jជ_•Jជ_⬅¹₄aជ^†_•aជ(aជ^†_•aជ_⫹2). Thus the representations of SU共2兲can be characterized by the eigenvalues of the occupation number operator; the spin value j is equal to (N₁⫹N₂)/2 where N₁ and N₂ are the eigenvalues of a₁^†a₁ and a₂^†a₂, respectively.

With these harmonic oscillator creation and annihilation operators, another definition of SU共2兲 coherent states is obtained by directly generalizing共4兲. We define a doublet of complex numbers (z₁,z₂) with the constraint 兩z₁兩²⫹兩z₂兩²⫽1; this gives three independent real parameters which define the sphere S³. Let us parameterize

z₁⫽cos␹^{ei␤1 and z}2⫽sin␹^ei␤2, 共10兲 where 0⭐␹⭐␲^{/2 and 0}⭐␤1,␤2⬍2␲. The integration measure on this space takes the form

d⍀S³⫽ 1

␲^{2dz1dz¯1dz2dz¯2}␦共兩z₁兩²⫹兩z₂兩²⫺1兲⫽ 1

2␲^2cos␹^sin␹^d␹^d␤1d␤2, 共11兲 where we have introduced a normalization factor so that 兰d⍀S³⫽1. The SU共2兲coherent state in the representation N is now defined as

兩z₁,z₂典^N⫽2 j⫽␦aជ^†•aជ,N

冑

^N!exp共zជ_•aជ^†_兲兩0,0典^⫽_N

兺

1,N₂ ⬘^FN₁^,N₂^兩N1,N2典^{j. 共12兲} In the second equation in共12兲, the 兺⬘ implies that only the terms satisfying the constraint a^†•a

⫽N⬅2 j are included or equivalently that

N₁⫹N₂⫽N. 共13兲

With 共13兲, the states兩N₁,N₂典j form a (2 j⫹1)-dimensional representation of SU共2兲. The expan- sion coefficients F_N

1,N₂(z₁,z₂) are analytic functions of (z₁,z₂) and are given by F_N

1,N₂⬅

冉

^N1N!!N2!

冊

^{1/2z1N1z2N2. 共14兲}

Equations 共12兲and 共14兲are similar to 共4兲and 共5兲, respectively. This will be generalized to the SU共3兲case in Sec. III. It is easy to check that 共12兲 provides the resolution of identity with the measure given in共11兲, namely,

冕

^{d⍀SU(2)兩z1,z2典j j具z1,z2兩⫽2 j1}⫹1 _m

兺

_⫽⫺^j _j ^{兩j,m典具j ,m兩. 共15兲}

Now we change variables from N₁ and N₂⫽2 j⫺N₁ to m⫽¹2(N₁⫺N₂), and define

␻⬅z₁

z₂⫽e^i␾cot␪

2. 共16兲

These parameters are related to the ones given in 共10兲 as ␪⫽2␹ ^and ␾⫽␤1⫺␤2. We now consider an unit sphere S²with its south pole touching the point␻⫽0. The sphere is characterized by (␪^,␾^{) where}␪ ^and␾are the polar and azimuthal angles, respectively. Using the stereographic projection, it is easy to verify that

兩z₁,z₂典j⫽共z₁兲^{2 j}m

兺

⫽⫺j

j

冑

共j⫹m^共兲^{2 j}!共^兲j^!⫺m兲!共␻兲^(m⫺j)兩j ,m典⫽兩nˆ共␪^,␾兲典j, 共17兲

where we have again ignored possible phase factors. Equation共17兲can also be written as 兩z₁,z₂典j⫽共z₁兲^{2 j}exp

冉

^zz21J⫺

冊

^{兩z1⫽1, z2⫽0典j, 共18兲}

where兩z₁⫽1, z₂⫽0典N⫽2 j⫽兩j , j典 and we have used the fact that J_⫺⫽a₂^†a₁. Equations 共17兲and 共18兲establish the equivalence between the group theoretical theoretical definition共7兲and the one using Schwinger bosons共12兲.

The stationary subgroup of a particular coherent state is defined as the subgroup H of the full group G which leaves that coherent state invariant up to a phase; the coherent states are functions

of the coset space G/H.² It is clear from the previous discussion that the stationary subgroup of the SU共2兲 coherent states is U共1兲; therefore the coherent states correspond to the coset space SU(2)/U(1)⫽S² which is parametrized by the angles (␪^,␾^).

III. SU„3…AND ITS REPRESENTATIONS

Let us first discuss a parametrization of SU共3兲matrices, i.e., 3⫻3 unitary matrices with unit determinant. To motivate this, let us first consider a parametrization of SO共3兲matrices. Consider a real vector of unit length of the form

pជ_⫽

冉

^{sinsincos␪␪cossin␪ ␾␾}

冊

^{. 共19兲}

The most general real vector q of unit length which is orthogonal to p is given by

qជ_⫽

冉

^{coscos␹␹coscos⫺␪␪coscossin␹␾␾⫺sin⫹sinsin␪ ␹␹cossin␾␾}

冊

^{. 共20兲}

Finally, we define a third unit vector rជ_⫽pជ_⫻qជ, i.e., r₁⫽p₂q₃⫺p₃q₂, etc. Then a 3⫻3 matrix whose columns are given by the vectors p,q and r is an SO共3兲matrix.

We will now generalize the previous construction to obtain an SU共3兲matrix. A complex vector of unit norm is given by

zជ_⫽

冉

^{sinsincos␪␪cossin␪e␾␾i␣ee3ii␣␣21}

冊

^{, 共21兲}

where 0⭐␪^,␾⭐␲^{/2 and 0}⭐␣1,␣2,␣3⬍2␲. Then the integration measure for zជ, which is equivalent to the sphere S⁵, is given by

d⍀S⁵⫽ 2

␲^{3dz1dz¯1dz2dz¯2dz3dz¯3}␦共兩z₁兩²⫹兩z₂兩²⫹兩z₃兩²⫺1兲

⫽ 1

␲^3sin

3␪^cos␪^cos␾^sin␾^d␪^d␾^d␣1d␣2d␣3, 共22兲

which has been normalized to make兰d⍀S⁵⫽1. The most general complex vector wជ of unit norm satisfying zជ_•wជ_⫽0 is given by

wជ_⫽

冉

^{eei(i(␤␤11⫺␣⫺␣12))coscos␹␹⫺coscosei(␤␪␪1⫺␣cossin3)␾␾cos⫺⫹e␹ei(i(sin␤␤22⫺␣⫺␣␪ 21))sinsin␹␹cossin␾␾}

冊

^{, 共23兲}

where 0⭐␹⭐␲^{/2 and 0}⭐␤1,␤2⬍2␲ just as in the integration measure for S³ in共11兲. We may now define a third complex vector of unit norm asvជ_⫽¯zជ_⫻wជ, where z¯ជ_⬅zជ^쐓. Then we can check that a 3⫻3 matrix whose columns are given by z, w¯ and v, i.e.,

S⫽

冉

^{zzz123 ww¯¯w¯123 ¯¯¯zzz231www312⫺⫺⫺¯¯¯zzz312www231}

冊

^共24兲

is an SU共3兲matrix.

The integration measure for the group SU共3兲is given by a product of共22兲and共11兲as^14,15 d⍀SU(3)⫽ 1

2␲^5sin

3␪^cos␪^cos␾^sin␾^cos␹^sin␹^d␪^d␾^d␹^d␣1d␣2d␣3d␤1d␤2, 共25兲 which is normalized so that兰d⍀SU(3)⫽1. To prove Eq.共25兲, we note that the matrix in共24兲can be written as a product of two SU共3兲matrices, i.e., S⫽A₃A₂, where

A₃⫽

冉

^{sinsincos␪␪cossin␪e␾␾i␣ee3ii␣␣21 coscos⫺␪␪sincossin␪␾e␾ie␣ei3i␣␣21 ⫺cossin␾␾ee⫺0⫺i␣i␣12⫺⫺i␣i␣23}

冊

^{, 共26兲}

and

A₂⫽

冉

^{100 ⫺sin␹ecos⫺i␤␹20e⫹⫺i␣i1␤⫹1i␣2⫹1␣3 sin␹eicos␤2⫺␹0i␣e1i⫺␤i1␣2⫺i␣3}

冊

^{. 共27兲}

The structure of the matrix A₃ is determined entirely by the three-dimensional complex vector which forms its first column; hence the integration measure corresponding to it is given by共22兲. The matrix A₂ is determined by the two-dimensional complex vector which forms its second column; its contribution to the integration measure is therefore given by共11兲. Note that although the parameter appearing in A₂ is␤2⫺␣1⫺␣2⫺␣2 instead of only ␤2 as in共10兲, this makes no difference in the product measure given in共25兲since the differentials d␣i already appear in the integration measure coming from A₃. Incidentally, this procedure generalizes to any SU(N); the integration measure is given by a product of measures for S^2N⫺1, S^2N⫺3, . . . , S³.¹⁴

In short, we have defined two complex vectors zជ_⫽(z₁,z₂,z₃) and wជ_⫽(w₁,w₂,w₃) in共21兲 and共23兲. These satisfy the constraints

¯zជ_•zជ_⫽兩z₁兩²⫹兩z₂兩²⫹兩z₃兩²⫽1,

共28兲 w¯ជ_•wជ_⫽兩w₁兩²⫹兩w₂兩²⫹兩w₃兩²⫽1,

and

zជ_•wជ_⫽z₁w₁⫹z₂w₂⫹z₃w₃⫽0. 共29兲 These constraints leave eight real degrees of freedom as required for SU共3兲. We will take zជ and wជ to transform respectively as the 3 and 3^쐓 representation of SU共3兲. Thus an SU共3兲transformation acts on the matrix S in Eq.共24兲by multiplication from the left.

Let us now define two triplets of harmonic oscillator creation and annihilation operators (a_i,b_i), i⫽1,2,3, satisfying

关a_i,a_j^†兴⫽␦i j, 关b_i,b_j^†兴⫽␦i j,

共30兲 关a_i,b_j兴⫽0, 关a_i,b^†_j兴⫽0.

We will often denote these two triplets by (aជ,bជ) and the two number operators by N_a(⬅aជ^†_•aជ) and N_b(⬅bជ^†_•bជ). Similarly, their vacuum state is denoted by兩0ជ_a,0ជ_b典. Henceforth, we will ignore the subscripts a,b and will denote the vacuum state by兩0ជ,0ជ典, and the eigenvalues of N_a, N_bby N and M, respectively.

Now let␭^a, a⫽1,2, . . . ,8 be the generators of SU共3兲in the fundamental representation; they satisfy the SU共3兲Lie algebra关␭^a,␭^b兴⫽i f^abc␭^c. Let us define the following operators

Q^a⫽a^†␭^aa⫺b^†␭*^ab, 共31兲 where a^†␭^aa⬅a_i^†␭i j

aa_j, and b^†␭*^ab⬅b_i^†␭i j*^ab_j. To be explicit, Q³⫽¹2共a₁^†a₁⫺a₂^†a₂⫺b₁^†b₁⫹b₂^†b₂兲,

Q⁸⫽ 1

2

冑

^{3共a1†a1⫹a2†a2⫺2a3†a3⫺b1†b1⫺b2†b2⫹2b3†b3兲,}

Q¹⫽¹2共a₁^†a₂⫹a₂^†a₁⫺b₁^†b₂⫺b₂^†b₁兲,

Q²⫽⫺i

2共a₁^†a₂⫺a₂^†a₁⫹b₁^†b₂⫺b₂^†b₁兲,

共32兲 Q⁴⫽¹2共a₁^†a₃⫹a₃^†a₁⫺b₁^†b₃⫺b₃^†b₁兲,

Q⁵⫽⫺i

2共a₁^†a₃⫺a₃^†a₁⫹b₁^†b₃⫺b₃^†b₁兲, Q⁶⫽¹2共a₂^†a₃⫹a₃^†a₂⫺b₂^†b₃⫺b₃^†b₂兲,

Q⁷⫽⫺i

2共a₂^†a₃⫺a₃^†a₂⫹b₂^†b₃⫺b₃^†b₂兲.

It can be checked that these operators satisfy the SU共3兲 algebra among themselves, i.e., 关Q^a,Q^b兴⫽i f^abcQ^c. Further,

关Q^a,a_i^†兴⫽␭ji

aa^†_j, 关Q^a,b_i^†兴⫽⫺␭*ji^ab_j^†,

关Q^a,a^†•a兴⫽0, 关Q^a,b^†•b兴⫽0, 共33兲 关Q^a,a^†•b^†兴⫽0, 关Q^a,a•b兴⫽0.

From Eqs.共33兲, it is clear that the three states a_i^†兩0ជ,0ជ典 ^{with (N⫽1, M⫽0) and b}i

†兩0ជ,0ជ典^with (N⫽0, M⫽1) transform respectively as the fundamental representation 共3兲 and its conjugate representation (3^쐓). By taking the direct product of N aជ^†’s and M bជ^†’s we can now form higher representations. We now define an operator

O_j

1j₂. . . j_M i₁i₂. . . i_N

⬅a_i

1

†a_i

2

† . . . a_i

N

†b_j

1

†b_j

2

† . . . b_j

M

† . 共34兲

Under SU共3兲 transformation the states defined as 兩␺^˜典(N, M )⬅O_j

1j₂. . . j_M i₁i₂. . . i_N

兩0ជ,0ជ典 will all have N_a

⫽N and N_b⫽M , and will transform among themselves. Further, 兩␺^˜典⫽N兩␺^˜典 ^{and N}b兩␺^˜典

⫽M兩␺^˜典. However, these do not form an irreducible representation because aជ_•bជ and aជ^†_•bជ^† are

SU共3兲 invariant operators 关see 共33兲兴. A general basis vector in the irreducible representation (N, M ) is obtained by subtracting the traces and completely symmetrizing in upper and lower indices.¹⁶ More explicitly, a state in (N, M ) representation is given by

兩␺典j₁, j₂, . . . , j_M i₁,i₂, . . . i_N

⬅

冋

^{Oji11ij22. . . i. . . jNM⫹L1l}

兺

^{1⫽N1 k}

兺

^{1M⫽1 ␦jilk11Oij11ij22..i.. jlk11⫺⫺11ilj1k⫹1⫹11..i. . . jN M}

⫹L_2l

兺

1,l₂⫽1

N _k

兺

1,k₂⫽1 M

␦_j

k₁ i_l

1␦_j

k₂ i_l

2O

j₁j₂.. j_k

1⫺1j_k

1⫹1.. j_k

2⫺1j_k

2⫹1. . . j_M i₁i₂..i_l

1⫺1i_l

1⫹1..i_l

2⫺1i_l

2⫹1..i_N

⫹L_3l

兺

1,l₂,l₃⫽1

N _k

兺

1,k₂,k₃⫽1 M

␦_j

k₁ i_l

1␦_j

k₂ i_l

2␦_j

k₃ i_l

3O

j₁j₂.. j_k

1⫺1j_k

1⫹1.. j_k

2⫺1j_k

2⫹1.. j_k

3⫺1j_k

3⫹1. . . j_M i₁i₂..i_l

1⫺1i_l

1⫹1..i_l

2⫺1i_l

2⫹1..i_l

3⫺1i_l

3⫹1..i_N

⫹. . .⫹L_Ql

兺

1,l₂,l₃,..,l_Q⫽1

N _k

兺

1,k₂,k₃,..,k_Q⫽1 M

␦_j

k₁ i_l

1␦_j

k₂ i_l

2..␦_j

k_Q i_l

Q

⫻O

j₁j₂.. j_k

1⫺1j_k

1⫹1.. j_k

2⫺1j_k

2⫹1.. j_k

Q⫺1j_k

Q⫹1. . . j_M i₁i₂..i_l

1⫺1i_l

1⫹1..i_l

2⫺1i_l

2⫹1..i_l

Q⫺1i_l

Q⫹1..i_N

册

^{兩0ជ,0ជ典, 共35兲}

where Q⫽Min(N, M ),

L_q⬅ 共⫺1兲^q共a^†•b^†兲^q

q!共N⫹M⫹1兲共N⫹M兲共N⫹M⫺1兲•••共N⫹M⫹2⫺q兲, 共36兲 and all the sums in共35兲are over different indices, i.e., l₁⫽l₂•••⫽l_q and k₁⫽k₂⫽•••⫽k_q. The coefficients in Eq.共36兲are chosen to satisfy the tracelessness condition

i_l, j

兺

_k⫽1 3

␦j_k i_l

兩␺典j₁, j₂, . . . , j_M i₁,i₂, . . . i_N

⫽0, for all l⫽1,2, . . . N, and k⫽1,2, . . . M . 共37兲

For future purposes, a more compact notation for describing all the states given above is to write O_j

1j₂. . . j_M i₁i₂. . . i_N

⬅共a₁^†兲^N1共a₂^†兲^N2共a₃^†兲^N3共b₁^†兲^M1共b₂^†兲^M2共b₃^†兲^M3, 共38兲 where (N_i, M_i) denote all the possible eigenvalues of the occupation number operators (a_i^†a_i,b_i^†b_i) satisfying

N₁⫹N₂⫹N₃⫽N and M₁⫹M₂⫹M₃⫽M . 共39兲 The action of共38兲on the vacuum is given by

O_M

1M₂M₃ N₁N₂N₃

兩0ជ,0ជ典^⫽共N₁!N₂!N₃! M₁! M₂! M₃!兲^1/2兩_M

1M₂M₃ N₁N₂N₃

典. 共40兲

We can now write the basis vectors of the representation (N, M ) as

兩␺典j₁, j₂, . . . , j_M i₁,i₂, . . . i_N

⬅兩␺典M₁M₂M₃ N₁N₂N₃

⫽

冋

^{OMN11NM22NM33⫹q}

兺

^{⫽Q1 Lq[}

兺

^␣ជ]q

册

^{N1C␣1N2C␣2N3C␣M31C␣1M2C␣2}

⫻^M3C_␣

3␣1!␣2!␣3!O_M

1⫺␣₁M₂⫺␣₂M₃⫺␣₃

N₁⫺␣₁N₂⫺␣₂N₃⫺␣₃ 兩0ជ,0ជ典^. 共41兲

In this equation, 关␣ជ_兴q denotes the sets of three non-negative integers (␣1,␣2,␣3) satisfying␣1

⫹␣2⫹␣3⫽q, and N_i⫺␣i⭓0, M_i⫺␣i⭓0 for i⫽1,2,3. The 兺[␣ជ]_q denotes a summation over all sets of three such integers. In the notation of Eq. 共41兲, the tracelessness condition共37兲 for the (N⫹1, M⫹1) representation takes the form

[

兺

␥ជ]₁

兩␺典M₁⫹␥₁M₂⫹␥₂M₃⫹␥₃

N₁⫹␥1N₂⫹␥2N₃⫹␥3 ⫽0. 共42兲

The definition in 共41兲 satisfies the condition given in 共42兲. This can be verified by using the identity

[

兺

␥ជ]_{1 [␣}

兺

_ជ]

q

␣1!␣2!␣3!^N1⫹␥1C_␣

1

N₂⫹␥2C_␣

2

N₃⫹␥3C_␣

3

M₁⫹␥1C_␣

1

M₂⫹␥2C_␣

2

M₃⫹␥3C_␣

3

⫻O_M

1⫹␥₁⫺␣₁M₂⫹␥₂⫺␣₂M₃⫹␥₃⫺␣₃ N₁⫹␥1⫺␣1N₂⫹␥2⫺␣2N₃⫹␥3⫺␣3

⫽

冋

^{共N⫹M⫹2⫺q兲[␣ជ}

兺

^{]q⫺1 ⫹共aជ†•bជ†兲[}

兺

^␣ជ]q

册

⫻␣1!␣2!␣3!^N1C_␣

1 N₂C_␣

2 N₃C_␣

3 M₁C_␣

1 M₂C_␣

2 M₃C_␣

3O_M

1⫺␣₁M₂⫺␣₂M₃⫺␣₃

N₁⫺␣1N₂⫺␣2N₃⫺␣3 . 共43兲

The dimension D(N, M ) of the representation (N, M ) can be obtained as follows. For the (N,0) representation, no tracelessness condition needs to be imposed, and the dimension is simply given by the number of states in Eq. 共40兲 which satisfy 兺iN_i⫽N and 兺iM_i⫽0. This gives D(N,0)⫽(N⫹1)(N⫹2)/2. Similarly, D(0,M )⫽( M⫹1)( M⫹2)/2. Now D(N, M ) is given by the number of states satisfying 兺iN_i⫽N, 兺iM_i⫽M , which is equal to the product D(N,0)D(0,M ), minus the number of states satisfying兺iN_i⫽N⫺1,兺iM_i⫽M⫺1, which is equal to D(N⫺1,0)D(0,M⫺1); the subtraction is because of the tracelessness condition. This gives

D共N, M兲⫽¹2共N⫹1兲共M⫹1兲共N⫹M⫹2兲. 共44兲 IV. SU„3…COHERENT STATES

We now observe that the states in Eq.共35兲 can be extracted from the following generating function,

兩zជ,wជ典(N, M )⬅

冑

^{N! M ! exp}共zជ_•aជ^†_⫹wជ_•bជ^†_兲兩0ជ,0ជ典^, 共45兲 where we have to project onto the subspace of states with aជ^†_•aជ_⫽N and bជ^†_•bជ_⫽M to obtain the representation (N, M ). More explicitly,

兩zជ,wជ典(N, M )⫽共zជ_•aជ^†_兲^N

冑

^{N! 共}

wជ_•bជ^†_兲^M

冑

^{M ! 兩0ជ,0ជ典⫽}_N

兺

^⬘

1,N₂,N_{3 M}

兺

^⬘

1, M₂, M₃

F_Nជ, Mជ共z₁,z₂,z₃;w₁,w₂,w₃兲兩_M

1M₂M₃ N₁N₂N₃

典^. 共46兲 In 共46兲, 兺⬘ implies that the occupation numbers (N_i, M_i) satisfy Eq. 共39兲, and F_Nជ, Mជ(zជ,wជ) are given by

F_Nជ, Mជ共zជ,wជ_兲⫽

冉

^{N1!N2!NN! M !3! M1! M2! M3!}

冊

^{1/2z1N1z2N2z3N3w1M1w2M2w3M3. 共47兲}

On expanding the right hand side of共46兲, the coefficients of z₁^N1z₂^N2z₃^N3w₁