aa r X i v : . [ qu a n t - ph ] F e b Classification of spin and multipolar squeezing
Emi Yukawa and Kae Nemoto
National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430,JapanE-mail: [email protected]
May 2015
Abstract.
We investigate various types of squeezing in a collective su(2 J + 1)system consisting of spin- J particles ( J > / J + 1) system can be classified into unitary equivalence classes, eachof which is characterized by a set of squeezed and anti-squeezed observables formingan su(2) subalgebra in the su(2 J + 1) algebra. The dimensionality of the unitaryequivalence class is fundamentally related to its squeezing limit. We also demonstratethe classification of the squeezing among the spin and multipolar observables in acollective su(4) system.PACS numbers: 03.65.Fd, 05.30.Ch, 42.50.Lc Keywords : spin squeezing, collective spin systems, su( N ) algebra Submitted to:
J. Phys. A: Math. Gen.
1. Introduction
Many quantum information protocols involve nonclassical states to achieve theirquantum advantages. For instance, quantum high precision measurements achievesensitivities beyond the standard quantum limit by utilising nonclassical states. Thestandard quantum limit is given by a coherent state, which satisfies the minimumuncertainty relation where quantum fluctuations are equally shared by any twoquadrature amplitudes. One way to break this limit is to squeeze a coherent state [1]. Asqueezed state exhibits quantum fluctuations below the standard quantum limit in onequadrature at the sacrifice of larger quantum fluctuations in the other, which is directlyapplicable to achieve high precision measurements. To apply squeezed states to highprecision measurements, it is important that squeezing can be achieved relatively easily.Fortunately, squeezing can be achieved via quadratic Hamiltonian, and hence it does notrequire higher-order optical nonlinearity such as Kerr effect [2, 3]. Both squeezing and lassification of spin and multipolar squeezing / J systems, the collective spin canbe treated as an su(2 J + 1) system, where its dimension is determined by the numberof the ensemble. This extension is relevant to current experiments of squeezing on spinensembles; for instance, squeezing in a spin-7 / J + 1) systems and to systematically classify them basedon unitary equivalent classes.Among the collective su(2 J + 1) systems, the su(2) collective system is simpleenough so that squeezing can be understood in comparison with optical squeezing. Therepresentation space based on the SU(2) coherent states is a sphere, i.e. the Blochsphere. Squeezing can be tracked on this two-dimensional space. Though it is compact,as the dimensionality of the Bloch sphere is the same as that of the phase space basedon the optical coherent states, there are similarities between the SU(2) squeezed statesand the optical squeezed states. When we extend the former to an ensemble of spin- J systems ( J > / J + 1) system has (2 J + 1) − J = 1, there are the eightindependent observables, which can be represented by three spin-vector componentsand five quadrupolar-tensor components, and squeezing can be implemented in terms ofthe su(2) subalgebra among these eight observables. Then, squeezing can be classifiedinto two classes with the different squeezing limits.In this paper, we generalize the classification to collective su(2 J + 1) systems, wherethe squeezing can be characterized by (2 J + 1) − lassification of spin and multipolar squeezing J +1) systems to show that the squeezing can be classified into the unitaryequivalence classes of (2 J + 1)-dimensional representations of the su(2) subalgebras. InSec. III, we derive quantum fluctuations for squeezed states of collective su(2 J + 1)systems with one-axis twisting and show their squeezing limits. In Sec. IV we apply ourclassification to a collective su(4) system to illustrate the unitary equivalence classes ofthe squeezing and their squeezing limits, and summarize the main results in Sec. V.Throughout this paper, a scalar, a vector, and a matrix are respectively represented bya normal letter, a bold letter, and a normal letter with a tilde, as in A , A , and ˜ A . Theoperator is denoted by a letter with a caret as in ˆ A .
2. Classification of squeezing in collective su(2J+1) systems
Let us identify the linearly independent observables whose quantum fluctuations can becontrolled via squeezing. Suppose there is a collective su(2 J + 1) system consisting of N spin- J particles. The particles can be fermions as well as bosons when the spatial degreesof freedom of each fermion are frozen and the spin degrees of freedom are separable fromthe spatial degrees of freedom as in ultracold fermions trapped in an optical lattice [13]or magnetic impurities in a crystal [14]. We consider a squeezed spin state (SSS) whichis generated from a coherent spin state (CSS) via a nonlinear interaction such as theone-axis twisting or the two-axis counter twisting [6].In a CSS, all particles are in the same single-spin state [6]. A single-spin statecan be expanded in terms of the rank- d multipoles ( d ∈ N ) and it can be describedby the spherical harmonics of degree d . In the case of a spin-1 / J particle, the 2 d + 1 components of the rank- d multipoles (1 ≤ d ≤ J ) are linearly independent of each other, while the multipoles ofthe rank higher than 2 J can be expressed in terms of the lower-rank multipoles and theidentity. Thus, the spin and multipoles up to the rank of 2 J , which are comprised of(2 J + 1) − J ( J + 1) observables in total, completely characterize a single spin- J state; hence they can be chosen as the generators of the su(2 J + 1) algebra. We definethe second-quantized forms of the single spin and multipolar observables asˆ λ n j ; J,k = J +1 X m,n =1 (˜ λ J,k ) mn ˆ c † n j ; J,m ˆ c n j ; J,n , (1)where (˜ λ J,k ) mn represents the mn -entry of the k -th spin or multipolar matrix ˜ λ J,k ofa single spin- J particle, and ˆ c n j ; J,m (ˆ c † n j ; J,m ) denotes the spin- J bosonic or fermionicannihilation (creation) operator of the spatial mode n j and the magnetic sublevel m z = J + 1 − m . Here, we define ˆ λ n j ; J,k in Eq. (1) so that the first three observables lassification of spin and multipolar squeezing λ J,k are normalized so that their trace norms satisfy || ˜ λ J,k || = J X m z = − J m z = 13 J ( J + 1)(2 J + 1) . (2)A CSS can be completely described by the collective observables of the singlespin and multipolar observables given in Eq. (1). Squeezing can redistribute quantumfluctuations in these collective observables. The second-quantized forms of the collectiveobservables ˆΛ J,k can be expressed asˆΛ
J,k = N X j =1 ˆ λ n j ; J,k . (3)The observables ˆΛ J,k in Eq. (3) satisfy the same commutation relations as ˜ λ J,k in Eq. (1).This implies that they also generate the su(2 J + 1) algebra and the matrices { ˜ λ J,k } can be regarded as the irreducible representation of { ˆΛ J,k } in the basis of {| J, m z i} ,which represents the basis of the single-spin magnetic sublevels with respect to thequantization axis along the z axis. Thus, a collective observable ˆ O J of the collectivesu(2 J + 1) system can be expressed by a (2 J + 1)-dimensional matrix representation ˜ O J in the representation space of V ( {| J, m z i} ) as follows:˜ O J = J ( J +1) X k =1 v J,k ˜ λ J,k , (4)where the real coefficients v J,k satisfy P J ( J +1) k =1 v J,k = 1.
We consider squeezing among three observables { ˆ O J,k } ( k = 1 , ,
3) of the collectivesu(2 J + 1) system, which form an su(2) subalgebra of the su(2 J + 1) algebra and satisfythe commutation relations given by[ ˆ O J, , ˆ O J, ± ] = ± f ˆ O J, , (5)where ˆ O J, ± ≡ ˆ O J, ± i ˆ O J, and f > f in Eq. (5) is not always f = 1, since ± f are equivalent to the structurefactors of the su(2 J + 1) algebra.The squeezing among an su(2) subalgebra { ˆ O J,k } can be classified based on theunitary equivalence class. The unitary equivalence class of the squeezing among { ˆ O J,k } can be determined by the (2 J + 1)-dimensional matrix representation of { ˆ O J,k } in thespace of V ( {| J, m z i} ) spanned by the basis {| J, m z i} . The unitary equivalence class isdefined as follows: Suppose { ˜ X k } and { ˜ X ′ k } are the n -dimensional matrix representationsof the semi-simple Lie algebra. Then, the representations { ˜ X k } and { ˜ X ′ k } belong to the lassification of spin and multipolar squeezing n ) transformation matrix ˜ U suchthat ˜ U ˜ X k ˜ U † = ˜ X ′ k for ∀ k .In our case, { ˜ O J,k } is the (2 J + 1)-dimensional matrix representation of thegenerators of the su(2) algebra, which is semi-simple; hence { ˜ O J,k } should becompletely reducible. The matrix representation { ˜ O J,k } and its representation space V ( {| J, m z i} ) can be decomposed into the direct sum of the lower dimensional irreduciblerepresentations of the su(2) generators and their representation spaces, respectively.Suppose the dimension of the l -th irreducible representation is 2 J l +1. Then, there existsan orthonormal basis set {| J l , m l i l } ( m l = − J l , · · · , J l ) such that the l -th irreduciblerepresentation of the su(2) algebra is given by the spin matrices { ˜ λ J l , , ˜ λ J l , , ˜ λ J l , } for aspin- J l particle (c.f. Eq. (1)). The state | J l , m l i l can be expressed as a linear combinationof | J, m z i ( m z = − J, · · · , J ), and {| J l , m l i l } and {| J l ′ , m l ′ i l ′ } ( l = l ′ ) are orthogonal toeach other. Then, the completely reducible representation of { ˜ O J,k } can be expressedas ˜ O J,k = f r M l =1 ˜ λ J l ,k , V ( {| J, m z i} ) = r M l =1 V ( {| J l , m l i l } ) . (6)In Eq. (6), r expresses the number of the irreducible representations and the “subspins” J l satisfy P rl =1 (2 J l + 1) = 2 J + 1. The structure constant f of { ˜ O J,k } defined in Eq. (5)is given by f = s J ( J + 1)(2 J + 1) P rl =1 J l ( J l + 1)(2 J l + 1) , (7)which can be derived from the irreducibility of { ˜ λ J l ,k } and the normalization conditionin Eq. (2). Here, we note that { ˜ λ J l ,k } and V ( {| J l , m l i l } ) are arranged so that J l satisfies0 ≤ J r ≤ · · · ≤ J ≤ J ≤ J, (8)and we define { ˜ λ J l =0 ,k } = { , , } .If two sets of the generators of the su(2) subalgebras, { ˆ O J,k } and { ˆ O ′ J,k } , belong tothe same unitary equivalence class, { ˆ O ′ J,k } and the representation space V ( {| J, m z i} )can be decomposed into˜ O ′ J,k = f ′ r ′ M l =1 ˜ λ J ′ l ,k , V ( {| J, m z i} ) = r ′ M l =1 V ( {| J ′ l , m ′ l i l } ) , (9)where f ′ = [ J ( J + 1)(2 J + 1) / P r ′ l =1 J ′ l ( J ′ l + 1)(2 J ′ l + 1)] / , m ′ l = − J ′ l , · · · , J ′ l , and r = r ′ ∧ ∀ l, J l = J ′ l . (10)Equation (10) implies that the structure constants f and f ′ are equal. If two sets ofthe generators of the su(2) subalgebras, { ˜ O J,k } and { ˜ O ′ J,k } , do not belong to the sameunitary equivalence class, Eq. (10) does not hold, since a unitary matrix transforms thebasis but it cannot change r and J l . The unitary equivalence classes of the su(2 J + 1)algebra can be systematically found via the Dynkin diagram of the su(2 J + 1) algebraas explained in 2.3. lassification of spin and multipolar squeezing J J-1 (a)
J-2 -J+2 -J+1 -J α α α α α J J-1 J-2 α α α l α l+1 α l+2 J-l+1 J-l J-l-1 J-l-2 l+1 dimension α l’+1 α l’+2 J-l’ J-l’-1 J-l’-2 α l’ J-l’+1 dimension (b)-(i)(b)-(ii) Figure 1. (Color Online) (a) Dynikin diagram of the su(2 J + 1) algebra. Thesimple root α k expresses the transition from m z = J − k to m z = J − k + 1.(b) Correspondences between the connected and disconnected simple roots and thelower dimensional irreducible representations of the su(2) generators. The filled circlesand the gray open circles represent the simple roots that are chosen and not chosen,respectively. (i) If the chosen simple roots from α to α l are connected, then they aresubstituted by the ( l + 1)-dimensional irreducible representation in the representationspace of V ( {| J, m z i} ) ( m z = J, · · · , J − l ). (ii) If a magnetic sublevel J − l ′ is isolatedfrom the connected simple roots, then it is substituted by the one-dimensional element,i.e., 0. The decomposition of the generators { ˜ O J,k } of the su(2) subalgebra in Eq. (6) can bederived from the Dynkin diagram of the su(2 J + 1) algebra. In the Dynkin diagram ofthe su(2 J + 1) algebra, the 2 J simple roots are connected as shown in Fig. 1 (a). Here,the k -th vertex represents the k -th simple root α k that corresponds to the raising matrix˜ A J,k from the k -th sublevel to the ( k +1)-th sublevel with respect to the quantization axisdetermined by the Cartan subalgebra. For the generators of the Cartan subalgebra, wechoose the z component of the spin vector ˜ λ J, and the other 2 J − z axis, which implies that ˜ A J,k raises thesublevel from m z = J − k to m z = J − k + 1 as follows:( ˜ A J,k ) mn ≡ r J ( J + 1)(2 J + 1) δ J − k +1 ,m δ J − k,n . (11)The matrix products of ˜ A J,k and their linear combinations reproduce the spin andmultipolar observables ˜ λ J,k .We can construct a complete irreducible representation by choosing 1 ≤ n ≤ J vertices from the 2 J vertices and substituting l -connected roots of α k , α k +1 , · · · , α k + l − ( l = 1 , · · · , J ) by the ( l + 1)-dimensional irreducible representation { ˜ λ J l ,k } ( k = 1 , ,
3) of the su(2) generators in the representation space of V ( {| J, m z i} )( m z = J − k + 1 , J − k, · · · , J − k − l + 1) as shown in Fig. 1 (b)-(i). If the magneticsublevel of m z is not involved by the connected simple roots, then it is substituted bythe one-dimensional element of 0. This procedure is equivalent to the decomposition of lassification of spin and multipolar squeezing V ( {| J, m z i} ) into the subspaces in Eq. (6) Arranging the irreducible representations sothat their dimensions satisfy Eq. (8), we can obtain the decomposition in Eq. (6). Sincethe Dynkin diagram does not depend on the choice of the basis, any (2 J +1)-dimensionalmatrix representation can be obtained by rotating one of the representations derivedfrom the Dynkin diagram via an SU(2 J + 1) unitary matrix.
3. Properties of squeezing determined by unitary equivalence classes
The properties of the squeezing reflect the structure of the unitary equivalence class, i.e.,the subspins and the initial coherent state. To confirm this, let us consider squeezingamong an su(2) subalgebra { ˆ O J,k } , which can be decomposed into Eq. (6) with thesubspins { J l } . A CSS [20, 21, 22, 23] can be expressed in terms of two parameters θ ∈ [0 , π ] and φ ∈ [0 , π ) as | θ, φ i tot ≡ " r M l =1 ζ l | θ, φ i l ⊗ N = N X n =0 N − n X n =0 · · · N − n −···− n r − X n r − =0 p N C n N − n C n · · · N − n −···− n r − C n r − × ζ n ζ n · · · ζ n r − r − ζ N − n −···− n r − r × h | θ, φ i ⊗ n ⊕ | θ, φ i ⊗ n ⊕ · · · ⊕ | θ, φ i ⊗ n r − r − ⊕ | θ, φ i ⊗ N − n −···− n r − r i , (12)where P rl =1 | ζ l | = 1 and the single particle states | θ, φ i l in Eq. (12) for J l = 0 and J l = 0 are defined in terms of the basis {| J l , m l i l } as ∀ J l = 0 , | θ, φ i l ≡ exp (cid:20) − θ e − iφ ˜ λ J l , + − e iφ ˜ λ J l , − ) (cid:21) | J l , J l i l , (13)with ˜ λ J l , ± ≡ ˜ λ J l , ± i ˜ λ J l , , and | θ, φ i l ≡ | J l = 0 , m l = 0 i l ( J l = 0), respectively. The CSS | θ, φ i tot in Eq. (12) satisfies the minimum uncertainty relation ∀ ν ∈ [0 , π ) , h (∆ O J,ν ) ih (∆ O J,ν + π ) i = f h ˆ O J, ⊥ i , (14)where h ˆ X i represents the expectation value of an observable ˆ X , the quantum fluctuationin ˆ X is defined as h (∆ X ) i = h ˆ X i − h ˆ X i , and ˆ O J,ν and ˆ O J, ⊥ are given byˆ O J, ⊥ ≡ ˆ O J, cos φ sin θ + ˆ O J, sin φ sin θ + ˆ O J, cos θ, (15)ˆ O J,ν ≡ ˆ O J, (cos φ cos θ cos ν − sin φ sin ν )+ ˆ O J, (sin φ cos θ cos ν + cos φ sin ν ) − ˆ O J, sin θ cos ν, (16)respectively. The expectation values in Eq. (14) can be obtained via the Schwinger-boson approach described in Appendix A as h ˆ O J, ⊥ i = f N X l,J l =0 J l | ζ l | , (17) lassification of spin and multipolar squeezing ∀ ν ∈ [0 , π ) , h (∆ O J,ν ) i = f N X l,J l =0 J l | ζ l | . (18)Equations (14), (17) and (18) imply that the squeezing among { ˆ O J,k } can suppress h (∆ O J,ν ) i below the coherent-spin-state value of f |h ˆ O J, ⊥ i| at the expense of h (∆ O J,ν + π ) i enhanced above f |h ˆ O J, ⊥ i| ; hence, the squeezing can be characterized bythe squeezing parameter ξ defined as ξ = N X l,J l =0 J l | ζ l | ! × min ν h (∆ O J,ν ) ih ˆ O J, ⊥ i , (19)where min ν h (∆ O J,ν ) i is the quantum fluctuations in Eq. (16) perpendicular to the O J, ⊥ plane and minimized with respect to the angle ν in Eq. (16). Equation (19) is equalto 1 for the CSS in Eq. (12) and it implies that a state giving ξ < {| ζ l | } of the initial CSS in Eq. (12) are given by | ζ l | = δ l,l with l suchthat J l = 0. The squeezing parameter ξ in Eq. (19) is characterized by the subspinsand the initial CSS, both of which reflect the structure of the unitary equivalence classof the spin and multipolar observables { ˆ O J,k } generating the su(2) subalgebra. Let us calculate the squeezing parameter ξ in Eq. (19) for an SSS generated via theone-axis twisting interaction [6]. We consider the one-axis twisting interactionˆ H OAT = ~ χ ˆ O J, (20)with the interaction energy χ , which distribute the quantum fluctuations in the O J, - O J, plane. A CSS of the N spin- J particles is given by | θ = π , φ = 0 i tot in Eq. (12).Defining the rescaled evolution time µ ≡ χf t , we can express the one-axis-twistedSSS | Ψ OAT ( J, N ; µ ) i tot at µ as | Ψ OAT ( J, N ; µ ) i tot = exp (cid:20) − i f ˆ O J, µ (cid:21) | θ = π , φ = 0 i tot . (21)In this case, the observable ˆ O J, ⊥ is given by ˆ O J, and its expectation value at time µ can be obtained in a manner similar to Eqs. (17) and (18) as h ˆ O J, i ( µ ) ≡ h Ψ OAT ( J, N ; µ ) | ˆ O J, | Ψ OAT ( J, N ; µ ) i tot = f N X l : J l =0 J l | ζ l | cos J l − µ h − | ζ l | (cid:16) − cos J l µ (cid:17)i N − , (22)as detailed in Appendix A. The quantum fluctuations in the plane perpendicular to ˆ O J, ⊥ can be simplified as a function of ν as h (∆ O J,ν ) i ( µ ) = f N X l : J l =0 J l | ζ l | [1 + A l (1 + cos 2 ν ) − B l sin 2 ν ] . (23) lassification of spin and multipolar squeezing A l and B l are defined as A l ≡ J l N − | ζ l | (cid:8) − cos J l − µ [1 − | ζ l | (1 − cos J l µ )] N − (cid:9) + 12 (cid:18) J l − (cid:19) { − cos J l − µ [1 − | ζ l | (1 − cos J l µ )] N − } , (24) B l ≡ (cid:26) J l ( N − | ζ l | cos J l µ (cid:18) J l − (cid:19) h − | ζ l | (cid:16) − cos J l µ (cid:17)i(cid:27) × sin µ J l − µ h − | ζ l | (cid:16) − cos J l µ (cid:17)i N − . (25)Equation (23) is periodic with respect to ν , so there exist the minimum and themaximum, i.e., the squeezed and anti-squeezed quantum fluctuations, respectively. Thesqueezing parameter ξ ( µ = 0) = 1 for the initial CSS in Eq. (12) and the spins are saidto be squeezed when ξ ( µ ) < µ ≪ N ≫
1, when the subspins { J l } in Eq. (6) and the coefficients {| ζ l | } of the initialcoherent state in Eq. (12) satisfy | ζ l | = δ l,l ( J l = 0). The quantum fluctuations in the O J, - O J, plane in Eq. (23) can be simplified as h (∆ O J l ,ν ) i ( µ ) = f J l N (cid:26) (cid:18) J l N − (cid:19) × (cid:20) (1 − cos J l N − µ )(1 + cos 2 ν ) − µ J l N − µ ν (cid:21)(cid:27) , (26)and the expectation value perpendicular to the O J, - O J, plane in Eq. (22) is h ˆ O J, i ( µ ) = f J l N cos JN − µ . (27)Here, we assume that µ and N satisfy α ≡ J l N µ ≫ β ≡ J l N µ ≪
1. Then,substituting Eqs. (26) and (27) into Eq. (19), we obtain the squeezing parameter for r = 1 up to the second order in β as: ξ ( µ ) ≃ α + 23 β + β α + O (max { β α , β } ) , (28)where ν ≃ − arctan α + π . The minimum of Eq. (28), i.e., the squeezing limit isachieved at µ = µ min = (12) / ( J l N ) − / are given by ξ ≡ ξ ( µ min ) ≃ (cid:18) J l N (cid:19) / + 12 J l N ∝ ( J l N ) − / , (29)which implies that the squeezing limit monotonically decreases with increasing J l .
4. Application to collective su( ) systems To examine the squeezing parameter in Eq. (19) for r >
1, especially the {| ζ l | } -dependence of the squeezing limit, let us consider a collective su(4) system consisting of lassification of spin and multipolar squeezing N spin-3/2 particles as an example. In this case, the observables that can completelycharacterize collective spin states are the spin vector, the quadrupolar tensor, and theoctupolar tensor. The Cartesian components of the spin vector ˆ λ n j ; J = ,k ( k = 1 , , λ n j ; ,k = N X j =1 4 X m,n =1 (˜ λ ,k ) mn ˆ c † n j ; ,m ˆ c n j ; ,n , (30)where ˜ λ ,k represent the spin-3 / J µ ( µ = x, y, z ) given by Eq. (B.1). Thematrix representations of the five independent components of the quadrupolar tensorand the seven independent components of the octupolar tensor [19] can be respectivelyexpressed in terms of ˜ J µ as( ˜ Q µν ) mn = √
156 ( ˜ J µ ˜ J ν + ˜ J ν ˜ J µ ) mn , (31)( ˜ D xy ) mn = √
156 ( ˜ J x − ˜ J y ) mn , (32)( ˜ Y ) mn = √
56 ( − ˜ J x − ˜ J y + 2 ˜ J z ) mn , (33)where ( µ, ν ) = ( x, y ) , ( y, z ) , ( z, x ) in Eq. (31), and( ˜ T αµ ) mn = 13 (2 ˜ J µ − ˜ J µ ˜ J ν − ˜ J η ˜ J µ ) mn , (34)( ˜ T βµ ) mn = √
159 ( ˜ J µ ˜ J ν − ˜ J η ˜ J µ ) mn (35)( ˜ T xyz ) mn = √
159 ( ˜ J x ˜ J y ˜ J z ) mn , (36)where ( µ, ν, η ) = ( x, y, z ), ( y, z, x ), and ( z, x, y ) and the overbars above the matrixproducts are defined as ˜ A ˜ B = ˜ A ˜ B + ˜ B ˜ A ˜ B + ˜ B ˜ A and ˜ A ˜ B ˜ C = ˜ A ˜ B ˜ C + ˜ B ˜ C ˜ A +˜ C ˜ A ˜ B + ˜ B ˜ A ˜ C + ˜ C ˜ B ˜ A + ˜ A ˜ C ˜ B with respect to the matrices ˜ A , ˜ B , and ˜ C . Here we notethat the matrix representations of the spin and multipolar observables in Eqs. (30)-(36)are normalized so that they satisfy the condition in Eq. (2). These fifteen spin andmultipolar observables in Eqs. (30)-(36) together form the su(4) Lie algebra. Then, theirreducible representations of the collective spin observables describing the symmetricspin state can respectively be given by the matrix representations of their single-spincounter parts in Eqs. (30)-(36), whose explicit expressions are given in Eqs. (B.1)-(B.3).We define the matrices { ˜ λ ,k } ≡ { ˜ J µ , ˜ Q µν , ˜ D xy , ˜ Y , ˜ T αµ , ˜ T βµ , ˜ T xyz } ( k = 1 , · · · ,
15) in theorder of Eqs. (30)-(36). Then, the matrix representation of any observable can beexpressed in terms of { ˜ λ ,k } ( k = 1 , · · · ,
15) as in Eq. (4).
There exist four unitary equivalence classes of the su(2) subalgebras in the su(4) algebra,which can be found as explained in Sec. 2.3. First, let us construct the Dynkindiagram and consider the relation between the simple roots and the spin and multipolar lassification of spin and multipolar squeezing α α α α α α (a) (b) (c)-(i)(c)-(ii)(c)-(iii)(c)-(iv) J z Y T α z Figure 2. (Color Online) (a) The root diagram of the su(4) algebra, (b) the Dynkindiagram of the the su(4) algebra, and (c) the four types of the unitary equivalenceclasses of the matrix representations of the su(2) subalgebras. In (c), the chosensimple roots and the omitted simple roots are indicated by the filled black circles andthe open grey circles, respectively. observables in Eqs. (30)-(36). In collective su(4) systems, the Dynkin diagram hasthree simple roots α , α , α as shown in Fig. 2 (b). Choosing the diagonal matrices˜ J z = ˜ λ , , ˜ Y = ˜ λ , , and ˜ T αz = ˜ λ , as the generators of the Cartan subalgebra, we canexpress the matrices ˜ A , , ˜ A , , and ˜ A , corresponding to the simple roots as˜ A , = √ J + + 12 ˜ Q + − √ T α + −
14 ˜ T β − , (37)˜ A , = 1 √ J + + 34 √ T α + + √
34 ˜ T β − , (38)˜ A , = √ J + −
12 ˜ Q − − √ T α + −
14 ˜ T β − , (39)where we define ˜ J ± ≡ ˜ J x ± i ˜ J y , ˜ Q ± ≡ ˜ Q zx ± i ˜ Q yz , ˜ T α ± = ˜ T αx ± i ˜ T αy , and ˜ T β ± = ˜ T βx ± i ˜ T βy ,respectively. The derivation of Eqs. (37)-(39) are detailed in Appendix C.Then, the four unitary equivalence classes of the su(2) subalgebras can be found,that is, the types (i)-(iv) as illustrated in Figs. 2 (c). The su(2) subalgebra { ˜ O ,k } ( k =1 , ,
3) of these four classes satisfy [ ˜ O , ± , ˜ O , ] = ± f ˜ O , ± , where ˜ O , ± = ˜ O , ± i ˜ O , .Suppose the matrices { ˜ O ,k } have the block-diagonalized forms as in Eq. (6); then theladder operator ˜ O , + and the observable ˜ O , should be expressed in terms of the linearcombinations of ˜ A ,k ( k = 1 , ,
6) and ˜ λ ,k ( k = 3 , , O , + = X k =1 , , c k ˜ A k , (40)and ˜ O , = d ˜ λ , + d ˜ λ , + d ˜ λ , , (41) lassification of spin and multipolar squeezing c k and d k are the solutions of [ ˜ O , ± , ˜ O , ] = ± f ˜ O , ± . The solutions, the numberof subspaces r , the subspins { J l } in Eq. (6), and the structure factor f are respectivelygiven by (i) ˜ O , + = r
310 ˜ A , + r
25 ˜ A , + r
310 ˜ A , , ˜ O , = ˜ λ , ,r = 1 , { J = 32 } , f = 1 (42)(ii) ˜ O , + = 1 √ A , ± ˜ A , ) , (43)˜ O , = 2 √
10 ˜ λ , + 1 √ λ , − √
10 ˜ λ , ,r = 2 , { J = 1 , J = 0 } , f = r , (44)(iii) ˜ O , + = 1 √ A , ± ˜ A , ) , ˜ O , = 1 √ λ , + 2 √ λ , ,r = 2 , { J = J = 12 } , f = √ , (45)(iv) ˜ O , + = ˜ A , , ˜ O , = 1 √
10 ˜ λ , + 1 √ λ , + r
25 ˜ λ , ,r = 3 , { J = 12 , J = J = 0 } , f = √ . (46)The type (i) squeezing in Eq. (42) is equivalent to the spin squeezing among { ˆ J x , ˆ J y , ˆ J z } and the type (iii) squeezing in Eq. (46) is equivalent to the quadrupole-octupolesqueezing among { ˆ T βz , ˆ T xyz , ˆ Y } and the quadrupole squeezing among { ˆ Q zx , ˆ Q yz , ˆ Y } . In the case of the type (i) in Eq. (42), r = 1 and the squeezing limit for the one-axistwisting is given by Eq. (29) as ξ ≃ (cid:18) N (cid:19) / + 13 N , (47)which is achieved at the evolution time of µ min = √ × N − / corresponding to t min = √ χ × N − / .In the case of the types (ii)-(iv) in Eq. (44)-(46), the squeezing limits depend onthe initial coherent state in Eq. (12) in general; however, the squeezing limits for thetypes (ii) in Eq. (44) and (iv) in Eq. (46) can be calculated in the same manner as thetype (i), when | ζ l | = δ l in the initial state in Eq. (12). They are given by(ii) ξ ≃ (cid:18) N (cid:19) / + 12 N , (48)(iv) ξ ≃ (cid:18) N (cid:19) / + 1 N , (49) lassification of spin and multipolar squeezing S qu eez i ng li m it s ξ m i n2 ζ | type (ii)type (iv) E vo l u ti on ti m e µ m i n ζ | type (ii)type (iv) Figure 3. (Color Online) (a) The | ζ | -dependence on the squeezing limit ξ and(b) the corresponding evolution time µ min for N = 10 . The horizontal dotted lines at ξ = 0 . . µ min = 0 . . | ζ | = 1. respectively. The minimum squeezing limits in Eqs. (48) and (49) are achieved at µ min = 12 / × N − / ( t min = / χ × N − / ) and µ min = 2 × / × N − / ( t min = / χ × N − / ), respectively. If | ζ l | = 0 for ∃ l >
0, the squeezing limits for types (ii)and (iv) cannot be obtained by the expression in Eq. (29). We numerically calculatethe | ζ | -dependences of the squeezing limits and their corresponding evolution timesand illustrate them in Figs. 3 (a) and (b). In Figs. 3, we plot the squeezing limit ξ and the evolution time µ min with respect to 1 − | ζ | . The squeezing limits for the types(ii) and (iv) monotonically decrease with increasing | ζ | . For | ζ | ≃
1, the squeezinglimits are almost equal to Eqs. (48) and (49), respectively; however, for | ζ | < .
2, theminimum squeezing limits sharply increase due to the decreases in the number of theSchwinger bosons which are nonlinearly interacting via the one-axis twisting interactionsin Eq. (20).In the case of the type (iii) in Eq. (45), r = 2 and J = J = 1 /
2, the | ζ | -dependence of the minimum squeezing limit is periodic because of the symmetry withrespect to the two subspaces. To see this, let us derive the expression for the squeezinglimit for the type (iii): ξ ( µ ) = 1 + ( N − P l =1 ∆ l ( µ ) P l =1 | ζ l | (1 − | ζ l | sin µ ) N − , (50)where ∆ l ’s ( l = 1 ,
2) are defined as∆ l ( µ ) = (cid:20) − (cid:16) − | ζ l | sin µ (cid:17) N − (cid:21) × − vuut " | ζ l | sin µ (1 − | ζ l | sin µ ) N − − (1 − | ζ l | sin µ ) N − . (51) lassification of spin and multipolar squeezing | ζ l | = δ l , the squeezing limit is given by Eq. (49) at µ min = 2 × / × N − / , which are same as those for the type (iv) with the initial stateof | ζ l | = δ l , while the evolution time t min = / χ × N − / is two times larger than thatfor the type (iv) with the initial state of | ζ l | = δ l . When the initial state is the equalsuperposition of the two subspaces, i.e., | ζ l | = , the squeezing limit can be obtainedby assuming α ≫ β ≪ ξ ≃ (cid:18) N (cid:19) / + 3 N ≃ (cid:18) N (cid:19) / , (52)at the evolution time of µ min = 2 × / × ( N/ − / ( t min = / χ × N − / ). Equation (52)is 6 / ≃ . / ≃ . | ζ l | = δ l , and 2 / ≃ . | ζ l | = δ l and the type (iii)with the initial state of | ζ l | = δ l . The | ζ | -dependence of the squeezing limit and thecorresponding evolution time for N = 10 are illustrated in Fig. 4 (a). The squeezinglimit reaches the maximum at | ζ | ≃ − π and π . The dependence of the squeezinglimit on the number of spins for | ζ | ≃ − π is shown in Fig. 4 (b), which can be wellfitted to ξ ≃ . ± .
00 + 0 . ± . N . ± . + 3 . ± . N (53)by the least squared method. Equation (53) implies that the scaling of the squeezinglimit with respect to N is 0 for | ζ | = π and 1 − π , although the squeezing limit is stillbelow the standard quantum limit of ζ = 1. The evolution time corresponding to thesqueezing limit for | ζ | = 1 − π can be well fitted to µ min ≃ (3 . ± . × N − . ± . , (54)with respect to the number of spins N by the least squared method.
5. Conclusion
In this paper, we consider the collective su(2 J + 1) systems and classify the squeezingamong the spin and multipolar observables generating the su(2) subalgebra of thesu(2 J + 1) algebra, based on the unitary equivalence class of the su(2 J + 1)-dimensionalrepresentations of the observables. The matrix representations of the observablesand their representation spaces can be decomposed into the direct sums of the lowerdimensional irreducible representations of the su(2) generators in Eq. (6). This impliesthat if two sets of observables belong to the same unitary equivalence class, they canbe decomposed into the same matrix representation in Eq. (6) whose bases can betransformed to each other via an SU(2 J +1) transformation; hence they are characterizedby the same subspins { J l } in Eq. (6) giving the structure factor f in Eq. (7). Theunitary equivalence class of the su(2) subalgebra in the su(2 J + 1) algebra can be foundby choosing vertices in the Dynkin diagram of the su(2 J + 1) algebra as shown in Fig. 1. lassification of spin and multipolar squeezing S qu eez i ng li m it ξ m i n2 E vo l u ti on ti m e µ m i n ζ | ξ min2 µ min S qu eez i ng li m it ξ m i n2 E v l u ti on ti m e µ m i n Number of spins N ξ min2 µ min Figure 4. (Color Online) (a) The | ζ | -dependence of the squeezing limit ξ andthe corresponding evolution time µ min for N = 10 . The maxima of ξ are achievedat | ζ | = 1 − π and π . (b) The N -dependence of the squeezing limit ξ and thecorresponding evolution time µ min for | ζ | = 1 − π . The fitting function for ξ and µ min are given by Eq. (53) and Eq. (54), respectively. The squeezing limits are determined by the dimensionality of the unitaryequivalence class of the observables and the initial CSS involved by the squeezing.Taking the one-axis-twisted SSS for example, we calculate the squeezing limit ξ ,which is given by the function in Eq. (19) in terms of the subspins { J l } in theirreducible representations in Eq. (6) and the coefficients {| ζ l | } of the initial CSS inEq. (12). When | ζ l | = δ l in Eq. (12), the squeezing limit ξ in Eq. (29) for theone-axis twisted SSS achieved to be proportional to ( J l N ) − / at the evolution timeof µ ≡ χf t ∝ ( J l N ) − / in the limit of J l N χf t ≫ J l N ( χf t ) ≪
1, whichimplies that the squeezing among the observables, of which matrix representations areirreducible, gives the minimum squeezing limit of the collective su(2 J + 1) consisting of N spin- J particles. In the case of | ζ l | < ∃ | ζ l = l | = 0, the analytical expressionsof the squeezing limits in Eq. (19) cannot be easily obtained due to the interferencebetween the representation spaces in Eq. (6).Finally, we apply our classification to the squeezing in the collective su(4) systemsand obtain the squeezing limits analytically or numerically. The squeezing can beclassified into one of four unitary equivalence classes as shown in Fig. 2. Their squeezinglimits depends on the coefficients {| ζ l | } in the initial coherent states in Eq. (12) as wellas the subspins { J l } , whose behaviors were numerically calculated as shown in Figs.3(a) and 4 (a). Since the subspins and the initial coherent sate reflect the structure ofthe unitary equivalence class of the spin and multipolar observables; hence the unitaryequivalence class of the observables can be considered as one of the systematical waysto classify and quantify the squeezing. lassification of spin and multipolar squeezing Acknowledgments
E. Y. thanks Prof. Mark Everitt, Prof. Todd Tilma, Dr. Shane Dooley, Mr. Itsik Cohen,and Ms. Marvellous Onuma-Kalu for fruitful discussions. This work is supported byMEXT Grant-in-Aid for Scientific Research(S) No. 25220601.
Appendix A. Schwinger-boson approach to calculate expectation values forEqs. (12) and (21)
The expectation values for the initial CSS in Eq. (12) and for the one-axis-twisted SSSin Eq. (21) can be simplified by the Schwinger boson approach.The observables { ˆ O J,k } can be decomposed into Eq. (6), which are matrix-represented by the direct sums of the spin matrices { ˜ λ J l ,k } for the spins J l . For eachof r subspaces, we can define the Schwinger boson operator ˆ a l ± (ˆ a † l ± ) which annihilates(creates) a boson in a mode ‘ l ± .’ The annihilation (creation) Schwinger-boson operatorsˆ a l ± (ˆ a † l ± ) satisfy[ˆ a ls , ˆ a l ′ s ′ ] = 0 , [ˆ a ls , ˆ a † l ′ s ′ ] = δ ll ′ δ ss ′ ( s, s ′ = ± ) , (A.1)since the r subspaces V ( {| J l , m l i l } ) ( l = 1 , · · · , r ) are orthogonal to each other. The l -th symmetric state | θ, φ i ⊗ n l l in Eq. (12) can be regarded as a CSS of the 2 J l n l spin-1 / l whose azimuth and polar angles are given by θ and φ ,respectively: | θ, φ i ⊗ n l l = N l X m =0 p N l C m cos N l − m θ m θ e − imφ × | n l + = N l − m, n l − = m i Sb , (A.2)where N l ≡ J l n l represents the number of the l -th Schwinger bosons, and | n l + , n l − i Sb isthe symmetric state of the n l + Schwinger bosons in the ‘ l +’ state and the n l − Schwingerbosons in the ‘ l − ’ state. The matrix representations ˜ λ J l ,k for the l -th subspace with J l = 0 can be mapped to the collective spin operators ˆΛ J l ,k :˜ λ J l , → ˆΛ J l , = 12 (ˆ a † l + ˆ a l − + ˆ a † l − ˆ a l + ) , (A.3)˜ λ J l , → ˆΛ J l , = i − ˆ a † l + ˆ a l − + ˆ a † l − ˆ a l + ) , (A.4)˜ λ J l , → ˆΛ J l , = 12 (ˆ a † l + ˆ a l + − ˆ a † l − ˆ a l − ) , (A.5)with the constraint ˆΛ J l , + ˆΛ J l , + ˆΛ J l , = J l n l ( J l n l + 1). For J l = 0, we defineˆΛ J l , = ˆΛ J l , = ˆΛ J l , = 0. The observables in Eqs. (15) and (16) can be expressedin terms of the Schwinger-boson representations in Eqs. (A.3)-(A.5) asˆ O J, ⊥ = f r M l =1 h ˆΛ J l , cos φ sin θ + ˆΛ J l , sin φ sin θ + ˆΛ J l , cos θ i , (A.6)ˆ O J,ν = f r M l =1 h ˆΛ J l , (cos φ cos θ cos ν − cos φ sin ν ) lassification of spin and multipolar squeezing
17+ ˆΛ J l , (sin φ cos θ cos ν + cos φ sin ν ) − ˆΛ J l , sin θ cos ν i . (A.7)Thus, the expectation values in Eqs. (15) and (16) for the CSS of Eq. (12) can beobtained as Eqs. (17) and (18), respectively.Next, let us simplify the expectation values for the one-axis-twisted SSSs in Eq. (21)in a manner similar to the case of the CSS. The one-axis twisting in Eq. (20) and | π , i ⊗ n l l in the initial CSS can be respectively expressed asˆ H OAT = ~ χf " r M l =1 ˆΛ J l , = ~ χf r M l =1 ˆΛ J l , , (A.8)and | π , i ⊗ n l l = 12 N l / N l X m =0 p N l C m | n l + = N l − m, n l − = m i Sb . (A.9)The l -th one-axis twisting interaction ~ χf ˆΛ J l , in Eq. (A.8) squeezes the l -th CSS inEq. (A.9). The one-axis-twisted SSS of the l -th Schwinger bosons at µ is given by | ψ OAT ( 12 , N l ; µ ) i l ≡ N l / N l X m =0 p N l C m e − imφ e − i (ˆ a † l + ˆ a l + − ˆ a † l − ˆ a l − ) µ × | n l + = N l − m, n l − = m i Sb . (A.10)Here, we note that for an observable ˆ X J l , two SSSs | ψ OAT ( , J l n l ; µ ) i l and | ψ OAT ( , J l n ′ l ; µ ) i l in the l -th subspace satisfy h ψ OAT ( 12 , J l n ′ l ; µ ) | ˆ X J l | ψ OAT ( 12 , J l n l ; µ ) i l ∝ δ n l n ′ l , (A.11)since the expectation value vanishes when the numbers of the Schwinger bosons in thetwo states are not equal, i.e., n l = n ′ l . Then, the expectation value of ˆ O J, can becalculated to give Eq. (22). The one-axis twisting redistribute the quantum fluctuationsin the O J, - O J, plane as follows:ˆ O J,ν = ˆ O J, cos ν − ˆ O J, sin ν = f r M l =1 ( ˆΛ J l , cos ν − ˆΛ J l , sin ν ) . (A.12)The quantum fluctuation in ˆ O J,ν with respect to the state | Ψ OAT ( J, N ; µ ) i tot in Eq. (21)is obtained by h (∆ O J,ν ) i ( µ ) = h Ψ OAT ( J, N ; µ ) | ˆ O J,ν | Ψ OAT ( J, N ; µ ) i tot − h Ψ OAT ( J, N ; µ ) | ˆ O J,ν | Ψ OAT ( J, N ; µ ) i . (A.13)Here, the first term of the right-hand side of Eq. (A.13) is given by h Ψ OAT ( J, N ; µ ) | ˆ O J,ν | Ψ OAT ( J, N ; µ ) i tot = f N X n =0 N − n X n =0 · · · N − n −···− n r − X n r − =0 N C n N − n C n · · · N − n −···− n r − C n r − × | ζ | n | ζ | n · · · | ζ r − | n r − | ζ r | N − n −···− n r − ) lassification of spin and multipolar squeezing × X l : J l =0 h ψ OAT ( 12 , J l n l ; µ ) | ( ˆΛ J, cos ν − ˆΛ J, sin ν ) | ψ OAT ( 12 , J l n l ; µ ) i l = f X l : J l =0 N X n l =0 N C n l | ζ l | n l (1 − | ζ l | ) N − n l × h ψ OAT ( 12 , J l n l ; µ ) | ( ˆΛ J, cos ν − ˆΛ J, sin ν ) | ψ OAT ( 12 , J l n l ; µ ) i l , (A.14)where the first equality is derived from Eq. (A.11) and the second equality is obtainedby the symmetry with respect to the subspace index, l . Similarly to Eq. (A.14), thesecond term in Eq. (A.15) can be calculated as h Ψ OAT ( J, N ; µ ) | ˆ O J,ν | Ψ OAT ( J, N ; µ ) i tot = f X l : J l =0 N X n l =0 N C n l | ζ l | n l (1 − | ζ l | ) N − n l × h ψ OAT ( 12 , J l n l ; µ ) | ( ˆΛ J, cos ν − ˆΛ J, sin ν ) | ψ OAT ( 12 , J l n l ; µ ) i l = 0 , (A.15)since h ψ OAT ( , J l n l ; µ ) | ˆΛ J,k | ψ OAT ( , J l n l ; µ ) i l = 0 for k = 2 ,
3. Substituting Eq. (A.10)into Eq. (A.14), we can simplify h (∆ O J,ν ) i ( µ ) in Eq. (A.13) as Eqs. (23)-(25). Appendix B. Matrix representations of a single spin-3/2 operators
The matrix representations of the spin-vector components ˜ J µ in Eq. (30), the fiveindependent components of the quadrupolar tensor, ˜ Q µν , ˜ D xy , and ˜ Y in Eqs. (31)-(33), and the seven independent components of the octupolar tensor, ˜ T αµ , ˜ T βµ , and ˜ T xyz in Eqs. (34)-(36), are given by˜ J x = 12 √ √ √
30 0 √ , ˜ J y = i −√ √ − −√
30 0 √ , ˜ J z = 12 − − , (B.1) lassification of spin and multipolar squeezing Q xy = i √ − −
11 0 0 00 1 0 0 ˜ Q yz = i √ − − ˜ Q zx = √ −
10 0 − ˜ D xy = √ ˜ Y = √ − − (B.2)and ˜ T αx = 14 −√ −√ −√
35 0 −√ , ˜ T αy = i √ −√ − √ − −√ , ˜ T αz = 12 − − , ˜ T βx = √ − −√ − √ √ − −√ − , ˜ T βy = i √ − √
31 0 √ −√ − −√ , ˜ T βz = √ −
11 0 0 00 − , ˜ T xyz := i √ − − . (B.3) Appendix C. Root diagram and simple roots of the su( ) algebra First, we chose ˜ λ , , ˜ λ , , and ˜ λ , as the Cartan subalgebra and obtain their adjointrepresentations (ad[˜ λ ,k C ]) mn ≡ f nk C m ( k C = 3 , ,
11 and m, n = 3 , , f nk C m is defined by [˜ λ k C , ˜ λ m ] = i P n f nk C m ˜ λ n . Here, the adjointrepresentations of ˜ λ ,k C can be simultaneously diagonalized; hence they have the same lassification of spin and multipolar squeezing A ,k = P k =3 , , c k ˜ λ ,k satisfying [˜ λ ,k C , ˜ A ,k ] = µ k C k ˜ A ,k , where µ k C k arethe eigenvalues of ad[˜ λ ,k C ] corresponding to the eigenvectors ˜ A ,k . Then, we obtaintwelve sets of eigenvalues α k ≡ ( µ k , µ k , µ k ), i.e., the roots, and the eigenvectors˜ A ,k ( k = 1 , · · ·
12) corresponding to the roots. Plotting these roots in the Cartesiancoordinate, we obtain the root diagram of the su(4) algebra in Fig. 2 (a). Here, theroots and their corresponding operators are given by α = √ , ˜ A , = √ J + + 12 ˜ Q + − √ T α + −
14 ˜ T β − = √ E , α = √ − , ˜ A , = 12 ˜ D + + 12 ˜ F + = √ E , α = , ˜ A , = √
54 ˜ T α − − √
34 ˜ T β + = √ E , α = − , ˜ A , = 1 √ J + + 34 √ T α + + √
34 ˜ T β − = √ E , α = −√ − , ˜ A , = 12 ˜ D + −
12 ˜ F + = √ E , α = −√ , ˜ A , = √ J + −
12 ˜ Q − − √ T α + −
14 ˜ T β − = √ E , α k = − α k , ˜ A k = ˜ A † k , ( k = 1 , · · · , where E mn denotes the matrix with 1 in the mn entry and 0s elsewhere and theladder operators are defined by ˜ J ± ≡ ˜ J x ± i ˜ J y , ˜ Q ± ≡ ˜ Q zx ± i ˜ Q yz , ˜ D ± = ˜ D xy ± i ˜ Q xy ,˜ T α ± = ˜ T αx ± i ˜ T αy , ˜ T β ± = ˜ T βx ± i ˜ T βy , and ˜ F ± = ˜ T βz ± i ˜ T xyz . References [1] Yuen H P 1976
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