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Characterization of quantum angular-momentum ?uctuations via principal components

? Departamento de Optica, Facultad de Ciencias F? ?sicas, Universidad Complutense, 28040 Madrid, Spain (Dated: February 2, 2008) We elaborate an approach to quantum ?uctuations of angular momentum based on the diagonalization of the covariance matrix in two versions: real symmetric and complex Hermitian. At di?erence with previous approaches this is SU(2) invariant and avoids any di?culty caused by nontrivial commutators. Meaningful uncertainty relations are derived which are nontrivial even for vanishing mean angular momentum. We apply this approach to some relevant states.

? Angel Rivas and Alfredo Luis?

arXiv:0710.4699v2 [quant-ph] 9 Jan 2008

PACS numbers: 03.65.Ca,42.50.St,42.25.Ja

I.

INTRODUCTION

Quantum ?uctuations and uncertainty relations play a key role in the fundamentals of quantum physics and its applications. In this work we focus on angular momentum variables. Beside mechanics, angular momentum operators are ubiquitous in areas such as quantum optics, matter-light interactions, and Bose-Einstein condensates. Basic observables such as light intensity, number of particles, and atomic populations are formally equivalent to angular momentum components [1–5]. This is also the case of the Stokes parameters representing light polarization and the internal state of two-level atoms identi?ed as spin 1/2 systems [5–7]. Angular momentum operators are also the generators of basic operations such as phase shifting, beam splitting, free evolution, as well as atomic interactions with classical ?elds [7, 8]. Moreover, angular momentum ?uctuations are crucial in diverse areas. This is for example the case of quantum metrology, implemented by very di?erent physical systems such as two beam interference with light or BoseEinstein condensates, or atomic population spectroscopy. This is because angular-momentum uncertainty relations determine the ultimate limit to the resolution of interferometric and spectroscopic measurements [1–3]. A dramatic example has been put forward in Ref. [3] concerning atomic clocks based on atomic population spectroscopy, whose signal to noise ratio is proportional to the square root of the duration of the measurement. In such a case, an atomic clock using 1010 atoms prepared in an state with reduced angular momentum ?uctuations would yield the same signal to noise ratio in a measurement lasting 1 second as an atomic clock with the same number of atoms in an state without reduced angular momentum ?uctuations in a measurement lasting 300 years. From a di?erent perspective it has been shown that angular momentum ?uctuations are useful in the analysis of many-body entanglement [4] and continuous-variable

polarization entanglement [5]. In comparison to other fundamental variables, such as position and momentum, the standard uncertainty relations for angular momentum run into two serious di?culties: basic commutators are operators instead of numbers, and there is lack of SU(2) invariance. Nontrivial commutation relations lead to uncertainty products bounded by state-dependent quantities. Among other consequences, these bounds become trivial for states with vanishing mean angular momentum. On the other hand, lack of SU(2) invariance is a basic di?culty since two states connected by a SU(2) transformation should be equivalent concerning quantum ?uctuations, in the same sense that phase-space displacements are irrelevant for position-momentum uncertainty relations. In this work we elaborate an approach to quantum ?uctuations for angular momentum variables which is SU(2) invariant and avoids the di?culties caused by nontrivial commutators. The analysis is based on the diagonalization of the covariance matrix, that we study in two versions: real symmetric (Sec. III) and complex Hermitian (Sec. IV). We examine the main properties of their eigenvalues (principal variances) and eigenvectors (principal components). In Sec. V we derive uncertainty relations involving principal variances that are meaningful even in the case of vanishing mean values. Finally, in Sec. VI we illustrate this approach applying it to some relevant examples. Section II is devoted to recall basic concepts and de?nitions.

II.

ANGULAR MOMENTUM OPERATORS AND SU(2) INVARIANCE

Let us consider arbitrary dimensionless angular momentum operators j = (j1 , j2 , j3 ) satisfying the commutation relations

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address: alluis@fis.ucm.es; URL: http://www.ucm. es/info/gioq

? Electronic

[jk , j? ] = i

n=1

?k,?,n jn ,

[j0 , j ] = 0,

(2.1)

2 where ?k,?,n is the fully antisymmetric tensor with ?1,2,3 = 1, and j0 is de?ned by the relation j 2 = j0 (j0 + 1) . (2.2) U ? j U = Rj , (2.7) with θ a real parameter, and u a unit three-dimensional real vector. It can be seen that the action of U on j is a rotation R of angle θ and axis u [10]

For the sake of completeness we take into account that j0 may be an operator. This is the case of two-mode bosonic realizations where j0 is proportional to the number of particles. More speci?cally, denoting by a1,2 the annihilation operators of two independent bosonic modes with [aj , a? j] = 1, [a1 , a2 ] = [a1 , a? ] = 0, we get that the following oper2 ators j0 = 1 ? a a1 + a? 2 a2 , 2 1 i ? a a1 ? a? 1 a2 , 2 2 1 ? a a1 + a? 1 a2 , 2 2 (2.3) 1 ? j3 = a a1 ? a? 2 a2 , 2 1 j1 =

j2 =

satisfy Eqs. (2.1) and (2.2) [9]. Concerning physical realizations, a1,2 can represent the complex amplitude operators of two electromagnetic ?eld modes. For material systems they can represent the annihilation operators for two species of atoms in two di?erent internal states, for example. In any case, the operator j0 is proportional to the total number of photons or atoms and j3 is proportional to the number di?erence. Therefore, these operators represent basic detection mechanisms such as the measurement of light intensity or number of atoms. In this regard we have the following correspondence |j, m = |n1 = j + m |n2 = j ? m , (2.4)

where Rt R = RRt = I , I is the 3 × 3 identity, and the superscript t denotes matrix transposition. So, the SU(2) invariance is just the mathematical statement corresponding to the fact that the conclusions which one could draw from an angular momentum measurement must be independent of which set of three orthogonal angular momentum components one chooses. One way to guarantee the cited invariance is obtained by using speci?c components of the angular momentum referred to the mean value j , the longitudinal j and transversal j⊥,k components with k = 1, 2. These components are the projections of j on a set of Cartesian axes adapted to j so that the longitudinal axis points in the direction of j [11, 12]. Therefore, by construction | j |= j , j⊥,k = 0. (2.8)

between the standard angular momentum basis |j, m of simultaneous eigenvectors of j3 and j0 , i. e., j3 |j, m = m|j, m , j0 |j, m = j |j, m , and the product of number states in the two modes |n1 |n2 with a? j aj |nj = nj |nj . Angular momentum operators also serve to describe the internal state of two-level atoms via the de?nitions j0 = 1 (|e e| + |g g |) , 2 i (|g e| ? |e g |) , 2 1 (|g e| + |e g |) , 2 (2.5) 1 j3 = (|e e| ? |g g |) , 2 j1 =

Among other properties j , j⊥ serve to properly de?ne SU(2) squeezing as reduced ?uctuations of a transversal component [1, 11]. However this approach breaks down in states with vanishing mean angular momentum. Finally we recall a basic relation showing that angularmomentum ?uctuations limit the resolution of interferometric and spectroscopic measurements [1–3]. This is because angular momentum components jn describe atomic and light free evolution, as well as the action of linear optical devices such as beam splitters and interferometers, via unitary transformations of the form Uφ = exp(iφjn ) acting in a given initial state ρ. The objective of interferometric and spectroscopic measurements is the accurate determination of the value of the phase shift φ. This is carried out by the measurement of a given observable A in ? the transformed state Uφ ρUφ . In a simple data analysis, the uncertainty ?φ in the inferred value of φ is related to the ?uctuations of the measured observable in the form [1] ?φ = ? A ?φ

?1

j2 =

?A =

where |e, g are the excited and ground states. This is formally an spin 1/2 where j0,3 represent atomic populations and j1,2 the atomic dipole [7]. Collections of two-level atoms are described by composition of the individual angular momenta. Throughout, by SU(2) invariance we mean that the density operators ρ and U ρU ? are fully equivalent concerning quantum ?uctuations, where U is any SU(2) unitary operator exponential of the angular momentum operators U = exp (iθu · j ) , (2.6)

1 ?A , ≥ | [A, jn ] | 2?jn

(2.9)

where the uncertainty relation ?A?jn ≥ | [A, jn ] |/2 has been used. Therefore, the accuracy of the detection is limited by the ?uctuations of the angular momentum component generating the transformation. In the optimum case ?jn ∝ j0 so that ?φ is inversely proportional to the mean number of particles. This ultimate limit is known as Heisenberg limit [1]. It is worth stressing that this conclusion applies exclusively to pure states, as required by the equality in the uncertainty product between A and jn . These results are supported by more involved analyses [13].

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III. PRINCIPAL COMPONENTS FOR REAL SYMMETRIC COVARIANCE MATRIX

and J 2 = j2, J

2

= j 2.

(3.9)

Complete second-order statistics of the operators j in a given state ρ is contained in the 3 × 3 covariance matrix associated with ρ. Due to the lack of commutativity we can propose two di?erent covariance matrices: real symmetric and complex Hermitian. In this section we focus on the real symmetric covariance 3 × 3 matrix M with matrix elements 1 (3.1) Mk,? = ( jk j? + j? jk ) ? jk j? , 2 where mean values are taken with respect to ρ and we ? have Mk,? = Mk,? = M?,k . Next we analyze the main properties of M . (i) The covariance matrix M allows us to compute the variances (?ju )2 of arbitrary angular-momentum components ju = u · j , where u is any unit real vector, in the form (?ju ) = ut M u.

2

(3.2)

(v) At most one principal variance can vanish for j0 = 0 since for j0 = 0 no state can be eigenvector of more than one angular-momentum component. (vi) The principal variances provide an SU(2) invariant characterization of ?uctuations. The invariance holds because under any SU(2) transformation (2.6), (2.7) we get that M transforms as M → RM Rt . Therefore, the covariance matrix RM Rt associated to the state U ? ρU has the same principal variances as the covariance matrix M associated to the state ρ. (vii) The principal variances are the extremes of the variances of arbitrary angular momentum components ju = u · j for ?xed ρ when the real unit vector u is varied. More speci?cally, from Eq. (3.2), taking into account that M is symmetric, and using a Lagrange multiplier λ for the constraint u2 = 1, we have that the extremes of ?ju when u is varied are given by the eigenvalue equation M u = λu, (3.10)

Similarly, M allows us to compute the symmetric correlation of two arbitrary components ju = u · j , jv = v · j , where u, v are unit real vectors, in the form 1 ( ju jv + jv ju ) ? ju jv = v t M u = ut M v . (3.3) 2 (ii) Since M is real symmetric the transformation that renders M diagonal is a rotation matrix Rd M=

t Rd

(?J1 )2 0 0

0 (?J2 )2 0

0 0 (?J3 )2

Rd .

(3.4)

so that from Eq. (3.5) the extremes coincide with the principal components. (viii) Next we examine the relation between the principal components and the longitudinal j and transversal j⊥,k components, with k = 1, 2. We can demonstrate that j is a principal component when the state ρ is invariant U ρU ? = ρ under the unitary transformation U = exp(iπj ). This is because for this transformation U ? j⊥,k U = ?j⊥,k , and U ? j⊥,k j U = ?j⊥,k j , so that U ρU ? = ρ → j⊥,k j = 0, (3.13) (3.12) U ?j U = j , (3.11)

The eigenvalues (?Jk )2 , k = 1, 2, 3, are the variances of the operators Jk = uk · j , where uk are the three real orthonormal eigenvectors of M M uk = (?Jk )2 uk . (3.5)

This implies that M is a positive semide?nite matrix. Following standard nomenclature in statistics we refer to J and ?J = (?J1 , ?J2 , ?J3 ) as principal components and variances, respectively. We stress that J and ?J depend on the system state ρ. (iii) For every ρ the principal components are uncorrelated 1 ( Jk J? + J? Jk ) ? Jk J? = 0, k = ?. (3.6) 2 (iv) The principal components J are related to j by the rotation Rd that diagonalizes M J = Rd j . (3.7) Thus, the three operators J are legitimate mutually orthogonal Hermitian angular-momentum components satisfying the standard commutation relations

3

[Jk , J? ] = i

n=1

?k,?,n Jn ,

(3.8)

and similarly for the opposite ordering j j⊥,k = 0. Then j is a principal component and the other two principal components are transversal. Invariance under U = exp(iπj ) is a very frequent symmetry. For bosonic two-mode realizations this is equivalent to symmetry under mode exchange a1 ? a2 for the ? modes for which j = (a? 1 a2 + a2 a1 )/2, where a1,2 are the corresponding annihilation operators. This is because for these modes U ? a1 U = ia2 and U ? a2 U = ia1 and we have mode exchange except for a global π/2 phase change. (ix) According to Eq. (2.9) the maximum principal variance of ρ provides an assessment of the resolution achievable with ρ in the detection of small phase shifts generated by angular momentum components. This includes all linear interferometric and spectroscopic measurements. Therefore, the maximum principal variance

4 provides an estimation of the usefulness of the corresponding state in quantum metrology. (x) Next we derive upper and lower bounds for the principal variances of states with j = 0. In such a case from Eqs. (2.2) and (3.9) we have trM = (?J1 ) +(?J2 ) +(?J3 ) = j0 (j0 + 1) . (3.14) If we arrange the principal variances in decreasing order ?J1 ≥ ?J2 ≥ ?J3 , and using Eq. (3.14) we get the following bounds for the principal variances

2 j0 ≥ (?J1 ) ≥ 1 3 2 1 3 2 2 2

More speci?cally, for u? = u we have that ju is not ? Hermitian, ju = ju , so that the variance must be rede?ned, for example in the form [14]

? (?ju )2 = ju ju ? | ju |2 .

(4.3)

? provides ?ju as Then M

2 ? u. (?ju ) = u? M

(4.4)

Note that for real u we have ? u = ut M u. u? M (4.5)

j0 (j0 + 1) ,

2 1 2

1 2

j0 (j0 + 1) ≥ (?J2 ) ≥

j0 , (3.15)

j0 (j0 + 1) ≥ (?J3 )2 ≥ 0.

Similarly, we can compute the correlation of two complex projections ju = u · j , jv = v · j where u, v are arbitrary unit complex vectors, in the form

? ? ? u. jv ju ? jv ju = v ? M

The upper bound for ?J1 holds because for an arbitrary 2 2 component jk ≤ j0 , while the lower bound is reached when all the principal variances are equal. The upper bound for ?J2 is reached when ?J1 = ?J2 and ?J3 = 0, 2 while the lower bound is reached when (?J1 )2 = j0 and ?J2 = ?J3 . Finally, the upper bound for ?J3 is reached when all the principal variances are equal, while the lower bound occurs for ?J3 = 0. (xi) The lower bound for ?J1 in Eq. (3.15) is, roughly speaking, of the order of j0 so that from Eq. (2.9) all pure states with j = 0 can reach maximum interferometric precision (Heisenberg limit). This explains why most optimum states for metrological applications satisfy j = 0 (see Sec. VI).

IV. PRINCIPAL COMPONENTS FOR COMPLEX HERMITIAN COVARIANCE MATRIX

(4.6)

? is Hermitian it becomes diagonal by means (ii) Since M of a 3 × 3 unitary matrix Ud ? ? ?1 )2 (?J 0 0 ? = U? ? 0 ?2 )2 (4.7) M (?J 0 ? Ud , d ?3 )2 0 0 (?J where the elements on the diagonal are the variances (4.3) ?k = uk · j , where uk are the three of the operators J ? complex orthonormal eigenvectors of M ? uk = (?J ?k )2 uk . M (4.8)

In this section we elaborate the statistical description of angular-momentum ?uctuations via the complex Her? with matrix elements mitian 3 × 3 covariance matrix M ? k,? = jk j? ? jk M j? , (4.1) ? ?,k . The two matrices M and M ? contain ?? = M with M k,? essentially the same information since they only di?er by a factor j ? k,? ? Mk,? = M i ?k,?,n jn . 2 n=1

3

? is positive semide?nite. We again This implies that M ? and ?J ? = (?J ?1 , ?J ?2 , ?J ?3 ) as principal comrefer to J ponents and variances, respectively. Since a global phase is irrelevant we regard always Ud as an SU(3) matrix. (iii) For every state ρ the principal components are uncorrelated ?? J ? ?? ? J k ? ? Jk J? = 0, k = ?. (4.9)

(iv) Standard commutation relations (2.1) are not preserved under transformations by unitary 3 × 3 matrices ? = U j. J (4.10) However, a slight modi?cation of (2.1) is actually preserved under the action (4.10) of SU(3) matrices. These are

3

(4.2)

?k , J ?? ] = i [J

n=1

?? , ?k,?,n J n

?] = 0, [j0 , J

(4.11)

In particular they coincide exactly for all states with j = 0. ? are di?erent matrices so we Nevertheless, M and M can exploit this di?erence by examining the most relevant features that they do not share in common. ? provides the properly (i) The covariance matrix M de?ned variances of complex linear combinations of angular-momentum components ju = u · j for arbitrary unit complex vectors u with u? u = 1.

which are equivalent to Eq. (2.1) for Hermitian operators. We have also ?? · J ? = j2, J ??· J ? = j 2. J (4.12)

The preservation of (4.11) under the action of SU(3) matrices in Eq. (4.10) can be demonstrated by direct computation of the commutators after expressing the most

5 general SU(3) matrix U as a suitable product of matrices belonging to the SU(2) subgroup of the form [15] U= cos θeiφ ? sin θe?i? 0 sin θei? cos θe?iφ 0 0 0 1 , (4.13) variation process for the complex Hermitian case takes place over a larger set of operators ju , with complex u, that includes as a particular case the projections on real u. (x) Because of Eqs. (4.14) and (4.15) for j = 0 the upper and lower bounds for principal variances in Eq. ?. (3.15) also hold replacing J by J ? (xi) It is questionable whether ?ju for ju = ju represents practical observable ?uctuations. For example, for ju = j1 + ij2 we have (?ju ) = (?j1 ) + (?j2 ) ? j3 ,

2 2 2

and permutations of lines and columns. Each matrix ? satisfying Eq. (4.11) into (4.13) transforms operators J ?′ = U J ? ful?lling the same comanother set of operators J mutation relations (4.11). (v) From Eq. (4.2) we can derive the equality of traces ? so that trM = trM ? ?J with ? J

2 2

(4.19)

= (?J ) = (?j ) = j0 (j0 + 1) ? J 2 , (4.14)

2

2

= J

2

= j 2.

(4.15)

?)2 > 0. This implies that Since J 2 ≤ j0 2 we have (?J there must be at least a nonvanishing principal variance. ? can vanish siOtherwise, two principal variances of M multaneously for the same state, as shown in Sec. VI for the SU(2) coherent states. ? are equal the (vi) Although the traces of M and M determinants are di?erent. This can be easily proven ? in the principal-component basis that by expressing M renders M diagonal ? = M so that ? = detM ? 1 detM 4

3

so we can have ?ju = 0 with ?j1,2 = 0, being this the case of the SU(2) coherent states (see Sec. VI). Never? contains all theless, from Eq. (4.14) we have that ?J angular momentum ?uctuations. In this regard it is worth recalling that non Hermitian operators can be related to experimental processes, as demonstrated by double homodyne detection where the statistics is given by projection on quadrature coherent states [16]. In our context, the eigenstates of ju = j1 + ij2 are SU(2) coherent states that de?ne by projection the SU(2) Q function [10]. This is an observable probability distribution function, via double homodyne detection of two ?eld modes for example [17].

V.

UNCERTAINTY RELATIONS

(?J1 ) ?i J3 /2 i J2 /2

2

i J3 /2 (?J2 )2 ?i J1 /2

?i J2 /2 i J1 /2 (?J3 )2

,

(4.16)

Variances are the most popular building blocks of uncertainty relations. For angular momentum, the standard procedure leads to ?j1 ?j2 ≥ 1 | j3 |, 2 (5.1)

(?Jk )2 Jk 2 ,

k=1

(4.17)

?. and detM ≥ detM ? are SU(2) invariant (vii) The principal variances of M ? → RM ? Rt , since under SU(2) transformations we have M t ? so that the covariance matrix RM R for the state U ? ρU ? for has the same eigenvalues as the covariance matrix M the state ρ. (viii) The principal variances are the extremes of the variances (?ju )2 of any complex combination of angular momentum components ju = u · j where u is a complex unit vector. More speci?cally, from Eq. (4.4), and introducing a Lagrange multiplier λ to take into account the constraint u? · u = 1, we get that the extremes of ?ju are given by the eigenvalue equation ? u = λu, M (4.18)

and cyclic permutations. This uncertainty relation faces two di?culties. On the one hand it is bounded by an state-dependent quantity that vanish for states with j = 0. On the other hand, it lacks SU(2) invariance so that it leads to di?erent conclusions when applied to SU(2) equivalent states [18]. These di?culties can be avoided by using the principal variances leading to meaningful SU(2) invariant relations which are nontrivial even for states with j = 0. A ?rst SU(2) invariant uncertainty relation can be de? ) [19] rived from the trace of M (or equivalently M trM = (?J ) = j0 (j0 + 1) ? J

2 2

≥ j0 ,

(5.2)

so that from Eq. (4.8) the extremes of (?ju )2 are the ?. principal variances ?J ? are more extreme (ix) The principal variances of M than the principal variances of M since from Eq. (4.5) the

where we have used that for any component jk 2 ≤ 2 2 jk ≤ j0 . 2 The minimum uncertainty states with (?J ) = j0 2 2 are obtained for maximum J = j0 . This is satis?ed exclusively by SU(2) coherent states [10, 19] |j, θ, u = U (θ, u)|j, j , (5.3)

6 where U is any SU(2) unitary operator (2.6) and |j, m are the simultaneous eigenvectors of j0 and j3 , with eigenvalues j and m, respectively. On the other hand, maximum uncertainty (?J )2 is obtained for J = 0. This is the case of the state |j, m = 0 for example (see Sec. VI). The uncertainty relation (5.2) is nontrivial even for j = 0. Nevertheless, this is not very informative about angular momentum statistics since this is actually just a function of the ?rst moments j . In order to proceed further deriving more meaningful uncertainty relations let us split the analysis in two cases j = 0 and j = 0.

A. 1. Case j = 0

Equation (5.7) holds irrespectively of whether j is a principal component or not. Furthermore, when j is a ? we have, respectively principal component of M or M (?J⊥,1 ) + (?J⊥,2 ) ≥ j0 , ?⊥,1 ?J

B.

2 2 2 2

?⊥,2 + ?J

≥ j0 .

(5.9)

Case j = 0

Product of variances

States with j = 0 arise very often in quantum metrological applications as explained in point (xi) of Sec. III [1, 20, 21]. In such a case the standard uncertainty products (5.1) are all trivial ?jk ?j? ≥ 0 since they do not establish any lower bound to the product of variances. Moreover, the components j , j⊥ are unde?ned.

1. Product of variances

For j = 0 an SU(2) invariant product of variances can be derived by applying the standard procedure to the longitudinal and transversal components (2.8), leading to just one nontrivial relation [12] ?j⊥,1 ?j⊥,2 ≥ 1 | j |, 2 (5.4)

while the other two are trivial ?j⊥,k ?j ≥ 0, for k = 1, 2. When J = j is a principal component we can show that the principal transversal variances ?J⊥,k provide the minimum uncertainty product ?j⊥,1 ?j⊥,2 ≥ ?J⊥,1 ?J⊥,2 1 ≥ | j |. 2 (5.5)

We can derive a suitable lower bound for the product ?J1 ?J2 of the two larger principal variances of M (or ? ) with ?J1 ≥ ?J2 ≥ ?J3 , valid for all states with M j = 0. To this end we begin with by considering the minimum of ?J2 for ?xed ?J1 . From the equality (3.14) we get (?J2 ) + (?J3 ) = j0 (j0 + 1) ? (?J1 ) ,

2 2 2

(5.10)

and for ?xed ?J1 the sum (?J2 )2 + (?J3 )2 is constant. Taking into account that ?J2 ≥ ?J3 we get that the minimum ?J2 occurs when ?J2 = ?J3 = 1 2 j0 (j0 + 1) ? (?J1 )

2

This is because the determinant of M is invariant under rotations of j , and the principal components are uncorrelated. Moreover they are the extreme variances in the transversal plane according to point (vii) in Sec. III.

2. Sum of variances

,

(5.11)

and then it holds that (?J2 ) (?J1 ) ≥

2 2

We can begin with by particularizing Eq. (2.2) to longitudinal and transversal components leading to (?j⊥,1 ) + (?j⊥,2 ) = j0 (j0 + 1) ? j 2 .

2 2

(5.12) The minimum of the right-hand side takes place when ?J1 reaches its extremes values in Eq. (3.15) so that (?J2 ) (?J1 ) ≥ min

2 2

1 2 (?J1 ) 2

j0 (j0 + 1) ? (?J1 )

2

.

(5.6)

1 2 1 j0 j0 , j0 (j0 + 1) 2 9

2

,

2 Since we have always j0 ≥ j 2 we get the following lower bound to the sum of transversal variances

(5.13) where “min” refers to the minimum of the alternatives. For j0 ≥ 2 we get that this is always (?J2 ) (?J1 ) ≥

2 2

(?j⊥,1 ) + (?j⊥,2 ) ≥ j0 ,

2

2

(5.7)

where the equality is reached by the SU(2) coherent states exclusively. Let us note that this relation is stronger than the similar one that can be derived from Eq. (5.4), (?j⊥,1 ) + (?j⊥,2 ) ≥ | j |, since j0 ≥ | j | always.

2 2

1 j0 2

2 , j0

(5.14)

and the equality is reached for states with maximum ?J1 and minimum ?J2

2 , (?J1 ) = j0 2

(?J2 ) = (?J3 ) =

2

2

(5.8)

1 j0 . 2

(5.15)

We will see in the next section that this is the case of the Schr¨ odinger cat states (6.14).

7

2. Sum of variances

On the other hand, the complex Hermitian covariance matrix is ? =1 M 2 j (j + 1) ? m2 ?im 0 im j (j + 1) ? m2 0 0 0 0 , (6.3)

Nontrivial bounds to sums of variances can be derived by particularizing Eq. (2.2) to three components ju , jv , jw obtained by projection on three mutually orthogonal (complex in general) unit vectors u, v , w (?ju )2 + (?jv )2 + (?jw )2 = j0 (j0 + 1) . (5.16)

with principal components ?⊥,1 = J ?⊥,2 = J

1 √ j 2 + 1 √ j 2 ?

2 ? Since for any projection, say jw , we have j0 ≥ jw jw = 2 (?jw ) we get

= =

1 √ 2 1 √ 2

(j1 + ij2 ) , (j1 ? ij2 ) , (6.4)

(?ju ) + (?jv ) ≥ j0 ,

2

2

(5.17)

? = j3 , J

where u, v are orthogonal complex unit vectors u? ·v = 0. As a byproduct we obtain that for j = 0 only one ? can vanish. principal variance ?J Finally we can appreciate that the above relations involve the trace of the covariance matrix being derived seemingly without resorting to commutation relations. Nevertheless, commutation relations are also at the hearth of these uncertainty relations since this is the ultimate reason forbidding the simultaneous vanishing of the ?uctuations of all angular momentum components. In this regard, as shown in the original Schr¨ odinger’s paper position-momentum uncertainty relations can be fruitfully related to the corresponding covariance matrix [22].

VI. EXAMPLES

?⊥,1,2 are proportional to the ladder operators so that J j± . The principal variances are ?⊥,1 )2 = (?J ?⊥,2 )2 = (?J

1 2 1 2

[j (j + 1) ? m(m ? 1)] , ? )2 = 0, (?J

[j (j + 1) ? m(m + 1)] , (6.5)

In this section we apply the preceding formalism to some relevant and illustrative quantum states.

A. States |j, m

Let us consider the simultaneous eigenstates |j, m of j0 and j3 with eigenvalues j and m respectively. This family includes the SU(2) coherent states for m = ±j and the limit of SU(2) squeezed coherent states for m = 0 [1, 10]. For two-mode bosonic realizations the case m = 0 is the product of states with the same de?nite number of particles in each mode [20]. For m = 0 we have j = 0 and there is a longitudinal component with j = j3 . In such a case U ρU ? = ρ for U = exp(iπj ) and j is a principal component. The real symmetric covariance matrix is directly diagonal in any basis containing the longitudinal component j3 = j M= 1 2 j (j + 1) ? m2 0 0 0 j (j + 1) ? m2 0 0 0 0 , (6.1)

which are larger and lesser, respectively, than the variances (6.2) of the real symmetric case, in accordance with point (ix) of Sec. IV. The SU(2) coherent states are minimum uncertainty states for the sum of three variances in Eq. (5.2), and for the product and sum of variances of transversal components in Eqs. (5.5), (5.7), and (5.9). The scaling of the largest principal variance as (?J1 )2 ∝ j agrees with the fact that the SU(2) coherent states are not optimum for metrological applications. Optimum states scaling as (?J1 )2 ∝ j 2 can be found below in this section. For SU(2) coherent states two of the principal variances in Eq. (6.5) vanish. This corresponds to the fact that they satisfy the double eigenvalue relation j± |j, ±j = 0 and j3 |j, ±j = ±j |j, ±j , so that |j, ±j are eigenstates of two of the principal components in Eq. (6.4). On the other hand, for the states with m = ±j the nonvanishing variances increase for decreasing |m| and for m = 0 the maximum scales as (?J1 )2 ∝ j 2 , which is consistent with the usefulness of these states in quantum metrology, in agreement with points (ix) and (xi) in Sec. III [20]. Moreover, the states m = 0 are far from the lower bounds of the uncertainty relations in Eqs. (5.2), (5.14), and (5.17).

B. SU(2) squeezed coherent states

Let us consider the SU(2) squeezed coherent states de?ned by the eigenvalue equations [1, 11] (j⊥,1 + iξj⊥,2 ) |ξ = 0, j0 |ξ = j |ξ , (6.6)

with principal variances (?J )2 = 0, 1 j (j + 1) ? m2 . 2 (6.2)

where ξ is a real parameter with ξ ≥ 0 without loss of generality. The ?rst of these equations corresponds to the case of zero eigenvalue among a larger family of eigenvalue equations [1]. The states |ξ are fully de?ned

8 by the eigenvalue relations (6.6). An approximate solution is provided below in Eq. (6.8). For ξ = 1 these states are the SU(2) coherent states while for ξ = 1 they are SU(2) squeezed coherent states being minimum uncertainty states of the uncertainty product (5.5) with reduced ?uctuations in the component j⊥,1 for ξ < 1. They satisfy the squeezing criterion suitable for interferometric and spectroscopic measurements approaching the Heisenberg limit [1, 11]. It is worth stressing that for any ξ the vanishing of the eigenvalue in Eq. (6.6) grants that j⊥,k , k = 1, 2, are actually transversal components for all parameters ξ . This can be readily seen by projecting Eq. (6.6) on |ξ . Furthermore, we can show that Eq. (6.6) grants also that the operators j⊥,k in Eq. (6.6) and j = ?i[j⊥,1 , j⊥,2 ] are principal components. To this end we can project Eq. (6.6) on j |ξ leading to ξ |j j⊥,1 |ξ + iξ ξ |j j⊥,2 |ξ = 0. (6.7) The commutation relations and j⊥,k = 0 imply that j j⊥,k = j⊥,k j so that the mean values in Eq. (6.7) are real quantities. Thus Eq. (6.7) implies that both mean values vanish and j is a principal component both ? . Moreover, by adding the projections on for M and M j⊥,1 |ξ and ?iξj⊥,2 |ξ we get that j⊥,k are uncorrelated in the sense that ξ |(j⊥,1 j⊥,2 + j⊥,2 j⊥,1 )|ξ = 0. Therefore, M is diagonal in the j⊥,k , j basis so they are the principal components of M . The vanishing of the eigenvalue in Eq. (6.6) is the only possibility dealing with principal components since otherwise the correlations between components are proportional to the eigenvalue, spoiling property (iii) in Sec. III [23]. All this suggests that Eq. (6.6) may be taken as the proper SU(2) invariant form of de?ning the SU(2) squeezed coherent states. The exact solution of Eq. (6.6) for arbitrary ξ is di?cult to handle [1]. For de?niteness we can consider the limit ξ → 0 retaining the ?rst nonvanishing power on ξ . In the basis |j, m of eigenvectors of j0 and j⊥,1 we have i |ξ ? N |j, 0 ? ξ 2 j (j + 1) (|j, 1 ? |j, ?1 ) , (6.8) The complex Hermitian covariance matrix is no longer diagonal in the j⊥,k basis ? = j (j + 1) M 2 ξ2 ?iξ 0 iξ 1 0 0 0 1 . (6.11)

The principal components are ?⊥,1 = j⊥,1 + iξj⊥,2 , J ?⊥,2 = j⊥,2 + iξj⊥,1 , J ? =j , J with principal variances ?⊥,1 )2 = 0, (?J ?⊥,2 )2 ? (?J ? )2 ? 1 j (j + 1). (?J 2

(6.12)

(6.13)

?1 is equivalent to the eigenvalue The vanishing of ?J equation (6.6).

C. Schr¨ odinger cat states

In this context a suitable example of Schr¨ odinger cat states are the coherent superposition of two opposite SU(2) coherent states. In the basis of simultaneous eigenvectors of j0 and a properly chosen j3 we have 1 |ψ = √ (|j, j + |j, ?j ) , 2 j0 |ψ = j |ψ , (6.14)

which for large j are also known as maximally entangled states, or NOON states, because of their form in the number basis of two-mode bosonic realizations, being also of much interest in metrological applications [21]. For j = 1/2 these are SU(2) coherent states while for ? coincide j ≥ 1 we have j = 0 and M and M ? = M =M j/2 + δj,1 /2 0 0 0 j/2 ? δj,1 /2 0 0 0 j2 . (6.15)

where N is a normalization constant. In this approximation, the principal variances of M are (?J⊥,2 )2 ? (?J )2 ? 1 2 j (j + 1), with J ? j (j + 1)ξ. (6.10)

1 (?J⊥,1 )2 ? 2 j (j + 1)ξ 2 ,

(6.9)

It can be seen that |ψ is a minimum uncertainty state for the product and sum of variances in Eqs. (5.14) and (5.17) for u, v = 1, 2. On the other hand, the sum of three variances takes the maximum value possible in Eq. (5.2). Moreover, the scaling of the maximum principal variance as (?J1 )2 ∝ j 2 con?rms the metrological usefulness of these states [21].

D. States |j, 0 + |j, 1

This is a minimum uncertainty state for the uncertainty product in Eq. (5.5) while for the sums of variances it behaves essentially as the state |j, m = 0 . The metrological usefulness of these states is con?rmed by the scaling of the maximum principal variance as (?J1 )2 ∝ j 2 , in accordance with point (ix) in Sec. III.

Finally, let us consider the following states expressed in the basis of eigenvectors of j0 and j3 as 1 |ψ = √ (|j, 0 + |j, 1 ) , 2 (6.16)

9 with applications in quantum metrology [1]. In our context these states provide an example where the longitudinal component is not principal. In this case j = 1 2 j (j + 1), 0, 1 , (6.17) derived from the diagonalization of the covariance matrix for the problem. We have considered two forms for the covariance matrix, real symmetric and complex Hermitian. We have related the principal variances with meaningful SU(2) invariant uncertainty relations. In particular we have derived nontrivial uncertainty relations for states with vanishing mean values of all angular-momentum components, for which all previously introduced variance products are trivially bounded by zero. We have found that the corresponding minimum uncertainty states are the maximally entangled states (NOON states or Schr¨ odinger cat states). Moreover, we have demonstrated that all pure states with vanishing mean angular momentum are optimum for metrological applications since they can reach the Heisenberg limit.

so that the longitudinal component is given by j = sin θj1 + cos θj3 , tan θ = j (j + 1). (6.18)

On the other hand, M is diagonal in the j basis M= and ? 0 j (j + 1) ? , 1 (6.20) so that no principal component coincides with j . ? j (j + 1) ? 1 i ? = 1? ?i 2j (j + 1) ? 1 i M 4 0 ?i j (j + 1) 1 4 j (j + 1) ? 1 0 0 0 2j (j + 1) ? 1 0 0 0 1 , (6.19)

Acknowledgment VII. CONCLUSIONS

In this work we have elaborated the assessment of angular-momentum ?uctuations via principal variances

A. L. acknowledges the support from project PR1A/07-15378 of the Universidad Complutense.

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