Characterization of quantum angular-momentum fluctuations via principal components

0.17 MB·English

by  Ángel Rivas

Characterization of quantum angular-momentum fluctuations via principal
components
A´ngel Rivas and Alfredo Luis
∗
Departamento de O´ptica, Facultad de Ciencias F´ısicas,
Universidad Complutense, 28040 Madrid, Spain
(Dated: February 2, 2008)
We elaborate an approach to quantum fluctuations of angular momentum based on the diago-
nalization of the covariance matrix in two versions: real symmetric and complex Hermitian. At
differencewithpreviousapproachesthisisSU(2)invariantandavoidsanydifficultycausedbynon-
trivial commutators. Meaningful uncertainty relations are derived which are nontrivial even for
vanishingmean angular momentum. Weapply this approach tosome relevant states.
8
PACSnumbers: 03.65.Ca,42.50.St,42.25.Ja
0
0
2
I. INTRODUCTION polarization entanglement [5].

n
In comparisonto other fundamental variables,such as
a
J Quantumfluctuationsanduncertaintyrelationsplaya position and momentum, the standard uncertainty rela-
key role in the fundamentals of quantum physics and its tionsforangularmomentumrunintotwoseriousdifficul-
9
applications. In this work we focus on angular momen- ties: basic commutators are operators instead of num-

] tumvariables. Besidemechanics,angularmomentumop- bers, and there is lack of SU(2) invariance. Nontriv-
h
erators are ubiquitous in areas such as quantum optics, ial commutation relations lead to uncertainty products
p
matter-lightinteractions,andBose-Einsteincondensates. bounded by state-dependent quantities. Among other
-
t Basic observables such as light intensity, number of par- consequences,theseboundsbecometrivialforstateswith
n
ticles, and atomic populations are formally equivalent to vanishingmeanangularmomentum. Onthe otherhand,
a
u angular momentum components [1–5]. This is also the lack of SU(2) invariance is a basic difficulty since two
q caseoftheStokesparametersrepresentinglightpolariza- states connected by a SU(2) transformation should be
[ tion and the internal state of two-level atoms identified equivalentconcerningquantumfluctuations, inthe same

asspin1/2systems[5–7]. Angularmomentumoperators sense that phase-space displacements are irrelevant for
2
v are also the generatorsof basic operations such as phase position-momentum uncertainty relations.
9 shifting, beam splitting, free evolution, as well as atomic Inthisworkweelaborateanapproachtoquantumfluc-
9 interactions with classical fields [7, 8]. tuationsforangularmomentumvariableswhichisSU(2)
6
Moreover, angular momentum fluctuations are crucial invariant and avoids the difficulties caused by nontrivial
4
indiverseareas. Thisisforexamplethecaseofquantum commutators. The analysis is based on the diagonal-
.
0 metrology, implemented by very different physical sys- ization of the covariance matrix, that we study in two
1 tems such as two beam interference with light or Bose- versions: real symmetric (Sec. III) and complex Her-
7
Einsteincondensates,oratomicpopulationspectroscopy. mitian (Sec. IV). We examine the main properties of
0
: Thisisbecauseangular-momentumuncertaintyrelations their eigenvalues (principal variances) and eigenvectors
v determinetheultimatelimittotheresolutionofinterfer- (principalcomponents). In Sec. V we derive uncertainty
i
X ometric and spectroscopic measurements [1–3]. A dra- relations involving principal variances that are meaning-
matic example has been put forward in Ref. [3] con- ful even in the case of vanishing mean values. Finally, in
r
a cerning atomic clocks based on atomic population spec- Sec. VI we illustrate this approach applying it to some
troscopy, whose signal to noise ratio is proportional to relevant examples. Section II is devoted to recall basic
the square root of the duration of the measurement. In concepts and definitions.
such a case, an atomic clock using 1010 atoms prepared
inanstatewithreducedangularmomentumfluctuations
would yield the same signal to noise ratio in a measure-
ment lasting 1 second as an atomic clock with the same
II. ANGULAR MOMENTUM OPERATORS
numberofatomsinanstatewithoutreducedangularmo-
AND SU(2) INVARIANCE
mentumfluctuationsinameasurementlasting300years.
From a different perspective it has been shown that an-
gular momentum fluctuations are useful in the analysis Let us consider arbitrary dimensionless angular mo-
of many-body entanglement [4] and continuous-variable mentum operators j = (j1,j2,j3) satisfying the commu-
tation relations
3
∗Electronic address: [email protected]; URL: http://www.ucm. [jk,jℓ]=i ǫk,ℓ,njn, [j0,j]=0, (2.1)
es/info/gioq
n=1
X
2
whereǫk,ℓ,nisthefullyantisymmetrictensorwithǫ1,2,3 = with θ a real parameter, and u a unit three-dimensional
1, and j0 is defined by the relation real vector. It can be seen that the action of U on j is a
rotation R of angle θ and axis u [10]
j2 =j0(j0+1). (2.2)
U jU =Rj, (2.7)
†
Forthesakeofcompletenesswetakeintoaccountthatj0
maybeanoperator. Thisisthecaseoftwo-modebosonic
where RtR = RRt = I, I is the 3 3 identity, and
realizations where j0 is proportional to the number of the superscript t denotes matrix trans×position. So, the
particles.
SU(2)invarianceisjustthemathematicalstatementcor-
Morespecifically,denotingbya1,2 theannihilationop- responding to the fact that the conclusions which one
eratorsoftwoindependentbosonicmodeswith[aj,a†j]= could draw from an angular momentum measurement
1, [a1,a2] = [a1,a†2] = 0, we get that the following oper- must be independent of which set of three orthogonal
ators angular momentum components one chooses.
One way to guarantee the cited invariance is obtained
1 1
j0 = 2 a†1a1+a†2a2 , j1 = 2 a†2a1+a†1a2 , by using specific components of the angular momentum
referred to the mean value j , the longitudinal j and
(cid:16) (cid:17) (cid:16) (cid:17) (2.3) transversal j componentshwiith k = 1,2. Thesekcom-
,k
i 1 ponentsaret⊥heprojectionsofj onasetofCartesianaxes
j2 = 2 a†2a1−a†1a2 , j3 = 2 a†1a1−a†2a2 , adaptedto j sothatthe longitudinalaxispointsinthe
(cid:16) (cid:17) (cid:16) (cid:17) direction ofh ji [11, 12]. Therefore, by construction
satisfy Eqs. (2.1) and (2.2) [9]. Concerning physical re- h i
alizations, a1,2 can represent the complex amplitude op- j = j , j =0. (2.8)
erators of two electromagnetic field modes. For material |h i| h ki h ⊥,ki
systemstheycanrepresenttheannihilationoperatorsfor
Among other properties j ,j serve to properly define
two species of atoms in two different internal states, for
example. In any case, the operator j0 is proportional to SU(2) squeezing as reducedk fl⊥uctuations of a transversal
component [1, 11]. However this approach breaks down
the total number of photons or atoms and j3 is propor-
tional to the number difference. Therefore, these oper- in states with vanishing mean angular momentum.
ators represent basic detection mechanisms such as the Finallywerecallabasicrelationshowingthatangular-
measurement of light intensity or number of atoms. In momentum fluctuations limit the resolution of interfero-
this regard we have the following correspondence metricandspectroscopicmeasurements[1–3]. Thisisbe-
causeangularmomentumcomponentsj describeatomic
n
j,m = n1 =j+m n2 =j m , (2.4) andlightfreeevolution,aswellastheactionoflinearop-
| i | i| − i
tical devices such as beam splitters and interferometers,
between the standard angular momentum basis j,m of via unitary transformations of the form U = exp(iφj )
| i φ n
simultaneous eigenvectors of j3 and j0, i. e., j3 j,m = actinginagiveninitialstateρ. Theobjectiveofinterfer-
| i
mj,m , j0 j,m = j j,m , and the product of number ometric and spectroscopic measurements is the accurate
| i | i | i
states in the two modes |n1i|n2i with a†jaj|nji=nj|nji. determination of the value of the phase shift φ. This is
Angular momentum operators also serve to describe carriedoutbythemeasurementofagivenobservableAin
the internal state of two-level atoms via the definitions the transformed state UφρUφ†. In a simple data analysis,
theuncertainty∆φintheinferredvalueofφisrelatedto
1 1
j0 = (e e + g g ), j1 = (g e + e g ), the fluctuations of the measured observable in the form
2 | ih | | ih | 2 | ih | | ih | [1]
(2.5)
i 1 1
j2 = 2(|gihe|−|eihg|), j3 = 2(|eihe|−|gihg|), ∆φ= ∂hAi − ∆A= ∆A 1 , (2.9)
∂φ [A,j ] ≥ 2∆j
(cid:12) (cid:12) |h n i| n
where e,g are the excited and ground states. This is (cid:12) (cid:12)
(cid:12) (cid:12)
formall|y ani spin 1/2 where j0,3 represent atomic pop- where the (cid:12)uncert(cid:12)ainty relation ∆A∆jn ≥ |h[A,jn]i|/2
ulations and j1,2 the atomic dipole [7]. Collections of has been used. Therefore, the accuracy of the detection
two-level atoms are described by composition of the in- is limited by the fluctuations of the angular momentum
dividual angular momenta. component generating the transformation. In the opti-
Throughout, by SU(2) invariance we mean that the mumcase∆jn j0 sothat∆φisinverselyproportional
∝h i
density operators ρ and UρU are fully equivalent con- to the mean number of particles. This ultimate limit is
†
cerningquantumfluctuations,whereU isanySU(2)uni- known as Heisenberg limit [1]. It is worth stressing that
tary operator exponential of the angular momentum op- this conclusion applies exclusively to pure states, as re-
erators quiredbytheequalityintheuncertaintyproductbetween
A and j . These results are supported by more involved
n
U =exp(iθu j), (2.6) analyses [13].
·
3
III. PRINCIPAL COMPONENTS FOR REAL and
SYMMETRIC COVARIANCE MATRIX
J2 =j2, J 2 = j 2. (3.9)
h i h i
Complete second-order statistics of the operators j in
(v) At mostone principalvariance canvanishfor j0 =
a given state ρ is contained in the 3 3 covariance ma- 6
× 0 since for j0 = 0 no state can be eigenvector of more
trix associatedwith ρ. Due to the lack ofcommutativity 6
than one angular-momentum component.
we can propose two different covariance matrices: real
(vi) The principal variances provide an SU(2) invari-
symmetric and complex Hermitian.
antcharacterizationoffluctuations. Theinvarianceholds
In this section we focus on the real symmetric covari-
because under any SU(2) transformation (2.6), (2.7) we
ance 3 3 matrix M with matrix elements
× get that M transforms as M RMRt. Therefore, the
1 covariance matrix RMRt asso→ciated to the state U ρU
M = ( j j + j j ) j j , (3.1) †
k,ℓ k ℓ ℓ k k ℓ
2 h i h i −h ih i hasthesameprincipalvariancesasthecovariancematrix
where mean values are taken with respect to ρ and we M associated to the state ρ.
have M = M = M . Next we analyze the main (vii) The principal variances are the extremes of the
k∗,ℓ k,ℓ ℓ,k
properties of M. variances of arbitrary angular momentum components
(i) The covariancematrixM allowsus to compute the ju =u j for fixed ρ when the real unit vector u is var-
·
variances (∆j )2 of arbitrary angular-momentum com- ied. Morespecifically,fromEq. (3.2),takingintoaccount
u
ponentsj =u j,whereuisanyunitrealvector,inthe that M is symmetric, and using a Lagrangemultiplier λ
u
form · for the constraint u2 = 1, we have that the extremes of
∆j when u is varied are given by the eigenvalue equa-
(∆j )2 =utMu. (3.2) u
u tion
Similarly, M allows us to compute the symmetric cor-
Mu=λu, (3.10)
relationoftwoarbitrarycomponentsj =u j,j =v j,
u v
· ·
where u, v are unit real vectors, in the form
so that from Eq. (3.5) the extremes coincide with the
1 principal components.
( j j + j j ) j j =vtMu=utMv. (3.3)
2 h u vi h v ui −h uih vi (viii)Nextweexaminetherelationbetweentheprinci-
pal components and the longitudinal j and transversal
(ii)Since M isrealsymmetricthe transformationthat
j components, with k = 1,2. Wekcan demonstrate
renders M diagonal is a rotation matrix R ,k
d th⊥at j is a principal component when the state ρ is
(∆J1)2 0 0 invariaknt UρU = ρ under the unitary transformation
M =Rdt 0 (∆J2)2 0 Rd. (3.4) U =exp(iπj ).†This is because for this transformation
0 0 (∆J3)2! k
The eigenvalues (∆J )2, k = 1,2,3, are the variances of U†j ,kU = j ,k, U†j U =j , (3.11)
k ⊥ − ⊥ k k
the operators J = u j, where u are the three real
k k · k and
orthonormal eigenvectors of M
Muk =(∆Jk)2uk. (3.5) U†j⊥,kjkU =−j⊥,kjk, (3.12)
This implies that M is a positive semidefinite matrix. so that
Followingstandardnomenclatureinstatisticswereferto
J and ∆J = (∆J1,∆J2,∆J3) as principal components UρU† =ρ j ,kj =0, (3.13)
→h ⊥ ki
and variances, respectively. We stress that J and ∆J
and similarly for the opposite ordering j j = 0.
depend on the system state ρ. ,k
Thenj isaprincipalcomponentandtheohthker⊥twioprin-
(iii)Foreveryρthe principalcomponentsareuncorre-
cipal cokmponents are transversal.
lated
InvarianceunderU =exp(iπj )isaveryfrequentsym-
1
( J J + J J ) J J =0, k =ℓ. (3.6) metry. For bosonic two-mode rekalizations this is equiva-
k ℓ ℓ k k ℓ
2 h i h i −h ih i 6
lent to symmetry under mode exchange a1 a2 for the
↔
(iv) The principal components J are related to j by modesforwhichj =(a†1a2+a†2a1)/2,wherea1,2 arethe
the rotation Rd that diagonalizes M correspondingannkihilationoperators. Thisisbecausefor
J =R j. (3.7) thesemodesU†a1U =ia2 andU†a2U =ia1 andwehave
d
mode exchange except for a global π/2 phase change.
Thus, the three operators J are legitimate mutually or- (ix) According to Eq. (2.9) the maximum principal
thogonalHermitianangular-momentumcomponentssat- variance of ρ provides an assessment of the resolution
isfying the standard commutation relations achievable with ρ in the detection of small phase shifts
generated by angular momentum components. This in-
3
[J ,J ]=i ǫ J , (3.8) cludes all linear interferometric and spectroscopic mea-
k ℓ k,ℓ,n n
surements. Therefore, the maximum principal variance
n=1
X
4
provides an estimation of the usefulness of the corre- More specifically, for u = u we have that j is not
∗ u
6
sponding state in quantum metrology. Hermitian, j = j , so that the variance must be rede-
u† 6 u
(x) Next we derive upper and lower bounds for the fined, for example in the form [14]
principalvariancesof states with j =0. In sucha case
from Eqs. (2.2) and (3.9) we haveh i (∆ju)2 =hju†jui−|hjui|2. (4.3)
trM =(∆J1)2+(∆J2)2+(∆J3)2 = j0(j0+1) . (3.14) Then M˜ provides ∆ju as
h i
If we arrange the principal variances in decreasing order (∆ju)2 =u†M˜u. (4.4)
∆J1 ≥ ∆J2 ≥ ∆J3, and using Eq. (3.14) we get the Note that for real u we have
following bounds for the principal variances
hj02i≥(∆J1)2 ≥ 31hj0(j0+1)i, u†M˜u=utMu. (4.5)
12hj0(j0+1)i≥(∆J2)2 ≥ 12hj0i, pleSximpirloarjelyc,tiwonescajn c=omupujte, tjhe=covrreljatwiohneoref tuw,ovcoamre-
31hj0(j0+1)i≥(∆J3)2 ≥0. (3.15) arbitrary unit comuplex ve·ctorsv, in the·form
cTohmepuopnpeenrtboj2undfojr2∆,Jw1hhiloeldthsebleocwauersebfoournadnisarrbeiatcrhaerdy hjv†jui−hjv†ihjui=v†M˜u. (4.6)
when all thhekpiri≤nchip0ail variances are equal. The upper (ii)SinceM˜ isHermitianitbecomesdiagonalbymeans
boundfor∆J2isreachedwhen∆J1 =∆J2and∆J3 =0, of a 3 3 unitary matrix d
whilethelowerboundisreachedwhen(∆J1)2 = j02 and × U
∆J2 =∆J3. Finally,theupperboundfor∆J3 ishreaiched (∆J˜1)2 0 0
whenalltheprincipalvariancesareequal,whilethelower M˜ =Ud† 0 (∆J˜2)2 0 Ud, (4.7)
bound occurs for ∆J3 =0. 0 0 (∆J˜3)2
(xi)Thelowerboundfor∆J1 inEq. (3.15)is,roughly  
speaking, of the order of j0 so that from Eq. (2.9) all wheretheelementsonthediagonalarethevariances(4.3)
pure states with j = 0hcain reach maximum interfer- of the operators J˜k = uk j, where uk are the three
ometric precisionh(Hieisenberg limit). This explains why complex orthonormaleigenv·ectors of M˜
mostoptimumstatesformetrologicalapplicationssatisfy
j =0 (see Sec. VI). M˜uk =(∆J˜k)2uk. (4.8)
h i
This implies that M˜ is positive semidefinite. We again
IV. PRINCIPAL COMPONENTS FOR refer to J˜and ∆J˜= (∆J˜1,∆J˜2,∆J˜3) as principal com-
ponentsandvariances,respectively. Sinceaglobalphase
COMPLEX HERMITIAN COVARIANCE
MATRIX is irrelevant we regard always d as an SU(3) matrix.
U
(iii) For every state ρ the principal components are
uncorrelated
In this section we elaborate the statistical description
omfitainagnu3lar-m3ocmoveanrtiaunmceflumcatturaixtioMn˜swviitahthmeactroimxpelleexmHenetrs- hJ˜k†J˜ℓi−hJ˜k†ihJ˜ℓi=0, k 6=ℓ. (4.9)
×
(iv)Standardcommutationrelations(2.1)arenotpre-
M˜ = j j j j , (4.1)
k,ℓ k ℓ k ℓ served under transformations by unitary 3 3 matrices
h i−h ih i
×
with M˜k∗,ℓ = M˜ℓ,k. The two matrices M and M˜ contain J˜= j. (4.10)
U
essentiallythesameinformationsincetheyonlydifferby
a factor j However, a slight modification of (2.1) is actually pre-
h i served under the action (4.10) of SU(3) matrices. These
i 3 are
M =M˜ ǫ j . (4.2)
k,ℓ k,ℓ k,ℓ,n n
− 2 h i 3
nX=1 [J˜k,J˜ℓ]=i ǫk,ℓ,nJ˜n†, [j0,J˜]=0, (4.11)
In particular they coincide exactly for all states with n=1
X
j =0.
h i which are equivalent to Eq. (2.1) for Hermitian opera-
Nevertheless, M and M˜ are different matrices so we
tors. We have also
canexploitthisdifferencebyexaminingthemostrelevant
features that they do not share in common. J˜ J˜=j2, J˜ J˜ = j 2. (4.12)
† ∗
(i) The covariance matrix M˜ provides the properly · h i ·h i h i
defined variances of complex linear combinations of Thepreservationof(4.11)undertheactionofSU(3)ma-
angular-momentum components j = u j for arbitrary trices in Eq. (4.10) can be demonstrated by direct com-
u
·
unit complex vectors u with u u=1. putation of the commutators after expressing the most
†
5
generalSU(3)matrix asasuitableproductofmatrices variation process for the complex Hermitian case takes
U
belonging to the SU(2) subgroup of the form [15] place over a larger set of operators j , with complex u,
u
that includes as a particular case the projections on real
cosθeiφ sinθeiϕ 0 u.
= sinθe−iϕ cosθe−iφ 0 , (4.13) (x) Because of Eqs. (4.14) and (4.15) for j = 0 the
U − 0 0 1! h i
upper and lower bounds for principal variances in Eq.
(3.15) also hold replacing J by J˜.
and permutations of lines and columns. Each matrix
(4.13) transforms operators J˜satisfying Eq. (4.11) into (xi) It is questionable whether ∆ju for ju† 6= ju repre-
anothersetofoperatorsJ˜ = J˜fulfillingthesamecom- sents practical observable fluctuations. For example, for
mutation relations (4.11).′ U ju =j1+ij2 we have
(v)FromEq. (4.2)wecanderivetheequalityoftraces 2 2 2
trM =trM˜ so that (∆ju) =(∆j1) +(∆j2) −hj3i, (4.19)
∆J˜ 2 =(∆J)2 =(∆j)2 = j0(j0+1) J 2, (4.14) scoasweeofcatnhehSavUe(2∆)jcuoh=er0enwtitshta∆tejs1,(2se6=e S0e,cb.eiVnIg).thNisevtehre-
h i−h i
(cid:16) (cid:17) theless, from Eq. (4.14) we have that ∆J˜ contains all
with
angular momentum fluctuations.
2 In this regardit is worth recalling that non Hermitian
J˜ = J 2 = j 2. (4.15)
h i h i h i operators can be related to experimental processes, as
(cid:12) (cid:12) demonstrated by double homodyne detection where the
Since J 2 j0 2(cid:12)(cid:12)we(cid:12)(cid:12)have(∆J˜)2 >0. Thisimpliesthat statistics is given by projection on quadrature coherent
h i ≤h i
there mustbe atleasta nonvanishingprincipalvariance. states[16]. Inourcontext,theeigenstatesofju =j1+ij2
Otherwise, two principal variances of M˜ can vanish si- are SU(2) coherent states that define by projection the
multaneously for the same state, as shownin Sec. VI for SU(2) Q function [10]. This is an observable probability
the SU(2) coherent states. distribution function, via double homodyne detection of
(vi) Although the traces of M and M˜ are equal the two field modes for example [17].
determinants are different. This can be easily proven
by expressing M˜ in the principal-component basis that
renders M diagonal
V. UNCERTAINTY RELATIONS
(∆J1)2 i J3 /2 i J2 /2
M˜ = i J3 /2 (h∆Ji2)2 −i Jh1 /i2 , (4.16) Variances are the most popular building blocks of un-
−i Jh2 /i2 i J1 /2 (h∆Ji3)2 ! certaintyrelations. Forangularmomentum,thestandard
h i − h i procedure leads to
so that
1
detM˜ =detM 1 3 (∆J )2 J 2, (4.17) ∆j1∆j2 ≥ 2|hj3i|, (5.1)
k k
− 4 h i
Xk=1 and cyclic permutations. This uncertainty relation faces
two difficulties. On the one hand it is bounded by
and detM detM˜.
(vii) The≥principal variances of M˜ are SU(2) invariant an state-dependent quantity that vanish for states with
j = 0. On the other hand, it lacks SU(2) invariance
sinceunderSU(2)transformationswehaveM˜ RM˜Rt, h i
→ so that it leads to different conclusions when applied to
so that the covariance matrix RM˜Rt for the state U ρU
† SU(2) equivalent states [18]. These difficulties can be
has the same eigenvaluesas the covariancematrix M˜ for
avoidedbyusingtheprincipalvariancesleadingtomean-
the state ρ.
ingfulSU(2)invariantrelationswhicharenontrivialeven
(viii) The principal variances are the extremes of the for states with j =0.
variances (∆j )2 of any complex combination of angular h i
u A first SU(2) invariantuncertainty relationcanbe de-
momentum components ju =u·j where u is a complex rived from the trace of M (or equivalently M˜) [19]
unit vector. More specifically, from Eq. (4.4), and intro-
dcouncsintrgaiantLaug†ranuge=m1u,lwtipeligeertλthtaotttahkeeeinxttoreamcecsouonft∆thjue trM =(∆J)2 =hj0(j0+1)i−hJi2 ≥hj0i, (5.2)
·
are given by the eigenvalue equation
where we have used that for any component j 2
k
M˜u=λu, (4.18) hjk2i≤hj02i. h i ≤
2
The minimum uncertainty states with (∆J) = j0
so that from Eq. (4.8) the extremes of (∆ju)2 are the are obtained for maximum J 2 = j02 . This is satishfiedi
principal variances ∆J˜. exclusively by SU(2) coherehntistatehs [1i0, 19]
(ix) The principal variances of M˜ are more extreme
thantheprincipalvariancesofM sincefromEq. (4.5)the j,θ,u =U(θ,u)j,j , (5.3)
| i | i
6
where U is any SU(2) unitary operator (2.6) and j,m Equation (5.7) holds irrespectively of whether j is a
arethesimultaneouseigenvectorsofj0andj3,withe|igeni- principal component or not. Furthermore, when jk is a
values j and m, respectively. On the other hand, max- principal component of M or M˜ we have, respectivkely
imum uncertainty (∆J)2 is obtained for J = 0. This
h i 2 2
is the case of the state j,m = 0 for example (see Sec. (∆J ,1) +(∆J ,2) j0 ,
| i ⊥ ⊥ ≥h i
VI). 2 2
The uncertainty relation (5.2) is nontrivial even for ∆J˜ ,1 + ∆J˜ ,2 j0 . (5.9)
⊥ ⊥ ≥h i
j =0. Nevertheless,this is notvery informativeabout (cid:16) (cid:17) (cid:16) (cid:17)
h i
angular momentum statistics since this is actually just a
function of the first moments j . In order to proceed B. Case hji=0
h i
further deriving more meaningful uncertainty relations
letussplitthe analysisintwocases j =0and j =0. Stateswith j =0ariseveryofteninquantummetro-
h i6 h i h i
logical applications as explained in point (xi) of Sec. III
[1,20,21]. Insuchacasethestandarduncertaintyprod-
A. Case hji=6 0 ucts (5.1) are all trivial ∆j ∆j 0 since they do not
k ℓ
≥
establish any lower bound to the product of variances.
1. Product of variances Moreover,the components j , j are undefined.
k ⊥
For j = 0 an SU(2) invariant product of variances
canbehdeiri6vedbyapplyingthestandardproceduretothe 1. Product of variances
longitudinalandtransversalcomponents(2.8),leadingto
just one nontrivial relation [12] We can derive a suitable lower bound for the product
∆J1∆J2 of the two larger principal variances of M (or
∆j⊥,1∆j⊥,2 ≥ 12|hjki|, (5.4) M˜j)=wi0t.h ∆J1 ≥ ∆J2 ≥ ∆J3, valid for all states with
h i
Tothisendwebeginwithbyconsideringtheminimum
whiletheothertwoaretrivial∆j ∆j 0,fork =1,2.
,k
When J = j is a principal c⊥ompokne≥nt we can show of ∆J2 for fixed ∆J1. From the equality (3.14) we get
tthheatmtihneimpukrmincuipnkacelrttraainntsyveprrsoadluvcatriances ∆J⊥,k provide (∆J2)2+(∆J3)2 =hj0(j0+1)i−(∆J1)2, (5.10)
1 and for fixed ∆J1 the sum (∆J2)2+(∆J3)2 is constant.
∆j ,1∆j ,2 ∆J ,1∆J ,2 j . (5.5) Taking into account that ∆J2 ∆J3 we get that the
⊥ ⊥ ≥ ⊥ ⊥ ≥ 2|h ki| minimum ∆J2 occurs when ≥
This is because the determinant of M is invariant under
1
rotations of j, and the principal components are uncor- ∆J2 =∆J3 = j0(j0+1) (∆J1)2 , (5.11)
2 h i−
related. Moreover they are the extreme variances in the
h i
transversalplane according to point (vii) in Sec. III. and then it holds that
1
2 2 2 2
(∆J2) (∆J1) (∆J1) j0(j0+1) (∆J1) .
≥ 2 h i−
2. Sum of variances
h (5i.12)
The minimum of the right-handside takes place when
We canbegin with by particularizingEq. (2.2)to lon- ∆J1 reaches its extremes values in Eq. (3.15) so that
gitudinal and transversalcomponents leading to
1 1
(∆j ,1)2+(∆j ,2)2 = j0(j0+1) j2 . (5.6) (∆J2)2(∆J1)2 ≥min 2hj0ihj02i,9hj0(j0+1)i2 ,
⊥ ⊥ h i−h ki (cid:20) (cid:21)
(5.13)
Since we have always j02 j2 we get the following where “min” refers to the minimum of the alternatives.
h i ≥ h i
lower bound to the sum of transvkersalvariances For j0 2 we get that this is always
h i≥
(∆j⊥,1)2+(∆j⊥,2)2 ≥hj0i, (5.7) (∆J2)2(∆J1)2 1 j0 j02 , (5.14)
≥ 2h i
where the equality is reached by the SU(2) coherent
(cid:10) (cid:11)
states exclusively. andtheequalityisreachedforstateswithmaximum∆J1
Letus note thatthis relationis strongerthanthe sim- and minimum ∆J2
ilar one that can be derived from Eq. (5.4), 1
(∆J1)2 = j02 , (∆J2)2 =(∆J3)2 = j0 . (5.15)
2 2 2h i
(∆j⊥,1) +(∆j⊥,2) ≥|hjki|, (5.8) Wewillseein(cid:10)th(cid:11)e nextsectionthatthisisthecaseofthe
since j0 j always. Schr¨odinger cat states (6.14).
h i≥|h ki|
7
2. Sum of variances On the other hand, the complex Hermitian covariance
matrix is
Nontrivialbounds to sums ofvariances canbe derived j(j+1) m2 im 0
by particularizing Eq. (2.2) to three components ju, jv, M˜ = 1 im− j(j+1) m2 0 , (6.3)
jw obtained by projection on three mutually orthogonal 2 −0 0 − 0!
(complex in general) unit vectors u, v, w
with principal components
2 2 2
(∆ju) +(∆jv) +(∆jw) = j0(j0+1) . (5.16)
h i
S(∆injcwe)f2orwaengyeptrojection,sayjw,wehavehj02i≥hjw†jwi= JJ˜˜⊥⊥,,21 == √√1122jj−+ == √√1122((jj11+−iijj22)),,
(∆ju)2+(∆jv)2 j0 , (5.17) J˜ =j3, (6.4)
≥h i k
Awhsearebuy,pvroadruecotrtwhoegoobntaalicnomthpaletxfournitjvec=tor0suo†n·lvy=on0e. jso.thTahteJ˜p⊥r,i1n,c2ipaarlevparroiapnocretsioanrael to the ladder operators
h i
principal variance ∆J˜can vanish. ±
invFoilnvaellythewetrcaacne oafpptrheeciactoevatrhiaantcethemaatbroixvebreeinlagtiodnes- (∆J˜⊥,1)2 = 21[j(j+1)−m(m+1)],
rived seemingly without resorting to commutation rela- (∆J˜⊥,2)2 = 12[j(j+1)−m(m−1)],
tions. Nevertheless, commutation relations are also at (∆J˜)2 =0, (6.5)
the hearthof these uncertainty relations since this is the k
ultimatereasonforbiddingthe simultaneousvanishingof which are larger and lesser, respectively, than the vari-
the fluctuations of all angular momentum components. ances(6.2)oftherealsymmetriccase,inaccordancewith
In this regard, as shown in the original Schr¨odinger’s point (ix) of Sec. IV.
paper position-momentum uncertainty relations can be The SU(2) coherent states are minimum uncertainty
fruitfully related to the correspondingcovariance matrix states for the sum of three variances in Eq. (5.2), and
[22]. for the product andsumof variancesoftransversalcom-
ponents in Eqs. (5.5), (5.7), and (5.9). The scaling of
the largest principal variance as (∆J1)2 j agrees with
∝
VI. EXAMPLES the fact that the SU(2) coherentstates are not optimum
for metrologicalapplications. Optimum states scaling as
In this section we apply the preceding formalism to (∆J1)2 j2 can be found below in this section.
∝
some relevant and illustrative quantum states. ForSU(2)coherentstatestwooftheprincipalvariances
in Eq. (6.5) vanish. This corresponds to the fact that
they satisfy the double eigenvalue relation j j, j = 0
A. States |j,mi and j3 j, j = j j, j , so that j, j are±e|ig±ensitates
| ± i ± | ± i | ± i
of two of the principal components in Eq. (6.4).
Let us consider the simultaneous eigenstates j,m of On the other hand, for the states with m = j the
jfa0mainlydijn3clwuditehs etihgeenSvUal(u2e)scjohaenrdenmt straestpesecftoirveml|y.=Tihijs nfoornmvan=is0hitnhgevmaraixainmceusminsccraeleasseasfo(r∆dJe1c)r2easijn2g6, w|m±h|icahnids
andthelimitofSU(2)squeezedcoherentstatesform=±0 consistentwiththeusefulness ofthese stat∝esinquantum
[1,10]. Fortwo-modebosonicrealizationsthecasem=0 metrology,in agreementwith points (ix) and(xi) in Sec.
isthe productofstateswiththe samedefinite numberof III [20]. Moreover, the states m = 0 are far from the
particles in each mode [20]. lower bounds of the uncertainty relations in Eqs. (5.2),
For m=0 we have j =0 and there is a longitudinal (5.14), and (5.17).
6 h i6
component with j = j3. In such a case UρU† = ρ for
U =exp(iπj ) andkj is a principal component.
The realsykmmetrikc covariancematrix is directly diag- B. SU(2) squeezed coherent states
onal in any basis containing the longitudinal component
j3 =j Letusconsiderthe SU(2)squeezedcoherentstatesde-
k fined by the eigenvalue equations [1, 11]
j(j+1) m2 0 0
M = 12 00 − j(j+10)−m2 00!, (6.1) (j⊥,1+iξj⊥,2)|ξi=0, j0|ξi=j|ξi, (6.6)
where ξ is a real parameter with ξ 0 without loss
with principal variances ≥
of generality. The first of these equations corresponds
1 to the case of zero eigenvalue among a larger family of
(∆J)2 =0, j(j+1) m2 . (6.2) eigenvalue equations [1]. The states ξ are fully defined
2 − | i
(cid:2) (cid:3)
8
by the eigenvalue relations (6.6). An approximate solu- ThecomplexHermitiancovariancematrixisnolonger
tion is provided below in Eq. (6.8). For ξ = 1 these diagonal in the j basis
,k
states are the SU(2) coherentstates while for ξ =1 they ⊥
are SU(2) squeezed coherent states being minim6 um un- j(j+1) ξ2 iξ 0
certaintystates of the uncertaintyproduct (5.5)with re- M˜ = iξ 1 0 . (6.11)
ducedfluctuationsinthecomponentj ,1forξ <1. They 2 −0 0 1!
satisfythesqueezingcriterionsuitable⊥forinterferometric
The principal components are
andspectroscopicmeasurementsapproachingtheHeisen-
berg limit [1, 11]. J˜ ,1 =j ,1+iξj ,2,
It is worth stressing that for any ξ the vanishing of ⊥ ⊥ ⊥
tahreeaecigtuenalvlaylutreaninsvEerqs.al(c6o.m6)pgornaennttsstfohrataljl⊥p,akr,akme=ter1s,2ξ., J˜⊥,2 =J˜j⊥=,2j+,iξj⊥,1, (6.12)
This can be readily seen by projecting Eq. (6.6) on ξ . k k
| i
Furthermore, we can show that Eq. (6.6) grants with principal variances
also that the operators j in Eq. (6.6) and j =
,k
i[j ,1,j ,2] are principal⊥components. To this endk we (∆J˜ ,1)2 =0,
c−an ⊥proje⊥ct Eq. (6.6) on jk|ξi leading to (∆J˜⊥,2)2 ≃(∆⊥J˜k)2 ≃ 12j(j+1). (6.13)
hξ|jkj⊥,1|ξi+iξhξ|jkj⊥,2|ξi=0. (6.7) The vanishing of ∆J˜1 is equivalent to the eigenvalue
The commutation relations and j = 0 imply that equation (6.6).
,k
j j = j j so that the meha⊥n vialues in Eq. (6.7)
,k ,k
harke ⊥reail quhan⊥titieksi. Thus Eq. (6.7) implies that both
meanvaluesvanishandj isaprincipalcomponentboth C. Schr¨odinger cat states
for M and M˜. Moreoverk, by adding the projections on
j ,1 ξ and iξj ,2 ξ we get that j ,k are uncorrelated In this context a suitable example of Schr¨odinger cat
in⊥th|eisense−that⊥ξ|(ji ,1j ,2+j ,2j⊥,1)ξ =0. states are the coherent superposition of two opposite
Therefore, M ish d|ia⊥gon⊥al in th⊥e j⊥ ,j| ibasis so they SU(2)coherentstates. Inthebasisofsimultaneouseigen-
,k
are the principal components of M⊥. Thke vanishing of vectors of j0 and a properly chosen j3 we have
the eigenvalue in Eq. (6.6)is the only possibility dealing
1
with principal components since otherwise the correla- ψ = (j,j + j, j ), j0 ψ =j ψ , (6.14)
| i √2 | i | − i | i | i
tions between components are proportionalto the eigen-
value, spoiling property (iii) in Sec. III [23].
which for largej are also knownas maximally entangled
All this suggests that Eq. (6.6) may be taken as
states,orNOONstates,becauseoftheirforminthenum-
the proper SU(2) invariant form of defining the SU(2)
ber basis of two-mode bosonic realizations, being also of
squeezedcoherentstates. TheexactsolutionofEq. (6.6)
much interest in metrological applications [21].
for arbitrary ξ is difficult to handle [1]. For definiteness
For j = 1/2 these are SU(2) coherent states while for
we can consider the limit ξ → 0 retaining the first non- j 1 we have j =0 and M and M˜ coincide
vanishing power on ξ. In the basis j,m of eigenvectors ≥ h i
| i
of j0 and j⊥,1 we have j/2+δj,1/2 0 0
ξ N j,0 iξ j(j+1)(j,1 j, 1 ) , (6.8) M =M˜ = 00 j/2−0δj,1/2 j02!. (6.15)
| i≃ | i− 2 | i−| − i
(cid:20) (cid:21)
p Itcanbeseenthat ψ isaminimumuncertaintystate
where N is a normalization constant. | i
for the product and sum of variances in Eqs. (5.14) and
Inthisapproximation,theprincipalvariancesofM are
(5.17)foru,v =1,2. Ontheotherhand,thesumofthree
(∆J⊥,1)2 ≃ 21j(j+1)ξ2, vMaoriraenocveesr,ttahkeesscthaleinmgaoxfimthuemmvaaxliumeupmospsirbinleciipnaElvqa.r(i5a.n2c)e.
(∆J⊥,2)2 ≃(∆Jk)2 ≃ 12j(j+1), (6.9) as (∆J1)2 ∝ j2 confirms the metrological usefulness of
these states [21].
with
J j(j+1)ξ. (6.10)
h ki≃ D. States |j,0i+|j,1i
This is a minimum uncertainty state for the uncer-
tainty product in Eq. (5.5) while for the sums of vari-
Finally, let us consider the following states expressed
ances it behaves essentially as the state j,m = 0 .
| i in the basis of eigenvectors of j0 and j3 as
The metrological usefulness of these states is confirmed
by the scaling of the maximum principal variance as 1
ψ = (j,0 + j,1 ), (6.16)
(∆J1)2 j2, in accordance with point (ix) in Sec. III. | i √2 | i | i
∝
9
with applications in quantum metrology [1]. In our con- derivedfromthediagonalizationofthecovariancematrix
text these states provide an example where the longitu- fortheproblem. Wehaveconsideredtwoformsfortheco-
dinal component is not principal. In this case variancematrix,realsymmetricandcomplexHermitian.
We have related the principal variances with meaningful
1
j = j(j+1), 0, 1 , (6.17) SU(2) invariant uncertainty relations.
h i 2
(cid:0)p (cid:1) In particular we have derived nontrivial uncertainty
so that the longitudinal component is given by
relations for states with vanishing mean values of all
angular-momentumcomponents,forwhichallpreviously
j =sinθj1+cosθj3, tanθ = j(j+1). (6.18)
k introduced variance products are trivially bounded by
On the other hand, M is diagonal inpthe j basis zero. We have found that the corresponding minimum
uncertainty states are the maximally entangled states
j(j+1) 1 0 0 (NOON states or Schr¨odinger cat states). Moreover, we
1 −
M = 0 2j(j+1) 1 0 , (6.19) have demonstrated that all pure states with vanishing
4 0 0 − 1! mean angular momentum are optimum for metrological
applications since they can reach the Heisenberg limit.
and
j(j+1) 1 i 0
1 −
M˜ = i 2j(j+1) 1 i j(j+1) ,
4 − − 
0 i j(j+1) 1
− p
 (6.20)
p
so that no principal component coincides with j .
k
Acknowledgment
VII. CONCLUSIONS
In this work we have elaborated the assessment of A. L. acknowledges the support from project PR1-
angular-momentum fluctuations via principal variances A/07-15378of the Universidad Complutense.
[1] B.Yurke,S.L.McCall, andJ.R.Klauder,Phys.Rev.A [12] A. Luis and N. Korolkova, Phys. Rev. A 74, 043817
33, 4033 (1986); M. Hillery and L. Mlodinow, ibid. 48, (2006).
1548(1993);C.BrifandA.Mann,ibid.54,4505(1996). [13] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72,
[2] Y.CastinandJ.Dalibard,Phys.Rev.A55,4330(1997); 3439 (1994); S. L. Braunstein, C. M. Caves, and G. J.
J. A. Dunningham and K. Burnett ibid. 61, 065601 Milburn,Ann.Phys.(N.Y.)247,135(1996);A.Monras,
(2000); 70, 033601 (2004); V. Meyer, M. A. Rowe, D. Phys. Rev.A 73, 033821 (2006).
Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and [14] J.-M.L´evy-Leblond,Ann.Phys.(N.Y.)101,319(1976);
D.J. Wineland, Phys.Rev.Lett. 86, 5870 (2001). V. V. Dodonov, E. V. Kurmyshev, and V. I. Man’ko,
[3] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Phys. Lett. 79A, 150 (1980).
Heinzen,Phys. Rev.A 50, 67 (1994). [15] P. H. Moravek and D. W. Joseph, J. Math. Phys. 4,
[4] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Na- 1363 (1963); K. Nemoto, J. Phys. A 33, 3493 (2000);
ture (London) 409, 63 (2001); G. T´oth, Phys. Rev. A K. Nemoto and B. C. Sanders, ibid. 34, 2051 (2001).
69, 052327 (2004). [16] U.LeonhardtandH.Paul,Prog.QuantumElectron.19,
[5] N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and 89 (1995); U. Leonhardt, Measuring the Quantum State
Ch.Silberhorn, Phys. Rev.A 65, 052306 (2002); N. Ko- of Light (Cambridge University Press, Cambridge, Eng-
rolkovaand R.Loudon, ibid.71, 032343 (2005). land,1997);D.-G.Welsch,W.Vogel,andT.Opatrny´,in
[6] A. Luis and L. L. S´anchez-Soto, in Progress in Optics, Progress in Optics, edited by E. Wolf (Elsevier Science,
edited by E. Wolf (Elsevier, Amsterdam, 2000), Vol. 41, Amsterdam, 1999), Vol. 39.
p.421, and references therein. [17] G.S.Agarwal, Phys.Rev.A57,671(1998); J.P.Amiet
[7] M.O.ScullyandM.S.Zubairy,Quantum Optics (Cam- and S. Weigert, J. Opt. B: Quantum Semiclassical Opt.
bridge UniversityPress, Cambridge, England, 1997). 2, 118 (2000); S.Weigert, ibid.6, 489 (2004).
[8] A. Luis and L. L. S´anchez-Soto, Quantum Semiclass. [18] K. Wodkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2,
Opt.7, 153 (1995). 458 (1985).
[9] J. Schwinger, Quantum Theory of Angular Momentum [19] R. Delbourgo, J. Phys. A 10, 1837 (1977).
(Academic Press, New York,1965). [20] M.J.HollandandK.Burnett,Phys.Rev.Lett.71,1355
[10] F.T.Arecchi,E.Courtens,R.Gilmore,andH.Thomas, (1993).
Phys.Rev.A 6, 2211 (1972). [21] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990); J.
[11] M.KitagawaandM.Ueda,Phys.Rev.A47,5138(1993). J. Bollinger, W. M. Itano, D. J. Wineland, and D. J.
10
Heinzen, Phys. Rev. A 54, R4649 (1996); S. F. Huelga, Mitchell, J. S. Lundeen,and A. M. Steinberg, ibid. 429,
C.Macchiavello,T.Pellizzari,A.K.Ekert,M.B.Plenio, 161 (2004).
andJ.I.Cirac,Phys.Rev.Lett.79,3865(1997);A.Luis, [22] E. Schr¨odinger, e-printarXiv:quant-ph/9903100.
Phys.Rev.A 64, 054102 (2001); 65, 034102 (2002); Ph. [23] R. N. Deb, N. Nayak, and B. Dutta-Roy, Eur. Phys. J.
Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gas- D 33, 149 (2005).
paroni,andA.Zeilinger,Nature429,158(2004); M.W.