Entanglement and spin squeezing properties for three bosons in two modes

0.63 MB·English

by  B. Zeng

Entanglement and spin squeezing properties for three bosons in two modes
B. Zeng,1 D. L. Zhou,2 Z. Xu,3 and L. You2,4
1Department of Physics, Massachusetts Institute of Technology, MA 02139, USA
2School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
3Center for Advanced Study, Tsinghua University, Beijing 10084, China and
4Interdisciplinary Center of Theoretical Studies, The Chinese Academy of Sciences, Beijing 10080, China
(Dated: February 1, 2008)
Wediscussthecanonicalformforapurestateofthreeidenticalbosonsintwomodes,andclassify
its entanglement correlation into two types, the analogous GHZ and the W types as well known
in a system of three distinguishable qubits. We have performed a detailed study of two important
entanglementmeasuresforsuchasystem,theconcurrenceC andthetripleentanglementmeasureτ.
Wehavealso calculated explicitly the spin squeezing parameter ξ and theresult shows that the W
5 stateisthemost“anti-squeezing”state,forwhichthespinsqueezingparametercannotberegarded
0 as an entanglement measure.
0
2 PACSnumbers: 03.65.Ud,03.67.Mn

n
a
I. INTRODUCTION in the following form
J

8 M
1 Quantum entanglement is an intriguing property of Ψ = λ a†a† 0 , (2)
composite systems. It refers to the inseparable corre- | i Xi=1 i i i| i
2 lations stronger than all classical counterparts. Recent
v studies indicate thatentanglementis notonly ofinterest wherea† 0 formsanorthonormalbasisinthesinglepar-
2 i| i
to the interpretation of the foundations of quantum me- ticle (boson) space.
8
0 chanics,butalsorepresentsausefulresourceforquantum The above two results are in fact simple extensions of
1 computation and quantum communication. Inseparable the well known result for two distinguishable particles;
0 correlations such as entanglement also exist in systems thatanarbitrarypure state canbe describedin termsof
5 of identical particles, e.g. electrons in quantum dots [1], the famous Schmidt decomposition
0 atomsinaBose-Einsteincondensate[2,3],andelectrons
/
h in quantum Hall liquids [4]. Even for the widely used ψ = λi i1i2 , (3)
| i | i
p parametric down conversion process, a complete treat- Xi p
- ment must take into account the indistinguishability of
t with real parameter λ satisfying λ = 1 and
n the down converted photons. Although the study of en- i i i
a tanglement has had a long history in systems of distin- im jn = δmnδij. A natural genePralization of the
h | i
u Schmidt decomposition to N >2 particles is
guishable particles, only recently did the entanglement
q
properties in a system of identical particles begin to at-
:
v tract much attention [1, 5, 6, 7, 8, 9, 10]. |ψi= λi|i1i2···iNi, (4)
i Xi p
X Quantum correlation among identical particles was
r noted by Schliemann et al. [1, 5] and they discussed with the sub-indices m (n) for the m-th (n-th) particle,
a the entanglement in a two-fermion system. It was ar- and i (j) the i-th (j-th) basis vector. However, even for
gued that the separability of a two fermion state should three two-stateparticlesorthree qubits with the Hilbert
be defined in terms of whether or not that state can be space = C2 C2 C2, the above Schmidt decom-
H ⊗ ⊗
expressedintermsofasumofsingleSlaterdeterminants position does not exist, pointing to a truly challenging
[1]. More generally, a two fermion pure state can always prospect for characterizing the multi-particle entangle-
be expressed in the following standard form [5] ment.
For a multi-particle system, the characterizationof its
k entanglementfor a pure state usually starts with certain
1
Ψ = z f† f† 0 , (1) canonicalformofitswavefunction. Inthisstudy,wecon-
| i q ki=1|zi|2 Xi=1 i a1(i) a2(i)| i sfotrrutchtrteheetwstoa-nsdtaatredifdoernmticoaflabnosaornbsit.raWryewfauvrtehfeurncchtiaorn-
P
acterize its correlation and squeezing properties based
where f† 0 and f† 0 represent the orthonormal
a1(i)| i a2(i)| i on the standard form. This paper is organized as fol-
basis states of a single particle (fermion). lows. In Sec. II, we survey the important results on the
The case oftwobosonswere consideredindependently standardformof a pure state wavefunction ofthree dis-
by Paˇskauskasand You [6] and Li et al. [7]. They found tinguishable qubits. We then present our result for the
asimilarstandardformfortwo-identicalbosons,namely, case of three bosons in two modes in Sec. III, which is
the wave functions of two bosons can always be written followed by a detailed discussion of entanglement types,
2
entanglement measures, and spin squeezing properties, III. THE STANDARD FORM OF AN
respectively in Sec. IV, Sec. V, and Sec. VI. Finally ARBITRARY PURE STATE FOR THREE
we discuss the relationship between spin squeezing and TWO-STATE BOSONS
pairwise entanglement in our system and conclude with
a summary. For three identical two-state bosons and in the first
quantizationrepresentation,the generalformofits wave
function reads
II. THE STANDARD FORM OF AN
ARBITRARY PURE STATE FOR THREE ψ = a000 +b(100 + 010 + 001 )
QUBITS | i | i | i | i | i
+c(011 + 101 + 110 )+d111 . (9)
| i | i | i | i
The Schmidt decomposition does not exist for a three After a single particle basis transformation
particle pure state, as provenby A. Peressome time ago
[12]. Wenowknowthatatleastfivenonlocalparameters 0 α0 +β 1 ,
| i → | i | i
areneededtocompletelyspecifytheLUequivalenttypes 1 β∗ 0 +α∗ 1 , (10)
of three qubits [14]. For example as was done by Linden | i → − | i | i
etal. [14]usingthemethodofgrouptheory,thefollowing the coefficients of transformed basis 000 , 011 , 100 ,
standard form for three qubits can be derived: and 111 become | i | i | i
| i
ψ = √λ 0 a00 + 1 a2 11 000 : aα3 dβ∗3 3bβ∗α2+3cαβ∗2,
ABC
| i | i(cid:16) | i p − | i(cid:17) |111i : aβ3+−dα∗3+−3bα∗β2+3cβα∗2,
+√1 λ 1 γ 1 a2 00 a11 | i
− | ih (cid:16)p − | i− | i(cid:17) 011 : aαβ2 dα∗β∗2 bβ∗β2+cαα∗2
| i − −
+f 01 +g 10 , (5) +2bαα∗β 2cββ∗α∗,
| i | ii 100 : aβα2+d−α∗β∗2+bα∗α2+cββ∗2
| i
whereaandf arerealnumbersandγ =(1 f2 g 2)1/2. 2bββ∗α 2cα∗αβ∗, (11)
We note this form is a superposition of six−ort−ho|n|ormal − −
basis states. In fact, it is found that at least five prod- As proven in Appendix A, we find that the following
uct orthonormal basis states are needed to express an proposition holds.
arbitrary state of three qubits. This is called the gener- Proposition 1: Byproperlychoosingαandβ wecan
alized Schmidt form of the canonicalform by Acin et al. make any one of the above four coefficients zero.
[11],laterAcinetal. [15]furtherdiscoveredthefollowing Proposition 1 leads to the following direct corollary
“least representation”, a superposition of five orthonor- with properly chosen phase factors for 0 and 1 :
mal basis states Corollary 1: The wave function o|fithree|idientical
bosonsintwomodescanbewritteninthestandardform
λ 000 +λ eiφ 100 +λ 101 +λ 110 +λ 111 , (6)
0 1 2 3 4
| i | i | i | i | i
ψ =r000 +s(100 + 010 + 001 )+t111 , (12)
where 0 φ π. This form is also called the “general- | i | i | i | i | i | i
≤ ≤
ized Schmidt decomposition” [15].
with r and t real.
If the LU equivalence is used to characterize a three
Our results in the next three sections will be basedon
qubitsystem,infinitely manydifferenttypesofentangle-
this standard form.
mentareneededduetothedifferentvaluesofthefivepa-
rameters in Eq. (6). To reduce the entanglement types,
Bennett et al. [13] introduced the concept of SLOCC
IV. ENTANGLEMENT TYPES
(stochastic LOCC) reducible. Dur et al. [16] further
found that there is only two SLOCC inequivalent types
Asdiscussedearlierthreedistinguishablequbitscanbe
of entanglement for three qubits [16]: the GHZ type
entangled in two different ways [16], denoted by a pure
1 statewavefunctionoftheGHZortheWtype. Forthree
GHZ = (000 + 111 ), (7) identical bosons, we give similar definitions for the two
| i √2 | i | i
different types as in the following.
and the W type Definition 1: Three two-state bosons are GHZ type
entangled if its wave function can be written as
1
W = (001 + 010 + 100 ), (8)
| i √3 | i | i | i ψ = ααα + βββ , (13)
| i | i | i
in contrast to what was known earlier for a system of under appropriate single particle transformations,where
two parties, where only one type of entanglement, i.e. α and β are linear independent but need not be or-
| i | i
the EPR type [17], characterizes all entangled states. thogonal and orthonormal.
3
Definition 2: Three two-statebosons are W type en- is well known, the concurrence and the quantity τ,
tangled iff the wave function can be written as introduced by Wootters et al. [18C, 19] are used to mea-
surepairwiseandternaryentanglementfortwoandthree
ψ = αββ + βαβ + ββα (14) qubits respectively. Here we will discuss these entangle-
| i | i | i | i
ment measures for our system of three two-state bosons.
under appropriate single particle transformations. α
| i Let us firstreview the definitions of the concurrence
and β are linear independent but need not be orthogo- C
| i and the quantity τ for an arbitrary pure state of three
nal and orthonormal.
qubits A, B, and C. The concurrence is defined as
Using proposition 1, it is straightforward to prove CAB
Proposition 2 (see Appendix B): When written in the
=max λ λ λ λ ,0 , (15)
standard form of Eq. (12), three two-state bosons are AB 1 2 3 4
C { − − − }
GHZ type entangled iff (r = 0, t = 0, and s = 0), or
6 6
(r = 0, t = 0, and s = 0), or (r = 0, t = 0, and s = 0); where λ1, λ2, λ3, and λ4 are the square roots of the
6 6 6 6 6
and they are W type entangled iff (r = 0, t = 0, and eigenvalues,indecreasingorder,ofthefollowingoperator
s=0), or (r =0, t=0, and s=0).
6 6 6
It is interesting to note that the parameters r and t ρ (σ σ )ρ∗ (σ σ ), (16)
are not symmetric with interchange to the middle term AB y ⊗ y AB y ⊗ y
inthe standardform(12). This observationis consistent
with ρ the reduceddensity matrix ofqubits A andB.
AB
with our proposition that the state is W type entangled
Similarly, one can define the concurrences , .
BC AC
iff s = 0 and t = 0. This point can be understood intu- C C
6 Wootters et al. [19] found that
itively as the basis 000 contains two 0 s, and is closer
| i | i
to the W state defined here than the basis 111 , thus it
is reasonable that only the state t=0 is W|entiangled. τABC := CA2(BC)−CA2B−CA2C
= 2 2 2 (17)
0.35 CB(AC)−CAB−CBC
r=0 = 2 2 2 ,
CC(AB)−CAC −CBC
0.3 t=0
where , , and are concurrences of
A(BC) B(AC) C(AB)
0.25 the purCe state Cψ with bCipartite partitions A(BC),
ABC
| i
B(AC) and C(AB).
0.2
Before presenting our results on entanglement for a
C
three boson pure state, we note that although and
0.15 C
τ have been customarily used for three distinguishable
particles [18, 19], they remain valid for the case of three
0.1
bosons. Thisissobecausewhenweconstructthedecom-
position of the two-qubit density matrix ρ that adopts
0.05
the minimum average pre-concurrence (and hence the
C
0 minimal concurrence of ρ), we start from the eigenvalue
0 0.1 0.2 0.3 0.4 0.5 0.6
decomposition of ρ [18], which is automatically sym-
s
metrized for a three-boson system. The quantity τ as
definedis alsoautomaticallyinvariantunderexchangeof
FIG. 1: The concurrences for a pure state of three bosons
particles.
|ψi as in Eq. (18) for r=0 (blue solid line) or t=0 (magenta
dashed line). Wenowcalculatefromthestandardform(12)forthree
bosons. For convenience, we rewrite Eq. (12) as
ψ =r000 +seiφ(100 + 010 + 001 )+t111 ,(18)
| i | i | i | i | i | i
V. ENTANGLEMENT MEASURES
where r, s, t, and φ are all realwith three of them being
The next task is to measure the entanglement of an independent due to normalization. A direct calculation
arbitrary pure state of three bosons in two modes. As leads to the following results
= 4t2s2+2t2r2+4s4+2 t4s4+t4r2s2 2s6t2+s4t2r2+s8+2r2s3t3cos(3φ)
C q −
p
4t2s2+2t2r2+4s4 2 t4s4+t4r2s2 2s6t2+s4t2r2+s8+2r2s3t3cos(3φ), (19)
− q − −
p
4
and exists no ternary entanglement. When r = 0, i.e., for a
GHZ type entangled state, we find
τ =4r2t2+4ts3ei3φ , (20)
| | = √2 s( 3 8s2 1) , (21)
where the reduced two party density matrices are iden- C (cid:12) p − − (cid:12)
tical ρ = ρ = ρ , and can be evaluated in the τ = 16 (cid:12)(cid:12)ts3 . (cid:12)(cid:12) (22)
AB BC AC
single particle basis directly, or more generally from the In this case, the conc(cid:12)(cid:12)urre(cid:12)(cid:12)nce vanishes for s = 0 or
two particle reduced density matrix of a general many s = 1/2, where no pairwise entCanglement exists. When
body system ∼ ρ(ij2k)l = ha†ia†jakali/2!. When s = 0, we s = √3/3, the concurrence takes the maximum C =
find = 0, i.e. there is no pairwise entanglement in the (√6 √2)/3. Another interesting feature is that there
stateCr000 +t111 . Nevertheless, there exists ternary is als−oa localmaximumats=√6/8with a concurrence
| i | i
entanglement τ =4r2t2. = √3/8. The concurrences for these two special cases
C
of r =0 and t=0 are shownin Fig. 1. The concurrence
for a general pure state is shown in three dimensional
C
graphs, as in Fig. 2 for φ=0 and in Fig. 3 for φ=π/2.
C
0.4
VI. SPIN SQUEEZING
0.2
Spin squeezing results from quantum correlations be-
0
tween individual atomic spins [20, 21]. Recent theoreti-
1
cal investigations have uncovered that spin squeezing is
0.5 a sufficient but not necessary condition for quantum en-
tanglement [3, 22, 23, 24, 25, 26, 27]. This has led some
0 effort to suggest using the spin squeezing parameter as
t
−1 a multi-atomic entanglement measure [22], as has been
−0.5 −0.5 fullydemonstratedinatwo-qubitsystem[23]. Wangand
0
0.5 Sanders [24] illustrated a quantitative relationship be-
−1 1 r tween the squeezing parameter and the concurrence for
the even and odd (multiple atom spin) states, and have
FIG. 2: The concurrencefor a purestate of threebosons |ψi further shown that spin squeezing implies pairwise en-
as in Eq. (18) when φ=0. tanglement for an arbitrary symmetric multi-qubit state
[24].
Inthissection,weinvestigatetherelationshipbetween
squeezing parameter and the pairwise concurrence en-
tanglementmeasures for an arbitrarypure state of three
two-state bosons. We start from the standard form Eq.
(18) and define the total “pseudo-spin” for three bosons
as S~ =(~σ +~σ +~σ )/2, a direct calculation then gives
1 2 3
3
C S~ = 3rscosφ,3rssinφ, (r2+s2 t2) . (23)
h i (cid:20) 2 − (cid:21)
0.4 1
Itiseasytocheckthatthesymmetricthreetwo-statebo-
son space consists only of the maximum total spin space
0.2 0.5
satisfying S2 = (3/2)(3/2+1)~2, which implies a geo-
0 0 metricBlochsphererepresentationalsoforthetotalspin
r
1 of three two state bosons. We define the unit vector
0.5 −0.5 zˆ S~ and choose a cartesian coordinate system with
t 0 −0.5 xˆ=∝(hsiniφ, cosφ,0), yˆ [cosφ(r2+s2 t2),sinφ(r2+
−1 −1 s2 t2), 2r−s]. Thislead∝stothearbitrary−transversespin
− −
directionbeing~n =xˆcos(θ)+yˆsin(θ)andS =S~ ~n .
⊥ ⊥ ⊥
FIG. 3: The concurrencefor a purestate of threebosons |ψi The squeezing parameter ξ is defined by ·
as in Eq. (18) when φ = π/2. Note it is symmetric with
4
respect to t→−t. ξ = (∆S ) . (24)
⊥ min
3
Whent=0,i.e.,foraW typeentangledstate,wefind After some tedious calculations, we find
= (√6 √2)s2 and τ = 0. In this case, the pairwise 4
eCntanglem−ent increases with the module of s, but there (∆S⊥)=Acos2θ+Bcosθsinθ+C, (25)
3
5
with expressions for A, B, and C given in Appendix C,
and
3
1
= (r2+s2 t2)2+4r2s2. (26)
u −
p
2
It is reasonably easy to find the minimum of Eq. (25)
since it is a simple trigonometric function of the form
(Acos2θ+Bsin2θ)/2+(A/2+C),whoseminimum can ξ 1
be found in terms of 2θ and the signs of A and B.
Wenowdiscussthreeimportantcasesforξ foranarbi-
trarypurestateofform(18). First,whens=0,wehave 0
ψ =r000 +t111 . Inthiscase,wefindthatξisalways 1
| i | i | i
1, independent of the values of r and t. This means that
r 0
these kind of entangled states are never spin-squeezed.
−1 1 0.5 0 t −0.5 −1
2.5
r=0
FIG. 5: The squeezing parameter ξ for a pure state of three
t=0
2 bosons |ψi as in Eq. (18) with φ=0.
1.5
ξ 3
1
2
0.5
ξ
1
0
0 0.2 0.4 0.6
s
0
1
FIG. 4: The squeezing parameter ξ for a pure state of three
bosons |ψi as in Eq. (18) with r=0 (blue solid line) or t=0 0
t 1
(magenta dashed line). 0.5
0
−1 −1 −0.5 r
Second, when t=0, we find
FIG. 6: The squeezing parameter ξ for a pure state of three
1 4s2+16s6
ξ = − , (27) bosons |ψi as in Eq. (18) with φ=π/2.
1 8s4
−
which has one minimum at s 0.4694 with a squeezing
≃
parameterξ 0.4738. Thedependenceofξ asafunction VII. CONCLUSION
≃
ofsisshowninFig. 4indashedline,wheresvariesfrom
0 to 1/√3. This result is independent of the value of φ.
To clarify the relationshipbetween spin squeezingand
Third when r =0, we find
pairwiseentanglement,letuspaysomeattentiontoFigs.
ξ =1+4s2 4 s2 3s4, (28) 1and4. Forstatesoft=0,i.e.,W-typeentangledstates,
− p − the concurrence is a monotonicallyincreasingquantity
C
which gives ξ = 1 for s = 0 and s = 1/2. Thus there with parameter s, while there exists a minimum of the
exists no squeezing in these two states. The maximum squeezing parameter ξ. Thus, we conclude that for W-
value of the squeezing parameter is ξ =7/3 in this case, typestatesthespinsqueezingisdrasticallydifferentfrom
corresponding to s = √3/3, i.e. a W state. The min- the pairwisequantumentanglement. For states ofr =0,
imum value of the squeezing parameter is ξ = 1/3 at i.e., GHZ tye-entangled states, we find several common
s=√3/6. The squeezing parameter ξ as a function of s featuresbetweenpairwiseentanglementandspinsqueez-
is plotted in Fig. 5, independent of φ in this case. ing: when s = 0 and s = 1/2, neither pairwise entan-
Finally, we use two three-dimensional figures to illus- glement nor spin squeezing exists. For s = √3/3, these
trate the squeezing parameter ξ as a function of r and t two quantities attain the maximum. There also exists
in Figs. 5 and 6 with both r and t varying from 0 to 1. another extreme point in these two quantities. How-
We have set φ=0 for Fig. 5 and φ=π/2 for Fig. 6. ever,wenotethattheparameterscorrespondingtothese
6
two extreme points are not the same. When s = √6/8, Take its complex conjugation, we obtain
the concurrence takes a local maximum value, while the
spin squeezing parameter takes a minimum value when a∗z∗ d∗z2 b∗+c∗z2z∗+2b∗zz∗ 2c∗z =0. (A5)
s = √3/6. This different dependence on the parame- After eli−minatin−g variable z∗ from Eqs−. (A4) and (A5),
ter s shows that also for GHZ-type entangled states, the
we are left with a fifth order polynomial equation for
spin squeezing parameter cannot be regarded as a mea-
the complex variable z. According to the fundamental
sure of pairwise entanglement. It is worthy to point out
theorem of Algebra, there exists at least one solution of
that on this point our result is consistent with several
equation (A4).
previous works [23, 24, 26]. Although for a collection of
Asimilarprocedurecanbeappliedtothecoefficientof
special states, there might exists a quantitative relation-
state 100 . Therefore we complete our proof of proposi-
ship between pairwise entanglement and spin squeezing, | i
tion 1.
these twoproperties in generalrefers to different aspects
of multi-party quantum correlation, and are not simply
related to each other.
In summary, we have obtained the canonical form of
APPENDIX B: PROOF OF PROPOSITION 2
anarbitrarypurestateforthreetwo-statebosons. Based
onthisform,wehaveclassifiedtheentanglementofthree
identical bosons in two modes into two types, GHZ and Proof:
W types, analogues to the case of three distinguishable Whenr =0,t=0,ands=0,the standardformitself
qubits [16]. We have completely studied two impor- is just the6GHZ t6ype entanglement.
tant entanglement measures, the concurrence and the
C When r = 0, t = 0, and s = 0, we can choose
tripleentanglementmeasureτ,andhavealsoinvestigated α = w0 +s1 /2w6 2and β =w6 0 +s1 /2w2,where
the spin squeezing property of our system by directly | i − | i | i | i | i | i
w satisfies w6 = s3/4t. Thus, we get GHZ type entan-
computing the spin squeezing parameter ξ. Our results
glement.
demonstratethatevenforpurestatesofasystemofthree
When r = 0, t = 0, and s = 0, we choose α =
bosonsintwomodes,thespinsqueezingparameterξcan-
a0 +b1 a6nd β =6 c0 +d1 , w6 here a=t2s2r/[|(tri2+
notbe regardedasanentanglementmeasure,incontrast
4s|3i)( t|+i2u3)v|2],ib=|v,ic=|rui(u3 t)/s(2u3 t),d=u.
to a system of two particles.
Andu−satisfies(tr2+4s3)u6+( t2r−2 4ts3)u3−+t2s3 =0,
This work is supported by NSF and CNSF.
v satisfies v3+u3 t = 0. Th−us We−will get GHZ type
−
entanglement.
APPENDIX A: PROOF OF PROPOSITION 1 When r = 0,t = 0,s = 0, the standard form itself is
6 6
just the W type entanglement.
Proof: For the coefficient of 000 we prove that the Whenr=0,t=0,s=0,wechoose α =r0 /3+s1 ,
| i 6 6 | i | i | i
solution to the equation β = 0 , and it becomes the W type.
| i | i
This completes our proof.
aα3 dβ∗3 bβ∗α2+cαβ∗2 =0, (A1)
− −
does exist. Without loss of generality, we assume α=0.
6
Divide the above equation by α3, we get
APPENDIX C: EXPRESSIONS FOR A, B, AND C
β∗ 3 β∗ β∗ 2
a d b +c =0. (A2)
− (cid:18) α (cid:19) − (cid:18) α (cid:19) (cid:18) α (cid:19)
Of course the solution to Eq. (A2) exits for the variable A = 128u2s3r2tcos3φ 96u2s3r2tcosφ 40u2s4r2
β∗/αfromthefundamentaltheoremofalgebra. Similarly − −
64u2st3r2cos3φ 24u2st5cosφ+48u2s3t3cosφ
we can prove that the coefficient of 111 can be elimi- − −
nated by the single particle transform|atioin Eq. (10). 24u2s5tcosφ 64u2s3t3cos3φ+32u2s5tcos3φ
− −
Now we consider the equation +32u2st5cos3φ+24u2s2r2t2+48u2st3r2cosφ
24u2sr4tcosφ+32u2sr4tcos3φ 8u2s2r4, (C1)
aαβ2 dα∗β∗2 bβ∗β2+cαα∗2 − −
− − B = 32ut3scos2φsinφ+32us3tcos2φsinφ
+2bαα∗β 2cββ∗α∗ =0. (A3) −
− +32ur2stcos2φsinφ 8ur2stsinφ
−
Without loss of generality, we assume β = 0. Let z = 8us3tsinφ+8ut3ssinφ, (C2)
α/β∗ and divide Eq. (A3) by β∗β2, we get6 C = −1+4s2+12stcosφ 16stcos3φ, (C3)
−
az dz∗2 b+czz∗2+2bzz∗ 2cz∗ =0. (A4)
− − −
7
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