Table Of ContentEPJ manuscript No.
(will be inserted by the editor)
N 1
W-like states of uncoupled spins
2
8
0 E. Ferraro1, A. Napoli1, M. A. Jivulescu1,2, A. Messina1
0
2 1 CNISM and Dipartimento di Scienze Fisiche ed Astronomiche, Universit`a di Palermo, via Archirafi
36, 90123 Palermo, Italy
n 2
Department of Mathematics, ”Politehnica” University of Timi¸soara, P-ta Victoriei Nr. 2, 300006
a
Timi¸soara, Romania
J
2
2
Abstract. Theexactdynamicsofadisorderedspinstarsystem,describingacen-
] tral spin coupled to N distinguishable and non interacting spins 1, is reported.
h 2
Exploitingtheirinteractionwiththecentralsinglespinsystem,wepresentpossible
p
conditional schemes for thegeneration of W-likestates, as well as of well-defined
-
t angular momentum states, of the N uncoupled spins. We provide in addition a
n
waytoestimatethecouplingintensitybetweeneachoftheN spinsandthecentral
a
one. Finally thefeasibility of ourprocedure is briefly discussed.
u
q
[
1 Introduction
1
v
Interacting spin models play a central role in many physical contexts providing a paradigm to
1
9 describe a wide range of different systems. In condensed matter physics, for example, they can
3 be explored to analyze many properties of magnetic compounds. It is indeed well known that
3 a suitable generalmodel of a magnet consists of N spins coupled by exchange interactionwith
. arbitrary range and strength.
1
0 Theinteresttowardspinsystems,andmoreinparticulartowardspindynamicsinsemicon-
8 ductor structures, has remarkably increased in the last few years also in connection with the
0 new emerging areas of quantum computation and information [1]-[3]. In these contexts, spin
: modelslikeHeisenbergspinchainsorspinstarsystems,describingforexampleasingleelectron
v
spin in a semiconductor quantum dot interacting with surrounding nuclear spins via hyperfine
i
X coupling mechanisms, have been extensively studied [4]-[14].
r Generally speaking spin models have proved to be promising candidates for the generation
a
andthecontrolofassignedquantumcorrelations,aswitnessedbythenumerouspapersrecently
appeared in literature[15]-[18].
In this paper we concentrate on the possibility of manipulating at demand the state of a
sampleofN uncoupledspinsexploitingtheircommoninteractionwithanothersinglespincalled
central system. The analysis we have developed, on the one hand provides possible procedures
to guide the system toward pure states characterized by fixed correlation conditions, on the
other hand suggests a way to to estimate the coupling strenght between each of the N spins
and the central one.
2 Disordered spin star system
Having in mind as objective the possibility of generating fixed entanglement conditions in a
system of N uncoupled spins 1, we exploit the interaction of each element of the system with
2
another spin 1 hereafter called central one. The system we are talking about is illustrated for
2
2 Will be inserted by theeditor
A
Fig. 1. Anillustration of a disordered spin star system
convenience in Figure 1 and it can be described adopting the following hamiltonian model:
H =H +H (1)
0 I
with
N
H =ω σj +ω σA, (2)
0 z 0 z
j=1
X
N
H = α (σAσj +σAσj ), (3)
I j + − − +
j=1
X
The Pauli operators labelled by the index A refer to the central spin, while the others char-
acterized by the index j (j = 1..N) refer to the N spins. The coupling strength between the
spin j and the central one is measured by the constant α and, generally speaking, α may be
j j
different from α for j = i. In realistic physical situations the coupling constants can change
i
6
forexample proportionallyto the distance fromthe centralsystem[19].We referto the system
described by eq. (1) as disordered spin star system.
Studying the dynamical properties of this model is for example of interest in contexts like
quantum dot coupled by hyperfine interaction with nuclear spins, or electronic spins bound to
phosphorus atoms in a matrix of silica or germanio in presence of defects [20].
Thesymmetrypropertiesofthehamiltonianmodelcanbesuccessfullyexploitedinorderto
analyzethedynamicsofthesystem.Itiseasytoconvinceoneselfthatthecomponentalongthe
z axes of the total angular momentum operator Sz = σ2zA + 21 Nj=1σzj = σ2zA +Jz is a constant
of motion. Then, starting from an eigenstate of S the system evolves in the correspondent
z
P
invariant Hilbert subspace. Let’s suppose in particular to prepare the system in the following
state
ψ (0) = ... ... ... , (4)
| {i1,...ip} i |↑Ai|↓ ↑i1 ↑ip ↓i
where the central spin, as well as p of the N spins, namely the i -th, i -th,... i th, are in their
1 2 p
respectiveup state defined asσi = withi=A,1,..,N,whereasthe othersarein their
down state with|↑σii = . z|↑i |↑i
|↓i z|↓i −|↓i
We in addition denote the state , ,..., , where all the uncoupled spins are down,
A
|↑ i|↓ ↓ ↓i
by ψ (0) .
0
| i
Will be inserted by theeditor 3
Taking into account the previous considerations we may claim that, at time instant t the
state of the system prepared in the state (4) can be written as
ψ (t) = a (t) ... ... ... +
| {i1,...ip} i j1<j2<...<jp j1,j2,...,jp |↑Ai|↓ ↑j1 ↑jp ↓i
+ b (t) ... ... ... ... , (5)
jP1<j2<...<jp+1 j1,j2,...,jp+1 |↓Ai|↓ ↑j1 ↓ ↑jp+1 ↓i
In equation (5) each iPndex j (i = 1,..,p+1) runs from 1 to N. Thus the first term of the
i
righthandside is a superpositionofallthe states in whichthe centralspin, aswell asp among
the N uncoupled spins, are in their up state. The second term of equation (5) is similarly a
linear combination of all the states in which the central system is in its down state whereas
p+1oftheN spinsareintheirupstate.Wenoticethatthestructureof ψ (t) canbededuced
0
| i
from that of ψ (t) simply substituting to the first sum of eq. (5) the term a(t)
| {i1,...ip} i |↑Ai
... ... .Insertingeq.(5)inthetime-dependentSchr¨odingerequationleadstothefollowing
|↓ ↓ ↓i
system of coupled equations for the probability amplitudes a (t) and b (t)
j1,j2,...,jp j1,j2,...,jp+1
N
ia˙ (t)= ∆a (t)+ α b (t) (6)
j1,j2,...,jp j1,j2,...,jp r O({j1,...,jp}∪{r})
r=1(rX6=j1,..jp)
N
ib˙ (t)= ∆b (t)+ α a (t) (7)
j1,j2,...,jp+1 − j1,j2,...,jp+1 r δr(j1,...,jp+1)
r=1(r∈X{j1,..jp})
where ∆=ω ω . In eq.(6) we have introduced the operator O which adds the index r to the
0
−
set of indices j ,...,j arranging them in increasing order. We point out that this operator
1 p
{ }
is well defined if r does not belong to the set j ,...,j , otherwise the probability amplitude
1 p
{ }
b (t) wouldhavep indices instead of (p+1) becoming senseless.The operatorδ
O({j1,...,jp}∪{r}) r
appearinginturnineq.(7)acts onthe family ofp+1indices,recoveringa setofpindices from
j ,j ,...,j by eliminating the index r. We have to mention that the above operator is
1 2 p+1
{ }
well defined if r belongs to the set j ,j ,...,j in order to assure the correct definition of
1 2 p+1
{ }
a probability amplitudes of the type a (t).
j1,j2,...,jp
This system of differential equations can be easily decoupled when the system is prepared
in the state ψ (0) . In this case eqs. (6) and (7) become
0
| i
N
ia˙(t)=∆a(t)+ α b (t) (8)
j j
j=1
X
ib˙ (t)= ∆b (t)+α a(t), (9)
j j j
−
and it is easy to show that a(t) fulfills the following Cauchy problem
a(0)=1 (10)
N
a¨(t)= ∆2+ α2 a(t), (11)
− r!
r=1
X
whose solution is
N N
∆
a(t)=cos α2+∆2 t i sin α2+∆2 t (12)
vuuXj=1 j − Nj=1α2j +∆2 vuuXj=1 j
t q t
Inserting eq. (12) in eq. (9) leads to a first ordPer non homogeneous linear differential equation
for b (t) which can be easily solved from the initial conditions b (0)=0 getting
j j
N
α
b (t)= i j sin α2+∆2 t . (13)
j − Nj=1α2j +∆2 vuuXj=1 j
q t
P
4 Will be inserted by theeditor
Whenthesystemispreparedinthestate(4),eqs.(6)and(7)maybestillmanagedinsuchaway
to decouple the probability amplitudes of the type a from those of type b .
j1,j2,...,jp j1,j2,...,jp+1
We do not present here such a procedure since in what follows we concentrate on the rich
dynamical properties of the system evolving in accordance with ψ (t) .
0
| i
3 Generation of W-like states
The results obtained in the previous Section suggest that measuring the central spin we have
thepossibilityofguidingthesystemofinterest,namelytheN uncoupledspins,towardalinear
coherent superposition of states characterized by the fact that only one spin is in the state
|↑i
whereas the others are in the state .
Starting from eq.(5) we may ind|e↓eid claim that a measure of the observable σA gives, with
z
probabilities N b (t)2,theeigenvalue 1.InthiscasetheN spinsareleftinthenormalized
j=1| j | −
state
P
1
Wgen = (α , ,..., +...+α ,..., ,..., +...+α , ,..., ).
1 j N
| i N α2+∆2 |↑ ↓ ↓i |↓ ↑ ↓i |↓ ↓ ↑i
j=1 j
q (14)
P
The state given by eq.(14) looks like the well known W-state [21]
1
W = ( , ,..., +...+ ,..., ,..., +...+ , ,..., ). (15)
| i √N |↑ ↓ ↓i |↓ ↑ ↓i |↓ ↓ ↑i
the only difference between the two states (14) and (15) being the weight of each component
in the superposition. In the W-state all the states of the superposition appear indeed with
the same probability. On the other hand the state (14) we have obtained with our procedure,
reduces to the W-state when the central spin does not distinguish the N spins around it, that
is when α α j =1,...,N. For these reasons we call the state Wgen a W-like state.
j
≡ ∀ | i
It is important to underline that the procedure we have discussed is a conditional one. In
otherwordswemayclaimtogeneratethestate Wgen onlyifthemeasurementofσA givesthe
| i z
eigenvalue 1. Starting from eq. (13) we can write the probability of success of our procedure
−
as follows
N α2 N
P = j sin2 α2+∆2 t . (16)
N α2+∆2 v j
j=1 j=1 j uj=1
X uX
t
Thus, appropriately choosing tPhe time instant t at which the measure of σA is performed,
z
we may generate the desired state with the highest probability that coincides with one in
correspondence to ∆=0. If indeed we measure the observable σA at time instant
z
π(2n+1)
t = , n=0,1,2,... (17)
n
2 N α2+∆2
j=1 j
q
P
the probability of success becomes P = N α2j and thus P =1 if ∆=0.
j=1 N α2+∆2
j=1 j
It is on the other hand of relevancePto analyze the behaviour of such a probability with
P
respect to imprecisions in setting the time instant at which the measurementof the observable
σA isperformed.Let’sfirstofallobservethattheprobabilityofsuccessP,asgivenbyequation
z
(16), is a periodic function of t, the period being T =π( Nj=1α2j +∆2)−12. This circumstance
suggests to choice the time instant at which to perform the measurement optimizing, as far as
P
possible,ourproposalontheexperimentalside too.Tothisendlet’s considerforsimplicitythe
case∆=0andindicatebyt∗ =t +δ thetimeinstantatwhichthemeasurementifperformed.
n n
If δ is small enough the probability of success does not appreciably get reduced. Let’s suppose
Will be inserted by theeditor 5
in particular that δ T10−1. Moreoverit is also reasonable to assume that the relative error
| |≤
δ is of the order of 10−3 thus implying that
tn
π
t 103δ 102 (18)
n
∼ ≤ N α2
j=1 j
q
If our model describes, for example, hyperfine interPaction of a localized electron with nuclei,
the order of magnitude of the coupling constant α can be estimated as 10−5eV [22], [23] in
j
correspondencetowhicht mustbelessthanorequaltoπ10−9sec.Thust 1nsiscompatible
n n
∼
witheq.(18)anditmayberealizedfixingn=100.Infigure2weplottheprobabilityofsuccess
Fig. 2. Probability of success to generate |Wgeni as function of x = δ in correspondence to ∆ = 0
tn
and n=100.
of our scheme givenby eq. (16) versus x= δ putting ∆=0 and t=t +δ. As foreseeable,
t100 100
this probability of success remains of experimental interest for x up to 3 10−3. Thus we may
·
conclude that the procedure is stable enough against unavoidable uncertainties in the time
instant at which the measurement of σA is done.
z
BeforeconcludingthissectionitisinterestingtoemphasizethatthecouplingbetweentheN
spins andthe centralsystemgeneratesentanglementbetweenanytwospins aroundthe central
one. In order to estimate the amount of such an entanglement and analyze its time evolution
we evaluate the relative concurrence function. It is easy to prove that, starting from the initial
state ψ (0) , the reduced density ρ (t) describing the system of the two spins i and j among
0 ij
| i
the N uncoupled ones, can be written in the form
0 0 0 0
0 b (t)2 b (t)b (t)∗ 0
ρij(t)=0b|(ti)∗b|(t) ib (tj)2 0 (19)
i j j
0 0 | 0 | |a(t)|2+ k6=i,j|bk(t)|2
when expressed in the standard two-spin basis , , P, .In equation (19)b (t) and
j
{|↑↑i |↑↓i |↓↑i |↓↓i}
a(t) are given by eqs. (13) and (12) respectively. Thus the concurrence function C (t) turns
ij
out to be
N
2α α
C (t)= | i|| j| sin2 α2+∆2 t (20)
ij N α2+∆2 v j
j=1 j uj=1
uX
Asexpected,thedegreeofentanPglementgetestablishetdbetweenthetwouncoupledspinsiand
j,oscillateswithtimet.InadditionthemaximumvalueofC (t)isproportionalto α α and
ij i j
| || |
6 Will be inserted by theeditor
when α = α k and ∆ = 0 reaches the value 2. This means that, increasing the number of
k ∀ N
spins aroundthe centralone,leadsto a weakandweakpairquantumcorrelationinthe system
of interest.
4 Generation of well-defined angular momentum states of the N uncoupled
spins
The W-state in eq.(15) is a multipartite entangled state of the N uncoupled spins around the
central one. On the other hand it coincides with a particular superposition of states in which
N 1 spins have projection down while only one has projection up. This state can be thus
−
obtained applying the collective ladder operator σ = σ(i) on the state , ,..., that is
+ i + |↓ ↓ ↓i
W N σ(i) , ,..., . In other words the W-state is a common eigenstate of J2 and J ,
| i∝ i=1 + |↓ ↓ ↓i P z
J= 1 N σ beingthecollectiveangularmomentumoperatoroftheN uncoupledspins,with
2 Pi=1 i
P J2 W = N(N +1)W (21)
| i 2 2 | i
J W =( N +1)W (22)
z| i −2 | i
Thus W J,M whit J = N and M = N +1.
| i≡| i 2 −2
Thisobservationsuggestsusthe possibilityto iterate ourprocedureinorderto generateall
the well defined angular momentum states J = N,M with M = N ...N.
| 2 i −2 2
Let’s indeed consider the spin star system in which the central spin interacts in the same
way with all the others uncoupled N spins, that is α α j. Under this condition the
j
≡ ∀
hamiltonianmodel(1)isinvariantbypermutationofanarbitrarycoupleofspinsamongtheN.
Moreover[σA2,H]=[J2,H]=[σA+J ,H]=[J2 ,H]=0,J beinganintermediateangular
z z int int
momentum resulting fromthe coupling of selectedatwill individual angularmomentum ofthe
N spins. These symmetry properties suggestto develop the dynamics of our system exploiting
the coupled angular momentum basis J,M,ν for the N spins instead of the factorized one
{| i}
previously used. The index ν runs from 1 to ν (J) and allows us to distinguish between
MAX
different states of the basis characterized by the same J and M.
Intheinitialcondition ψ (0) theN spinsaroundthecentraloneareinthecoupledangular
0
momentum state J = N,|M = i N,ν = 1 . In what follows we do not indicate anymore the
| 2 −2 i
the index ν remaining it equal to one. Thanks to the symmetry properties of our system at a
generic time instant t we can rewrite the state of the total system in the form
N N N N
ψ(t) =A (t) , +B (t) , +1 (23)
1 A 1 A
| i |↑ i| 2 − 2 i |↓ i| 2 − 2 i
where
A =cos(p αt), (24)
1 1
B = isin(p αt), (25)
1 1
−
with
N N N N
p = +1 +1 √N. (26)
1
s 2 2 − − 2 − 2 ≡
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
Thusasbefore,ifweassumethecentralspinAmeasuredinthestate theN uncoupled
A
spins are projected onto the W-state W N, N +1 . As demonst|r↓ateid in the previous
| i ≡ |2 −2 i
Section the probability of success to generate the W-state coincides with sin2(p αt) and it
1
is equal to 1 if the measurement is performed at time instants t = π(2n+1) obtained from
n 2αp1
equation (17) putting α α j and ∆ = 0. Suppose now to iterate the procedure preparing
j
≡ ∀
once again the central spin in the up state. The new initial condition is then
N N
ψ(0) = , +1 (27)
A
| i |↑ i| 2 − 2 i
Will be inserted by theeditor 7
that as easily demonstrable evolves as
N N N N
ψ(t) =A (t) , +1 +B (t) , +2 (28)
2 A 2 A
| i |↑ i| 2 − 2 i |↓ i| 2 − 2 i
with
A (t)=cos(p αt) (29)
2 2
B (t)= isin(p αt) (30)
2 2
−
where p = N(N +1) ( N +1)( N +2). Eq. (28) immediately implies that, measuring
2 2 2 − −2 −2
the spin A inqits down state, makes the N spins to collapse onto the angular momentum state
N, N+2 .Itispossibleatthispointtoconvinceoneselfthat,iteratingk-timesourprocedure,
|2 −2 i
we generate the coupled angular momentum state N, N +k . As far as the probability of
|2 −2 i
success P that after k measurementsthe N uncoupled spins areleft in the state N, N +k ,
k |2 −2 i
it is easy to prove that it is given by
k
P = sin2(p αt) (31)
k i
i=1
Y
where p = N(N +1) ( N +i 1)( N +i).
i 2 2 − −2 − −2
Thus, ifqat the i th step we have the possibility of choosing the time instant at which
−
performing the measurement act on the central spin A in such a way that t(i) = (2n+1)π, at
n 2αpi
leastinprinciplewemayclaimthatthedesiredstate N, N +k oftheN spinsaregenerated
|2 −2 i
with certainty.
5 Estimating the order of magnitude of the coupling constant α
j
Aswearegoingtoprovethe dynamicsofoursystemcanbealsosuccessfullyexploitedinorder
to estimate the coupling intensity between each of the N distinguishable and non interacting
N spins and the central one.
Let’s indeed consider for simplicity the case ∆ = 0 and, under this condition, concentrate
on the behaviour of the probability P (t) to find the system, at a generic time instant t, in the
0
initialcondition ψ (0) . The results obtainedin the previousSectionimmediately implies that
0
| i
such a probability is given by
N
P (t)= a(t)2 =cos2 α2 t . (32)
0 | | v j
uj=1
uX
t
Thus, the probability of recovering the N uncoupled spins as well as the central one in their
initial state is a periodic function of t, the period being in inverse relation to the quantity
N α2. If, as reasonable, we assume that all the coupling constants are of the same order,
j=1 j
analyzing the temporal behaviour of P (t), we have the possibility of estimating the order of
0
P
magnitudeofeachα .Itisimportanttostressthatthepossibilityofknowingatleasttheorder
j
of magnitude of the coupling constants α play a central role for example in all the cases in
j
which the spin of an electron localized in a quantum dot is used as realization of a quantum
bit [22]. In these cases indeed the spin relaxation mechanism is mainly connected with its
interaction with bulk nuclear spins.
Let’smoreoverobservethattheknowledgeofthefrequencyofthefunctionP (t)givenbyeq.
0
(32)canbealsoexploitedinordertoestimatehowmuchdisorderedthespinstarsystemmodel
(1) is. Let’s suppose indeed that the N spins of interest have been prepared in a W-like state
8 Will be inserted by theeditor
following the procedure previously discussed. At this point, if we measure the observable σj
z
finding the eigenvalue +1, then the total system is projected onto the state , ,..., ,...,
j
|↓ ↓ ↑ ↓i
in which only the j-th spin is in its upper state whereas the others are in their respective
downstate.The probabilityofsuchanevent,directly obtainablestartingfromeq.(14),exactly
α2
coincides with the quantity j .
N α2
j=1 j
We may thus conclude that knowing the probability of success to find the j-th spin in the
P
up state when the system of N spins is prepared in a W-like state, allows us to give an
|↑i
estimation of the interaction strength between the j-th spin and the central one.
6 Conclusion
Generally speaking the possibility of establishing on demand fixed entanglement conditions
in a multipartite system is an interesting objective both in its own and also in view of its
applicative potentialities. In this paper in particular we have concentrated on a multipartite
system composed by N not interacting spins 1. In order to guide this system toward assigned
2
entangled states, we have exploited the interaction between each of the N subsystems with
a single spin 1. The disordered spin star system thus obtained has been successfully used to
2
generate W-like states as well as well-defined angular momentum states of the N uncoupled
spins.
The study of the exact dynamics of the disorderedspin star system reported in this paper,
has provided the possibility to envisage a way to estimate at least the order of magnitude of
the coupling strength between each of the N uncoupled spins and the central one. To gain
this information is, for example, of particular relevance when electron spin relaxation plays
an important role. It is indeed appropriate to remark that our system can be adopted to
describe hyperfine interaction of a single electron spin with nuclei in quantum dots and that
this interaction mechanism may be the dominant source of electron spin relaxation.
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