Anomalous Entanglement in Chaotic Dirac Billiards

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by  J. G. G. S. Ramos

Anomalous Entanglement in Chaotic Dirac Billiards
J. G. G. S. Ramos1, I. M. L. da Silva2, A. L. R. Barbosa2
1 Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Joa˜ao Pessoa Para´ıba, Brazil
2 Departamento de F´ısica,Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil
(Dated: January 26, 2015)
We present analytical and numerical results that demonstrate the presence of anomalous entan-
5 glementbehaviorontheDiracBilliards. Weinvestigatethestatisticaldistributionofthecharacter-
1 istic entangled measures, focusing on the mean, on the variance and on the quantum interference
0 terms. We show a quite distinct behavior of the Dirac Billiard compared with the non-relativist
2 (Schr¨odinger) ones. Particularly, we show a very plausible Bell state and a sharp amplitude of
n quantum interference term on entangled electrons left from the Dirac Billiards. The results have
a remarkable relevance to the novel quantum dots build of materials like graphene or topological
J insulators.

2
PACSnumbers: 73.23.-b,73.21.La,05.45.Mt
2

]
l The entanglement is one the most fundamental effect (Dirac) billiard20,21, which henceforth we call chaotic
l
a on quantum mechanics, with no classical analog1. Two Dirac Billiard (DB)22. Through this device, the wave
h ormoreparticlesareentanglediftheysupportanonlocal functionsoftheelectronsaredescribedbymasslessDirac
-
correlationwhichcannotbe acquiredbythe dynamics of equation of the corresponding relativistic quantum me-
s
e classicalmechanics2. Becauseitsnon-classicalcharacter- chanics, instead of Schr¨odinger equation. The two cat-
m istics, the control of entangled states has attracted the egories of billiards show blunt differences on the corre-
. interest of numerous science communities3. The tech- spondingelectronictransportstatistics,whichleadsusto
t
a nological applications of the effect has a broad range as suspect that a subtle difference occurs also on the quan-
m quantumcomputation,teleportation,telecommunication tum entanglement statistical moments.
- and cryptography2–4.
d
A large number of mechanism to entangle electronic In this work, we analyze the concurrence and entan-
n
o particles, with or without interaction, can be found in glement statistics of two non-interacting electrons, with
c the literature4–6, and the quantum chaotic devices are coherentphasetransport,insideachaoticDB.Twoleads,
[ a promising option7,8. In a recent work, Beenakker bothwithtwoopenchannels,connectstheDBwithelec-

1 et al.9 proposed the possibility to entangle two non- tronic source and drain at the Dirac point16,18. We cal-
v interacting electrons using, as an orbital entangler, a culated the exacts distributions of concurrence ( ) for
C
3 quantum(Schr¨odinger)chaoticbilliardor,aswewillcall systems with or without TRS. We show a drastic dis-
6 henceforth,theSchr¨odingerBilliard(SB).Theyobtained similarity on the characteristic distributions of the DB
6 the averages and variances of concurrence and entangle- compared with the SB results10. We highlight the two
5
ment. However the results were found to be nearly in- maindifferencebetweenDBandSBfocusingonplausible
0
variant under the presence or absence of time-reversal experiments. Particularly, first we show that it is more
.
1 symmetry(TRS).Thismeansthatthequantuminterfer- probable the production of maximally entangled elec-
0
ence corrections (weak-localization) of two arbitrary en- tronic state ( =1 or Bell state) than of separable state
5 C
tanglement measures are approximately null, in contrast ( =0)forexperimentusingaDB,preciselytheopposite
1 C
with anotherphysicalobservablesofelectronic transport to the behavior of SB, as indicated in Fig.(1). Secondly,
:
v as conductance and shot-noise power. Nevertheless, the we show a weak localization correction of concurrence
i
X Ref. [10] has argued that the two first moments of en- with the same order of magnitude of the mean (semi-
tanglementmeasuresdo notcapturethe fullinformation classical) term, while for a SB it is approximately null9.
r
a about quantumdynamics,andthe complete information Clearly,we indicate that the averageand the variance of
aboutfundamentalsymmetriesofnatureemergesonlyon concurrence carry relevant information about quantum
the distribution probabilities. More recently, the effects mechanics statistics of the DB, unlike what happens in
of tunneling barriers11,12 on statistic of concurrence and SB.Ouranalyticalresultsareinaccordancewithnumer-
jointprobabilitydistributionofconcurrenceandsquared ical simulation from the random matrix theory18,19,23.
norm12,13 for SB were also studied.
The novel materials, including Dirac materials14 Scattering Model - The setup consists of a DB con-
(grapheneandtopologicalinsulators),introducenewfun- nected to two leads, both with two open channels. We
damental symmetries15,16 that can affect the entangled consider the system in the absence of electron-electron
electrons17. In a recent investigation, the Ref.[18] an- interactions and at zero temperature. The DB is repre-
alyzes how the sublattices or chiral symmetries19 (SLS) sented by the massless Dirac Hamiltonian with SLS15.
affecttheelectronictransportthroughachaoticquantum The Hamiltonian satisfy the following anti-commutation
2
relation15,19 5
β = 1
= σ σ , σ = 1M 0 . (1) 4 DB
H − zH z z 0 1M SB
(cid:20) − (cid:21)
3
The -matrixhas dimension2M 2M,with 1M denot- C)
ing aHM ×M identity matrix. W×e can interpret the M P(12
number of 1’s and 1’s in σ as the number of atoms
z
in the sublattices A−and B respectively16, in a total of
1
2M atoms in the DB. The Hamiltonian model for the
scattering matrix process can be written as24
0
(ǫ)=1 2πi (ǫ +iπ ) 1 . (2) 5 β = 2
† † −
S − W −H WW W
4
The -matrix has dimension 4 4, indicating a total of
S ×
two open channels in each terminal, each one originated
C) 3
from A or B sub-lattices. The 2M 4 matrix rep- (
× W P2
resents all deterministic couplings of the DB resonances
2
to the open channels of the two terminals. The scatter-
ing matrix is unitary † = 1 due to the conservation 1
S S
of electronic charge. It is convenient to represent the -
S
matrix as a function of transmission, t, and reflection, r, 0
0 0.2 0.4 0.6 0.8 1
blocks as
C
r t
S = t r′′ , (3) F(βIG=.11):orTwhiethdoiusttr(ibβu=tio2n)sTofRSco.nTchuerrEenqcse.(1(P1)β(aCn)d)(f1o2r)tahreeDdeBpiwctiethd
(cid:20) (cid:21) togetherwiththecorrespondingonesfortheSB10.
where t, t, r and r have dimension 2 2. From Eqs.(1)
′ ′
×
and (2), the -matrix also satisfies the relation
S
ProbabilityDistributions-Toobtainthedistributionof
=Σ Σ , Σ = 12 0 , (4) DB’s concurrence, we start using the joint distribution,
S zS† z z 0 12 defined for the two characteristic eigenvalues, which was
(cid:20) − (cid:21)
calculated in Ref.[23] for β = 1,2 . The result can be
at the Dirac point, zero energy (ǫ=0). Substituting the { }
written as
Eq.(4) in Eq. (3), we conclude that r = r , r = r and
† ′ ′†
t′ =We−tf†o.llow the previously mentioned RMT model for Pβ(φ1,φ2) = cβ|sin(φ1+φ2)sin(φ1−φ2)|β
the matrix and, consequently, for the matrix to × sinβ−1(2φ1)sinβ−1(2φ2). (8)
H S
investigate the entanglement between two no interact-
The relation between the variables φ and the transmis-
ing electrons as described in Refs.[7,9]. Firstly, the con- i
sion eigenvalues τ is τ = sin2(2φ ), with φ [o,π/2],
currence of two electrons, after they left the mesoscopic i i i i ∈
while normalization constants assume the values c = 1
device with SLS, can be written in terms of the trans- 1
and c =6.
mission, tt , eigenvalues, τ and τ , both encoded in the 2
† 1 2
The Eqs.(5) and (8) can be used to calculate the con-
scattering matrix, Eqs.(3) and (4), as
currence distribution ( ) following the definition
β
P C
τ (1 τ )τ (1 τ )
1 1 2 2
=2 − − . (5)
C p τ1+τ2−2τ1τ2 β( ) = δ 2 τ1(1−τ1)τ2(1−τ2) , (9)
The electronic states, after they left the DB, are sepa- P C * "C− p τ1+τ2−2τ1τ2 #+
rableornonentangledif =0,while,if =1,theparti-
C C with ... denotingtheensembleaverageperformedwith
clesaremaximallyentangled(Bellstate). Forintermedi- h i
theeigenvaluesdistributionsofEq. (8). TheEq. (9)can
ate values of between 0 and 1, the states are known as
non-separableCor partly entangled11. The entanglement be rewritten in the integral form as
and concurrence are related through the equation25
tan(2φ )tan(2φ )
1 2
( ) = δ 2
1+√1 2 Pβ C Z Z (cid:20)C− tan2(2φ1)+tan2(2φ2)(cid:21)
ε( )=h −C , (6)
(φ ,φ )dφ dφ (10)
C 2 ! × Pβ 1 2 1 2
with
The double integral over variables φ and φ in Eq.
1 2
h(x)= xlog (x) (1 x)log (1 x). (7) (10) can be performed through the transformation of
− 2 − − 2 −
3
5 5
β = 1 β = 2
4 4
3 3
)
C
(
P
2 2
1 1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
C C
10 10
β = 1 β = 2
8 8
6 6
)
ε
(
Q
4 4
2 2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ε ε
sFyIsGte.m2s:wiTthhe(βdi=str1i)buatnidonwsiothfocuotnc(βur=ren2c)eT(RPSβ.(CT)h)eahnidsteongtraanmgliesmtehnetn(uQmβe(rεi)c)sifmroumlatEioqns.s(1o1f)t,h(e12H)amanidlto(n1i6a)namreoddeelpwicittehd1(0c6onrteianluizoautsiolninseosf)tfhoer
correspondingrandom -matrix,theEq. (2). Theanalyticalandnumericalresultsagreesnicely.
S
variables z = tan2(2φ ) and an appropriate expansion The Fig.(1) shows the distributions of concurrence,
i i
the delta function12 Eqs. (11) and (12), for the DB. We plot in the same
place the distributions for the SB, which were obtained
tan(2φ )tan(2φ )
1 2
δ C−2tan2(2φ )+tan2(2φ ) = F+(C)z2δ[z1−f+(C)z2]in Ref.[10], for a direct comparison. The systems exhibit
(cid:20) 1 2 (cid:21) a quite distinct behavior. We observe,as the main char-
+ ( )z2δ[z1 f ( )z2]acteristic of DB, a major probability to find the elec-
F− C − − C
trons in the maximally entangled states ( = 1), con-
with
C
trary to the behavior of the SB. However, the preserved
(f ( )+1)2 f ( ) TRSsupportselectronswiththeprobabilityofseparable
( ) ± C ± C ;
F± C ≡ ±f±(C)∓p1 sFtiagt.e(1s)(.CT=he0m) seiagnnivfiaclaunetloyficnocnrceuarsreedn,caescdaenpbiceteodbtianinthede
2 2 2√1 2
f ( ) −C ± −C . from Eqs.(11) and (12) and renders
± C ≡ 2
C
After a some algebra, we obtain the following expres- 0.4675 β =1
(13)
sionsforthe probabilitydistributionofconcurrencewith hCi≈ 0.5708 β =2
(cid:26)
preserved TRI (β =1)
Using Eq.(13), the weak localization of concurrence can
1 1 √2 √1 2 be calculated with the same algebra and renders
( ) = 1+ −C (11)
P1 C 21 2  2 s 1 2 = 0.1033. (14)
−C  −C hCiwl hCi1−hCi2 ≈−
Notice the weak localization term has the same mag-
 2
arccoth 1+ C 1 nitude order of the main (semiclassical) term, Eq.(13),
  
× vu 2(1−C2) 1+ √11−C22 −  while for the SB it is approximately null9. The TRI,
u −C  unlike what happens in SB, has a strong influence over
t (cid:16) (cid:17)
and, for broken TRI (β =2), it renders a simple expres- average of concurrence in DB. For the variance of con-
sion currence we obtain
√1 2 0.1117 β =1
( ) = −C . (12) var[ ] (15)
P2 C (1+ )(1 2) C ≈ 0.1034 β =2
C −C (cid:26)
4
Like weak localization, the variance of concurrence has where 2M is the number of resonances inside the DB,
the same order of magnitude of the its mean, Eq.(13). including both the two sub-lattices degrees of freedom.
Lastly, we obtain the distribution of entanglement Also, λ = 2M∆/π is the variance, related to the
β
Q
from Eqs. (11) and (12) using the following appropriate electronic single-particle level spacing, ∆. The =
W
change of variables10 ( , ) matrix is a 2M 4 deterministic matrix, de-
1 2
W W ×
scribing the coupling of the resonances states of the
ln(2) 1 (ε)2 chaotic DB with the propagating modes in the two ter-
(ε) = −C (ε). (16)
Qβ (ε) arctanh[ 1 (ε)2]Pβ minals. This deterministic matrix satisfies non-direct
C p −C process, i.e., the orthogonality condition = 1δ
The analytical results for the dpistributions of entangle- holds. Accordingly, we consider the relatWiop†nWσqz Σπzp=,q
W
ment are cumbersome, but it is depicted in the Fig.(2). ,indicatingthescatteringmatrixissymmetric(4). We
W
The averageof entanglement renders the results consider the system on the Dirac point, ǫ = 0, and, to
ensure the chaotic regime and consequently the univer-
0.382 β =1 salityofobservables,thenumberofresonancesinsideDB
ε (17)
h i≈ 0.485 β =2 is large (M 4).
(cid:26) ≫
while its weak localization amplitude term is The numerical simulations produce the Fig.(2), which
showsthedistributionsofconcurrenceandentanglement
ε = ε ε 0.10. (18) obtainedthrough106 realizationscomparedwiththe an-
h iwl h i1−h i2 ≈−
alyticalresults,Eqs.(11), (12)and(16), forsystemswith
For the variance of entanglement we obtain orwithout TRS. We use the matrices,with dimension
T
100 100 (M =100), andthe corresponding matrices
var[ε] 0.121 β =1 (19) with×dimension 200 200 (200 resonances). H
≈ 0.122 β =2 ×
(cid:26)
Conclusions - We present a complete statistical study
Numeric Simulation - In order to confirm the analyti- of the entangled electronic measures. We show a very
calresults,theEqs.(11)and(12),weemployanumerical peculiar quantum behavior if the production of orbital
simulationusingthe -matrixformulationofEq.(2). The entanglement is raised on the Dirac billiards quantum
S
ensemble of Hamiltoninans satisfies the SLS of Eq.(1). dots. The analytical expressions for the distributions of
ThenumericalsimulationwasdevelopedinRef.[18]. The concurrence were obtained in the presence or absence of
anti-commutationrelation,Eq.(1),impliesaHamiltonian TRS.WecomparetheresultsfortheDiracbilliardswith
member of the ensemble rewritten as19 previous ones of the Schr¨odinger billiards, Refs. [9,10],
and found significant statistics differences between these
0
= T . (20) devices. Clearly, we indicate that the average, the vari-
H (cid:18)T† 0 (cid:19) ance of concurrence and entanglement carry relevant in-
formationaboutquantummechanicsstatisticsoftheDB,
Here,the -matrixblockofthe -matrixhasdimension
T H unlikewhathappensinSB.Finally,ouranalyticalresults
M M. The Random Matrix Theory establishes that
× areconfirmedbynumericsimulationfromrandommatrix
the entries of -matrix can be chosen as a member of a
T theory.
Gaussian distribution23
This work was partially supported by CNPq, CAPES
βM
P( ) exp Tr( ) , (21) and FACEPE (Brazilian Agencies).
†
T ∝ −2λ2 T T
(cid:26) (cid:27)
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