Chaotic scattering through coupled cavities

0.55 MB·English

by  Kazutaka Takahashi

Chaotic scattering through coupled cavities
Kazutaka Takahashi
Department of Physics, Tokyo Institute of Technology, Tokyo 152–8551, Japan
Tomosuke Aono
7 Department of Physics, Ben–Gurion University of the Negev, Beer–Sheva 84105, Israel
0 (Dated: February 6, 2008)
0
WestudythechaoticscatteringthroughanAharonov-Bohmringcontainingtwocavities. Oneof
2
the cavities has well-separated resonant levels while the other is chaotic, and is treated by random

n matrixtheory. Theconductancethroughtheringiscalculatedanalyticallyusingthesupersymmetry
a method and the quantum fluctuation effects are numerically investigated in detail. We find that
J
the conductance is determined by the competition between the mean and fluctuation parts. The

5 dephasingeffectactsonthefluctuationpartonly. TheBreit-Wignerresonantpeakischangedtoan
antiresonancebyincreasingtheratioofthelevelbroadeningtothemeanlevelspacingoftherandom
] cavity,andtheasymmetricFanoformturnsintoasymmetricone. Fortheorthogonalandsymplectic
l
l ensembles, the period of the Aharonov-Bohm oscillations is half of that for regular systems. The
a
conductance distribution function becomes independent of the ensembles at the resonant point,
h
- which can be understood by the mode-locking mechanism. We also discuss the relation of our
s results to therandom walk problem.
e
m
PACSnumbers: 05.45.Gg,73.21.La,73.23.-b,05.60.Gg
.
t
a
m I. INTRODUCTION double slit system,” a lot of interesting phenomena such
as the AB oscillations and the Fano effects can be ob-
-
d servedbytheinterferenceofwavefunctionstransmitting
Starting from the study of atomic nuclei, chaotic scat-
n through the two arms [12, 13, 14, 15]. In the context of
tering has been a topic of intensive research in a large
o
chaotic scattering, it is interesting to apply the known
c variety of systems such as atoms, molecules, quantum
analysis based on RMT [16, 17, 18, 19, 20, 21, 22, 23]
[ devices,andmicrowavecavities[1]. Afundamentalques-
to the AB ring system. We study how the interference
tion to be asked is how much information is reflected in
2 effects appear and the conductance behaves as the func-
the scattering through randommedia suchas disordered
v
tion of the controllable parameter such as the magnetic
4 andclassicallychaoticsystems. Oneofthemostremark-
flux through the ring.
6 able and promising ideas is to introduce the statistical
5 conceptintotheanalysis. Theensembleaverageoverdif- Our formulation is rather general and the application
0 ferentrealizationsofthesampleisconsideredtocalculate of our result is not limited to the QD systems. It is
1 several statistical quantities. A large number of systems knownthatmicrowavesinanirregularshapedcavitybe-
6
exhibit universal behavior determined by the symmetry havechaotically andthe statisticalproperties canbe de-
0
/ ofthe systems. Forthissituation,randommatrixtheory scribed by RMT [24, 25]. Based on the formal anal-
at (RMT) [2, 3, 4] has been used to understand the result ogy between the Helmholtz and Schr¨odinger equations,
m andhasplayedanimportantroleasastandardanalytical the classicalwavesaresimulatedasquantummechanical
tool. wave functions. Compared with the mesoscopic systems
-
d Recently, the experimental stage of the chaotic scat- in nanoscale,the cavity system is easier to fabricate and
n is ideal for an experimental study. We can also observe
teringhasbeenshiftedfromnaturaltoartificialsystems.
o the Fano effect in this system [26].
Typical examples are mesoscopic systems [5, 6, 7, 8, 9]
c
: such as quantum dots (QDs) and disordered wires. Re- How canwe define the statisticalmodel for the hybrid
v
centdevelopmentofnanotechnologymakesitpossible to system? For the system of two QDs attached to each
Xi fabricatemesoscopicquantumhybridsystemsthatcould armof the AB ring,Gefen et al. [12]consideredthe case
r notberealizedbeforeandalotofinterestinginterference wheneachdothasasingleregularlevel. As asimple but
a phenomenahavebeenobservedundercontrollableexter- nontrivial extension, we treat the case when one of the
nal parameters. Due to the interference of wave func- dots has regular levels and the other has random levels.
tions, a system made from parts such as the QD, lead, RMT is applied to the random dot.
andquantumpointcontactcannotbetreatedseparately. Thismodelcanbeviewedasamixedsystemofchaotic
Suchsystemsshownewinterestingphenomenawhichare and integrable levels. In single dot systems, such struc-
absentinsingleisolatedsystems. Typicalexperimentally ture is employed as an idea to explain anomalous phe-
fabricated systems are the QD on the Aharonov-Bohm nomena such as critical statistics [27] and fractal con-
(AB) ring [10], the side-coupled QD [11], and so on. ductance [28]. It is known in the open QD system that
ThemodeltreatedinthispaperistwoQDsputonthe the several specific levels couple with the lead strongly
two arms of the AB ring. In this so-called “mesoscopic while the other levels couple weakly via strong coupled
2
levels [29]. Thus it is too simple to treat the dot as a
single random matrix and we need to consider the inter-
nal structure more seriously. Although our model is not
directly related to such phenomena, it is instructive and
usefultoconsiderthepresentringsystemasthesituation
wherethestrongandweakcouplingscoexist. Inthissys-
tem, the regular transmission in the one arm is affected
by the random ones in the other arm, and vice versa.
From a point view of RMT, special attention is paid
to the universality of the statistical quantities. A natu-
ral question to be asked in the present model is how the
universal level correlations described by RMT are mod-
FIG. 1: Schematic drawing of our model. Dot 1 with a reso-
ified by the regular contribution. Naive expectation is
nant level E1 and dot 2 with a random Hamiltonian H2 are
thattheeffectoftheregularlevelscanbesafelyremoved
connected by leads. w denotes a dot-lead coupling matrix.
by the proper scaling (unfolding) [30]. It is known that
the effective theory is written in terms not of the mi-
croscopic parametersbut of the transmissioncoefficients
[16]. However, in the present system, the effect is am-
II. MODEL
plifiedbymultiple scatteringsthroughtheringandgives
highly nontrivial results.
A. Random Hamiltonian approach
Nowthatourmodelhasbeendescribed,wemustrefer
to the work by Clerk et al.[31]. They considered many
resonant levels in a single dot and RMT was employed WeconsidertheABringsystemdepictedinFig.1. The
for their distribution. The regular component to the S upper dot (dot 1) has a single resonant level, and the
matrix expressing the direct nonresonant path through lower dot (dot 2) has random levels and is treated by
the dot was used to find the Fano resonances. For each RMT. It is knownfrom scatteringtheory that the S ma-
resonance, the Fano parameter q was calculated and the trix of the system is written as [6, 7, 16, 23]
statistical distribution of q was defined over the reso-
nances. On the other hand, in our case, only the single 1
S = 1 2πiw† w
resonant level is present regularly and it is affected by − E H +iπww†
random levels. Thus our attention is fixed on the single −
1 iπw† 1 w
regular resonance. To discuss the statistical properties = − E+−H , (1)
of the transport we must prepare different realizations 1+iπw†E+1−Hw
of the random dot. The ensemble average is defined in
terms of such realizations.
whereH denotes the Hamiltonianmatrix for dots andw
the dot-lead coupling matrix. H can be written as
Theoutlineofthispaperisthefollowing. TheABring
model is defined in Sec.II. We define the randomHamil-
tonian model in Sec.IIA. A model based on the random E 0
H = 1 , (2)
SmatrixisalsodefinedinSec.IIBandtherelationtothe 0 H
2
(cid:18) (cid:19)
random Hamiltonian model is discussed. In Sec.III, we
calculate the average of the S matrix based on the ran-
where E is the fixed energy level for the dot 1 and H
dom Hamiltonian model. As a result the mean part of 1 2
is the random Hamiltonian for the dot 2. The size of
the conductance is calculated. It is not enough to calcu-
H , N, is taken to be infinity to find the universal re-
late the conductance including the quantum fluctuation 2
sult. We note that the total size of H is 1+N. It is a
effect andwe developthe supersymmetry method [32] in
straightforward task to extend the size of the upper dot
Sec.IVA to calculate the full conductance. The results
Hamiltoniantoarbitraryvaluesandhereweconsiderthe
oftheconductanceareshowninSec.IVB. Wealsostudy
minimal size 1. As another simplification, we consider
the AB oscillations in Sec.IVC and the Fano effect in
the 2 2 (unitary) matrix S, which means that the left
Sec.IVD. The fluctuation effects can be best seen in the ×
and right leads have a single channel, respectively. It
conductance distribution functions, which are studied in
is believed that the quantum interference effect becomes
Sec.V. Since realistic situations are not ideal and phase
maximal in this case [35]. Then the dot-lead coupling
breaking effect is present [25, 33], it is important to con-
matrixw isthe(1+N) 2matrixandcanbewrittenas
siderthedephasingeffecttheoretically. Weconsideritin ×
Sec.VI using a simple imaginary-potential model. Sec-
tion VII is devoted to discussions and conclusions. Part w(1) w(1L) w(1R)
w = =
oftheresultswerepublishedinapreliminaryreport[34]. w(2) w(2L) w(2R)
(cid:18) (cid:19) (cid:18) (cid:19)
3
w(1L) w(1R) derive the explicit representation using the matrix ele-
w(2L) w(2R) ments. Defining each S matrix elements as
 1 1 
= ww2((22...LL)) ww2((22...RR)) , (3) S =(cid:18)rt rt′′ (cid:19), Si =(cid:18)rtii rt′ii′ (cid:19) (i=1,2), (9)
 N N  we obtain, for example,
 
where (1L) refers to the coupling between the dot 1 and
lead L, and so on. t = 4 t1(1+s2+r2+r2′)+t2(1+s1+r1+r1′)
The conductance measures the transmission from the
h i
left to right lead and is defined by [6, 7, 16] × 9+3(r1+r1′ +r2+r2′)+s1+s2
3h(r r +r′r′)+(r r′ +r r′) 4(t t′ +t t′)
g = S12 2 , (4) − 1 2 1 2 1 2 2 1 − 1 2 2 1
h| | i −1
(r +r′)s (r +r′)s +s s , (10)
where denotes the ensemble averaging of the random − 1 1 2− 2 2 1 1 2
Hamilthoniian H . We employ the Gaussian ensemble [3] i
2 where s = detS = r r′ t t′ (i =1,2). Thus the total
and the probability density is given by i i i i− i i
transmissiontisnotequaltot +t ,ratherincludingnon-
1 2
π2 lineareffectsduetomultiplescatteringthroughthering.
P(H)=Cexp trH2 , (5) Suchmultiple scatteringeffects areputtogetherwithin-
−2N∆2
(cid:18) (cid:19) terferenceduetorandomscatteringandgivehighlynon-
trivial results for the conductance g = t2 .
where ∆ is the mean level spacing, and C is the normal-
h| | i
izationconstant. Inthefollowingcalculations,wemainly Another way of representing the total S matrix is to
considerunitarysymmetry,whichmeansthatH isHer- separate the S matrix of the system into the upper and
2
mitian and no additional condition is imposed. lowerdot parts andthe left andright fork parts [12, 13].
Choosing the fork matrices in a proper way, we can find
The result of the conductance depends on the choice
the same expression of t as in Eq.(10).
of the dot-lead coupling w. Although this matrix w has
4N degrees of freedom, there is no need to specify them The conductance can be calculated by taking the en-
completely. After the averaging, the effective degrees of sembleaverageoverS2determinedbytherandomHamil-
freedombecomes finite. Generally,it is 6andwe restrict tonian H2. Instead of doing that, we may disregard the
our discussion to the special case of 4 (see below). detailed structure of S2 and impose randomness directly
on S , simulated by the circular ensembles [3]. It is well
2
known that the random S matrix approach is equivalent
B. Random S matrix approach to the random Hamiltonian approachif we use the Pois-
son kernel [17]
Equation (1) is a useful formula to relate the Hamil-
1
tonian to the S matrix and can be used for the present Pβ(S)dµβ(S)∝ det(1 S S †)2β+2−βdµβ(S), (11)
coupled system. It is convenient to express the S matrix
| − h i |
in terms of the K matrix defined by
wheredµ (S)denotestheinvariantmeasurefortheSma-
β
1 iK trixandisusedasthemeasureforthecircularensemble.
S = − . (6) β is the index for the universality class. β = 1,2, and 4
1+iK
for the unitary, orthogonal, and symplectic case, respec-
K is expressed as the sum of contributions from dot 1 tively. S is the averaged value of S which is treated
h i
and 2: as an input parameter and is determined by the random
Hamiltonian model. The total S matrix is constructed
K =K +K , by the sum rule (8) and the conductance is expressedby
1 2
1 t2 where t is given by Eq.(10). By taking the circu-
K =πw† w, | |
E+ H lar ensemble average with the weight Pβ(S2), we obtain
− the conductance g which is the same as that obtained
πw(1)†w(1) 1
K = , K =πw(2)† w(2). (7) by the random Hamiltonian approach. The equivalence
1 E+ E 2 E+ H
− 1 − 2 of both approaches was shown in Ref.[22]. The random
S matrix approach has a great advantage for numerical
Thissimple relationimpliesthe sumruleofthe Smatrix
calculations because there is no need to take the ther-
1 S 1 S 1 S modynamic limit N and one may consider 2 2
− = − 1 + − 2, (8) → ∞ ×
1+S 1+S1 1+S2 random matrices S2.
Alternatively, we can parametrize the S matrix in
where S (S ) is the Smatrix for dot 1 (2). Itis instruc- terms of the K matrix (7). Then the expression of the
1 2
tive and useful in the followingnumericalcalculations to conductance becomes much simpler than Eq.(10) as we
4
showinSec.V. Thedisadvantageofthisparametrization Since w(2L) and w(2R) are N 1 matrices, we
isthattheKmatrixisHermitianandthematrixelements see that the relation w(2L)†w(2L×) w(2R)†w(2R)
| || | ≥
arenotcompact,whichisinconvenientforthe numerical w(2L)†w(2R) w(2R)†w(2L) holds. The equal sign holds
calculation. Thus we employ the S matrix parametriza- w| hen w(2L) |=| w(2R) or N| = 1, the latter is the case
tion (10) with compact variables for most of the numer- for w(1). Thus we need the additional parameter for the
ical calculations. parametrization of γ . Assuming the symmetry of the
2
left and right coupling again, we obtain the form with
the level width Γ as
2
III. AVERAGED S MATRIX
NΓ2 1 aeiφ
γ = . (19)
2 2 ae−iφ 1
AswehaveshowninEq.(7),theKmatrixiswrittenas (cid:18) (cid:19)
the sumofthe regular(dot1)andrandom(dot2)parts. The parametera reflects the abovementionedinequality
Thus,togettheaveragedKmatrix,wemayconsiderthe and 0 a 1. We note that the same phase φ appears
≤ ≤
ensemble averaging of the random part. We know from inγ andγ ,butthe signis oppositetoeachother. This
1 2
RMT thatthe averagedGreenfunction for the Gaussian phase affects the transmission part of the S matrix and
unitary ensemble is given by [3] can be identified with the AB flux through the ring [12,
13].
1 π
= e−iz, (12) Using this parametrization, we can write
E+ H N∆
(cid:28) − 2(cid:29) 1 iX 1 aeiφ
K = Φ , (20)
where h i ǫ − 2 ae−iφ 1
(cid:18) (cid:19)
πE where
cosz = . (13)
2N∆ E E πΓ
1 2
ǫ= − , X = . (21)
N is taken to be infinity while E/∆ is kept finite. Then Γ1 ∆
we have e−iz i and the averaged K matrix is given
The energy ǫ represents the distance from the resonance
→ −
by
point and X is the ratio of the level width to the mean
level spacing of the dot 2. Thus this model is described
1 iπ
hKi= E+ E γ1− N∆γ2, (14) by four parameters ǫ, X, a, and φ.
− 1 For the random dot, the elements of the dot-lead
where γ = πw(i)†w(i) (i = 1,2). It is important to note coupling w(2) distribute randomly and the summation
i
that the result depends on the dot-lead couplings w(1,2) N w(2L)∗w(2R) can be smallwhen the randomphases
i=1 i i
through γ . of w(2L) and w(2R) almost cancel out. This means a is
1,2
P
For the regular dot, the most general form of γ is vanishingly small. On the other hand, the summation
1
can be finite when the left and right dot-lead couplings
πw(1L)∗w(1L) πw(1L)∗w(1R) are correlated mutually. This results in direct nonreso-
γ =
1 πw(1R)∗w(1L) πw(1R)∗w(1R) nant reaction [16]. We first discuss the case of a= 0 for
(cid:18) (cid:19)
1 Γ √Γ Γ e−iφ simplicity. The averagedK matrix takes the form
= 1L 1L 1R , (15)
2 √Γ1RΓ1Leiφ Γ1R 1 iX
(cid:18) (cid:19) K = Φ . (22)
h i ǫ − 2
where Γ (Γ ) turns out to be the level width for the
1L 1R
left (right) coupling of the dot to the lead and φ is a The finite-a effect is discussed afterwards.
phase. We assume the symmetric coupling Γ = Γ Now we go back to the S matrix. The averagedS ma-
1L 1R
for simplicity and use trix is simply obtained by using the averagedK matrix,
1 K
γ1 =Γ1Φ, (16) S = −h i
h i 1+ K
h i
where 1 X 2i
= − 2 Φ. (23)
1 1 e−iφ 1+ X − 1+ X 1+ X ǫ+i
Φ= . (17) 2 2 2
2 eiφ 1
(cid:18) (cid:19) This is justified by the s(cid:0)addle-p(cid:1)o(cid:2)i(cid:0)nt anal(cid:1)ysis of(cid:3)the non-
linear sigma model described below. We define g =
This matrix satisfies Φ2 =Φ and is diagonalizedas Φ 0
→ S 2, which is the conductance if we can disregard
diag(0,1). |h 12i|
the quantum fluctuations. It is given by
Onthe otherhand, forthe randomdot, the formofγ
2
is slightly complicated. It is written as 2
Γ1
1 1+X
γ2 =(cid:18)ππww((22RL))††ww((22LL)) ππww((22RL))††ww((22RR)) (cid:19). (18) g0 = 1+ X2 2(E−E(cid:16)1)2+2 (cid:17)1+Γ1X2 2. (24)
(cid:0) (cid:1) (cid:16) (cid:17)
5
TheresultshowsthatthelevelwidthΓ forthedot1and This is easilysolvedwith the proper boundarycondition
1
the conductance are renormalized by the factor 1/(1+ as
X/2).
Forlateruse,wedefinethetransmissioncoefficientsas πE πE 2
Q=e−izΛ = iΛ 1 iΛ, (30)
T = 1 S S † 2N∆ − s − 2N∆ →−
(cid:18) (cid:19)
−h ih i
2
= 2X 2X 1+Γ2X2 Φ. wahgeerneewraeltsooolukttiohne liinmciltudNin→g t∞hekseaedpdinleg-pEo/in∆t fimnainteif.oAlds,
1+ X2 2 − 1+ X2 2(E−E(cid:16)1)2+(cid:17)1+Γ2X2 2 we can write
(cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:17) (25) Q= iσ, σ =VΛV¯, (31)
−
This matrix can be diagonalized to find the eigenvalues
where V is the 4 4 supermatrix and satisfies VV¯ = 1.
2X 2X ×
T = , T = . (26) The symmetry of V is determined in the standard way
1 1+ X 2 2 1+ X 2+ 1 [32].
2 2 ǫ2
Now the generating function reads
Note that 0 (cid:0)T T(cid:1) , T = T(cid:0) at E(cid:1) E , and
2 1 2 1 1
≤ ≤ | − | → ∞
T = 0 at E = E . At X = 2, T takes the maximum
2 1 1
value, T =1, and the transmission through the random Z = σe−F,
1 h i D
dot becomes ideal. Z
1 0
In conclusion of this section, we found the averaged S F = str4(1+N)ln 1+ E−0E1 π Q iΛπww†
matrix (23) and the conductance (24). Of course, this is (cid:20) (cid:18) N∆ (cid:19) (cid:21)
notthefinalresultoftheaveragedconductance. Wejust 1 iπ
cfraolmcultahteedotrhigeinmaeladnefipnairttiogn0 =(4)|.hS1W2ie|2mwuhstichexiasmdiinffeertehnet = str8ln(cid:20)1+iΛ(cid:18)E−E1γ1− N∆γ2σ(cid:19)(cid:21), (32)
fluctuation part δg =g−g0 =h|S12|2i−|hS12i|2. where we assumed γ1 = Γ1Φ and γ2 = NΓ2/2. Since
the matrix sizes of σ and γ are 4 and 2, respectively, the
total size of the superspace in the last expression is 8.
IV. CONDUCTANCE
We finally obtain
A. Supersymmetry method 1 iX
F = str ln 1+i Φ Λ σ
8
ǫ − 2
(cid:20) (cid:18) (cid:19) (cid:21)
We derive the nonlinear sigma model for the coupled
1 T Λσ+σΛ
system to calculate the fluctuation part of the conduc- = str ln 1+ 2 . (33)
tance. Accordingtothesupersymmetrymethod[16,32], 2 8 " 1 T 2 #
− 2
the generating function for the product of Green func-
tions G(E) = 1/(E H +iπww†) and G†(E) is defined The nonlinear sigma model (33) can be written in terms
−
by of the transmission matrix T (25) and the microscopic
fundamentalparameterX doesnotappearintheexpres-
Z = (ψ¯,ψ)exp iψ¯ E+iΛπww† H ψ , (27) sionexplicitly. Thisisamanifestationoftheuniversality
D − [3, 4, 16].
Z h (cid:16) (cid:17) i
where ψ has 4(1 + N) components coming from su- Equation (33) is for systems with unitary symmetry.
persymmetry (bosons and fermions), retarded-advanced Inthe samewaywecanderivethe nonlinearsigmamod-
structure, and Hamiltonian space. Λ = diag(1, 1) in els for the orthogonal and symplectic symmetry classes.
retarded-advanced space. Following the standard−proce- Thenσ becomesan8 8supermatrixandtheadditional
×
dure,we introducethe Hubbard-StratonovitchfieldQ to symmetryduetotime-reversalinvarianceisimposed[32].
write the averagedgenerating function as
Z = Qexp str ln E+iΛπww† B. Conductance
4(1+N)
h i D −
Z (cid:26) (cid:20)
E 0 1
1 str Q2 , (28) In the nonlinear sigma model approach, the averaged
− 0 N∆Q − 2 4 conductance is calculated by performing the integration
(cid:18) π (cid:19)(cid:21) (cid:27)
ofthe σ matrix. Themeanpartofthe conductanceg in
where Q is a 4 4supermatrix. “str” denotes supertrace 0
× Eq.(24)iseasilyobtainedbyneglectingthefluctuationof
andthe subscriptindicatesthesizeofsuperspace. When
theσmatrixasσ =Λ. Tofindthefluctuationpartofthe
w =0, the saddle-point equation is written down as
conductance δg = S 2 S 2, we must take into
12 12
N∆ 1 accountthe contribhu|tion| fir−om|htheis|addle-point manifold
Q= . (29)
π E+ N∆Q parametrizedbytheV matrix. Thiscalculationishighly
− π
6
FIG. 2: Conductance vs X =πΓ2/∆. The thick (thin) lines FIG. 3: Conductance vs ǫ = (E−E1)/Γ1. The thick (thin)
are analytical results of the total conductance g (mean part lines are the total conductance g (the mean part g0). Inset :
g0). Inset : Comparison of the analytical (denoted by lines) Fluctuation part δg=g−g0.
and numerical (dots) results for the fluctuation part δg =
g−g0.
complicated and we refer to the Appendix for details.
We finally obtain
T +T 1 X2 1 X2 1
δg = 1 2 − 4 − 4 − ǫ2
4 − 2X ! 2X !
2
T T T +T T T T
1 2 1 2 1 2 1
ln ,
× T T 2 − T T T
(cid:18) 1− 2(cid:19) (cid:20) (cid:18) 1− 2(cid:19) 2(cid:21)
(34)
whereT aretheeigenvaluesofthetransmissionmatrix
1,2 FIG.4: Numericalresultsoftheconductanceg(ǫ)fororthog-
T given by Eq.(26).
onalandsymplecticensembles. Theresultforthesymplectic
We first examine the two limiting cases, ǫ and
case is normalized to unity.
| | → ∞
ǫ = 0. The limit ǫ means that dot 1 is detached
| | → ∞
from the system and the S matrix is givenby S =1. In
1
this case, T = T = 2X/(1+X/2)2 and we recover the
1 2
known result [36] the random S matrix S is treated statistically by using
2
the Poisson kernel (11). We take the ensemble average
δg = T1 + T12. (35) over more than 106 samples of the S matrix.
3 6 In Fig. 2, X dependence of the conductance is shown
for several values of ǫ. g shows a peak at X = 0 while
In the other limit ǫ = 0 (E = E ) the energy coin- 0
1 δg takesamaximumatX =2asshownbythethinlines
cideswiththe levelindot1andthe perfecttransmission
and the inset in Fig. 2, respectively. As ǫ g (δg)
through dot 1 is achieved. Then T2 =0 and we obtain is monotonically decreasing (increasing) an→d∞the 0result
T rapidly approachesEq.(35). The numerical result agrees
1
δg = . (36) with Eq.(34) in a highly accurate way, which shows the
4
equivalence of the random Hamiltonian and random S
We see that Eq.(35) is larger than Eq.(36), which means matrix approach.
that the fluctuation effects are reduced as we approach ǫ dependence of the conductance is shown in Fig. 3.
the resonantpoint. For intermediate values ofǫ, Eq.(34) A resonance peak appears at ǫ = 0, reflecting transport
cannot be written in terms of T only in contrast to through the regular dot 1. This peak structure, how-
1,2
Eqs.(35) and (36). This is because the source term to ever, changes qualitatively as a function of X. For small
calculatetheconductancedependsonǫandX explicitly, X the peak is convex and the peak height decreases on
althoughthe nonlinearsigmamodelitselfcanbe written increasing X. When X = 2, g is independent of ǫ. In-
in terms of T, as shown in the Appendix . creasing X further, we find that the peak turns into an
These results are checked by numerical calculations. antiresonanceand g decreases monotonically. The result
We use the formula (10) for the transmissionmatrix. S for X =2 correspondsto that ofthe circularunitary en-
1
isgivenbyS =(1 iK )/(1+iK )withK =Φ/ǫ,and semble because S = 0, and the Poisson kernel P (S )
1 1 1 1 2 β 2
− h i
7
In systems with unitary symmetry, since the scatter-
ing in the random dot randomizes the phase of the am-
plitude,the resultbecomesindependentoftheAB phase
φ. Thisisnotthecaseforthe orthogonalandsymplectic
systems and the oscillations can be observed. However,
theperiodoftheoscillationisdifferentfromthatforregu-
larsystems. Thiscanbe understoodfromthe expression
of the transmission t in Eq.(10). The phase is included
in that expression as
At eiφ+Bt e−iφ
1 2
t= , (37)
C D(t t′e2iφ+t t′e−2iφ)
− 1 2 2 1
FIG. 5: Conductance vs φ for X = 2. The lower figure is where A, B, C, and D are phase independent contri-
for the orthogonal ensemble and upper for symplectic. No butions. If we neglect the multiple scattering effect the
oscillations are observed for the unitary case. totaltransmissionisapproximatedbyt t eiφ+t e−iφ.
1 2
∼
Then the conductance is given by
g t eiφ+t e−iφ 2
1 2
becomes unity. As we see in the inset of Fig.3, δg at ∼ | |
= t 2+ t 2+t t∗e2iφ+t∗t e−2iφ. (38)
the resonant point is relatively small and the quantum | 1| | 2| 1 2 1 2
fluctuation effect smooths the resonance.
We see thatthe thirdandfourthterms ofthe righthand
Forcomparisonwecalculategasafunctionofǫforthe
side give oscillations with the period π. However, these
orthogonal and symplectic ensembles numerically. For
terms vanish after the random averaging. The contribu-
the orthogonal case, the Hamiltonian has time-reversal
tions going around the ring twice give oscillations with
invarianceandthematrixelementsarereal. Forsymplec-
theperiodπ/2andsurviveaftertheaveraging. Suchcon-
tic, the Hamiltonian becomes a quaternion real matrix
tributions comefromexpandingthe denominator. Thus,
[3]. TheresultsareshowninFig.4. When0<X <2,the
in the orthogonal and symplectic systems, g depends on
resonanceisenhanced(reduced)fortheorthogonal(sym-
the AB phase due to the multiple scattering inside the
plectic) ensembles. At X = 2, the orthogonal ensemble
ring. The period of the AB oscillations becomes half of
givesaresonancewhile the symplecticensemble givesan
that for the regular systems. This effect can be inter-
antiresonance. When X =10, we see that antiresonance
preted as a kind of the Altshuler-Aronov-Spivak effect
isreduced(enhanced)fortheorthogonal(symplectic)en-
[37] for cylinder systems. In a ring system, it was dis-
semble in contrast to the case of X <2.
cussed in Ref.[38] that the period of the oscillation be-
Awayfromthe resonance,the quantumfluctuationef-
comes half a flux quantum by the self-averagingeffect.
fect becomes larger as the number of degrees of freedom
We showthe numericalresults in Fig.5 for the orthog-
of random variables increases. We note that the number
onal and symplectic ensembles. The period of the oscil-
of degreesbecomes maximumwhen β =4 and minimum
lations is π/2 as we discussed above and the difference
when β = 1. We can also see that the conductance at
between these two results is that the conductance be-
the resonant point is independent of the choice of the
comesminimum(maximum) fororthogonal(symplectic)
ensemble. This result is discussed in detail in Sec.V.
atφ=0. Thiscanbeunderstoodbythe standardmech-
anism of weak localization [39].
C. Aharonov–Bohm oscillations
D. Fano effect
For regular ring systems, it is well known that the AB
oscillations are observed by applying the magnetic flux
through the ring. Since the flux is a tunable parameter The Fano effect is induced by the correlation of the
it is an important method to control the system. Our resonant and direct path [31, 40]. The direct path can
interest is how the effect of the AB flux can be observed be described by the parameter a in Eq.(19). If we keep
in the present random system. Can the AB oscillations thisparameterinEq.(20),theaveragedSmatrixisgiven
survive after the random averaging? by
1
S =
h i 1+ 1+aX 1+ 1−aX + 1+ 1−acos2φX i
2 2 2 ǫ
(cid:0) (cid:1)(cid:0) (cid:1) (cid:16) (cid:17)
8
1 1−a2X2 1−acos2φXi ie−iφ aXeiφ
× − 4ieiφ −aXe−2iφ ǫ 1 1−−aǫ2X2 −1−acos2φXi !, (39)
−ǫ − − 4 − 2 ǫ
in δg as shown in the inset of Fig.3, the asymmetry is
oppositetothatofg . Asaresult,thetotalconductance
0
becomes symmetric. This result means that the Fano
effect appears not on g but on the mean part g and
0
the fluctuation part δg, respectively. We note that the
asymmetricformisobtainedwhenReq =0. Theeffectof
6
theimaginarypartofqkeepstheconductancesymmetric.
WecanconcludethattherealpartoftheFanoparameter
doesnotaffectthetotalconductance. Weconfirmedthat
thesymmetricconductanceisobtainedfortheorthogonal
and symplectic classes as well.
FIG. 6: Conductance vs ǫ = (E −E1)/Γ1 at a = 0.7 and
φ=−π/8. Thethick(thin)linesarethetotalconductanceg V. CONDUCTANCE DISTRIBUTION
(themeanpartg0). Inset: Fluctuationpartδg=g−g0. The FUNCTIONS — MODE-LOCKING EFFECT
total conductance g is obtained numerically and the mean
part g0 is plotted by using Eq.(40) Intheprevioussectionwefocusedontheaveragedcon-
ductance. It is wellknownthat disorderedsystems show
strongfluctuation effects, whichmeanthat the squareof
the conductance and even the higher moments become
The mean part of the conductance is derived from this
relevanttocharacterizethesystem. Todiscusstheeffects
expression as
of the fluctuations, here we calculate the conductance
distribution function P(g) = δ(g S 2) . We show
a2X2 h −| 12| i
g = the analytical results when ǫ = 0 which show universal-
0 1+ 1+aX 2 1+ 1−aX 2 ity among the ensembles. We also report the numerical
2 2
(cid:0) (cid:1) ǫ(cid:0) ǫ1+qΓ1(cid:1)2 results.
| − | , (40) The expression of the conductance distribution be-
×(E E )2+ (1+1−ac2os2φX)2 Γ2 comes simpler if we use the K matrix representation as
− 1 (1+1+aX)2(1+1−aX)2 1
2 2 we mentioned in Sec.IIB. K is a Hermite matrix and
K =K +K with
where q is the Fano parameter 1 2
ie2iφ K = 1 1 e−iφ , K = a1 beiφ . (42)
q = aX . (41) 1 2ǫ(cid:18)eiφ 1 (cid:19) 2 (cid:18)b†e−iφ a2 (cid:19)
ThustheFanoeffectappearswhena=0. Theadditional a arereal,andb depends onthe universalityclassand
1,2
6
condition φ=0 is required to obtain a finite realpart of is expressed as
6
q. Then the asymmetric conductance form is obtained.
Atthelimit E E ,g hasafinitecontributionin b for β =1,
1 0 0
| − |→∞
contrast with Eq.(24). This means that there is a direct b= b +ib for β =2, (43)
0 1

regularcouplingbetweentheleftandrightleadsthrough b +b e +b e +b e for β =4,
 0 1 1 2 2 3 3
the random dot.
This Fano effect also appears on δg. The numerical where b  are real and e quaternion matrices de-
0,1,2,3 1,2,3
result in Fig.6 shows that the Fano parameter for δg is fined by e = iσ with the Pauli matrix σ (j = 1,2,3).
j j j
the same as Eq.(41). Since the antiresonance appears The conductance is expressed in this parametrization as
1+4b0ǫcos2φ+4|b|2ǫ2 for β =1,
[b0cos2φ−a1+2a2+(1−a1a2+|b|2)ǫ]2+[1+(a1+a2)ǫ]2
g = 1+4(b0ǫcos2φ−b1sin2φ)+4|b|2ǫ2 for β =2, (44)
 1tr[b0cos2φ−b1sin2φ−a11+2+a42b+0ǫ(1c−osa21φa+2+4||bb||22ǫ)2ǫ]2+[1+(a1+a2)ǫ]2 for β =4,
2 [(b0cos2φ−rσ3sin2φ)−a1+2a2+(1−a1a2+|b|2)ǫ]2+[1+(a1+a2)ǫ]2

9
where b2 = β−1b2 and r2 = b2+b2+b2. We note that the conductance for β = 4 is normalized to unity. This
| | i=0 i 1 2 3
expression is averagedby the generalized circular ensemble (Poisson kernel)
P
(β+2)/2 β−1
1
P (S )dµ (S ) da da db , (45)
β 2 β 2 ∝( X + 2( a a + b2) 2+(a +a )2) 1 2 i
2 X − 1 2 | | 1 2 iY=0
(cid:2) (cid:3)
whereweused S =(1 X/2)/(1+X/2). Thenumeri- consideredasfollows. InEq.(42),allthematrixelements
2
h i −
calresultsusingthe Metropolisalgorithm[41]areshown ofK aredivergentata=0. Whenφ=0,thisdiverging
1
in Fig.7 at φ=0. term belongs to the member of the orthogonalensemble
The resultsatlargeǫareinterpretedasthe singleran- and affects the variables a and b in the second term
1,2 0
dom dot case. This case was discussed in Ref.[18] using K which are common to all the ensembles. Then the
2
the randomHamiltonianapproachandthe analyticalre- effective modes are locked on those for the orthogonal
sult for the unitary systemwas obtained. In the random class and the conductance (47) becomes independent of
S matrix approach, the case of the perfect transmission the rest of the parameters b .
1,2,3
X =2 was obtained in Ref.[20] as
Equation (49) with X = 2 appears in the problem of
the classical random walk [42] and is known as the arc-
β
P(g)= g−1+β/2, (46) sine law. Consider the one-dimensional classical random
2
walk starting at the origin. The walker can move to ei-
and other cases were discussed in Ref.[21]. The case of ther one of its two nearest neighbor sites with the equal
ǫ = 10 is enough to find a large-ǫ result and we find a probabilityp=0.5. After theN-stepwalk,wecountthe
good agreement with their results. number of the events which the walker was in the posi-
Inthecaseofǫ=1,wecanclearlyseehowtherandom tiveaxisM. Thenthe distributionfunctionofg =M/N
dot significantly affects the distribution function. If we approaches Eq.(49) with X = 2 as N . It can
→ ∞
increase X, a single peak at small X turns into a broad be considered that the walker at the positive (negative)
one and a different peak aroundg =0 is formed at large direction corresponds to the transmission (reflection) to
X. the left (right)leadinourmodel. Due to the presenceof
Itisinterestingtoseetheresultsatǫ=0whicharein- theresonantpaththroughthedot1,aparticletransmit-
dependentofthechoiceoftheensemble. Inthiscase,the ted through the dot 2 can go to either left or right lead
conductance distribution can be calculated analytically. with equal probability. The particle reflected by the dot
The conductance is written as 2 can go either way as well. Thus the particle entered
from a lead forgets where it came from. Such a process
1
g = . (47) can be interpreted as a random-walk-like one and gives
1+ b0− a1+2a2 2 the same distribution function. It is interesting that the
asymmetric random walk with the probability p = 0.5
and the distribution fu(cid:0)nction is ob(cid:1)tained from the ex- can be described by our model with X = 2. Sinc6e the
pression analytic form is not known in the asym6metric random
walk, our result may be useful for understanding the re-
β−1
sult. Itisalsoknownthatthesamedistributionfunction
P(g)=C da da db
1 2 i appears in the problem of the continuous-time quantum
Z i=0
Y walk [43].
1
δ g When the phase φ is finite, K does not belong to the
× − 1+ b0− a1+2a2 2! member of the orthogonal ensem1ble and the results can
(β+2)/2 depend on the choice of the ensembles. We numerically
(cid:0) 1 (cid:1)
, foundthattheorthogonalandunitarycasesareindepen-
×( X + 2( a a + b2) 2+(a +a )2) dent of φ and the result (49) is kept unchanged. For the
2 X − 1 2 | | 1 2
symplectic case, we found that the result depends on φ
(48)
(cid:2) (cid:3)
and Eq.(49) does not maintain anymore. The numerical
where C is the normalization constant. We perform the resultforX =2andǫ=0isshowninFig.8. Remarkably,
integrals and obtain all plotted curves give the averagedconductance g =0.5
and the phase dependence appears only for the conduc-
1 1 tance fluctuations. We also see that plotted curves has
P(g)= . (49)
π g(1 g) 2(1 g)+ Xg a nonanalytic point at around g = 0.5, which implies a
− X − 2 nontrivialmechanismduetothephasecoherenteffect. It
This result agreespwith the numerical ones in Fig.7. The isnotclearhowthishappensandfurtherstudyisneeded
reason why this result becomes independent of β can be to clarify the underlying mechanism.
10
FIG. 8: Conductance distribution function of the symplectic
system for several values of φ. Weset X =2 and ǫ=0.
FIG.9: Conductancedistributionfunctionforunitarysystem
for several valuesof p=1/∆τ.
ThismethodisequivalentwiththatofRef.[18]wherethe
imaginary part of the Hamiltonian was introduced. In
the supersymmetry method, this effect can be described
by the additional term of the sigma model [18]
1
F = strσΛ. (51)
τ
∆τ
This term makes the massless “diffusion” modes on the
saddle-point manifold massive and reduces the quantum
fluctuations. See the Appendix for details.
It is wellknownin the S matrix approachthat the de-
phasingeffect canbe describedby the Bu¨ttiker’s voltage
FIG.7: Ensembledependenceoftheconductancedistribution
probe model [44]. A fictitious voltage probe eliminat-
functionsat φ=0. Thecurvesat ǫ=0arewellfittedbythe
ing the phase coherence is attached to the dot and is
analytical result (49)
described by an enlarged S matrix.
BrouwerandBeenakkershowedthatthevoltage-probe
model at a certain limit becomes equivalent to the
VI. DEPHASING imaginary-potentialmodel and found the modified Pois-
son kernel in the random S matrix approach [35]. Here
we investigate this limit using the imaginary-potential
Inthe Hamiltonianapproach,the dephasingeffectcan
model. Since the dephasing effect to the regular dot is
be modeledbyintroducingthe imaginarypartto the en-
trivial, we include the effect in the random dot only.
ergy
In Fig.9, the numericalresults of the conductance dis-
i tribution function using the random Hamiltonian model
ǫ ǫ+ . (50) are shown. We add the dephasing term, p = 1/∆τ with
→ 2τ