Table Of ContentResonance width oscillation in the bi-ripple ballistic electron waveguide
Hoshik Lee1,2 and L. E. Reichl1
1 The Center for Complex Quantum Systems, The University of Texas at Austin, Austin, Texas 78712
2Department of Physics, Temple University, Philadelphia, Pennsylvania 19122
(Dated: January 27, 2009)
Interferenceofquasiboundstatesisstudiedinaballisticelectronripplewaveguidewithtworipple
cavities whose distance apart can be varied. This system is the waveguide analog of Dicke’s model
fortwointeractingatomsinaradiationfield. Dicke’smodelhasresonanceswhosewidthschangein
an oscillatory manner as the distance between the atoms is varied. Resonances that form in a bi-
ripplewaveguidebehaveinamannerthathassomesimilaritytoDicke’ssystem,butalsoimportant
9
differences. We numerically investigate the behavior of resonance widths in the waveguide as the
0
distancebetweenthetworipplecavitieschangesandwefindthattheresonancewidthsoscillatewith
0
variation of distance, but the coupling does not decrease as it does in Dicke’s system. We discuss
2
differencesbetweenourwaveguidesystemandothersystemsshowingtheanalogousofDickeeffect.
n We also study S-matrix pole trajectories and find that they rotate in counterclockwise direction on
a a circle in the complex energy plane.
J
7 PACSnumbers: 72.10.-d,42.50.Gy,73.63.Kv,73.23.Ad
2
] Interference is a distinctive property of waves in both pendent atoms. There is also an antisymmetic collective
l
l quantum and classical mechanics. It plays an especially state which is weakly coupled to the photon fields. Its
a
important role in quantum mechanics because it demon- decayiscalledsubradiancebecauseitdecaysmoreslowly
h
- strates the wave nature of particles. Matter-wave inter- than independent atoms. The different coupling to the
s ference has been observed in atomic, nuclear, and solid photon fields results in broadening and narrowing of the
e
statesystems. However,thesesystemsareoftentoodiffi- resonance widths, which is called Dicke effect. The anal-
m
culttocontroltoallowinvestigationofavarietyofinter- ogous effect can be found in mesoscopic systems because
.
t ference phenomena. Recent developments of nano tech- quantum dots or quasi bound states in a waveguide be-
a
nology now enable us to study interference phenomena have like atoms. The Dicke effect in mesoscopic systems
m
using ballistic electron waveguides, carbon nanotubes, was first found by Shahbazyan and Raikh [8] for two
-
quantum dots. Indeed, quantum dots are often called channel resonant tunneling in a system with two impu-
d
n ”artificialatoms”becausetheyhavediscreteenergylevels rities. In their system, the bound state of each impurity
o duetotheconfinementoftheelectronwavefunction. One are indirectly coupledthrough the electron wavefunction
c can change the electron state inside a quantum dot with in external leads. The analogy to Dicke effect has since
[
external controls such as gate voltages. The Coulomb been found in several other mesoscopic systems, such as
1 blockade conductance profile shows the discreteness of quantumdotscoupledviaacommonphononfield[9]and
v energy levels inside a quantum dot. a quantum wire with side coupled quantum dots [10].
1 Ballistic electron waveguides have been used to study The Dicke effect has also been seen in the spontaneous
8
the interaction of quantum systems with their environ- radiation from collectively interacting quantum dots in
2
4 ment. They have provided an important tool for study- an experiment by M. Scheibner et. al. in 2007 [11].
. ing the effects of quantum chaos on scattering phenom- In a previous study [12], we found the analog of the
1
ena [1, 2, 3, 4, 5, 6]. In waveguides, Coulomb blockade Dicke effect in a multi-ripple ballistic electron waveg-
0
does not occur because discrete energy levels don’t ex- uide. Qusaiboundstatesineachcavityofthewaveguide
9
0 ist. However, even though there is no discrete energy interact through the internal leads to form the collec-
: spectrum, quasi bound states can build up in waveg- tive states which result in different coupling to the ex-
v
uides and significantly affect the electron transmission. ternal leads. We found the broadening and narrowing
i
X Since there exist both non-resonant and resonant chan- of resonance widths in electron transmission (conduc-
r nels for electron transmission, resonances show the Fano tance) as more ripple cavities are added to the waveg-
a
profile, indicating the interaction between resonant and uide. However, we could not determine how the reso-
non-resonant channels. nance widths change as the distance between ripple cav-
Dicke proposed that spontaneous radiation from two ities varies, which is one of the key features of the Dicke
interacting atoms would have a much longer wavelength effect. In this brief report, we study the dependence of
thanthedistancebetweenthetwoatoms [7]. Thisradia- resonance widths on the distance between the two cavi-
tionisassociatedwithatransitionfromasymmetriccol- ties of a bi-ripple waveguide the Generalized Scattering
lectivestateoftheatomstothegroundstate. Itiscalled Matrix Method(GSM) [13, 14].
superradiance, because the symmetric state is strongly We consider a waveguide system in which two ripple
coupledtotheexternalphotonfieldsanditsdecayoccurs cavities are connected to each other by a flat waveg-
more rapidly than the decay of excited states of inde- uide whose length is WL. The length of each ripple
2
(a) (b) (c)
1 1 1
0.8 0.8 0.8
mission0.6 mission0.6 mission0.6
Trans0.4 Trans0.4 Trans0.4
0.2WWWLLL===000 0.2WWWLLL===111000˚A˚A˚A 0.2WWWLLL===111555˚A˚A˚A
0 0 0
1.327 1.328 1.329 1.327 1.328 1.329 1.327 1.328 1.329
Energy(E1) Energy(E1) Energy(E1)
(d) (e) (f)
1 1 1
0.8 0.8 0.8
FthIeGt.w1o: rAippblie-ricpapvlietiewsa.veguidewithaflatwaveguidebetween Transmission000...246WWWLLL===222000˚A˚A˚A Transmission000...246WWWLLL===222555˚A˚A˚A Transmission000...246WWWLLL===333000˚A˚A˚A
0 0 0
1.327 1.328 1.329 1.327 1.328 1.329 1.327 1.328 1.329
Energy(E1) Energy(E1) Energy(E1)
cavity is W, and the upper wall of each cavity is de-
scribed by an analytic function, y = d−acos(2πx/W)
(See Fig. 1). The outer ends of each cavity is connected FIG. 2: (Color Online) Electron transmission profiles of the
to a semi-infinite lead whose height is L = (d−a). We lowest resonance for bi-ripple electron waveguides with vari-
use R-matrix theory to calculate the S-matrix for a sin- ousdistancesWLoftheflatwaveguidebetweenthetworipple
gle ripple cavity [12], [1]. R-matrix theory was orig- cavities. AsWLchanges,thewidthofthetworesonancesand
theresonanceenergieschange. Thedashedlineistheelectron
inally developed by Wigner and Eisenbud in 1950s for
transmission for a waveguide with a single ripple cavity.
the study of nuclear scattering. Recently, R-matrix the-
ory has been used to study electron transmission in bal-
listic electron waveguides [1, 5, 6, 12]. We consider
a waveguide built in a 2DEG (two dimensional elec- ()!’!*
) WL=0
tron gas) made of a GaAs/AlGaAs heterostructure at ’
a very low temperature. We then use the GaAs effec- WL=10˚A
tive mass of the electron m∗ = 0.061me, where me is WL=15˚A
the free electron mass. We use parameters a = 13.846 WL=20˚A
˚A, d = 47.269 ˚A, and W = 300 ˚A for the waveguide /! WL=25˚A
in Fig. 1. E1 = 2h¯m2∗πL22 = 0.5034074eV is the thresh- -).- WL=30˚A
old energy to open the first propagating mode in the 1) !!’
semi-infinite waveguide. We use E as the unit of en- 0
1
ergy through this report. We only consider the incident
electronenergiesE whichallowonepropagatingmodeso
that E ≤E ≤4E .
1 1
Since electrons freely propagate in the flat waveg- !!&
)
uide between the ripple cavities, the electron wavefunc- !"#$%% !"#$%&
+,)-).- /
tion acquires the phase factor sWL = eik1WL, where !
(cid:113)
k = 2m∗/¯h2(E−E ). The S-matrix for phase ac-
1 1
quisition is a diagonal matrix,
FIG.3: (Coloronline)S-matrixpolesforthebi-ripplewaveg-
(cid:18) (cid:19)
s 0 uide with different values of WL. Each pole rotates on a cir-
S = WL . (1)
WL 0 s cle (whose center is the S-matrix pole of the waveguide(red
WL
*) with a single cavity) in counterclockwise direction as WL
For a single ripple waveguide, the S-matrix is written in increases. The pair of poles has π phase difference so that
terms of transmission and reflection coefficients such as they are located in the opposite direction on the circle.
(cid:18)r t(cid:48) (cid:19)
S = 1 1 . (2)
1 t r(cid:48)
1 1 where U = (1 − r(cid:48)s r s )−1 and U = (1 −
1 1 WL 2 WL 2
The S-matrix for the second ripple waveguide S2 can be r1(cid:48)sWLr2sWL)−1.
obtained in the same way. If two ripple waveguides are Fig.2showsthetransmission(conductance)ofanelec-
identical,S andS aresame. Weconsideronlyidentical tron through the bi-ripple waveguide for different flat
1 2
ripples in this report. The overall scattering matrix S waveguide lengths WL. The dashed line is the electron
T
is obtained by the GSM [13, 14], transmission for a waveguide with a single ripple cavity.
We can see the resonance width narrowing and broaden-
(cid:18)r +t(cid:48)s U r s t t(cid:48)s U r(cid:48) (cid:19) ing (Dicke effect) depending on WL.
S = 1 1 WL 2 2 WL 1 1 WL 2 2 ,
T t2sWLU1t1 r2(cid:48) +t2sWLU1r1(cid:48)sWLt(cid:48)2 Fig. 3 shows how the S-matrix poles change position
(3) inacomplexenergyplane. Thetwopolesinducethetwo
3
resonances in the transmission plot (Fig. 2). The two
x 10−5
polesrotateonacircle(whichiscenteredatthepolepo-
0
sitionforawaveguidewithasinglecavity(red*inFig.3)
as the distance WL is varied. The behavior of the poles
indicatesthatnotonlythewidthsbutalsotheresonance
energy(realpartofthepole)changeswithWL. Thepair
of poles is located on opposite sides of the circle, which
)1
means they have a π phase difference. As WL increases, E
oneofthepolesgetsclosertotherealaxiswhiletheother E (
m −10
movesawayfromtherealaxis(SeeFig.3). Asoneofthe I
poles gets closer to the real axis, its corresponding res-
onance becomes sharper (long-lived state, subradiance)
while the other resonance becomes broader (short-lived
state, superradiance). The narrowing and broadening of
the resonance width is shown in Fig. 2 (b) ∼ (f). −18
As the poles rotate, it becomes possible for one of the 1.3277 1.3278
Re E (E )
poles to reach the real axis and then the width of the 1
resonance collapses, indicating that the lifetime of the
resonance has become infinite. When this happens, the
FIG.4: (Coloronline)S-matrixpoletrajectoriesforbi-ripple
resonance state is completely decoupled from external
waveguide as WL varies from 0 ˚Ato 250 ˚A.
leads. This phenomena has been called “bound state in
continuum” (BIC) in other studies [15, 16].
Let us express the S-matrix pole position for a waveg-
functionwhichgivestheoscillatorybehaviorofthewidth
uide with a single ripple cavity as E = E −iΓ (red *
0 0 in Fig. 5. This behavior of the coupling parameter in
intheFig.reffig3). SinceS-matrixpolesforthebi-ripple
our 1D system is different from that seen in higher di-
waveguiderotateonacircleinthecomplexenergyplane,
mension. In Ref. [8], the coupling between a collective
the position of the poles can be written as a sinusoidal
state of two impurities and external electron wavefunc-
function of WL such that,
tion (a 2D system) is a Bessel funtion (J (s /λ )) (See
0 12 f
Eq. (19)inRef.[8]). Inthedoublequantumdotscoupled
E =E −iΓ and E =E −iΓ (4)
1 + + 2 − −
through the common phonon fields [17] (a 3D system),
where the coupling parameter is a zeroth order Bessel function
(sin (Qd)/ Qd). In these 2D and 3D systems, the cou-
E = E +Γ sin(kL+δ ) (5) pling parameter decays with the distance between two
+ 0 0 0
E = E −Γ sin(kL+δ ) atoms, as is the case for the Dicke system (a 3D sys-
− 0 0 0
tem). This dependence of the coupling on zeroth order
and Bessel functions (which have their largest values at long
wavelength), means that in 2D and 3D systems the su-
Γ+ = Γ0+Γ0cos(kL+δ0) = Γ0(1+α) (6) perradiance occurs predominantly with the wavelengths
Γ = Γ −Γ cos(kL+δ ) = Γ (1−α). much longer than the atomic distance. However, in our
− 0 0 0 0
1D waveguide system (the transverse direction is con-
The quantity α = cos(kL+δ ) is a measure of the cou- strained), the coupling parameter (α) does not decay.
0
pling between the cavities, k is a wave number at a res- Therefore, the superradiant resonance exists regardless
onance energy, and a phase factor, δ is used because a ofthewavelengthoftheelectron. Itispossiblethatasu-
0
pole is not on the real axis when WL = 0. Fig. 4 shows perradiant resonance appears with a very large distance
the pole trajectories in the complex energy plane as WL between two cavities, but it could not exceed the coher-
increases up to 250 ˚A. It confirms that the pair of poles ence length of electrons in the waveguide. In 2DEGs
always stays on a circle and it verifies Eq. 5 and 6. made of GaAs/AlGaAs, for example, at a very low tem-
As was shown in Refs. [1] and [12], a sequence of res- perature the coherence length is about 10µm.
onances occurs as energy in the interval E ≤E≤4E is As we have seen, the two S-matrix poles induce two
1 1
increased. Fig.5(a)showsthewidthofthenarrowreso- resonances in the electron transmission. The behav-
nanceasafunctionofthedistanceWLfortheresonance ior of these poles is associated with the symmetry of
at energy E = 1.3278E and 5 (b) shows the width of the scattering wavefunction at resonance energies. In
1
the resonance at energy E = 1.5830E . The resonance Dicke’smodel,subradiantresonanceisrelatedtotheanti-
1
widths are clearly sinusoidal functions of WL, which in- symmetric collective state of the atomic system and the
dicates that the coupling between the two cavities varies superradiant resonance is related to the symmetric col-
sinusoidally with increasing WL. Fig. 5 (b) oscillates lective state. In the bi-ripple waveguide, as the pair of
slightly faster than 5(a) because of its higher resonance polesrotateinacounterclockwisedirection(withincreas-
energy. InEq.6,thecouplingparameterαisasinusoidal ing distance between the two ripple cavities) , they each
4
creasing WL.
x 10−4
3
(a) In conclusion, we have studied the behavior of quasi
2 boundstatesinabi-ripplewaveguideasthedistancebe-
)1
(E!− 1 twwideethnsthaendtwpoorsiiptipolnescaovfittiheescrheasonngaens.ceWseasfosoucnidattehdatwtihthe
0
the quasi bound state poles change in a sinusoidal way
−1 with variation of the distance between the ripple cavi-
0 50 100 150 200 250
WL(˚A) ties. In 2D and 3D models of the Dicke effect, the cou-
x 10−4 pling parameter decays when the distance between two
10
(b) atoms (impurities or quantum dots) is longer than the
)1 5 wavelength of photon or electron. In the ripple waveg-
(E− uide system, the coupling parameter does not decay be-
! 0 cause the waveguide is a quasi 1-D system. We also
studied the trajectories of S-matrix poles in the com-
−5 plex energy plane. We found that the pair of poles that
0 50 100 150 200 250
WL(˚A) give rise to the super- and subradiant resonances in the
electron transmission, rotate on a circle centered on the
S-matrix pole for the waveguide with a single cavity and
FIG.5: (Coloronline)TheresonancewidthΓ forabi-ripple arephaseshiftedbyπ. Therefore,superradiantandsub-
−
waveguide. Γ− oscillates as WL is varied. The blue line (a) radiant resonances appear in oscillatory manner as the
is for the resonance near E = 1.3278E1. The red line (b) is distance between cavities is changed. Furthermore, as
for the resonance near E =1.5830E .
1 the S-matrix poles rotate, the symmetry of the electron
stateassociatedwiththesuperradiant(subradiant)reso-
nance alternates between symmetric and anti-symmetric
maintain the symmetry of their corresponding resonance
with increasing distance between the two ripple cavities.
wavefunction. The antisymmetric state is superradiant
when α < 0 and the symmetric state is superradiant TheauthorswishtothanktheRobertA.WelchFoun-
when α > 0. Therefore, superradiant resonances appear dation (Grant No. F-1051) for support of this work. We
in a sinusoidal manner as WL increases and the symme- alsothanktoDr. KyungsunNaforheradviceanduseful
try of their corresponding scattering wavefunctions al- discussions, and we thank the Texas Advanced Comput-
ternates between symmetric and antisymmetric with in- ing Center (TACC) for use of their facilities.
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