Entangled Terahertz Photon Emission from Topological Insulator Quantum Dots

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by  Li-kun Shi

Entangled Terahertz Photon Emission from Topological Insulator Quantum Dots
Li-kun Shi,1 Kai Chang,2 and Chang-Pu Sun1
1Beijing Computational Science Research Center, Beijing 100193, China
2SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
Weproposean experimentallyfeasible schemefor generatingentangledterahertzphotonpairsin
topological insulatorquantumdots(TIQDs). Wedemonstratetheoretically that ingenericTIQDs:
1) the fine structure splitting, which is the obstacle to produce entangled photons in conventional
semiconductorquantumdots,isinherentlyabsent for one-dimensional massless excitons duetothe
time-reversal symmetry; 2) the selection rules obey winding number conservation instead of the
conventional angular momentum conservation between edge states with a linear dispersion. With
thesetwoadvantages,theentanglementoftheemittedphotonsduringthecascadeinourschemeis
robust against unavoidable disorders and irregular shapes of theTIQDs.
6
1
PACSnumbers: 03.67.Bg,71.70.Gm,73.21.La,78.67.-n
0
2

b Introduction.— Entanglement is the heart of quan- ture by etching technique, and it is weakly coupled to
e tum physics and quantum information technology[1–3]. superconducting (SC) leads on two sides at p- or n-type
F
Semiconductor quantum dots (QDs), named as artifi- regime,respectively[seeFig. 1(a)]. Thelow-energyelec-

1 cial atoms, are one of the most promising sources for tron states in the isolated HgTe QD are described by a
quantum light, e.g., single photon and entangled pho- four-bandeffectiveHamiltonianinthebasis e, , hh, ,
] | ↑i | ↑i
l ton pair [4–21], because, comparing with real atoms e, , and hh, [23, 24]:
l | ↓i | ↓i
a and other sources, QDs possess an unique advantage
h of being compatible with well-developed semiconduc- H = H↑(kˆ) 0 +V (r), (1)
s- tor integration[17, 19] as well as on-demand and non- 4×4 (cid:20) 0 H↓(kˆ)(cid:21)
e probabilistic feature[5, 6, 18] of biexciton cascade emis-
m where kˆ = (kˆ ,kˆ ) is the in-plane wavevector, H (kˆ) =
sion in QDs. The fine structure splitting (FSS), the x y ↑
t. energy difference existing between two bright exciton H↓∗(−kˆ)=ǫ(kˆ)I2×2+di(kˆ)σi, andσi(i=1,2,3)are the
a statesinQDswithunavoidableirregularshapes,strongly Pauli matrices; ǫ(kˆ) = C Dkˆ2, d (kˆ) = Akˆ , and
1,2 x,y
m −
spoils the entanglement of the emitted photons[8]. To d (kˆ) =M Bkˆ2. The parameters A, B, C, D, and M
3
- overcome the FSS brought by breaking of rotational depend on t−he thickness of the HgTe layer. M is used to
d
n symmetry, various methods such as sophisticated QD describe the band insulator with a positive gap (M >0)
o growth control[9, 10, 14, 17, 19], external electric[13], and non-trivialtopologicalinsulator with a negative gap
c magnetic[8], and strain fields[15, 16, 20, 21], have been (M <0). The hard-wall confining potential V (r) is em-
[
used to manipulate and/or suppress the FSS of excitons ployed to describe potential difference between the vac-

3 in semiconductor QDs. However, for practical devices uumandtheTIQDwithRasitsradius. Theeigenstates
v with large numbers of QDs differ from dot to dot, it is of the QD ψ [25, 26] are linear combinations of de-
j,↑↓
2 | i
a great challenge to scale all the QDs at the same time. generateeigenstatesofJˆ whichisconservedalongthe z
2 z
7 Therefore,QDshavingnoFSSintrinsically,regardlessof axis. Note, we refer to ψj,↑↓ as spin-up (down) states,
4 random alloy distributions and other uncontrollable ef- asthey arethe eigenstat|esforithe upper (lower)blockof
0 fects,areessentialtorealizeintegratedentangledphoton the 4 4 Hamiltonian.
1. emitters. For×a given w↑↓ = j 1, the lowest energy state
∓
0 In this Letter, we show emitters composed of topolog- w, = ψ with its energy ǫ in the non-
6 ical insulator quantum dots (TIQDs) can produce tera- |triv↑ia↓ligap|[Mw±, 1,M↑↓i]l,oiwsamasslessedgestawt,e↑↓[27,28]. Here
1 −
hertz photon pairs with robust entanglement. We first we go beyond Refs. [27, 28] and use winding numbers to
:
v present the theory in detail with a HgTe QD to illus- classify the edge states: 2w = (j 1/2)+(j 3/2)
↑↓
Xi trate our main ideas and notations. Then we prove by is the total winding number in the∓basis e, ∓ and
| ↑↓i
the time-reversal symmetry (TRS) and winding number hh, as we will explain below. The edge states lo-
r | ↑↓i
a conservation, one-dimensional (1D) massless excitons in calize on the edge and possess a linear energy disper-
generic TIQDs with random disorders and morphology sion ǫ = w ∆ǫ + ǫ . The energy level spacing
w,↑↓ 0
± ·
fluctuations are free from FSS, and the entanglement of ∆ǫ = A[1 (D/B)2]1/2R−1[27] depends on QD’s ra-
the produced photon pairs from the cascade process is dius, a−nd ǫ −is the constant sets the Dirac point. w,
0
robust. and w, = Θ w, are always degenerate due t|o th↑ei
|− ↓i | ↑i
Topological insulator quantum dots.—We consider a TRS. Here Θ = iσ K is a time-reversal operator and
y
−
superconductor-TIQD hybrid system[22] as an example: K is the operation of complex conjugation. Due to the
aTIQDisfabricatedinaCdTe/HgTe/CdTeheterostruc- TRS, the backscattering between w, and w, are
| ↑i |− ↓i
2
σ w, ]= H (kˆ)[iΘ w, ]=ǫ [iΘ w, ]. We
y ↑↓ w,↑↓
| ↑↓i − | ↑↓i | ↑↓i
call ǫ conduction level and ǫ valance level for conve-
c v
nience. H = ǫ a† a + U n n is the
α α σ=↑↓ α,σ α,σ α α,↑ α,↓
Hamiltonian for thPe conduction (α = c) or valance
(α = v) level. Two pairs of degenerate edge states
are obtained by the creation or annihilation operator:
a† 0 = a 2 = , where 0 is the empty
α,↑↓| ic α,↑↓| iα |↓↑iα | iα
state and 2 = a† a† 0 is the two-electron occu-
| iα α,↑ α,↓| iα
pied state for the conduction/valance (α = c/v) level.
We denote = 1/2, and = 1/2,
|↑↓ic |± ↑↓i |↑↓iv |∓ ↑↓i
since the spin andangularmomentum indices are locked
forthe correspondinglevel. We willprovelater: 1)when
thereisoneelectronintheconductionlevelandoneelec-
troninthevalancelevel,theresultingtwo-particlestates
and are degenerate exciton states with
|↑ic|↓iv |↓ic|↑iv
thesameCoulombrepulsionU betweenthetwoelectrons
due to the TRS; 2) U = U describes the charging en-
c,v
ergy by electron edge states in the conduction/valance
level.
Superconducting p-n junction as a light-emitting
diode.— We coupled the HgTe QD weakly to a s-wave
FIG. 1: (color online). (a) Sketch of the proposed p-n junc-
tion with SC source and drain electrodes weakly coupled to SC p-n junction, with pair potential ∆s,d and chemical
a HgTe QD whose Fermi level is controlled by a gate elec- potential µs,d for its source (drain) lead [see Fig. 1 (b)].
trode VG. (b) Energy diagram for the structure shown in eVsd =µs µd isthebiasvoltageappliedacrossthejunc-
(a): ∆ is the SC gap; µs and µd are the Fermi energies of tion. µs,d−is aligned to the conduction (valance) level
the source and drain electrodes; E is the Fermi level and
F ǫ . Once the charge transfers through the p-n junc-
c,v
c (v) stands for conduction (valance) level we defined in the
tion, biexciton state 2 0 , which has two electrons
HgTe QD; Red (blue) bars inside the QD region are spin-up | ic| iv
in the conduction level and no electron in the valance
(down)edgestates,wdenotestheirwindingnumbers;Shaded
circles with red (blue) arrows are spin-up (down) electrons; level, would go through a two-step cascade process [see
The white circles on the drain side are holes; The ellipses Fig. 1 (c)]: 1) 2 0 1 1 , the biexciton un-
| ic| iv → | ic| iv
enclose two shaded (white) circles are electron- (hole-) type dergoes a spontaneous transition to 1 1 , one of the
Cooper pairs; Dashed arrows show the charge transfers from two degenerate exciton states | ica|ndiv ; 2)
the source to drain electrode through the HgTe QD as elec- |↑ic|↓iv |↓ic|↑iv
1 1 0 2 , the exciton state 1 1 decays to
tron singlets. The thin grey areas are barriers between the | ic| iv → | ic| iv | ic| iv
0 2 . During the cascade, two photons with energy
leads and the QD. (c) The initial state of the QD |2ic|0iv |~ωic| iveV ,whichisinterahertz( meV),areproduced.
goes through two indistinguishable pathes with the interme- 0 sd
≃ ∼
diatestate(cid:12)σ+(cid:11) +(cid:12)σ−(cid:11) ,and becomesan en- Hereweuseawaveguidetoproducephotonspropagating
(cid:12) |↓ic|↑iv (cid:12) |↑ic|↓iv
tangled final state (cid:0)(cid:12)σ+σ−(cid:11)+(cid:12)σ−σ+(cid:11)(cid:1) 0 2 /√2. σ± is a along the z axis only[19, 31]. The transitionis described
right-handed(left-ha(cid:12)nded)circ(cid:12)ularly po|lairciz|edivphoton carry- by the following Hamiltonian:
ing one unit of angular momentum ~. Dashed arrows are
possible decay pathes. ± H =g ˆb† a† a +ˆb† a† a +h.c., (2)
int qz,+ v,↑ c,↑ qz,− v,↓ c,↓
Xqz (cid:16) (cid:17)
which conserves the J , as a right-handed (left-handed)
not allowed, therefore edge states possess a long spin z
liisfebtilmocek[-2d9i,a3g0o]n.aTl hanedmwomillenntoutmcomuaptlreixthPe∝sp∂inH-u4×p4/a∂nkd ucinrictuloafrlaynpguollaarrizmedompehnottuomn ˆb†qz~,±. |T0ih=e c|oσu±pilicnognvsteryesnognthe
spin-down states, thus the transition between the edge g e(~ω0/2ǫV)1/2 can be t±uned by changing the vol-
∼
stateswithoppositespinsisforbidden. Forphotonsprop- ume of the waveguide. After each biexciton cascade, the
agatingin z direction, circularpolarizedtransitions only final two-photon state entangled with opposite circular
exist between the states with the same spin while con- polarizations is
serving j.
ψ = σ+σ− + σ−σ+ /√2. (3)
We assume the Fermi energy of the HgTe QD is be- | i
(cid:0)(cid:12) (cid:11) (cid:12) (cid:11)(cid:1)
tween two energy levels ǫc = ǫ±1/2,↑↓ and ǫv = ǫ∓1/2,↑↓. Thisfinalstatemean(cid:12)sthetwos(cid:12)equentiallyproducedpho-
They are evenlyawayfromthe Dirac point, andthe cor- tons in the z direction are polarization entangled: if the
responding edge states have an additional TRS symme- first emitted photon is σ+ (σ−), then the second emit-
try iΘ w, = w, because near the Dirac point ted photon is σ− (σ+). For a HgTe QD with a radius
| ↑↓i |− ↑↓i
keeping the linear terms in k one has K[σ H (kˆ)σ R=150nm, both the biased voltage eV 2.4meV and
y ↑↓ y sd
· ≃
3
the charging energy U 0.8meV[28] are smaller than Zero FSS in generic TIQDs.— We now prove in
≃
∆ 3meV (for Nb). In this case,we neglect the tran- generic TIQDs, two-particle states and
s,d ≈ |↑ic|↓iv |↓ic|↑iv
sitions creating a single quasiparticle in the SC leads. are degenerate exciton states without FSS. and
|↑ic|↓iv
The emission intensity of circular polarization entangled are products of edge states which satisfy: 1)
|↓ic|↑iv
photons equals to charge current through the junction. = Θ (α=c,v)due tothe TRS;2)morespecifi-
|↓iα | ↑iα
The weak coupling between the HgTe QD and the SC cally,anadditionalTRS symmetry =iΘ near
|↑↓ic |↑↓iv
leads will slightly split the energy levels for two-particle the Dirac point. These two relations hold, not only for
states 2 0 and 0 2 and shifts the frequencies of HgTeQDs,butalsoforgenericTIQDsthatarecomposed
emitted| ipch|otiovns fro|mict|heivcentral frequency ω [12, 26]. of other materials.
0
However,edge states and excitonstates in the HgTe QD The two-particle Hamiltonian for the conduction and
isunaffectedbythisweakcoupling,andthepolarizations valance level is H = ǫ a† a + Vˆ,
cv α=c,v σ=↑↓ α α,σ α,σ
and energies of the photons are uncorrelated. The delay where Vˆ is the CoulomPb interaPction between the two
time τX between the first and second emission events is electrons. The bright exciton states must be the lin-
much shorter than the emission time of the pair τXX ear combinations of the two-particle states
(τX/τXX 0.09)[12], thus the two photons in a pair and [26]. In the basis , |↑ic|↓iv,
are time- a∼nd polarization-correlated. In our proposal, the o|ff↓-idci|a↑giovnal term of H is {|↑icH|↓iv |↓ic|↑iv}=
the SC gaps forbid single quasiparticle creation in the Vˆ =0[26]ascvVˆ cohm↑|mchu↓t|evswcivth|↓tihce|↑tiivme-
SC leads, so the entangled photons are produced pair hr↑ev|cehr↓sa|vlop|e↓riact|o↑rivΘ2 = 1. Two diagonal terms are also
by pair and could be identified efficiently in the time the same as Vˆ− = Vˆ
domain. This pair-by-pair feature could also appear in h↑|ch↓|v |↑ic|↓iv h↓|ch↑|v |↓ic|↑iv ≡
U. Therefore and are degenerate exci-
a large non-trivialgap(about 0.3eV) TIQD[32] with the |↑ic|↓iv |↓ic|↑iv
ton states with the same energy ǫ +ǫ +U. Besides,
c v
aid of the Pauli exclusion principle[4]. U = Vˆ =U (α=c,v)[26].
α h↑|αh↓|α |↑iα|↓iα
SuperradianceandJosephsoneffecthasbeendiscussed
On the other side, the exciton FSS arises from the
in similar SC p-n systems[12, 33]. Here we neglect these
exchange interaction between two exciton states. Note
effects because the absence of a static in-plane electric
that, by swapping its electron with the hole, is
field does not allow the emission of a single photon |↑ic|↓iv
transformed to . The exchange energy[34] E
fromCooperpairtransferthroughtheQDviaJosephson Vˆ |↓ic=|↑i0v leads to vanishing FSS betwexee∝n
effect[12]. Ourproposalisbasedonthebiexcitoncascade h↑|ch↓|v |↓ic|↑iv
the two exciton states. The physical reason behind the
process which is not related to the Josephson effect.
zeroFSSis thatthe backscatteringis forbiddenby TRS.
It is necessary to emphasize that, the unavoidable non-
magneticdisordersandmorphologyvariationsofaTIQD
willnotbreakTRS.Thereforethe FSS vanishesbetween
exciton states and in generic TIQDs as
|↑ic|↓iv |↓ic|↑iv
longas the TRSsymmetry exist. This is the significance
between the TIQDs and the conventionalsemiconductor
QDs studied before.
Winding number conservation.— One may intuitively
concludeSO(2)symmetryinanactualTIQDwillbebro-
ken,sothatJ isnotconservedandtheoriginalselection
z
rulesdonothold. Therefore,thefinalphotonstate ψ′ =
| i
c (σ+σ− + σ−σ+ )/√2 + c (σ+σ+ + σ−σ− )/√2
0 1
| 2i | 2 i | i | i
with c + c = 1 possesses the reduced fidelity
0 1
| | | |
=[1+ c /c 2]−1withrespecttoBellstate. Toquanti-
1 0
F | |
tatively analyze the magnitude of c /c 2, we have per-
1 0
| |
formed finite element method simulations[26] on about
103 different HgTe QDs with random shapes. We gener-
ate these QDs by fluctuating their boundaries as r(θ)=
r [1+δ f(θ)] in the polar coordinate system, where the
0 r
FinIgGfi.n2i:te(ceolelomreonntlimnee)t.hoCda.lcTuhlaeteTdIQ|cD1/sca0|r2efgoern1e0ra3tTedIQbDysfluucs-- mean radius is r0 = 02πr(θ)dθ/2π while the random
functionf(θ) [ 1,1]Rsatisfiesf(θ)=f(θ+2π),where
tuating their radius randomly as r0[1+δrf(θ)] in the polar ∈ −
coordinate system; r0 = 150nm, f(θ) [ 1,1] is a random δr is a dimensionless number characterizing the maxi-
function. The results are classified by∈δ −which denotes the mum fluctuation percentage (typically less than 10% in
r
maximumfluctuationpercentage. Theinsetshowsthedensity experiments). The simulation results (see Fig. 2) show
distributions of an edge state in one of the randomly shaped that the irregularities of the TIQDs’ configurations only
TIQDs. cause neglectable deviations (. 10−4) of selection rules
4
from that with a perfect circular shape. in this system respect winding number conservationn=
We have a topological explanation for this robustness n′ 1. Ontheotherhand,theedgestatesinaTIQDare
±
ofselectionrulesagainstrandomnessoftheTIQDs’mor- naturallyquasi1Dstates(seeinsetinFig. 2)withlinear
phologies. Consider a 1D system on a closed loop with dispersions regardless of the TIQD’s shape due to their
linear dispersion (see Fig. 3). The Hamiltonian for the massless feature, which also originates from the topolog-
ical property of the TIQD[29, 30]. The winding number
foreachoftheedgestatesinaspecificbasis,e.g., e, ,
{| ↑↓i
hh, , is well defined. Therefore the selection rules
| ↑↓i}
between edge states in TIQDs respect the conservation
of the topological winding number. The zero FSS to-
getherwith the winding number conservationduring the
cascadeprocessinTIQDs resultsin the finaltwo-photon
state, whose entanglement is robust against disorders,
morphology fluctuations and asymmetry of the TIQD.
Conclusion.— In this Letter, we proposed a realiz-
able scheme to produce photon pairsat terahertzregime
withrobustentanglementingenerictopologicalinsulator
FIG. 3: (color online). A 1D system on a closed loop with quantumdots(TIQDs). Weproveone-dimensionalmass-
lineardispersion. ϕ kˆ ϕ isthewindingnumbercounting less excitons composed of edge states with linear disper-
n θ n
h | | i
the total number of counterclockwise turns that ϕ makes sions in TIQDs are free from the fine structure splitting
n
| i
around the origin. (FSS) due to the time-reversal symmetry. The entan-
glement of the emitted photon pairs from the cascade
systemreadsHˆ =∆ǫ kˆ +V (θ)anddependsonasingle process is robust against disorders and morphology fluc-
θ
·
parameter θ [0,2π]. Both the variations of the radius tuationsinTIQDs,becauseoftheabsenceofFSSandthe
∈
r(θ) and the disorder potential V (θ) break SO(2) sym- selection rules between edge states respecting the wind-
metry. Although J is no longer conserved, eigenstate ing number conservation. Both merits root in the topol-
z
of the system θ ϕ = exp[iφ (θ)]/√2π with energy ogy of the proposed TIQD. These two findings have not
n n
h | i
n ∆ǫ+ǫ has a well defined winding number been discussed in previous literatures as far as we know.
0
·
NotethatourresultsarevalidneartheDiracpointwhere
2π
w = ϕ θ θ i∂ ϕ =n, (4) the dispersions of edge states are linear. This discovery
n n θ n
Z0 h | ih |− | i pavesanewwaytowardsthefutureapplicationsandinte-
grations of entangled photon pair emitters, and possible
because its phase term φ (θ) = nθ + δ(θ) must sat-
n
isfy φ (θ+2π) = φ (θ) +2πn (n Z) to match the novelopticaldevicesbasedonwindingnumberconserva-
n n
∈ tion, by utilizing the massless edge states of topological
periodical boundary condition. The phase-shift term
δ(θ) = [ǫ θ + θV (θ′)dθ′]/∆ǫ with δ(2π) = 0 is in- materials. Our scheme,which canbe implemented using
0 0 currentexperimentaltechniques,isfreefromtherequire-
dependent of nR, which is possible only for a massless
mentstofabricatequantumdotswithhighsymmetryand
1D system. The winding number characterize the total
sophisticated regulations to eliminate FSS.
number of counterclockwise turns that the wavefunction
makes around the origin. The coupling between eigen- K.C. is supported by NSFC Grants No.11434010 and
states ϕn and ϕn′ with different winding numbers n the Grant Nos. 2015CB921503, and 2011CB922204-2
and n′|byicircula|rlyipolarized photon with angular mo- fromtheMOSTofChina. C.-P.S.acknowledgessupport
mentum 1 is proportional to fromthe NSFC GrantNos. 11421063and11534002,and
± the National 973 program (Grant Nos. 2012CB922104
2π
Cn±,n′ =hϕn|rˆ±|ϕn′i=Z hϕn|θir(θ)e±iθhθ|ϕn′i and 2014CB921403).
0
2π
=r0δn,n′±1+r0δr f(θ)ei(−n+n′±1)θdθ/2π.
Z
0
(5)
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Supplementary Material: Entangled Terahertz Photon Emission from Topological
Insulator Quantum Dots
Li-kun Shi,1 Kai Chang,2 and Chang-Pu Sun1
1Beijing Computational Science Research Center, Beijing 100193, China
2SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
I. Eigenstates in a Circular HgTe Quantum Dot
The low-energy electron states in the HgTe QD are described by a four-band effective Hamiltonian in the basis
e, , hh, , e, , and hh, :
| ↑i | ↑i | ↓i | ↓i
6 H (kˆ) 0
1 H = ↑ +V (r), (1-1)
0 4×4 (cid:20) 0 H↓(kˆ)(cid:21)
2
where kˆ = (kˆ ,kˆ ) is the in-plane wavevector, H (kˆ) = H∗( kˆ) = ǫ(kˆ)I +d (kˆ)σi, and σi(i=1,2,3) are the
b x y ↑ ↓ − 2×2 i
e Pauli matrices; ǫ(kˆ) = C Dkˆ2, d (kˆ) = Akˆ , and d (kˆ) = M Bkˆ2. The hard-wall confining potential V (r) is
1,2 x,y 3
F − −
employed to describe potential difference between the vacuum and the TIQD with R as its radius. The four-band

1 Hamiltonian is derivedfromthe full multi-band Hamiltonianwith strong spin-orbitcoupling and bandmixing effects
included. The total angular momentum Jˆ = Lˆ +Sˆ , where Lˆ is the orbit azimuthal angular momentum,
z,↑↓ z z,↑↓ z
l] and Sˆz,↑↓ is the spin projection onto the z direction, which is 1/2 for e, and 3/2 for hh, . One can verify
l ± | ↑↓i ± | ↑↓i
a [H ,Jˆ ]=0.
↑↓ z,↑↓
h The eigenstate of Jˆ in the basis e, , hh, , e, , hh, satisfying Jˆ ψ =j ψ has the form
- z {| ↑i | ↑i | ↓i | ↓i} z,,↑↓| j,↑↓i | j,↑↓i
s
e ψ (r,φ)= c J (kr)exp[i(j 1/2)φ],c J (kr)exp[i(j 3/2)φ] T (1-2)
m j,↑↓ { 1,↑↓ j−1/2 ∓ 2,↑↓ j−3/2 ∓ }
. inthepolarcoordinate. Forbulkstateswithnoboundaryconstraintsonk,theeigenequationisHˆ ψ =ǫ ψ ,
t ↑↓ j,↑↓ j,↑↓
a which leads to [c ,c ]T =0 and the relation between k and ǫ: | i | i
m M1,↑↓ 1,↑↓ 2,↑↓
- det =0, (1-3)
d M1,↑↓
n
where
o
c
M B k2 ǫ iAk
[ M1,↑↓=(cid:20) − i+Ak − M +∓B−k2 ǫ(cid:21). (1-4)
3 ± − −
v As the bulk spectrum has two branches, for a given value of ǫ there are two non-trivial solutions k
i
2
(i = 1,2) for the momentum. For eigenstates in the QD, we are able to satisfy their vanishing boundary
2
condition ψ (R) = 0 by combining two linearly independent degenerate solutions in the bulk ψ (r) =
7 j,↑↓ j,↑↓
4 2 [c J (k r),c J (k r)]T and requiring the coefficients to satisfy [c ,c ]T = 0. For
i=1 1i,↑↓ j∓1/2 i 2i,↑↓ j∓3/2 i M1,↑↓ 1i,↑↓ 2i,↑↓
0 Pconvenience we define γ2i,↑↓ as
.
1
0 γ2i,↑↓ =c2i,↑↓/c1i,↑↓ =±iAki/(−M +B−ki2−ǫ). (1-5)
6
1 Therefore
:
v 2
Xi ψj,↑↓(r)= c1i,↑↓[Jj∓1/2(kir),γ2i,↑↓Jj∓3/2(kir)]T, (1-6)
r Xi=1
a
then ψ (R)=0 leads to [c ,c ]T =0 and the equation to fix the eigenenergy:
j,↑↓ 2,↑↓ 11,↑↓ 12,↑↓
M
det =0, (1-7)
2,↑↓
M
where
J (k R) J (k R)
= j∓1/2 1 j∓1/2 2 . (1-8)
M2 (cid:20) γ21,↑↓Jj∓3/2(k1R) γ22,↑↓Jj∓3/2(k2R)(cid:21)
2
II. Emissions from the Superconducting p-n Junction
The weak coupling between the QD and the SC leads induce a pair potential ∆ = exp[iφ ]Γ /2 for the
c,v s,d s,d
conduction(valance)level,i.e., the proximityeffect. φ is the phaseofthe corresponding∆ , andΓ characterize
s,d s,d s,d
the broadening of the conduction (valance) level proportionalto square of the tunneling amplitude (assuming Γ
s,d
≪
∆ , i.e., the coupling between the QD and SC leads is weak). The effective low-energy Hamiltonian for the
s,d
| |
conduction (α=c) or valance (α=v) level is
H˜ =ǫ˜ a† a +Un n
α α α,σ α,σ α,↑ α,↓
σX=↑↓
+∆ a† a† +∆∗a a , (2-1)
α α,↑ α,↓ α α,↓ α,↑
where ǫ˜ = ǫ µ counts from µ . By diagonalizing H˜ , beside two degenerate states a† 0 with energy
c,v c,v − s,d s,d α α,↑↓| iα
ǫ˜ , there are excited andgroundstates being linear superpositionof empty state 0 and fully occupied two-particle
α | iα
state 2 , i.e.,
| iα
= exp[ iφ ] u 0 + u 2 , (2-2)
|±iα − − α | α,±|| iα | α,∓|| iα
with
ǫ˜ =ε (ε2 + ∆ 2)1/2, (2-3)
α,± α± α | α|
where ε =ǫ˜ +U/2 and u =(1/√2) [1 ε /(ε2 + ∆ 2)1/2]1/2.
α α | α,±| × ± α α | α|
FIG. 1: (a) Sketch of the energy level splittings for two-particle states in the HgTe QD weakly coupled to SC electrodes. The
p-n junction is biased with a voltage Vs,d. The cascade produces two photons with opposite circular polarization σ±. (b)
Emission intensity of time- and polarization-correlated photons emitted from the biexciton cascade. The central frequency
~ω0 ≃eVsd. The dominant cascade process is |−ic|+iv →|1ic|1iv →|−ic|+iv. Parameters: ˜ǫc =ǫ˜v =0, U =3, Γph =0.02
(in units of ∆ =∆ ).
c v
The emission intensity computed versus photon frequency ω is shown in Fig. 1 (b). The emission intensity is
i(ω ) = wp (ω )ρ with the transition rate wp (ω ) = (2π/~) b;q ,p H a;0 2δ[ǫ˜ ~ω˜ +
qz ab,p a→b qz a a→b qz qz|h z | int| i| ab − qz
iΓph]. WePwill explain below the symbols in i(ωqz) and wap→b(ωqz): P
a, b are one of six possible states in the QD + + , + , + , , , [see Fig.
{| ic| iv | ic|−iv |−ic| iv |−ic|−iv |↑ic|↓iv |↓ic|↑iv}
1 (a)]. These six states are direct products of two eigenstates (one from the conduction level and another one from
the valance level) and can always be expanded by in the basis 2 2 ,0 0 ,2 0 ,0 2 , , .
{| ic| iv | ic| iv | ic| iv | ic| iv |↑ic|↓iv |↓ic|↑iv}
In our work we only use electron notations and do not introduce hole operators. For example, 2 0 means there
| ic| iv
are two electrons in the conduction level and no electron in the valance level (biexciton); 1 1 means there is one
| ic| iv
electron in the conduction level and one electron in the valance level (exciton); 0 2 means there is no electron in
| ic| iv
the conduction level and two electrons in the valance level (no exciton).
3
p=σ± denotesthepolarization;˜ǫ =ǫ˜ ǫ˜ istheenergydifferencebetweenstatesaandb(Notethatǫ˜ orǫ˜ isthe
ab a b a b
energyfortwo-particle state); ω˜ =ω ω−isthefrequencyshiftcausedbyproximityeffect; andΓ =2πν g 2 is
qz qz− 0 ph ph| |
thebroadeningoftheemissionpeakswithν thephotonDOSassumedtobeindependentofenergyandpolarization.
ph
The probabilities ρ are determinedby the stationarysolutionof the master equationρ˙ = [Wp ρ Wp ρ ],
a a b,p b→a b− a→b a
Wp = dω′wp (ω′). P
a→b a→b
AttheRfirstglance,itisweirdtoseetheinitialstateoftheQDisthesamewiththefinalstateafterbiexcitoncascade.
In fact, the state + is the superposition of two parts: 2 0 with two electrons in the conduction level and
|−ic| iv | ic| iv
0 2 withtwoelectronsinthevalancelevel. Thereforetheinitialstateisactuallythe 2 0 partofthe +
| ic| iv | ic| iv |−ic| iv
and the final state is the 0 2 part of the + . However, with induced pair potential ∆ , 2 0 and
| ic| iv |−ic| iv c,v | ic| iv
0 2 areno longertwo-particleeigenstates. Onehas tocalculatethe processlike + 1 1 +
| ic| iv |−ic| iv →| ic| iv →|−ic| iv
instead of 2 0 1 1 0 2 , while the frequencies of the emitter photons are shifted from the central
frequency ~|ωic=| i(v2ǫ→+| Uic)| iv(ǫ→+|ǫic+| Uiv)=(ǫ +ǫ +U) (2ǫ +U)=ǫ ǫ =eV , where 2ǫ +U is the energy
0 c c v c v v c v sd c
− − −
for 2 0 state; 2ǫ +U is the energy for 2 0 state; (ǫ +ǫ +U) is the energy for 1 1 state.
| ic| iv v | ic| iv c v | ic| iv
III. Coulomb Interactions between Two Particle States
The two-particle Hamiltonian for the conduction and valance level is
H = ǫ a† a +Vˆ, (3-1)
cv α α,σ α,σ
αX=c,vσX=↑↓
where Vˆ is the Coulomb interaction between the two electrons. Any exciton state must be the linear combinations
of the two-particle basis , , , , in which one electron is in the conduction level and
{|↑ic|↓iv |↓ic|↑iv |↑ic|↑iv |↓ic|↓iv}
another in the valance level. These two-particle states are products of edge states which satisfy: 1) = Θ
|↓iα | ↑iα
(α=c,v) due to the TRS; 2) anadditional symmetry =iΘ as the conduction andvalance levelare evenly
|↑↓ic |↑↓iv
away from and near the Dirac point. These two relations hold, not only for HgTe QDs, but also for generic TIQDs
that are composed of other materials and can have irregular shapes.
In the two-particle basis , , , ,
{|↑ic|↓iv |↓ic|↑iv |↑ic|↑iv |↓ic|↓iv}
ǫ +ǫ +U 0 0 0
c v
 0 ǫc+ǫv+U 0 0 
H = , (3-2)
cv 0 0 ǫ +ǫ +U′ Vˆ
 c v h↑|ch↑|v |↓ic|↓iv 
 0 0 Vˆ ǫ +ǫ +U′ 
 h↓|ch↓|v |↑ic|↑iv c v 
where
U = Vˆ = Θ Θ Vˆ Θ Θ = Vˆ , (3-3)
h↑|ch↓|v |↑ic|↓iv h ↓|ch− ↑|v | ↓ic|− ↑iv h↓|ch↑|v |↓ic|↑iv
U′ = Vˆ = Θ Θ Vˆ Θ Θ = Vˆ .
h↑|ch↑|v |↑ic|↑iv h ↓|ch− ↓|v | ↓ic|− ↓iv h↑|ch↑|v |↓ic|↓iv
All but two off-diagonal terms in H are zero. For example,
cv
H = Vˆ =I I =0, (3-4)
h↑|ch↓|v cv|↓ic|↑iv h↑|ch↓|v |↓ic|↑iv 1− 2
in which = [ ]/√2, = [ ]/√2 where we assign 1 and 2
|↑ic|↓iv |↑1ic|↓2iv −|↑2ic|↓1iv |↓ic|↑iv |↓1ic|↑2iv −|↓2ic|↑1iv
explicitly to label two identical electrons;
I = V = V Θ
1 h↑1|ch↓2|v 12|↓1ic|↑2iv h↑1|ch↓2|v 12| ↑1ic|↑2iv
= Θ2 V Θ = I =0, (3-5)
↑1 ch↓2|v 12| ↑1ic|↑2iv − 1
(cid:10) (cid:12)
(cid:12)
I = V = iΘ V iΘ
2 h↑1|ch↓2|v 12|↓2ic|↑1iv h ↑1|vh↓2|v 12| ↓2iv|↑1iv
= Θ V Θ Θ2 = I =0. (3-6)
h ↑1|vh↓2|v 12| ↓2iv ↑1 v − 2
(cid:12) (cid:11)
NotethatV =V istheCoulombinteractionbetweent(cid:12)heelectron1and2andV commuteswiththetime-reversal
12 21 12
operator Θ2 = 1 for fermions. Other zero off-diagonal terms can be obtained by similar analysis.
−
Amongthefourtwo-particlestates , , , , and aredarkstates. The
{|↑ic|↓iv |↓ic|↑iv |↑ic|↑iv |↓ic|↓iv} |↑ic|↑iv |↓ic|↓iv
optical transitions within and are not allowed because the spin-up (down) states are fully occupied
|↑ic|↑iv |↓ic|↓iv
in ( ) state, while transitions between opposite spin states are forbidden.
|↑ic|↑iv |↓ic|↓iv
4
These two dark states are irrelevant to our cascade process, and do not couple with other two bright states.
Therefore, the bright exciton states must be linear combinations of two-particle bright states and .
|↑ic|↓iv |↓ic|↑iv
In the two-particle basis , , as we analyzed above,
{|↑ic|↓iv |↓ic|↑iv}
ǫ +ǫ +U 0
H = c v , (3-7)
cv (cid:20) 0 ǫc+ǫv+U (cid:21)
therefore and are degenerate bright exciton states with the same energy ǫ +ǫ +U.
|↑ic|↓iv |↓ic|↑iv c v
The Coulomb repulsion between two electrons in the same conduction/valance level is
U = Vˆ = iΘ iΘ Vˆ iΘ iΘ =U , (3-8)
c h↑|ch↓|c |↑ic|↓ic h ↑|vh ↓|v | ↑iv| ↓iv v
while the Coulombrepulsionbetween one electronin the conduction leveland anotherelectron in the valance levelis
U = Vˆ = iΘ Vˆ iΘ =U , (3-9)
h↓|ch↑|v |↓ic|↑iv h ↓|vh↑|v | ↓iv|↑iv v
therefore U =U.
c,v
IV. Notes on Finite Element Method Simulations
To obtain the edge states in the deformed HgTe QD, we still start from the four-band effective Hamiltonian in the
basis e, , hh, , e, , and hh, :
| ↑i | ↑i | ↓i | ↓i
H (kˆ) 0
H = ↑ +V (r), (4-1)
4×4 (cid:20) 0 H (kˆ)(cid:21)
↓
buthere V (r) isthe hard-wallpotentialwith its boundarydescribedby r(θ)=r [1+δ f(θ)]inthe polarcoordinate
0 r
2π
system,withthemeanradiusr = r(θ)dθ/2πandtherandomfunctionf(θ) [ 1,1]satisfiesf(θ)=f(θ+2π).
0 0 ∈ −
δr is a dimensionless number charaRcterizing the maximum fluctuation percentage. Each random function for each of
the QDs is generated independently as
N
f(θ)=(1/ ) c sin(nθ+ϕ )/√π, (4-2)
i i
N
nX=1
where N is generated randomly in the interval [10,100]; each c (ϕ ) is generated randomly and independently in the
i i
interval [0,2π]; = N c2. This generating scheme makes f(θ) to be a white noise. It is reasonable to model the
N n=1 i
fluctuation of a QD’sPboundary f(θ) as the white noise. Because the boundary of a QD formed by etching is caused
by atomic dissociation process which is mesoscopically regulated and microscopically random.
When the Hamiltonian of a deformed QD is generated, we solve the Schr¨odinger equation for the QD with finite
2
element method, and obtain the wavefunctions for the edge states and . Then we can calculate c /c as
|↑↓ic |↑↓iv | 1 0|
2 2 2
c rˆ rˆ
1 = h↑|v +|↑ic = h↓|v −|↓ic . (4-3)
(cid:12)c (cid:12) (cid:12) rˆ (cid:12) (cid:12) rˆ (cid:12)
(cid:12) 0(cid:12) (cid:12)h↑|v −|↑ic(cid:12) (cid:12)h↓|v +|↓ic(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)