Mesoscopic one-way channels for quantum state transfer via the Quantum Hall Effect

0.18 MB·English

by  T. M. Stace

Mesoscopic one-way channels for quantum state transfer via the Quantum Hall Effect
T.M. Stace,1,2 C.H.W. Barnes,1 and G.J. Milburn3
1Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
2DAMTP, University of Cambridge, Wilberforce Rd, CB3 0WA, UK∗
3Centre for Quantum Computer Technology, University of Queensland, St Lucia, QLD 4072, Australia
4
(Dated: February 2, 2008)
0
0
We show that the one-way channel formalism of quantum optics has a physical realisation in
2
electronic systems. In particular, we show that magnetic edge states form unidirectional quantum

n channelscapableof coherentlytransportingelectronicquantuminformation. Usingtheequivalence
a between one-way photonic channels and magnetic edge states, we adapt a proposal for quantum
J state transfer to mesoscopic systems using edge states as a quantum channel, and show that it is

3 feasible with reasonable experimental parameters. We discuss how this protocol may be used to
2 transfer information encoded in number, charge or spin states of quantum dots, so it may prove
useful for transferring quantuminformation between parts of a solid-state quantum computer.

]
l
l PACSnumbers: 03.67.Hk,73.43.Jn,73.23.-b
a
h
-
There is growing interest in using mesoscopic chan-
s
e nelsforquantuminformationprocessingtasks[1,2],and
m
mesoscopicanaloguesofone-way channelshavebeencon-
. sideredabstractly[3,4]. One-wayquantumchannelspre-
t
a serve quantum coherence, and have the additional prop-
m
ertythatforwardandreversepropagatingmodesaredis-
- tinguishable. Thesechannelsareusefulfordescribingthe
d
n dynamics of a system coupled to a non-classical source FIG.1: Quantumdotscoupledtooneanotherandtonearby
o [5] and have been proposed for use in a quantum state edge states. Dotted lines indicate tunnel contact. The grey
c transfer(QST)protocol[6]. Theprototypicalexampleof rectangles indicate Ohmic contacts. Dots are labelled as
[ “atom”or“cavity”tomakecleartheanalogywiththeatom-
a one-way channel comes from quantum optics: an op-
optical scheme [6].
1 tical fibre with a Faraday isolator. A magnetic field in
v the Faradayisolator,whichbreakstime-reversalsymme-
2 try in the channel, correlates propagationdirection with
4 the possibility of transferring quantum information via
polarisation, making counter-propagating modes distin-
4 edge states over distances of 10 µm or more.
guishable [5]. To date, mesoscopic realisations of this
1
Using edge states as one-way quantum channels, we
0 kind of channel have not been discussed.
show how to implement a proposal for QST in a meso-
4
Here we propose magnetic edge states [7] of a 1D wire
scopic system. Cirac et al. [6] describe a protocol for
0
as a physical realisation of one-way quantum channels,
/ transferring the quantum state of one two-level atom in
t and discuss their application for QST in a mesoscopic
a a cavity, connected by optical fibre, to another identi-
m system. In the quantum Hall effect (QHE) a magnetic cal system using shaped control pulses applied to each
field appliednormalto the wire induces the formationof
- atom. Here, we map this protocol to a system consist-
d edgestatesalongeachedgeofthewire,quantisedinunits
ing of quantum dots connected by magnetic edge states.
n of the cyclotron energy [8, 9]. States on each edge prop-
Thismayproveusefulfortransferringquantuminforma-
o
agate in a definite direction, so backscattering between
c tion between parts of a solid-state quantum computer.
: counter-propagatingmodes is suppressed. This accounts Wealsoconsidersourcesoferrorintheprotocolandcon-
v
for the conductance plateaus observed in the QHE [8].
i clude that these are not debilitating.
X
Boundedgestatesareformedbycreatingalocalpoten- The QHE is understood by solving the Schr¨odinger
r tial minimum (maximum) using surface gates to define equation for an electron in a lateral potential, U(y), and
a
a quantum(anti)dot. Resonanttunnelling via(anti)dots a magnetic field, B [7]. The eigenmodes are of the form
coupledto extendededgestateshasbeenexperimentally ψ (x,y) = eikxχ (y). For a linear lateral potential,
n,k n
observed [10, 11, 12]. The tunnelling rates are tuneable U(y) = eEy, the eigenenergies are ε = ~v k+(n+
n,k 0
using external magnetic and electric fields. 1)~ω where v = E/B > 0, ω = eB/m is the
Coherentsuperpositions ofsingle electrons indifferent c2ycloBtron freque0ncy a−nd n Z+, prBoducing subebffands of
∈
edgestateshavebeenobservedexperimentally[13]. High edge states. Thus, ideally, edge states are dispersionless,
visibility fringes ( 0.6 at 20 mK) were seen in a Mach- ω =v k, with group velocity v .
0 0
∼
Zender interferometer [14] consisting of edge states con- The system of dots in tunnel contact with magnetic
nectedbytunnelbarriers. Thisexperimentdemonstrates edge states is shownschematically in Fig.1. The system
2
Hamiltonian is H =H +H +H [15],
tot dots MES int
4
H
dots = ω c†c +Ω (c†c +c†c )+Ω (c†c +c†c ),
~ i i i 1 1 2 2 1 2 4 3 3 4
X
i=1
H (a) (b)
MES = dωg(ω)ωb†(ω)b(ω),
~ Z
FIG. 2: (a) QSTfor charge qubits. (b) QSTfor spin qubits.
H
int = dω(κ (ω)(b†(x ,ω)c +c†b(x ,ω))
~ Z 1 0 2 2 0
+κ (ω)(b†(x ,ω)c +c†b(x ,ω))). coupling gates are equal, γ1,2 = γ, the channel is dis-
2 1 4 4 1 persionlessanddoesnotscatter,thentheHamiltonianis
Here c† creates a dot i electron, b†(ω) creates an edge formally identical to that of [6].
state eilectron with energy ~ω, b†(x,ω) = b†(ω)eiωx/v0, The physical correspondence to [6] is shown in Fig. 1.
We identify an electron on dot 1 (or 3) with an atomic
g(ω) is the density of states, κ(ω) is the tunnelling rate
excitation and an electron on dot 2 (or 4) with a cav-
between dots and an edge states, x are locations of
0,1
ity photon. The computational basis for each dot is the
the dots and Ω (t) aretuneable tunnelling ratesbetween
i absence, 0 , or presence, 1 = c† 0 of an electron.
the dots. Tunnelling rates decrease exponentially with | ii | ii i| ii
Tunnelling rates, Ω (t), between dots may be controlled
distance from the dot, so we only consider the nearest, i
viaexternalgates. GatepulsesthatimplementQSThave
resonant subband.
been derived previously [6, 17].
Non-linearities in U(y) introduce dispersion. A small
Finally,weidentifyopticalfibremodesin[6]withedge
quadratic term in the lateral potential, U(y) = eEy +
states. Inbothmesoscopicandopticalsystems,thepres-
1m ω2y2, results in the dispersion relation ω = v k+
2 eff 0 0 ence of a magnetic field breaks time-reversal symmetry
αk2, where α = ω2~/(2ω2m ). As discussed later, for
0 B eff of the system so that counter-propagating modes are in
typical experimentally relevant parameters, α is small,
principledistinguishable. Inthisway,edgestatesarethe
and we treat this as a source of error in the protocol.
electronicanaloguesofopticalone-waychannels,andthe
Input-output relations. The input-output formalism
theoryforone-wayquantumchannels[5]maybeapplied.
relates the output channel of a quantum system to its
Thus the QST protocoldescribedin [6]canbe usedto
input. Formally,ifg(ω)ω andκ (ω) γ /2π werecon-
i ≡ i transfer the electronic state of dot 1 to dot 3 using edge
stant over relevant frequencies aroundpthe dot energies,
states as a one-way quantum channel. In this scheme
ω , then the Markov approximation applies [5], and the
i the computational basis states are the eigenstates of the
input-output relation for the first system of dots is
dot number operator, 0 , 1 . Therefore a superpo-
i i
{| i | i }
sition such as ψ = c 0 + c 1 can be transferred
bout(x0,t)=bin(x0,t)+i√γ1c2(t), (1) coherently from|diot 1 to0|diot 31u|siing this protocol, i.e.
Ψ = ψ 0 0 0 Ψ = 0 0 ψ 0 .
where x0 is the location of the first pair of dots and | ii | i1| i2| i3| i4 →| fi | i1| i2| i3| i4
b(x,t)= ∞ dωb(ω)eiω(t−x/v0). Asimilarrelationholds The dynamics of the density matrix for the system of
−∞ dots is given by the master equation [17],
for the seRcond pair. For a dispersionless channel, the in-
put to the second system is given by ρ˙ = i[H ,ρ]/~+γ [c ]ρ+γ [c ]ρ
sys 1 2 2 4
− D D
γ [c†,c ρ]+[ρc†,c ] [ρ], (4)
bin(x1,t)=bout(x0,t L/v0). (2) − { 4 2 2 4 }≡L
−
where γ = √γ1γ2 and [c]ρ cρc† (c†cρ+ρc†c)/2.
where L = x x . By virtue of the fact that back- D ≡ −
1 − 0 A simple pulse shape that effects ideal quantum state
scattering is strongly suppressed in edge states, we can
transfer is Ω (t) = γ sech(γ t/2)/2 [17], which we will
i i i
neglect the effect of the second system on the first.
useforevaluatingvarioussourcesoferrorintheprotocol.
For a dispersing channel, the input to the second sys-
Forthispulseshape,thesystemofquantumdotsremains
tem is related to the output of the first by [16]
in a pure state, given by
bin(x1,t)=δν(t)∗bout(x0,t−τ) (3) |ψ(t)i = c0|0000i+c1{α(t)|1000i+α(−t)|0010i
+β(t)(0100 0001 )/√2 , (5)
where τ = L/v , δ (t) = (iπν)−1/2e−t2/iν, ν = 4ατ/v2 | i−| i }
0 ν 0
and the convolution f(t) g(t)= dt′f(t)g(t t′). α(t)=(1 tanh(γt/2))/2 and β(t)=isech(γt/2)/√2.
∗ − −
Number eigenstate QST. WitRihin the Markov ap- Charge QST. Encoding a qubit in the number state
proximation,this modelfor quantumdotsin tunnelcon- ofelectronsisimpracticalduetotheconservationofelec-
tact with edge states parallels the atom-optical system tronnumber. However,theschemeoutlinedaboveserves
presented in [6]. In particular, in the ideal case where as a primitive to implement QST for a charge qubit en-
the dot energies are equal (ω = ω¯), the channel-dot codedinthepositionofanelectroninadouble-well. The
i
3
charge qubit QST protocol simply repeats the primitive p δω1 δω2 δγ ǫs ǫD
protocoltwice, as illustrated in Fig.2(a). Qubit A is en- Fp 4.29δω12 1.00δω22 0.25δγ2 1.00ǫs 0.02ǫ2D
coded in the position of an electron distributed between
TABLE I: Lowest order terms in infidelity due to systematic
dots 1aand1b. To transferthe state ofqubit Ato qubit
errors. Dimensionless parameters: δωi = (ωi−ω¯)/γ1, δγ =
B, the protocol first transfers the state of dot 1a to dot 1−γ2/γ1, ǫs =1−e−L/LS and ǫD =ατγ2/v02.
3a,thenthestateofdot1btodot3b. Theidealprotocol
is unitary [6], so it may be concatenated as described.
Spin QST. The primitive transfer protocoldescribed nels implicitly assumes a dispersionless channel. There
above may also be used to transfer a spin qubit from aredifficultiesingeneralisingtheinput-outputformalism
one dot to another, achieved using a similar two step to include dispersion, so we employ a different approach
process as described above. This relies on the spin de- toestimatetheseerrors. Aftertheelectronhastunnelled
pendenceofthetunnellingrates[12],duetothedifferent intothechannel,thechannelisinasuperpositionofzero
spatial configuration of spin-polarised edge states. Sup- and one electrons. The amplitude for an electron to be
pose the spin-up states, , occupy the outer edges, at position x at time t is found from Eqs. (1) and (2) to
| ↑i
as shown in Fig. 2(b). Since states are in closer be 0b(x,t)φ = γ/2β(t τ ), where φ is the chan-
| ↑i h | | i − x | i
proximity to one another than states, the tunnelling nel state and τ =p(x x )/v is the propagation time
|↓i x − 0 0
rate, γ↑, between states may be much larger than to position x. Dispersion convolves this amplitude with
| ↑i
γ↓. Therefore, the gate pulses to transfer a electron thechanneltransferfunctionaccordingtoEq.(3),sothe
| ↑i
from dot 1 to dot 3 are much faster than those required transmission fidelity is then
to transfer a electron. If an electron were in a state
| ↓i ∞
ψ 1 =c↓ 1+c↑ 1, then applying fast gate pulses, = φ˜φ 2 = dtβ˜∗(t)β(t)2, (6)
d|eitermine|d↓iby γ↑,|w↑oiuld have a small effect on the F |h | i| |Z−∞ |
| ↓i
component of ψ for which tunnelling rates are propor-
| i where β˜ = δ β. We expand β˜(t) in powers of ν then
tionaltoγ↓. Afterthis,the componentistransferred ν ∗
|↓i evaluate Eq. (6) yielding = 1 ǫ2 /45 + ..., where
from dot 1 to dot 3 using pulse rates determined by γ↓. F − D
ǫ =νγ2/4 is the dimensionless dispersion strength.
However, turning on the coupling between dots 3 and 4 D
Table I lists the various dimensionless parameters in
leads to a leakage error in the component which was
|↑i which errors may occur, along with the associated infi-
previously transferred. To circumventthis, the com-
|↑i delities, which are computed as described earlier. The
ponentisswappedfromdot3toanancilla,dot3aduring
protocol is quadratically sensitive to all parameter vari-
the transfer of the component. Once this process is
| ↓i ations except for scattering. Clearly, when the dots are
completed the component is swapped back to dot 3.
|↑i slightly out of resonance with one another (δω = 0) or
Error sources. Awayfromtheidealcase,thetransfer i 6
if γ =γ , there is a correspondingerrorinthe protocol.
fidelity is reduced, so we analyse the sensitivity of the 1 6 2
Foragivenerrorrate,wemayusethetabulatedresults
protocol to small perturbations from the ideal. For each
to estimate the tolerable deviations of the parameters
parameter in the system, labelled generically as p, the
from ideal. Time and energy scales for the protocol are
transmission fidelity due to variations in p is given by
determined by γ, so we estimate this quantity for a typ-
= 1 , where Ψ ∆ρ Ψ and ∆ρ
p p p f p f p
F − F F ≡ h | | i ≡ ical experimentalsystem. Suppose the coupling between
Ψ Ψ ρ ( ) is the error in the final state, ρ ( ),
f f p p
| ih |− ∞ ∞ dot 3 and the edge state channel is operated with a tun-
ofthe systemdue to errorsin p. willbe largestwhen
Fp nelling probability of 2. This is given by 2 = (hgγ)2
theinitialstateofdot1is 1 1,soweusetheinitialstate T T
| i [19],wheregisthedensityofedgestatesinenergyspace.
Ψ = 1 0 0 0 . The desired final state is then
i 1 2 3 4
| i | i | i | i | i Roughly, g 1/∆E, where ∆E is the bandwidth of the
Ψf = 0 1 0 2 1 3 0 4. Since the errors are presumed ≈
| i | i | i | i | i tunnelling interaction given by
small, they can be analysed independently, by solving
Eq. (4) for small perturbations of each parameter. ∆E =∆kdE/dk =(2π/l )~v =hv /l , (7)
c 0 0 c
There is an amplitude for an electron in an edge state
to be scattered into nearby states. This is analogous to wherel isthelengthoverwhichtheboundandextended
c
photon scattering, so we model it in the same way [5]. states are in contact, shown in Fig. 1. Therefore, 2 =
T
Eq. (4) is modified by the replacement γ γ√1 ǫ , (l γ/v )2 1,soγ v /l . Thus,theinverseofthetime
→ − s c 0 ≤ ≤ 0 c
where ǫ is the scattering probability for the channel. it takes an electron to pass the dot gives a bound on γ.
s
The scattering probability depends on the transmission To estimate v = E/B, we take B 1 T (set by
0
length, L, according to ǫ = 1 e−L/Ls, where L is an external source), a−nd E which we esti∼mate from the
s s
−
the scattering length [18]. L depends strongly on the potential profile at the edge of the wire due to exter-
s
temperature and purity of the sample, but for T <5 K, nal gate voltages and screening effects [20]. An upper
L &100 µm has been reported [13, 18]. estimate for E is given by the slope of the lateral poten-
s
Accountingforchanneldispersionismorecomplicated, tial in the incompressible region[20]. The voltageacross
since the originalformulationof one-wayquantumchan- the incompressible region is of order V = ~ω /e and
i B
4
ω = 176 GHz for bulk GaAs in a 1 T field [21]. The CVCP for financial support. CHWB acknowledges the
B
widthofthe incompressibleregionisaboutthe magnetic EPSRC for funding. GJM acknowledges the support of
length, l = ~/eB = 25.6 nm in a 1 T field. Then the ARC through the IREX grant. We would like to
B
E V /l pand we find that v 67 km s−1. Thus thank Sean Barrett, Hsi-Sheng Goan and Andre Luiten
max i B 0
≈ ≤
we estimate v = 104 m s−1 for the group velocity, con- for useful conversations.
0
sistent with previous results [10].
Thecontactlengthisoforderthedotsize,soforl .1
c
µm and v = 104 ms−1 we have γ 100 µeV. This
0 ≤ ∗ Electronic address: [email protected]
correspondstoanedgeperfectlytransmittedintothedot
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AB
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