Heralded quantum repeater based on the scattering of photons off single emitters in weak-coupling regime

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by  Guo-Zhu Song

Heralded quantum repeater based on the scattering of photons off single emitters in
weak-coupling regime
Guo-Zhu Song, Mei Zhang, Qing Ai, Guo-Jian Yang, and Fu-Guo Deng
∗
Department of Physics, Applied Optics Beijing Area Major Laboratory,
Beijing Normal University, Beijing 100875, China
(Dated: February 2, 2016)
We propose a heralded quantum repeater based on the scattering of photons off single emitters
in one-dimensional waveguides. We show the details by implementing nonlocal entanglement gen-
eration, entanglementswapping, and entanglementpurification moduleswith atoms in waveguides,
anddiscussthefeasibilityoftherepeaterwithcurrentlyachievabletechnology. Theschemefeatures
a filtering mechanism that the faulty events can be discarded by detecting the polarization of the
6
photons. That is, our protocols either succeed with 100% fidelity or fail in a heralded way, which
1
0 is advantageous for implementing realistic long-distance quantum communication. Moreover, ad-
2 ditional atomic qubits are not required, but only a single-photon medium. Our scheme is scalable
and attractive since it can be realized in solid-state quantum systems. With the great progress
n
on controlling atom-waveguide systems, this repeater may be very useful in quantum information
a
processing in the future.
J

0 PACSnumbers: 03.67.Pp,03.67.Bg,03.67.Hk,42.50.Pq
3

] I. INTRODUCTION sented a heralded high-efficiency quantum repeater with
h
p atomicensemblesassistedbyfaithfulsingle-photontrans-
- mission. Later, Li, Yang, and Deng introduced another
t An important task of quantum communication is to
n heralded quantum repeater for quantum communication
transmit information between distant parties securely,
a networkbasedonquantumdotsembeddedinopticalmi-
such as quantum key distribution [1–3], quantum secret
u crocavities, resorting to effective time-bin encoding [19].
q sharing[4–6],andquantumsecuredirectcommunication
Thebuildingblocksofquantumrepeatersareexperimen-
[ [7–9]. The creation of long-distance entanglement plays
tally realized by some research groups, and remarkable
an important role in quantum communication. In order
1 progress has been reported [20–24].
toexchangeprivateinformationandavoidanexponential
v
decayofphotonsoverlongdistance,theschemeforquan- In the past decade, the interaction between photons
4
5 tumrepeaterwasproposedby Briegelet al. [10]in1998. and atoms in high-quality optical microcavities has be-
0 Its main idea is to generate the entangled pairs in small comeoneofthe mostimportantmethods forimplement-
0 segmentsfirst,avoidingtheexponentialdecayofphotons ingquantumcomputationandquantuminformationpro-
0 with the transmission distance, and then use entangle- cessing. Somesignificantachievements[25–29]havebeen
2. mentswapping[11]andentanglementpurification[12,13] made inphoton-atomsystems in boththeory andexper-
0 to create a long-distance entangled quantum channel. iment. With strong coupling and high-quality cavities,
6 they can obtain a high-fidelity quantum computation.
Therearesomeinterestingproposalsforimplementing
1 In 2005, an interesting proposal [30] was proposed to
a quantum repeater, by utilizing different physical sys-
v: tems. For example, in 2001, Duan et al. [14] suggested realize the coupling between a single quantum emitter
i andaphotoninone-dimensional(1D)waveguides,which
X an original proposal to set up a quantum repeater with
can be considered as a bad cavity. In their scheme, the
atomic ensembles, called it the Duan-Lukin-Cirac-Zoller
r coupling between the emitter and the cavity is stronger
a (DLCZ) protocol. It exploits atomic ensembles as quan-
than the atomic decay rate, but weaker than the cavity-
tum memories, and Raman transitions in atomic ensem-
loss rate, and the atomic spontaneous emission into the
bles are used to produce the nonclassical correlation be-
waveguidebecomesthe maineffect,calledthePurcellef-
tween atomic excitations and emitted photons. In 2006,
Klein et al. [15] put forward a robust scheme for quan- fect. In 2007,a similar proposalwas presented to realize
this coupling using nanoscalesurfaceplasmons [31]. Itis
tum repeaters with decoherence-free subspaces. In 2007,
worth noting that the significant nonlinear optical prop-
using the two-photon Hong-Ou-Mandel interferometer,
Zhaoetal. [16]proposedarobustquantumrepeaterpro- erty can still be obtained in the weak-coupling regime
(low-Q regime).
tocol. In 2012, Wang, Song, and Long [17] presented an
efficient scheme for robust quantum repeaters based on In contrast to strong-coupling regime, the weak-
spatial entanglement of photons and quantum-dot spins coupling regime is less technically demanding and still
in optical microcavities. In 2015, Li and Deng [18] pre- allows for interesting quantum state manipulation and
quantum information processing, such as entanglement
generation [32–34], efficient optical switch [35], quan-
tum logicgates[36], andquantumstate transfer[37–39].
∗Correspondingauthor: [email protected] However,withemitterdecayandfinitecouplingstrength,
2
the physical device is restricted to finite P (Purcell fac- (a) e (b) e- e+
tor), so that the scattering of photons off single emit- M L R
ters may not happen at all. To solve this problem, in g g g
- +
2012, Li et al. [39] proposed a simple scattering setup
to realize a high-fidelity Z gate on an atom assisted by
a single-photon medium, in which the faulty events can
in 1 BS
be heralded by detecting the polarization of the photon
M
pulse. 0 x out 2
Inthispaper,wepresentaheraldedquantumrepeater
thatallowsthecreationoftheentangledstateoveranar- FIG.1: (a) The basic structurefor aphoton mirror in which
bitrarylargedistancewithatolerabilityoferrors. Inour a two-level atom (an emitter marked by the black dot) is
scheme,sinceatomscanprovidelongcoherencetime, we coupled to a 1D waveguide (marked by the cylinder). Here
the atom has a ground state g and an excited state e ,
chooseafour-levelatomastheemitter. Withthescatter- | i | i
and its position is x = 0. In an ideal situation, an incident
ing of photons off single emitters in 1D waveguides, the
photon (purple, the upper left wave packet) is fully reflected
parties in quantum communication can realize nonlocal
(blue, the lower left wave packet) when it resonates with the
entanglement creation against collective noise, entangle-
atom, butthereisatransmittedcomponent (black,theright
ment swapping, and entanglement purification. More- wave packet) in a practical scattering. Note that, having no
over, our protocols can turn errors into the detection of connectionwiththeemitter,theincidentphotongoesthrough
photon polarization, which can be discarded. The pre- theatomwithnoeffect(green,theupperrightwavepacket).
diction of faulty events ensures that our repeater either (b) A heralded setup to realize a high-fidelity Z gate on an
succeeds with perfect fidelity or fails in a heralded way, atominthe1Dwaveguide. Differentfrom theemitterinFig.
which is advantageousfor quantum information process- 1(a), the atom has two degenerate ground states g± and
| i
ing. two degenerate excited states e± coupled to the waveguide
| i
(marked by the cylinder). BS is a 50 : 50 beam splitter, M
The paper is organized as follows: In Sec. II, we in-
is a fully reflected mirror, and the black lines represent the
troduce the transport property of a photon scattering
pathsof thetraveling photon.
with a two-level atom coupled to a 1D waveguide, and
then show the protocols for implementing nonlocal en-
tanglementcreation,entanglementswapping,andentan-
glement purification. In Sec. III, we discuss a few rele- the system in real space as
vantissuesrelatedtotheimplementationofourrepeater
in a realistic system, and end up with a summary. H′ = ¯hZdkωka†kak +¯hgZdk(akσ+eikxa +h.c.)
iγ
′
+¯h(ω )σ , (2)
II. QUANTUM REPEATER BASED ON THE a− 2 ee
SCATTERING CONFIGURATION
where x is the position of the atom, σ = e e, and
a ee
| ih |
ω = ck (c is the group velocity of the propagating
A. The scattering of photons off single emitters in k | |
electromagnetic modes and k is its wave vector). γ is
a 1D waveguide ′
thedecayrateoftheatomoutofthewaveguide(e.g.,the
emission into the free space). Because we only care the
Letusconsideraquantumsystemcomposedofasingle
interactionsofthe near-resonantphotonswiththe atom,
emittercoupledtoelectromagneticmodesina1Dwaveg-
we could make the approximation that left- and right-
uide, as shown in Fig. 1(a). We first choose a simple
propagating photons form completely separate quantum
two-level atom as the emitter, consisting of the ground
fields [30]. Under this approximation,the operatora in
state g and the excited state e with the frequency k
Eq. (2) can be replaced by (a +a ).
differe|ncie ω . Under the Jaynes|-Ciummings model, the k,R k,L
a To get the reflection and transmission coefficients of
Hamiltonian for the interactions between a set of waveg-
single-photon scattering, we assume that a photon with
uide modes and a two-level atom reads:
the energy E is propagating from the left. The state of
k
the system is described by
1
H = ¯hωka†kak + 2¯hωaσz + ¯hg(a†kσ−+akσ+), (1)
X X
k k E = c e,vac + dx[φ (x)c (x)
| ki e| i Z L †L
awthoerrseoafkthaendwaav†kegaureidtehemaondneihwiiltahtiofrneqaunednccyreaωti,onreosppeecr-- +φR(x)c†R(x)]|g,vaci, (3)
k
tively. σ , σ , and σ are the inversion, raising, and where vac represents the vacuum state of photons, c
z + − | i e
lowering operators of the two-level atom, respectively. g is the probability amplitude of the atom in the excited
is the coupling strength between the atom and the elec- state, and c (x) (c (x)) is a bosonic operator creating
† †
L R
tromagnetic modes of the 1D waveguide, assumed to be a left-going (right-going) photon at position x. φ (x)
R
same for all modes. One can rewrite the Hamiltonian of and φ (x) are the probability amplitudes of right- and
L
3
left-traveling photon, respectively. Note that the photon into
propagatesfromtheleft,φ (x)andφ (x)couldtakethe
R L g ψ H g φ H + g φ V ,
forms | +i| i| i → | +i| ti| i | +i| ri| i (7)
g ψ H g φ H g φ V ,
t r
φ (x) = eikxθ( x)+teikxθ(x), | −i| i| i → | −i| i| i−| −i| i| i
R − (4)
φ (x) = re ikxθ( x). where V = (R L )/√2 is the vertical linear-
L − − polariza|tioin state| oif−ph|otoins. It is interesting that the
Here t and r are the transmission and reflection coef- scattering process generates a vertical-polarized compo-
ficients, respectively. The Heaviside step function θ(x) nent, with which a heralded setup can be constructed to
equals 1 when x is larger than zero and 0 when x is realize a high-fidelity Z gate on an atom, as shown in
smaller than zero. By solving the time-independent Fig. 1(b), similar to that in Ref. [39].
Schr¨odinger equation H E =E E , one can obtain The input photon in spatial state ψ with H or V
| ki k| ki (from port 1) is split by a 50 : 50|beiam spl|ittier (B|S)i
1 into two halves that scatter with the atom simultane-
r = ,
−1+γ′/γ1D −2i∆/γ1D (5) ously. Then, the transmitted and reflected components
travelback and exit the beam splitter from port1. Note
t = 1+r,
that, due to quantum destructive interference, there is
where∆=ω ω is the photondetuning withthe two- no photon component coming out from port 2. Finally,
level atom, akn−d γa = 4πg2/c is the decay rate of the discarding the output with unchanged photon polariza-
1D
atom into the waveguide. tion(i.e.,the faultyevents),one cangetthe high-fidelity
Provided that the incident photon resonates with the Za gate as follows:
emitter (i.e., ∆ = 0), one can easily obtain the reflec-
µ ψ H Z µ φ V ,
tthioenPcuorecffielcliefnacttror=. A−s1/w(e1+kn1o/wP,)i,nwthheeraetPom=-wγa1vDe/gγu′idies |µiaa|ψi|Vi → −Zaa|µiaa|φrri|Hi, (8)
| i | i| i → − | i | i| i
system, the spontaneous emission rate γ into the 1D
1D where µ isanarbitrarysuperpositionstateoftheatom
waveguide can be much larger than the emission rate γ a
′ | i
in the basis 0 g , 1 g . As mentioned
intoallotherpossiblechannels[30,31]. Consideringthat a a +
the Purcell factor P can exceed 103 in realistic systems above, φr is{|thie s≡pa|ti−aliw|avie ≡fun|ctioin} of the reflected
| i
photoncomponentafterthescatteringprocess,asshown
[31], one canget the reflectioncoefficient r= 1 for this
− in Fig. 1(a). When P , the perfect scattering pro-
system. Thatis,whenthephotoniscoupledtotheemit-
→ ∞
cessleadsto φ = ψ . Intheimperfectsituationwith
ter, the atom acts as a photon mirror [30], which puts a r
| i −| i
a finite P, we get φ = ψ , the output photon with
π-phase shift on reflection. However, when the photon r
| i 6 −| i
unchanged polarization is detected and the correspond-
is decoupled from the emitter, it transmits through the
inggatefails,whichcanbe discarded. Thatis,the setup
atom with no effect.
works for the Z gate on an atom in a heralded way.
Let us consider a four-level atom with the degenerate
groundstates g andexcitedstates e as the emitter
in a 1D waveg|ui±dei, as shown in Fig. 1|(b±)i. For the emit-
B. Robust entanglement creation against collective
tceoru,ptlhede ttoratnhseittiownoseolefc|tgr+oim↔agn|ee+tiic manodde|gs−ai ↔a|ned−iaare, noise
k,R k,L
with the absorption (or emission) of right (R) and left
With the scattering property of a photon scattering
(L) circular polarization photons, respectively. Assum-
withafour-levelatomcoupledtoa1Dwaveguide,wecan
ing thatthe spatialwavefunction ofthe incident photon
designa robust scheme for the entanglement creationon
is ψ , with the scattering properties in the practical sit-
| i two nonlocal stationary atoms a and b, as shown in Fig.
uation discussed above, one can get
2. Suppose that the single photon medium and the two
g ψ R g φ R , stationaryatomsin1Dwaveguidesareinitiallyprepared
+ +
| i| i| i → | i| i| i in the superposition states ψ ph = 1 (H + V ) and
g ψ L g φ L , | 0i √2 | i | i
| −i| i| i → | −i| i| i (6) ϕ = 1 (0 + 1 ) (here i= a,b),respectively,andthe
g ψ R g ψ R , | ii √2 | i | i i
| −i| i| i → | −i| i| i state of the system composed of the photon and the two
g ψ L g ψ L .
| +i| i| i → | +i| i| i atoms is
Here φ isthespatialstateofthephotoncomponentleft 1
inthe|wiaveguideafterthescatteringprocess. Inpractice, Ω0 = (H + V ) (0 + 1 )a (0 + 1 )b. (9)
| i 2√2 | i | i ⊗ | i | i ⊗ | i | i
there is alwaysa transmitted part [31], thus φ = φ +
t
| i | i
φ , where φ and φ refer to the transmitted and Our scheme works with the following steps.
r t r
| i | i | i
reflected parts of the photon, respectively. When the First,the H and V componentsoftheinputphoton
| i | i
Purcell factor P is infinite, φ is normalized. Whereas, are spatially split by a polarizing beam splitter (PBS).
| i
if the input photon is in the horizontalliner-polarization In detail, the photon in state H passes through both
| i
state H = (R + L )/√2, the transformations turn PBS1 andPBS2 towardsthesetuptoscatterwithatom
| i | i | i
4
D M the output port becomes
6
b
D
5
PBS– Ω2 = Ω + Ω ,
M | i | 2Si | 2Li
D4 2 BS2 M |Ω2Si = 2√12(γ|Vi+δ|Hi)(|0i+|1i)a(|0i+|1i)b,
1
D3 PBS– 1 PBS4 PBS3 M |Ω2Li = 2√2(γ|Vi+δ|Hi)(|0i−|1i)a(|0i+|1i)b.
D (12)
M DL 2
TR
Third, coming out from the noisy channel, the early
channel partofthephotoninstate Ω isreflectedbytheoptical
| 2Si
device TR and goes through a time-delay device, while
M the late part in state Ω transmits through TR into
M | 2Li
a PBS3. Afterthat,thecomponentsin H and V ofthe
| i | i
S L late partaresplitinto two halvesthatscatter with atom
M bandtravelbacktoPBS3simultaneously. Subsequently,
D1 input PBS1 PBS2 BS1 theearlypartandthelatepartarerejoinedinPBS4,and
they are separated into two paths 1 and 2. The state of
the whole system evolves into
FIG. 2: Schematic setup for the creation of maximally en-
tangled states on two nonlocal atoms a and b in 1D waveg- 1
uides. PBSi(i=1,2,3,4)representsapolarizingbeamsplit- Ω3 = γ(H + V )1(0 0 + 1 1 )ab
| i 2√2 | i | i | i| i | i| i
ter which transmits the horizontal polarized photon H and
reflects the vertical polarized photon V . PBS tr|anismits 1
photonswithpolarization + andreflec|tsiphotons±withpolar- −2√2γ(|Hi−|Vi)1(|0i|1i+|1i|0i)ab
ization|−i,where|±i=(||Hii±|Vi)/√2. Di (i=1,2,··· ,6) 1
isasingle-photondetectorandDLisatime-delaydevice. TR + δ(H + V )2(0 0 + 1 1 )ab
isanopticaldevicewhichcanbecontrolledexactlyasneeded 2√2 | i | i | i| i | i| i
to transmit or reflect a photon. 1
+ δ(H V )2(0 1 + 1 0 )ab. (13)
2√2 | i−| i | i| i | i| i
Finally, the two parts in paths 1 and 2 go through
a, while the componentin state V is reflected andgoes PBS , and the photon is detected by one of the four
| i ±
directly to the channel, having no interaction with atom single-photon detectors D3, D4, D5, and D6. If the de-
a. After the scatteringprocess,the partinteractingwith tector D4 or D5 clicks, we should put a σx operation on
atomaentersintothechannel,butalittlelaterthanthe atomb. Ifthe detectorD3 orD6 clicks,nothingneedsto
otherpart. Thestateofthewholesystemattheentrance be done. Eventually, the state of the system composed
of the channel becomes Ω1 . Here of atoms a and b collapses to the maximally entangled
| i
state
|Ω1i = |Ω1Si+|Ω1Li, 1
1 φ+ = (0 0 + 1 1 ) . (14)
|Ω1Si = 2√2|Vi⊗(|0i+|1i)a⊗(|0i+|1i)b, (10) | iab √2 | i| i | i| i ab
1
Ω = V (0 1 ) (0 + 1 ) , Oursetupfortherobustentanglementcreationontwo
| 1Li 2√2| i⊗ | i−| i a⊗ | i | i b nonlocal atoms has some interesting features. First, the
earlypartand the late partof the photonin the channel
where Ω and Ω represent the two parts of the aresonearthattheysufferfromthesamecollectivenoise
| 1Si | 1Li
photon going through the short path (S) and the long [3,40–43],andanarbitraryqubiterrorcausedbythelong
path (L) to the channel, respectively. noisychannelcanbeperfectlysettled. Second,thefaulty
Second, as the two parts in the channel are near and interactions between the photon and two atoms can be
theirpolarizationstatesarebothin V ,theinfluencesof heraldedby the detectors D1 and D2. Indetail, if one of
| i
the collective noise in the quantumchannelon these two the two detectors clicks, the event of the entanglement
parts are the same one [40–43], which can be described creation fails, which can be discarded. Moreover, the
by photonlossinthelongquantumchannelcanbeheralded
bythe photondetectorsD3, D4, D5,andD6. Third,the
V γ V +δ H , (11) probability of the entanglement creation is independent
| i → | i | i
of the noise parameters. These good features make our
where γ 2 + δ 2 = 1. After the photon travels in the setup have good applications in quantum repeaters for
| | | |
long quantum channel, the state of the whole system at long-distance quantum communication.
5
p p
D 2 1 PBS QWP
1 1 1
input
a c QWP d b
2
PBS PBS
2 3
PBS
4
D PBS–
2
D
3
FIG.3: Schematicdiagram showingtheprincipleofentanglement swapping. QWPi (i=1,2)representsaquarter-waveplate,
which is used to implement theconversion of the photon polarization.
C. Entanglement swapping Ψ1 . Here,
| i
(H + V )
Ψ1 = | i | i p1 (00 11 )cd (00 + 11 )ab
Theatomicentangledstatecanbeconnectedtolonger | i 4√2 (cid:2) | i−| i ⊗ | i | i
communication distance via local entanglement swap- +(00 + 11 ) (00 11 )
cd ab
| i | i ⊗ | i−| i
ping. AsshowninFig. 3,thetwopairsofnonlocalatoms (cid:3)
(H V )
acandbdarebothinitiallypreparedinthemaximallyen- | i−| i p1 (01 10 ) (01 + 10 )
cd ab
tangledstates φ+ = 1 (0 0 + 1 1 ) and φ+ = − 4√2 (cid:2)| i−| i ⊗ | i | i
| iac √2 | i| i | i| i ac | ibd
√12(|0i|0i+ |1i|1i)bd, respectively. With the Bell-state +(|01i+|10i)cd⊗(|01i−|10i)ab(cid:3). (16)
measurement on local atoms cd and single-qubit oper-
ations, the two nonlocal atoms ab can collapse to the
The first and the second parts of the photon in Eq. (16)
maximallyentangledstate φ+ = 1 (0 0 + 1 1 ) ,
| iab √2 | i| i | i| i ab are detected by the detectors D3 and D2, respectively.
which indicates that the nonlocal entanglement for a
Second, after the measurement on photon p1, a
longercommunicationis realized. The principleofquan-
Hadamard operation H (e.g., using a π/2 microwave
a
tum swapping is shown in Fig. 3, and the details are
pulse or optical pulse [44, 45]) is performed on the two
described as follows.
localatomscanddinthewaveguides,respectively. Sub-
sequently, we input another photon p2 into the setup,
First, suppose that the input photon p1 is preparedin which is initially prepared in the state 1 (H + V ) .
thesuperpositionstate|ψip1 = √12(|Hi+|Vi)p1,andthe Photon p2 has the same process as pho√t2on| pi1, fo|r isimp2-
initial state of the whole system composed of photon p1 plicity, we omit the process of photon p2. When the two
and the four atoms acbd is |Ψ0i. Here parts of photon p2 arrive in PBS4 simultaneously, the
state of the whole system becomes
(H + V )
|Ψ0i = 2√12(|Hi+|Vi)p1 ⊗(|0i|0i+|1i|1i)ac |Ψ2i = | i4√|2 i p2(cid:2)(|00i−|11i)cd⊗(|00i−|11i)ab
⊗(|0i|0i+|1i|1i)bd. (15) −(|00i+|11i)cd⊗(|01i−|10i)ab(cid:3)
(H V )
| i−| i p2 (01 10 )cd (00 + 11 )ab
− 4√2 | i−| i ⊗ | i | i
(cid:2)
The injecting photon p1 passes through PBS1, which +(01 + 10 )cd (01 + 10 )ab .
| i | i ⊗ | i | i
transmitsthephotoninstate H andreflectsthephoton (cid:3) (17)
| i
instate V . Thephotoninstate V isreflectedbyPBS1
| i | i
andPBS2 intothe scatteringsetupcomposedofatomc, Then, photon p2 travels through the PBS and is de-
±
whiletheotherpartinstate H goesthroughQWP1and tected. If the photon detector D3 clicks, it corresponds
| i
is reflected by PBS3 into the scattering setup to scatter to the first part in Eq. (17). If D2 clicks, it means the
with atom d. Then, the two parts of the photon p1 are second part in Eq. (17) is probed.
rejoinedinPBS4 andtravelthroughPBS . Afterthat, Third, with the outcomes of the detectors for photons
±
the state of the whole system is changed from Ψ0 to p1 and p2, one can see that the four Bell states of the
| i
6
atoms a and b are completely distinguished. Finally, the First, both Alice and Bob prepare an optical pulse in
parties can perform corresponding operations (see Table thesuperpositionstate 1 (H + V )andlettheirpulses
√2 | i | i
I) on atom a to complete the quantum swapping. Af- pass through the equipments shown in Fig. 4. To sim-
ter that, the state of the two nonlocal atoms a and b in plify the discussion, we just discuss the interactions in
a longer distance collapses to the maximally entangled Alice, and Bob need complete the same process simulta-
state φ+ ab. neously. For Alice, the H and V components of the
Itis|imiportantto notethatthe wronginteractionsbe- inputphoton1arespatia|llyisplitb|yiPBS1. Indetail,the
tween photon and atoms are heralded by the photon de- component in V is reflected by both PBS1 and PBS2
tectors for the quantum swapping in our scheme. In de- tothescatterin|gsietupcomposedofatoma1,whereasthe
tail, if detector D1 clicks, the interactions between pho- componentin H goesthroughQWP2andisreflectedby
ton and two atoms in 1D waveguides are faulty, which PBS3tothes|catiteringsetupcontainingatoma1. Subse-
could be discarded. If detector D2 or D3 does not click, quently, the two parts of photon 1 are rejoined at PBS4
the photon loss occurs during the whole process. There- and travel through PBS . Meanwhile, another photon
fore, with the prediction of the faulty events, the parties 2 has the same process as±photon 1 simultaneously.
canobtainahigh-fidelitynonlocalatomicentangledstate Second,aftertheaboveinteractions,photon1andpho-
in a longer distance. ton2areprobedbysingle-photondetectorsandthereare
two kinds of measurement results. In detail, if the two
pairs of nonlocal two-atom systems are initially in the
TABLE I: The operations on atom a corresponding to the
state φ+ φ+ ,the evolutionofthe wholesystem
outcomes of thephoton detectors D2 and D3. | ia1b1| ia2b2
is
First click Second click Operations on atom a
1
DD33 DD23 σIz 4(|Hi+|Vi)1(|Hi+|Vi)2(|00i+|11i)a1b1(|00i+|11i)a2b2
1
D2 D2 σx → 4(|Hi+|Vi)1(|Hi+|Vi)2(|0000i+|1111i)a1b1a2b2
D2 D3 σzσx
1
+4(|Hi−|Vi)1(|Hi−|Vi)2(|0011i+|1100i)a1b1a2b2.
(19)
D. Entanglement purification
FromEq. (19), one cansee that either the single-photon
detectors D3D6 click or the detectors D2D5 click. Simi-
In Sec. IIB and Sec. IIC, we only talk about the
larly, the measurement of the other three cases is shown
influence of noise on flying photons in long quantum
in Table II. With the outcomes of four detectors, we
channel. In the practical situation, the errors also occur can distill φ+ φ+ and ψ+ ψ+ from
in stationary atoms embedded in 1D waveguides, which | ia1b1| ia2b2 | ia1b1| ia2b2
the four cases discussed above.
will decrease the entanglement of the nonlocal two-atom
systems. Using entanglement purification, we can dis-
till some high-fidelity maximally entangledstates from a TABLE II: The results of the four single-photon detectors
mixed entangled state ensemble. correspondingtotheinitialentangledstatesofthefouratoms.
Now, we start to explain the principle of our purifi-
Initial states Photons measurement
cation protocol for nonlocal atomic entangled states, as-
sistedbythescatteringofphotonsoffsingleatomsin1D (a1b1) (a2b2) Detector click
waveguides,as shown in Fig. 4. φ+ φ+ D3D6 or D2D5
| i | i
Suppose that the initial mixed state sharedby two re- φ+ ψ+ D3D5 or D2D6
| i | i
mote parties, say Alice and Bob, can be written as ψ+ φ+ D3D5 or D2D6
| i | i
ρ =F φ+ φ+ +(1 F)ψ+ ψ+ , (18) |ψ+i |ψ+i D3D6 or D2D5
ab ab ab
| i h | − | i h |
where |ψ+iab = √12(|0i|1i+|1i|0i)ab. The subscripts a Third,to recoverthe entangledstates ofatomsa1 and
andbrepresentthesingleatomsin1Dwaveguidesowned b1,AliceandBobshouldperformaHadamardoperation
by the two parties Alice and Bob, respectively. F is the Ha on the two nonlocal atoms a2 and b2 in the waveg-
initialfidelityofthestate φ+ . Byselectingtwopairsof uides, respectively. Then, Alice and Bob measure the
| i
nonlocalentangledtwo-atomsystems,thefouratomsare states of the two atoms a2 and b2, and compare their
inthestates φ+ φ+ withtheprobabilityofF2, results with the help of classical communication. If the
φ+ ψ+ | iaa1nbd1| ψ+ia2b2 φ+ with a probabil- resultsarethe sameones,nothingneedstobe done;oth-
io|tfy(o1iaf1Fb1F(|1)2−,iFrae2)sb,2paenctdiv|eψ|l+y.iaiO1abu11br|1ψ|e+ntiaai2anb2g2bl2ewmitehntappurroibfiacbatiliiotny eTrawbilsee,IIa,σozneopcearnatsieoentnheaetdtshteorebearpeuttwooncaatsoems ian1.thFerorme-
proto−col for nonlocal entangled atom pairs works with servedentangledpairs a1 and b1. One is φ+ with a
the following steps. probability of F2, and the other one is |ψ+ia1b1 with a
| ia1b1
7
D1 1 PBS1 QWP2
input
D
Alice 3
D PBS–
a1 2 a2
PBS
4
PBS QWP PBS
2 1 3
b1 PBS QWP PBS b2
2’ 1’ 3’
PBS
4’
Bob
D PBS–
5
D4 2 D6
input
PBS QWP
1’ 2’
FIG. 4: Schematic setup showing the principle of the atomic entanglement purification protocol based on the scattering of
photons off single emitters.
probability of (1 F)2. Therefore, in the filtered states, polarizationofoutputphotonischangedbut φ =rψ
r
the probability of−φ+ is (r < 1); the other one is that the scattering| evient b|ei-
| ia1b1 | |
tweenatomsandphotonsdoesnothappenatall. Forthe
′ F2 lattercase,theoutputphotonswithunchangedpolariza-
F = . (20)
F2+(1 F)2 tion are detected at the entrance, and the correspond-
−
ing quantum computation is discarded. Assuming that
That is, after the purification process, the fidelity of
the linear optical elements are perfect in our protocols,
φ+ becomes F′. When F > 1, one can easily get
| ′ ia1b1 2 the heralded mechanism ensures that the faulty events
F >F. cannot influence the fidelity of our scheme, but decrease
theefficiency,becausethesuccessprobabilityp isdeter-
s
mined by the quality of the atom-waveguide systems.
III. DISCUSSION AND SUMMARY As mentioned above, a high Purcell factor is needed
in our scheme, which can effectively improve the per-
We have proposed a heralded scheme for quantum re- formance of our protocols. In the last decade, a great
peater, including robust nonlocal entanglement creation progress has been made in the emitter-waveguide sys-
against collective noise, entanglement swapping, and en- tems in experiment. In 2006, Chang et al. [46] pre-
tanglement purification modules. The key element in sented a scheme that a dipole emitter is coupled to a
our protocol is the scattering process between photons nanowireora metallic nanotip,in whicha Purcellfactor
and atoms in 1D waveguides. In the following section, P = 5.2 102 is obtained for a silver nanowire. Sub-
×
wewilldiscussthe performanceofourquantumrepeater sequently, using the surface plasmons of a conducting
under practicalconditions,defining p = ψ φ 2 asthe nanowire, Chang et al. [31] obtained an effective Pur-
s r
success probability of the scattering even|th in| thi|e proto- cell factor, which can reach 103 in realistic systems. In
col of Z gate on an atom, as shown in Fig. 1(b). Here addition, short waveguide lengths of only 10 to 20 unit
ψ and φ are the spatial wave functions of the inci- cells were found by Manga Rao and Hughes [47] to pro-
r
| i | i
dent photon and the reflected photon component after duce very large Purcell factors in 2007. Later, based on
the scattering event, respectively. For perfect scatter- modifying photon fields around an emitter using high-
ing event (i.e., P ), φ = ψ , and the success finesse optical cavities, Akimov et al. [48] demonstrated
r
→ ∞ | i −| i
probability p is 100%. Whereas, in realistic situations, a broadband approach for manipulating photon-emitter
s
with finite P, φ = ψ , it includes two cases: one interactions. In 2008, photonic crystal waveguides were
r
is the successfu|l evien6 t o−f|imiperfect processes, where the exploited by Hansen et al. [49] to enable a single atom
8
P = 100 and ∆ = 0.1γ for atom-waveguide systems,
1D
thesuccessprobabilityp ofthescatteringprocessinFig.
s
1.0 1(b)canreach94.3%. Indetail,fortheentanglementcre-
0.8 ationprotocolinourscheme,theincidentphotonscatters
s with the atoms twice in 1D waveguides, so the success
p 0.6 probabilityis (94.33%)2 =88.98%,while for both entan-
0.4 glement swapping and entanglement purification proto-
(a)
0.2 cols, the scattering process occurs four times, and the
success rates for them are both (94.33%)4 = 79.18%.
0.0
0 20 40 60 80 100 Considering the influence of other factors, such as pho-
P ton loss in fiber and defective linear optical elements,
the efficiency of our scheme may be low, but this issue
1.0 can be well solved by the highly efficient single-photon
source[54], whichprovides10000single photons per sec-
0.8
s ond. Thus, we are able to accomplish our scheme in a
p 0.6
short time.
0.4
(b) Compared with other schemes, the protocols we
0.2
presentforthe heraldedquantumrepeaterhavesomein-
0.0 teresting features. (1) Different from the previous works
-2 -1 0 1 2
for quantum repeaters [14–16, 55], in our scheme, the

D faultyeventscausedbyfrequencymismatches,weakcou-
pling, atomic decay into free space, or finite bandwidth
FIG. 5: The success probability ps of the scattering process
of the incident photon can be turned into detection of
vs the Purcell factor P and the detuning parameter ∆/γ .
1D
(a) The success probability ps vs the Purcell factor P when the output photon polarization, and it just affects the
thedetuning∆=0. (b)Thesuccessprobabilitypsvsthepa- efficiency of our protocols but not the fidelity. In other
rameters ∆/γ . The dotted (green), dashed-dotted (blue), words, our scheme either succeeds with a perfect fidelity
1D
dashed(red),andsolid(black)linescorrespondtoP=1,P=5, or fails in a heralded way, which is an advantageous fea-
P=10, and P=50, respectively. ture for quantum communication. (2) In contrast to
strong-couplingregime[26,27,29,56,57],ourschemefo-
cuses on Purcell regime [37, 58], where a four-level atom
is either embedded in the waveguide or side-coupled to
to exhibit nearly perfect emission into the guided modes the waveguide, which can be thought of as a bad cav-
(γ1D ≫γ′)andtoactasahighlyreflectivemirror,where ity. It is interesting that the nonlinear optical property
thelight-mattercouplingstrengthislargelyenhanced. In canstill be obtainedin the weak-couplingregime(low-Q
2012, Goban [50] reported the experimental implemen- regime),which providesan exciting possibility for realis-
tation of a state-insensitive, compensated optical trap tic long-distance quantum communication and quantum
for single Cs atoms, which provides the precise atomic informationprocessing[32,35–37,58–61]. (3)Motivated
spectroscopy near dielectric surfaces. Moreover,in 2013, by recent experimental progress,our scheme for the her-
Hung et al. [51] proposed a protocol in which one atom aldedquantumrepeaterisfeasibleinotherrelatedquan-
trapped in single nanobeam structure could provide a tum systems, such as superconducting quantum circuit
resonant probe with transmission t2 10−2. A similar coupled to transmission lines [62–65], quantum dot em-
scheme was realized in experiment| b|y≤Goban et al. [52] bedded in a nanowire [66], and photonic crystal waveg-
in2014. Recently,due tothe couplingofasingleemitter uide with quantum dots [49, 67].
to a dielectric slot waveguide, a high Purcell factor was
also obtained by Kolchin et al. [53]. Insummary,wehaveproposedaheraldedquantumre-
peaterbasedonthe scatteringofphotonsoffsingleemit-
AsillustratedinSec. IIA,wecanobtainthereflection
ters in1Dwaveguides. The informationofthe entangled
coefficientforanincidentphotonscatteringwithanatom
states is encoded on four-level atoms embedded in 1D
in the 1D waveguide. On resonance, r = 1/(1+1/P),
− waveguides. Asourprotocolscantransformfaultyevents
andwegettherelationbetweenthesuccessprobabilityp
s
(i.e., r2)andthePurcellfactorP,asshowninFig. 5(a). into the detection of photon polarization, we present a
| | differentwayforconstructingquantumrepeatersinsolid-
Moreover,the scatteringqualityisalsoinfluencedbythe
state quantumsystems. With the significantprogresson
nonzerophotonicdetuning,andthe detailsaredescribed
manipulating atom-waveguide systems, our quantum re-
in Fig. 5(b), where p is plotted as a function of the de-
s
peater may be very useful for quantum communication
tuning parameter ∆/γ . From Fig. 5, one can see that
1D in the future.
the success probability p would exceed 90% on the con-
s
dition that the Purcell factor P 50 and the detuning
≥
parameter ∆/γ 0.13, which could be easily realized
1D ≤
in realistic systems [31]. For instance, when we choose
9
ACKNOWLEDGMENTS Key Basic Research Program of China under Grant No.
2013CB922000, the Youth Scholars Program of Beijing
This workwassupportedby the NationalNaturalSci- Normal University under Grant No. 2014NT28, and the
ence Foundation of China under Grant No. 11474026, Open Research Fund Program of the State Key Labo-
No. 11174040, No. 11475021, and No. 11505007, ratory of Low-Dimensional Quantum Physics, Tsinghua
the Fundamental Research Funds for the Central Uni- University Grant No. KF201502.
versities under Grant No. 2015KJJCA01, the National
[1] A.K. Ekert,Phys. Rev.Lett. 67, 661 (1991). [26] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,
[2] C. H. Bennett, G. Brassard, and N. D. Mermin, Phys. Phys. Rev.Lett. 78, 3221 (1997).
Rev.Lett. 68, 557 (1992). [27] T.Wilk,S.C.Webster,H.P.Specht,G.Rempe,andA.
[3] X.H.Li,F.G.Deng,andH.Y.Zhou,Phys.Rev.A78, Kuhn,Phys. Rev.Lett. 98, 063601 (2007).
022321 (2008). [28] D.L.Moehring,P.Maunz,S.Olmschenk,K.C.Younge,
[4] M. Hillery, V. Buzek, and A. Berthiaume, Phys. Rev. A D.N.Matsukevich,L.M.Duan,andC.Monroe,Nature
59, 1829 (1999). (London) 449, 68 (2007).
[5] A.Karlsson,M.Koashi,andN.Imoto,Phys.Rev.A59, [29] Y. Li, L. Aolita, and L. C. Kwek, Phys. Rev. A 83,
162 (1999). 032313 (2011).
[6] L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, Phys. [30] J. T. Shen and S. Fan, Opt.Lett. 30, 2001 (2005).
Rev.A 69, 052307 (2004). [31] D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D.
[7] G. L. Long and X. S. Liu, Phys. Rev. A 65, 032302 Lukin, Nat.Phys. 3, 807 (2007).
(2002). [32] K. Kojima, H. F. Hofmann, S. Takeuchi, and K. Sasaki,
[8] F.G.Deng,G.L.Long,andX.S.Liu,Phys.Rev.A68, Phys. Rev.A 68, 013803 (2003).
042317 (2003). [33] T. C. H. Liew and V. Savona, Phys. Rev. A 85,
[9] F. G. Deng and G. L. Long, Phys. Rev. A 69, 052319 050301(R)(2012).
(2004). [34] M. C. Kuzyk, S. J. van Enk, and H. Wang, Phys. Rev.
[10] H. J. Briegel, W. Du¨r, J. I. Cirac, and P. Zoller, Phys. A 88, 062341 (2013).
Rev.Lett. 81, 5932 (1998). [35] M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M.
[11] M.Z˙ukowski,A.Zeilinger,M.A.Horne,andA.K.Ekert, Hafezi, A. Zibrov, V. Vuletic, and M. D. Lukin, Phys.
Phys.Rev.Lett. 71, 4287 (1993). Rev. Lett.102, 203902 (2009).
[12] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. [36] B. C. Ren, G. Y. Wang, and F. G. Deng, Phys. Rev. A
Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 91, 032328 (2015).
(1996). [37] E. Waks and J. Vuckovic, Phys. Rev. Lett. 96, 153601
[13] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. (2006).
Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 [38] J. H. An, M. Feng, and C. H. Oh, Phys. Rev. A 79,
(1993). 032303 (2009).
[14] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, [39] Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, Phys.
Nature(London) 414, 413 (2001). Rev. Lett.109, 160504 (2012).
[15] A. Klein, U. Dorner, C. M. Alves, and D. Jaksch, Phys. [40] D. Kalamidas, Phys. Lett. A 343, 331 (2005).
Rev.A 73, 012332 (2006). [41] T.Yamamoto,J.Shimamura,S.K.O¨zdemir,M.Koashi,
[16] B.Zhao,Z.B.Chen,Y.A.Chen,J.Schmiedmayer,and and N.Imoto, Phys.Rev.Lett. 95, 040503 (2005).
J. W. Pan, Phys. Rev.Lett. 98, 240502 (2007). [42] X.H.Li,F.G.Deng,andH.Y.Zhou,Appl.Phys.Lett.
[17] T. J. Wang, S. Y. Song, and G. L. Long, Phys. Rev. A 91, 144101 (2007).
85, 062311 (2012). [43] T. Yamamoto, K. Hayashi, S. K. Ozdemir, M. Koashi,
[18] T. Li and F. G. Deng,Sci. Rep. 5, 15610 (2015). and N.Imoto, Nat.Photon. 2, 488 (2008).
[19] T. Li, G. J. Yang, and F. G. Deng, Phys. Rev. A 93, [44] J. Berezovsky,M.H.Mikkelsen, N.G. Stoltz,L.A.Col-
012302 (2015). dren, and D. D.Awschalom, Science 320, 349 (2008).
[20] Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. [45] D. Press, T. D. Ladd, B. Y. Zhang, and Y. Yamamoto,
Pan, Phys.Rev.Lett. 90, 207901 (2003). Nature (London) 456, 218 (2008).
[21] A.Kuzmich,W.P.Bowen,A.D.Boozer,A.Boca,C.W. [46] D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D.
Chou, L. M. Duan, and H. J. Kimble, Nature (London) Lukin, Phys.Rev.Lett. 97, 053002 (2006).
423, 731 (2003). [47] V.S.C.MangaRaoandS.Hughes,Phys.Rev.Lett.99,
[22] Z. S. Yuan, Y. A. Chen, B. Zhao, S. Chen, J. Schmied- 193901 (2007).
mayer, and J. W. Pan, Nature (London) 454, 1098 [48] A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang,
(2008). A. S. Zibrov, P. R.Hemmer, H. Park, and M. D. Lukin,
[23] N.Curtz,R.Thew,C.Simon,N.Gisin,andH.Zbinden, Nature 450, 402 (2007).
Opt.Express 18, 22099 (2010). [49] T. Lund-Hansen, S. Stobbe, B. Julsgaard, H.
[24] N. Sangouard, C. Simon, H. de Riedmatten, and N. Thyrrestrup, T. Su¨nner, M. Kamp, A. Forchel, and P.
Gisin, Rev. Mod. Phys. 83, 33 (2011). Lodahl, Phys.Rev. Lett.101, 113903 (2008).
[25] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev. [50] A.Goban,K.S.Choi,D.J.Alton,D.Ding,C.Lacrouˆte,
Mod. Phys. 82, 1041 (2010). M.Pototschnig,T.Thiele,N.P.Stern,andH.J.Kimble,
10
Phys.Rev.Lett. 109, 033603 (2012). Rev. A 65, 022304 (2002).
[51] C. L. Hung, S. M. Meenehan, D. E. Chang, O. Painter, [61] A. Auff`eves-Garnier, C. Simon, J. M. G´erard, and J. P.
and H. J. Kimble, New J. Phys. 15, 083026 (2013). Poizat, Phys.Rev.A 75, 053823 (2007).
[52] A.Goban,C.L.Hung,S.P.Yu,J.D.Hood,J.A.Muniz, [62] A.A.AbdumalikovJr.,O.Astafiev,A.M.Zagoskin,Yu.
J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. A. Pashkin, Y. Nakamura, and J. S. Tsai, Phys. Rev.
E. Chang, O. Painter, and H. J. Kimble, Nat. Commun. Lett. 104, 193601 (2010).
5, 3808 (2014). [63] I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B.
[53] P.Kolchin,N.Pholchai,M.H.Mikkelsen,J.Oh,S.Ota, Peropadre,andP.Delsing,Phys.Rev.Lett.107,073601
M. S. Islam, X. B. Yin, and X. Zhang, Nano Lett. 15, (2011).
464 (2015). [64] I. C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P.
[54] M. Hijlkema, B. Weber, H. P. Specht,S. C. Webster, A. Delsing,andC.M.Wilson,Phys.Rev.Lett.108,263601
Kuhn,and G. Rempe,Nat. Phys. 3, 253 (2007). (2012).
[55] Z.B.Chen,B.Zhao,Y.A.Chen,J.Schmiedmayer,and [65] A.F.vanLoo,A.Fedorov,K.Lalumi´ere,B.C.Sanders,
J. W. Pan, Phys. Rev.A 76, 022329 (2007). A. Blais, and A.Wallraff, Science 342, 1494 (2013).
[56] H.J. Kimble, Nature(London) 453, 1023 (2008). [66] M.Munsch,J.Claudon,J.Bleuse,N.S.Malik,E.Dupuy,
[57] N. H. Lindner and T. Rudolph, Phys. Rev. Lett. 103, J.M.G´erard,Y.Chen,N.Gregersen,andJ.Mork,Phys.
113602 (2009). Rev. Lett.108, 077405 (2012).
[58] C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, [67] M. Arcari, I. S¨ollner, A. Javadi, S. Lindskov Hansen, S.
D. Ding, M. P. van Exter, and D. Bouwmeester, Phys. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D.
Rev.Lett. 104, 160503 (2010). Song, S. Stobbe, and P. Lodahl, Phys. Rev. Lett. 113,
[59] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 093603 (2014).
2881 (1992).
[60] X. S. Liu, G. L. Long, D. M. Tong, and L. Feng, Phys.