Fermat's Library | Quantum repeaters with individual rare-earth ions at telecommunication wavelengths annotated/explained version.

Quantum repeaters with individual rare-earth ions at telecom-

munication wavelengths

F. Kimiaee Asadi, N. Lauk, S. Wein, N. Sinclair, C. O’Brien, and C. Simon

Institute for Quantum Science and Technology, and Department of Physics & Astronomy, University of Calgary, 2500 University Drive NW,

Calgary, Alberta T2N 1N4, Canada

August 30, 2018

We present a quantum repeater scheme that is

based on individual erbium and europium ions.

erbium ions are attractive because they emit

photons at telecommunication wavelength, while

europium ions oﬀer exceptional spin coherence

for long-term storage. Entanglement between

distant erbium ions is created by photon detec-

tion. The photon emission rate of each erbium

ion is enhanced by a microcavity with high Pur-

cell factor, as has recently been demonstrated.

Entanglement is then transferred to nearby eu-

ropium ions for storage. Gate operations be-

tween nearby ions are performed using dynam-

ically controlled electric-dipole coupling. These

gate operations allow entanglement swapping to

be employed in order to extend the distance

over which entanglement is distributed. The de-

terministic character of the gate operations al-

lows improved entanglement distribution rates

in comparison to atomic ensemble-based proto-

cols. We also propose an approach that utilizes

multiplexing in order to enhance the entangle-

ment distribution rate.

1 Introduction

Entanglement is a key requirement for many applica-

tions of quantum science. These include, for example,

quantum key distribution [1], global clock networks [2],

long-baseline telescopes [3], and the quantum internet

[4, 5]. However, due to transmission loss the direct

transmission of entanglement over distances of more

than several hundred kilometers is practically impos-

sible using current technology. The use of a quantum

repeater has been suggested to reduce (or eliminate)

the impact of loss in order to establish entanglement be-

tween distant locations. [6]. In many quantum repeater

schemes, entanglement is ﬁrst distributed between two

locations that are separated by a short distance, referred

to as an elementary link. Then, the range of entangle-

ment is extended to successively longer distances by per-

forming entanglement swapping operations between the

entangled states that span each elementary link. Due to

the availability and diversity of component systems and

the strong light-matter coupling oﬀered by atomic en-

sembles, many quantum repeater proposals use sources

of entanglement, ensemble-based quantum memories,

and linear optics-based entanglement swapping oper-

ations [7]. However, the success probability of linear

optics-based entanglement swapping (without auxiliary

photons) cannot exceed 50%, which has a compounding

eﬀect for more complex quantum networks. The use of

auxiliary photons to improve the entanglement swap-

ping probability is possible [8, 9], but adds complex-

ity and compounds errors, thereby restricting their use

in practice. Single-emitter-based quantum repeaters,

on the other hand, oﬀer the possibility to outperform

ensemble-based repeaters by using deterministic swap-

ping operations [10]. Impressive demonstrations of cer-

tain parts of a single-emitter quantum repeater schemes

have been performed using atom-cavity systems [11, 12],

color centers in diamond [13, 14], trapped ions [15, 16],

donor qubits in silicon [17], as well as quantum dots

[18, 19].

Over the years, crystals doped with rare earth (RE)

ions have attracted considerable attention for their use

in electromagnetic signal processing applications that

range from quantum memories [20, 21] to biological

imaging [22]. Narrow optical and spin homogeneous

linewidths, their convenient wavelengths as well as their

ability to be doped into solid-state crystals are some of

the most desired properties of RE ions. Compared to

nitrogen-vacancy centers in diamond [23] and quantum

dots [24], they also exhibit smaller spectral diﬀusion

[25–27], and have a reduced sensitivity to phonons.

Among the diﬀerent RE ions, Er

is attractive due

to its well-known optical transition (around 1.5 µm)

in the conventional telecommunication wavelength win-

dow, in which absorption losses in optical ﬁbers are min-

imal. Another unique aspect of certain RE ions is the

presence of hyperﬁne levels that feature long lifetimes,

which allows for long-term quantum state storage [28–

31]. In particular, a coherence lifetime of six hours

Accepted in Quantum 2018-08-27, click title to verify 1

arXiv:1712.05356v3 [quant-ph] 29 Aug 2018

was reported in a europium-doped yttrium orthosili-

cate crystal (

151

SiO

) [32]. Motivated by these

properties, we propose and analyze a quantum repeater

protocol that is based on individual RE ions in which

ions are used to establish entanglement over ele-

mentary links and

151

ions are employed to store

this entanglement.

One disadvantage of RE ions is their weak light-

matter coupling, which has mostly precluded their use

as single quantum emitters. However, optical detec-

tion and addressing of single RE ions has recently been

shown by multiple groups [33–37]. Moreover, very re-

cently Dibos et al. [34] demonstrated an enhancement

in the emission rate of a single Er

ion in Y

SiO

a factor of more that 300 using a silicon nanophotonic

cavity. Strong coupling of ensembles of Nd ions was pre-

viously demonstrated using nanophotonic cavities fabri-

cated from Y

SiO

[38] and yttrium orthovanadate [39]

hosts. In light of these results, for our scheme we pro-

pose to couple single RE ions to a high-ﬁnesse cavity in

order to enhance the light-matter coupling and thus to

increase the collection eﬃciency as well as the indistin-

guishability of the emitted single photons.

The paper is organized as follows. In Sec. 2 we intro-

duce our proposal and discuss the required components

as well as the underlying principles. The entanglement

distribution rates and possible implementations are dis-

cussed in Secs. 3 and 4. We conclude and provide an

outlook in Sec. 5.

2 Quantum repeater protocol

The goal of our repeater scheme is to generate entan-

glement between nodes that are separated by a long

distance by swapping entanglement that is established

between nodes that are separated by smaller distances.

Each node consists of a Y

SiO

crystal containing an

ion that features an enhanced emission rate of

its telecommunication transition due to a high-ﬁnesse

cavity, and a nearby

151

ion that acts as a quan-

tum memory due to its long spin coherence lifetime.

Entanglement is transferred, or gates are performed,

between nearby atoms using electric dipole-dipole cou-

pling [40, 41]. Our proposal has some similarity to a

scheme that involves the excitation of electron spins in

nitrogen-vacancy centers to generate entanglement be-

tween nodes, nuclear spins of carbon atoms for storage,

and gates that are performed using a magnetic dipole-

dipole coupling [42]. We emphasize that our (electric

dipole) approach to implementing gates is quite diﬀer-

ent than previous approaches, as are many of the physi-

cal properties of the system (in addition to the emission

wavelength).

In the ﬁrst step, entanglement between pairs of neigh-

boring Er

ions is created (see Fig.1 (a)) which illus-

trates this step for the i-th and i+1-th ions) by ﬁrst cre-

ating local entanglement between the spin state of each

ion and a spontaneously-emitted single photon. Then,

by performing a joint Bell-state measurement (BSM) on

photons that are spontaneously-emitted by neighboring

ions, the spin state of these ions is projected onto a

maximally-entangled state. This procedure is the same

as that used to create entanglement between two remote

nitrogen-vacancy centers in diamond [13, 43].

(i) (i+1) (i+2)(i-1)

= 1 / 2

= −1 / 2

13/2

15/2

Initialize

Entangle

Er-Er

Initialize

Map

Er-Eu

Initialize

Entangle

Er-Er

Swap

Er-Eu

Fail

= 3 / 2

= −3 / 2

Figure 1: In each cavity, the blue and black dots represent

individual

151

and Er

ions, respectively. In order to dis-

tribute entanglement to the nodes that are situated at each end

of a long channel, ﬁrst a) entanglement between two neighbor-

ing Er

ions is created via the detection of two photons; b)

the quantum state of each Er

ion is then mapped to a nearby

151

ion using an electric dipole interaction; c) using two-

photon detection, Er

ions are used again to establish entan-

glement between the other neighboring Er

ions. Finally, we

extend the entanglement distance by performing entanglement

swapping using an electric dipole interaction between nearby

ions (red circle). d) Flow-chart that depicts our scheme. e)

Simpliﬁed energy level structure of Er

and

151

ions in

the presence of an external magnetic ﬁeld. The electronic Zee-

man levels of Er

, and nuclear hyperﬁne levels of

151

are split. We encode qubit states |↑i and |↓i in the m

= −

and m



= −

and m



levels for the Er

(

151

) ion, respectively.

Accepted in Quantum 2018-08-27, click title to verify 2

Next, the quantum state of each Er

ion is mapped

to a nearby

151

ion for storage by exploiting a non-

vanishing permanent electric dipole moment, a common

feature of many RE ions that are doped into solids.

This mapping is achieved using a mutual electric dipole-

dipole interaction between close-lying Er

and

151

ions. The small nuclear magnetic moment of

151

results in a magnetic-dipole coupling that is orders of

magnitude smaller than the electric-dipole coupling,

thus our scheme allows for much shorter gate durations

in comparison to those based on magnetic interactions

[42]. Another advantage of the proposed scheme is the

ability to dynamically control the interaction optically

by bringing the ion to the excited state and back to the

ground state. This allows for the realization of deter-

ministic two-qubit gate operations.

The mapping allows the Er

ions to be re-initialized

so that new elementary links can be created between

them. Fig.1(c) illustrates this processes for the (i − 1)th

and (i)th nodes. Immediately after generating entan-

glement between (the other) neighboring Er

ions, the

entanglement distance is extended by performing en-

tanglement swapping between each of the closely-lying

and

151

ions. As a result, the outer nodes

becoming entangled. Fig. 1 (d) depicts a ﬂow chart of

our scheme. Photon loss and errors might cause some

steps in the protocol to fail.

As will be discussed in Sec. II A–C, the generation

of entanglement between Er

ions, the mapping of

entanglement to the

151

ions, and the entangle-

ment swapping steps, all rely on the detection of sin-

gle photons that are spontaneously emitted from the

ions. However, the lifetime of the

15/2

↔

13/2

telecommunication transition of Er

SiO

is rela-

tively long T

= 11.4 ms [44] with radiative lifetime

being even longer T

rad

= 54 ms [45], which necessi-

tates the need for a high-ﬁnesse cavity to enhance the

emission rate. Nonetheless, the cavity will also signiﬁ-

cantly increase the probability of collecting the emitted

photons as compared to emission into free space. We

emphasize that the repetition rate of the protocol (i.e.,

the number of attempts of the protocol per unit time)

is limited by the communication time between distant

nodes, which means that there is no need for very fast

emission. Next we discuss each of the steps of the pro-

tocol in detail.

2.1 Entanglement generation

Our scheme considers several remote cavities that each

contain a single pair of close-lying Er

and

151

ions. Note that there may be other RE ions within each

cavity, but we assume that we can address a single such

close-lying pair. See also section4 below. An externally-

applied magnetic ﬁeld splits the degenerate electronic

ground levels of Er

via the Zeeman eﬀect. We refer

to the resultant m

= −

and m

Zeeman levels

as the qubit states |↑i and |↓i, respectively (see Fig. 1

(e)).

To generate entanglement between distant Er

ions

that are separated by a long distance L

, we follow

the scheme of Barret and Kok [13, 43] (see also Fig.

1 (d)). First, each Er

ion is prepared in one of the

qubit states (here m

= −

; denoted |↑i). After this

initialization step, a

microwave (MW) pulse rotates

each Er

ion into a superposition of |↑i and |↓i states.

The application of a brief laser pulse that is resonant

with the |↑i ↔ |e

i transition, followed by spontaneous

emission, will entangle each qubit state with the emit-

ted photon number. That is, when the qubit state is

|↑i (|↓i) there will be 1 (0) emitted photon(s). The

spontaneously-emitted photons are then directed to a

beam splitter located in between the ions using opti-

cal ﬁbers. A single-photon detection at one of the two

beam splitter output ports projects the Er

ions onto

an entangled state.

A possible loss in the ﬁber can lead to a situation

where both Er

ions emit a photon but only one pho-

ton is detected while the other is lost. In this case, the

ions are left in a product state rather than an entangled

state. To exclude this possibility, immediately after the

ﬁrst excitation-emission step of each Er

ion, a π MW

pulse inverts each qubit state. Then a second excitation

pulse is applied. The detection of two consecutive single

photons at the beam splitter will leave the qubits in an

entangled state:

|ψi

√

(|↑↓i ± |↓↑i). (1)

Here the + (-) sign corresponds to the case in which the

same (diﬀerent) detector(s) received a photon.

2.2 Controlled logic

After successful entanglement of the two distant Er

ions we transfer this entanglement to the neighboring

151

ions for a long-term storage. Eﬃcient entan-

glement mapping between neighboring rare-earth ions

can be employed by performing CNOT gate operations

and single qubit rotations and read-out. Here we ﬁrst

explain the underlying mechanism and general scheme

for how to implement a CNOT gate in our system, and

in Sec.2.3 we will discuss in more detail how to swap

entanglement between the ions using this gate. Due to

a lack of site symmetry when doped into a crystal, a

RE ion can have a permanent electric dipole moment

that is diﬀerent depending on whether the ion is in its

ground or optically excited state. The diﬀerence in the

Accepted in Quantum 2018-08-27, click title to verify 3

1 2 34

e, Eu

↓, Eu

e, Er

↓, Er

Figure 2: General pulse sequence of π-rotations to perform

a controlled logic gate between nearby Er

and

151

ions. The numbers indicate the sequential time ordering of

the pulses. For this gate, the Er

spin state acts as the con-

trol qubit and the

151

spin state acts as the target. A π

pulse excites the Er

ion if it is in the state |↑i. When this

occurs, pulses 2, 3 and 4 are not resonant with the

151

ion, leaving its state unaﬀected. Pulse 5 then brings the Er

ion back to its original state. On the other hand, when Er

in the state |↓i, pulses 1 and 5 will be ineﬀective and hence it

will remain in the ground state. Instead pulses 2, 3 and 4 will

now be resonant with the

151

ion and (optically) swap its

spin-state

permanent dipole moments aﬀects the optical transition

frequency of other nearby RE ions via the Stark eﬀect

due to a modiﬁed local electric ﬁeld environment. It is

possible to dynamically control the shift in the transi-

tion frequency of one ion by optically exciting its neigh-

boring ion. Based on this interaction, we can perform a

controlled-NOT (CNOT) operation between nearby RE

ions [46]. The modiﬁcation of the transition frequency

∆ν of a

151

ion by an Er

ion due to the mutual

electric dipole-dipole interaction is given by [47]:

∆ν =

∆µ

4π

((ˆµ

·ˆµ

) − 3 (ˆµ

·ˆr) (ˆµ

·ˆr)) ,

(2)

where r is the distance between the ions, and ∆µ is the

change of the permanent electric dipole moment of each

ion, h is the Planck constant, 

is vacuum permittivity,

and  is the dielectric constant.

To perform a CNOT gate between the nearby Er

(control) ion and Eu (target) ion, a sequence of ﬁve

π pulses is applied (see Fig.(2)). First a π pulse is ap-

plied to the Er

ion on resonance with the |↑i ↔ |e

transition. From here, two cases must be considered:

either the state of the Er

ion is |↑i or it is |↓i.

If the Er

ion is in state |↑i, it will be excited by

pulse 1. This changes the permanent electric dipole

moment of the Er

ion, and thus its local electric ﬁeld.

Consequently, the transition frequency of the nearby

151

ion will be shifted by ∆ν. For the case that

this frequency shift is large enough (that is, the

151

ion is suﬃciently close to the Er

ion), the

151

ion will be unaﬀected by pulses 2, 3 and 4, thereby

remaining in its initial ground state (see Sec. 4.2 for

more discussion). Finally, pulse 5 will bring Er

ion

back to its initial state. Hence, in this case, the pulse

sequence does not modify the initial state of the ion pair

system.

On the other hand, if the Er

ion is initially in |↓i,

then pulses 1 and 5 will have no eﬀect on the Er

ion.

Since the Er

ion will not be excited, optical pulses

2, 3 and 4 will be resonant with the transitions of the

151

ion, and the pulse sequence will optically ﬂip

the two spin states of the

151

ion.

2.3 Entanglement mapping and distribution

Both the ground and excited states of

151

have

three doubly-degenerate nuclear hyperﬁne levels. With

the application of a magnetic ﬁeld, each doublet



= ±

, ±



will be split. For our proposal, we

denote the m

= −

and

hyperﬁne levels [32] as the

151

ion qubit states, |↑i and |↓i, respectively (see

Fig. 1 (e)). To map the state of each Er

ion onto a

nearby Eu ion, we ﬁrst perform a CNOT gate between

them. In our scheme, the

151

ion is initially pre-

pared in one of the ground state levels (here |↑i) using

optical pumping. In this special case, pulse 4 does not

aﬀect the system and can be neglected. This reduces

the total gate time and improves the state mapping ﬁ-

delity (see Sec. 4.2).

After considering the phases that are acquired by

performing the CNOT gate on the ions in neighbor-

ing nodes (labeled here as 1 and 2 instead of (i) and

(i + 1)), the ﬁnal state is

|ψi

Er,Eu

√



|↑↓i

,Er

|↑↓i

,Eu

±e

iφ

|↓↑i

,Er

|↓↑i

,Eu



, (3)

where

φ = (ω

↓,Er

− ω

↓,Er

)τ + (ω

− ω

)τ

+ (ω

↓,Eu

− ω

↓,Eu

)(τ

+ τ

)

+ k

↓Eu

− k

↓Eu

, (4)

and x is the distance the photon travels between each

151

ion and the beam splitter, k

↓Eu

↓,Eu

is the

wavenumber, τ

is the time elapsed after application of

the j

pulse and τ is the total time duration that is

needed to perform a CNOT gate. Since these phases

are known, they can be compensated by applying local

operations on the ions.

To conclude the mapping step, a

microwave (MW)

pulse is applied to rotate each Er

qubit. This is fol-

lowed by a state measurement of both nearest-neighbor

ions, see also Sec. IV below. This projects the

Accepted in Quantum 2018-08-27, click title to verify 4

151

ions onto an entangled state. Depending on the

outcome of these measurements, the entangled state be-

tween remote

151

ions would be |ψ

i or |ψ

−

i. For

example, in the case that we begin with |ψi

(given

in Eq. 1), after performing the gate, MW rotation, and

measurement, if both Er ions are found in the state

|↑i, the entangled state between remote

151

ions is

√



|↑↓i

,Eu

+ |↓↑i

,Eu



Once entanglement is established in neighboring ele-

mentary links, we perform a joint measurement on both

ions at each intermediate node to distribute entangle-

ment (see Fig. 1 (c)). In our scheme, it is possible to

perform entanglement swapping deterministically using

the permanent electric dipole-dipole interaction.

To perform the desired entanglement swapping, ﬁrst

a CNOT gate is applied in which Er

serves as the con-

trol qubit and

151

as the target qubit (similarly as

before). Here, both ions are in a superposition state

and so all 5 pulses of the CNOT gate sequence are re-

quired. Then the Er

ion is measured in the diagonal

(X) basis. Next, another CNOT operation is performed

but now the target and control qubits are exchanged.

Finally the Er

ion is measured in the logical (Z) ba-

sis. This ‘reverse’ CNOT gate is performed in order to

avoid directly measuring the spin state of

151

ion

optically (the optical lifetime of

151

is longer than

the Er

spin coherence lifetime). Fig.(2) shows the

pulse sequence needed to perform the ﬁrst CNOT gate

between the ions.

Based on the outcomes of the measurements on each

ion, the outer nodes will be projected onto one of

the four Bell states. To be more precise, when perform-

ing entanglement swapping between Er

and Eu

, if only

one of the state measurements of Er

is |↑i, the state of

i−1

and Eu

i+1

will project onto the |ψ

i or the |ψ

−

Bell state. On the other hand, if both measurements of

are |↑i or both |↓i, the state will project onto the

|φ

i or the |φ

−

i Bell state.

To verify or utilize the entanglement that is gener-

ated between the distant ions, a measurement must be

performed on the endpoint nodes. Our protocol results

in an entangled state of the Er

ion at the ﬁrst node

with the

151

ion at the last node. A measurement

of the Er

ion can be performed by direct optical exci-

tation. A measurement of the

151

ion spin can be

performed by using the nearby Er

as a readout ion.

This is done by mapping the

151

state to Er

using gate operations (also see section 4.3 for further

details).

3 Entanglement generation rates and

multiplexing

In our scheme, the success probability of generating an

entangled state between two neighboring Er

ions that

are separated by a distance L

is p

, where

= e

−

att

is the transmission probability of a photon

through optical ﬁber and L

att

≈ 22 km (which cor-

responds to a loss of 0.2 dB/km), η

is the detection

eﬃciency, and p is the success probability of emitting a

single photon into a cavity mode. Similarly, the success

probability of mapping the state of an Er

ion onto

a nearby

151

ion is p

≈ pη

, because it requires

one optical read-out of each Er

ion. Therefore, the

average time to generate entanglement in an elementary

link and perform the state mapping steps is

hT i





, (5)

where c = 2 × 10

is the speed of light in ﬁber.

The average time that is required to distribute entan-

glement over two neighboring elementary links, which

corresponds to a distance L = 2L

, can be estimated

as follows. First, entangled pairs of

151

ions are

generated over an elementary link and, once successful,

ions are then generated in a neighboring elemen-

tary link. Thus, the probability of establishing entan-

glement for both links is





−1

, (6)

where P

= p

and P

= p

represent the success

probability of generating entanglement between Er

and

151

ions, respectively. Then, entanglement is

extended by performing a BSM between the close-lying

and

151

ions at the center node. The success

probability for the entanglement swapping step is p

≈

because it requires two optically-induced spin read-

outs of an Er

ion. Consequently, the average time to

distribute entanglement over a distance 2L

hT i

. (7)

Accordingly, the average time to distribute an entan-

gled pair over a distance L = 2

, with n denoting the

number of nesting levels, is [7, 10]:

hT i





n−1

. (8)

The factor 3/2 for each of the next nesting levels, which

is a good approximation for the exact result [7], can be

understood in following way.

Accepted in Quantum 2018-08-27, click title to verify 5

In contrast to the ﬁrst nesting level, a successful en-

tanglement distribution for higher nesting levels does

not require waiting for a success in one link before the

establishment of entanglement in another link can be

attempted; rather, the establishment of entanglement

can be attempted in both links simultaneously. For ex-

ample, if a goal is to distribute entanglement over the

distance 4L

, entanglement must be generated in two

neighboring links each of length 2L

before entangle-

ment swapping is performed. If the average waiting

time for a success in one link of length 2L

is hT i

entanglement will be established in one of two links af-

ter hT i

/2. Then, after another hT i

time dura-

tion, entanglement will be established in the other link.

Hence, the average time to establish entanglement in

the two neighboring links is 3 hT i

/2. The same ar-

guments hold for the next nesting levels, resulting in a

factor





n−1

for n nesting levels.

Multiplexing can be employed to signiﬁcantly en-

hance the rate of entanglement distribution. Referring

to the encoding of several individual qubits, each into

their own distinguishable modes, multiplexing has been

utilized for some quantum repeater proposals [7, 48–

50]. As outlined in Fig. 3, we consider an array of

m cavities in which each cavity emits a photon that

features a distinguishable resonance frequency from the

rest. This can be accomplished using frequency trans-

lation (see Sec. IV for more details). Using a coupler

device, each photon is directed into a common ﬁber and

is directed towards a BSM station that is composed of

similarly-designed decouplers and single-photon detec-

tors. Beneﬁting from the distribution of entanglement

into m parallel modes, the probability that at least one

entangled state is distributed over the entire channel is

1−(1−P

)

, which can be made unity for a suﬃciently

high m. Here, P

= (2/3)

n−1

is the success prob-

ability of distributing entanglement over a distance L

using a qubit that is encoded into a single mode. Fur-

ther details of this set-up are described in Sec.4.4.

The entanglement distribution rate of our scheme is

plotted as a function of distance for n = 3 in Fig. 4

and compared to that employing the well-known DLCZ

scheme [51], which uses ensemble-based memories, as

well as that using direct transmission with a single-

photon source which produces photons at 10 GHz. This

10 GHz photon rate is an optimistic rate for the direct

transmission of photons. The rate that we assume for

a single photon source is much faster than the rate for

the proposed repeater because they have diﬀerent lim-

itations. Even though the photon rate of our scheme

(which depends on the cavity characteristics and the

optical lifetime of the Er

ion) is much lower, the time

scale for the repeater is actually determined by the

communication time L

/c which is even longer. Note

MOTIVATION

Why+using+rare-earth+ions?+

• Storing(and(retrieving(photons(at(conventional(telecom(wavelength(

•

Long(spin(coherence(times,(up(to(6(hours,(are(feasible(

• Provide(scalab le(architecture

PROPOSAL

TEXT+BOXES

Contact+Information+

Email:((faezeh.kimiaeeasadi@ucalgary.ca(

Email:((nikolai.lauk@ucalgary.ca

Quantum+repeaters+with+single+rare-earth+ions+at+

telecommunication+wavelengths

Faezeh+Kimiaee+Asadi,+Nikolai+Lauk,+Stephen+Wein,+Chris+O’Brien,+Neil+Sinclair+and+Christoph+Simon

Department(of(Physics,(Institute(for(Quantum(Science(and(Technology,(University(of(Calgary,(Canada

1.+Creating+heralded+entanglement+between+distant+Er+ions+using+Barrett-Kok+scheme++

2.+Mapping+Er+states+to+nearby+Eu+ions+using+dipole-dipole+interaction

3.+Er+ions+are+used+again+to+create+new+entangled+pairs

4.+Performing+deterministic+entanglement+swapping+between+Er +and+Eu+ions+

TEXT+BOXES

ENTANGLEMENT+GENERATION+RATES+AND+MULTIPLEXING

Tel ecom(

C-band

1535(nm

15/2

=1/2

= 1/2

| "i

| #i

13/2

cavity(enhanced(

transition

|ei

= 3/2

=3/2

| "i

| #i

Erbium Europium

|ei

| #i

| "i

⌫

µ

4⇡✏✏

(ˆµ

· ˆµ

 3(ˆµ

· ˆr)(ˆµ

· ˆr))

dipole-dipole(shift

⌫

10 MHZ

10 kHZ

1nm

10 nm

State(after(performing(CNOT(gate(

and(measuring(Er(ions(in(X-basis

| i = |"#i

,Er

±|#"i

,Er

| i = |"#i

,Eu

±|#"i

,Eu

|ei

| #i

| "i

⌫

• Perform(CNOT(gate(using(Er(as(

control(qubit(

• Measure(Er(in(X-basis(

• Perform(CNOT(gate(using(Eu(as(

control(qubit(

• Measure(Er(in(X-basis

Procedure(analogous(to(Step(1.

• Initialize(Er(state(

• Excite(and(let(emit(a(photon(

• Flip(Er(qubit(and(excite(again

State(after(detecting(two(photons

General+scheme+to+generate+and+distribute+entanglement+

Initialize(

Entangle(

Er-Er(

Map(

Er-Eu

Initialize(

Entangle(

Er-Er(

Swap

Fail

Possible+imperfections:+

transmission(efficiency((

detector(efficiency((

probability(of(emittin g(into(cavity(mode(

small(deviation(from(the(rotation(angle



⌘

Er-Er

⇡

(1 

3

)⌘

⌘

Probability(to(entangle(Er(ions

Probability(to(map(Er(state(to(Eu

Entanglement(swapping

⇡ (1 



)p⌘

⇡ (1 

⌘

Using(these(quantities(we(can(calculate:

1.+Average+time+required+to+entangle+two+Eu+ions+separated+by+the+distance+

hT i

✓

Er-Er

◆

Eu-Eu

2.+Average+time+required+to+distribute+entanglement+over+the+distance++

hT i

✓

◆

✓

Er-Er

Eu-Eu



(1  P

Er-Er

)(1  P

Eu-Eu

)

Er-Er

+ P

Eu-Eu

 P

Er-Er

Eu-Eu

◆

3.+Average+time+required+to+distribute+entanglement+over+the+distance++

hT i

✓

◆

n1

Enhancing+the+rate+by+parallelizing+and+multiplexing

Success(probability(

for(m(parallel(links

P =1 (1  P

)

Rate+comparison:+

• direct(transmission(

• proposed(scheme(with(8(

elementary(links(and(

• parallelized(version(for(m=100((

Here(we(assumed(

p = ⌘

= 90%.

OUTLOOK

• Quantify(decoherence(effects(

• Calculate(overall(fidelity(of(the(distributed(entangled(state(

• Using(proposed(scheme(it(might(be(possible(to(design(a(

scalable(quantum(computing(architecture(for(rare-earth(ions

m1

MUX

DEMUX

Figure 3: Our multiplexed scheme consists of 2

+1 nodes that

span the total channel distance L. Each node consists of an

array of m cavities that emit photons which feature diﬀering

carrier frequencies. The carrier frequency of each photon is de-

termined by a frequency translation device. A coupling element

(COUPLER) ensures that each photon traverses a common

channel to a Bell-state measurement station that consists of

a beam splitter, decoupling element (DECOUPLER), and 2m

single photon detectors. This set-up allows m entanglement

generation protocols to be operated in parallel.

that the direct transmission scheme can also be inter-

preted as the Pirandola bound [52] for a repetition rate

of 10/1.44=6.9 GHz. The performance of our proto-

col and the DLCZ scheme with m = 100 multiplexed

channels is also shown Fig. 4. For more information

on other approaches see Ref. [7]. In this review paper,

the multiplexed DLCZ outperformed the other repeater

protocols considered (see Fig. 18 of ref.[7]). As will

be discussed in Sec. IV B, the use of even more than

m = 100 parallel spectral channels is possible.

Fig. 4 shows that our use of deterministic gates is

advantageous for the rate. The scaling of the DLCZ

scheme with distance is slightly better than that of our

scheme due to the requirement of detecting only one

photon for each elementary entanglement creation step

(rather than two photons for our scheme). However,

this comes at the signiﬁcant expense of requiring phase

stabilization for the long-distance ﬁber links.

4 Implementation

For our scheme, we consider ion beam-milled Y

SiO

photonic crystal cavity systems that have been weakly-

doped with Er

and

151

ions [38, 39, 53]. Co-

doped crystals may be grown from the melt [53], or in-

dividual RE ions may be implanted into single Y

SiO

crystals [54]. After a milling step, the output of the

cavity can be coupled to an optical waveguide using,

e.g., microscopy [55], bonding [56] or a pick-and-place

technique [57], with the latter having been used to het-

erogeneously interface InAs/InP quantum dots with Si

waveguides. Despite the lack of on-demand control of

Accepted in Quantum 2018-08-27, click title to verify 6

Figure 4: A comparison of the entanglement distribution rate

for various schemes. Direct transmission scheme using a 10

GHz single-photon source (E) is compared with original DLCZ

protocol (A) and our scheme (B) using 8 elementary links, each

of length L

. Also shown are the rates that correspond to the

multiplexed versions of DLCZ (C) as well as our scheme (D)

with m = 100 using the same number of elementary links. We

assume that the detection eﬃciency and the success probability

of emitting a single photon by an ion is p = η

= 0.9, and in the

case of DLCZ, a storage eﬃciency of η

= 0.9. The repetition

rate for each repeater protocol is set by the communication

time L

/c.

the position and relative orientation of each ion [54], a

suitable Er-Eu ion pair can be identiﬁed by performing

laser-induced ﬂuorescence spectroscopy. This involves

exciting each Er

(

151

) ion with a laser and mea-

suring the spectral shift of the resultant ﬂuorescence

from other

151

(Er

) ions due to the aforemen-

tioned electric-dipole coupling (see Sec. 2.3) [53]. Note

that measurements using a 0.02%Er:1%Eu:Y

SiO

bulk

crystal revealed the optical transition frequency of sets

of Er

ions that lie within approximately one nanome-

ter from adjacent Eu

ions [53]. These results suggest

that the transition frequencies of suitable Er-Eu pairs

can be rapidly distinguished from other spectator ions.

The magnetic ﬁeld applied in the D

− D

plane at

135 degrees relative to the D

axis results in the decay of

the excited Er

ion back into the initial spin state via

spontaneous emission with a probability of higher than

90%[58], and a Zeeman-level lifetime of about 130ms

was measured for a magnetic ﬁeld of 1.2 mT at 2.1 K

temperature[58]. At large external magnetic ﬁelds of 1

T or more and temperatures below 3 K, one-phonon di-

rect process is the dominant spin-relaxation mechanism

and the temperature and magnetic ﬁeld dependence of

the relaxation rate could be approximated by [59]:

R(B) = R

+ α

coth(

gµ

2kT

) (9)

where α

is the anisotropic constant, µ

is the Bohr

magneton, and k is the Boltzmann constant. The ﬁeld

independent contribution R

can be attributed to cross-

relaxations with paramagnetic impurities in the crystal

and hence depends strongly on the crystal purity. Ex-

trapolating from Refs [58, 59], the spin relaxation time

would be about 40ms at 20mK for an external magnetic

ﬁeld of 1T at the 135

◦

in D

− D

plane. At this mag-

netic ﬁeld, however, the splitting of the Zeeman levels

would be too large to address it with microwave pulses

and optical Raman pulses should be applied instead.

For an ensemble-doped Er

SiO

the spin coher-

ence lifetime at mK temperatures and external mag-

netic ﬁelds of few hundred mT can be as short as ∼ 7 µs

[60]. However, for an ensemble-doped Er

SiO

crystal spin ﬂip-ﬂop processes are one of the main

sources of decoherence, the spin coherence lifetime of

a single Er

ion in Y

SiO

is largely determined by

spin-spin magnetic dipole interactions. The magnetic

moments of the constituent spins of Y

SiO

are small:

−0.137µ

, −0.5µ

, and −1.89µ

for

Si, and

O respectively. Compared to yttrium ions,

Si and

O have low isotopic natural abundances, so we assume

the contribution of these isotopes to be negligible and

only consider the Er-Y interactions. In a large enough

magnetic ﬁeld (a few hundred milli-tesla or more) com-

pared to the Er-Y coupling strength, the magnetic mo-

ment of Er

will detune the closest-proximity Y ions

from resonance with those further away. This well-

known eﬀect referred to as the “frozen core” has been

observed and results in weaker decoherence of the Er

ion by nearby Y ions [61–63].

Hence, for a single Er

in Y

SiO

, the presence of a

strong magnetic ﬁeld may increase the spin coherence

lifetime into the milliseconds range. To further increase

the spin coherence lifetime, dynamical decoupling im-

mediately after the second optical excitation step is

necessary. While this has not been demonstrated in

SiO

, it is a widely-employed method, and has

been used to extend coherence lifetimes in Ce-doped yt-

trium aluminum garnet [64] as well as nitrogen-vacancy

centers [42].

Spin polarization of up to 90% in Er

SiO

has

been realized by using stimulated emission and spin-

mixing methods [65]. The eﬃciency of spin polarization

is determined by a competition between decay of the

ground-level population relative to the optical pumping

eﬃciency of the ion. Since the latter will be enhanced

due to the cavity-induced decay rate, we expect a near-

unity Er

-spin polarization for our scheme. Since the

ground state lifetime of

151

is several hours, spin

polarization can be achieved by performing continuous

optical pumping of all but one of the ground states.

In Y

SiO

, RE ions can occupy two crystallographi-

cally inequivalent sites with C

symmetry. Due to this

Accepted in Quantum 2018-08-27, click title to verify 7

lack of symmetry, the orientation of the dipole moments

and their magnitudes, and hence their dipole-dipole in-

teraction strengths, are unknown. Previous measure-

ments of the linear Stark shift of Er

SiO

[66] al-

low the calculation of the projection of the electric-

dipole moment diﬀerence onto the direction of the

externally applied electric ﬁeld to be approximately

0.84 × 10

−31

Cm [67]. The dipole moment diﬀerence

for

151

SiO

can be as high as ∆µ

= 0.81 ×

−31

Cm [68], resulting in the shift of the transition fre-

quency of 10 and 0.01 MHz for r

= 1 and 10 nm, re-

spectively. As a comparison, the magnetic dipole-dipole

interaction between Er

magnetic moment, which can

be as high as µ

= 14.65 µ

[69], and

151

Eu magnetic

moment with its intrinsic value of µ

= 3.42 µ

[70]

is much weaker and amounts to 363 and 0.363 kHz for

= 1 and 10 nm, respectively.

4.1 Cavity

The cavity serves three main purposes in this proposal.

It improves the quantum eﬃciency, enhances the single-

photon indistinguishability, and increases the rate of

emission. To achieve these beneﬁts, the cavity

must provide a signiﬁcant Purcell enhancement to the

i −→ |↑i transition of Er

In the context of RE ions, Purcell factors of several

hundred have been achieved [34, 38], and up to 10

seems to be a reasonable goal [71]. The Purcell factor

can be written: P = (γ/γ

)C, where 1/γ is the excited

state lifetime, γ

is the radiative decay rate, and C is

the cavity cooperativity. The cavity-enhanced quan-

tum eﬃciency (probability of emitting a photon into

the cavity mode) is then given by: p = ηP/(1 + ηP ),

where η = βγ

/γ is the Er

spin-conserving quantum

eﬃciency, and β is the probability of an excited ion

to relax into the initial spin state via the spontaneous

emission. For Er

SiO

, β = 0.9, γ = 2π × 14 Hz,

and γ

= 2π × 3 Hz [45], resulting in η = 0.19. For

P = 100, a cavity quantum eﬃciency of p = 0.95 is pos-

sible. Increasing the Purcell factor to P = 1000 allows

p = 0.995.

The single-photon indistinguishability is a metric

that quantiﬁes the quality of interference between pho-

tons that originate from the same quantum emitter.

Without a cavity, the single-photon indistinguishabilty

can be deﬁned as I

= T

/2T

[72]. For an ensemble

of 0.0015% Er

SiO

we have T

= 11.4ms [44] and

optical T

is around 200µs at 1 T and a temperature of

a few K [73]. This would imply I

= 0.009 without a

cavity, which would require signiﬁcant spectral ﬁltering

to achieve successful entanglement generation. With

a cavity, the single-photon indistinguishability can be

approximated by: I

= (1 + ηP )/(ζ + 1 + ηP ), where

ζ = 2T

= 2T

−1 is the dephasing ratio. There-

fore, with Purcell factors P = 1000 and P = 20, 000 the

single-photon indistinguishability would be I

= 0.63

and I

= 0.97, respectively. In a large magnetic ﬁeld,

however, T

= 4 ms was measured at a few K [73]. For

an ensemble-doped Er

SiO

spin ﬂip-ﬂop processes

are the dominant decoherence mechanism and applying

a large magnetic ﬁeld can freeze these processes. As a

result we have a long optical T

. The ﬂip-ﬂop process

can also be suppressed by reducing the Er

concentra-

tion. Therefore in our system, where we assume very

low dopant concetration, we still can expect to have op-

tical coherence times of the order of a few ms even at

lower magnetic ﬁelds (hundreds of mT). With P = 100

and T

= 4ms, I

= 0.82 is possible, which could be fur-

ther improved by attempting to spectrally ﬁlter the nar-

row 2π × 1.4 kHz bandwidth photons. With P = 1000,

= 0.98 could be achieved without spectrally ﬁltering

the 2π × 14 kHz photons (corresponding to a duration

of 11 µs).

4.2 CNOT gate

To perform the CNOT gate, it is necessary for Er

remain excited for a time that is long enough to apply

three π-pulses to

151

. This implies that each Er

ion must be excited to a diﬀerent Zeeman level (|e

than the level that is coupled to the cavity (|e

i). This

can be done if the Zeeman splitting between |e

i and

i is much larger than the cavity linewidth. In this

section, for Er

we use |ei

= |e

To quantify the ﬁdelity of the CNOT gate and state

transfer procedures, we use a model nine-level system

where both Er

and

151

are represented by three-

level systems with two lower-energy spin states (|↑i, |↓i)

and a single excited state (|ei) (see Fig. 2). We use the

dipole-interaction Hamiltonian:

H = ∆νσ

Er↑

−

Er↑

Eu↑

−

Eu↑

k,l

Ω

k,l



k,l

+ σ

−

k,l



(10)

where ∆ν is the dipole interaction strength and Ω

k,l

the Rabi frequency for transition {k, l} for l ∈ {↑, ↓},

k ∈ {Er, Eu}. Without loss of generality, we choose

∆ν > 0 and Ω

k,l

> 0. The master equation is

˙ρ = −i[H, ρ] +

k,l

D(σ

−

k,l

)ρ +

D(σ

k,l

−

k,l

)ρ

D(σ

z,k

)ρ,

(11)

where D(σ)ρ = σρσ

†

−σ

†

σρ/2−ρσ

†

σ/2. The operators

are deﬁned as σ

k,l

= |eihl|

, σ

−

k,l

= (σ

k,l

)

†

, and σ

z,k

(|↑ih↑|

−|↓ih↓|

)/2. The rate γ

k,l

is the decay rate for

Accepted in Quantum 2018-08-27, click title to verify 8

transition {k, l}, γ

is the optical pure dephasing rate,

and χ

is the spin decoherence rate for ion k.

To solve for the Er-Eu state after applying the ﬁve-

pulse CNOT gate sequence, we ﬁrst assume that each

π-pulse is a square pulse and that they are applied to the

system sequentially with no time delay between pulses.

In this case, the time taken to apply each π-pulse is

given by T

k,l

= π/Ω

k,l

. We also assume that Ω

k,l

and

∆ν are much larger than any dissipative rate so that we

can treat the dissipation perturbatively.

We then deﬁne the zero-order (reversible) superop-

erator L

where L

ρ = −i[H, ρ]. Then we deﬁne the

ﬁrst-order (irreversible) perturbation superoperator L

as L

ρ = ˙ρ − L

ρ. From this, we deﬁne the rotation

superoperator R

k,l

(θ) corresponding to the pulse where

Ω

k,l

6= 0 and Ω

= 0 for all k

6= k and l

6= l. This

superoperator is given by

k,l

(θ) = e

Ω

k,l

1 +

Ω

k,l

dτe

−L

, (12)

which is accurate to ﬁrst-order in γ

, γ

, χ



Ω

k,l

, ∆ν for all k

and l

To simulate the entanglement swapping between

151

and Er

, we begin with an initial superpo-

sition state ρ

where |ψi = (|↑i + |↓i)

(|↑i + |↓i)

and ρ

= |ψihψ|. Then after applying the CNOT pulse

scheme, the ﬁnal state is

ρ = R

Er,↑

(θ)R

Eu,↑

(θ)R

Eu,↓

(θ)R

Eu,↑

(θ)R

Er,↑

(θ)ρ

where ideally θ = π; however, we set θ = π +  to

simulate a π-pulse with a small rotation deviation || 

π caused by a non-ideal pulse area. The total time taken

to apply this sequence is T

gate

= 2T

Er,↑

+2T

Eu,↑

Eu,↓

For a high ﬁdelity gate, the pulse durations must be

small so that the gate is fast and qubit states do not

dephase during the sequence. This favors larger Rabi

frequencies. On the other hand, it is also necessary that

the three Eu pulses do not excite the

151

spin state

if Er

is excited. One way to achieve this is to set Ω

Eu,l

to satisfy the detuning condition Ω

Eu,l

/∆ν

 1 so that

there is little chance for oﬀ-resonant excitation. How-

ever, since ∆ν cannot be arbitrarily large (the separa-

tion between Er and Eu cannot be arbitrarily small), the

detuning criteria necessitates a very slow gate, which

cannot provide a high ﬁdelity.

Alternatively, the detuning condition can be circum-

vented if Ω

Eu,l

is chosen so that the Eu pulses per-

form an eﬀective 2π rotation on the Eu spin state when

Er is excited, but still perform a π-pulse when Er is

not excited. This can be accomplished by requiring

that Ω

Eu,l

satisﬁes the eﬀective Rabi frequency relation

∆ν

+ Ω

Eu,l

= 2Ω

Eu,l

. This sets Ω

Eu,l

= ∆ν/

√

3. As

a consequence of ﬁxing the Rabi frequency, an accurate

characterization of the dipole interaction strength for

each Er-Eu pair is necessary to achieve a high ﬁdelity.

This is because any mischaracterization δν from the

true value ∆ν − δν will cause a deviation from the de-

sired 2π rotation. To account for this, we also consider a

perturbation of the ﬁdelity for deviation ∆ν → ∆ν −δν

where we assume |δν|  ∆ν ∝ Ω

k,l

The eﬀective 2π pulse leaves a relative phase between

the

151

ground-state spins of ϕ = −π(2 −

√

3)/2.

If Er

is in |↓i, then only π-rotations are applied to

151

. Hence, in this case, there is no acquired rel-

ative phase from the detuned pulses. However, if the

state is |↑↓i,

151

will be aﬀected by the third pulse

performing an eﬀective 2π rotation, and so |↑↓i will ac-

quire a relative phase of ϕ. Likewise, if the state is

|↑↑i,

151

will be aﬀected by the second and fourth

pulses performing an eﬀective 2π rotation, and so |↑↑i

will acquire a relative phase of 2ϕ. Hence, in the ab-

sence of dissipation and using perfect square π-pulses,

the expected ﬁnal state is

|ψ

i =



|↓↓i + |↓↑i+ e

iϕ

|↑↓i + e

i2ϕ

|↑↑i



(13)

Since the expected acquired phase ϕ = −π(2 −

√

3)/2

is known, and independent of the dipole interaction

strength ∆ν, it can be tracked or corrected and so we

use |ψ

i as the ﬁnal state when calculating the ﬁdelity.

We use the above expressions for ρ and |ψ

i to com-

pute the ﬁdelity F

CNOT

= |hψ

|ρ |ψ

i|. For simplicity,

we choose to set Ω

Er,l

= Ω

Eu,l

= Ω = ∆ν/

√

3; however,

Ω

Er,l

is not restricted by the dipole interaction strength

and could be made larger than Ω to further decrease

the total gate time and increase ﬁdelity. The solution

CNOT

to ﬁrst-order in γ

k,l

, γ

, χ

 Ω ∝ ∆ν for all

k, l and second-order in   π and ξ = δν/∆ν  1 is

CNOT

' 1 − T

CNOT

Γ − 

−

13π

ξ −

43π

128

(14)

Here T

CNOT

= 5π/Ω = 5π

√

3/∆ν is the total gate time

and Γ is the eﬀective dissipation rate:

Γ '

[31γ

+ 17γ

+ 8χ

+ 11γ

+ 8γ

+ 17χ

] ,

(15)

where we deﬁne γ

= γ

k↑

+ γ

k↓

. To obtain the reverse

CNOT ﬁdelity, it is only necessary to swap the dissipa-

tive parameters for Er

and

151

For a dipole interaction strength of ∆ν = 2π × 46

kHz corresponding to an Er-Eu separation of about 6

nm, the CNOT gate time can be as small as T

CNOT

94 µs. To estimate the ﬁdelity, we use the parameters

= 2π × 3 Hz, γ

= 2π × 8 Hz, γ

= 2π × 1.3

Hz, and γ

= 2π × 19 Hz estimated from Ref. [45];

Accepted in Quantum 2018-08-27, click title to verify 9

also χ

' 2π × 80 Hz and χ

' 0. In this case,

for a small π-pulse over-rotation of  ' π/64 and a

dipole interaction strength over-estimation of 2% (ξ '

0.02), the ﬁdelity is F

CNOT

= 0.986. The reverse CNOT

ﬁdelity is slightly smaller due to the spin dephasing of

Er: F

R-CNOT

= 0.980.

This method can also be used to compute the ﬁdelity

of state transfer from Er

151

. In this case,

we remove the fourth pulse (R

Eu,↑

) and use the initial

state |ψi = (|↑i + |↓i)

|↑i

. The expected ﬁnal state

is |ψ

i = |↓↓i + e

i2ϕ

|↑↑i. In this case, the ﬁdelity is

similar to the CNOT gate (Eq. (14)):

' 1 − T

Γ −



−

3π

ξ −

21π

256

(16)

where T

= 4π/Ω = 4π

√

3/∆ν. The eﬀective dissipa-

tion rate was found to be very similar to Eq. (15), but

depends less strongly on

151

dissipation:

Γ '

[29γ

+ 16γ

+ 9χ

+ 11γ

+ 7γ

+ 9χ

] .

(17)

Using the same parameters as above, the ﬁdelity of state

transfer is F

= 0.989.

4.3 State measurement

The spin readout of Er

is performed by optical exci-

tation. Since only the |↑i−|e

i transition is coupled to

the cavity, optical excitation will result in a presence or

absence of a photon emission depending on the state of

the Er

ion. The measurement of the

151

ion can

be performed through its nearby Er

ion as a readout

ion, provided that the Er

ion is initially prepared in

the |↑i state. Performing spin readout of the

151

ion in the Z basis is then achieved by optically exciting

the

151

ion (|↑i to |ei transition) followed by ex-

citing the Er

ion (|↑i to |e

i transition). The Er

ion will excite to |e

i (remain in (|↑i) if the state of the

151

ion is |↓i (|↑i) due to the permanent electric

dipole-dipole interaction which shifts the Er

optical

resonance. To readout in the X basis, a

MW pulse

should be applied to the

151

ion in order to rotate

the ground-state spins before the optical excitation step.

State measurement of each Er

ion requires the de-

tection of an emitted photon. Due to the Purcell eﬀect,

as discussed in Sec. 4.1, the emission of a single photon

from the ion is highly preferential into the cavity mode.

For a high quality cavity, the probability that the pho-

ton emits into the cavity tends to unity. Therefore, the

detection probability will be limited by coupling losses

and single photon detectors, which can have detection

eﬃciency as high as 95%, as has been demonstrated in

superconducting detectors [74, 75]. To do better than

this limit, it is necessary to pump the Er

ion into a

cycling transition such that many photons will be emit-

ted by the cavity, and eventually detected. Using such a

cycling transition, detection probability can be as high

as 98.7% [71]. The detection eﬃciency is not 100% in

this case because there is a small chance for the ion to

decay into a diﬀerent state than the initial state, thus

ending the photon cycling [76]. This chance grows lin-

early with the number of cycles before detection.

4.4 Spectrally-multiplexed implementation

Our spectral multiplexing scheme relies on the possibil-

ity that many spectral channels can be operated in par-

allel. This requires that, in one node, diﬀerent cavities

emit photons that feature diﬀerent carrier frequencies.

This can be accomplished by frequency translation.

Noise-free translation over tens of gigahertz [50, 77]

can be achieved by using voltage-swept, commercially-

available, waveguide electro-optic modulators that can

be optically-coupled to the output port of each cavity.

After frequency translation, the output can be coupled

to a common spatial mode (e.g. a waveguide or ﬁber)

by using a tunable ring resonator ﬁlter that features

resonance linewidths as narrow as 1 MHz [78]. Arrayed

waveguides or ﬁber-Bragg gratings may also be used,

however they are bulky and their resonance linewidths

are currently not at the MHz level. The modulators

and ﬁlters may be fabricated on a single chip, oﬀer-

ing the possibility of low loss and up to 10

spectral

modes. The Bell-state measurement station consists of

a beamsplitter, two sets of ring resonator ﬁlters which

are identical to that used at the nodes, and an array of

superconducting nanowire-based photon detectors, cho-

sen due to their combination of high-eﬃciency and low

noise properties [75].

5 Conclusion

Our proposal for a quantum repeater which is based on

individual RE ions promises the deterministic establish-

ment of high-ﬁdelity entanglement over long-distances

at a rate which exceeds that corresponding to the di-

rect transmission of photons. Our scheme utilizes some

of the most desirable features of RE-ion-doped crys-

tals, speciﬁcally emission within the low-loss telecom-

munications window (Er

) and the hours-long nuclear

spin coherence lifetime (

151

SiO

) that is needed

to perform long-distance transmission and swapping of

entanglement. Moreover, control logic gates between

close-lying individual

151

and Er

ions allow the

quasi-deterministic swapping of entanglement by means

of a permanent electric dipole-dipole interaction. The

multiplexed version of our scheme improves the entan-

glement distribution rate by at least a factor of 100 over

Accepted in Quantum 2018-08-27, click title to verify 10

that of the single-mode version of our repeater.

Looking forward, it is interesting to consider the pos-

sibility of employing individual

167

ions instead of

Er-Eu ion pairs for a telecommunication wavelength

quantum repeater. In the presence of strong magnetic

ﬁelds,

167

SiO

features a nuclear spin coherence

lifetime in the one-second range [79], allowing the pos-

sibility of entanglement generation and storage using

the same ion, pairs (or small ensembles) of Er

ions.

One of the main challenges for future work in this direc-

tion is to devise a scheme whereby individual Er

ions

may be addressed and coupled within a single cavity.

This could be achieved by using magnetic dipole-dipole

interactions in a similar spirit to what has been demon-

strated using nitrogen vacancy centers and carbon spins

in diamond [42], or by using the cavity mode to mediate

the interaction. Another interesting direction is the pos-

sibility of long-term storage using host-ion spins such as

yttrium in Y

SiO

[80].

ACKNOWLED GMENTS

The authors would like to thank M. Afzelius, P. Barclay,

P. A. Bushev, C. W. Thiel and W. Tittel for helpful

discussions. This work was supported by the Natural

Sciences and Engineering Research Council (NSERC)

through its Discovery Grant program, the CREATE

grant ’Quanta’, and through graduate scholarships, by

the University of Calgary through an Eyes High post-

doctoral fellowship, by Alberta Innovates Technology

Futures (AITF) through graduate scholarships, and

by the Defense Advanced Research Projects Agency

(DARPA) through the Quiness program (Contract No.

W31P4Q-13-1-0004).

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