2007.11157.pdf

Teleportation Systems Towards a Quantum Internet

Raju Valivarthi,

1, 2

Samantha Davis,

1, 2

Cristi ́an Pe ̃na,

1, 2, 3

Si Xie,

1, 2

Nikolai Lauk,

1, 2

Lautaro Narv ́aez,

1, 2

Jason P. Allmaras,

Andrew D. Beyer,

Yewon Gim,

2, 5

Meraj Hussein,

George Iskander,

Hyunseong Linus

Kim,

1, 2

Boris Korzh,

Andrew Mueller,

Mandy Rominsky,

Matthew Shaw,

Dawn Tang,

1, 2

Emma E.

Wollman,

Christoph Simon,

Panagiotis Spentzouris,

Neil Sinclair,

1, 2, 7

Daniel Oblak,

and Maria Spiropulu

1, 2

Division of Physics, Mathematics and Astronomy,

California Institute of Technology, Pasadena, CA 91125, USA

Alliance for Quantum Technologies (AQT), California Institute of Technology, Pasadena, CA 91125, USA

Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

AT&T Foundry, Palo Alto, CA 94301, USA

Institute for Quantum Science and Technology, and Department of Physics & Astronomy,

University of Calgary, Calgary, AB T2N 1N4, Canada

John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138, USA

(Dated: July 29, 2020)

Quantum teleportation is essential for many quantum information technologies including long-

distance quantum networks. Using fiber-coupled devices, including state-of-the-art low-noise super-

conducting nanowire single photon detectors and off-the-shelf optics, we achieve quantum teleporta-

tion of time-bin qubits at the telecommunication wavelength of 1536.5 nm. We measure teleportation

fidelities of

≥

90% that are consistent with an analytical model of our system, which includes realis-

tic imperfections. To demonstrate the compatibility of our setup with deployed quantum networks,

we teleport qubits over 22 km of single-mode fiber while transmitting qubits over an additional 22

km of fiber. Our systems, which are compatible with emerging solid-state quantum devices, provide

a realistic foundation for a high-fidelity quantum internet with practical devices.

I. INTRODUCTION

Quantum teleportation [1], one of the most captivat-

ing predictions of quantum theory, has been widely in-

vestigated since its seminal demonstrations over 20 years

ago [2–4]. This is due to its connections to fundamental

physics [5–14], and its central role in the realization of

quantum information technology such as quantum com-

puters and networks [15–19]. The goal of a quantum

network is to distribute qubits between different loca-

tions, a key task for quantum cryptography, distributed

quantum computing and sensing. A quantum network

is expected to form part of a future quantum internet

[20–22]: a globally distributed set of quantum proces-

sors, sensors, or users there-of that are mutually con-

nected over a network capable of allocating quantum re-

sources (e.g. qubits and entangled states) between loca-

tions. Many architectures for quantum networks require

quantum teleportation, such as star-type networks that

distribute entanglement from a central location or quan-

tum repeaters that overcome the rate-loss trade-off of

direct transmission of qubits [19, 23–26].

Quantum teleportation of a qubit can be achieved by

performing a Bell-state measurement (BSM) between the

qubit and another that forms one member of an entan-

gled Bell state [1, 18, 27]. The quality of the teleporta-

tion is often characterized by the fidelity

〈

〉

of the teleported state

with respect to the state

〉

accomplished by ideal generation and teleportation [15].

This metric is becoming increasingly important as quan-

tum networks move beyond specific applications, such as

quantum key distribution, and towards the quantum in-

ternet.

Qubits encoded by the time-of-arrival of individual

photons, i.e. time-bin qubits [28], are useful for net-

works due to their simplicity of generation, interfacing

with quantum devices, as well as independence of dy-

namic transformations of real-world fibers. Individual

telecom-band photons (around 1.5

m wavelength) are

ideal carriers of qubits in networks due to their ability

to rapidly travel over long distances in deployed optical

fibers [17, 29–31] or atmospheric channels [32], among

other properties. Moreover, the improvement and grow-

ing availability of sources and detectors of individual

telecom-band photons has accelerated progress towards

workable quantum networks and associated technologies,

such as quantum memories [33], transducers [34, 35], or

quantum non-destructive measurement devices [36].

Teleportation of telecom-band photonic time-bin

qubits has been performed inside and outside the labora-

tory with impressive results [29–31, 37–42]. Despite this,

there has been little work to increase

beyond

∼

90% for

these qubits, in particular using practical devices that al-

low straightforward replication and deployment of quan-

tum networks (e.g. using fiber-coupled and commercially

available devices). Moreover, it is desirable to develop

teleportation systems that are forward-compatible with

emerging quantum devices for the quantum internet.

In the context of Caltech’s multi-disciplinary multi-

institutional collaborative public-private research pro-

gram on Intelligent Quantum Networks and Technologies

(IN-Q-NET) founded with AT&T as well as Fermi Na-

arXiv:2007.11157v2 [quant-ph] 28 Jul 2020

tional Accelerator Laboratory and Jet Propulsion Labo-

ratory in 2017, we designed, built, commissioned and de-

ployed two quantum teleportation systems: one at Fer-

milab, the Fermilab Quantum Network (FQNET), and

one at Caltech’s Lauritsen Laboratory for High Energy

Physics, the Caltech Quantum Network (CQNET). The

CQNET system serves as an R&D, prototyping, and

commissioning system, while FQNET serves as an ex-

pandable system, for scaling up to long distances and is

used in multiple projects funded currently by DOE’s Of-

fice of High Energy Physics (HEP) and Advanced Scien-

tific Research Computing (ASCR). Material and devices

level R&D in both systems is facilitated and funded by

the Office of Basic Energy Sciences (BES). Both systems

are accessible to quantum researchers for R&D purposes

as well as testing and integration of various novel de-

vices, such as for example on-chip integrated nanopho-

tonic devices and quantum memories, needed to up-

grade such systems towards a realistic quantum inter-

net.Importantly both systems are also used for improve-

ments of the entanglement quality and distribution with

emphasis on implementation of protocols with complex

entangled states towards advanced and complex quan-

tum communications channels. These will assist in stud-

ies of systems that implement new teleportation proto-

cols whose gravitational duals correspond to wormholes

[43], error correlation properties of wormhole teleporta-

tion, on-chip codes as well as possible implementation of

protocols on quantum optics communication platforms.

Hence the systems serve both fundamental quantum in-

formation science as well as quantum technologies.

Here we perform quantum teleportation of time-bin

qubits at a wavelength of 1536.5 nm with an average

≥

90%. This is accomplished using a compact setup

of fiber-coupled devices, including low-dark-count sin-

gle photon detectors and off-the-shelf optics, allowing

straight-forward reproduction for multi-node networks.

To illustrate network compatibility, teleportation is per-

formed with up to 44 km of single-mode fiber between

the qubit generation and the measurement of the tele-

ported qubit, and is facilitated using semi-autonomous

control, monitoring, and synchronization systems, with

results collected using scalable acquisition hardware. Our

systems, which operates at a clock rate of 90 MHz, can

be run remotely for several days without interruption and

yield teleportation rates of a few Hz using the full length

of fiber. Our qubits are also compatible with erbium-

doped crystals, e.g. Er:Y

SiO

, that are used to develop

quantum network devices like memories and transduc-

ers [44–46]. Finally, we develop an analytical model of

our system, which includes experimental imperfections,

predicting that the fidelity can be improved further to-

wards unity by well-understood methods (such as im-

provement in photon indistinguishability). Our demon-

strations provide a step towards a workable quantum net-

work with practical and replicable nodes, such as the

ambitious U.S. Department of Energy quantum research

network envisioned to link the U.S. National Laborato-

ries.

In the following we describe the components of our sys-

tems as well as characterization measurements that sup-

port our teleportation results, including the fidelity of our

entangled Bell state and Hong-Ou-Mandel (HOM) inter-

ference [47] that underpins the success of the BSM. We

then present our teleportation results using both quan-

tum state tomography (QST) [48] and projection mea-

surements based on a decoy state method [49], followed

by a discussion of our model. We conclude by consid-

ering improvements towards near-unit fidelity and GHz

level teleportation rates.

II. SETUP

Our fiber-based experimental system is summarized

in the diagram of Fig. 1. It allow us to demonstrate

a quantum teleportation protocol in which a photonic

qubit (provided by Alice) is interfered with one member

of an entangled photon-pair (from Bob) and projected

(by Charlie) onto a Bell-state whereby the state of Al-

ice’s qubit can be transferred to the remaining member

of Bob’s entangled photon pair. Up to 22 (11) km of

single mode fiber is introduced between Alice and Char-

lie (Bob and Charlie), as well as up to another 11 km

at Bob, depending on the experiment (see Sec. III). All

qubits are generated at the clock rate, with all of their

measurements collected using a data acquisition (DAQ)

system. Each of the Alice, Bob, Charlie subsystems are

further detailed in the following subsections, with the

DAQ subsystem described in Appendix A 1.

A. Alice: single-qubit generation

To generate the time-bin qubit that Alice will teleport

to Bob, light from a fiber-coupled 1536.5 nm continuous

wave (CW) laser is input into a lithium niobate intensity

modulator (IM). We drive the IM with one pulse, or two

pulses separated by 2 ns. Each pulse is of

∼

65 ps full

width at half maximum (FWHM) duration. The pulses

are produced by an arbitrary waveform generator (AWG)

and amplified by a 27 dB-gain high-bandwidth amplifier

to generate optical pulses that have an extinction ratio

of up to 22 dB. We note that this method of creating

time-bin qubits offers us flexibility not only in terms of

choosing a suitable time-bin separation, but also for syn-

chronizing qubits originating from different nodes in a

network. A 90/10 polarization-maintaining fiber beam

splitter combined with a power monitor (PWM) is used

to apply feedback to the DC-bias port of the IM so as

to maintain a constant 22 dB extinction ratio [50]. In

order to successfully execute the quantum teleportation

protocol, photons from Alice and Bob must be indistin-

guishable in all degrees of freedom (see Sec. III B). Hence,

the optical pulses at the output of the IM are band-pass

filtered using a 2 GHz-bandwidth (FWHM) fiber Bragg

Charlie

DAQ

Alice

CIR

VOA

POC

AMP

Bob

PWM

90|10

TDC

Data Analysis

and Storage

Clock

FBG

FIS

LAS

FIS

CIR

SPDC

SHG

MZI

EDFA

PWM

AMP

Cryostat

PBS

50|50

BPF

FBG

1536 nm

signal

idler

POC

FIS

AWG

Clock

90|10

PWM = Powermeter

PBS = Polarizing

Beam Splitter

Controller

POC = Polarization

MZI = Mach-Zehnder

Interferometer

LAS = Laser

IM = Intensity Modulator

HPF = High Pass Filter

EDFA = Erbium Doped

Fiber Amplifier

FIS = Fiber Spool

SPDC = Spontaneous

Parametric Down Conversion

SNSPD = Superconducting

Nanowire Single Photon Detector

VOA = Variable Optical

Attenuator

SHG = Second Harmonic

Generation

TDC = Time-To-Digital

Converter

FBG

1536 nm

1510 nm

768 nm

1536 nm

AMP = Amplifier

AWG = Arbitrary

Waveform Generator

BS = Beam Splitter

CIR = Circulator

FBG = Fiber Bragg Grating

BPF = Band Pass Filter

Bandwidth: 20 nm

Cryostat

50|50

PBS

HPF

1510 nm

HPF

1510 nm

FIG. 1. Schematic diagram of the quantum teleportation system consisting of Alice, Bob, Charlie, and the data acquisition

(DAQ) subsystems. See the main text for descriptions of each subsystem. One cryostat is used to house all SNSPDs, it is

drawn as two for ease of explanation. Detection signals generated by each of the SNSPDs are labelled 1-4 and collected at

the TDC, with 3 and 4 being time-multiplexed. All individual components are labeled in the legend, with single-mode optical

fibers (electronic cables) in grey (green), and with uni- and bi-chromatic (i.e. unfiltered) optical pulses indicated.

grating (FBG) centered at 1536.5 nm to match the spec-

trum of the photons from the entangled pair-source (de-

scribed in Sec. II B). Furthermore, the polarization of

Alice’s photons is determined by a manual polarization

controller (POC) in conjunction with a polarizing beam

splitter (PBS) at Charlie. Finally, the optical pulses from

Alice are attenuated to the single photon level by a vari-

able optical attenuator (VOA), to approximate photonic

time-bin qubits of the form

〉

√

−

〉

where the late state

〉

arrives 2 ns after the early state

〉

is real and set to be either 1, 0, or 1

√

2 to

generate

〉

, or

〉

= (

〉

)

√

2, respec-

tively, depending on the experiment. The complex rel-

ative phase is absorbed into the definition of

〉

. The

duration of each time bin is 800 ps.

B. Bob: entangled qubit generation and

teleported-qubit measurement

Similar to Alice, one (two) optical pulse(s) with a

FWHM of

∼

65 ps is (and separated by 2 ns are) cre-

ated using a 1536.5 nm CW laser in conjunction with a

lithium niobate IM driven by an AWG, while the 90/10

beam splitter and PWM are used to maintain an extinc-

tion ratio of at least 20 dB. An Erbium-Doped Fiber

Amplifier (EDFA) is used after the IM to boost the pulse

power and thus maintain a high output rate of photon

pairs.

The output of the EDFA is sent to a Type-0 period-

ically poled lithium niobate (PPLN) waveguide for sec-

ond harmonic generation (SHG), upconverting the pulses

to 768.25 nm. The residual light at 1536.5 nm is re-

moved by a 768 nm band-pass filter with an extinction

ratio

≥

80 dB. These pulses undergo spontaneous para-

metric down-conversion (SPDC) using a Type-II PPLN

waveguide coupled to a polarization-maintaining fiber

(PMF), approximately producing either a photon pair

pair

〉

, or the time-bin entangled state

〉

(

〉

)

√

2, if one or two pulses, respectively, are

used to drive the IM.

The ordering of the states refers to so-called signal and

idler modes of the pair of which the former has parallel,

and the latter orthogonal, polarization with respect to

the axis of the PMF. As before, the relative phase is

absorbed into the definition of

〉

. Each photon is sep-

arated into different fibers using a PBS and spectrally

filtered with FBGs akin to that at Alice. Note the band-

width of the FBG is chosen as a trade-off between spec-

tral purity and generation rate of Bob’s photons [51].

The photon in the idler mode is sent to Charlie for

teleportation or HOM measurements (see Sec. III B), or

to the MZI (see below) for characterizations of the en-

tangled state (see Sec. III A), with its polarization de-

termined using a POC.The photon in the signal mode

is sent to a Mach Zehnder interferometer (MZI) by way

of a POC (and an additional 11 km of single-mode fiber

for some measurements), and is detected by supercon-

ducting nanowire single photon detectors (SNSPDs) [52]

after high-pass filtering (HPF) to reject any remaining

768.25 nm light. The MZI and detectors are used for

projection measurements of the teleported state, charac-

terization of the time-bin entangled state, or measuring

HOM interference at Charlie. The time-of-arrival of the

photons is recorded by the DAQ subsystem using a time-

to-digital converter (TDC) referenced to the clock signal

from the AWG.

All SNSPDs are installed in a compact sorption fridge

cryostat [53], which operates at a temperature of 0.8 K

for typically 24 h before a required 2 h downtime. Our

SNSPDs are developed at the Jet Propulsion Laboratory

and have detection efficiencies between 76 and 85%, with

low dark count rates of 2-3 Hz. The FWHM temporal

resolution of all detectors is between 60 and 90 ps while

their recovery time is

∼

50 ns. A detailed description

of the SNSPDs and associated setup is provided in Ap-

pendix A 2.

The MZI has a path length difference of 2 ns and is

used to perform projection measurements of

〉

and (

〉

iφ

〉

)

√

2, by detecting photons at three

distinct arrival times in one of the outputs, and varying

the relative phase

[28]. Detection at the other out-

put yields the same measurements except with a relative

phase of

. Using a custom temperature-feedback

system, we slowly vary

for up to 15 hour time intervals

to collect all measurements, which is within the cryostat

hold time. Further details of the MZI setup is described

in Appendix A 3.

C. Charlie: Bell-state measurement

Charlie consists of a 50/50 polarization-maintaining

fiber beam splitter (BS), with relevant photons from the

Alice and Bob subsystems directed to each of its inputs

via a PBSs and optical fiber. The photons are detected

at each output with an SNSPD after HPFs, with their

arrival times recorded using the DAQ as was done at

Bob. Teleportation is facilitated by measurement of the

−

〉

= (

〉

−|

〉

)

√

2 Bell state, which cor-

responds to the detection of a photon in

〉

at one de-

tector followed by the detection of a photon in

〉

at the

other detector after Alice and Bob’s (indistinguishable)

qubits arrive at the BS [54]. Projection on the

−

〉

state corresponds to teleportation of

〉

up to a known

local unitary transformation, i.e. our system produces

−

iσ

〉

, with

being the Pauli

-matrix.

III. EXPERIMENTAL RESULTS

Prior to performing quantum teleportation, we mea-

sure some key parameters of our system that underpin

the teleportation fidelity. Specifically, we determine the

fidelity of the entangled state produced by Bob by mea-

suring the entanglement visibility

ent

[55], and also de-

termine to what extent Alice and Bob’s photons are in-

distinguishable at Charlie’s BS using the HOM effect [47].

A. Entanglement visibility

The state

pair

〉

(and hence the entangled state

〉

) described in Sec.

II B is idealized.

In real-

ity, the state produced by Bob is better approximated

by a two-mode squeezed vacuum state

TMSV

〉

√

−

∑

∞

√

〉

after the FBG filter and neglect-

ing loss [56]. Here,

is the number of photons per tem-

poral mode (or qubit),

is the emission probability of

a single pair per mode (or qubit), with state ordering

referring to signal and idler modes. However,

TMSV

〉

approximates a photon pair for

p <<

1, with

≈

mean number of pairs per mode (or qubit), conditioned

on measurement of a pair such that the

= 0 term is

eliminated. As a compromise between the pair-creation

rate

∝

and the quality of entanglement, here and hence-

forth we set the mean photon number of our pair source

to be

= (8

−

per time bin, which is fea-

sible because of the exceptionally low dark counts of our

SNSPDs. Measurement of

is outlined in Appendix B.

We generate

〉

and measure

ent

by directing the

idler photon to the second input port of the MZI, slightly

modifying the setup of Fig. 1. The idler photon is de-

layed compared to the signal, allowing unambiguous mea-

surement of each qubit. We vary

and project each qubit

of the entangled state onto phase-varied superpositions of

〉

and

〉

by accumulating coincidence events of photons

at both the outputs of the interferometer [55].

The results shown in Fig. 2 are fit proportional to

ent

sin (

ωT

+ Φ), where

ent

= (

−

)

(

with

(

)

denoting the maximum (minimum) rate of

coincidence events [55],

and Φ are unconstrained con-

stants, and

is the temperature of the MZI, finding

ent

= 96

3%.

The deviation from unit visibility is mainly due to non-

zero multi photon emissions [57], which is supported by

an analytical model that includes experimental imperfec-

tions [58]. Nonetheless, this visibility is far beyond the

3 required for non-separability of a Werner state [59]

and the locality bound of 1

√

2 [55, 60]. Furthermore, it

predicts a fidelity

ent

= (3

ent

+ 1)

4 = 97

2% with

respect to

〉

[59], and hence is sufficient for quantum

teleportation.

24.4

24.5

24.6

24.7

24.8

24.9

Interferometer Temperature (°C)

10000

20000

30000

40000

50000

60000

Coincidences / (30 min)

CQNET/FQNET Preliminary 2020

ent

: 96.4 ± 0.3%

FIG. 2. Entanglement visibility. The temperature of the in-

terferometer is varied to reveal the expected sinusoidal vari-

ations in the rate of coincidence events. A fit reveals the

entanglement visibility

ent

= 96

3%, see main text for

details. Uncertainties here and in all measurements are cal-

culated assuming Poisson statistics.

B. HOM interference visibility

The BSM relies on quantum interference of photons

from Alice and Bob. This is ensured by the BS at Charlie,

precise control of the arrival time of photons with IMs,

identical FBG filters, and POCs (with PBSs) to provide

the required indistinguishabiliy. The degree of interfer-

ence is quantified by way of the HOM interference visibil-

ity

HOM

= (

−

)

, with

(

)

denoting the rate of

coincident detections of photons after the BS when the

photons are rendered as distinguishable (indistinguish-

able) as possible [47]. Completely indistinguishable sin-

gle photons from Alice and Bob may yield

HOM

= 1.

However in our system, Alice’s qubit is approximated

from a coherent state

〉

= e

−|

2

∑

∞

√

〉

with

α <<

1, akin to how Bob’s pair is approximated

from

TMSV

〉

(see Sec. III A), with

being

Alice’s mean photon number per mode (or qubit) [56].

Therefore, the contribution of undesired photons from

Alice and Bob lowers the maximum achievable

HOM

below unity, with a further reduction if the interfering

photons are not completely indistinguishable. The de-

pendence of

HOM

with varied

and

, including

effects of losses or distinguishable photons in our system

is analytically modelled in Ref. [58], and briefly discussed

in Sec. IV.

We measure

HOM

by slightly modifying the setup of

Fig. 1: We prepare

〉

with

= 2

−

and

Bob as

pair

〉

and direct Alice’s photon and Bob’s idler

to Charlie, with Bob’s signal bypassing the MZI to be di-

rectly measured by an SNSPD. Alice’s IM is used to intro-

duce distinguishability by way of a relative difference in

arrival time ∆

of Alice and Bob’s photons at Charlie’s

BS. Using Charlie’s SNSPDs and the third detector at

Bob, a three-fold coincidence detection rate is measured

for varying ∆

, with results shown in Fig. 3a. Since

the temporal profiles of our photons are approximately

Gaussian, we fit our results to

−

HOM

exp(

−

∆

2

2

)],

where A is the maximum coincidence rate when the pho-

tons are completely distinguishable and

= 300 ps is the

temporal duration of the optical pulses [47, 61], find-

ing

HOM

= 70

9%. The maximum

HOM

for this

experiment is 83

5% if the photons were completely in-

distinguishable [58], with the difference ascribed to slight

distinguishability between our photons as supported by

the further measurements and analytical modelling in

Sec. IV. Improvements to our system to remove this

distinguishability is discussed in Sec. V.

To test our system for quantum teleportation over long

distances, we introduce the aforementioned 22, 11, and

11 km lengths of single-mode fiber between Alice and

Charlie, Bob and Charlie, and in the path of Bob’s signal

photon, respectively, repeat our measurement of

HOM

and fit the results as before (see Fig. 3b). We find

HOM

= 63

9%, which is consistent with the maxi-

mum

HOM

we expect when including the impact of the

additional 5.92 (2.56) dB loss between Charlie and Alice

(Bob) as well as the effect of photon distinguishability

(analyzed in Sec. IV). This suggests that the additional

fiber importantly does not introduce any further distin-

guishability (that we cannot account for), thereby sup-

porting our system’s use in quantum networking. Over-

all, the presence of clear HOM interference suggests our

system (with or without the additional fiber) introduces

relatively little imperfections that can negatively impact

the BSM and hence the fidelity of quantum teleportation.

C. Quantum teleportation

We now perform quantum teleportation of the time-

bin qubit basis states

〉

and

〉

, so as to mea-

sure the teleportation fidelities,

, an

, respec-

tively, of the teleported states with respect to their ideal

counterparts, up to the local unitary introduced by the

600

400

200

400

600

(ps)

100

150

200

250

300

Three-fold coincidences / (10 min)

CQNET/FQNET Preliminary 2020

HOM

: 70.9 ± 1.9%

600

400

200

400

600

(ps)

Three-fold coincidences / (10 min)

HOM

: 63.4 ± 5.9%

FIG. 3. Hong-Ou-Mandel (HOM) interference. A relative dif-

ference in arrival time is introduced between photons from Al-

ice and Bob at Charlie’s BS. HOM interference produces a re-

duction of the three-fold coincidence detection rate of photons

as measured with SNSPDs after Charlie’s BS and at Bob. A

fit reveals a)

HOM

= 70

9% and b)

HOM

= 63

when lengths of fiber are added, see main text for details.

BSM (see Sec. II C). Since measurement of

〉

in our

setup by symmetry is equivalent to any state of the

form (

〉

iφ

〉

)

√

2 (and in particular the remaining

three basis states (

〉−|

〉

)

√

2 and (

〉±

〉

)

√

2), we

may determine the average teleportation fidelity

avg

(

+ 4

)

6 of any time-bin qubit.

First, we prepare

〉

and

〉

with

= 3

−

with Bob’s idler bypassing the MZI to be detected by

a single SNSPD. We measure

= 95

1% and

1%, conditioned on a successful measurement of

−

〉

at Charlie, with fidelity limited by multipho-

ton events in Alice and Bob’s qubits and dark counts

of the SNSPDs [58]. We then repeat the measurement

with

= 9

−

after inserting the aforementioned

44 km length of fiber as before to emulate Alice, Charlie

and parts of Bob being separated by long distances. This

gives

= 98

1% and

= 98

2%, with no reduc-

tion from the additional fiber loss owing to our low noise

SNSPDs.

Next, we prepare

〉

with

= 9

−

, in-

sert the MZI and, conditioned on the BSM, we measure

= (1 +

)

2 = 84

5% by varying

. Here,

= 69

9% is the average visibility obtained by

fits to the resultant interference measured at each out-

put of the MZI, as shown in Fig. 4a. The reduction in

fidelity from unity is due to multiphoton events and dis-

tinguishability, consistent with that inferred from HOM

interference, as supported by further measurements and

analytical modelling in Sec. IV.

The measurement is repeated with the additional long

fiber, giving

= 58

7% and

= 79

9% with

results and corresponding fit shown in Fig. 4b. The re-

duced fidelity is likely due to aforementioned polarization

variations over the long fibers, consistent with the reduc-

tion in HOM interference visibility, and exacerbated here

owing to the less than ideal visibility of the MZI over long

measurement times (see Sec. A 3).

The results yield

avg

= 89

1% (86

3%) without

(with) the additional fiber, which is significantly above

the classical bound of 2

3, implying strong evidence of

quantum teleportation [62], and limited from unity by

multiphotons events, distinguishability, and polarization

variations, as mentioned [58].

To glean more information about our teleportation sys-

tem beyond the fidelity, we reconstruct the density matri-

ces of the teleported states using a maximum-likelihood

QST [48] described in Appendix C. The results of the

QST with and without the additional fiber lengths are

summarized in Figs. 8 and 9, respectively. As can be

seen, the diagonal elements for

〉

are very close to the

expected value indicating the preservation of probabili-

ties for the basis states of

〉

and

〉

after teleportation,

while the deviation of the off-diagonal elements indicate

the deterioration of coherence between the basis states.

The decoherence is attributed to multiphoton emissions

from our entangled pair source and distinguishability,

consistent with the aforementioned teleportation fideli-

ties of

〉

, and further discussed in Sec. IV. Finally,

we do also extract the teleportation fidelity from these

density matrices, finding the results shown in Fig. 5,

and

avg

= 89

1% (88

3%) without (with) the fiber

spools, which are consistent with previous measurements

given the similar

used for QST.

We point out that the 2

3 classical bound may only be

applied if Alice prepares her qubits using genuine single

photons, i.e.

= 1

〉

, rather than using

α <<

〉

as we

do in this work [63]. As a way to account for the photon

statistics of Alice’s qubits we turn to an analysis using

decoy states.

1. Teleportation fidelity using decoy states

To determine the minimum teleportation fidelity of

qubits prepared using single photons, we use a decoy

24.2

24.4

24.6

24.8

25.0

Interferometer Temperature (°C)

100

200

300

400

Three-fold coincidences / (12 min)

CQNET/FQNET Preliminary 2020

+, 1

: 69.9 ± 1.2%

100

200

300

400

+, 2

: 69.5 ± 1.2%

24.0

24.2

24.4

24.6

24.8

Interferometer Temperature (°C)

Three-fold coincidences / (40 min)

+, 1

: 63.2 ± 9.6%

+, 2

: 54.1 ± 6.3%

FIG. 4. Quantum teleportation of

〉

. Teleportation is per-

formed b) with and a) without an additional 44 km of single-

mode fiber inserted into the system. The temperature of the

inteferometer is varied to yield a sinusoidal variation of the

three-fold coincidence rate at each output of the MZI (blue

and red points). A fit of the visibilities (see Sec. III A) mea-

sured at each output (

) of the MZI gives an average

visibility

= (

)

2 of a) 69

91% without the

additional fiber and b) 58

7% with the additional fiber.

state method [49] and follow the approach of Refs.

[29, 64].

Decoy states, which are traditionally used

in quantum key distribution to defend against photon-

number splitting attacks, are qubits encoded into co-

herent states

〉

with varying mean photon number

. Measuring fidelities of the teleported qubits

for different

, the decoy-state method allows us to cal-

culate a lower bound

on the teleportation fidelity if

Alice had encoded her qubits using

= 1

〉

We prepare decoy states

〉

, and

〉

with vary-

ing

, as listed in Table I, and perform quantum telepor-

tation both with and without the added fiber, with tele-

portation fidelities shown in Table I. From these results

we calculate

as shown in Fig. 5, with

avg

≥

(

avg

≥

2%) without (with) the added fiber, which

significantly violate the classical bound and the bound

of 5

6 given by an optimal symmetric universal cloner

[65, 66], clearly demonstrating the capability of our sys-

| +

Average

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Fidelity

99.2±0.6%

97.9±1.3%

90.3±5.9%

93.0±3.9%

95.2±1.2%

95.9±1.3%

85.0±1.6%

88.5±1.1%

CQNET/FQNET Preliminary 2020

Single-photon fidelity from DSM

Fidelity from QST

| +

Average

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Fidelity

98.6±0.9%

98.4±0.9%

84.5±3.3%

89.2±2.2%

98.6±0.6%

96.2±1.9%

83.1±5.0%

87.9±3.3%

Single-photon fidelity from DSM

Fidelity from QST

FIG. 5. Quantum teleportation fidelities for

〉

, and

〉

, including the average fidelity. The dashed line rep-

resents the classical bound. Fidelities using quantum state

tomography (QST) are shown using blue bars while the min-

imum fidelities for qubits prepared using

= 1

〉

and

, including the associated average fidelity

avg

, respec-

tively, using a decoy state method (DSM) is shown in grey.

Panels a) and b) depicts the results without and with addi-

tional fiber, respectively. Uncertainties are calculated using

Monte-Carlo simulations with Poissonian statistics.

tem for high-fidelity teleportation. As depicted in Fig.

5 these fidelities nearly match the results we obtained

without decoy states within statistical uncertainty. This

is due to the suitable

, as well as low

and SNSPD

dark counts in our previous measurements [58].

IV. ANALYTICAL MODEL AND SIMULATION

As our measurements have suggested, multi-photon

components in, and distinguishability between, Alice and

Bob’s qubits reduce the values of key metrics includ-

ing HOM interference visibility and, consequently, quan-

tum teleportation fidelity. To capture these effects in

our model, we employ a Gaussian-state characteristic-

function method developed in Ref. [58], which was en-

qubit

without long fiber

with long fiber

(

−

)

(%)

(

−

)

(%)

〉

3.53

95.2

26.6

95.7

1.5

1.24

86.7

9.01

98.4

1.1

52.8

3.4

〉

3.53

95.9

32.9

98.6

0.7

1.24

90.5

9.49

98.4

1.6

52.8

3.4

〉

9.38

84.7

1.1

29.7

73.6

3.0

2.01

83.2

3.6

10.6

82.21

3.9

52.8

3.4

TABLE I. Teleportation fidelities with (right column) and

without (center column) the 44 km-length of fiber for Alice’s

qubit states prepared with varying

. Mean photon numbers

and fidelities for vacuum states with fiber are assumed to be

zero and 50%, respectively.

abled by work in Ref. [67]. This approach is well-suited

to analyze our system because the quantum states, oper-

ations, and imperfections (including losses, dark counts,

etc.) of the experiment can be fully described using

Gaussian operators, see e.g. Ref. [68]. We now briefly

outline the model of Ref. [58], and employ it to estimate

the amount of indistinguishability

between Alice and

Bob’s qubits in our measurements of HOM interference

and quantum teleportation.

Distinguishability in any degree-of-freedom may be

modelled by introducing a virtual beam splitter of trans-

mittance

into the paths of Alice and Bob’s relevant pho-

tons. As shown in Fig. 6, indistinguishable components

of incoming photon modes are directed towards Charlie’s

BS where they interfere, whereas distinguishable compo-

nents are mixed with vacuum at the BS and do not con-

tribute to interference. Here

= 1 (

= 0) corresponds

to the case when both incoming photons are perfectly

indistinguishable (distinguishable). Now we may calcu-

late the probability of a three-fold coincidence detection

event

between

(Charlies’ detectors), and

FIG. 6. Schematic depiction of distingushability between Al-

ice and Bob’s photons at Charlie’s BS. Distinguishability is

modeled by means of a virtual beam splitter with a transmit-

tance

. Indistinguishable photons contribute to interference

at the Charlie’s BS while distinguishable photons are mixed

with vacuum, leading to a reduction of HOM visibility and

teleportation fidelity. See main text for further details.

(detects Bob’s signal photon) for a given qubit state

from Alice and Bob:

= Tr

{

(

I

−

(

〉〈

)

⊗

3

1

2

3

)

⊗

(

I

−

(

〉〈

)

⊗

3

1

2

3

)

⊗

(

I

−

(

〉〈

)

}

(1)

where the ˆ

and

operators refer to modes, which origi-

nate from Alice and Bob’s virtual beam splitters and are

directed to

and

, respectively, and ˆ

corresponds

to Bob’s idler mode, which is directed to

, see Fig. 6.

This allows the derivation of an expression for the HOM

interference visibility

HOM

(

) = [

(0)

−

(

)]

(0)

(2)

consistent with that introduced in Sec.

III B. Since

Alice and Bob ideally produce

= (

〉〈

)

⊗

(

TMSV

〉〈

TMSV

), and recognizing that all operators in

are Gaussian, we analytically derive

(

) = 1

−

exp(

−

2[1+(1

−

2

)

1 +

−

1 +

exp(

−

)

1 +

−

exp(

−

)

1 + (1

−

)

+ 2

exp(

−

2[1+(1

−

2

)(1

−

)

]

1+(1

−

)

1 + (1

−

)

2 +

(3)

for varied

, where

and

are the transmission ef-

ficiencies of the signal and idler photons, including de-

tector efficiencies. We similarly calculate the impact of

distinguishability on the teleportation fidelity of

〉

(

) =

(

ζ,φ

max

)

[

(

ζ,φ

max

) +

(

ζ,φ

min

)]

(4)

where

max

(

min

) is the phase of the MZI added into

the path of the signal photon, corresponding to maximum

(minimum) three-fold detection rates.

To compare the model to our measurements, we use the

experimental mean photon numbers for the photon-pair

source

= 1

−

and

= 4

−

as deter-

mined by the method described in Appendix B. We then

measure the teleportation fidelity of

〉

and HOM inter-

ference visibility (keeping the MZI in the system to en-

sure

remains unchanged) for different values

. The

results are plotted in Fig. 7. The data is then fitted to

the expressions

HOM

(

) and

(

) derived in our model

and graphed in Fig. 7. The fitted curves are in very good

agreement with our experimental values and consistently

yield a value of

= 90% for both measurements types.

This implies that we have only a small amount of resid-

ual distinguishability between Alice and Bob’s photons.

Potential effects leading to this distinguishability are dis-

cussed in Sec. V.

Overall, our analytic model is consistent with our ex-

perimental data [58] in the regime of

1, which

is the parameter space most often used in quantum net-

working protocols (e.g. key distribution). Our model,

thus, offers a practical way to determine any underlying

distinguishability in a deployed network where a full char-

acterization of the properties of Alice and Bob’s photons

may not be possible.

V. DISCUSSION AND OUTLOOK

We have demonstrated quantum teleportation systems

for photonic time-bin qubits at a quantum channel- and

device-compatible wavelength of 1536.5 nm using a fiber-

based setup comprising state-of-the-art SNSPDs and off-

the-shelf components. We measure an average fidelity

above 90% using QST and a decoy state analysis with up

to 44 km of single mode fiber in our setup. Our results

are further supported by an analytical model in conjunc-

tion with measurements of the entanglement and HOM

interference visibilities.

The decoy state analysis indicates that the maximum

teleportation fidelity is currently restricted by that of

the teleported qubits prepared in superposition states,

for which a 10% distinguishability between the qubits

undergoing BSM and the contribution of multiple pho-

ton pairs play the largest role. Our model predicts that

the average fidelity will increase to 95% with completely

indistinguishable photons, while fidelities close to unity

can be achieved with lowered mean number of photon

pairs. Alternatively, we may replace our SNSPDs with

efficient photon-number resolving (PNR) SNSPDs [69] to

allow postselection of multiphoton events at the MZI or

BSM [70]. Both approaches must be accompanied by

increased coupling efficiencies of the pair source beyond

the current

∼

1% either to maintain realistic teleporta-

tion rates (above the system noise and current rate of

phase drift of the MZI), or to derive any advantage from

PNR capability.

As suggested by the width of our HOM interference

FIG. 7. Evaluation of photon indistinguishability using an

analytical model. Panel a) depicts the quantum teleportation

fidelity of

〉

while panel b) shows the HOM interference vis-

ibility, each with varied mean photon number

of Alice’s

qubits. Fits of analytical models the data reveal

= 90% in-

distinguishability between Alice and Bob’s photons at Char-

lie’s BS. Bob produces

photon pairs on average,

and

are the probabilities for an individual idler (signal) photon

to arrive at Charlie’s BS and be detected at Bob’s detector,

respectively.

fringe – which predicts an average photon bandwidth of

/σ

∼

1.5 GHz (see Sec. III B), i.e. less than the

2 GHz bandwidth of our FBGs – the indistinguishabil-

ity in our system could be limited by the large differ-

ence in the bandwidth between the photons originating

from the SPDC (

100 GHz) and those generated at Al-

ice by the IM (15 GHz), leading to nonidentical filtering

by the FBG. This can be improved by narrower FBGs

or by using a more broadband pump at Alice (e.g. us-

ing a mode locked laser or a higher bandwidth IM, e.g

50 GHz, which is commercially available). Alternatively,

pure photon pairs may be generated by engineered phase

matching, see e.g. Ref. [71]. Distinguishability owing to

nonlinear modulation during the SHG process could also

play a role [72]. The origin of distinguishability in our

system, whether due to imperfect filtering or other device

imperfections (e.g. PBS or BS) will be studied in future

work. Coupling loss can be minimized to less than a

few dB overall by improved fiber-to-chip coupling, lower-

loss components of the FBGs (e.g. the required isolator),

spliced fiber connections, and reduced losses within our

MZI. Note that our current coupling efficiency is equiva-

lent to

∼

50 km of single mode fiber, suggesting that our

system is well-suited for quantum networks provided loss

is reduced.

While the fidelities we demonstrate are sufficient for

several applications, the current

∼

Hz teleportation rates

with the 44 km length of fiber are still low. Higher repe-

tition rates (e.g. using high-bandwidth modulators with

wide-band wavelength division multiplexed filters and

low-jitter SNSPDs [73]), improvements to coupling and

detector efficiencies, enhanced BSM efficiency with fast-

recovery SNSPDs [74], or multiplexing in frequency [64]

will all yield substantial increases in teleportation rate.

Note that increased repetition rates permits a reduction

in time bin separation which will allow constructing the

MZI on chip, providing exceptional phase stability and

hence, achievable fidelity. Importantly, the aforemen-

tioned increases in repetition rate and efficiency are af-

forded by improvements in SNSPD technology that are

currently being pursued with our JPL, NIST and other

academic partners.

Upcoming system-level improvements we plan to in-

vestigate and implement include further automation by

the implementation of free-running temporal and polar-

ization feedback schemes to render the photons indistin-

guishable at the BSM [29, 30]. Furthermore, several elec-

trical components can be miniaturized, scaled, and made

more cost effective (e.g. field-programmable gate arrays

can replace the AWG).We note that our setup prototype

will be easily extended to independent lasers at different

locations, also with appropriate feedback mechanisms for

spectral overlap [75, 76]. These planned improvements

are compatible with the data acquisition and control sys-

tems that were built for the systems and experiments at

FQNET and CQNET presented in this work.

Overall, our high-fidelity teleportation systems achiev-

ing state-of-the-art teleporation fidelities of time-bin

qubits serve as a blueprint for the construction of quan-

tum network test-beds and eventually global quantum

networks towards the quantum internet. In this work,

we present a complete analytical model of the telepora-

tion system that includes imperfections, and compare it

with our measurements. Our implementation, using ap-

proaches from High Energy Physics experimental systems

and real-world quantum networking, features near fully-

automated data acquisition, monitoring, and real-time

data analysis. In this regard our Fermilab and Caltech

Quantum Networks serve as R& D laboratories and pro-

totypes towards real-world quantum networks. The high

fidelities achieved in our experiments using practical and

replicable devices are essential when expanding a quan-

tum network to many nodes, and enable the realization

of more advanced protocols, e.g. [18, 77, 78].

ACKNOWLEDGEMENTS

R.V., N.L., L.N., C.P., N.S., M.S. and S.X. acknowl-

edge partial and S.D. full support from the Alliance

for Quantum Technologies (AQT) Intelligent Quantum

Networks and Technologies (IN-Q-NET) research pro-

gram.

R.V., N.L., L.N., C.P., N.S., M.S. S.X. and

A.M. acknowledge partial support from the U.S. De-

partment of Energy, Office of Science, High Energy

Physics, QuantISED program grant, under award num-

ber de-sc0019219. A.M. is supported in part by the

JPL President and Directors Research and Develop-

ment Fund (PDRDF). C.P. further acknowledges par-

tial support from the Fermilab’s Lederman Fellowship

and LDRD. D.O. and N.S. acknowledge partial sup-

port from the Natural Sciences and Research Council of

Canada (NSERC). D.O. further acknowledges the Cana-

dian Foundation for Innovation, Alberta Innovates, and

Alberta Economic Development, Trade and Tourisms

Major Innovation Fund. J.A. acknowledges support by

a NASA Space Technology Research Fellowship. Part

of the research was carried out at the Jet Propulsion

Laboratory, California Institute of Technology, under a

contract with the National Aeronautics and Space Ad-

ministration (80NM0018D0004). We thank Jason Trevor

(Caltech Lauritsen Laboratory for High Energy Physics),

Nigel Lockyer and Joseph Lykken (Fermilab), Vikas

Anant (PhotonSpot), Aaron Miller (Quantum Opus), In-

der Monga and his ESNET group at LBNL, the groups of

Wolfgang Tittel and Christoph Simon at the University

of Calgary, the groups of Nick Hutzler, Oskar Painter,

Andrei Faraon, Manuel Enders and Alireza Marandi

at Caltech, Marko Loncar’s group at Harvard, Ar-

tur Apresyan and the HL-LHC USCMS-MTD Fermilab

group; Marco Colangelo (MIT); Tian Zhong (Chicago);

AT&T’s Soren Telfer, Rishi Pravahan, Igal Elbaz, Andre

Feutch and John Donovan. We acknowledge the enthusi-

astic support of the Kavli Foundation on funding QIS&T

workshops and events and the Brinson Foundation sup-

port especially for students working at FQNET and

CQNET. M.S. is especially grateful to Norm Augustine

(Lockheed Martin), Carl Williams (NIST) and Joe Broz

(SRI, QED-C); Hartmut Neven (Google Venice); Amir

Yacoby and Misha Lukin (Harvard); Ned Allen (Lock-

heed Martin); Larry James and Ed Chow (JPL); the

QCCFP wormhole teleportation team especially Daniel

Jafferis (Harvard) and Alex Zlokapa (Caltech), Mark Ka-

sevich (Stanford), Ronald Walsworth (Maryland), Jun

Yeh and Sae Woo Nam (NIST); Irfan Siddiqi (Berkeley);

Prem Kumar (Northwestern), Saikat Guha (Arizona),

Paul Kwiat (UIUC), Mark Saffman (Wisconcin), Jelena

Vuckovic (Stanford) Jack Hidary (X), and the quantum

networking teams at ORNL, ANL, and BNL, for produc-

tive discussions and interactions on quantum networks

and communications.

Appendix A: Detailed description of experimental

components

1. Control systems and data acquisition

Our system is built with a vision towards future repli-

cability, with particular emphasis on systems integra-

tion. Each of the Alice, Bob and Charlie subsystems

is equipped with monitoring and active feedback stabi-

lization systems (e.g. for IM extinction ratio), or has ca-

pability for remote control of critical network parameters

(e.g. varying the qubit generation time). Each subsystem

has a central classical processing unit with the following

functions: oversight of automated functions and work-

flows within the subsystem, data acquisition and man-

agement, and handling of input and output synchroniza-

tion streams. As the quantum information is encoded

in the time domain the correct operation of the classical

processing unit depends critically on the recorded time-

of-arrival of the photons at the SNSPDs. Thus signifi-

cant effort was dedicated to build a robust DAQ subsys-

tem capable of recording and processing large volumes of

time-tagged signals from the SNSPDs and recorded by

our TDCs at a high rate. The DAQ is designed to en-

able both real-time data analysis for prompt data quality

monitoring as well as post-processing data analysis that

allows to achieve the best understanding of the data.

The DAQ system is built on top of the standalone

Linux library of our commercial TDC. It records time

tags whenever a signal is detected in any channel in co-

incidence with the reference 90 MHz clock. Time tags

are streamed to a PC where they are processed in real-

time and stored to disk for future analysis. A graphical

user interface has been developed, capable of real-time

visualization and monitoring of photons detected while

executing teleportation. It also allows for easy control of

the time-intervals used for each channel and to configure

relevant coincidences between different photon detection

events across all TDC channels. We expect our DAQ sub-

system to serve as the foundation for future real-world

time-bin quantum networking experiments (see Sec. V).

2. Superconducting nanowire single photon

detectors

We employ amorphous tungsten silicide SNSPDs man-

ufactured in the JPL Microdevices Laboratory for all

measurements at the single photon level (see Sec. II B)

[52]. The entire detection system is customized for opti-

mum autonomous operation in a quantum network. The

SNSPDs are operated at 0.8 K in a closed-cycle sorption

fridge [53]. The detectors have nanowire widths between

140 to 160 nm and are biased at a current current of 8

to 9

A. The full-width at half maximum (FWHM) tim-

ing jitter (i.e. temporal resolution) for all detectors is

between 60 and 90 ps (measured using a Becker & Hickl

SPC-150NXX time-tagging module). The system detec-

tion efficiencies (as measured from the fiber bulkhead of

the cryostat) are between 76 and 85 %. The SNSPDs fea-

ture low dark count rates between 2 and 3 Hz, achieved by

short-pass filtering of background black-body radiation

through coiling of optical fiber to a 3 cm diameter within

the 40 K cryogenic environment, and an additional band-

pass filter coating deposited on the detector fiber pigtails

(by Andover Corporation). Biasing of the SNSPDs is fa-

cilitated by cryogenic bias-Ts with inductive shunts to

prevent latching, thus enabling uninterrupted operation.

The detection signals are amplified using Mini-Circuits

ZX60-P103LN+ and ZFL-1000LN+ amplifiers at room

temperature, achieving a total noise figure of 0.61 dB

and gain of 39 dB at 1 GHz, which enables the low sys-

tem jitter. Note that FWHM jitter as low as 45 ps is

achievable with the system, by biasing the detectors at

approximately 10

A, at the cost of an elevated DCR on

the order of 30 cps. Using commercially available com-

ponents, the system is readily scalable to as many as 64

channels per cryostat, ideal for star-type quantum net-

works, with uninterrupted 24/7 operation. The bulkiest

component of the current system is an external helium

compressor, however, compact rack-mountable versions

are readily available [53].

3. Interferometer and phase stabilization

We use a commercial Kylia 04906-MINT MZI, which

is constructed of free-space devices (e.g mirrors, beam

spliters) with small form-factor that fits into a hand-held

box. Light is coupled into and out of the MZI using

polarization maintaining fiber with loss of

∼

2.5 dB. The

interferometer features an average visibility of 98.5% that

was determined by directing

〉

with

= 0

07 into one

of the input ports, measuring the fringe visibility on each

of the outputs using an SNSPD. The relative phase

controlled by a voltage-driven heater that introduces a

small change in refractive index in one arm of the MZI.

However, this built-in heater did not permit phase stabil-

ity sufficient to measure high-fidelity teleportation, with

the relative phase following the slowly-varying ambient

temperature of the room. To mitigate this instability,

we built another casing, thermally isolating the MZI en-

closure from the laboratory environment and controlled

the temperature via a closed-loop feedback control sys-

tem based on a commercial thermoelectric cooler and a

LTC1923 PID-controller. The temperature feedback is

provided by a 10 kΩ NTC thermistor while the set-point

is applied with a programmable power supply. This con-

trol system permits us to measure visbilities by slowly

varying

over up to 15 hour timescales. We remark

that no additional methods of phase control were used

beyond that of temperature.

Appendix B: Estimation of mean number of photon

pairs and transmission efficiencies of signal and idler

photons

Using a method described in Ref. [55], we measure

the mean number of photon pairs produced by Bob

as a function of laser excitation power before the PPLN

waveguide used for SHG. To this end, we modify the

setup of Fig. 1 and direct each of Bob’s signal and idler