# An elementary 158 km long quantum network connecting room temperature
quantum memories

Dounan Du  Department of Physics and Astronomy, Stony Brook University, Stony
Brook, NY 11794, USA  Paul Stankus  Brookhaven National Laboratory, Upton, NY,
11973, USA  Olli-Pentti Saira  Brookhaven National Laboratory, Upton, NY,
11973, USA  Mael Flament  Qunnect Inc., 141 Flushing Av. Suite 1110, Brooklyn,
NY, 11205, USA  Steven Sagona-Stophel  Department of Physics and Astronomy,
Stony Brook University, Stony Brook, NY 11794, USA  Mehdi Namazi  Qunnect
Inc., 141 Flushing Av. Suite 1110, Brooklyn, NY, 11205, USA  Dimitrios
Katramatos  Brookhaven National Laboratory, Upton, NY, 11973, USA  Eden
Figueroa  Department of Physics and Astronomy, Stony Brook University, Stony
Brook, NY 11794, USA  Brookhaven National Laboratory, Upton, NY, 11973, USA

###### Abstract

First-generation long-distance quantum repeater networks require quantum
memories capable of interfacing with telecom photons to perform quantum-
interference-mediated entanglement generation operations. The ability to
demonstrate these interconnections using real-life fiber connections in a
long-distance setting is paramount to realize a scalable quantum internet.
Here we address these significant challenges by observing Hong-Ou-Mandel (HOM)
interference between indistinguishable telecom photons produced in two
independent room temperature quantum memories, separated by a distance of 158
km. We obtained interference visibilities after long-distance propagation of
$\rm\boldsymbol{V=(38\pm 2)\%}$ for single-photon level experimental inputs.
This first-of-its-kind quantum network prototype connecting quantum
laboratories in Stony Brook University and Brookhaven National Laboratory is
envisioned to evolve into a large-scale memory-assisted entanglement
distribution quantum network, the basis for inter-city quantum communication.

##  I Introduction

Quantum information technologies, in which quantum mechanics’ physical laws
govern the control and transmission of information, show significant promise
for revolutionary new capabilities that are fundamentally different from what
is possible with classical technology alone Acin2018 . Any quantum processing
that takes place across multiple locations will require the preservation,
transport, and sharing of quantum states across significant distances. Thus a
widespread quantum network can usher in a new age of quantum information
processing, with never-before-seen capabilities in communication, sensing, and
computing Simon2017 . Prime examples of such quantum network protocols are
Measurement-Device-Independent Quantum Key Distribution (MDI-QKD) CurtisLo2002
, and entanglement distribution using quantum repeaters Lloyd2001 ,

Over the last decade, the experimental development of such networks’
components has been widely pursued. Prototypes have demonstrated elementary
quantum functionality, such as direct entanglement distribution Hensen2016 ;
Valivarthi2016 ; Sun2016 ; Valivarthi2020 , quantum-state transfer
Puigibert2020 ; Cao2020 , and short-distance interference-mediated
entanglement generation between light-matter quantum nodes Slodicka2013  . As
we move forward in implementing large quantum networks, we can list a set of
general properties that we would want to have to build an extended network to
be practical Loock2020 . These follow the discussions presented, for example,
in the DOE-ASCR workshop “Quantum Networks for Open Science” ASCRQN2019 , the
NSF “Quantum InterConnects (QuICs) for Next-Generation Information
Technologies” NSFQINT2019  and the DOE “Quantum Internet Blueprint Workshop”
DOEQINT2020 .

  * •

Portability. The essential components of each node, particularly quantum
memories and frequency converters, should all be portable, meaning that they
can be manufactured centrally and then deployed in the field with minimal
installation complexity.

  * •

Room temperature operation. It will be of great advantage in any system to be
built across a wide area and utilize a wide variety of locations to minimize
environmental demands by the equipment. In particular, if all the components
can work at room temperature without the need for cryogenic partitions or
consumables.

  * •

Agnostic/signal independence. Just as the operation of a digital data network
is essentially independent of what data is being carried over it, the quantum
network should have the capability to transport different kinds of quantum
states. Such a network is termed agnostic, e.g. not fixed to any one specific
computational basis.

  * •

Transparent. The quantum network will also be greatly simplified if it is
transparent, i.e, all-optical, with no optical-to-electrical-to-optical
conversions across which it would be difficult to maintain entanglement at
room temperature.

  * •

Compatible with telecom infrastructure. The building of a quantum network will
be greatly advanced if it can take advantage of the physical channels,
particularly optical fibers, that have already been highly developed and
widely deployed for data networks. In the case of using entangled photon
pairs, this will require a photon-photon transduction capability between the
wavelengths used by the sources, the memories, and existing telecom industry
fiber channels.

One of the most promising avenues to build first-generation quantum networks
is the use of atomic-ensemble-based optical quantum memories. These systems
have recently shown to operate with high-efficiency Wang2019  and long storage
times Yang2016 . Furthermore, elementary demonstrations of small quantum
networks have also been performed recently JWP2019 ; JWP2019_2 . This work has
focused on implementing the DLCZ protocol DLCZ2001 , by using Class 1
memories, which emit one photon first and another photon later on-demand,
relying on quantum interference to generate entanglement Cabrillo1999 . Room-
temperature quantum technology offers a very exciting solution to build these
Class 1 memory-based quantum repeater networks Novikova2012 . Recent studies
have demonstrated the potential of room-temperature quantum devices for this
purpose, including the demonstration of noiseless room-temperature quantum
memories for single-photons and polarization qubits Namazi2017 ;
Finkelstein2018 ; Kaczmarek2018  with coherence times of several seconds in
ambient conditions Katz2018 . Furthermore, preliminary quantum networks using
room-temperature memories are already available Namazi2017_2 ; Flament2019 .

Here, we demonstrate the implementation of Class 1 room-temperature long-
distance quantum network of two quantum light-matter interfaces and benchmark
the inter-connectivity between its components at telecom wavelengths. We
interconnected the quantum devices in a configuration akin to the one needed
to perform the Cabrillo entanglement scheme Cabrillo1999 . This configuration
requires two sources of telecom qubits (herein called Alice and Bob)
generating single-photon level pulses. We have chosen Alice and Bob to be two
room-temperature quantum memories performing transduction between the 795 and
1324 nm wavelengths Radnaev2010 . After transmission in long-distance fiber
links, the qubits are sent to the measurement station, where we investigate
the indistinguishability of the two single-photon wave-packets produced in the
memory using Hong-Ou-Mandel (HOM) interference HOM . We measure the two-
detector coincidence rate as a function of the relative arrival time between
the qubits.

We characterize our elementary long-distance quantum network in two different
configurations. First, we show Mach-Zender interferometry of the memory-
produced qubits in an interferometer with two 70 km long arms. Next, we
demonstrate the degree of indistinguishability of the telecom polarization
qubits transduced in two independent quantum memories using HOM interference
experiments, with one arm of the interferometer being 70 km, and the second
one being 88km, in order to achieve phase independency between the qubits.

##  II Telecom Compatible Quantum Memories

The first step in our procedure is to demostrate that our rubidium vapor
quantum memory platform is compatible with telecom operation. This desired
property has been demonstrated in proof of principle experiments with
cryogenically cooled crystals Saglamyurek2015 ; Jin2015 ; Falamarzi2019 , but
not yet at room temperature.

###  II.1 Two-photon absorption and telecom laser rubidium stabilization

Stabilization of a 1367 nm diode laser to the rubidium transitions is achieved
in the following manner. We apply an optical-optical double resonance (OODR)
scheme to obtain the $\rm{}^{85}Rb$ spectrum in the vecinity of $\rm 1.3\mu
m$. Figure 2a shows the energy scheme. The $\rm 5S_{1/2}\rightarrow 5P_{3/2}$
transition($\ket{1}\rightarrow\ket{3}$) is $\rm 780nm$ and the $\rm
5P_{3/2}\rightarrow 6S_{1/2}$ $\ket{3}\rightarrow\ket{4}$ is $\rm 1367nm$. A
$\rm 780nm$ pump laser is used to increase the population in the $\rm
5P_{3/2}$ level. The pump laser is locked to the $\rm 5S_{1/2}F=3\rightarrow
5P_{3/2}F^{\prime}=4$ transition via a Doppler-free saturated absorption
spectroscopy (SAS) system. The $\rm 1367nm$ laser is conterpropagating with
the pump laser beam in a $\rm 7cm$ long Rubidium reference cell. The cell is
maintained at $\rm 60^{\circ}C$. A InGaAs balanced amplified photodetector is
used to obtain the spectrum signal. We scan the $\rm 1367nm$ laser while
keeping the pump laser locked to obtain the OODR spectrum. We generate the
error signal by modulating the $\rm 1367nm$ laser at $\rm 100kHz$ and use a
standard locking loop, as shown in Figure 1.

Figure 1: The laser stabilization setup. SAS: saturated absorption
spectroscopy setup; LD: laser diode; BS: beam splitter; LF: long pass filter;
BPD: balanced amplified photodetector; FG1: function generator generating
100kHz laser current modulation signal; FG2: function generator used to
generate 10Hz signal for laser Piezo scanning; LPF: low-pass filter; OSC:
monitor oscilloscope; PI: proportional integral controller.

###  II.2 Telecom Frequency Conversion in Rubidium (Theory and Simulation)

Figure 2:  a. Diamond atomic scheme use for frequency conversion and the two-
photon absorption spectroscopy. In our setup $\rm\ket{1}=^{85}Rb:5S_{1/2}F=3$,
$\rm\ket{2}=^{85}Rb:5P_{1/2}F=2$,$\rm\ket{3}=^{85}Rb:5P_{3/2}F=4$ and
$\rm\ket{4}=^{85}Rb:6S_{1/2}F=1$. b. A simulation of the frequency conversion
process using the simulation procedure described below. The two pump fields
are kept at CW, the $\rm 795nm$ is modulated to have a Gaussian envelope using
an AOM. We make an assumption that the photo detector is placed at the output
port of the Rubidium cell.

A diamond configuration 4-level system is used for frequency conversion, see
Figure 2a. The signal field ($795nm$) is the input which encodes the qubit
information. The near resonance Pump I ($780nm$) and Pump II ($1367nm$) are
strong CW pump fields. The telecom field ($1324nm$) is the generated field
which encodes the same qubit information as that of the signal field. The Pump
fields are given as

$\vec{E}_{I}^{phy}(z,t)=\vec{E}_{I}(z)e^{-i(\omega_{I}t-\vec{k}_{I}\cdot\vec{z})}+c.c.\\\
$ (1)
$\vec{E}_{II}^{phy}(z,t)=\vec{E}_{II}(z)e^{-i(\omega_{II}t-\vec{k}_{II}\cdot\vec{z})}+c.c.$
(2)

Here the $\vec{E}_{I}^{phy}(z,t)$ is the physical field that can be measured.
The absolute value of $\vec{E}_{I}(z)$ is half of that of the physical field.

$|\vec{E}_{I}(z)|=\frac{1}{2}|\vec{E}_{I}^{phy}(z,t)|$ (3)

Generally $\vec{E}_{I}(z)$ is complex to include phase shifts. In this model
we assume it to be real to simplify the modeling. Since the signal and telecom
field encode qubits, they are of single photon level. They are given as

$\hat{\vec{E}}_{s}^{phy}(z,t)=\vec{e}_{s}(\frac{\hbar\omega_{s}}{2V\epsilon_{0}})^{\frac{1}{2}}\hat{\psi}_{s}(z,t)e^{-i(\omega_{s}t-\vec{k}_{s}\cdot\vec{z})}+h.c.\\\
$ (4)
$\hat{\vec{E}}_{t}^{phy}(z,t)=\vec{e}_{t}(\frac{\hbar\omega_{t}}{2V\epsilon_{0}})^{\frac{1}{2}}\hat{\psi}_{t}(z,t)e^{-i(\omega_{t}t-\vec{k}_{t}\cdot\vec{z})}+h.c.\\\
$ (5)

Here $\hat{\psi}(z,t)$ is the wave envelope.

This model works on a reduced 4 level Hilbert space. The Hamiltonian is

$\hat{H}=\hat{H}_{0}+\hat{V}$ (6)

The $\hat{H}$ is the free Hamiltonian of the four system without the
interaction with all the fields. If we use the free Hamiltonian energy
eigenstates $\ket{n}$ as our basis, the free Hamiltonian have the form

$\hat{H}_{0}=\begin{bmatrix}\hbar\omega_{1}&0&0&0\\\ 0&\hbar\omega_{2}&0&0\\\
0&0&\hbar\omega_{3}&0\\\ 0&0&0&\hbar\omega_{4}\\\ \end{bmatrix}$ (7)

The incident fields cause a perturbation $\hat{V}$ of the form

$\hat{V}=-\vec{\hat{d}}\cdot\vec{E}^{phy}$ (8)

where

$\vec{E}^{phy}=\vec{E}^{phy}_{I}+\vec{E}^{phy}_{II}+\vec{E}^{phy}_{s}+\vec{E}^{phy}_{t}$
(9)

and $\vec{\hat{d}}$ is the dipole operator. From our diamond configuration,
the nonzero element of dipole operator are

$\vec{\hat{d}}=\begin{bmatrix}0&\vec{d}_{12}&\vec{d}_{13}&0\\\
\vec{d}_{21}&0&0&\vec{d}_{24}\\\ \vec{d}_{31}&0&0&\vec{d}_{34}\\\
0&\vec{d}_{42}&\vec{d}_{43}&0\\\ \end{bmatrix}$ (10)

To proceed, rotating wave approximation is used to simplify the Hamiltonian.
First decompose the $E$ field into positive and negative rotating part. Pump I
physical field can be written as

$\vec{E}_{I}^{phy}(z,t)=\vec{E}_{I}^{+}(z,t)+\vec{E}_{I}^{-}(z,t)$ (11)

where

$\vec{E}_{I}^{+}(z,t)\equiv\vec{E}_{I}(z)e^{-i(\omega_{I}t-\vec{k}_{I}\cdot\vec{z})}$
(12)

and $\vec{E}_{I}^{-}(z,t)$ equal the complex conjugate part. Similar
decomposition can be done with the dipole operator. We write the dipole
operator from equation (10) as

$\begin{split}\vec{\hat{d}}=&\vec{d}_{12}\ket{1}\bra{2}+\vec{d}_{21}\ket{2}\bra{1}+\\\
&\vec{d}_{13}\ket{1}\bra{3}+\vec{d}_{31}\ket{3}\bra{1}+\\\
&\vec{d}_{24}\ket{2}\bra{4}+\vec{d}_{42}\ket{4}\bra{2}+\\\
&\vec{d}_{34}\ket{3}\bra{4}+\vec{d}_{43}\ket{4}\bra{3}\end{split}$ (13)

We can adjust the energy scale so that the energy of level 1 is the zero
energy reference point. In interaction picture, the first operator on the
right becomes

$\vec{d}_{12}\ket{1}\bra{2}\rightarrow
U^{\dagger}\vec{d}_{12}\ket{1}\bra{2}U=\vec{d}_{12}\ket{1}\bra{2}e^{-i\omega_{21}}$
(14)

It contains a positively rotating phase factor, which means its expectation
value contains the same phase factor, so we write

$\begin{split}\vec{d}_{12}^{+}&\equiv\vec{d}_{12}\ket{1}\bra{2}\\\
\vec{d}_{12}^{-}&\equiv\vec{d}_{21}\ket{2}\bra{1}=\vec{d}_{12}\ket{2}\bra{1}\end{split}$
(15)

Notice we assume that the matrix element of the dipole operator is real. We
can achieve this by choosing the phase factor of the energy eigenkets. This is
always possible because we have $4$ unknown phase factors and $4$
equations(difference of corresponding phase factor is zero), one for each
dipole matrix element from the upper right triangular section of the matrix.
Therefor a solution always exists. With our decomposition of the fields and
the dipole operator, we rewrite perturbation given by equation (8) as

$\begin{split}\hat{V}=-&(\vec{d}_{12}^{+}+\vec{d}_{12}^{-}+\vec{d}_{13}^{+}+\vec{d}_{13}^{-}+\vec{d}_{24}^{+}+\vec{d}_{24}^{-}+\vec{d}_{34}^{+}+\vec{d}_{34}^{-})\cdot\\\
&[\vec{E}_{I}^{+}(z,t)+\vec{E}_{I}^{-}(z,t)+\vec{E}_{II}^{+}(z,t)+\vec{E}_{II}^{-}(z,t)\\\
&\vec{E}_{s}^{+}(z,t)+\vec{E}_{s}^{-}(z,t)+\vec{E}_{t}^{+}(z,t)+\vec{E}_{t}^{-}(z,t)]\end{split}$
(16)

Notice that the dipole operator rotating with the frequency corresponds to the
energy difference between levels while the field rotating with the frequency
of the laser. We then make two approximations. First, only the laser detuning
frequency is small enough to make a significant contribution in the
Schrödinger equation; second, between different levels the frequency
difference is large so that they contribute little in the Schrödinger
equation. Hence laser Pump I only couples level 1 and 3, the same for other
fields. With the above approximations, we can rewrite equation (16) as

$\begin{split}\hat{V}=-[&\vec{d}_{12}^{-}\cdot\vec{E}_{s}^{+}(z,t)+\vec{d}_{13}^{-}\cdot\vec{E}_{I}^{+}(z,t)+\\\
&\vec{d}_{24}^{-}\cdot\vec{E}_{t}^{+}(z,t)+\vec{d}_{34}^{-}\cdot\vec{E}_{II}^{+}(z,t)+h.c]\end{split}$
(17)

For numerical simulation, we transform the system in rotating frame. In
Schrödinger picture, we rewrite the perturbation from equation (17) in matrix
form

$\hat{V}=-\leavevmode\resizebox{359.90538pt}{}{$\begin{bmatrix}0&\vec{d}_{12}\cdot\vec{E}_{s}(z)e^{i(\omega_{s}t-\vec{k}_{s}\cdot\vec{z})}&\vec{d}_{13}\cdot\vec{E}_{I}(z)e^{i(\omega_{I}t-\vec{k}_{I}\cdot\vec{z})}&0\\\
\vec{d}_{12}\cdot\vec{E}_{s}(z)e^{-i(\omega_{s}t-\vec{k}_{s}\cdot\vec{z})}&0&0&\vec{d}_{24}\cdot\vec{E}_{t}(z)e^{i(\omega_{t}t-\vec{k}_{t}\cdot\vec{z})}\\\
\vec{d}_{13}\cdot\vec{E}_{I}(z)e^{-i(\omega_{I}t-\vec{k}_{I}\cdot\vec{z})}&0&0&\vec{d}_{34}\cdot\vec{E}_{II}(z)e^{i(\omega_{II}t-\vec{k}_{II}\cdot\vec{z})}\\\
0&\vec{d}_{24}\cdot\vec{E}_{t}(z)e^{-i(\omega_{t}t-\vec{k}_{t}\cdot\vec{z})}&\vec{d}_{34}\cdot\vec{E}_{II}(z)e^{-i(\omega_{II}t-\vec{k}_{II}\cdot\vec{z})}&0\end{bmatrix}$}$
(18)

Notice that the matrix element is rotating at laser frequencies, which is of
the order of Terahertz. If a computer is to simulate the model, the time steps
should be shorter than the laser period, which would require too much
resources. We need to eliminate the time dependence of the elements. This is
achieved by going into the co-rotating frame of the lasers.

To understand the co-rotating frame, let’s recall the interaction picture. In
interaction picture, the states and operators are transformed by an unitary
operator $U$, which gives a rotating phase factor to energy eigen state as
$e^{-i(\omega_{n}t)}\ket{n}$, where $\omega_{n}$ is the transition frequency
of level $n$ relative to the reference level, which is the ground state
usually.

The transformation to the rotating frame is very similar to the transition to
interaction picture, except that the unitary operator brings a phase factor
rotating at the corresponding laser frequency instead of the atomic transition
frequency. In this 4 level system, we can choose level 4 as our reference
level, so that the unitary operator is

$U_{R}^{\dagger}=\begin{bmatrix}e^{-i[(\omega_{I}+\omega_{II})t-(\vec{k}_{I}+\vec{k}_{II})\cdot\vec{z}]}&0&0&0\\\
0&e^{-i(\omega_{t}t-\vec{k}_{t}\cdot\vec{z})}&0&0\\\
0&0&e^{-i(\omega_{II}t-\vec{k}_{II}\cdot\vec{z})}&0\\\ 0&0&0&1\end{bmatrix}$
(19)

Here the time related phase factor is enough to remove the time dependence.
However, it is convenient to also remove the $z$ dependence of the
Hamiltonian. The trick is to ensure that the $\vec{k}$ terms always have a
opposite sign as the $t$ terms is the phase factor.

Equation (19) is the Hermitian conjugate of the operator $U_{R}$ that
corresponds to the $U$ in interaction picture transformation, since it is the
Hermitian conjugate acts on the states

$\ket{\tilde{n}}=U_{R}^{\dagger}\ket{n}$ (20)

where the tilde means this is the ket in the rotating frame. The Hamiltonian
transforms in such a way to preserve the Shrödinger equation of the rotating
frame states.

$\tilde{H}=U_{R}^{\dagger}HU_{R}+i\hbar(\partial_{t}U_{R}^{\dagger})U_{R}$
(21)

We want to preserve the Schrödinger euqation because the master equation of
the density matrix is based on the Schrödinger equation.

The Hamiltonian in rotating frame is

$\tilde{H}=\hbar\begin{bmatrix}\omega_{1}+\omega_{II}+\omega_{I}&-e^{i\phi}\Omega_{s}&-\Omega_{I}&0\\\
-e^{-i\phi}\Omega_{s}&\omega_{2}+\omega_{t}&0&-\Omega_{t}\\\
-\Omega_{I}&0&\omega_{3}+\omega_{II}&-\Omega_{II}\\\
0&-\Omega_{t}&-\Omega_{II}&\omega_{4}\end{bmatrix}$ (22)

where

$\phi=(-\vec{k}_{I}+\vec{k}_{II}+\vec{k}_{s}-\vec{k}_{t})\cdot\vec{z}+(\omega_{I}-\omega_{II}-\omega_{s}+\omega_{t})t$
(23)

and

$\begin{split}&\Omega_{I}(z)\equiv\frac{\vec{E}_{I}(z)\cdot\vec{d}_{13}}{\hbar}\\\
&\Omega_{II}(z)\equiv\frac{\vec{E}_{II}(z)\cdot\vec{d}_{34}}{\hbar}\end{split}$
(24)

are the laser Rabi frequencies,

$\begin{split}&\Omega_{s}(z)\equiv\frac{1}{\hbar}(\frac{\hbar\omega_{s}}{2V\epsilon_{0}})^{\frac{1}{2}}\vec{e}_{s}\cdot\vec{d}_{12}\\\
&\Omega_{t}(z)\equiv\frac{1}{\hbar}(\frac{\hbar\omega_{t}}{2V\epsilon_{0}})^{\frac{1}{2}}\vec{e}_{t}\cdot\vec{d}_{24}\end{split}$
(25)

are the single photon Rabi frequencies.

We can further simplify the Hamiltonian. First the generated telecom photon
must satisfy the energy conservation and momentum conservation

$\omega_{s}+\omega_{II}=\omega_{t}+\omega_{I}$ (26)
$\vec{k}_{s}+\vec{k}_{II}=\vec{k}_{t}+\vec{k}_{I}$ (27)

use these relation we find the phase factor in equation (23) is $\phi=0$.
Second, the calculation is not changed if we change the reference energy
level. This is achieved by adding multiples of identity operator to
$\tilde{H}$. Notice we can do this even if we are in rotating frame but not in
Schrödinger picture. This is because the transformation given by equation (21)
does not change the diagonal elements of the original $H$, up to some
constants introduced by the second term on the right of equation (21). If we
add $\alpha I$ to $\tilde{H}$, it is the same if we add $\alpha I$ to $H$
first, then perform the transformation given by equation (21). Here, we want
to use the laser detunings to express frequencies on the diagonal elements, so
we add $-(\omega_{1}+\omega_{II}+\omega_{s})\hat{I}$ to $\tilde{H}$

$\tilde{H}=\hbar\begin{bmatrix}0&-\Omega_{s}&-\Omega_{I}&0\\\
-\Omega_{s}&-\Delta_{s}&0&-\Omega_{t}\\\
-\Omega_{I}&0&-\Delta_{I}&-\Omega_{II}\\\
0&-\Omega_{t}&-\Omega_{II}&-\Delta_{41}\end{bmatrix}$ (28)

where $\Delta_{s}=\omega_{s}-(\omega_{2}-\omega_{1})$,
$\Delta_{I}=\omega_{I}-(\omega_{3}-\omega_{1})$,
$\Delta_{II}=\omega_{II}-(\omega_{4}-\omega_{3})$ are the one-photon detunings
and $\Delta_{41}=\Delta_{I}+\Delta_{II}$ is the two-photon detuning.

From equation (20) the density matrix in the rotating frame is

$\tilde{\rho}=U_{R}^{\dagger}\rho U_{R}$ (29)

or express $\rho$ in terms of $\tilde{\rho}$,

$\rho=U_{R}\tilde{\rho}U_{R}^{\dagger}$ (30)

where $\rho$ is the original density matrix in Schrödinger Picture. The
evolution of the rotating frame density matrix is governed by the master
equation

$\dot{\tilde{\rho}}=-i[\tilde{H},\tilde{\rho}]+\sum_{a>0}(L_{a}\tilde{\rho}L_{a}^{\dagger}-\frac{1}{2}L_{a}^{\dagger}L_{a}\tilde{\rho}-\frac{1}{2}\tilde{\rho}L_{a}^{\dagger}L_{a})$
(31)

The polarization of the medium is

$\vec{P}=n\text{Tr}(\rho\vec{\hat{d}})$ (32)

and the light propagation is governed by

$(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}})\vec{E}=\mu_{0}\frac{\partial^{2}\vec{P}}{\partial t^{2}}$ (33)

substitute Eq(4),(5) and (32) into Eq(33), matching the terms with the same
rotating frequency

$(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z})\vec{\psi}_{s}(z,t)=i\frac{\omega_{s}nV}{\epsilon_{0}}\tilde{\rho}_{21}\vec{d}_{12}$
(34) $(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z})\vec{\psi}_{t}(z,t)=i\frac{\omega_{t}nV}{\epsilon_{0}}\tilde{\rho}_{42}\vec{d}_{24}$
(35)

These are the equations couple the atomic states to the field envelopes.
Together with Eq (31), the three equations fully describe the conversion
process. We conduct a numerical simulation to investigate the frequency
response of the conversion process. In the simulation we apply several
simplifications. We only consider an effective four-level system and ignore
the effects from all hyperfine structures. We also completely ignore the
Doppler effect from the motion of the atoms. We simulate a $7cm$ long cell,
the beam diameter is $0.29mm$ and the atomic density is $4\times
10^{17}/m^{3}$. We set the $780nm$ and $1367nm$ pump to be CW light and the
signal $795nm$ to be a Gussian pulse. We found that the conversion efficiency,
which is defined as the ratio of the converted $1324nm$ intensity to the
incident $795nm$ light intensity, is very sensitive to the choice of the
dipole moments. Since we ignore all the hyperfine levels, we do not expect an
accurate conversion efficiency from the numerical simulation. Figure 2b shows
a typical conversion process from the numerical simulation. We calculate the
conversion efficiency as a function of the detunings of the two pump fields,
the results is show in Fig. 3.

Figure 3: Simulation results. a. Conversion output power as a function of the
two pump field detunings. Here $\rm\Delta_{I}$ is the $\rm 780nm$ pump and the
$\rm\Delta_{II}$ is the $\rm 1367nm$ pump. The colorbar is in arbitrary unit.
In this plot $\rm\Delta_{s}=0MHz$ b. Conversion output power as a function of
the signal detuning. In this plot $\rm\Delta_{I}=0MHz$ and
$\rm\Delta_{II}=200MHz$. The conversion power in both figure is in arb.u
because they are highly dependent on the dipole moment value.

###  II.3 Telecom Frequency Conversion in Rubidium (Experiments and
Charaterization)

Figure 4: The frequency conversion setup. An AOM is used to modulate the field
envelope of $\rm 795nm$ signal. The $\rm 795nm$ signal is combined with
$780nm$ pump, and is combined with the $\rm 1367nm$ pump by a long pass
filter. We use a half-wave plate and a quarter-wave plate before the second
Glan Laser Polarizer to compensate the birefringence introduced by the
Rubidium cell. An $\rm InGaAs$ avalanche photodetector is used to detect the
generated $\rm 1324nm$ field.

The frequency conversion setup is shown in Fig. 4. The two pump fields and the
signal $795nm$ beam propagate col-linearly. The $795nm$ and the $1367nm$
fields are H polarized and the $780nm$ light is V polarized. We use a $71.8mm$
long Thorlabs Rubidium reference cell(GC25075-RB) maintained at a temperature
of $\rm 100^{\circ}C$. The cell contains $\rm 5-10mg$ nature composed vapor of
$\rm{}^{85}Rb$ and $\rm{}^{87}Rb$. The attenuation of the $1367nm$ pump field
is achieved by three processes. The generated $1324nm$ is orthogonal to the
$1367nm$ pump field. A pair of Glan Laser Polarizer is used to filter out H
polarized $1367nm$ field. A band-pass filter($1320nm$ center wavelength, FWHM
$12nm$) is then used. Finally, the pumps are slightly misaligned from the
signal to provide a small spatial separation between the $1367nm$ pump and the
generated $1324nm$ beam.

We first measure the frequency conversion efficiency by locking the $795nm$
laser to Rubidium Saturated Absorption Spectrum at transitions
$\rm{}^{85}Rb:5S_{1/2}F=3\rightarrow 5P_{1/2}F^{\prime}=2$ and the $780nm$
laser at $\rm{}^{85}Rb:5S_{1/2}F=3\rightarrow 5P_{3/2}F^{\prime}=4$. We then
lock the $1367nm$ laser to the $\rm 5P_{3/2}F^{\prime}=4-6S_{1/2}F^{\prime}=3$
transition using the two-photon absorption spectrum. We can also modulate the
$795nm$ into pulse envelopes using the AOM. The conversion efficiency is
measured to be around $0.1\%$. We then lock the $795nm$ to
$\rm{}^{85}Rb:5S_{1/2}F=3\rightarrow 5P_{3/2}F^{\prime}=4$ transition and scan
the $780nm$ pump across transition $\rm{}^{85}Rb:5S_{1/2}F=3\rightarrow
5P_{1/2}F^{\prime}=2$ at 10.1 Hz and scan the $1367nm$ pump across transition
$\rm 5P_{3/2}F^{\prime}=4-6S_{1/2}F^{\prime}=3$ at 10 Hz. We record the
reading of the photodetector in an oscilloscope and obtain the conversion
intensity as a function of the pump fields detunings, as shown in Figure 5.

Figure 5: The measured conversion intensity as a function of the pump fields
laser Pizeo voltage offset. The colorbar is in arbitrary unit. In a coarse
approximation, the Pizeo voltage offset can be regarded as linear to the
frequency detuning. However, more in depth calibration should be done to
translate the voltage offset to frequency detunins. The zero point of the
voltage offset does not necessary corresponds to the zero detuning position.
The graininess is due to the limited data points density.

##  III Long Distance Quantum Network Infrastructure

###  III.1 Long Distance Fiber Infrastructure

A practical quantum network, used to interconnect quantum systems over large
distances in real-world configurations, needs to co-exist and be compatible to
the largest degree possible with existing classical telecommunication network
infrastructure. This is the litmus test for bringing quantum networking
technology out of the confines of a laboratory without the requirement of a
new infrastructure that can be prohibitively costly. Furthermore, the co-
existence of quantum and classical signals is an open research topic. While it
is feasible to create fully transparent, all-optical, quantum networks
utilizing the available dark fiber infrastructure and use separate channels
for quantum and classical traffic, extensive research is required to determine
if and how it would be possible to have quantum and classical signals share
the same optical fibers while avoiding significant cross-talk and reliably
maintaining the quantum information over long distances.

The SBU/BNL quantum network prototype interconnects quantum laboratories at
SBU and BNL, taking advantage of the existing optical fiber infrastructure
within the campuses and a commercially available long-distance optical fiber
links. There are four single-mode fiber strands available that are used for
quantum and classical signals, enabling a wide range of experiments. We
utilize dedicated dark fibers for the quantum signals while all classical
signals are carried in separate classical-signal-only fibers, except for the
polarization reference frames, which co-propagate with the qubits. The long-
distance fiber link provides connectivity between the main networking hub at
each campus with an average attenuation of approx. 22.7dB at 1310nm. It
consists of a long segment and a shorter one, comprising multiple fusion-
spliced shorter segments, patched together in a co-location room near the SBU
campus. At SBU and BNL, the long-distance fibers are patched to the local
infrastructure fibers that interconnect the various campus buildings. At both
SBU and BNL, these fibers comprise multiple segments, fusion and/or
mechanically spliced, as well as fiber patches between distribution frames. At
SBU, the fiber path runs from the university’s main networking hub (ECC
building) to the Physics building’s QIT lab. At BNL, the fiber path also runs
from the main networking hub (ITD building) to the QIST lab in the
Instrumentation Division building. In addition to the overall physical length
one must also account for the patch fibers within the networking hubs and the
patch fibers from the fiber panels within each quantum lab to the quantum
memories and detectors. Overall, the physical fiber length from Alice to Bob
through Charlie is estimated at approximately 140km; however, the total
measured loss is 23dB, corresponding to a perfect fiber length (with a typical
loss of 0.35dB/km at 1310nm) to over 160km.

Figure 6: Overview of the testbed on Long Island, New York. It consists of 4
commercially available fibers (rented from Crown Castle) connecting the main
fiber exchange points in Stony Brook University (ECC Engineering Building) and
Brookhaven National Laboratory (SDCC Building). From ECC, several fiber
strands connect to the Quantum Internet Laboratory (QIT) in the Physics and
Astronomy Building. From SDCC, several strands connect to the Quantum
Information Science and Technology Laboratory (QIST) in the Instrumentation
Building. Two quantum memories and frequency conversion setups (Alice and Bob)
are located on the SBU QIT laboratory and are connected independently to the
network. The interference setup and telecom compatible single photon nano-wire
detectors (Charlie station) are located in the QIST laboratory in BNL. The
other two fibers are used to transport classical timing triggers, network
status, and sequencing information.

###  III.2 Polarization Compensation

Since in quantum networking applications, the quantum information is often
encoded in the polarization state (SOP) of photons, a particular bottleneck in
these long-distance applications is that of polarization fluctuations, which
is rarely of concern in classical telecommunication networks. If fibers were
perfect (e.g. polarization-maintaining), the signal SOP would remain mostly
constant, eliminating the need for compensation. However, millions of miles of
deployed non-polarization maintaining fiber infrastructure already exist and
should be utilized given the requiremens needed for entanglement
communication. In these optical fibers the SOP of the light propagating varies
along its length due to random birefringence induced by thermal changes,
mechanical stress, or fiber core irregularities and other changing
environmental and material inhomogeneities - influencing both polarization
axes simultaneously. These fluctuations can also lead to variations in the
optical path length, resulting in fluctuations/drift of power. We have
established the dynamic compensation of polarization drift to maintain quantum
information integrity. Here we use a quantum-state compensator for flying
photons to maintain their polarization, ensuring that transmitted qubits’
purity is optimal at their destination/measurement station. The SOP can be
changed using compensation retardation plates or by actively modifying the
birefringence in the optical fiber, as was performed here. Our experiment uses
a prototype of a machine learning enhanced polarization compensating device
built by Qunnect Inc (Qu-APC module). The module requires a triggered string
of reference photons interspersed between the qubit carrying photons. The
method has the advantage of measuring the polarization of photons at the same
transmission wavelength as the qubits. The polarization controller itself
consists of three motorized rotating paddles with fiber loops acting as wave-
plates on each of the Alice and Bob branches to rotate the light’s
polarization-state with feedback from a polarimeter capable of remapping the
light’s SOP down to -70dB. The retardation plates effectively rotate around
the optical beam (fiber) with respect to each other, inducing strain on the
fiber, and changing the polarization of the photons travelling through.

Figure 7: Principle of operation for polarization compensation. The initial
qubit polarization is represented by the green vector mapped onto a sphere. As
the H qubit (left) transmits through the fiber, fluctuations in the
birefringence cause the polarization to drift (defined as the product of the 3
Stokes vectors shown in black) from its initial state (trending towards red).
The compensation device modulates the fiber stress to compensate for this
drift and restore the initial state (right).

Using rotating wave-plates allows to easily correct for the random
polarization drifts by letting the state randomly “walk-off” within a specific
interval before correcting for it. Therefore, at every acquisition cycle,
Alice sends a “calibration start” trigger along with CW polarized light at
-65bB (receiving end). Polarization compensation setup then adjusts both
branches to the same chosen polarization at the measurement station. Once the
two have converged within an acceptable margin of the desired state, it sends
back to Alice a “calibration termination”. Then Alice confirms as she sends
qubits back instead of CW light (corresponding to the “calibration stop”). The
more frequent the calibration is applied, the less time it takes to converge,
but the less network uptime is available. For optimal results and maximal
network uptime, the re-calibration was re-applied every 5 minutes by which
time the drift was too large to continue the interference with sufficient
visibility. Figure 7 shows the effect of the polarization compensation on an
interference observed after the 68km propagation in both branches.

Figure 8: Top: State compensation for out-of-cycle polarization basis
realignment showing the three normalized Stokes vectors S1, S2, and S2 as a
function of time. Here the Qu-APC device is active and cycling its
compensation routine. Grey periods correspond to when the compensation system
is active (network down-time), whereas periods in between correspond to free
drift. Bottom: Compensation score indicating the state of the drift and the
expected need for recalibration. We observe that without compensation, on
average, the available window for experiments is on the order of 2-5minutes
before the drift impends the results (red). With the calibration above
threshold (green), the post-stabilized visibility at the interferometer was
roughly constant at 88%.

###  III.3 Charlie station at Brookhaven

At the Charlie station in BNL, a series of additional wave-plates are used to
compensate for polarization rotation caused by propagation in the lab optical
fibers after the compensation apparatus. Then the pulses retrieved from the
two paths interfere at a $\rm 50:50$ NPBS beamsplitter, and then go to two
single-photon detectors placed in another adjacent laboratory through 10m
light-shielded fibers. These detectors generate a signal every time they
record a hit. Data is then analyzed to calculate the coincidence rate between
the two output arms of the interferometer.

##  IV Long Distance Quantum Interference Experiments

Having quantum memories operating at telecom wavelengths and a long-distance
fiber infrastructure that does not degrade the initial quantum states, allow
us to connect the two quantum light matter interfaces. We now show quantum
interference experiments using the large interferometer connecting SBU and
BNL. Firstly, we present a single photon (SP) Mach-Zehnder interferometer
demostrating the preservation of quantum state trasmited over a 140 km
network. Secondly, we present a two-photon (TP) Hong-Ou-Mandel interference
experiment, showing the indistinguishability of the telecom photons produced
in the quantum memories and the possibility to create entanglement between
them. We mention that first experiments connecting telecom operational light-
matter interfaces and telecom optical links have been shown recently Bock2018
; Leent2020 ; Yu2020 . However, the experiments presented here are covering an
unprecedented distance, one order of magnitude larger that anything attempted
before.

Figure 9: Basic schematic diagrams of the long-distance interference
experiments. Top: The 1367 nm coherent beam at SBU is split and sent down the
two long fiber runs; for single-photon running the initial beam is heavily
attenuated. At the receiving end at BNL the two lines are combined through a
beam splitter whose outputs go to two detectors. The paths are equal length
for the MZ configuration, and an extra 18 km loop is added to one path for the
HOM configuration. Bottom: The input 795 nm beam is split and then sent to two
quantum memories, where it is combined with 1367 nm and 780 nm beams (not
shown) and converted to 1324 nm light, which is then sent on to the two long
fiber runs.

##  V Single Photon (SP) Mach-Zehnder interferometry

The simplest experiment to demonstrate interference after transmission on two
long-distance links is to split a low-intensity beam at the transmitting
station, and then re-combine them through a splitter at the receiving station.
The basic arrangement is shown in Figure 9 and corresponds to a Mach-Zehnder
interferometer with one spatial mode in the fibers.

With a narrow-band laser source we can write the initial state $I$ of the
field before the initial symmetric beamsplitter as a coherent, polarized,
quasi-monochromatic state $\ket{\alpha}_{I,s}$ where $s$ labels the
polarization of the occupied mode. We can then write the state of the two
modes $A$ and $B$ after the splitter as

$\Psi_{AB}=\frac{1}{\sqrt{2}}\ket{\alpha}_{A,s}\,\otimes\,\frac{1}{\sqrt{2}}\ket{\alpha}_{B,s}$
(36)

After transmission the state of the modes at points $C$ and $D$ are still
coherent but attenuated

$\Psi_{CD}=\frac{t_{AC}}{\sqrt{2}}\ket{\alpha}_{C,p}\,\otimes\,\frac{e^{i\phi}\;t_{BD}}{\sqrt{2}}\ket{\alpha}_{D,q}$
(37)

where (i) the intensities along the $A\rightarrow C$ and $B\rightarrow D$ are
reduced by (real) transmission coefficients $T_{AC}={t_{AC}}^{2}$ and
$T_{BD}={t_{BD}}^{2}$, respectively; (ii) the factor $e^{i\phi}$ encodes any
phase difference resulting from any length differences between the two long
fibers; and (iii) the polarizations of the modes occupied after transmission
are denoted $p$ and $q$; in the absence of polarization compensation the
transforms $s\rightarrow p$ and $s\rightarrow q$ will generally be random.

The state in the modes $E$ and $F$ after the final beamsplitter then becomes

$\Psi_{EF}=\left(\frac{t_{AC}}{2}\ket{\alpha}_{E,p}+\frac{e^{i\phi}\;t_{BD}}{2}\ket{\alpha}_{E,q}\right)\,\otimes\,\left(\frac{t_{AC}}{2}\ket{\alpha}_{F,p}-\frac{e^{i\phi}\;t_{BD}}{2}\ket{\alpha}_{F,q}\right)$
(38)

From standard photo-detection theory we can now calculate the rates of
detectable photons in $E$ and $F$ at any specific time as

$R_{E}(t)=\frac{|\alpha|^{2}}{2}\frac{T_{AC}+T_{BD}}{2}(1+\mathcal{V}\cos\phi(t))\hskip
36.135ptR_{F}(t)=\frac{|\alpha|^{2}}{2}\frac{T_{AC}+T_{BD}}{2}(1-\mathcal{V}\cos\phi(t))$
(39)

where $|\alpha|^{2}$ is the original beam intensity in photons per coherence
time, $(T_{AC}+T_{BD})/2$ is the average transmission through the two fiber
runs. This is the standard Mach-Zehnder interference for a macroscopic beam in
a single spatial mode, where the intensities at the two outputs are directly
anti-correlated. It is interesting to note, that even though the entangled
state is travelling through both fibers, the loss in photon intensity
corresponds to only the average loss in a single fiber and not the product of
two losses.

Here $\mathcal{V}$ is the visibility of the M-Z interference pattern

$\mathcal{V}\equiv\frac{2}{t_{AC}/t_{BD}+t_{BD}/t_{AC}}\;|\langle
p|q\rangle|=\frac{1}{\cosh(\log(t_{AC}/t_{BD}))}\;|\langle p|q\rangle|$ (40)

The first factor relates to the balance between the transmission losses along
the two fiber runs, and the second factor is the magnitude of the overlap
between the two final polarization states $p$ and $q$. If the beams are
balanced on arrival and the polarization states are the same then the
visibility will be unity; as either of these is not satisfied then the
visibility will decrease.

###  V.1 Single-photon interference

The anti-correlated intensities described in Equation 39 are true for any
coherent state, with any intensity $|\alpha|^{2}$. However, for beams at the
single-photon level the M-Z interferometer can be seen as transporting an
entangled state along macroscopic distances.

The coherent beam state as defined by Glauber can be expanded in a Fock basis
as

$\ket{\alpha}=e^{-|\alpha|^{2}}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}$
(41)

with the average occupation number of state $\ket{\alpha}$ being $\langle
n\rangle=|\alpha|^{2}$. For a purely monochromatic beam this would be the
number of photons in the mode over infinite time; for a quasi-monochromatic
beam it can be taken to represent the number of photons in the beam per
interval of the coherence time, as above.

For a weak, highly attenuated initial beam $|\alpha|^{2}\ll 1$ and so we can
expand Equation 36 as

$\Psi_{AB}=\frac{e^{-|\alpha|^{2}}}{\sqrt{2}}\left(\ket{0}_{A,s}+\alpha\ket{1}_{A,s}+\mathcal{O}(\alpha^{2})\right)\,\otimes\,\frac{e^{-|\alpha|^{2}}}{\sqrt{2}}\left(\ket{0}_{B,s}+\alpha\ket{1}_{B,s}+\mathcal{O}(\alpha^{2})\right)$
(42)

Limiting our observations to the cases where at least one photon is detected
effectively post-selects against the $\ket{0}_{A,s}\otimes\ket{0}_{B,s}$
branch of the wavefunction. Expanding the reduced wavefunction to leading
order in $\alpha$ we have

$\Psi_{AB}=\frac{\alpha}{\sqrt{2}}\left(\ket{1}_{A,s}\otimes\ket{0}_{B,s}\,+\,\ket{0}_{A,s}\otimes\ket{1}_{B,s}\right)$
(43)

Equation 43 is then an entangled state, a superposition between a single
photon having gone down the $A$ channel and the $B$ channel. If we can observe
an interference between the two terms at the receiving end then we confirm
that the entangled superposition is preserved in transit, i.e. no “which-way”
information was available to the outside world.

If the state we $\Psi_{AB}$ in Eq. 43 propagated through the fiber
transmission and the beamsplitter without decoherence or interaction with the
outside world, then as in Sec. V we can derive the (complementary)
probabilities of seeing a photon at $E$ versus $F$, conditioned on a photon
having been observed at one or the other:

$P_{E}(t)=\frac{1}{2}(1+\mathcal{V}\cos\phi(t))\hskip
36.135ptP_{F}(t)=\frac{1}{2}(1-\mathcal{V}\cos\phi(t))$ (44)

Here the visibility $\mathcal{V}$ is the same as shown in Eq. 38. If we can
show that these probabilities are observed in the single-photon limit then it
will confirm that transmission over the long fibers preserves the entangled
state in Equation 43. By contrast, if the state were to de-cohere, de-polarize
or otherwise interact non-reversibly with the environment then the
interference would not manifest and the observation probabilities would reduce
to $P_{E}=P_{F}=1/2$.

It may not be straightforward to measure the probabilities of Eq. 44 directly
if the photon rates are low and the phase difference $\phi(t)$ is changing
quickly. We can instead create a correlation observable, following the simple
idea that if photons are, say, generally being steered toward output $E$ then
to photons observed close together in time will both tend to register at $E$;
and this will generate a net suppression in observing $E$-$F$ pairs which are
close together in time.

With individual photon detection we measure the standard normalized
correlation function

$G^{(2)}(\Delta t)\equiv\frac{\langle P_{E}(t)P_{F}(t+\Delta
t)\rangle_{t}}{\langle P_{E}(t)\rangle_{t}\,\langle P_{F}(t+\Delta
t)\rangle_{t}}$ (45)

With Equation 44 this becomes for the Mach-Zehnder configuration

$G^{(2)}(\Delta
t)=1-\mathcal{V}^{2}\langle\cos(\phi(t))\cdot\cos(\phi(t+\Delta
t))\rangle_{t}=1-\mathcal{V}^{2}/2\textrm{ for }\Delta t\rightarrow 0\textrm{
or }1,\textrm{ for large }\Delta t$ (46)

and so if the coherence between the two branches of the superposition is
preserved then we would expect to observe a dip feature in the $G^{(2)}(\Delta
t)$, with a maximum depth of $1/2$ for unit visibility; and then a return to
$G^{(2)}=1$ when $\Delta t$ is much larger than the time scale on which
$\phi(t)$ is fluctuating. It should be noted that this anti-correlation
between two photon detections is not a two-photon interference effect; it is
rather the correlation in time between two single-photon interference effects.
We will examine true two-photon interference in Section VI below.

Figure 10: Measured $G^{(2)}$ correlation function between single-photon
detections at the two outputs of the final beamsplitter in the Mach-Zehnder
configuration. Here we are observing 1324 nm photons which have been produced
by a 795 nm photon being split and then converted in two quantum memories, as
in Figure 9. The dip follows the expectation from Equation 46, with a
visibility of about $\mathcal{V}=0.9$ and a feature width of about a
millisecond.

Figure 10 shows the $G^{(2)}(\Delta t)$ correlation function between single-
photon hits detected at the two output channels of the final beamsplitter, in
the M-Z configuration, as a function of the time separation between the
detected hits. A clear anti-correlation is seen for small $\Delta t$, and a
return to $G^{(2)}=1$ at large $\Delta t$, exactly as expected from Equation
46. The feature’s width measures the timescale on which the induced phase
difference $\phi(t)$ is varying, here on the order of a milliseconds. These
data were taken with polarization optimization at the receiving end. The dip
has a minimum value of about $G^{(2)}=0.6$, indicating to a very high
visibility of $\mathcal{V}\sim 0.9$ and so a correspondingly good match
between the final polarizations.

The interference results in Figure 10 were measured with 1324 nm light, as
shown in the lower panel of Figure 9. Thus, in this experiment the first
entangled state of the type shown in Equation 43 is present with the 795 nm
photons at the initial mode points $I$ and $J$. That superposition state is
then put through frequency conversion to 1324 nm light in the two quantum
memory vapor cells, to emerge at mode points $A$ and $B$ before being sent
onto the long fiber runs. The fact that the Mach-Zehnder interference is
observed with high visibility at the receiving end indicates that the
entangled superposition is well preserved through both the conversion process
and the long-distance transmission stage.

##  VI Two-photon (TP) Hong-Ou-Mandel interferometry

To demonstrate that our memories are capable of being entangled by
measurement, we must demonstrate that the quantum information is preserved
after frequency conversion, retrieval and long distance transmission. The
application of the Cabrillo entanglement scheme requires that the temporal
envelope, optical frequency, and polarization of both photon streams produced
in the Alice and Bob stations remain indistinguishable at the input of the
interference experiment. Therefore, we calibrate the relevant parameters of
the quantum memories to ensure that Alice and Bob have identical FWM
bandwidths and conversion efficiencies. Two independent filtering systems (to
filter out the auxiliary frequency conversion fields) located after the atomic
interfaces, each consisting of several consecutive wavelength filters, are
calibrated to have identical transmissions for Alice and Bob. After carefully
matching all the auxiliary fields parameters, we proceed to frequency convert
qubits at the few-photon level in the two memories and couple their output
into the long distance fibers.

Figure 11: Correlation $G^{(2)}(\Delta t)$ measurements between single-photon
detections of 1324 nm light produced through frequency conversion, as per the
lower panel of Figure 9. These were taken in the HOM configuration with the
extra delay loop on one path; this delay is much longer than the coherence
time of the 1324 nm beam and so ensures that the photons arriving at the
receiver are mutually incoherent. We observe the Hong-Ou-Mandel (HOM)
interference, manifested as a depletion of observed pairs at the beamsplitter
output ports, when the detection time difference is on the order of the beam
coherence time. Upper: The magnitude of the HOM dip is small when the
polarization compensation is deliberately set badly. Middle: Without any
polarization adjustement the HOM dip reaches a minimum of of $G^{(2)}=0.75$,
which is typical for a random alignment. Lower: With optimization in
polarization compensation the HOM minimum is as low as $G^{(2)}=0.65$
indicating a high degree of indistinguishability between the arriving photons.

The mean number of photons per pulse at the output of each memory is measured
to be $\rm\langle n\rangle_{Alice}=20$ and $\rm\langle n\rangle_{Bob}=20$. As
indicatd in Fig. 9, in this experiment we add a 18km long delay on the BNL
side in order to scramble the phase coherence and measure the second order
correlation. The losses through both the long distance setups is measured to
be $\rm\approx 23dB$ for one link and $\approx 26dB$ for the longer arm. We
obtain average photon numbers after long distance propagation of $\rm\langle
n\rangle_{Alice}\approx 0.1$ and $\rm\langle n\rangle_{Bob}\approx 0.06$ at
the nanowire detectors in BNL. The HOM coincidence rate is measured versus the
arrival time of the photons in the two detectors. The coincidences within a
temporal region of interest (ROI) are post-selected with a width of $0.1\mu
s$. We observe the desired modulation in the coincidence rate, exhibiting a
minimum for the initial identical polarization and reaching a maximum
corresponding to uncorrelatd photons beyond the coherence time of the FWM
process. The interference visibility is measured to be $V=(38\pm 2.0)\%$. We
conclude that our memories do not significantly affect the
indistinguishability of the photons.

##  VII Discussion and Outlook

Our experimental scheme reflects the original idea of the Cabrillo scheme
Cabrillo1999 , where entanglement is produced based on a measurement
projection, applied in a long distance 158 km network. In our case, the two
atomic quantum memories are initially prepared in the in $\ket{1}$ states
(ground state in Fig. 2). The 795 nm weak excitation pulse, together with the
FWM pump beams excites the atoms to state $\ket{4}$ (excited state in Fig. 2).
The atoms can then decay to the intermediate state $\ket{2}$ by spontaneous
emission (creating the 1324 nm photons), with a small probability given by the
efficiency of the FWM process. After the photons propagation in the long
distance links, the detection of one photon in either detector after the HOM
interference projects the state of the two memories to
$\ket{\phi}=\frac{1}{\sqrt{2}}(\ket{12}+e^{i\theta}\ket{21})$. The phase
$\theta$ is due to the phase difference of the weak excitation pulses applied
to the two memories and the memory-detector path difference. Future
experiments will aim to store these entangled state of the memories. An
important constrain will be to have long coherence times on the order of
hundreds of microseconds, allowing for the long-distance transmission of the
telecom photons and the heralding information transmission. With these
conditions it will possible to verify the entanglement created between the
memories by retrieving their state into an entangled photonic state and
reconstructing its density matrix.

##  VIII Aknowledgements

This work is supported by the Department Of Energy ASCR grant: “Inter-campus
network enabled by atomic quantum repeater nodes”, the Department Of Energy
CESER grant: “A Prototype for Ultimate Secure Transmission” and the Brookhaven
National Laboratory LDRD grant, “Building a quantum repeater prototype
connecting BNL to NYC”.

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