# Generation of multiphoton entangled quantum states with a single silicon
nanowire

Ming Zhang111missing State Key Laboratory for Modern Optical Instrumentation,
Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key
Laboratory for Sensing Technologies, Zhejiang University, Zijingang Campus,
Hangzhou 310058, China.  Lan-Tian Feng111These authors contributed equally to
this work. Zhi-Yuan Zhou  Yang Chen  Key Laboratory of Quantum Information,
University of Science and Technology of China, CAS, Hefei, 230026, People’s
Republic of China.  Synergetic Innovation Center of Quantum Information $\&$
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China.  Hao Wu  Ming Li  State Key Laboratory for Modern Optical
Instrumentation, Centre for Optical and Electromagnetic Research, Zhejiang
Provincial Key Laboratory for Sensing Technologies, Zhejiang University,
Zijingang Campus, Hangzhou 310058, China.  Guo-Ping Guo  Guang-Can Guo  Key
Laboratory of Quantum Information, University of Science and Technology of
China, CAS, Hefei, 230026, People’s Republic of China.  Synergetic Innovation
Center of Quantum Information $\&$ Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China.  Dao-Xin
<EMAIL_ADDRESS>State Key Laboratory for Modern Optical
Instrumentation, Centre for Optical and Electromagnetic Research, Zhejiang
Provincial Key Laboratory for Sensing Technologies, Zhejiang University,
Zijingang Campus, Hangzhou 310058, China.  Xi-Feng<EMAIL_ADDRESS>Key
Laboratory of Quantum Information, University of Science and Technology of
China, CAS, Hefei, 230026, People’s Republic of China.  Synergetic Innovation
Center of Quantum Information $\&$ Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China.

Multiphoton entanglement plays a critical role in quantum information
processing, and greatly improves our fundamental understanding of the quantum
world. Despite tremendous efforts in either bulk media or fiber-based devices,
nonlinear interactions in integrated circuits show great promise as an
excellent platform for photon pair generation with its high brightness,
stability and scalability Caspani2017 . Here, we demonstrate the generation of
bi- and multiphoton polarization entangled qubits in a single silicon nanowire
waveguide, and these qubits directly compatible with the dense wavelength
division multiplexing in telecommunication system. Multiphoton interference
and quantum state tomography were used to characterize the quality of the
entangled states. Four-photon entanglement states among two frequency channels
were ascertained with a fidelity of $0.78\pm 0.02$. Our work realizes the
integrated multiphoton source in a relatively simple pattern and paves a way
for the revolution of multiphoton quantum science.

Optical quantum states with entanglement shared among several modes are
critical resources for studies in quantum communication Kimble2008 ,
computation Walther2005 ; Humphreys2013 , simulation Aspuru-Guzik2012 , and
metrology Afek2010 . Therefore, the controllable and scalable realization of
multiphoton quantum states would enable a practical and powerful
implementation of quantum technologies. For generating and manipulating photon
pairs, nonlinear interactions in integrated circuits show great promise as an
excellent platform with its high brightness, stability and scalability.
Recently, optical integrated kerr frequency combs in high-refractive-index
glass platform were used to generate time-bin bi- and multiphoton entangled
qubits Reimer2016 , which started a new time to exploit multiphoton entangled
states with the help of quantum photonic integrated circuits.

The silicon-on-insulator (SOI) photonic circuit is a very promising platform
for realizing complex quantum states for its strong third-order optical
nonlinearity, low nonlinear noise, small footprint, mature fabrication
technics and compatibility with complementary metal oxide semiconductor (CMOS)
electronics Bogaerts2005  as well as telecom techniques. Multiple biphoton
entanglement sources have been manipulated and further used for large-scale
integration and quantum information processings Li2017 ; Silverstone2014 ;
Harris2014 ; Wang2016 ; Wang2017 ; Paesani2017 . Nevertheless, whether the
silicon photonic circuit can be used to manipulate multiphoton entangled
source is still unexplored.

Figure 1: Schematic showing the generation and measurement of the multiphoton
entanglement states from a silicon nanowire. A pulse erbium laser with a
repetition frequency $100MHz$ was involved as the pump light. After forward
and backward filtering with $100GHz$ bandwidth, it was input into a Sagnac
loop for the generation of polarization entanglement states. One DWDM system
was used to divide photon pairs into multiple frequency channels, and normal
architectures for polarization state tomography were used to ascertain the
quality of the entangled quantum states.

In this study, we use the silicon-on-insulator (SOI) photonics platform to
explore the generation of multiphoton entangled states. A single silicon
nanowire waveguide, which has a length of $1\ cm$ and transverse dimension of
$\sim 450\ nm\times 220\ nm$, is employed to generate photon pairs. The
external pump laser is coupled to the waveguide with grating couplers. The
waveguide has a propagation loss of $1\ dB$, and the total insertion loss of
this structure is $11\ dB$, including $5\ dB$ coupling loss each at the input
and output grating coupler.

Polarization encoding was chosed to demonstrate the multiphoton entanglement
generation, since it is popular for various quantum information applications
Kok2007 . The experimental setup for generation and measurement of the
multiphoton polarization entangled states is shown in Fig. 1. A pulsed erbium
laser with a repetition frequency $100\ MHz$ was involved for the degenerate
spontaneous four wave mixing process in the silicon nanowire. After forward
and backward filtering with $100\ GHz$ bandwidth, the pump light was passed
through a polarization controller (PC) and a fiber circulator, and then went
into a Sagnac loop Takesue2008 . The Sagnac loop consists of half-wave plates
(HWPs), quarter-wave plates (QWPs), a polarization beam splitter (PBS) and the
chip of silicon nanowire. The combination of the HWP and QWP is used to
modulate the optical polarization for maximum grating coupling. In the Sagnac
loop, pump beam was split into clockwise and counterclockwise circulation
directions by the PBS, and the HWP before the PBS is used for optical power
modulation. In each direction, time correlated signal and idler photons are
generated and their frequencies are equally separated from the central pump
frequency. In the clockwise circulation direction, the photon pair generated
is $\left|V_{s}V_{i}\right\rangle$, while in the counterclockwise circulation
direction, the photon pair generated is $\left|H_{s}H_{i}\right\rangle$ after
the PBS. The biphoton quantum state at the output port of the loop can be
expressed as

$\left|\Phi\right\rangle=\frac{1}{\sqrt{1+\eta^{2}}}(\left|H_{s}H_{i}\right\rangle+\eta
e^{i\delta}\left|V_{s}V_{i}\right\rangle),$ (1)

where $\eta$ is determined by the ratio of the pump power in the two
circulation directions, and $\delta$ depends on the initial phase of the pump
beam and the birefringence experienced by the pump beam in the $H$ and $V$
polarizations. By modulating the pump power ratio and relative phase, a
maximally polarization entangled Bell state

$\left|\Phi\right\rangle=\frac{1}{\sqrt{2}}(\left|H_{s}H_{i}\right\rangle+\left|V_{s}V_{i}\right\rangle)$
(2)

can be produced.

At the output port of the fiber circulator, the pump laser was blocked by
post-filters with $200\ GHz$ bandwidth. One 40-channel dense wavelength
division multiplexing (DWDM) system was used to separate the signal and idler
photons, whose frequencies are equally separated from the central pump
frequency.. Each channel has a $100\ GHz$ bandwidth, then we can select any
combination of photon pairs with a frequency detuning about $2\ THz$ from the
central pump frequency. The generated photon pairs were finally detected by
superconducting nanowire single-photon detectors (SCONTEL, dark count rate
$100\ Hz$, detector efficiency $85\%$ at C band).

The entangled photon pairs are frequency multiplexed and generated over all
DWDM frequency channels Li2017 . We selected five signal-idler channels to
ascertain the effectiveness and stability of our system (Supplementary Table
1). Through inputting $120\ \mu W$ pump light into the single cycle of the
Sagnac loop, we recorded two-photon coincidences between different signal-
idler channel combinations (Supplementary Fig. 1). It shows that the crosstalk
is negligible between different frequency channels, even for the adjacent
ones.

Next, we modulated the polarization of the pump light before the PBS of the
Sagnac loop to obtain the maximally entangled states (Eq. 2). Combinations of
QWP, HWP and PBS after the DWDM were used to constitute a normal polarization
state tomography architecture James2001  for characterizing the degree of
entanglement. We took the signal-idler channels 1 for testing the quality of
the entanglement. We first measured the two-photon polarization interference
fringes. The interference is expected to be proportional to
$1+Vsin(2\pi(x-x_{c})/T)$, where $V$ is the fringe visibility, $x_{c}$ is the
initial phase, and $T$ is the oscillation period. The fringe visibility $V$ is
defined as $V=(d_{max}-d_{min})/(d_{max}+d_{min})$, where $d_{max}$, $d_{min}$
are the maximum and minimum fitting data, respectively. Through setting the
angle of the HWP in the signal channel as $0^{\circ}$ or $45^{\circ}$, and
modulating the angle of the HWP in the idler channel, we obtained two
interference fringes (Fig. 2a). The raw visibilities in the $0^{\circ}$ (solid
red line) and $45^{\circ}$ (solid black line) bases were $96.1\pm 3.2\%$ and
$93.0\pm 3.2\%$, respectively, which, being greater than
$\frac{1}{\sqrt{2}}\approx 70.7\%$, confirming entanglement through the
violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. In this case,
we estimated the photon pair generation rate of $270\ kHz$ per channel (0.0027
pairs per double pulse), accounting for system and detection losses of $16\
dB$ (see methods).

Figure 2: Quantum interference fringes of two-photon and four-photon
polarization entangled states. Two-fold and four-fold coincidences as a
function of the idler polarizer angles when the signal polarizer angles were
kept at $0^{\circ}$ (solid red line) and $45^{\circ}$ (solid black line),
respectively. Four-photon entangled states unfold totally different
interference patterns from the two-photon entangled states. The error bar was
obtained from the square root of the experiment data.

Now we consider the multiphoton cases. In principle, the generated multiphoton
entanglement state for our experimental setup can be expressed as

$\left|\Phi^{2n}\right\rangle=\frac{1}{\sqrt{2^{n}}}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)\otimes\cdots\otimes(\left|H_{sn}H_{in}\right\rangle+\left|V_{sn}V_{in}\right\rangle).$
(3)

That is, combination of different correlated signal-idler channels could be
used to produce multiphoton entangled states. For example, by selecting the
signal-idler channels 1 and 5, we can get two two-photon entangled states,
given by
$\left|\Phi_{1}\right\rangle=\frac{1}{\sqrt{2}}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)$
and
$\left|\Phi_{5}\right\rangle=\frac{1}{\sqrt{2}}(\left|H_{s5}H_{i5}\right\rangle+\left|V_{s5}V_{i5}\right\rangle)$,
respectively. By post-selecting four-photon events with one photon on each
frequency channel, these two states are multiplied, resulting in four-photon
polarization entangled states, which is given by

$\left|\Phi^{4}\right\rangle=\frac{1}{2}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)\otimes(\left|H_{s5}H_{i5}\right\rangle+\left|V_{s5}V_{i5}\right\rangle).$
(4)

To prove this state, we measured the four-photon quantum interference fringes,
which generally cannot be present for two completely independent two-photon
qubit states. By setting the pump power to $600\ \mu W$, we obtained a four-
fold coincidence rate of $0.34\ Hz$, which corresponds to a calculated
generation rate of $340\ kHz$, taking into account the system and detection
losses of $15\ dB$ (see methods). The interference is expected to be
proportional to $1+(1-\sqrt{1-V^{2}})/V)sin^{2}(2\pi(x-x_{c})/T$. By setting
the angle of the HWPs in the signal channels as $0^{\circ}$ or $45^{\circ}$,
and modulating the angle of the HWPs in the idler channels simultaneously, we
obtained two four-photon interference fringes (Fig. 2b). The raw visibilities
in the $0^{\circ}$ (solid red line) and $45^{\circ}$ (solid black line) bases
were $96.5\pm 1.5\%$ and $99.1\pm 0.4\%$, respectively. Four-photon entangled
states unfold totally different interference patterns from the two-photon
entangled states and the high interference visibilities prove the existence of
the multiphoton entanglement. Note that, for the measurement of quantum
interference fringes, the QWPs were moved out from the setup.

Figure 3: Quantum state characterization through state tomography. A quantum
state can be fully described by its density matrix $\widehat{\rho}$. The real
(Re) and imaginary (Im) parts of the ideal density matrices of two- and four-
photon entangled qubit states are shown in a and c, respectively. The measured
density matrix of the two-photon states b agrees well with the ideal case. The
measured density matrix of the four-photon states d reaches a fidelity of
$0.78\pm 0.02$, which is completely satisfied for quantum information
processings.

Quantum state tomography James2001  was also used to fully characterize the
entangled states. Using this method, we could obtain the experimental state
density matrix and thus study how close of the measured states to the ideal
entangled states. Firstly, we performed the two-photon quantum state
tomography for the frequency channels 1. 16 data at different measurement
bases were acquired to construct the state density matrix. The ideal density
matrix of the maximally entangled states (Eq. 2) and the measured density
matrix of the output states were displayed in Figs. 3a, and 3b, respectively.
We used the maximum-likelihood-estimation method to construct the density
matrix with the experimental results. The fidelity is defined as
$F=Tr(\widehat{\rho}_{exp}\widehat{\rho}_{th})$, where $Tr$ is the trace,
$\widehat{\rho}_{exp}$ is the measured density matrix, and
$\widehat{\rho}_{th}$ is the ideal density matrix. We estimated the fidelity
of $0.94\pm 0.01$, confirming that the generated quantum states were of high
quality and very close to the ideal maximally entangled states. The error of
the fidelity was obtained by 100 times Monto Carlo calculation with the
experimental data subject to Gaussian statistics. The deviation of the
fidelity from unity was mainly due to the unideal rotating the angle of the
wave plates. It is worth pointing out that the quantum states are frequency
multiplexed and entanglement is in any signal-idler channels. As a test, we
also performed the two-photon quantum state tomography for signal-idler
channels 3 and 5, and found the fidelities of $0.97\pm 0.01$ and $0.95\pm
0.01$, respectively (Supplementary Fig. 1).

For the four-photon entangled states, 256 data of different measurement bases
were obtained to construct the state density matrix. The ideal density matrix
of the maximally entangled states (Eq. 4) and the measured density matrix of
the output states from signal-idler channels 1 and channel 5 were displayed in
Figs. 4c, and 4d, respectively. We obtained a fidelity of $0.78\pm 0.02$
without background and accidence subtraction. The high visibility clearly
proved the entanglement and is satisfied for further practical applications.

Since it was demonstrated recently that the frequency encoding can be used for
single-photon frequency shifting Fan2016  and construction of high-fidelity
quantum gates Lukens2017 ; Lu2018  for high-dimensional quantum information
processing, our presented multiphoton entangled source is fully compatible
with these post-processing systems and it will constitutes an indispensable
block for the frequency-based quantum information processing. Furthermore, the
multiphoton entangled state (Eq. 3) can be converted to the Greenberger-Horne-
Zeilinger (GHZ) state through post operation and selection, which is usually
used as the resource for linear-optical quantum computation WangX2016 .

In addition, the entangled photon numbers can be increased Reimer2016 .
Further integration of multiphoton manipulation block in one integrated
circuits Matthews2009  will also lead to more compact and stable systems with
higher performance, resulting better detection rates and higher fidelity. A
scheme to decrease coupling loss by using high coupling efficiency etched
facet tapers Cardenas2014  and low-loss filter system would increase the
brightness of four-fold coincidence greatly, even to the useful kilohertz
range.

In conclusion, we have experimentally shown that a single silicon nanowire can
be used for multiphoton source manipulation. Due to its strong third-order
nonlinearity, we achieved a high brightness photon pair source with a very low
pump power. The fidelity of the four-photon entangled states is high enough
for practical applications. The multiphoton entangled source is directly
compatible with the dense wavelength division multiplexing communication
system and frequency-based post-processing system, thus provided a scalable
and practical platform for quantum information processing.

## Methods

System efficiency. We ascertained the efficiencies of all departments with
laser light measurements. The grating coupler has coupling loss of $5\ dB$.
The system for state manipulation and state measurement has loss of $4.3\ dB$.
The post-filters and WDMs have inherent loss of $6\ dB$ and both detectors
have efficiency of $85\%\ (-0.7\ dB)$. The total loss is $16\ dB$ for two-
photon state measurements and $15\ dB$ for four-photon state measurements
where we decreased the loss of $1\ dB$ in post-processing system through finer
adjustment.

Optical apparatus. We used pulsed erbium laser (repetition frequency $100\
MHz$) to generate the pump light (Supplementary Fig. 2). After filtered, the
left laser was amplified by an erbium-doped fiber amplifier and filtered again
before input into the source manipulation setup. Fiber alignment was
maintained using a piezo-controlled four-dimensional displacement table for
position and coupling angle adjustment. The coupling angle was set as
$10^{\circ}$ for fiber-chip coupling. Two cascaded off-chip post-filters
($100\ dB$ extinction ratio) were used to remove the pump photons, and one
DWDM system ($30\ dB$ extinction ratio for adjacent channel and $50\ dB$ for
non-adjacent channel) was used to separate the signal and idler photons. The
correlated photons were recorded by superconducting nanowire single-photon
detectors. The electrical signals were collected and analyzed through time-
correlated single photon counting (TCSPC) system, and the coincidence window
was set as $0.8\ ns$ for two-photon state measurements. For four-photon state
measurements, the electrical signals were analyzed by the UQD-Logic and the
coincidence window was set as $1\ ns$.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(NSFC) (Nos. 61590932, 11774333, 61725503, 61431166001), Anhui Initiative in
Quantum Information Technologies (No. AHY130300), the Strategic Priority
Research Program of the Chinese Academy of Sciences (No. XDB24030600), the
National Key R & D Program (No. 2016YFA0301700), Zhejiang Provincial Natural
Science Foundation of China (Z18F050002), and the Fundamental Research Funds
for the Central Universities. This work was partially carried out at the USTC
Center for Micro and Nanoscale Research and Fabrication.

## Author contributions

All authors contributed extensively to the work presented in this paper. M.Z.,
Z.Y.Z., H.W., M.L. and D.X.D. prepared the samples, L.T.F., M.Z., D.X.D. and
X.F.R. performed the measurements, data analyses and discussions. Y.C., G.P.G.
and G.C.G. conducted theoretical analysis. X.F.R. and D.X.D. wrote the
manuscript and supervised the project.

## Additional information

Supplementary information is available in the online version of the paper.
Correspondence and requests for materials should be addressed to D.X.D or
X.F.R.

## Competing financial interests

The authors declare no competing financial interests.

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