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Complex optical photon states with entanglement shared among several modes are critical to improving our fundamental understanding of quantum mechanics and have applications for quantum information processing, imaging, and microscopy. We demonstrate that optical integrated Kerr frequency combs can be used to generate several bi- and multiphoton entangled qubits, with direct applications for quantum communication and computation. Our method is compatible with contemporary fiber and quantum memory infrastructures and with chip-scale semiconductor technology, enabling compact, low-cost, and scalable implementations. The exploitation of integrated Kerr frequency combs, with their ability to generate multiple, customizable, and complex quantum states, can provide a scalable, practical, and compact platform for quantum technologies.
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5. Y. V. Shtogun, L. M. Woods, . J. Phys. Chem. Lett. 1, 13561362
(2010).
6. W. Gao, A. Tkatchenko, Phys. Rev. Lett. 114, 096101 (2015).
7. A. M. Reilly, A. Tkatchenko, Phys.Rev.Lett.113,055701(2014).
8. N. Ferri, R. A. DiStasio Jr., A. Ambrosetti, R. Car, A. Tkatchenko,
Phys. Rev. Lett. 114, 176802 (2015).
9. A. J. Stone, The Theory of Intermolecular Forces (Oxford Univ.
Press, ed. 2, 2013).
10. E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956).
11. S. Grimme, WIREs Comput. Mol. Sci. 1, 211228 (2011).
12. A. Tkatchenko, M. Scheffler, Phys.Rev.Lett.102,073005(2009).
13. C. A. Silvera Batista, R. G. Larson, N. A. Kotov, Science 350,
1242477 (2015).
14. P. Loskill et al., Adv. Colloid Interf ace Sci. 179182,107113(2012).
15. P. Loskill et al., J. R. Soc. Interface 10, 20120587 (2013).
16. S. Tsoi et al., ACS Nano 8, 1241012417 (2014).
17. G. A. Rance, D. H. Marsh, S. J. Bourne, T. J. Reade,
A. N. Khlobystov, ACS Nano 4, 49204928 (2010).
18. C. Wagner et al., Nat. Commun. 5, 5568 (2014).
19. D. B. Chang, R. L. Cooper, J. E. Drummond, A. C. Young, Phys.
Lett. A 37, 311312 (1971).
20. M. Boström, B. E. Sernelius, Phys. Rev. B 61,22042210 (2000).
21. J. F. Dobson, A. White, A. Rubio, Phys. Rev. Lett. 96, 073201
(2006).
22. J. F. Dobson, T. Gould, G. Vignale, Phys. Rev. X 4, 021040
(2014).
23. A. J. Misquitta, J. Spencer, A. J. Stone, A. Alavi, Phys. Rev. B
82, 075312 (2010).
24. A. J. Misquitta, R. Maezono, N. D. Drummond, A. J. Stone,
R. J. Needs, Phys. Rev. B 89, 045140 (2014).
25. A. Tkatchenko, R. A. DiStasio Jr., R. Car, M. Scheffler, Phys.
Rev. Lett. 108, 236402 (2012).
26. J. Applequist, K. R. Sundberg, M. L. Olson, L. C. Weiss, J. Chem.
Phys. 70, 1240 (1979).
27. W. L. Bade, J. Chem. Phys. 27, 1280 (1957).
28. A. G. Donchev, J. Chem. Phys. 125, 074713 (2006).
29. A. Ambrosetti, A. M. Reilly, R. A. DiStasio Jr., A. Tkatchenko,
J. Chem. Phys. 140, 18A508 (2014).
30. A. Tkatchenko, A. Ambrosetti, R. A. DiStasio Jr., J. Chem. Phys.
138, 074106 (2013).
31. A. P. Jones, J. Crain, V. P. Sokhan, T. W. Whitfield,
G. J. Martyna, Phys. Rev. B 87, 144103 (2013).
32. A. M. Reilly, A. Tkatchenko, J. Phys. Chem. Lett. 4, 10281033
(2013).
33. A. Ambrosetti, D. Alfè, R. A. DiStasio Jr., A. Tkatchenko, J. Phys.
Chem. Lett. 5, 849855 (2014).
34. R. A. DiStasio Jr., O. A. von Lilienfeld, A. Tkatchenko, Proc. Natl.
Acad. Sci. U.S.A. 109, 1479114795 (2012).
35. R. A. DiStasio Jr., V. V. Gobre, A. Tkatchenko, J. Phys. Condens.
Matter 26, 213202 (2014).
36. J. F. Dobson, J. Wang, B. P. Dinte, K. McLennan, H. M. Le,
Int. J. Quantum Chem. 101,579598 (2005).
37. V. V. Gobre, A. Tkatchenko, Nat. Commun. 4, 2341 (2013).
38. R.-F. Liu, J. G. Ángyán, J. F. Dobson, J. Chem. Phys. 134,
114106 (2011).
39. M. Dion, H. Rydberg, E. Schröder, D. C. Langreth,
B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
40. K. Berland et al., Rep. Prog. Phys. 78, 066501 (2015).
41. J. Dai, J. Yuan, P. Giannozzi, Appl. Phys. Lett. 95, 232105 (2009).
42. P. A. Denis, Chem. Phys. Lett. 508,95101 (2011).
43. J. Tao, J. P. Perdew, J. Chem. Phys. 141, 141101 (2014).
44. L. J. Lis, M. McAlister, N. Fuller, R. P. Rand, V. A. Parsegian,
Biophys. J. 37, 667672 (1982).
45. K. Autumn et al., Nature 405, 681685 (2000).
46. See supplementary materials on Science Online.
ACKNOWL EDGMENTS
Supported by a startup grant from Cornell University (R.A.D.),
European Research Council Starting Grant VDW-CMAT, and DFG/
SFB-951 HIOS project A10. A.A. thanks F. Toigo and P. L. Silvestrelli
for useful discussions. This research used resources of the
Argonne Leadership Computing Facility at Argonne National
Laboratory, which is supported by the Office of Science of the U.S.
Department of Energy under contract DE-AC02-06CH11357.
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/351/6278/1171/suppl/DC1
Theoretical Methods
Supplementary Text
Tables S1 to S3
10 December 2015; accepted 31 January 2016
10.1126/science.aae0509
REPORTS
QUANTUM OPTICS
Generation of multiphoton entangled
quantum states by means of
integrated frequency combs
Christian Reimer,
1
*Michael Kues,
1
*Piotr Roztocki,
1
Benjamin Wetzel,
1,2
Fabio Grazioso,
1
Brent E. Little,
3
Sai T. Chu,
4
Tudor Johnston,
1
Yaron Bromberg,
5
Lucia Caspani,
6
§ David J. Moss,
7
|| Roberto Morandotti
1,8
Complex optical photon states with entanglement shared among several modes are
critical to improving our fundamental understanding of quantum mechanics and have
applications for quantum information processing, imaging, and microscopy. We
demonstrate that optical integrated Kerr frequency combs can be used to generate
several bi- and multiphoton entangled qubits, with direct applications for quantum
communication and computation. Our method is compatible with contemporary fiber
and quantum memory infrastructures and with chip-scale semiconductor technology,
enabling compact, low-cost, and scalable implementations. The exploitation of
integrated Kerr frequency combs, with their ability to generate multiple, customizable,
and complex quantum states, can provide a scalable, practical, and compact platform
for quantum technologies.
Multi-entangled states of light hold an-
swerstofundamentalquestionsinquan-
tum physics and are the cornerstone of a
range of applications, including quantum
communications (1), computation (24),
and sensing and imaging with a resolution be-
yond the classical limit (5). Thus, the controllable
realization of multiple quantum states in a com-
pact platform would enable a practical and pow-
erful implementation of quantum technologies.
Although applications of frequency combs have
been mostly classical thus far, their distinctive
architecture, based on multiple interacting modes
and the phase characteristics of the underlying
nonlinear processes, has the potential to offer
new and powerful ways to achieve the gener-
ation of multiple, customizable, and complex
states of nonclassical light. The quantum pro-
perties of frequency combs have recently begun
to be investigated, revealing their potential for
the generation of large quantum states (68).
However, the continuous-variable nonclassical
states (squeezed vacuum) that have been dem-
onstrated with this approach have not yet achieved
the quality (amount of squeezing) required for
optical quantum computation (9). For the gen-
eration of single photons and continuous- and
discrete-variable quantum states (qubits), a wide
variety of second- and third-order nonlinear
sources, optical fibers, and gases, as well as single
quantum emitters, have been exploited (10,11).
Recent progress has focused on transferring both
classical frequency combs (12)andquantum
sources (13) to integrated optical platforms. Such
integrated approaches provide the advantages
of compact, scalable, mass-producible, and low-
cost devices (14). Demonstrated integrated de-
vices include sources of heralded single photons
(1517) and entangled photon pairs (18), in prin-
ciple allowing implementations of quantum al-
gorithms (19,20). Here we show the parallel
generation of bi- and multiphoton entangled
states in a compact, integrated quantum fre-
quency comb source.
Our quantum frequency comb is generated in a
CMOS (complementary metal-oxide semiconductor)
compatible, high-refractive-index glass in a four-
port microring resonator architecture [details
on device fabrication and characteristics are
presented in (21)]. The weak and anomalous
dispersion of our device enables broadband phase
matching for spontaneous four-wave mixing
(SFWM), thereby generating a broad frequency
1176 11 MARCH 2016 VOL 351 ISSUE 6278 sciencemag.org SCIENCE
1
Institut National de la Recherche ScientifiqueÉnergie
Matériaux Télécommunications, 1650 Boulevard Lionel-
Boulet, Varennes, Québec J3X 1S2, Canada.
2
Department of
Physics and Astronomy, University of Sussex, Falmer,
Brighton BN1 9RH, UK.
3
State Key Laboratory of Transient
Optics and Photonics, Xi'an Institute of Optics and Precision
Mechanics, Chinese Academy of Science, Xi'an, China.
4
Department of Physics and Materials Science, City
University of Hong Kong, Tat Chee Avenue, Hong Kong,
China.
5
Department of Applied Physics, Yale University, New
Haven, CT 06520, USA.
6
School of Engineering and Physical
Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.
7
School of Electrical and Computer Engineering, RMIT
University, Melbourne, Victoria 3001, Australia.
8
Institute of
Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, Chengdu 610054, China.
*These authors contributed equally to this work. Corresponding
author. E-mail: michael.kues@emt.inrs.ca (M.K.); morandotti@
emt.inrs.ca (R.M.) Present address: Racah Institute of Physics,
The Hebrew University of Jerusalem, Jerusalem 91904, Israel.
§Present address: Institute of Photonics, Department of Physics,
University of Strathclyde, Glasgow G4 0NW, UK. ||Present address:
Center for Micro-Photonics, Swinburne University of Technology,
Hawthorn, Victoria 3122,Australia.
RESEARCH
on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from
comb of photons emitted at thecavityresonances.
The ring resonators characteristics, with a high
quality (Q) factor of around 240,000 (803 MHz
linewidth and 200 GHz free spectral range; fig. S1),
lead to high field enhancement and allow for low-
power operation, while simultaneously enabling the
direct generation of bright and narrow-bandwidth
photons, without the need for spectral filtering
of the photons (22,23). The device is pumped
with a passive mode-locked fiber laser (repeti-
tion rate, 16.8 MHz), which is spectrally filtered
to excite a single ring resonance at 1556.2 nm
(192.65 THz) with a pulse duration of 570 ps
coupled into the resonator. The pulses are di-
rectly filtered by the resonator, resulting in a
perfect match between the spectral bandwidth
of the pump pulses and the excited resonator
mode. This in turn leads to the generation of
pure single-mode photons in each resonance (13),
confirmed by single-photon autocorrelation mea-
surements [fig. S2 and (21)].
Using a high-resolution tunable C-band
wavelength filter and a grating-based spectrum
analyser, the single-photon spectrum was char-
acterized at the output of the resonator (21).
Figure 1 shows the measured single-photon count
rate and the calculated photon-pair production
rate per pulse as a function of wavelength. A very
broad frequency comb of photons is emitted,
covering the full S, C, and L bands defined by the
International Telecommunication Union (wave-
lengths ranging from 1470 to 1620 nm). The SFWM
process generates a spectrum that is symmetric
in frequency, whereas the spectral asymmetry in
the measured photon counts can be explained by
Raman scattering, which could be further reduced
by cooling the chip (24). As a result of the broad
phase-matching condition, achieved through the
close-to-zero waveguide dispersion, the emitted
comb exhibits a flat and broadband spectrum
with uniform pair production rates, ranging from
0.02 to 0.04 pairs per pulse, over the full mea-
sured comb.
To demonstrate an entangled quantum fre-
quency comb, we chose time-bin entanglement,
which, among several intrinsic advantages, is
particularly suitable for information processing
and transmission (25) because of its robustness
(e.g., with respect to polarization fluctuations); it
thus can be preserved even over long propaga-
tion distances in standard fiber networks (26).
Starting from the single-mode photon pairs,
we generated time-binentangled qubits by pass-
ing the pulsed pump laser through a stabilized
unbalanced fiber interferometer with a 11.4-ns
delay (longer than the pulse duration of the laser),
thereby producing double pulses of equal power
with a defined relative phase difference. The tem-
poral separation of the two pulses can be arbitrarily
chosen, as long as it is larger than the temporal
duration of the single photons; this approach thus
offers considerable flexibility. The double pulses
were then coupled into the integrated microring
resonator (Fig. 2), where an average pump power
of 0.6 mW was chosen so that the probability of
creating a photon pair from both pulses simulta-
neously was low enough to be negligible.
This pump configuration transforms the orig-
inally single-mode photon pairs into entangled
states, where the photons are in a superposition
of two temporal modes. In particular, the en-
tangled state jytimebin 1
ffiffi
2
pðjSs;SiiþjLs;Li
is generated, where the signal (s) and idler (i)
photons are in a quantum superposition of the
short (S)andlong(L)timebins(21). Most im-
portantly, these entangled qubits are generated
over all the microring resonances, thus leading to
a quantum frequency comb of time-binentangled
photon pairs.
To characterize the degree of entanglement,
the generated signal and idler photons were each
passed individually through a different fiber
interferometer with an imbalance identical to
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1
2
3
4
5
6
1480 1500 1520 1540 1560 1580 1600
0.01
0.02
0.03
S-Band C-Band
Pump
Entangl. Visibility
s1-i1: 83.0% (98.5%)
s2-i2: 84.1% (93.2%)
s3-i3: 86.5% (96.0%)
s4-i4: 82.4% (96.0%)
s5-i5: Stabilization
s6-i6: 83.1% (99.5%)
Single photon count rate (kHz)
SFWM photon pairs per pulse
Wavelength (nm)
s6 s5 s4 s3 s2s1 i1 i2 i3 i4 i5 i6
Pump
L-Band
SFWM
Asymmetry
Fig. 1. Measured single-photon spectrum of the integrated quantum frequency comb. Single-photon
spectrum (red circles) emitted by the microring resonator, measured using a grating-based spectrum
analyzer and a high-resolution digital tunable filter in the C band (bottom inset). The S, C, and L bands are
indicated. The red curve shows the symmetric contribution generated through SFWM, whereas the blue
curve shows the spectral asymmetry, which can be explained by Raman scattering.The channels used in the
entanglement measurements are shown in the bottom inset; the measured raw entanglement visibilities
(with background-corrected values in parentheses) for the individualchannel pairs are shown in the top inset.
Fig. 2. Quantum frequency comb. A pulsed laser is passed through an unbalanced fiber Michelson
interferometer, generating double pulses with a phase difference ϕ. The pulses are fed into the microring
resonator, exciting one microring resonance and generating time-binentangled photon pairs on a fre-
quency comb through SFWM. For the purposes of analysis [entanglement verification (Fig. 3, A to D) or
quantum state tomography (Fig. 4, A and B)], each photon of the spectrally filtered photon pair is
individually passed through an interferometer with the temporal imbalance equal to the time-bin sep-
aration and then detected with a single-photon detector. For the four-photon measurements (Fig. 3E and
Fig. 4, C to D), four frequency modes that are symmetric to the excitation field are collected, passed
through the interferometers, and spectrally filtered before detection.
RESEARCH |REPORTS
that used for the pump laser (Fig. 2). This setup
allowed the measurement of the quantum inter-
ference between the signal and idler photons. For
the resonances closest to the excitation frequency,
we measured a photon coincidence rate of 340 Hz,
which gives an estimated pair production rate of
302 kHz per channel (0.018 pairs per double pulse),
accounting for system and detection losses of
14.75 dB (21). We selected five different frequency
channel pairs within the C band (marked in Fig. 1)
and recorded quantum interference with raw
visibilities above 82.4%, which, being greater than
1
ffiffi
2
p71% (Fig. 1, top inset, and Fig. 3), confirm en-
tanglement through the violation of the Clauser-
Horne-Shimony-Holt (Bell-like) inequality (27). Af-
ter subtracting the measured background (Fig. 3),
the visibility was found to be above 93.2% on all
channel pairs (Fig. 1, top inset).
In addition to confirming time-bin entangle-
ment, the quantum interference also reveals the
phase characteristic of the nonlinear generation
process. This phase dependency has been well
described in theory (28) and has been exploited,
for example, in optical squeezing (29). Here we
show that this phase dependency is also man-
ifested at the single-photon level, with a clear
difference between second- and third-order non-
linear interactions. When spontaneous parametric
down-conversion (SPDC) in second-order non-
linear media is used to generate the entangled
photon pairs, quantum interferenceisexpected
to be proportional to 1 Vcos(a+b+ϕ)(26),
where Vis the fringe visibility, ϕis the pump
interferometer phase, and aand bare the phases
for the signal and idler interferometers, respec-
tively. In contrast, for photons generated through
SFWM (as in this work), quantum interference is
expected to be of the form 1 Vcos(a+b2ϕ)for
degenerate SFWM or 1 Vcos(a+bϕ
1
ϕ
2
)for
nondegenerate SFWM, where ϕ
1
and ϕ
2
are the
phases of the two pump fields (21). As shown in
Fig. 3, we confirmed this important difference in
phase dependency between SPDC and SFWM
through five separate quantum interference mea-
surements, where the phases of the interferom-
eters are either tuned separately or simultaneously
in a symmetric and an antisymmetric way. The
expected behavior for SPDC and SFWM is plotted
in Fig. 3, A to E, with dashed and solid lines,
respectively. The measurements confirm the dif-
ference between the two processes, which can be
explained through the additional photon involved
in SFWM. In the generation of entangled photon
pairs, the phase of the excitation photon (or photons)
can be used to adjust the quantum state. If a single
photon is involved (as in second-order processes),
only the phase of this photon can be used as a
control parameter. In third-order nonlinear pro-
cesses, however, two photons generate the quan-
tum state, enabling an additional control parameter
(the relative phase between the two photons).
Although in this study only a single excitation field
was used to demonstrate the effect, exploiting
two distinct nondegenerate pump fields could
lead to an additional degree of freedom for the
generation of all-optical reconfigurable quan-
tum states.
The distinctive multimode characteristic of the
frequency comb architecture presented here can
be extended to create multiphoton entangled
quantum states. By selecting two different signal-
idler pairs, we can generate two-photon qubit
states, given by jy11
ffiffi
2
pðjSs1;Si1 ei2ϕjLs1;Li1
and jy21
ffiffi
2
pðjSs2 ;Si2 ei2ϕjLs2 ;Li2 .Bypost-
selecting four-photon events with one photon
on each frequency channel, these two states are
multiplied, resulting in a four-photon time-bin
entangled state, given by jy4photoni¼jy1ijy2
1
2ðjSs1;Si1 ;Ss2 ;Si2 ei2ϕjSs1;Si1 ;Ls2;Li2 ei2ϕj
Ls1 ;Li1 ;Ss2 ;Si2 ;ei4ϕjLs1;Li1 ;Ls2 ;Li2 .Forthe
generation of a four-photon state, the coherence
length of both photon pairs has to be the same
and must be matched to the excitation fields
coherence time. This requirement is intrinsically
fulfilled through the resonant characteristics
(equal resonance bandwidths) of the ring cavity,
in combination with the excitation scheme de-
scribed above. By setting the pump power to 1.5 mW,
we measured a quadruple detection rate of 0.17 Hz,
which corresponds to a calculated generation rate
of 135 kHz, taking into account the system and
detection losses of 14.75 dB. We then performed
four-photon quantum interference measurements
(Fig. 3F). Four-photon interference generally is
not present for two completely independent
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2
Phase (π)
0 0.8 10.60.40.2 1.2 1.81.61.4
0
1
Normalized counts
0.5
0
1
Normalized counts
0.5
0
1
Normalized counts
0.5
0
1
Normalized counts
0.5
0
1
Normalized counts
0.5
Measured signal
Measured background Theory SPDC
Entangl.
Visibility
Theory SFWM
φ=0
α=β∈[0,2π]
φ∈[0,2π]
α=β=0
α∈[0,2π]
φ=β=0
α=β=φ∈[0,2π]
α=β=–φ∈[0,2π]
2
Phase (π)
0 0.8 10.60.40.2 1.2 1.81.61.4
0
1
Normalized counts
0.5
2-Photon quantum interference
4-Photon quantum interference
Measured Theory 4-Photon Theory 2-Photon
Fig. 3. Entanglement
and phase control by means
of SFWM. (Ato E)To
demonstrate the difference
between the phase character-
istics of SPDC and SFWM, five
different quantum interference
measurements were per-
formed. Three interferometer
phases were adjusted: ϕ,a,
and b, being the phases of the
pump, signal, and idler interfer-
ometers, respectively
(conditions are specified in
each panel). The error bars
represent the standard devia-
tion of seven measurements.
(F) Four-photon entanglement
measurement with all photon
phases tuned simultaneously,
showing clear four-photon
quantum interference with a
visibility of 89%. The solid line
indicates the expected function;
the dashed line shows the
cosine interference in the two-
photon case.
RESEARCH |REPORTS
two-photon qubit states. The interference is ex-
pected to be proportional to 3 + cos(4a4ϕ)+
4cos(2a2ϕ), where ais the phase of all four
entangled photons and ϕis the pump interfer-
ometer phase (21). Our data follow the expected
relation, having a visibility of 89% without compen-
sation for background noise or losses. Further-
more, we repeated the four-photon measurement
by selecting different combinations of four modes,
always finding four-photon entanglement [fig. S3
and (21)].
Lastly, to fully characterize the entangled states,
we performed quantum state tomography (30).
This method measures the state density matrix,
fromwhichitispossibletoextractimportantchar-
acteristics such as the fidelity, which describes
how close the measured state is to the ideal en-
tangled state (21). We first measured the two-
photon qubits generated on comb lines that were
symmetric with respect to the pump wavelength
(Fig.4,AandB)andfoundafidelityof96%,con-
firming that our generated quantum states are
of high quality and very close to the ideal en-
tangled state. For the four-photon entangled
state (Fig. 4, C and D), we obtained a fidelity of
64% without compensation for background
noise or interf erometer imperfections, which is
comparable to the fidelity measured for non-
integrated four-photon states used for practical
applications (3).
Key characteristics of our quantum frequency
comb include the intrinsic and simultaneous
operation over many modes, the generation of
high-purity photon pairs, the high-quality bi- and
multiphoton entanglement shared among these
modes, and the inherent compatibility with fiber
technology. Because of these features, it has ver-
satile and immediate applications in areas such as
quantum communications and quantum compu-
tation. For example, our source can be implemented
into both single-photon and entanglement-based
quantum communication protocols. The broad-
band nature of the quantum comb is particularly
attractive for multichannel applications, where
the amount of transmitted data can be increased
through the use of multiple, equally well-performing
channels. We repeated the two-photon tomography
measurement after adding 40 km of fiber and
measured a fidelity of 87% [fig. S4 and (21)], dem-
onstrating that the entanglement is preserved
after long fiber propagation.
Furthermore, two-photon time-binentangled
qubits have been used successfully for linear
universal quantum computation (4,25), and
the parallel generation and processing of mul-
tiple qubits can directly enhance such proto-
cols where the information capacity scales with
the number of comb lines used, as theoretically
predicted (25). Even though the demonstrated
multiphoton entangled states are separable,
because they are generated as a product of bi-
photon Bell states, it is conceivable thatthrough
the use of multiple excitation fields (6) or con-
trolled phase gates (31)nonseparable multiphoton
cluster states could be constructed and used for
measurement-based quantum computation (3).
Further device integration of the frequency
comb will lead to more compact and stable sys-
tems with higher performances, resulting in
better detection rates. All the components that
were used in our setup, such as the laser, filters,
interferometers, and detectors (connected via
optical fibers), could be integrated on a single
chip (13)toreducesizeandlosses(currentlyat
14.75 dB). An easily realized decrease in losses by
5 dB would increase the four-photon detection
rate by a factor of 100, and an achievable loss
reduction of 10 dB would increase it to the useful
kilohertz range.
Our results indicate that integrated quantum
frequency comb sources based on third-order non-
linearities can open up new venues for the gen-
eration and control of complex quantum states,
thus providing a scalable and practical platform
for optical quantum information processing.
REFERENCES AND NOTES
1. H. J. Kimble, Nature 453, 10231030 (2008).
2. D. Deutsch, Proc. R. Soc. A Math. Phys. Eng. Sci. 400,97
(1985).
3. P. Walther et al., Nature 434, 169176 (2005).
4. P. C. Humphreys et al., Phys. Rev. Lett. 111, 150501
(2013).
5. M. Kolobov, Rev. Mod. Phys. 71, 15391589 (1999).
6. M. Chen, N. C. Menicucci, O. Pfister, Phys. Rev. Lett. 112,
120505 (2014).
7. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, O. Pfister,
Phys. Rev. Lett. 107, 030505 (2011).
8. J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, N. Treps,
Nat. Photonics 8, 109112 (2013).
9. N. C. Menicucci, Phys. Rev. Lett. 112, 120504 (2014).
10. M.D.Eisaman,J.Fan,A.Migdall,S.V.Polyakov,Rev. Sci.Instrum.
82, 071101 (2011).
11. W. Wieczorek et al., IEEE J. Sel. Top. Quantum Electron. 15,
17041712 (2009).
SCIENCE sciencemag.org 11 MARCH 2016 VOL 351 ISSUE 6278 1179
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ρ
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ρ
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0
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2-Photon qubit state - measured
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|LLSS>
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Re ( ) Im ( )
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4-Photon qubit state - ideal
4-Photon qubit state - measured
Re ( )
ρ
Im ( )
ρ
Re ( )
ρ
Im ( )
ρ
Fig. 4. Quantum state characterization by means of tomography. A quantum state can be fully described by its density matrix r
^
. The real (Re) and imaginary
(Im) parts of theideal density matrices of a two- and four-photon entangled qubit stateare shown in (A) and (C), respectively, represented in the time-bin basis
(|SSi,|SLi,|LSi,|LLi)and(|SSSSi,|SSSLi,|LLLLi).The measured density matrix of the two-photon state (B) agrees very well with the ideal state, confirmed by
a fidelity of 96%. The measured density matrix of the four-photon entangled qubit state (D) reaches a fidelity of 64%, which is comparable in quality to other
nonintegrated four-photon states (3).
RESEARCH |REPORTS
12. T. J. Kippenberg, R. Holzwarth, S. A. Diddams, Science 332,
555559 (2011).
13. D. Bonneau, J. W. Silverstone, M. G. Thompson, in Silicon
Photonics III, L. Pavesi, D. J. Lockwood, Eds. (Springer, ed. 3,
2016), pp. 4182.
14. D. J. Moss, R. Morandotti, A. L. Gaeta, M. Lipson, Nat. Photonics 7,
597607 (2013).
15. S. Azzini et al., Opt. Express 20, 2310023107
(2012).
16. C. Reimer et al., Nat. Commun. 6, 8236 (2015).
17. N. C. Harris et al., Phys. Rev. X 4, 041047 (2014).
18. D. Grassani et al., Optica 2, 88 (2015).
19. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, J. L. OBrien, Science
320, 646649 (2008).
20. J. C. F. Matthews, A. Politi, A. Stefanov, J. L. OBrien,
Nat. Photonics 3, 346350 (2009).
21. Materials and methods are available as supplementary
materials on Science Online.
22. Z. Ou, Y. Lu, Phys. Rev. Lett. 83, 25562559 (1999).
23. C. Reimer et al., Opt. Express 22, 65356546
(2014).
24. A. S. Clark et al., Opt. Express 20, 16807 (2012).
25. T. Pittman, Physics 6, 110 (2013).
26. J. Brendel, N. Gisen, W. Tittel, H. Zbinden, Phys. Rev. Lett. 82,
25942597 (1999).
27. J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett.
23,880884 (1969).
28. C. C. Gerry, P. L. Knight, Introductory Quantum Optics
(Cambridge Univ. Press, 2005).
29. Z. Y. Ou, S. F. Pereira, H. J. Kimble, K. C. Peng, Phys. Rev. Lett.
68, 36633666 (1992).
30. D.F.V.James,P.G.Kwiat,W.J.Munro,A.G.White,Phys. Rev. A
64, 052312 (2001).
31. G. Vallone, E. Pomarico, P. Mataloni, F. De Martini, V. Berardi,
Phys. Rev. Lett. 98, 180502 (2007).
ACKNOW LEDGMEN TS
This work was supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) through the Steacie
Memorial Fellowship and Discovery Grants programs and by
the Australian Research Council Discovery Projects program.
C.R. and P.R. acknowledge the support of a NSERC Vanier Canada
Graduate Scholarship and a NSERC Alexander Graham Bell
Canada Graduate Scholarship (Masters), respectively. M.K.
acknowledges support from FRQNT (Fonds de Recherche du
QuébecNature et Technologies) through the MELS fellowship
program (Merit Scholarship Program for Foreign Students;
Ministère de lÉducation, de lEnseignement Supérieur et de la
Recherche du Québec). M.K. also acknowledges funding from
the European Unions Horizon 2020 research and innovation
program under Marie Sklodowska-Curie grant agreement no.
656607. We acknowledge support from the People Programme
(Marie Curie Actions) of the European Unions 7th Framework
Programme for Research and Technological Development
[B.W. under Research Executive Agency (REA) grant agreement
no. 625466 and L.C. under REA grant agreement no. 627478].
F.G. acknowledges support from Mitacs through the Accelerate
Program. S.T.C. acknowledges support from the CityU SRG-Fd
(Strategic Research Grant for fundable Competitive Earmarked
Research Grant) program #7004189. We thank R. Helsten
for the mechanical design of the interferometer housing;
P. Saggu for assisting in the measurement of the single-photon
spectrum; L. La Volpe for assisting in the evaluation of the
tomography measurement; J. Azaña, T. A. Denidni, S. O. Tatu,
and L. Razzari for providing some of the required experimental
equipment; and A. Tavares, A. Yurtsever, and M. A. Gauthier
for helpful discussions. Special thanks go to QuantumOpus
and N. Bertone of OptoElectronics Components for their help
and for providing us with state-of-the-art photon detection
equipment.
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/351/6278/1176/suppl/DC1
Materials and Methods
Supplementary Text
Figs. S1 to S4
References (3237)
12 November 2015; accepted 5 February 2016
10.1126/science.aad8532
SPIN MODELS
Simple universal models capture all
classical spin physics
Gemma De las Cuevas
1
*and Toby S. Cubitt
2
Spin models are used in many studies of complex systems because they exhibit
rich macroscopic behavior despite their microscopic simplicity. Here, we prove that
all the physics of every classical spin model is reproduced in the low-energy sector
of certain universal models,with at most polynomial overhead. This holds for classical
models with discrete or continuous degrees of freedom. We prove necessary and
sufficient conditions for a spin model to be universal and show that one of the simplest
and most widely studied spin models, the two-dimensional Ising model with fields,
is universal. Our results may facilitate physical simulations of Hamiltonians with
complex interactions.
The description of systems with many inter-
acting degrees of freedom is a ubiquitous
problem across the natural and social sci-
ences. Be it electrons in a material, neurons
interacting through synapses, or specula-
tive agents in a market, the challenge is to sim-
plify the system so that it becomes tractable
while capturing some of the relevant features of
therealsystem.Spinmodelsareonewayofad-
dressing this challenge. Originally introduced
in condensed matter physics in order to study
magnetic materials (14), they have permeated
many other disciplines, including quantum grav-
ity (5), error-correcting codes (6), percolation the-
ory (3), graph theory (7), neural networks (8),
protein folding (9), and trading models in stock
markets (10).
Spin models are microscopically simple, yet
their versatile interactions lead to a very wide
variety of macroscopic behavior. Formally, a spin
model is specified by a set of degrees of freedom,
the spins,and a cost function, or Hamilto-
nian,Hthat specifies the interaction pattern as
well as the type and strength of interactions
among the spins. (In physics, the Hamiltonian
specifies the energy of each possible spin con-
figuration; in other contexts, this energy value
mayquantifyamoreabstractcostassociated
with a configuration.)
Thisdefinitionencompassesawiderangeof
models, including attractive and/or repulsive in-
teractions, regular and irregular interaction pat-
terns, models in different spatial dimensions,
models with different symmetries (for example,
conventionalspin models with global symme-
tries versus models with local symmetries, such
as lattice gauge theories), many-body interactions
[such as vertex models and edge models (11)], and
more.Wewillusethewordmodelto refer to a
(generally infinite) family of spin Hamiltonians.
Different Hamiltonians within the same model
are typically related in some natural way. For ex-
ample, the two-dimensional (2D) Ising model
with fieldsis the family of Hamiltonians of the
form
HGðsÞ¼Xhi;jiJijsisjþXirisið1Þ
where s=s
1
,s
2
,...,s
n
is a configuration of Ising
(two-level) spins sif1;1gon a 2D square lat-
tice, hi;jidenotes neigbouring spins, and J
ij
and r
i
are real numbers specifying the coupling
strengths and local fields, respectively.
Here, we show that there exist certain spin
models, which we call universal, whose low-energy
sector can reproduce the complete physics of
any other classical spin model. What does it
mean to reproduce the complete physics? In-
formally, we say that a spin model with Ha-
miltonian Hsimulates a target Hamiltonian
Hif ( i) t he energy levels of Hbelow a threshold
Dreproduce the energy levels of H;(ii)thereisa
fixed subset Pof the spins of Hwhich we call
the physical spins”—whose configuration for
each energy level below Dreproduces the spin
configuration of the corresponding energy level
of H; and (iii) the partition function of Hre-
produces that of H. From the partition function,
one can derive all equilibrium thermodynamical
properties of the system. A universal model is
then a model that can simulate any other spin
model.
We denote spin degrees of freedom
discrete or continuousby a string of spin
states s=s
1
,s
2
, ..., s
n
.Forq-level Ising spins
(discrete degrees of freedom with a finite
number qof distinct states), we can label the
states arbitrarily by integers: sif1;;qg.For
continuous spins, a spin state is represented by a
1180 11 MARCH 2016 VOL 351 ISSUE 6278 sciencemag.org SCIENCE
1
Max Planck Institute for Quantum Optics, Hans-Kopfermann-
Strasse 1, 85748 Garching, Germany.
2
Department of Computer
Science, University College London, Gower Street, London WC1E
6EA, UK.
*Corresponding author. E-mail: gemma.delascuevas@mpq.mpg.de
RESEARCH |REPORTS
DOI: 10.1126/science.aad8532
, 1176 (2016);351 Science et al.Christian Reimer
integrated frequency combs
Generation of multiphoton entangled quantum states by means of
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