# Ultrashort dissipative Raman solitons in Kerr resonators driven with phase-
coherent optical pulses

Zongda Li1,2 Yiqing Xu1,2 Sophie Shamailov1,2 Xiaoxiao Wen3 Wenlong Wang3
Xiaoming Wei3 Zhongmin Yang3 Stéphane Coen1,2 Stuart G. Murdoch1,2 Miro
Erkintalo1,2<EMAIL_ADDRESS>1Department of Physics, University of
Auckland, Auckland 1010, New Zealand  2The Dodd-Walls Centre for Photonic and
Quantum Technologies, New Zealand  3School of Physics and Optoelectronics,
South China University of Technology, Guangzhou 510640, China

###### Abstract

External driving of passive, nonlinear optical resonators has emerged over the
past decade as a novel route for the generation of ultrashort optical pulses
and corresponding broadband frequency combs. Whilst the pulse formation
dynamics in such systems differ dramatically from those manifesting themselves
in conventional mode-locked lasers, the demarcation between the two
traditionally distinct paradigms has recently begun to blur, with
demonstrations of hybrid systems incorporating both external driving and
active media shown to offer specific advantages. Here we explore a new pathway
for ultrashort pulse generation at the interface of externally-driven passive
resonators and lasers. By leveraging the nonlinear Raman gain inherent to
fused silica, we achieve deterministic generation of low-noise dissipative
solitons with durations well below 100 fs via phase-coherent pulsed driving of
resonators made of standard, commercially-available optical fibre. We explore
and explain the physics of the new dissipative Raman soliton states,
identifying scaling laws that govern the pulses’ characteristics and that
allow output repetition rates to be scaled at will without influencing the
soliton duration. The scheme explored in our work enables the shortest ever
pulses generated in resonators (active or passive) made from a single
commercially-available optical fibre, and it has the potential to be
transferred into a chip-scale format by using existing dispersion-engineered
silica microresonators.

##  I Introduction

Coherent external driving of passive, Kerr nonlinear optical resonators with
continuous wave (cw) or pulsed laser light allows for the generation of
ultrashort pulses of light known as dissipative Kerr cavity solitons (CSs) [1,
2, 3, 4, 5, 6, 7]. Whilst such pulses were first observed in macroscopic ring
resonators made of standard single-mode optical fibre [2, 3], their
application potential was only unlocked by later realisations in monolithic
Kerr microresonators [4, 5, 6, 8, 9, 10]. In particular, dissipative Kerr
solitons are the time-domain representations of the coherent and broadband
microresonator frequency combs, whose burgeoning applications range from
telecommunications [11, 12] and artificial intelligence [13, 14] to astronomy
[15, 16] and distance measurements [17, 18].

Why are applications of CSs restricted to microresonators and not fibre
resonators? In addition to the potential of on-chip integration [19], the
answer lies in the fundamental scaling laws that govern the solitons’
characteristics [20]. Specifically, fibre resonators typically have long
lengths and modest nonlinearity and finesse, resulting in solitons with
picosecond-scale durations. While durations below 200 fs can be achieved using
judicious fibre-based dispersion management [21, 22, 23], these numbers are
still substantially larger than what can be realised in state-of-the-art
integrated microresonator systems. Conversely, the microresonator platforms
that permit dissipative solitons with durations measured in tens of
femtoseconds typically have very large (hundreds of gigahertz) free-spectral
range [10, 24], which makes them unsuitable for applications that require
frequency combs with fine spacing (e.g. high-resolution spectroscopy). There
is therefore a need to break the scaling laws that govern dissipative solitons
in driven Kerr resonators in order to concomitantly realise ultrashort pulse
durations and repetition rates in the sub-GHz to GHz range.

CSs are fundamentally different from pulses generated in mode-locked lasers
[2, 25]: they exist in _passive_ resonators, gaining their energy via
instantaneous, phase-sensitive Kerr nonlinear interactions with the quasi-cw
background on top of which they sit. Mode-locked lasers, in contrast, operate
via stimulated emission provided by an active medium, with typical gain
relaxation timescales orders of magnitude larger than other characteristic
timescales of the system. Recently, however, the boundary between the active
and passive paradigms of ultrashort pulse (and broadband comb) generation has
begun to blur, with studies considering pulse generation in hybrid systems
that incorporate both external coherent driving as well as material gain [26,
27, 28, 29]. But, of course, even resonators that are considered purely
_passive_ can be associated with nonlinear optical processes that are akin to
phase-insensitive stimulated emission – such as stimulated Brillouin and Raman
scattering. These processes have been leveraged in a number of studies to
achieve cw lasing in microresonator systems [30, 31]. They are also known to
influence the dynamics and characteristics of CSs [32, 33, 34], as well as
give rise to new forms of dissipative solitons [35, 36] and soliton
interactions [37].

In this Article, we experimentally and theoretically explore a new paradigm of
ultrashort pulse generation at the interface of externally-driven passive
resonators and lasers. Building upon a recent serendipitous discovery [38], we
show that the broadband stimulated Raman scattering (SRS) gain inherent to
fused silica, in conjunction with phase-coherent pulsed pumping, allows for a
new form of ultrashort dissipative Raman soliton, with record durations well
below 100 fs achieved in fibre resonators made from standard
telecommunications fibre and components. In stark contrast to early
implementations of synchronously-pumped fibre Raman oscillators [39, 40, 41,
42], we leverage a carefully chosen resonator dispersion and resonant, phase-
coherent pumping to obtain phase-locked operation that completely overcomes
the low coherence and comparatively broad pulses associated with earlier
systems. We show that the resultant ultrashort Raman solitons can be excited
deterministically with perfect fidelity, and that they can be sustained in
resonators of varying lengths, thus offering a route to broadband frequency
combs with bespoke line spacing. Our current experiments focus on fibre
resonators and the generation of sub-GHz combs that could find immediate use
in applications that benefit from fine comb spacing (e.g. dual-comb
spectroscopy), but we envisage further that the Raman soliton states explored
in our work will enable substantially improved performance of GHz combs in
integrated silica microresonators [31, 43, 44].

##  II Results

Figure 1: Experimental demonstration of dissipative Raman soliton generation.
(a) Experimental setup. cw, continuous wave; PM, phase modulator; FS,
frequency shifter; EDFA, erbium-doped fibre amplifier; PC, polarization
controller; EOC, electro-optic comb generator; PBS, polarizing beam splitter;
circ., circulator; M, mirror; IPC, in-line polarization controller; PD,
photodetector; OSA, optical spectrum analyzer; ESA, electronic spectrum
analyzer. (b) Optical spectra measured at the cavity output as a function of
the desynchronization per round trip. Open white circles indicate the
calculated wavelength that is group-velocity-matched to the input pulse train.
(c) RF spectra corresponding to (b). (e) Example RF spectra measured with the
desynchronization in the Raman soliton regime (blue), outside of that regime
(orange), and the input pump pulse train (pink). (e) Example optical spectrum
measured (blue solid curve) for a desynchronization $\Delta t\approx
400\leavevmode\nobreak\ \mathrm{fs}$; dashed red curve shows corresponding
simulation result. (f) Blue solid curve shows the simulated temporal intensity
profile of the spectrum shown in (b), whilst the black dashed curve shows the
input pump profile for reference. All results obtained in a resonator with
round trip length and finesse approximately 2.65 m and 150, respectively. See
Methods for other parameters.

We begin by describing our experimental platform [Fig. 1(a)] and by presenting
illustrative results of Raman soliton formation. Our setup is built around a
passive fibre Fabry-Pérot resonator with zero-dispersion wavelength at 1633 nm
that is formed from two dielectrically-coated mirrors that surround a single
segment of non-zero dispersion-shifted optical fibre (Corning MetroCor) [see
Methods]. We chose a Fabry-Pérot (rather than a ring) architecture to allow
more easily control the resonator length, yet note that the Raman solitons
also manifest themselves in ring-type resonators [38]. The mirrors that form
our resonator impose an upper limit of about 500 for the system’s finesse, but
our setup also gives us the option to introduce additional micro-bend losses
when required [see Methods], thus permitting us to systematically study how
the Raman soliton characteristics depend upon the resonator finesse. We
emphasise that our resonator does not include any active gain medium.

The resonator is driven with a train of roughly 5 ps duration pulses generated
from a narrow-linewidth cw laser via electro-optic comb (EOC) generation
referenced to an RF signal generator; the frequency of the RF generator sets
the repetition rate of the pulse train [see Methods]. Before the pulses are
launched into the resonator, they are amplified with a C-band erbium-doped
optical amplifier and their polarization is aligned along one of the
eigenmodes of the resonator. In the first experiments to be discussed, the
central wavelength of the cw laser is set to about 1562.5 nm which is in the
region of normal group-velocity dispersion of the resonator. This wavelength
is actively stabilized to coincide with a cavity resonance (zero detuning) by
applying the Pound-Drever-Hall technique on a low-power cw probe beam that is
orthogonally polarized with respect to the picosecond driving pulses [21, 45].
This ensures that our experiment operates in the coherent regime with constant
relative phase between the picosecond driving pulses and the field circulating
inside the resonator.

The desynchronization $\Delta
t=t_{\mathrm{R}}(\omega_{\mathrm{P}})-t_{\mathrm{P}}$ between the period of
the driving pulse train ($t_{\mathrm{P}}$) and the resonator round-trip time
at the driving frequency [$t_{\mathrm{R}}(\omega_{\mathrm{P}})$] is a key
control parameter for the dynamics of interest. Figure 1(b) shows experimental
measurements of the optical spectrum at the output of a 2.65-m-long resonator
as a function of the desynchronization, achieved by changing the RF clock
signal that drives the EOC generator. As can be seen, when the magnitude of
the desynchronization is sufficiently large, new spectral components that are
red-shifted with respect to the injected pulse train are generated inside the
resonator. These components originate from stimulated Raman scattering, as
evidenced by their frequency down-shift of about 10 THz compared to the pump
[46]. The exact frequencies $\omega$ of the components’ spectral maxima are
determined by a group-velocity-matching condition: the components that are
most efficiently amplified are those with a group-velocity that yields a
round-trip time equal to the period of the input pulse train, i.e.,
$t_{\mathrm{R}}(\omega)=t_{\mathrm{P}}$ [see circles in Fig. 1(b) and
Methods]. In other words, whilst the injected pulse train is desynchronous
with respect to the pump frequency, the Raman components that emerge are those
for which the driving is synchronous. We note that such group-velocity-
matching is reminiscent of Raman-induced energy transfer between higher-order
modes in resonators [35] and single-pass fibre systems [47, 48], but occurs
here by virtue of the desynchronization at the pump frequency.

A key feature of our scheme is that, whilst the injected pump field lies in
the _normal_ dispersion regime, the group-velocity-matched Raman components
reside in the _anomalous_ dispersion regime, thus permitting soliton
formation. Indeed, the measurements depicted in Fig. 1(b) reveal that, for
desynchronizations within a small interval, considerable spectral broadening
takes place. Concomitantly with the broadening of the optical spectrum, the RF
spectrum collapses to a narrow structure around the fundamental beat tone
[Fig. 1(c)], signalling the emergence of a low-noise soliton state. Figure
1(d) and (e) show the RF and optical spectra measured for such a state,
respectively. The RF spectrum replicates the corresponding spectrum of the
injected pulse with little additional noise [see Supplementary Fig. 2 for the
phase noise spectra], whilst the optical spectrum shows a broad spectral
envelope that bridges the group-velocity-matched component at 1665 nm and the
injected pump component at 1562.5 nm. Also shown in Fig. 1(e) is the optical
spectrum obtained from numerical simulations of the generalized Lugiato-
Lefever equation [34] using parameters corresponding to the experiments [see
Methods]. The simulation results are in excellent agreement with the
experimentally measured spectrum. The simulations additionally reveal that, in
the time-domain, the intracavity field corresponds to a 64-fs-long pulse that
sits atop a low-level pedestal [Fig. 1(f)]. We must emphasise that the results
in Fig. 1 were obtained with the pump peak power set to about 100 W, which
implies that the average pump power required to support the Raman solitons in
the experiments is as low as 40 mW. We also highlight that the Raman solitons
are generated deterministically and with perfect fidelity: every time the pump
desynchronization is tuned into the appropriate regime, a soliton is formed
[see Supplementary Fig. 1].

Figure 2: Numerically simulated dynamics of Raman soliton formation. (a) and
(b) respectively show the round trip to round trip evolution of the
intracavity temporal and spectral intensity profile, demonstrating the
formation of a dissipative Raman soliton. The parameters are the same as those
used in Fig. 1(d) and (f) [see Methods]. The temporal dynamics are shown in
the reference frame of the group-velocity-matched component. The dashed blue
rectangle in (a) highlights the Raman soliton. GVM in (b) highlights the
group-velocity-matched component as predicted from Eq. (2). (c, d) Intracavity
profile (c) before and (d) after Raman soliton formation. The dashed
horizontal lines show the SRS threshold [see Methods]. The shaded rectangle in
(d) highlights the Raman soliton (RS). (e) Phase-mismatch [$\Delta\phi$, left-
hand side of Eq. (1)] as a function of desynchronization $\Delta t$ with (blue
curve) and without (orange curve) taking into account the nonlinear phase
shift of a propagating soliton with peak power $P_{\mathrm{s}}\approx
600\leavevmode\nobreak\ \mathrm{W}$, i.e., $q=\gamma
P_{\mathrm{s}}/2=0.08\leavevmode\nobreak\ \mathrm{rad\,m^{-1}}$, where
$\gamma=2.6\leavevmode\nobreak\ \mathrm{W^{-1}km^{-1}}$ is the Kerr
nonlinearity coefficient. The shaded blue region indicates the Raman soliton
existence range observed in experiments [see Fig. 1(b)].

To gain more insights, Figs. 2(a) and (b) respectively show the temporal and
spectral growth dynamics of the dissipative Raman soliton of Fig. 1(d) and (f)
as obtained from numerical simulations starting from an empty cavity [see also
Supplementary Video]. The dynamics can be divided into three regimes. In the
first regime, a temporally broad and spectrally narrow pulse forms inside the
resonator at the wavelength of the injected pulse train. In the absence of
SRS, this state would be the steady-state of the system. However, because of
SRS, the intracavity field at the injected wavelength amplifies a spectral
component at the group-velocity-matched frequency (regime II). In the time-
domain, the SRS gain is above threshold only over a small temporal interval
around the peak of the intracavity field at the pump wavelength [see Fig. 2(c)
and Methods], due to which the amplification results in the break-up of the
intracavity field about its temporal peak. From this break-up, an ultrashort
soliton is seen to emanate (regime III), reaching a steady-state where the
soliton sits at the leading edge of the intracavity pump pulse. Because of the
pump-desynchronization (and the finite response time of SRS), the soliton is
able to deplete the intracavity pump pulse in a way that pushes the Raman gain
completely below threshold, thus enabling low-noise, steady-state operation
whilst preventing the formation of a second soliton [Fig. 2(d)].

Figure 3: Scaling laws of dissipative Raman solitons. (a) Solid and dashed
curves respectively show experimentally measured and numerically simulated
optical spectra versus frequency shift from the pump for three different pump
wavelengths as indicated. The resonator used had a round trip length of about
1 m and finesse of 400. Insets shows the group-velocity dispersion profile of
the resonator. Pumping further away from the zero-dispersion wavelength yields
an overall broader spectrum. (b) Raman soliton spectra measured for constant
round trip length (2.65 m) but for various values of the cavity finesse
(indicated in the legend). The spectral width increases with decreasing
finesse. (c) Raman soliton spectra measured in three different resonators with
approximately constant product between resonator finesse and length. (d)–(f)
RF spectra corresponding to Raman soliton pulse trains generated in the
different resonators used in (c). The results in (c)–(f) highlight that
similar Raman solitons can be generated in various resonator configurations,
thus allowing to realise ultrashort pulse trains with bespoke repetition.

Why does the low-noise dissipative Raman soliton only manifest itself for some
specific pump desynchronizations? Extensive simulations reveal that, in
addition to group-velocity-matching, the Raman component must be phase-matched
to the intracavity field at the pump frequency. The two conditions can be
written as [see Methods]

$\displaystyle\frac{\Delta
t}{L}\Omega_{\mathrm{R}}+\hat{D}(\Omega_{\mathrm{R}})+q=0$ Phase-matching, (1)
$\displaystyle\frac{\Delta t}{L}+\hat{D}_{1}(\Omega_{\mathrm{R}})=0$ GV-
matching. (2)

Here $L$ is the resonator round trip length,
$\Omega_{\mathrm{R}}=\omega_{\mathrm{R}}-\omega_{\mathrm{P}}$ is the angular
frequency shift between the Raman component and the pump, $q$ is a nonlinear
phase shift, and $\hat{D}(\Omega)$ is the reduced dispersion of the fibre that
forms the resonator:

$\hat{D}(\Omega)=\sum_{k\geq 2}\frac{\beta_{k}}{k!}\Omega^{k},$ (3)

with $\beta_{k}$ the Taylor series expansion coefficients of the fibre
propagation constant $\beta(\omega)$ evaluated at the pump frequency
$\omega_{\mathrm{P}}$. Finally, $\hat{D}_{1}(\Omega)=d\hat{D}/d\Omega$ is the
group-velocity-mismatch relative to the pump frequency. To provide a physical
basis for the significance of phase-matching, we note that the condition given
by Eq. (1) essentially implies that the pump wavelength is equal to the
wavelength of resonant dispersive waves emitted by the Raman soliton [49].
This observation suggests that the emergence of the low-noise Raman states can
be interpreted as a form of injection-locking between the solitons’ dispersive
wave and the residual pump background.

For typical resonator dispersion, and for a given nonlinear phase shift $q$,
Eqs. (1) and Eqs. (2) can be simultaneously satisfied only for a single
desynchronization $\Delta t$. As an example, in Fig. 2(e) we plot the phase-
mismatch [left-hand side of Eq. (1)] as a function of desynchronization
$\Delta t$ for resonator parameters corresponding to our experiments with the
nonlinear phase shift set to zero (orange curve) and to that of the nonlinear
phase shift of a propagating soliton with peak power
$P_{\mathrm{s}}=600\leavevmode\nobreak\ \mathrm{W}$ [see Fig. 1(f)]. As can be
seen, the phase-mismatch crosses zero at $\Delta t\approx
400\leavevmode\nobreak\ \mathrm{fs}$, which is precisely the value
corresponding to the experiments and simulations shown in Fig. 1(d) and (f).
In practice, the steady-state Raman soliton states can be sustained over a
small range of desynchronizations (rather than a single value) due to the
solitons’ ability to compensate for small phase-mismatches by adjusting their
peak power (and hence the nonlinear phase shift $q$).

Equations (1) and (2) can be leveraged to qualitatively understand how the
Raman soliton characteristics depend upon the resonator dispersion.
Specifically, ignoring the nonlinear phase shift ($q=0$), the conditions allow
to predict the soliton’s spectral shift from the input pump
($\Omega_{\mathrm{R}}$); moreover because the soliton spectrum bridges the
interval from its central frequency to the pump frequency, the parameter
$\Omega_{\mathrm{R}}$ is a key influencer of the solitons’ bandwidth (and
hence duration). Assuming that dispersion can be truncated at third order
($\beta_{k}=0$ for $k>3$), Eqs. (1) and (2) yield (for $q=0$)

$\Omega_{\mathrm{R}}=-\frac{3\beta_{2}}{2\beta_{3}}.$ (4)

We thus see that a large group-velocity dispersion coefficient $\beta_{2}$ at
the pump frequency $\omega_{\mathrm{P}}$ is conducive for obtaining Raman
solitons with broad bandwidths (short durations). Pragmatically, $\beta_{2}$
can be increased simply by shifting the pump frequency away from the zero-
dispersion wavelength of the resonator. Figure 3(a) shows results obtained in
a 1-m-long resonator that illustrate these points. Here the solid curves show
optical spectra measured for three different pump center wavelengths, hence
different values of the group-velocity dispersion coefficient $\beta_{2}$. As
predicted by Eq. (4), shifting the pump to shorter wavelengths (with larger
$\beta_{2}$) shifts the Raman soliton center frequency away from the pump,
whilst concomitantly yielding an overall broader spectrum. Also shown as
dashed curves in Fig. 3(a) are results from numerical simulations.

Figure 4: Multi-soliton operation. (a) Numerical simulation results, showing
the sequential emergence of new Raman solitons in a 1-m-long resonator as the
input peak power is continuously increased [for other parameters, see
Methods]. (b) Overlaid temporal intensity profiles of five Raman solitons
indicated by the colored arrows in (a). All of the profiles overlap,
highlighting that all of the solitons exhibit identical characteristics. (c)
Experimentally measured optical spectra for various input peak powers in a
1-m-long resonator. Line plots corresponding to the blue, orange, and green
dashed curves are shown in (d); these highlight the 3 dB and 1.8 dB increase
in the soliton spectral amplitude when respectively transitioning from one to
two, and from two to three solitons. Note: spectra in (c) are plotted in
linear scale and focus on the Raman soliton for clarity, whilst the spectra in
(d) are plotted in log scale and include the pump frequency. (e) Results from
dispersive Fourier transform measurements made for a 0.5-m-long resonator that
show the optical spectrum for a two-soliton state each round trip. Jitter in
experimental parameters causes the spectral interference to jitter, which
results in averaged spectra with no visible fringes [see (d)]. (f) Numerical
simulation of a three-soliton state [parameters as in (a) with pump power 80
W], showing how timing jitter with 5 fs standard deviation in the pump pulse
repetition rate gives rise to jitter in the relative separation of the Raman
solitons, with the trailing solitons experiencing larger jitter.

Theoretical analyses [see Methods] reveal that, in addition to
$\Omega_{\mathrm{R}}$, the soliton’s overall bandwidth also depends upon the
resonator’s characteristic dispersive timescale
$\tau_{\mathrm{s}}=\sqrt{|\beta_{2}|L\mathcal{F}/(2\pi)}$, where $\mathcal{F}$
is the finesse of the resonator; smaller $\tau_{\mathrm{s}}$ generally yield
broader soliton spectra. This point is demonstrated in Fig. 3(b), in which we
show measurements performed under conditions where the resonator length is
kept constant but finesse is varied [see Methods].

The scaling laws discussed above suggest that different pump-resonator
configurations for which the parameters $\Omega_{\mathrm{R}}$ and
$\tau_{\mathrm{s}}$ are constant can support Raman solitons with identical
characteristics. In particular, for a given chromatic dispersion, solitons
with identical bandwidth (hence duration) can be sustained across systems for
which the product between the round trip length $L$ and finesse $\mathcal{F}$
is an invariant. This scaling is illustrated in Fig. 3(c). Here we show
experimentally measured and numerically simulated optical spectra at the
output of resonators with three different round trip lengths: 0.50 m, 1.02 m,
and 2.65 m. For each case, we fine-tune the cavity finesse so as to obtain an
approximately constant product $L\mathcal{F}\approx 250\leavevmode\nobreak\
\mathrm{m}$. As can be seen, the optical spectra measured display high
similarity, thus corroborating the scaling law. On the other hand, the
corresponding RF spectra [Fig. 3(d)–(f)] make it clear that the Raman solitons
are generated at very different repetition rates due to the different lengths
of the resonators considered. It is worth emphasising that, based on numerical
modelling, the Raman solitons of Fig. 3(c) and (d) have a duration of about 55
fs: the results accordingly highlight how the scheme can be used to generate
ultrashort pulses across resonator length scales (and hence repetition rates).

In obtaining the results shown in Figs. 3(b) and (c), the peak power of the
injected pulses was increased with reducing finesse so as to maintain net
Raman gain at the group-velocity-matched component. Extensive simulations show
that, whilst the Raman soliton existence depends on the pump power, their
characteristics do not: provided that the input power is sufficiently large to
provide net Raman gain, the same soliton characteristics ensue. However, as
illustrated by the numerical simulation results shown in Figs. 4(a) and (b),
as the pump power is increased, the Raman gain available may exceed the amount
that a single soliton can deplete [see Fig. 2(d)], resulting in the sequential
appearance of new (identical) solitons trailing the first one. The Raman
soliton characteristics also appear to be insensitive to the detuning between
the pump carrier frequency and a cavity resonance, which is in stark contrast
to conventional dissipative Kerr cavity solitons [20]. In fact, for the Raman
solitons, experiments and simulations suggest the detuning plays a
comparatively similar role as the input pump power: both influence the
intracavity field at the pump frequency prior to Raman soliton formation,
which in turn dictates the net gain available for soliton (or multi-soliton)
formation.

Multi-soliton operation can be readily observed also in our experiments. The
pseudocolor plot in Fig. 4(c) shows the measured evolution of the Raman
soliton spectrum as a function of injected pump power, whilst Fig. 4(d) shows
selected line spectra in more detail. As can be seen, the spectral density
around the soliton feature sequentially increases in discrete, equal steps. It
is interesting to note that, whilst the presence of two temporally-separated
solitons would be expected to yield interference fringes across the optical
spectrum, such fringes are only observed around a narrow region about the
pump. To gain more insights, we used the dispersive Fourier transform
technique [see refs. [50, 51, 52] and Methods] to measure the optical spectrum
at the resonator output from round trip to round trip. The results, shown in
Fig. 4(e), reveal that spectral interference is present across the entire
spectrum but with the fringes moving about erratically over timescales of
several photon lifetimes. As the optical spectra shown in Fig. 4(d) are
obtained from time-averaged measurements, the interference pattern is largely
washed away.

The drifting of the spectral interference pattern [Fig. 4(e)] suggests that
the solitons’ relative phase and/or temporal separation is fluctuating in our
experiments, in a manner similar to previous observations of vibrating soliton
molecules in mode-locked lasers [53, 54]. Numerical simulations reveal that,
in our configuration, this phenomenon originates from noise in one or more of
the system parameters (e.g. timing jitter or amplitude noise of the pump
pulses). Specifically, because the temporal position of a trailing soliton is
dictated by regeneration of the Raman gain from the depletion induced by all
of the leading solitons, any fluctuation in the background field at the pump
frequency (which acts as the Raman pump) results in jitter of the solitons’
relative position. The same line of thought suggests that, in a $N$ soliton
complex, the first (leading) soliton will experience the smallest jitter
whilst the last ($N^{\mathrm{th}}$) soliton will experience the largest jitter
(being impacted by the gain depletion of all the leading solitons). This
hypothesis is corroborated by our simulations [Fig. 4(f)].

##  III Discussion

Our work demonstrates that phase-coherent pulsed driving of “passive” silica
Kerr resonators enables the generation of coherent and broadband optical
spectra and corresponding ultrashort pulses thanks to stimulated Raman
scattering. In stark contrast to conventional temporal CSs of driven Kerr
resonators, this new type of dissipative Raman solitons can exhibit pulse
durations well below 100 femtoseconds even when realised in macroscopic fibre
resonators with modest finesse and nonlinearity. To the best of our knowledge,
these are the shortest pulses ever generated in resonators made from a single
piece of commercially-available optical fibre (active or passive). We have
demonstrated that the ultrashort Raman solitons can be deterministically
generated in resonators of varying lengths, thus providing an attractive
avenue for the generation of optical frequency combs with desired line
spacing. We have studied the solitons’ scaling laws, identifying conditions
that are conducive for increasing (reducing) the solitons’ bandwidth
(duration): simulations indicate that durations below 40 fs should be
attainable using commercially available fibers [see Supplementary Fig. 3].
Finally, whilst our experiments have been performed in fibre-based resonators,
we expect that the dissipative Raman solitons can also be generated in
synchronously-pumped chip-scale silica resonators, thus enabling an
interesting new avenue for the generation of ultra-broadband frequency combs
with small comb line spacing in photonic integrated platforms [31, 43, 44].

## Methods

### Additional experimental details and parameters.

The fibre Fabry-Pérot resonators used in our experiments are formed around two
conventional FC/PC fibre connectors with standard single-mode-fibre (SMF-28)
core that are dielectrically coated to produce high reflectivity over a broad
band. The resonators are formed by connecting the two coated connectors with a
segment of Corning MetroCor non-zero dispersion-shifted optical fibre.
Resonators of various lengths can be straightforwardly achieved by simply
changing the length of the fibre segment connecting the two mirrors, and it is
predominantly for this reason that we chose a Fabry-Pérot rather than a ring
architecture. We measure the finesse of the resonators by scanning a narrow-
linewidth cw laser across multiple cavity free-spectral ranges and by fitting
Lorentzian resonance shapes onto each adjacent resonance pair. For each pair,
we obtain a statistical sample of the finesse by dividing the resonance
separation with the average resonance linewidth; the final estimate for the
finesse is obtained as the average of about 1000 resonance pairs. We measured
the mirror transmission (i.e., coupling) coefficient into the resonator to be
about $\theta=0.002$. We note that the coupling is not lossless: the maximum
resonator finesse that we have obtained is around 500, which is considerably
smaller than the finesse accounting solely for the coupling
($\mathcal{F}_{\mathrm{max}}=\pi/\theta\approx 1600$). We suspect the losses
arise mostly due to imperfect contact between the connectors of the mirrors
and the MetroCor fibre, and from the mode-mismatch between SMF-28 and
MetroCor. We also note that the finesse is systematically tuned in our
experiments by placing a small segment of the optical fibre that forms the
resonator within an inline polarization controller, which introduces micro-
bends into the fibre, and thus increases the resonator losses.

The dispersion in the resonators used in our work arise from the optical fibre
and from the dielectric mirrors. We estimate the overall resonator dispersion
from desynchronization scan data similar to the one shown in Fig. 1(b).
Specifically, we extract the dispersion coefficients by fitting the group-
velocity-matching condition [Eq. (2)] to the experimentally observed positions
of the Raman components as the desynchronization is varied. We then further
constrain the dispersion coefficients by fitting numerical simulations of the
Raman solitons to selected experimental measurements. This two-step procedure
yields the following second- and third-order dispersion coefficients at the
wavelength of 1562.5 nm: $\beta_{2}=7.15\leavevmode\nobreak\
\mathrm{ps^{2}km^{-1}}$ and $\beta_{3}=1.36\leavevmode\nobreak\
\mathrm{ps^{3}km^{-1}}$. (Higher-orders of dispersion are negligible.) These
coefficients predict the resonator zero-dispersion wavelength to be 1633 nm.
We estimate (based on the quoted mode-field diameter) that the MetroCor fibre
has a Kerr nonlinearity coefficient $\gamma=2.6\leavevmode\nobreak\
\mathrm{W^{-1}km^{-1}}$.

The resonators are pumped with a train of pulses that are generated by passing
cw laser light through an electro-optic-comb generator comprised of one
amplitude modulator and two phase modulators in cascade [55]. The resulting
electro-optic frequency comb is passed through 300 metres of dispersion
compensating fibre and then through a nonlinear (soliton) compression scheme
consisting of a 1-km-long segment of SMF-28. The pulses generated in this
manner pass through a nonlinear amplifying loop mirror to remove residual low-
level backgrounds. Before the pulses are launched into the resonators, they
are amplified with an erbium-doped fibre amplifier. The repetition rate of the
pump pulse train (hence the pump periodicity $t_{\mathrm{P}}$) is defined by
an electronic clock signal that is fed into the modulators that make the
electro-optic comb generator; adjusting this clock frequency allows to control
the desynchronization $\Delta t=t_{\mathrm{R}}-t_{\mathrm{P}}$, where
$t_{\mathrm{R}}$ is the reciprocal of the resonator free-spectral range at the
pump wavelength. To calibrate the desynchronization (i.e., to determine the
point where $\Delta t=0$), we adjust the clock frequency until the spectrum
around the pump frequency is symmetric [38]. We estimate that the uncertainty
in the calibration is about 2 fs. The carrier frequency of the driving pulse
train is locked at a fixed detuning from a cavity resonance using the Pound-
Drever-Hall (PDH) technique. Specifically, a small portion of the cw laser is
phase modulated and frequency shifted, and then coupled into a cavity mode
that is orthogonally polarized with respect to the main pump. (The frequency-
shifter compensates for the resonator birefringence and allows to control the
value of the detuning experienced by the main pump.) At the cavity output, the
cw component is isolated using a polarizing beam-splitter, detected, and fed
to an active controller that generates the PDH signal and actuates the
frequency of the driving cw laser.

The dispersive Fourier transform measurements shown in Fig. 4 were obtained by
first spectrally filtering the resonator output to attenuate the pump
component, then passing the resulting signal through a 100-m-long segment of
dispersion compensating fibre with dispersion
$\beta_{2}=144.2\leavevmode\nobreak\ \mathrm{ps^{2}km^{-1}}$ to realise the
frequency-to-time mapping. The output was then recorded in real-time on a 12.5
GHz photodetector and digitized on a 12 GHz real-time oscilloscope. A single,
long time trace was recorded, divided into segments that span a single round
trip, and finally concatenated vertically to yield the pseudo-color plot in
Fig. 4(e).

### Numerical model.

Our numerical simulations are based on the generalized Lugiato-Lefever
equation that includes pump-cavity desynchronization, higher-order dispersion,
and stimulated Raman scattering. The equation describes the evolution of the
slowly-varying intracavity electric field envelope $E(t,\tau)$ according to
[34]

$\displaystyle t_{\mathrm{R}}\frac{\partial E}{\partial t}=$
$\displaystyle\left[-\alpha-i\delta_{0}-\Delta
t\frac{\partial}{\partial\tau}+iL\hat{D}\left(i\frac{\partial}{\partial\tau}\right)\right]E+\sqrt{\theta
P_{\mathrm{P}}}S(\tau)$ $\displaystyle+i\gamma
L\left[(1-f_{\mathrm{R}})|E|^{2}+f_{\mathrm{R}}h_{\mathrm{R}}(\tau)\ast|E|^{2}\right]E.$
(5)

Here the continuous variable $t$ is a “slow” time that describes the evolution
of the field envelope $E(t,\tau)$ at the scale of the cavity photon lifetime,
whilst $\tau$ is a “fast” time variable that describes the field profile over
a single round trip. $\alpha=\pi/\mathcal{F}$ is half the power dissipated by
the cavity per round trip, $\delta_{0}$ is the phase detuning between the
injected pump pulse train and the closest cavity resonance (stabilised at
$\delta_{0}=0$ in our experiments), $L$ is the cavity length, $\Delta t$ and
$\hat{D}$ are the pump-cavity desynchronization and dispersion operator as
defined in the main text, and $\theta$ is the power coupling coefficient of
the coupler through which the pump pulses are injected into the cavity.
$P_{\mathrm{P}}$ is the peak power of the pulses driving the cavity, whilst
$f(\tau)$ describes the functional form of the pulse envelope. It is
noteworthy that the (fast time) reference frame of Eq. (5) has been chosen as
the reference frame that is synchronous with the pump pulse train (but not the
cavity round trip time), due to which (i) the desynchronization term $\Delta
t$ appears explicitly in Eq. (5) and (ii) the driving field profile $S(\tau)$
does not depend on the slow time $t$. The second row of Eq. (5) describes the
nonlinear effects: $\gamma$ is the Kerr nonlinearity coefficient,
$f_{\mathrm{R}}$ is the Raman fraction, and $h_{\mathrm{R}}(\tau)$ is the
time-domain Raman response, with the convolution
$h_{\mathrm{R}}(\tau)\ast|E|^{2}$ reflecting the delayed nature of the Raman
nonlinearity.

All of the simulation results shown in our work were obtained by numerically
integrating Eq. (5) using the well-known split-step Fourier method, and (apart
from one exception to be discussed below) with parameters close to accepted
literature values or values measured for the corresponding experiments (quoted
in the main text or in the section above). For desynchronizations, we find
that multiplying the value obtained by solving Eqs. (1) and (2) with $q=0$ by
1.1 typically yields a value within the soliton existence regime. The
simulations assume that the pump pulses have a Gaussian profile, such that

$S(\tau)=e^{-\tau^{2}/\tau_{0}^{2}},$ (6)

with the width parameter $\tau_{0}$ estimated as 5 ps.

The Raman response function used in our simulations corresponds to the
analytical multi-vibrational model described in ref. [56]. Interestingly,
however, we find that reaching quantitative agreement with our experiments
requires us to set the Raman fraction as $f_{\mathrm{R}}=0.09$, which is half
of the accepted value for fused silica. This discrepancy is not limited to a
single set of parameters: we find that the value $f_{\mathrm{R}}=0.09$ allows
to obtain good agreement with experiments performed across numerous resonators
with different lengths and finesse as well as both in Fabry-Pérot and ring
geometries, with all other parameters corresponding to measured or accepted
values. In contrast, using the accepted value $f_{\mathrm{R}}=0.18$ does not
yield simulation results that are quantitatively similar to our experiments.
We suspect that the use of a reduced Raman fraction effectively lumps together
the individual influences of a variety of higher-order effects that are
present in the experiments but are ignored in Eq. (5), such as wavelength-
dependent nonlinearity (self-steepening), wavelength-dependent losses, and
polarization effects. All of such higher-order effects would tend to reduce
the efficiency of stimulated Raman scattering, which we hypothesise can be
effectively described by simply using a reduced Raman fraction of
$f_{\mathrm{R}}=0.09$.

### Raman gain.

The net Raman amplitude gain per round-trip experienced by a signal with
carrier frequency down-shifted by $\Omega$ from a pump pulse can be
approximately written as [46]

$g\approx\frac{g_{\mathrm{R}}}{2}P_{\mathrm{P}}-\alpha.$ (7)

Here the coefficient $g_{\mathrm{R}}$ is related to the resonator and Raman
response parameters via $g_{\mathrm{R}}=2\gamma
Lf_{\mathrm{R}}\text{Im}[\tilde{h}_{\mathrm{R}}(\Omega)]$, where
$\tilde{h}_{\mathrm{R}}(\Omega)$ is the Fourier transform of the Raman
response function $h_{\mathrm{R}}(\tau)$, and $P_{\mathrm{P}}$ is the power of
the Raman pump. By setting $g=0$, we can readily derive an expression for the
threshold pump power to achieve net Raman gain:

$P_{\mathrm{th}}=\frac{2\alpha}{g_{\mathrm{R}}}.$ (8)

The thresholds shown in Fig. 2(c) and (d) were computed using Eq. (8) with
$\Omega$ chosen as the group-velocity-matched frequency.

### Matching conditions.

We present here a brief derivation of the phase-matching and group-velocity-
matching conditions given by Eqs. (1) and (2), respectively. When evaluated at
a time corresponding to the period of the input pulse train, the phase
accumulated by the intracavity field at the pump frequency after one round
trip is given by
$\phi_{\mathrm{P}}=\beta(\omega_{\mathrm{P}})L-\omega_{\mathrm{P}}t_{\mathrm{P}}$,
where $\beta(\omega)$ is the propagation constant of the resonator. Similarly,
the phase accumulated by the Raman soliton, with carrier frequency $\omega$,
is $\phi_{\mathrm{S}}=\beta(\omega)L-\omega t_{\mathrm{P}}+qL$, where $q$
describes the soliton’s nonlinear phase shift. Equating
$\phi_{\mathrm{P}}=\phi_{\mathrm{S}}$ yields

$\beta(\omega_{\mathrm{P}})L-\omega_{\mathrm{P}}t_{\mathrm{P}}=\beta(\omega)L-\omega
t_{\mathrm{P}}+qL.$ (9)

Expanding the propagation constant as a Taylor series about the pump frequency
$\omega_{\mathrm{P}}$ yields

$-\omega_{\mathrm{P}}t_{\mathrm{P}}=\beta_{1}(\omega-\omega_{\mathrm{P}})L+\hat{D}(\Omega)L-\omega
t_{\mathrm{P}}+qL,$ (10)

where $\Omega=\omega-\omega_{\mathrm{P}}$ and
$\hat{D}(\Omega)=\beta(\omega)-\beta(\omega_{\mathrm{P}})-\beta_{1}\Omega$.
Rearranging, we obtain

$\hat{D}(\Omega)L+qL=(\omega-\omega_{\mathrm{P}})t_{\mathrm{P}}-\beta_{1}(\omega-\omega_{\mathrm{P}})L.$
(11)

Using the fact that $\beta_{1}L=t_{\mathrm{R}}$, where $t_{\mathrm{R}}$ is the
resonator round trip time at the pump frequency, we have

$\hat{D}(\Omega)L+qL=(\omega-\omega_{\mathrm{P}})\left[t_{\mathrm{P}}-t_{\mathrm{R}}\right]=-\Omega\Delta
t.$ (12)

Rearranging and dividing by $L$ yields the phase-matching Eq. (1).

For the group-velocity-matching condition, we first note that the resonator
round trip time at frequency $\omega$ can be written as

$t_{\mathrm{R}}(\omega)=\beta_{1}(\omega)L,$ (13)

where $\beta_{1}(\omega)=d\beta/d\omega$. Again expanding the propagation
constant as a Taylor series about the pump frequency $\omega_{\mathrm{P}}$
yields

$t_{\mathrm{R}}(\omega)=t_{\mathrm{R}}(\omega_{\mathrm{P}})+\hat{D}_{1}(\Omega)L,$
(14)

where $\hat{D}_{1}(\Omega)=d\hat{D}/d\Omega$. Equating this round trip time
with the pump periodicity $t_{\mathrm{P}}$ yields

$\hat{D}_{1}(\Omega)L=t_{\mathrm{P}}-t_{\mathrm{R}}(\omega_{\mathrm{P}})=-\Delta
t.$ (15)

Rearranging and dividing by $L$ yields the group-velocity-matching Eq. (2).

When truncating the dispersion at third-order ($\beta_{k}=0$ for $k>3$) and
setting the nonlinear phase-shift $q=0$, it is straightforward to solve Eqs.
(1) and (2) to yield Eq. (4). In this case, one finds that the corresponding
desynchronization $\Delta t=3\beta_{2}^{2}L/(8\beta_{3})$.

### Scaling laws.

The generalized Lugiato-Lefever Eq. (5) can be written in a dimensionless form
by introducing the following variable transformations [34]

$\displaystyle t\rightarrow\frac{\alpha
t}{t_{\mathrm{R}}},\qquad\tau\rightarrow\frac{\tau}{\tau_{\mathrm{s}}},\qquad
E\rightarrow E\sqrt{\frac{\gamma L}{\alpha}},$ (16)

where the fast time normalization timescale
$\tau_{\mathrm{s}}=\sqrt{\frac{|\beta_{2}|L}{2\alpha}}$. Applying these
changes of variables transforms Eq. (5) into the following dimensionless form:

$\displaystyle\frac{\partial E}{\partial t}=$ $\displaystyle\left[-1-i\Delta-
d_{1}\frac{\partial
E}{\partial\tau}+i\hat{D}_{\mathrm{N}}\left(i\frac{\partial}{\partial\tau}\right)\right]E+\sqrt{X}f\left(\tau\tau_{\mathrm{s}}\right)$
$\displaystyle+i\left[(1-f_{\mathrm{R}})|E|^{2}+f_{\mathrm{R}}\Gamma(\tau,\tau_{\mathrm{s}})\ast|E|^{2}\right]E.$
(17)

Here, the normalized detuning $\Delta=\delta_{0}/\alpha$, desynchronization
$d_{1}=\Delta t/(\alpha\tau_{\mathrm{s}})$, driving peak power $X=\gamma
L\theta P_{\mathrm{P}}/\alpha^{3}$, Raman response function
$\Gamma(\tau,\tau_{\mathrm{s}})=\tau_{\mathrm{s}}h_{\mathrm{\tau}}(\tau\tau_{\mathrm{s}})$
and dispersion operator

$\hat{D}_{\mathrm{N}}=\sum_{k\geq
2}d_{k}\left(i\frac{\partial}{\partial\tau}\right),$ (18)

with the normalized dispersion coefficients
$d_{k}=L\beta_{k}/(\alpha\tau_{\mathrm{s}}^{k}k!)$.

For a given Raman response, the solutions and dynamics of Eq. (17) depend upon
the following normalized variables: the detuning $\Delta$, desynchronization
$d_{1}$, dispersion $\hat{D}_{\mathrm{N}}$, fast time normalization timescale
$\tau_{\mathrm{s}}$, driving peak power $X$, and the pump pulse profile
$S(\tau)$. Extensive simulations show that the impact of the detuning
$\Delta$, the driving peak power $X$, and the pump pulse profile $S(\tau)$ are
limited to influencing whether Raman solitons exist, but not the soliton
characteristics. Rather, the soliton characteristics appear to depend upon the
desynchronization $d_{1}$, dispersion $\hat{D}_{\mathrm{N}}$ and the timescale
$\tau_{\mathrm{s}}$. On the other hand, the desynchronization $d_{1}$ is
straightforward to adjust (and optimise) in experiments by simply adjusting
the clock frequency that defines the repetition rate of the pump pulse train;
this parameter can therefore be considered unimportant from the point of view
of system design. Assuming that the dispersion can be truncated at third-
order, we can therefore conclude that the main resonator parameters that
influence the attainable Raman soliton characteristics are the normalization
timescale $\tau_{\mathrm{s}}$ and the third-order dispersion coefficient
$d_{3}$. On the other hand, the latter coefficient can be written as
$d_{3}=-1/(\Omega_{\mathrm{R}}\tau_{\mathrm{s}})$ with $\Omega_{\mathrm{R}}$
defined by Eq. (4), thus highlighting that the soliton characteristics are
dictated by $\tau_{\mathrm{s}}$ and $\Omega_{\mathrm{R}}$.

## Acknowledgements

We acknowledge financial support from the Marsden Fund of the Royal Society of
New Zealand.

## Author Contributions

Z.L. performed all of the experiments and most of the simulations with the
help of Y.X. and M.E. S.S. performed numerical simulations of soliton
characteristics. X.W., W.W., X.W. and Z.Y. fabricated the dielectric mirrors
used in the resonators. S.C. assisted with the analysis of results. S.G.M.
helped supervise the experiments and interpret the results. M.E. supervised
the overall project and wrote the manuscript with Z.L.

## Data availability

The data that support the plots within this paper and other findings of this
study are available from M.E. upon reasonable request.

## Competing financial interests

The authors declare no competing financial interests.

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Supplementary Figure 1: Experimental demonstration of deterministic Raman
soliton generation. (a) The desynchronization is changed according to a
triangular waveform by changing the clock frequency that defines the input
pulse repetition rate. (b) and (c) show the optical and electronic spectra
measured as the desyncrhonization is changed according to (a). Every time the
desynchronization enters the soliton existence regime, a coherent Raman
soliton emerges deterministically and spontaneously. All parameters as in Fig.
1 of the main manuscript. Note that the absolute desynchronization values
needed for soliton existence differ somewhat from those observed in Fig. 1 of
the main manuscript (by about 10 fs). We ascribe this to uncertainty in
calibrating the zero-desynchronization point as well as to desynchronization
drifts that occur during the comparatively long overall measurement times
involved [e.g. 30 minutes for data shown in Fig. 1(b) and (c) of the main
manuscript.]. Supplementary Figure 2: Single-sideband phase noise measurements
for the Raman soliton state (black), the input pump (green), and an unstable
state obtained by setting the desynchronization out of the Raman soliton
regime (red). The purple curve shows the noise floor of the electronic
spectrum analyzer. The Raman soliton phase noise is similar to the phase noise
of the pump, further supporting the notion that the Raman soliton noise
properties are derived from the pump. Supplementary Figure 3: Numerical
simulation results, showing an ultrashort dissipative Raman soliton with
duration as short as 32 fs. (a) and (b) show simulated spectral (a) and
temporal (b) intensity profiles of the total intracavity field. Inset in (b)
shows the Raman soliton in more detail. The simulations use the generalized
Lugiato-Lefever equation displayed as Eq. 5 in the main manuscript. The
simulation parameters are the same as those quoted in the main manuscript
expect for the following adjustments. The group-velocity dispersion
coefficient was set to $\beta_{2}=14.56\leavevmode\nobreak\
\mathrm{ps^{2}km^{-1}}$ which corresponds to a driving wavelength of 1495 nm
for the MetroCor fibre used in our work, the resonator was assumed to be 5-cm-
long and to have a finesse of 400, and the Gaussian pump pulses were assumed
to have an $\exp(-2)$ width of $\tau_{0}=800\leavevmode\nobreak\ \mathrm{fs}$
and a peak power of 2.5 kW. The desynchronization was set to $\Delta t=32.15$
fs per round trip. These pump parameters (centre wavelength in particular)
cannot be achieved with equipment available to us, but they are not
fundamentally prohibitive.


# Local complementation rule for continuous-variable four-mode unweighted
graph states

Jing Zhang† State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China

###### Abstract

The local complementation rule is applied for continuous-variable (CV) graph
states in the paper, which is an elementary graph transformation rule and
successive application of which generates the orbit of any graph states. The
corresponding local Gaussian transformations of local complementation for
four-mode unweighted graph states were found, which do not mirror the form of
the local Clifford unitary of qubit exactly. This work is an important step to
characterize the local Gaussian equivalence classes of CV graph states.

Entanglement lies at the heart of quantum mechanics and plays a crucial role
in quantum information processing. Recently, special types of multipartite
entangled states, the so-called the graph states one ; two , have moved into
the center of interest. A graph quantum state is described by a mathematical
graph, i.e. a set of vertices connected by edges. A vertex represents a
physical system, e. g. a qubit (2-dimensional Hilbert space), qudit
(d-dimensional Hilbert space), or CV (continuous Hilbert space). An edge
between two vertices represents the physical interaction between the
corresponding systems. An interesting feature is that many entanglement
properties of graph states are closely related to their underlying graphs.
They not only provide an efficient model to study multiparticle entanglement
one , but also find applications in quantum error correction three ; four ,
multi-party quantum communication five  and most prominently, serve as the
initial resource in one-way quantum computation six . Considerable efforts
have been stepped toward generating and characterizing cluster state with
linear optics experimentally seven ; eight ; nine ; ten . The principle
feasibility of one-way quantum computing model has been experimentally
demonstrated through photon cluster state successfully seven ; ten .

Most of the concepts of quantum information and computation have been
initially developed for discrete quantum variables, in particular two-level or
spin-$\frac{1}{2}$ quantum variables (qubits). In parallel, quantum variables
with a continuous spectrum, such as the position and momentum of a particle or
amplitude and phase quadrature of an electromagnetic field, in informational
or computational processes have attracted a lot of interest and appears to
yield very promising perspectives concerning both experimental realizations
and general theoretical insights eleven ; twelve , due to relative simplicity
and high efficiency in the generation, manipulation, and detection of CV
state. Although up to six-qubit single-photon cluster states have been created
via postselection using nonlinear and linear optics, the deterministic,
unconditional realization of optical cluster states would be based on
continuous variables. CV cluster and graph states have been proposed thirteen
, which can be generated by squeezed state and linear optics thirteen1 ;
thirteen2 , and demonstrated experimentally for four-mode cluster state
forteen . The one-way CV quantum computation was also proposed with CV cluster
state fifteen . Moreover, the protocol of CV anyonic statistics implemented
with CV graph states is proposed fifteen1 .

One of the interesting issues on entanglement is how to define the equivalence
of two entangled states. The transformations of qubit graph states under local
Clifford operations were studied by Hein one  and Van den Nest sixteen . They
translate the action of local Clifford operations on qubit graph states into
transformations on their associated graphs, that is, to derive transformations
rules called the local complement rule, stated in purely graph theoretical
terms, which completely characterize the evolution of graph states under local
Clifford operations. The corresponding local Clifford unitary is a single and
simple form. The successive application of this rule suffices to generate the
complete orbit of any qubit graph state under local Clifford operations. In
this paper, the local complement rule for CV four-mode unweighted graph state
is applied and the corresponding local Clifford transformations (also called
local Gaussian transformation for CV) for four-mode graph state were found.
The local Gaussian equivalence classes of CV four-mode unweighted graph states
can be obtained by this way. It was shown that the corresponding local
Gaussian unitary can not exactly mirror that for qubit, which is not a single
form compared with qubit. This result shows the complexity of CV quantum
systems and stimulate the research on the local Gaussian equivalence of CV
graph states. Although only focusing on the CV four-mode unweighted graph
states, this work makes an important step in the direction of addressing the
general question ”What are the graph transformation rules that describe local
unitary equivalence of any CV graph states?”.

The CV operations are reviewed firstly that follow the standard prescription
given in Ref.seventeen . The Pauli $X$ and $Z$ operators of qubit are
generalized to the Weyl-Heisenberg group, which is the group of phase-space
displacements. For CVs, this is a Lie group with generators
$\hat{x}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$ (quadrature-amplitude or
position) and $\hat{p}=-i(\hat{a}-\hat{a}^{\dagger})/\sqrt{2}$ (quadrature-
phase or momentum) of the electromagnetic field as the CV system. These
operators satisfy the canonical commutation relation $[\hat{x},\hat{p}]=i$
(with $\hbar=1$). In analogy to the qubit Pauli operators, the single mode
Pauli operators are defined as $X(s)=exp[-is\hat{p}]$ and
$Z(t)=exp[it\hat{x}]$ with $s,t\in\mathbb{R}$. The Pauli operator $X(s)$ is a
position-translation operator, which acts on the computational basis of
position eigenstates $\\{|q\rangle;q\in\mathbb{R}\\}$ as
$X(s)|q\rangle=|q+s\rangle$, whereas $Z$ is a momentum-translation operator,
which acts on the momentum eigenstates as $Z(t)|p\rangle=|p+t\rangle$. These
operators are non-commutative and obey the identity
$X(s)Z(t)=e^{-ist}Z(t)X(s)$. The Pauli operators for one mode can be used to
construct a set of Pauli operators $\\{X_{i}(s_{i}),Z_{i}(t_{i});i=1,...,n\\}$
for n-mode systems. This set generates the Pauli group $\mathcal{C}_{1}$. The
clifford group $\mathcal{C}_{2}$ is the normalizer of the Pauli group, whose
transformations acting by conjugating, preserve the Pauli group
$\mathcal{C}_{1}$; i.e., a gate U is in the Clifford group if
$\emph{UR}\emph{U}^{-1}\in\mathcal{C}_{1}$ for every
$\emph{R}\in\mathcal{C}_{1}$. The clifford group $\mathcal{C}_{2}$ for CV is
shown seventeen  to be the (semidirect) product of the Pauli group and linear
symplectic group of all one-mode and two-mode squeezing transformations.
Transformation between the position and momentum basis is given by the Fourier
transform operator $F=exp[i(\pi/4)(\hat{x}^{2}+\hat{p}^{2})]$, with
$F|q\rangle_{x}=|q\rangle_{p}$. The action $FRF^{-1}$ of the Fourier transform
on the Pauli operators is

$\displaystyle F:X(s)$ $\displaystyle\rightarrow$ $\displaystyle Z(s),$
$\displaystyle Z(t)$ $\displaystyle\rightarrow$ $\displaystyle X(-t).$ (1)

This is the generalization of the Hadamard gate for qubits. The phase gate
$P(\eta)=exp[i(\eta/2)\hat{x}^{2}]$ with $\eta\in\mathbb{R}$ is a squeezing
operation for CV and the action on the Pauli operators is

$\displaystyle P(\eta):X(s)$ $\displaystyle\rightarrow$ $\displaystyle
e^{-is^{2}\eta/2}Z(s\eta)X(s),$ $\displaystyle Z(t)$
$\displaystyle\rightarrow$ $\displaystyle Z(t),$ (2)

in analogy to the phase gate of qubit eighteen . The controlled operation C-Z
is generalized to controlled-$Z(C_{Z})$. This gate
$C_{Z}=exp[i\hat{x}_{1}\bigotimes\hat{x}_{2}]$ provides the basic interaction
for two mode 1 and 2, and describes the quantum nondemolition (QND)
interaction. This set $\\{X(s),F,P(\eta),C-Z;s,\eta\in\mathbb{R}\\}$ generates
the Clifford group. Transformations in the Clifford group do not form a
universal set of gates for CV quantum computation. However, Clifford group
transformation (Gaussian transformations) together with any higher-order
nonlinear transformation (non-Gaussian transformation) acting on a single-mode
form a universal set of gates seventeen . The local Gaussian group only was
concerned here, which can be obtained by repeated application of Fourier and
phase gates. In the following, another type of the phase gate will be used
$P_{X}(\eta)=FP(\eta)F^{-1}=exp[i(\eta/2)\hat{p}^{2}]$ and the action on the
Pauli operators is

$\displaystyle P_{X}(\eta):X(s)$ $\displaystyle\rightarrow$ $\displaystyle
X(s),$ $\displaystyle Z(t)$ $\displaystyle\rightarrow$ $\displaystyle
e^{-it^{2}\eta/2}X(-t\eta)Z(t),$ (3)

where $P_{X}(\eta)^{\dagger}=P_{X}(\eta)^{-1}=P_{X}(-\eta)$.

A graph quantum state is described by a mathematical graph $G=(V,E)$, i.e. a
finite set of $n$ vertices $V$ connected by a set of edges $E$ ninteen . An
$\left\\{a,c\right\\}$-path is a order list of vertices
$a=a_{1},a_{2},\ldots,a_{n-1},a_{n}=c$, such that for all $i$, $a_{i}$ and
$a_{i+1}$ are adjacent. A connected graph is a graph that has an
$\left\\{a,c\right\\}$-path for any two $a,c\in V$. Otherwise it is referred
to as disconnected. The neighborhood $N_{a}\subset V$ is defined as the set of
vertices $b$ for which $\left\\{a,b\right\\}\in E$. When a vertex a is deleted
in a graph G, together with all edges incident with a, one obtains a new
graph, denoted by $G-a$. For a subset of vertices $U\subset V$ of a graph
$G=(V,E)$ let us denote with $G-U$ the graph that is obtained from $G$ by
deleting the set $U$ of vertices and all edges which are incident with an
element of $U$. Similarly, an subgraph $G[C]$ of a graph $G=(V,E)$, where
$C\subset V$, is obtained by deleting all vertices and the incident edges that
are not contained in $C$. The preparation procedure of CV graph states
thirteen  can exactly mirror that for qubit graph states only using the
Clifford operations: first, prepare each mode (or graph vertex) in a phase-
squeezed state, approximating a zero-phase eigenstate (analog of Pauli-X
eigenstates), then, apply a QND interaction (C-Z gate) to each pair of modes
$(j,k)$ linked by an edge in the graph. All C-Z gates commute. Thus, the
resulting CV graph state becomes, in the limit of infinite squeezing,
$g_{a}=(\hat{p}_{a}-\sum_{b\in N_{a}}\hat{x}_{b})\rightarrow 0$, where the
modes $a\in V$ correspond to the vertices of the graph of $n$ modes, while the
modes $b\in N_{a}$ are the nearest neighbors of mode $a$. This relation is as
a simultaneous zero-eigenstate of the position-momentum linear combination
operators. The stabilizers $G_{a}(\xi)=exp[-i\xi g_{a}]=X_{a}(\xi)\prod_{b\in
N_{a}}Z_{b}(\xi)$ with $\xi\in\mathbb{R}$ for CV graph states are analogous to
$n$ independent stabilizers $G_{a}=X_{a}\prod_{b\in N_{a}}Z_{b}$ for qubit
graph states. Note that the CV graph states that is discussed here are
unweighted since the QND interactions all have the same strength. For the CV
weighted graph states generated by the different QND interaction strength, the
stabilizers become $G_{a}(\xi)=X_{a}(\xi)\prod_{b\in
N_{a}}Z_{b}(\Omega_{ab}\xi)$, where $\Omega_{ab}$ is the interaction strength
between mode a and b. The CV weighted graph states are more complex, which is
not considered in this paper.

Figure 1:  The connected four-vertex graphs for an successive application of
the local complementation. The rule is successively applied to the vertex,
which is circle in the figure.

The action of the local complement rule, can be described as: letting
$G=(V,E)$ be a graph and $a\in V$ be a vertex, the local complement of $G$ for
$a$, denoted by $\lambda_{a}(G)$, is obtained by complementing the subgraph of
$G$ generated by the neighborhood $N_{a}$ of $a$ and leaving the rest of the
graph unchanged. The successive application of this rule suffices to generate
the complete orbit of any graph. Here, the corresponding local Gaussian
unitary for CV four-mode graph state were examined. The corresponding four-
mode graph state $|\lambda_{a}(G)\rangle$ by local complement of a graph $G$
at some vertex $a\in V$, is given by a local Gaussian unitary operation

$\displaystyle|\lambda_{a}(G)\rangle=U_{\lambda_{a}}|G\rangle,$ (4)

where $U_{\lambda_{a}}$ is local Gaussian operation. A form of the local
Gaussian unitary comprising two types of phase gate is defined

$\displaystyle U_{LG_{a}}=P_{Xa}(1)\prod_{b\in N_{a}}P_{b}(-1),$ (5)

which mirrors the form of qubit local Clifford operation for local
complementation. Fig.1 depicts connected four-mode graphs by such a successive
application of the local complement rule. The four independent stabilizers of
the first graph state $|G^{(1)}\rangle$ are given by

$\displaystyle G_{1}^{(1)}(\xi)$ $\displaystyle=$ $\displaystyle
X_{1}(\xi)Z_{2}(\xi),$ $\displaystyle G_{2}^{(1)}(\xi)$ $\displaystyle=$
$\displaystyle X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi),$ $\displaystyle
G_{3}^{(1)}(\xi)$ $\displaystyle=$ $\displaystyle
X_{3}(\xi)Z_{2}(\xi)Z_{4}(\xi),$ $\displaystyle G_{4}^{(1)}(\xi)$
$\displaystyle=$ $\displaystyle X_{4}(\xi)Z_{3}(\xi).$ (6)

with $G_{i}^{(1)}(\xi)|G^{(1)}\rangle=|G^{(1)}\rangle$ in the limit of
infinite squeezing, where $i=1,...,4$. Applying the local Gaussian unitary
$U_{LG_{3}}$ to the vertex 3, I can compute the four independent stabilizers
of the resulting graph state $|G^{(2)}\rangle$ by Eqs. (2,3,4,5), for example
calculating $G_{2}^{(2)}(\xi)$,

$\displaystyle|G^{(2)}\rangle$ $\displaystyle=$
$\displaystyle|\lambda_{3}(G^{(1)})\rangle$ (7) $\displaystyle=$
$\displaystyle U_{LG_{3}}G_{2}^{(1)}(\xi)|G^{(1)}\rangle$ $\displaystyle=$
$\displaystyle
U_{LG_{3}}G_{2}^{(1)}(\xi)U_{LG_{3}}^{-1}U_{LG_{3}}|G^{(1)}\rangle$
$\displaystyle=$
$\displaystyle[e^{i\xi^{2}/2}Z_{2}(-\xi)X_{2}(\xi)]Z_{1}(\xi)$
$\displaystyle[e^{-i\xi^{2}/2}X_{3}(-\xi)Z_{3}(\xi)]U_{LG_{3}}|G^{(1)}\rangle$
$\displaystyle=$ $\displaystyle X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)U_{LG_{3}}$
$\displaystyle[Z_{2}(-\xi)X_{3}(-\xi)]|G^{(1)}\rangle$ $\displaystyle=$
$\displaystyle X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)U_{LG_{3}}$
$\displaystyle[Z_{2}(-\xi)X_{3}(-\xi)]G_{3}^{(1)}(\xi)|G^{(1)}\rangle$
$\displaystyle=$ $\displaystyle
X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)Z_{4}(\xi)|\lambda_{3}(G^{(1)})\rangle$
$\displaystyle=$ $\displaystyle G_{2}^{(2)}(\xi)|G^{(2)}\rangle$

to obtain

$\displaystyle G_{1}^{(2)}(\xi)$ $\displaystyle=$ $\displaystyle
X_{1}(\xi)Z_{2}(\xi),$ $\displaystyle G_{2}^{(2)}(\xi)$ $\displaystyle=$
$\displaystyle X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)Z_{4}(\xi),$ $\displaystyle
G_{3}^{(2)}(\xi)$ $\displaystyle=$ $\displaystyle
X_{3}(\xi)Z_{2}(\xi)Z_{4}(\xi),$ $\displaystyle G_{4}^{(2)}(\xi)$
$\displaystyle=$ $\displaystyle X_{4}(\xi)Z_{2}(\xi)Z_{3}(\xi),$ (8)

which exactly correspond to the stabilizers of No.2 graph state in Fig.1. The
complete orbit of the first graph can be obtained by applying the local
complement rule repeatedly to the vertices and the corresponding local
Gaussian unitary is shown in the following forms: $No.1\stackrel{{\scriptstyle
U_{LG_{3}}}}{{\longrightarrow}}No.2\stackrel{{\scriptstyle
U_{LG_{3}}^{2}F^{2}_{1}U_{LG_{2}}^{\dagger}}}{{\longrightarrow}}No.3\stackrel{{\scriptstyle
U_{LG_{3}}^{\dagger}}}{{\longrightarrow}}No.4\stackrel{{\scriptstyle
U_{LG_{1}}}}{{\longrightarrow}}No.5\stackrel{{\scriptstyle
U_{LG_{2}}^{2}F^{2}_{1}U_{LG_{3}}^{\dagger}}}{{\longrightarrow}}No.6\stackrel{{\scriptstyle
U_{LG_{1}}^{\dagger}}}{{\longrightarrow}}No.7\stackrel{{\scriptstyle
U_{LG_{3}}}}{{\longrightarrow}}No.8\stackrel{{\scriptstyle
U_{LG_{4}}^{\dagger}}}{{\longrightarrow}}No.9\stackrel{{\scriptstyle
U_{LG_{1}}}}{{\longrightarrow}}No.10\stackrel{{\scriptstyle
U_{LG_{2}}^{\dagger}}}{{\longrightarrow}}No.11$. Here the complete orbit means
the local complement rule is applied on the graph until exhaust all
possibilities.

Figure 2:  The set of four-vertex graphs is equivalent to Fig.1 under local
Gaussian transformation and graph isomorphisms. The graph No.7, which is
repeated and placed in the dash-line box behind the No.10, is used for
generating the graph No.11 directly.

Notice the difference in the Gaussian operations of $2\rightarrow 3$, and
$5\rightarrow 6$. In the qubit case, these would have been of identical form.
This shows the added richness of CV graph states over their qubit
counterparts. Note that Hein et al. one  classify the equivalence of the graph
states by considering the local complementation and additional graph
isomorphisms, which corresponds to the permutations of the vertices. Fig.2
shows another set of graphs, which are not equivalent to any graph in the
equivalence class represented in Fig.1 only considering the local
complementation. However, they belong to the same equivalence class when
considering both, local Gaussian unitary and graph isomorphisms. The
corresponding local Gaussian unitary in Fig.2 is shown in the following forms:
$No.1\stackrel{{\scriptstyle
U_{LG_{1}}}}{{\longrightarrow}}No.2\stackrel{{\scriptstyle
U_{LG_{3}}^{2}F^{2}_{1}U_{LG_{2}}^{\dagger}}}{{\longrightarrow}}No.3\stackrel{{\scriptstyle
U_{LG_{1}}^{\dagger}}}{{\longrightarrow}}No.4\stackrel{{\scriptstyle
U_{LG_{2}}}}{{\longrightarrow}}No.5\stackrel{{\scriptstyle
U_{LG_{2}}^{2}F^{2}_{3}U_{LG_{1}}^{\dagger}}}{{\longrightarrow}}No.6\stackrel{{\scriptstyle
U_{LG_{4}}^{2}F^{2}_{2}U_{LG_{3}}^{\dagger}}}{{\longrightarrow}}No.7\stackrel{{\scriptstyle
U_{LG_{4}}^{\dagger}}}{{\longrightarrow}}No.8\stackrel{{\scriptstyle
U_{LG_{3}}}}{{\longrightarrow}}No.9\stackrel{{\scriptstyle
U_{LG_{1}}^{\dagger}}}{{\longrightarrow}}No.10;$ $No.7\stackrel{{\scriptstyle
U_{LG_{2}}^{\dagger}}}{{\longrightarrow}}No.11$.

Figure 3:  The set of four-vertex graphs is not equivalent to Fig.1 and 2
under local Gaussian transformation and graph isomorphisms. The graph No.1,
which is placed in the dash-line box, is used repeatedly by the local
complementation.

The set of graphs in Fig.3, usually called GHZ (Greenberger-Horne-Zeilinger)
entangled states, is not equivalent with Fig.1 and 2 under local Gaussian
transformation and graph isomorphisms. The local Gaussian unitary is applied
to four-mode graph states in Fig.3, which is written above the arrows of the
following diagram: $No.1\stackrel{{\scriptstyle
U_{LG_{1}}^{\dagger}}}{{\longrightarrow}}No.2\stackrel{{\scriptstyle
U_{LG_{1}}}}{{\longrightarrow}}No.1\stackrel{{\scriptstyle
U_{LG_{2}}^{\dagger}}}{{\longrightarrow}}No.3\stackrel{{\scriptstyle
U_{LG_{2}}}}{{\longrightarrow}}No.1\stackrel{{\scriptstyle
U_{LG_{3}}^{\dagger}}}{{\longrightarrow}}No.4\stackrel{{\scriptstyle
U_{LG_{3}}}}{{\longrightarrow}}No.1\stackrel{{\scriptstyle
U_{LG_{4}}^{\dagger}}}{{\longrightarrow}}No.5$. Fig. 4 lists the graphs with
up to four vertices that are not equivalent under local Gaussian
transformation and graph isomorphisms.

Figure 4:  The connected graphs with up to four vertices are not equivalent
under local Gaussian transformation and graph isomorphisms.

In summary, the local complement rule was extended for CV graph states and the
corresponding local Gaussian transformations of four-mode unweighted graph
states were given. Thus the local Gaussian equivalence classes of CV four-mode
unweighted graph states can be obtained. It was shown that the corresponding
local Clifford unitary can not exactly mirror that for qubit and demonstrate
the complexity of CV quantum systems. It is worth remarking that, whether the
local complementation for any CV graph states can be implemented completely by
the local Gaussian transformations and the general form of the corresponding
local Gaussian unitary can be found, still need be further investigated. This
work not only contribute to a deeper and more complete understanding of CV
multipartite entanglement, but also stimulate the research on CV graph states
theoretically and experimentally.

†Corresponding author’s email address<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  I ACKNOWLEDGMENTS

J. Zhang thanks K. Peng and C. Xie for the helpful discussions. This research
was supported in part by NSFC for Distinguished Young Scholars (Grant No.
10725416), National Basic Research Program of China (Grant No. 2006CB921101),
NSFC (Grant No. 60678029), Program for the Top Young and Middle-aged
Innovative Talents of Higher Learning Institutions of Shanxi and NSF of Shanxi
Province (Grant No. 2006011003).

##  II Reference

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