arXiv:2003.06685v1 [physics.optics] 14 Mar 2020
Quantum diffusion of microcavity solitons
Chengying Bao,
1
Myoung-Gyun Suh,
1
Boqiang Shen,
1
Kemal S ̧afak,
2
Anan Dai,
2
Heming Wang,
1
Lue Wu,
1
Zhiquan Yuan,
1
Qi-Fan Yang,
1
Andrey B. Matsko,
3
Franz X. K ̈artner,
4
,
5
and Kerry J. Vahala
1
,
∗
1
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
2
Cycle GmbH, Hamburg 22607, Germany.
3
Jet Propulsion Laboratory, California Institute of Technology, Pa
sadena, California 91109, USA.
4
Center for Free-Electron Laser Science, Deutsches Elektronen
-Synchrotron, Hamburg 22607, Germany.
5
Department of Physics and the Hamburg Center for Ultrafast
Imaging, University of Hamburg, Hamburg 22761, Germany.
∗
Corresponding author: vahala@caltech.edu
Coherently-pumped (Kerr) solitons in an ideal
optical microcavity are expected to undergo ran-
dom quantum motion that determines fundamen-
tal performance limits in applications of soliton
microcombs. Here, this diffusive motion and its
impact on Kerr soliton timing jitter is studied
experimentally. Typically hidden below techni-
cal noise contributions, the quantum limit is dis-
cerned by measuring counter-propagating soli-
tons. Their relative motion features only weak
interactions and also presents excellent common
mode suppression of technical noise. This is in
strong contrast to co-propagating solitons which
are found to have relative timing jitter well below
the quantum limit of a single soliton on account
of strong mutual motion correlation. Good agree-
ment is found between theory and experiment.
The results establish the fundamental limits to
timing jitter in soliton microcombs and provide
new insights on multi-soliton physics.
Recently, coherently pumped solitons
1
,
2
have been re-
alized in optical microcavities
3
. Unlike earlier temporal
optical solitons, these new solitons are able to regenerate
through Kerr-induced parametric amplification
4
,
5
, and
strong resonant build-up in the high-Q microcavity en-
ables access to optical nonlinearity at low power levels
6
.
These desirable features make these devices well suited
as chip-scale frequency comb sources or microcombs
7
.
The random motion of solitons in these systems (tim-
ing jitter) is of central importance in many of their ap-
plications. But while the quantum limit of this motion
has been studied theoretically
8
, its measurement has not
been possible on account of technical noise masking of
fundamental fluctuations. It is also unclear if the pre-
dicted quantum timing jitter level is achievable in prac-
tice.
Here, we experimentally observe the quantum diffusion
of microcavity solitons as well their overall timing jitter
dynamics. Technical noise suppression has been reported
for both counter-propagating (CP)
9
and co-propagating
(CoP)
10
soliton pairs in a single microcavity. This sug-
gests that measurement of the relative motion of such
a soliton pair could provide a way to observe quantum
noise. However, dispersive waves
11
–
14
are known to sta-
bilize the relative positions of CoP solitons
15
,
16
and to
enable the existence of complex structures called soli-
ton crystals
17
. These interactions are shown to interfere
with the observation of the intrinsic quantum noise asso-
ciated with a single soliton’s motion. On the other hand,
CP solitons feature much weaker interactions that rely
upon optical backscattering
9
, which suggests that obser-
vation of weak quantum fluctuations could be possible
in these systems. We use this feature of CP solitons to
observe the quantum noise limit of soliton motion. More-
over, the different soliton interaction dynamics in CP
and CoP systems are also studied. The motions are also
numerically examined using the coupled Lugiato-Lefever
equations
9
,
18
.
Two resonator types were used for CP soliton gener-
ation as illustrated in Fig.
1
A (3 mm and 7 mm di-
ameters with 21.9 GHz and 9.4 GHz repetition rates,
respectively). Coupling to the resonators uses a tapered
fiber
19
,
20
that is in direct contact with the resonator. CP
solitons are generated by counter-pumping using a single
pump laser as shown in the experimental setup in Fig.
1
B. The amplified spontaneous emission (ASE) of the
pump was filtered by a 100 GHz optical filter. Counter-
pumping frequencies could be adjusted by acousto-optic
modulation, however, in the measurement the pump fre-
quencies were equal. The optical spectra of a single CP
soliton from each resonator type is presented in Figs.
1
C,
D. The generated soliton streams were amplified, disper-
sion compensated by pulse shapers and then conveyed
to a balanced optical cross-correlator (BOC) on optical
fibers (see Fig.
1
A and Supplement for BOC operation).
BOC has been used for the characterization of timing jit-
ter in mode-locked lasers with attosecond resolution
21
,
22
.
While CoP solitons are known to feature strong inter-
actions, CP solitons interact more weakly through op-
tical backscattering, which can cause a form of optical
trapping
23
. Nonetheless, as shown here, this weak trap-
ping still permits diffusive transport of the CP solitons
over an observable time scale. In the measurement, an
oscilloscope was used to record the BOC output signal
over the time window during which the CP solitons rela-
tive delay remains within the BOC operational range (see
the inset of Figs.
1
E). The Allan deviation, which is nor-
mally used for frequency stability evaluation
24
, is used
2
21.9 GHz CP soliton
Balanced-
Xcorrelator
Oscilloscope
B
Pump
PS
PS
EDFA
EDFA
FBG
FBG
AOM
AOM
1510
1530
1550
1570
1590
Wavelength (nm)
C
EDFA
E
Power (dBm)
-60
-40
-20
DM
DM
Forward SFG
Backward SFG
OSC/SSA
BOC & CP soliton motion
PPKTP
ccw soliton
cw soliton
T
R3
T
R2
T
R1
A
cw-soliton
ccw-soliton
microresonator
D
Time (2 s/div)
Delay (200 fs/div)
Time (
s)
Allan deviation (fs)
10
0
10
2
10
4
10
6
10
-1
10
0
10
1
10
2
Exp. (w/o ASE)
Analytical model (1X)
Time (
s)
10
0
10
1
10
2
0
1
0.1
0.2
D (ps
2
/s)
ASE power (a. u.)
0
Exp. (w/ weak ASE)
Exp. (w/ strong ASE)
Power (dBm)
-60
-40
-20
-80
1520
1530
1540
1570
1580
1550
1590
9.4 GHz CP soliton
Allan deviation (fs)
1.6X
2.2X
1.0X
10
0
10
1
Wavelength (nm)
split single soliton
Exp.
Sim.
Flr. 1
Flr. 2
Model (Exp.)
Model (Sim.)
F
circ.
circ.
50/50
FIG. 1:
Counter-propagating solitons and measurmeent of quantum l
imited motion. A
Solitons un-
dergo random motion in the presence of noise. This can be measured
by a balanced optical cross-correlator using
counter-propagating (CP) solitons as inputs. DM: dichroic mirror,
SFG: sum frequency genenration, OSC: oscil-
loscope, SSA: signal source analyzer.
B
Detailed experimental setup for the measurement of CP soliton mot
ion.
The dashed line indicates the injection position for the split soliton mea
surement. AOM: acousto-optical modula-
tors, FBG: fiber Bragg grating, PS: pulse shaper, EDFA: erbium-d
oped fiber amplifier.
C, D
Optical spectra for
the 21.9 GHz and 9.4 GHz CP solitons.
E
Allan deviation of the measured (21.9 GHz device) and simulated CP
soliton motion. The solid and dashed lines are the predicted Allan deviat
ion from the analytical model using exper-
imentally measured and simulated parameters. The inset shows an ex
ample of the measured CP soliton motion on
a long time scale. As discussed further in the Supplement, the circle p
oints indicate the measurement floor with-
out (flr. 1) and with an additional 5 m fiber inserted into the one of th
e fiber paths (flr. 2).
F
The Allan deviation
of the CP solitons (9.4 GHz device) in the presence and absence of ad
ditional ASE. The solid and dashed lines are
1
×
, 1.6
×
and 2.2
×
of the analytical model, respectively. The inset summarizes the cha
nge of the Allan variance dif-
fusion coefficient (D, see main text) when increasing the ASE power in
jected into one direction of the pump. The
solid line in the inset is a linear fit and the dashed line is the theoretical qu
antum limited value for D.
here instead to analyze the measured relative temporal
motion of the CP solitons (see Supplement for the cal-
culation of the Allan deviation). Over a short time scale
(
<
30
μ
s), the calculated Allan deviation (
σ
) increases
with averaging time (
τ
A
) and scales as
σ
2
∼
D
τ
A
where
D is introduced as a diffusion coefficient. The measured
Allan deviation is close to the theoretical prediction (an-
alytical model) based on the quantum-limited diffusive
motion of the solitons
8
. The measured Allan deviation is
well above the measurement noise floor. This noise floor
was characterized by splitting a single soliton train into
the two arms of the measurement system as illustrated by
3
Frequency (Hz)
Analytical model (1X)
Exp. (w/o ASE)
9.4 GHz
Exp. (w/ strong ASE)
Exp. (w/ weak ASE)
4.8X
2.6X
Frequency (Hz)
A
B
Jitter spectral density (fs
2
/Hz)
Jitter spectral density (fs
2
/Hz)
10
-7
10
-6
10
-5
10
-4
10
-3
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
3
10
4
10
5
10
3
10
4
10
5
10
-2
Analytical model (Exp.)
Experiment
21.9 GHz
Simulation
Analytical model (Sim.)
1.0X
FIG. 2:
Measured and theoretical jitter spectral density. A
Measured jitter spectral density of the CP soli-
tons in the 21.9 GHz (green line) and the simulated jitter spectral de
nsity (purple line). Both are close to the the-
oretically predicted jitter spectral density.
B
The measured jitter spectral density for the 9.4 GHz device for ca
ses
without ASE and with added ASE into one propagation direction. The a
nalytical theory is also plotted and the
dashed lines are 4.8 and 2.6 times the analytical theory. The theoret
ical spectral density plotted is 4
×
the value of
Eqn.
1
to account for two independent solitons and single-side-band expe
rimental and numerical spectral plots.
the dashed line in Fig.
1
B. Two tests of the noise floor
(floor 1 and floor 2 in the figure) were performed and
are discussed in the Supplement. Numerical simulation
of the relative CP soliton quantum motion (see Supple-
ment) also agrees well with the measurement and the an-
alytical model (Fig.
1
E). Such good agreement between
the measured timing jitter and the analytical theory was
also observed in the 9.4 GHz device (Fig.
1
F).
In the data, a roll-over of the Allan deviation is ob-
served with increasing averaging time. This behavior in-
dicates that the quantum-limited soliton diffusion is con-
strained. Simulations show that this results from weak
mutual trapping of the CP solitons that is caused by
optical backscattering
23
. From the data (and simula-
tion), the quantum-noise diffusive behavior is observed
when the corresponding temporal fluctuations are much
smaller than the trap scale, which can be no smaller than
the soliton pulse width (100s of fs).
Ultimately, on a time scale exceeding 10s of ms, an-
other fluctuation behavior is apparent in the Allan de-
viation data. This drifting-like motion is attributed to
forced motion of the CP solitons
23
driven by random
differential variation in the fiber paths that convey the
optical pumps. These slow temporal phase changes cre-
ate slow, random variations in the counter-pump phases
and drive relative motion of the CP solitons through the
backscatter process. This is analogous to a systematic
modulation of relative CP soliton motion that was re-
cently reported using non-degenerate counter-pumps
23
.
As a further test of the theory and measurements, we
injected broadband ASE noise from an independent op-
tical amplifier (i.e., without signal input) into one of the
pumping directions and then measured the change of the
CP soliton relative motion. This creates non-common-
mode noise in the CP soliton motion. Two representa-
tive examples of Allan deviation with additional ASE are
plotted in Fig.
1
F, and these indicate a noisier soliton
motion (larger D) with increasing ASE power. The diffu-
sion coefficient D is observed to increase nearly linearly
with the input ASE power (the error bars correspond to
multiple measurements - more than 5). The y-intercept
of the linear fit is also close to the theoretically predicted
quantum limit indicated by the horizontal dashed line in
the inset of Fig.
1
F.
Relative jitter spectral density of CP solitons was also
analyzed and compared with the analytical model
8
. This
jitter spectral density is obtained by Fourier transform
of the temporal motion captured in a 4 ms observation
time window and is shown in Fig.
2
. Consistent with
Allan deviation data and analysis, the measured jitter
spectral density rolls off as 1/
ω
2
and matches the ana-
lytical theory at higher offset frequencies. On the other
hand, the relative jitter spectral density is suppressed for
lower offset frequencies (e.g.,
<
10 kHz) in Figs.
2
A, B,
which is consistent with the observed roll-over of the Al-
lan deviation in Fig.
1
. Figure
2
B also shows an increase
of the jitter spectral density with increasing input ASE
power levels. Similarly, the numerically simulated jitter
spectral density is also in agreement with the analytical
theory (Fig.
2
A). The close agreement of the measure-
ments with simulations and the analytical theory in Figs.
1
,
2
further confirms that the measured results reflect the
quantum limit of soliton motion. These results also sug-
gest that the timing jitter predicted in ref.
8
is achievable
with sufficient technical noise suppression.
In contrast to the CP solitons, the CoP solitons are ob-