##  I Title

All-optical polarization and amplitude modulation of second harmonic
generation in atomically thin semiconductors

##  II Author list

Sebastian Klimmer,1 Omid Ghaebi,1 Ziyang Gan,2 Antony George,2,3 Andrey
Turchanin,2,3 Giulio Cerullo,4 Giancarlo Soavi1,3

##  III Affiliations

1Institute of Solid State Physics, Friedrich Schiller University Jena, 07743
Jena, Germany

2Institute of Physical Chemistry, Friedrich Schiller University Jena, 07743
Jena, Germany

3Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena,
Germany

4Dipartimento di Fisica, Politecnico di Milano, 20133 Milan, Italy

##  IV Abstract

###### Abstract

Second harmonic generation is of paramount importance in several fields of
science and technology, including frequency conversion, self-referencing of
frequency combs, nonlinear spectroscopy and pulse characterization. Advanced
functionalities are enabled by the modulation of the harmonic generation
efficiency, which can be achieved with electrical or all-optical triggers.
Electrical control of the harmonic generation efficiency offers large
modulation depth at the cost of low switching speed, in contrast to all-
optical nonlinear devices which provide high speed and low modulation depth.
Here we demonstrate all-optical modulation of second harmonic generation in
MoS2 with close to 100 % modulation depth and speed limited only by the
fundamental pulse duration. This result arises from the combination of the D3h
crystal symmetry and the deep sub-wavelength thickness of the sample and it
can therefore be extended to the whole family of transition metal
dichalcogenides, thus providing a large flexibility in the design of advanced
nonlinear optical devices such as high-speed integrated frequency converters,
broadband autocorrelators for ultra-short pulse characterization and tunable
nanoscale holograms.

##  V Main text

Stemming from the first demonstration of optical harmonic generation [1],
nonlinear optics has been in the spotlight of science and technology for more
than half-century. In particular, second harmonic generation (SHG) is a
second-order nonlinear process widely used for frequency conversion, self
referencing of frequency combs [2], crystal symmetry and Rashba effect studies
[3, 4], sensing [5], interface spectroscopy [6] and ultra-short pulse
characterization [7]. Besides free-space applications, there is an increasing
interest towards the realization of micro-scale integrated nonlinear devices.
Here, a major challenge comes from the centro-symmetric nature of silicon (Si)
and silicon nitride (Si3N4), which forbids second-order nonlinearities. Large
efforts have been devoted to the integration of nonlinear crystals such as
lithium niobate [8, 9] or to symmetry breaking in Si and Si3N4, for instance
via strain [10], electric fields [11] or the photogalvanic effect [12].

Two-dimensional (2D) materials, such as graphene and transition metal
dichalcogenides (TMDs), hold great promise for nonlinear optical applications.
They have strong and broadband optical response [13, 14], combined with the
possibility of harmonic generation enhancement at excitonic resonances in TMDs
[15] and at multi-photon resonances in the graphene’s Dirac cone [16, 17]. In
addition, thanks to their flexibility and mechanical strength [18], they are
easy to integrate on photonic platforms. Various functionalised devices for
sensing and frequency conversion have been demonstrated on fibers [19],
waveguides [20] and microrings [21], while direct patterning of TMDs has been
used to realize atomically thin meta-lenses [22, 23] and nonlinear holograms
[24, 25]. In addition, harmonic generation in 2D materials can be efficiently
tuned by external electrical [26, 27, 16, 28] or all-optical excitation [29,
30], offering an additional degree of freedom for the design of advanced
nanoscale devices.

However, all the electrical and all-optical schemes that have been proposed to
date for SHG modulation in 2D materials have significant downsides. On one
hand, electrical modulation has been demonstrated in tungsten diselenide
(WSe2) monolayers [26] by tuning the oscillator strength of neutral and
charged exciton resonances through electrostatic doping, and in molybdenum
disulfide (MoS2) homo-bilayers [27], by breaking the naturally occurring
inversion symmetry through electrical gating, in the latter case with large
modulation depth up to a factor of 60. However, electronics is intrinsically
slower compared to optics and photonics. On the other hand, all-optical SHG
modulation has been achieved by quenching of the exciton oscillator strength
following ultrafast optical excitation in MoS2 [29, 30]. This approach offers
high modulation speed, limited in principle only by the excited state/exciton
lifetime ($\sim$ tens of ps). However, the largest depth in all-optical SHG
modulation reported to date in TMDs is 55 % [29], with a strong dependence on
the excitation wavelength and fluence. In addition, this scheme for all-
optical SHG modulation is only effective for excitation and frequency
conversion above-gap or at excitonic resonances, while it is not applicable
for below-gap excitation, leading to a naturally limited spectral bandwidth.

Here we demonstrate a novel approach for the all-optical control of the second
harmonic (SH) polarization in MoS2 and show that this can be used for all-
optical modulation of the SH efficiency with modulation depth close to 100 %
and speed limited only by the fundamental frequency (FF) pulse duration. Our
method relies solely on symmetry considerations in combination with the deep
sub-wavelength thickness of the sample and thus does not require resonant
enhancement or above-gap excitation for its implementation. Moreover, the same
approach can be extended to any 2D material belonging to the D3h symmetry
group, thus for instance to any material of the TMDs’ family. Our findings
provide a new strategy for the tuning of the fundamental properties of light
(polarization and amplitude) in the nonlinear regime and in the 2D thickness
limit, and thus pave the way to the design of novel advanced functionalities
in high-speed frequency converters, nonlinear all-optical modulators and
transistors [31, 32], interferometric autocorrelators for ultra-short pulse
characterization and tunable atomically thin holograms [24].

### Nonlinear optical characterization

Figure 1: MoS2 symmetry properties, optical selection rules and all-optical
SHG modulation scheme. (a) Top view of a MoS2 cystal. The arrows inside the
hexagon highlight the D3h three-fold rotational symmetry. (b) Schematic of the
resulting SH polarization for different combinations of FF along AC
(horizontal arrow) and ZZ directions (vertical arrow). (c-d) Sketch of the
all-optical SH polarization modulation. The control pulse is polarized along
the ZZ direction while the probe pulse is polarized along the AC direction of
the MoS2 sample. (c) When the delay between the control and probe pulses is
larger than the FF pulse duration, both will generate a SH signal polarized
along the AC direction. (d) For zero-delay between the probe and control
pulses the SH signal will be emitted along the ZZ direction. Figure 2: All-
optical modulation set-up and SHG characterization. (a) Sketch of the setup
used for the experiments. For the FF we used an OPO tunable between $\sim$1.3
and 2.0 µm. The FF beams are separated into two perpendicular replicas inside
a Mach-Zehnder interferometer and subsequently focused onto the MoS2 sample
with a 40x microscope objective. The back-scattered SH signal is spectrally
filtered and detected with a single-photon avalanche diode. (b) Power
dependence of the SHG signal for all wavelengths used in our experiments. The
grey dashed line is a guide to the eye to indicate a slope of 2, typical of
the SHG process. (c) Polar plot of the normalized SH intensity as a function
of the excitation polarization angle $\theta$ with SH polarization detection
always parallel to the FF. Blue circles show experimental data and the solid
blue line indicates the $\text{cos}^{2}[3(\theta-\phi_{0})]$ fit.

For the experiments, we used high quality monolayer MoS2 flakes, fabricated
with a modified chemical vapour deposition method [33, 34] on thermally
oxidized Si/SiO2 substrates. In contrast to thin films, where surface SHG is
allowed due to an out-of-plane component of the second-order susceptibility
[35, 6], TMDs belong to the D3h symmetry group and thus have only one non-
vanishing in-plane component of the nonlinear optical susceptibility (see
Methods) [35, 36, 37, 38, 39, 40].

$\displaystyle\chi^{(2)}\equiv\chi_{yyy}^{(2)}=-\chi_{yxx}^{(2)}=-\chi_{xyx}^{(2)}=-\chi_{xxy}^{(2)},$
(1)

where x and y refer to the in-plane Cartesian coordinates of the SH
polarization and of the two FFs. A sketch of the hexagonal lattice for MoS2 is
shown in Fig. 1(a), where the Cartesian coordinates are defined with respect
to the two main lattice orientations: the armchair (AC) and zigzag (ZZ)
directions. In this framework, the SH intensity I2ω as a function of the FF
for any TMD along the AC and ZZ directions can be written as

$\displaystyle I_{AC}^{2\omega}$ $\displaystyle\sim$ $\displaystyle\lvert
E_{AC}^{2}-E_{ZZ}^{2}\rvert^{2},$ (2) $\displaystyle I_{ZZ}^{2\omega}$
$\displaystyle\sim$ $\displaystyle\lvert 2E_{AC}E_{ZZ}\rvert^{2},$ (3)

where EAC and EZZ correspond to the FF fields with polarization along the AC
and ZZ directions respectively [41, 42, 43, 39]. Thus, the SHG from two
electric fields with the same polarization (either along AC or ZZ) will always
result in an emitted SH intensity with polarization along the AC direction, as
depicted in Fig. 1(b). This is indeed the case of all the SHG experiments
performed to date on 2D materials [15, 36, 43, 39]. On the other hand, two
ultrashort FFs with perpendicular polarization (along the AC and ZZ
directions) and with the same amplitude will generate a SH signal along the AC
direction if they do not overlap in time (Fig. 1(c)), while they will generate
a SH signal along the ZZ direction at zero-delay (Fig. 1(d)), thus leading to
an ultrafast $90^{\circ}$ polarization switch within the FFs pulse duration.
Finally, for circularly polarized FFs, the emitted SH has opposite circular
polarization due to valley-dependent selection rules (see analysis of equation
(4)) [44, 45, 46].

Figure 3: Ultrafast polarization and amplitude all-optical switching. (a) SH
intensity for 1480 nm FF wavelength (black line) measured along the zigzag
direction as a function of the delay between the two perpendicularly polarized
FFs (illustrated in the inset). The blue line corresponds to a Gaussian fit of
the autocorrelation curve. The grey shaded area represent the noise level at
$\sim$3.5 fW of our experiments. (b) Normalized SH intensity and noise level
for all FF wavelengths used in our experiments. (c) Double logarithmic plot of
the emitted SH power as a function of the incident FF power along the AC
direction. The grey dashed line is a guide to the eye representing a slope of
1.

The SHG measurements were performed with the set-up shown in Fig. 2(a) and
described in the Methods. In order to realize all-optical polarization
switching and SH modulation it is crucial to first characterize the relative
orientation between the FFs and the MoS2 sample. To do so, we first performed
SHG experiments using a tunable near-IR optical parametric oscillator (OPO) as
a FF. The emitted SHG power for the FF wavelengths used in our experiments
(between 1360 nm and 1560 nm) is shown in Fig. 2(b). The slope of 2 in the
double logarithmic plot is a further proof of a genuine SHG process. The
crystal orientation of the MoS2 sample was determined for each FF wavelength
by SH polarization dependent experiments with an additional polarizer in front
of the detector to measure the SH parallel to the excitation polarization [47,
36, 43, 39]: the SH intensity is proportional to
$\text{cos}^{2}[3(\theta-\phi_{0})]$, where $\theta$ is the FF polarization
angle and $\phi_{0}$ is the rotation of the AC direction relative to the
p-polarization in the laboratory frame. Fig. 2(c) is an example of the SH
polar plot for a FF wavelength of 1400 nm and shows that the AC direction is
tilted by $\phi_{0}=13.6^{\circ}+n\cdot 60^{\circ}$ ($n$ integer) with respect
to p-polarization in the lab coordinates. In addition, Fig. 1(c) confirms the
absence of any detectable strain in our sample, since an uniaxial strain would
result in a symmetric attenuation along its direction of action [38, 39]. The
small asymmetry of the lobes along the two AC directions ($\sim$ 15∘/70∘ and
$\sim$ 190∘/250∘) is attributed to polarization scrabbling effects due to the
use of a dichroic mirror in reflection geometry [48]. Finally, based on the
results in Fig. 2(b), we determine the modulus of the complex second-order
nonlinear susceptibility at the FF wavelengths used in our experiments. To do
so, we estimate the optical losses of the setup from the SH emission at the
sample position to the detection on the Single Photon Avalanche Detector
(SPAD) and calculate the SH tensor element $\chi^{(2)}$ of MoS2, as described
in the Methods. We thus obtained effective second-order susceptibility values
for our FF wavelengths 1360 nm, 1400 nm, 1480 nm and 1560 nm of $\sim$ 282.1
pm/V, $\sim$ 153.7 pm/V, $\sim$ 44.7 pm/V, $\sim$ 24.2 pm/V respectively. The
highest value obtained at 1360 nm FF wavelength is due to exciton resonant SH
enhancement [50, 15]. All values are in good agreement with those previously
reported by experiments performed at similar FF wavelengths [42, 36, 51, 50,
52] and predicted by theory [53, 54]. It is worth noting that for single layer
TMDs, where interlayer interference [49] is absent, SHG is insensitive to the
phase of the nonlinear optical response.

Figure 4: Phase-locked all-optical SH modulation along armchair and zigzag
directions. (a) SH power for 1480 nm FF wavelength along AC (blue) and ZZ
(red) directions as a function of the relative delay between the two
perpendicularly polarized FFs. The inset shows the Poincaré-sphere with
polarization directions on its orthodromes. The red line indicates the change
in polarization when tuning the delay between the two FFs with a birefringent
delay line. (b) Schematic image of the modified common-path delay generator.
The two perpendicular components of an incident pulse, polarized at 45∘ with
respect to the AC/ZZ directions, are delayed by two birefringent alpha barium
borate crystals with perpendicular optical axis, whereby the variable
thickness of the second one allows to control the delay with interferometric
precision. (c) Zoom-in of (a), showing the detected SH power for 1480 nm FF
along the AC (blue) and ZZ directions (red) close to zero delay.

### Nonlinear all-optical modulation

Having defined the AC and ZZ directions of our sample, we now demonstrate all-
optical SH polarization and amplitude modulation. We separate the FF beam into
two perpendicular replicas, align them along the sample’s AC and ZZ directions
using a half-waveplate and control their relative delay with a motorized
mechanical stage (see Methods for details). For large delays (i.e., longer
than the FF pulse duration) between the two perpendicular FFs, SH will be
emitted by each individual FF along the AC direction following equation (2).
Instead, at zero-delay, when the two FFs overlap perfectly in time, the SH
intensity along the AC direction will go to zero and the SH signal will be
emitted only along the ZZ direction. Fig. 3(a) shows the SH average power
emitted along the ZZ direction as a function of the delay between the two
perpendicularly polarized FFs and for a FF wavelength of 1480 nm. The Gaussian
fit, blue curve in Fig. 3(a), has a FWHM of $\sim$ 250 fs, corresponding to
the autocorrelation function of our OPO pulse with duration of $\sim$ 175 fs.
Moreover, the SH signal along the ZZ direction is now ideally background free,
demonstrating the potential of ultrafast SH polarization switching and close
to 100 % amplitude modulation of our approach.

We further note that this result is solely based on symmetry considerations
and thus provides ultra-broadband response, not limited to above-gap or
resonant exciton pumping. Indeed, we obtained the same result for all the FF
wavelengths used in our experiments, as shown in Fig. 3(b). The possibility to
emit SH along perpendicular directions (AC and ZZ) with the same efficiency is
a unique feature that arises from the combination of symmetry and deep sub-
wavelength thickness of TMDs, which relaxes the phase-matching constraints
typical of harmonic generation. This result could have an immediate
application in background free ultra-broadband and ultra-short pulse
characterization. For instance, in the most advanced commercial systems [55]
for ultra-short pulse characterization one has to switch between collinear and
non-collinear geometries in order to collect either the interferometric or the
background-free intensity autocorrelation signals respectively. In contrast,
in our approach both signals are accessible using the same geometry and by
simply switching the SH detection from AC to ZZ. Further, following equation
(3), the power scaling of the emitted SH along the ZZ direction is linear with
respect to each of the FF intensities. This is confirmed by the power-
dependent SHG measurements reported in Fig. 3(c), where we show the emitted SH
power along the ZZ direction at zero delay between the two FFs and as a
function of the AC polarized FF power.

To gain more insight into the temporal evolution of the emitted SH
polarization and amplitude, we finally scan the delay between the two
perpendicularly polarized FFs with interferometric precision and measure the
emitted SH along both AC and ZZ directions, as shown in Fig. 4(a). In order to
control the delay between two perpendicular pulses with the desired sub-
optical cycle precision we used the common-path delay generator sketched in
Fig. 4(b) and described in the Methods. As expected, for delays longer than
our pulse duration the SH power is emitted only along the AC direction, blue
curve in Fig. 4(a), and no signal is detected along the ZZ direction, red
curve in Fig. 4(a). Instead, for delays close to zero we observe a strong
ultrafast modulation of the SH power emitted along the AC direction. This can
be better appreciated looking at Fig. 4(c), which shows the emitted SH power
along AC and ZZ directions at 1480 nm for delays between -10 fs and +10 fs.

It is useful to note that for delays much shorter compared to the pulse
duration our interferometric measurement is the analogue of tuning the
polarization of one single FF pulse along the orthodrome of the Poincaré-
sphere, inset in Fig. 4(a). This corresponds to a rotation of the FF
polarization from $-45^{\circ}$ with respect to the AC-ZZ directions (at zero
delay) to left/right circular polarization (at delay $\tau=\pm\frac{T}{4}$,
where $T$ is the FF optical cycle) to $+45^{\circ}$ with respect to the AC-ZZ
directions at delay $\tau=\pm\frac{T}{2}$. This result is consistent with the
theoretical SH polarization $\textbf{P}^{(2)}$ generated by an arbitrary
elliptically polarized FF [46] after a simple basis transformation to account
for the rotation by $-45^{\circ}$ with respect to AC/ZZ directions:

$\displaystyle\textbf{P}^{(2)}=\epsilon_{0}\chi^{(2)}\lvert\textbf{E}\rvert^{2}\begin{pmatrix}\widehat{ZZ}\pm
i\,\text{sin}(2\vartheta)\widehat{AC}\end{pmatrix},$ (4)

Here $\vartheta=0^{\circ}$ denotes a linearly polarized FF at $45^{\circ}$
with respect to the AC/ZZ direction and $\vartheta=45^{\circ}$ corresponds to
a circularly polarized FF. This clearly shows that the SH component emitted
along the AC direction oscillates with a period of $\frac{T}{2}$ as a function
of the FF polarization, in contrast to the SH emitted along the ZZ direction.
This underpins the interferometric precision required in order to fully
capture the modulation along the AC direction. The experimental results show a
weak modulation also for the SH emitted along the ZZ direction, although this
is not expected from theory. This could arise from weak strain (i.e., below
the limit detectable by our SHG polarization measurements [56, 39]), small
deviations in the alignment of the detection polarizer with respect to the
AC/ZZ directions or from additional terms in the $\chi^{(2)}$ arising from the
valley degree of freedom [57, 48]. Looking at Fig. 4(c) one can indeed
appreciate that at zero delay (FF at $-45^{\circ}$ with respect to AC/ZZ
directions) the SH is emitted only along the ZZ direction, while at
$\frac{T}{4}$ delay the emitted SH components along AC and ZZ are identical,
as expected for circular polarization.

### Discussion

In conclusion, we have demonstrated all-optical polarization switching and
amplitude modulation of SHG in MoS2. Our approach surpasses all previously
reported electrical and all-optical attempts of SH tuning in terms of
modulation depth and speed, providing $90^{\circ}$ polarization switch, close
to 100 % modulation depth and speed limited only by the FF pulse duration.
Moreover, our method is intrinsically broadband since it only relies on the
crystal symmetry of TMDs. We thus foresee a direct impact of our results on a
variety of photonic devices, such as high-speed frequency converters,
nonlinear all-optical modulators and transistors [31, 32], autocorrelators for
ultra-short pulse characterization and atomically thin optically tunable
nonlinear holograms [24].

##  VI Methods

Polarization dependent SH intensity. The vectorial components of the second-
order polarization $P^{(2)}(2\omega)$ for a material with D3h symmetry (such
as TMDs) are given by

$\displaystyle\begin{pmatrix}P^{(2)}_{x}(2\omega)\\\ P^{(2)}_{y}(2\omega)\\\
P^{(2)}_{z}(2\omega)\end{pmatrix}=\epsilon_{0}\begin{pmatrix}0&0&0&0&0&\chi^{(2)}_{xxy}\\\
\chi^{(2)}_{yxx}&\chi^{(2)}_{yyy}&0&0&0&0\\\
0&0&0&0&0&0\end{pmatrix}\begin{pmatrix}E^{2}_{x}(\omega)\\\
E^{2}_{y}(\omega)\\\ E^{2}_{z}(\omega)\\\ 2E_{y}(\omega)E_{z}(\omega)\\\
2E_{x}(\omega)E_{z}(\omega)\\\ 2E_{x}(\omega)E_{y}(\omega)\end{pmatrix}$

where $\chi^{(2)}_{yyy}=-\chi^{(2)}_{xxy}=-\chi^{(2)}_{yxx}=\chi^{(2)}$. If we
now consider a TMD oriented in such way that the ZZ and AC directions lie
along the x and y Cartesian coordinates respectively and we neglect the z
(out-of-plane) direction, we obtain the following expression:

$\displaystyle\begin{pmatrix}P^{(2)}_{ZZ}(2\omega)\\\
P^{(2)}_{AC}(2\omega)\end{pmatrix}=\epsilon_{0}\chi^{(2)}(2\omega,\omega,\omega)\begin{pmatrix}2E_{ZZ}(\omega)E_{AC}(\omega)\\\
E^{2}_{ZZ}(\omega)-E^{2}_{AC}(\omega)\end{pmatrix}$

Finally, since the SH intensity is proportional to the absolute square value
of the second-order polarization, we retrieve equations (2) and (3) of the
main text:

$\displaystyle I_{AC}^{2\omega}=\lvert P^{(2)}_{AC}\rvert^{2}\sim\lvert
E^{2}_{AC}-E^{2}_{ZZ}\rvert^{2}$ $\displaystyle I_{ZZ}^{2\omega}=\lvert
P^{(2)}_{ZZ}\rvert^{2}\sim\lvert 2E_{AC}E_{ZZ}\rvert^{2}$

SHG setup. For the FF we used an OPO (Levante IR from APE) pumped by an
ytterbium-doped mode locked laser (FLINT12, Light Conversion) with a
repetition rate of 76 MHz and 100 fs pulse duration and generating pulses
tunable between $\sim$1.3 µm and 2.0 µm. The FF is then separated into two
perpendicular replicas whose relative delay is tuned with two different
approaches: a computer controlled motorized stage (M-414.2PD, PI) in a Mach-
Zehnder interferometer configuration and a commercial common-path birefringent
interferometer (GEMINI, NIREOS) [58]. Compared to standard home-built
interferometers, the GEMINI provides sub-wavelength interferometric stability
with precise control on the delay between the two replicas with attosecond
precision. The polarization of the FFs was tuned using a half-waveplate
(AHWP05M-1600, Thorlabs) and the power on the sample was controlled by two
polarizers (LPNIR050 and WP12L-UB, both Thorlabs). Finally, the two collinear
and perpendicularly polarized FFs were focused on the sample using a custom
built microscope equipped with a 40x reflective objective (LMM-40X-P01,
Thorlabs). The back-scattered SH signal is spectrally separated using a
dichroic mirror (DMSP950, Thorlabs), further spectrally purified by filters
(FELH0650, FESH850, FESH0950, all Thorlabs) and detected with a single-photon
avalanche diode (C11202-050, Hamamatsu). The SH polarization was measured
using a wire grid polarizer (WP12L-UB, Thorlabs).

Estimate of the optical losses of the setup. In order to quantify the SH
signal generated directly at the sample position, optical losses of the
different components of the setup must be considered. While the transmission
coefficients for the investigated SH wavelengths of the filters and the
dichroic mirror (all > 96 %) were taken from the manufacturer’s website, the
values for polarizers and the microscope objective were determined
experimentally. A transmission of $\sim$ 79 % was determined for the wire grid
polarizer while for the reflective objective we determined a transmission of
50 %. Last, the responsivity of the single-photon avalanche diode was taken
into account, which ranges depending on the investigated SH wavelength between
$\sim$ 17 % and $\sim$ 31 %. In total, we estimated our optical losses from
the SH emission to the detector to be $\sim$ 86-92 % depending on the
wavelength.

Calculation of the second-order nonlinear susceptibility. The sheet-SH tensor
element $\chi_{S}^{(2)}$ can be calculated from the FF and SH average powers
using the equation [42]:

$\displaystyle\chi_{S}^{(2)}=\sqrt{\frac{c^{3}\epsilon_{0}f\pi
r^{2}t_{FWHM}(1+n_{2})^{6}P_{SHG}(2\omega)}{16\sqrt{2}S\omega^{2}P_{FF}^{2}(\omega)}},$
(5)

where $c$ is the speed of light, $\epsilon_{0}$ is the permittivity of free-
space, $f$ = 76 MHz is the pump laser repetition rate, $r\sim$ 1.85 µm is the
focal spot radius, $t_{FWHM}\sim$ 200 fs is the pulse full width at half
maximum, $n_{2}\sim$ 1.45 is the substrate refractive index, $S=0.94$ is a
shape factor for Gaussian pulses, $\omega$ is the FF angular frequency, and
$P_{SHG}(2\omega)$ \- $P_{FF}(\omega)$ are the SH and FF average powers. The
effective bulk-like second-order susceptibility $\chi_{eff}^{(2)}$ can be
subsequently calculated from equation (5) as
$\chi_{eff}^{(2)}=\frac{\chi_{S}^{(2)}}{d_{MoS_{2}}}$, where the thickness
$d_{MoS_{2}}$ of MoS2 is 0.65 nm [18, 46].

Pulse duration of the FFs. In order to prove that our method is solely limited
by the pulse duration of the FFs, we performed a standard characterization of
temporal profile of our OPO source at different wavelengths (Extended Data
Fig. 1). For the measurements we used a home built autocorrelator based on a
Michelson interferometer equipped with a motorized and computer controlled
translation stage (HPS60-20X-M5, Optosigma). Two identical and temporally
delayed replicas of the OPO pulse were then focused onto a 1 mm thick
$\beta$-barium borate crystal (BBO-652H, Eksma Optics) in non-collinear
geometry and the background free SHG autocorrelation intensity was detected on
a Si photodetector (DET10A2, Thorlabs). From this we obtained values for the
autocorrelation in the range of 217 fs to 310 fs with gaussian fits,
corresponding to pulse durations between 150 fs and 220 fs.

##  VII References

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##  VIII Acknowledgments

The authors greatly acknowledge Habib Rostami and Fabrizio Preda for helpful
discussions. This work was supported by the European Union’s Horizon 2020
Research and Innovation program under Grant Agreement GrapheneCore3 881603
(G.S.). This publication is part of the METAFAST project that received funding
from the European Union’s Horizon 2020 Research and Innovation program under
Grant Agreement No. 899673 (G.S. and G.C.). The authors acknowledge the German
Research Foundation DFG (CRC 1375 NOA project numbers B2 (A.T.) and B5 (G.S.))
and the Daimler und Benz foundation for financial support (G.S.).

##  IX Author Contributions Statement

S.K. and G.S. conceived the experiments. S.K. and O.G. performed the all-
optical modulation measurements. Z.G., A.G. and A.T. fabricated and provided
the high quality MoS2 sample. S.K., G.C and G.S. wrote the manuscript with
contribution from all the authors. All the authors participated to the
discussion and commented on the manuscript.

##  X Competing Interests Statement

The authors declare no competing interest.

##  XI Figure Legends/Captions (for main text figures)

MoS2 symmetry properties, optical selection rules and all-optical SHG
modulation scheme. (a) Top view of a MoS2 cystal. The arrows inside the
hexagon highlight the D3h three-fold rotational symmetry. (b) Schematic of the
resulting SH polarization for different combinations of FF along AC
(horizontal arrow) and ZZ directions (vertical arrow). (c-d) Sketch of the
all-optical SH polarization modulation. The control pulse is polarized along
the ZZ direction while the probe pulse is polarized along the AC direction of
the MoS2 sample. (c) When the delay between the control and probe pulses is
larger than the FF pulse duration, both will generate a SH signal polarized
along the AC direction. (d) For zero-delay between the probe and control
pulses the SH signal will be emitted along the ZZ direction.

All-optical modulation set-up and SHG characterization. (a) Sketch of the
setup used for the experiments. For the FF we used an OPO tunable between
$\sim$1.3 and 2.0 µm. The FF beams are separated into two perpendicular
replicas inside a Mach-Zehnder interferometer and subsequently focused onto
the MoS2 sample with a 40x microscope objective. The back-scattered SH signal
is spectrally filtered and detected with a single-photon avalanche diode. (b)
Power dependence of the SHG signal for all wavelengths used in our
experiments. The grey dashed line is a guide to the eye to indicate a slope of
2, typical of the SHG process. (c) Polar plot of the normalized SH intensity
as a function of the excitation polarization angle $\theta$ with SH
polarization detection always parallel to the FF. Blue circles show
experimental data and the solid blue line indicates the
$\text{cos}^{2}[3(\theta-\phi_{0})]$ fit.

Ultrafast polarization and amplitude all-optical switching. (a) SH intensity
for 1480 nm FF wavelength (black line) measured along the zigzag direction as
a function of the delay between the two perpendicularly polarized FFs
(illustrated in the inset). The blue line corresponds to a Gaussian fit of the
autocorrelation curve. The grey shaded area represent the noise level at
$\sim$3.5 fW of our experiments. (b) Normalized SH intensity and noise level
for all FF wavelengths used in our experiments. (c) Double logarithmic plot of
the emitted SH power as a function of the incident FF power along the AC
direction. The grey dashed line is a guide to the eye representing a slope of
1.

Phase-locked all-optical SH modulation along armchair and zigzag directions.
(a) SH power for 1480 nm FF wavelength along AC (blue) and ZZ (red) directions
as a function of the relative delay between the two perpendicularly polarized
FFs. The inset shows the Poincaré-sphere with polarization directions on its
orthodromes. The red line indicates the change in polarization when tuning the
delay between the two FFs with a birefringent delay line. (b) Schematic image
of the modified common-path delay generator. The two perpendicular components
of an incident pulse, polarized at 45∘ with respect to the AC/ZZ directions,
are delayed by two birefringent alpha barium borate crystals with
perpendicular optical axis, whereby the variable thickness of the second one
allows to control the delay with interferometric precision. (c) Zoom-in of
(a), showing the detected SH power for 1480 nm FF along the AC (blue) and ZZ
directions (red) close to zero delay.

##  XII Data Availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author on reasonable request.
Source data are provided with this paper.


††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# The effect of fast noise on the fidelity of trapped-ions quantum gates

Haim Nakav  Physics of Complex Systems, Weizmann Institute of Science and
AMOS, Rehovot 7610001, Israel  Ran Finkelstein  Physics of Complex Systems,
Weizmann Institute of Science and AMOS, Rehovot 7610001, Israel  Current
address : Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Lee Peleg  Physics of Complex Systems,
Weizmann Institute of Science and AMOS, Rehovot 7610001, Israel  Nitzan
Akerman  Physics of Complex Systems, Weizmann Institute of Science and AMOS,
Rehovot 7610001, Israel  Roee Ozeri  Physics of Complex Systems, Weizmann
Institute of Science and AMOS, Rehovot 7610001, Israel

###### Abstract

High fidelity single and multi-qubit operations compose the backbone of
quantum information processing. This fidelity is based on the ability to
couple single- or two-qubit levels in an extremely coherent and precise
manner. A necessary condition for coherent quantum evolution is a highly
stable local oscillator driving these transitions. Here we study the effect of
fast noise, that is noise at frequencies much higher than the local oscillator
linewidth, on the fidelity of one- and two-qubit gates in a trapped-ion
system. We analyze and measure the effect of fast noise on single qubit
operations including resonant $\pi$ rotations and off-resonant sideband
transitions . We further analyze the effect of fast phase noise on the Mølmer-
Sørensen two-qubit gate. We find a unified and simple way to estimate the
performance of all of these operations through a single parameter given by the
noise power spectral density at the qubit response frequency. While our
analysis focuses on phase noise and on trapped-ion systems, it is relevant for
other sources of fast noise as well as for other qubit systems in which spin-
like qubits are coupled by a common bosonic field. Our analysis can help in
guiding the deign of quantum hardware platforms and gates, improving their
fidelity towards fault-tolerant quantum computing.

##  I Introduction

Quantum coherence is the fundamental resource in any quantum information
processing task. Much of the remarkable progress achieved in the field over
the past two decades was achieved by careful examination and reduction of
dephasing mechanisms through improvements in control technology, and
mitigation of environmental noise [1, 2]. Yet, single and two-qubit gates are
still often limited by the finite coherence between the qubit and the
classical local oscillator (LO) used to control and measure the quantum
system.

In quantum information processing, phase noise typically limits the coherence
time of quantum registers and compromises the fidelity of quantum gates. Many
studies investigated this effect, and techniques to mitigate it were proposed
and implemented. These include dynamical decoupling [3, 4, 5, 6, 7, 8, 9], and
decoherence-free subspaces [10, 11, 12, 13, 14]. These methods were designed
to mitigate the effects of slow phase noise ($\sigma_{z}$ errors), which
refers to noise within the linewidth of the oscillator as it acts on the qubit
transition. However, noise that is faster than this linewidth affects qubit
performance differently. In some cases, which will be discussed in this paper,
such noise acts as a bit flip error, rather than a $\sigma_{z}$ error,
rendering the above-mentioned methods ineffective.

Fast LO noise can result from various mechanisms. One example of such a source
is a side effect of reducing the magnitude of slow phase noise. In order to
narrow the linewidth of an oscillator often an external servo loop is added to
suppress the slow phase noise of the LO with respect to a stable reference.
While subduing the slow phase noise, the servo loop generates excess noise at
the higher frequencies close to its unity gain response [15]. This noise
feature, which is prevalent mainly in narrow linewidth lasers, is sometimes
referred to as servo bump and results in fast phase noise.

Traditionally, noise at these high frequencies was thought to average out over
the much longer time scale of typical quantum evolution. However, it was
recently realized that such noise limits various quantum operations [16, 17,
18, 19, 20, 21, 22] if it overlaps with a frequency at which the quantum
system resonantly responds.

As an example, we consider a single qubit rotation on the Bloch sphere shown
in Fig. 1(a). Phase or frequency instabilities lead to fast fluctuations of
the Rabi vector in the Bloch sphere equatorial plane, resulting in randomly
modified trajectories [purple trajectories on the Bloch sphere in Fig. 1(a)].
Specifically, the fast noise frequency components that will not average out
and lead to significant rotation errors are those that overlap the Rabi
frequency. Fig 1(b) showcases an example of such a modified spectrum. In fact,
the effect of fast noise goes beyond impacting single qubit rotations and can
have deleterious effects on the fidelity of two-qubit gates. For example,
Starting from $\mid\uparrow\uparrow\rangle$, the Mølmer-Sørensen (MS) gate
shown in Fig. 1(c,d) under fully coherent evolution forms a perfect Bell-
state. However, the presence of fast noise that overlaps the carrier
transition leads to incoherent errors and reduced fidelity.

Here, we simulate and measure the effect of fast noise on the fidelity of
single- and two-qubit gates in trapped ion qubits. Single-qubit rotations are
performed via resonant pulses and their fidelity was directly measured for
different phase-noise spectra. Two-qubit entanglement is performed using the
MS gate that uses electromagnetic fields that are close to resonance with the
sideband of a common ion-phonon mode. We find that the magnitude of the error
induced by fast phase noise in quantum gates predominantly scales with the
overlap of the noise power spectral density (PSD) with the relevant response
frequency. That is, the Rabi frequency in the case of single qubit gates and
the detuning from carrier transition in the case of sideband transitions and
MS gates. While this study focuses on the effect of phase noise on trapped-ion
qubit gates, our results are broadly relevant for any fast noise sources, such
as amplitude noise as well. Our results are also relevant for other quantum
computing technologies in which two-qubit gates are realized through coupling
to a common bosonic mode, such as in superconducting qubits coupled through a
microwave resonator.

Figure 1: The effect of fast noise on the fidelity of trapped-ions quantum
gates. (a) Illustration of the Bloch sphere of a qubit where the Rabi vector
orientation fluctuates due to fast phase noise. The orange dashed line is the
noise-free trajectory for this qubit operation, whereas the purple lines are
trajectories of the qubit in the presence of fast noise. (b) Typical laser
phase power spectral densities (PSD). The red curve represents a typical PSD
for a free-running laser which is characterized by a white frequency noise or,
equivalently, $1/f^{2}$ phase noise. The purple curve represents the PSD of
the same noise process after applying an external feedback loop that
suppresses the slow noise while increasing the noise at the edge of its
bandwidth. The equivalent time traces are shown in the inset. (c) Mølmer-
Sørensen (MS) gate in the presence of fast noise can result in incoherent
coupling to other levels resulting in leakage of population to other motional
states, decreasing the gate fidelity. (d) The MS interaction dynamics,
visualized through the two populations without phase noise (dashed lines) and
with noise (solid lines). At gate time (green area), we expect to obtain a
Bell state, due to phase noise the qubits reach a slightly erroneous state
which includes an incoherent mixture of Bell states.

##  II Experimental and numerical setup

We perform stochastic master equation simulations and controlled experiments
to evaluate the effect of fast noise on the fidelity of quantum operations.
Specifically, here we focus on the PSD typical of oscillators where fast noise
is amplified by a frequency-stabilization feedback. For spectral frequencies
far from the carrier typical oscillators exhibit a white frequency noise
spectrum or equivalently, a brown phase noise spectrum
$S(\omega)=S_{0}/\omega^{2}$ [23]. To obtain a random vector with such a PSD
and a target root-mean square (RMS) we use a vector of Gaussian distributed
independent samples as a seed for a generalized Gauss-Markov stochastic model
[24, 25]. This noise is then convoluted with the impulse response of a phase-
locked loop system. The gain and positions of the system’s poles can be tuned
to shape the spectral response and the resulting output PSD. This results in a
spectral region of amplified noise or servo-bump, around the unity gain
frequency of the simulated system as shown in Fig. 1(b). The random time-
series generated through this process are used for both the experiments and
the simulations presented below. For the simulations, we numerically solve the
master equation with a time-dependent phase in the $\sigma_{x}$ drive term. In
each iteration we sample a different random phase vector, while all vectors
are drawn from the same PSD. The numerical simulations are performed using the
QuTiP[26] python package. For the different configurations described below, we
use different expectation values obtained from the time-evolution of the full
density matrix. These values are averaged over the different noise
realizations to obtain the results presented here.

For the experiment, we use an external cavity diode laser stabilized to a
high-finesse cavity. We further utilize the cavity as an extremely narrow
filter and take light transmitted through it to perform injection locking on
an additional diode, yielding the desired power with low noise. We synthesize
phase noise using an arbitrary waveform generator and inject it to the
experiment through modulation with an acousto-optical modulator (AOM). The
laser is then used to drive the optical qubit transition
$|S_{1/2};m_{j}{=}1/2\rangle-|D_{5/2};m_{j}{=}3/2\rangle$ in a single
${}^{88}\mathrm{Sr}^{+}$ ion trapped in a linear Paul trap.

##  III Single qubit gates

We begin by studying the most simple case of single-qubit rotations in the
presence of fast phase noise. In Fig. 2(a) we show the results of continuously
driving the qubit transition with a Rabi frequency $\Omega=2\pi\times 100$ kHz
and an additional synthesized phase noise characterized by a peak in the PSD
around 200 KHz. The experimental measured excited state population (circles)
exhibits Rabi oscillations with a decoherence rate which agrees well with the
stochastic numerical simulation (solid line) accounting for the the same noise
PSD [27]. We further study numerically the $\pi$-pulse error for a wide range
of noise spectra and find that the gate error is linearly dependent on the
noise PSD at the Rabi frequency. The relevant PSD in this case is the PSD of
the electric field (rather than the phase) driving the transition,
$\mathrm{E}=\mathrm{E_{0}}\cos[\omega t+\phi(t)]$, where $\phi(t)$ is the
time-dependent phase which includes the noise term. To evaluate the field PSD
in units of Rabi frequency we normalize the entire spectrum such that the area
under the carrier peak is $\Omega^{2}$. We term the new spectral density as
the Rabi PSD (RSPD). The dependence of a $\pi$-pulse error vs. the RPSD is
shown in Fig. 2(c) where we see a linear dependence. For pulses that are long
as compared with the inverse of relevant noise features, this dependence makes
sense, as the RPSD at the Rabi frequency generates an effective coupling
between the Rabi-dresses states. This effect was widely studied in the context
of noise spectroscopy [28, 17, 29].

For short pulses however, such as $\pi$ rotations, we find a slightly
different picture. In Fig. 2(b) we plot the noise PSD and the calculated $\pi$
pulse infidelity $1-F_{\pi}$ for different Rabi frequencies. We find that the
largest infidelity does not occur when the Rabi frequency exactly overlaps the
peak in the PSD, but rather at lower Rabi frequencies. Considering the $\pi$
pulse length $t_{\pi}=\pi/\Omega$ we find that the effective bandwidth of such
a short pulse is on the order of the Rabi frequency and thus samples, in
practice, the entire relevant region of the noise PSD equally. However, the
pulse length is still inversely proportional to the Rabi frequency such that
longer pulses sample the noise for a longer time. This combined integrated
response leads to a shift in the maximal infidelity towards lower Rabi
frequencies [30]. The sensitivity of short gate fidelity to noise integrated
over a wide spectral window, renders the proportionality factor between the
gate error and the RPSD at the Rabi frequency, seen in Fig. 2(c) to be on the
order of $10^{-2}$, dependent on the details of the noise spectrum.

Figure 2: Single qubit rotation infidelity due to fast phase noise. (a)
Simulation (solid line) and measurements (filled circles) of Rabi oscillations
for an ion driven by a noisy oscillator. Experimental error bars are smaller
than the marker size. (b) The infidelity of a single qubit $\pi$ rotation
calculated at different Rabi frequencies (orange markers), overlaid on the
phase noise PSD characterizing the simulated driving laser. (c) The infidelity
of a single qubit $\pi$ rotation at a fixed Rabi frequency as a function of
the Rabi PSD at the Rabi frequency multiplied by the gate time.

##  IV Two qubit gates

_Sideband transitions.–_ Here we consider the Mølmer-Sørensen gate [31, 32]
which is a composite operation in which the spin and motion of trapped ions
are entangled during the gate through driving of motional sideband
transitions. We thus start by disassembling this operation into its primitive
constituents. We first study numerically and experimentally the interplay of
fast noise and coherent driving of sideband transitions in a single trapped
ion. Beyond their role in the MS gate, off-resonant coupling fields are widely
used in experiments with atoms and molecules. Such fields are used in
generating dressed states, trapping, cooling and local addressing of atomic
qubits. It is thus of general relevance to understand the effect of fast noise
on sideband transitions.

We begin by studying the coupling of noise to the spin degree of freedom. We
effectively remove any coupling to motional degrees of freedom by detuning the
laser central frequency by a few hundreds of KHz from both the carrier
transition as well as from any sideband transition. The synthesized noise PSD
here has a broad peak around typical trap frequencies $\nu\approx 700$ KHz.
Following a drive of variable time, we image the ion to determine its spin
state. In Fig. 3 we plot the excited spin state population as a function of
drive time for different levels of synthesized noise, each averaged over 31
realizations. We observe incoherent spin pumping due to fast noise overlapping
the carrier transition. We find good agreement with the numerical simulation,
which for a sufficient number of realizations converges well to a simple
pumping rate model $\mathrm{P_{e}}=0.5[1-\mathrm{exp}(-\Gamma t)]$. In Fig.
3(b) we plot the pumping rate $\Gamma$ as a function of the RPSD at the
detuning frequency from the carrier transition $\Delta$. We find that the
pumping rate is proportional to the RPSD at the carrier transition frequency,
_i.e._ $\Gamma\simeq 2\cdot\mathrm{PSD}(f=\Delta)$.

In a second experiment, we effectively trace out the spin degree of freedom
and measure the occupation of the motional modes. We drive the blue sideband
transition for an integer number of spin cycles, and then perform thermometry
by driving the blue sideband transition again with a noise-free laser and
infer the occupation of motional states from sideband Rabi thermometry [33,
34]. In Fig 4, we plot the measured average number of phonons $\bar{n}$
following an integer number of blue sideband cycles. We find excess heating in
the presence of fast noise, which grows as a function of the drive time. As a
reference, we repeat the measurement in the absence of synthesized noise and
find negligible amount of residual heating, which may be attributed to
inherent noise in our laser.

This heating effect is another outcome of the incoherent carrier coupling in
the presence of fast phase noise and driving sideband transition. As seen in
the inset of Fig. 4, fast noise drives population incoherently on the carrier
transition which results in transfer of the ion to higher phonon states. Our
numerical simulation in this regime (solid line in Fig. 4) reveals the
interplay between coherent sideband driving and incoherent population of
vibrational modes. As a result, the contrast of coherent Rabi oscillations is
reduced along with a constant increase in the average number of phonons. When
this number reaches $\bar{n}=1$, the distribution becomes thermal, and we can
assign an effective temperature to the ion. Our analysis shows that further
driving of the ion would lead to a linear increase in temperature, which we
expect to saturate when the motional spread of the ion exceeds the limit of
the Lamb-Dicke regime such that carrier and sideband excitations are
suppressed.

Figure 3: Incoherent pumping due to carrier coupling. (a) Simulation (solid
line) and measurements (filled circles) of fraction of atoms in the excited
state as a function of off-resonant drive time for drives with different power
spectral densities. (b) Incoherent pumping rate as a function of the Rabi
frequency power spectral density at the detuning from the carrier transition.
Figure 4: Heating due to fast noise in driven sideband transitions. In the
main panel we plot the average number of occupied motional states $\bar{n}$
inferred from thermometry following an integer number of blue sideband driving
cycles. The red markers show measurements following evolution with added
synthesized noise while the blue markers show measurements with no added noise
(dashed line is a guide to the eye). The inset shows the relevant level scheme
where unwanted carrier transitions (orange dashed arrows) are induced due to
fast noise. Figure 5: Simulated Mølmer-Sørensen gate error under noisy drive.
We calculate and plot the gate infidelity as a function of the RPSD at the
trap frequency $\nu$ (filled circles). We find that infidelity depends
linearly in the RPSD at this specific frequency with a proportionality factor
close to unity. This is shown by the linear fit to the numerical results
(dashed line). The standard error is smaller than the marker size.

_Full numerical simulation of two qubit gates.–_ We now combine our
observation on both incoherent spin and motional dynamics to study the effect
of fast phase noise on the MS gate. We use a similar numerical noise PSD where
the noise overlaps with the trap frequency and a Hamiltonian with Rabi
frequency $\Omega=20$ kHz, $\nu=200$ kHz, and the Lamb-Dicke parameter
$\eta=0.15$. We solve the stochastic master equation for two qubits and their
common motional state modeled as a quantum harmonic oscillator with 30 modes
and find the gate fidelity as the overlap between the resulting density matrix
and the target Bell-state. This is averaged over 1000 realizations of the same
phase noise PSD. We repeat this for different noise amplitudes and plot the
gate fidelity vs. the RPSD at the carrier multiplies with the gate time in Fig
5. Once again, we find that this single parameter quantifies the performance
of this two-qubit gate. We fit this trend with a linear model and find that
the fidelity is proportional to the RPSD at the trap frequency with a
proportionality factor near unity $1-F\simeq T\cdot RPSD(f=\nu)$. The nearly
unity proportionality factor results from the fact that MS gates are long as
compared with the inverse spectral width of typical noise features.

Similarly to our findings in a single ion, the mechanism behind this
infidelity can be either due to incoherent spin pumping or heating of the ion.
Since the MS gate is insensitive to the ion’s temperature in the Lamb-Dicke
regime, the leading source of infidelity is the incoherent spin pumping due to
noise overlapping the carrier transition. This mechanism results in bit-flip
errors during the gate that can further propagate. However, repeated operation
of the MS gate as needed in a quantum circuit could eventually lead to
considerable heating and larger gate errors.

##  V Discussion

In this paper, we study the effect of fast noise on the fidelity of quantum
operations. For a broad class of single and two qubit operations including
single qubit rotations with resonant transitions, off-resonant driving of
multi-level single qubits, and two-qubits entangling gates, we identify a
single parameter which quantifies the rate of errors or decoherence during the
drive. This parameter is the noise Rabi PSD at the relevant frequencies, that
is the Rabi frequency for resonant rotations and the detuning from the carrier
transition for off-resonant drives and the MS gate.

We find that the infidelity of single qubit rotations via resonant drive is
set by the spectral overlap of the Rabi frequency and the noise PSD. For off-
resonant drives or operations on sideband transitions, we identify two main
channels for errors. The first one is due to a incoherent spin-pumping on the
carrier transition and the second is coupling to higher excited motional
modes, which takes the system out of the Hilbert space of exchanging a single
motional quanta and results in effective heating. The latter becomes more
significant with more drive cycles as the motional excitation accumulates. We
show that the heating mechanism has a negligible contribution to errors in a
single operation of a two-qubit gate, whereas the incoherent spin pumping
plays a pivotal role.

Assuming a trapped-ions quantum register with a carrier Rabi frequency of
$\Omega=2\pi\times$100 kHz and a motional mode with a Lamb-Dicke parameter
$\eta=0.05$, a MS gate using this mode will last 100 $\mu$s. Our findings
indicate that in order to achieve an error below $10^{-4}$ we require the
phase noise overlapping the carrier, in terms of RPSD to be under $\thicksim
1[\frac{Hz^{2}}{Hz}]$. In terms of dBc this represents a requirement of -100
dBc/Hz on the RPSD in this frequency range. For weak noise the RPSD is
proportional to the phase PSD and we have the -100 dBc/Hz requirement on phase
PSD as well. This is not necessarily an easy goal to achieve. As an example,
using a narrow linewidth laser operating on an optical qubit, with 100 kHz
wide servo bump overlapping with the trap frequency, no more than $\simeq
10^{-5}$ of the laser intensity can be contained within this servo bump.

The fact that the gate error is well approximated by the RPSD at a single
frequency results from the fact that typically the gate time is longer than
the correlation time of the noise at these frequencies. In our case, the servo
bump is spectrally wider than the Fourier width of the gate. This is not
always true in trapped ion systems, and in particular we can find short single
qubit $\pi$-times that deviate from this assumption. In these cases a proper
overlap integral is necessary to estimate the gate error [30].

Our analysis indicates that for full quantum control of operations at the high
fidelity frontier it is important to characterise and study such fast noise
mechanisms in any system. We note that our analysis is valid for any noise at
the relevant spectral window which may overlap characteristic Hamiltonian
energy scale and that the intricate dynamics coupling spin and motion under a
noisy drive in the MS gate are relevant to any system where a bosonic mode
mediates interaction between two spin-like qubits, such as gates between
superconducting qubits mediated by a resonator. Our findings, analysis and
detailed numerical calculations may thus guide the design and further
improvement of quantum hardware and tailored quantum gates.

##  VI acknowledgements

This work was supported by the Israeli Science Foundation and the Goldring
Family Foundation. We thank Yotam Shapira and Tom Manovitz for fruitful
discussions.

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