[a]Andrea Sofia Triolo

# Calibration of the Upgraded ALICE Inner Tracking System

, for the ALICE Collaboration

###### Abstract

The ALICE Experiment has replaced its Inner Tracking System with a 7-layer
pixel-only tracker made out of more than 24000 monolithic active pixel sensor
chips, in order to fulfill the requirements of the physics program of the LHC
Run 3. The upgraded Inner Tracking System (ITS2) has been installed in the
ALICE experiment during the LHC long shutdown 2 and has started to take data
with the beginning of Run 3 in July 2022, with proton-proton collisions at
$\sqrt{s}$ = 13.6 TeV. With its 12.5 billion pixels it is the largest pixel
detector installed in a high energy physics experiment to date. To guarantee
stable operation and a consistently high data quality, a regular calibration
of the detector has to be performed. The main part of the calibration program
consists of a tuning and subsequent measurement of the pixel thresholds and a
determination of the noisy channels. In particular the complexity of the
threshold scan depends linearly on the number of pixels, which is why the
threshold scan of the ITS2 is an unprecedented challenge. This work describes
the architecture of the calibration framework, which has been developed using
the detector control system of the ITS2 and the ALICE data processing layer.
Results of first threshold and noise calibrations done in situ are shown as
well.

##  1 The Upgraded ALICE Inner Tracking System

ALICE (A Large Ion Collider Experiment) is one of the four largest experiments
at the Large Hadron Collider (LHC) at CERN, and it is dedicated to the study
of strongly interacting matter.
ALICE completed a major upgrade in 2022, aimed to fulfill the requirements of
the Run 3 physics program of the LHC [1]. The main physics topics addressed
and discussed in the ALICE Upgrade Letter of Intent [2] require the
measurement of particles at low transverse momenta, together with novel
measurements of jets and their constituents. Moreover, many of the
measurements in Pb-Pb collisions are characterized by a very small signal-
over-background ratio, and they require also a significant improvement in
vertexing and tracking efficiency at low transverse momentum. The upgrade will
provide an increase of statistics of about two orders of magnitude with
respect to the period previous to the Long Shutdown 2.
One of the several features of the upgrade was the replacement of the beam
pipe with a smaller one. Thanks to this, the detectors were brought closer to
the interaction point. Consequently, the innermost detectors themselves were
modified to suit the new configuration. In particular, the Inner Tracking
System was entirely changed during the upgrade.
A schematic view of the ITS2 is shown in Figure 1. It is the innermost
detector of ALICE, and its main purposes are the localization of the primary
and secondary vertices and the reconstruction of particle tracks.

Figure 1: Schematic view of the ITS2 (top) and the constituent staves
(bottom): Inner Barrel stave on the left, Outer Barrel stave on the right [1].

The ITS2 consists of 7 cylindrical layers of Monolithic Active Pixel Sensors
(MAPS) named ALPIDE (ALice Pixel Detector), with radial coverage from 2.2 cm
to 39 cm. The 7 layers are divided into Inner Barrel (IB, layers 0-2) and
Outer Barrel (OB, layers 3-6), which is subsequently divided in Middle Layers
(ML, layers 3-4) and Outer Layers (OL, layers 5-6). Each layer is azimuthally
segmented into units named staves, for a total of 192 staves. Figure 1 bottom
shows two schematics of an IB stave and an OB (ML) stave. The IB staves
consist only of 9 chips in a row, instead the OB staves are composed of 112
chips for the ML or 196 chips for the OL, grouped in 4 rows of 7 chips each
called modules, then the ML staves are composed by 4 modules, the OL staves by
7 modules.
With a total of 24120 chips, and an area of 10 m2 covered by the detector, the
ITS2 is the largest MAPS-based detector in High-Energy Physics to date.
With this new setup, another important feature achieved was the reduction of
the material budget of the ITS: now the values of the material budget are
0.36% $X_{0}$/layer for the IB and 1.1% $X_{0}$/layer for the OB. This will
allow the tracking performance and momentum resolution to be significantly
improved, especially for the IB layers, where resolution may be more likely
affected by multiple Coulomb scattering in the material.

##  2 ALPIDE chips

The ALPIDE chip (Figure 2) is a CMOS MAPS developed on purpose for the ALICE
Upgrade [3]. It is implemented in a 180 nm CMOS imaging process and fabricated
on substrates with a high-resistivity (1-6 k$\Omega$cm) epitaxial layer. Its
total power consumption is 40 mW/cm2, and the input capacitance of the front-
end is below 2 fF.
The signal-sensing elements are n-well diodes with a diameter of approximately
2 $\upmu$m and an area typically 100 times smaller than the pixel cell area.
The particular manufacturing process also provides a deep p-well layer that is
exploited to implement the full CMOS circuitry in the active sensor area
without compromising the sensing diodes charge collection.

Figure 2: On the left: Photo of an ALPIDE chip. On the right: General
architecture of the ALPIDE chip.

The ALPIDE chips measure 15 $\times$ 30 mm2 and include a matrix of
512$\times$1024 pixel cells, each one measuring 29.24 $\times$ 26.88
$\upmu$m2. Analog biasing, control, readout, and interfacing functionalities
are implemented in a peripheral region of 1.2 $\times$ 30 mm2. The periphery
of the chip contains fourteen 8-bit analog DACs for the biasing of the pixel
front-ends. Each pixel cell contains a sensing diode, a front-end amplifier
and shaping stage, a discriminator, and a digital section. A hit is registered
on a pixel only if the current signal induced in the input transistor and
amplified is greater than a configurable threshold. In every pixel there is a
pulse injection capacitor for the injection of test charges in the input of
the front-end.
The sensor fulfils the experimental requirements of detection efficiency above
99% and a spatial resolution of 5 $\upmu$m.

##  3 ITS2 Calibration procedure

To guarantee stable operation and consistently high data quality, a regular
calibration of the detector has to be performed. The ITS2 configuration and
calibration are managed by the ITS2 Detector Control System (DCS). Data
extracted from the ITS through the O2 First Level Processor (FLP) are sent to
the Event Processing Nodes (EPN), which consist of 250 servers, each with 2
AMD Rome 32 core CPUs, 512 GB RAM, 8 AMD MI50 GPUs with 32 GB memory each, as
well as a network interface. Here data are stored, reconstructed and processed
online. While the experiment is not taking data, the EPN cluster is used as an
analysis facility, reading data from the disk buffer for additional processing
and physics analysis (offline processing).
The data readout and subsequent processing, including calibration, are
integrated in the ALICE O2 [4], the ALICE official computing framework for Run
3.

A general ITS Calibration procedure involves the tuning of the pixel
thresholds and the masking of the noisy pixels. The calibration is needed to
guarantee a stable operational point and efficiency of the detector. There are
three main calibration procedures for the ITS: threshold tuning, threshold
scan and noise calibration. From the ITS2 Detector Control System (DCS) it is
possible to configure the chips and control the scan execution, while the data
readout and subsequent processing are integrated into the ALICE O2. Other
scans are available for detector studies.
Data collected from the ITS2 are reconstructed and processed on the EPNs. The
calibration of the ITS2 is a challenging procedure, especially from the
computational point of view: with a total of 24120 chips and more than 12.5
billion pixels, the number of channels to be analyzed is enormous. To give an
example, the number of EPNs required to perform the calibration is 40, out of
250 EPNs available in the ALICE farm: a threshold scan of the full detector
results in about 3 $\times$ 1013 hits, corresponding to 100 TB of raw hit
data. To collect this huge amount of data the time needed is about 1 hour, and
for this reason a scan of the full detector is generally used as reference and
not performed on daily basis. Instead, at each beam dump, a threshold scan
using only $\sim$2% of pixels distributed uniformly on each chip is performed.

###  3.1 Threshold tuning

The purpose of the threshold tuning is to set the operating point of the
detector. The threshold is influenced by the setting of two DACs, named VCASN
and ITHR. Setting the threshold of the detector means setting these two DACs.
The target value chosen for the threshold setting is 100 e-, resulting from a
compromise between having a good detection efficiency and a low fake-hit rate
(FHR). As shown in Figure 3, the chip threshold should be maintained between
50 and 150 e- to keep the FHR below 10-6 hits/events/pixel and the detection
efficiency above 99%, and thus 100 e- is taken as the optimal value at which
the ALPIDE is fully efficient.

Figure 3: Experimental results obtained during beam tests. Detection
efficiency and fake-hit rate are shown as a function of the chip threshold for
different chips irradiation.

Figure 3 also shows the importance of performing recurring threshold scans: if
the threshold changes, a new tuning would be required to bring the threshold
again at 100 e- .
The tuning of the two DACs is performed by injecting an analogue pulse with
the fixed amplitude of 100 e- and measuring the number of hits registered in
the pixels. Then the DAC setting is changed and the injection of the fixed
charge and the hit counting are repeated. This process is repeated 50 times
for each DAC.
For each of the two DACs a separate scan is required. The first DAC to tune is
VCASN, which provides a coarse tuning of the threshold. Then the ITHR tuning
is performed to fine-adjust the threshold. For each tuning, the pixel response
is fitted with an error function to extract the 50% point, which corresponds
to DAC value to set to obtain a 100 e- threshold.
The threshold tuning is performed at chip level, because the setting of VCASN
and ITHR can only be made for the entire chip and not for the single pixels,
so what is set is the average threshold for each chip. For this reason, only
about 1% of pixels for each chip are scanned for the tuning, for a total of
$\sim$98 million pixels selected uniformly in the whole detector: only these
pixels are enough to correctly find the VCASN and ITHR values for a correct
threshold tuning.

###  3.2 Threshold scan

The threshold scan is performed to measure the threshold for each pixel of the
detector.
This measurement is performed by injecting analog pulses ranging from 0 to 500
electrons into the pixels. In this case, the values of VCASN and ITHR found
thanks to the threshold tuning are kept fixed for the chip: the purpose is
indeed to test that the tuning is correct. The pixel response will result in
an error function, that can be fitted to extract the 50% point (threshold) and
the $\sigma$ (ENC noise).
Because of the huge amount of pixels constituting the detector, the threshold
extraction for each pixel costs a lot in terms of time and resource
consumption. For this reason about 2% of the pixels of the entire detector,
taken uniformly from only 11 rows on each chip, are considered for this
measurement, which means $\sim$271 million pixels for the whole detector; the
scan with less pixels requires around 5 minutes to be completed, and the
quality of the scan is not compromised from the reduced sample of pixels,
since the chosen pixels are evenly distributed inside each chip and all the
pixels behave almost the same way.
In Figure 4 it is possible to see the two distributions of the in-pixel
threshold for each chip of the detector, before and after the tuning to 100
e-. In the first case (top plot) it is clear that the threshold distribution
is not uniform across all chips. In the bottom plot, obtained after the
threshold tuning of the entire detector, it is evident that the average of
every chip aligns with the target threshold of 100 e- at which the ALPIDE
chips are known to be fully efficient, with an RMS of about 20 e- .

Figure 4: Top: Threshold distribution as a function of the chip ID for the
full ITS2, before the tuning. Bottom: Threshold distribution as a function of
the chip ID for the full ITS2, after the threshold tuning to 100 e- .

The difference between the pixel thresholds before and after the tuning is
directly visible in Figure 5. In this plot, the black line refers to the not
tuned case, and the red line refers to the tuned case, and both lines are
normalized to the number of pixels considered in the scan. From the comparison
of the two distributions it is clear that the tuning of the thresholds shifts
the average point towards the target value (100 electrons) and reduces the
number of pixels in the tail at large thresholds (>250 electrons) of about 2
orders of magnitude. A residual tail remains since we do not have control on
every single pixel for the tuning, but only on the average per chip, which is
our typical estimate of the chip threshold.

Figure 5: Threshold distribution with and without tuned threshold of ITS2. In
the tuned case, the thresholds are tuned to 100 electrons.

From the threshold scan it is also possible to measure the ENC noise. In
Figure 6, the typical ENC noise is of only 5 electrons and it is stable for
all the chips, both before and after the tuning, so it is independent of the
tuning of the detector.
Both threshold and noise measurements obtained are comparable with those
obtained during the production phase.

Figure 6: ENC Noise distribution as a function of the chip ID for the full
ITS2.

###  3.3 Noise calibration

The purpose of the noise calibration is to tag and mask the noisy pixels of
the detector, once the threshold is tuned.
A noise calibration run consists in a standard data taking run without beam.
In order to mask the noisy pixels, a FHR threshold of 10-2 hits/events/pixel
for the IB and 10-6 hits/events/pixel for the OB is set. As a result, all the
pixels exceeding these values are tagged as noisy and masked to ensure a
detection efficiency above 99%. The difference between the two values of FHR
chosen is driven by the choice to avoid losses of efficiency in the Inner
Barrel. Moreover, considering the lower occupancy of the OB layers, the number
of hits registered in the OB would be dominated by the noise if such a strict
cut would not be applied.
With these requirements, the number of pixels tagged as noisy and masked in
the entire detector is around the 0.015% out of 12.5 billion pixels (also
including the pixels masked after the digital and analog scan that will be
discussed in the next section), and they mostly come from the OB. Figure 7
shows the average Fake-hit rate (FHR) for each layer after applying the mask.
The overall value in average is stable over time and remains around 10-8
hits/events/pixel, well below the 10-6 hits/events/pixel imposed by design to
achieve a good track reconstruction performance.

Figure 7: ITS2 average FHR trend as a function of date after masking 0.015% of
pixels of the entire detector. Each point shows a separate run from Run 3. The
Average FHR was obtained by taking a mean value among each chip (HIC) FHR
value for IB (OB) layer respectively. Errors are calculated as corresponding
values of RMS/$\sqrt{N_{entries}}$

##  4 Monitoring scans: digital/analog scan, pulse-shape scan, DAC scan

In addition to the main calibration scans discussed above, few other scans are
available for monitoring and detector studies, for example some of them can be
used to detect chip performance degradation due to radiation.

The Digital and Analog scans are used to mask the pixels with a problematic
readout. To spot the problematic pixels that have to be masked, the totality
of pixels of the detector are pulsed 50 times (digitally or analogically, with
an injected charge), and then a classification of the pixels is performed
based on their response to the pulses: pixels with 0 hits registered are
marked as "Dead", those with a number of registered hits between 0 and 49 are
marked as "Inefficient", the ones with 50 hits registered are marked as "OK"
and pixels with more than 50 hits registered are "Noisy". The pixels in the
ALPIDE chips are organized in "double columns", two adjacent columns of pixels
which share the same readout logic. If a double column contains more than 50
"Noisy" pixels, all its pixels are masked, in order to minimize the risk that
the ALPIDE chip goes into busy status. The latter can happen because of a too
large number of noisy pixels in a single double column, and also because it is
assumed that the readout logic of that double column is not working properly.
The amount of pixels masked this way is included in the 0.015% of pixels
masked for the entire detector.

The pulse-shape scan is used only for monitoring: it is possible to probe the
shape of the pulse signal with a small strobe window to monitor the influence
of this on the in-pixel threshold and to measure the time over threshold. The
measurement is done sampling the pulse signal with different delays as a
function of its amplitude and the delay of the trigger.

The DAC scans can be performed to monitor the on-chip DACs output. It is
possible to perform a DAC scan for each of the 14 8-bit DACs present on the
chip periphery. The scan is done by providing at the input of each DAC a
digital value between 0 and 255 and reading the analog output. This scan could
be performed for monitoring purposes, to verify that the linearity between the
digital input and the analog output of the DACs is maintained. Figure 8 shows
a stable linearity of the DACs for all the chips of the Inner Barrel, meaning
that the DACs were not modified by the radiation absorbed ($\sim$ 20 krad) in
around 5 months of proton-proton collisions.

Figure 8: DAC scans for all the DACs for all the chips of the Inner Barrel.
The analogue voltage and current values in output from the DAC are plotted as
a function of the input digital values.

##  5 Conclusions

The ALICE Experiment has replaced its Inner Tracking System during the Long
Shutdown 2, in order to fulfill the requirements of the Run 3 physics program
of the LHC. The Upgraded detector is composed of 7 layers of monolithic active
pixel sensor chips named ALPIDE. With an area of 10 m2 covered by the detector
and 12.5 billion pixels, the ITS2 is the largest successfully operating MAPS-
based detector in high-energy physics.
The calibration of the detector consists of the tuning and measurement of the
in-pixel threshold, and in spotting and masking the noisy pixels. The
calibration procedure is very challenging because of the large number of
channels that have to be calibrated: a threshold scan of the full detector
results in about 3 $\times$ 1013 hits, corresponding to 100 TB of raw hit
data.
From the start of the LHC Run 3 in July 2022 to the beginning of December
2022, ALICE accumulated a total absorbed dose of about 20 krad, a recorded
luminosity of about 18 pb-1 from proton-proton collisions, around 700 hours of
data taking and a number of interaction recorder of the order of 1012. ITS2
threshold and noise remained stable during this period. The threshold is
stable around 100 e- , and the fake-hit rate is stable around 10-8
hits/events/pixel by masking only 0.015% of the total number of pixels. These
stable results are comparable to single chip tests of threshold and noise
performed during the various tests in the lab during the production phase.

## References

  * [1] ALICE Collaboration, Technical Design Report for the Upgrade of the ALICE Inner Tracking System,  J. Phys. G: Nucl. Part. Phys. 41 087002\.
  * [2] ALICE Collaboration, Letter of Intent for the Upgrade of the ALICE Experiment, CERN-LHCC-2012-012, LHCC-I-022, 2012
  * [3] G.Aglieri Rinella on behalf of the ALICE Collaboration, The ALPIDE pixel sensor chip for the upgrade of the ALICE Inner Tracking System, NIM A 845 2017 583-587
  * [4] Anton Alkin, Giulio Eulisse, Jan Fiete Grosse-Oetringhaus, Peter Hristov, Maja Kabus, ALICE Run 3 Analysis Framework, EPJ Web Conf. 251 03063 (2021)


# Mechanical Bistability in Kerr-modified Cavity Magnomechanics

Rui-Chang Shen  Interdisciplinary Center of Quantum Information, State Key
Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, Zhejiang
University, Hangzhou 310027, China  Jie Li<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China  Zhi-Yuan Fan  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Modern Optical Instrumentation, and
Zhejiang Province Key Laboratory of Quantum Technology and Device, School of
Physics, Zhejiang University, Hangzhou 310027, China  Yi-Pu Wang
<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information, State Key
Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, Zhejiang
University, Hangzhou 310027, China  J. Q. You<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China

###### Abstract

Bistable mechanical vibration is observed in a cavity magnomechanical system,
which consists of a microwave cavity mode, a magnon mode, and a mechanical
vibration mode of a ferrimagnetic yttrium-iron-garnet (YIG) sphere. The
bistability manifests itself in both the mechanical frequency and linewidth
under a strong microwave drive field, which simultaneously activates three
different kinds of nonlinearities, namely, magnetostriction, magnon self-Kerr,
and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr
nonlinearity is first predicted and measured in magnomechanics. The system
enters a regime where Kerr-type nonlinearities strongly modify the
conventional cavity magnomechanics that possesses only a dispersive
magnomechanical coupling. Three different kinds of nonlinearities are
identified and distinguished in the experiment. Our work demonstrates a new
mechanism for achieving mechanical bistability by combining magnetostriction
and Kerr-type nonlinearities, and indicates that such Kerr-modified cavity
magnomechanics provides a unique platform for studying many distinct
nonlinearities in a single experiment.

Introduction.—Bistability, or multistability, discontinuous jumps, and
hysteresis are characteristic features of nonlinear systems. Bistability is a
widespread phenomenon that exists in a variety of physical systems, e.g.,
optics Walls1 , Walls2 , Lugiato , electronic tunneling structures Goldman ,
magnetic nanorings Zhu , thermal radiation Biehs , a driven-dissipative
superfluid Ott , and cavity magnonics YP18 . Its presence requires
nonlinearity in the system. To date, bistability has been studied in various
mechanical systems, including nano- or micromechanical resonators Capasso ,
Zant03 , Badzey , piezoelectric beams Cottone , mechanical morphing structures
Chen , and levitated nanoparticles Ricci . Bistable mechanical motion finds
many important applications: It is the basis for mechanical switches switch1 ,
switch2 , memory elements Badzey2 , Zant09 , logic gates Murali , vibration
energy harvesters Cottone , Harne , and signal amplifiers Badzey , Ricci ,
etc.

Different mechanisms can bring about nonlinearity in the system leading to
bistable mechanical motion. Most commonly, a strong drive can induce
bistability of a mechanical oscillator, of which the dynamics is described by
the Duffing equation Badzey , Cleland , Buks , Katz . It can also be caused by
the Casimir force Capasso , nanomechanical effects on Coulomb blockade Zant03
, magnetic repulsion Neri , and intrinsic nonlinearity in the optomechanical
coupling Seok , etc.

In this Letter, we introduce a mechanism to induce mechanical bistability,
distinguished from all the above mechanisms, by exploiting rich nonlinearities
in the ferrimagnetic yttrium-iron-garnet (YIG) in cavity magnomechanics (CMM).
In the CMM Tang , APE , Davis , Yuan , the magnons are the quanta of
collective spin excitations in YIG, which can strongly couple to microwave
cavity photons by the magnetic-dipole interaction, leading to cavity
polaritons S1 , S2 , S3 , S4 , S5 , S6 . They can also couple to deformation
vibration phonons of the ferrimagnet via the magnetostrictive force Kittle ,
Tang , Davis . Such a dispersive magnon-phonon coupling provides necessary
nonlinearity, enabling a number of theoretical proposals, including the
preparation of entangled states Jie18 , Jie19b , Tan , Jie20 , QST , squeezed
states Jie19a , HFW , nsr , mechanical quantum-ground states Ding , HFW2 , FRe
, slow light Xiong , Ullah , a thermometer Davis2 , quantum memory JJ ,
Twamley , exceptional points JH , and parity-time-related phenomena YXL , Sun
, Wu , Ding3 , etc. In contrast, the experimental studies on this system are,
by now, very limited: Only magnomechanically induced transparency and
absorption Tang  and mechanical cooling and lasing Davis  were demonstrated.

Here we report an experimental observation of bistable mechanical vibration of
a YIG sphere in the CMM. We show that both the frequency and linewidth of the
mechanical mode exhibit a bistable feature as a result of the combined effects
of the dispersive magnetostrictive interaction Tang , Davis , the magnon self-
Kerr YP16 , YP21 , and the magnon-phonon cross-Kerr nonlinearities. Three
different kinds of nonlinearities are simultaneously activated by applying a
strong drive field on the YIG sphere, owing to the rich nonlinear interactions
among excitations in YIG. Their respective contributions to the mechanical
frequency and linewidth, which exhibit a bistable feature, are analysed and
distinguished. Our work studies rich and delicate nonlinear effects in the
Kerr-modified CMM, which offers a compact platform for studying the interplay
among distinctly different nonlinearities. The combination of the
magnetostriction and Kerr-type nonlinearities reveals a new mechanism for
achieving mechanical bistability.

FIG. 1: Kerr-modified CMM. (a) Left panel: Schematic of the CMM system. A 0.28
mm-diameter YIG sphere is placed (free to move) in a horizontal 0.9 mm-inner-
diameter glass capillary and at the antinode of the magnetic field of the
cavity mode $\rm{TE}_{\rm{101}}$. The cavity has two ports: Port 1 is
connected to a microwave source (MW) to load the drive field, and Port 2 is
connected to a vector network analyzer (VNA) to measure the reflection of the
probe field with the power of $-5$ dBm. The cavity decay rates associated with
the two ports are $\kappa_{\rm{1}}/2\pi=0.22~{}\rm{MHz}$ and
$\kappa_{\rm{2}}/2\pi=1.05~{}\rm{MHz}$, respectively. We set the direction of
the bias magnetic field $B_{\rm{0}}$ as the $z$ direction, and the vertical
direction as the $x$ direction. Right panel: Schematic of the coupled three
modes. (b) Left panel: Measured reflection spectra under a red-detuned drive.
The frequency of the cavity mode $\omega_{\rm{a}}/2\pi=7.653~{}\rm{GHz}$, and
of the drive field $\omega_{\rm{d}}/2\pi=7.645~{}\rm{GHz}$ (black arrow). By
increasing the drive power, the magnon frequency shift is negative:
$\omega_{\rm{m}}/2\pi=7.658~{}\rm{GHz}$ at the power
$P_{\rm{d}}=4.7~{}\rm{dBm}$ (upper panel) and
$\omega_{\rm{m}}/2\pi=7.640~{}\rm{GHz}$ at $P_{\rm{d}}=29.7~{}\rm{dBm}$ (lower
panel). The green arrow indicates the direction in which the magnon frequency
(green dashed line) shifts by increasing the power. Right panel: Zoom-in on
the shaded areas in the left panel shows detailed spectra of the
magnomechanically induced resonances, where
$\Delta_{\rm{pd}}=\omega_{\rm{p}}-\omega_{\rm{d}}$. The thin black lines are
the fitting curves. (c) Left panel: Measured reflection spectra under a blue-
detuned drive. By adjusting the bias magnetic field, the magnon frequency is
tuned close to the drive frequency $\omega_{\rm{d}}/2\pi=7.66~{}\rm{GHz}$
under $P_{\rm{d}}=4.7~{}\rm{dBm}$. Increasing the power to $23.7~{}\rm{dBm}$,
the magnon frequency $\omega_{\rm{m}}/2\pi=7.645~{}\rm{GHz}$. Right panel:
Zoom-in on the shaded areas in the left panel shows detailed spectra of the
magnomechanically induced resonances. The system parameters can be extracted
by fitting the reflection spectra, which are
$\kappa_{\rm{a}}/2\pi=2.78~{}\rm{MHz}$, $\kappa_{\rm{m}}/2\pi=2.2~{}\rm{MHz}$,
$g_{\rm{ma}}/2\pi=7.37~{}\rm{MHz}$, and
$\omega_{\rm{b}}/2\pi=11.0308~{}\rm{MHz}$,
$\kappa_{\rm{b}}/2\pi=550~{}\rm{Hz}$, $g_{\rm{mb}}/2\pi=1.22~{}\rm{mHz}$. The
effective coupling $G_{\rm{mb}}/2\pi=45.8~{}\rm{kHz}$ (42.7 kHz) under the
red-detuned (blue-detuned) drive power of $29.7~{}\rm{dBm}$ (23.7 dBm). Also,
we observe another nearby mechanical mode and the parameters are
$\omega^{\prime}_{\rm{b}}/2\pi=11.0377~{}\rm{MHz}$,
$\kappa^{\prime}_{\rm{b}}/2\pi=180~{}\rm{Hz}$, and
$g^{\prime}_{\rm{mb}}/2\pi=0.62~{}\rm{mHz}$. Due to their similar behaviors,
we focus on the lower-frequency mode which has a stronger magnomechanical
coupling.

Kerr-modified CMM.—The CMM system consists of a microwave cavity mode, a
magnon mode, and a mechanical vibration mode, see Fig. 1(a). The magnon and
mechanical modes are supported by a 0.28 mm-diameter YIG sphere. The magnon
mode couples to the cavity magnetic field by the magnetic-dipole interaction,
and to the vibration mode by the magnetostrictive interaction. Under a strong
pump, the Hamiltonian of the CMM system is given by SM

$\displaystyle H/\hbar$
$\displaystyle=\omega_{\rm{a}}a^{\dagger}a+\omega_{\rm{m}}m^{\dagger}m+\omega_{\rm{b}}b^{\dagger}b+g_{\rm
ma}(a^{\dagger}m+am^{\dagger})$ (1) $\displaystyle+g_{\rm
mb}m^{\dagger}m\left(b+b^{\dagger}\right)+H_{\rm{Kerr}}/\hbar+\\!\\!\sqrt{\kappa_{\rm{1}}}\varepsilon_{\rm{d}}\left(a^{\dagger}e^{-i\omega_{\rm{d}}t}+{\rm
H.c.}\right),$

where $a$, $m$, and $b$ ($a^{{\dagger}}$, $m^{{\dagger}}$, and
$b^{{\dagger}}$) are the annihilation (creation) operators of the cavity mode
at frequency $\omega_{\rm{a}}$, the magnon mode at frequency
$\omega_{\rm{m}}$, and the mechanical mode at frequency $\omega_{\rm{b}}$,
respectively, $g_{\rm{ma}}$ is the cavity-magnon coupling strength, and
$g_{\rm{mb}}$ is the bare magnomechanical dispersive coupling strength. This
dispersive coupling is typically weak, but can be significantly enhanced by
applying a pump field on the magnon mode Jie18 . In our experiment, this is
realized by strongly driving the cavity, which linearly couples to the magnon
mode. The driving Hamiltonian is described by the last term, where
$\kappa_{1}$ is the cavity decay rate associated with the driving port (Port
1), and $\varepsilon_{\rm{d}}=\sqrt{P_{\rm{d}}/(\hbar\omega_{\rm{d}})}$, with
$P_{\rm{d}}$ ($\omega_{\rm{d}}$) being the power (frequency) of the microwave
drive field.

The novel part of the Hamiltonian, with respect to the conventional CMM, is
the Kerr nonlinear term $H_{\rm{Kerr}}$ activated by the strong pump field SM

$\displaystyle H_{\rm{Kerr}}/\hbar$ $\displaystyle=$ $\displaystyle
K_{\rm{m}}m^{\dagger}mm^{\dagger}m+K_{\rm{cross}}m^{\dagger}mb^{\dagger}b,$
(2)

where $K_{\rm{m}}$ is the magnon self-Kerr coefficient, and $K_{\rm{cross}}$
is the magnon-phonon cross-Kerr coefficient. The magnon self-Kerr effect is
caused by the magnetocrystalline anisotropy YP16 , YP21 , and the cross-Kerr
nonlinearity originates from the magnetoelastic coupling by including the
second-order terms in the strain tensor Landau , Tero ,

$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
l_{j}}+\frac{\partial u_{j}}{\partial l_{i}}+\sum_{k}\frac{\partial
u_{k}}{\partial l_{i}}\frac{\partial u_{k}}{\partial l_{j}}\right),$ (3)

where $u_{i}$ are the components of the displacement vector. The first-order
terms lead to the conventional dispersive interaction Hamiltonian Fan , $\hbar
g_{\rm mb}m^{\dagger}m\left(b+b^{\dagger}\right)$. Under a moderate drive
field, the second-order terms are negligible Tang , Davis , but can no longer
be neglected when the drive becomes sufficiently strong, as in our experiment,
yielding an appreciable magnon-phonon cross-Kerr nonlinearity. As will be seen
later, the cross-Kerr nonlinearity is indispensable in the model for fitting
the mechanical frequency shift.

Both the magnon self-Kerr and magnon-phonon cross-Kerr terms, as well as the
dispersive term, cause a magnon frequency shift
$\delta\omega_{\rm{m}}=2K_{\rm{m}}|M|^{2}+K_{\rm{cross}}|B|^{2}+2g_{\rm
mb}{\rm Re}[B]$, where $O\,{=}\,\langle o\rangle$ ($o\,{=}\,m,b,a$) denote the
average of the modes. In our experiment, the dominant contribution is from the
self-Kerr nonlinearity SM , which gives a bistable magnon frequency shift YP18
. Also, the cross-Kerr nonlinearity causes a mechanical frequency shift
$\delta\omega_{\rm{b}}=K_{\rm{cross}}|M|^{2}$. Using the Heisenberg-Langevin
approach, we obtain the equation for the steady-state average $M$ SM

$\displaystyle\begin{aligned}
\eta_{\rm{a}}\kappa_{\rm{1}}&g_{\rm{ma}}^{2}\varepsilon_{\rm{d}}^{2}=|M|^{2}\times\\\
&\left[\left(\Delta_{\rm{m}}-\eta_{\rm{a}}g_{\rm{ma}}^{2}\Delta_{\rm{a}}+2K_{\rm{m}}|M|^{2}\right)^{2}+\left(\frac{\kappa_{\rm{m}}}{2}+\eta_{\rm{a}}g_{\rm{ma}}^{2}\frac{\kappa_{\rm{a}}}{2}\right)^{2}\right],\end{aligned}$
(4)

where $\Delta_{\rm{a\,(m)}}=\omega_{\rm{a\,(m)}}-\omega_{\rm{d}}$,
$\eta_{\rm{a}}=\frac{1}{\Delta_{\rm{a}}^{2}+(\kappa_{\rm{a}}/2)^{2}}$, and
$\kappa_{\rm{a\,(m)}}$ is the linewidth of the cavity (magnon) mode. In
deriving Eq. (4), we neglect contributions from the mechanical mode to the
magnon frequency shift, because of a much smaller mechanical excitation number
compared to the magnon excitation number for the drive powers used in this
work. It is a cubic equation of the magnon excitation number $|M|^{2}$. In a
suitable range of the drive power, there are two stable solutions, leading to
the bistable magnon and phonon frequency shifts by varying the drive power.

The magnomechanical dispersive coupling gives rise to an effective
susceptibility of the mechanical mode SM

$\displaystyle\begin{aligned}
\chi_{\rm{b,eff}}(\omega)=\left(\chi_{\rm{b}}^{-1}(\omega)-2i|G_{\rm{mb}}|^{2}\left(\chi_{\rm{ma}}(\omega)-\chi_{\rm{ma}}^{*}(-\omega)\right)\right)^{-1},\end{aligned}$
(5)

where $\chi_{\rm{b}}(\omega)$ is the natural susceptibility of the mechanical
mode, but depends on the modified mechanical frequency
$\tilde{\omega}_{\rm{b}}=\omega_{\rm{b}}+K_{\rm{cross}}|M|^{2}$, which
includes the cross-Kerr induced frequency shift. The effective dispersive
coupling is $G_{\rm{mb}}=g_{\rm{mb}}M$, and
$\chi_{\rm{ma}}(\omega)=\left[\chi_{\rm{m}}^{-1}(\omega)+g_{\rm{ma}}^{2}\chi_{\rm{a}}(\omega)\right]^{-1}$,
where $\chi_{\rm{m}}(\omega)$ and $\chi_{\rm{a}}(\omega)$ are the natural
susceptibilities of the magnon and cavity modes, with the magnon detuning in
$\chi_{\rm{m}}(\omega)$ modified as
$\tilde{\Delta}_{\rm{m}}=\Delta_{\rm{m}}+2K_{\rm{m}}|M|^{2}$, which includes
the dominant magnon self-Kerr induced frequency shift. See SM  for the
explicit expressions of the susceptibilities.

The effective mechanical susceptibility yields a frequency shift of the phonon
mode (the so-called “magnonic spring” effect Davis , in analogy to the
“optical spring” in optomechanics MA )

$\displaystyle\delta\omega_{\rm{b}}=-{\rm{Re}}\left[2i|G_{\rm{mb}}|^{2}\left(\chi_{\rm{ma}}(\omega)-\chi_{\rm{ma}}^{*}(-\omega)\right)\right]+K_{\rm{cross}}|M|^{2},$
(6)

where we write together the frequency shift induced by the cross-Kerr
nonlinearity. Moreover, it leads to a mechanical linewidth change

$\displaystyle\delta\Gamma_{\rm{b}}={\rm{Im}}\left[2i|G_{\rm{mb}}|^{2}\left(\chi_{\rm{ma}}(\omega)-\chi_{\rm{ma}}^{*}(-\omega)\right)\right].$
(7)

Clearly, this linewidth change is only caused by the dispersive coupling,
distinguished from the frequency shift caused by the self-Kerr or cross-Kerr
nonlinearity. By applying a red- or blue-detuned drive field, we can choose to
operate the system in two different regimes, where either the magnomechanical
anti-Stokes or Stokes scattering is dominant. This yields an increased
($\delta\Gamma_{\rm{b}}>0$) or a reduced ($\delta\Gamma_{\rm{b}}<0$)
mechanical linewidth, corresponding to the cooling or amplification of the
mechanical motion Davis , MA .

Red-detuned drive.—In our system, due to the strong coupling
$g_{\rm{ma}}>\kappa_{\rm{m}},\kappa_{\rm{a}}$, the magnon and cavity modes
form two cavity polariton (hybridized) modes (Fig. 1(b) and 1(c)) S1 , S2 , S3
, S4 , S5 , S6 . Here, a red (blue)-detuned drive means that the drive
frequency is lower (higher) than the frequency of the cavity-like polariton
mode, i.e., the “deeper” polariton in the spectra close to the cavity
resonance. For the red (blue)-detuned drive, we show the anti-Stokes (Stokes)
sidebands associated with two mechanical modes for two drive powers in the
zoom-in plots of Fig. 1(b) (Fig. 1(c)). When the detuning between the drive
field and the deeper polariton matches the mechanical frequencies, the anti-
Stokes (Stokes) sidebands are manifested as the magnomechanically induced
transparency (absorption) Tang .

FIG. 2: Bistable mechanical frequency and linewidth under a red-detuned drive.
(a) The mechanical frequency shift versus the drive power. The red (green)
curve is the fitting of the frequency shift induced by the dispersive coupling
(cross-Kerr effect) using Eq. (6), and the black curve is the sum of the two
contributions. (b) The mechanical linewidth variation versus the drive power.
The black curve is the fitting of the linewidth change using Eq. (7). In both
figures, the blue (orange) triangles are the experimental data obtained via
forward (backward) sweep of the drive power.

The microwave cavity mode $\rm{TE}_{\rm{101}}$ has a frequency
$\omega_{\rm{a}}/2\pi\,\,{=}\,\,7.653~{}\rm{GHz}$, and the lower-frequency
mechanical mode has a natural frequency
$\omega_{\rm{b}}/2\pi\,\,{=}\,\,11.0308~{}\rm{MHz}$ and linewidth
$\kappa_{\rm{b}}/2\pi\,\,{=}\,\,550~{}\rm{Hz}$, and a stronger magnomechanical
coupling $g_{\rm{mb}}/2\pi\,\,{=}\,\,1.22~{}\rm{mHz}$. We drive the cavity
with a red-detuned microwave field at frequency
$\omega_{\rm{d}}/2\pi=7.645~{}\rm{GHz}$. By adjusting the bias magnetic field,
we tune the magnon frequency to be
$\omega_{\rm{m}}/2\pi\,\,{=}\,\,7.658~{}\rm{GHz}$ at the drive power
$P_{\rm{d}}\,\,{=}\,\,4.7~{}\rm{dBm}$ (see Fig. 1(b)). We have the [110] axis
of the YIG sphere aligned parallel to the static magnetic field, which yields
a negative self-Kerr coefficient $K_{\rm{m}}/2\pi\,\,{=}\,{-}6.5~{}\rm{nHz}$.
This means a negative magnon frequency shift
$\delta\omega_{\rm{m}}=2K_{\rm{m}}|M|^{2}$ by increasing the power, and the
magnon frequency reduces to $\omega_{\rm{m}}/2\pi\,\,{=}\,\,7.640~{}\rm{GHz}$
when the power increases to $P_{\rm{d}}\,\,{=}\,\,29.7~{}\rm{dBm}$ (Fig.
1(b)). Under these conditions, the magnon excitation number $|M|^{2}$ shows a
bistable behavior by varying the power.

Equation (6) indicates that the magnomechanical dispersive coupling results in
a mechanical frequency shift, and so does the cross-Kerr nonlinearity. This is
confirmed by the experimental data in Fig. 2(a). It shows that the cross-Kerr
plays a dominant role because of a large magnon excitation number, and both
the frequency shifts caused, respectively, by the cross-Kerr and the
dispersive coupling show a bistable feature with the forward and backward
sweeps of the drive power. This is because both of them originate from the
bistable magnon excitation number $|M|^{2}$, c.f. Eq. (6). Note that for the
spring effect, the bistability of $|M|^{2}$ is mapped to the magnon frequency
shift $\tilde{\Delta}_{\rm{m}}$, then to the polariton susceptibility
$\chi_{\rm{ma}}(\omega)$, and finally to the mechanical frequency.

By increasing the drive power from $4.7~{}\rm{dBm}$ to $19.7~{}\rm{dBm}$, the
cross-Kerr causes a maximum frequency shift of $-4.6~{}\rm{kHz}$ (Fig. 2(a),
the green line). The fitting cross-Kerr coefficient is
$K_{\rm{cross}}/2\pi=-5.4~{}\rm{pHz}$. Under the red-detuned drive, the
magnonic spring effect yields a negative frequency shift
$\delta\omega_{\rm{b}}=\rm{Re}\left[\chi_{\rm{b,eff}}^{-1}(\omega)-\chi_{\rm{b}}^{-1}(\omega)\right]<0$
(Fig. 2(a), the red line), and the maximum frequency shift is
$-200~{}\rm{Hz}$. Adding up these two frequency shifts gives the total
mechanical frequency shift (Fig. 2(a), the black line), which fits well with
the experimental data (Fig. 2(a), triangles) when the power is not too strong.

Another interesting finding is the bistable feature of the mechanical
linewidth (Fig. 2(b)). The magnomechanical backaction leads to the variation
of the mechanical linewidth
$\delta\Gamma_{\rm{b}}=-\rm{Im}\left[\chi_{\rm{b,eff}}^{-1}(\omega)-\chi_{\rm{b}}^{-1}(\omega)\right]$.
For a red-detuned drive, the anti-Stokes process is dominant, resulting in an
increased mechanical linewidth $\delta\Gamma_{\rm{b}}>0$ and the cooling of
the motion. The bistable mechanical linewidth is also induced by the bistable
$|M|^{2}$ (see Eq. (7)), similar to the mechanical frequency. The theory fits
well with the experimental results, and the discrepancy appears only in the
high-power regime, because the significant heating effect at strong pump
powers can reduce the $Q$ factor of the mechanical mode heat1 , heat2 , which
is not included in our model.

Blue-detuned drive.—When a blue-detuned drive is applied, the system enters a
regime where the Stokes scattering is dominant. The magnomechanical parametric
down-conversion amplifies the mechanical motion with the characteristic of a
reduced linewidth. Furthermore, the mechanical frequency shift induced by the
spring effect will move in the opposite direction compared with the red-
detuned drive.

To implement a blue-detuned drive, we drive the cavity with a microwave field
at frequency $\omega_{\rm{d}}/2\pi=7.66~{}\rm{GHz}$, and tune the magnon
frequency close to the drive frequency (see Fig. 1(c)). We attempted to make
the Stokes sideband of the drive field resonate with the “deeper” polariton at
a high pump power, such that the Stokes scattering rate is maximized and the
magnomechanical coupling strength $G_{\rm mb}$ becomes strong. However, to
meet the drive conditions for a bistable magnon excitation number $|M|^{2}$,
the drive frequency is restricted to a certain range, which hinders us to make
the Stokes sideband and the “deeper” polariton resonate. Therefore, we only
achieve this at lower drive powers, giving a faint magnomechanically induced
absorption (Fig. 1(c), upper panels).

From the red to the blue detuning, we only adjust the magnon and the drive
frequencies. Because the direction of the crystal axis is unchanged, the
magnon self-Kerr coefficient $K_{\rm{m}}$ is still negative, so again a
negative frequency shift by increasing the power (green arrow in Fig. 1(c)).
For the power up to 23.7 dBm, the frequency of the cavity-like polariton is
always lower than the drive frequency, so the system is operated under a blue-
detuned drive.

FIG. 3: Bistable mechanical frequency and linewidth under a blue-detuned
drive. (a) The mechanical frequency shift and (b) the mechanical linewidth
change versus the drive power. The curves and the triangles are shown in the
same manner as in Fig. 2.

Figure 3 displays the bistable mechanical frequency shift
$\delta\omega_{\rm{b}}$ and linewidth change $\delta\Gamma_{\rm{b}}$. For the
frequency shift, both the contributions from the cross-Kerr and the dispersive
coupling should be considered, but the former plays a dominant role (Fig.
3(a), the green line), as in the case of the red-detuned drive, yielding a
frequency shift of $-6.5$ kHz at the power of 23.7 dBm. Differently, the
spring effect induced frequency shift (370 Hz at $23.7$ dBm) is positive (Fig.
3(a), the red line). The opposite frequency shifts by the spring effect in the
blue and red-detuned drives agree with the finding of Ref. Davis , but no
bistability was observed in their work.

The reduced mechanical linewidth $\delta\Gamma_{\rm{b}}<0$ under a blue-
detuned drive is confirmed by Fig. 3(b). However, unlike the bistable curve in
the red-detuned drive case (Fig. 2(b)), $\delta\Gamma_{\rm{b}}$ manifests the
bistability in an alpha form by sweeping the drive power. This is the result
of the trade-off between the growing coupling strength $G_{\rm mb}$ (which
enhances the Stokes scattering rate, yielding an increasing
$|\delta\Gamma_{\rm{b}}|$) and the larger detuning between the Stokes sideband
and the “deeper” polariton (Fig. 1(c)) (which reduces the Stokes scattering
rate, resulting in a decreasing $|\delta\Gamma_{\rm{b}}|$) by raising the
drive power. These two effects are balanced when the power is in the range of
12 dBm to 15 dBm.

Conclusions.—We have observed bistable mechanical frequency and linewidth and
the magnon-phonon cross-Kerr nonlinearity in the CMM system. The mechanical
bistability results from the magnomechanical backaction on the mechanical mode
and the strong modifications on the backaction due to the magnon self-Kerr and
magnon-phonon cross-Kerr nonlinearities. Three different kinds of
nonlinearities are simultaneously activated in the YIG sphere, and their
respective effects are identified and distinguished in the experiment by
measuring the magnon frequency shift, the mechanical frequency shift, and the
mechanical linewidth, respectively note .

Our work indicates that the YIG-based CMM system under a strong drive,
manifested as the Kerr-modified CMM, provides a unique platform for studying
rich nonlinear effects and the interplay among them. The mechanical
bistability induced by the combined effects of the magnetostriction and Kerr-
type nonlinearities in this work reveals a new mechanism for achieving
bistable mechanical motion. It promises a wide range of applications, such as
in mechanical switches, memories, logic gates, and signal amplifiers.

Acknowledgments. This work was supported by the National Natural Science
Foundation of China (Grants Nos. 11934010, U1801661, 12174329, 11874249),
Zhejiang Province Program for Science and Technology (Grant No. 2020C01019),
and the Fundamental Research Funds for the Central Universities (No.
2021FZZX001-02).

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