# A first demonstration of active feedback control and multi-frequency imaging
techniques for kinetic inductance detectors

M. Rouble  Department of Physics and Trottier Space Institute, McGill
University, Montreal, QC, H3A 2T8, Canada  G. Smecher  t0.technology inc,
Montreal, QC, Canada  M. Adamič  Department of Physics and Trottier Space
Institute, McGill University, Montreal, QC, H3A 2T8, Canada  A. Anderson
Fermi National Accelerator Laboratory, PO BOX 500, Batavia, IL 60510  P. S.
Barry  School of Physics & Astronomy, Cardiff University, The Parade, Cardiff,
CF24 3AA, United Kingdom  K. Dibert  University of Chicago, 5640 South Ellis
Avenue, IL, 60637, USA  M. Dobbs  Department of Physics and Trottier Space
Institute, McGill University, Montreal, QC, H3A 2T8, Canada  Canadian
Institute for Advanced Research, CIFAR Program in Gravity and the Extreme
Universe, Toronto, ON, M5G 1Z8, Canada  K. Fichman  University of Chicago,
5640 South Ellis Avenue, IL, 60637, USA  J. Montgomery  Department of Physics
and Trottier Space Institute, McGill University, Montreal, QC, H3A 2T8, Canada

###### Abstract

RF-ICE is a signal processing platform for the readout of large arrays of
superconducting resonators. Designed for flexibility, the system’s low digital
latency and ability to independently and dynamically set the frequency and
amplitude of probe tones in real time has enabled previously-inaccessible
views of resonator behaviour, and opened the door to novel resonator control
schemes. We introduce a multi-frequency imaging technique, developed with RF-
ICE, which allows simultaneous observation of the entire resonance bandwidth.
We demonstrate the use of this technique in the examination of the response of
superconducting resonators to variations in applied readout current and
thermal loading. We observe that, used in conjunction with a conventional
frequency sweep at sufficiently large amplitude to induce resonance
bifurcation, the multi-frequency imaging technique reveals a resonator
response which is not captured by the frequency sweep measurement alone. We
demonstrate that equivalent resonant frequency shifts can be achieved using
either thermal, optical, or readout loading, and use this equivalence to
counteract a change in thermal loading by digitally modulating the readout
current through a resonator. We develop and implement a proof-of-concept
closed-loop negative electro-quasiparticle feedback algorithm which first sets
and then maintains the resonant frequency of a lumped element kinetic
inductance detector while the loading on it is varied. Although this simple
implementation is not yet suitable to deploy at scale, it demonstrates the
utility of this feedback technique to improve linearity while addressing
amplifier distortion, resonator response non-uniformity, and crosstalk. It can
be applied to kinetic inductors in non-bolometric operation, and sets the
stage for future developments.

##  1 Introduction

As new generations of experiments look to answer new science goals and extend
performance, the need for increased sensitivity is important. With
established, state-of-the-art detection technologies such as transition-edge
sensors (TESs) surpassing the quantum limit of individual sensitivity, further
gains are to be made by increasing the number of detectors fielded per
instrument. With each detector providing an effectively independent parallel
measurement, the observation time required to reach a given sensitivity goal
is reduced the more detectors that are operated in parallel. However,
achieving these higher detector counts presents certain practical challenges.
This is especially true for TESs, whose bolometric nature requires thermal
isolation, placing stringent requirements on fabrication capabilities, and
whose operation via multiplexing requires additional and often extensive
hardware situated at cryogenic stages. [1][2][3]

Another superconducting detector technology, the kinetic inductance detector
(KID), is gaining popularity in large-scale deployments. While a TES’s
detection mechanism is an increase in resistance caused by the temperature
change associated with photon absorption, a KID is a superconducting resonator
whose charge carriers are disrupted by the absorption of energy (either
thermal or from incident photons), altering the material’s complex impedance
and thereby its resonant properties. In essence a resistor, a TES requires
additional cryogenic circuitry to enable the multiplexing of the signals from
many detectors along one readout line. In contrast, KID resonators can be
fabricated in parallel along a feedline on a single wafer, with frequency-
domain multiplexing naturally achieved via variation in the resonant
frequencies of each device. This dramatic simplification in both fabrication
and operation not only allows very high multiplexing factors, but also a high
degree of flexibility in their design. KIDs can be made to directly absorb sky
photons or to operate as bolometers sensing a temperature change via thermal
link with an external absorber, enabling a broad range of detectable
wavelengths. The addition of optical elements on the same cryogenic wafer is
also possible, such as a broadband optical coupler connected to an array of
KIDs in an on-chip spectrometer. [4][5][6]

##### Operational challenges

These advances in detector design and capability must be accompanied by
advances in the technology and techniques used to operate them. In a typical
frequency-multiplexing scheme, each detector requires at least one dedicated
readout channel, which synthesizes and demodulates a carrier sinusoid at or
near its resonant frequency (typically GHz for microwave KIDs). Modern
synthesis and signal processing hardware is capable of producing the necessary
carrier frequency arrays, but processing constraints still impose limits on
the instantaneous bandwidth that a system can achieve. This in turn drives the
design of frequency-multiplexed KID arrays to pack as many resonators as
possible into a given readout bandwidth, which can be difficult to reliably
achieve. Slight material or fabrication process variations can cause
unpredictability in the frequency placement of resonators, which, when many
resonators must be densely arranged on a single array, may lead to overlap or
collisions between detectors’ resonant features. This increases undesirable
crosstalk of signals between them and reduces the yield of operable detectors
per wafer.

Because their detection mechanism is primarily the change of their resonant
frequency in response to changes in absorbed loading, if the responsivities of
individual KIDs are not identical, even in the ideal case of identically-
varying loading, resonators may drift into each other’s influence range,
collide, or even cross over one another. This is especially problematic when
resonators on an array are deliberately differentially loaded as in a
spectrometer, where atmospheric molecular lines may deposit drastically larger
loading on some resonators than on others. A system which purely reads out
resonators, such as by using carrier frequencies which are either static or
which vary to track their target resonators, can only mitigate this problem,
using thorough characterization of the array responsivities and careful
bookkeeping of individual detector positions at all times. Correcting this
problem requires continuously controlling the frequency placement of all
resonators on the array as the loading on them varies, such as by varying the
readout power deposited on each resonator in order to keep its total loading
effectively constant.

##### Resonator feedback control

This control can be achieved passively for thermal kinetic inductance
detectors (TKIDs) [5], where placing each resonator on a thermally-isolated
island allows the dissipation of readout current in the resonator to heat the
island. This thermal loading varies as the resonator moves in frequency due to
absorbed optical loading, changing the impedance at the frequency of the
readout drive tone and modulating the current through it. Similar to the
electrothermal feedback which enables most astrophysical TES readout systems
[7][8], this technique has been demonstrated in TKIDs to improve the linearity
and response time of the detector. [9]

Further improvements can be achieved by invoking active, rather than passive,
resonator control schemes. Rapidly digitally altering the readout current that
is sent to a resonator keeps it locked in place in frequency as the absorbed
power on it varies. When this technique is applied to TKIDs, the gain of the
feedback loop can be made much larger than in a passive system, with
correspondingly larger improvements in linearity and response time. Such
improvements would be particularly useful for TKIDs, whose bolometric
operation allows them to observe sky frequencies below the superconducting gap
energy, but with an inherently long detector time constant.

However, the fabrication of thermally-isolated islands is complex and
reintroduces many of the challenges facing the development of large TES
arrays, so we therefore consider whether such a control scheme may be applied
to conventional KID arrays. Non-thermally isolated KIDs are easier to
fabricate, and therefore lend themselves more readily to deployment in larger
arrays. Due to their integration with the large thermal mass of the silicon
wafer, it is likely not practical to control their temperature with the
absorption of readout current. However, we introduce the idea and demonstrate
that it is possible to exert active resonator control on non-thermally-
isolated KIDs through the dissipation of readout current to energize the
quasiparticle distribution.

We provide a demonstration of this control technique using the RF-ICE readout
platform, and describe the hardware, firmware, software, and algorithmic
advancements that enabled it.

##  2 RF-ICE

RF-ICE is a readout platform designed for high-channel-count, frequency-
multiplexed operation of kinetic inductance detectors at GHz frequencies.
Developed on the established ICE general-purpose signal processing
motherboard, which it pairs with the commercially-available Analog Devices
AD9082-FMCA-EBZ mixed-signal front-end daughterboard [10], RF-ICE utilizes a
modular design to operate up to 1024 superconducting resonators per module,
with an arbitrary number of modules operated in parallel. The present
generation of this readout framework operates on the ICE-based hardware [11]
and will deploy on the SPT-SLIM on-chip spectrometer pathfinder experiment.
[12] The readout framework is also compatible with the forthcoming t0 Control
and Readout System (CRS) 111www.t0.technology RFSoC readout hardware, on which
it will deploy as the readout system for the fourth-generation South Pole
Telescope experiment, SPT-3G+. [13][14] A detailed overview of the ICE-based
RF-ICE hardware and firmware can be found in [15]. This paper focuses on the
work using the ICE-based hardware; more detail on its implementation using the
CRS RFSoC hardware can be found in [16] (these proceedings).

Each RF-ICE motherboard houses two readout modules, each comprising two
synthesizers, one demodulator, and an internal pathway for low-latency digital
feedback. Both the synthesizer and demodulator use polyphase filterbanks to
generate and digitize carrier sinusoids, with up to 1024 operating per carrier
comb. The instantaneous bandwidth spanned by each module is approximately 500
MHz, which can be placed anywhere within a 0 to 3 GHz analog bandwidth using a
numerically-controlled oscillator internal to each module.

In operation with superconducting resonators, RF-ICE uses a series of
attenuators on the cryogenic stages at the input to the resonator array, and a
series of amplifiers (cryogenic and room temperature) on the output line to
scale the amplitude of the returning carrier comb to maximize usage of the
dynamic range of the ADC. The input-line attenuation reduces the scale of
room-temperature Johnson-Nyquist noise to below that of the attenuator
immediately preceding the resonators on the coldest stage. The dominant source
of white noise from the readout is the LNA’s input noise. The DAC contributes
a 1/f spectrum which is dominant at low frequencies. The overall system noise
is dominated by the detector.

###  2.1 Resonator characterization

Developed alongside prototype detectors for the SPT-3G+ and SPT-SLIM
experiments, RF-ICE’s control interface provides software routines for many
typical aspects of resonator characterization, including resonance finding,
parameter fitting, on- and off-resonance noise measurement, and resonator bias
point optimization, as well as an extensive bookkeeping framework for mapping
physical resonators to digital readout channels for large arrays of
resonators.

Figure 1: Example output per-detector status digest from the RF-ICE array
initialization software algorithm. The algorithm performs localized frequency
sweep measurements, verifies and updates the bias amplitude and frequency,
captures noise timestreams at low and high sampling rates, and activates the
carrier comb in preparation for the start of an observation. With the
exception of the high sampling rate noise data (which can only be streamed at
2 MSPS from one channel at a time), all operations are performed in
simultaneously for all resonators in the array.  The gathered data is saved
and processed into products useful for both quick-look heads-up displays for
individual device debugging (such as above) and for longer-term tracking of
array and device performance across multiple observation sessions or time
periods.

RF-ICE uses each module’s 1024 independent readout channels to accomplish the
detector and array characterization tasks that would traditionally be
performed with a network and spectrum analyzer. It parallelizes operations
across multiple channels, improving efficiency relative to traditional network
and spectrum analyzer techniques. This streamlines the characterization of
large numbers of resonators in the lab, and allows the rapid initialization of
detector arrays at the beginning of an observation. To facilitate
commissioning of new arrays and instruments, the software routines produce a
wide range of data products and heads-up displays, organized at the level of
the individual resonator and of the array as a whole. An example resonator-
level display is shown in Figure 1, featuring a collection of data products
captured by the array initialization algorithm. The algorithm captures this
data for all resonators in the array simultaneously (excepting the high
sampling rate noise timestream which is taken one channel at a time), then
collates them into various representations and formats for preliminary
inspection and further analysis. Shown here for an example detector, these
displays provide readily-accessible summaries of system status for the user
and enable rapid debugging.

###  2.2 Multi-probe: RF-ICE multi-frequency imaging mode

To implement the measurement routines described in the previous section, RF-
ICE uses an operational mode which is akin to a system of network and spectrum
analyzers, operated in parallel. Each measurement in the routines is performed
by synthesizing and recording carriers, each at a given frequency. In this
‘traditional’ mode, measurements spanning a frequency range are made
sequentially in time as well as in frequency, as in a network analyzer.
Measurement sets are performed in parallel across multiple channels.

A multi-channel readout platform such as RF-ICE can re-imagine this paradigm
and measure multiple frequency points across a range simultaneously, rather
than sequentially. This allows the user to capture a snapshot of the system’s
behaviour at every frequency of interest at a given moment in time, rather
than seeing only the frequency of the probe as it sweeps across. We refer to
this technique as a multi-probe measurement. This simple change in operational
mode can reveal a surprising wealth of previously-inaccessible information
about a given system.

Figure 2 shows an example of a resonator imaged using a multi-probe
measurement, as a demonstration of the technique. A number of very low
amplitude ‘watcher’ channels are placed within a bandwidth of interest. The
positioning of the channels in frequency space is arbitrary, but they are
typically placed approximately evenly across the bandwidth. In order to not
perturb the system under test, the amplitudes of the watcher channels must be
kept very small, generally at least a factor of 100 smaller than a typical
readout probe tone. On all devices studied for this work, it was
experimentally verified that at the default small amplitude, the presence of
the watcher tones had no measurable effect on the resonance. Once the watcher
channel frequencies are set, a batch of samples is captured from all channels
in the module simultaneously through the standard RF-ICE data acquisition
pathway. Each channel’s data are averaged into a single complex voltage value,
which is saved to disk along with the channel frequency. Fig. 2 shows an array
of such values, plotted as the absolute value of the complex I,Q voltage pairs
versus the channel frequencies at which they were acquired. In this way, the
multi-probe measurement captures an arbitrary bandwidth at a single moment in
time.

Figure 2: Illustration of the multi-probe measurement technique. ‘Watcher’
channels at very low amplitude are spaced across some bandwidth of interest
(here, across a resonator). A single data capture is taken from all channels
simultaneously, and the measured voltages (orange points) are displayed versus
the watcher channel frequencies (indicated by dashed lines). This technique
allows the imaging of the entire resonator’s response at a given moment in
time, as opposed to the conventional frequency sweep technique, in which the
displayed data is sequential in both frequency and time.

This is of particular use for observing superconducting resonators at high
readout probe amplitudes, as it is well-documented that the absorbed power of
the probe affects the resonator’s behaviour. [17][18][19] In traditional
network analyzer mode, the user sees the system at the probe frequency only,
and thus the final measurement product convolves the response of the resonator
to the probe current with the motion of the probe itself across the resonance.
This leads to an incomplete picture of the resonator’s dynamics, with the
resonator exhibiting complicated nonlinear behaviour which is often
challenging to model analytically from within this framework.

This distinction is well-illustrated by performing a frequency sweep
measurement, while using additional channels to monitor the entire resonance
bandwidth. Figure 3(a) shows the traditional network analyzer view of a probe
sweeping across a resonator at sufficient amplitude to trigger typical
bifurcation behaviour, such as is characterized in detail in [17]. The
resonance exhibits a hysteric dependence on the direction of the probe sweep.
The multi-probe view (Figure 3(b)), obtained by placing a number of small-
amplitude watcher channels at frequencies spanning the resonance while the
single large-amplitude probe channel is swept across, provides a much greater
amount of information about the processes underway.

(a) Frequency sweep measurements of a resonator, in ascending (blue) and
descending (orange) frequency order, made using a traditional network
analyzer-like single-tone probe. The amplitude of the probe channel, whose
measured output voltage amplitude is plotted here versus its frequency in each
step of the measurement, is sufficiently large to trigger characteristic
bifurcation behaviour in the resonator. The frequency sweep was performed in
discrete steps, with the probe frequency held stationary at each frequency
step for 1-2 seconds before jumping to the next frequency. At each frequency
step, with the probe frequency constant, a multi-probe measurement (Figure
3(b)) of the same resonance was captured. A subset of these multi-probe
measurements is plotted in Fig. 3(b), as individual frames corresponding to
the indicated probe frequency.

(b) Multi-probe measurements of the same resonance as in Figure 3(a) made
during the same frequency sweep measurement. The panels are ordered by time,
with each corresponding to one of the probe frequency steps plotted in Fig
3(a). Upwards and downwards frequency sweep directions are indicated by the
same colours as in 3(a). The location of the probe channel is indicated in
each panel by a vertical dashed line. The probe itself is not shown in this
view, as it does not enter within the bandwidth of any of the watcher
channels. Measured voltages from the watcher channels trace the shape of the
resonance and reveal its response to the stimulus of the probe tone as it is
swept across the bandwidth. The discontinuity which is characteristic of
bifurcation in a traditional sequential frequency sweep measurement is here
seen as the sudden shape change in the upward sweep between panels 3 and 4, as
the resonance snaps from its typical inverted Lorentzian shape into a
distorted state with some watcher channels experiencing positive gain. On the
downward sweep (reading the panels in descending order), the onset of the
distortion is gradual: in panel 6, the resonance is displays a frequency shift
due to probe current flowing through it. As the probe moves further into the
resonance, the magnitude of the effect grows until panel 3 when the resonance
snaps back into its regular state in panel 2 (the discontinuity seen in the
downward sweep). As expected from the traditional frequency sweep
measurements, the jump between the two states occurs at a lower frequency on
the downward sweep than the upward, pushing the resonance further into the
distorted state than is accessible on the upward sweep. Note that the voltage
values reported by the probe channel do not trace the same shape reported by
the watcher channels – the probe channel does not observe the positive gain,
and instead measures a further decreased output signal amplitude where the
largest positive gain occurs in the watchers. In this state, the driven
resonator is parametrically amplifying the watcher channels. The exploitation
of parametric amplifiers is presently an active field of research, and so the
appearance of this behaviour in resonators intended for use as kinetic
inductance detectors may be of interest as a medium for further study.

Each panel of 3(b) corresponds to one measurement position of the probe tone
(indicated by a vertical dashed line and corresponding to one of the numbered
points in 3(a)). On the upward sweep, as the probe moves toward the resonance,
we see the resonance shift to slightly lower frequency as the readout current
through it increases. The eventual abrupt transition (occurring between panels
3 and 4 in the upward sweep and 3 and 2 in the downward sweep) corresponds to
the bifurcation discontinuity seen in the probe view, although unlike in the
conventional frequency sweep view, here the resonance is seen to snap into a
distorted or excited state, with the watcher channels receiving positive gain.
As the probe moves out of the resonance on the other side, the excited state
relaxes back to the usual resonance curve.

Notably, the shape traced by the probe channel does not match the shape
reported by the watcher channels. Indeed, the lowest amplitude sections of the
probe’s data correspond to the frequency steps at which the watcher channels
receive the highest gain. At these probe frequencies, the amplitude of the
probe current flowing through the resonator is sufficient for parametric
amplification of the watcher channels to become dominant, in a manner which is
qualitatively reminiscent of the 4-wave mixing described in [20]. The multi-
probe view has allowed us to directly observe and report on this remarkable
behaviour in these devices.

This multi-frequency view of resonator hysteretic switching behaviour suggests
that the probe channel’s amplitude is distributed amongst the nearby watcher
channels, increasing their amplitudes while decreasing that of the probe
itself. As the probe moves toward the resonance, the amount of current flowing
through it increases. This increase is compounded by the motion of the
resonance toward the probe frequency due to the nonlinear increase in kinetic
inductance. This positive feedback eventually causes the resonance to jump
into the excited state, which is seen as an apparent bifurcation in the
sequential frequency sweep measurement. [17] On the downward sweep, the probe
not only pushes the resonance away as it moves toward it, but also loses
amplitude to the distortion of the resonance and to the watcher channels. This
causes it to appear to trace out a deeper part of the resonance.

A deeper treatment of this phenomenology will be presented in a later work.
For the purposes of this work, we use this new multi-probe capability to
measure the detectors’ response to light, temperature, and readout power
(Section 3.1), and, in Section 3.2, develop a new control scheme via negative
electro-quasiparticle feedback for kinetic inductance detectors.

##  3 Resonator control via readout-quasiparticle excitation

###  3.1 Resonator circuit model under optical, thermal, and readout current
loading

The absorption of heat, light, and readout power impacts the number and
distribution of quasiparticles in a superconducting resonator. In steady
state, the density of quasiparticles is constant, so the rates of generation
from all sources are balanced by the rate of recombination:

$\frac{dn_{qp}}{dt}=0=\Gamma_{th}+\Gamma_{opt}+\Gamma_{read}-\Gamma_{r}$ (1)

Solving this for $n_{qp}$ then allows the estimation of the complex
conductivity $\sigma=\sigma_{1}-j\sigma_{2}$ of the superconductor [21]:

$\sigma_{1}=\sigma_{N}\frac{2\Delta_{0}}{hf}\frac{n_{qp}}{N_{0}\sqrt{2\pi
k_{B}T\Delta_{0}}}K_{0}(\xi)$ (2)

and

$\sigma_{2}=\sigma_{N}\frac{\pi\Delta_{0}}{hf}[1-\frac{n_{qp}}{2N_{0}\Delta_{0}}(1+\sqrt{\frac{2\Delta_{0}}{\pi
k_{B}T}}e^{-\xi}I_{0}(\xi))]$ (3)

where $\sigma_{N}$ is the normal-state conductivity for the sample, $I_{0}$
and $K_{0}$ are modified zeroeth-order Bessel functions of the first and
second kind respectively, $\xi=\frac{hf}{2k_{B}T}$, $\Delta_{0}$ is the zero-
temperature gap energy, $h$ is the Planck constant, $N_{0}$ is the single spin
density of states for aluminum, and $k_{B}$ is the Boltzmann constant.

From this with the geometry of the superconductor, we can compute the kinetic
inductance and dissipative impedance via the surface impedance [22][23]:

$Z_{s}=\sqrt{\frac{j2\pi f\mu_{0}}{\sigma}}(\mathrm{tanh}(t\sqrt{j2\pi
f\mu_{0}\sigma}))^{-1}$ (4)

where $t$ is the thickness of the superconductor, $\mu_{0}$ is the
permittivity of the material, $f$ is the frequency at which the impedance is
being evaluated, and $\sigma$ is the complex conductivity.

Putting all this together, we obtain values for the resistance and kinetic
indutance as

$R=\mathrm{Re}(Z_{s})\frac{l}{w}\quad\text{and}\quad
L_{k}=\frac{\mathrm{Im}(Z_{s})}{2\pi f}\frac{l}{w}$ (5)

(where $l$ and $w$ are the length and width of the inductor) based on the
expected density of quasiparticles in a resonator under a given set of loading
conditions.

The devices studied in this work are lumped-element KIDs, with an aluminum
inductor whose superconducting impedance we derive above, and a physical
planar capacitor (either interdigitated or parallel-plate) which is typically
niobium and which is assumed to not contribute to loading-dependent effects.
[24][25][26][27] The resonator is coupled to the feedline with a small
coupling capacitor. A circuit model for this type of resonator in its environs
within the readout system is shown in Figure 4. The resonator (or array of
resonators) is in parallel with the input impedance of the cryogenic LNA, and
forms the load on the last-stage cryogenic attenuator.

Figure 4: Circuit model of a single lumped element resonator and surrounding
readout electronics. On resonance, the impedance of the resonator is small but
non-zero (typically $\lesssim 10\Omega$), and purely real. Readout drive
current amplitude varies widely between resonator designs, and is difficult to
calculate precisely due to the many frequency-, geometry-, and temperature-
dependent impedances in the system. For the resonators studied in this work,
estimates of current amplitude through the resonator are on the order of 0.1-1
$\mu$A, giving an approximate dissipated power on-resonance of $\lesssim$ pW.
This is comparable to the expected sky loading, and suggests that modulating
the readout drive current can effectively compensate for a modulation in sky
loading.

A complete solution of the impedance of the system involves a frequency- and
geometry-dependent simulation for the specifics of each detector on the wafer
and the nearby electrical connections, in order to account for signal
reflections and propagation effects. Nonetheless, considering the lumped
elements in a standard circuit analysis provides useful intuition and a
reasonable degree of accuracy. In this framework, we model the resonator as a
inductance (comprised of the kinetic component computed as above and a
geometric component which does not change with loading) with small series
resistance (arising from the real component of the complex conductance), in
parallel with the lumped element capacitor:

$Z_{RLC}=\big{[}(j2\pi fL+R)^{-1}+(\frac{1}{j2\pi fC})^{-1}\big{]}^{-1}$ (6)

all of which is in series with the coupling capacitance $C_{c}$ such that the
total resonant branch has impedance

$Z_{res}=\frac{1}{j2\pi fC_{c}}+Z_{RLC}$ (7)

The transfer function for this system from the point labeled $V_{in}$ to the
output of the LNA is:

$V_{out}=V_{res}G_{LNA}=V_{in}\frac{R_{3}||(Z_{res}||Z_{LNA})}{(R_{3}||(Z_{res}||Z_{LNA}))+R_{2}}G_{LNA}$
(8)

To understand the impact of a change in loading (whether from optical,
thermal, or readout dissipation), we evaluate the change in quasiparticle
density, and propagate this through to a measurable change in the transfer
function. The resonant frequency is a function of all four lumped elements in
the resonator, but can be approximated as $2\pi f_{r}=(LC)^{-1/2}$, making it
easy to see that an increase in the kinetic inductance will result in a
decrease in resonant frequency.

The response of superconducting resonators to optical and thermal loading is
well-studied (see, for example, [28], [21], [29]) and will not be discussed
here in detail. The impact of readout power absorption on the quasiparticle
population has been studied in several works ([18][19][23], among others).
Readout photons generally have energies much lower than the superconducting
gap energy of aluminum, and therefore cannot individually directly break
Cooper pairs into quasiparticles. There is nonetheless substantive readout
current dissipation in a resonator, which affects the charge carrier
distribution.

The impact of readout power absorption on a resonator may be described using a
model of a driven, non-equilibrium distribution which may be approximated to
considerable accuracy as an equilibrium thermal distribution at elevated
effective temperature, $T_{eff}$. [19][30] In this framework, we may expect
that loading a resonator via readout power absorption will result in a
frequency shift analogous to that incurred by increased thermal loading.

This equivalence is demonstrated in Figure 5, which compares the response of a
resonator under a change in thermal and optical loading to its response after
altering the amplitude of a drive tone (at fixed frequency) to achieve the
same frequency shift. In the device shown, the increased readout current has
the additional effect of increasing the resonator’s quality factor, possibly
due saturation of two-level systems (TLS), as described in [31]. The ability
of a modulation in the readout current to counteract a frequency shift due to
loading change shows that readout probe power is viable for negative electro-
quasiparticle feedback, to lock a detector’s resonant frequency in place as
sky signal changes.

Figure 5: A resonator’s response, measured using the multi-probe technique,
showing equivalent frequency shifts when loading on it is varied by either a
temperature change (solid lines), an optical loading change (dotted lines), or
by varying the amplitude of a readout tone at fixed frequency (dashed lines).
The relative shift of the device’s resonator frequency is well matched between
the different loading sources. Unlike increased loading from thermal or
optical sources, an increased drive current generally results in an enhanced
resonance Q factor, possibly due to saturation of TLS [31], an effect which is
particularly prominent on this device.

In thermally-isolated KIDs operating as bolometers, the dissipation of readout
current can be used as passive electrothermal feedback (ETF) to linearize the
detector response.[9] In this architecture, which resembles ETF in transition-
edge sensor bolometers [8], the dissipation contributes to the total thermal
load on the KID, made possible by its isolation from the bulk of the
substrate. As the resonance moves in response to thermal load from absorbed
optical load, the changing impedance at the drive tone frequency (located
above the resonant frequency) modulates the amount of current flowing through
and readout power dissipated in the resonance. This elegant negative feedback
stabilizes the resonant frequency of the device, improving its linearity and
response time.

The devices studied in this work are direct-absorption KIDs and are not
thermally isolated from the wafer substrate. We therefore expect that the
temperature of the device does not meaningfully differ from its surroundings,
so we cannot employ the same ETF scheme described above. Instead, control of
the resonant frequency occurs through the excitation of the quasiparticle
distribution, rather than true thermal loading.

To illustrate this, consider a slightly modified rate equation for the total
population of quasiparticles in the system:

$\frac{dn_{qp}}{dt}=\Gamma_{T_{eff}}+\Gamma_{opt}-\Gamma_{r}$ (9)

where $T_{eff}$ is the effective temperature of the driven quasiparticle
distribution, as described in [19]. $T_{eff}$ is a function of the physical
temperature of the superconductor (the bath temperature $T_{b}$) and the
absorbed power per volume in the resonator, with higher powers leading to
higher effective temperatures, and $T_{eff}>T_{b}$. A change in the rate of
quasiparticle production from optical sources, $\Gamma_{opt}$, may therefore
be countered by the opposite change in $\Gamma_{T_{eff}}$, which may be
achieved via modulation of the readout current through the resonator.

In Figure 6, we demonstrate that this technique works by modulating the
amplitude of a readout drive tone at static frequency, using a software
algorithm that implements feedback, in order to counteract the effect of a
temperature change on a resonator. Other resonators in the array which are not
controlled in this way exhibit a shift to lower frequency as the temperature
of the wafer is increased, but the controlled resonator remains fixed in
place.

Figure 6: Proof-of-concept demonstration of stabilizing the resonant frequency
of a single resonator via readout-quasiparticle excitation during a cryogenic
stage temperature change, which we use as an easy-to-control proxy for
incident optical power variation. A fixed-frequency drive tone is placed 200
kHz above the resonant frequency of resonator 0013. The stage temperature is
increased, which causes a negative frequency shift in the other, undriven
resonances, but which is counteracted by decreasing the amplitude of the drive
tone on 0013. As a result, 0013 remains at the same resonant frequency, while
the other resonances drift to lower frequencies. The applied drive tone
amplitudes are recorded, and can be used to reconstruct the applied
temperature change. All four resonances are imaged here using simultaneous
multi-probe captures taken every few seconds throughout the temperature sweep.
Although under realistic operational conditions this control would be achieved
using a single channel per resonance to both monitor position and apply drive
control, for this simple demonstration, these multi-channel measurements of
0013 were used to digitally adjust the drive tone amplitude in feedback to
keep its resonant frequency static.

This demonstration serves as a proof-of-concept for an active quasiparticle
excitation control technique that can be used for KIDs, whether or not they
are thermally-isolated. However, the utility of the open-loop control
algorithm implemented in the demonstration is limited, as it is dominated by
networking delays and makes use of multiple readout channels per resonance in
order to monitor the resonance’s position and calculate the drive tone
amplitude. In a deployment context, this must be achieved at low latency and
using a single readout channel to both monitor and control the behaviour of
the target resonator. An RF-ICE-based early-stage implementation of such a
closed-loop active control system using quasiparticle excitation is described
in Section 3.2.

###  3.2 Active Resonator Control

The new active resonator control (ARC) system presented here uses a digital
feedback mechanism to modulate the applied readout drive current in response
to a changing signal on a detector. This fixes the device’s resonant frequency
in place under conditions where it would otherwise drift to a lower or higher
value as the loading on it is changed. It can be applied to TKIDs in
bolometric operation (true heating), or non-thermally-isolated resonators
(quasiparticle excitation).

A detailed discussion of feedback theory will be covered in a separate work.
Here we present a phenomenological overview of an early-stage implementation
of a closed-loop ARC system using quasiparticle excitation, its use to
stabilize resonators under temperature or incident optical loading changes,
and discuss motivating factors for the development of such a system.

####  3.2.1 Implementation

The control system will strive to keep the total density of quasiparticles in
the resonator roughly constant, and so the resonator must first be initialized
to an elevated density before beginning the feedback control loop. This
initialization involves the placement of a drive tone at a frequency at a
chosen offset below the un-biased resonant frequency of the target device. The
amplitude of the tone is increased, increasing the current through the
resonator and driving up its effective quasiparticle distribution temperature
until the resonant frequency is pulled in to reach the drive tone. The
optimization of the initial placement of the drive tone with respect to the
resonance, and its starting amplitude before ramping up, depend on the
properties of the resonator and the expected signal power changes. The
initialized amplitude sets the dynamic range of the feedback, and so must be
sufficient to compensate for the anticipated range of variations in signal
loading.

Once the resonator is initialized, the control loop begins. The final complex
output voltage from the initialization routine is saved and kept as the target
value, to which the feedback will lock the system. At each iteration of the
loop, the drive channel measures a voltage and computes an error term with
respect to the target voltage. This error term is passed into a proportional
controller, which determines the new drive amplitude. Because the measured
voltage is directly proportional to the drive amplitude (as can be seen in to
the transfer function for the circuit given in Section 3.1), either the target
value or the measured value must be scaled by the drive amplitude at each
iteration.

A proof of concept implementation of this technique is shown in Figure 7. In
this example, the controller is implemented in software on the readout control
computer, and so the latency and instability of the loop are large and
strongly dominated by communication over the network to the RF-ICE board.
Nonetheless, in the example shown, the controller is able to recover the
starting resonant frequency of the target resonance after a 20 mK stage
temperature change, while uncontrolled resonances in the array experienced an
average frequency shift of 4 kHz. Future implementations, under development
now, will move the control loop into low-latency FPGA firmware, so that it can
operate across the full detector bandwidth.

Figure 7: Proof-of-concept demonstration of closed-loop active resonator
control, using a single readout channel to stabilize the resonant frequency of
KID when the cryogenic stage temperature (again used as a simple proxy for
optical loading variation) is abruptly lowered 20 mK from its starting point.
In this implementation, the feedback controller resides in software on a
control PC, and so the control loop is severely limited by variable delays in
network communication to and from the RF-ICE readout hardware. The system is
nonetheless able to recover the starting resonant frequency of the target
resonator by modulating the drive amplitude sent into the system, and feeding
back on measurements of the output carrier voltage, $V_{meas}$, in comparison
with the target voltage, $V_{targ}$, established by the initialization
routine. Because in this implementation the control lags the response of the
resonator, this system is analogous to slow or intermittent tone-tracking,
albeit with the advantage that the carrier frequencies can be fixed to avoid
in-band IMD and maintain uniform spacing to avoid collisions and variations in
crosstalk. Future implementations will use a low-latency digital feedback
pathway onboard the readout hardware itself, reducing the latency of the
controller below the response time of the detector and allowing it to exert
much greater control over the resonance. In this case, the resonant frequency
no longer varies under changing loading, and the primary science signal is
instead in the feedback drive current.

####  3.2.2 Discussion

Generally speaking, the designers of a readout system operating KIDs must
choose where to place carrier tones with respect to the resonant frequencies
of the detectors and have a strategy for keeping them at the chosen point, as
deviations from the ideal bias frequency will result in reduced sensitivity of
the output voltage to changes in signal on the detector. They must also choose
the amplitude with which to bias the resonators, which scales the signal
amplitude relative to system noise sources, and devise a way to mitigate
dynamical effects within the array, such as increased overlap or collisions
between neighbouring resonances in response to differential loading.

##### Intermodulation distortion in a passive readout system

Every electronic system includes elements with nonlinearities. This leads to
intermodulation distortion (IMD) products that increase with the number and
amplitude of carriers that are multiplexed together in a detector module. IMD
occurs when signals at two or more frequencies pass through a nonlinear
element, such as an amplifier, and results in the re-distribution of signal
power to non-signal frequencies. Of particular concern are the odd-order IMD
products, which tend to fall near the signal frequencies. To maximize
sensitivity over system noise sources, kinetic inductance detectors typically
use large carrier amplitudes to bias the detectors. As the impact of IMD
scales with both the number of carriers in the system and with the amplitude
of these carriers, care must be taken when operating high-multiplexing-factor
KID arrays to avoid spectral contamination and/or reduced sensitivity.

In a frequency-multiplexed readout system, which necessarily involves the
summing of many frequencies along a single conductive path, care must be taken
to mitigate the impacts of IMD. IMD arises from the nonlinear mixing of
frequencies produced by a wide variety of sources in the system, such as the
assigned frequency of each carrier in the comb, the sample rate of the ADC,
the IceBoard’s upconverted clock frequency, the switching frequency of buck
converters on the IceBoard and the AD9082 mezzanine, and the frequency
precision of the direct digital synthesizer (DDS).

When operating as a passive readout system, RF-ICE adopts a mitigation
strategy for IMD products which places all third-order IMD products outside
the several-hundred-hertz-wide bandwidth occupied by the sky signal in every
readout channel during operation. This is accomplished by ensuring that all
frequencies in the system are integer multiples of a base frequency greater
than the sky signal bandwidth. The chosen base frequency for RF-ICE is
$f_{BASE}\simeq 476.8$ Hz. All other key frequencies listed above are integer-
divisible by this number, which is itself integer-divisible by the DDS
frequency precision, which sets the smallest achievable difference between two
frequencies for the system.

With this strategy,  IMD products occur at integer multiples of $f_{BASE}$ and
land either on top of the carriers themselves or outside the channel
bandwidth. Carrier frequencies may be regularly updated (in integer steps of
$f_{BASE}$) to track the position of a target resonance. Without this
frequency scheduling, the positions of the carriers are arbitrary in relation
to one another. When few carriers are active, the arbitary placement and base
frequency aligned cases are equivalent, as there are so few IMD spurs produced
that it is unlikely that one will fall within a given carrier bandwidth. At
higher active channel counts however, the arbitrary placement case introduces
a very large number of IMD products at small individual amplitudes, which
contribute an excess noise to the system. This excess can then be deliberately
whitened with an additional modulation step, trading narrowband spectral
contamination for a general increase in white noise level. [32]

In operation, loading changes result in shifts in the resonant frequency of
the detectors. The user has the choice of continuously following the resonance
by updating the carrier frequency as needed to maintain a constant bias point
with respect to the resonance (tone tracking), intermittently updating the
carrier locations to maintain a proximity to the resonant frequency (quasi-
static), or leaving the carriers in place (true static). In the static and
quasi-static cases, carrier frequencies can maintain base-frequency alignment
by quantizing any frequency adjustments made to integer multiples of
$f_{BASE}$. This alleviates the risk of in-band IMD and so is not negatively
impacted by the use of large carrier amplitudes, but is at the expense of
being continually slightly detuned from the resonance. When tone tracking, the
carriers may be maintained at the ideal tuning point, but IMD products may
appear at any frequency and are made larger by the use of larger carrier
amplitudes.

None of these strategies can prevent collisions (two resonances overlapping)
in the presence of differential loading changes. In a tone-tracking system,
differential loading is also accompanied by changes in carrier spacing,
resulting in dynamically changing crosstalk and IMD spurs which may be
difficult to account for in analysis. Choosing between these strategies
requires a detailed understanding of the science targets of an experiment and
the characteristics of the observations needed to achieve them. The motivation
is also clear for the exploration of new strategies which might allow both
continuous optimal tuning and large carrier amplitudes, without spectral
contamination from IMD.

##### Benefits of ARC

The ability to control the resonant frequencies of the devices in an array
offers advantages which are apparent even in this early stage of development.
In a densely-packed array, it can prevent collisions between nearby resonators
which either have different responsivities or which are receiving different
amounts of loading, by locking them to their assigned frequencies. It also
allows the user some freedom of choice in the selection of the precise
resonant frequencies for all the detectors in the array, as this is determined
by the application of the drive tones over a range of 10s of kHz (as we have
demonstrated here), and perhaps further. While it is not likely that this
technique can be used to disentangle resonances which are significantly
overlapping in their undriven state, this may still allow some relaxation of
fabrication constraints when designing dense arrays. Further, this allows the
readout system to maintain a base-frequency carrier schedule with the
resonances ‘pulled’ by the readout current to perfectly match the carrier
placements, as described above. For thermal KIDs, passive electrothermal
feedback can stabilize the location of the resonance in frequency space.
However, the technique requires that the drive tone be located above the
resonant frequency in order to achieve negative feedback, and so by definition
cannot maintain the ideal bias frequency.

In a system where noise from a source such as the cryogenic LNA is not
negligible on-resonance, increasing the amplitude of the carrier tones
increases the system’s sensitivity. The amplitude of carrier tones in an
active control system are typically larger than those achievable in a passive
system, as the resonators are being biased at or beyond the onset of
nonlinearity seen in traditional sequential frequency sweep measurements.
Given the additional headroom above such a noise source, a system employing
active control might choose to relax its requirements on LNA noise
performance, and/or choose a unit with increased dynamic range, which would in
turn reduce concerns about IMD.

Although the simple, software-based ARC demonstrated here is too slow to be
used in a practical observational context, it demonstrates the possibility of
using digitally-modulated readout current to control and improve the operation
of kinetic inductance detectors. In a fast, strong feedback regime such as is
widely employed on TESs in astronomical observations, the response time of the
detector is reduced as a function of the feedback loop gain, widening its
useful science bandwidth. [8] Further, the feedback linearizes the device
response, and extends its dynamic range. The development of such a fast active
control system, building on the principles demonstrated here and implemented
using the existing firmware modules employed for digital feedback in
transition edge sensors [33][34] is currently in progress, and will be
reported on in a later work.

##  4 Conclusion

The ambitions of future experiments to field ever-higher numbers of detectors
and in increasingly creative designs are motivating the growing adoption of
kinetic inductance detectors in astrophysics and cosmology. A promising but
still comparatively new detector technology to be used at scale, considerable
work is currently underway to address the challenges these KID deployments
face and improve their performance on the sky.

The development of new tools and techniques is an important part of this
process. With the introduction of the multi-probe imaging technique, we have
reported on the remarkable behaviour of these superconducting resonators in
their response to absorbed readout power. The picture this measurement
technique provides of the interaction of these resonators with the dissipation
of readout current is much more complete than what can be obtained using
conventional sequential frequency sweep measurements. It remains in good
agreement with existing models of inductive nonlinearity as a non-equilibrium
excitation of the quasiparticle distribution, and suggests parametric
conversion of the large probe tone into the amplitudes of secondary observer
tones as a novel explanation for the hysteretic switching behaviour which is
well-documented in these devices.

With this new technique and the insights it has provided us, we have
demonstrated the ability to exercise resonant-frequency control over kinetic
inductance detectors without the need for thermal isolation. Although still in
an early phase of development for these devices, this type of feedback control
was a key development in the large-scale use of transition-edge sensors for
millimeter-wavelength cosmology. Future implementations of the basic feedback
control algorithm demonstrated here will move the controller on-board the
readout hardware, where the ICE/CRS low-latency digital feedback pathway will
greatly expand its gain and bandwidth. This holds the promise of improved
response time, dynamic range, system linearity, and sensitivity for future
systems fielding this type of control architecture on large arrays of kinetic
inductance detectors.

##  5 Acknowledgements

The McGill authors acknowledge funding from the Natural Sciences and
Engineering Research Council of Canada, the Canadian Foundation for
Innovation, and Canadian Institute for Advanced Research. The detectors used
in this work made use of the Pritzker Nanofabrication Facility of the
Institute for Molecular Engineering at the University of Chicago, which
receives support from Soft and Hybrid Nanotechnology Experimental (SHyNE)
Resource (NSF ECCS-2025633), a node of the National Science Foundation’s
National Nanotechnology Coordinated Infrastructure.

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2017 Vol. X No. XX, 000–000

11institutetext:  Department of Astronomy, Beijing Normal University, Beijing
100875, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: Key Laboratory of Earth and Planetary Physics, Institute of
Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
33institutetext: College of Earth Sciences, University of Chinese Academy of
Sciences, Beijing, China
Received ——-; accepted ——

# Sunspots rotation and magnetic transients associated with flares in NOAA AR
11429 ∗ 00footnotetext: $*$ Supported by the National Natural Science
Foundation of China.

Jianchuan Zheng 11 Zhiliang Yang 11 Jianpeng Guo 11 2 2 Kaiming Guo 11 Hui
Huang 11 Xuan Song 11 Weixing Wan 22 3 3

###### Abstract

We analyze sunspots rotation and magnetic transients in NOAA AR 11429 during
two X-class (X5.4 and X1.3) flares using the data from the Helioseismic and
Magnetic Imager on board the _Solar Dynamics Observatory_. A large leading
sunspot with positive magnetic polarity rotated counterclockwise. As expected,
the rotation was significantly affected by the two flares. The magnetic
transients induced by the flares were clearly evident in the sunspots with
negative polarity. They were moving across the sunspots with speed of order
$3-7\ \rm km\ s^{-1}$. Furthermore, the trend of magnetic flux evolution of
these sunspots exhibited changes associated with the flares. These results may
shed light on the understanding of the evolution of sunspots.

###### keywords:

Sunspots; Sun: rotation; Sun: magnetic fields

##  1 Introduction

Sunspots are concentrations of strong magnetic field on the solar surface,
consisting of a dark umbra and a fibrous penumbra (Solanki 2003). It is
generally accepted that eruption events such as flares occurring in upper
atmosphere are associated with magnetic field generated from the base of
convection zone, emerging as bipolar magnetic regions at solar surface, then
extending to corona (Solanki et al. 2006). So the visible sunspots play an
important role in studying a variety of phenomena on the Sun.

The evolution of sunspots includes formation, movement, deformation and
disappearance. Rotation is an important feature during evolution. In general,
there are two kinds of rotation: one sunspot rotates around its umbral centre
and one sunspot rotates around the other (Yan & Qu 2012). The rotation of
sunspots and spot-groups have been studied for many decades (see, e.g.,
Evershed 1910; St. John 1913; Maltby 1964; Brown et al. 2003; Zhang et al.
2007; Yan et al. 2009; Min & Chae 2009). Yan et al. (2008b) found 182
significantly rotating sunspots among 2959 active regions. Some authors
believed the rotational motion of sunspots may be involved with energy build-
up and later release by flares (e.g., Stenflo 1969; Barnes & Sturrock 1972;
Hiremath et al. 2005; Zhang et al. 2008). Furthermore, the relationships
between the sunspot rotation and coronal consequence (Brown et al. 2003; Tian
et al. 2008), flare productivity (Yan et al. 2008a; Zhang et al. 2008; Ruan et
al. 2014), direction of the rotation (Zheng et al. 2016), and magnetic
helicities (Vemareddy et al. 2012) have been investigated. The association of
flares with abnormal rotation rates (Hiremath et al. 2005) has been found.
Specifically, Wang et al. (2014) reported two sunspots in AR 11158 rotating
along with a X2.2 flare. Liu et al. (2016) reported a non-uniform rotation
induced by a M6.5 flare in NOAA AR 12371, and Bi et al. (2016) found an abrupt
reverse rotation of sunspot in NOAA AR 12158 caused by back reaction of X1.6
flare.

Besides, the photospheric magnetic field changes during the flares. The
magnetic field is distorted enough to store energy powering flare. After a
flare taking place, the distorted field relax and restructure (Sakurai & Hiei
1996). There are two kinds of changes of magnetic field based on observations:
the first is rapid short-term changes (Patterson & Zirin 1981; Zhao et al.
2009), which is due to flare-induced spectral line changes (Patterson 1984;
Qiu & Gary 2003). It is generally believed that the rapid short-term changes
are not the real changes of magnetic field (see, Ding et al. 2002). However,
Harker & Pevtsov (2013) also analyzed the magnetic transients during the M7.9
flare in NOAA AR 11429 on 13 March 2012 and suggested that the magnetic
transients represented a real change in the photospheric magnetic filed.
Another is irreversible changes from pre-flare to post-flare (e.g., Spirock et
al. 2002; Wang et al. 2002; Wang 2006; Song & Zhang 2016). The irreversible
changes were first noticed by Wang et al. (1994). They showed that magnetic
shear increased after flares with vector magnetic field data. Cameron & Sammis
(1999) reported a change in the line-of-sight (LOS) field during the X9.3
flare on 24 May 1990 with the videomagnetograph data from the Big Bear Solar
Observatory (BBSO). Sudol & Harvey (2005) and Petrie & Sudol (2010) analyzed
the changes of the LOS magnetic field during some X- and M- class flares with
Global Oscillation Network Group (GONG) magnetograms. The median duration of
changes was about 15 minutes and the median absolute value of changes was
about 69 G. Wang & Liu (2010) demonstrated that the change of LOS magnetic
field is due to increase of horizontal magnetic field near the polarity
inversion line. With the seeing-free data of vector magnetograms, Liu et al.
(2012) found that the photospheric transverse magnetic field enhancement was
associated with the M6.6 flare in NOAA AR 11158. Wang et al. (2012) observed
the same phenomenon associated with the X2.2 flare in this active region.
These results suggested that the magnetic field near the polarity inversion
line could become more horizontal after a flare.

In this study, we analyze the evolution of sunspots in NOAA AR 11429. We will
explore the rotation of sunspots along with the flares, i.e., how do the
flares and the rotation of sunspots affect each other. The magnetic transients
induced by flares are studied in prior works. We further study the motion of
the magnetic transients to understand how the flares influence sunspots. The
paper is organized as follows. In Section 2 we describe the data used and the
entire structure of the active region. Section 3 shows data processing. We
summarize and discuss the results in Section 4.

##  2 Observational Data

The 45 s cadence and spatial sampling of $0^{\prime\prime}.5$ pixel-1 full-
disk continuum intensity images and line-of-sight magnetograms observed by the
Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board the _Solar
Dynamics Observatory_ (_SDO_) are used to analyze the variation of sunspots in
NOAA AR 11429. The images from the Atmospheric Imaging Assembly (AIA; Lemen et
al. 2012) on board _SDO_ are used to investigate the chromospheric and coronal
context: AIA 304 $\rm\AA$ formed in transition region and chromosphere, and
AIA 1600 $\rm\AA$ formed in upper photosphere and transition region.

AR 11429 appeared on the eastern limb on 3 March 2012, and then rapidly became
complicated reverse-polarity $\beta\gamma\delta$ active region. The evolution
and development of AR 11429 from 5 March to 12 March are displayed in Figure
1. It is interesting to note that the sunspots in this active region were
moving away from each other, i.e., the leading sunspots moving westward and
the following sunspots moving eastward. Several major eruptions were clearly
visible in this active region. Typically, three X-class flares occurred: one
(X1.1) on 5 March and two of interest (X5.4 and X1.3) on 7 March. The GOES
soft X-ray flux (Figure 3a) indicates that the X5.4 flare started at 00:02 UT,
peaked at 00:27 UT, and ended at 00:40 UT, and the X1.3 flare started at 01:05
UT, peaked at 01:14 UT, and ended at 01:23 UT. The evolution of five prominent
sunspots, two positive polarity sunspots (P1 and P2) and three negative
polarity sunspots (N1, N2 and N3), were associated with these two flares, as
be discussed in the next section.

##  3 Analysis and Results

###  3.1 Sunspots rotating associated with flares

To study the rotation, we take the rotation of individual sunspots as a
simplified solid-body rotation (i.e., a sunspot rotates around its umbral
center), and track the variation of the ellipses that best fit individual
sunspots on time-series intensity images (see Bi et al. 2016). Take sunspot P1
as an example. The best-fit ellipses on the intensity images at 23:29:23 UT of
6 March and 01:28:38 UT of 7 March are shown in Figure 2(a) and Figure 2(b),
respectively. Obviously, the major axis rotates counterclockwise about
$7^{\circ}$. In the present study, the changes of the angle between the major
axis and the horizontal direction of the best-fit ellipses are used to
describe the rotation of the sunspot. The results together with the GOES soft
X-ray flux are shown in Figure 3.

Figure 3(b) shows that P1 rotated mainly counterclockwise from $\sim$23:30 UT
of 6 March to $\sim$01:30 UT of 7 March. Before the start of the X5.4 flare,
the rotation was very fast with speed up to $7.92^{\circ}\ \rm hr^{-1}$. From
start to peak of the flare, the rotation speed decreased slowly to about
$4.89^{\circ}\ \rm hr^{-1}$ and the rotation trend kept counterclockwise. Just
after the peak of the flare, the rotation trend reversed from counterclockwise
to clockwise. Its speed was about $4.07^{\circ}\ \rm hr^{-1}$. After the
flare, the rotation direction returned to counterclockwise with the rotation
speed of about $6.56^{\circ}\ \rm hr^{-1}$, and the rotation speed increased
gradually until the beginning of the X1.3 flare. Later, the rotation trend
reversed to clockwise again with the rotation speed of about $6.44^{\circ}\
\rm hr^{-1}$ and slowed down. These results indicate that the rotation of
sunspot P1 was significantly affected by both flares. The relative mean
intensity profiles in different AIA wavelengths, obtained from the co-aligned
with HMI images, are also plotted in Figure 3. The intensity in the AIA
wavelengths of 304 $\rm\AA$ (green curve) and AIA 1600 $\rm\AA$ (blue curve)
started rising around 00:05 UT, peaked at about 00:07 UT, and then decreased
during the impulsive phase of the X5.4 flare. At about 01:05 UT, the
intensities started rising again and peaked at about 01:14 UT. It provides
further support for the view that solar flares are often related to the
rotation of sunspots.

The rotation of the sunspot N1 was complicated (Figure 3c). Before the peak of
the X5.4 flare, the rotation trend was mainly counterclockwise, and then
reversed to clockwise. The situation around the X1.3 flare was similar to that
around the former flare. The intensity of AIA wavelengths increased and the
rotation changed significantly during the impulsive phases of the X5.4 and
X1.3 flares. As shown in Figure 3(d), sunspot N2 rotated clockwise from
$\sim$00:15 UT to $\sim$00:40 UT. The relative mean intensity of 1600 $\rm\AA$
in this sunspot started rising at $\sim$00:05 UT and reached its maximum at
$\sim$00:20 UT. The relative mean intensity of 304 $\rm\AA$ started rising at
$\sim$00:05 UT and reached its maximum at $\sim$00:30 UT. So the X5.4 flare
affected the rotation of sunspot N2. The sunspot N3 remained stable during the
two flares (Figure 3e). The rotation speed was poorly correlated with the
intensity of AIA wavelengths, and therefore the X5.4 and X1.3 flares.

###  3.2 The motion of Magnetic Transient

Magnetic transients induced by flares have been well investigated by Patterson
(1984) and Qiu & Gary (2003). They analyzed the cause, location and level of
the magnetic transients. Here, we find that there are some magnetic transients
moving through sunspots N1, N2 and N3, and there is no apparent magnetic
transient motion in the sunspots P1 and P2. Because the polarity of the
sunspots N1, N2 and N3 is negative (black regions in the magnetograms), the
magnetic transients means that the polarity is reversed.

Figure 4 shows the changes of the magnetic field in sunspot N1. At 01:09 UT of
7 March, the magnetic transients marked by red arrows in the umbra of the
sunspot can be clearly seen. These magnetic transients (the positive magnetic
field) moved toward the southeast direction. At 01:14 UT, the area with
positive magnetic field became larger while the strength became weaker, as
shown in Figure 4(a2). The green line in Figure 4(a3) indicates the moving
trajectory of the positive polarity points. To investigate the kinematic
evolution of the reversed polarity, the time-distance plots obtained along the
green trajectory line are shown in Figure 4(d). A white streak is clearly
visible. It started at $\sim$01:07 UT and ended at $\sim$01:16 UT, consistent
with the time of the X1.3 flare. It might imply that the reversed magnetic
polarity was caused by this flare. The distance corresponding to the streak
was about 5 arcsec. Thus, the moving speed of the magnetic transients can be
estimated to be about $6.59\ \rm km\ s^{-1}$. Similarly, some magnetic
transients (i.e., the reversed magnetic field) were presented in sunspots N2
and N3, and probably associated with the X5.4 flare, as shown in Figures 5 and
6. They were moving toward southeast with speed about $3.69\ \rm km\ s^{-1}$
and $3.22\ \rm km\ s^{-1}$ respectively.

In order to check whether the magnetic transients were co-spatial with flare
ribbons, we further examined the co-aligned AIA 304 $\rm\AA$ and AIA 1600
$\rm\AA$ images corresponding to HMI magnetograms, which are displayed in
panels (b1)–(b3) and panels (c1)–c(3) in Figures 4, 5 and 6 respectively. The
flare ribbons can be seen in these panels. They were separating from each
other. The locations of sunspots N1, N2 and N3 in AIA images, marked by
magnetogram contours at -800 G, were moving with flare ribbons. This implies
the magnetic transients were co-spatial with flare ribbons.

We calculated magnetic flux of sunspots N1, N2 and N3 for the regions marked
by the red rectangular boxes in Figure 1. The magnetic flux of sunspots with
negative polarity together with GOES soft X-ray flux are shown in Figure 7.
Before the start of the X5.4 flare, the absolute value of magnetic flux in
sunspot N1 decreased slowly. From the start to peak of the flare, it decreased
gradually. Then, the value kept almost unchanged. At the end of the flare, it
increased slightly. During the period of the X1.3 flare, the absolute value of
magnetic flux decreased first and then increased abruptly. These results
suggest that the changes of trend of magnetic flux in sunspot N1 were caused
by both the X5.4 flare and the X1.3 flare.

The magnetic flux of sunspot N2 mainly increased from 23:30 UT of 6 March to
01:30 UT of 7 March. There was an obvious change near the peak of X5.4 flare.
Specifically, the absolute value decreased first and then increased, as shown
in Figure 7(c). The absolute magnetic flux value of sunspot N3 (Figure 7d)
almost kept unchanged before the start of the X5.4 flare. Similarly, during
the X5.4 flare , the magnetic flux decreased rapidly, and then increased
monotonically. These results suggest that the changes of trend of magnetic
flux in sunspots N2 and N3 were mainly due to the X5.4 flare.

##  4 Summary and Discussion

AR 11429 is a complicated active region with a reversed polarity structure, in
contrast with the nature of solar cycle 24. It has a large leading sunspot
with positive magnetic polarity (P1) exhibiting significant counterclockwise
rotation. The rotation underwent some changes during two X-class (X5.4 and
X1.3) flares. It is possible that the flares were triggered by the sunspot
rotation (see Zhang et al. 2008; Vemareddy et al. 2016), and in turn they
reacted back and affected the sunspot P1 rotation.

The magnetic transients induced by the two X-class flares can be seen in
sunspots with negative magnetic polarity (N1, N2 and N3). They moved outward
with respect to the center of the active region and moved through the sunspots
quickly with speed up to about $6.59\ \rm km\ s^{-1}$. The magnetic transients
lasted for about ten minutes within the period of the X5.4 and X1.3 flare. As
expected, the magnetic transients, which are clearly illustrated in sunspots
N1, N2 and N3 were moving away from the polarity inversion line shown in
Figures 4, 5 and 6. The magnetic transients occurred during the X5.4 and X1.3
flares were co-spatial with flare ribbons. This is consistent with Maurya et
al. (2012), who found that magnetic transients in AR 11158 during a X2.2 flare
persisted for a few minutes and showed co-spatial with flare ribbons, which
were separating out with a mean velocity of 8 $\rm km\ s^{-1}$.

During the evolution of sunspots (N1, N2 and N3), changes in the trend of
magnetic field evolution were presented, and are associated with the two
X-class flares. Similar results in five $\delta$ sunspots were found by Wang
(2006). The increasing and decreasing tendency of the magnetic flux in active
region might be attributed to magnetic flux emergence and magnetic
cancellation (van Driel-Gesztelyi & Green 2015).

The evolution of sunspots with complicated structure can be affected by many
factors such as interaction among sunspots within same active region, physical
change induced by flare, and interaction between magnetic field and plasma. In
this study, we have found that the motion and the trend of magnetic field
evolution of sunspots are significantly affected by flares. Other influences
on the evolution of sunspots will be examined in future studies.

###### Acknowledgements.

The authors thank the referees for their corrections and valuable suggestions
to improve the paper This work is supported by the National Natural Science
Foundations of China (41231068, 41374187, 41531073, and 41674147). We
acknowledge the _SDO_ /HMI and _SDO_ /AIA teams for providing the data. The
GOES soft X-ray flux data is available at NASA/GSFC Solar Data Analysis Center
(SDAC).

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Figure 1: The HMI intensity images and magnetograms show the evolution of AR
11429 during 5 - 12 March 2012. The field of view is $200^{\prime\prime}\times
200^{\prime\prime}$. The main sunspots are labeled with P/N*. The white/black
areas in magnetograms represent positive/negative magnetic polarity. The red
rectangular boxes are the selected regions for further study. Figure 2: HMI
intensity images before and after X5.4 class flare. The green dashed lines
represent the ellipse fit to the sunspots P1. The blue and yellow dashed lines
are the major axes of the ellipses. Figure 3: (a) Evolution of GOES soft X-ray
emission from 23:30 UT on 6 March to 01:30 UT on 7 March. The vertical dash
lines represent the GOES flare start, peak and end times. (b)–(e) Time
profiles of orientation angle (between the major axis and horizontal
direction) of sunspot P1, N1, N2 and N3, respectively. The green and blue
curves are profiles of relative mean intensity within each sunspot region with
respect to the quiet Sun in AIA 304 $\rm\AA$ and AIA 1600 $\rm\AA$
respectively. Figure 4: The reversed magnetic field moving through sunspot N1.
The red arrows in (a1) and (a2) indicate the location of the reversed magnetic
field and the green line in (a3) indicates the trajectory of the moving
reversed magnetic field. (b1)–(b3) The co-aligned AIA 304 $\rm\AA$ images
corresponding to HMI magnetograms. The blue contours represent the magnetic
field of sunspot N1 at -800 G. (c1)–(c3) The co-aligned AIA 1600 $\rm\AA$
images overlaid by magnetogram contours of sunspot N1. (d) Time-slice plot of
the trajectory. The blue dashed line shows the evolution of the reversed
magnetic field. The red curve overlaid on the time-slice plot is the GOES soft
X-ray flux. Figure 5: Same as Figure 4, but for sunspot N2. Figure 6: Same as
Figure 4, but for sunspot N3. Figure 7: Magnetic flux of sunspot N1, N2 and
N3. (a) Evolution of GOES soft X-ray emission from 23:30 UT on 6 March to
01:30 UT on 7 March. The vertical dash lines represent the GOES flare start,
peak and end times. (b)–(d) Time profiles of magnetic flux of sunspot N1, N2
and N3, respectively.