# Comparison of three different camera systems monitoring the meteor activity
over Hungary in 2020-2023

Lívia Deme, Krisztián Sárneczky, Antal Igaz, Balázs Csák, Nándor Opitz, Nóra
Egei, József Vinkó (HUN-REN CSFK Konkoly Observatory) HUN-REN CSFK Konkoly
Observatory, Konkoly Thege út 15-17, Budapest, 1121 Hungary
Email<EMAIL_ADDRESS>
###### Abstract

We present statistical analysis of visual meteor data taken with networks of
meteor cameras operating in Hungary between 2020 and 2023. We use three
different camera systems: a set of traditional MetRec-based video cameras, a
self-developed automated DSLR camera system and a network of newly installed
AllSky7 camera stations. Similarities and differences between the data
produced by the three systems, aimed at recording different types of meteor
phenomena, are presented and discussed.

Received 2023 December 19

##  1 Introduction

11footnotetext: HUN-REN Research Centre for Astronomy and Earth Sciences,
Konkoly Observatory, Konkoly Thege út 15-17, Budapest, 1121 Hungary
Email<EMAIL_ADDRESS>
Dust grains from cometary tails and/or ejected asteroid material, ranging from
several microns to roughly meter-sized bodies in diameter, are responsible for
most of the observed meteor events when entering Earth’s atmosphere (see e.g.
[Ye & Jenniskens, 2022] and references therein). Meteor populations are
usually divided into two basic categories: sporadic meteors having no
preferred direction, i.e. they appear at random positions on the sky all year
round, and meteor showers arriving from a specific direction, called the
radiant. Meteoroid orbits causing many of the currently known meteor showers
can be traced back to a parent body, usually a comet or an asteroid
([McIntosh, 1991, Jenniskens, 1994, Ye & Jenniskens, 2022]). Sporadic meteors
are thought to be originated from old cometary tails and/or interplanetary
dust trails that have been dissolved since their ejection, providing a more-
or-less stationary background of incoming sporadic meteor flux.

Since the visual meteor activity, i.e. the hourly or nightly rate of
observable meteors are usually increased by more than an order of magnitude
during the peak date of showers, meteor showers have been studied more
frequently in the literature. Still, there is an increasing number of papers
dealing with the statistical properties of sporadic meteors (see e.g. [Jones &
Brown, 1993], [Rendtel, 2006], [Wiegert et al., 2009], [Dubietis & Arlt, 2010]
and references therein).

Despite the lack of a specific radiant, several preferred directions of
sporadic meteors can be identified (e.g. [Campbell-Brown & Jones, 2006]): most
of the sporadic meteors arrive from the apex of Earth’s orbit, while the
helion and antihelion sources are located in the ecliptic roughly
perpendicular to the apex-antapex direction. Beside these sources, two
additional ones at $\sim 60$ degrees north and south of the ecliptic, named
northern and southern toroidal sources, were identified from radar
observations ([Jones & Brown, 1993]). The apex and the antihelion sources
provide the majority of the visually observable sporaric meteor population
([Rendtel, 2006]).

In this paper we use data from different meteor camera systems operating in
Hungary to reveal the visual meteor activity between 2020 and 2023. Here we
concentrate on the statistical properties of sporadic meteors and the
brightest events, mostly fireballs. The measured properties of recent meteor
showers will be presented in a subsequent paper. In the next section we
briefly summarize the technical parameters of the camera systems applied in
this study, followed by the description of the data taken by the different
cameras, the obtained results and our conclusions.

##  2 Meteor camera systems in Hungary

In Hungary, the deployment of video camera systems for automated detection of
visual meteors was started more than 20 years ago. Most of those cameras are
still operating. They consist of an analogue video camera (either PAL, or
NTSC) attached to a PC having a Matrox frame-grabber video card that samples
the analogue video stream, and takes rapid snapshots of digital frames ($\sim
25$ frames per second) when a moving object, presumably a meteor, appears in
the field-of-view. Meteor detection and measurement is handled by the ”Meteor
Recognizer” (MetRec, https://metrec.org) software. The sensitivity limit is
typically in between 4 and 5 magnitude, depending on the type and manufacturer
of the particular camera. These cameras are integrated within the camera
network of the International Meteor Organization (IMO). In this paper we use
three of them: HUKON, HUPIS and HUHOD. The basic parameters for these three
cameras are summarized in Table 1. MetRec uses a well-defined set of reference
stars for computing astrometry and calibrated photometry for each meteor
event.

More recently, starting in 2017, our team at Konkoly Observatory developed and
installed a new system, named Konkoly Meteor Observatory Network (KoMON),
which is based on a combination of a digital video camera with a more modern
DSLR digital camera. The moving object is recognized by the video camera uses
a custom Python-based software then it triggers the DSLR camera to take
digital frames for 10 seconds. An LCD panel, placed in front of the CMOS chip
of the DSLR camera, chops the incoming light by a pre-programmed de Bruijn
sequence ([Howie et al., 2017]) to separate the likely meteors from other,
slow-moving objects (airplanes, birds, treeleafs, insects, satellites, etc).
The KoMON system has been designed to be sensitive to the brightest,
relatively slow meteors, like fireballs. Such meteors might be missed by the
conventional video camera systems, because they might saturate the camera
causing a failure of the recognition algorithm. Thus, the KoMON system detects
only the brightest end of the meteor brightness distribution function (BDF),
and, by design, it likely detects the slower events instead of the fastest
ones. The system parameters are collected in Table 2.

At the time of writing this paper, proper astrometry and photometry is not yet
implemented for the KoMON system. Thus, the cameras in the KoMON network are
capable of recording only the moment (within $\sim$1 seconds) of the beginning
of a meteor event and its angular speed along the projected trajectory. An
integrated digital image (taken with 10 seconds exposure time) as well as
digital video frames are also stored for each triggered event. Due to the lack
of calibrated digital photometry, we were only able to use a visual brightness
estimate for the recorded meteors that appeared brighter than the stars on the
corresponding digital frame. This way we can still classify meteors that are
brighter than $\sim 0$ visual magnitude, beside the lack of more precise
photometric information.

Finally, starting in mid-2021, several stations of the AllSky7 camera system
(https://allsky7.net) have been deployed at various sites in Hungary. These
stations consist of an integrated unit of 7 cameras providing practically all-
sky coverage, calibrated astrometry and broad brightness sensitivity range.
The average limiting magnitude for each unit is $\sim 4$ mag. The parameters
for the stations that are at our group’s disposal are summarized in Table 3.

Name | Longitude | Latitude | Altitude | Center Az | Center Alt | Field-of-view
---|---|---|---|---|---|---
| (deg.) | (deg.) | (m) | (deg.) | (deg.) | (deg. x deg.)
HUKON | 18.963793 | 47.499622 | 490 | 317.7 | 40.9 | 80.7 x 56.4
HUPIS | 19.895140 | 47.917229 | 947 | 348.1 | 50.8 | 80.9 x 56.8
HUHOD | 20.312500 | 46.418499 | 79 | 328.7 | 50.8 | 80.3 x 55.8
Table 1: Parameters of the MetRec-based video cameras Site name | Longitude | Latitude | Altitude | Direction | Field-of-view
---|---|---|---|---|---
| (deg.) | (deg.) | (m) |  | (deg. x deg.)
Konkoly | 18.963793 | 47.499622 | 490 | W | 90 x 80
Piszkesteto | 19.895140 | 47.917229 | 947 | Z,N,E,S,W | 360 x 90
Gothard | 16.6031 | 47.2578 | 232 | Z,N,E,S | 270 x 90
Devavanya | 20.9356 | 47.0814 | 84 | Z,N,E,S,W | 360 x 90
Becsehely | 16.7918 | 46.4479 | 180 | Z,N,E | 180 x 90
Table 2: Camera parameters for the KoMON stations Site name | Camera name | Longitude | Latitude | Altitude | Field-of-view
---|---|---|---|---|---
|  | (deg.) | (deg.) | (m) |
Konkoly | AMS18 | 18.963793 | 47.499622 | 490 | All-sky
Becsehely | AMS71 | 16.7918 | 46.4479 | 180 | All-sky
Piszkesteto | AMS72 | 19.895140 | 47.917229 | 947 | All-sky
Fehergyarmat | AMS97 | 22.5178 | 47.9854 | 111 | All-sky
Hortobagy | AMS98 | 21.1451 | 47.5941 | 121 | All-sky
Table 3: Camera parameters for the AllSky7 stations

##  3 Data

The three different camera systems provided different types of data for us.
The most complete high-level dataset was obtained by the MetRec-based cameras,
after post-processing and cleaning the nightly data by visual inspection.
MetRec is able to classify each recorded meteor based on its projected trail
on the sky. If a meteor trail is close to a radiant of a known meteor shower,
and the meteor angular velocity matches to the known velocity of that meteor
shower, then the meteor is classified as a member of that particular shower.
The sporadic meteors belonging to the antihelion source are classified as
"ANT". Two other ecliptical showers, the Northern and Southern Taurids (NTA
and STA, respectively) are also thought to belong to the sporadic meteor
population. All other meteors that cannot be assigned to any meteor showers
are classified by MetRec as SPO. Many of those meteors are coming roughly from
the apex direction. We extracted and analyzed the following data from the
logfile of each night: date, time (both in UT), classification and magnitude
for each event. We also used the nightly statistics computed by the auxiliary
program CheckLog: the total number of meteors, the number of sporadic meteors
and the effective observing time for each recorded night. These numbers were
then summed up for each month to create a monthly statistics from 2019
December to 2023 October.

The KoMON system provided a more limited dataset. We used only the UT dates
and time for each event recorded by each camera since 2019 December up to 2023
October. Simultaneous detections were treated as a single event, using a
maximum time difference of $\pm 1$ second for multiple detections of the same
meteor. We also visually inspected all frames to identify potential fireballs
by selecting meteors that appeared brighter than any of the stellar objects on
each frame (see Section 2). Finally, we corrected the number of detected
meteors for the number of active cameras at each site, resulting in a number
of detections per a single DSLR camera FoV on each observed night.

The AllSky7 system uses a custom Python-based software for meteor detection,
astrometry and brightness estimates. Neither classification of meteor showers,
nor calibrated photometry is done at the individual sites, though. The meteor
shower classification is part of the server-based multi-station analysis of
the AllSky7 datasets. Since the primary goal of the present study is meteor
statistics, we extracted and used only the date and time of each event
recorded by the AllSky7 stations at each site.

##  4 Results

In this section we describe our results based on the data obtained by the
three camera systems.

###  4.1 The population index of sporadic meteors

Based on the magnitude information and classification for each meteors by
MetRec, we were able to determine the nightly variation of the population
index for the sporadic meteors, after summing up the number of events labeled
as SPO, ANT, NTA or STA.

The population index is determined from the observed cumulative BDF for a
sample of meteors:

$N(m)=N_{0}\cdot r^{m}~{}~{}~{}(m<m^{*}),$ (1)

where $m$ is the magnitude, $N(m)$ is the number of meteors brighter than $m$
magnitude, $N_{0}$ is a normalization constant, $m^{*}$ is the magnitude limit
for detection completeness and $r$ is the population index. From Equation (1)
the population index can be derived simply as $r=N(m+1)/N(m)$ for $m<m^{*}$.

We adopted $m^{*}$ as the magnitude of the peak of the meteor BDF histogram
using 1 magnitude-wide bins, and derived the population index using the
numbers for 2 magnitude bins below the peak. Figure 1 illustrates the
methodology: the left panel shows the histogram of the BDF of all sporadic
meteors from the HUKON camera, which turned out to be the most stable one from
photometric point of view. The right panel plots the cumulative BDF for the
same dataset. $m^{*}=1$ is estimated as the magnitude of the maximum of the
BDF in the left panel. The blue line is a least-squares fit of Equation (1) to
the points within the interval of $[m^{*}-2<m<m^{*}]$.

Figure 1: The method used for population index determination. Left panel: the
observed number of meteors ($n(m)$) as a function of magnitude. Right panel:
the cumulative BDF ($N(m)$) against magnitude. The blue line indicates the
fitted model from Eq. 1.

Figure 2 shows the variation of the nightly population indices for the three
MetRec cameras as a function of Julian dates. Ignoring the few outliers
(probably due to low number of recorded meteors and/or inferior weather
conditons), most of the indices are within the range of $2<r<4$. The black
horizontal line indicates the mean value of $<r>=2.85\pm 0.48$ for the
combined sample of the three cameras. This is in good agreement with result of
[Rendtel, 2006] who obtained $<r>=2.95\pm 0.15$ as a yearly average.

Note, however, that the population index determination suffers from known
issues. For example, [Molau, 2015] discussed several improvements which may
result in somewhat different population indices for the same dataset. He
obtained $<r>\approx 2.5$ as a mean value for sporadic meteors, which is very
similar to the results of [Jenniskens, 1994] (2.2 - 2.5) and [Vida et al.,
2020] ($<r>=2.55\pm 0.06$). More recently [Betzler, 2023] suggested a
different statistical function (the q-exponential function) to represent the
cumulative meteor BDF. He obtained $<r>=3.63\pm 0.01$ based on a more extended
magnitude range of $-5<m<0$. The differences between these reported population
indices illustrate that despite the improvement of the methodology, these
estimates may still suffer some sort of systematic errors. Nevertheless, in
the following we apply our results above, $<r>=2.85\pm 0.48$ for consistency.
Contrary to [Betzler, 2023], we did not detect significant variation in the
population index of sporadic meteors along a year (Figure 2).

Figure 2: The variation of the nightly population index of SPO meteors
recorded by HUKON during 2020-2023.

###  4.2 The hourly rate of sporadic meteors

Figure 3: Left: The hourly rates of SPO meteors as a function of J.D. (upper
panel) and solar longitude (lower panel). Right: Hourly meteor rates by HUKON
for all sporadic meteors, including SPO, ANT, NTA and STA (upper panel) and
meteor showers (lower panel). Major showers are labeled by their IMO codes.

The rates of sporadic meteors were estimated from the HUKON, HUPIS and HUHOD
data in the following way. We adopt the sporadic hourly rate
($\mathrm{HR_{spo}}$) definition from [Dubietis & Arlt, 2010] as

$\mathrm{HR_{spo}}~{}=~{}\frac{n_{spo}}{t_{eff}}\cdot F\cdot r^{\Delta m},$
(2)

where $n_{spo}$ is the number of sporadic meteors per night, $r$ is the
population index, $\Delta m$ is the magnitude difference between the camera
sensitivity limit and a reference value, $t_{eff}$ is the effective
observation time in hours and $F$ is the correction factor between the camera
field-of-view (FoV) and that of a human observer. For the latter we adopt a
conservative estimate that a human observer sees $\sim 1/3$ of the visible
hemisphere of the sky. From Equation 2 it is seen that we do not correct the
rate for the zenith distance of a fiducial ”radiant” of sporadic meteors,
unlike e.g. [Rendtel, 2006]. It is a common practice to define $\Delta
m=6.5-Lm$, where $Lm$ is the limiting magnitude (the sensitivity limit) for a
particular camera ([Jenniskens, 1994]).

We adopted the following nominal limiting magnitudes: $Lm=4.5$ (HUKON and
HUPIS) and $Lm=4.0$ (HUHOD). Since these limiting magnitudes may change during
the night due to variable weather conditions, we also attempted to use the
nightly averaged values from the ”MAG” files provided by MetRec. The limiting
magnitude estimates by MetRec have a 1 minute resolution, thus, they could be
used to model the variable weather conditions during each night. However, it
was found that while using a variable $\Delta m$ based on these data the
night-to-night scattering of the calculated $\mathrm{HR_{spo}}$ values
increases by a large amount, which is not physically expected for the true
rate of sporadic meteors. Therefore, we decided to apply a single $\Delta m$
value for each camera as listed above.

Figure 3 shows the results. In the upper left panel we plot
$\mathrm{HR_{spo}}$ for all three cameras against Julian days. The annual
oscillation, noted previously by many observers (e.g. [Campbell-Brown & Jones,
2006]; [Rendtel, 2006]) is evident from these data. [Dubietis & Arlt, 2010]
even found correlation between $\mathrm{HR_{spo}}$ and the solar activity
cycle. Our data do not cover $\sim 20$ years, which would be necessary to
confirm the existence of a $\sim 11$ year-long cycle, but the annual variation
is recovered consistently by all three cameras.

The lower left panel of Figure 3 displays the same data against solar
longitude. Again, the increase of $\mathrm{HR_{spo}}$ during the Fall season
(September - November) is visible each year. This is fully consistent with the
results of [Dubietis & Arlt, 2010] who also found a $\sim 50$ percent increase
of $\mathrm{HR_{spo}}$ in September with respect to the rate measured in
April. This annual variation of $\mathrm{HR_{spo}}$ seems to be consistent
with the seasonal visibility of the apex direction during the 2nd half of the
night.

[Rendtel, 2006] attributed the annual variation of $\mathrm{HR_{spo}}$ he
found in the IMO database (VMDB) to the polluting effect of misclassified
shower meteors (see Figure 6 in [Rendtel, 2006]). We also tested this effect,
but found that this is less significant in our data. In the upper right panel
of Figure 3 we plot the nightly $\mathrm{HR_{spo}}$ derived from the HUKON
data as a function of solar longitude. It is seen that the SPO meteors
(plotted with red symbols) dominate the majority of sporadic events, and the
increase of the sporadic activity within $\lambda=100$ \- 200 deg is mostly
due to SPO meteors. The antihelion source (plotted with blue symbols) provides
only a small fraction of the total sporadic rate, and the annual variation is
less visible in those data.

The lower right panel displays the inferred rates for meteor showers. These
are only approximate ZHRs as they are not corrected for radiant position to
keep consistency with the measurement of the sporadic rates, and they are used
only for comparison. Indeed, the appearance of several of the richest showers
can be easily identified, and they are labeled. Comparing the top right panel
with the bottom right figure, it is seen that the increase of the sporadic
rate is not generally due to the polluting effect of showers. For example, in
the late August - late October season ($\lambda=150$ \- 200) , when
$\mathrm{HR_{spo}}$ is the highest in our data, the activity from showers was
negligible, at least from 2020 to 2023. On the other hand, a pollution effect
can be identified during the Geminids in mid-December, but this is localized
to a single night during the shower maximum, around December 13. Thus, we
conclude that the increase of $\mathrm{HR_{spo}}$ during September - November
is real. It correlates well with the seasonal visibility of the apex direction
where most of the sporadic meteors are coming from, but it might also be
connected with some sort of enhancement in the distributions of meteoroid
parameters (mass, particle size, velocity, spatial density, etc) that
contribute to the appearance of visually identified meteors, observed locally
from the northern hemisphere ([Dubietis & Arlt, 2010]).

Regarding the median value of $\mathrm{HR_{spo}}$ and its standard deviation,
we measure $26.3\pm 6.8$, $32.1\pm 9.2$ and $26.9\pm 9.1$ for HUPIS, HUKON and
HUHOD, respectively. For the combined sample it is $28.4\pm 8.4$. These values
are generally higher than those in recent literature: they are a factor of
$\sim 3$ higher than the results of [Dubietis & Arlt, 2010]
($\mathrm{HR_{spo}}\sim 10\pm 2$ from data by visual observers), and still
somewhat higher than the estimate by [Rendtel, 2006] based on video meteor
observations ($\mathrm{HR_{spo}}\sim 22\pm 3$). Since $\mathrm{HR_{spo}}$ as
defined in Equation 2 depends critically on the assumed FoV of a visual
observer as well as the proper statistical accounting of the unobserved faint
meteors, the difference between our results and that of [Rendtel, 2006] is not
significant. Note that if we had corrected the camera FoV to the whole area of
the full sky hemishere ($2\pi$ steradian) instead of the FoV of a human
observer, then the calculated rates should have been scaled up to 3 times
higher values, resulting in $\sim 85$ sporadic meteors/hour.

###  4.3 The frequency of slow fireballs as potential impactors

Figure 4: The ratio of ”fireballs” (meteors brighter than 0 mag in this paper)
to the number of all meteors recorded by the KoMON system as a function of
J.D. (top panel) and solar longitude (bottom panel)

Fireballs are spectacular events representing the brightest end of the BDF of
visual meteors. Traditionally, meteors brighter than Venus ($m<-4$ mag) are
called fireballs. Most of them produce flares during or at the end of their
visible path, but the definition is related only to the visual brightness
(i.e. brighter than $-4$ magnitude) and not the occurrence of any flares.
Since fireballs are the most probable impactors among meteors, there is
significant interest in studying their frequency in any meteor stream.

The KoMON system, as mentioned in Section 2, was designed to capture mostly
the brightest and relatively slow meteors, in order to detect potential
impactors. Even though its limited capabilities, we were able to visually
identify the brightest events (brighter than $\sim 0$ magnitude) by visually
inspecting every recorded frame, starting from 2019 December up to 2023
October. The reason for extending the magnitude limit to 0 mag, instead of
$-4$ mag, was twofold. First, since there is no calibrated photometry
available in the KoMON system, only relative brightness estimates with respect
to the brightest visible stars could be made. Second, since the real fireballs
are rare, extending the magnitude limit from $-4$ mag down to 0 mag increases
the number of events in the sample, thus, it becomes more suitable for
statistical studies. In the following, when we refer to fireballs, we mean
meteors brighter than 0 magnitude.

Figure 4 plots the ratio of fireballs to all events recorded by the KoMON
cameras as a function of Julian dates (upper panel) and solar longitude (lower
panel). Meteor numbers belonging to the same month are summed, in order to get
better statistics for these relatively rare events. Thus, each symbol
represent the ratio of the monthly sum of fireballs to the sum of all recorded
meteors. As mentioned in Section 2, multiple detections have been removed,
thus, they do not introduce a bias in the number of fireballs. The mean
fireball ratio for the whole sample turned out to be about 30 - 50 percent
($0.37\pm 0.12$). The actual numbers go below or above this range in a few
months when very few meteors were detected by the KoMON cameras for a number
of reasons (weather, technical issues, etc).

Figure 5: The number of fireball events per month scaled up to all-sky as a
function of J.D. (upper panel) and solar longitude (lower panel). The Geminids
and Perseids are clearly identifiable as local maxima.

Figure 5 displays the number of fireballs per month, normalized to a single
camera and scaled up to all-sky FoV, again, as a function of J.D. (upper
panel) and solar longitude (lower panel). These numbers can be regarded as the
number of fireballs in each month for the whole sky above Hungary. Again, we
used the monthly sums to reduce the fluctuations due to low number statistics.
As expected, meteor showers are clearly identifiable in these plots. For
example, Geminids in 2019 December produced especially high number of
fireballs. Later Perseids at $\lambda\sim 145$ deg showed the highest number
of fireballs recorded by the KoMON system. The mean value is estimated as
$N_{fb}=2.94\pm 1.91$ fireballs per month.

Figure 6: The comparison of the KoMON fireball numbers normalized to a single
camera (black lines with dots) and the MetRec monthly meteor numbers corrected
for 0 magnitude (color curves). See text for explanation.

Finally, we compare the KoMON fireball numbers to the meteor numbers taken by
the MetRec-based cameras after statistically selecting only the brightest
meteors. Figure 6 shows the comparison for those cameras that are either at
the same site (Piszkesteto - HUPIS, Konkoly - HUKON) or relatively nearby
(Devavanya - HUHOD). Since the camera FoV-s in the two systems are similar
($81^{o}$x$56^{o}$ for a MetRec camera and $90^{o}$x$80^{o}$ for a single
KoMON camera), no correction for this factor was made. The KoMON data taken
with multiple units at the same site were normalized to a single camera. The
numbers from the MetRec-based cameras are corrected for the magnitude
difference between the camera limiting magnitude and 0 mag, thus, applying the
factor of $r^{\Delta m}$, where $r$ is the population index. We found that if
we set $r=3.0$ and $\Delta m=-5.0$, $-5.0$ and $-4.5$ for HUPIS, HUKON and
HUHOD, respectively, they give the best agreement between the MetRec and the
KoMON data. It is seen in Figure 6 that the two datasets are generally
consistent. Thus, the KoMON system is able to capture most of the meteors at
the brightest tail of the meteor BDF if they appear within the camera field-
of-view.

Figure 7: The total number of KoMON fireballs corrected for the $-4$ magnitude
limit (assuming a population index of $r=1.5$, see text) plotted with filled
red circles and the number of observed fireballs brighter than $-4$ magnitude
reported by visual observers from Hungary (green open triangles).

It is interesting how many true fireballs we can see if we use the original
$-4$ mag limit as a cutoff. Using the same population index of $r=3.0$, the
normalized KoMON fireball numbers must be corrected with $r^{-4}\approx
0.012$. Such a correction would predict an average of $\sim 0.036$ fireball
per month, i.e. a single fireball in every 2 years. This is clearly way below
the actual number of detected fireballs, either by our cameras or by visual
observers. Thus, the population index may be different from 3. This is
expected because most of the true fireballs are due to shower meteors that
have lower population index. Assuming $r=1.5$ one would get the true fireball
rate as $\sim 0.6$ per camera per month, which is a more reasonable estimate.

We have made an independent test in order to confirm this assumption by
comparing the predicted number of KoMON fireballs brighter than $-4$ magnitude
using the correction factor of $r^{-4}$ with $r=1.5$ with the number of
actually observed fireballs brighter than $-4$ magnitude over Hungary. For the
former we selected all fireballs detected by the KoMON system, while for the
latter data we used the Hungarian mirror of the AMS fireball report webpage
(https://mcse.imo.net) to collect the observed fireballs over Hungary as
reported by visual observers between 2020 and 2023. Multiple detections were
filtered out, so each fireball was counted only once. The results are plotted
together in Figure 7. It is seen that the two independent datasets are
consistent with each other. It confirms the hypothesis that the magnitude
distribution of bright fireballs can be modeled with the population index of
$r\sim 1.5$, which is significantly less than that of the sporadic meteors.
Also, based on the detections by the KoMON system one can predict the true
number of very bright fireballs reasonably well.

###  4.4 Comparison of MetRec-based cameras with AllSky7 units

Figure 8: Left panel: monthly meteor detections for the AllSky7 cameras. Right
panel: comparison of monthly meteor detections by AMS18 and HUKON, after
correction for field-of-view difference. Figure 9: Ratio of AMS and MetRec
meteor numbers.

Since the AllSky7 units provide all-sky coverage, limiting magnitude
comparable to that of the MetRec-based cameras and non-stop operation, they
are expected to become the next generation of autonomous meteor cameras when
the MetRec-based video cameras are retired. Thus, it is useful to compare the
performance of the two camera systems in order to test whether the AllSky7
cameras delivered datasets similar to MetRec, both in quantity and quality.

Because the Python-based software of the AllSky7 cameras are still under
continuous development (Hankey, personal communication), in this paper we use
only the total number of meteors recorded by the AllSky7 stations at our
disposal (Table 3) as a test parameter. The meteor numbers were collected from
the camera servers by summing up the number of ”reduced” meteor events on each
night, after visually inspecting all frames and removing the likely artificial
records (usually due to airplanes).

In the left panel of Figure 8 we plot the sum of the recorded meteors in each
month since the deployment of the individual stations as a function of Julian
date. It is seen at first glance that the summed meteor numbers from the
different stations are more-or-less consistent with each other, even though
the individual stations are distributed at $\sim 100$ \- 200 km from each
other, thus, they may experience different observing/weather conditions. On
the other hand, it is also visible that the monthly meteor numbers show a
declining trend for all cameras during the first $\sim 800$ days after
deployment. This decline of the recorded meteor numbers seems to be less
pronounced for AMS72 during the 1st year of its operation, but then it also
develops the same declining trend in its recorded meteors.

In the right panel of Figure 8 the data from AMS18 are compared with the
monthly sum of meteors captured by HUKON. These two cameras are both placed at
the same site, on the rooftop of the headquarters of Konkoly Observatory in
Budapest, Hungary, thus, weather-related differences are not expected. The
HUKON data were corrected for the smaller field-of-view of the camera, thus,
they represent the expected number of meteors as if HUKON were an all-sky
camera, like AMS18. It is seen that the mean level of the HUKON data is
relatively constant each year (except, of course, the peak amplitudes due to
the different activity levels of the richest meteor showers). On the contrary,
the meteor numbers from AMS18 show a notable decline with respect to the HUKON
data. Since these two cameras are at the same site, this illustrates that the
decline is not due to the general and continuous worsening of the observing
conditions (e.g. climate change) over Hungary. Instead, it is suspected that
this effect is caused by some kind of technical issues affecting the AllSky7
camera stations.

In Figure 9 we plot the ratio of the monthly meteor numbers from three
different AllSky7 station and the nearest MetRec-based cameras. The decline in
the AMS18/HUKON data is clearly visible, as above. The AMS72/HUPIS ratio seems
to be more stable, but the second part of the curve (the data taken in 2023)
shows wide oscillations preventing the observability of any long-term trend.
The AMS98/HUHOD ratio spans a shorter range than the other two, but it
suggests the same declining trend as the AMS18 data.

Thus, it is concluded that all AllSky7 stations (except maybe AMS72) managed
by our group show a general long-term decline in the recorded monthly meteor
numbers, while such a decline is not visible in the data of our MetRec-based
cameras. According to the developer of the AllSky7 system (Hankey, personal
communication), this seems to be a software-related issue due to multiple
reasons. One is related to the applied screening criterion: previously the
software was designed to select meteors whose tracks contain at least three
measured points, while later it was changed to keep only those that have at
least 5 points. Also, sometimes the screening AI unintentionally misses some
of the faint and short meteors. Since the AllSky7 system was specifically
designed to be triggered by meteors that can be simultaneously measured from
many sites, these changes resulted in better performance of the system as a
whole. However, our analysis above shows that with the present conditions the
data from the individual AllSky7 stations are less suitable for statistical
studies than those from the MetRec-based cameras.

##  5 Conclusion

In this paper we presented a statistical study of the data from three
different meteor camera system operating in Hungary from 2020 to 2023. Based
on the results detailed in the previous sections, we draw the following
conclusions.

  1. 1.

We measured the population index of all sporadic meteors detected by our
MetRec-based cameras, and obtained $<r>=2.85\pm 0.48$ as the mean value and
standard deviation. This is consistent with the results by [Jenniskens, 1994],
[Rendtel, 2006] and [Vida et al., 2020]. Unlike [Betzler, 2023], we did not
detect annual variation in the population index of sporadic meteors.

  2. 2.

The hourly rate of sporadic meteors was estimated based on the data provided
by MetRec. We got $\mathrm{HR_{spo}}=28.4\pm 8.4$ sporadic meteors/hour for
the combined sample from all three cameras (HUPIS, HUKON and HUHOD). This is
consistent with the results of [Rendtel, 2006] based on video meteor data, but
seems to be a factor of 3 higher than the estimates by [Dubietis & Arlt, 2010]
from visual observations.

  3. 3.

The frequency of ”fireballs” (defined as meteors brighter than 0 mag in this
paper for statistical reasons) was studied by the KoMON camera system. We
obtained a monthly rate of $N_{fb}\approx 2.94\pm 1.91$ events per month.
Comparing these data with those from the MetRec-based cameras after scaling
those down to 0 limiting magnitude we got reasonable, order-of-magnitude
agreement. Note that after applying the brightness correction to $-4$ mag
using a population index of $r\sim 1.5$, we found good agreement with the
numbers of true fireballs reported by visual observers from Hungary. This is
consistent with the expectations, because most fireballs likely belong to
meteor showers, whose population index is substantially lower than that of the
sporadic meteors.

  4. 4.

Comparing the monthly total meteor numbers taken by our AllSky7 stations with
similar data from MetRec (after scaling the latter to all-sky) we found an
unexpected decrease of the AllSky7 data with respect to those from MetRec. All
our AllSky7 stations show the same trend, except maybe AMS72. We tested that
it is not related to any long-term worsening of the observing conditions, and
it is not likely due to the varying contamination of non-meteor events in the
AllSky7 data. All the data suggest that it is probably a software-related
reason, which has been confirmed by the developer.

Acknowledgements: This work was supported by the project ”Cosmic Effects and
Risks” GINOP 2.3.2-15-2016-0003 by the Hungarian National Research Development
and Innovation Office, based on funding provided by the European Union.
Special thanks are due to Nagykanizsa Amateur Astronomers Association
(especially Zsolt Perkó and Attila Gazdag), and the Bárdos Lajos Primary
School, Fehérgyarmat (especially Zoltán Pásztor) for their kind contributions.

## References

  * Betzler, 2023 Betzler A. S. (2023).  “The Monthly Rates of TV Sporadic Meteors Between 2014 and 2021”.  Earth Moon and Planets, 127:1, 1.
  * Campbell-Brown & Jones, 2006 Campbell-Brown M. D. and Jones J. (2006).  “Annual variation of sporadic radar meteor rates”.  Monthly Notices of the Royal Astronomical Society, 367:2, 709–716.
  * Dubietis & Arlt, 2010 Dubietis A. and Arlt R. (2010).  “Periodic Variability of Visual Sporadic Meteor Rates”.  Earth Moon and Planets, 106:2-4, 105–111.
  * Howie et al., 2017 Howie R. M., Paxman J., Bland P. A., Towner M. C., Sansom E. K., and Devillepoix H. A. R. (2017).  “Submillisecond fireball timing using de Bruijn timecodes”.  Meteoritics and Planetary Science, 52:8, 1669–1682.
  * Jenniskens, 1994 Jenniskens P. (1994).  “Meteor stream activity I. The annual streams”.  Astronomy and Astrophysics, 287, 990–1013.
  * Jones & Brown, 1993 Jones J. and Brown P. (1993).  “Sporadic meteor radiant distributions - Orbital survey results”.  Monthly Notices of the Royal Astronomical Society, 265, 524\.
  * McIntosh, 1991 McIntosh B. A. (1991).  “Debris from Comets: the Evolution of Meteor Streams”.  In Newburn, R. L. J., Neugebauer M., and Rahe J., editors, IAU Colloq. 116: Comets in the post-Halley era, volume 167 of Astrophysics and Space Science Library, page 557.
  * Molau, 2015 Molau S. (2015).  “Population index reloaded”.  In International Meteor Conference Mistelbach, Austria, page 11.
  * Rendtel, 2006 Rendtel J. (2006).  “Visual Sporadic Meteor Rates”.  WGN, Journal of the International Meteor Organization, 34:3, 71–76.
  * Vida et al., 2020 Vida D., Campbell-Brown M., Brown P. G., Egal A., and Mazur M. J. (2020).  “A new method for measuring the meteor mass index: application to the 2018 Draconid meteor shower outburst”.  Astronomy and Astrophysics, 635, A153.
  * Wiegert et al., 2009 Wiegert P., Vaubaillon J., and Campbell-Brown M. (2009).  “A dynamical model of the sporadic meteoroid complex”.  Icarus, 201:1, 295–310.
  * Ye & Jenniskens, 2022 Ye Q. and Jenniskens P. (2022).  “Comets and meteor showers”.  arXiv e-prints, page arXiv:2209.10654.


††thanks: Equal contribution††thanks: Equal contribution

# Measurement-Induced State Transitions in a Superconducting Qubit: Within the
Rotating Wave Approximation

Mostafa Khezri  Alex Opremcak  Zijun Chen  Kevin C. Miao  Google Quantum AI,
Goleta, CA  Matt McEwen  Google Quantum AI, Goleta, CA  Department of Physics,
University of California, Santa Barbara, CA, USA  Andreas Bengtsson  Theodore
White  Ofer Naaman  Daniel Sank  Google Quantum AI, Goleta, CA  Alexander N.
Korotkov  Google Quantum AI, Goleta, CA  Department of Electrical and Computer
Engineering, University of California, Riverside, CA  Yu Chen  Vadim
Smelyanskiy  Google Quantum AI, Goleta, CA

###### Abstract

Superconducting qubits typically use a dispersive readout scheme, where a
resonator is coupled to a qubit such that its frequency is qubit-state
dependent. Measurement is performed by driving the resonator, where the
transmitted resonator field yields information about the resonator frequency
and thus the qubit state. Ideally, we could use arbitrarily strong resonator
drives to achieve a target signal-to-noise ratio in the shortest possible
time. However, experiments have shown that when the average resonator photon
number exceeds a certain threshold, the qubit is excited out of its
computational subspace in a process we refer to as a measurement-induced state
transition (MIST). These transitions degrade readout fidelity, and constitute
leakage which precludes further operation of the qubit in, for example, error
correction. Here we study these transitions experimentally with a transmon
qubit by measuring their dependence on qubit frequency, average resonator
photon number, and qubit state, in the regime where the resonator frequency is
_lower_ than the qubit frequency. We observe signatures of resonant
transitions between levels in the coupled qubit-resonator system that exhibit
noisy behavior when measured repeatedly in time. We provide a semi-classical
model of these transitions based on the rotating wave approximation and use it
to predict the onset of state transitions in our experiments. Our results
suggest the transmon is excited to levels near the top of its cosine potential
following a state transition, where the charge dispersion of higher transmon
levels explains the observed noisy behavior of state transitions. Moreover, we
show that occupation in these higher energy levels poses a major challenge for
fast qubit reset.

##  I Introduction

Superconducting qubits typically use the dispersive interaction with a linear
resonator for state measurement [1, 2, 3]. This interaction is modeled by the
Jaynes-Tavis-Cummings (JTC) Hamiltonian [4, 5] that can generally describe the
coupling between a multi-level nonlinear system (qubit) and a linear
resonator, where the frequency of the resonator depends on the state of the
nonlinear system. The resonator frequency is probed to infer the state of the
qubit. Dispersive readout has been the method of choice for most
superconducting qubit hardware due to its speed, accuracy, and quantum non-
demolition character [6, 7, 8, 9]. These properties are particularly important
for cyclic error correcting protocols that require the qubit state to remain
in a known computational state after measurement and reset are complete, and
where the cycle time needs to be much shorter than the characteristic
coherence time of the qubits [10, 11, 12, 13].

A straightforward approach for making dispersive readout faster is to increase
the strength of the resonator drive pulse, which increases the average
(resonator) photon number. However, multiple experiments have found that when
the average photon number exceeds a certain threshold, the qubit undergoes a
state transition that degrades readout fidelity and leaves the qubit in a
state outside of the computational subspace [14, 15, 16], thus limiting
measurement speed. Ref. [17] found that, when the resonator frequency is
_higher_ than the qubit frequency, state transitions are _not_ predicted by
JTC interactions within the rotating wave approximation (RWA). Ref. [18] then
showed that in this regime the transitions are mediated by non-RWA
interactions instead, and are caused by resonances within the overall
structure of the JTC energy ladder.

Dispersive readout can also be performed when the resonator frequency is
_lower_ than the qubit frequency. This regime is of practical significance
since a frequency-tunable qubit can be brought into resonance with the
resonator to remove leakage outside of the computational subspace and reset
the qubit to ground state [19]. Such leakage removal reset schemes are
typically easier to calibrate and also faster than other methods where the
qubit and resonator are detuned, e.g., in the case where resonator frequency
is higher than the qubit’s maximum frequency [20, 21, 22, 23]. Furthermore,
evidence shows that fast and accurate removal of leakage is a necessary
component for achieving performant error correcting codes [24].

In this work we study measurement-induced state transitions (MIST) using a
transmon qubit [25] when the resonator frequency is _lower_ than the qubit
frequency. We begin by characterizing the dependence of state transitions on
qubit frequency, average photon number, and qubit state. We then provide a
semi-classical model that predicts the onset of transitions versus qubit
frequency and photon number with no fitting parameters. Similar to Ref. [18],
we find that the state transitions are caused by energy resonances within the
JTC energy ladder, but in this regime they are mainly mediated by RWA
interaction terms and can occur at relatively small photon numbers $\sim 5$.
Additionally, we find that transmon offset charge changes these resonance
conditions, thus explaining the fluctuations in these transitions when
measured repeatedly in time. Finally, we show that when a qubit undergoes a
measurement-induced state transition, the leaked state is resistant to reset
and therefore poses complications for further operations of the system,
specifically cyclic error correcting codes.

##  II Experiment

Our circuit consists of a frequency-tunable transmon qubit that is
dispersively coupled to a Purcell-filtered resonator [26, 27, 15], with bare
resonator frequency $\omega_{r}/(2\pi)=4.750~{}\text{GHz}$, Purcell filter
frequency $\omega_{f}\approx\omega_{r}$, qubit-resonator coupling strength
$g/(2\pi)\approx 130~{}\text{MHz}$, and resonator decay rate
$\kappa=1/(22~{}\text{ns})$ as shown in Fig. 1(a). The qubit has a maximum
frequency $\omega_{q,\text{max}}/(2\pi)=6.340~{}\text{GHz}$, nonlinearity
$\eta\equiv\omega_{q}-\omega_{21}\approx 2\pi\times 195~{}\text{MHz}$, and is
operated at frequencies above the resonator in these experiments
($\Delta=\omega_{q}-\omega_{r}>0$), in contrast to Ref. [18] where $\Delta<0$.
The qubit and resonator are controlled using external voltage sources $V_{q}$
and $V_{r}$, respectively, which are responsible for single-qubit gates (XY),
qubit frequency tuning (Z), and resonator drive pulses. Readout signals
scattered off of the resonator-filter system are amplified, digitized, and
post-processed by the receiver, providing measurement of qubit states
$|0\rangle$ and $|1\rangle$. In what follows, we refer to any state outside of
the 01-subspace as an outlier.

Figure 1: Experimental setup and initial observations. (a) Circuit schematic
of the qubit-resonator system. (b) Pulse sequence used for characterizing
state transitions. The x- and y-axis are not drawn to scale; see text for
timing information. (c) Conditional outlier probability versus average photon
number with the qubit prepared in $|0\rangle$ (blue curves) or $|1\rangle$
(red curves) at two different qubit probe frequencies (solid vs dashed). (d)
With the qubit probe frequency fixed at 5.64 GHz, we repeat the experiment
from panel (c) in time. Each semi-transparent curve corresponds to a unique
realization of the experiment. The opaque curves correspond to the data from
panel (c) with the qubit probe frequency at 5.64 GHz. Error bars are smaller
than the marker size.

We characterize state transitions using the pulse sequence depicted in Fig.
1(b). At the beginning of each measurement repetition, the qubit is tuned to
its resonator frequency for 1 $\mu\text{s}$ to prepare the qubit in state
$|0\rangle$ (waveform not shown) [26]. Next, the qubit is moved to its idle
frequency and an $X$-gate ($I$-gate) is used to prepare the qubit in state
$|1\rangle$ ($|0\rangle$). The qubit is then tuned to its probe frequency, at
which point the resonator is driven using a probe tone that is resonant with
the corresponding dressed resonator frequency for a duration of
$1~{}\mu\text{s}$ ($\gg 1/\kappa$) to ensure that the system reaches a steady
state. See Appendix A for information on how the dressed resonance frequencies
were measured in these experiments. At the end of the resonator drive, the
system idles for 400 ns to allow the resonator photons to decay. Then, the
qubit is tuned back to its idle frequency, where the system idles for an
additional $10/\kappa=220~{}\text{ns}$ to ensure the resonator has returned to
vacuum. Finally, the qubit is measured to extract the conditional
probabilities of finding the qubit in $|0\rangle$, $|1\rangle$, or an outlier
state, given the initial state, i.e., $P(\text{final}|\text{initial})$. For
each combination of the initial qubit state, qubit probe frequency, and
resonator probe power, the experiment is repeated 5,000 times to estimate
conditional probabilities.

In Fig. 1(c), we show the conditional outlier probability versus the average
photon number with the qubit prepared in $|0\rangle$ (blue) or $|1\rangle$
(red). See Appendix B for information about conversion between resonator probe
power and average photon number. We show data with the qubit probe frequency
at 5.59 GHz (solid lines, $\circ$ markers) and 5.64 GHz (dashed lines,
$\times$ markers). In Fig. 1(d), we repeat the experiment from Fig. 1(c) in
time, with the qubit probe frequency fixed at 5.64 GHz. The semi-transparent
curves, which follow the same color, marker, and line style conventions
described in Fig. 1(c), correspond to three additional realizations of this
experiment, taken consecutively with one minute between each realization,
where each realization takes $\sim 1$ second to complete. The opaque curves
correspond to the data from Fig. 1(c) with the qubit probe frequency at 5.64
GHz. We observe several interesting trends: (i) the onset of state transitions
typically occurs at a lower average photon number for $|1\rangle$ than
$|0\rangle$, (ii) features in outlier probability are strongly dependent on
the qubit probe frequency, and (iii) features in outlier probability change
significantly on timescales $\sim 1$ minute.

##  III Model

We model our experiment using a semi-classical approach where the resonator is
described by a coherent state (classical field) that directly drives the
transmon via RWA interactions. The Hamiltonian of our model is ($\hbar=1$, see
Appendix D)

$\displaystyle H=$ $\displaystyle\sum_{k}(E_{k}-k\omega_{r})|k\rangle\langle
k|$ (1) $\displaystyle+$ $\displaystyle\frac{\alpha
e^{i(\omega_{r}-\omega_{d})t}}{|\alpha|}\sum_{k}\text{Re}\left(\sqrt{|\alpha|^{2}-k}\right)g_{k,k+1}|k+1\rangle\langle
k|$ $\displaystyle+$ H.c.

where $E_{k}$ ($|k\rangle$) are the transmon eigenenergies (eigenstates) and
$g_{k,k+1}\equiv g\langle k|\hat{n}|k+1\rangle/\langle 0|\hat{n}|1\rangle$ are
the normalized transmon charge matrix elements in its eigenbasis, where
$g=k_{\text{eff}}\sqrt{\omega_{q}\omega_{r}}/2$ is the qubit-resonator
coupling strength defined via coupling efficiency $k_{\text{eff}}$. We note
that $E_{k}$, $|k\rangle$, and $g_{k,k+1}$ depend on offset charge. Lastly,
$\alpha$ describes the coherent state of the resonator and evolves as

$\dot{\alpha}(t)=-i(\tilde{\omega}_{r}-\omega_{d})\alpha(t)-\frac{\kappa}{2}\,\alpha(t)-i\varepsilon,$
(2)

where $\kappa$ is the resonator’s energy decay rate, $\varepsilon$ and
$\omega_{d}$ denote the amplitude and frequency of the resonator drive,
respectively, and $\tilde{\omega}_{r}$ is the dressed resonator frequency
corresponding to the initial qubit state. Note that the average photon number
in the coherent state is $\bar{n}=|\alpha(t)|^{2}$ which fluctuates by
$\pm\sqrt{\bar{n}}$, and this uncertainty is always present in our model when
predicting the onset of readout transitions. The term
$\text{Re}\left(\sqrt{|\alpha|^{2}-k}\right)$ is used to turn off the
interaction when $\bar{n}<k$, which mimics the behavior of the full JTC model
at small photon numbers. See Appendix D for details of the model.

Figure 2: Energy structure of an RWA strip with $\omega_{q}>\omega_{r}$ for
$\bar{n}=|\alpha|^{2}=50$. Thick lines show the bare energies, thin lines show
the instantaneous eigenenergies, and the offset charge is scanned along the
length of each line in the range corresponding to one period $-0.5\leq
n_{g}\leq 0.5$. Note that eigenenergies at the bottom of the well have a
noticeable offset charge dispersion compared to their bare energies. Curved
arrows show the coupling between bare levels, and straight vertical arrows
show the energy difference between them. The panel shows resonance (i.e.,
avoided crossing) between $|0\rangle$ and $|9\rangle$ at multiple offset
charge values. Not all levels are annotated or shown. Here we use
$\Delta/2\pi=1$ GHz, $\eta/2\pi=200$ MHz, and $g/2\pi=120$ MHz.

We use our model to simulate the behavior of the qubit during readout as
follows. First, we solve Eq. (2) for the coherent state evolution starting
with $\alpha(t=0)=0$, where we use a resonant drive $\omega_{d}=\omega_{r}$ to
replicate the experimental procedure (see Appendix A). We then plug this
solution into Eq. (1) and numerically solve the Schrödinger equation for the
state of the transmon, starting from initial states $|0\rangle$ or
$|1\rangle$. Here, the resonator field plays the role of a classical drive
that ramps up the interactions between levels as photon number increases,
where solving the Schrödinger equation for $\sim 20$ transmon levels is fast.
Finally, we look at the populations of the instantaneous eigenstates of the
system; a state transition in our model is identified by loss of the initial
population, and that is what we will compare later with experiments. See
Appendix D for an example of such population loss.

Fig. 2 shows an example of the energy structure of the model in Eq. (1). Our
model effectively describes the evolution within an RWA strip [18], i.e.,
levels that are coupled via excitation preserving interaction terms of the JTC
ladder. Note that when $\omega_{r}<\omega_{q}$, the RWA strip bends over
itself at $k_{\text{bend}}\approx(\omega_{q}-\omega_{r})/\eta$ and levels in
the bottom of the transmon cosine potential can become on resonance with
levels near the edge of the cosine potential, i.e., the qubit may borrow a few
excitations from the resonator and jump upward several levels. The coupling
between these resonant levels is provided by a multi-step process involving
intervening virtual levels. This is the main mechanism that leads to state
transitions and transfers the qubit population from the computational subspace
to higher states.

Note that the large offset charge dispersion of transmon levels near the edge
of the cosine potential leads to a significant role of offset charge for the
resonance conditions studied here. Additionally, we observe that the offset
charge dispersion of these higher levels affects not only their own
eigenenergies but also the eigenenergies of levels near the _bottom_ of the
well. This is depicted in Fig. 2, where thick lines show the bare energies of
the system and thin lines show the instantaneous eigenenergies, and the offset
charge is scanned along the length of each line over a full period of
$n_{g}\in[-0.5,0.5]$ to show the dispersion of each level.

In the full JTC energy ladder, different RWA strips are detuned by integer
multiples of the resonator frequency and are coupled to each other via non-RWA
terms. Due to the energy structure of the system in this parameter regime, in
contrast to the case of Ref. [18], it is less likely for different RWA strips
to become on resonance with each other at small photon numbers. We have
checked that with the addition of non-RWA terms the eigenenergies and the
location of resonances change but not significantly. Therefore, in this work
we focus on dynamics within RWA and defer study of non-RWA effects to a future
work.

Our main goal with this model is computational speed and applicability: we
need to predict the occurrence of state transitions versus qubit frequency and
photon number to avoid them during readout, and to inform parameters for new
designs. As such, the model presented in this paper was used to optimize the
readout in Refs. [13, 28]. However, this is an approximate model that is no
longer applicable after a transition occurs, as the resonator is no longer
described by a coherent state. Ref. [29] performed fully quantum simulations
of the system with a very large Hilbert space, which can be used to study the
dynamics during and after a state transition occurs. They also provided a
semiclassical method for studying these systems based on evolution of the
resonator that is dressed by the qubit. Recently, Ref. [30] studied the same
problem using a directly driven transmon model, where they used the
alternative approach of Floquet solutions, with an emphasis on identifying
signatures of chaos in the transmon. They found results similar to our work
such as resonance between transmon levels due to the drive, increased offset
charge dispersion of lower levels in presence of the drive, and the important
role of offset charge in calculating system parameters.

##  IV Experiment vs model

Figure 3: Probability of remaining in the initial state for ground (top row)
and excited (bottom row) state as a function of qubit-resonator detuning and
resonator average photon number, as observed in experiment (left column) and
the model (right column). Red filled circles mark the photon numbers at which
state transitions occur in the model, with error bars $\pm\sqrt{\bar{n}}$, and
they are re-drawn on the experimental data for easier comparison. Solid red
lines show an exponential fit to the filled circles as
$n_{\text{fit}}=Ae^{B\Delta}$, and the dashed red lines show the boundary of
state transitions defined as $n_{\text{fit}}-\sqrt{n_{\text{fit}}}$. The model
uses transmon charging energy of $194$ MHz, $k_{\text{eff}}=0.048$,
$1/\kappa=22$ ns, $\omega_{r}/2\pi=4.750$ GHz, $\varepsilon/2\pi=45$ MHz,
$\omega_{d}=\omega_{r}$, and the coherent state is evolved for 100 ns under
the drive. In the model we truncate the transmon at 20 eigenstates, tune
transmon junction energy to sweep the qubit frequency, and take a uniform
average over the offset charge in the range $n_{g}\in[-0.5,0]$. In the
experimental data for $\text{P}(1|1)$, the vertical features extending down to
the lowest photon numbers in the 1.2-1.6 GHz range are due to resonant
interactions with two level system defects.

We capture the trends in state transitions by repeating our experiment over a
range of qubit frequencies for initial states $|0\rangle$ and $|1\rangle$. The
result is presented in the left column of Fig. 3, where the heatmap shows the
probability of remaining in the prepared state. To compare with the
experimental data, we simulate the model over the same range of qubit
frequencies and initial states, using experimentally measured parameters of
the qubit-resonator system. To capture the effect of the offset charge, the
model heatmaps are uniformly averaged over simulations for $n_{g}\in[-0.5,0]$
(the behavior is symmetric around $n_{g}=0$) in steps of $0.05$. The model
heatmaps are shown in the right column of Fig. 3. Note that the experiment is
sampling the offset charge in a manner that depends on its noisy behavior [31,
32], and therefore repeating the same experiment in time can yield different
results from run to run, in agreement with the observations in Fig. 1(d).

Comparing the model with the experimental data in Fig. 3 shows good
qualitative and quantitative agreement both in detuning and photon number at
which state transitions occur, where features group together in frequency near
bands of resonance. Roughly speaking, each resonance band corresponds to a
detuning range which bends the RWA strip at a specific level. This is because
the eigenenergies within an RWA strip depend strongly on the level at which
the strip bends over itself, which is determined by the qubit-resonator
detuning. As such, it is evident that the multi-level energy structure of the
qubit circuit plays an important role in resonance conditions, which depends
on the details of the transmon circuit model [33], estimates of the circuit
parameters such as charging energy, and also dressing by other nearby elements
(e.g., couplers). Note that some of the experimentally observed transitions in
between the resonance bands and at higher photon numbers, which are absent in
the model, can originate from non-RWA interactions ignored in this model;
however, as evident from Fig. 3, they are not the main effect that limits the
allowed photon number for the readout in this regime. We also compared the
model with experiments performed on different qubit-resonator systems that
were fabricated with coupling efficiencies in the range 0.025–0.04 and
anharmonicities in the range 190–280 MHz, and we observed similar agreement.

As we show in Sec. V, measurement-induced transition to higher states is
mostly immune to typical reset protocols [19] and leaves the qubit excited for
a long time, which is detrimental to cyclic operations of quantum circuits.
Therefore it is of practical interest to predict the photon number at which
transitions set in. To do that, in our simulations we identify the photon
number for the onset of state transitions where the initial probability drops
below a given threshold, and only keep the points that monotonically increase
with detuning. These points are marked with red filled circles in Fig. 3 with
error bars indicating $\pm\sqrt{\bar{n}}$, and are re-drawn on the
experimental data for easier comparison. By fitting a phenomenological
exponential model $n_{\text{fit}}=Ae^{B\Delta}$ to these points, (solid red
lines in Fig. 3), we define the boundary of state transitions as
$n_{\text{fit}}-\sqrt{n_{\text{fit}}}$ (dashed red lines in Fig. 3) and avoid
operating beyond this boundary. The boundary depends on the qubit-resonator
detuning $\Delta$, coupling between them $g$, and the qubit anharmonicity
$\eta$, as these are the main parameters that shape the eigenenergies of an
RWA strip. Note that for our parameters, the photon number at which
transitions set in is $\sim 3\times$ lower for excited state than for ground
state, and that sets the photon limit for dispersive measurement of unknown
states.

##  V Qubit reset following a state transition

Our theory suggests that state transitions may leave the transmon in energy
levels near the top of the cosine potential at the end of the measurement
process. These higher transmon energy levels are difficult to reset as: (i)
their transition frequencies are extremely sensitive to offset charge (with
charge dispersions $\sim$100 MHz), (ii) their transition frequencies can be
significantly lower than $\omega_{q}$ and $\omega_{r}$, and (iii) the number
of excitations needed to be removed from the qubit to achieve reset is large
($\sim$ 5-10). These considerations pose major complications for microwave-
and resonator-assisted qubit reset schemes [26, 20, 21, 22].

Figure 4: Qubit reset following a state transition. (a) Pulse sequence used
for characterizing qubit reset error following a measurement-induced state
transition. The x- and y-axis are not drawn to scale; see text for timing
information. (b) Reset error versus qubit probe frequency and average photon
number. The left (right) column of images corresponds to the prepared state
$|0\rangle$ ($|1\rangle$), with a time before reset $\tau$ = 0.1 $\mu$s (top),
0.5 $\mu$s (middle), and 1 $\mu$s (bottom). The dashed red line in each image
is the state transition boundary corresponding to the prepared qubit state.

We characterize our ability to reset the qubit following a state transition
using the multi-level reset (MLR) gate developed in Ref. [19]. In the absence
of a resonator probe pulse, we measure a reset error, defined as the
probability of not finding the qubit in $|0\rangle$, of $<0.5\%$ when the
qubit is initially prepared in state $|0\rangle$, $|1\rangle$, or $|2\rangle$.
To characterize reset error following a state transition, we modify our
original pulse sequence [Fig. 1(b)] by placing an idle gate, with duration
$\tau$, followed by an MLR gate before measurement of the conditional
probabilities [see Fig. 4(a)]. In Fig. 4(b), we show the measured reset error
(color scale) versus the average photon number and qubit probe frequency, at
idle gate durations of $\tau$ = 0.1 $\mu$s (top row), $\tau$ = 0.5 $\mu$s
(middle row), and $\tau$ = 1 $\mu$s (bottom row). The left (right) column
corresponds to the prepared state $|0\rangle$ ($|1\rangle$). We detect
significant levels of reset error ($\gtrsim 4\%$) above the state transitions
boundary (dashed red lines) for $\tau=0.1,~{}0.5,~{}1~{}\mu\text{s}$. For
comparison of time scales, the average qubit relaxation time over this
frequency range is $T_{1\rightarrow 0}\approx 20\,\mu s$, and relaxation time
for higher transmon levels is roughly $T_{k\rightarrow k-1}\sim
T_{1\rightarrow 0}/k$ [34]. This demonstrates that state transitions produce
long-lived leakage states that are immune to the MLR gate, indicating
significant qubit population above state $|2\rangle$. Such leakage states are
detrimental to error correcting protocols [24] and were intentionally avoided
in Ref. [13] by choosing readout parameters that fall below state transitions
boundary.

##  VI Conclusion

In conclusion, we have characterized measurement-induced state transitions in
a transmon qubit when the readout resonator frequency is lower than the qubit
frequency. With $\omega_{r}<\omega_{q}$, we find that state transitions are
caused by resonances in the JTC energy ladder that are mediated by RWA
interactions: the transmon’s negative anharmonicity enables the qubit to
borrow a few excitations from the resonator and transition to a higher state
in an excitation preserving manner. These findings are unlike the ones in Ref.
[18], which studied a transmon-resonator system with similar parameters, but
with $\omega_{q}<\omega_{r}$, and found that non-RWA interactions within the
JTC Hamiltonian were needed to explain state transitions.

We provide a semi-classical model that predicts the onset of state transitions
versus qubit frequency, average photon number, and qubit state with no fitting
parameters. We also observe that due to large offset charge dispersion of the
higher transmon levels, the role of offset charge becomes important in
explaining experimental observations in two ways: (i) it explains why
experimental features vary in time, and (ii) averaging simulations over offset
charge is required to properly compare them with experiments. Our results show
that the onset of state transitions is exponentially suppressed with increased
detuning, though the resonator dispersive shift becomes quadratically smaller.
We also establish that the excited state is more susceptible to state
transitions versus average photon number, and that sets the photon limit for
dispersive measurement. We also note that following a state transition event,
fast reset protocols fail to reset the qubit, and that can be detrimental to
error correcting algorithms.

###### Acknowledgements.

We are grateful to the Google Quantum AI team for building, operating, and
maintaining software and hardware infrastructure used in this work.

##  Appendix A Dressed resonance tracking

To keep the range of average photon number, $\bar{n}$, constant as the qubit
probe frequency is varied in our experiments, we must account for changes in
the dressed resonance frequencies, $\omega_{r,|0\rangle}$ and
$\omega_{r,|1\rangle}$. We characterize this with resonator spectroscopy
versus qubit frequency as shown in Fig. 5(a). In this experiment, the qubit is
prepared in $|0\rangle$ or $|1\rangle$, and then rapidly tuned (with risetime
$\sim$ 1-2 ns) to the frequency of interest. Next, a variable frequency
resonator spectroscopy pulse is applied to the resonator-filter system (see
Fig. 1(a) for circuit information). Note that this spectroscopy pulse uses a
small power to prevent the resonator from exhibiting nonlinear behavior due to
its coupling to the qubit. The scattered signal is then heterodyne-detected by
the readout receiver, allowing for extraction of the amplitude and phase
($\phi$) versus the spectroscopy frequency ($f$). On resonance, the rate of
change $\phi$ with respect to $f$, denoted by $\Delta\phi/\Delta f$, is
maximized. The color scale in Fig. 5(a) is $|\Delta\phi/\Delta f|$, and is
evaluated along each fixed vertical line cut (i.e., at a fixed qubit
frequency). To extract models for the dressed resonator frequencies, we fit
the locations of maxima in $|\Delta\phi/\Delta f|$ versus qubit frequency to
the dispersive model for a transmon-resonator system [33]:

$\displaystyle\omega_{r,|0\rangle}$ $\displaystyle=$
$\displaystyle\omega_{r}-\frac{g^{2}}{\Delta}$ (3)
$\displaystyle\omega_{r,|1\rangle}$ $\displaystyle=$
$\displaystyle\omega_{r,|0\rangle}-2\chi,$ (4)

where

$\displaystyle\chi=\frac{g^{2}}{\Delta}\times\bigg{(}\frac{\eta}{\Delta-\eta}\bigg{)}\times\bigg{(}\frac{\omega_{r}}{\omega_{q}}\bigg{)}.$
(5)

The blue (red) solid line in Fig. 5(a) is a fit to Eq. (3) for
$\omega_{r,|0\rangle}$ (Eq. (4) for $\omega_{r,|1\rangle}$). Finally, to
maintain a resonant probe tone versus qubit probe frequency during our
experiments, we drive the resonator at the extracted value for
$\omega_{r,|0\rangle}$ ($\omega_{r,|1\rangle}$) when the qubit is prepared in
$|0\rangle$ ($|1\rangle$).

Figure 5: Resonator frequency tracking and photon number estimates. (a)
Resonator spectroscopy versus qubit frequency with the qubit prepared in
$|0\rangle$ (left panel) and $|1\rangle$ (right panel). The blue (red) solid
line is a model of the dressed resonator frequency $\omega_{r,|0\rangle}$
($\omega_{r,|1\rangle}$) as a function of qubit frequency. (b) Control and
measurement sequence used for $\bar{n}$ calibration. The x- and y-axis are not
drawn to scale; see text for timing information. (c) AC Stark spectroscopy
versus resonator probe power. The $\times$-shaped markers are the extracted
qubit (center) frequencies versus resonator probe power. The data are well
described by a linear model (see Eq. (6)).

##  Appendix B Estimating average photon number

To compare our experimental results with the model, we need to present our
data versus average photon number, $\bar{n}$, rather than resonator probe
power, $P_{\text{probe}}$. To calibrate the conversion between
$P_{\text{probe}}$ and $\bar{n}$, we use the ac Stark effect [35, 36]. The
waveforms for this experiment are depicted in Fig. 5(b). At the beginning of
each experiment, the qubit is prepared in $|0\rangle$ (waveform not shown).
Then, the qubit is biased to the frequency of interest, at which point the
resonator is driven at frequency $\omega_{r,|0\rangle}$ (see previous section
for further detail). After the steady state is reached (at
$t=500~{}\text{ns}\gg 1/\kappa$), a 100 ns long, variable-frequency, qubit
spectroscopy pulse is applied to determine the ac-Stark-shifted qubit
frequency, $\omega_{q}(\bar{n})$. At the end of the spectroscopy pulse, the
resonator idles for $10/\kappa$ to allow the resonator return to vacuum. This
is followed by a terminal measurement of the qubit $|1\rangle$ state
population, $P(1|0)$. For sufficiently low $\bar{n}$, the qubit center
frequency changes linearly with photon number as

$\omega_{q}(\bar{n})=\omega_{q}(\bar{n}=0)-2\chi\bar{n},$ (6)

where $\omega_{q}(\bar{n}=0)$ is the qubit frequency in absence of an applied
drive and $\chi=(\omega_{r,|0\rangle}-\omega_{r,|1\rangle})/2>0$ . In Fig.
5(c), we show the results from this experiment versus resonator probe power.
The $\times$-shaped markers are the extracted qubit (center) frequencies at
each resonator probe power, where $P(1|0)$ is the color scale. The extracted
qubit frequencies are linear in the power (see solid white line), thus
providing us with a conversion between $P_{\text{probe}}$ and $\bar{n}$ by
dividing the measured qubit frequency shift by $2\chi$. To avoid repeating
this experiment at each qubit probe frequency, we assume the following are
true: (i) $P_{\text{probe}}$ is constant (relative to power reading of the
instruments) over the resonator frequency ranges used, and (ii) the resonator
decay rate $\kappa$ is independent of the qubit frequency and state. Under
these assumptions, the $\bar{n}$ calibration can be readily extended to the
qubit $|1\rangle$ state by driving the resonator at frequency
$\omega_{r,|1\rangle}$.

Figure 6: Characterizing state transitions versus qubit-resonator detuning and
average photon number. (a) In-phase (I) and quadrature (Q) component
distributions with the qubit initially prepared in state $|0\rangle$,
$|1\rangle$, and $|2\rangle$. (b) IQ shots following the state transitions
experiment described in Fig. 1 with the qubit initially prepared in
$|0\rangle$. (c) Same as panel (b), but with the qubit initially prepared in
$|1\rangle$. The IQ distributions in panels (b) & (c) were taken at a qubit
probe frequency of 5.59 GHz and an average photon number $\bar{n}\approx 35$
photons. (d) Conditional probabilities versus qubit-resonator detuning and
average photon number. Vertical dashed lines correspond to probed frequencies
of Fig.1(c)-(d). In the panels corresponding to $\text{P}(0|1)$ and
$\text{P}(1|1)$, the vertical features extending down to the lowest photon
numbers in the 1.2-1.6 GHz range are due to resonant interactions with two
level system defects. See text for further detail.

##  Appendix C Characterizing state transitions

Characterizing measurement-induced state transitions relies on our ability to
detect qubit outlier states. In Fig. 6(a), we show the in-phase (I) and
quadrature (Q) component distributions with the qubit prepared in states
$|0\rangle$, $|1\rangle$, and $|2\rangle$. The black circular lines
surrounding the IQ distributions for the $|0\rangle$ and $|1\rangle$ states
correspond to three-sigma deviations from the IQ centers and are used to
define the outlier boundaries – any shots landing outside of these regions are
defined as outliers. In Fig. 6(b), we show an example of the IQ distributions
with the qubit prepared in $|0\rangle$ following the state transitions
experiment described in Fig. 1. The IQ distributions correspond to a qubit
probe frequency of 5.59 GHz and an average photon number $\bar{n}\approx 35$
photons. We display the same data in Fig. 6(c), but with the qubit initially
prepared in $|1\rangle$.

In Fig. 6(d), we show conditional probabilities versus qubit-resonator
detuning and average photon number. The left (right) column of panels
corresponds to the initial state $|0\rangle$ ($|1\rangle$). The color scale of
each panel represents the conditional probability of finding the qubit in
particular final state, where the top, middle, and bottom rows correspond to
final states $|0\rangle$, $|1\rangle$, and outlier, respectively. The dashed
vertical lines correspond to the qubit probe frequencies used in Fig. 1(c).
The white circles overlaid on each panel indicate the qubit-resonator detuning
and average photon number used to produce the conditional IQ distributions
displayed in Fig. 6(b)-(c). Note that the data of Fig. 6(d) and Fig. 3 were
taken on the same device, but at different times. This highlights
reproducibility of our experiment and also shows the possible differences due
to change of offset charge and two level system defects.

##  Appendix D Theory of direct drive of transmon

In this section we discuss the derivation of the direct drive model used in
the main text. To model our system we start from the full Hamiltonian in the
lab frame ($\hbar=1$), ignoring the Purcell filter

$\displaystyle H=$ $\displaystyle
4E_{C}(\hat{n}-n_{g})^{2}-E_{J}\cos(\hat{\varphi})$
$\displaystyle+\omega_{r}\hat{a}^{\dagger}\hat{a}+\varepsilon
e^{-i\omega_{d}t}\hat{a}^{\dagger}+\varepsilon^{*}e^{i\omega_{d}t}\hat{a}$
$\displaystyle+ig^{\prime}\hat{n}(\hat{a}-\hat{a}^{\dagger}).$ (7)

The terms in the first line of Eq. (7) denote the transmon Hamiltonian where
$E_{C}$ is the capacitive energy, $E_{J}$ is the junction energy, and $n_{g}$
is the offset or background charge [25]. The second line includes the terms
for the resonator at frequency $\omega_{r}$ and its drive, where $\varepsilon$
and $\omega_{d}$ are the resonator drive amplitude and frequency respectively,
and $\hat{a}$ denotes the resonator annihilation operator. The third line
shows the charge-charge interaction between the transmon and the resonator,
where the coupling strength that is usually measured in the lab is
$g=k_{\text{eff}}\sqrt{\omega_{q}\omega_{r}}/2$, redefined here via the
coupling efficiency $k_{\text{eff}}$, and relates to the interaction strength
in this representation as $g^{\prime}\approx(32E_{C}/E_{J})^{1/4}g$.

Next, we rewrite the Hamiltonian in the joint basis of transmon eigenbasis and
resonator Fock basis, i.e., $|k,n\rangle$, where $|k\rangle$ is transmon
eigenstate and $|n\rangle$ is the resonator Fock state. We also use a rotating
frame that moves the transmon to the frame of the resonator and moves the
resonator to the frame of its drive, i.e., $|k,n\rangle\rightarrow
e^{ik\omega_{r}t}e^{in\omega_{d}t}|k,n\rangle$. For the charge-charge
interactions, we only keep the excitation preserving terms, i.e., we employ
RWA and ignore fast rotating terms, yielding

$\displaystyle H_{\text{r.f.}}^{\text{RWA}}=$
$\displaystyle\sum_{k,n}(E_{k}-k\omega_{r})|k,n\rangle\langle k,n|$
$\displaystyle+(\omega_{r}-\omega_{d})\hat{a}^{\dagger}\hat{a}+\varepsilon\hat{a}^{\dagger}+\varepsilon^{*}\hat{a}$
$\displaystyle+\hat{a}e^{i(\omega_{r}-\omega_{d})t}\sum_{k,n}g_{k,k+1}|k+1,n\rangle\langle
k,n|+\text{H.c.},$ (8)

where $E_{k}$ and $|k\rangle$ are the transmon eigenenergies and eigenstates
respectively, both of them depend on offset charge, and $g_{k,k+1}\equiv
g\langle k|\hat{n}|k+1\rangle/\langle 0|\hat{n}|1\rangle$ are the normalized
transmon charge matrix elements in its eigenbasis. Note that here
$a=\sum_{k,n}\sqrt{n}|k,n-1\rangle\langle k,n|$.

We now utilize the dressed coherent state picture [37, 38, 17] where it was
shown that the qubit and its coupled driven resonator approximately form a
coherent state made of their joint eigenstates. In this picture, the resonator
is dressed and evolves according to evolution equation for a coherent state,
and the qubit is in an eigenladder that is formed due to its interaction with
the resonator. Therefore, we use an approximation that replaces the resonator
operators with their classical coherent state counterpart as
$\hat{a}\rightarrow\alpha(t)$. This yields a decoupled evolution equation for
the coherent state of the resonator, where the resonator field directly drives
and dresses the transmon:

$\displaystyle H_{\text{eff}}=$
$\displaystyle\sum_{k}(E_{k}-k\omega_{r})|k\rangle\langle k|$ (9)
$\displaystyle+\alpha(t)e^{i(\omega_{r}-\omega_{d})t}\sum_{k}g_{k,k+1}|k+1\rangle\langle
k|+\text{H.c.},$ $\displaystyle\dot{\alpha}(t)=$
$\displaystyle-i(\tilde{\omega}_{r}-\omega_{d})\alpha(t)-\frac{\kappa}{2}\,\alpha(t)-i\varepsilon.$
(10)

Eq. (10) describes the evolution of the coherent state of the resonator where
the average photon occupation is $\bar{n}=|\alpha(t)|^{2}$ and fluctuates by
$\pm\sqrt{\bar{n}}$. Note that here we implicitly assume the resonator is
dressed by the qubit, i.e., $\tilde{\omega}_{r}$ here refers to the _dressed_
resonator frequency and Eq. (10) describes the evolution of the dressed
resonator.

As the last step, in order to mimic the behavior of the full JTC ladder at
small photon numbers, we modify the interaction term in second line of Eq. (9)
as

$\frac{\alpha}{|\alpha|}e^{i(\omega_{r}-\omega_{d})t}\sum_{k}\text{Re}\left(\sqrt{|\alpha|^{2}-k}\right)g_{k,k+1}|k+1\rangle\langle
k|+\text{H.c.}\,.$ (11)

In the RWA JTC ladder, only levels that have a fixed number of total
excitations can couple to each other, i.e.,
$|0,\bar{n}\rangle\leftrightarrow|k,\bar{n}-k\rangle$ with total excitation of
$\bar{n}$ here. The above modified drive term prevents interaction between
states at the bottom of the JTC ladder where there is not enough total
excitations to couple all the levels, i.e., the interaction is turned off when
$\bar{n}<k$. We have checked that with this modification we can exactly
reproduce the spectrum of the full qubit-resonator JTC ladder within RWA,
specially at small photon numbers.

A reminder that the charge matrix elements $g_{k,k+1}$ inside the transmon
cosine well grow roughly as $\sqrt{k}$, and outside of the cosine well they
exponentially decrease. For simulations, this means if we choose a cut off
value for the number of transmon levels that includes a handful of levels
outside of the cosine well, we should be safe since the interaction with
levels higher than those is exponentially suppressed.

For simplicity and also to match the experimental conditions, in the
simulations we use $\omega_{d}=\omega_{r}$. Using a small detuning of a few
MHz between resonator and drive, as is the typical case in the dispersive
readout, does not change the overall outcome of this model and our results as
long as experimental average photon numbers are properly calibrated. Also note
that in simulations we use a square pulse for the resonator drive, i.e.,
$\varepsilon(t\geq 0)=\varepsilon$, and use a fixed drive amplitude
corresponding to a large steady state photon that sweep over all smaller
photon numbers as the resonator is populated, i.e., we do not use different
drive amplitudes corresponding to different steady state photon numbers.

Figure 7: Example of evolution for a directly driven transmon. (a) Spectrum of
the system as a function of average photon number in the coherent state
$\bar{n}=|\alpha(t)|^{2}$, also called “fan diagram”. Each color belongs to a
given transmon level, and not all simulated levels are drawn for brevity.
Dashed lines show the eigenenergies for $|0\rangle$ (lower dashed line) and
$|9\rangle$ (upper dashed line). (b) Avoided crossing between $|0\rangle$ and
$|9\rangle$ at $\bar{n}\approx 40$, with splitting of 25 MHz at the crossing.
(c) Population in the instantaneous eigenstate of the system as a function of
average photon number in the coherent state, where the system was initialized
in $|0\rangle$. A population exchange occurs at the avoided crossing between
$|0\rangle$ and $|9\rangle$, which marks a readout transition event. All
parameters are the same as Fig. 3 of the main paper, while the qubit-resonator
detuning and the offset charge are fixed at $\Delta/2\pi=1.1$ GHz and
$n_{g}=0.2$ respectively.

Let us now provide an example of the time evolution of the directly driven
transmon, which shows the energy resonances in the system and population
exchange between resonant levels. The procedure for evolving the system via
Schrödinger equation is outlined in the main text, and here we show an example
for a single qubit frequency and offset charge, reiterating that the behavior
will be different for other values of these parameters.

Fig. 7(a) shows the spectrum of Eq. (9) with the modification of (11) as a
function of average photon number in the coherent state $\bar{n}$ (time is
only a sweeping parameter here such that $\bar{n}=|\alpha(t)|^{2}$). This
spectrum of the qubit in the rotating frame of the resonator was named “fan
diagram” in Ref. [18]. Fig. 7(b) shows eigenenergies of $|0\rangle$ and
$|9\rangle$, where an avoided crossing occurs between them at around
$\bar{n}_{\text{cross}}=40$, with a splitting of $2g_{\text{eff}}/2\pi=25$ MHz
at the crossing. For these parameters, let us use perturbation theory to
estimate the effective coupling between these two levels that is mediated via
“virtual” levels in between them as

$g_{\text{eff}}\approx\frac{g_{0,1}g_{1,2}...g_{8,9}}{\Delta_{1,0}\Delta_{2,0}...\Delta_{8,0}}\bar{n}_{\text{cross}}^{9/2},$
(12)

where $\Delta_{k,0}$ is the detuning between bare levels $k$ and $0$. One may
expect that the detuned levels in between $|0\rangle$ and $|9\rangle$ would
suppress the effective coupling between them significantly, however, using the
parameters of Fig. 7 for Eq. (12) we find $g_{\text{eff}}/2\pi\approx 32$ MHz,
which is comparable to the observed value of $12.5$ MHz in Fig. 7 and is not
suppressed. Note that as discussed in Refs. [17, 18], the so called critical
photon number $n_{\text{crit}}\equiv(\Delta/g)^{2}/4$ [1] is not the photon
number at which the readout transitions occur. Nevertheless, in order to have
large enough effective coupling between resonant levels, we need photon
numbers to be larger than $n_{\text{crit}}$. This can be seen by noting that
the effective coupling of Eq. (12) can be rewritten as
$g_{\text{eff}}\propto(\bar{n}_{\text{cross}}/n_{\text{crit}})^{4}g\sqrt{\bar{n}_{\text{cross}}}$.

Fig. 7(c) shows the population in the instantaneous eigenstate of the system
as a function of average photon number, where the system was initialized in
$|0\rangle$. At the avoided crossing, there is a Landau-Zener type population
exchange between $|0\rangle$ and $|9\rangle$, where the loss of population in
the initially prepared state $|0\rangle$ signifies a state transition event.
The speed at which the system goes through these avoided crossings is set by
the resonator decay rate $\kappa$, meaning an avoided crossing with coupling
larger or comparable to $\kappa$ will result in notable population exchange
(between diabatic states) and therefore a measurement-induced state
transition.

Let us further discuss the effect of resonator energy decay in the full system
of Eq. (7), where the resonator loses photons at rate $\kappa$, which is
usually described via master equation in the Lindblad form [39]. This
evolution can also be thought as a probabilistic mixture of scenarios
described by Kraus operators [40], where a photon loss is described by a
stochastic application of annihilation operator $\hat{a}$ onto the system
state vector. Additionally, the system of a transmon coupled to a driven
resonator forms a dressed coherent state, i.e., a coherent state made of joint
eigenstates of the qubit and resonator. Therefore, a photon loss event in the
full system corresponds to application of the annihilation operator onto a
(dressed) coherent state, which by definition retains that state. This means
that the evolution of the full system prior to state transitions (i.e., prior
to change of the qubit state) can be described in the eigenbasis of the system
with resonator having a coherent state, as we have assumed in this work.
Another perspective on the same problem that validates our semi-classical
approximation is to note that a photon loss moves the system from one RWA
strip to another, and with large enough photons in the system these RWA strips
are approximately similar copies of each other and yield the same evolution
for the qubit.

## References

  * Blais  _et al._ [2004] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
  * Wallraff  _et al._ [2004] A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, G. S.M., and R. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature (London) 431, 162 (2004).
  * Blais  _et al._ [2021] A. Blais, A. Grimsmo, S. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Jaynes and Cummings [1963] E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proceedings of the IEEE 51, 89 (1963).
  * Tavis and Cummings [1968] M. Tavis and F. Cummings, Exact solution for an $n$-molecule—radiation-field hamiltonian, Phys. Rev. 170, 379 (1968).
  * Johnson  _et al._ [2012] J. Johnson, C. Macklin, D. Slichter, R. Vijay, E. Weingarten, j. Clarke, and I. Siddiqi, Heralded state preparation in a superconducting qubit, Phys. Rev. Lett. 109, 050506 (2012).
  * Walter  _et al._ [2017] T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff, Rapid high-fidelity single-shot dispersive readout of superconducting qubits, Phys. Rev. Appl. 7, 054020 (2017).
  * Google Quantum AI and collaborators [2019] Google Quantum AI and collaborators, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Jurcevic  _et al._ [2021] P. Jurcevic, A. Javadi-Abhari, L. Bishop, L. I., D. Bogorin, M. Brink, L. Capelluto, O. Günlük, T. Itoko, N. Kanazawa, A. Kandala, G. Keefe, K. Krsulich, W. Landers, E. Lewandowski, D. McClure, G. Nannicini, A. Narasgond, H. Nayfeh, E. Pritchett, M. Rothwell, S. Srinivasan, N. Sundaresan, C. Wang, K. Wei, C. Wood, J.-B. Yau, E. Zhang, O. Dial, J. Chow, and J. Gambetta, Demonstration of quantum volume 64 on a superconducting quantum computing system, Quantum Sci. Technol. 6, 025020 (2021).
  * Fowler  _et al._ [2012] A. Fowler, M. Mariantoni, J. Martinis, and A. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  * Google Quantum AI and collaborators [2021] Google Quantum AI and collaborators, Exponential suppression of bit or phase errors with cyclic error correction, Nature 595, 383 (2021).
  * Krinner  _et al._ [2022] S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. Norris, C. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, Realizing repeated quantum error correction in a distance-three surface code, Nature 605, 669 (2022).
  * Google Quantum AI and collaborators [2023] Google Quantum AI and collaborators, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  * Reed  _et al._ [2010a] M. Reed, L. DiCarlo, B. Johnson, L. Sun, D. Schuster, L. Frunzio, and R. Schoelkopf, High-fidelity readout in circuit quantum electrodynamics using the jaynes-cummings nonlinearity, Phys. Rev. Lett. 105, 173601 (2010a).
  * Sank [2014] D. Sank,  _Fast, Accurate State Measurement in Superconducting Qubits_ , Ph.D. thesis, University of California, Santa Barbara (2014).
  * Lescanne  _et al._ [2019] R. Lescanne, L. Verney, Q. Ficheux, M. H. Devoret, B. Huard, M. Mirrahimi, and Z. Leghtas, Escape of a driven quantum josephson circuit into unconfined states, Phys. Rev. Applied 11, 014030 (2019).
  * Khezri  _et al._ [2016] M. Khezri, E. Mlinar, J. Dressel, and A. Korotkov, Measuring a transmon qubit in circuit qed: Dressed squeezed states, Phys. Rev. A 94, 012347 (2016).
  * Sank  _et al._ [2016] D. Sank, Z. Chen, M. Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, T. White, J. Wenner, A. Korotkov, and J. Martinis, Measurement-induced state transitions in a superconducting qubit: Beyond the rotating wave approximation, Phys. Rev. Lett. 117, 190503 (2016).
  * McEwen  _et al._ [2021] M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. Satzinger, C. Quintana, P. Klimov, D. Sank, C. Gidney, A. Fowler, F. Arute, K. Arya, B. Buckley, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, S. Demura, A. Dunsworth, C. Erickson, B. Foxen, M. Giustina, T. Huang, S. Hong, E. Jeffrey, S. Kim, K. Kechedzhi, F. Kostritsa, P. Laptev, A. Megrant, X. Mi, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Niu, A. Paler, N. Redd, P. Roushan, T. White, J. Yao, P. Yeh, A. Zalcman, Y. Chen, V. Smelyanskiy, J. Martinis, H. Neven, J. Kelly, A. Korotkov, A. Petukhov, and R. Barends, Removing leakage-induced correlated errors in superconducting quantum error correction, Nat. Commun. 12, 1761 (2021).
  * Geerlings  _et al._ [2013] K. Geerlings, Z. Leghtas, I. M. Pop, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Demonstrating a driven reset protocol for a superconducting qubit, Phys. Rev. Lett. 110, 120501 (2013).
  * Magnard  _et al._ [2018] P. Magnard, P. Kurpiers, B. Royer, T. Walter, J.-C. Besse, S. Gasparinetti, M. Pechal, J. Heinsoo, S. Storz, A. Blais, and A. Wallraff, Fast and unconditional all-microwave reset of a superconducting qubit, Phys. Rev. Lett. 121, 060502 (2018).
  * Zhou  _et al._ [2021] Y. Zhou, Z. Zhang, Z. Yin, S. Huai, X. Gu, X. Xu, J. Allcock, F. Liu, G. Xi, Q. Yu, H. Zhang, M. Zhang, H. Li, X. Song, Z. Wang, D. Zheng, S. An, Y. Zheng, and S. Zhang, Rapid and unconditional parametric reset protocol for tunable superconducting qubits, Nat. Commun. 12, 5924 (2021).
  * Marques  _et al._ [2023] J. F. Marques, H. Ali, B. M. Varbanov, M. Finkel, H. M. Veen, S. L. M. van der Meer, S. Valles-Sanclemente, N. Muthusubramanian, M. Beekman, N. Haider, B. M. Terhal, and L. DiCarlo, All-microwave leakage reduction units for quantum error correction with superconducting transmon qubits, Phys. Rev. Lett. 130, 250602 (2023).
  * Miao  _et al._ [2022] K. C. Miao, M. McEwen, J. Atalaya, D. Kafri, L. P. Pryadko, A. Bengtsson, A. Opremcak, K. J. Satzinger, Z. Chen, P. V. Klimov, C. Quintana, R. Acharya, K. Anderson, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. C. Bardin, A. Bourassa, J. Bovaird, L. Brill, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, J. Campero, B. Chiaro, R. Collins, P. Conner, A. L. Crook, B. Curtin, D. M. Debroy, S. Demura, A. Dunsworth, C. Erickson, R. Fatemi, V. S. Ferreira, L. F. Burgos, E. Forati, A. G. Fowler, B. Foxen, G. Garcia, W. Giang, C. Gidney, M. Giustina, R. Gosula, A. G. Dau, J. A. Gross, M. C. Hamilton, S. D. Harrington, P. Heu, J. Hilton, M. R. Hoffmann, S. Hong, T. Huang, A. Huff, J. Iveland, E. Jeffrey, Z. Jiang, C. Jones, J. Kelly, S. Kim, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, L. Laws, K. Lee, B. J. Lester, A. T. Lill, W. Liu, A. Locharla, E. Lucero, S. Martin, A. Megrant, X. Mi, S. Montazeri, A. Morvan, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, M. Newman, J. H. Ng, A. Nguyen, M. Nguyen, R. Potter, C. Rocque, P. Roushan, K. Sankaragomathi, C. Schuster, M. J. Shearn, A. Shorter, N. Shutty, V. Shvarts, J. Skruzny, W. C. Smith, G. Sterling, M. Szalay, D. Thor, A. Torres, T. White, B. W. K. Woo, Z. J. Yao, P. Yeh, J. Yoo, G. Young, A. Zalcman, N. Zhu, N. Zobrist, H. Neven, V. Smelyanskiy, A. Petukhov, A. N. Korotkov, D. Sank, and Y. Chen, Overcoming leakage in scalable quantum error correction, arXiv:2211.04728 (2022).
  * Koch  _et al._ [2007] J. Koch, T. Yu, J. Gambetta, A. Houck, D. Schuster, J. Majer, A. Blais, M. Devoret, S. Girvin, and R. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Phys. Rev. A 76, 042319 (2007).
  * Reed  _et al._ [2010b] M. Reed, B. Johnson, A. Houck, L. DiCarlo, J. Chow, D. Schuster, L. Frunzio, and R. Schoelkopf, Fast reset and suppressing spontaneous emission of a superconducting qubit, Appl. Phys. Lett. 96, 203110 (2010b).
  * Jeffrey  _et al._ [2014] E. Jeffrey, D. Sank, J. Mutus, T. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. O’Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. Cleland, and J. Martinis, Fast accurate state measurement with superconducting qubits, Phys. Rev. Lett. 112, 190504 (2014).
  * Bengtsson  _et al._ [2023] A. Bengtsson, A. Opremcak, M. Khezri, D. Sank, A. Bourassa, K. J. Satzinger, S. Hong, C. Erickson, B. J. Lester, K. C. Miao, A. N. Korotkov, J. Kelly, Z. Chen, and P. V. Klimov, Model-based optimization of superconducting qubit readout, arXiv:2308.02079 (2023).
  * Shillito  _et al._ [2022] R. Shillito, A. Petrescu, J. Cohen, J. Beall, M. Hauru, M. Ganahl, A. Lewis, G. Vidal, and A. Blais, Dynamics of transmon ionization, Phys. Rev. Appl. 18, 034031 (2022).
  * Cohen  _et al._ [2023] J. Cohen, A. Petrescu, R. Shillito, and A. Blais, Reminiscence of classical chaos in driven transmons, PRX Quantum 4, 020312 (2023).
  * Ristè  _et al._ [2013] D. Ristè, C. C. Bultink, M. J. Tiggelman, R. N. Schouten, K. W. Lehnert, and L. DiCarlo, Millisecond charge-parity fluctuations and induced decoherence in a superconducting transmon qubit, Nat. Commun. 4, 1913 (2013).
  * Christensen  _et al._ [2019] B. Christensen, C. Wilen, A. Opremcak, J. Nelson, F. Schlenker, C. Zimonick, L. Faoro, L. Ioffe, Y. Rosen, J. DuBois, B. Plourde, and R. McDermott, Anomalous charge noise in superconducting qubits, Phys. Rev. B 100, 140503 (2019).
  * Khezri [2018] M. Khezri,  _Dispersive Measurement of Superconducting Qubits_ , Ph.D. thesis, University of California, Riverside (2018).
  * Peterer  _et al._ [2015] M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, Coherence and decay of higher energy levels of a superconducting transmon qubit, Phys. Rev. Lett. 114, 010501 (2015).
  * Schuster  _et al._ [2005] D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. M. Girvin, and R. J. Schoelkopf, ac stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field, Phys. Rev. Lett. 94, 123602 (2005).
  * Schuster  _et al._ [2007] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Resolving photon number states in a superconducting circuit, Nature 445, 515 (2007).
  * Sete  _et al._ [2013] E. Sete, A. Galiautdinov, E. Mlinar, J. Martinis, and A. Korotkov, Catch-disperse-release readout for superconducting qubits, Phys. Rev. Lett. 110, 210501 (2013).
  * Govia and Wilhelm [2016] L. C. G. Govia and F. K. Wilhelm, Entanglement generated by the dispersive interaction: The dressed coherent state, Phys. Rev. A 93, 012316 (2016).
  * Walls and Milburn [2008] D. Walls and G. Milburn,  _Quantum Optics_ (Springer, Berlin, 2008).
  * Lidar  _et al._ [2001] D. Lidar, Z. Bihary, and K. Whaley, From completely positive maps to the quantum markovian semigroup master equation, Chem. Phys. 268, 35 (2001).