2022/08/192022/11/19

radio continuum: stars — radio continuum: general — stars: individual (FRB
20201124A)

#  Detection of a bright burst from the repeating FRB 20201124A at 2 GHz

Sota Ikebe11affiliation: Department of Astronomy, School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
22affiliation: Mizusawa VLBI Observatory, National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan ikebe-
<EMAIL_ADDRESS>Kazuhiro Takefuji33affiliation: Usuda Deep Space
Center, Japan Aerospace Exploration Agency, 1831-6 Oomagari, Kamiodagiri,
Saku, Nagano 384-0306, Japan Toshio Terasawa44affiliation: Institute for
Cosmic Ray Research, University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa, Chiba
277-8582, Japan Sujin Eie11affiliationmark:  22affiliationmark:
55affiliation: Institute of Astronomy and Astrophysics, Academia Sinica, 11F
of AS/NTU Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei
10617, Taiwan, R.O.C. Takuya Akahori22affiliationmark:  66affiliation:
Operation Division, Square Kilometre Array Observatory, Jodrell Bank
Observatory, Lower Withington, Macclesfield, Cheshire SK11 9FT, UK Yasuhiro
Murata33affiliationmark:  77affiliation: Japan Aerospace Exploration Agency,
Institute of Space and Astronautical Science, 3-1-1, Yoshinodai, Sagamihara,
Kanagawa 252-5210, Japan 88affiliation: Department of Space and Astronautical
Science, SOKENDAI (The Graduate University for Advanced Studies), Sagamihara
252-5210, Japan Tetsuya Hashimoto99affiliation: Department of Physics,
National Chung Hsing University, No. 145, Xingda Rd., South Dist., Taichung
40277, Taiwan (R.O.C.) Shota Kisaka1010affiliation: Department of Physics,
Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima
739-8526, Japan Mareki Honma11affiliationmark:  22affiliationmark:
1111affiliation: Mizusawa VLBI Observatory, National Astronomical Observatory
of Japan, 1-2-2 Mizusawa-Hoshigaoka, Oshu, Iwate 023-0861, Japan Shintaro
Yoshiura22affiliationmark:  Syunsaku Suzuki22affiliationmark:  Tomoaki
Oyama1111affiliationmark:  Mamoru Sekido1212affiliation: National Institute of
Information and Communications Technology, 4-2-1 Nukui-Kitamachi, Koganei,
Tokyo 184-8795, Japan Kotaro Niinuma1313affiliation: Graduate School of
Sciences and Technology for Innovation, Yamaguchi University, 1677-1 Yoshida,
Yamaguchi, 753-8512, Japan 1414affiliation: The Research Center for Time
Studies, Yamaguchi University, 1677-1, Yoshida, Yamaguchi 753-8511, Japan
Hiroshi Takeuchi77affiliationmark:  88affiliationmark:  Yoshinori
Yonekura1515affiliation: Center for Astronomy, Ibaraki University, 2-1-1,
Bunkyo, Mito, Ibaraki 310-8512, Japan Teruaki Enoto1616affiliation: Department
of Physics, Graduate School of Science, Kyoto University, Kitashirakawa
Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan 1717affiliation: Cluster for
Pioneering Research, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

###### Abstract

We present a detection of a bright burst from FRB 20201124A, which is one of
the most active repeating FRBs, based on S-band observations with the 64-m
radio telescope at the Usuda Deep Space Center/JAXA. This is the first FRB
observed by using a Japanese facility. Our detection at 2 GHz in February 2022
is the highest frequency for this FRB and the fluence of $>$ $189$ Jy ms is
one of the brightest bursts from this FRB source. We place an upper limit on
the spectral index $\alpha$ $=$ $-2.14$ from the detection of the S band and
non-detection of the X band at the same time. We compare an event rate of the
detected burst with ones of the previous research, and suggest that the power-
law of the luminosity function might be broken at lower fluence, and the
fluences of bright FRBs distribute up to over 2 GHz with the power-law against
frequency. In addition, we show the energy density of the burst detected in
this work was comparable to the bright population of one-off FRBs. We propose
that repeating FRBs can be as bright as one-off FRBs, and only their brightest
bursts could be detected so some of repeating FRBs intrinsically might have
been classified as one-off FRBs.

##  1 Introduction

Fast Radio Bursts (FRBs), short-duration ($\ll$ 1 sec) bright radio
transients, have still unknown progenitors and emission mechanisms (Petroff et
al., 2022). Detection of FRBs at different frequencies and over wider
bandwidths are helpful for studying the intrinsic emission properties and the
frequency dependence of potential propagation effects due to the circumburst
environment. Some proposed emission models have a limited range of radio
frequency and/or spectral index (e.g., Kumar et al. (2017); Metzger et al.
(2019)). Simultaneous observation covering a wide range of frequencies for FRB
20180916B discovered the frequency dependence of the activity window, which
has provided potentially an important insight into the nature of these sources
(Pastor-Marazuela et al., 2021; Bethapudi et al., 2022). Some FRBs have been
observed at various frequencies, such as FRB 20121102 from 400 MHz to 8 GHz,
(Josephy et al., 2019; Gajjar et al., 2018), FRB 20180916B from 110 MHz to 5
GHz (Chawla et al., 2020; Pleunis et al., 2021b; Bethapudi et al., 2022).
However, in general, most of FRBs have been detected at 400 – 800 MHz or 1.4
GHz at the rest frame.

A subset of FRBs have shown repetition, which rules out solely catastrophic
progenitors. There is a possibility that one-off FRBs are detected only once
due to limited observation time or sensitivity, and whether all FRBs are
capable of repeating or not has been actively debated (e.g., Ravi (2019); Ai
et al. (2021); Gardenier et al. (2021)). Some research (e.g., Hashimoto et al.
(2020a)) propose that FRBs are classified into two populations, that is, not
all FRBs repeat. It is suggested that repeating FRBs have some different
trends from one-off FRBs, such as the duration time, bandwidth, redshift
evolution, and especially energy; repeating FRBs tend to be fainter than one-
off FRBs (Hashimoto et al., 2020a, b; Pleunis et al., 2021a; Kim et al.,
2022).

The repeaters can provide an opportunity to conduct follow-up observations at
higher radio frequencies. Long-term follow-up observations over a time scale
of years can constrain periodic activity. Some repeating FRBs, such as FRB
121102 (Cruces et al., 2021) and FRB 180916B (CHIME/FRB Collaboration, 2020),
have shown periodicity of the burst activity window. The repeating FRB
20201124A entered a period of high activity in April of 2021 (CHIME/FRB
Collaboration, 2021; Lanman et al., 2022), at which time several observatories
recorded tens to thousands of bursts from the source. Recently, it again
entered an active phase in January–March 2022 (Ould-Boukattine et al., 2022a,
b; Atri et al., 2022). However, no detection has been reported in the high
frequency band above 2 GHz from this source.

Here we report a detection of a bright FRB from the repeater FRB 20201124A
source at 2.2 – 2.3 GHz, which is the highest-frequency detection from FRB
20201124A to date, using the 64-m radio dish of Usuda Deep Space Center
(UDSC)/JAXA. This is the first FRB detected in Japan. The remainder of this
paper is structured as follows. In Section 2, we present our observational
setup and analyses. In Section 3, we describe the detected burst in detail,
dispersion measure (DM) determination, digital artifact, and then estimate the
spectral index. We compare the detected burst with the previous bursts from
FRB 20201124A and other FRBs in Section 4, and summarize this paper in Section
5.

##  2 Observations and Analyses

Following the recent reports of the 2022 reactivation of FRB 20201124A as
cited in the introduction, we conducted an observing session for this FRB
source using UDSC/JAXA for 8 hours on February 18, 2022 (MJD 59628),
07:11:00–15:14:00 UT at the S (2194 – 2322 MHz) and X bands (8374 – 8502 MHz).
We also observed the 3C147 as a flux calibrator before and after the FRB
observation. We used the polynomial expression for 3C147 studied by Perley and
Butler (2017) to speculate the flux densities in the S and X bands to be 15.2
and 4.7 Jy, respectively, and determined values of the system equivalent flux
density (SEFD) 122 Jy (7% uncertainty) for the S band, and 173 Jy (40%
uncertainty) for the X band. These uncertainties resulted from atmospheric
fluctuation.

We observed right-handed circular polarized waves using a multi-channel
digital A/D sampler , ADS3000+ (Takeuchi et al., 2006), with a sampling rate
of 64 MHz and 4 bit quantization, and divided the observed full bandwidths of
128 MHz $\times\ 2$ (S and X bands) to 4 $\times\ 2$ channels of 32 MHz
bandwidth (effectively 30 MHz width after the elimination of the band edges).
The data in each channel are coherently dedispersed with a trial value of DM =
413 pc cm-3, which is from Xu et al. (2021), and then incoherently summed over
4 channels to make the averaged time series for the S band and X band,
separately (see Enoto et al. (2021) for more details).

Figure 1: Dynamic spectrum of a burst from FRB 20201124A at 2 GHz band before
dedispersion. Central frequencies of channel(ch) 0, ch 1, ch 2, and ch 3 are
2210, 2242, 2274, and 2306 MHz, respectively. Figure 2:  Upper panel: Time
series of squared antenna voltages at the S band with 100 $\mu$s time
integration. Lower panel: Waterfall plot of the detected FRB event after
dedispersion with the best-fit DM of 411.0 pc cm-3. Time is converted to TDB
(solar-system barycentric dynamical time) using TEMPO2 (Hobbs et al., 2006)
for the infinite frequency. The frequency resolution of the waterfall plot is
1 MHz.

We searched the averaged time series of 1 ms integration for burst candidates
having signal-to-noise ratios (S/N) $\geq$ 10 both in the S and X band, namely
those having peak fluxes $>$ 3.5 Jy for the S band and $>$ 5.0 Jy for the X
band.

##  3 Results

After the analysis mentioned above, we found an FRB candidate having S/N $>$
300 at the S band (Takefuji et al., 2022) and none at the X band. For this
candidate Figure 1 shows a dynamic spectrum before dedispersion covering a 60
ms time window starting at UTC 14:21:34.040. This dynamic spectrum shows a
clear detection of a strong radio burst with a dispersive delay. The
dispersive delay apparently follows a $\nu^{-2}$ law, which is a typical
characteristic of FRBs, being caused by propagation through the
interstellar/intergalactic plasma. Intensity variation against frequency may
be attributed to interstellar scintillation.

To determine the optimal DM, we averaged the dedispersed data every 100 $\mu$s
and calculated the structure parameter (Gajjar et al. (2018)) defined as,

${\rm{Structure\
Parameter}}=\frac{1}{n}\sum^{n}_{i}\left|\frac{S_{i}-S_{i+1}}{\Delta
t}\right|,$ (1)

where $n$ is the number of total bins which include the bursts, $S_{i}$ is the
$i$-th flux, and $\Delta t$ is the integration time. We calculated the
structure parameters by changing DM. Then we fitted the structure parameters
with the Gaussian distribution and determined the structure-parameter-
maximizing-DM of 411.0 $\pm$ 0.5 pc cm-3 as the optimal DM111We searched burst
candidates again by dedispersion with the optimal DM value of 411.0 pc cm-3
and, in the end, there was only the one FRB already described in this
section.. This is consistent with values reported in the previous work within
uncertainties (Xu et al., 2021).

In the upper panel of Figure 2 we present the burst profiles integrated over
ch 0 – 3 separately after the dedispersion using our best estimate DM. The
lower panel of Figure 2 shows the dynamic spectrum, e.g., the waterfall plot,
after the dedispersion. The bright main component has a pulse width (FWHM) of
1.5 ms which is typical for repeating FRBs (Petroff et al., 2022). There is a
faint sub-structure at 3 ms prior to the main component, with a peak intensity
of $\sim$ 10% of the peak of main component. Similar sub-structures with a
timescale of a few ms are also reported for FRB 20201124A (e.g., Marthi et al.
(2022); Main et al. (2022)).

Figure 3:  The S/N time profile for ch 0 with the integration time of 1 ms.
The filled area shows digitization artifacts. The inset show an enlarged
portion of digitization artifacts.

We should note one caveat about the flux/fluence determination. Since the
previous observations of FRB 20201124A showed that the majority of bursts from
this FRB have flux density $\lesssim$ several tens of Jy (e.g., Xu et al.
(2021)), we had optimized the digital system to this flux density level.
Figure 3 shows the burst profile for ch 0, and we see a dip in the noise floor
in the neighboring interval of the burst peak (filled area), which represents
digital artifact due to a finite bit size of the digitization system (Jenet
and Anderson, 1998) and results in the flux/fluence underestimations. For ch 1
there is also a dip although shallower than ch 0 but significant. We see no
dip for ch 2 and ch 3 and concluded that these channels are free from the
digitization artifact. Table 1 summarizes our estimation of the flux/fluence
separately for ch 0 – 3. If we incoherently sum up these four channels, we get
estimations for the peak flux $>$ 114 Jy and for fluence $>$ 189 Jy ms for the
frequency band, 2194 – 2322 MHz. It should be noted again that only right-
handed circular polarized waves were observed. The mean value of V/I of FRBs
detected in Hilmarsson et al. (2021) was -2±9 %, where V and I are stokes
parameters. From this result, the intensity of a right-handed circular
polarized wave is as high as that of a left-handed circular polarized wave.
Hence, FRBs that should be detected if we also carried out left-handed
circular polarized wave observations have a comparable intensity of right-
handed circular polarized wave, so could also be detected by this observation.
To summarize, we conclude that only right-handed circular polarized wave
observation would not miss FRBs, namely, would not affect the detection
number. The total fluence would be higher than the value obtained from this
observation.

Assuming that fluence can be expressed as $F_{\nu}\propto\nu^{\alpha}$, we
calculate the spectral index of $\alpha$ for the detected burst from the
simultaneous observations of the S and X bands. We obtain the lower limit of
fluence, 189 Jy ms (7 % uncertainty), for the S band due to the digitization
artifact, and the upper limit of fluence, 7.5 Jy ms (40 % uncertainty), for
the X band due to the non-detection by assuming that the duration in the X
band is the same as that in the S band. Considering uncertainties we get the
upper limit of $\alpha$ to be $-2.14$. This upper limit is at variance with
the spectral index of the mean of bright FRBs detected with the Australian
Square Kilometre Array Pathfinder (ASKAP) (Macquart et al., 2019), and
magnetar emissions (e.g., Levin et al. (2010); Eie et al. (2021)). However it
is consistent with a mean index of main pulse and interpulse for giant pulses
from the Crab pulsar between 2.3 GHz and 8.4 GHz (S. Eie in preparation). This
comparison could reveal clues to the emission mechanism of FRB 20201124A.
Although further investigations are necessary, observations of FRB 20201124A
might support the emission mechanism models like Crab giant radio pulses more.

Flux/Fluence estimation for a burst of FRB 20201124A on 18 February 2022.  ch
central flux**footnotemark: $*$ fluence**footnotemark: $*$ frequency 0 2210
MHz $>$ 262 Jy $>$ 444 Jy ms 1 2242 $>$ 109 $>$ 178 2 2274 34 55 3 2306 51 79
**footnotemark: $*$7% uncertainly

##  4 Discussion

###  4.1 Comparison with the previous bursts from FRB 20201124A

During the previous active phases of FRB 20201124A, some radio telescopes,
such as the Five-hundred-meter Aperture Spherical radio Telescope (FAST) and
Effelsberg 100-m radio telescope, conducted follow-up observations (e.g., Xu
et al. (2021); Hilmarsson et al. (2021)). FAST detected 1863 bursts from the
FRB 20201124A source from April 1 to June 11 in 2021, covering the range from
1.0 GHz to 1.5 GHz (Xu et al., 2021). The observed fluences distributed up to
$\sim$ 70 Jy ms. Hilmarsson et al. (2021) reported 20 bursts detected with
Effelsberg 100-m radio telescope at $\sim$ 1.36 GHz during observations
conducted on April 9 2021. The brightest event among the 20 bursts detected by
Effelsberg 100-m radio telescope showed a fluence of $\sim$ 30 Jy ms.
Therefore, the burst detected in this work, whose flux and fluence is the
lower limit determination due to the digitization artifact at $\sim$ 2 GHz, is
one of the brightest events from this FRB source (e.g., 334 Jy ms reported in
Herrmann et al. (2021), and $>771$ Jy ms in Ould-Boukattine et al. (2022b) at
1.3 GHz). We also note that the detected frequency ($\sim$ 2.2 – 2.3 GHz) is
highest.

We contrast the event rate of FRBs speculated from the observation result of
this work with the FAST result, which detected large quantities of FRBs (Xu et
al., 2021). We observed for 8 hours and detected one burst, giving an event
rate of 1/8 ${\rm hr^{-1}}$. Since the observed frequency and threshold are
different, we correct for these differences. Assuming all FRBs follow the form
of $F_{\nu}\propto\nu^{\alpha}$ over all frequencies, we convert the threshold
of Xu et al. (2021) to the fluence at our frequency, 2258 MHz. Moreover, the
single power-law of luminosity function, $N(>F)\propto F^{\beta+1}$ where $N$
is a cumulative number per an hour, is simply assumed to extrapolate event
rates for the threshold of each telescope. In our calculation, we used
$\alpha=-2.14$, which is the upper limit estimated in Section 3, and
$\beta=-4.6\pm 1.3\pm 0.6$, which is from Lanman et al. (2022). Figure 4 shows
cumulative event rates of FRBs of which fluence is higher than thresholds at
the same frequency. Our event rate is notably larger than that of extrapolated
from Xu et al. (2021), $1.0\times 10^{-11}\ {\rm hr^{-1}}$, although it is
within errors of extrapolated result from Xu et al. (2021). We also compare
the event rates from Marthi et al. (2022), Atri et al. (2022), and Ould-
Boukattine et al. (2022b) by performing the same procedures. Marthi et al.
(2022) observed last year, whereas Atri et al. (2022) and Ould-Boukattine et
al. (2022b) observed this year, namely in the same active period as this work.
As a consequence, the event rate extrapolated from Xu et al. (2021) is
inconsistent with that of Marthi et al. (2022) even though they carried out
their observations during the same active period last year. This suggests that
the single power-law distribution of the luminosity function which we assume
to estimate event rates is not correct. That is, the power-law of the
luminosity function might be broken at lower fluence. In addition, the
consequence that event rates of Atri et al. (2022) and Ould-Boukattine et al.
(2022b) in the same active phase we observed are roughly equivalent to our
event rate implies a possibility that bright FRBs distribute up to over 2 GHz
with the power-law.

Figure 4:  Cumulative event rates of FRBs having fluence larger than threshold
for each telescope. Blue, purple, green, red, black points represent
cumulative event rates of Xu et al. (2021), Marthi et al. (2022), Atri et al.
(2022), Ould-Boukattine et al. (2022b), this work, respectively. Xu et al.
(2021) and Marthi et al. (2022) observed last year, whereas Atri et al. (2022)
and Ould-Boukattine et al. (2022b) observed this year, namely in the same
active period as this work. The blue solid line represents the event rate by
extrapolating luminosity function $N(>F)\propto F^{\beta+1}$. Gray region
corresponds the range obtained from the uncertainty of $\beta$ and event rate
of Xu et al. (2021).

###  4.2 A possibility that repeating FRBs are observed as one-off FRBs

We compare the properties of the burst of FRB 20201124A observed on 18
February 2022 (this work) to the sample collected from the FRBCAT project
(Petroff et al., 2016). More detail about the sample of FRBs is described in
https://www.frbcat.org/. FRBs are detected at different frequencies and
different redshifts so, for a fair comparison, Hashimoto et al. (2020a)
estimate energy densities at a unified rest-frame frequency of 1.83 GHz and
rest-frame intrinsic duration. Detailed procedures are described by Hashimoto
et al. (2020a). After excluding FRBs without a solution to DM-derived
redshift, i.e., ${\rm DM}_{\rm obs}-{\rm DM}_{\rm halo}-{\rm DM}_{\rm
ISM}-{\rm DM}_{\rm host}<0$, the comparison sample includes a total of 11
repeating FRB sources with 144 repeats and 77 one-off FRBs222${\rm DM}_{\rm
obs}$, ${\rm DM}_{\rm halo}$, ${\rm DM}_{\rm ISM}$ and ${\rm DM}_{\rm host}$
represent an observed DM, a DM contributed from the dark matter halo hosting
the Milky Way, the interstellar medium in the Milky Way, and a FRB host
galaxy.. Hashimoto et al. (2020b) confirms that the DM-derived redshifts
estimated by these procedures are consistent with spectroscopic redshifts
within uncertainties (see also Figure 1).

We performed the same procedures to the FRB in this work. The lower limit of
the energy density ($E_{\nu}$) of FRB 20201124A is calculated using $z_{\rm
spec}=0.0979$ (Fong et al., 2021), the observed fluence of $F_{\nu}>444$ Jy ms
at 2210 MHz (ch 0), and the spectral index of $\alpha<-2.14$. The calculated
lower limit is $\log(E_{\nu}/{\rm erg~{}Hz}^{-1})=32.24$ at 1.83 GHz. The
average fluence at 2194 – 2322 MHz (ch 0 – 3) is 189 Jy ms, which places the
lower limits of $\log(E_{\nu}/{\rm erg~{}Hz}^{-1})=31.89$. We note that the
instrumental pulse broadening effects, including dispersion smearing and
finite time sampling, are negligible due to the coherent dedispersion
described in Section 2 and 100 $\mu$s time sampling. Therefore, the rest-frame
intrinsic duration of the burst detected in this work is $1.5/(1+z_{\rm
spec})\sim 1.4$ ms.

Figure 5 shows the rest-frame intrinsic duration as a function of the energy
density of FRB 20201124A (this work) along with those of the comparison
sample. The samples of FRBs used in this work were obtained by various
telescopes so include observational biases. Considering observational results,
FRBs are classified into two populations, repeating and one-off FRBs. There is
a possibility that this is caused by observational biases and FRBs
intrinsically consist of only one population. However, some research propose
that repeating and one-off FRBs have distinct physical origins. Namely, this
classification a physical difference to some extent (e.g., Hashimoto et al.
(2020a); Chen, H.-Y. et al. (2022)). To mitigate a possible observational bias
due to targeted follow-up campaigns of repeating FRBs, the first-detected FRB
for each repeating FRB source (magenta diamond in Figure 5) can be compared
with one-off FRBs (blue markers). The first-detected repeating FRBs show lower
energy densities than those of one-off events on average. A similar result is
also reported for energies of the CHIME FRB sample (Kim et al., 2022). One
possible reason is the narrower bandwidths of repeating FRBs compared to
apparent one-offs (e.g., Pleunis et al. (2021a)).

The detected burst from the FRB 20201124A source shows one of the highest
energy densities of repeating FRBs. The energy density is comparable to those
of the bright population of one-off FRBs. The other two repeating FRBs show
high energy densities comparable to that of FRB 20201124A; FRB 171019, FRB
181017.J1705+68 (FRB 20181017A). The former was originally reported as one of
the bright one-off FRBs detected with ASKAP at $\sim$ 1.2 GHz (Shannon et al.,
2018). Afterward, two repetitions were detected from this FRB source with the
Green Bank Telescope in observations centered at 820 MHz (Kumar et al., 2019).
The two repetitions were observed as $\sim$ 590 times fainter than the ASKAP-
discovered burst, indicating the importance of deep follow-up observations to
identify repeating FRB sources. Another is FRB 181017.J1705+68 (FRB 20181017A)
with $\log(E_{\nu}/{\rm erg~{}Hz}^{-1})>32.3$. Two repeating FRBs were
detected from this source with CHIME at $\sim$500 – 550 MHz so far (CHIME/FRB
Collaboration, 2019). Our result highlights that repeating FRBs can be as
bright as the bright population of one-off FRBs.

Ravi (2019) argued that the volumetric occurrence rate of one-off FRBs likely
exceeds the event rate of cataclysmic progenitor candidates, suggesting that
one-off FRBs are significantly contaminated by repeating FRB sources.
Repeating FRBs are fainter than one-off events on average (e.g., Hashimoto et
al. (2020a); Kim et al. (2022)). Therefore, some repeating FRBs might have
been missed due to sensitivity limitations of current radio telescopes, which
would mislead the FRB classification. The bright repeating FRB detected in
this work might support the hypothesis that some of or a significant fraction
of one-off events are a part of repeating bursts (e.g., Ravi (2019); Kumar et
al. (2019); Chen, B. H. et al. (2022)). However, only their brightest rare
population could be detected and thus misclassified as one-off FRBs.
Monitoring the other repeating FRB sources and deep follow-up observations of
one-off FRB sources are highly encouraged to prove this hypothesis.

Figure 5:  Rest-frame intrinsic duration as a function of energy density at
rest-frame 1.83 GHz of FRBs. FRB 20201124A detected in this work is shown as a
dark red star with an arrow. The arrow indicates the lower limit of the energy
density. The lower limit of the energy density is calculated by the observed
fluence at 2210 MHz (ch 0). A translucent red star indicates the lower limit
of the energy density derived by the fluence averaged over 2194 – 2322 MHz (ch
0 – 3). The comparison samples (dots and open triangles) are collected from
the FRBCAT project (Petroff et al., 2016). Red and blue colors indicate
repeating and one-off FRBs, respectively. The repeats of each repeating FRB
source are individually demonstrated. The first-detected FRB for each
repeating FRB source is highlighted in a magenta diamond. Open triangles
indicate upper limits on the rest-frame intrinsic duration whereas dots are
the measured values. The other two repeating FRBs that have high energy
densities comparable to that of FRB 20201124A in this work are FRB 171019, FRB
181017.J1705+68 (FRB 20181017A).

##  5 Summary

We conducted follow-up observations of the FRB 20201124A source using the
UDSC/JAXA radio telescope and detected one bright FRB. This is the first FRB
detection in Japan. The frequency of the detection, 2.2 – 2.3 GHz, is the
highest end of the multi-wavelength coverage of this repeating FRB so far
reported. In addition, the fluence at 2 GHz is comparable to the brightest
events detected from this FRB source at 1.3 GHz. By calculating the upper
limit of the spectral index of the detected burst, $\alpha=-2.14$, we compare
the event rate of the detected burst in this work with those of Marthi et al.
(2022), Atri et al. (2022), Ould-Boukattine et al. (2022b), and extrapolated
from Xu et al. (2021) under the assumption of single power-law of the
luminosity function and find that the event rates of the former three are
close to ours and notably larger than the event rates extrapolated from Xu et
al. (2021). It is suggested that the power-law of the luminosity function
might be broken at lower fluence, and bright FRBs distribute up to over 2 GHz
with the power-law. In addition, from comparison of energy densities, we
insist that some of the repeating FRBs intrinsically are classified as one-off
FRBs due to the limited observation time or limited sensitivity ; that is, the
classification based on observations may not represent the intrinsic physical
mechanisms completely.

We are very grateful to the anonymous referee for many insightful comments. We
thank the Usuda 64-m antenna operation support team in Space Tracking and
Communication Center and ISAS, JAXA. This work was supported in part by JSPS
KAKENHI Grant Nos. 21J00416(SY), 21H01078, 22K03681 (SK), 22H01267 (YM, TE,
MH, SK, YY). SY is supported by JSPS Research Fellowships for Young
Scientists. TH acknowledges the support of the National Science and Technology
Council of Taiwan through grants 110-2112-M-005-013-MY3, 110-2112-M-007-034-,
and 111-2123-M-001-008-.

## References

  * Ai et al. (2021) Ai, S., Gao, H., & Zhang, B. 2021, ApJ, 906, L5
  * Atri et al. (2022) Atri, P., Bilous, A., van Leeuwen, J., et al. 2022, Astron. Telegram, 15197, 1
  * Bethapudi et al. (2022) Bethapudi, S., Spitler, L. G., Main, R. A, et al. 2022, arXiv:2207.13669
  * Chawla et al. (2020) Chawla, P., Andersen, B. C., Bhardwaj, M. 2020, ApJ, 896, 41
  * Chen, B. H. et al. (2022) Chen, B. H., Hashimoto, T., Goto, T., et al. 2022, MNRAS, 509, 1227. doi:10.1093/mnras/stab2994
  * Chen, H.-Y. et al. (2022) Chen, H.-Y., Gu, W.-M., Sun, M., et al. 2022, arXiv:2209.13943
  * CHIME/FRB Collaboration (2019) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K., et al. 2019, ApJ, 885, L24. doi:10.3847/2041-8213/ab4a80
  * CHIME/FRB Collaboration (2020) CHIME/FRB Collaboration, Amiri, M., Andersen, B. C., et al. 2020, Nature, 582, 351
  * CHIME/FRB Collaboration (2021) CHIME/FRB Collaboration 2021, Astron. Telegram, 14497, 1
  * Cruces et al. (2021) Cruces, M., Spitler, L. G., Scholz, P., et al. 2021 MNRAS500, 448-463
  * Eie et al. (2021) Eie, S., Terasawa, T., Akahori, T., et al. 2021, PASJ, 73, 1563-1574
  * Enoto et al. (2021) Enoto, T., Terasawa, T., Kisaka, S., et al. 2021, Science, 372, 187-190, Supplementary Material
  * Fong et al. (2021) Fong, W.-. fai ., Dong, Y., Leja, J., et al. 2021, ApJ, 919, L23. doi:10.3847/2041-8213/ac242b
  * Gajjar et al. (2018) Gajjar, V., Siemion, A. P. V., Price, D. C., et al. 2018, ApJ, 863, 2. doi:10.3847/1538-4357/aad005
  * Gardenier et al. (2021) Gardenier, D. W., Connor, L., van Leeuwen, J., Oostrum, L. C., & Petroff, E. 2021, A&A, 647, A30
  * Hashimoto et al. (2020a) Hashimoto, T., Goto, T., Wang, T.-W., et al. 2020, MNRAS, 494, 2886-2904
  * Hashimoto et al. (2020b) Hashimoto, T., Goto, T., On, A. Y. L., et al. 2020, MNRAS, 498, 3927
  * Herrmann et al. (2021) Herrmann, W. 2021, Astron. Telegram, 14556, 1
  * Hilmarsson et al. (2021) Hilmarsson, G. H., Spitler, L. G., Main, R. A., et al. 2021, MNRAS, 508, 5354. doi:10.1093/mnras/stab2936
  * Hobbs et al. (2006) Hobbs, G. B., Edwards, R. T., Manchester, R. N. 2006, MNRAS, 369, 655. doi:10.1111/j.1365-2966.2006.10302.x
  * Jenet and Anderson (1998) Jenet, F. A., and Anderson, S. B. 1998, PASP, 110, 1467-1478
  * Josephy et al. (2019) Josephy, A., Chawla, P., Fonseca, E., et al. 2019, ApJ, 882, L18
  * Kim et al. (2022) Kim, S. J., Hashimoto, T., Chen, B. H., et al. 2022, MNRAS, 514, 5987. doi:10.1093/mnras/stac1689
  * Kumar et al. (2017) Kumar, P., Lu, W., & Bhattacharya, M. 2017, MNRAS, 468, 2726
  * Kumar et al. (2019) Kumar, P., Shannon, R. M., Osłowski, S., et al. 2019, ApJ, 887, L30. doi:10.3847/2041-8213/ab5b08
  * Lanman et al. (2022) Lanman, A. E., Andersen, B. C., Chawla, P., et al. 2022, ApJ, 927, 59
  * Levin et al. (2010) Levin, L., Bailes, M., Bates, S., et al. 2010, ApJ, 721, 33-37
  * Macquart et al. (2019) Macquart, J.-P., Shannon, R. M., Bannister, K. W., et al. 2019, ApJ, 872, L19. doi:10.3847/2041-8213/ab03d6
  * Main et al. (2022) Main, R. A., Hilmarsson, G. H., Marthi, V. R., et al. 2022, MNRAS, 509, 3172-3180
  * Marthi et al. (2022) Marthi, V. R., Bethapudi, S. ,Main, R. A., et al. 2022, MNRAS, 509, 2209-2219
  * Metzger et al. (2019) Metzger, B. D., Margalit, B., & Sironi, L. 2019, MNRAS, 485, 4091
  * Ould-Boukattine et al. (2022a) Ould-Boukattine, O. S., Kirsten, F., Nimmo, K., et al. 2022, Astron. Telegram, 15190, 1
  * Ould-Boukattine et al. (2022b) Ould-Boukattine, O. S., Kirsten, F., Nimmo, K., et al. 2022, Astron. Telegram, 15192, 1
  * Pastor-Marazuela et al. (2021) Pastor-Marazuela, I., Connor, L., van Leeuwen, J., et al. 2021, Nature, 596, 505
  * Perley and Butler (2017) Perley, R. A., and Butler, B. J. 2017, ApJS, 230, 7
  * Petroff et al. (2016) Petroff, E., Barr, E. D., Jameson, A., et al. 2016, PASA, 33, e045. doi:10.1017/pasa.2016.35
  * Petroff et al. (2022) Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2022, A&A Rev., 30, 2
  * Pleunis et al. (2021a) Pleunis, Z., Good, D. C., Kaspi, V. M. et al. 2021, ApJ, 923, 1
  * Pleunis et al. (2021b) Pleunis, Z., Michilli, D., Bassa, C. G. et al. 2021, ApJ, 911, 3
  * Ravi (2019) Ravi, V. 2019, Nature Astronomy, 3, 928. doi:10.1038/s41550-019-0831-y
  * Shannon et al. (2018) Shannon, R. M., Macquart, J.-P., Bannister, K. W., et al. 2018, Nature, 562, 386. doi:10.1038/s41586-018-0588-y
  * Takefuji et al. (2022) Takefuji, K., Murata, Y., Ikebe, S., et al. 2022, Astron. Telegram, 15285, 1
  * Takeuchi et al. (2006) Takeuchi, H., Kimura, M., Nakajima, J., et al. 2006, PASP, 850, 1739-1748
  * Xu et al. (2021) Xu, H., Niu, J. R., Chen, P., et al. 2021, arXiv:2111.11764


# Active resonator depletion with short microwave pulses

Sashank Kaushik Sridhar  Indian Institute of Technology Madras,
Chennai-600036, India  David P. DiVincenzo<EMAIL_ADDRESS>Peter
Grünberg Institute (PGI-2), Forschungszentrum Jülich, D-52425, Jülich, Germany

###### Abstract

We propose a physical model to explain the phenomenon of photon depletion in
superconducting microwave resonators in the dispersive regime, coupled to
Josephson junction qubits, via short microwave pulses. We discuss the
conditions for matching the amplitude and phase of the pulse optimally within
the framework of the model, allowing for significant reductions in reset times
after measurement of the qubits. We consider how to deal with pulses and
transient dynamics within the input-output formalism, along with a
reassessment of the underlying assumptions for a wide-band pulse.

##  I Introduction

Rapid measurements and rapid reset are crucial elements in the operation of
superconducting quantum computers. Modern circuit-QED architectures largely
rely on dispersive readout of qubits [1], i.e., measuring conditional shifts
in the resonance peaks of microwave resonators dispersively coupled to the
qubits. These superconducting microresonators typically have very high quality
factors Q and low decay rates $\kappa$, due to the small value of the
capacitive coupling to a Coplanar Waveguide (CPW) transmission line, also
called a feedline [2]. After measurement, the resonator must be depleted so as
to not affect the qubit state in further operations (via the AC Stark shift
[1]). However, typical depletion times are limited by the same decay rate,
leading to very slow passive reset. One proposal to tackle this is by sending
microwave pulses in through the feedline to interact with the photons in the
resonator and deplete it faster. The experiments [3, 4] have been promising,
although limited to an empirical understanding of the underlying physical
processes. While there have been efforts to understand the numerical
optimisation of the pulses better [5], our work focuses on establishing the
theory and the equations governing the evolution of the systems.

We provide a two-pronged explanation for this phenomenon; a classical model
and a quantum model, with perfect agreement between the two. Such an endeavour
also acts as a foot-in-the-door to explore the often neglected “short-pulse”
regime. We also express what ‘photon depletion’ in such a system really means,
so as to root the theory on solid footing for further explorations. We find
that the depletion time is entirely dependent on the pulse-width, and the
amplitude and phase of the pulse can be fixed based on the derived matching
conditions. Although limited to the dispersive approximation of the
superconducting qubit-resonator system, the analytical expressions can be used
as groundwork for further extensions that deal with nonlinear resonators and
highly non-classical states of radiation.

##  II Proposed Setup

We assume a series LC-oscillator to model the linear response of a gap-coupled
$\lambda/2$-transmission line resonator operating in its fundamental mode [6],
with the coupling strength controlled by adjusting the inductance or
capacitance. The time-dependent voltage of this LC resonator would correspond
to the peak voltage at the antinode in the transmission line resonator. The
resonator is coupled to a superconducting qubit within the dispersive
approximation with the Hamiltonian,

$\hat{\mathcal{H}}_{disp}=\hbar\omega_{r}^{\prime}\hat{a}^{\dagger}\hat{a}+\frac{\hbar\omega_{q}^{\prime}}{2}\hat{\sigma}_{z}+\hbar\chi\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z}$

where $\omega_{r}^{\prime}$ and $\omega_{q}^{\prime}$ are the effective
resonator and qubit frequencies respectively, and $\chi$ is the dispersive
shift (see Eq. (44) in [7]). When the qubit is measured,
$\langle\hat{\sigma}_{z}\rangle$ collapses to $\pm 1$, thus the resonator
frequency is higher or lower than $\omega_{r}^{\prime}$ by $\chi$ depending on
the measured qubit state. Furthermore, the resonator coupled to the qubit is
further capacitively coupled to the transmission lines via the circulator
ports. We deal with these coupled entities in our model by treating their
response as equivalent to a series LC-resonator with a resonant frequency that
is determined by the qubit state, since the effect of the capacitive coupling
to the feedline is also just a shift in the resonance peak [6]. This resonator
is then connected to the B-port of a circulator [6], with the input feedline
on the A-port and an output feedline on the C-port (see Fig. 1). The input
line is driven with a time-varying current source, which is meant to represent
the action of an arbitrary waveform generator. The output feedline carries the
output signal from the resonator to a detector circuit that reads out the
signal after amplification (usually done with a JPA [8] followed by a high
electron-mobility transistor).

##  III Classical Model

To understand the depletion phenomenon in the classical sense, we consider the
voltage amplitude of the resonator node and posit that this must go to zero
when depletion of photons occurs. This statement would become clearer when
dealing with the quantum model.
In our analysis, we assume that the transmission line is excited in the TEM
mode (practically quasi-TEM [9]), which reduces the modelling to a 1D problem.
We consider the flux $\Phi(x,t)$ as the main quantity of the transmission
line, as the voltage and current at any given position and instant can be
derived from partial derivatives of this flux variable. We assume $I(t)$ as
the time-varying current from the source, $l(c)$ is the distributed inductance
(capacitance) of the line and $v=1/\sqrt{lc}$ is the characteristic phase
velocity of the line which has a linear dispersion relation in the TEM mode.

From the Telegrapher’s equations of a lossless transmission line [6],

$\frac{\partial^{2}\Phi(x,t)}{\partial
t^{2}}=v^{2}\frac{\partial^{2}\Phi(x,t)}{\partial x^{2}}$ (1)

Figure 1: Schematic used in the model, with an arbitrary waveform generator
acting as a time-varying source that is directly coupled to a finite
transmission line. The output transmission line goes to a detector circuit and
the required non-reciprocity is enforced by an ideal circulator coupled to the
input, output and resonator nodes.

This gives a general solution consisting of a superposition of right and left-
propagating parts,

$\Phi(x,t)=\phi^{+}f^{+}(t-x/v)+\phi^{-}f^{-}(t+x/v)$

Taking the current source end of the line as $x=0$ and the circulator port as
$x=L_{c}$, we can assign boundary conditions to the flux.

$\frac{\Phi^{\prime}(0,t)}{l}=-I(t)$ (2)

where $\Phi^{\prime}\equiv\frac{\partial\Phi}{\partial x}$ and
$\dot{\Phi}\equiv\frac{\partial\Phi}{\partial t}$ are as per standard
notations.
Assuming no residual excitations in the input feedline prior to turning on the
current source, we then see that only the right-moving solution is allowed by
the current source boundary condition due to causality.

$I(t)=\frac{\phi^{+}}{vl}\frac{\mathrm{d}f^{+}}{\mathrm{d}t}(t-0/v)=\frac{\phi^{+}}{vl}\frac{\mathrm{d}f^{+}}{\mathrm{d}t}((t+x/v)-x/v)=-\frac{\Phi^{\prime}(x,t+x/v)}{l}=I_{in}(x,t+x/v)$
(3)

Thus, we can uniquely define the waveform that propagates down the line from
this boundary condition.
Here, we make the important assumption about the driving current that it has
finite support in time,

$I(t)=0\,,\quad\text{if }t<t_{0}-t_{w}\text{ or }t>t_{0}$ (4)

where $t_{0}<0$ and $t_{w}$ is the pulse-width. We assume
$t_{0}-t_{w}=-L_{c}/v$, so the pulse reaches the resonator at $t=0$.

The circulator acts as a directional element that blocks counter-propagation
of signals, i.e., signals only go cyclically from port A to port B, B to C and
C to A. The pulse is directed through the circulator to the resonator, where
the boundary conditions depend on both the input and output signals,

$\displaystyle I_{B}(t)=I_{in}(L_{c},t)-I_{out}(L_{c},t)$ (5) $\displaystyle
V_{B}(t)=V_{in}(L_{c},t)+V_{out}(L_{c},t)$ (6)

where $V_{B}(t)$ and $I_{B}(t)$ are the net voltage at port B and net current
flowing into the resonator respectively.
The charge in the capacitor of the resonator can be obtained from the
equation,

$\ddot{q_{r}}=-\omega_{r}^{2}q_{r}-\frac{V_{B}(t)}{L_{0}}$

where, $L_{0}$($C_{0}$) is the inductance (capacitance) of the resonator and
$\omega_{r}=1/\sqrt{L_{0}C_{0}}$ is the resonant frequency.
Using the fact that $\dot{q_{r}}=I_{B}(t)$ and $V_{in}(x,t)=Z_{0}I_{in}(x,t)$
(similarly for $V_{out}(x,t)$), we get

$\displaystyle\ddot{q_{r}}=-\omega_{r}^{2}q_{r}-\kappa\dot{q_{r}}-\frac{2V_{in}(L_{c},t)}{L_{0}}$
(7)
$\displaystyle\implies\ddot{V_{r}}=-\omega_{r}^{2}V_{r}-\kappa\dot{V_{r}}-2\omega_{r}^{2}Z_{0}I(t-L_{c}/v)$
(8)

where $\kappa=Z_{0}/L_{0}$ and $Z_{0}=\sqrt{\frac{l}{c}}$ is the
characteristic impedance of the feedline.

Thus, the resonator behaves as a damped harmonic oscillator with a time-
dependent drive. It is generally assumed that $\kappa<<\omega_{r}$ for high-Q
resonators that are weakly out-coupled to the feedline.
This can now be solved by taking the Laplace transform:

$s^{2}\tilde{V}_{r}[s]+sV_{r}(0^{-})-\dot{V_{r}}(0^{-})=-\omega_{r}^{2}\tilde{V}_{r}[s]+s\kappa\tilde{V}_{r}[s]+\kappa
V_{r}(0^{-})-2\omega_{r}^{2}Z_{0}\tilde{I}_{in}[s]$

where $\tilde{I}_{in}[s]$ is the Laplace transform of $I_{in}(L_{c},t)$, and
$I_{in}(L_{c},t)=I(t-L_{c}/v)$. Note that the sign convention used here is
opposite to what is the norm in most classical circuit-theory literature, so
as to stay consistent with the physics convention of $e^{-i\omega t}$.

Simplifying this gives,

$\tilde{V}_{r}[s]=\frac{\frac{\kappa}{2}V_{r}(0^{-})+\dot{V_{r}}(0^{-})}{s^{2}-s\kappa+\omega_{r}^{2}}-\frac{(s-\frac{\kappa}{2})V_{r}(0^{-})}{s^{2}-s\kappa+\omega_{r}^{2}}-2\omega_{r}^{2}Z_{0}\frac{\tilde{I}_{in}[s]}{s^{2}-s\kappa+\omega_{r}^{2}}$

Taking the limit $\kappa<<\omega_{r}$, the denominator reduces to
$(s+\kappa/2)^{2}+\omega_{r}^{2}$, giving us the time-domain expression:

$\displaystyle V_{r}(t)=e^{-\kappa
t/2}\Bigg{[}\left\\{\frac{\kappa}{2\omega_{r}}V_{r}(0^{-})+\frac{\dot{V_{r}}(0^{-})}{\omega_{r}}\right\\}\sin{\omega_{r}t}+V_{r}(0^{-})\cos{\omega_{r}t}\Bigg{]}+i\omega_{r}Z_{0}\left(\int_{0}^{t}{\mathrm{d}\tau
e^{(\frac{\kappa}{2}+i\omega_{r})(\tau-t)}I_{in}(L_{c},\tau)}\,-\,c.c\right)$
(9) $\displaystyle\implies V_{r}(t)=V_{nat}(t)+V_{dr}(t)$ (10)

Figure 2: a) Transient voltage at the resonator node for the pulse
$I_{in}(L_{c},t)=0.5\sin(\omega_{r}t+\theta_{p})$ for
$0\,\mathrm{ns}<t<10\,\mathrm{ns}$, with $\omega_{r}=2\pi\times
10\,\mathrm{GHz}$ and $\theta_{p}=\pi$, showing an almost linear drop till
$t_{w}$ after which it stays at a constant amplitude, b) the time variation of
$\arg\\{\tilde{I}[t]\\}$ defined in Eq. (14) for $\theta_{p}=\pi$, which
settles to $0$ and is thus matched with a resonator voltage that has a phase
of $(2n-1)\pi$ at $t=0\,\mathrm{ns}$.

If we take the reference time $t_{fill}<0$ when the resonator is excited with
photons from measuring the qubit(s), with an initial phase $\theta_{fill}$,

$V_{nat}(t)=V_{r}(t_{fill})e^{-\kappa(t-t_{fill})/2}\sin{(\omega_{r}(t-t_{fill})+\theta_{fill})}$

For $t\geq t_{w}$, we can write $V_{dr}(t)$ as:

$V_{dr}(t)=-2\omega_{r}Z_{0}e^{-\kappa
t/2}\mathrm{Im}\left(e^{-i\omega_{r}t}\tilde{I}_{in}\left[s=\frac{\kappa}{2}+i\omega_{r}\right]\right)=2\omega_{r}Z_{0}e^{-\kappa
t/2}\left|\tilde{I}_{in}\left[s=\frac{\kappa}{2}+i\omega_{r}\right]\right|\sin{(\omega_{r}t-\theta_{I})}$

where,
$\theta_{I}=\arg\left\\{\tilde{I}_{in}\left[s=\frac{\kappa}{2}+i\omega_{r}\right]\right\\}$.

By definition of the pulse in Eq. (4), $V_{dr}(t)$ attains a constant
amplitude with an overall exponential decay at $t=t_{w}$. Beyond this point,
if the amplitude of $V_{dr}(t)$ matches that of $V_{nat}(t)$ and the phases
differ by $(2n-1)\pi$, then the voltage in the resonator goes to zero, i.e.,
the resonator gets depleted. This lets us define a set of ’matching’
conditions:

$\displaystyle\left|\tilde{I}_{in}\left[s=\frac{\kappa}{2}+i\omega_{r}\right]\right|=\frac{\exp{(\kappa
t_{fill}/2)}}{2\omega_{r}Z_{0}}V_{r}(t_{fill})$ (11)
$\displaystyle\theta_{I}=(2n-1)\pi+\omega_{r}t_{fill}-\theta_{fill}$ (12)

This gives a good physical picture of a high-Q resonator acting as a filter
for the pulse, with an effective resonance peak at
$\omega=\omega_{r}-i\kappa/2$. Furthermore, the matching conditions give a
very clear picture of the protocol to follow. The resonator can initially be
excited for a qubit read-out at $t_{fill}$, through the same input feedline
and current source, with a weak microwave probe tone that is detected with a
heterodyne measurement [10]. It is possible to calculate and characterise the
resultant amplitude and phase of the resonator voltage at the end of the
protocol. One can thus send in the pulses that are matched to these quantities
to fully deplete the photons. The transient dynamics for a particular choice
of pulse are shown in Fig. 2.

Figure 3: Schematic describing the relevant instants of time in our model.
Note that the input and output lines are connected to the same port of the
resonator via a circulator. At $t=t_{fill}$, the qubit(s) is(are) measured,
exciting the resonator voltage. Subsequently, we see a decaying voltage from
the resonator that is characterised by $\kappa$. At $t=t_{0}-t_{w}=-L_{c}/v$,
the pulse is turned on and fully emitted by $t=t_{0}$. We set $t=0$ as the
reference for when the pulse is incident on the resonator, and the interaction
occurs till $t_{w}$, the pulse width. The output waveform is as given in
Section III.1.

###  III.1 Output Signal

One can verify the occurence of the depletion by detecting the output signal,
calculated from Eq. (5),

$I_{B}(t)=I_{in}(L_{c},t)-I_{out}(L_{c},t)=C_{0}\dot{V_{r}}(t)$ (13)

From Eq. (9),

$\dot{V_{r}}(t)=-\frac{\kappa}{2}V_{r}(t)+\omega_{r}^{2}\Bigg{[}\frac{V_{r}(t_{fill})}{\omega_{r}}e^{\kappa(\Delta
t)/2}\cos{(\omega_{r}(\Delta
t)+\theta_{fill})}+Z_{0}\left(\int_{0}^{t}{\mathrm{d}\tau
e^{(\frac{\kappa}{2}+i\omega_{r})(\tau-t)}I(\tau-L_{c}/v)}+c.c\right)\Bigg{]}$

where $\Delta t=t-t_{fill}$. We define,

$\tilde{I}[t]=\int_{0}^{t}{\mathrm{d}\tau
e^{(\frac{\kappa}{2}+i\omega_{r})\tau}I(\tau-L_{c}/v)}$ (14)
$\displaystyle\therefore\dot{V_{r}}(t)$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}V_{r}(t)+\omega_{r}\Bigg{[}V_{r}(t_{fill})e^{\kappa(\Delta
t)/2}\cos{(\omega_{r}(\Delta
t)+\theta_{fill})}+2Z_{0}\omega_{r}\big{|}\tilde{I}[t]\big{|}e^{-\kappa
t/2}\cos{\left(\omega_{r}t+\arg\\{\tilde{I}[t]\\}\right)}\Big{]}$
$\displaystyle=$
$\displaystyle\omega_{r}\Bigg{[}V_{r}(t_{fill})e^{\kappa(\Delta
t)/2}\cos{(\omega_{r}(\Delta
t)+\theta_{fill}+\theta_{r})}+2Z_{0}\omega_{r}\big{|}\tilde{I}[t]\big{|}e^{-\kappa
t/2}\cos{\left(\omega_{r}t+\arg\\{\tilde{I}[t]\\}+\theta_{r}\right)}\Bigg{]}$

where $\theta_{r}=\sin^{-1}\left(\frac{\kappa}{2\omega_{r}}\right)$.
Therefore, the out-flowing signal is given by:

$\displaystyle V_{out}(L_{c},t)\,(0\leq t\leq
t_{w})=Z_{0}\left(I(t-L_{c}/v)-C_{0}\dot{V_{r}}(t)\right)$
$\displaystyle=Z_{0}I(t-L_{c}/v)-\frac{\kappa}{\omega_{r}}\left[V_{r}(t_{fill})e^{\kappa(\Delta
t)/2}\cos(\omega_{r}(\Delta
t)+\theta_{fill}+\theta_{r})+2Z_{0}\omega_{r}\big{|}\tilde{I}[t]\big{|}e^{-\kappa
t/2}\cos\left(\omega_{r}t+\arg\\{\tilde{I}[t]\\}+\theta_{r}\right)\right]$

The output would show a weak, underdamped sinusoidal oscillation from the
photons present in the resonator prior to the pulse reaching it, followed by
the reflected pulse and no signal after that (amplitudes cancel out from the
matching conditions). The output contains a phase-shifted version of the
voltage transient of the resonator, which gives a clear indication of whether
the depletion has occured or not. The coefficient of $\kappa/\omega_{r}$
however leads to a very weak contribution from the resonator output as
compared to the reflected pulse, so the output signal must be analysed by
subtracting out the input pulse.

It is important to note that the same results can be derived by solving for
the normal modes of the transmission line. We use this very technique in the
next section to derive the photon picture, so we avoid mentioning it in the
classical model.

###  III.2 Motivation for Quantum Model

From the classical theory, depletion can be explained under the high-Q cavity
limit as interference between the pulse component that gets filtered into the
resonator and the standing wave mode present within it. This model can already
give us rich insights about the phenomenon and related observables, but it is
still unclear how the voltage amplitude connects to the photon number
depletion. Moreover, the model would be inherently insufficient in explaining
the depletion of photons in squeezed states of the resonator, which would be
useful for precision readout [11]. Thus, a more general quantum model would
serve to generalise and broaden the scope, further allowing one to potentially
probe the classicality of the radiation states in the resonator.

##  IV Quantum Model

###  IV.1 Quantisation in CQED

In vacuum quantum electrodynamics [12], the EM fields are spatially
represented in a Fourier basis, with the basis functions reinterpreted as
field operators and represented in terms of creation and annihilation
operators. The multimode EM radiation can then be viewed as an ensemble of
photons [13]. These spatial basis modes change depending on the system
properties and geometric constraints, such as in cavity QED [14, 15].

We utilise the method for quantisation as stated in [16] and [17], and use
[18] and the Appendix of [19] as a reference for the Input-Output Theory.
Recent work on Input-Output theory with quantum pulses [20] provides
motivation for a generalised framework to accommodate multi-mode quantum
radiation and its interaction with classical and quantum scatterers in the
presence of different loss channels. Along similar lines, we show the
treatment of short pulses in a superconducting transmission line as multi-mode
coherent states. A detailed discussion of the photon number in these short
pulses can be found in Appendix A.

Figure 4: Discretised representation of a lossless transmission line with
infinite coupled LC oscillators. The inductances and capacitances are taken as
differential elements that depend linearly on their size $\Delta x$, which
goes to $0$ in the continuum limit, while $N\to\infty$, and it is ultimately
coupled to the readout resonator, which is modeled by a finite inductance
$L_{0}$ and capacitance $C_{0}$. The flux variable for each node is marked by
the node index in parentheses.

First we will consider the system only till the $N^{th}$ node of the line,
without considering any coupling to the resonator. With the Kirchhoff current
equations at each node acting as the Hamilton equations of motion, we can
construct the system Hamiltonian,

$\mathcal{H}=\sum_{m=1}^{N}{\frac{q_{m}^{2}}{2c\Delta
x}}+\sum_{m=2}^{N}{\frac{(\phi_{m}-\phi_{m-1})^{2}}{2l\Delta x}}-I(t)\phi_{1}$
(15)

where $(\phi_{i},q_{i})$ are canonically conjugate variables of the flux and
charge at each node. The summations can be rewritten as matrices, with the
inverse capacitance matrix proportional to $\mathbb{I}$. The inverse
inductance matrix is symmetric and is guaranteed to be diagonalisable.
After diagonalisation,

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{0}+\sum_{j=1}^{N-1}{\left(\frac{\tilde{q}_{j}^{2}}{2c\Delta
x}+\frac{c_{j}\tilde{\phi}_{j}^{2}}{2l\Delta
x}-I(t)\mathbf{O}_{j1}\tilde{\phi}_{j}\right)}$ (16) $\displaystyle=$
$\displaystyle\mathcal{H}_{0}+\sum_{j=1}^{N-1}{\mathcal{H}_{j}}$ (17)
$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\frac{\tilde{q}_{0}^{2}}{2c\Delta
x}-I(t)\frac{\tilde{\phi}_{0}}{\sqrt{N}}$ (18) $\displaystyle\omega_{j}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{c_{j}}}{\Delta
x\sqrt{lc}}=\frac{v\sqrt{c_{j}}}{\Delta
x}=\frac{v}{L_{c}}(N\sqrt{c_{j}})=vk_{j}$ (19)

where, $c_{j}$ is the eigenvalue of the inverse inductance matrix,
$(\tilde{\phi}_{j},\tilde{q}_{j})$ are the transformed canonically conjugate
variables of the normal modes and $\mathbf{O}_{jm}$ is the element of the
orthogonal matrix which implements the canonical transformation
$\overrightarrow{\mathbf{\tilde{\phi}}}=\mathbf{O}\,\overrightarrow{\phi}$,
where $\overrightarrow{\phi}$ and $\overrightarrow{\mathbf{\tilde{\phi}}}$ are
the vectors of flux variables at each node and in each normal mode,
respectively.

Taking the continuum limit for our discrete model, i.e., $N\to\infty$ and
$\Delta x\to 0$ such that $m\Delta x=x$ and $N\Delta x=L_{c}$,

$\displaystyle\mathbf{O}_{jm}$ $\displaystyle=$ $\displaystyle
O_{j}\cos(jm\pi/N)\approx\sqrt{\frac{2}{N}}\cos\left(\frac{jx\pi}{L_{c}}\right)$
$\displaystyle\sqrt{c_{j}}$ $\displaystyle\approx$
$\displaystyle\frac{j\pi}{N}\implies k_{j}=\frac{j\pi}{L_{c}}$
$\displaystyle\therefore\mathbf{O}_{jm}$ $\displaystyle\approx$
$\displaystyle\sqrt{\frac{2}{N}}\cos(k_{j}x)$

Numerics for $N=21$ in Fig. 5 gives results in line with the continuum limit.
We show the square-root of the eigenvalue spectrum (proportional to the mode
frequencies) and the first three eigenmodes, ordered by the node indices. The
zeroth mode corresponds to the DC term, i.e., the sum over all the charges or
fluxes of the line. Its contribution in the following can be neglected by
choosing the pulse such that its time integral goes to zero. Fig. 5 shows an
initial linear behaviour, beyond which it tapers out. This behaviour is
typical of periodic systems with discrete translational symmetries, like cubic
lattices. When taking a continuum limit, the linear regime becomes the
relevant part of the spectrum as $\omega_{N-1}\to\infty$.

We can now diagonalize the Hamiltonian by defining the creation and
annihilation operators in the normal mode basis:

$\displaystyle\tilde{\phi}_{j}(j\geq 1)=\sqrt{\frac{\hbar
Z_{0}}{2\sqrt{c_{j}}}}\,(\hat{a}_{j}+\hat{a}_{j}^{\dagger})$ (20)
$\displaystyle\tilde{q}_{j}(j\geq
1)=\frac{1}{i}\sqrt{\frac{\hbar\sqrt{c_{j}}}{2Z_{0}}}\,(\hat{a}_{j}-\hat{a}_{j}^{\dagger})$
(21)
$\displaystyle\therefore\mathcal{H}_{j}=\hbar\omega_{j}(\hat{a}_{j}^{\dagger}\hat{a}_{j}+1/2)-\sqrt{\frac{\hbar
Z_{0}\omega_{c}}{\pi\omega_{j}}}I(t)(\hat{a}_{j}+\hat{a}_{j}^{\dagger})$ (22)

This represents a multi-mode harmonic oscillator with a linear drive term,
perfectly analogous to a multi-mode driven cavity [15]. The driving can be
represented by a displacement operator, giving rise to a coherent state in
each mode of the transmission line.

$\displaystyle\dot{\hat{a}}_{j}(t)=i\omega_{j}\hat{a}_{j}+i\sqrt{\frac{Z_{0}\omega_{c}}{\pi\hbar\omega_{j}}}I(t)\hat{\mathbb{I}}$
(23)
$\displaystyle\implies\langle\hat{a}_{j}(t)\rangle=\alpha_{j}(t)=\sqrt{N}C_{j}(t)=i\sqrt{\frac{Z_{0}\omega_{c}}{\pi\hbar\omega_{j}}}e^{-i\omega_{j}t}\int_{t_{0}-t_{w}}^{t}{\mathrm{d}\tau
e^{i\omega_{j}\tau}I(\tau)}$ (24)
$\displaystyle\implies\alpha_{j}(t)=\alpha_{j}(t_{0})e^{-i\omega_{j}(t-t_{0})}\quad(t\geq
t_{0})$ (25)

where $t_{0}$ and $t_{w}$ have been defined in (4), $C_{j}$ is the normalised
coefficient of the annihilation operator of each mode (refer to Eq. (47)) and
$N$ is the total average photon number of the multi-mode coherent state.

Thus, at $t=t_{0}$ the pulse is fully formed, and the multimode coherent state
amplitude is fixed and evolves freely under the Harmonic oscillator
Hamiltonian until $t=0$, after which the interaction with the resonator takes
place.

Figure 5: a) Square root of the eigenvalue of each normal mode which is
proportional to the normal mode frequency and shows characteristic behaviour
of periodic systems, b) The normalised eigenmode with an eigenvalue $0$,
corresponding to the “DC” mode which is a normalised sum over all the node
variables, whose EOM provides a statement of charge/flux conservation; c) and
d) are the first and second eigenmodes of the transmission line, where one can
see the standing waves even in the discretised line. All data is plotted for
$N=21$.

###  IV.2 Coupling to resonator

With the pulse displacing the transmission line modes to a multi-mode coherent
state, we can now write the full Hamiltonian with the coupling to the
resonator:

$\displaystyle\mathcal{H}=\mathcal{H}_{0}+\sum_{j=1}^{\infty}\left(\mathcal{H}_{j}+\mathcal{H}_{coup,j}\right)+\mathcal{H}_{r}$
(26)
$\displaystyle\sum_{j=1}^{\infty}\mathcal{H}_{coup,j}=-\sqrt{\frac{2}{N}}\frac{\phi_{r}}{L_{0}}\sum_{j=1}^{\infty}(-1)^{j}\tilde{\phi}_{j}=-\frac{\phi_{r}\Phi(L_{c},t)}{L_{0}}$
(27)
$\displaystyle\mathcal{H}_{r}=\frac{\phi_{r}^{2}}{2L_{0}}+\frac{q_{r}^{2}}{2C_{0}},\,\omega_{r}=\frac{1}{\sqrt{L_{0}C_{0}}}$
(28)

Thus, we can see that there is a coupling term between the resonator mode and
every mode of the transmission line, with the $(-1)^{j}$ term coming from the
parity of the eigenmode at the end of the line
($\mathbf{O}_{jN}=(-1)^{j}\sqrt{2/N}$ ). The complete Hamiltonian would also
include the coupling to the output feedline modes, but we neglect that in our
expression for brevity, keeping the coupling bidirectional here and enforcing
the non-reciprocity in the equations of motion (an in-depth treatment of non-
reciprocal elements can be found in [21]).

Writing these in second quantization:

$\displaystyle\mathcal{H}_{r}=\hbar\omega_{r}(\hat{a}_{r}^{\dagger}\hat{a}_{r}+1/2)$
(29)
$\displaystyle\mathcal{H}_{coup,j}=-\hbar(-1)^{j}\sqrt{\frac{Z_{0}\omega_{r}\omega_{c}}{2\pi\omega_{j}L_{0}}}(\hat{a}_{j}+\hat{a}_{j}^{\dagger})(\hat{a}_{r}+\hat{a}_{r}^{\dagger})$
(30)

We can simplify the Hamiltonian by taking the Rotating Wave Approximation for
$\mathcal{H}_{coup,j}$. This is justified by Eq. (45); the resonator loss rate
$\kappa$ goes inversely as the inductance $L_{0}$ of the LC resonator in our
model, which can be tuned to satisfy $\kappa<<\omega_{r}$ in the dispersive
approximation.

$\displaystyle\mathcal{H}_{coup,j}$ $\displaystyle=$
$\displaystyle-i\hbar(f_{j}\hat{a}_{j}\hat{a}_{r}^{\dagger}-f_{j}^{*}\hat{a}_{j}^{\dagger}\hat{a}_{r})$
(31) $\displaystyle f_{j}$ $\displaystyle=$
$\displaystyle-i(-1)^{j}\sqrt{\frac{Z_{0}\omega_{r}\omega_{c}}{2\pi\omega_{j}L_{0}}}$
(32)

The key feature is that the coupling terms come into the picture only when
$\langle\Phi(L_{c},t)\rangle$ is non-zero. Despite the photon states of each
mode being excited at $t=t_{0}-t_{w}$, they do not interact with the resonator
modes until $t=0$ due to causality (see Eq. (E58) in the appendices of [19]).
However, the resonator mode starts to interact with the modes of the output
feedline as soon as the measurement happens at $t_{fill}$. A visual
representation of this can be viewed in Fig. 3.

###  IV.3 Depletion of photons from the resonator mode

We define $\hat{b}_{j}$ to be the annihilation operator of the standing wave
modes in the output feedline, distinct from $\hat{a}_{j}$. The coupled
equations of motion for the second quantised operators are given by,

$\displaystyle\dot{\hat{a}}_{r}=-i\omega_{r}\hat{a}_{r}-\sum_{j=1}^{\infty}f_{j}\hat{b}_{j}\quad(t_{fill}\leq
t<0)$ (33)
$\displaystyle\hat{b}_{j}(t)=f_{j}^{*}e^{-i\omega_{j}t}\int_{t_{fill}}^{t}\mathrm{d}\tau
e^{i\omega_{j}\tau}\hat{a}_{r}(\tau)$ (34)
$\therefore\dot{\hat{a}}_{r}=-i\omega_{r}\hat{a}_{r}-\sum_{j=1}^{\infty}|f_{j}|^{2}\int_{t_{fill}}^{t}\mathrm{d}\tau
e^{-i(\omega_{j}-\omega_{r})(t-\tau)}[e^{i\omega_{r}(t-\tau)}\hat{a}_{r}(\tau)]$
(35)

We can now define the loss rate of the cavity $\kappa$ from Fermi’s Golden
rule, as per Eq. E24 in [19].

$\kappa(\omega_{r})=2\pi\sum_{j}|f_{j}|^{2}\delta(\omega_{r}-\omega_{j})$ (36)

The frequency dependence of $\kappa$ can be neglected under the Markov
approximation [22], supported further by the restriction to an LC-resonator
with a single resonant mode. From this, we end up with

$\sum_{j=1}^{\infty}|f_{j}|^{2}e^{-i(\omega_{j}-\omega_{r})(t-\tau)}=\kappa\delta(t-\tau)$
(37)

Plugging this into Eq. (35),

$\dot{\hat{a}}_{r}=-(i\omega_{r}+\kappa/2)\,\hat{a}_{r}(t)$ (38)

When the pulse is incident on the resonator,

$\dot{\hat{a}}_{r}=-(i\omega_{r}+\kappa/2)\,\hat{a}_{r}(t)-\sum_{j=1}^{\infty}f_{j}\hat{a}_{j}(t)\quad(0\leq
t<t_{w})$ (39)
$\begin{split}\implies\langle\hat{a}_{r}(t)\rangle=\alpha_{r}(t)=\,&e^{-(i\omega_{r}+\kappa/2)t}\alpha_{r}(0^{-})\\\
-\,&e^{-(i\omega_{r}+\kappa/2)t}\sum_{j=1}^{\infty}f_{j}e^{-i\omega_{j}L_{c}/v}\int_{0}^{t}\mathrm{d}\tau
e^{(i\omega_{r}+\kappa/2)\tau}\left(i\sqrt{\frac{Z_{0}\omega_{c}}{\pi\hbar\omega_{j}}}\right)e^{-i\omega_{j}\tau}\left(\int_{0}^{t_{w}}\mathrm{d}\tau^{\prime}e^{i\omega_{j}\tau^{\prime}}I(\tau^{\prime}-L_{c}/v)\right)\end{split}$
(40)

We define $s_{0}=i\omega_{r}+\kappa/2$. Using (31) and the fact that
$e^{-i\omega_{j}L_{c}/v}=e^{-ij\pi}=(-1)^{j}$,

$\alpha_{r}(t)=e^{-s_{0}t}\left[\alpha_{r}(0^{-})-\sqrt{\frac{2L_{0}}{\hbar\omega_{r}}}\int_{0}^{t}\mathrm{d}\tau
e^{s_{0}\tau}\int_{0}^{t_{w}}\mathrm{d}\tau^{\prime}I(\tau^{\prime}-L_{c}/v)\sum_{j=1}^{\infty}|f_{j}|^{2}e^{-i\omega_{j}(\tau-\tau^{\prime})}\right]$
(41)

Substituting Eq. (37),

$\alpha_{r}(t)=e^{-s_{0}t}\left[\alpha_{r}(0^{-})-\kappa\sqrt{\frac{2L_{0}}{\hbar\omega_{r}}}\int_{0}^{t}\mathrm{d}\tau
e^{s_{0}\tau}I(\tau-
L_{c}/v)\right]=e^{-s_{0}t}\left[\alpha_{r}(0^{-})-\kappa\sqrt{\frac{2L_{0}}{\hbar\omega_{r}}}\tilde{I}(t)\right]$
(42)

Thus at $t=t_{w}$, for full depletion:

$\alpha_{r}(0^{-})=\kappa\sqrt{\frac{2L_{0}}{\hbar\omega_{r}}}\tilde{I}_{in}\left[s=\frac{\kappa}{2}+i\omega_{r}\right]$
(43)

Depletion is thus shown to occur within the generalised Input-Output
framework, as is elucidated by the matching condition defined above.

###  IV.4 Consistency with Classical picture

We now possess the required information to justify our original claim that
photon depletion in the classical model can be viewed as the drop in the
voltage amplitude of the resonator node.
From the quantisation of the resonator mode, we can get $\langle
q_{r}(t)\rangle$, and from $\langle q_{r}(t)\rangle=C_{0}\langle
V_{r}(t)\rangle$,

$\langle
V_{r}(t)\rangle=\sqrt{\frac{2\hbar\omega_{r}}{C_{0}}}\mathrm{Im}\\{\alpha_{r}(t)\\}$
(44)

Rewriting $|\alpha_{r}(0^{-})|$ as $|\alpha_{r}(t_{fill})|e^{\kappa
t_{fill}/2}$ and defining $\theta_{fill}=\pi-\arg\\{\alpha_{r}(t_{fill})\\}$,

$\langle
V_{r}(t)\rangle=\sqrt{\frac{2\hbar\omega_{r}}{C_{0}}}|\alpha_{r}(t_{fill})|e^{-\kappa(t-t_{fill})/2}\sin(\omega_{r}(t-t_{fill})+\theta_{fill})+2\kappa\sqrt{\frac{L_{0}}{C_{0}}}|\tilde{I}(t)|\sin(\omega_{r}t-\arg\\{\tilde{I}(t)\\})$

Using Eq. (36) and taking the approximation $\omega_{c}<<\omega_{r}$,

$\begin{split}\kappa&=\sum_{j=1}^{\infty}\frac{\omega_{c}\omega_{r}Z_{0}}{\omega_{j}L_{0}}\delta(\omega_{r}-\omega_{j})\\\
&\approx\frac{\omega_{r}Z_{0}}{L_{0}}\int_{0}^{\infty}\mathrm{d}\omega\frac{\delta(\omega_{r}-\omega)}{\omega}\\\
&=\frac{Z_{0}}{L_{0}}\end{split}$ (45)

Substituting Eq. (45),

$\langle
V_{r}(t)\rangle=V_{r}(t_{fill})e^{-\kappa(t-t_{fill})/2}\sin(\omega_{r}(t-t_{fill})+\theta_{fill})+2Z_{0}\omega_{r}e^{-\kappa
t/2}|\tilde{I}(t)|\sin(\omega_{r}t-\arg\\{\tilde{I}(t)\\})$ (46)

Thus, we recover the classical equation for the resonator voltage, along with
perfect agreement in the loss rate of the resonator. This establishes the
self-consistency of the model and allows further extensions to non-classical
regimes.

###  IV.5 The short-pulse regime

A salient feature of the wide-band Input-Output theory is the coupling
coefficient $f_{j}$ not being taken as a constant, as shown in Eq. (32). It is
shown to vary as $\sqrt{\frac{1}{\omega_{j}}}$ along with a $(-1)^{j}$ factor
arising from the parity at the $N^{th}$ node for each eigenmode. It must be
noted that the nature of this coupling coefficient would change if instead of
the series LC-resonator in Fig. 1 and Fig. 4, we model the resonator as a
parallel LC-resonator that is capacitively coupled to the transmission line,
leading to a proportionality to $\sqrt{\omega_{j}}$ which causes a divergence
in the total photon number of the pulse. This divergence has been studied in
depth [23] and is dealt with by looking at the exact nature of the canonical
transformation and thereby defining an ultraviolet cutoff for the modes under
consideration. Moreover, the divergence in zero point fluctuations for this
multi-mode system is resolved when accounting for the detectors having finite
bandwidths.

As seen in the classical model and its correspondence with the quantum
picture, the relevant observables are the voltage operators of the
transmission lines and the resonator. One can analyse the power spectral
density of the voltage pulse and follow the steps in Appendix A to calculate
the average number of photons per mode. The question then arises - where are
these photons?

When a microwave pulse is mixed with an LO and low-pass filtered, the pulse’s
central frequency gets downshifted to allow the ADC to convert and detect the
signal. The bandwidth of the detector plays a key role in characterising the
quantum noise and accurately detecting the signal. Usually, this is given by
the sampling rate of the ADC, which needs to be at least twice the bandwidth
of the pulse to detect it accurately (from the Shannon-Nyquist Sampling
Theorem). The idea of time-resolved photon counting begins to break down for
such broadband detectors, as the expression for the photon flux
$\phi=P/\hbar\omega_{central}$ no longer remains valid.

We can, however, conclude that the off-resonant photons lie strictly within
the pulse because of causality. This is easily demonstrated by calculating a
Fourier Transform of the detected signal at different time-steps. When the
resonator depletes passively before the arrival of the depletion pulse, the
spectrum shows a Lorentzian peak that converges to the narrow resonator
spectrum as more samples are included. When the reflected pulse arrives at the
detector, we see a wide-band function superimposed on the resonant peak, which
converges to the spectrum of the output pulse after a time $t_{w}$. Clearly,
the photons constituting the broader band arrived only after the pulse was
detected.

The existence of this uncertainty relationship between time and frequency in
the detection scheme drives home the idea that these short pulses clearly
exhibit pure wave-like behaviour in these systems under the classical limit,
with no information about where the photons are, other than within the
confines of the pulse width.

##  V Conclusion and Outlook

We have thus defined a possible physical explanation for the depletion
phenomenon and provided a set of matching conditions to engineer the pulses
for maximum depletion. The flexibility of the model lies in the freedom to
choose a pulse shape. In prior literature, specific pulse shapes have achieved
better results than others [4]. The most important utility of this protocol is
the elimination of a large bottleneck in the operation of superconducting
quantum computers. There is a major reduction in the depletion time as
compared to passive resonator loss. After the measurement is completed, the
pulse is conditioned on the measured qubit state, taking the time $L_{c}/v$ to
reach the resonator and $t_{w}$ to deplete it. Thus, when the pulses are
shorter than, but comparable in their width to the length of the transmission
line, i.e., $vt_{w}\lesssim L_{c}$, the total time it takes to deplete the
resonator after measurement is $L_{c}/v+t_{w}\sim t_{w}$. It is practical to
construct transmission lines with delay times $\sim 100/\omega_{r}$, and with
$1/\kappa\sim 10^{4}/\omega_{r}$, there is potential for improving depletion
times by more than a factor of $100$ (compared to current long-pulse
experiments [3] with reductions by $\sim 6/\kappa$). Thus, for
$\omega_{r}\approx 10\mathrm{GHz}$, we can reduce depletion times from
$10/\kappa\approx 10\mathrm{\mu s}$ to tens of nanoseconds, now comparable to
the gate times of the qubits.

The presence of higher modes in transmission line resonators has not been
dealt with in this work, however a generalisation can be made to higher modes
by simply coupling independently to higher frequency LC-resonators or limiting
the pulse width to minimise the frequency components at higher modes. Possible
extensions for this theory may include other kinds of cavities and opto-
mechanical systems.

##  Appendix A Photon number of a short Gaussian pulse

We discuss a technique to derive the photon number of each mode of a wide-band
Gaussian pulse. This can easily be generalised to other pulse shapes and
system-dependent eigenmode bases.

Based on the definition of photons in Multi-mode Quantum Optics, we can write
the overall creation operator for the pulse as a superposition of densely-
spaced discrete harmonic modes $\hat{a}^{\dagger}_{\omega_{i}}$:

$\displaystyle\hat{a}^{\dagger}_{pulse}=\displaystyle\sum_{i}C_{i}\hat{a}^{\dagger}_{\omega_{i}}$
(47) $\displaystyle C_{i}=$
$\displaystyle\displaystyle\frac{\sqrt{\delta\omega}\left[\exp\\{-(\omega_{i}-\omega_{p})^{2}/4W^{2}\\}+\exp\\{-(\omega_{i}+\omega_{p})^{2}/4W^{2}\\}\right]}{\sqrt{c_{n}}}$
(48)

where $\omega_{p}$ is the center frequency, $W$ is the bandwidth of the pulse,
$\omega_{i}$ is the frequency of the mode $i$, $\delta\omega$ is the
difference between frequencies of successive modes and $c_{n}$ is the
normalisation constant.

This expression comes from the Fourier transform of a pulse with a Gaussian
profile. In the narrow-band regime, as is considered in the Appendix of [24],
it is implicitly assumed that $\omega_{p}/W>>1$, and hence one can directly
take the weight of each mode to be,

$C_{i}=\frac{\sqrt{\delta\omega}\exp\\{-(\omega_{i}-\omega_{p})^{2}/4W^{2}\\}}{(2\pi
W^{2})^{1/4}}$ (49)

and safely integrate over just one of the Gaussians in the spectrum.

Here, we will assume $\omega_{p}$ to be comparable in magnitude to $W$, i.e.,
the pulse is short, so that there is significant contribution to the photon
number from the low-frequency modes (even the ’zero-frequency’ mode) due to
the overlap between the two Gaussians.

The weights $C_{i}$ are normalised to unity in the limit of continuum of modes
as,

$\begin{split}\sum|C_{i}|^{2}\approx\displaystyle\frac{1}{c_{n}}\int_{0}^{\infty}\mathrm{d}\omega\left(e^{-(\omega-\omega_{p})^{2}/4W^{2}}+e^{-(\omega+\omega_{p})^{2}/4W^{2}}\right)^{2}=1\\\
\implies\displaystyle\frac{1}{c_{n}}\int_{0}^{\infty}\mathrm{d}\omega\left(e^{-(\omega-\omega_{p})^{2}/2W^{2}}+e^{-(\omega+\omega_{p})^{2}/2W^{2}}+2e^{-\omega^{2}/2W^{2}}e^{-\omega_{p}^{2}/2W^{2}}\right)=1\end{split}$
(50)

We solve this integral by exploiting the even nature of the integrand and
mirroring the limits to cover the full real line. We end up with:

$c_{n}=\sqrt{2\pi W^{2}}\left(1+e^{-\omega_{p}^{2}/2W^{2}}\right)$ (51)

The pulse is constructed with multiple modes of coherent photons, with each
mode contributing an average photon number $n_{i}=|C_{i}|^{2}N$, where $N$ is
the total number of photons in the pulse. Thus,
$\displaystyle\sum_{i}n_{i}=N\displaystyle\sum_{i}|C_{i}|^{2}=N$.
For a single mode of coherent radiation, the average energy contained in the
mode is given by $n_{i}\hbar\omega_{i}$, so the total energy in the pulse is
given by:

$\langle
E\rangle=\displaystyle\sum_{i}n_{i}\hbar\omega_{i}=N\displaystyle\sum_{i}\hbar|C_{i}|^{2}\omega_{i}$
(52)

Figure 6: a) Plot of $|C(\omega)|^{2}$ vs $\omega$ for a unit amplitude pulse
with $\omega_{p}=1.414\times 10^{10}$ and $W=1\times 10^{10}$, which shows
clear difference in behaviour at different CEO phases due to varying
contributions from the low-frequency photons. b) Plot of $F(\theta,z)$ vs $z$
for different CEO phases. As shown, it clearly settles to the expected narrow
band value for large $z$.

Taking continuum of modes:

$\begin{split}\langle
E\rangle&=\frac{N\hbar}{c_{n}}\int_{0}^{\infty}\omega\mathrm{d}\omega\left(e^{-(\omega-\omega_{p})^{2}/2W^{2}}+e^{-(\omega+\omega_{p})^{2}/2W^{2}}+2e^{-\omega^{2}/2W^{2}}e^{-\omega_{p}^{2}/2W^{2}}\right)\\\
&=\frac{N\hbar}{c_{n}}\int_{0}^{\infty}\omega\mathrm{d}\omega\left(e^{-(\omega-\omega_{p})^{2}/2W^{2}}+e^{-(\omega+\omega_{p})^{2}/2W^{2}}\right)+\displaystyle\frac{2N\hbar
e^{-\omega_{p}^{2}/2W^{2}}}{c_{n}}\int_{0}^{\infty}\omega
e^{-\omega^{2}/2W^{2}}\mathrm{d}\omega\\\
&=\left\\{\frac{2N\hbar\omega_{p}}{c_{n}}\sqrt{\frac{\pi
W^{2}}{2}}\mathrm{erf}\left(\frac{\omega_{p}}{W\sqrt{2}}\right)+\frac{2N\hbar
W^{2}}{c_{n}}\mathrm{exp}\left(\frac{-\omega_{p}^{2}}{2W^{2}}\right)\right\\}+\frac{4N\hbar
W^{2}}{c_{n}}\mathrm{exp}\left(\frac{-\omega_{p}^{2}}{2W^{2}}\right)\\\
&=N\hbar\omega_{p}\left\\{\frac{\mathrm{erf}\left(\frac{\omega_{p}}{W\sqrt{2}}\right)+\frac{3W\sqrt{2}}{\sqrt{\pi}\omega_{p}}\mathrm{exp}\left(\frac{-\omega_{p}^{2}}{2W^{2}}\right)}{\mathrm{exp}\left(\frac{-\omega_{p}^{2}}{2W^{2}}\right)+1}\right\\}\end{split}$
(53)

Let $z=\frac{\omega_{p}}{W\sqrt{2}}$,

$\implies\langle E\rangle=\displaystyle
N\hbar\omega_{p}\left\\{\frac{\mathrm{erf}(z)+\frac{3}{z\sqrt{\pi}}\mathrm{exp}(-z^{2})}{\mathrm{exp}(-z^{2})+1}\right\\}$
(54)

One can calculate the energy $\bar{E}$ of a pulse with classical methods, so
it is usually a known quantity. Therefore,

$N=\frac{\bar{E}}{\hbar\omega_{p}}\left\\{\frac{\mathrm{exp}(-z^{2})+1}{\mathrm{erf}(z)+\frac{3}{z\sqrt{\pi}}\mathrm{exp}(-z^{2})}\right\\}$
(55)

We can now generalise the expression by introducing a non-zero carrier-
envelope offset (CEO) phase $\theta$ [25]:

$\begin{split}N&=\frac{\bar{E}}{\hbar\omega_{p}}\left\\{\frac{\mathrm{cos}(2\theta)\mathrm{exp}(-z^{2})+1}{\mathrm{erf}(z)+\frac{(2cos(\theta)+1)}{z\sqrt{\pi}}\mathrm{exp}(-z^{2})}\right\\}\\\
&=\frac{\bar{E}}{\hbar\omega_{p}}F(\theta,z)\end{split}$ (56)

where $\theta$ is the CEO phase. The value of $F(\theta,z)$ approaches $1$ for
large $z$, as the pulse becomes longer in time and approaches the narrow-band
limit. The plots for $|C_{i}|^{2}$ and $F(\theta,z)$ are given in Fig. 6.

## 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, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162 (2004).
  * Bultink  _et al._ [2016] C. C. Bultink, M. A. Rol, T. E. O’Brien, X. Fu, B. C. S. Dikken, C. Dickel, R. F. L. Vermeulen, J. C. de Sterke, A. Bruno, R. N. Schouten, and L. DiCarlo, Active resonator reset in the nonlinear dispersive regime of circuit qed, Phys. Rev. Applied 6, 034008 (2016).
  * McClure  _et al._ [2016] D. T. McClure, H. Paik, L. S. Bishop, M. Steffen, J. M. Chow, and J. M. Gambetta, Rapid driven reset of a qubit readout resonator, Phys. Rev. Applied 5, 011001 (2016).
  * Boutin  _et al._ [2017] S. Boutin, C. K. Andersen, J. Venkatraman, A. J. Ferris, and A. Blais, Resonator reset in circuit qed by optimal control for large open quantum systems, Phys. Rev. A 96, 042315 (2017).
  * Pozar [2011] D. Pozar,  _Microwave Engineering, 4th Edition_ (Wiley, 2011).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Yurke  _et al._ [1988] B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whittaker, A. D. Smith, A. H. Silver, and R. W. Simon, Observation of 4.2-k equilibrium-noise squeezing via a josephson-parametric amplifier, Phys. Rev. Lett. 60, 764 (1988).
  * Wen [1969] C. Wen, Coplanar waveguide: A surface strip transmission line suitable for nonreciprocal gyromagnetic device applications, IEEE Transactions on Microwave Theory and Techniques 17, 1087 (1969).
  * Salathé  _et al._ [2018] Y. Salathé, P. Kurpiers, T. Karg, C. Lang, C. K. Andersen, A. Akin, S. Krinner, C. Eichler, and A. Wallraff, Low-latency digital signal processing for feedback and feedforward in quantum computing and communication, Phys. Rev. Applied 9, 034011 (2018).
  * Barzanjeh  _et al._ [2014] S. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, Dispersive qubit measurement by interferometry with parametric amplifiers, Phys. Rev. B 90, 134515 (2014).
  * Dirac [1927] P. A. M. Dirac, The quantum theory of the emission and absorption of radiation, Proc. R. Soc. Lond. A 114, 243–265 (1927).
  * Titulaer and Glauber [1966] U. M. Titulaer and R. J. Glauber, Density operators for coherent fields, Phys. Rev. 145, 1041 (1966).
  * Walther  _et al._ [2006] H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Reports on Progress in Physics 69, 1325 (2006).
  * Kimble [1998] H. J. Kimble, Strong interactions of single atoms and photons in cavity QED, Physica Scripta T76, 127 (1998).
  * Wallquist  _et al._ [2006] M. Wallquist, V. S. Shumeiko, and G. Wendin, Selective coupling of superconducting charge qubits mediated by a tunable stripline cavity, Phys. Rev. B 74, 224506 (2006).
  * Vool and Devoret [2017] U. Vool and M. Devoret, Introduction to quantum electromagnetic circuits, International Journal of Circuit Theory and Applications 45, 897–934 (2017).
  * Gardiner and Zoller [2000] C. W. Gardiner and P. Zoller,  _Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics_ (Springer, 2000).
  * Clerk  _et al._ [2010] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  * Kiilerich and Mølmer [2019] A. H. Kiilerich and K. Mølmer, Input-output theory with quantum pulses, Phys. Rev. Lett. 123, 123604 (2019).
  * Parra-Rodriguez  _et al._ [2019] A. Parra-Rodriguez, I. L. Egusquiza, D. P. DiVincenzo, and E. Solano, Canonical circuit quantization with linear nonreciprocal devices, Phys. Rev. B 99, 014514 (2019).
  * Walls and Milburn [2012] D. Walls and G. Milburn,  _Quantum Optics_, Springer Study Edition (Springer Berlin Heidelberg, 2012).
  * Parra-Rodriguez  _et al._ [2018] A. Parra-Rodriguez, E. Rico, E. Solano, and I. L. Egusquiza, Quantum networks in divergence-free circuit QED, Quantum Science and Technology 3, 024012 (2018).
  * DiVincenzo and Solgun [2013] D. P. DiVincenzo and F. Solgun, Multi-qubit parity measurement in circuit quantum electrodynamics, New Journal of Physics 15, 075001 (2013).
  * Telle  _et al._ [1999] H. Telle, G. Steinmeyer, A. Dunlop, J. Stenger, D. Sutter, and U. Keller, Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation, Appl. Phys. B 69, 327 (1999).
  * Zmuidzinas [2012] J. Zmuidzinas, Superconducting microresonators: Physics and applications, Annual Review of Condensed Matter Physics 3, 169 (2012), https://doi.org/10.1146/annurev-conmatphys-020911-125022 .