  * June 2021

# Is there really a Hubble tension?

Mohamed Rameez  Tata Institute of Fundamental Research, Homi Bhabha Road,
Mumbai 400005, India<EMAIL_ADDRESS>Subir Sarkar  Department of
Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
<EMAIL_ADDRESS>
###### Abstract

The heliocentric redshifts ($z_{\mathrm{hel}}$) reported for 150 Type Ia
supernovae in the Pantheon compilation are significantly discrepant from their
corresponding values in the JLA compilation. Both catalogues include
corrections to the redshifts and magnitudes of the supernovae to account for
the motion of the heliocentric frame relative to the ‘CMB rest frame’, as well
as corrections for the directionally coherent bulk motion of local galaxies
with respect to this frame. The latter is done employing modelling of peculiar
velocities which assume the $\Lambda$CDM cosmological model but nevertheless
provide evidence for residual bulk flows which are discordant with this model
(implying that the observed Universe is in fact anisotropic). Until recently
such peculiar velocity corrections in the Pantheon catalogue were made at
redshifts exceeding 0.2 although there is no data on which to base such
corrections. We study the impact of these vexed issues on the $4.4\sigma$
discrepancy between the Hubble constant of $H_{0}=67.4\pm 0.5$ km s-1Mpc-1
inferred from observations of CMB anisotropies by Planck assuming
$\Lambda$CDM, and the measurement of $H_{0}=73.5\pm 1.4$ km s-1Mpc-1 by the
SH0ES project which extended the local distance ladder using Type Ia
supernovae. Using the same methodology as the latter study we find that for
supernovae whose redshifts are discrepant between Pantheon and JLA with
$\Delta z_{\mathrm{hel}}>0.0025$, the Pantheon redshifts favour $H_{0}\simeq
72$ km s-1Mpc-1, while the JLA redshifts favour $H_{0}\simeq 68$ km s-1Mpc-1.
Thus the discrepancies between SNe Ia datasets are sufficient to undermine the
claimed ‘Hubble tension’. We further note the systematic variation of $H_{0}$
by $\sim$6–9 km s-1Mpc-1 across the sky seen in multiple datasets, implying
that it cannot be measured locally to better than $\sim$10% in a model-
independent manner.

††: CQG

Keywords: Cosmology, Hubble constant, Type Ia supernovae

##  1 Introduction

The intrinsic magnitudes of nearby Type Ia supernovae (SNe Ia) to which
distances are known independently are characterised by a large scatter.
However by exploiting the empirical (wavelength-dependent) correlation between
the intrinsic supernova magnitude and the timescale of the luminosity decline
[35], this scatter can be reduced to $\sim$0.1-0.2 mag, making them
‘standardisable candles’.111This too has however been questioned by the recent
evidence for luminosity evolution of a sub-class of SNe Ia with redshift [26].
Recently, the magnitude-redshift relation of SNe Ia in the nearby ($z<0.15$)
Universe has been leveraged with the local Cepheid-calibrated distance ladder
to measure the Hubble constant to increasingly high precision [37] in what is
claimed to be a model-_independent_ manner. The single largest source of
uncertainty in determining $H_{0}$ is now said to be the mean luminosity of
the SNe Ia calibrators [38].

Such measurements have been used to argue [38] that there is a $4.4\sigma$
‘Hubble tension’ between $H_{0}$ in the late Universe and its value inferred
from the Cosmic Microwave Background (CMB) assuming the standard flat
$\Lambda$CDM cosmological model [1]. This is said to be robust with respect to
choices of independent calibrators [39] so has stimulated an avalanche of
explanations, many involving speculative new physics.

Today, publicly available SNe Ia data such as the SDSS-II/SNLS3 Joint
Lightcurve Analysis (JLA) [4] as well as the subsequent Pantheon [40]
compilations come with observables already “corrected” to account for the
effects of peculiar velocities in the local Universe. However, as we describe
in Section 2, apart from modifying the magnitudes (redshifts) of low redshift
SNe Ia by up to 0.2 mag ($\sim$20 %), these corrections appear to be
arbitrary, error-prone and discontinuous within the data, while the datasets
are discrepant with respect to each other in key observables. In Section 3 we
show that these discrepancies in heliocentric redshifts between JLA and
Pantheon are on their own large enough that a significant ‘Hubble tension’
cannot be claimed between the late and early Universe determinations of
$H_{0}$. We conclude (Section 4) with a discussion of the model dependence of
these corrections.

##  2 Data quality issues

Figure 1: The directions of the 58 SNe Ia in Table 1 in Galactic coordinates.
The directions of the CMB dipole (star), the SMAC bulk flow [24] (triangle),
and the 2M++ bulk flow [7] (inverted triangle) are also shown. Note that these
SNe Ia with discrepant redshifts (all from SDSS-II) are in the hemisphere
_opposite_ to the CMB dipole and bulk flow directions.

For 58 SNe Ia from SDSS-II [28] that are in common between the JLA and the
Pantheon catalogues, the quoted heliocentric redshifts differ by between 5 to
137 times the quoted uncertainty in the redshift measurement. Their data is
given in Table 1, while their distribution on the sky is shown in Figure 1.
Many more SNe Ia in common between the catalogues have smaller shifts in their
$z_{\mathrm{hel}}$ values. The uncertainty on the spectroscopic redshift
measurement of the host galaxy is quoted as 0.0001–0.0002 in SDSS DR4, and
$0.0005$ for redshifts which were measured by the authors themselves [28].
Note that the the $\sigma_{z_{\mathrm{spec}}}$ arising from peculiar
velocities mentioned in Ref.[28] is not a measurement uncertainty but rather
the expected dispersion with respect to theoretical predictions. The quoted
redshifts cannot have changed unless if they have been remeasured, a process
that has not been documented in Ref.[40]. In fact the JLA $z_{\mathrm{hel}}$
values are in exact agreement with other public sources such as VizieR,
whereas the Pantheon $z_{\mathrm{hel}}$ values are not independently
verifiable. While some of these discrepancies may be due to the slight
difference between the redshift of the supernova and of its host galaxy [45],
no such distinction is made in the disseminated datasets and the covariances
provided with the data are certainly not large enough to account for the
shifts.

It has been noted that the magnitudes of these SNe Ia are also inconsistent
between the two catalogues.222https://github.com/dscolnic/Pantheon/issues/5
These shifts seem to have been introduced on 27 November 2018 when new files
were uploaded to purportedly rectify previously reported errors in the
peculiar velocity corrections.333https://github.com/dscolnic/Pantheon/issues/2
Such inconsistencies in publicly available SNe Ia data have also been noted
earlier with respect to the choice of light curve fitter [2].

Table 1: Discrepant redshifts in JLA and Pantheon: While the names of the 58 SNe Ia listed below (from SDSS-II) differ by survey-specific prefixes, the fact that they are the same can be verified from their Right Ascension and Declination. The JLA and Pantheon redshifts are taken from, respectively, https://github.com/cmbant/CosmoMC/blob/master/data/jla_lcparams.txt and from https://github.com/dscolnic/Pantheon/blob/master/lcparam_full_long_zhel.txt . The significance of the shifts are calculated assuming the host galaxy redshift measurement uncertainty of $\sigma_{z}=0.0005$ — however for 34 of these the uncertainty may be 10 times larger ($\sigma_{z}=0.005$), being derived from the spectroscopic features of the SNe Ia [28]. Name in JLA | $z_{\mathrm{hel}|\mathrm{JLA}}$ | Name in Pantheon | $z_{\mathrm{hel}|\mathrm{Pantheon}}$ | $z_{\mathrm{diff}}$ | Shift
---|---|---|---|---|---
SDSS12881 | 0.233 | 12881 | 0.237838 | 0.004838 |  9.68 $\sigma$
SDSS12927 | 0.175 | 12927 | 0.189638 | 0.014638 |  29.28 $\sigma$
SDSS13044 | 0.121 | 13044 | 0.125735 | 0.004735 |  9.47 $\sigma$
SDSS13136 | 0.366 | 13136 | 0.371627 | 0.005627 |  11.25 $\sigma$
SDSS13152 | 0.207 | 13152 | 0.203311 | 0.003689 |  7.38 $\sigma$
SDSS13305 | 0.201 | 13305 | 0.214557 | 0.013557 |  27.11 $\sigma$
SDSS13727 | 0.221 | 13727 | 0.226402 | 0.005402 |  10.80 $\sigma$
SDSS13796 | 0.145 | 13796 | 0.148518 | 0.003518 |  7.04 $\sigma$
SDSS14261 | 0.281 | 14261 | 0.285517 | 0.004517 |  9.03 $\sigma$
SDSS14331 | 0.214 | 14331 | 0.220905 | 0.006905 |  13.81 $\sigma$
SDSS14397 | 0.371 | 14397 | 0.386084 | 0.015084 |  30.17 $\sigma$
SDSS14437 | 0.144 | 14437 | 0.149098 | 0.005098 |  10.20 $\sigma$
SDSS14481 | 0.255 | 14481 | 0.243249 | 0.011751 |  23.50 $\sigma$
SDSS15057 | 0.299 | 15057 | 0.246586 | 0.052414 |  104.83 $\sigma$
SDSS15203 | 0.216 | 15203 | 0.204218 | 0.011782 |  23.56 $\sigma$
SDSS15287 | 0.274 | 15287 | 0.237419 | 0.036581 |  73.16 $\sigma$
SDSS15301 | 0.248 | 15301 | 0.17963 | 0.06837 |  136.74 $\sigma$
SDSS15365 | 0.178 | 15365 | 0.187733 | 0.009733 |  19.47 $\sigma$
SDSS15383 | 0.312 | 15383 | 0.315791 | 0.003791 |  7.58 $\sigma$
SDSS15440 | 0.253 | 15440 | 0.262051 | 0.009051 |  18.10 $\sigma$
SDSS15461 | 0.18 | 15461 | 0.185954 | 0.005954 |  11.91 $\sigma$
SDSS15704 | 0.365 | 15704 | 0.370275 | 0.005275 |  10.55 $\sigma$
SDSS15868 | 0.242 | 15868 | 0.250516 | 0.008516 |  17.03 $\sigma$
SDSS15872 | 0.203 | 15872 | 0.20629 | 0.00329 |  6.58 $\sigma$
SDSS15897 | 0.17 | 15897 | 0.174692 | 0.004692 |  9.38 $\sigma$
SDSS15901 | 0.199 | 15901 | 0.204563 | 0.005563 |  11.13 $\sigma$
SDSS16072 | 0.277 | 16072 | 0.285523 | 0.008523 |  17.05 $\sigma$
SDSS16073 | 0.146 | 16073 | 0.154541 | 0.008541 |  17.08 $\sigma$
SDSS16116 | 0.15 | 16116 | 0.156305 | 0.006305 |  12.61 $\sigma$
SDSS16185 | 0.097 | 16185 | 0.101255 | 0.004255 |  8.51 $\sigma$
SDSS16206 | 0.152 | 16206 | 0.15954 | 0.00754 |  15.08 $\sigma$
SDSS16232 | 0.367 | 16232 | 0.37532 | 0.00832 |  16.64 $\sigma$
SDSS17220 | 0.172 | 17220 | 0.178821 | 0.006821 |  13.64 $\sigma$
SDSS17552 | 0.25 | 17552 | 0.253014 | 0.003014 |  6.03 $\sigma$
SDSS17809 | 0.282 | 17809 | 0.288624 | 0.006624 |  13.25 $\sigma$
SDSS18325 | 0.255 | 18325 | 0.258369 | 0.003369 |  6.74 $\sigma$
SDSS18602 | 0.135 | 18602 | 0.138175 | 0.003175 |  6.35 $\sigma$
SDSS18617 | 0.322 | 18617 | 0.326919 | 0.004919 |  9.84 $\sigma$
SDSS18721 | 0.393 | 18721 | 0.402456 | 0.009456 |  18.91 $\sigma$
SDSS18740 | 0.157 | 18740 | 0.154249 | 0.002751 |  5.50 $\sigma$
SDSS18787 | 0.193 | 18787 | 0.190054 | 0.002946 |  5.89 $\sigma$
SDSS18804 | 0.192 | 18804 | 0.198237 | 0.006237 |  12.47 $\sigma$
SDSS18940 | 0.22 | 18940 | 0.212127 | 0.007873 |  15.75 $\sigma$
SDSS19002 | 0.268 | 19002 | 0.27081 | 0.00281 |  5.62 $\sigma$
SDSS19027 | 0.295 | 19027 | 0.2923 | 0.0027 |  5.4 $\sigma$
SDSS19341 | 0.228 | 19341 | 0.236507 | 0.008507 |  17.01 $\sigma$
SDSS19632 | 0.308 | 19632 | 0.314512 | 0.006512 |  13.02 $\sigma$
SDSS19818 | 0.293 | 19818 | 0.304775 | 0.011775 |  23.55 $\sigma$
SDSS19953 | 0.119 | 19953 | 0.123087 | 0.004087 |  8.17 $\sigma$
SDSS19990 | 0.246 | 19990 | 0.24967 | 0.00367 |  7.34 $\sigma$
SDSS20040 | 0.285 | 20040 | 0.287713 | 0.002713 |  5.43 $\sigma$
SDSS20048 | 0.182 | 20048 | 0.185096 | 0.003096 |  6.19 $\sigma$
SDSS20084 | 0.131 | 20084 | 0.139557 | 0.008557 |  17.11 $\sigma$
SDSS20227 | 0.284 | 20227 | 0.276958 | 0.007042 |  14.08 $\sigma$
SDSS20364 | 0.215 | 20364 | 0.218249 | 0.003249 |  6.50 $\sigma$
SDSS21062 | 0.147 | 21062 | 0.13848 | 0.00852 |  17.04 $\sigma$
sn1997dg | 0.0308 | 1997dg | 0.03396 | 0.00316 |  6.32 $\sigma$
sn2006oa | 0.06255 | 2006oa | 0.059931 | 0.002619 |  5.238 $\sigma$

The supernova catalogues also provide $z_{\mathrm{cmb}}$, the boosted redshift
in the ‘CMB rest frame’ in which the CMB is supposed to look isotropic,
assuming that its dipole asymmetry is entirely due to our motion with respect
to this frame, and further corrected using a model of the local peculiar (non-
Hubble) velocity field to account for the motion of the SNe Ia with respect to
this frame.444Note that the sortable table at
https://archive.stsci.edu/prepds/ps1cosmo/scolnic_datatable.html reports the
_same_ values for both $z_{\mathrm{CMB}}$ and $z_{\mathrm{hel}}$. The
inconsistencies in the peculiar velocity corrections made in the JLA catalogue
have been discussed elsewhere [11], in particular that while relying on the
flow model [24], the corrections applied extend well beyond the extent of the
survey ($z\sim 0.04$) on which this model is based, and moreover abruptly fall
to zero at $z\sim 0.06$, even though the same model [24] reports a residual
bulk velocity of $687\pm 203$ km s-1 beyond $z\sim 0.04$ — which is over 4
times larger than the uncorrelated velocity dispersion of $c\sigma_{z}=150$ km
s-1 included in the JLA error budget for cosmological fits. (There are other
inconsistencies as well, e.g. SDSS2308 has the same $z_{\mathrm{cmb}}$ and
$z_{\mathrm{hel}}$ despite being at a redshift of 0.14.)

Significantly more egregious errors are seen in the first version of the
Pantheon compilation on Github [41] wherein peculiar velocity corrections were
used to modify the redshifts of SNe Ia all the way up to $z\gtrsim 0.2$
although no survey has yet gone to such depths so the information required to
make such corrections is simply not available. An illustrative example is
SN2246, with $z_{\mathrm{cmb}}=0.19422$ which has been corrected by a peculiar
velocity of $444.3$ km s-1. This object is $117^{0}$ away from the CMB dipole,
and $115^{0}$ away from the direction of the external bulk flow reported by
Ref.[7].

This issue is now said [41] to have been fixed by not making _any_ peculiar
velocity corrections for $z>0.08$ in Pantheon. However the impact of this
major change on the determination of $H_{0}$ or other cosmological parameters
[40] has not been documented. In Ref.[37] the sample of SNe Ia is limited to
to $0.023<z<0.15$ in order not to be affected by the coherent flow in the
local volume, but these authors too account for such flows using the same
model [7].

Peculiar velocities affect the distance modulus by $(5/\mathrm{log}10)(v/cz)$
mag. Thus the peculiar velocity corrections shift the distance moduli of SNe
Ia at redshifts smaller than the extent of the flow model by between 0.03 and
0.4 mag. Neither of the models used for correcting the supernova observables
in JLA or Pantheon — respectively Ref.[24] and Ref.[7] — report convergence
between the correlated flow and the ‘CMB rest frame’. Instead Ref.[7] provides
definitive ($>$5$\sigma$) evidence for a residual bulk flow of
$V_{\mathrm{ext}}=159\pm 23$ km s-1 originating from sources beyond the extent
($z=0.067$) of the model. This means that by choosing to make corrections up
to the extent of the flow model and leaving the rest uncorrected, abrupt and
arbitrary discontinuities of $\sim$0.07 ($\sim$0.03) mag are introduced in the
distance moduli of JLA (Pantheon) supernovae at these threshold redshifts.
These discontinuities which introduce bias, are handled by adding just a
variance term in the error budgets of the SNe Ia catalogues. We wish to draw
attention to the following problematic issues in this regard:

  * •

The modified data now encode a picture of the Universe in which a sphere of
radius 170, 285 or 340 Mpc (depending upon whether the flow model is taken to
extend to $z=$ 0.04, 0.067 or 0.08) is smashing into the rest of the Universe
— which is arbitrarily treated as at rest.

  * •

Out of the 740 (1048) SNe Ia that make up JLA (Pantheon) catalogue, 632 (890)
(including all the objects in Table 1) are in the direction _opposite_ to bulk
flow reported in Ref.[24] and Ref.[7].

  * •

Corrections for peculiar velocities were introduced in supernova cosmology in
2011 [12] after low-to-intermediate redshift supernovae were first observed
[28]. These are _all_ in the hemisphere opposite to the CMB dipole (see Figure
1). It is this set of SDSS-II SNe Ia, which crucially fill in the ‘redshift
desert’ between low and high redshift objects, that are now alarmingly
discrepant between JLA and Pantheon.

##  3 The impact on the ‘Hubble tension’

Errors in redshift measurements as small as $\Delta z\sim 0.0001$ can have
significant impact on the value of inferred cosmological parameters such as
$H_{0}$ [14]. Since some of the shifts in $z_{\mathrm{hel}}$ reported in Table
1 are as high as 0.1, we study the impact of these shifts on the measured
value of the Hubble constant. Following ref. [37], the distance modulus is

$\mu=25+5\mathrm{log}_{10}(D/\mathrm{Mpc}),$ (1)

where $D$ is the luminosity distance in Mpc, given in the cosmographic Taylor
expansion (which is accurate to better than 7% for $z<1.3$ [11] as in the JLA
catalogue) by:

$D(z)=\frac{cz}{H_{0}}\left[1+\frac{1}{2}(1-q_{0})z-\frac{1}{6}(1-q_{0}-3q^{2}_{0}+j_{0}-\Omega_{k})z^{2}+\ldots\right].$
(2)

For each SNe Ia, the observational distance modulus is constructed as the
difference in magnitudes of an apparent and absolute flux,
$\mu^{\mathrm{obs}}=m-M$ , where $m$ is corrected for the specific light curve
shape and colour by adding to it: $\alpha x_{1}-\beta c$. Here $\alpha$ and
$\beta$ are parameters assumed to be constants for all SNe Ia, while $x_{1}$,
$c$ and $m$ are provided separately for each SNe Ia from the SALT2 template
fit [4]. We then employ the standard ‘$\chi^{2}$ statistic’

$\chi^{2}=\sum_{i}\frac{(\mu_{i}-\mu_{i}^{\mathrm{obs}})^{2}}{(\sigma_{\mu_{i}})^{2}+(\sigma_{\mu}^{\mathrm{int}})^{2}}.$
(3)

Here $\sigma_{\mu i}$ is the uncertainty for the $i^{th}$ SNe Ia provided with
the catalogue, while $\sigma_{\mu}^{\mathrm{int}}$ is the (unknown) intrinsic
scatter.

Figure 2: Top left: Posteriors on $H_{0}$ from the SDSS-II SNe Ia in JLA which
have $z_{\mathrm{Pantheon}}-z_{\mathrm{JLA}}>0.0025$, using JLA redshifts
(blue) and Pantheon redshifts (pink). Since the Pantheon magnitudes are also
discrepant from JLA [41], the posterior using both Pantheon redshifts and
magnitudes are also shown (in green). The vertical grey band shows the CMB
determination [1]. Top right: The same with
$z_{\mathrm{Pantheon}}-z_{\mathrm{JLA}}>0.0005$. The bottom panel shows the
same but for $|z_{\mathrm{JLA}}-z_{\mathrm{Pantheon}}|>0.0025$ (left) and
$|z_{\mathrm{JLA}}-z_{\mathrm{Pantheon}}|>0.0005$ (right), thus illustrating
that the discrepancy persists regardless of whether the quoted JLA or Pantheon
redshift is larger.

Following ref.[37], we set the deceleration parameter $q_{0}=-0.55$, the jerk
parameter $j_{0}=1$ and the curvature parameter $\Omega_{k}=0$ in the scans.
While precise measurements of $H_{0}$ require the calibration of the absolute
supernova luminosity $M$ using a local distance ladder, we fix $M$ to -19.10
[37] with the specific aim of studying the impact of redshift errors on
$H_{0}$ since none of the 19 SNe Ia with Cepheid calibrations for the host
galaxies reported in Ref.[37] are included in either the JLA or Pantheon
compilations. This choice affects only the absolute value of $H_{0}$, and not
the relative shift introduced by the choice of redshift. We use the Emcee code
[19] to perform a Markov Chain Monte Carlo scan in likelihood over the
parameter space of $H_{0},\alpha,\beta$ and $\sigma_{\mathrm{int}}$ (the
intrinsic dispersion), all with flat priors. The observed shift in $H_{0}$ is
found to be robust with respect to alternative parametrisations of the
likelihood which are provided in a Jupyter notebook [36]. Out of the 58 SNe Ia
listed in Table 1, 45 have $z_{\mathrm{Pantheon}}-z_{\mathrm{JLA}}>0.0025$,
while 13 have $z_{\mathrm{Pantheon}}-z_{\mathrm{JLA}}<-0.0025$. Figure 2 (top
panel) shows that for supernovae with $\Delta z>0.0025$, the Pantheon
redshifts favour $H_{0}\sim 72$ km s-1Mpc-1, while the JLA redshifts favour
$H_{0}\sim 68$ km s-1Mpc-1. If we change the criterion to an absolute
difference $|\Delta z|>0.0025$, Figure 2 (bottom panel) shows that the
Pantheon redshifts now favour $H_{0}\sim 71$ km s-1Mpc-1, while the JLA
redshifts favour $H_{0}\sim 69$ km s-1Mpc-1. If all 117 SNe in JLA with
$0.023<z_{\mathrm{CMB}}<0.15$, which are also in Pantheon, are used in the
scans, as shown in Figure 3 (left panel), the redshift discrepancies average
out to give consistent values of $H_{0}$ as long as the JLA magnitudes are
employed — nevertheless the magnitude discrepancies between the two catalogues
are large enough to shift $H_{0}$ by 10% or more. Indeed Figure 3 (right
panel) shows that the 178 JLA SNe Ia in the relevant redshift range indicate a
low $H_{0}$ compared to the 237 SNe in the Pantheon compilation in the same
redshift range. The magnitudes of the SNe in both JLA [4] and Pantheon [40]
were also corrected for bias assuming the $\Lambda$CDM model. The JLA
catalogue provides sufficient detail to enable these corrections to be
_reversed_. For the 117 SNe in JLA with $0.023<z_{\mathrm{CMB}}<0.15$, which
are also in Pantheon, the impact of these corrections on the inferred value of
$H_{0}$ is $<2$% as shown in Figure 3 (left panel).

Ref.[38] finds $H_{0}=74.03\pm 1.42$ km s-1Mpc-1 and emphasises the difference
from $H_{0}\sim 67.4\pm 0.5$ km s-1Mpc-1 inferred from Planck data on CMB
anisotropies assuming flat $\Lambda$CDM [1]. However this ignores the various
inconsistencies we have pointed out above, so it cannot be claimed [37, 38]
that this ‘Hubble tension’ is significant.

Figure 3: Left: Posteriors on $H_{0}$ from the 117 SNe Ia in JLA with
$0.023<z_{\mathrm{CMB}}<0.15$ which are also included in Pantheon, using JLA
redshifts (blue) and Pantheon redshifts (pink). Removing the magnitude bias
corrections in JLA makes $<2\%$ difference (ochre). The posterior using both
Pantheon redshifts and magnitudes are also shown (in green). The vertical grey
band shows the CMB determination [1]. Right: The same for the 178 SNe in JLA
and 237 SNe in Pantheon in the same redshift range, using redshifts and
magnitudes from the corresponding catalogues.

##  4 Discussion

In general relativity, metric expansion in the late-time Universe is an
average effect, arising from the coarse-graining of physics at smaller scales
[6]. This differs from the metric expansion in the FLRW solution to the field
equations, wherein due to the isotropy and homogeneity _imposed_ on the
stress-energy tensor, all clocks remain synchronised and space expands
isotropically and homogeneously, described by a single scale factor. However
the observed inhomogenous Universe can only be approximately described by a
FLRW metric. Fitting a Hubble diagram of observables _not_ corrected for
peculiar velocities, as was done in supernova cosmology analyses until 2011,
and employing peculiar velocity corrections after transforming to the CMB
frame, as was done subsequently [12], simply amounts to different choices of
corresponding 2-spheres within the ‘null fitting’ procedure described in
Ref.[18]. Whether the dynamical evolution of the representative FLRW model can
be precisely related to the average evolution of the inhomogeneous Universe is
contingent on the resolution of a number of open questions [8].

Motivated by such considerations it has been argued that peculiar velocities
should rather be thought of as variations in the expansion rate of the
Universe [30](see also Refs.[46, 48]). Assuming the CMB dipole arises due to
our motion with respect to a frame in which the Universe is isotropic, all
matter in the local Universe seems to be flowing in a similar direction [7],
however convergence of this flow to the hypothetical CMB rest frame has not
been found [10]. This adds additional credence to the possibility that the CMB
dipole is not purely kinematic [42, 43]. The Universe we observe appears to be
anisotropic, and lacking an observationally consistent standard of rest.

A statistically significant variation of $\sim$9 km s-1Mpc-1 across the sky
was found in the Hubble constant measured in the Hubble Key Project [30].
Recently a similar anisotropy has been seen using ROSAT and Chandra data on
X-ray clusters [31, 32]. Such a systematic variation of $H_{0}$ across the sky
is as expected according to a cosmographic Taylor series expansion of the
luminosity distance for a universe _without_ exact symmetries [23]. This study
finds that the monopole of the generalised Hubble parameter $\Theta/3$ is
modified by a quadrupolar contribution ($90^{0}$ separation between poles)
$e^{\mu}e^{\nu}\sigma_{\mu\nu}$ from the shear tensor of the observer
congruence $\sigma_{\mu\nu}$ describing the anisotropic expansion of space
[23]. It appears therefore that $H_{0}$ cannot be measured in a model-
_independent_ manner in the locally inhomogeneous and anisotropic universe to
a precision better than $\sim$10%.

To gain apparently better precision, Ref.[38] follows Ref.[37] who employed
peculiar velocity corrections based on the flow model [7], presumably as was
also done for Pantheon [40]. However Ref.[7] infers the velocity field from
the density field using linear perturbation theory around an _assumed_ FLRW
background. These measurements cannot thus be said to be model-independent.
Data thus ‘corrected’ have then been used to insist that the Hubble expansion
is indeed isotropic [44] and also to argue that local structure has no impact
on the measurement of the Hubble constant [27].

While other astronomical probes e.g. strong gravitational lensing are said to
provide independent evidence for the ‘Hubble tension’, there appears to be a
similar directional dependence to the $\sim$15% relative variation in the
value of $H_{0}$ derived from the six lenses [47]. However systematic
uncertainties in these measurements may have been underestimated [16, 5].

Within the concordance $\Lambda$CDM model, the effect of inhomogeneities is
studied by linearising the field equations around a maximally symmetric
solution (see ref. [9]). Further making restricted ‘gauge’ choices [3]
motivated by the idea that the Universe began with only scalar density
perturbations left over from inflation, the usual perturbed FLRW framework is
arrived at. However, as emphasized in Ref.[3] this eliminates known physical
phenomena, and solutions to the linearized field equations can only be
linearisations of the solutions to the fully nonlinear equations [33, 15].
From a general relativity perspective there is in fact locally inhomogeneous
expansion beyond that expected in linear perturbation theory around a
maximally symmetric background [21]. Studies of the impact of peculiar
velocities [25, 13, 34] using Newtonian N-body simulations cannot capture such
physics. A recent numerical relativity simulation [29] suggests significantly
larger deviations from an isotropic Hubble law in the late Universe.

It has been argued that systematic calibration offsets within the distance
ladder can account for the ‘Hubble tension’ [17, 22]. In fact the discrepancy
we have established between the determinations of $H_{0}$ using the JLA and
Pantheon catalogues corresponds precisely to the “systematic bias of 0.1–0.15
mag in the intercept of the Cepheid period-luminosity relations of SH0ES
galaxies” [17]. Indeed independent calibrations of SNe Ia [20] prefer lower
values of $H_{0}$ which are consistent with the early Universe. We emphasise
that any measurement with a claimed uncertainty smaller than that of Ref.[30]
must be scrutinised for its understanding of, and correction for, peculiar
velocities. It is clear that the corrections that have been applied so far are
rather arbitrary. The fact that even the observed (uncorrected) heliocentric
redshifts undergo unexplained changes from one catalogue (JLA) to another
(Pantheon) thus inducing significant variations in the inferred value of
$H_{0}$, undermines the “precision cosmology” programme. This calls for the
_blinded_ testing of the isotropy of the Hubble diagram with forthcoming data
from e.g. the Legacy Survey of Space and Time (LSST) to be conducted at the
Vera C. Rubin Observatory (https://www.lsst.org/).

## Acknowledgements

We thank the anonymous referees for helpful comments and suggestions which
have led to improvement of our paper.

## References

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# Reply to "Comment" on "Regular evaporating black holes with stable cores"

Alfio Bonanno  INAF, Osservatorio Astrofisico di Catania, via S.Sofia 78 and
INFN, Sezione di Catania, via S. Sofia 64, I-95123,Catania, Italy  Amir-Pouyan
Khosravi  Frank Saueressig  Institute for Mathematics, Astrophysics and
Particle Physics (IMAPP),
Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

###### Abstract

We reply to the “Comment” on “Regular evaporating black holes with stable
cores” by R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M.
Visser. As a key result, we show that the regime of mass-inflation identified
in the comment connects smoothly to the late-time attractors discovered in our
works [A. Bonanno _et. al._ , Regular black holes with stable cores, Phys.
Rev. D 103, 124027 (2021) and Regular evaporating black holes with stable
cores, Phys. Rev. D 107, 024005 (2023)]. Hence, the late-time stability of
regular black holes is not affected by this intermediate phase.

###### pacs:

04.60.-m, 11.10.Hi

##  I Introduction

Regular black holes are a phenomenologically interesting alternative to the
black hole solutions arising from general relativity. In this context, it is
important to clarify whether these alternatives remain regular once small
perturbations are included. Extrapolating from the Reissner-Nordstrom
solution, the presence of a Cauchy horizon could lead to a dynamical
instability of the regular geometry through the mass-inflation effect [1, 2,
3]. Our works [4, 5] established that specific classes of regular black hole
solutions, including the Hayward geometry [6], do not exhibit an exponential
growths of the mass function at asymptotically late times, despite the
presence of a Cauchy horizon. The comment [7] discusses this conclusion in a
critical way. In this reply, we clarify the global picture arising from the
various claims made in the literature.

###  I.1 What we have done

We reported a stability analysis of static, regular black hole geometries [4]
and dynamical regular black hole geometries emitting Hawking radiation [5].
Following the lines of the Ori model [3], we modelled the perturbations by an
infinitely thin shell impacting on the Cauchy horizon. As our main result, we
established the existence of late-time attractors which tame the instability
due to the mass-inflation effect. We also identified the conditions on the
regular geometry that give rise to the modified dynamics. The results of our
dynamical analysis superseed earlier claims [8], which argued the presence of
the mass-inflation instability based on a non-dynamical analysis employing the
Dray-t’Hooft-Redmount (DTR) relations [9].

###  I.2 Why is our analysis trustworthy?

Our dynamical analysis is based on the Ori-model [3]. Thus, we work with the
same simplifying assumptions that were used when establishing the mass-
inflation effect for the Reissner-Nordstrom black hole [1]. We carefully
cross-checked that we reproduce the instability encountered in this case.
Analyzing the same model in a regular black hole background then allows a
direct comparison of the dynamics in the vicinity of the Cauchy horizon. This
showed beyond any doubt that certain regular black hole geometries, when
perturbed, do not exhibit an exponential growth of the mass function at late
times. Since it is this exponential growth that underlies the notion of mass-
inflation, it is fair to say that the mass-inflation effect is absent for
these geometries.

###  I.3 On extending the stability analysis
to early times

The central claim made in [10] and iterated in the comment [7] is that the
mass-inflation effect destabilizes the regular black hole geometry _before_
the late-time attractors identified in [4, 5] are reached. To substantiate
this assertion, the Ori model is extended into the early-time domain. In this
case, it is imperative to pay attention to the initial conditions which need
to be imposed sufficiently close to the Cauchy horizon so that the
perturbative solution is valid. Otherwise, the extrapolation _lacks
reliability_. This is underscored by Poisson and Israel, and collaborators [2,
1, 11, 12, 13], who emphasize that conclusions drawn from this model rest upon
three pivotal assumptions: 1) the applicability of the optical geometric limit
for both inward and outward propagating gravitational waves, 2) the
permissibility of neglecting the matter flow from the collapsing star and 3)
the existence of the Cauchy horizon $r_{0}$ at $v=\infty$, with $v$ being the
advanced time. These simplifying assumptions do not necessarily hold at early
times.

Figure 1:  Log-log plot of the Misner-Sharp mass $M_{+}(v)$ obtained from
integrating the Ori-model based on the Hayward geometry for several initial
conditions. The exponential growths of the mass function at intermediate time-
scales ($30\lesssim v\lesssim 60$) connects smoothly to the late-time
attractor quenching the mass-inflation effect at $v>70$.

###  I.4 Connecting the scaling regimes

An interesting point raised by the comment [7] is that there could be an
intermediate regime which is characterized by the black hole having shed a
sufficient amount of the initial perturbations so that the dynamics can be
captured by the Ori model while the dynamics is not yet controlled by the
late-time attractor discovered in our works. It is argued that this phase
exhibits an exponential growths of the mass function. This may be fatal for
two reasons:

  1. 1.

The spacetime curvature may reach values where the model is no longer
applicable.

  2. 2.

The mass-function may be driven to values where the dynamics fails to connect
to the late-time attractor.

The first point is not applicable, since the Ori model is actually used to
infer the dynamical build-up of a curvature singularity at the Cauchy horizon.
On this basis, the model is not used outside the traditional analysis carried
out for the Reissner-Nordstrom geometry.

Clarifying the second point requires a robust numerical analysis of the basin-
of-attraction for the late-time attractor. This is currently missing in the
literature. On this basis, we revisited the dynamics of the Hayward model with
$l=1/2$ and initial conditions set at $R(v)-r_{0}=0.01$. This guarantees that
we are close enough to the Cauchy horizon, so that a perturbative analysis is
on secure ground. The numerical integration of the dynamics then shows that
there are solutions where the exponential increase of the mass function
smoothly connects to the late-time attractor. These solutions are shown in
Fig. 1. This establishes that the dynamics at intermediate time-scales is by
no means fatal for the regular black hole solution.

###  I.5 Towards realistic models of black holes

Our study focuses on an idealized situation where effects related to the
formation of the black hole, accretion, or its interaction with the cosmic
microwave background are neglected. This is the typical setting used when
discussing, e.g., black hole thermodynamics and constitutes a standard
assumption in the field. While it is interesting to investigate such effects,
this discussion is beyond the scope of our initial works.

##  II Conclusion

The comment [7] identified a transient phase of exponential growth in the
black hole mass function which could potentially destabilize regular black
hole geometries. We show that this phase connects smoothly to the late-time
attractor discovered in our works [4, 5]. Hence our conclusions about the
late-time stability of regular black hole solutions, also reviewed in [14],
remain valid.

## References

  * Poisson and Israel [1989] E. Poisson and W. Israel, Inner-horizon instability and mass inflation in black holes, Phys. Rev. Lett. 63, 1663 (1989).
  * Poisson and Israel [1990] E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D. 41, 1796 (1990).
  * Ori [1991] A. Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys. Rev. Lett. 67, 789 (1991).
  * Bonanno  _et al._ [2021] A. Bonanno, A.-P. Khosravi, and F. Saueressig, Regular black holes with stable cores, Phys. Rev. D 103, 124027 (2021), arXiv:2010.04226 [gr-qc] .
  * Bonanno  _et al._ [2023] A. Bonanno, A.-P. Khosravi, and F. Saueressig, Regular evaporating black holes with stable cores, Phys. Rev. D 107, 024005 (2023), arXiv:2209.10612 [gr-qc] .
  * [6] S. A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96, 031103 (2006), arXiv:gr-qc/0506126 [gr-qc].
  * Carballo-Rubio  _et al._ [2022] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, Comment on ”Stability properties of Regular Black Holes”, (2022), arXiv:2212.07458 [gr-qc] .
  * Carballo-Rubio  _et al._ [2018] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, On the viability of regular black holes, JHEP 07, 023, arXiv:1805.02675 [gr-qc] .
  * Barrabes and Israel [1991] C. Barrabes and W. Israel, Thin shells in general relativity and cosmology: The lightlike limit, Phys.Rev.D 43, 1129 (1991).
  * Carballo-Rubio  _et al._ [2021] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, Inner horizon instability and the unstable cores of regular black holes, JHEP 05, 132, arXiv:2101.05006 [gr-qc] .
  * Anderson  _et al._ [1993] W. G. Anderson, P. R. Brady, W. Israel, and S. M. Morsink, Quantum effects in black hole interiors, Phys. Rev. Lett. 70, 1041 (1993), arXiv:gr-qc/9210013 .
  * Bonanno  _et al._ [1994] A. Bonanno, S. Droz, W. Israel, and S. M. Morsink, Structure of the inner singularity of a spherical black hole, Phys. Rev. D 50, 7372 (1994), arXiv:gr-qc/9403019 .
  * Bonanno  _et al._ [1995] A. Bonanno, S. Droz, W. Israel, and S. M. Morsink, Structure of the spherical black hole interior, Proc. Roy. Soc. Lond. A 450, 553 (1995), arXiv:gr-qc/9411050 .
  * Bambi [2023] A. Bonanno and F. Saueressig,  _Stability properties of Regular Black Holes in Regular Black Holes (C. Bambi (Ed.))_, Springer Series in Astrophysics and Cosmology (Springer Singapore, 2023) arXiv:2307.13249 [gr-qc] .


# Phase-modulated Autler-Townes splitting in a giant-atom system within
waveguide QED

Wei Zhao  Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China  Yan Zhang  Center for Quantum Sciences
and School of Physics, Northeast Normal University, Changchun 130024, China
Zhihai Wang<EMAIL_ADDRESS>Center for Quantum Sciences and School of
Physics, Northeast Normal University, Changchun 130024, China

###### Abstract

The nonlocal emitter-waveguide coupling, which gives birth to the so called
giant atom, represents a new paradigm in the field of quantum optics and
waveguide QED. In this paper, we investigate the single-photon scattering in a
one-dimensional waveguide on a two-level or three-level giant atom. Thanks to
the natural interference induced by the back and forth photon
transmitted/reflected between the atom-waveguide coupling points, the photon
transmission can be dynamically controlled by the periodic phase modulation
via adjusting the size of the giant atom. For the two-level giant-atom setup,
we demonstrate the energy shift which is dependent on the atomic size. For the
driven three-level giant-atom setup, it is of great interest that, the Autler-
Townes splitting is dramatically modulated by the giant atom, in which the
width of the transmission valleys (reflection range) is tunable in terms of
the atomic size. Our investigation will be beneficial to the photon or phonon
control in quantum network based on mesoscopical or even macroscopical quantum
nodes involving the giant atom.

##  I INTRODUCTION

In the field of quantum optics, the study of the interaction between light and
matter is one of the long-lived subjects. Recently, the light-matter
interaction in waveguide structures has attracted much attention, which leads
to lots of theoretical and experimental works in waveguide QED community
xg2017 ; cj2021 ; dr2017 ; ljy2020 ; gz2017 ; gz2018 ; gz2019 ; ii2020 , such
as dressed or bound states hz2010 ; cj2019 ; ts2016 ; gc2016 ; es2017 ; pt2017
; gc2019 , phase transitions qj2011 ; mf2017 ; lq2019 , single-photon devices
jt2005 ; de2007 ; lz2008 , exotic topological and chiral phenomena mr2014 ;
vy2014 ; cg2016 ; im2016 ; sm2019 ; mb2019 , where the wavelength of light (or
microwave field) is usually tens or hundreds of times larger than the size of
the natural/artificial atoms constituting the matter pg1983 ; dl2003 ; aw2004
; rm2005 ; sh2013 . Therefore, the light-matter interaction is usually
modelled by the dipole approximation, where the atoms are regarded as point-
like dipoles dw2018 .

However, in recent years, the artificial superconducting transmon qubit
coupled by the acoustic waves sd1986 ; dm2007 ; rm2017 , of which the size is
comparable to the wavelength of the phonons, is named as “giant atom” and has
been successfully realized in experiments mv2014  . Alternatively, the giant-
atom model can also be realized in the superconducting transmission line
setup, where the capacitive or inductive coupling allows more than one
coupling points between the microwave field and the qubit bk2020 ; AM2020 .
Moreover, using the cold atomic system, a theoretical scheme for the
realization of giant atom has been proposed in dynamical state-dependent
optical lattices AG2019 . In the giant-atom community, a lot of new phenomena
not existing in the conventional small atomic system have been predicted, such
as frequency dependent relaxation af2014 , non-exponential decay lg2017 ;
ga2019 ; sg2020 , tunable bound state lg2020 ; xw2020  as well as decoherence
free subspace af2018 ; ac2020  (For a recent review, see Ref. review ). The
underlying physics behind these phenomena is the interference and retarded
effect during the photon/phonon propagating process between the different
coupling points.

On the other hand, the dynamical control of the single-photon transmission is
a hot topic in constructing quantum networks hj2008 ; sr2012 ; pl2017 ,
motivated by the photon based quantum information processing. Photons provide
a reliable transmission of quantum information and the waveguide is often seen
as photonic channels in quantum network, with atoms (or artificial atoms)
acting as quantum nodes. Along this line, people have proposed lots of schemes
to realize single-photon device, in which the propagating of the photon in
channels is controlled on demand by adjusting the nature of the quantum nodes
jt2005 ; de2007 ; lz2008 ; jt2009 ; pl2010 ; wb2013 ; zh2014 ; wz2017 .
Furthermore, for a three-level quantum node, the electromagnetically induced
transparency (EIT) AA2010  and Autler-Townes splitting (ATS) phenomena have
been investigated both theoretically and experimentally PM2011 ; XG2016 ;
WX2016 ; JL2018 . Meanwhile, it is also possible to control photon by photon,
to realize an all optical routing is2014  or microwave-photon detector ki2016
. Combined with the interference effect, it motivates us to study how to
modulate the single photon scattering by the giant atom.

In this paper, we tackle this issue in a one-dimensional waveguide with a two-
level or three-level giant atom. For the two-level atom setup, we find that
the change in the size of the atom can control its energy shift due to the
interference effect between the backward and forward photon in the waveguide.
In the driven three-level giant-atom system, we demonstrate the controllable
phase-modulated ATS physics. Our results show that, the size of the giant
atom, which serves as a controller, can be used to tune the width of the
transmission valleys (reflection range) in a periodical manner. The underlying
physical principal is further revealed in the viewpoint of quantum open system
based on the dressed state representation.

The rest of the paper is structured as follows: In Sec. II, we present the
model and discuss the single photon scattering in a two-level giant-atom
system. In Sec. III, we demonstrate the Autler-Townes splitting behavior for a
driven three-level giant-atom with $\Lambda$-type transition. In Sec. IV, we
discuss the effects of the dissipation for both of two-level and three-level
giant atom setups. At last, we end up with a brief conclusion in Sec. V.

##  II Two-level giant atom

Figure 1: Schematic configuration for a linear waveguide coupled to a giant
atom at the points $x=0$ and $x=x_{0}$.

As schematically shown in Fig. 1, the system we consider is composed of a
linear waveguide and a giant atom, which is actually a two-level system. The
giant atom is connected to the waveguide via two points with $x=0$ and
$x=x_{0}$, respectively. The Hamiltonian $H_{1}$ of the system can be divided
into three parts, i.e., $H_{1}=H_{s}+H_{\omega}+V_{1}$. The first part $H_{s}$
is the free Hamiltonian of the giant atom (Hereafter, we set $\hbar=1$).

$H_{s}=\omega_{e}\left|e\right\rangle\left\langle e\right|,$ (1)

where $\omega_{e}$ is the transition frequency between the ground state
$\left|g\right\rangle$ and the excited state $\left|e\right\rangle$. As a
reference, we have set the frequency of the ground state
$\left|g\right\rangle$ as $\omega_{g}=0$.

The second part $H_{\omega}$ of the Hamiltonian $H_{1}$ represents the free
Hamiltonian of the waveguide, and is expressed as

$H_{\omega}=\int
dx\\{-iv_{g}C_{R}^{\dagger}\left(x\right)\frac{d}{dx}C_{R}\left(x\right)+iv_{g}C_{L}^{\dagger}\left(x\right)\frac{d}{dx}C_{L}\left(x\right)\\},$
(2)

where $v_{g}$ is the group velocity of photons traveling in the waveguide.
$C_{R}^{\dagger}(x)[C_{L}^{\dagger}(x)]$ is the bosonic creation operator for
the right-going (left-going) photon at position $x$.

For the third part $V_{1}$ of the Hamiltonian $H_{1}$, we describe the
interaction between the waveguide and the giant atom. Within the rotating wave
approximation, the Hamiltonian $V_{1}$ can be expressed as

$\begin{split}V_{1}=&f\int
dx\delta(x)\left[\sigma^{+}C_{R}\left(x\right)+\sigma^{+}C_{L}\left(x\right)+{\rm
H.c.}\right]\\\ &+f\int
dx\delta\left(x-x_{0}\right)\left[\sigma^{+}C_{R}\left(x\right)+\sigma^{+}C_{L}\left(x\right)+{\rm
H.c.}\right],\end{split}$ (3)

where $f$ is the coupling strength between the waveguide and the two-level
giant atom. $\sigma^{+}=(\sigma^{-})^{\dagger}=|e\rangle\langle g|$ is the
raising operator of the atom. The Dirac-$\delta$ function in the Hamiltonian
$V_{1}$ indicates that the giant atom has a length of $x_{0}$ and connects to
the waveguide via its head and tail, that is, $x=0$ and $x=x_{0}$.

It is noted that, the total excitation of the atom and the photon in the
waveguide is conserved. In the following section, we will restrict ourselves
in the single excitation subspace, to investigate how to control the single
photon scattering state via adjusting the frequency of the photon and the size
of the two-level giant atom.

In the single-excitation subspace, the eigenstate of the system can be written
as

$\begin{split}\left|E\right\rangle=&\int
dx\left[\phi_{R_{1}}\left(x\right)C_{R}^{\dagger}\left(x\right)+\phi_{L_{1}}\left(x\right)C_{L}^{\dagger}\left(x\right)\right]\left|G\right\rangle\\\
&+u_{e}\sigma^{+}\left|G\right\rangle,\end{split}$ (4)

where $|G\rangle$ represents that the waveguide is in the vacuum states while
the giant atom is in the ground state $|g\rangle$.
$\phi_{R_{1}}\left(x\right)$ and $\phi_{L_{1}}\left(x\right)$ are single-
photon wave functions of the right-going and left-going modes in the
waveguide, respectively. $u_{e}$ is the excitation amplitude of the giant
atom. Solving the stationary Schödinger equation
$H_{1}\left|E\right\rangle=E\left|E\right\rangle$, the amplitudes equation can
be obtained as

$\displaystyle-iv_{g}\frac{d}{dx}\phi_{R_{1}}\left(x\right)+fu_{e}M$
$\displaystyle=$ $\displaystyle E\phi_{R_{1}}\left(x\right),$ (5a)
$\displaystyle iv_{g}\frac{d}{dx}\phi_{L_{1}}\left(x\right)+fu_{e}M$
$\displaystyle=$ $\displaystyle E\phi_{L_{1}}\left(x\right),$ (5b)
$\displaystyle\omega_{e}u_{e}+fN$ $\displaystyle=$ $\displaystyle Eu_{e}.$
(5c)

where $M=\delta(x)+\delta(x-x_{0})$ and
$N=\phi_{R_{1}}(0)+\phi_{R_{1}}(x_{0})+\phi_{L_{1}}(0)+\phi_{L_{1}}(x_{0})$.

Next, we consider the scattering behavior when a single photon with wave
vector $k$ is incident from the left side of the waveguide. In this case, the
wave functions of $\phi_{R_{1}}\left(x\right)$ and
$\phi_{L_{1}}\left(x\right)$ can be expressed as

$\begin{split}\phi_{R_{1}}\left(x\right)=&e^{ikx}\\{\theta\left(-x\right)+A_{1}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\\\
&+t_{1}\theta\left(x-x_{0}\right)\\},\end{split}$ (6)
$\phi_{L_{1}}\left(x\right)=e^{-ikx}\left\\{r_{1}\theta\left(-x\right)+B_{1}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\right\\},$
(7)

with

$\theta\left(x\right)=\begin{cases}1&x>0\\\ \frac{1}{2}&x=0\\\
0&x<0\end{cases}.$ (8)

We explain the expression of the above wave functions physically as follows.
When the right-going photon incident from the region $x<0$ reaches the first
connection point $x=0$ between the giant atom and the waveguide, it can be
transmitted or reflected, with the amplitudes of $A_{1}$ and $r_{1}$,
respectively. The photon transmitted at the first connection point at $x=0$
will travel freely in the waveguide until it reaches the second connection
point at $x=x_{0}$, it will be then reflected or transmitted secondly, with
the amplitudes $B_{1}$ and $t_{1}$, respectively.

Now, we substitute Eq. (6-8) into the amplitude Eq. (5), it yields the
dispersion relation $E=v_{g}k$, and

$t_{1}=\frac{i\Delta v_{g}-2if^{2}\sin\left(kx_{0}\right)}{i\Delta
v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)}.$ (9)

where $\Delta=E-\omega_{e}$ is the detuning between the atom and the
propagating photon in the waveguide.

Furthermore, the transmission rate $T_{1}=\left|t_{1}\right|^{2}$ can be
obtained as

$T_{1}=\frac{\left(\Delta
v_{g}-2f^{2}\sin\left(kx_{0}\right)\right)^{2}}{\left(\Delta
v_{g}-2f^{2}\sin\left(kx_{0}\right)\right)^{2}+4f^{4}\left(1+\cos\left(kx_{0}\right)\right)^{2}},$
(10)

Figure 2: (a) The transmission rate $T_{1}$ as functions of $x_{0}$ and
$\Delta$. (b) The transmission rate $T_{1}$ as a function of $\Delta$. (c) The
transmission rate $T_{1}^{\prime}$ as a function of $\Delta$. The parameters
are set as $\omega_{e}=3$ GHz, $v_{g}=3\times 10^{8}$ m/s,
$f/\sqrt{v_{g}\omega_{e}}=0.05$. The other parameters is
$\gamma_{e}=10^{-3}\omega_{e}$ for (c).

In the small atom scenario ($x_{0}=0$) which is studied in Ref. jt2005 , it is
obvious that the incident photon will be completely reflected ($T_{1}=0$) when
it is resonant with the atom, that is, $\Delta=0$. However, for the giant atom
in our setup, the incident photon will propagate back and forth in the spatial
regime covered by the giant atom, leading to an interference effect. As a
result, as shown in Fig. 2(a), the transmission rate can be controlled by
adjusting the photon-atom detuning $\Delta$ and the size of the giant atom
$x_{0}$.

Furthermore, considering the atomic spontaneous emission, where we replace
$\omega_{e}$ by $\omega_{e}-i\gamma_{e}$, the transmission amplitudes
$t_{1}^{\prime}$ becomes

$t_{1}^{\prime}=\frac{i\left(\Delta+i\gamma_{e}\right)v_{g}-2if^{2}\sin\left(kx_{0}\right)}{i\left(\Delta+i\gamma_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)},$
(11)

where $\gamma_{e}$ is the spontaneous emission of the giant atom. As shown in
Fig. 2(b) and (c), we plot the transmission rates $T_{1}$ and
$T_{1}^{\prime}=\left|t_{1}^{\prime}\right|^{2}$ as functions of the detuning
$\Delta$, respectively, to compare the results with and without spontaneous
emission. It shows that the spontaneous emission breaks the complete
transmission and makes the transmission valleys become wider and shallower.
However, the locations of the valleys are not changed. It implies that the
environment nearly doesn’t change the energy structure of the system.

Figure 3: (a) The detuning $\Delta_{r}$ as a function of $x_{0}$ for
$\omega_{e}=3$ GHz, $v_{g}=3\times 10^{8}$ m/s,
$f/\sqrt{v_{g}\omega_{e}}=0.05$. The solid line is given by the solution of
the transcendental equation in Eq. (12) and the empty circles are the results
from the numerical fitting. (b) The fitting parameters $S$ in Eq. (13) as a
function of the atom-waveguide coupling strength $f$.

More interestingly, in the giant-atom situation ($x_{0}\neq 0$), the detuning
for complete reflection is determined by the transcendental equation

$\Delta_{r}=\frac{2f^{2}\sin\left(k_{r}x_{0}\right)}{v_{g}},$ (12)

where $\Delta_{r}$ is the detuning between the atom and the propagating photon
when the incident photon is completely reflected, and $k_{r}$ is the
corresponding photonic wave vector, $\Delta_{r}$ is dependent on $k_{r}$ via
$\Delta_{r}=v_{g}k_{r}-\omega_{e}$. It is well known that the atomic frequency
is determined by the location of the complete reflection of the incident
photon jt2005 . Therefore, in Fig. 2(b), we can observe an atomic frequency
shift which originates from the interference effect as discussed above,
compared with the small atom system. Since $\left|\sin k_{r}x_{0}\right|\leq
1$, Eq. (12) implies that only when the frequency of the incident photon
satisfies $|\Delta_{r}|\leq 2f^{2}/v_{g}$, that is,
$\omega_{e}+2f^{2}/v_{g}\geq v_{g}k_{r}\geq\omega_{e}-2f^{2}/v_{g}$, it is
possible to be completely reflected. Within this regime, we plot the
dependence of $\Delta_{r}$ on the size of giant atom $x_{0}$ in Fig. 3(a)
(solid line), which shows a sinusoidal shape. The exact sinusoidal shape by
the numerical fitting (empty circles) is also shown in the figure and the
fitting function is obtained by

$\Delta_{r}\approx S\sin(\omega_{e}x_{0}/v_{g}),$ (13)

where the fitting parameters $S\approx 2f^{2}/v_{g}$ is plotted as a function
of atom-waveguide coupling strength $f$ in Fig. 3(b). We recall that, when two
identical small atoms are coupled to the waveguide at the position $x=0$ and
$x_{0}$, the photon will also propagate back and forth between the two atoms.
However, as shown in Appendix, we can not observe the shift, that is $T=0$
occurs at $\Delta=0$. The reason is that, small atoms do not interfere with
themselves. However, in giant atoms, the interference within an atom promotes
the apparent frequency shift in its energy levels.

In experiments, the giant atom setup can be realized by the superconducting
circuits, and the transition frequency is in the order of GHz bk2020 ; AM2020
, while the intrinsic decoherence time of the superconducting qubit is
achieved by $20{\rm\mu s}$ FY2017  or even longer AJ2017 . The natural line
width of the giant atom is then $\gamma=0.05$ MHz . In Fig. 3(a), the energy
shift is with the regime of $|\Delta_{r}|=5\times 10^{-3}\omega_{e}=15$ MHz,
for some $x_{0}$ so that $\Delta_{r}\gg\gamma$, and $\Delta_{r}$ is about
$300$ times larger than $\gamma$. Therefore, the energy shift of the giant
atom can be larger than its natural line width and observed experimentally by
increasing the atom-waveguide coupling strength.

##  III THREE-LEVEL giant atom

In this section, let us consider the single-photon scattering on a three-level
giant atom driven in $\Lambda$ type. As schematically shown in Fig. 4, the
atom is characterized by the ground state $\left|g\right\rangle$, excited
state $\left|e\right\rangle$ and metastable state $\left|f\right\rangle$.
Here, there are two types of physical processes in this system. One is similar
to that in two-level giant atom system, that is, the interplay between the
multiple backward and forward photons of the transmission and reflection in
the waveguide; the other is peculiar for such a three-level atomic system,
i.e., that the strong driving field couples the states $|e\rangle$ and
$|f\rangle$ and a weak field (the incident photon in the waveguide) probes the
states $|e\rangle$ and $|g\rangle$ will induce the ATS phenomenon. The
incorporation between these two processes may provide a possibility of the
extra and interesting single-photon scattering spectrum for the giant-atom
setup.

The Hamiltonian $H_{2}$ of the current system can be divided into three parts,
i.e., $H_{2}=H_{w}+H_{a}+V_{2}$. Here, $H_{w}$ is the free Hamiltonian of the
waveguide which is given in Eq. (2). $H_{a}$ is the Hamiltonian of the giant
atom

$\begin{split}H_{a}=&\omega_{e}\left|e\right\rangle\left\langle
e\right|+\omega_{f}\left|f\right\rangle\left\langle f\right|\\\
&+\eta(\left|e\right\rangle\left\langle
f\right|e^{-i\omega_{d}t}+\left|f\right\rangle\left\langle
e\right|e^{i\omega_{d}t}),\end{split}$ (14)

where $\omega_{e}$ and $\omega_{f}$ ($\omega_{f}<\omega_{e}$) are the
frequencies of the state $|e\rangle$ and $|f\rangle$, respectively. As a
reference, we have set $\omega_{g}=0$. $\eta$ and $\omega_{d}$ are,
respectively, the strength and frequency of the classical field, which drives
$|f\rangle\leftrightarrow|e\rangle$ transition. In the rotating frame, the
time independent Hamiltonian becomes

$\tilde{H}_{a}=\omega_{e}\left|e\right\rangle\left\langle
e\right|+\delta\left|f\right\rangle\left\langle
f\right|+\eta(\left|e\right\rangle\left\langle
f\right|+\left|f\right\rangle\left\langle e\right|),$ (15)

where $\delta=\omega_{f}+\omega_{d}$.

Figure 4: Schematic configuration for a linear waveguide coupled to a
$\Lambda$-type three-level giant atom at the points $x=0$ and $x=x_{0}$.

The third part $V_{2}$ of the Hamiltonian $H_{2}$ describes the interaction
between the waveguide and the giant atom. Within the rotating wave
approximation, it can be expressed as

$\begin{split}V_{2}&=f\int
dx\delta(x)\left\\{\left[C_{R}^{\dagger}(x)+C_{L}^{\dagger}(x)\right]\left|g\right\rangle\left\langle
e\right|+{\rm H.c.}\right\\}\\\ &+f\int
dx\delta(x-x_{0})\left\\{\left[C_{R}^{\dagger}(x)+C_{L}^{\dagger}(x)\right]\left|g\right\rangle\left\langle
e\right|+{\rm H.c.}\right\\},\end{split}$ (16)

where $f$ is the coupling strength between the waveguide and giant atom.

In the single-excitation subspace, the eigenstate of the system can be
expressed as

$\begin{split}\left|E\right\rangle=&\int
dx\left[\phi_{R_{2}}(x)C_{R}^{\dagger}+\phi_{L_{2}}(x)C_{L}^{\dagger}\right]\left|\emptyset,g\right\rangle\\\
&+\lambda_{e}\left|\emptyset,e\right\rangle+\lambda_{f}\left|\emptyset,f\right\rangle,\end{split}$
(17)

where $|\emptyset,m\rangle(m=e,g,f)$ represents that the waveguide is in the
vacuum state while the giant atom is in the state $|m\rangle$.
$\phi_{R_{2}}\left(x\right)$ and $\phi_{L_{2}}\left(x\right)$ are single-
photon wave functions of the right-going and left-going modes in the
waveguide, respectively. $\lambda_{e}$ and $\lambda_{f}$ are the excitation
amplitudes of the giant atom in the excited state $\left|e\right\rangle$ and
metastable state $\left|f\right\rangle$, respectively. Similar to the
discussion in two-level atom setup, for a right-going incident photon with
wave vector $k$, the photon amplitude can be expressed as

$\begin{split}\phi_{R_{2}}\left(x\right)=&e^{ikx}\\{\theta\left(-x\right)+A_{2}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\\\
&+t_{2}\theta\left(x-x_{0}\right)\\},\end{split}$ (18)
$\phi_{L_{2}}\left(x\right)=e^{-ikx}\left\\{r_{2}\theta\left(-x\right)+B_{2}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\right\\},$
(19)

where $t_{2}$ and $r_{2}$ are the transmission and reflection amplitudes while
$A_{2}$ ($B_{2}$) are the amplitudes for finding a right (left)-going photon
inside the regime of the giant atom. Then, the Schödinger equation
$\tilde{H}|E\rangle=E|E\rangle$ (where $\tilde{H}=H_{w}+\tilde{H}_{a}+V_{2}$)
yields

$t_{2}=\frac{(E-\delta)\left[i\left(E-\omega_{e}\right)v_{g}-2if^{2}\sin\left(kx_{0}\right)\right]-iv_{g}\eta^{2}}{(E-\delta)\left[i\left(E-\omega_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)\right]-iv_{g}\eta^{2}},$
(20)
$r_{2}=\frac{2f^{2}(E-\delta)\left(1+\cos\left(kx_{0}\right)\right)e^{ikx_{0}}}{(E-\delta)\left[i\left(E-\omega_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)\right]-iv_{g}\eta^{2}}.$
(21)

Then the transmission rate $T_{2}=|t_{2}|^{2}$ and reflection rate
$R_{2}=|r_{2}|^{2}$ satisfy $T_{2}+R_{2}=1$ due to the neglect of the natural
relaxations in the atom.

In Fig. 5, we plot the transmission rate $T_{2}$ as functions of the detuning
$\Delta=E-\omega_{e}$ between the incident photon and atomic
$|g\rangle\leftrightarrow|e\rangle$ transition and the size of the giant atom
$x_{0}$. Here, we illustrate the result for resonantly driving the atom by the
field $\eta$ in Fig. 5(a) and (d), and non-resonantly in Fig. 5(b),(c) and
(e). All of the results are characterized by two narrow transmission valleys
($T_{2}\simeq 0$) and a relatively wide transmission windows ($T_{2}\simeq 1$)
between them around the two-photon resonance.

First, we focus on the transmission window ($T_{2}\simeq 1,R_{2}\simeq 0$). It
can be observed in Eqs. (21) that $R_{2}$=0 when
$E-\delta=E-\omega_{f}-\omega_{d}=0$, which implies the two-photon resonance
condition $\omega_{e}-E=\omega_{e}-\omega_{f}-\omega_{d}$. This transmission
window is caused by the usual ATS mechanism, which is true for both of the
small and giant atom setup. Moreover, we can also observe that $R_{2}=0$ when
$1+\cos{kx_{0}}=0$, that is $kx_{0}=(2m+1)\pi$ where $m$ is an integer, and
the dependence of the position is peculiar for the giant atom setup. Note that
$kx_{0}$ is the accumulated phase as the travelling photon moves from one
coupling point to the other. Therefore, such a ATS transmission is also
related to the position-dependent phase introduced by the interference effect
from the back and forth photons inside the regime covered by the giant atom.

Figure 5: The transmission rate $T_{2}$ as functions of the detuning $\Delta=E-\omega_{e}$ and the size of the giant atom $x_{0}$ for resonant driving. The parameters are set as $\omega_{e}=3$ GHz, $v_{g}=3\times 10^{8}$ m/s, $\omega_{f}=0.7\omega_{e}$, $f/\sqrt{v_{g}\omega_{e}}=0.05$, $\eta=5f=0.08\omega_{e}$. The other parameters are $\omega_{d}=0.3\omega_{e}$, $\omega_{d}=0.33\omega_{e}$, $\omega_{d}=0.27\omega_{e}$ for (a), (b) and (c), respectively. (d) The transmission rate $T_{2}$ for the lines labelled by “A” (solid) and “B” (dashed) in (a). (e) The transmission rate $T_{2}$ for the line labelled by “C” (solid) and “D” (dashed) in (b) and (c), respectively. Table 1: The values of $x_{0}$, $\Delta_{2}$ and corresponding values of $|G_{\pm,{k}}|$ for the two valleys in Fig. 6 (d) and (e). Line Label | curve | $\Delta_{2}/\omega_{e}$ | $x_{0}$ | $|G_{-,\omega_{-}/v_{g}}|/f$ | $|G_{+,\omega_{+}/v_{g}}|/f$
---|---|---|---|---|---
A | solid in Fig. 5(d) | $0$ | $1.48$ | $1.2069$ | $0.1783$
B | dashed in Fig. 5(d) | $0$ | $2.43$ | $0.2574$ | $1.1892$
C | dashed in Fig. 5(e) | $-0.03$ | $3.98$ | $1.4813$ | $1.2469$
D | solid in Fig. 5(e) | $0.03$ | $3.98$ | $0.7944$ | $1.0316$

Next, we discuss the two valleys, which represent the complete reflection
$(T_{2}\simeq 0,R_{2}\simeq 1)$. Based on Eq. (20), the complete reflection
occurs when

$\displaystyle E_{\pm}$ $\displaystyle=$
$\displaystyle\omega_{e}+f^{2}\sin(kx_{0})/v_{g}-\frac{\Delta_{2}}{2}$ (22)
$\displaystyle\pm\frac{\sqrt{\left(\Delta_{2}+2f^{2}\sin\left(kx_{0}\right)/v_{g}\right)^{2}+4\eta^{2}}}{2},$

where $\Delta_{2}=\omega_{e}-\omega_{f}-\omega_{d}$ is the detuning between
the driving field and the atomic $|f\rangle\leftrightarrow|e\rangle$
transition. This fact can be explained intuitively in the dressed state
presentation. The eigenfrequencies of the driving Hamiltonian $\tilde{H}_{a}$
in Eqs. (15) are

$\omega_{\pm}=\omega_{e}-\frac{\Delta_{2}}{2}\pm\frac{\sqrt{\Delta_{2}^{2}+4\eta^{2}}}{2}.$
(23)

and the corresponding states are

$\displaystyle|\psi_{+}\rangle$ $\displaystyle=$
$\displaystyle\cos{\frac{\theta}{2}}|e\rangle+\sin{\frac{\theta}{2}}|f\rangle,$
(24) $\displaystyle|\psi_{-}\rangle$ $\displaystyle=$
$\displaystyle-\sin{\frac{\theta}{2}}|e\rangle+\cos{\frac{\theta}{2}}|f\rangle.$
(25)

For the resonantly driving ($\Delta_{2}=0$), we will have $\theta=\pi/2$. For
the case of non-resonantly driving, we will have $\theta={\rm
atan}(2\eta/\Delta_{2})$ for $\Delta_{2}>0$ and $\theta=\pi+{\rm
atan}(2\eta/\Delta_{2})$ for $\Delta_{2}<0$.

Comparing Eqs. (23) with (22), we find that the incident photon will be
completely reflected when it is “nearly” resonant with $\omega_{\pm}$. This
fact is verified by the results shown in Fig. 5. Here, we use the phrase
“nearly” to imply that $E_{\pm}$ is not exactly equal to $\omega_{\pm}$, but
is slightly modulated by $x_{0}$. As a result, the photon transmission shows a
periodic phase modulation in terms of $x_{0}$, which is clearly demonstrated
in Figs. 5 (a), (b) and (c) for both of resonantly and non-resonantly driving
situations.

It is also shown in Fig. 5 that the two valleys discussed above possess
different widths. For example, see the horizontal lines labelled by “A”, “B”,
“C” and “D” in Figs. 5 (a), (b) and (c). This can be explained by the
different effective decay rates of the eigenstates with frequency
$\omega_{\pm}$ in the viewpoint of quantum open system by regarding the
waveguide as the effective environment. To this end, we rewrite the
interaction Hamiltonian $V_{2}$ in the momentum space by performing the
Fourier transformation (in terms of $|\psi_{\pm}\rangle$) as

$V_{2}=\sum_{k}\\{[C_{L}^{\dagger}(k)+C_{R}^{\dagger}(k)]|g\rangle[G_{+,k}\langle\psi_{+}|+G_{-,k}\langle\psi_{-}|]+{\rm
H.c.}\\}$ (26)

where

$G_{+,k}=f\cos{\frac{\theta}{2}}(1+e^{ikx_{0}}),G_{-,k}=-f\sin{\frac{\theta}{2}}(1+e^{ikx_{0}}),$
(27)

characterize the coupling strength between the $k$th mode in the waveguide and
the giant atom. We note that, $|G_{+,k}|$ and $|G_{-,k}|$ actually reflect the
width of the two valleys when the value of $k$ is taken to satisfy
$v_{g}k=\omega_{\pm}$.

In Table I, we list the values of $|G_{+,k}|$ and $|G_{-,k}|$ for the
horizontal lines in Fig. 5(a) (b) and (c), along which the transmission rates
are plotted as functions of the detuning $\Delta$ in Fig. 5 (d) and (e). It
shows a good agreement for the valley width between the table and the curves.
For the resonant driving, which valley is wider depends on the size of the
giant atom $x_{0}$, as shown in Fig. 5 (a) and (d). However, for the non-
resonant driving, as shown in Fig 5 (b), (c) and (e), the left valley is
nearly always narrower than the right one when $\Delta_{2}>0$ and wider when
$\Delta_{2}<0$. In this sense, the modification to the photon transmission by
the atomic size is more sensitive for the resonant driving setup.

Figure 6: The transmission rate $T_{2}^{\prime}=|t_{2}^{\prime}|^{2}$ as a
function of $\Delta$ within the atomic spontaneous emission for resonant
driving (a) and non-resonant driving (b). The parameters are set as
$\omega_{e}=3$ GHz, $v_{g}=3\times 10^{8}$ m/s, $\omega_{f}=0.7\omega_{e}$,
$f/\sqrt{v_{g}\omega_{e}}=0.05$, $\eta=5f=0.08\omega_{e}$,
$\gamma_{e}=\gamma_{f}=10^{-3}\omega_{e}$. The frequency of the driving field
is set as $\omega_{d}=0.3\omega_{e}$ in (a).

##  IV Effects of dissipation

When the natural relaxation of atoms is taken into consideration, the
transmission amplitude is expressed as

$t_{2}^{\prime}=\frac{\Delta_{f}\left[i\left(E-\omega_{e}+i\gamma_{e}\right)v_{g}-2if^{2}\sin\left(kx_{0}\right)\right]-iv_{g}\eta^{2}}{\Delta_{f}\left[i\left(E-\omega_{e}+i\gamma_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)\right]-iv_{g}\eta^{2}},$
(28)

where $\Delta_{f}=E-\delta+i\gamma_{f}$, and $\gamma_{e(f)}$ is the
spontaneous emission of the excited (metastable) state
$\left|e(f)\right\rangle$. The corresponding transmission rate is plotted in
Fig. 8. Compared with the ideal situation without relaxation, it is found that
the valley becomes wider and shallower by comparing Fig. 5(d),(e) and Fig.
6(a),(b).

In the previous discussion, we have only considered the effect of spontaneous
emission on the transmission rate. It should be noted that, in the presence of
the natural relaxation, the conservation relation
$|t_{1(2)}|^{2}+|r_{1(2)}|^{2}=1$ does not hold on any more. Therefore, it is
beneficial to discuss the reflection rate. Based on Eqs. (6) and (18), and the
Schödinger equation, the reflection amplitudes of two- and three-level giant
atom are expressed by

$\displaystyle r_{1}$
$\displaystyle=\frac{2f^{2}\left[1+\cos(kx_{0})\right]e^{ikx_{0}}}{i\left(E-\omega_{e}+i\gamma_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)},$
(29) $\displaystyle r_{2}$
$\displaystyle=\frac{2f^{2}\Delta_{f}\left[1+\cos\left(kx_{0}\right)\right]e^{ikx_{0}}}{\Delta_{f}\left[i\left(E-\omega_{e}+i\gamma_{e}\right)v_{g}-2f^{2}\left(1+e^{ikx_{0}}\right)\right]-iv_{g}\eta^{2}}.$
(30)

In Fig. 7, we plot the reflection rates $R_{1}=\left|r_{1}\right|^{2}$ and
$R_{2}=\left|r_{2}\right|^{2}$ as functions of $\Delta$, respectively.
Comparing the results with and without spontaneous emission, it shows that the
spontaneous emission breaks the complete reflection and makes the reflection
peaks become wider. However, the locations of the peaks relative to the
transmission valleys remain unchanged.

Figure 7: (a) The reflection rate $R_{1}$ as a function of $\Delta$. (b) The
reflection rate $R_{2}$ as a function of $\Delta$. The parameters are set as
$\omega_{e}=3$ GHz, $v_{g}=3\times 10^{8}$ m/s, $\omega_{d}=0.3\omega_{e}$,
$\omega_{f}=0.7\omega_{e}$,$f/\sqrt{v_{g}\omega_{e}}=0.05$,
$\eta=5f=0.08\omega_{e}$, $\gamma_{e}=\gamma_{f}=10^{-3}\omega_{e}$.

##  V conclusion and remarks

In this paper, we have studied the single-photon transmission in a one-
dimensional linear waveguide system coupled with two-level or three-level
giant atom. Generally, in the giant-atom regime, the backward and forward
photons propagating in the waveguide can lead to an interference effect. Thus,
for the case of two-level giant atom, we have shown that the complete
reflection occurs, but not for the case that the incident photon is exactly
resonant to the atom. In other words, the natural interference leads to an
effective small but nonzero frequency shift, and the shift can be
approximately regarded as a sinusoidal function of the atomic size. For the
driven three-level giant atom, we obtain an ATS line shape, and the
transparency window can be controlled by phase modulation in terms of the size
of the giant atom. In addition, the location and width of the transmission
valleys are tunable by adjusting the atomic size.

Throughout the paper, we have considered the tunability from the viewpoint of
the changing coupling distance, i.e., the giant atomic size. Considering the
superconducting transmission line as the linear waveguide, this distance can
be tuned by the coupled inductance or capacitance bk2020 ; AM2020 , to adjust
the propagating phase $kx_{0}$. Alternatively, one also can also the frequency
of the atom, so that the wave vector $k$ of the resonant mode in the waveguide
will be shifted. As a result, the propagating phase will be changed, and
produce the similar single-photon scattering spectrum. However, the period on
$x_{0}$, which is relative to $k$ [as shown in Eqs. (10,20,21)] will be
correspondingly modulated.

Beyond the specific model here, our work demonstrates how to use the interplay
between (among) the different physical processes to modify the photon
transmission in the waveguide. In the giant atom scenario, one of the
processes is provided by interference effect during the photon propagation,
while the other arises from the inner energy-level coupling, which is induced
by, for example, the classical driving. Motivating by the interplay mechanism,
we hope that the giant atom can be useful in designing photon or phonon based
quantum device, which goes beyond the conventional small atom setup.

###### Acknowledgements.

This work is supported supported by National Key R$\&$D Program of China (No.
2021YFE0193500), by National Natural Science Foundation of China (Grant No.
11875011, No. 12047566) and the Fundamental Research Funds for the Central
Universities (Grant No. 2412019FZ045).

##  Appendix A Two small atoms

In this appendix, we give the results when the giant atom is replace by two
small atoms separated by $x_{0}$. Similar to the giant atom setup, we also
divide the Hamiltonian of the system into three parts, i.e.,
$H_{3}=H_{e}+H_{\omega}+V_{3}$. The first part $H_{e}$ of the two atoms is

$H_{e}=\omega_{e}(\left|e\right\rangle_{1}\left\langle
e\right|+\left|e\right\rangle_{2}\left\langle e\right|),$ (31)

where $\omega_{e}$ is the transition frequency between the ground state
$\left|g\right\rangle$ and the excited state $\left|e\right\rangle$.

For the third part of the Hamiltonian $H_{3}$, we describe the interaction
between the two small atoms and the waveguide. Within the rotating wave
approximation, it can be expressed as

$\begin{split}V_{3}=&f\int
dx\delta\left(x\right)\left[\sigma_{1}^{+}C_{R}\left(x\right)+\sigma_{1}^{+}C_{L}\left(x\right)+{\rm
H.c.}\right]\\\ &+f\int
dx\delta\left(x-x_{0}\right)\left[\sigma_{2}^{+}C_{R}\left(x\right)+\sigma_{2}^{+}C_{L}\left(x\right)+{\rm
H.c.}\right].\end{split}$ (32)

where $f$ is the coupling strength between the waveguide and two small atoms.
The Dirac-$\delta$ function in the Hamiltonian $V_{3}$ indicates that the two
small atoms are located at $x=0$ and $x=x_{0}$, respectively, and interact
with the linear waveguide at these two points.

The eigenstate in the single-excitation subspace can be written as

$\begin{split}\left|E\right\rangle=&\int
dx\left[\phi_{R_{3}}\left(x\right)C_{R}^{\dagger}\left(x\right)+\phi_{L_{3}}\left(x\right)C_{L}^{\dagger}\left(x\right)\right]\left|G\right\rangle\\\
&+u_{1}\sigma_{1}^{+}\left|G\right\rangle+u_{2}\sigma_{2}^{+}\left|G\right\rangle,\end{split}$
(33)

where $|G\rangle$ represents that the waveguide is in the vacuum states, while
two atoms are in the ground state $|g\rangle$. $\phi_{R_{3}}\left(x\right)$
and $\phi_{L_{3}}\left(x\right)$ are single-photon wave functions of the
right-going and left-going modes in the waveguide respectively. $u_{1}$ and
$u_{2}$ are the excitation amplitudes of the two atoms, respectively. Solving
the stationary Schödinger equation
$H_{3}\left|E\right\rangle=E\left|E\right\rangle$, the amplitudes equation can
be obtained as

$\displaystyle-
iv_{g}\frac{d}{dx}\phi_{R_{3}}\left(x\right)+f\left[\delta\left(x\right)u_{1}+\delta\left(x-x_{0}\right)u_{2}\right]$
$\displaystyle=$ $\displaystyle E\phi_{R_{3}}\left(x\right),$ $\displaystyle
iv_{g}\frac{d}{dx}\phi_{L_{3}}\left(x\right)+f\left[\delta\left(x\right)u_{1}+\delta\left(x-x_{0}\right)u_{2}\right]$
$\displaystyle=$ $\displaystyle E\phi_{L_{3}}\left(x\right),$
$\displaystyle\omega_{e}u_{1}+f\left[\phi_{R_{3}}\left(0\right)+\phi_{L_{3}}\left(0\right)\right]$
$\displaystyle=$ $\displaystyle Eu_{1},$ (34c)
$\displaystyle\omega_{e}u_{2}+f\left[\phi_{R_{3}}\left(x_{0}\right)+\phi_{L_{3}}\left(x_{0}\right)\right]$
$\displaystyle=$ $\displaystyle Eu_{2}.$ (34d)

Next, we consider the scattering behavior when a single photon with wave
vector $k$ is incident from the left side of the waveguide. Similar to the
discussion in two-level giant-atom setup, the wave function
$\phi_{R_{3}}\left(x\right)$ and $\phi_{L_{3}}\left(x\right)$ can be expressed
as

$\begin{split}\phi_{R_{3}}\left(x\right)=&e^{ikx}\\{\theta\left(-x\right)+A_{3}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\\\
&+t_{3}\theta\left(x-x_{0}\right)\\},\end{split}$ (35)
$\phi_{L_{3}}\left(x\right)=e^{-ikx}\left\\{r_{3}\theta\left(-x\right)+B_{3}\left[\theta\left(x\right)-\theta\left(x-x_{0}\right)\right]\right\\},$
(36)

Then, we substitute Eqs.(35) and (36) into Eqs.(34), it yields the dispersion
relation $E=v_{g}k$. Furthermore the transmission rate
$T_{3}=\left|t_{3}\right|^{2}$ can be obtained as

$T_{3}=\frac{\left(\Delta^{2}v_{g}\right)^{2}}{\left(\Delta^{2}v_{g}-\frac{2f^{4}}{v_{g}}\sin^{2}\left(kx_{0}\right)\right)^{2}+\left(2f^{2}\Delta+\frac{f^{4}}{v_{g}}\sin\left(2kx_{0}\right)\right)^{2}}.$
(37)

where $\Delta=E-\omega_{e}$ is the detuning between the atom and the
propagating photon in the waveguide. It clearly shows that, the incident
photon will completely reflected when $\Delta=0$ which is different from that
of a single giant atom, as discussed in the main text.

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