# JWST view of three infant galaxies at $z=8.3$ and implications for
reionization

Zhiyuan Ma Department of Astronomy, University of Massachusetts, Amherst, USA, 01003  Bangzheng Sun Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA  Cheng Cheng Chinese Academy of Sciences South America Center for Astronomy, National Astronomical Observatories, CAS, Beijing 100101, China  Haojing Yan Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA  Fengwu Sun Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009, USA  Nicholas Foo School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Eiichi Egami Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009, USA  José M. Diego Instituto de Física de Cantabria (CSIC-UC), Avda. Los Castros s/n, 39005, Santander, Spain  Seth H. Cohen School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Rolf A. Jansen School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Jake Summers School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Rogier A. Windhorst School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Jordan C. J. D’Silva International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Anton M. Koekemoer Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Dan Coe Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Association of Universities for Research in Astronomy (AURA) for the European Space Agency (ESA), STScI, Baltimore, MD 21218, USA  Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD 21218, USA  Christopher J. Conselice Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK  Simon P. Driver International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  Brenda Frye Department of Astronomy/Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009, USA  Norman A. Grogin Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Madeline A. Marshall National Research Council of Canada, Herzberg Astronomy & Astrophysics Research Centre, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Mario Nonino INAF-Osservatorio Astronomico di Trieste, Via Bazzoni 2, 34124 Trieste, Italy  Rafael Ortiz III School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Nor Pirzkal Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Aaron Robotham International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  Russell E. Ryan, Jr. Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Christopher N. A. Willmer Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009, USA  Nathan J. Adams Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK  Nimish P. Hathi Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Hervé Dole Université Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France  S. P. Willner Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Daniel Espada Departamento de Física Teórica y del Cosmos, Campus de Fuentenueva, Edificio Mecenas, Universidad de Granada, E-18071, Granada, Spain  Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, E-18071, Granada, Spain  Lukas J. Furtak Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel  Tiger Yu-Yang Hsiao Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St., Baltimore, MD 21218, USA  Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Qiong Li Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK  Wenlei Chen Department of Physics, Oklahoma State University, 145 Physical Sciences Bldg, Stillwater, OK 74078, USA  Jean-Baptiste Jolly Max-Planck-Institut für Extraterrestrische Physik (MPE), Giessenbachstraße 1, D-85748 Garching, Germany  Chian-Chou Chen Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan

###### Abstract

New JWST/NIRCam wide-field slitless spectroscopy provides redshifts for two
$z>8$ galaxies located behind the lensing cluster MACS J0416.1$-$2403\. Both
galaxies are strong [O 3] $\lambda$5007 emitters. For one galaxy, “Y1,” the
existing redshift $z=8.31$, based on ALMA measurements of [O 3] 88 $\mu$m and
[C 2] 157.7 $\mu$m lines, is confirmed. JWST/NIRCam images resolve this galaxy
into three components of similar colors, and the whole system extends over
$\sim$3.4 kpc. The other galaxy, “JD,” is at $z=8.34$ instead of the
previously claimed $z=9.28$. It has a companion, “JD-N,” at the same redshift
with projected separation $\sim$2.3 kpc. All objects are only moderately
magnified and have intrinsic $M_{\rm UV}$ ranging from $-19.66$ to $-20.85$
mag. Their eight-band NIRCam spectral energy distributions show that the
galaxies are all very young with ages $\lesssim$11 Myr and stellar masses
about $10^{8}$ ${\rm M_{\odot}}$. These infant galaxies are actively forming
stars at rates of a few tens to a couple of hundred $\rm{M_{\odot}}$ yr-1, but
only one of them (JD) has a blue rest-frame UV slope. This slope indicates a
high Lyman-continuum photon escape fraction that could contribute
significantly to the cosmic hydrogen-reionizing background. The other two
systems have much flatter slopes largely because their dust extinction is
twice as high as JD’s albeit only $A_{V}\sim 0.90$ mag. The much lower
indicated escape fractions show that even very young, actively star-forming
galaxies can have negligible contribution to reionization when they quickly
form dust throughout their bodies.

##  1 Introduction

Young, star-forming galaxies are long thought to be the major drivers of the
cosmic hydrogen reionization because they should have strong UV emission and
are sufficiently abundant at $z>6$ (e.g., Yan & Windhorst, 2004; Bouwens et
al., 2006; Finkelstein et al., 2012; Robertson, 2022). Their exact
contribution to the ionizing photon background, however, still depends on how
effectively their Lyman continuum (LyC; $\lambda<912$Å) photons can escape and
reach the surrounding intergalactic medium (IGM). Direct measurement of this
escape fraction ($f_{\rm esc}$) for star-forming galaxies at $z>6$ is
impossible, because the line-of-sight IGM H 1 absorption at such redshifts
wipes out any LyC photons. As the alternative, $f_{\rm esc}$ has been measured
for the “analogs” at lower redshifts ($z\lesssim 3$–4) where the IGM hydrogen
is fully ionized (e.g. Steidel et al., 2001, 2018; Vanzella et al., 2012;
Siana et al., 2015; Grazian et al., 2016; Izotov et al., 2016, 2018, 2021;
Saldana-Lopez et al., 2022; Griffiths et al., 2022; Flury et al., 2022), and
the correlations between $f_{\rm esc}$ and various observables are sought
after in order to provide some viable routes to indirectly measure $f_{\rm
esc}$ at $z>6$. Among all possible correlations, the one with the rest-frame
UV slope (commonly denoted as $\beta$; $f_{\lambda}\propto\lambda^{\beta}$) is
the most promising (e.g., Zackrisson et al., 2013; Chisholm et al., 2022).

There have been ample studies of the UV slopes of galaxies at $z>6$ using the
Hubble Space Telescope (HST) deep survey data (e.g., Hathi et al., 2008;
Bouwens et al., 2010; Dunlop et al., 2012, 2013; Finkelstein et al., 2012),
and the investigation has been extended to higher redshifts and fainter limits
by the James Webb Space Telescope (JWST; e.g., Topping et al., 2022; Cullen et
al., 2023; Nanayakkara et al., 2023; Morales et al., 2024; Austin et al.,
2024). However, the vast majority of those are based on photometric samples of
candidates that have not yet been spectroscopically confirmed. Thanks to the
growing number of JWST spectroscopic programs, the situation is now quickly
changing (e.g., Tang et al., 2023; Fujimoto et al., 2023; Saxena et al.,
2024). Nevertheless, there are still few studies using confirmed $z>6$
galaxies. In particular, there seems to be a lack of extremely young galaxies
(age $\lesssim 30$ Myr) in the samples. Such galaxies presumably should have
the bluest UV slopes because their UV emission must be dominated by O and B
stars.

This work presents a case study of three $z\approx 8.3$ galaxies in the MACS
J0416.1$-$2403 cluster (hereafter “MACS0416”) field. This is one of the six
Hubble Frontier Fields (HFF; Lotz et al., 2017) and one of the targets of the
Reionization Lensing Cluster Survey (RELICS; Coe et al., 2019). A few
investigations have used HST to search for lensed high-redshift galaxies
behind this cluster (e.g., Coe et al., 2015; Infante et al., 2015; Laporte et
al., 2015), and some candidates with photometric redshifts $z_{\rm ph}\gtrsim
8$–9 have been found. Among them, two have reported spectroscopic redshifts at
$z>8$, both based on ALMA spectroscopy. One is MACSJ0416.1_Y1 (hereafter “Y1”
for simplicity), which was first selected as a $z\approx 8$ candidate (Laporte
et al., 2015, $z_{\rm ph}=8.57$) and was later confirmed at $z=8.31$ through
the detections of the [O 3] 88 $\mu$m line (Tamura et al., 2019) and the [C 2]
157.7 $\mu$m line (Bakx et al., 2020). The other is MACS0416.1-JD (hereafter
“JD”), which was also first discovered as a $z\approx 8$ candidate by Coe et
al. (2015, object “FFC2-1151-4540”) with $z_{\rm ph}=8.1$. It was
independently recovered by Laporte et al. (2015, object “MACSJ0416.1_Y2”) but
with $z_{\rm ph}=9.29$ (see also Infante et al. 2015; Laporte et al. 2016).
Laporte et al. (2021) reported detection of the [O 3] 88 $\mu$m line at
$z=9.28$ and called the object “JD” because its colors qualify it as a
$J$-band dropout. The present work was motivated by these two objects. As we
will show, the new JWST data confirm the redshift of Y1 at $z=8.31$ but change
that for JD to $z=8.34$. In addition, we identify a neighbor to JD that is at
nearly the same redshift. Interestingly, they all have ages $\lesssim 11$ Myr
inferred from their spectral energy distributions (SEDs), and yet none of them
shows extremely blue UV slopes.

This Letter is organized as follows. Section 2 describes the JWST data, and
Section 3 presents the photometric and spectroscopic results. The analysis is
given in Section 4, and Section 5 is a brief summary. We adopt a flat
$\Lambda$ cold-dark-matter cosmology with $h=0.7$, $\Omega_{\Lambda}=0.7$, and
$\Omega_{M}=0.3$. All magnitudes are in the AB system, and all coordinates are
in the ICRS frame (equinox 2000).

##  2 JWST NIRCam Observations and Data Reduction

Figure 1: Cutouts of Y1 (top) and JD (bottom) in the eight NIRCam bands using
the 30mas images. North is up and East is left. The size of the Y1 images is
1″$\times$1″, while that of the JD images is 2″$\times$2″. The three resolved
Y1 components (Y1-E, Y1-C and Y1-W) are labeled in the F200W image. JD has a
close companion, JD-N, whose colors also meet the requirements of a $J$-band
dropout. Both objects are labeled in the F200W image.

###  2.1 NIRCam Imaging

MACS0416 is one of the targets observed by the JWST GTO program Prime
Extragalactic Areas for Reionization and Lensing Science (PEARLS; PI: R.
Windhorst; PID 1176; Windhorst et al. (2023)). It was imaged by NIRCam in
eight filters, F090W, F115W, F150W, and F200W in the short wavelength channel
(SW) and F277W, F356W, F410M, and F444W in the long wavelength channel (LW).
The observations were carried out in three epochs: 2022 Oct 7 (Ep1), 2022 Dec
29 (Ep2), and 2023 Feb 10 (Ep3). The observations were described by Yan et al.
(2023, their Table 1), who gave details of the data reduction. In Yan et al.
(2023), the images were stacked on a per-epoch basis for the transient
studies. For this work, we combined the data in all three epochs for JD, which
reach the total integration times of 8761 seconds in F150W, F200W, F277W, and
F356W, 10909 seconds in F115W and F410M, and 11338 seconds in F090W and F444W.
In Ep3, Y1 is contaminated by a strong diffraction spike of a bright star, and
therefore we only combined Ep1 and Ep2 data for the study of this object. The
total integration times are 5841 seconds in F150W, F200W, F277W, and F356W,
and 7559 seconds in F090W, F115W, F410M, and F444W, respectively. As Yan et
al. (2023), we created stacks at scales of both 0$\farcs$06 (“60mas”) and
0$\farcs$03 (“30mas”), which have the magnitude zero point of 26.581 and
28.087, respectively. These images are all aligned to the HFF HST images.

###  2.2 NIRCam Wide-field Slitless Spectroscopy

MACS0416 was also observed by the NIRCam instrument in its wide-field slitless
spectroscopy (WFSS) mode. This mode has two settings, Grism R and Grism C,
which disperse light in either the direction of detector rows (“R”) or columns
(“C”) in the LW channel, both having spectral resolution $R\approx 1600$ at
$\sim$4 $\mu$m. The field was observed by the programs PID 2883 (“MAGNIF:
Medium-band Astrophysics with the Grism of NIRCam in Frontier Fields,” PI: F.
Sun) and PID 3538 (“Unveiling the properties of high-redshift
low/intermediate-mass galaxies in Lensing fields with NIRCam Wide Field
Slitless Spectroscopy,” PI: E. Iani).

The MAGNIF observations of MACS0416 were carried out on 2023 Aug 20 in Grism C
using the F480M filter with total integration time of $\sim$3100 seconds.
These observations covered only JD with Y1 $\sim$3″ outside the field. The
observations of PID 3538 in this field were carried out on 2023 Dec 22–24 and
2024 Jan 17 in both Grisms R and C. Four filters were used, F300M, F335M,
F410M, and F460M with total integration times in each band 1761 and 1546
seconds respectively for the two grisms.

To reduce the grism data, we retrieved the Level 1b “uncal” files from the
Mikulski Archive for Space Telescopes (MAST) and ran them through the
calwebb_detector1 routine of the JWST data reduction pipeline (version 1.13.4
under the context of jwst_1223.pmap) to obtain the “rate.fits” files. We then
followed the procedures described by Sun et al. (2023) to further process the
data. All single exposures were registered to the GAIA DR3 astrometry. There
is a systematic offset between the HFF astrometry and that of GAIA in this
field, which can be corrected by R.A.(GAIA) $=$ R.A.(HFF) $+$ 0$\farcs$21, and
Decl.(GAIA) $=$ Decl.(HFF) $-$ 0$\farcs$08\. This was taken into account when
extracting the spectra.

##  3 Data Analysis

###  3.1 Overview

From their high-resolution (beam size 81.1$\times$112.2 mas) ALMA [O 3] 88
$\mu$m image of Y1, Tamura et al. (2023) resolved the system into three knots
(“O1,” “O2,” and “O3”). Using this as a guide, these authors separated its HST
WFC3 image into three components (“E,” “C,” and “W” from east to west) that
coincide with the three [O 3] 88 $\mu$m knots. These three components are
clearly distinguished in the NIRCam SW images (Figures 1 and 2), and the
entire system extends $\sim$0$\farcs$73 along the long axis.

JD has a close companion object 0$\farcs$91 away to its northeast, which was
also selected by Laporte et al. (2015) as a Y-band dropout (their
“MACSJ0416.1_Y3”). As we will show below, its colors meet the requirements of
a $J$-band dropout as well, and its redshift is nearly the same as that of JD.
We refer to it as “JD-N” in this work to indicate its association with JD.

Figure 2: Color images of Y1 and JD, at the same scales as in Figure 1. The
color scheme is F090W+F115W+F150W as blue, F200W+F277W as green, and
F356W+F410M+F444W as red. In each image, the red arrows show the NIRCam grism
WFSS dispersion direction in the PID 3538 observation in Grism C (GC) and
Grism R (GR), respectively; the yellow arrow in the JD image shows the
dispersion direction of GC utilized by the MAGNIF observations.

###  3.2 Photometry and SEDs

To construct the SEDs of our targets, we used the photometry on the 60mas
images. These images were convolved to match their point spread functions
(PSFs) to that of the F444W image, which has the coarsest spatial resolution
(PSF full width at half-maximum, or FWHM, 0$\farcs$145). WebbPSFs (Perrin et
al., 2014, 2015) were used in the convolution. Matched-aperture photometry was
done by running SExtractor (Bertin & Arnouts, 1996) in the dual-image mode
with F444W as the detection band, and we adopted the MAG_ISO magnitudes.

We obtained SEDs for Y1’s three components using the 30mas images. As they are
severely blended, they cannot be separated automatically. Therefore, we
manually defined the segmentation map for the three components based on the
30mas F200W image, where they are the most clearly detected and separated.
With this segmentation map (illustrated in Figure 3), we used Photutils
(Bradley et al., 2023) to extract the flux for each component in each band on
the PSF-matched 60mas images.

Table 1 summarizes the photometry for all these sources. From their colors
(large magnitude breaks between F115W and F150W and being ${\rm S/N}<2$ in
F090W), JD and JD-N indeed qualify as $J$-band dropouts.

Figure 3: Illustration of the Y1 segmentation (in red polygons) that separate its three components for individual flux extraction in the imaging data. The background is the 30mas F200W image. Table 1: NIRCam photometry (without correcting for lensing magnification) | Y1 | Y1-E | Y1-C | Y1-W | JD | JD-N
---|---|---|---|---|---|---
R.A. (deg) | 64.0391682 | 64.0392323 | 64.0391780 | 64.0391192 | 64.0479851 | 64.0480694
Decl (deg) | -24.0931764 | -24.0931976 | -24.0931800 | -24.0931886 | -24.0816685 | -24.0814279
F090W |  $>$29.96 |  $>$29.96 |  $>$29.96 |  $>$29.96 | 30.41 $\pm$ 2.54 | 30.29 $\pm$ 2.78
F115W | 26.89 $\pm$ 0.15 | 28.39 $\pm$ 0.09 | 28.68 $\pm$ 0.11 | 28.17 $\pm$ 0.08 | 27.66 $\pm$ 0.21 | 27.71 $\pm$ 0.26
F150W | 26.14 $\pm$ 0.07 | 27.56 $\pm$ 0.04 | 27.69 $\pm$ 0.04 | 27.37 $\pm$ 0.04 | 26.65 $\pm$ 0.08 | 26.52 $\pm$ 0.09
F200W | 26.07 $\pm$ 0.06 | 27.51 $\pm$ 0.04 | 27.54 $\pm$ 0.03 | 27.29 $\pm$ 0.03 | 26.85 $\pm$ 0.08 | 26.43 $\pm$ 0.07
F277W | 25.96 $\pm$ 0.02 | 27.53 $\pm$ 0.01 | 27.43 $\pm$ 0.01 | 27.26 $\pm$ 0.01 | 26.97 $\pm$ 0.03 | 26.30 $\pm$ 0.02
F356W | 25.58 $\pm$ 0.01 | 27.27 $\pm$ 0.01 | 27.15 $\pm$ 0.01 | 27.06 $\pm$ 0.01 | 26.91 $\pm$ 0.03 | 25.94 $\pm$ 0.01
F410M | 25.90 $\pm$ 0.03 | 27.57 $\pm$ 0.02 | 27.41 $\pm$ 0.02 | 27.28 $\pm$ 0.02 | 27.06 $\pm$ 0.05 | 25.96 $\pm$ 0.03
F444W | 24.84 $\pm$ 0.01 | 26.58 $\pm$ 0.01 | 26.30 $\pm$ 0.01 | 26.34 $\pm$ 0.01 | 26.22 $\pm$ 0.02 | 25.32 $\pm$ 0.01

###  3.3 Spectroscopic Identifications

Figure 2 shows the color composites of these two systems, where the WFSS
dispersion directions are indicated. The NIRCam WFSS mode is very suitable for
the detection of emission lines, which is the main focus here. To optimize the
line detection, we subtracted a continuum estimated (following Kashino et al.
2023) by running a median filter $1\times 51$ pixels in size with a 9-pixel
“hole” at the center along each row or column.

As Y1 consists of three components, it would be ideal to extract the spectra
for each component. In practice, however, the individual extraction could only
split the signal into two parts, which largely coincide with Y1-E and Y1-W,
respectively. This is due to both the faintness of Y1-C at 3–5 $\mu$m and the
coarse resolution at these wavelengths. For simplicity, we refer to the
separate extractions as Y1-E and Y1-W even though both could include some
contribution from Y1-C. As shown in Figure 2, the PID 3538 Grism R dispersion
direction is very close to the spatial extension of the Y1 system, causing
severe spectral contamination between the two components. For this reason, we
only used the Grism C data. The extracted 2D and 1D spectra are shown in
Figure 4.

For both Y1-E and Y1-W, the [O 3] $\lambda\lambda$4959,5007 lines are clearly
seen in the F460M data. The detection of the [O 2] $\lambda$3727 line in the
F335M data is marginal in the 1D spectrum but convincing in the 2D spectrum.
To determine the redshifts, we fitted a Gaussian profile to the [O 3]
$\lambda$5007 line, which is the strongest, to obtain its observed central
wavelengths. The redshifts thus derived are $z=8.309\pm 0.002$ for Y1-E and
$z=8.312\pm 0.002$ for Y1-W. We also fitted Gaussians to the weaker lines,
using the initial guesses of the line centers at where they should be under
these redshifts. The results based on the best-fit central wavelengths thus
obtained are all in excellent agreement with the redshifts based on the [O 3]
$\lambda$5007 line.

For JD and JD-N, we extracted the spectra using the data from both MAGNIF and
PID 3538. However, we discarded the Grism C data from the latter because its
dispersion direction is almost parallel to the position angle of the two
objects (Figure 2). The spectra are shown in Figure 4. JD-N shows two lines,
one strong and the other marginal, in the F460M data from PID 3538 Grism R.
Based on Gaussian fitting as above, these two lines coincide with the [O 3]
$\lambda\lambda$4959,5007 doublet at $z=8.346\pm 0.002$. The MAGNIF F480M
Grism C data reveal only one line, which should be [O 3] $\lambda$5007\. The
[O 3] $\lambda$4959 line falls outside the filter transmission range and is
not detected. For JD, the data from both programs show only one strong line.
Given its possible redshift range, the line must be [O 3] $\lambda$5007 at
$z=8.341\pm 0.002$. The non-detection of the [O 3] $\lambda$4959 line in the
MAGNIF data is again because it is outside of the filter transmission range.
On the other hand, the non-detection of the $\lambda$4949 line in the PID 3538
is likely because of limited sensitivity: given the expected ratio of
$\sim$1:3 of the [O 3] doublet, the [O 3] $\lambda$4959 line flux is
comparable to the noise level of the data.

Table 2 summarizes the line measurements. The line widths were converted to
velocities by $\Delta v=c\Delta\lambda/\lambda_{c}$, where $\lambda_{c}$ and
$\Delta\lambda$ are the mean and FWHM of the Gaussian fit, respectively, and
$c$ is the speed of light. The total line intensities were obtained by
integrating the fitted Gaussian profile within a $4\times$FWHM wavelength
range centered at $\lambda_{c}$. The associated errors were estimated using
the non-smoothed 1D spectra. When calculating the observed equivalent widths
(EW), the F410M magnitudes in Table 1 were used as the continuum flux density.
To account for continuum contributions from Y1-C in the spectra of Y1-E and
Y1-W, half the F410M flux density of Y1-C was added to the Y1-E and Y1-W
continua. Not surprisingly, these objects all have very large [O 3] EW values,
which can largely account for the brightening of their SEDs in F444W. For
example, the measured [O 3] $\lambda$5007 EW of JD implies that this line
alone increases the object’s F444W brightness (as compared to that in F410M)
by $-0.83\pm 0.06$ mag, which is almost the same as the observed
$m_{410}-m_{444}=-0.84\pm 0.05$ mag.

Table 2: Summary of NIRCam WFSS results  Target | Program | Grating | $z$ | $I_{5007}$ | $\Delta v_{5007}$ |  EW5007 | $I_{4959}$ | $\Delta v_{4959}$ |  EW4959 | $I_{3727}$ | $\Delta v_{3727}$ |  EW3727
---|---|---|---|---|---|---|---|---|---|---|---|---
Y1-E | 3538 | F335M+F460M/C | 8.309±0.002 | 1.10±0.12 | 338±26 | 14879±1221 | 0.43±0.07 | 366±91 | 5727±733 | 0.18±0.08 | 541±171 | 1338±542
Y1-W | 3538 | F335M+F460M/C | 8.312±0.002 | 1.11±0.10 | 310±32 | 12519±913 | 0.45±0.11 | 346±78 | 4967±848 | 0.24±0.10 | 424±71 | 1505±310
JD | MAGNIF | F480M/C | 8.339±0.002 | 0.55±0.12 | 249±35 | 7394±1805 | … | … | … | … | … | …
| 3538 | F460M/R | 8.341±0.002 | 0.64±0.10 | 217±23 | 8635±1124 | … | … | … | … | … | …
JD-N | MAGNIF | F480M/C | 8.346±0.002 | 1.13±0.11 | 274±35 | 5479±423 | … | … | … | … | … | …
| 3538 | F460M/R | 8.346±0.002 | 1.09±0.12 | 316±20 | 5325±356 | 0.39±0.15 | 409±91 | 1870±442 | … | … | …
Figure 4: Extracted 2D and 1D spectra from the NIRCam WFSS observations. For
each object, the top panel shows the 2D spectrum with background and continuum
subtracted, and the bottom panel shows the resultant 1D spectrum and includes
the object label. Identified emission lines are marked. The 2D spectra of JD
and JD-N are displayed after smoothing with a Gaussian kernel with $\sigma=1$,
while the 1D spectra are based on the unsmoothed 2D spectra, then smoothed
using a Gaussian kernel with $\sigma=0.6$. The 2D spectra of Y1 are not
smoothed to show the separation of the two components, but the 1D spectra are
smoothed with $\sigma=0.6$. With only a few exposures in each setup, stacking
has limited ability to remove outlying pixels, leaving some narrow spikes in
the 1D spectra. Fortunately, the emission lines of interest are not
contaminated by such spikes.

##  4 Discussion

###  4.1 Redshifts, environment, and magnification

The redshifts of Y1-E and Y1-W agree with each other within the uncertainties,
and their rest-frame relative velocity is only $97\pm 64$ km s-1. The average
is $z=8.311\pm 0.003$, which is in very good agreement with the previously
reported $z=8.3118\pm 0.0003$ based on [O 3] 88 $\mu$m (Tamura et al., 2019)
and $z=8.31132\pm 0.00037$ based on [C 2] 157.5 $\mu$m (Bakx et al., 2020) for
Y1. The apparent [O 3] 88 $\mu$m line flux measured by (Tamura et al., 2019)
is 0.66 Jy km s-1, which corresponds to $7.92\times 10^{-18}$ erg cm-2 s-1 at
the observed frequency (364.377 GHz). Therefore, the line flux ratio of [O 3]
$\lambda$5007 and 88 $\mu$m is 2.79.

For JD, our redshift of $z=8.341\pm 0.002$ disagrees with the previously
reported $z=9.28$ by Laporte et al. (2021), which was based on their claimed
detection of the [O 3] 88 $\mu$m line. That redshift would give no obvious
identification for the emission line observed at 4.68 µm. We have reduced the
same ALMA Band 7 data used by these authors but do not detect a line (Appendix
A). At $z=8.346\pm 0.002$, the velocity offset from JD to JD-N is $160\pm 64$
km s-1.

According to the lens model of Diego et al. (2023), the magnification factors
for Y1 and JD/JD-N are $\mu=1.21$ and 2.26, respectively. Based on their
magnitudes in F150W (sampling the rest-frame UV range of $\sim$1430–1790Å at
these redshifts), their intrinsic absolute UV magnitudes after correcting the
lensing magnifications are $M_{\rm UV}=-20.85$, $-19.66$ and $-19.80$ mag for
Y1, JD, and JD-N, respectively. The observed size of Y1 as a whole is only
marginally affected by the magnification, and its corresponding physical size
along the long axis is $\sim$3.4 kpc. The separation of JD and JD-N, on the
other hand, is affected significantly by lensing. Their separation in the
source plane is $\sim$0$\farcs$51, which corresponds to 2.4 kpc. Therefore, JD
and JD-N very likely form a interacting pair.

###  4.2 Stellar populations

To understand the stellar populations of these galaxies, we fitted their SEDs
using the Bagpipes software (Carnall et al., 2018), which utilizes the stellar
population synthesis models of Bruzual & Charlot (2003) with the Kroupa (2001)
initial mass function. The fitting was done at fixed redshifts of 8.31 and
8.34 for Y1 and JD/JD-N, respectively. An exponentially declining star
formation history (SFH) in the form of SFR $\propto e^{-t/\tau}$ was adopted.
The option to include nebular emission lines was enabled, and we used the
Calzetti dust-extinction law (Calzetti et al., 1994; Calzetti, 2001) with
$A_{V}$ ranging from 0 to 2.0 mag. The metallicity was allowed in the range of
$0\leq Z_{*}/Z_{\odot}\leq 2.5$, and the ionization parameter could vary in
$-2.5<{\rm log}(U)<-0.5$. The results of the SED fitting are summarized in
Table 3, and Figure 5 shows the 16th to 84th percentile range for the
posterior spectra superposed on the SEDs. The “corner plots” showing the
posterior for fitted parameters are given in Figure 6. Overall, Y1, JD, and
JD-N have rather similar physical properties, and the three components of Y1
are also quite similar to each other.

The existence of the oxygen lines in these objects (as well as the [C 2] 157.7
$\mu$m line in Y1) indicates that they have already acquired a substantial
amount of metals, and this is further supported by the metallicities dervied
from the SED analysis ($Z/Z_{\odot}\approx 0.4$–0.6). Interestingly, all the
objects are $\lesssim 11$ Myr old, which means that the only route for their
metal enrichment was through core-collapse supernovae. Consistent with their
young ages, the galaxies have low stellar masses of 0.8–6.4 $\times
10^{8}\leavevmode\nobreak\ {\rm M_{\odot}}$ (after correcting for lensing
magnifications), which can be achieved by SFRs on the order of a few tens to
$\sim$100 ${\rm M_{\odot}}$ yr-1 as derived (see also Figure 6).

Table 3: Bagpipes posterior parameters of the fitted stellar populations Object | $A_{V}$ | Age/Myr | $Z/Z_{\sun}$ | ${\rm log}(U)$ | ${\rm log}(\mu M/\mathrm{M_{\sun}})$ |  $\mu$SFR/($\mathrm{M_{\odot}\ yr}^{-1}$)
---|---|---|---|---|---|---
Y1 | $0.80^{+0.06}_{-0.11}$ | $2.68^{+0.81}_{-1.09}$ | $0.60^{+0.19}_{-0.30}$ | $-2.04^{+0.09}_{-0.15}$ | $8.89^{+0.06}_{-0.14}$ | $291.69^{+299.73}_{-127.88}$
Y1-E | $0.85^{+0.08}_{-0.11}$ | $2.86^{+1.08}_{-1.15}$ | $0.60^{+0.21}_{-0.25}$ | $-1.73^{+0.13}_{-0.22}$ | $8.27^{+0.08}_{-0.14}$ | $63.09^{+72.83}_{-27.12}$
Y1-C | $0.95^{+0.07}_{-0.08}$ | $2.44^{+0.68}_{-0.96}$ | $0.57^{+0.15}_{-0.17}$ | $-1.62^{+0.12}_{-0.13}$ | $8.37^{+0.07}_{-0.07}$ | $107.42^{+65.41}_{-37.10}$
Y1-W | $0.95^{+0.09}_{-0.16}$ | $3.32^{+1.24}_{-1.53}$ | $0.58^{+0.30}_{-0.24}$ | $-1.43^{+0.22}_{-0.21}$ | $8.46^{+0.12}_{-0.16}$ | $87.19^{+112.34}_{-44.32}$
JD | $0.41^{+0.13}_{-0.20}$ | $3.57^{+1.09}_{-1.63}$ | $0.44^{+0.35}_{-0.18}$ | $-1.02^{+0.34}_{-0.35}$ | $8.26^{+0.16}_{-0.19}$ | $51.37^{+86.37}_{-25.17}$
JD-N | $0.94^{+0.10}_{-0.10}$ | $11.47^{+2.20}_{-1.88}$ | $0.36^{+0.13}_{-0.08}$ | $-1.99^{+0.17}_{-0.12}$ | $8.96^{+0.06}_{-0.08}$ | $80.23^{+22.97}_{-13.55}$
Figure 5: Illustration of the SED fitting with Bagpipes for each entry in
Table 1. The blue points with error bars show the observed flux densities, and
the orange points show the synthesized magnitudes from the 50th-percentile
Bagpipes fit. The posterior spectra corresponding to the 16th and 84th
percentile range are shown in yellow in each panel.

Figure 6: Corner plots showing the posterior distributions of the fitted parameters corresponding to the SED fitting illustrated in Figure 5. The inset in the upper right corner in each panel shows the SFH, where the shaded region corresponds to the 16th to 84th percentile range, and the black curve is the median value. Table 4: Rest-frame UV slopes and $f_{\rm esc}$ Name | $\beta^{150,200}$ | $f_{\rm esc}^{150,200}$ | $\beta^{150,277}$ | $f_{\rm esc}^{150,277}$ | $\beta^{\rm PL}$ | $f_{\rm esc}^{\rm PL}$
---|---|---|---|---|---|---
Y1 |  $-$1.769 | 0.019 |  $-$1.731 | 0.017 |  $-$1.417 | 0.007
Y1-E |  $-$1.838 | 0.023 |  $-$1.955 | 0.032 |  $-$1.304 | 0.005
Y1-C |  $-$1.515 | 0.009 |  $-$1.610 | 0.012 |  $-$1.200 | 0.004
Y1-W |  $-$1.761 | 0.018 |  $-$1.834 | 0.022 |  $-$1.206 | 0.004
JD |  $-$2.630 | 0.210 |  $-$2.479 | 0.138 |  $-$1.967 | 0.033
JD-N |  $-$1.718 | 0.016 |  $-$1.673 | 0.014 |  $-$1.204 | 0.004

###  4.3 UV slopes and Lyman-continuum photon escape fractions

Using the photometry reported in Table 1, we calculated the rest-frame UV
slope $\beta$ for these sources. The calculation was done using two different
filter pairs, (F150W, F200W) and (F150W, F277W), which approximately
correspond to using the rest-frame wavelength pairs of (1600Å, 2100Å) and
(1600Å, 3000Å), respectively. As an alternative, we also used the best-fit
template spectrum (corresponding to the 50th percentile of the posterior
distribution) of the SED and fitted a power law between 1300Å and 1850Å in the
rest-frame to obtain this slope. These results are listed in Table 4. The
slopes derived using the best-fit template spectra ($\beta^{\rm PL}$) are all
significantly flatter than those obtained using the photometry in the blue
passband pairs ($\beta^{150,200}$ and $\beta^{150,277}$), which cautions that
care must be taken when interpreting UV slopes obtained by different methods
(e.g., Dunlop et al., 2013; Austin et al., 2024). The UV slopes are related to
the Lyman continuum photon escape fractions ($f_{\rm esc}$), which are
critical in understanding the sources of the cosmic hydrogen reionization. For
each $\beta$ value, we calculated $f_{\rm esc}$ following Equation 11 of
Chisholm et al. (2022),

$\displaystyle f_{\rm esc}=(1.3\pm 0.6)\times 10^{-4}\cdot 10^{(-1.22\pm
0.1)\beta}\,.$ (1)

The derived values are reported in Table 4.

Somewhat surprisingly, only JD has a blue slope of $\beta\lesssim-2.0$, and
its inferred $f_{\rm esc}$ ranges from $\sim$3.3% to 21.0% depending on the
exact $\beta$ value adopted. In terms of their stellar populations, the three
objects are quite similar to each other, and the only major difference is that
JD has only half as much reddening ($A_{\rm V}\sim 0.4$ mag) as Y1 and JD-N
($\sim$0.8–0.9 mag). This should be the main factor that affects their UV
slopes and hence their $f_{\rm esc}$. This also suggests that a significant
fraction of very young galaxies at high redshifts might not contribute to
reionization because they formed a considerable amount of dust that dispersed
throughout their bodies in only a few Myr.

##  5 Summary

New JWST data show that three $z\approx 8.3$ galaxies behind the lensing
cluster MACS0416 are strong [O 3] $\lambda$5007 emitters. Their line
equivalent widths can easily explain their apparent flux excess in F444W. Two
galaxies, Y1 and JD, had prior spectroscopic redshifts through ALMA detections
of the [O 3] 88 $\mu$m line and/or [C 2] 157.7 $\mu$m line for JD. We confirm
the redshift 8.31 for Y1 but find $z=8.34$ for JD as opposed to the previously
claimed $z=9.28$. Y1 is a merging system and is resolved into three components
extending $\sim$3.4 kpc along the long axis. The three components all have
very similar properties. The third object, JD-N, is a previously discovered
candidate at $z\approx 8$ and is now confirmed to have the same redshift as
JD; the two are only $\sim$0$\farcs$51 ($\sim$2.4 kpc) apart in the source
plane and therefore very likely merging or at least interacting. These three
objects are magnified by only a moderate amount and have intrinsic $M_{\rm
UV}$ ranging from $-19.66$ to $-20.85$ mag. Our SED analysis show that they
are all very young systems in the making, with ages $\lesssim$11 Myr, arguably
the youngest galaxies ever reported at $z>6$. Their stellar masses are on the
order of $10^{8}$ ${\rm M_{\odot}}$. They are actively forming stars, with
SFRs of a few tens to a couple of hundred $\rm{M_{\odot}}$ yr-1. However, only
JD has a blue rest-frame UV slope ($\beta$ ranging from $-1.97$ to $-2.63$
depending on how it is derived), indicative of a high Lyman-continuum photon
escape fraction ($f_{\rm esc}$ could be as high as 21%). In contrast, Y1 and
JD-N have $\beta\lesssim-1.7$ – $-1.8$, implying $f_{\rm esc}\lesssim 3\%$
despite their higher SFRs than JD, due to their twice as high dust extinction.
This suggests that even very young, very actively star-forming galaxies at
high-$z$ could have negligible contribution to the ionizing background if they
form dust throughout their bodies too quickly (over a few Myr time scale). Y1
and JD-N are examples that dust formation and pollution processes at $z>8$
could indeed be very fast.

We dedicate this work to the memory of our dear colleague Mario Nonino, a kind
and gentle person and an example for many. This project is based on
observations made with the NASA/ESA/CSA James Webb Space Telescope and
obtained from the Mikulski Archive for Space Telescopes, which is a
collaboration between the Space Telescope Science Institute (STScI/NASA), the
Space Telescope European Coordinating Facility (ST-ECF/ESA), and the Canadian
Astronomy Data Centre (CADC/NRC/CSA). We thank the Program Coordinator, Tony
Roman, for his expert help scheduling the PEARLS program. The authors also
acknowledge the JWST GO-3538 team led by PI E. Iani for developing their
observing program with a zero-exclusive-access period. ZM is supported by the
NSF grants No. 1636621, 2034318, and 2307448. CC is supported by the National
Natural Science Foundation of China, No. 11933003 and 12173045 as well as the
CAS through a grant to the CAS South America Center for Astronomy (CASSACA).
BS and HY acknowledge the support from the NSF grant AST-2307447. SHC, RAW,
and RAJ acknowledge support from NASA JWST Interdisciplinary Scientist grants
NAG5-12460, NNX14AN10G, and 80NSSC18K0200 from GSFC. FS and CNAW acknowledges
funding from the JWST/NIRCam contract NASS-0215 to the University of Arizona.
FS acknowledges support for program #2883 provided by NASA through a grant
from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127. DE acknowledges support from a Beatriz Galindo senior
fellowship (BG20/00224) from the Spanish Ministry of Science and Innovation,
projects PID2020-114414GB-100 and PID2020-113689GB-I00 financed by
MCIN/AEI/10.13039/501100011033, project P20-00334 financed by the Junta de
Andalucía, and project A-FQM-510-UGR20 of the FEDER/Junta de Andalucía-
Consejería de Transformación Económica, Industria, Consejerá, Universidades.

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##  Appendix A Null line detection in the ALMA data for JD

Figure A.1: Spectrum of JD from the ALMA Band 7 data based on our own
reduction. The grey regions indicate frequency ranges not covered by the
observations. The spectrum was extracted at the position shown in Figure 2 of
Laporte et al. (2021) using an aperture of 1$\farcs$00$\farcs$35 in diameter.
The dashed vertical line indicates 329.69 GHz, where Laporte et al. (2021)
claimed a line detection.

We reduced the same ALMA band 7 data of JD (PID: 2019.1.00061.S, PI: R. Ellis)
used by Laporte et al. (2021). We utilized the default ALMA data reduction
script ScriptForPi.py to obtain the visibility file and performed tclean in
the Common Astronomy Software Applications package (CASA). The cleaning was
done using natural weighting and frequency bin 16 MHz. The final cube has an
rms of 0.23 mJy beam-1. We extracted the spectrum at the position shown by
(Laporte et al., 2021, their Figure 2) with an aperture of 1$\farcs$0 in
diameter, and the result is shown in Figure A.1. No emission line is detected
at the frequency where Laporte et al. (2021) saw a detection.

##  Appendix B Age estimates using different SFHs

For a given stellar population synthesis model, the age estimate could be
affected by the adopted SFH. The SED analysis presented in Section 4.2 uses an
exponentially declining SFH, also known as the “$\tau$ model”. To test the
robustness of the very young ages thus obtained, we also ran Bagpipes using
some other SFHs offered with the package:

  * •

Delayed $\tau$:

$\quad\mathrm{SFR}(t)\ \propto\ \begin{cases}\ \ \ (t-T_{0})\
\mathrm{exp}\bigg{(}-\dfrac{t-T_{0}}{\tau}\bigg{)}&\quad t>T_{0}\\\ \ \ \
0&\quad t<T_{0}.\end{cases}$ (B1)

  * •

Log-normal:

$\mathrm{SFR}(t)\ \propto\ \frac{1}{t}\
\mathrm{exp}\bigg{(}-\frac{\big{(}\mathrm{ln}(t)-T_{0}\big{)}^{2}}{2\tau^{2}}\bigg{)}$
(B2)

  * •

Double power law:

$\mathrm{SFR}(t)\ \propto\
\Bigg{[}\bigg{(}\frac{t}{\tau}\bigg{)}^{\alpha}+\bigg{(}\frac{t}{\tau}\bigg{)}^{-\beta}\Bigg{]}^{-1}$
(B3)

Table B.1 compares the best-fit stellar mass and age parameters obtained using
the $\tau$ model with those obtained using these alternative SFHs. For both
the $\tau$ and the delayed $\tau$ models, the age parameter is among the
direct outputs. This is not the case when using the log-normal or the double
power law models, however, because there is no clear definition of age in
either SFH. To obtain an estimate in these two cases, we derived an age proxy
denoted as ${\rm Age_{N}}$, which is the time for the galaxy to gain 100% of
its stellar mass starting from the time when it had N% of its total stellar
mass. For the demonstration purpose here, $N$ was set to 10, 50, and 90. As
shown by this comparison, these alternative SFHs resulted in comparable, young
ages for all three objects.

Table B.1: Comparison of best-fit galaxy properties with different SFH |  | Exponential | Delayed $\tau$ | Log-norm | Double PL
---|---|---|---|---|---
Y1 | ${\rm log}(\mu M/\mathrm{M_{\sun}})$ | $8.89^{+0.06}_{-0.14}$ | $8.86^{+0.09}_{-0.11}$ | $8.80^{+0.06}_{-0.07}$ | $8.82^{+0.07}_{-0.01}$
| Age/Myr | $2.68^{+0.81}_{-1.09}$ | $4.15^{+1.04}_{-2.19}$ | … | …
| Age10/Myr | … | … | 3.75 | 3.62
| Age50/Myr | … | … | 1.29 | 1.70
| Age90/Myr | … | … | 1.00 | 1.00
JD | ${\rm log}(\mu M/\mathrm{M_{\sun}})$ | $8.26^{+0.16}_{-0.19}$ | $8.28^{+0.15}_{-0.16}$ | $8.19^{+0.12}_{-0.13}$ | $8.24^{+0.12}_{-0.11}$
| Age/Myr | $3.57^{+1.09}_{-1.63}$ | $4.96^{+2.27}_{-2.53}$ | … | …
| Age10/Myr | … | … | 6.46 | 3.30
| Age50/Myr | … | … | 2.16 | 1.43
| Age90/Myr | … | … | 1.00 | 1.00
JD-N | ${\rm log}(\mu M/\mathrm{M_{\sun}})$ | $8.96^{+0.06}_{-0.08}$ | $9.00^{+0.06}_{-0.07}$ | $9.05^{+0.06}_{-0.08}$ | $8.99^{+0.06}_{-0.08}$
| Age/Myr | $11.47^{+2.20}_{-1.88}$ | $24.39^{+5.2}_{-5.77}$ | … | …
| Age10/Myr | … | … | 20.40 | 15.22
| Age50/Myr | … | … | 6.28 | 5.72
| Age90/Myr | … | … | 1.01 | 1.02


# Fast high-fidelity geometric quantum control with quantum brachistochrones

Yang Dong  Ce Feng  Yu Zheng  Xiang-Dong Chen  Guang-Can Guo  Fang-Wen Sun
<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, 230026, People’s Republic of China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, People’s
Republic of China

(August 27, 2024)

###### Abstract

We experimentally demonstrate fast and high-fidelity geometric control of a
quantum system with the most brachistochrone method on hybrid spin registers
in diamond. Based on the time-optimal universal geometric control, single
geometric gates with the fidelities over $99.2{\%}$ on the spin state of
nitrogen-vacancy center are realized with average durations shortened by
$74.9{\%}$, comparing with conventional geometric method. The fidelity of the
fast geometric two-qubit gate exceeds $96.5{\%}$ on the hybrid spin registers.
With these fast high-fidelity gates available, we implement quantum
entanglement-enhanced phase estimation algorithm and demonstrate the
Heisenberg quantum limit at room-temperature. By comparing with the
conventional geometric circuit, the measurement bandwidth and sensitivity is
enhanced by $3.5$ and $2.9$ times. Hence, our results show that high-fidelity
quantum control based on a fast geometric route will be a versatile tool for
broad applications of quantum information processing in practice.

Quantum information processing (QIP), which can provide an unprecedented
supremacy Arute et al. (2019); Alexeev et al. (2021) over classical
counterpart in searching algorithm Abrams and Lloyd (1999); Harrow et al.
(2009), simulation Buluta and Nori (2009); Georgescu et al. (2014), metrology
Giovannetti et al. (2004, 2006); Pirandola et al. (2018); Pezze et al. (2018);
Dong et al. (2016); Liu et al. (2015), and secure communication Xu et al.
(2020a), relies seriously on high quality quantum operations Rong et al.
(2015); Dong et al. (2018a, 2019); Bourassa et al. (2020). Meanwhile, the
geometric phase Zu et al. (2014); Nagata et al. (2018); Yuan et al. (2018) is
currently a central topic not only in quantum physics but also in various
scientific fields including optics, electronics, nanotechnology, chemical
reactions, and materials science. With the built-in noise-resilience feature,
the geometric phase based holonomic quantum gate is believed to be an ideal
way to build a fault-tolerant universal quantum computer Duan et al. (2001).
Based on adiabatic or nonadiabatic cyclical paths, it has been proposed as a
promising method to realize high-fidelity and robust manipulation in theory
Duan et al. (2001); Sjöqvist et al. (2012); Xu et al. (2012); Liu et al.
(2019); Johansson et al. (2012), along with experimental demonstration in
various physical platforms, including superconducting circuits Abdumalikov Jr
et al. (2013); Yan et al. (2019); Xu et al. (2020b), nuclear magnetic
resonance Feng et al. (2013); Zhu et al. (2019), and solid-state defect in
diamond Zu et al. (2014); Arroyo-Camejo et al. (2014); Nagata et al. (2018);
Huang et al. (2019); Dong et al. (2021).

However, the duration times of these geometric operations are still much
longer than dynamical cases, especially for adiabatic geometric quantum
computation (GQC) protocol, resulting in more decoherence degrading quantum
states over time Vitanov et al. (2017); Guéry-Odelin et al. (2019); Yan et al.
(2019); Johansson et al. (2012); Zhu et al. (2019); Zheng et al. (2016); Li et
al. (2020). Besides fighting decoherence Guéry-Odelin et al. (2019); Barry et
al. (2020); Dong et al. (2021, 2021), long operation durations will severely
reduce the detection sensitivity and bandwidth for quantum sensing based on
full GQC. Usually, time optimal control (TOC) method can provide a useful
toolbox to accelerate operations. However, finding accurate TOC geometric
operations is difficult because the closed evolution path, the optimization of
fidelity and time should be satisfied simultaneously Rezakhani et al. (2009);
Chen et al. (2010); Ban et al. (2012); Daems et al. (2013); Torosov et al.
(2011).

Here, by employing quantum brachistochrone equation (QBE) Wang et al. (2015);
Carlini et al. (2006); Carlini and Koike (2012); Geng et al. (2016), we
implement fast and high-fidelity geometric quantum gates based on the TOC
method. In the experiment, we demonstrate a single-loop brachistochrone non-
adiabatic holonomic quantum computation (B-NHQC) in a prototype hybrid quantum
system: a two-qubit register consisting of nitrogen-vacancy (NV) electron and
a nuclear spin in diamond. Geometric single-qubit gates with fidelities over
$99.2\%$ on electron spin are realized with the average cutoff duration
shortened by $74.9\%$, comparing with conventional NHQC method. The fast two-
qubit geometric gate is also realized with a fidelity of $96.5\%$. Based on
the fast and high fidelity single- and two-qubit gates, we further illustrate
the power of the B-NHQC scheme by implementing the entanglement-enhanced phase
estimation algorithm. In additional to the demonstration of the Heisenberg
quantum limit (HQL) with the two entangled qubits Giovannetti et al. (2004,
2006); Pirandola et al. (2018); Pezze et al. (2018); Dong et al. (2016); Liu
et al. (2015), the best detection sensitivity and bandwidth are further
improved by $2.9$ and $3.5$ times, respectively, which benefits from fast
geometric operation mainly. Our results, thus, suggest that B-NHQC based gates
can provide a novel and promising route to achieve fast and precise universal
quantum control and make an important step towards practical high performance
application of QIP.

Figure 1: (a) A geometric picture of the quantum gates in the Bloch sphere
representation shows the evolution paths of B-NHQC (green line) and
conventional NHQC (red line) schemes. (b) Comparison of time costs for target
gate operator $R(\hat{z},\theta)$ with $\theta\in\left[{0,\pi}\right]$. The
maximum of the Rabi frequency of the Gaussian $4\sigma$ truncated pulse Motzoi
et al. (2009); Abdumalikov Jr et al. (2013) is $12.76$ MHz for NHQC protocol
SM . (c) Comparison of the experimental gate time for $\theta=\pi/8$, $\pi/4$,
and $\pi$. (d) State evolutions during
$T=R\left({\hat{z},\frac{\pi}{4}}\right)$ with B-NHQC and the conventional
NHQC paths. The initial state is
$\left|b\right\rangle=\left({\left|0\right\rangle+\left|1\right\rangle}\right)/\sqrt{2}$.
The time of the geometric gate with the TOC is $191.1$ ns ($75.4\%$) shorter
than that of the conventional scheme.

We construct the geometric quantum gate Arroyo-Camejo et al. (2014); Zu et al.
(2014); Zhu et al. (2019); Dong et al. (2021) with the NV center in diamond
Maze et al. (2008); Xiang et al. (2013); Dong et al. (2018a); Lu et al.
(2020), a promising candidate for a scalable quantum register. By applying a
magnetic field along the axis of the NV center, the electron spin-triplet
states form a $V$-type energy level system (see Supplemental Material for
details SM ). We encode
$\left|{{m_{s}}=-1}\right\rangle\equiv\left|0\right\rangle$ and
$\left|{{m_{s}}=1}\right\rangle\equiv\left|1\right\rangle$ as the qubit basis
states, and use $\left|{{m_{s}}=0}\right\rangle\equiv\left|a\right\rangle$ as
an ancillary state for the geometric manipulation of the qubit. Two microwave
(MW) fields with frequencies ${\omega_{1(2)}}$ and initial phases
${\phi_{1(2)}}$ are used to couple the sequential transitions
$\left|0\right\rangle\leftrightarrow\left|a\right\rangle$
($\left|a\right\rangle\leftrightarrow\left|1\right\rangle$) with corresponding
Rabi frequencies ${\Omega_{1(2)}}$, as described by SM

$\displaystyle{H}=\frac{{\Omega(t)}}{2}{e^{i\phi_{2}}\left|b\right\rangle}\langle
a|+H.c.\text{,}$ (1)

where
${\left|b\right\rangle={e^{i\phi}}\sin\frac{\theta}{2}|0\rangle{+\cos}\frac{\theta}{2}|1\rangle}$
is a bright state with
$\tan\frac{\theta}{2}=\frac{{\Omega_{1}}}{{\Omega_{2}}}$. Here,
$\phi={\phi_{1}}-{\phi_{2}}$ and
$\Omega(t)=\sqrt{\Omega_{1}^{2}(t)+\Omega_{2}^{2}(t)}$. Therefore, the bright
state ${\left|b\right\rangle}$ interacts with the state
$\left|a\right\rangle$, while decouples from the dark state
${\left|d\right\rangle=\cos\frac{\theta}{2}|0\rangle-{e^{-i\phi}}\sin\frac{\theta}{2}|1\rangle}$.

The NHQC scheme Dong et al. (2021); SM  can be realized with a single-loop
scenario by engineering the quantum system to evolve along with a red-slice-
shaped path in the Bloch sphere, as shown in Fig. 1(a). In the qubit
computational basis $\left\\{{|0\rangle,|1\rangle}\right\\}$, the gate
operation is
${U_{G}}(\gamma,\theta,\phi)={e^{i(\gamma/2)}}{e^{-i(\gamma/2)\vec{n}\cdot\vec{\sigma}}}$,
which describes a rotation around the
$\vec{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ axis by a
$\gamma$ angle. Therefore, the duration times are same for all gates, as shown
in Figs. 1(b) and 1(c).

However, in realistic physical systems, the ever-present decoherence from the
environment degrades the qualities of quantum states and operations over time.
Therefore, generating the fastest possible cyclic path evolution by TOC
becomes a preferable choice for realizing precise quantum controls Ban et al.
(2012); Wang et al. (2015); Carlini et al. (2006); Carlini and Koike (2012).
And the accurate path with the minimal time cost can be obtained by solving
the QBE together with the Schrödinger equation. Here, the QBE can be written
as $\dot{F}=-i[H,F]$, where $F=\partial L_{C}/\partial H$ and
$L_{C}=\sum_{j}\lambda_{j}f_{j}(H)$ with the Lagrange multiplier
$\lambda_{j}$. One physically relevant constraint is the finite operation
speed Lloyd (2000); Lam et al. (2021) which is described as
${f_{1}}\left({H(t)}\right)\equiv\frac{1}{2}\left[{\operatorname{Tr}\left({H{{(t)}^{2}}}\right)-\frac{{\Omega{{(t)}^{2}}}}{2}}\right]=0,$
where ${\Omega(t)}$ is the Rabi frequency. The other constraint is
${f_{2}}\left({H(t)}\right)\equiv Tr(H(t){\sigma_{z}})=0$ since we can not
directly implement the independent ${\sigma_{z}}$ operation in the experiment.
A simple minimum-time solution Liu et al. (2020); Chen et al. (2020) to this
QBE is ${\phi_{2}}(t)=2(\gamma-\pi)t/\tau$, with the minimum gate duration
time of $\tau_{min}=2\sqrt{{\pi^{2}}-{{(\pi-\gamma)}^{2}}}/\Omega$, which
increases with the geometric phase, as shown in Figs. 1(b) and 1(c). Under
this condition, the single loop geometric gate $U_{G}$ can be implemented by a
green-spherical-crown-shaped path as shown in Fig. 1(a), which is less time
consuming than the conventional NHQC scheme as shown in red.

By setting $\Omega(t)=12.5$ MHz, we experimentally demonstrate
$R(\hat{z},\theta)$ operation with $\theta=\pi/8$, $\pi/4$, and $\pi$ based on
the B-NHQC. The time cost is significantly shorter than that with the NHQC
scheme, on average, by $74.9\%$, as shown in Fig. 1(c). Especially, we
implement the target $T={U_{G}}\left({\pi/4,0,0}\right)$ gate operation with
both methods as shown in Fig. 1(d). The duration is only $24.6\%$ of that with
the NHQC scheme. We characterize the single-qubit B-NHQC gates through a
standard quantum process tomography (QPT) method Dong et al. (2021) on six
specific geometric gates $\left\\{{I,X/2,X,Y/2,Y,T}\right\\}$, with an average
fidelity of 0.984(4) SM . The major contribution to the QPT infidelity comes
from state preparation and detection errors Zu et al. (2014); Arroyo-Camejo et
al. (2014); Kleißler et al. (2018); Huang et al. (2019). Therefore, we
implement $X$ and $Y$ gates successively on the electron spins and perform QPT
when the gates are repeated for $N$ times. The average fidelity Zu et al.
(2014); Rong et al. (2015, 2014) of $N$ B-NHQC gates can be calculated as:
$F_{N}=\left[1+\left(1-\varepsilon_{if}\right)(1-p)^{N}\right]/2$, where
${\varepsilon_{if}}$ describes errors in the state preparation and measurement
and $p$ is the average error per gate Zu et al. (2014); Rong et al. (2014), as
shown in Fig. 2 (a). We estimate the average fidelity of the $X$, $Y$ gates to
be ${F_{X}}=1-p_{X}=0.9922(4)$ and ${F_{Y}}=0.9923(4)$ from the experimental
data, which exceed the threshold of the necessary fidelity for the realization
of the state-of-art error correction codes based on surface codes Kleißler et
al. (2018).

Figure 2: (a) The decay of the fidelity of the B-NHQC X,Y gates obtained via QPT. The golden line shows the decay with the gate fidelity of $F=0.9$. (b) The quantum circuit of $N$ B-NHQC C-Y gates. (c) Two-qubit state fidelities after $N$ applications of the C-Y operation. Table 1: Summary of the gate fidelity and duration for various QC schemes. Scheme | DD protected QC Zhang et al. (2014); Zhang and Suter (2015) | SUPCODE QC Rong et al. (2014) | GRAPE QC Rong et al. (2015) | AGQC Huang et al. (2019) | B-NHQC
---|---|---|---|---|---
Single-bit fidelity | 0.98 | 0.9961(2) | 0.999952(6) | 0.991(2) | 0.9922(4)
Two-bit fidelity | 0.90 | – | 0.9920(1) | 0.94(2) | 0.965(4)
Single-bit gate duration ($\mu$s) | 0.355 | 5.063 | 0.34 | 1 | 0.063
Two-bit gate duration ($\mu$s) | 50 | – | 0.696 | 2 | 0.354

Universal geometric control of qubits always requires a nontrivial two-qubit
gate. Here, in the construction of two-qubit gate, we use the electron spin
state as target state and the nitrogen nuclear spin (${}^{14}\text{N}$) as the
control qubit SM . In the experiment, we implement a controlled-rotation gate
with MW selective operation, which is equivalent to a controlled-NOT gate
Jelezko et al. (2004); Zhang et al. (2014); Zhang and Suter (2015). A typical
controlled-Y (C-Y) gate can be represented as
${U_{\text{C-Y}}}=\left({\begin{array}[]{cc}\mathbf{1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{{{R_{y}}(\pi)}}\end{array}}\right)$, where
$\mathbf{1},\mathbf{0}$, and $\mathbf{{{R_{y}}(\pi)}}$ represent $2\times 2$
matrices corresponding to the unit operator, zero matrix, and a rotation
matrix around the $y$-axis with
$\mathbf{R_{y}}\left(\pi\right)={e^{-i\pi{\mathbf{\sigma}_{y}}/2}}$. Hence,
the electron spin will be rotated if the ${}^{14}{\text{N}}$ nuclear spin
state is $\left|{{m_{I}}=1}\right\rangle$. We implement geometric $I$ gate if
the ${}^{14}{\text{N}}$ nuclear spin state is
$\left|{{m_{I}}=-1}\right\rangle$ with selective MW field driving instead of
doing nothing. In this way, the coherence of this hybrid system can be
protected by the dressed states Golter et al. (2014). Using this method, the
geometric entangled state between the electron and ${}^{14}{\text{N}}$ nuclear
spin is prepared and measured in the Bell state
$\left|\Phi\right\rangle=\left({\left|{00}\right\rangle+\left|{11}\right\rangle}\right)/\sqrt{2}$
with a state fidelity of ${F_{s}}=0.947(8)$ by the quantum state tomography
(QST) method SM . By applying the B-NHQC C-Y gate repeatedly, the state of the
hybrid system is coherently transferred among entangled and separable states,
as shown in Fig. 2(b). Fig. 2(c) shows the fidelity of the final state to the
ideal state for up to $10$ applications of the C-Y operation with no
entanglement in the hybrid systems for even number of gates. The result
reveals ${F_{s}(N)}=AF_{g}^{N}+B$, where $A$, $B$ and two-qubit gate fidelity
${F_{g}}$ are fit parameters Chow et al. (2012). Here, we can obtain that the
two-qubit gate fidelity is ${F_{g}}=0.965(4)$.

In Table I, we make a comparison between various QC schemes with B-NHQC in
experiment. The power of the MW or radio-frequency drives is almost the same
for these schemes. Since the B-NHQC scheme is the fastest geometric operation
and the longest sequence duration is much less than the decoherence time,
decoherence effects are expected to be negligible and the fluctuations of
control fields will be the major error. Fortunately, these control errors can
be directly mitigated by combining B-NHQC with composite-pulse architecture
Zhu et al. (2019); Rong et al. (2014); Genov et al. (2017).

Figure 3: Demonstration of the HQL with entanglement-enhanced phase estimation
algorithm with NHQC and B-NHQC gates. (a) Phase estimation schemes of the
independent quantum probes. (b)-(c) Phase relation of single probe with NHQC
and B-NHQC protocol. (d) Phase estimation schemes of two entangled quantum
probes. (e)-(f) Phase relation of the two entangled probes with NHQC and
B-NHQC gates. The phase gate ${Z_{\varphi}}$ is realized by adding a phase to
the drive field for all operations.

We further illustrate the advantage of QIP based on the full fast B-NHQC gates
by implementing entanglement-enhanced parameter estimation algorithm
Giovannetti et al. (2004, 2006); Pirandola et al. (2018); Pezze et al. (2018);
Dong et al. (2016); Liu et al. (2015) with the hybrid solid-state spin
registers. By using independent sensors to estimate a parameter, we can
achieve at best the standard quantum limit. However, it is believed that using
entangled quantum sensor qubits, such as NOON states Giovannetti et al. (2004,
2006); Pirandola et al. (2018); Pezze et al. (2018); Dong et al. (2016); Liu
et al. (2015), are able to achieve the fundamental HQL scaling. Fig. 3 shows
the quantum circuit to prepare quantum states and probe the phase $\varphi$
Giovannetti et al. (2004, 2006); Pirandola et al. (2018); Pezze et al. (2018);
Dong et al. (2016); Liu et al. (2015), which can encode the quantity to be
measured. For conventional phase estimation procedure in Fig. 3(a), the
interferometer signal is the expectation value $P=({{1+\cos\varphi}})/{2}$ SM
, as shown in Figs. 3(b) and 3(c) with the photon-shot-noise at the output
denoted by error bars. Moreover, the electron and nuclear spin can be prepared
in an entangled state. Shown in Fig. 3(d), we implement a phase gate on both
sensor qubits, which brings the system to the two-qubit NOON state
$\left({\left|{00}\right\rangle+{e^{{-}2{\text{i}}\varphi}}\left|{11}\right\rangle}\right)/\sqrt{2}$.
The interferometer signal is $P=({{1+\cos 2\varphi}})/{2}$, as shown in Figs.
3(e) and 3(f) with NHQC and B-NHQC protocol, respectively.

The quantum sensor is most sensitive to variations of the phase
($\Delta\varphi$) at the point of the maximum slope, with the sensitivity
limited by the uncertainty in the interferometer signal measurement ($\Delta
P$). The uncertainty of the hybrid spin register to small variations in phase,
as depicted in the measurements, is given by
$\Delta{\varphi_{\min}}=\sigma_{S}^{n}/dP$, where $\sigma_{S}^{n}$ is the
standard deviation of the interferometer measurement after $n$ averages and
$dP$ is the slope of the interferometer signal variation with phase
Giovannetti et al. (2004, 2006); Pirandola et al. (2018); Pezze et al. (2018);
Maze et al. (2008). By harnessing the entanglement of the electron and nuclear
spins, the phase estimation uncertainties are reduced by factors of $1.93(1)$
for the NHQC and $1.99(1)$ for the B-NHQC protocol compared with single qubit,
which is close to $2$ to demonstrate the HQL. Because of reducing gate
duration time significantly with the B-NHQC protocol, the results of the phase
estimation is better than those from NHQC protocol, as shown in Figs. 3(c) and
3(f). The visibility SM  of the interferometer signal of the B-NHQC circuit is
larger than the conventional scheme mainly by mitigating quantum decoherence
effect with fast quantum control. Also, the instability of experimental
environment is suppressed for fast quantum circuit. As a result, the
fluctuation ($\sigma_{S}^{n}$) of interferometer signal is reduced in the
experiment, which is denoted with error bars in Fig. 3. Here, average error
bars levels are $\sigma_{S,NHQC}^{n}=0.044(4)$ and
$\sigma_{S,B-NHQC}^{n}=0.031(3)$.

Moreover, to comprehensively and systematically analyze the experimental
results, the duration time of both protocols should be taken into
consideration to discuss the measurement sensitivity Maze et al. (2008); Barry
et al. (2020); Dong et al. (2021). The best measurement sensitivity is given
by $S=\Delta{\varphi_{\min}}\sqrt{T}_{t}$, where $T_{t}$ is the total
measurement time. The phase estimation sensitivity enhancement is a direct and
effective indicator, which can be calculated by

$\displaystyle\kappa=\frac{{{S_{NHQC}}}}{{{S_{B-NHQC}}}}{\text{
}}=\frac{{{{\left({\sigma_{S}^{n}{{\sqrt{T}}_{t}}/dP}\right)}_{NHQC}}}}{{{{\left({\sigma_{S}^{n}{{\sqrt{T}}_{t}}/dP}\right)}_{B-NHQC}}}}\text{.}$
(2)

Here, the slope ($dP$) is proportional to the visibility. The total single
measurement time for the B-NHQC circuit includes the optical initial time
(${T_{ini}}=3$ $\mu$s), the duration of entanglement-enhanced phase estimation
algorithm
(${T_{a}}=2T\left({{{}^{14}{\text{N}}},\frac{\pi}{2}}\right)+2T\left({e,\pi}\right)\approx
79.1$ $\mu$s), and quantum state readout time ${T_{r}}=2$ $\mu$s. So the
sensing time is mostly spent on the entanglement-enhanced phase estimation
algorithm, which is more prominent for NHQC circuit (${T_{a}}\approx 287.0$
$\mu$s). Therefore, the sensitivity enhancement is $\kappa=2.9$, which
benefits from fast geometric operation mainly
(${T_{t,NHQC}}/{T_{t,B-NHQC}}=3.5$). Because of the reduction in time cost,
the signal detection bandwidth is improved by $3.5$ times with the B-NHQC
circuit.

In summary, fast and high-fidelity quantum coherent control is of fundamental
significance in quantum computation, simulation, and metrology. We have
implemented a robust B-NHQC based single- and two-qubit gate with hybrid spin
registers in diamond, which paves the way for full fast geometric QIP in
practice. And benefiting from short quantum control time, the measurement
sensitivity and bandwidth in the entanglement-enhanced phase estimation is
improved significantly, which is far beyond the conventional estimation
paradigm. It will be further applied in high-resolution nuclear magnetic
resonance detection at the nanoscale Zaiser et al. (2016); Aslam et al.
(2017); Barry et al. (2020); Dong et al. (2021). Beyond the demonstration in
our work, the B-NHQC protocol can be applied to other promising physical
platforms, such as atoms, ions, transmon or flux qubits Xiang et al. (2013),
for fast and high-fidelity QIP.

We thank Zheng-Yuan Xue for valuable discussions. This work is supported by
the National Key Research and Development Program of China (Grant No.
2017YFA0304504), the National Natural Science Foundation of China (Grants No.
91850102 and No. 12005218), the Anhui Initiative in Quantum Information
Technologies (Grant No. AHY130000), the Science Challenge Project (Grant No.
TZ2018003), the Fundamental Research Funds for the Central Universities (Grant
No. WK2030000020) and Excellent Youth Foundation of Hebei Scientific Committee
(Grant No. F2019516002).

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