# HST/WFC3 H$\alpha$ Direct-Imaging Detection of a Point-like Source in the
Disk Cavity of AB Aur

Yifan Zhou These authors contributed equally to this work.  Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400 Austin, TX 78712, USA  51 Pegasi b Fellow  Aniket Sanghi These authors contributed equally to this work.  Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400 Austin, TX 78712, USA  Brendan P. Bowler Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400 Austin, TX 78712, USA  Ya-Lin Wu Department of Physics, National Taiwan Normal University, Taipei 116, Taiwan  Center of Astronomy and Gravitation, National Taiwan Normal University, Taipei 116, Taiwan  Laird M. Close Department of Astronomy, University of Arizona, Tucson, AZ 85719, USA  Feng Long Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA  Kimberly Ward-Duong Department of Astronomy, Smith College, Northampton, MA 01063, USA  Zhaohuan Zhu Adam L. Kraus Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400 Austin, TX 78712, USA  Katherine B. Follette Department of Physics and Astronomy, Amherst College, Amherst, MA 01003, USA  Jaehan Bae Department of Astronomy, University of Florida, Gainesville, FL 32611, USA

###### Abstract

Accreting protoplanets enable the direct characterization of planet formation.
As part of a high-contrast imaging search for accreting planets with the
Hubble Space Telescope (HST) Wide Field Camera 3, we present H$\alpha$ images
of AB Aurigae (AB Aur), a Herbig Ae/Be star harboring a transition disk. The
data were collected in two epochs of direct-imaging observations using the
F656N narrow-band filter. After subtracting the point spread function of the
primary star, we identify a point-like source located at a P.A. of
$182.5^{\circ}\pm 1.4^{\circ}$ and a separation of $600\pm 22$ mas relative to
the host star. The position is consistent with the recently identified
protoplanet candidate AB Aur b. The source is visible in two individual epochs
separated by ${\sim}50$ days and the H$\alpha$ intensities in the two epochs
agree. The H$\alpha$ flux density is $F_{\nu}=1.5\pm 0.4$ mJy, $3.2\pm 0.9$
times of the optical continuum determined by published HST/STIS photometry. In
comparison to PDS 70 b and c, the H$\alpha$ excess emission is weak. The
central star is accreting and the stellar H$\alpha$ emission has a similar
line-to-continuum ratio as seen in AB Aur b. We conclude that both planetary
accretion and scattered stellar light are possible sources of the H$\alpha$
emission, and the H$\alpha$ detection alone does not validate AB Aur b as an
accreting protoplanet. Disentangling the origin of the emission will be
crucial for probing planet formation in the AB Aur disk.

††software: Astropy (Robitaille et al., 2013), Numpy (van der Walt et al.,
2011), Scipy (Virtanen et al., 2020), Matplotlib (Hunter, 2007), pyKLIP (Wang
et al., 2015)

##  1 Introduction

Two of the most fundamental goals of planet formation studies are to
understand how and when planets gain mass. The planetary mass assembly process
can be directly constrained by characterizing actively accreting protoplanets.
So far, the discoveries of protoplanets are limited to PDS 70 b and c, two gas
giants directly imaged within the same planetary system (Keppler et al., 2018;
Haffert et al., 2019). A plethora of follow-up studies on this system
confirmed the presence of circumplanetary disks (Isella et al., 2019; Benisty
et al., 2021), estimated the mass accretion rates (e.g., Wagner et al., 2018;
Haffert et al., 2019; Hashimoto et al., 2020; Zhou et al., 2021), investigated
the planet-disk interactions (e.g., Bae et al., 2019), and constrained the
planetary orbits (Wang et al., 2021). Despite this steady progress, these
studies are confined to one protoplanetary system and only represent a single
outcome of the planet formation process observed at an instant in time.

To expand the sample of accreting planets, we launched the Hubble Accreting
Luminous Protoplanets in H-Alpha (HALPHA) Survey (Program ID: 16651111Program
information can be found at https://www.stsci.edu/cgi-bin/get-proposal-
info?id=16651&observatory=HST, PI: Zhou) to search for accreting planets in
ten transition disk systems. We are exploiting the recently demonstrated
optical and ultraviolet (UV) high-contrast imaging capibility of Wide Field
Camera 3 (WFC3) UVIS (Zhou et al., 2021; Sanghi et al., 2022) and using the
narrow-band F656N (H$\alpha$) filter to look for H$\alpha$-emitting planets.
Accretion onto planets produces strong hydrogen line emission, reduces planet-
star contrasts, and can in principle improve search efficiency. Any detections
result in H$\alpha$ luminosities ($L_{\mathrm{H\alpha}}$) and accretion rate
estimates of candidate planets.

The strategy of observing transition disk systems is motivated by models
showing that the disk gaps and cavities are consistent with being sculpted by
multiple giant planets (Dodson-Robinson & Salyk, 2011; Close, 2020). These
models are broadly supported by the discoveries and characteristics of PDS 70
b and c (Bae et al., 2019). We identified ten systems that harbor giant
cavities demonstrated in ALMA dust continuum images (Francis & van der Marel,
2020). The host stars are generally faint or located in the northern
hemisphere and, consequently, are not easily accessible by ground-based
visible-light adaptive-optics (AO) systems.

Figure 1: Primary-subtracted images (upper) and the SNR maps (lower) of AB Aur
observed by HST/WFC3/UVIS in the F656N filter. The left column shows the
primary-subtracted natively sampled images (pixel scale = 40 mas) and the
right column shows the reconstructed images (pixel scale = 20 mas). The upper
panels show the intensity maps (in unit of $\mathrm{e^{-}s^{-1}}$). These
images, as well as the upper panels of Figure 2, the left panel of Figure 3,
and Figure 4 are smoothed using a Gaussian kernel with
$\mbox{FWHM}=0.07^{\prime\prime}$ ($1.0\times$ PSF FWHM). Smoothing improves
visualization by reducing high-frequency noise but is not applied in the
sensitivity or contrast curve estimates. The lower panels are the SNR maps.
For all four images, the color maps have a linear stretch. The central
$r=0.3^{\prime\prime}$ radius circle is masked out, because in this region the
detection sensitivity is poor due to high-contrasts and saturation. In the
upper-left panel, an ellipse marks the transition disk cavity identified by
Francis & van der Marel (2020). In the lower-left panel, gray dashed lines
indicate the spiral disk structures (S1 to S4) detected in our images. The
point-like source at $0.6^{\prime\prime}$ south of AB Aur is visible in all
panels.  Figure 2: Primary-subtracted images and SNR maps in Epochs 1 (left
four panels) and 2 (right four panels). The intensity and SNR maps obtained
from processing the natively sampled and reconstructed images are organized in
the same way as Figure 1. The point source is consistently recovered in both
epochs, although the detection SNRs are below 5 in the second epoch due to
shorter integration time. Systematic residuals near the diffraction spikes are
also prominent in the second epoch.

The first target of the survey is AB Aurigae (AB Aur), a Herbig Ae/Be star
(DeWarf et al., 2003) with a spectral type of A0, a mass of $2.4\pm
0.2M_{\odot}$, and an isochronal age of $6.0^{+2.5}_{-1.0}$ Myr222The age is
estimated with the PARSEC isochrones (Bressan et al., 2012) and the
observationally constrained $L_{\mathrm{bol}}$ and spectral type of AB Aur
(Herczeg & Hillenbrand, 2014).. It is actively accreting at a rate of
$\dot{M}\approx 10^{-7}M_{\odot}\mathrm{yr^{-1}}$ (Garcia Lopez et al., 2006)
and exhibits significant variability (${>}10\%$) in both broadband photometry
and the H$\alpha$ line (Harrington & Kuhn, 2007; Cody et al., 2013). AB Aur
hosts a transition disk that includes a highly structured inner disk with
multiple spirals (e.g., Boccaletti et al., 2020; Jorquera et al., 2022) and a
cavity with an outer edge at 156 AU (Francis & van der Marel, 2020). Several
spiral structures have been identified in the inner disk and their presence
has been attributed to ongoing planet formation (Fukagawa et al., 2004;
Oppenheimer et al., 2008; Boccaletti et al., 2020). Specifically, Tang et al.
(2017) identified two spirals in high-resolution ALMA 12CO $J{=}2{-}1$
emission observations and suggested that the spiral morphology could be
explained by tidal disturbance caused by a companion located at 60 to 80 AU
($0.4^{\prime\prime}$ to $0.6^{\prime\prime}$ angular separation) and a
${\sim}180^{\circ}$ position angle (P.A.). Recently, Currie et al. (2022)
presented evidence for an embedded protoplanet, referred to as AB Aur b, in
visible and infrared images near the location predicted by Tang et al. (2017).
In this _Letter_ , we report results from two epochs of HST/WFC3 F656N
(H$\alpha$) high-contrast imaging observations of AB Aur.

##  2 Observations

###  2.1 HST Observations

AB Aur was observed by HST/WFC3/UVIS on UT 2022-02-07 (Epoch 1) and 2022-03-28
(Epoch 2) for three (Orbits 1 to 3) and two orbits (Orbits 4 and 5),
respectively. A 0.5-pixel (20 mas), four-point-box dithering strategy was
adopted in all five orbits to enable the reconstruction of Nyquist-sampled
images. In every orbit, HST sequentially pointed at each position and took
eleven identical 2.7 s exposures in the F656N filter (H$\alpha$ narrow-band;
$\lambda_{\mathrm{c}}=6561.5\,\text{\AA}$, $\Delta\lambda=17.9\,\text{\AA}$).
The exposures were captured by the c512c subarray, which has a native pixel
scale of 40 mas and a field of view of $20^{\prime\prime}\times
20^{\prime\prime}$. Due to guide star acquisition failures, the first frame of
Epoch 1 and the first two frames of Epoch 2 did not acquire the target. We
discarded these images in our analysis. In total, the HST observations
comprised 217 raw frames, amounting to 586 s of on-source exposure time.

HST’s roll angles (the V3 axis orientation) were $246.3^{\circ}$,
$271.3^{\circ}$, $246.3^{\circ}$, $260.6^{\circ}$, and $260.6^{\circ}$ in
Orbits 1 to 5, respectively. The angular differentials between any two
distinct roll angles are $10.7^{\circ}$, $14.3^{\circ}$, and $25^{\circ}$. At
a separation of $0.6^{\prime\prime}$, these correspond to spatial
displacements of 108, 149, and 262 mas, or 1.6, 2.2, and 3.7 times the full
width at half maximum (FWHM) of the F656N point spread function (PSF),
respectively. These large azimuthal displacements enable the use of angular
differential imaging (ADI, e.g., Marois et al. 2006) to subtract the PSF of
the host star between any pair of images with different telescope rolls.

The 2022-03-28 observations were a repeat of the first and third orbits of the
2022-02-07 observations that suffered from a fine-guiding sensor failure
resulting in 0.1 to 0.3 pixel pointing errors. This issue impairs image
reconstruction accuracy at small scales (${<}15\,\mbox{mas}$) but is not
detrimental to this study. Therefore, we present results from all five orbits.

###  2.2 High-resolution Spectroscopy of AB Aur

The HST images do not constrain the continuum or resolve the H$\alpha$ line.
To precisely characterize the H$\alpha$ emission from the host star, we
obtained an optical échelle spectrum of AB Aur using the Tull Coudé
spectrograph (Tull et al., 1995) on the 2.7-m Harlan J. Smith Telescope at
McDonald Observatory on UT 2019-09-26. The exposure time was 600 s and
conducted with a $1.2^{\prime\prime}$ slit, resulting in a resolving power of
$R=60,000$ and a wavelength span from $3870\,\text{\AA}$ to
$10,450\,\text{\AA}$ in a total of 56 orders. The wavelength calibration was
carried out with a ThAr emission lamp spectrum.

The Tull spectrum is reduced with a custom pipeline. We adopt the order that
spans from $6547.7\,\text{\AA}$ to $6656.7\,\text{\AA}$ and contains the
H$\alpha$ line. The spectrum is normalized by the continuum determined by the
best-fitting fifth order polynomial of the line-excluded region
($\lambda<6550\text{\AA}$ or $\lambda>6580\text{\AA}$). We compute the
synthetic photometry in the F656N filter using the pysynphot package and find
that the band-average flux is 2.58 times of the continuum.

##  3 Data Reduction

Figure 3: The injection-and-recovery image (left) and the $5\sigma$ contrast
curve (right). The injection-and-recovery test is conducted on the natively
sampled images. Injected artificial planets with the same brightness as the
one measured in the point-like source are consistently recovered and their
recovered photometry is used in correcting companion flux loss in primary
subtraction and calibrating the contrast curve. The inset in the left panel is
the inner $r=0.7^{\prime\prime}$ circular region of the injection-and-recovery
image, where the artificial sources have the same angular separations as AB
Aur b. In the right panel, the $5\sigma$ contrast measurements are shown in
squares, which are connected by a spline interpolation (solid line). The loss
of sensitivity at $3.6^{\prime\prime}$ to $4.0^{\prime\prime}$ is due to a
known filter ghost.

Our data reduction starts with the flc files downloaded from the HST archive.
The nominal bias, dark, flat field calibrations, as well as charge transfer
efficiency correction are performed by the CalWFC3 pipeline. We first identify
cosmic-ray (CR) affected pixels using the data quality flag (DQ=1024) and
replace these pixels with two-dimensional linear interpolations of the
respective neighboring pixels. Every set of four-point dithered and CR-
corrected frames are processed by a Fourier-interlacing pipeline (Lauer, 1999;
Zhou et al., 2021) to construct one $2\times$ finer sampled image. This
pipeline is optimized for point-source PSFs and distinct from the drizzling
method that is part of the regular data reduction performed at STScI.

We then independently perform a complete set of primary subtraction procedures
on the natively sampled (pixel scale${=}40$ mas) and reconstructed (pixel
scale${=}20$ mas) image sets. First, the image cube is registered by the
primary star centroids determined by the photutils package’s centroid_2dg
function (Robitaille et al., 2013). Then, the images are split into two annuli
that have inner/outer radii of $0.3^{\prime\prime}/0.7^{\prime\prime}$ and
$0.7^{\prime\prime}/2.5^{\prime\prime}$, respectively. We subtract the host
star PSF in each annulus using the Karhunen-Loève Image Projection (KLIP,
Soummer et al. 2012) algorithm with a custom pipeline. For each image, all
frames obtained at a different roll angle are used as references and the
number of KLIP components is equal to the number of reference frames. We have
also experimented with limiting the KLIP components to 5, 10, and 15 and find
that the variation has negligible effect on the subtraction results. After
primary subtraction, we rotate all frames to align their $y$-axes to the true
north and co-add the aligned frames to form the final image. The co-adding
step is also performed for the two epochs individually. In total, we create
six primary-subtracted images for two pixel scales each with three sets of co-
adds (the combined set: Figure 1; individual epochs: Figure 2).

We convert the primary-subtracted images into signal-to-noise ratio (SNR) maps
for point-source detections. The SNR is defined as the ratio between the flux
integrated in a 1 FWHM radius aperture and the standard deviation of fluxes
along the remaining azimuthal region sampled by non-overlapping apertures.
Biases induced by small-number statistics were corrected based on Mawet et al.
(2014). The SNR maps are shown in Figures 1 and 2 under the respective
primary-subtracted images. Substantial residuals appear near the diffraction
spikes in the reconstructed images, which are related to the telescope
pointing errors.

A point-like source emerges in every PSF-subtracted image at an identical
location ${\sim}600$ mas south of the primary star. We apply the KLIP forward
modeling method (Pueyo, 2016) on the natively sampled images to precisely
determine the astrometry and photometry of this source and correct measurement
biases introduced by primary subtraction. Synthetic WFC3 UVIS2 F656N PSFs are
generated with the TinyTim package (Krist et al., 2011) as the point-source
model. We inject a negative-flux PSF to the original frames and optimize the
P.A., separation, and flux of the injection so that the residual sum of
squares in a 3-pixel radius circular aperture minimizes.

To estimate the astrometric and photometric uncertainties, as well as the KLIP
throughput, we conduct injection-and-recovery tests on the natively sampled
images. Positive-flux TinyTim PSFs are injected at P.A.s of $0^{\circ}$, $\pm
45^{\circ}$, $\pm 90^{\circ}$, and $\pm 135^{\circ}$ spanning eight equal-
interval separations between $0.62^{\prime\prime}$ and $4.26^{\prime\prime}$
(Figure 3). These PSFs have the same flux as the point-like source. We
subtract the primary PSF and then measure the positions and fluxes of the
injected sources. At a specified separation, the average ratio between the
injected and recovered fluxes is adopted as the throughput, and the standard
deviations of the recovered fluxes, P.A.s, and separations are adopted to be
the photometric and astrometric uncertainties.

Figure 3 includes the $5\sigma$ contrast curve that characterizes our
detection sensitivity at angular separations spanning $0.4^{\prime\prime}$ to
$4.5^{\prime\prime}$. In deriving the contrast curve, we calculate the mean
and standard deviation of the fluxes in non-overlapping 1-FWHM-radius
apertures centered at a given separation. At 600 mas, the resolution element
containing the detected point source is excluded. We follow the SNR definition
in Mawet et al. (2014) to convert the mean and standard deviation into a flux
corresponding to $\mbox{SNR}=5$ and normalize this value by the flux of AB
Aur. Finally, we divide the contrast by the KLIP throughput determined by
positive PSF injections (Figure 3) to correct for flux loss in primary
subtraction.

##  4 Results

We detect a point-like source near AB Aur in the primary-subtracted image
(Figure 4). In the natively sampled frames, the detection SNRs are 5.4, 3.8,
and 6.4 in Epoch 1, Epoch 2, and the combined image, respectively; in the
reconstructed images, the SNRs are 5.6, 4.4, and 7.8. The consistent results
in natively sampled and reconstructed images, as well as in individual epochs,
further strengthen the confidence in the detection. The source is located at a
separation of $600\pm 22$ mas ($93\pm 3$ AU in projected physical distance)
away from AB Aur with a P.A. of $182.5^{\circ}\pm 1.4^{\circ}$. This position
is consistent with the protoplanet candidate AB Aur b reported in Currie et
al. (2022). Photometry on the combined image yields $64\pm 16$
$\mathrm{e}^{-}$/s, corresponding to a band-averaged flux density of
$F_{\nu}=1.5\pm 0.4$ mJy or $F_{\lambda}=1.0\pm 0.3\times 10^{-15}$
erg/s/cm2/Å. Fitting a 2D Gaussian to the source yields a FWHM of 150 mas in
the tangential direction and 127 mas in the radial direction, which are 2.1
and 1.8 times the FWHM of a F656N PSF. The FWHMs of recovered artificial
companions are between 85 to 120 mas. The larger size of the detected source
suggests it may trace slightly extended emission.

The H$\alpha$ emission from the companion is consistent between the two epochs
separated by 50 days, but our constraint on variability is weak. Photometric
results of the individual epochs are $55\pm 18$ $\mathrm{e}^{-}$/s and $76\pm
30$ $\mathrm{e}^{-}$/s. The corresponding flux densities are $F_{\nu,1}=1.3\pm
0.4$ mJy and $F_{\nu,2}=1.8\pm 0.7$ mJy or $F_{\lambda,1}=0.9\pm 0.3\times
10^{-15}$ erg/s/cm2/Å and $F_{\lambda,2}=1.2\pm 0.5\times 10^{-15}$
erg/s/cm2/Å. Based on the joint flux uncertainty between the two epochs, we
reject variability with amplitude greater than 50%. Using the same HST images,
we precisely determine the H$\alpha$ variability of the star AB Aur to be
$13.1\pm 0.2\%$, consistent with the known variability of the host star
(Harrington & Kuhn, 2007; Cody et al., 2013).

Four spiral-like scattered-light features are detected at $\mathrm{SNR}$
values between 2 to 3. In Figure 1, we mark these structures as S1 to S4.
These spiral features have all been previously identified in high-contrast
imaging observations of AB Aur (e.g., Fukagawa et al., 2004; Boccaletti et
al., 2020). Notably, the leading knots of S1 and S2 point towards the position
of the point-like source.

Figure 4: A zoomed-in ($2^{\prime\prime}\times 2^{\prime\prime}$) natively
sampled primary-subtracted image highlighting the detection of the point-like
source. The source is at a P.A. of $182^{\circ}$ and a separation of $600$ mas
relative to the host star. This detection is at the location of AB Aur b, a
candidate embedded planet recently reported by Currie et al. (2022).

We do not detect any sources at $2.75^{\prime\prime}$ or $3.72^{\prime\prime}$
reported in Currie et al. (2022). The $5\sigma$ upper limits in our combined
WFC3 frames are $0.098$ mJy and $0.091$ mJy, respectively, and are 4.8 and 22
times the flux measured by HST/STIS (Currie et al., 2022). Therefore, our
observations do not have sufficient sensitivity to determine the nature of the
two sources.

Figure 5: The SED of AB Aur b (upper left), PDS 70 b (upper right), a
comparison of the two SEDs (lower left), and a comparison to the stellar
H$\alpha$ emission of AB Aur (lower right). The SED of AB Aur b includes
optical to NIR measurements from Currie et al. (2022, the blue squares and
solid line) and the H$\alpha$ flux from the combined HST data in this work
(orange square). All AB Aur b fluxes are de-reddened based on $A_{V}=0.5$ mag
and $R_{V}=3.1$. The best-fitting blackbody (3800 K) is shown in a green-
dashed line. In contrast, the NIR SED of PDS 70 b (gray circles, data source:
Müller et al. 2018; Mesa et al. 2019; Wang et al. 2020; Stolker et al. 2020)
is best-fit by a 1200 K blackbody (black dashed line). In the bottom-left
panel, we normalize the two SEDs by their respective H$\alpha$ flux, so the
H$\alpha$ points (orange square for AB Aur b, black circle for PDS 70 b)
overlap. AB Aur b is bluer and has a stronger optical continuum. In the lower
right panel, flux is normalized to the continuum (for AB Aur b, it is the
HST/STIS flux). The H$\alpha$ of AB Aur b is consistent with the stellar value
within $1\sigma$.

##  5 Discussion

Rings and spirals in transition disks may introduce ambiguous protoplanet
signals in high-contrast imaging observations. Emission from these structures
may appear clumpy and even point-like after PSF-subtraction (Follette et al.,
2017; Rameau et al., 2017). For example, the protoplanet candidates LkCa 15
bcd (Kraus & Ireland, 2012; Sallum et al., 2015) and HD 100546 bc (Currie et
al., 2015) were later found to be consistent with disk features (LKCa 15:
Currie et al. 2019; HD 100546: Follette et al. 2017 and Rameau et al. 2017).
Because H$\alpha$ emission from an accreting planet distinguishes the planet
from the disk in their spectral energy distributions (SED), H$\alpha$ images
can provide an independent way of identifying protoplanets. For PDS 70 b and
c, their H$\alpha$ flux densities are over 100 times that of the estimated
optical continuum, which help confirm these objects as accreting protoplanets.
In contrast, in the HD 100546 case, non-detections of H$\alpha$-emitting point
sources call into question the candidates as accreting protoplanets (Follette
et al., 2017; Rameau et al., 2017). For AB Aur b, Currie et al. (2022)
presented several lines of evidence to support the protoplanet identification:
orbital motion has been detected, and the source’s spectrum and polarimetric
intensities differ from the rest of the disk. However, the companion resides
in a region of extended scattered light and photospheric features appear to be
absent in the SED (Currie et al., 2022). Below, we discuss whether the
H$\alpha$ detection adds support to the protoplanet interpretation.

The H$\alpha$ emission at the position of AB Aur b has been detected by
Subaru/VAMPIRES (Currie et al., 2022) and HST/WFC3 (this work). However, it is
unclear whether the VAMPIRES detection is physically associated with AB Aur b
due to uncertainties in the instrument’s astrometric calibration (Currie et
al., 2022). The astrometry of the WFC3 detection is well-calibrated and
consistent with measurements in other bands. Therefore, we only adopt the WFC3
measurement for this discussion. The WFC3 band-averaged H$\alpha$ flux density
of $f_{\nu}{=}1.5\pm 0.4$ mJy is higher than the optical continuum measured by
HST/STIS ($f_{\nu}{=}0.47\pm 0.05$ mJy; $\lambda{=}0.57\mu$m,
$\Delta\lambda{=}0.43\,\mu$m) and Subaru/VAMPIRES ($f_{\nu}{=}1.13\pm 0.37$
mJy; $\lambda{=}0.647\mu$m), but the difference is moderate to marginal,
respectively. The line-to-continuum flux ratio (defined as the band-averaged
F656N flux divided by the continuum flux) is $3.2\pm 0.9$ or $1.3\pm 0.6$,
depending on whether the STIS or the VAMPIRES flux is adopted for the
continuum, much lower than the flux ratios ${>}100$ estimated for the PDS 70
planets, as well as several accreting planetary mass companions (Zhou et al.,
2014; Eriksson et al., 2020; Stolker et al., 2021). AB Aur b’s low line-to-
continuum flux ratio is inconsistent with the accretion shock models that
explains the observations of PDS 70 b and c (e.g., Aoyama et al., 2018), which
produce pronounced H$\alpha$ lines. As a result, the H$\alpha$ detection does
not strengthen the interpretation that AB Aur b is an accreting protoplanet.

If the observed H$\alpha$ emission is indeed from an accretion shock, the
shock also produces strong optical continuum emission that reduces the line-
to-continuum flux ratio. Follow-up characterization of this source to measure
ultraviolet and optical excess emission would provide useful comparisons to
model SEDs and enable a diagnosis of the accretion shock interpretation (e.g.,
Zhu, 2015). Assuming that the observed F656N flux is entirely produced by the
accretion shock of AB Aur b, we can determine the instantaneous mass accretion
rate of the protoplanet based on the observed H$\alpha$ luminosity. Adopting a
line-of-sight extinction of $A_{V}=0.5$ mag (the same as the star, Garcia
Lopez et al. 2006) and assuming no additional extinction opacity, we find a
H$\alpha$ line luminosity of $L_{\mathrm{H\alpha}}=2.2\pm 0.7\times
10^{-5}L_{\odot}$. At this $L_{\mathrm{H\alpha}}$, gas in the accretion flow
should not lead to significant absorption (Marleau et al., 2022). The
$L_{\mathrm{H\alpha}}$ value corresponds to a total accretion luminosity of
$\log(L_{\mathrm{acc}}/L_{\odot})=-2.8\pm 0.3$ by assuming the planetary
surface shock model (Aoyama et al., 2018; Aoyama et al., 2021) or
$\log(L_{\mathrm{acc}}/L_{\odot})=-3.5\pm 0.3$ using the empirical relation
for classic T Tauri stars (Alcalá et al., 2017), respectively. Adopting a
planetary mass of $9M_{\mathrm{Jup}}$ and radius of $2.7R_{\mathrm{Jup}}$
(Currie et al., 2022), these accretion luminosities can be translated to mass
accretion rates of $\log(\dot{M}/(M_{\mathrm{Jup}}\mathrm{yr^{-1}}))$ between
$-5.8$ and $-6.6$. These estimates roughly agree with the result
$\log(\dot{M}/(M_{\mathrm{Jup}}\mathrm{yr^{-1}}))=-6.0$ obtained from SED
fitting (Currie et al., 2022).

Alternatively, the observed H$\alpha$ flux can also be stellar light scattered
by a compact disk structure or an envelope surrounding the protoplanet. In
this case, the optical SED of the candidate protoplanet, which requires
additional multi-band photometry to precisely determine, should mimic the SED
of the star. Remarkably, the line-to-continuum ratios of AB Aur (2.58) and AB
Aur b ($3.2\pm 0.9$, adopting HST/STIS photometry as the continuum) are
consistent within $1\sigma$ (Figure 5, bottom right panel). We note that this
agreement might be serendipitous because AB Aur’s H$\alpha$ emission is
variable (Harrington & Kuhn, 2007) and our comparison is based on asynchronous
observations (the stellar spectrum is obtained 15 months earlier than the HST
images). Our HST observations do not measure continuum flux and thus cannot
simultaneously constrain the stellar line-to-continuum flux ratios. Continuous
and simultaneous H$\alpha$ monitoring of AB Aur and AB Aur b can help test
this scattering scenario. Currie et al. (2022) concluded that the scattered
light could not account for the infrared emission, because AB Aur b was
undetected in infrared polarized images. If the H$\alpha$ flux is scattered
light, the disk structure or dust envelope should predominantly scatter at
short wavelengths such that scattered stellar emission is only significant in
optical bands.

Finally, we emphasize the striking difference in the SED shape between AB Aur
b and the PDS 70 planets. In Figure 5, we compare the SEDs of AB Aur b and PDS
70 b and find that AB Aur b is significantly brighter and bluer. Fitting
single blackbody models to the SEDs of AB Aur b and PDS 70 b (excluding the
H$\alpha$ points) yield effective temperatures/radii of 3800 K/1.6
$R_{\mathrm{Jup}}$ and 1200 K/2.7 $R_{\mathrm{Jup}}$, respectively. These
results correspond to over 1 dex difference in their bolometric luminosities
($\log(L_{\mathrm{bol}}/L_{\odot})=-2.3$ for AB Aur b 333This value is higher
than the one ($\log(L_{\mathrm{bol}}/L_{\odot})=-2.695\pm 0.095$) reported by
Currie et al. (2022), because we adopted a blackbody rather than a
photospheric model SED to calculate $L_{\mathrm{bol}}$. and
$\log(L_{\mathrm{bol}}/L_{\odot})=-3.8$ for PDS 70 b). We can compare the
bolometric luminosity of AB Aur b to evolutionary track predictions while
safely ignoring contributions from an accreting circumplanetary disk, because
the expected disk luminosity is at least one order of magnitude
lower444$\log(L_{\mathrm{CPD}}/L_{\odot})\approx-3.5$. This is estimated using
Equation (5) of Zhu (2015), assuming $M=9M_{\mathrm{Jup}}$,
$R_{\mathrm{in}}=7.5R_{\mathrm{Jup}}$ (Currie et al., 2022), and
$\log(\dot{M}/(M_{\mathrm{Jup}}\mathrm{yr^{-1}})=-5.8$. Assuming a hot-start
model (e.g., Burrows et al., 1997; Mordasini et al., 2017) and that AB Aur b
has a planetary mass ($M{<}13M_{\mathrm{Jup}}$), AB Aur b has to be younger
than 3 Myr to have such a high $L_{\mathrm{bol}}$. In contrast,
$L_{\mathrm{bol}}$ of the PDS 70 planets are consistent with $1$ to $4$
$M_{\mathrm{Jup}}$ planets at an age of ${\sim}$5 Myr (Wang et al., 2020;
Stolker et al., 2020). Because the isochronal age of AB Aur
($6.0^{+2.5}_{-1.0}$ Myr) is older than 3 Myr, the planetary interpretation of
AB Aur b and the inferred young age might imply delayed planet formation in
the AB Aur disk.

##  6 Summary

1\. We observed the Herbig Ae/Be star AB Aur using HST/WFC3/UVIS in the F656N
(H$\alpha$) band in two epochs separated by 50 days. After subtracting the
primary-star PSF, we detected a point-like source $0.6^{\prime\prime}$ away
from the star in both epochs. In the most optimal reduction, the detection
SNRs are 5.6, 4.4, and 7.8 in Epoch 1, Epoch 2, and the combined data-sets,
respectively.

2\. The P.A. and separation of the companion are $182.5^{\circ}\pm
1.4^{\circ}$ and $600\pm 22$ mas relative to the host star. This location is
consistent with the astrometry of AB Aur b, a recently reported protoplanet
candidate. The candidate companions at separations of $2.75^{\prime\prime}$
and $3.72^{\prime\prime}$ are below our detection limits.

3\. The band-averaged H$\alpha$ flux densities are $0.9\pm 0.3$, $1.2\pm 0.5$,
and $1.0\pm 0.3$ $\times 10^{-15}$erg/s/cm2/Å ($F_{\lambda}$) or $1.3\pm 0.4$,
$1.8\pm 0.7$, $1.5\pm 0.4$ mJy ($F_{\nu}$) in Epoch 1, Epoch 2, and the
combined images, respectively.

4\. In comparison to PDS 70 b and c, the H$\alpha$-to-continuum flux ratio of
AB Aur b is significantly lower, suggesting that an accretion shock that
produces strong H$\alpha$ line that has flux density over 100 times the
continuum is not visible from AB Aur b. Both planetary accretion and scattered
stellar emission from either a compact disk structure or an envelope
surrounding AB Aur b can contribute to the H$\alpha$ flux, and the origin of
the H$\alpha$ emission remains unconstrained.

5\. Assuming that the observed H$\alpha$ emission is entirely powered by
accretion onto the protoplanet and there is no gas or dust extinction from the
disk nor from the accretion flow itself, we estimate an mass accretion rate of
$\log(\dot{M}/M_{\mathrm{Jup}}\mathrm{yr^{-1}})=-5.8$ to $-6.6$, depending on
the assumed $L_{\mathrm{H\alpha}}-L_{\mathrm{acc}}$ relations. The accretion
rate estimate suffers from significant systematic uncertainty due to the
unknown origin of the H$\alpha$ emission and unconstrained extinction.

The detection of H$\alpha$ emission from the candidate protoplanet AB Aur b
offers exciting opportunities to understand the mass assembly processes of
giant planets forming in a highly structured disk. These interpretations are
conditioned on definitively resolving the origin of the H$\alpha$ emission.
Disentangling the planetary accretion versus stellar scattering of H$\alpha$
emission possibilities could not only prove to be crucial in understanding the
origin of AB Aur b, but also for the interpretation of future discoveries of
protoplanets embedded in the disks of their host stars.

We thank the referee for a constructive report that improves the rigor of the
manuscript. We acknowledge excellent observing support from STScI staffs,
especially J. Green and A. Armstrong. We thank Thayne Currie for the CHARIS
spectrum and Gabriel-Dominique Marleau for helpful comments. Y.Z. acknowledges
support from the Heising-Simons Foundation 51 Pegasi b Fellowship. B.P.B.
acknowledges support from the National Science Foundation grant 450
AST-1909209, NASA Exoplanet Research Program grant 451 20-XRP20_2-0119, and
the Alfred P. Sloan Foundation. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. The observations and
data analysis works were supported by program HST-GO-16651. Supports for
Program numbers HST-GO-16651 were provided by NASA through a grantfrom the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555.

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# Deterministic spin-wave interferometer based on Rydberg blockade

Ran Wei  Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Bo Zhao<EMAIL_ADDRESS>Institute for
Theoretical physics, University of Innsbruck, A-6020 Innsbruck, Austria
Institute for Quantum Optics and Quantum Information of the Austrian Academy
of Science,
A-6020 Innsbruck, Austria  Youjin Deng<EMAIL_ADDRESS>Hefei National
Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Yu-Ao Chen  Fakultät für Physik, Ludwig-Maximilian-Universität,
Schellingstrasse 4, 80798 München, Germany  Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany  Jian-Wei Pan
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, China

###### Abstract

The spin-wave (SW) NOON state is an $N$-particle Fock state with two atomic
spin-wave modes maximally entangled. Attributed to the property that the phase
is sensitive to collective atomic motion, the SW NOON state can be utilized as
a novel atomic interferometer and has promising application in quantum
enhanced measurement. In this paper we propose an efficient protocol to
deterministically produce the atomic SW NOON state by employing Rydberg
blockade. Possible errors in practical manipulations are analyzed. A feasible
experimental scheme is suggested. Our scheme is far more efficient than the
recent experimentally demonstrated one, which only creates a heralded second-
order SW NOON state.

###### pacs:

42.50.-p, 42.50.Dv, 32.80.Ee, 32.80.Qk, 37.25.+k, 03.75.Dg

##  I Introduction

The NOON state, an $N$-particle Fock state with two modes maximally entangled,
has attracted many interests since it has the potential to enhance the
measurement precision by employing quantum entanglement lee2002 . Attributed
to the property of superresolution and supersensitivity, the NOON state has
been experimentally realized in various photonic systems walther2004 ;
mitchell2004 ; nagata2007 ; resch2007 . Recently, a new type of NOON state -
the atomic spin wave (SW) NOON state - was proposed, and a heralded second-
order SW NOON state as well, was experimentally demonstrated yuao2010 . The
scheme yuao2010  employs Raman transitions to generate the atom-photon
entanglement and the SW NOON state is created in a herald way by detecting the
photons. The SW NOON state can be used as an atomic SW interferometer and can
in principle be implemented in a scalable way. However, owing to the
probabilistic nature, this SW interferometer works in a very low efficiency
and thus cannot be put into practical measurement.

In recent years, the Ryberg atom draw extensive concern in quantum information
processing molmer2010 . It has large size and can exhibit large electric
dipole moment. This property introduces strong interactions between two
Rydberg atoms. Consequently, in a small volume, when an atom is excited to the
Rydberg state $|r\rangle$, the energy level of state $|r\rangle$ for other
atoms will be shifted by $\Delta_{e}$. Therefore, the probability for other
atoms being excited to $|r\rangle$ is suppressed by a factor of
$1/\Delta_{e}^{2}$. In the limit $\Delta_{e}\rightarrow\infty$, only one atom
is excited to $|r\rangle$. This is the so-called Rydberg blockade mechanism.
The Rydberg blockade has been proposed to deterministically implement quantum
computer and quantum repeater jaksch2000 ; lukin2001 ; saffman2005a ;
saffman2005b ; saffman2009 ; markus2009 ; zhaobo2010 ; yanghan2010 ;
isenhower2010 .

In this paper, we propose an efficient way to implement the SW interferometer
by deterministically generating the SW NOON state with Rydberg blockade. An
elaborate error analysis shows that the $20$th-order SW NOON state can be
generated with $91\%$ fidelity under realistic parameters, and accordingly a
high fidelity SW interferometer with $F\approx 82\%$ can be realized. This
Rydberg-based SW interferometer is much more efficient than the one based on
photon detection and might be used as an inertial sensor, for measuring
position and displacement, or further, for measuring acceleration and platform
rotation. The remaining of this paper is organized as follows. Sec. II
describes an envisioned setup and presents the scheme to generate and measure
the SW NOON state. Error analysis in practical implementations is given in
Sec. III. Experimental realization is suggested in Sec. IV, and finally we
conclude in Sec. V.

##  II Protocol

We envision a setup as illustrated in Fig. 1(a). An ensemble of $N$ atoms is
confined in a volume $V$, where the blockade mechanism is effective. In other
words, the scale of $V$ is smaller than the blockade radius. The working
atomic energy levels are chosen to be of the double-$\Lambda$ type, as shown
in Fig. 1(b). They are labeled as the ground state $|g\rangle$, the Rydberg
state $|r_{a}\rangle$, $|r_{b}\rangle$, and the metastable state
$|s_{a}\rangle$, $|s_{b}\rangle$. The atoms are coupled by four types of
classic light pulses propagating along two spatial modes $a,b$, whose wave
vectors are denoted as $\bm{k}_{gr_{a}}$, $\bm{k}_{r_{a}s_{a}}$,
$\bm{k}_{gr_{b}}$ and $\bm{k}_{r_{b}s_{b}}$ respectively. They will also be
used to denote the corresponding light pulses if no ambiguity arises. These
light pulses couple $|g\rangle$ and $|r_{a}\rangle$, $|r_{a}\rangle$ and
$|s_{a}\rangle$, $|g\rangle$ and $|r_{b}\rangle$, and $|r_{b}\rangle$ and
$|s_{b}\rangle$ respectively, as illustrated in Fig. 1(b).

Before giving the detailed scheme, we shall first introduce some definitions.
We define a collective ground state $|\bm{0}\rangle\equiv|g...g\rangle$, a
collective operator
$M_{\bm{k},\epsilon}^{\dagger}\equiv\frac{1}{\sqrt{N}}\sum\limits_{j}^{N}e^{i\bm{k}\cdot\bm{r}_{j}}|\epsilon_{j}\rangle\langle
g|$, and $|\bm{1},\bm{k}\rangle_{\epsilon}$
$(\epsilon=r_{a},r_{b},s_{a},s_{b})$ to describe a collective state with wave
vector $\bm{k}$,

$|\bm{1},\bm{k}\rangle_{\epsilon}\equiv\frac{1}{\sqrt{N}}\sum\limits_{j}^{N}e^{i\bm{k}\cdot\bm{r}_{j}}|g...\epsilon_{j}...g\rangle=M_{\bm{k},\epsilon}^{\dagger}|\bm{0}\rangle.$
(1)

Namely, state $|\bm{1},\bm{k}\rangle_{\epsilon}$ is a coherent superposition
of states which have a specific atom at $|\epsilon\rangle$ with the position-
dependent phase under the wave vector $\bm{k}$. The same applies to the
higher-order collective state
$|\bm{\ell},\bm{k}\rangle_{\epsilon}\equiv\frac{1}{\sqrt{\ell!}}(M_{\bm{k},\epsilon}^{\dagger})^{\ell}|0\rangle_{\epsilon}$,
with $\ell$ a positive integer. On this basis, a $\ell$th-order SW NOON state
can be written as

$\left|\mathrm{NOON}\right\rangle_{\ell}=\frac{1}{\sqrt{2}}\left(\left|\bm{\ell},\bm{k}\right\rangle_{s_{a}}+\left|\bm{\ell},\bm{k}\right\rangle_{s_{b}}\right).$
(2)

Figure 1: (Color Online) $\bm{(a)}$ An ensemble of N atoms trapped in volume
V. The atoms are coupled by four types of light pulses, propagating along two
spatial modes $a,b$. $\bm{(b)}$ The double-$\Lambda$ type energy levels. An
effective energy shift $\Delta_{e}$ is introduced because of the strong
interaction between the atoms at the Rydberg states.

We first consider the ideal case by making the following assumptions. (1), the
atom number is exactly known, i.e., $\Delta N=0$; (2), the Rydberg blockade
mechanism is perfect, i.e., $\Delta_{e}\rightarrow\infty$; (3), the lifetime
of the Rydberg state is infinite and thus no spontaneous decay occurs; (4),
the atomic cloud remains still during the whole process. On this basis, our
scheme to generate a $\ell$th-order SW NOON state can be described as

  1. 1.

Prepare an ensemble at the ground state $|\bm{0}\rangle$.

  2. 2.

Apply sequentially a collective $\pi$ pulse $\bm{k}_{gr_{a}}$ and a single-
atomic $\pi/2$ pulse $\bm{k}_{r_{a}s_{a}}$. The former flips one of the $N$
atoms from $|\bm{0}\rangle$ to the Rydberg state
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ and the latter flips
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ to the equal superposition of the
first-order SW state $|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$ and
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$, where
$\bm{k}_{\epsilon_{1}\epsilon_{2}\epsilon_{2}}\equiv\bm{k}_{\epsilon_{1}\epsilon_{2}}-\bm{k}_{\epsilon_{2}\epsilon_{3}}$
$(\epsilon_{1},\epsilon_{2},\epsilon_{3}=g,r_{a},r_{b},s_{a},s_{b})$.
Accordingly, one obtains

$i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}},$

where a relative phase shift $\pi/2$ is introduced.

  3. 3.

Apply successively three collective $\pi$ pulses $\bm{k}_{gr_{b}}$,
$\bm{k}_{gr_{a}}$ and $\bm{k}_{gr_{b}}$, which leads to

$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}-|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$
$\displaystyle\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}+|\bm{0}\rangle$
$\displaystyle\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}.$

  4. 4.

Apply in order a collective $\pi$ pulse $\bm{k}_{gr_{a}}$ and a single-atomic
$\pi$ pulse $\bm{k}_{r_{b}s_{b}}$, and a collective $\pi$ pulse
$\bm{k}_{gr_{b}}$ and a single-atomic $\pi$ pulse $\bm{k}_{r_{a}s_{a}}$, which
results in

$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}-|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}-i|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}+|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{2},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}.$

  5. 5.

Repeatedly apply a sequence of four collective $\pi$ pulses $\bm{k}_{gr_{a}}$,
$\bm{k}_{r_{b}s_{b}}$, $\bm{k}_{gr_{b}}$, $\bm{k}_{r_{a}s_{a}}$ for $\ell-2$
times, and one obtains

$\displaystyle|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{\bm{\ell}-1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{r_{b}s_{b}}\rangle_{r_{b}}.$

  6. 6.

Apply a collective $\pi$ pulse to flip the atom from
$|\bm{\bm{\ell}-1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{r_{b}s_{b}}\rangle_{r_{b}}$
to $|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$ and take into account the
normalized factor, and one obtains a $\ell$th-order SW NOON state

$|\Psi\rangle_{\ell}=(|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}})/\sqrt{2}.$
(3)

According to the above procedure, the generation of a $\ell$th-order SW NOON
state needs totally $4\ell+2$ light pulses, the number of which is linear to
$\ell$. Note that one needs two $\pi$ pulses $\bm{k}_{gr_{a}}$ and
$\bm{k}_{r_{a}s_{a}}$ to produce a first-order SW state
$|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$. Accordingly, two $\ell$th-order
SW states $|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$ and
$|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$ would consume $4\ell$ light
pulses. The $\ell$th-order SW NOON state is the superposition of two $\ell$th-
order SW states at the $a$ and $b$ modes. Thus, we consider the above protocol
close to being optimal, albeit the possibility of further improvement is not
entirely excluded.

Here we demonstrate how the SW NOON state can be utilized as an atomic
interferometer. Let’s assume that, after the $\ell$th-order SW NOON state is
prepared, the atomic cloud moves to a new position with a displacement
$\Delta\bm{x}$. To measure $\Delta\bm{x}$, we apply a sequence of operations
reverse to the generation procedure, until the last operation, i.e., the
collective $\pi$ pulse $\bm{k}_{gr_{a}}$. Detailed calculations show that we
obtain the superposition state

$\displaystyle|\Psi^{\prime}\rangle_{\ell}$ $\displaystyle=$
$\displaystyle(ie^{i(\bm{k}_{gr_{a}s_{a}})\cdot\Delta\bm{x}}(1+e^{i\ell\Delta\bm{k}\cdot\Delta\bm{x}})|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$
(4) $\displaystyle+$ $\displaystyle
e^{i\bm{k}_{gr_{b}}\cdot\Delta\bm{x}}(1-e^{i\ell\Delta\bm{k}\cdot\Delta\bm{x}})|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}})/\sqrt{2},$

where $\Delta\bm{k}\equiv-\bm{k}_{gr_{a}s_{a}}-\bm{k}_{gr_{b}s_{b}}$. Note
that, by applying an ionizing electric field, the Rydberg state
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ will be ionized and a free electron
will fly out of the atomic ensemble. Thus, the state (4) can be measured onto
the $|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ basis, and the average result
will reflect the phase shift $\ell\Delta\bm{k}\cdot\Delta\bm{x}$. Since the
wave vectors of the light pulses are known, this gives the displacement
$\Delta\bm{x}$. The phase shift is proportional to the order $\ell$, and thus
the larger $\ell$ would bring $\Delta\bm{x}$ the better precision.

##  III Error analysis

In actual implementations, errors can always occur. For instance, the precise
number $N$ of atoms in the ensemble is normally unknown, and the atom number
$N$ also varies for different experimental trials. This leads to an
uncertainty $\Delta N$ of the atom number, which is $\Delta N\simeq\sqrt{N}$
for large $N$. Since the collective Rabi frequency $\Omega_{c}$ of the $\pi$
pulse $\bm{k}_{gr_{\lambda}}$ definition  is related to the atom number $N$ as
$\Omega_{c}\propto\sqrt{N}$, $\Delta N$ would induce an imprecision in
$\Omega_{c}$ as $\Delta\Omega_{c}/\Omega_{c}\simeq 1/(2\sqrt{N})$. This means
that, when a collective $\pi$ pulse $\bm{k}_{gr_{\lambda}}$ is applied to flip
one of the atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$, there exists a
probability $p\simeq\pi^{2}/(16N)$ that the flip fails. To generate a
$\ell$th-order SW NOON state, the total error introduced by $\Delta N$ is
about $\pi^{2}\ell/(8N)$. In lab, one can prepare an ensemble of $N\approx
400$ atoms, and thus the error is about $\pi^{2}\ell/(8N)\approx 6\%$ for
order $\ell=20$.

Aside from the error induced by the uncertainty of the atom number, the
imperfect blockade mechanism and the finite lifetime of the Rydberg state also
introduces errors. Attributed to these factors, each operation in our scheme
is implemented with a non-unity probability. We step by step analyze all the
operations from Step $1$ to Step $6$, and find that, these non-unity
probabilities can be categorized into five types, denoted as $P^{I}$,
$P^{II}$, $P^{III}$, $P^{IV}_{q}$, $P^{V}_{q}$, and the generated $\ell$th-
order SW NOON state should be rewritten approximately as

$\displaystyle\sqrt{\mathcal{P}_{\ell}(P^{I}P^{II}P^{III})^{\ell}}|\Psi\rangle_{\ell},$
(5)

where $\mathcal{P}_{\ell}=\prod_{q=1}^{\ell}P^{IV}_{q}P^{V}_{q}$. Symbol $q$
stands for the order of the SW state during the generation process, and it
increases from $1$ to $\ell$ as one produces the $\ell$th-order SW NOON state.
Accordingly, the probability for preparing the $\ell$th-order SW NOON state is

$\displaystyle P(\ell)=\mathcal{P}_{\ell}(P^{I}P^{II}P^{III})^{\ell}.$ (6)

The total error accumulated by these operations is the probability that one
fails to generate the $\ell$th-order SW NOON state, thus it reads
$E(\ell)=1-P(\ell)$. (The error induced by the uncertainty of atom number is
not included in $E(\ell)$.) Before evaluating $E(\ell)$, we shall first
analyze the origins of these probabilities.

The probability $P^{I}$ is introduced by the imperfect blockade that occurs
between the atoms of the same mode when the pulse $\bm{k}_{gr_{\lambda}}$
flips one of the atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. In other words, there is
an error that two atoms are excited to the Rydberg state
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ due to the non-infinite
energy shift. This mechanics is described by the following equations,

$\displaystyle i\dot{c}_{0}$ $\displaystyle=-\frac{\sqrt{N}\Omega}{2}c_{1},$
(7) $\displaystyle i\dot{c}_{1}$
$\displaystyle=-i\frac{\gamma}{2}c_{1}-\frac{\sqrt{N}\Omega}{2}c_{0}-\frac{\sqrt{2N}\Omega}{2}c_{2},$
(8) $\displaystyle i\dot{c}_{2}$
$\displaystyle=(\Delta_{e}-i\gamma)c_{2}-\frac{\sqrt{2N}\Omega}{2}c_{1},$ (9)

where $c_{0},c_{1},c_{2}$ stand for the amplitudes of $|\bm{0}\rangle$,
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$,
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. Symbols $\gamma/2$ and
$\gamma$ are the decay rates of
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ and
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. Symbol $\Delta_{e}$ is
the effective finite energy shift, and $\sqrt{N}\Omega,\sqrt{2N}\Omega$ are
the corresponding two collective Rabi frequencies, which have been assumed to
be real. Since the amplitudes for the states of more than two atoms being
excited are significantly suppressed due to the Rydberg blockade, we have
neglected them here and in the following. Besides, we have assumed the number
of atoms $N\gg 1$ and the coupling light pulses are all in resonance. The
initial condition describing this mechanics is
$c_{0}(0)=1,c_{1}(0)=0,c_{2}(0)=0$. After applying the collective $\pi$ pulse
$\bm{k}_{gr_{\lambda}}$ with the operation time $\Delta
t=\pi/(\sqrt{N}\Omega)$, one can express the probability for generating
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ from $|\bm{0}\rangle$, as
$P^{I}=|c_{1}(\Delta t)|^{2}$.

The probability $P^{II}$ characterizes the imperfect blockade that takes place
between the atoms of the different modes during $\Delta t$. That is to say,
there is an error that the pulse $\bm{k}_{gr_{\lambda}}$ would flip one of the
atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ when another atom has
already been excited to
$|\bm{1},\bm{k}_{gr_{\bar{\lambda}}}\rangle_{r_{\bar{\lambda}}}$. Accordingly,
this mechanics is governed by the following equations,

$\displaystyle i\dot{c}_{0}$ $\displaystyle=-\frac{\sqrt{N}\Omega}{2}c_{1},$
(10) $\displaystyle i\dot{c}_{1}$
$\displaystyle=(\Delta_{e}-i\frac{\gamma}{2})c_{1}-\frac{\sqrt{N}\Omega}{2}c_{0}.$
(11)

The initial condition describing this mechanics is $c_{0}(0)=1,c_{1}(0)=0$,
and one can express the probability for holding the atoms at the ground state,
as $P^{II}=|c_{0}(\Delta t)|^{2}$.

The probability $P^{III}$ is contributed by the decay rate of the Rydberg
state. The finite lifetime will inevitably cause some loss when the atom is
still at the Rydberg state during $\Delta t$, thus the probability for the
atom remaining at the Rydberg state is $P^{III}=e^{-\gamma\Delta t}$.

These three types ($P^{I}$, $P^{II}$, $P^{III}$) are all determined by a
shared Rabi frequency $\Omega$ or a shared operation time $\Delta t$. Note
that there is tradeoff between the imperfect Rydberg blockade and the loss
caused by the decay, and a simple argument is that if we enhance the the
magnitude of the Rabi frequency to shorten the operation time, which reduces
the loss from the Rydberg state, it will be associated with more errors from
the imperfect blockade. Therefore, there is an optimal Rabi frequency to
maximize the value of $P^{I}P^{II}P^{III}$. By numerically solving Eqs. (7-9)
and Eqs. (10-11), one can easily obtain this maximal value.

The probability $P^{IV}_{q}$ reflects an error that one of the atoms at
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
would be flipped back to
$|\bm{q-2},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
when the pulse $\bm{k}_{r_{\lambda}s_{\lambda}}$ is applied to flip the atom
from
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
to $|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$. This
mechanics is described by the following equations,

$\displaystyle i\dot{\widetilde{c}}_{0}$
$\displaystyle=-\frac{\sqrt{q}\widetilde{\Omega}}{2}\widetilde{c}_{1},$ (12)
$\displaystyle i\dot{\widetilde{c}}_{1}$
$\displaystyle=-i\frac{\gamma}{2}\widetilde{c}_{1}-\frac{\sqrt{q}\widetilde{\Omega}}{2}\widetilde{c}_{0}-\frac{\sqrt{2(q-1)}\widetilde{\Omega}}{2}\widetilde{c}_{2},$
(13) $\displaystyle i\dot{\widetilde{c}}_{2}$
$\displaystyle=(\Delta_{e}-i\gamma)\widetilde{c}_{2}-\frac{\sqrt{2(q-1)}\widetilde{\Omega}}{2}\widetilde{c}_{1},$
(14)

where $\widetilde{c}_{0},\widetilde{c}_{1},\widetilde{c}_{2}$ are the
amplitudes of $|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$,
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$,
$|\bm{q-2},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$.
Symbols $\sqrt{q}\widetilde{\Omega},\sqrt{2(q-1)}\widetilde{\Omega}$ are the
corresponding two collective Rabi frequencies, which have also been assumed to
be real. The initial condition describing this mechanics is
$\widetilde{c}_{0}(0)=0,\widetilde{c}_{1}(0)=1,\widetilde{c}_{2}(0)=0$. After
applying the collective $\pi$ pulse $\bm{k}_{r_{\lambda}s_{\lambda}}$ with the
operation time $\Delta\widetilde{t}_{q}=\pi/(\sqrt{q}\widetilde{\Omega})$, one
can express the probability for producing the $q$th-order SW state
$|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$, as
$P^{IV}_{q}=|\widetilde{c}_{0}(\Delta\widetilde{t}_{q})|^{2}$.

The origin of $P^{V}_{q}$ is similar to $P^{III}$, it reflects the probability
that the atom remains at the Rydberg state during $\Delta\widetilde{t}_{q}$,
and thus $P^{V}_{q}=e^{-\gamma\Delta\widetilde{t}_{q}}$. The value of
$P^{IV}_{q}P^{V}_{q}$ is determined by a shared Rabi frequency
$\widetilde{\Omega}$ or a shared operation time $\Delta\widetilde{t}_{q}$.
Likewise, one can calculate the maximal value of $P^{IV}_{q}P^{V}_{q}$ by
numerically solving Eqs. (12-14) with $q$ from $1$ to $\ell$.

To evaluate $E(\ell)$, we choose the parameters as, the atom number $N=400$,
the lifetime of the Rydberg state $\tau=1/(2\pi\gamma)=300$ $\mu s$ and $400$
$\mu s$, and the energy shift $\Delta_{e}$ varying from $20$ $MHz$ to $400$
$MHz$. Accordingly, Eq.(6) can be calculated in a numerical way. We obtain the
error $E(\ell)$ versus the energy shift $\Delta_{e}$, shown in Fig.2.

Figure 2: (Color Online) The figure demonstrates the error $E(\ell)$ versus
the energy shift $\Delta_{e}$ under various order $\ell$ after generating the
SW NOON state. The solid data and the open one respectively denote the
lifetime of the Rydberg state with $\tau=300$ $\mu s$ and $\tau=400$ $\mu s$.

From the figure, we see that the larger the energy shift, the smaller the
error, and the error vanishes as $\Delta_{e}$ tends to infinity. This is an
anticipated result since the error $E\sim\Omega^{2}/\Delta_{e}^{2}$. However,
in actual experiment, $\Delta_{e}$ cannot be unlimitedly large. An intrinsic
limitation originates from the average distance of two Rydberg atoms, which
should be larger than the radius of each Rydberg atom. In the limit of high
density where the Rydberg atoms remarkably overlap, our blockade model is
inappropriate, and a more elaborate mechanism should be taken into account.
This mechanism goes beyond the extent of our paper and will not be discussed.
Besides, as one readily expects, the figure shows that the error is suppressed
as the lifetime of the Rydberg state becomes longer, and is intensified when
the order $\ell$ of the SW NOON state increases.

##  IV Experimental realization

To design an atomic interferometer with sufficiently high precision and
relatively high fidelity, we use the $20$th-order SW NOON state for the
practical application. The interferometer can be implemented by cold alkali
atoms. By choosing the suitable laser polarization, the two spacial modes $a$
and $b$ can be individually addressed. The energy shift is isotropic due to
the property of repulsive van der Waals interaction. The lifetime of the
Rydberg state with $\tau=300\sim 400$ $\mu s$ is achievable by exciting the
atoms to the Rydberg $s$ state with a principal quantum number $n=100$
saffman2005a . In our scheme, the energy shift $\Delta_{e}$ of the Rydberg
state can be expressed as $\Delta_{e}=-n^{11}(c_{0}+c_{1}n+c_{2}n^{2})/r^{6}$
singer2005 , where the terms $1/r^{8}$ and $1/r^{10}$ are neglected due to the
dominating long-range property. For Rubidium,
$c_{0}=13,c_{1}=-0.85,c_{2}=0.0034$ singer2005 , and thus an ensemble of atoms
with the radius $R=3.8$ $\mu m$ enables the energy shift $\Delta_{e}\geq 300$
$MHz$, which ensures the error $E(20)<3\%$, as illustrated in Fig.2. In a
volume of $4\pi/3R^{3}$, a density of $1.7\times 10^{12}$ $cm^{-3}$ allows
$N\approx 400$ atoms in an ensemble. Based on these estimated parameters
above, we suggest to employ the one-dimensional optical lattice as the
experimental setup, where the size of the ensemble can be controlled by tuning
the angle between the trapping light fields fallani2005 . Finally, we should
point out that, to detect the displacement of atomic cloud by the
interferometer, the reverse operations to those in the generation procedure
should be considered, and thus the total error is doubled. Fortunately, the
field ionization can be implemented with near-unity detection efficiency
guerlin2007 . Therefore, taking into account the error induced by the
uncertainty of atom number, our proposed atomic SW interferometer with a high
precision ($\ell=20$) can reach a high fidelity as $F\approx
1-2\times(6\%+3\%)=82\%$.

##  V Summary

By employing Rydberg blockade, we have demonstrated an efficient scheme to
deterministically produce the atomic SW NOON state, of which, a direct
application is the atomic SW interferometer. Possible errors in practical
manipulations are analyzed, and the experimental realization also is
suggested. Our proposed atomic SW interferometer is far more efficient than
the recent experimentally demonstrated one, and holds promise in the practical
application.

##  VI Acknowledgement

This work is supported by the NNSFC, the NNSFC of Anhui (under Grant No.
090416224), the CAS, the National Fundamental Research Program (under Grant
No. 2011CB921304), and the SFB FOQUS of FWF.

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