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With the advance of AI, road object detection has been a prominent topic in computer vision, mostly using perspective cameras. Fisheye lens provides omnidirectional wide coverage for using fewer cameras to monitor road intersections, however with view distortions. To our knowledge, there is no existing open dataset prepared for traffic surveillance on fisheye cameras. This paper introduces an open FishEye8K benchmark dataset for road object detection tasks, which comprises 157K bounding boxes across five classes (Pedestrian, Bike, Car, Bus, and Truck). In addition, we present benchmark results of State-of-The-Art (SoTA) models, including variations of YOLOv5, YOLOR, YOLO7, and YOLOv8. The dataset comprises 8,000 images recorded in 22 videos using 18 fisheye cameras for traffic monitoring in Hsinchu, Taiwan, at resolutions of 1080$\times$1080 and 1280$\times$1280. The data annotation and validation process were arduous and time-consuming, due to the ultra-wide panoramic and hemispherical fisheye camera images with large distortion and numerous road participants, particularly people riding scooters. To avoid bias, frames from a particular camera were assigned to either the training or test sets, maintaining a ratio of about 70:30 for both the number of images and bounding boxes in each class. Experimental results show that YOLOv8 and YOLOR outperform on input sizes 640$\times$640 and 1280$\times$1280, respectively. The dataset will be available on GitHub with PASCAL VOC, MS COCO, and YOLO annotation formats. The FishEye8K benchmark will provide significant contributions to the fisheye video analytics and smart city applications., Comment: CVPR Workshops 2023

We investigate speeding up of relaxation of Markovian open quantum systems with the Liouvillian exceptional point (LEP), where the slowest decay mode degenerate with a faster decay mode. The degeneracy significantly increases the gap of the Liouvillian operator, which determines the timescale of such systems in converging to stationarity, and hence accelerates the relaxation process. We explore an experimentally relevant three level atomic system, whose eigenmatrices and eigenspectra are obtained completely analytically. This allows us to gain insights in the LEP and examine respective dynamics with details. We illustrate that the gap can be further widened through Floquet engineering, which further accelerates the relaxation process. Finally, we extend this approach to analyze laser cooling of trapped ions, where vibrations (phonons) couple to the electronic states. An optimal cooling condition is obtained analytically, which agrees with both existing experiments and numerical simulations. Our study provides analytical insights in understanding LEP, as well as in controlling and optimizing dissipative dynamics of atoms and trapped ions.

Automatic dubbing, which generates a corresponding version of the input speech in another language, could be widely utilized in many real-world scenarios such as video and game localization. In addition to synthesizing the translated scripts, automatic dubbing needs to further transfer the speaking style in the original language to the dubbed speeches to give audiences the impression that the characters are speaking in their native tongue. However, state-of-the-art automatic dubbing systems only model the transfer on duration and speaking rate, neglecting the other aspects in speaking style such as emotion, intonation and emphasis which are also crucial to fully perform the characters and speech understanding. In this paper, we propose a joint multi-scale cross-lingual speaking style transfer framework to simultaneously model the bidirectional speaking style transfer between languages at both global (i.e. utterance level) and local (i.e. word level) scales. The global and local speaking styles in each language are extracted and utilized to predicted the global and local speaking styles in the other language with an encoder-decoder framework for each direction and a shared bidirectional attention mechanism for both directions. A multi-scale speaking style enhanced FastSpeech 2 is then utilized to synthesize the predicted the global and local speaking styles to speech for each language. Experiment results demonstrate the effectiveness of our proposed framework, which outperforms a baseline with only duration transfer in both objective and subjective evaluations., Comment: TASLP

Abridged - Stars with ZAMS masses between 140 and $260 M_\odot$ are thought to explode as pair-instability supernovae (PISNe). During their thermonuclear runaway, PISNe can produce up to several tens of solar masses of radioactive nickel, resulting in luminous transients similar to some superluminous supernovae (SLSNe). Yet, no unambiguous PISN has been discovered so far. SN2018ibb is a H-poor SLSN at $z=0.166$ that evolves extremely slowly compared to the hundreds of known SLSNe. Between mid 2018 and early 2022, we monitored its photometric and spectroscopic evolution from the UV to the NIR with 2-10m class telescopes. SN2018ibb radiated $>3\times10^{51} \rm erg$ during its evolution, and its bolometric light curve reached $>2\times10^{44} \rm erg\,s^{-1}$ at peak. The long-lasting rise of $>93$ rest-frame days implies a long diffusion time, which requires a very high total ejected mass. The PISN mechanism naturally provides both the energy source ($^{56}$Ni) and the long diffusion time. Theoretical models of PISNe make clear predictions for their photometric and spectroscopic properties. SN2018ibb complies with most tests on the light curves, nebular spectra and host galaxy, potentially all tests with the interpretation we propose. Both the light curve and the spectra require 25-44 $M_\odot$ of freshly nucleosynthesised $^{56}$Ni, pointing to the explosion of a metal-poor star with a He-core mass of 120-130 $M_\odot$ at the time of death. This interpretation is also supported by the tentative detection of [Co II]$\lambda$1.025$\mu$m, which has never been observed in any other PISN candidate or SLSN before. Powering by a central engine, such as a magnetar or a black hole, can be excluded with high confidence. This makes SN2018ibb by far the best candidate for being a PISN, to date., Comment: Accepted in A&A, the revised version includes a PISN rate estimate and an additional test with PISN models. 47 pages, main text 41 pages, 38 figures, 16 Tables

This study examines the anisotropic dissolution of the basal plane gibbsite ({\gamma}-Al(OH)3) nanoplates in sodium hydroxide solution using in situ atomic force microscopy (AFM) and density functional theory (DFT) calculations. In the surface-reaction controlled regime, in situ AFM measurements reveal anisotropic dissolution of hillocks and etch pits on the gibbsite basal plane, with preferred dissolution directions alternating between layers. The mirror-symmetric pattern of dissolution preference between adjacent gibbsite aluminum hydroxide sheet, observed along the crystallographic a-c plane, results from the matching symmetry between the structures of the adjacent (001) and (002) crystal planes. Consequently, the overall dissolution rate of gibbsite nanoplates exhibits crystallographic a-c plane symmetry, as the rate of parallel steps is governed by the slower ones. DFT calculations suggest that the anisotropic dissolution is partially due to the orientation and strength of Al-OH-Al linkages pair within gibbsite surface structure. These findings offer a comprehensive understanding of anisotropic dissolution behavior of gibbsite and illuminate the mechanisms behind preferential dissolution., Comment: Supporting information is available at https://drive.google.com/file/d/1JXldLu0XwG2VYxQ7YgNZwvFdPuwyPJX4/view?usp=sharing

Pitch estimation is to estimate the fundamental frequency and the midi number and plays a critical role in music signal analysis and vocal signal processing. In this work, we proposed a new architecture based on a learning-based enhancement preprocessor and a combination of several traditional and deep learning pitch estimation methods to achieve better pitch estimation performance in both noisy and clean scenarios. We test 17 different types of noise and 4 SNRdb noise levels. The results show that the proposed pitch estimation can perform better in both noisy and clean scenarios with short response time., Comment: From ICCE-Taiwan

Quantum correlations, both spatial and temporal, are the central pillars of quantum mechanics. Over the last two decades, a big breakthrough in quantum physics is its complex extension to the non-Hermitian realm, and dizzying varieties of novel phenomena and applications beyond the Hermitian framework have been uncovered. However, unique features of non-Hermitian quantum correlations, especially in the time domain, still remain to be explored. Here, for the first time, we experimentally achieve this goal by using a parity-time (PT )-symmetric trapped-ion system. The upper limit of temporal quantum correlations, known as the algebraic bound, which has so far not been achieved in the standard measurement scenario, is reached here by approaching the exceptional point (EP), thus showing the unexpected ability of EPs in tuning temporal quantum correlation effects. Our study, unveiling the fundamental interplay of non-Hermiticity, nonlinearity, and temporal quantum correlations, provides the first step towards exploring and utilizing various non-Hermitian temporal quantum effects by operating a wide range of EP devices, which are important for both fundamental studies and applications of quantum EP systems., Comment: 4 figures and 8 pages

In this paper, we consider a class of reflected stochastic differential equations for which the constraint is not on the paths of the solution but on its law. We establish a small noise large deviation principle, a large deviation for short time and the Malliavin derivative. To prove large deviation principles, a sufficient condition for the weak convergence method, which is suitable for Mckean-Vlasov stochastic differential equation, plays an important role.

The controllable synthesis of iron oxides particles is a critical issue for materials science, energy storage, biomedical applications, environmental science, and earth science. However, synthesis of iron oxides with desired phase and size are still a time-consuming and trial-and-error process. This study presents solutions for two fundamental challenges in materials synthesis: predicting the outcome of a synthesis from specified reaction parameters and correlating sets of parameters to obtain products with desired outcomes. Four machine learning algorithms, including random forest, logistic regression, support vector machine, and k-nearest neighbor, were trained to predict the phase and particle size of iron oxide based on experimental conditions. Among the models, random forest exhibited the best performance, achieving 96% and 81% accuracy when predicting the phase and size of iron oxides in the test dataset. Premutation feature importance analysis shows that most models (except logistic regression) rely on known features such as precursor concentration, pH, and temperature to predict the phases from synthesis conditions. The robustness of the random forest models was further verified by comparing prediction and experimental results based on 24 randomly generated methods in additive and non-additive systems not included in the datasets. The predictions of product phase and particle size from the models are in good agreement with the experimental results. Additionally, a searching and ranking algorithm was developed to recommend potential synthesis parameters for obtaining iron oxide products with desired phase and particle size from previous studies in the dataset., Comment: See link below for Supporting Information. https://docs.google.com/document/d/1boiUQtlBoc4nvXPoVDMkc4RCg2isWtZ5/edit?usp=sharing&ouid=108731997922646321851&rtpof=true&sd=true

Naked eye recognition of age is usually based on comparison with the age of others. However, this idea is ignored by computer tasks because it is difficult to obtain representative contrast images of each age. Inspired by the transfer learning, we designed the Delta Age AdaIN (DAA) operation to obtain the feature difference with each age, which obtains the style map of each age through the learned values representing the mean and standard deviation. We let the input of transfer learning as the binary code of age natural number to obtain continuous age feature information. The learned two groups of values in Binary code mapping are corresponding to the mean and standard deviation of the comparison ages. In summary, our method consists of four parts: FaceEncoder, DAA operation, Binary code mapping, and AgeDecoder modules. After getting the delta age via AgeDecoder, we take the average value of all comparison ages and delta ages as the predicted age. Compared with state-of-the-art methods, our method achieves better performance with fewer parameters on multiple facial age datasets., Comment: Accepted by CVPR2023; 8 pages, 3 figures

When discovered, SN~2017egm was the closest (redshift $z=0.03$) hydrogen-poor superluminous supernova (SLSN-I) and a rare case that exploded in a massive and metal-rich galaxy. Thus, it has since been extensively observed and studied. We report spectroscopic data showing strong emission at around He~I $\lambda$10,830 and four He~I absorption lines in the optical. Consequently, we classify SN~2017egm as a member of an emerging population of helium-rich SLSNe-I (i.e., SLSNe-Ib). We also present our late-time photometric observations. By combining them with archival data, we analyze high-cadence ultra-violet, optical, and near-infrared light curves spanning from early pre-peak ($\sim -20\,d$) to late phases ($\sim +300\,d$). We obtain its most complete bolometric light curve, in which multiple bumps are identified. None of the previously proposed models can satisfactorily explain all main light-curve features, while multiple interactions between the ejecta and circumstellar material (CSM) may explain the undulating features. The prominent infrared excess with a blackbody luminosity of $10^7$--$10^8\,L_{sun}$ detected in SN~2017egm could originate from the emission of either an echo of a pre-existing dust shell, or newly-formed dust, offering an additional piece of evidence supporting the ejecta-CSM interaction model. Moreover, our analysis of deep $Chandra$ observations yields the tightest-ever constraint on the X-ray emission of an SLSN-I, amounting to an X-ray-to-optical luminosity ratio $\lesssim 10^{-3}$ at late phases ($\sim100-200\,d$), which could help explore its close environment and central engine., Comment: 25 pages, 14 Figures, 4 Tables; accepted for publication in ApJ (Mar. 2023)

The direct detection of core-collapse supernova (SN) progenitor stars is a powerful way of probing the last stages of stellar evolution. However, detections in archival Hubble Space Telescope images are limited to about one per year. Here, we explore whether we can increase the detection rate by using data from ground-based wide-field surveys. Due to crowding and atmospheric blurring, progenitor stars can typically not be identified in pre-explosion images alone. Instead, we combine many pre-SN and late-time images to search for the disappearance of the progenitor star. As a proof of concept, we implement our search for ZTF data. For a few hundred images, we achieve limiting magnitudes of about 23 mag in the g and r band. However, no progenitor stars or long-lived outbursts are detected for 29 SNe within z<0.01, and the ZTF limits are typically several magnitudes less constraining than detected progenitors in the literature. Next, we estimate progenitor detection rates for the Legacy Survey of Space and Time (LSST) with the Vera C. Rubin telescope by simulating a population of nearby SNe. The background from bright host galaxies reduces the nominal LSST sensitivity by, on average, 0.4 mag. Over the ten-year survey, we expect the detection of about 50 red supergiant progenitors and several yellow and blue supergiants. The progenitors of SNe Ib and Ic are detectable if they are brighter than -4.7 mag or -4.0 mag in the LSST i band, respectively. In addition, we expect the detection of hundreds of pre-SN outbursts depending on their brightness and duration., Comment: Submitted to ApJ. Comments are welcome

Polarized upconversion luminescence (PUCL) of lanthanide (Ln3+) ions has been widely used in single particle tracking, microfluidics detection, three-dimensional displays, and so on. However, no effective strategy has been developed for modulating PUCL. Here, we report a strategy to regulate PUCL in Ho3+-doped NaYF4 single nanorods based on the number of upconversion photons. By constructing a multiphoton upconversion system for Ho3+, we regulate the degree of polarization (DOP) of PUCL from 0.6 for two-photon luminescence to 0.929 for three-photon upconversion luminescence (UCL). Furthermore, our strategy is verified by cross-relaxation between Ho3+ and Yb3+, excitation wavelength, excitation power density, and local site symmetry. And this regulation strategy of PUCL has also been achieved in Tm3+, where DOP is ranged from 0.233 for two-photon luminescence to 0.925 for four-photon UCL. Besides, multi-dimensional anti-counterfeiting display has been explored with PUCL. This work provides an effective strategy for the regulation of PUCL and also provides more opportunities for the development of polarization display, optical encoding, anti-counterfeiting, and integrated optical devices.

We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown which shows how spin squeezing and Schrodinger cat states can be generated using the measurement.

We study the systematical one-loop decomposition of the dimension-7 long-range $0\nu\beta\beta$ decay operators. We find that there are 3 genuine one-loop topologies and 8 diagrams. The procedure to determine the SM quantum number assignments for both internal and external fields is presented. The Majorana neutrino mass in long-range $0\nu\beta\beta$ models is discussed. We also present a one-loop $0\nu\beta\beta$ decay model which produces Majorana neutrino mass at three-loop level. The phenomenological predictions for light neutrino mass and $0\nu\beta\beta$ decay half-life time including both mass mechanism and long-range contribution are studied., Comment: 42 pages, 19 figures

Fast radio bursts (FRBs) are highly dispersed millisecond-duration radio bursts, of which the physical origin is still not fully understood. FRB 20201124A is one of the most actively repeating FRBs. In this paper, we present the collection of 1863 burst dynamic spectra of FRB 20201124A measured with the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The current collection, taken from the observation during the FRB active phase from April to June 2021, is the largest burst sample detected in any FRB so far. The standard PSRFITs format is adopted, including dynamic spectra of the burst, and the time information of the dynamic spectra, in addition, mask files help readers to identify the pulse positions are also provided.

Narrow transient emission lines (flash-ionization features) in early supernova (SN) spectra trace the presence of circumstellar material (CSM) around the massive progenitor stars of core-collapse SNe. The lines disappear within days after the SN explosion, suggesting that this material is spatially confined, and originates from enhanced mass loss shortly (months to a few years) prior to explosion. We performed a systematic survey of H-rich (Type II) SNe discovered within less than two days from explosion during the first phase of the Zwicky Transient Facility (ZTF) survey (2018-2020), finding thirty events for which a first spectrum was obtained within $< 2$ days from explosion. The measured fraction of events showing flash ionisation features ($>36\%$ at $95\%$ confidence level) confirms that elevated mass loss in massive stars prior to SN explosion is common. We find that SNe II showing flash ionisation features are not significantly brighter, nor bluer, nor more slowly rising than those without. This implies that CSM interaction does not contribute significantly to their early continuum emission, and that the CSM is likely optically thin. We measured the persistence duration of flash ionisation emission and find that most SNe show flash features for $\approx 5 $ days. Rarer events, with persistence timescales $>10$ days, are brighter and rise longer, suggesting these may be intermediate between regular SNe II and strongly-interacting SNe IIn.

Change detection (CD) aims to find the difference between two images at different times and outputs a change map to represent whether the region has changed or not. To achieve a better result in generating the change map, many State-of-The-Art (SoTA) methods design a deep learning model that has a powerful discriminative ability. However, these methods still get lower performance because they ignore spatial information and scaling changes between objects, giving rise to blurry or wrong boundaries. In addition to these, they also neglect the interactive information of two different images. To alleviate these problems, we propose our network, the Scale and Relation-Aware Siamese Network (SARAS-Net) to deal with this issue. In this paper, three modules are proposed that include relation-aware, scale-aware, and cross-transformer to tackle the problem of scene change detection more effectively. To verify our model, we tested three public datasets, including LEVIR-CD, WHU-CD, and DSFIN, and obtained SoTA accuracy. Our code is available at https://github.com/f64051041/SARAS-Net.

Generalized Category Discovery (GCD) aims to recognize both known and novel categories from a set of unlabeled data, based on another dataset labeled with only known categories. Without considering differences between known and novel categories, current methods learn about them in a coupled manner, which can hurt model's generalization and discriminative ability. Furthermore, the coupled training approach prevents these models transferring category-specific knowledge explicitly from labeled data to unlabeled data, which can lose high-level semantic information and impair model performance. To mitigate above limitations, we present a novel model called Decoupled Prototypical Network (DPN). By formulating a bipartite matching problem for category prototypes, DPN can not only decouple known and novel categories to achieve different training targets effectively, but also align known categories in labeled and unlabeled data to transfer category-specific knowledge explicitly and capture high-level semantics. Furthermore, DPN can learn more discriminative features for both known and novel categories through our proposed Semantic-aware Prototypical Learning (SPL). Besides capturing meaningful semantic information, SPL can also alleviate the noise of hard pseudo labels through semantic-weighted soft assignment. Extensive experiments show that DPN outperforms state-of-the-art models by a large margin on all evaluation metrics across multiple benchmark datasets. Code and data are available at https://github.com/Lackel/DPN., Comment: Accepted by AAAI 2023

We propose a novel framework for 3D hand shape reconstruction and hand-object grasp optimization from a single RGB image. The representation of hand-object contact regions is critical for accurate reconstructions. Instead of approximating the contact regions with sparse points, as in previous works, we propose a dense representation in the form of a UV coordinate map. Furthermore, we introduce inference-time optimization to fine-tune the grasp and improve interactions between the hand and the object. Our pipeline increases hand shape reconstruction accuracy and produces a vibrant hand texture. Experiments on datasets such as Ho3D, FreiHAND, and DexYCB reveal that our proposed method outperforms the state-of-the-art., Comment: BMVC 2022 Spotlight

Despite recent progress in Multiple Object Tracking (MOT), several obstacles such as occlusions, similar objects, and complex scenes remain an open challenge. Meanwhile, a systematic study of the cost-performance tradeoff for the popular tracking-by-detection paradigm is still lacking. This paper introduces SMILEtrack, an innovative object tracker that effectively addresses these challenges by integrating an efficient object detector with a Siamese network-based Similarity Learning Module (SLM). The technical contributions of SMILETrack are twofold. First, we propose an SLM that calculates the appearance similarity between two objects, overcoming the limitations of feature descriptors in Separate Detection and Embedding (SDE) models. The SLM incorporates a Patch Self-Attention (PSA) block inspired by the vision Transformer, which generates reliable features for accurate similarity matching. Second, we develop a Similarity Matching Cascade (SMC) module with a novel GATE function for robust object matching across consecutive video frames, further enhancing MOT performance. Together, these innovations help SMILETrack achieve an improved trade-off between the cost ({\em e.g.}, running speed) and performance (e.g., tracking accuracy) over several existing state-of-the-art benchmarks, including the popular BYTETrack method. SMILETrack outperforms BYTETrack by 0.4-0.8 MOTA and 2.1-2.2 HOTA points on MOT17 and MOT20 datasets. Code is available at https://github.com/pingyang1117/SMILEtrack_Official, Comment: Our paper was accepted by AAAI2024

Novel category discovery aims at adapting models trained on known categories to novel categories. Previous works only focus on the scenario where known and novel categories are of the same granularity. In this paper, we investigate a new practical scenario called Fine-grained Category Discovery under Coarse-grained supervision (FCDC). FCDC aims at discovering fine-grained categories with only coarse-grained labeled data, which can adapt models to categories of different granularity from known ones and reduce significant labeling cost. It is also a challenging task since supervised training on coarse-grained categories tends to focus on inter-class distance (distance between coarse-grained classes) but ignore intra-class distance (distance between fine-grained sub-classes) which is essential for separating fine-grained categories. Considering most current methods cannot transfer knowledge from coarse-grained level to fine-grained level, we propose a hierarchical weighted self-contrastive network by building a novel weighted self-contrastive module and combining it with supervised learning in a hierarchical manner. Extensive experiments on public datasets show both effectiveness and efficiency of our model over compared methods. Code and data are available at https://github.com/Lackel/Hierarchical_Weighted_SCL., Comment: Accepted by EMNLP 2022

We present the discovery and analysis of SN\,2022oqm, a Type Ic supernova (SN) detected $<1$\,day after explosion. The SN rises to a blue and short-lived (2\,days) initial peak. Early-time spectral observations of SN\,2022oqm show a hot (40,000\,K) continuum with high-ionization C and O absorption features at velocities of 4000\,km\,s$^{-1}$, while its photospheric radius expands at 20,000\,\kms, indicating a pre-existing distribution of expanding C/O material. After $\sim2.5$\,days, both the spectrum and light curves evolve into those of a typical SN Ic, with line velocities of $\sim10,000$\,km\,s$^{-1}$, in agreement with the photospheric radius evolution. The optical light curves reach a second peak at $t\approx15$\,days. By $t=60$\,days, the spectrum of \oqm\ becomes nearly nebular, displaying strong \ion{Ca}{2} and [\ion{Ca}{2}] emission with no detectable [\ion{O}{1}], marking this event as Ca-rich. The early behavior can be explained by $10^{-3}$\,\msun\ of optically thin circumstellar material (CSM) surrounding either (1) a massive compact progenitor such as a Wolf-Rayet star, (2) a massive stripped progenitor with an extended envelope, or (3) a binary system with a white dwarf. We propose that the early-time light curve is powered by both interaction of the ejecta with the optically thin CSM and shock cooling (in the massive-star scenario). The observations can be explained by CSM that is optically thick to X-ray photons, is optically thick in the lines as seen in the spectra, and is optically thin to visible-light continuum photons that come either from downscattered X-rays or from the shock-heated ejecta. Calculations show that this scenario is self-consistent., Comment: 36 pages, 23 figures. Comments are welcome ido.irani@weizmann.ac.il or idoirani@gmail.com. Accepted to ApJ

We analyze a scheme that uses quantum nondemolition measurements to induce squeezing of a spinor Bose-Einstein condensate in a double well trap. In a previous paper [Ilo-Okeke et al. Phys. Rev. A \textbf{104}, 053324 (2021)], we introduced a model to solve exactly the wavefunction for all atom-light interaction times. Here, we perform approximations for the short interaction time regime, which is relevant for producing squeezing. Our approach uses a Holstein-Primakoff approximation for the atoms while we treat the light variables exactly. It allows us to show that the measurement induces correlations within the condensate, which manifest in the state of the condensate as a superposition of even parity states. In the long interaction time regime, our methods allow us to identify the mechanism for loss of correlation. We derive simple expressions for the variances of atomic spin variables conditioned on the measurement outcome. We find that the results agree with the exact solution in the short interaction time regime. Additionally, we show that the expressions are the sum of the variances of the atoms and the measurement. Beyond the short interaction time regime, our scheme agrees qualitatively with the exact solution for the spin variable that couples to light.