diff --git "a/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt" "b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt" @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf,len=970 +page_content='High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Milner,1, ∗ Lingfeng Yan,1 Ross B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson,1 Christian Sanner,1 and Jun Ye1, † 1JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA We report on the observation of a high-density, band insulating state in a three-dimensional optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Filled with a nuclear-spin polarized degenerate Fermi gas of 87Sr, the 3D lattice has one atom per site in the ground motional state, thus guarding against frequency shifts due to contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At this high density where the average distance between atoms is comparable to the probe wavelength, standard imaging techniques suffer from large systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To spatially probe frequency shifts in the clock and measure thermodynamic properties of this system, accurate imaging techniques at high optical depths are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using a combination of highly saturated fluorescence and absorption imaging, we confirm the density distribution in our 3D optical lattice in agreement with a single spin band insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining our clock platform with this high filling fraction opens the door to studying new classes of long-lived, many-body states arising from dipolar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Optical lattice clocks integrate quantum many-body physics and precision metrology to achieve state-of-the-art measurement precision [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To advance clock performance, one wishes to probe as many atoms as feasible for the longest possible coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Improvements in both precision and accuracy of optical lattice clocks, with increased atom num- bers, have been enabled by the development of high-fidelity, microscopic imaging of the atomic cloud to spatially resolve clock shifts [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The combination of high density and long coherence time will allow characterization of novel systematic effects such as that arising from dipolar interactions between atoms on neighboring lattice sites [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lattice thermom- etry [12] and studies of novel physics such as SU(N) mag- netism [13, 14] will also benefit from accurate imaging at high density where these phenomena emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To optimize atom number while minimizing interaction- related dephasing, a clock platform based on a 3D lattice geometry and Fermi-degenerate matter has been developed [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following nuclear spin polarization [16, 17], the Pauli exclusion principle mandates there is at most one atom per lat- tice site in the ground motional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To ensure this ground state motional occupation during lattice loading we operate with kBT < kBTF < ℏωbg, where T, TF , ℏωbg refers to the atomic temperature, Fermi temperature and lattice bandgap respectively [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At the highest density affordable with one fermion per lattice site, this system realizes an insulating state of matter where tunnelling is suppressed [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining this high-density system with spin-orbit coupling generated from clock addressing will enable exploring cluster state gen- eration and tunable spin models [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Differential frequency shifts across the optical lattice en- coding potential systematic effects can be spatially resolved by combining in situ imaging and narrow-line clock spec- troscopy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract these frequency shifts, two subse- quent images of the ground and excited state density distribu- tions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus for our clock platform, accurate in situ imaging at high density is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In our lattice where the average distance between atoms (407 nm) is comparable to the probe wavelength (461 nm), imaging with a weak, reso- nant probe is strongly perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Both collective effects medi- ated by dipolar interactions [22] and systematic defects such as lensing of the probe beam [23, 24] introduce errors to the reconstructed density distribution at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To mitigate these systematic effects, different techniques can be used to reduce the absorption cross section and make the cloud ”optically thin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' These techniques can be broadly divided into two categories: dispersive imaging at large de- tuning from resonance [25–27] and saturated imaging at high intensity [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For dispersive imaging extracting informa- tion about the atomic density often requires spatially filtering the scattered and unscattered light in the Fourier plane of the imaging system, demanding precise fabrication and alignment of custom optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Additionally, careful studies of dispersive imaging show that residual systematic effects at finite detun- ing are non-negligible and can be addressed using differential measurement schemes at opposite detuning [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To address these imaging errors in this work, we use both highly saturated fluorescence and absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this Letter, we report on the observation of a band in- sulating state in our 3D optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using highly saturated imaging to mitigate imaging errors, with a satura- tion parameter far greater than the optical depth, we accu- rately confirm the density distribution in our 3D optical lattice in good agreement with thermodynamic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We ex- tend previous work using high intensity fluorescence imaging [28], confirming the accuracy of this imaging technique in a new high density regime with a degenerate Fermi gas of 87Sr [16, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' With atomic densities as high as 6×1014 atoms/cm3, we observe systematic agreement with atom counts obtained via time-of-flight absorption imaging and identify the range where the extracted atomic density distribution is not blurred by our imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Our high intensity imaging scheme is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The combination of atomic level structure and relatively large mass of 87Sr is particularly well suited for our imaging tech- nique, providing a cycling transition with a large scattering rate while avoiding significant motional effects from the imag- ing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The transition from 1S0 to 1P1 with linewidth Γ = 2π × 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 MHz provides a large photon scattering rate with minimal depumping to dark states during the imaging time [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' During a 1 µs pulse at full saturation about 100 photons per atom are scattered and the atoms accelerate at a = ℏkΓ 2m where k is the imaging light wavenumber and m is the atomic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The net momentum transfer amounts to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='03343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='atom-ph] 9 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Schematic of our clock platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Vertical and horizontal imaging systems with numerical apertures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 respectively provide measurements of the 2D density distribution ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Accounting for the lattice spacing a = 407 nm, ˜na2 is determined from highly saturated absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To mitigate imaging errors, the atoms are highly saturated and each scatters photons with a maximum rate of Γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Measurements from our high resolution imaging system in- tegrated along gravity are presented in panel (a), where the density distribution is extracted for thermodynamic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Images from the horizontal imaging system in panel (b) are just used to determine our atom cloud aspect ratio for our inverse Abel transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' a Doppler shift of kaτ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='8 MHz which is much less than the transition linewidth Γ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Finally, the linear displace- ment for a 1 µs pulse at full saturation is just aτ 2 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This linear displacement and corresponding Doppler shift can be largely cancelled in fluorescence imaging by retroreflect- ing the incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The spread in transverse position due to random momentum transfer from spontaneous emission is ℏk 6mt3/2� Γ/2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 µm over a 1 µs pulse duration and small compared to our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm imaging resolution [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using highly saturated absorption imaging, we measure the column density distribution ˜n in our optical lattice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ac- counting for the lattice spacing a = 407 nm corresponding to the 87Sr magic wavelength at 813 nm, the scaled column density ˜na2 is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Saturated absorption and fluorescence imaging are benefi- cial in comparison to standard imaging techniques in a num- ber of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this highly saturated regime the scattering rate is largely immune to beam intensity, frequency, and pointing fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the saturation intensity Isat = 40 mW/cm2 for the imaging transition, a Gaussian probe beam with 20 mW of optical power and a 100 µm waist corresponds to a peak intensity of I ∼ 3000 Isat, within the typical constraints of a standard imaging laser system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given that the probe beam is attenuated through the atom cloud, a saturation parameter I/Isat much greater than the optical depth is required to fully saturate the imaging transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note parallels between flu- orescence and absorption imaging at high saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In both cases, the extracted atom number is determined by a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging, this corresponds to the number of collected photons per atom and for saturated ab- sorption imaging the number of missing photons per atom in the probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, both fluorescence and saturated ab- sorption imaging can be calibrated via a single absolute atom number measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For images in our 3D lattice, we de- termine our atom number via clock excitation fraction fluctu- ations arising from quantum projection noise (QPN) [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging, only a single image of collected fluorescence in an arbitrary direction is required, minimizing fringing and simplifying image processing substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Flu- orescence imaging also avoids limited dynamic range issues suffered from high intensity absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Strategies such as multiple measurements at varying intensity to deter- mine the atomic density in different regions of the cloud may be taken to confront this issue [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The primary disad- vantage of fluorescence imaging in comparison to absorption imaging is that the signal-to-noise is generally worse [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To optimize signal-to-noise ratio (SNR) in fluorescence imaging, the photon collection efficiency and therefore the numerical aperture (NA) of the imaging system, must be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In our experiment, the vertical and horizontal imaging systems have numerical apertures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, corresponding to col- lection efficiencies of approximately 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Al- ternatively, if spatial resolution is not required then the pulse duration can be extended enhancing the number of detected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To motivate the development of our high intensity imaging technique, systematic errors associated with standard in situ imaging techniques at high density are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging at I ∼ Isat and high intensity fluorescence imaging are presented side-by-side for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To study these systematic errors at high density, we prepare a sample with optical depth > 200 by producing a degenerate Fermi gas with 10 nuclear spin components, ≈ 2 × 105 atoms and a T/TF of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 in a crossed dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The errors associated with low intensity absorption imaging can be seen twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' First, the reconstructed optical depth from absorp- tion detection in the upper left panel is far too low, two orders of magnitude less than the expected value of ∼ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This erroneously low optical depth is attributed to effects such as enhanced forward emission and lensing of probe light [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Secondly, the reconstructed optical depth in the upper right panel increases after a 500 µs time-of-flight expansion con- clusively demonstrating the density dependence of these ob- served systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In comparison, saturated fluorescence imaging yields a far larger reconstructed optical depth and diffuses following ex- pansion as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We compare this reconstructed 2D den- sity distribution with the expected distribution corresponding to a Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using independently measured experimental values, we calculate this distribution with no free parameters [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total atom number and reduced temperature T/TF are determined from time-of-flight absorption imaging at low density with an optical density ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The trapping frequencies are extracted from parametric confinement modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using 皖A·23 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A comparison of high intensity fluorescence and standard ab- sorption imaging (I ∼ Isat) at optical depths exceeding 200 in our highly degenerate Fermi gas is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In situ absorption imaging at low intensity yields strikingly erroneous measurements at high den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The calculated 2D Fermi gas distribution according to our ex- perimental parameters is shared for comparison in qualitative agree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' these parameters, we calculate both an in situ and 500 µs time- of-flight Fermi gas profile for comparison with our measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We observe qualitative agreement between measure- ment and calculation at these extremely high optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Intrigued by the measurements presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2, we un- dertake a quantitative study on the fidelity of our saturated imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We present a calibration method for flu- orescence detection, using the total number of collected fluo- rescence photons for comparison with an accurate atom num- ber reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging at low density following time-of-flight expansion serves as an appropriate calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following expansion for 7 ms, the optical depth is ∼ 1 and systematic imaging errors can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To inde- pendently calibrate the atom number in our 10 spin Fermi gas, we prepare a thermal sample and use measured density fluctuations to determine the effective absorption cross sec- tion [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a) we ensure this calibration shows systematic agreement with atom numbers between approxi- mately 1 × 105 and 4 × 105, varied by increasing our final evaporation trap depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the 3 µs pulse duration used, the fitted calibration is in reasonable agreement with calculation using the measured quantum efficiency and imaging system numerical aperture [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To ensure that the imaging transition is fully saturated, the laser intensity at 1 µs pulse duration is increased until the collected photon number plateaus, as seen in the figure inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To perform accurate spatially resolved measurements, we must also determine the blurring induced by our imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Just as collective effects introduce errors to the re- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (a) Calibration method for in situ fluorescence detection us- ing atom counts from time-of-flight absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Collected photon counts from both the vertical and horizontal imaging systems are plotted, with solid and dashed lines representing fits to the hori- zontal and vertical measurements respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inset: Collected pho- ton count with vertical imaging system as a function of I/Isat at 1 µs pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (b) Peak column density as a function of fluo- rescence pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Measurements are normalized by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='9×1011 atoms/cm2, the column density at the shortest pulse duration of 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Images at 500 ns and 2 µs in inset are plotted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The error bars denote the standard error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' constructed density distribution, any systematic changes to ˜n introduced by our imaging pulse must be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To cal- ibrate this blurring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b), we extend the fluorescence pulse duration and examine the peak column density as atoms diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows averaged images from 500 ns and 2 µs pulse durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that we observe no atom loss or molecular formation over the full 2 µs range, confirmed by the detected photon count increasing linearly with pulse dura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To minimize blurring, we carefully retroreflect our probe beam by optimizing the backcoupled light through the probe optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At pulse durations up to 1 µs, we confirm that the peak column density decreases by < 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Motivated by the calibration reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3, we directly determine the 3D density distribution in a deep optical lattice via saturated in situ absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We form a cubic lat- tice with trap depths of approximately 60, 70, and 50 Er in three orthogonal directions, where Er is the lattice photon re- coil energy ≈ h × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following forced evaporation with 10 nuclear spin states we spin polarize using a focused beam detuned from the 3P1 intercombination line to form a state-dependent potential, removing nearly all but the mF = - 9/2 atoms [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Clock spectroscopy confirms ≈ 90% spin purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' An additional step of spin purification is applied by 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (a) The three-dimensional density distribution and the corre- sponding lattice filling fraction are determined from in situ absorp- tion image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) and the use of an inverse Abel transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (b) A linecut along z = 0 provides the data points in circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Errorbars are both the statistical uncertainty of the Abel transformation and atom number uncertainty added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We start with a pre- diction based on HTSE calculation, using independently measured values for the temperature, atom number, and harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The best fit to the data results in a 10% reduction of the measured as- pect ratio ωy/ωx and 5% reduction of the measured T/TF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The red line captures this fit, with temperature uncertainty in the shaded band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The blue dashed line is a fit to Gaussian in qualitative disagreement with na3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' coherently driving the mF = -9/2 atoms into the excited clock state and removing any residual spins with a resonant imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging directly provides us with the col- umn density distribution ˜n, integrated through the vertical axis along gravity as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Based on our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b) analysis, we choose a pulse duration of 1 µs to minimize blur- ring and a saturation intensity of 54(4), substantially larger than peak optical density of ∼ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To spatially probe the band insulator plateau we use an imaging magnification of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='8 to achieve an effective pixel size of 412 nm, roughly equal to the lattice constant a = 407 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that our effective pixel size is smaller than our optical resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm, thus our imaging system is optically oversampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract the 3D density distribution, we use an inverse Abel transform [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given our vertical imaging is not along an axis of cylindri- cal symmetry, n must be appropriately scaled by the aspect ratio of the spatial density distribution [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The aspect ra- tio is independently calibrated using the absorption imaging measurement in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At this high magnification, the SNR in fluorescence imag- ing for a 1 µs pulse duration is limited by a combination of read noise and photon shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We found that even after extensive averaging the extracted 3D density distribution us- ing an inverse Abel transform was sensitive to small fluctua- tions in ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, saturated absorption imaging with a superior SNR provides a more robust technique to characterize the 3D density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This extracted 3D density distribution is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To judge the fidelity of our measured 3D density distri- bution, we compare the line cut at z = 0 with calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To estimate the density distribution, we use a High Temperature Series Expansion (HTSE) calculation in the atomic limit [12, 14, 37, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The ingredients of this cal- culation include values for the atomic temperature, harmonic confinement, and total atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the density dis- tribution only depends on the ratio of the respective harmonic confinements, the measured aspect ratios from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1 are used for our HTSE calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total atom number N is de- termined from quantum projection noise measurements [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To estimate the temperature including heating during lattice loading, we measure the reduced temperature T/TF in time- of-flight after a round-trip from the lattice back to the dipole trap and determine an entropy-per-particle increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='25(6) kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inferring an entropy increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='13(3) kB in a single lattice loading sequence, we estimate a T/TF of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='165(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Although we did not perform a cross-dimensional thermaliza- tion measurement to directly verify thermal equilibrium, the uncertainty in our temperature is included in the shaded band of the HTSE calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b) [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that the extended plateau region is larger than our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm imaging res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To further quantify the imaging fidelity, we compare na3 to a Gaussian fit in clear disagreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In conclusion, we report on the observation of a spin- polarized, band insulating state in our 3D optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This has been enabled by characterizing saturated in situ imaging techniques to accurately determine our density dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Broadly, the saturated imaging techniques in this work will be applicable for studies of SU(N) magnetism and thermodynamics in the Mott-insulating regime [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' With the high filling fraction demonstrated in this work, many-body states arising from dipolar interactions can be generated be- tween atoms on neighboring lattice sites [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We thank D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kedar for maintain- ing the ultrastable clock laser used in this work and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Aep- pli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Miklos, and Y.' metadata={'source': 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Supplemental material to High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock Density diffusion Here we provide supplemental analysis to the data pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In panel A of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S1, we plot the in- tegrated counts along the x axis of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We see an asymmetry emerge along the direction of the probe beam as the pulse duration is extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This asymmetry suggests that the observed density diffusion may arise from inhomogeni- ety between the incident and retroreflected beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' While the power is certainly mismatched, this could also be due to ei- ther imperfect spatial alignment or mode mismatch given the divergence of the probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We also plot the total counts in each image as a function of pulse duration in panel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The linear character of the counts over the full pulse duration range suggests that we do not ob- serve appreciable atom loss or pumping to dark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The counts at each pulse duration are normalized to the counts at 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows the Gaussian RMS width of the cloud as a function of pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Signal-to-noise comparison In the main text of the paper we refer to both saturated ab- sorption and fluorescence imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We provide a quantita- tive comparison of the signal-to-noise ratio (SNR) between the two techniques here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We express our signal-to-noise for a detection pixel in terms of the normalized variance V(N)/N, where N denotes the number of atoms within the respective detection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging the SNR is simply determined by the shot noise associated with the number of detected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To calculate the total atom number, we first convert the fluorescence counts detected on our camera to the number of collected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Then, using the collection effi- ciency of our imaging system and scattering rate of our atomic transition we determine the conversion of detected photons per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' On our CCD camera, we measure na counts in a given pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using the quantum efficiency q of the imaging system, and the camera conversion gain g in units of counts per photo electron, we infer na qg photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At full saturation, the atomic scattering rate is Γ 2 and the number of photons scat- tered per atom is Psc = Γ 2 × τ, where τ is the pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Finally, we denote the collection efficiency as Y , determined by the numerical aperture of our imaging system and by ra- diation pattern anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining terms, the total atom number is N = na gqY Psc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using error propagation, we deter- mine the variance VF l(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VF l(N) = � ∂N ∂na �2 V(na) = � 1 gqY Psc �2 gna (1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Panel (a) shows the integrated counts from the images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b) of the main text along the x axis as a function of pulse du- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total counts at each pulse duration is plotted in panel (b), normalized by the counts at 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the detected photon count increases linearly with pulse duration, we observe minimal atom loss or molecular formation over the full 2 µs range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows the Gaussian RMS width of the cloud as a function of pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Here, we have used the fact that the distribution of gener- ated photo electrons ne is binomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, V(na) = V(g × ne) = g2V(ne) = g2ne = gna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining terms: VF l(N)/N = 1 qY Psc (2) The SNR associated with absorption imaging is more com- plicated given the formula for the atom number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3 has both logarithmic and linear terms and involves two images na and nb with and without atoms present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Here, A and σ0 refer to the effective pixel size accounting for the imaging system magnification and effective atomic absorption cross section, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Similar to fluorescence imaging, an appropriate error propagation of the na and nb terms determines Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We summarize the formulas here and point a reader to reference [1] for a full derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='03343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='atom-ph] 9 Jan 2023 2 N = A σ0 log( nb na ) + 2 Γτgq (nb − na) (3) VAbs(N) = g ˜A2( 1 na + 1 nb ) + g ˜B2(na + nb) + 4g ˜A ˜B (4) ˜A = A σ0 , ˜B = 2 qgτΓ (5) We compare the different techniques in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S2 using the experimentally relevant parameters for our imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In both cases, a 1 µs resonant pulse is used with a numerical aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and a quantum efficiency of 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the fluo- rescence SNR in blue, the transition is assumed to be fully sat- urated and scatters photons with a rate of Γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the I/Isat = ∼ 55 we use for our inverse Abel measurements, the SNR in absorption imaging is superior to fluorescence imaging in regions where the column density is higher than 2 atoms/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Particularly given our peak density of ˜na2 = ∼ 20 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a), absorption imaging provides a better SNR in the regions of high density where we extract our peak filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At a critical OD of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='17, fluorescence detection under our experi- mental parameters provides a superior SNR at all imaging in- tensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note these calculations neglect technical noise, in particular camera readout noise, which can be accounted for by offsetting V(na) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This contribution will disproportionately reduce the SNR of fluorescence imaging, as the fluorescence counts are substantially lower than the ab- sorption counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To probe fine spatial details in our atomic cloud, an imag- ing resolution smaller than the length scale of these spatial features is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To achieve this condition, a sufficiently large numerical aperture imaging system must be utilized and aberrations must be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this case, the imaging res- olution is fundamentally limited by diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We verified the diffraction-limited performance of our NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 objec- tive lens by propagating a point source at 461 nm through a test setup (including all imaging path optics and vacuum viewports) and measuring the point-spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' While absorption and fluorescence imaging rely on the same light scattering process (they only collect different parts of the scattered EM field [2]), the signal amplitudes for these two methods scale differently with the NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' When collect- ing fluorescence, the solid angle coverage of the imaging sys- tem proportionally affects the signal down to the lowest spa- tial frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is not the case for absorption imaging, where the amplitude of spatial frequency components below the NA-dependent bandwidth is constant as the NA is further increased (assuming the lens fully covers the probe beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In other words, for fluorescence imaging, most of the signal light gets collected in the outer ring fraction of the lens aperture, which renders it particularly susceptible to lens imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 0 20 40 60 80 100 I/Isat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 Var(N)/N Absorption beam intensity 1 s pulse duration, NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 Saturated fluorescence OD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='28 atoms/a2 OD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2, 2 atoms/a2 OD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, 5 atoms/a2 OD = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, 10 atoms/a2 OD = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2, 20 atoms/a2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' SNR comparison between absorption and fluorescence imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The relevant imaging parameters from the main figures of the paper are used for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For absorption imaging the atom count variance scales inversely proportional with intensity in the non-saturated limit I ≪ Isat, and proportional with intensity in the high saturation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The variance is for both imaging methods proportional to 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In the fully saturated regime (and assuming no technical noise) the normalized variance for fluorescence imaging is independent of atomic column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To avoid imaging defects at the high densities used in clock operation, an I/Isat > 50 was used in all imaging measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The black dashed line indicates the intensity used for our inverse Abel measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' HTSE calculation To accurately model the density distribution in our 3D lat- tice, we use a High Temperature Series Expansion (HTSE) calculation in the atomic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The general Hamiltonian for SU(N) symmetric fermions in a 3D lattice in the atomic limit takes the following form: HAL = U 2 � i,σ̸=σ′ ˆni,σˆni,σ′ + � i,σ Viˆni,σ (6) On a lattice site i, there are just two competing energy scales: an interaction energy U between particles and a po- sition dependent energy offset Vi according to the harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' By using the local density approximation µ = V (x, y, z)−µ0, where V (x, y, z) = 1 2m(ω2 xx2+ω2 yy2+ω2 zz2) and µ0 corresponds to the peak chemical potential in the lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the spin-polarized system in this work, U = 0 and the calculations are substantially simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ultimately, we want to express the density distribution n(µ, T, r) in terms of the chemical potential, atomic tempera- ture, and position in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' On a lattice site i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' we express the Grand partition function Z and Grand potential Ω : Z(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = N=1 � σ=0 �N σ � e−βµσ (7) 3 Ω = −kBTln(Z) From here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' we determine the entropy and occupancy per lattice site i: s(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = −∂Ω ∂T = kB ln(Z) + ∆s ∆s = kB Z βµe−βµ (8) n(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = −∂Ω ∂µ = 1 Z(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T)e−βµ (9) We accurately determine the total atom number Nlat from in situ absorption imaging and total entropy Slat via time-of- flight fitting to a non-interacting Fermi-Dirac profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Simi- larly, we express the entropy s and occupation n on a given lattice site using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 8 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 9 expressed in terms of T and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We then determine global fitting parameters T and µ0 to ensure the integrated entropy and occupancy over all lat- tice sites equals our experimentally measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' After determining µ0 and T to realize the equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 9, we cal- culate n(µ, T, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A linecut of n(µ, T, r) at z = 0 is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inverse Abel transform We outline our reconstruction procedure here using mea- surements of the atomic cloud aspect ratios and an inverse Abel transform: First, we use saturated absorption images along a vertical axis aligned with z and a horizontal axis aligned with x corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b) to determine the aspect ratios ωx/ωy and ωx/ωz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we perform an inverse Abel transform on the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) image to reconstruct an initial three-dimensional density dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given there is no axis of cylindrical symmetry in our system geometry, the reconstructed density from the in- verse Abel transform must be appropriately re-scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Treating our system as an ellipsoid with radii rx, ry, rz and N atoms the density is nlat = N/Vlat where Vlat = 4 3πrxryrz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We extract the inverse Abel transform for the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) image along the x axis, given the largest Band in- sulator plateau will occur along the axis with the weakest har- monic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The density distribution from this pro- cedure assumes a volume of VAbel = 4 3πrxrxry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus we scale the extracted density by nAbel/nlat = rz rx = ωz/ωx us- ing the measured aspect ratio from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given excess noise around the origin, the x = 0 point is interpolated with the neighboring point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This reconstruction proce- dure was cross-checked with simulated density distributions to ensure its fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The three-point Abel transform method was used for this work, which has been independently studied to verify its fidelity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' QPN calculation To calibrate our atom number, we analyze quantum projec- tion fluctuations using the narrow-linewidth clock transition between the 1S0 and 3P0 states in 87Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using a clock laser stabilized to our 8 mHz linewidth silicon reference cavity, ro- tation noise due to laser instability can be neglected in these measurements [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Additionally, fluctuations in total counts are < 2% and not a limiting systematic for determining the atom number calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Referenced in many texts [5], by preparing atoms in a superposition of 1S0 to 3P0 the variance V of the measured excitation fraction is related to the mean atom number ¯N and mean excitation ¯pe by: VQP N = ¯pe(1 − ¯pe) ¯N (10) To determine this variance, we do many subsequent mea- surements of pe under identical operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For a measurement i to determine pi e, two fluorescence counts ˜Ci g and ˜Ci e are read off a region of interest of our camera includ- ing our atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' These counts are subtracted by two averaged dark frames ¯Bg and ¯Be to yield Ci g = ˜Ci g− ¯Bg, Ci e = ˜Ci e− ¯Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We would like to determine the coefficient a that satisfies N i e = aCi e/τ, N i g = aCi g/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We can immediately see that the excitation fraction has no dependence on this coefficient: pi e = �aCi e �aCie + �aCig (11) However, the total atom number N i = a(Ci e + Ci g)/τ = aCi t/τ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 10, we see a measurement of the variance VQP N, the mean excitation ¯pe, and the mean total counts ¯Ct can determine a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VQP N = ¯pe(1 − ¯pe) a ¯Ct/τ (12) The coefficient a can be interpreted as the ”atoms per count per pulse duration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In principle, with knowledge of the quan- tum efficiency, gain, scattering rate, numerical aperture, and radiation pattern one could calculate this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Practically, assumptions about the radiation pattern based on the quanti- zation axis and probe light polarization make this calculation more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In practice, it is much more straightforward to directly measure a than to individually measure each of these values with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The observed variance of the excitation fraction Vpe has contributions from quantum projection noise (QPN), photon shot noise (PSN), and camera readout noise (RN): Vpe = VQP N + VP SN + VRN (13) Here g is the detector gain in units of counts per electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4 VP SN = ¯pe(1 − ¯pe) ¯Ct × g (14) VRN = R2 ¯Ct 2 (2¯p2 e − 2¯pe + 1) (15) VP SN can be understood intuitively considering the ratio VQP N/VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The number of signal electrons (equivalently the number of collected photons multiplied by the camera quantum efficiency) per atom determines the relative scaling of VQP N and VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VQP N VP SN = 1 g × a (16) 105 3 × 104 4 × 104 6 × 104 Ct (counts) 10 5 Var(pe) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Readout noise calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A π pulse on our optical clock transition is used so pe ≈ 1 and Vpe = R2 ¯ Ct2 + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We use 4 pulse durations between 5 and 20 µs to vary Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We fit R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 and C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='73 × 10−6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='02 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To determine a we need to accurately calibrate VRN and VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We see at pe = 1, VP SN, VQP N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, measur- ing Vpe at pe = 1 will independently determine VRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We wish to fit R and ensure it is consistent with the cameras specified readout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract this value, we use 4 pulse durations between 5 and 20 µs to vary Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In practice, we fit Vpe = R2 ¯Ct 2 + C (17) We fit R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 and C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='73 × 10−6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='02 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For our circular ROI there are X = 889 pixels in the masked radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the calibrated gain g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='59 counts/e- and readout noise r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='4 e- respectively , Rcalc = √Xgr = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='7 in agreement with R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that the gain and readout noise of the camera are close to specifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dark counts over our 30 ms exposure are < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 e- and considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we wish to determine aQP N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To do so, we perform a second measurement at pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The variance of this dataset contains contributions from VQP N, VP SN, and VRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Us- ing the measured R value, we subtract the VRN contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we fit the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S4 to: Vpe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5) a ¯Ct/τ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5) ¯Ct × g (18) We fit aQP N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is in reasonable agree- ment with the calculated value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='43 assuming Γ/2 scatter- ing into 4 π while also accounting for the measured quantum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 105 4 × 104 6 × 104 Ct (counts) 2 × 10 5 3 × 10 5 4 × 10 5 Var(pe) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' aQP N calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The atoms in our optical lattice are placed in a superposition of the ground and clock states with a π/2 pulse so pe ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 for these measurements and Vpe is fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We determine aQP N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' READOUT NOISE Here, we derive the readout noise term used in our variance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The expressions used are somewhat different than other literature, given that we use averaged dark frames ¯Be and ¯Bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Recall, pe = Ce Ce+Cg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To determine the readnoise contribution to the excitation fraction, we perform standard error propagation: VRN = � ∂pe ∂Ce �2 V(Ce) + � ∂pe ∂Cg �2 V(Cg) (19) Here, ∂pe ∂Cg = Ce (Ce + Cg)2 = pe (Ce + Cg) (20) 5 ∂pe ∂Ce = Cg (Ce + Cg)2 = 1 − pe (Ce + Cg) (21) To determine V(Ce) consider an X pixel region-of-interest for which we extract Cg, Ce in two separate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Each pixel contains r read noise in electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The single pixel read noise in units of counts is thus g × ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total noise in this region of interest is summed in quadrature pixel-by-pixel V(Cg), V(Ce) = � X (ri × g)2 = Xr2g2 = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Plugging terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 19: VRN = R2 ¯Ct 2 (2¯p2 e − 2¯pe + 1) (22) Imaging system parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a) In Table 1 is a summary of the imaging parameters for the measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4, a 1 µs pulse duration was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b), we vary the pulse length be- tween 500 ns and 2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Atom number fluctuations in time- of-flight absorption imaging for these measurements have a standard deviation less than 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Table 1 Vertical imaging system Numerical aperture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='23 Pulse duration 3 µs Total photons scattered per atom at full saturation 287 Collection efficiency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 % Camera quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='85 Imaging system quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='65 Calculated photon count per atom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='06 Measured photon count per atom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='91(1) Horizontal imaging system Numerical aperture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='10 Pulse duration 3 µs Total photons scattered per atom at full saturation 287 Collection efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='25 % Camera quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='78 Imaging system quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 Calculated photon count per atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='402 Measured photon count per atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='445(3) [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 118, 263202 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Itano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bergquist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bollinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gilligan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Heinzen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Moore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Raizen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wineland, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 47, 3554 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'}