Absorption Imaging Lab

The standard OD formula $-\ln(I_{\rm out}/I_{\rm in})$ underestimates atom number when $I/I_{\rm sat} \gtrsim 0.1$, because saturation reduces the absorption cross section. The corrected formula adds back the "consumed photons" term:

This is exact within the two-level model. In practice, use $I/I_{\rm sat} \sim 0.1$–$0.5$ to balance photon count (SNR) against saturation distortion. At $I = I_{\rm sat}$ uncorrected OD can be badly biased unless the saturation term is included; at $I = 0.1\,I_{\rm sat}$ the correction is usually small but still worth tracking. If you can't apply the correction (e.g., unknown $I_{\rm sat}$ for mixed states), use $I \ll I_{\rm sat}$.

For high-field imaging (Paschen-Back regime, B > 100 G), the cycling transition breaks down and the effective cross section must be computed from the actual magnetic sublevel populations. Imaging at the stretched states $|F, m_F=\pm F\rangle$ restores a near-cycling cross section.

At large probe photon counts (N_ph > 1000/pixel), shot noise $\delta\text{OD} = \sqrt{2/N_{\rm ph}}$ becomes negligible — but OD noise floors around 0.01–0.1 remain due to imaging fringes: interference patterns from dust, optical surfaces, or air currents that shift slightly between the probe and reference frames.

PCA / eigenface removal: take a library of reference images (no atoms), decompose into principal components, and subtract the best fit from each probe image. This suppresses common fringe modes, reducing effective noise to ~0.002–0.01 OD in well-optimized setups. Implementation: ref_lib @ pinv(ref_lib) @ probe gives the optimal projection.

Practical tips: take probe and reference within the same camera frame (jitter-limited by vibration, not drift), enclose beam path, use antireflection-coated optics, keep the probe pulse < 50 μs to freeze air fluctuations, and take the reference frame immediately (within 1 ms) after the probe frame.

For single atoms in tweezers, absorption imaging is typically not used — the OD of one atom in a ~μm² area is OD ~ 10⁻⁴ to 10⁻³, well below any noise floor. Instead, fluorescence imaging (collecting photons scattered by the atom during a detection pulse) is the standard, giving ~50–200 detected photons per atom and a detection fidelity > 99%.

Absorption imaging is most powerful for ensemble experiments: MOTs (10⁶–10⁹ atoms), BECs (10³–10⁶ atoms), and Fermi gases, where the column density OD is 0.1–5 across a ~10–100 μm cloud. For atom numbers below ~500 in a MOT-scale trap, fluorescence wins even for ensembles.

• Reinaudi et al. (2007) — "Strong saturation absorption imaging of dense clouds of ultracold atoms." Opt. Lett. 32, 3143–3145. Derives the saturation-corrected OD formula used in this calculator. Essential reading before imaging any cloud with OD > 1 or I > 0.1 I_sat. DOI →
• Ockeloen et al. (2010) — "Detection of small atom numbers through image processing." Phys. Rev. A 82, 061606(R). Introduces PCA/eigenface fringe removal for absorption images. Demonstrates detection of ~10 atoms in a BEC with standard CCD cameras. Includes MATLAB code outline. DOI →
• Hueck et al. (2017) — "Calibrating high intensity absorption imaging of ultracold atoms." Rev. Sci. Instrum. 88, 123702. Rigorous treatment of saturation correction beyond the two-level model, including sublevel populations and polarization effects. Provides calibration procedure for I_sat in the actual experimental setup. DOI →
• Schmidutz et al. (2014) — "Quantum Joule-Thomson Effect in a Saturated Homogeneous Bose Gas." Phys. Rev. Lett. 112, 040403. Appendix contains a concise, pedagogical summary of the three-image absorption technique and noise sources — useful as a quick reference alongside this calculator. DOI →
• Ketterle, Durfee & Stamper-Kurn (1999) — "Making, probing and understanding Bose-Einstein condensates." In: Bose-Einstein Condensation in Atomic Gases, Proc. Int. School of Physics "Enrico Fermi." arXiv:cond-mat/9904034. Sections 3–5 give a complete treatment of absorption imaging, cross sections, and cloud reconstruction for BEC experiments. The standard reference for BEC groups. arXiv →

• Steck atomic data pages (steck.us/alkalidata) — comprehensive tables of σ₀, I_sat, hyperfine A/B coefficients, transition dipole matrix elements, and Clebsch-Gordan factors for the cycling transition. Values used in this calculator are taken directly from these tables.
• Steck, "Rubidium-87 D Line Data" (2019) — steck.us/alkalidata/rubidium87numbers.pdf — Table 7 gives cycling-transition σ₀ = 2.907×10⁻¹³ m² and I_sat = 1.669 mW/cm² for the F=2→F'=3 D2 line used in this calculator.
• NIST Atomic Spectra Database (physics.nist.gov/asd) — Einstein A-coefficients and oscillator strengths for transitions not covered by Steck (e.g., Sr, Yb, Dy, Er).
• Loftus et al. (2004) — "Narrow line cooling and momentum-space crystals." Phys. Rev. A 70, 031401(R). Contains saturation intensity and cross section data for the Sr ¹S₀→³P₁ intercombination line, useful if imaging Sr on the narrow line. DOI →

• Metcalf & van der Straten, Laser Cooling and Trapping (1999) — Springer. Chapter 2 derives the resonant cross section σ₀ = 3λ²/2π from the oscillator model and discusses the role of degeneracy, polarization, and hyperfine pumping in real atoms.
• Foot, Atomic Physics (2005) — Oxford University Press. Chapter 8 "Doppler cooling" and Chapter 9 contain derivations of the scattering force, relevant for understanding the probe-atom interaction during the imaging pulse.
• Pethick & Smith, Bose-Einstein Condensation in Dilute Gases (2002, 2008) — Cambridge University Press. Appendix covers absorption imaging as a diagnostic tool for BEC density profiles.
• Inguscio & Fallani, Atomic Physics: Precise Measurements and Ultracold Matter (2013) — Oxford University Press. Chapter 4 covers time-of-flight expansion and absorption imaging together, useful for connecting this calculator to the TOF thermometry tool.