Follow the photons from atom to threshold.
Single-atom imaging is not one number. Work through the chain in order: how many photons the atom scatters, how many your optics collect, how camera noise broadens the distributions, and how much the bright/dark histograms overlap.
Choose species, detuning, saturation, and exposure to compute the scattering rate.
Set NA, transmission, QE, and camera model to turn scattered photons into detected signal.
See the combined signal-to-noise ratio update live from your current inputs.
See how SNR grows with exposure and where the SNR=10 and SNR=50 targets fall.
Learn what the bright/dark peaks and their overlap mean for detection fidelity.
Choose a threshold and inspect the false-positive/false-negative tails directly.
See the underlying scattering-rate, SNR, and fidelity equations this tool is built on.
More NA helps twice
Sentence: Larger NA increases collected photons without increasing atom heating.
ηcoll = (1 - cosθ)/2
Failure mode: assuming higher camera gain improves photon collection.
Thresholds are physics
Sentence: The optimal threshold trades false positives against false negatives.
F = 1 - (P0→1 + P1→0)/2
Failure mode: quoting SNR when state loss dominates.
Survival is separate
Sentence: A measurement can identify an atom correctly while still ejecting it.
Freadout ≠ Psurvival
Failure mode: multiplying every error into one unlabeled number.
01 Imaging Beam
Atom & Imaging Transition
Two-level scattering rate
On resonance (Δ=0) a two-level atom saturates at Γ/2. Off resonance the Lorentzian denominator reduces the rate:
02 Collection & Camera
Objective & Optics
Camera Noise Model
03 Results
Noise Budget
| Source | σ² (photons²) | Fraction | Contribution |
|---|---|---|---|
| Set parameters above to compute noise budget. | |||
04 SNR vs Exposure Time
SNR as a function of imaging duration
05 Imaging Histogram & Detection Fidelity
What is the imaging histogram?
Run your experiment ~500–2000 times. Each shot, sum the camera electrons over the region of interest (typically a 3–7 px ROI around one tweezer site). Plot the distribution of those counts. You get a bimodal histogram: a narrow "dark" peak when the site is empty (only background scatter + camera noise), and a broader "bright" peak when an atom is present (atom fluorescence + background). The overlap between the two peaks is the fundamental limit on how well you can distinguish atom present from atom absent, this is your imaging fidelity.
Anatomy of a single-atom imaging histogram
Peak models
For $N_{\rm det} \gg 1$ detected photons, both peaks are well-approximated by Gaussians (Central Limit Theorem on Poisson photon statistics + Gaussian read noise):
Why σ_bright > σ_dark
In the shot-noise-dominated model, noise grows as $\sqrt{N}$. More photons in the bright peak → larger absolute fluctuations → the bright peak is wider than the dark peak. This asymmetry means the optimal threshold is not exactly at the midpoint — it shifts slightly toward the brighter side to equalize miss and false-alarm probabilities.
Error probabilities & fidelity
Given threshold $\theta$, define two error types:
$P_{\rm false}(\theta) = 1 - \Phi\!\left(\frac{\theta - \mu_{\rm dark}}{\sigma_{\rm dark}}\right) \quad$ [dark peak above θ]
$$\mathcal{F}_{\rm detect} = 1 - \frac{P_{\rm miss} + P_{\rm false}}{2}$$ Optimal threshold: minimise $P_{\rm miss} + P_{\rm false}$.
Equal $\sigma$: $\theta_{\rm opt} = (\mu_{\rm bright} + \mu_{\rm dark})/2$.
Unequal $\sigma$: solve numerically.
For equal peak widths and ${\rm SNR} = (\mu_{\rm bright}-\mu_{\rm dark})/\sigma$:
$\mathcal{F} \approx \Phi\!\left(\tfrac{\rm SNR}{2}\right)$. With unequal widths, compute $P_{\rm miss}$ and $P_{\rm false}$ separately.
Interactive Histogram Simulator
Alkali (⁸⁷Rb, ¹³³Cs)
Imaging transition: D2 cycling transition (780 nm for Rb, 852 nm for Cs). Closed cycling transition with repumper keeps the atom in the imaging cycle. R_sc → Γ/2 at saturation (~3 MHz for Rb87).
Photon counts: With NA=0.5, η≈5%, 5ms exposure → ~750 detected photons. σ_bright ≈ 40e⁻ (EMCCD with excess noise). Well-separated from dark peak (~50 e⁻).
Main challenge: Photon recoil heating (~181 nK per one-way recoil for Rb87; roughly twice that per scatter cycle without cooling). At R_sc = 3×10⁶/s, the atom heats by ~1 mK/s. Limit imaging to <10 ms or use concurrent Sisyphus/gray molasses cooling (recapture imaging, EIT imaging).
Background: Stray light from near-resonant beams contributes to dark peak. Typical dark peak: μ_dark = 50–200 e⁻ depending on background suppression.
Alkaline-earth-like (¹⁷¹Yb, ⁸⁸Sr)
Imaging transition: Broad ¹P₁ transition (399 nm for Yb, 461 nm for Sr, Γ/2π = 29/32 MHz). Very high R_sc ≥ Γ/2 at low saturation. Alternative: narrow ³P₁ (556 nm Yb, 689 nm Sr) for gentler imaging.
J=0 ground state advantage: Qubit states (nuclear spin) are largely spectator degrees of freedom for electronic cycling transitions such as ¹S₀ → ¹P₁. Atoms in the metastable ³P₀ state are invisible to 399/461 nm, enabling shelved (non-destructive) readout of one qubit basis while leaving the other undisturbed.
Magic wavelength: 759 nm tweezer is magic for ¹S₀/³P₀, no differential light shift on the qubit. The imaging beam (399 nm) shifts ¹S₀ but not ³P₀, so imaging the un-shelved population gives state discrimination.
Histogram: Narrower dark peak (lower background at UV wavelengths far from tweezer), tight bright peak. Excellent separation even at 1ms exposure. Atom Computing Yb171: reports >99.9% imaging fidelity.
Two-Gaussian fit procedure
Given your measured histogram H(n), fit the sum of two Gaussians:
Quality check: is $A_0/A_1 \approx$ expected loading fraction? Is $\sigma_{\rm bright}/\sigma_{\rm dark} \approx \sqrt{\mu_{\rm bright}/\mu_{\rm dark}}$? (this tests whether shot noise dominates as expected)
When peaks are NOT clean Gaussians
Real histograms show deviations from two clean Gaussians in several situations:
- Atom loss during imaging: Some fraction of bright shots become mid-sequence dark → bright peak develops a low-side tail. Fix: reduce imaging power or duration, or use concurrent cooling to prevent loss.
- Multiple occupancy: If two atoms load into one site, you get a third peak at ~2×μ_bright. Set loading conditions (MOT density, tweezer depth) to suppress. Good diagnostic: multi-peak structure at integer multiples of μ_bright.
- Background drift: If background changes shot-to-shot (laser power fluctuations), both peaks broaden. σ_measured² = σ_intrinsic² + σ_drift². Use real-time background subtraction.
- Non-Poissonian photon statistics: Sub-Poissonian emission (from e.g. shelving protocols) or super-Poissonian (from intensity noise) → peak widths deviate from $\sqrt{\mu}$ scaling. Check: plot σ² vs μ, should be linear with slope 1 (Poisson) or slope F² (EMCCD).
- Charge-transfer smear (EMCCD): Very fast pixel readout + high scattering rates → charge smear across pixels. Solved by choosing appropriate readout speed or using sCMOS.
Bootstrap fidelity uncertainty
After fitting, estimate statistical uncertainty via bootstrapping: resample your histogram N_shots times with replacement, refit, and take the std of the fidelity distribution. Typical uncertainty: δF ≈ 0.01–0.1% for 1000 shots, 0.001–0.01% for 10,000 shots. Note: systematic error from threshold choice is usually larger than statistical error for well-separated peaks.
Definitions used in neutral-atom QC papers
Different papers define "fidelity" slightly differently. For qubit experiments, the total SPAM (state preparation and measurement) error bundles several contributions:
$\varepsilon_{\rm meas} = (P_{\rm miss} + P_{\rm false})/2$, from histogram overlap [what this section computes]
$\varepsilon_{\rm prep}$, optical pumping error, atom number fluctuations, tweezer loading fraction
$\mathcal{F}_{\rm SPAM} = 1 - \varepsilon_{\rm SPAM} \approx 1 - \varepsilon_{\rm meas} - \varepsilon_{\rm prep}$
In Rb87 Harvard experiments: $\varepsilon_{\rm meas} \lesssim 0.1\%$, $\varepsilon_{\rm prep} \lesssim 0.2\%$ → $\mathcal{F}_{\rm SPAM} > 99.5\%$
Key Papers — Single-Atom Imaging & Detection
06 Fluorescence Histogram — Threshold Preview
Each frame adds simulated imaging shots drawn from the real Poisson distributions for your settings. Green = atom present (N_sig photons) · Gray = empty tweezer (N_bg background) · dashed = optimal threshold θ. Detection fidelity = 1 − P(miss) − P(false alarm).
07 Physics & Formulas
Geometric collection fraction η_geo
A high-NA objective collects photons into a cone defined by the half-angle θ_max = arcsin(NA/n), where n is the refractive index of the medium (n=1 in air/vacuum).
Note that isotropic emission into 4π steradians means even NA=0.95 captures only ~26% of photons geometrically. High-NA objectives (0.5–0.85) are typical for tweezer experiments.
Noise variance per pixel per frame
Each camera type has a distinct noise model. The total variance σ²_total determines the denominator of SNR = N_sig / σ_total.
where $G$ = EM gain, $\sigma_{\rm read}$ = read noise (e$^-$ RMS)
For EMCCD the read noise is divided by EM gain G, making it negligible at high gain (G ≳ 50). The penalty is the √2 excess noise factor that increases shot noise variance by 2×. For sCMOS cameras (1–2 e⁻ read noise), read noise is the dominant term at low photon counts.
Binary atom detection
We distinguish "atom present" (Poisson mean μsig) from "no atom" (Poisson mean μbg) by choosing a threshold θ. The optimal threshold minimises total error probability.
Detection fidelity: $\mathcal{F} = 1 - (P_{\rm miss} + P_{\rm false})/2$.
If the bright and dark peaks have equal variance and ${\rm SNR}=(\mu_{\rm sig}-\mu_{\rm bg})/\sigma$, then $\mathcal{F}\approx\Phi({\rm SNR}/2)$. The calculator below uses the bright and dark variances separately.
For SNR ≥ 10, fidelity exceeds 99.9%. The key insight is that fidelity scales as erfc(SNR), doubling SNR dramatically reduces error. Background suppression (e.g. dark-field or EIT imaging) improves fidelity by reducing μ_bg without reducing μ_sig.
Photon recoil vs trap depth
Every scattered photon imparts a recoil kick ℏk. At scattering rate R_sc the atom heats at rate dE/dt = E_rec × R_sc (in a σ⁺/σ⁻ Sisyphus beam, this is partially cancelled; in a σ beam near resonance it accumulates).
Heating rate: $\dot{T} \approx 2E_{\rm rec} R_{\rm sc}/k_{\rm B}$ [absorption + spontaneous emission, no cooling]
Max imaging time ($U_0$ = trap depth): $t_{\rm max} \sim U_0/(E_{\rm rec} R_{\rm sc})$
Example: $U_0/k_{\rm B} = 1$ mK, $R_{\rm sc} = 50$ kHz $\Rightarrow t_{\rm max} \approx 55$ ms
In practice, tweezer imaging uses repump + imaging beams in a gray molasses or Λ-enhanced dark SPOT configuration to scatter >1000 photons without losing the atom. This pushes fidelity above 99.5% in optimized tweezer experiments (Bergamini 2004; Blodgett et al. 2023; Manetsch et al. 2025).