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 Fluorescence Histogram — Single-Atom Detection
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).
05 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% (Bergamini 2004; Liu 2019).
06 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
Shot noise grows as $\sqrt{N}$. More photons in the bright peak → larger absolute fluctuations → the bright peak is always 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: $\varepsilon_{\rm meas} \lesssim 0.1\%$, $\varepsilon_{\rm prep} \lesssim 0.2\%$ → $\mathcal{F}_{\rm SPAM} > 99.5\%$