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Sandbox Physics

Q008 · Split / recombine / count

Atom Interferometer Gravimeter

Split, redirect and recombine an atomic wave. Scan the final laser phase, fit the recorded counts and see why a longer wait increases both sensitivity and ambiguity.

Interactive modelAtom Interferometer Gravimeter
Signal phase · model00
Visibility · model11

02 / COUNT AT THIS SETTING

Which internal state returns?

Initial state · A

0

Transferred state · B

0

Not detected

0

Latest 60 trials at this setting. Blue: A · amber: B · grey: missed. Each trial starts with a fresh particle.

Fraction in A · detected trials

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Pointwise 95% Wilson interval. Misses are retained in the raw records; the fraction conditions on detection.

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03 / READ THE FRINGE

The fringe moves. Gravity leaves a phase.

Fraction in A versus scan setting ϕL/π\phi_L/\pi

Blue points: recorded counts. Thin lines: Wilson intervals. Amber: model. Dashed blue: a fixed-period harmonic fit to the records. Click the chart to choose a setting.

Estimated from the records

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Equivalent gravity above reference

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Only one representative of a periodic family. A single fringe cannot select the absolute branch.

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At least eight settings with ten detections each are needed. An unresolved fringe gives no phase estimate. Errors use a conservative variance bound and a local approximation, not an instrument accuracy claim.

04 / CONNECT THE PATHS TO THE PHASE

Three pulses close the two branches.

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Time runs to the right; height runs upward. Both centers move on the same vertical axis in the apparatus. Their separation is enlarged 12 times around their mean; the readout reports the actual separation. Dotted lines mark the three pulses.

The first pulse splits, the middle pulse exchanges momenta, and the last recombines. State-selective detection counts the two internal outputs at one collection region.

05 / KEEP THE EVIDENCE

Save, change one thing, compare.

Up to eight frozen samples. Saving freezes the run; the next acquisition starts a new run. Export includes active and saved trials with all preparation settings.

Model, conventions & sources
PA=12[1+C0e−σϕ2/2cos⁡Δϕ],PB=1−PAP_A=\tfrac12[1+C_0e^{-\sigma_\phi^2/2}\cos\Delta\phi],\quad P_B=1-P_A
Δϕ=keff(g−gc)T2+ϕL\Delta\phi=k_{\rm eff}(g-g_c)T^2+\phi_L

Downward is positive for gravity and effective wavevector. Nominal rubidium-87, wavelength 780 nm, twice the optical wavevector, reference gravity 9.81 metres per second squared. The launch velocity is chosen to return near the starting height after the sequence. Uniform gravity, ideal instantaneous pulses and independent Gaussian phase noise. Finite pulse duration, correlated vibration, gradients and wavefront errors are omitted.

McGuirk et al. · 2001

Noise is averaged analytically before independent binary sampling. Detection loss is equal for both states and is recorded without a hidden outcome. The harmonic regression fixes the known period; it does not measure that period. It is unconstrained and descriptive, so finite samples can give fitted contrast above one.

Physics tutorial

Atom Interferometer Gravimeter

BackgroundA controlled phase becomes a countable output imbalance.

Why it mattersA single count is random. A recorded scan reveals a reproducible fringe.

Start with the essentials

Focus question
Measure a fall with three flashes of light.
One-sentence intuition
Compare settings, retain raw records, and fit only the recorded counts.

Core mathematical model

Acceleration phase

Δϕ=(keffg−α)T2+ϕ3\Delta\phi=(k_{\rm eff}g-\alpha)T^2+\phi_3

Downward acceleration and effective wavevector are positive. Alpha is the angular-frequency chirp, defined as effective wavevector times the reference-plus-offset chirp acceleration.

Pulse geometry

vr=ℏkeff/m,dmax⁡=vrTv_r=\hbar k_{\rm eff}/m,\qquad d_{\max}=v_rT

The first pulse adds recoil, the middle pulse exchanges it, and the final pulse closes the centers. Separation is vertical in the apparatus.

Ambiguity interval

Δgperiod=2πkeffT2\Delta g_{\rm period}=\frac{2\pi}{k_{\rm eff}T^2}

The fitted gravity offset is one representative modulo this interval. Choosing the correct fringe requires additional information.

Laser phase finite difference

ϕL(0)−2ϕL(T)+ϕL(2T)=(keffg−α)T2\phi_L(0)-2\phi_L(T)+\phi_L(2T)=(k_{\rm eff}g-\alpha)T^2

The constant and initial-velocity terms cancel for uniform acceleration and instantaneous pulses. Real vibration, finite pulses and wavefront effects are omitted.

Loss and phase stability

C=C0e−σϕ2/2,P∅=1−ηC=C_0e^{-\sigma_\phi^2/2},\quad P_{\varnothing}=1-\eta

Independent Gaussian phase noise is averaged analytically. Equal loss reduces accepted counts, not conditional visibility.

Common difficulties

A path is not a trajectory

Typical misconceptionThe colored lines reveal the route taken by each detected particle.

Better mental modelThey show coherent branches or wavepacket centers; no which-path record is generated.

An ideal instrument

Typical misconceptionA narrow fit error establishes real instrument accuracy.

Better mental modelThe stated uncertainty is statistical within this model. Omitted apparatus effects and ambiguity remain.

Run the experiment

  1. 01

    Predict

    Try the first two presets and predict which exit gains counts.

    What to observe: The presets set conditions but do not secretly acquire data.
  2. 02

    Measure

    Detect once, then collect 200 trials. Scan all 21 positions.

    What to observe: Misses are retained; the frequency and its pointwise Wilson interval use accepted detections.
  3. 03

    Compare

    Save the scan, change coherence or pulse spacing, and scan again. Export both runs.

    What to observe: Low contrast suppresses the phase estimate; more time narrows the gravity ambiguity interval.