Q008 · Split / recombine / count

Atom Interferometer Gravimeter

How can three laser pulses measure gravity with falling atoms? Three laser pulses split each atom's wave, steer the two halves and recombine them. Gravity adds a phase that grows with the square of the time between pulses: double the time and the phase grows fourfold. Scanning the last pulse's phase and fitting the output counts reads it off. Longer waits raise the sensitivity but make it harder to tell which fringe you are on.

Split an atomic matter wave, steer the two parts back together and recombine them. Scan the phase of the last laser pulse, fit the recorded counts, and see why a longer wait between pulses makes the measurement more sensitive but also more ambiguous.

Signal phase · model $0$

Visibility · model $1$

01 / Each particle takes two branches and is detected once

A longer wait leaves a larger phase.

Record a phase scan. Double the pulse separation and record again: the signal phase grows fourfold.

Vertical fountain · collinear Raman pulses · recoil separation enlarged 12 times

Final pulse phase

\phi_L/\pi=0

No counts yet. A scan records 100 trials at each of 21 settings.

Inspect the pulse sequence

Sequence progress · model only

t=T

Moving this preview does not collect data. Each “Send” runs the complete sequence.

Try a comparison

Pulse separation

T/\mathrm{ms}

80

Gravity above reference

\delta g/(\mathrm{\mu m\,s^{-2}})

20

Chirp-equivalent acceleration

\delta g_c/(\mathrm{\mu m\,s^{-2}})

0

Coherence, noise & detection

Coherence before recombination

C_0

1

Independent phase noise

\sigma_\phi/\mathrm{rad}

0.15

Detection probability

\eta

0.95

Sampling seed

s

2026

Turning the scan knob keeps the data. Changing the preparation starts a new sample, so save first if you want a comparison. Presets do not take data.

The model is an ideal short-pulse Raman interferometer in uniform gravity, with independent phase noise and the same detection loss for both outputs. Only the one-dimensional paths of the wave-packet centers are computed. The Lab makes no accuracy claim about finite pulses, vibration, wavefront errors or gravity gradients.

02 / Count at this setting

Which internal state returns?

Model reference

Initial state · A

Transferred state · B

Not detected

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

Fraction in A · detected trials

\text{—}

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

\text{—}

03 / Read the fringe

Gravity adds a phase, and the fringe moves.

Fraction in A versus scan setting $\phi_L/\pi$

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

\text{—}

Equivalent gravity above reference

\text{—}

This is only one value from a periodic family of solutions. A single fringe cannot pick the absolute branch.

\text{—}

At least eight settings with ten detections each are needed. An unresolved fringe gives no phase estimate. The errors come from a conservative variance bound and a local approximation; they do not describe instrument accuracy.

04 / Connect the paths to the phase

Three pulses bring the two branches back together.

\text{—}

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 / Save the evidence

Save a sample, change one setting, then 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

P_A=\tfrac12[1+C_0e^{-\sigma_\phi^2/2}\cos\Delta\phi],\quad P_B=1-P_A

\Delta\phi=k_{\rm eff}(g-g_c)T^2+\phi_L

Gravity and the effective wavevector are positive downward. The atoms are nominal rubidium-87 at a wavelength of 780 nm; the effective wavevector is twice the optical one, and the reference gravity is 9.81 metres per second squared. The launch velocity brings the atoms back near the starting height after the sequence. Gravity is uniform, the pulses are ideal and instantaneous, and the phase noise is independent and Gaussian. Finite pulse duration, correlated vibration, gradients and wavefront errors are left out.

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.

Start with the essentials

Focus question: How can three flashes of light measure a fall?
One-sentence intuition: Compare settings, keep the raw counts and fit only the counts you recorded.

Core mathematical model

Phase set by the fall

\Delta\phi=(k_{\rm eff}g-\alpha)T^2+\phi_3

Downward acceleration and the effective wavevector count as positive. The chirp term is an angular-frequency chirp: the effective wavevector times the chirp acceleration, which is the reference value plus an offset.

Recoil speed and largest separation

v_r=\hbar k_{\rm eff}/m,\qquad d_{\max}=v_rT

The first pulse adds a recoil kick, the middle pulse swaps it between the branches, and the last pulse brings the two centers together again. The branches separate vertically in the apparatus.

Gravity spacing between equivalent fringes

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

The fit returns one gravity offset, but shifting it by any whole number of these intervals fits equally well. Picking the right fringe takes extra information.

Second difference of the laser phase

\phi_L(0)-2\phi_L(T)+\phi_L(2T)=(k_{\rm eff}g-\alpha)T^2

With uniform acceleration and instantaneous pulses, the constant term and the initial-velocity term cancel. Real vibration, finite pulse length and wavefront effects are left out.

Detector loss and phase noise

C=C_0e^{-\sigma_\phi^2/2},\quad P_{\varnothing}=1-\eta

Independent Gaussian phase noise is averaged analytically. Loss that is equal for both outputs reduces the number of detected atoms, not the visibility among those detected.

Common difficulties

A path is not a trajectory

Typical misconceptionThe colored lines show which way each detected atom went.

Better mental modelThey show the coherent branches of the wave or the centers of the wave packets. Nothing records which path an atom took.

A small fit error is not real accuracy

Typical misconceptionA small fit error proves the real instrument is accurate.

Better mental modelThe quoted uncertainty is only the statistical error within this model. Apparatus effects the model leaves out, and the ambiguity, are still there.

Run the experiment

01
Predict
Try the first two presets and predict which exit gains counts.
What to observe: A preset only sets the conditions; it does not record any data.
02
Measure
Detect once, then collect 200 trials. Scan all 21 positions.
What to observe: Missed atoms stay in the record. The output frequency and its pointwise Wilson interval use only the detected atoms.
03
Compare
Save the scan, change coherence or pulse spacing, and scan again. Export both runs.
What to observe: When the contrast is low, no phase estimate is given. A longer pulse spacing narrows the gravity ambiguity interval.