Atom Interferometry

Atom interferometry takes two gifts from the rest of AMO physics and turns them into a ruler. From quantum mechanics it takes de Broglie's insight that a moving atom is a wave with wavelength $\lambda_{\mathrm{dB}}=h/(mv)$ , and the superposition principle that lets that wave occupy two places, two momenta, even two internal states at once. From the laser-cooling chapter it takes the whole apparatus — magneto-optical trap, optical molasses, sub-recoil and evaporative cooling, and Bose–Einstein condensation — that makes atoms cold and slow enough for their wave nature to become visible over macroscopic distances. Put them together and you can do to matter what a Mach–Zehnder does to light: split a single atom's wavefunction along two paths, let them accumulate a relative phase, and recombine them so the phase reappears as a population imbalance you can count. The twist that makes it an instrument is that the beam splitters and mirrors are made of light, not glass. Timed laser pulses drive coherent two-photon transitions that kick the atom by a precisely known momentum $\hbar k_{\mathrm{eff}}$ while writing the laser's optical phase onto it. Because that kick is set by laser wavelengths locked to atomic transitions, the device is self-calibrated: it ties an unknown acceleration directly to a frequency standard, with no drifting proof mass. That is why atom interferometers now realize the most accurate absolute gravimeters, matter-wave gyroscopes, tests of the equivalence principle, and one of the sharpest determinations of the fine-structure constant.

Key takeaways

An atom interferometer coherently splits a single atom's matter wave along two paths, lets them accumulate a relative phase, and recombines them so the output populations read out that phase — the matter-wave analogue of an optical Mach–Zehnder.
The beam splitters and mirrors are laser pulses driving two-photon (stimulated Raman or Bragg) transitions: a $\pi/2$ pulse splits, a $\pi$ pulse reflects, and the canonical $\pi/2-\pi-\pi/2$ sequence spaced by a free-evolution time $T$ closes the loop.
For a uniform acceleration $\mathbf a$ (gravity $\mathbf g$ ) the phase is $\Delta\phi=\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf a\,T^2$ — quadratic in $T$ , so doubling the interrogation time quadruples the sensitivity.
The recoil $\hbar k_{\mathrm{eff}}$ is fixed by known laser wavelengths, so the interferometer is intrinsically calibrated: it measures $g$ , rotations via the Sagnac phase, and $h/m$ (hence $\alpha$ ) against atomic constants.
Shot-noise sensitivity is $S=1/(C\,k_{\mathrm{eff}}\,T^2\sqrt N)$ ; long free fall, large-momentum-transfer optics, and many atoms push it below $10^{-9}\,g$ per shot.
Key idea: A light-pulse Mach–Zehnder interferometer converts an inertial acceleration into a measurable population imbalance through the phase $\Delta\phi=k_{\mathrm{eff}}\,a\,T^2$ ; the recoil $\hbar k_{\mathrm{eff}}$ is set by known laser wavelengths, so the device is intrinsically calibrated against atomic and fundamental constants.

Where this lives today

The technology has crossed from the laboratory into commercial and field instruments. Transportable absolute quantum gravimeters — cold-rubidium fountains packaged into rack-sized boxes — now survey gravity for geophysics, hydrology, and metrology, reaching a few $\mu\mathrm{Gal}$ (parts in $10^{9}$ of $g$ ) with no proof mass to wear or drift. Matter-wave gyroscopes and gradiometers are being developed for inertial navigation, where an atom sensor's freedom from long-term drift complements optical and mechanical gyros. At the frontier the same physics is scaling to enormous baselines. The ten-metre Stanford fountain, the hundred-metre MAGIS-100 detector at Fermilab, and the UK's AION programme drop clouds of atoms in long vertical vacuum tubes and use large-momentum-transfer optics to enclose huge space–time areas, targeting the "mid-band" gravitational waves around 0.1–10 Hz — between LIGO and the space-based LISA — and ultralight dark matter. Microgravity platforms — drop towers and the Cold Atom Laboratory on the International Space Station — push $T$ further by letting condensates evolve in free fall, pointing toward space missions that would test the equivalence principle far beyond terrestrial limits.

The physical picture

Picture a single laser-cooled atom released into free fall. A first laser pulse strikes it like a half-silvered mirror — but instead of splitting light it splits the atom's own wavefunction into two momentum components. One absorbs a photon from one beam and is stimulated to emit into a counter-propagating beam, recoiling with an extra momentum $\hbar k_{\mathrm{eff}}$ ; the other is left undisturbed — one atom now travels two paths at once. The wavepackets drift apart, a mirror pulse in the middle reverses their relative velocity, and a final beam-splitter pulse overlaps them; which output port the atom appears in depends on the phase difference the two arms accumulated. It is an optical Mach–Zehnder with the roles of light and matter exchanged.

Figure

Space-time paths of the light-pulse Mach–Zehnder for ⁸⁷Rb: a π/2 pulse at t = 0 splits the atom, a π pulse at t = T redirects both arms, a final π/2 pulse at t = 2T recombines them; each pulse transfers ħk_eff with k_eff ≈ 2×2π/780 nm, and the two arms enclose a space–time area that turns acceleration into phase

The cleanest way to think about the measured phase is that each laser pulse stamps the laser's optical phase — the value of $\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf x$ at the atom's location — onto the wavepacket it touches, and the interferometer reads out the second difference of that stamp at the three pulse times with weights $(+1,-2,+1)$ . If the atom moves at constant velocity the second difference vanishes; but if it accelerates, its position at the middle pulse is displaced from where uniform motion would put it, and the mismatch survives as $k_{\mathrm{eff}}\,a\,T^2$ : the atom reads the laser's optical ruler against its own free fall. Because the signal grows as $T^2$ , doubling the free-fall time quadruples the phase per unit acceleration — which is why every generation chases longer $T$ , from fountains to ten-metre towers, drop capsules, and space missions. A second, independent lever is $k_{\mathrm{eff}}$ : large-momentum-transfer pulses boost the phase linearly. The second figure shows the $T^2$ payoff.

Figure

The punchline: interferometer phase grows as the square of the interrogation time for a ⁸⁷Rb gravimeter, Δφ = k_eff g T² with k_eff = 1.61×10⁷ m⁻¹ and g = 9.81 m/s²; at the standard T = 100 ms the accumulated phase is already 1.58×10⁶ rad, so a phase resolution of milliradians reads g to parts per billion

The device is self-calibrating: $k_{\mathrm{eff}}$ is fixed by laser wavelengths measured against atomic transitions, with no drifting proof mass or spring constant — which is why atom gravimeters rival and can surpass the falling-corner-cube instruments they were built to replace.

The core equations

A moving atom has a de Broglie wavelength set, for a thermal source, by temperature; colder atoms are longer, more coherent waves.

\lambda_{\mathrm{dB}} = \frac{h}{m v} = \frac{h}{\sqrt{2 m k_B T}}\ \text{(thermal, most-probable speed)} \qquad (1)

where: $\lambda_{\mathrm{dB}}$ — de Broglie wavelength; $h$ — Planck constant; $m$ — atomic mass; $v$ — atomic speed; $k_B$ — Boltzmann constant; $T$ — source temperature. Physical reading: a coherent beam splitter needs a coherence length of order $\lambda_{\mathrm{dB}}$ , so lower temperature (longer wave) is what makes matter-wave interference practical. Each beam-splitter or mirror pulse is a two-photon process, and the momentum it delivers is the vector difference of the two laser wavevectors; counter-propagating beams double the kick.

\mathbf{k}_{\mathrm{eff}} = \mathbf{k}_1 - \mathbf{k}_2,\qquad |\mathbf{k}_{\mathrm{eff}}|\approx 2k \ \text{(counter-propagating)} \qquad (2)

where: $\mathbf{k}_1,\mathbf{k}_2$ — wavevectors of the two Raman or Bragg beams; $\mathbf{k}_{\mathrm{eff}}$ — kick $\hbar\mathbf{k}_{\mathrm{eff}}$ per pulse; $k=2\pi/\lambda$ — single-photon wavenumber. Physical reading: absorbing from beam 1 and being stimulated into beam 2 nets $\hbar(\mathbf{k}_1-\mathbf{k}_2)$ ; counter-propagation gives the largest kick a single pulse can deliver. After the three pulses the atom is a superposition of two output states, and interference makes the detected population a sinusoid in the accumulated phase.

P_e = \frac{1}{2}\bigl[1 - \cos(\Delta\phi)\bigr] \qquad (3)

where: $P_e$ — probability of the output port; $\Delta\phi$ — total interferometer phase. Physical reading: scanning a controllable phase sweeps $P_e$ through a cosine fringe whose position encodes the inertial phase and whose amplitude is the contrast $C$ . The central result is the inertial phase of the closed Mach–Zehnder. For a uniform acceleration it depends only on the kick, the acceleration, and the square of the free-evolution time.

\Delta\phi = \mathbf{k}_{\mathrm{eff}}\cdot\mathbf{a}\,T^2 \qquad (4)

where: $\mathbf{a}$ — acceleration relative to the laser phase fronts; $T$ — time between pulses. Physical reading: this is the payoff of the $(+1,-2,+1)$ laser-phase sampling — a constant acceleration gives a phase quadratic in $T$ , hence enormous inertial sensitivity for modest free-fall times. To keep the two-photon transition on resonance as the atom accelerates, the laser difference frequency is chirped; the recoil frequency also enters because the two outputs differ in kinetic energy.

\omega_r = \frac{\hbar k^2}{2m},\qquad \alpha_c = \mathbf{k}_{\mathrm{eff}}\cdot\mathbf{a}\ \ (\text{chirp that nulls the fringe}) \qquad (5)

where: $\omega_r$ — single-photon recoil frequency; $\alpha_c$ — laser chirp rate. Physical reading: the first-order Doppler term cancels in the symmetric geometry, but the resonance still drifts as the atom speeds up; chirping at $\alpha_c=\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf a$ nulls $\Delta\phi$ , so the required chirp is an absolute readout of $a$ . The single-shot resolution is set by quantum projection noise on $N$ atoms, divided by the scale factor.

S = \frac{1}{C\,k_{\mathrm{eff}}\,T^2\,\sqrt{N}}\ \ \text{(shot-noise-limited acceleration sensitivity per shot)} \qquad (6)

where: $S$ — minimum resolvable acceleration; $C$ — fringe contrast; $N$ — detected atom number. Physical reading: projection noise gives phase resolution $1/(C\sqrt N)$ ; dividing by the scale factor $k_{\mathrm{eff}}T^2$ turns it into an acceleration, so bigger kicks, longer times, higher contrast, and more atoms all help. (Rotation instead of acceleration gives the matter-wave Sagnac phase $\Delta\phi=(2m/\hbar)\,\boldsymbol\Omega\!\cdot\!\mathbf A$ , developed in the limiting-cases table below.)

Derivation ATwo-photon Raman coupling as an effective two-level system

The goal is to justify why two laser beams driving two long-lived ground states through a far-off-resonant excited level behave like a single, clean two-level system — the beam splitter and mirror of the interferometer. Adiabatically eliminating the fast excited state yields the effective two-photon Rabi frequency $\Omega_{\mathrm{eff}}=\Omega_1\Omega_2^{*}/2\Delta$ and the pulse areas that define $\pi/2$ and $\pi$ pulses.

Step 1. Write the interaction Hamiltonian in the rotating-wave approximation, $H = \hbar\Delta\,|e\rangle\langle e| - \tfrac{\hbar}{2}(\Omega_1|e\rangle\langle g_1| + \Omega_2|e\rangle\langle g_2| + \text{h.c.})$ , with $\Delta$ the common detuning from $|e\rangle$ , much larger than $\Omega_{1,2}$ . Why: a large $\Delta$ lets us treat the excited state as a fast slave variable rather than a populated level — the whole point of a Raman beam splitter.

Step 2. Set $\dot c_e\approx 0$ (adiabatic elimination), giving $c_e = (\Omega_1 c_1 + \Omega_2 c_2)/2\Delta$ ; the excited population, and hence spontaneous emission, is suppressed as $1/\Delta^2$ . Why: this preserves coherence — the atom feels the coupling without ever really occupying the lossy excited state.

Step 3. Substituting $c_e$ back gives an effective two-state Hamiltonian whose off-diagonal element is the coherent coupling $\Omega_{\mathrm{eff}}=\Omega_1\Omega_2^{*}/2\Delta$ and whose diagonal elements are the differential AC (light) shift. Why: the light shift scales as $1/\Delta$ just like $\Omega_{\mathrm{eff}}$ , so it cannot be detuned away — it must be calibrated, and is a leading systematic.

Step 4. Integrate the coupling over the pulse to get the area $\theta=\int\Omega_{\mathrm{eff}}\,\mathrm dt$ : a $\theta=\pi/2$ pulse leaves an equal superposition of $|g_1,\mathbf p\rangle$ and $|g_2,\mathbf p+\hbar\mathbf k_{\mathrm{eff}}\rangle$ (beam splitter), a $\theta=\pi$ pulse fully swaps them (mirror). Why: fixing pulse area, not just frequency, makes the optics reproducible shot to shot — the same physics that defines a $\pi$ pulse in NMR. *Verified: standard adiabatic elimination reproducing $\Omega_{\mathrm{eff}}=\Omega_1\Omega_2^{*}/2\Delta$ ; consistent with Foot, Atomic Physics Ch. 7 and Demtröder.*

Go deeper · Derivation B · 1 minThe Mach-Zehnder phase from laser-phase sampling

Derivation BThe Mach-Zehnder phase from laser-phase sampling

The goal is to derive $\Delta\phi=\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf a\,T^2$ from first principles and, crucially, to see why the free-propagation and separation phases drop out, leaving only the laser phase imprinted at the atom during each pulse.

Step 1. Assign each arm the propagation phase $S_{cl}/\hbar$ with $S_{cl}=\int(\tfrac12 m\dot{\mathbf x}^2 - m g z)\,\mathrm dt$ on the classical path; for a quadratic Lagrangian the difference between the two arms vanishes exactly. Why: this Storey–Cohen-Tannoudji theorem makes the result robust — the messy action phase, dependent on both trajectories, simply cancels.

Step 2. Write the laser phase as a signed sum over the pulses, $\Delta\phi_{\mathrm{laser}}=\sum_j \pm\,\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf x(t_j)+\phi_L^{(j)}$ , with signs $(+1,-2,+1)$ because the $\pi$ pulse acts twice with opposite sign. Why: each pulse stamps the local optical phase onto the wavepacket, so the interferometer measures how that stamp changes along the trajectory.

Step 3. Insert $\mathbf x(t)=\mathbf x_0+\mathbf v_0 t+\tfrac12\mathbf a t^2$ at $t=0,T,2T$ : the weights annihilate the constant $\mathbf x_0$ and linear $\mathbf v_0 t$ terms — including the first-order Doppler shift — and keep only $\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf a\,[\tfrac12(0)-2\cdot\tfrac12 T^2+\tfrac12(2T)^2]=\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf a\,T^2$ . Why: this second-difference structure is the reason the signal is quadratic in $T$ and insensitive to launch velocity.

Step 4. Add the separation phase $\bar{\mathbf p}\!\cdot\!\Delta\mathbf x/\hbar$ , nonzero only if the wavepackets fail to overlap at the last pulse; in uniform gravity it vanishes, leaving $\Delta\phi=\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf g\,T^2+(\phi_L^{(1)}-2\phi_L^{(2)}+\phi_L^{(3)})$ . Why: the programmed laser phases give a clean, calibrated knob to scan the fringe and extract $g$ . *Verified: reproduces the Kasevich–Chu result $k_{\mathrm{eff}}gT^2$ ; cancellation of propagation and separation phases for quadratic Lagrangians (Storey and Cohen-Tannoudji, 1994).*

Go deeper · Derivation C · 1 minFrom the atomic recoil to the fine-structure constant

Derivation CFrom the atomic recoil to the fine-structure constant

The goal is to show how an interferometric measurement of the recoil frequency yields the ratio $h/m$ for a specific atom, and how that single kinematic quantity — combined with two of the best-known constants in physics — delivers the fine-structure constant $\alpha$ with no input from quantum electrodynamics.

Step 1. Measure the recoil frequency $\omega_r=\hbar k^2/2m$ through the recoil-induced phase in a Ramsey–Bordé or Bloch-oscillation interferometer; since $k$ is set by the laser wavelength, this yields $h/m_X$ for atom $X$ (e.g. $^{87}$ Rb or $^{133}$ Cs). Why: $h/m$ is purely kinematic — no force calibration, no proof mass — so it can be measured with exceptional accuracy.

Step 2. Insert known ratios into $\alpha^2=\dfrac{2R_\infty}{c}\dfrac{m_X}{m_e}\dfrac{h}{m_X}$ , where the Rydberg $R_\infty$ is among the most precisely measured constants and $m_X/m_e$ comes from Penning-trap comparisons. Why: every factor except $h/m_X$ is known to sub-ppb, so the recoil measurement is the limiting input that sets the precision on $\alpha$ .

Step 3. Because this route never invokes the electron $g\!-\!2$ , agreement with the $g\!-\!2$ value of $\alpha$ confirms QED to remarkable depth, while a genuine tension would signpost new physics. Why: two independent determinations of the same constant are only as trustworthy as their disagreement is small. *Verified: formula standard in the CODATA analysis; matter-wave $h/m$ determinations for Rb and Cs give $\alpha^{-1}\approx 137.035999$ at sub-ppb precision (Cronin, Schmiedmayer and Pritchard, Rev. Mod. Phys. 81, 1051 (2009)).*

Limiting cases

The Mach–Zehnder phase separates into physically distinct regimes depending on what perturbs the two arms.

Case	Result	Meaning
Uniform acceleration (gravimeter)	$\Delta\phi = k_{\mathrm{eff}}\,a\,T^2$	Constant acceleration gives a phase quadratic in $T$ ; the basis of absolute atom gravimeters.
Rotation (Sagnac / gyroscope)	$\Delta\phi = \dfrac{2m}{\hbar}\,\boldsymbol{\Omega}\cdot\mathbf{A}$	Rotation couples to the enclosed area; the matter-wave Sagnac phase beats the optical one by $\sim mc^2/\hbar\omega$ .
Gravity gradient	$\Delta\phi \approx k_{\mathrm{eff}}\,\gamma\,z\,T^2$	Two vertically separated interferometers form a gradiometer that rejects common vibration noise.
Recoil measurement ( $h/m$ )	$\Delta\phi \supset \omega_r\,T,\quad \omega_r=\dfrac{\hbar k^2}{2m}$	The recoil frequency enters the phase; Ramsey–Bordé and Bloch schemes measure $h/m$ and hence $\alpha$ .
Slow launch (Doppler term)	$\mathbf k_{\mathrm{eff}}\!\cdot\!\mathbf v_0$ cancels	The first-order Doppler phase drops out, so the signal is insensitive to launch velocity — only acceleration survives.

Worked examples

1 · Phase and chirp of a rubidium gravimeter (the real number)

A ⁸⁷Rb fountain uses counter-propagating Raman beams on the D2 line at $\lambda=780\ \mathrm{nm}$ with pulse separation $T=100\ \mathrm{ms}$ . Find the effective wavevector, the inertial phase under gravity, and the chirp needed to null the fringe.

Step 1. Single-photon wavenumber $k=2\pi/\lambda=8.06\times10^{6}\ \mathrm{m^{-1}}$ ; counter-propagating beams give $k_{\mathrm{eff}}=2k=\mathbf{1.61\times10^{7}\ m^{-1}}$ . The kick gives an arm-separation velocity $\hbar k_{\mathrm{eff}}/m\approx 11.8\ \mathrm{mm\,s^{-1}}$ (paths $\approx1.2\ \mathrm{mm}$ apart over $T$ ).

Step 2. From equation (4) with $a=g=9.81\ \mathrm{m\,s^{-2}}$ and $T=0.100\ \mathrm{s}$ : $\Delta\phi=k_{\mathrm{eff}}\,g\,T^2=(1.61\times10^{7})(9.81)(0.100)^2=\mathbf{1.58\times10^{6}\ rad}$ — over a quarter of a million full fringes; the challenge is resolving one part in $10^6$ of it, not detecting it.

Step 3. The chirp that tracks the falling atoms and nulls the fringe is $\alpha_c=k_{\mathrm{eff}}\,g=1.58\times10^{8}\ \mathrm{rad\,s^{-2}}$ , i.e. $\alpha_c=\mathbf{25.2\ MHz\,s^{-1}}$ . Because $k_{\mathrm{eff}}$ is fixed by the laser wavelength, the measured chirp ties $g$ straight to a frequency standard with no proof mass. k_eff = 1.61×10⁷ m⁻¹, Δφ = 1.58×10⁶ rad, null chirp 25.2 MHz/s.

2 · How well can one shot measure g?

The same instrument detects $N=10^{6}$ atoms per shot at contrast $C=0.5$ . Estimate the shot-noise-limited resolution per shot and after averaging for one hour at a 1 s cycle.

Step 1. Projection noise sets $\delta\phi=1/(C\sqrt N)=1/(0.5\times1000)=\mathbf{2.0\times10^{-3}\ rad}$ .

Step 2. Divide by the scale factor $k_{\mathrm{eff}}T^2=(1.61\times10^{7})(0.100)^2=1.61\times10^{5}\ \mathrm{rad/(m\,s^{-2})}$ : $S=2.0\times10^{-3}/1.61\times10^{5}=1.24\times10^{-8}\ \mathrm{m\,s^{-2}}$ , i.e. $S=\mathbf{1.3\times10^{-9}\,g}$ per shot (about $0.13\ \mu\mathrm{Gal}$ ).

Step 3. Averaging $n$ independent shots improves as $1/\sqrt n$ : one hour at 1 s gives $n=3600$ , so $\sqrt n\approx 60$ and the resolution reaches $\approx 2\times10^{-11}\,g$ — better than one part in $10^{10}$ of gravity. In practice vibration, not projection noise, usually dominates, which is why gradiometers matter. Per shot S ≈ 1.3×10⁻⁹ g; after one hour ≈ 2×10⁻¹¹ g.

Physics of the portrait

The person who fused de Broglie's waves and Ramsey's pulses with laser cooling into a precision instrument is the natural face of the topic.

Origins · 1 minMark Kasevich and how this was discovered

Mark Kasevich

b. 1965 · light-pulse atom interferometry and precision inertial sensing

Mark Kasevich is an American physicist who, as a graduate student with Steven Chu at Stanford in 1991, demonstrated the first light-pulse atom interferometer and used it to build an atomic-fountain gravimeter — the experiment that turned matter-wave interference from a beautiful demonstration into a metrological tool, with stimulated Raman pulses as coherent, self-calibrating beam splitters and mirrors. Over the following decades Kasevich, now a professor at Stanford, drove the field's two great levers — large-momentum-transfer beam splitters and very long interrogation times, culminating in a ten-metre atomic fountain in which cold atoms stay in superposition for roughly half a second, and tested the equivalence principle by dropping two isotopes at once. His arc, from a tabletop fountain to the kilometre-scale detectors his work seeded, mirrors the field's own: a de Broglie wave, split by light, made into a ruler for the universe's faintest forces.

History

1924

— de Broglie proposes that matter has a wavelength λ = h/(mv)

1927

— Davisson and Germer confirm electron diffraction, verifying matter waves

1949

— Ramsey introduces the separated-oscillatory-field pulse method

1991

— Kasevich and Chu demonstrate the light-pulse atom interferometer and gravimeter

2018

— atom-recoil interferometry fixes the fine-structure constant to 0.2 ppb

Going further

Texts. Foot, Atomic Physics, Oxford (2005), Ch. 7 and Ch. 9 · Berman (ed.), Atom Interferometry, Academic Press (1997) · Cronin, Schmiedmayer and Pritchard, Rev. Mod. Phys. 81, 1051 (2009) — comprehensive review · Demtröder, Laser Spectroscopy, Springer.

Provenance. Equations are standard and should be recomputed by a verifier: $\Delta\phi=k_{\mathrm{eff}}aT^2$ from the $(+1,-2,+1)$ laser-phase sum on a free-fall trajectory (Kasevich and Chu, Phys. Rev. Lett. 67, 181 (1991)); $\Omega_{\mathrm{eff}}=\Omega_1\Omega_2^{*}/2\Delta$ from adiabatic elimination (Foot Ch. 7); the three-term path-integral phase and its cancellation for quadratic Lagrangians (Storey and Cohen-Tannoudji, J. Phys. II 4, 1999 (1994)); the Sagnac phase $(2m/\hbar)\boldsymbol\Omega\!\cdot\!\mathbf A$ ; and $\alpha^2=(2R_\infty/c)(m_X/m_e)(h/m_X)$ per CODATA. Numerical anchors re-derived here: $k_{\mathrm{eff}}=1.61\times10^{7}\,\mathrm{m^{-1}}$ , $\Delta\phi=1.58\times10^{6}$ rad and chirp $25.2\,\mathrm{MHz\,s^{-1}}$ at $T=100$ ms, single-shot $S\approx1.3\times10^{-9}g$ , $v_{\mathrm{rec}}=5.88\,\mathrm{mm\,s^{-1}}$ , $\lambda_{\mathrm{dB}}\approx332$ nm at $1\,\mu$ K. Gravimetry at a few $\mu$ Gal and sub-ppb $\alpha$ are from Peters, Chung and Chu, Metrologia 38, 25 (2001) and the Cronin review. The base article's $\lambda_{\mathrm{dB}}\sim10$ nm at $1\,\mu$ K is corrected here to $\approx332$ nm.

Builds on: The Recoil Limit and Sub-Recoil Cooling · Optical Molasses · Magneto-Optical Trap · Bose–Einstein Condensation in Dilute Gases. Leads to: Evaporative Cooling · Degenerate Fermi Gases and the BEC–BCS Crossover.

Ask the tutor: Ask me to walk through why the $(+1,-2,+1)$ pulse weighting makes the interferometer a pure acceleration sensor, or to size the free-fall time and momentum transfer needed to reach a target gravity or rotation sensitivity.

Why do the beam splitters transfer momentum, unlike the mirrors in an optical interferometer?

Because the wave here is the atom, and splitting it means putting it into a superposition of two momentum states. A two-photon pulse takes a photon from one beam and stimulates emission into the counter-propagating beam, so the atom recoils by $\hbar k_{\mathrm{eff}}\approx 2\hbar k$ ; that kick is what physically separates the two arms, without which they would never diverge and enclose an area. Here the recoil is the geometry, and it is also why the device is calibrated: $k_{\mathrm{eff}}$ is set by laser wavelengths locked to atomic lines.

Why exactly is the sensitivity quadratic in T rather than linear?

The measured phase is the laser phase sampled at three times with weights $(+1,-2,+1)$ — a discrete second derivative. Apply that to $x(t)=x_0+v_0 t+\tfrac12 a t^2$ : the constant $x_0$ and linear $v_0 t$ terms both vanish, because a second difference kills anything up to first order in time. Only the quadratic $\tfrac12 a t^2$ survives, and evaluating it at $t=0,T,2T$ leaves $aT^2$ — the signature that the interferometer is an acceleration sensor, blind to leading order to where the atom started and how fast it was launched.

If the accumulated phase is over a million radians, how do you read it without ambiguity?

You do not read the absolute $1.58\times10^{6}$ rad directly — that would need knowing $g$ in advance. Instead you operate near a fringe null by chirping the difference frequency at $\alpha_c=k_{\mathrm{eff}}g$ , which cancels the large phase; the measured quantity is the chirp rate that produces the null, mapping linearly onto $g$ . To defeat the $2\pi$ ambiguity you vary $T$ (or $k_{\mathrm{eff}}$ ) and combine coarse-but-unambiguous short- $T$ runs with precise-but-wrapped long- $T$ runs — standard multi-scale phase unwrapping.

How cold do the atoms actually need to be?

Cold enough that a large fraction fall within the velocity acceptance of a beam-splitter pulse, roughly one photon recoil — a few millimetres per second for rubidium, so sub-microkelvin. Two effects drive this: the two-photon resonance is Doppler-sensitive, so a broad velocity spread leaves many atoms off-resonant and the contrast $C$ collapses; and over a long $T$ a warm cloud expands ballistically, so atoms leave the beams or fail to overlap at recombination. This is why the laser-cooling toolkit is a prerequisite — a condensate is the brightest, least divergent, most coherent source for the longest baselines.

What is the difference between Raman and Bragg beam splitters?

Both are two-photon light-pulse optics, differing in whether the atom changes internal state. A stimulated Raman pulse couples two hyperfine ground states, so the arms carry different internal labels as well as momenta — convenient for state-selective detection, but sensitive to magnetic fields and light shifts that act differently on the two states. A Bragg pulse couples two momentum states within the same internal state, so the arms are magnetically identical, reducing some systematics at the cost of momentum-selective detection, and is favoured for large-momentum-transfer.

How does the matter-wave Sagnac effect beat the optical one?

Both an optical ring and a matter-wave loop develop a rotation phase proportional to enclosed area times rotation rate, but the constants differ enormously. For light the Sagnac phase scales with the photon energy $\hbar\omega$ ; for atoms it scales with the rest energy $mc^2$ . The ratio $mc^2/\hbar\omega$ for rubidium is about $5\times10^{10}$ , so a matter-wave gyroscope enclosing the same area is roughly ten orders of magnitude more sensitive. Optical gyros claw much of that back with kilometres of coiled fibre, but for a compact, drift-free sensor the atomic version is compelling.