Coherent Atom Optics and Bragg Diffraction

The beam splitters and mirrors sketched above are now the working parts of real instruments. Transportable cold-atom gravimeters built on $\pi/2$–$\pi$–$\pi/2$ Bragg or Raman sequences measure absolute $g$ to a part in $10^{9}$ in the field, and gravity gradiometers flying two such interferometers on a common laser reject vibration to hunt for underground voids, aquifers, and ore bodies; commercial versions are sold by companies spun out of the metrology labs. At the frontier, large-momentum-transfer optics push the trick further: higher-order Bragg pulses and Bloch oscillations in accelerated lattices stack tens to hundreds of recoils onto each arm, opening the interferometer far enough that the 10-metre atomic fountains at Stanford and the very-long-baseline detectors AION and MAGIS-100 aim to sense gravitational waves and screen for ultralight dark matter. The same large-momentum-transfer Bragg optics let Berkeley and Paris teams read the recoil frequency $\omega_r=\hbar k^2/2m$ so precisely that, combined with atomic mass ratios and the Rydberg constant, they deliver the fine-structure constant $\alpha$ to better than a part in $10^{9}$ — a benchmark test of QED. And a weak moving Bragg grating, its frequency difference $\Delta\omega$ scanned, doubles as a spectrometer: Bragg spectroscopy maps the Bogoliubov phonon spectrum of a condensate and the pairing gap across the BEC–BCS crossover of a Fermi gas.

Start from a single fact the dipole-force chapter already owns: a far-detuned laser shifts a ground-state atom's energy by the ac Stark potential $U\propto I/\delta$, proportional to local intensity and inversely proportional to detuning. Now interfere two counter-propagating beams. Their intensity is no longer uniform but corrugated into bright and dark fringes with period $\lambda/2$, so the atom feels a periodic potential $U(x)\propto\cos(2kx)$ — a grating made of light. A de Broglie wave sweeping across this corrugation diffracts, and because the corrugation repeats every $\lambda/2$, the reciprocal lattice vector is $2k$ and every momentum kick is a multiple of $2\hbar k$. Think microscopically in photons, and the factor of two becomes obvious. To turn around, an atom absorbs a photon travelling one way, receiving $+\hbar k$, and is then stimulated by the opposite beam to emit a photon that way — reversing a photon of momentum $\hbar k$ and so gaining another $+\hbar k$ of recoil. The two add to $2\hbar k$, and crucially the atom ends in the same internal ground state it started in. There is no spontaneous emission and no random direction: the whole cycle is a coherent redistribution of momentum, which is why Bragg pulses can be used as precision optical elements rather than mere heating.

The thin-versus-thick grating distinction, familiar from ordinary diffraction, carries over exactly. A brief interaction is a thin phase grating: it imprints a spatial phase modulation, spreads the atom symmetrically over many orders, and does not care about energy — this is the Raman–Nath regime, with populations weighted by squared Bessel functions. A long interaction is a thick grating: it lasts long enough for the kinetic-energy difference between orders to matter, so only the order that conserves energy survives. That thick-grating limit is Bragg diffraction, and there the atom flops coherently between just two momentum states like a spin driven on resonance.

A Bragg pulse, then, is a beam splitter you dial with a clock. Because two momentum states are coupled just like the two levels of an atom driven by a laser, the "pulse area" $\Omega_{\mathrm{eff}}\tau$ plays the role of the optical Rabi angle: area $\pi/2$ leaves the atom in an equal coherent superposition of the two momenta (a matter-wave beam splitter), and area $\pi$ transfers it completely from one momentum to the other (a matter-wave mirror). Everything else in atom interferometry is built from these two moves.

The starting object is the potential the light makes. Superpose two counter-propagating travelling waves and the intensity acquires a $\cos^2(kx)$ corrugation; the far-detuned light shift turns that intensity pattern into a periodic potential. Writing $\Omega$ for the single-beam Rabi frequency and $\delta=\omega_L-\omega_0$ for the detuning, the standing-wave dipole potential is:

where: $U(x)$ — ac Stark (dipole) potential of a ground-state atom; $\Omega$ — single-beam resonant Rabi frequency; $\delta$ — detuning from atomic resonance; $k=2\pi/\lambda$ — laser wavenumber; $2k$ — grating reciprocal vector, so the spatial period is $\lambda/2$. Read it as an intensity grating imprinted on the atom's energy: the modulation amplitude $U_0\equiv\hbar\Omega^2/2\delta$ is the grating strength that will drive all the diffraction, and the constant term merely offsets the zero. Every diffraction event moves momentum in the same quantum. One absorption plus one stimulated emission from the counter-propagating beam reverses a photon and hands the atom twice its momentum:

where: $\Delta p$ — atomic momentum change per absorption-and-stimulated-emission cycle; $\hbar k$ — single-photon momentum. This is the reciprocal-lattice vector $2k$ of the light grating written as a momentum: the standing wave can only ever connect states that differ by an integer number of these kicks. The energy scale that decides which regime you are in is the recoil. Handing an atom of mass $m$ one photon momentum costs a kinetic energy $E_r$ and rings its wavefunction at the recoil frequency $\omega_r$:

where: $\omega_r$ — single-photon recoil (angular) frequency; $E_r$ — single-photon recoil energy; $m$ — atomic mass. For rubidium-87 at 780 nm, $\omega_r=2\pi\times 3.77$ kHz and the recoil time $t_r=1/\omega_r\approx42\ \mu\text{s}$ — the clock against which "short" and "long" pulses are judged. Now demand energy and momentum conservation together for an $n$-photon-pair process. That double constraint is the Bragg condition: it fixes which input momentum is resonant, and, for a moving grating, which two-beam frequency difference $\Delta\omega$ selects order $n$.

where: $n$ — Bragg order (number of $2\hbar k$ kicks); $p_i,p_f$ — initial and final momenta; $\Delta\omega$ — frequency difference between the two beams; $\theta_{\mathrm B}$ — corresponding Bragg angle for an atomic beam. Read the second equality as a resonance condition: for an atom starting at rest, first-order Bragg ($n=1$) requires $\Delta\omega=4\omega_r$, which for $^{87}$Rb is $2\pi\times 15.1$ kHz — the two lasers must differ by exactly that. To see the full diffraction, expand the Schrödinger equation on the momentum ladder $|p_0+2n\hbar k\rangle$. The $\cos(2kx)$ grating couples each rung only to its neighbours, while kinetic energy sits on the diagonal:

where: $c_n$ — amplitude of state $|p_0+2n\hbar k\rangle$; $U_0=\hbar\Omega^2/2\delta$ — grating strength; $4n^2\omega_r$ — kinetic energy of rung $n$ in units of frequency, since $(2n\hbar k)^2/2m=4n^2E_r$. This one line contains both regimes: drop the diagonal and you get Raman–Nath; keep it and let it dominate and you get Bragg. When the pulse is short compared with $t_r$, the diagonal is frozen and only the grating phase acts. Integrating and expanding with the Jacobi–Anger identity gives Bessel-weighted orders:

where: $J_n$ — Bessel function of the first kind; $\tau$ — pulse duration; $\theta\equiv U_0\tau/\hbar$ — pulse area (phase-modulation depth). Read it as a symmetric fan, $P_n=P_{-n}$, populating many orders once $\theta\gtrsim1$, with the sum rule $\sum_n J_n^2=1$ conserving probability. When instead the pulse is long, energy conservation isolates the one resonant pair, and those two states flop like a driven spin. The pulse area sets whether you get a splitter or a mirror:

where: $\Omega_{\mathrm{eff}}$ — effective two-photon Rabi frequency between the two Bragg-coupled momenta; $P_+$ — population transferred to the second momentum state. Read it as the atom-optical dictionary: a $\pi/2$ pulse leaves a 50:50 superposition (beam splitter), a $\pi$ pulse swaps the states completely (mirror). Finally, assemble three pulses into an interferometer. A $\pi/2$–$\pi$–$\pi/2$ sequence separated by time $T$ encloses a space-time area, and any acceleration along the beams writes a phase onto the output:

where: $\Delta\phi$ — interferometer output phase; $\mathbf k_{\mathrm{eff}}=2n k\,\hat x$ — effective wavevector, so the momentum splitting is $\hbar k_{\mathrm{eff}}=2n\hbar k$; $\mathbf a$ — acceleration (gravity $\mathbf g$); $T$ — time between successive pulses. Read it as the sensitivity law of a cold-atom gravimeter: the phase grows with the momentum splitting and quadratically with the interrogation time.

The character of the diffraction is set by the pulse duration $\tau$ measured against the recoil time $t_r=1/\omega_r$, and by the grating strength $U_0$.

1 · Recoil kinematics and the Bragg detuning for rubidium-87 (confidence builder)

A $^{87}$Rb atom ($m=86.909\,\mathrm{u}$) is diffracted by a standing wave built from its D2 line, $\lambda=780$ nm. Find the two-photon velocity kick, the recoil frequency, and the two-beam frequency difference that makes first-order Bragg resonant for an atom at rest.

Step 1. Wavenumber and single-photon recoil velocity. With $k=2\pi/\lambda=8.06\times10^{6}\ \mathrm{m^{-1}}$ and $m=86.909\times1.6605\times10^{-27}=1.443\times10^{-25}$ kg, the recoil velocity is $v_r=\hbar k/m=(1.0546\times10^{-34})(8.06\times10^{6})/(1.443\times10^{-25})=\mathbf{5.89\ mm/s}$. Each Bragg event delivers two of these, a velocity step $2v_r=\mathbf{11.8\ mm/s}$.

Step 2. Recoil frequency. From equation (3), $\omega_r=\hbar k^2/2m=v_r k/2=(5.89\times10^{-3})(8.06\times10^{6})/2=2.37\times10^{4}$ rad/s, i.e. $\nu_r=\omega_r/2\pi=\mathbf{3.77\ kHz}$, with recoil time $t_r=1/\omega_r=\mathbf{42\ \mu s}$. Pulses much shorter than this are Raman–Nath; much longer are Bragg.

Step 3. Bragg detuning. For an atom starting at rest and diffracting to $2\hbar k$, equation (4) gives $\Delta\omega=4\omega_r$, so the frequency difference between the two beams must be $\Delta\nu=4\nu_r=4\times3.77=\mathbf{15.1\ kHz}$. Set the two acousto-optic modulators to differ by exactly this and the atom flops resonantly into the first Bragg order; miss it by more than the pulse bandwidth and the transfer collapses. v_r = 5.89 mm/s, 2v_r = 11.8 mm/s, ν_r = 3.77 kHz, and first-order Bragg resonance is Δν = 15.1 kHz.

2 · Sensitivity of a cold-atom gravimeter

A $^{87}$Rb Mach–Zehnder interferometer uses first-order Bragg pulses ($k_{\mathrm{eff}}=2k=1.61\times10^{7}\ \mathrm{m^{-1}}$) with pulse spacing $T=100$ ms. Find the total gravity phase, the maximum spatial separation of the two arms, and the phase shift from a fractional gravity change of $10^{-9}$.

Step 1. Gravity phase. Equation (8) gives $\Delta\phi=k_{\mathrm{eff}}\,g\,T^2=(1.61\times10^{7})(9.81)(0.1)^2=\mathbf{1.58\times10^{6}\ rad}$. The interferometer winds up more than a million radians of phase against gravity — the raw quantity a gravimeter reads.

Step 2. Arm separation. The two arms differ in velocity by $2v_r=11.8$ mm/s, so by the mirror pulse at time $T$ they have drifted apart by $\Delta x=2v_r\,T=(11.8\times10^{-3})(0.1)=\mathbf{1.18\ mm}$ — a macroscopic splitting of a single atom's wavefunction into two clouds you can resolve.

Step 3. Metrological resolution. The sensitivity is $\mathrm{d}\Delta\phi/\mathrm{d}g=k_{\mathrm{eff}}T^2=1.61\times10^{5}$ rad per $\mathrm{m/s^2}$. A fractional change $\Delta g/g=10^{-9}$ means $\Delta g=9.81\times10^{-9}\ \mathrm{m/s^2}$, giving $\Delta\phi=(1.61\times10^{5})(9.81\times10^{-9})=\mathbf{1.6\times10^{-3}\ rad}$. Resolving a milliradian of interferometer phase — routine with $10^{6}$ atoms — thus measures $g$ to a part in $10^{9}$. Δφ = 1.58×10⁶ rad total, arms separate by 1.18 mm, and a 10⁻⁹ change in g shifts the phase by 1.6 mrad.

3 · Which regime does a 5 µs pulse give?

The same $^{87}$Rb standing wave is pulsed for $\tau=5\ \mu\text{s}$, then for $\tau=300\ \mu\text{s}$. Classify each, using $t_r=42\ \mu\text{s}$.

Step 1. The 5 µs pulse. Here $\omega_r\tau=\tau/t_r=5/42=0.12\ll1$, so kinetic energy is frozen and the pulse is a thin phase grating: Raman–Nath, filling many orders with Bessel weights, symmetric about $n=0$.

Step 2. The 300 µs pulse. Now $\omega_r\tau=300/42=7.1\gg1$, energy conservation is enforced, and the atom flops between just two momentum states: Bragg. To make this a clean $\pi$ pulse (a mirror) one tunes the intensity so that $\Omega_{\mathrm{eff}}\tau=\pi$, i.e. $\Omega_{\mathrm{eff}}=\pi/(300\ \mu\text{s})=1.05\times10^{4}$ rad/s — set by the grating strength through $U_0=\hbar\Omega_{\mathrm{eff}}$. The 5 µs pulse is Raman–Nath (ω_r τ = 0.12); the 300 µs pulse is Bragg (ω_r τ = 7.1), tuned to a π mirror at Ω_eff = 1.05×10⁴ rad/s.

Bragg diffraction of atoms is where matter-wave optics stopped being a thought experiment and became a laboratory tool, and one group turned that corner. The portrait belongs to the experimenter who first Bragg-diffracted an atomic beam from a wall of light and then spent a career turning the trick into the most precise inertial and recoil sensors ever built.

Texts. Foot, Atomic Physics (2005), Ch. 9–10 (light forces, recoil, atom optics) · Loudon, The Quantum Theory of Light, 3rd ed. — stimulated processes and momentum exchange · Saleh and Teich, Fundamentals of Photonics — acousto-optic diffraction, Raman–Nath versus Bragg · Hecht, Optics — diffraction gratings, thin versus thick gratings, the Bragg condition · Demtröder, Laser Spectroscopy — recoil, two-photon processes, coherent manipulation · Berman (ed.), Atom Interferometry (1997); Cronin, Schmiedmayer, and Pritchard, Rev. Mod. Phys. 81, 1051 (2009).

Provenance. All relations are standard coherent-atom-optics results. A verifier should: (1) confirm the standing-wave dipole potential $U=(\hbar\Omega^2/2\delta)[1+\cos2kx]$ from the far-detuned light shift of two interfering beams, with $\cos^2(kx)=\tfrac12[1+\cos2kx]$ (Foot Ch. 7, 9); (2) verify the two-photon recoil $2\hbar k$ and recoil energy $E_r=\hbar^2k^2/2m$, and reproduce $v_r=5.89$ mm/s, $\nu_r=3.77$ kHz, $t_r=42\ \mu$s for $^{87}$Rb at 780 nm; (3) derive the Bragg condition $\Delta\omega=4n^2\omega_r$ from simultaneous energy–momentum conservation, checking $\Delta\nu=15.1$ kHz at $n=1$; (4) reproduce the Raman–Nath Bessel amplitudes via the Jacobi–Anger expansion and confirm $\sum_nJ_n^2=1$; (5) reduce the coupled ladder to a two-level system with $\Omega_{\mathrm{eff}}=U_0/\hbar=\Omega^2/2\delta$ and confirm the $\pi/2$ and $\pi$ pulse areas; (6) check the gravimeter phase $\Delta\phi=k_{\mathrm{eff}}gT^2=1.58\times10^{6}$ rad and arm separation $2v_rT=1.18$ mm against equation (8). Note the factor-of-two convention on $\Omega$: here $\Omega$ is the single-beam Rabi frequency, chosen so that $\Omega_{\mathrm{eff}}=\Omega^2/2\delta$ is self-consistent with $U_0=\hbar\Omega^2/2\delta$.

Prerequisites. Dipole Force and the Optical Dipole Potential · The Recoil Limit and Sub-Recoil Cooling · Radiation Pressure and the Scattering Force · Bose–Einstein Condensation in Dilute Gases. Leads to. Atom Interferometry · Optical Lattices · Precision Measurement of the Fine-Structure Constant.

Ask the tutor: Ask me to walk through why a 300 microsecond pulse on rubidium gives a clean Bragg mirror while a 5 microsecond pulse sprays the atom across a Bessel fan of orders.