Feshbach Resonances

Quantum scattering theory hands ultracold physics a single, almost miraculous number. When two atoms collide at a microkelvin or below, their de Broglie wavelength dwarfs the range of the interatomic potential, and every detail of that potential collapses into one length, the s-wave scattering length $a$ , recalled here from aq-optics-collisions-scattering-length. Dilute-gas interactions are then completely specified by $a$ : the mean-field energy of a condensate, the pairing in a Fermi gas, the stability of a soliton all follow from it. But nature hands you whatever value the potential happens to produce, and you cannot change it. A Feshbach resonance turns that fixed number into a dial. Two atoms colliding in an energetically open channel — internal states that can separate to infinity — are allowed to couple resonantly to a bound molecular state in a closed channel, whose threshold sits above the collision energy so the atoms cannot fly apart in it. Because the two channels are built from different Zeeman sublevels, they carry different magnetic moments, so a magnetic field slides the molecule up or down relative to the open threshold. Sweep the field, the molecular level crosses threshold, and at that crossing the scattering length diverges through $\pm\infty$ . A benchtop electromagnet becomes a knob on the strength and even the sign of the force between atoms — the master control tool of modern cold-atom physics, and the gateway to molecular condensates, the BEC–BCS crossover, and universal few-body physics.

Key takeaways

Two ultracold atoms collide in an open channel; a closed-channel bound molecule tuned to their collision energy resonantly enhances the scattering and rewrites the s-wave scattering length $a$ .
Open and closed channels have different magnetic moments $\delta\mu$ , so a field $B$ tunes the molecular level through threshold and drives $a(B)$ across $\pm\infty$ .
Near the resonance field $B_0$ , $a(B)=a_{\mathrm{bg}}\left(1-\Delta/(B-B_0)\right)$ : it diverges at $B_0$ and passes through zero at $B_0+\Delta$ .
The divergence does not mean an infinite cross section — at finite collision energy unitarity caps $\sigma\le4\pi/k^2$ .
On the $a>0$ side a universal "halo" dimer exists with binding energy $E_b=\hbar^2/(m a^2)$ , adiabatically connected to a Cooper pair on the $a<0$ side.
Key idea: because an open-channel collision resonantly couples to a closed-channel molecule whose energy is Zeeman-tunable, a magnetic field turns the fixed scattering length into a fully adjustable knob $a(B)=a_{\mathrm{bg}}\left(1-\Delta/(B-B_0)\right)$ that sweeps interactions from strongly attractive through zero to strongly repulsive.

The core insight

A collision in an energetically open channel resonantly couples to a bound molecular state in a closed channel; because the two channels differ in magnetic moment, an external field tunes that state through the collision threshold and drives the s-wave scattering length $a(B)=a_{\mathrm{bg}}\left(1-\Delta/(B-B_0)\right)$ through $\pm\infty$ .

Where this lives today

Feshbach tuning is now a standard fixture on essentially every ultracold-atom apparatus, wired into the same rack as the trapping lasers. The showcase application is the BEC–BCS crossover in a two-spin-state ⁶Li or ⁴⁰K gas: a slow field ramp across the broad resonance carries the system continuously from a condensate of tightly bound molecules ( $a>0$ ), through the unitary regime at $B_0$ , to a BCS superfluid of Cooper pairs ( $a<0$ ) — one experiment spanning two textbook limits of superfluidity, with the unitary Fermi gas between them a clean benchmark whose thermodynamics (the Bertsch parameter) is checked against quantum Monte Carlo. The same tool associates ultracold molecules: an adiabatic sweep converts free atom pairs into halo dimers, which STIRAP then transfers to the rovibrational ground state, giving dense gases of polar molecules for quantum chemistry and dipolar physics. And near unitarity the three-body problem reveals the Efimov effect — a geometric ladder of trimers with universal ratio $e^{\pi/s_0}\approx22.7$ in $a$ — seen as resonant peaks in $L_3$ , a tabletop confirmation of discrete scale invariance.

The physical picture

Picture two potential energy curves plotted against the distance $R$ between the atoms. The lower is the open channel: the atoms in the hyperfine states they were prepared in, free to separate to infinity where the curve flattens to the collision threshold, set at $E=0$ . The upper is a closed channel: a different pair of internal states whose asymptote lies above the collision energy, so a slow atom pair can never come apart along it — but its well holds one or more vibrational bound states, genuine diatomic molecules. As the atoms collide with a whisper of energy just above threshold, quantum mechanics lets them virtually hop onto the closed curve, linger for an instant in the bound state, and hop back. When that state sits exactly at threshold, the detour becomes overwhelmingly probable and the collision is resonantly amplified.

Figure

Two-channel model. The open channel where atoms are free at infinity sets the collision threshold at E = 0, while the higher closed channel holds a bound molecular state. A magnetic field slides that state at the rate δμ ≈ 2μ_B ≈ 2.8 MHz per gauss, and when the level reaches threshold the scattering length diverges

Because the molecule and the free-atom pair are built from different spin states, they feel a field differently — their moments differ by $\delta\mu$ — so a field $B$ slides the molecular level relative to threshold at a few megahertz per gauss. Cross the field where the molecule sits at threshold and the scattering length swings from $+\infty$ to $-\infty$ , turning strong attraction into strong repulsion. This is what a cold-atom experimenter means by "dialing in the interactions," and the second figure shows the dial for a real atom.

Figure

The knob. Magnetic tuning of the s-wave scattering length across the broad lithium-6 resonance at B₀ = 832.2 G with background a_bg = −1405 a₀ and width Δ = −262.3 G. The length diverges at B₀, passes through zero at 569.9 G, is positive on the molecular side and negative on the BCS side

On the repulsive $a>0$ side, just past the pole, a genuine but absurdly weakly bound molecule exists: a "halo dimer" so diffuse that the two atoms spend most of their time outside the potential that binds them. Its size grows as $a$ and its binding energy shrinks as $1/a^2$ , so as $B\to B_0$ the molecule swells without limit and dissolves back into the continuum. The universal punchline is that near resonance every microscopic detail of the short-range potential has washed out — two different species tuned to the same $a$ collide, pair, and bind identically. That is what makes a cold-atom gas a quantitative, tabletop model for strongly interacting quantum matter.

The core equations

The one equation an experimenter carries to the lab is the shape of the dial: how the scattering length depends on field near a resonance. Two parameters beyond the background, and its structure — one pole, one zero — is fixed by the bound-state crossing.

a(B)=a_{\mathrm{bg}}\left(1-\frac{\Delta}{B-B_0}\right) \qquad (1)

where: $a(B)$ — s-wave scattering length versus field; $a_{\mathrm{bg}}$ — background scattering length set by the open-channel potential; $B_0$ — resonance field, where $a$ diverges; $\Delta$ — resonance width, separating the pole at $B_0$ from the zero at $B_0+\Delta$ . Physically: far from $B_0$ the second term dies and $a\to a_{\mathrm{bg}}$ ; approaching $B_0$ the pole drives $|a|\to\infty$ ; at $B_0+\Delta$ the interaction switches off. That pole exists because the closed-channel molecule is Zeeman-tuned through threshold: its energy moves linearly with field, at a slope equal to the differential magnetic moment — the fact that makes magnetic control possible at all.

E_c(B)=\delta\mu\,(B-B_c),\qquad \delta\mu=\mu_{\mathrm{closed}}-\mu_{\mathrm{open}} \qquad (2)

where: $E_c$ — closed-channel bound-state energy from the open threshold; $\delta\mu$ — difference in magnetic moment between the channels (of order a Bohr magneton $\mu_B$ ); $B_c$ — bare crossing field before the coupling shift to $B_0$ . If $\delta\mu$ were zero the molecule and threshold would move together and no field could tune them into resonance. On the $a>0$ side the coupling leaves a real bound state fixed by $a$ alone — the universal halo dimer, whose binding energy is the cleanest signature of universality in the subject.

E_b=\frac{\hbar^2}{m a^2}\qquad (a>0,\ a\gg r_0) \qquad (3)

where: $E_b$ — binding energy of the weakly bound molecule; $\hbar$ — reduced Planck constant; $m$ — atomic mass (reduced mass $m/2$ for identical atoms); $a$ — scattering length; $r_0$ — potential range. Valid only in the universal window $a\gg r_0$ ; $E_b$ knows nothing about the potential — only about $a$ . The scattering length is also the direct input to condensate mean-field physics: it sets the contact coupling in the Gross–Pitaevskii equation, so tuning $a$ tunes a condensate's energy, size, and stability.

g=\frac{4\pi\hbar^2 a}{m} \qquad (4)

where: $g$ — contact interaction strength; $a$ — scattering length; $m$ — atomic mass. Positive $a$ repels (a stable condensate); negative $a$ attracts (collapse, or a 1D bright soliton); $a=0$ is an ideal gas. The elastic cross section grows with $a^2$ but not forever: at finite collision energy it saturates at the unitarity bound, capped by the wavevector rather than by $a$ — the most misunderstood point in the subject.

\sigma=\frac{4\pi a^2}{1+k^2 a^2}\ \xrightarrow[\ k|a|\to\infty\ ]{}\ \frac{4\pi}{k^2} \qquad (5)

where: $\sigma$ — total elastic s-wave cross section for distinguishable particles (identical bosons carry a factor of 2, giving $8\pi/k^2$ ); $k$ — relative collision wavevector, $E=\hbar^2 k^2/2\mu$ ; $a$ — scattering length. Once $k|a|\gg1$ the $a^2$ cancels and $\sigma$ depends only on how slow the atoms are. Finally, the price of large $|a|$ is fast inelastic loss: three atoms recombine into a dimer plus a fast atom, dumping binding energy into the gas, and the loss climbs steeply.

L_3\propto a^4\qquad (a>0) \qquad (6)

where: $L_3$ — three-body recombination coefficient (loss rate $\dot n/n = -L_3 n^2$ ); $a$ — scattering length. The $a^4$ scaling is why "infinite interaction" is never free — near resonance the gas heats and empties, and every experiment is a race against $L_3$ .

Derivation AWhy the scattering length diverges when a bound state hits threshold

The goal is to show, from a two-channel model, that coupling an open-channel collision to a single closed-channel bound state produces exactly the pole-plus-zero structure of equation (1), with the divergence where the molecular level crosses threshold. We treat the collision as two coupled subspaces — an open channel with a scattering continuum and a closed channel holding one bound molecule of energy $E_c(B)$ — and use Feshbach's projection to fold the closed channel into an effective open-channel interaction, then read off the scattering length.

Step 1. Split the Hilbert space with projectors $P$ (open) and $Q$ (closed), so the open-channel dynamics acquires an energy-dependent effective potential $V_{\mathrm{eff}}(E)=V_{PP}+V_{PQ}\,(E-H_{QQ})^{-1}\,V_{QP}$ . Why: eliminating the closed channel exactly is legitimate because at low energy the atoms cannot escape along it — it feeds back only as a virtual, resonant correction.

Step 2. Keep only the closest bound state $|\phi_c\rangle$ , so the second term reduces to a separable resonant piece $V_{PQ}|\phi_c\rangle\langle\phi_c|V_{QP}/(E-E_c(B))$ with a pole at $E=E_c(B)$ . Why: near threshold every other closed-channel state is far away and non-resonant; the physics is carried by the one molecular level being tuned toward $E=0$ .

Step 3. Take the zero-energy limit that defines the scattering length, giving $a(B)=a_{\mathrm{bg}}-C/E_c(B)$ with coupling $C>0$ and $a_{\mathrm{bg}}$ the background from $V_{PP}$ alone. Why: at $E\to0$ the resonant denominator becomes $-E_c(B)$ , so the correction to $a$ grows without bound as the molecule approaches threshold.

Step 4. Insert the Zeeman tuning $E_c(B)=\delta\mu\,(B-B_c)$ from equation (2), turning the energy pole into a field pole and shifting the crossing from the bare $B_c$ to the dressed $B_0$ . Why: the field enters only through the molecular energy, so the divergence tracks the field where $E_c=0$ .

Step 5. Absorb the constants into a width $\Delta=C/(a_{\mathrm{bg}}\,\delta\mu)$ to recover $a(B)=a_{\mathrm{bg}}\left(1-\Delta/(B-B_0)\right)$ , with a pole at $B_0$ and a zero at $B_0+\Delta$ . Why: this is the canonical experimental form — $B_0$ and $\Delta$ encode the crossing field and the coupling, and the sign of $\Delta$ fixes which side carries $a>0$ .

Go deeper · Derivation B · 1 minThe universal binding energy of a halo dimer

Derivation BThe universal binding energy of a halo dimer

The goal is to prove that on the $a>0$ side a shallow bound state exists with binding energy $E_b=\hbar^2/(m a^2)$ and mean size $\langle r\rangle=a/2$ , both fixed by $a$ alone with no reference to the potential's shape. The trick is that when $a\gg r_0$ , the dimer wavefunction lives almost entirely where the potential has already vanished, so free-space quantum mechanics with one boundary condition does all the work.

Step 1. Write the zero-angular-momentum radial equation outside the range, where the potential is zero, for a bound state at $E=-E_b$ : the reduced wavefunction $u=r\psi$ obeys $u''=\kappa^2 u$ with $\kappa=\sqrt{2\mu E_b}/\hbar$ and reduced mass $\mu=m/2$ . Why: for $r>r_0$ there is no force, so the exact bound state must be a free decaying exponential — the potential's complexity is pushed to the boundary at $r_0$ .

Step 2. Select the normalizable solution $\psi(r)=N e^{-\kappa r}/r$ , whose decay length $1/\kappa$ is the physical size and whose mean radius is $\langle r\rangle=1/(2\kappa)$ . Why: only the falling exponential is square-integrable, and its single scale $\kappa$ must be everything the dimer knows about itself.

Step 3. Impose the boundary condition that defines the scattering length — the logarithmic derivative of $r\psi$ extrapolates to $-1/a$ at the origin — which forces $\kappa=1/a$ . Why: the short-range physics that fixes how free waves scatter must also fix how the bound state matches on, so a large positive $a$ means a shallow, enormous molecule.

Step 4. Substitute $\kappa=1/a$ into $E_b=\hbar^2\kappa^2/2\mu$ to get $E_b=\hbar^2/(2\mu a^2)=\hbar^2/(m a^2)$ , and read off $\langle r\rangle=1/(2\kappa)=a/2$ . Why: the binding energy depends only on $a$ , the sharpest statement of universality — and it is why $a\to\infty$ dissolves the molecule ( $E_b\to0$ , $\langle r\rangle\to\infty$ ) exactly at $B_0$ .

Go deeper · Derivation C · 1 minThe unitarity bound on the cross section

Derivation CThe unitarity bound on the cross section

The goal is to reconcile the diverging scattering length with the finite cross section measured on resonance, by showing that the low-energy s-wave amplitude gives a cross section that grows as $a^2$ only while $k|a|\ll1$ and then saturates at $4\pi/k^2$ , a bound set by probability conservation rather than by $a$ . This resolves the most common misconception in the subject: that " $a=\infty$ " ought to mean an infinite scattering rate.

Step 1. Write the s-wave amplitude in the effective-range expansion, keeping leading terms, $f_0(k)=1/(-1/a-ik)$ , where the $-ik$ is demanded by the optical theorem and is not optional. Why: unitarity — conservation of probability flux — forces the imaginary part $-ik$ , and it is precisely this term that caps the cross section when $a$ blows up.

Step 2. Form $|f_0|^2=1/(1/a^2+k^2)$ and multiply by $4\pi$ for the total elastic cross section $\sigma=4\pi/(1/a^2+k^2)=4\pi a^2/(1+k^2a^2)$ . Why: for distinguishable particles the s-wave cross section is $4\pi|f_0|^2$ ; the algebra just recovers the form in equation (5).

Step 3. Take the limits. For $k|a|\ll1$ , $\sigma\to4\pi a^2$ , the familiar growth; for $k|a|\gg1$ , including on resonance where $a\to\infty$ , the $a$ cancels and $\sigma\to4\pi/k^2$ . Why: once $a$ exceeds the collision wavelength $1/k$ , making it larger changes nothing — the atoms already scatter with unit probability, and the only scale left is their own wavelength. *The two limits confirm the structure: $\sigma\to4\pi a^2$ as $k\to0$ , and $\sigma\to4\pi/k^2$ as $a\to\infty$ ; for identical bosons symmetry doubles the amplitude, giving $8\pi/k^2$ .*

Limiting cases

Case	Result	Meaning
Far off resonance	$a(B)\to a_{\mathrm{bg}}$	Away from $B_0$ interactions revert to the ordinary background value.
On resonance (unitarity)	$a\to\pm\infty,\ \sigma=4\pi/k^2$	The length diverges but the cross section is capped by the collision energy; thermodynamics becomes universal.
Zero crossing	$a(B_0+\Delta)=0$	Interactions switch off — a nearly ideal gas, prized for atom interferometry and optical clocks.
Weakly bound dimer ( $a>0,\ a\gg r_0$ )	$E_b=\hbar^2/(m a^2),\ \langle r\rangle=a/2$	A universal halo molecule whose binding energy vanishes as $a\to\infty$ .
Broad vs narrow	$\zeta=\dfrac{a_{\mathrm{bg}}\,\delta\mu\,\Delta}{\hbar^2/(m\,\bar a^2)}$	Broad, entrance-channel-dominated resonances ( $\zeta\gg1$ ) stay universal over a wide range; narrow ones ( $\zeta\ll1$ ) retain closed-channel character.

Common trapWhen the scattering length diverges, the scattering rate also diverges

It does not. The cross section $\sigma=4\pi a^2/(1+k^2a^2)$ grows as $a^2$ only while $k|a|\ll1$ ; once $k|a|\gg1$ — and certainly on resonance, where $a\to\infty$ — the $a$ cancels and $\sigma$ saturates at $4\pi/k^2$ (or $8\pi/k^2$ for identical bosons), set purely by how cold the gas is. So "infinite scattering length" means the collision is as strong as quantum mechanics allows at that energy, not infinitely strong. Two traps ride along: large $|a|$ brings fast three-body recombination, $L_3\propto a^4$ , so the strongly interacting regime is also the lossy one; and a real molecule exists only on the $a>0$ side — on the $a<0$ side the "bound state" is virtual, not something you can hold.

Worked examples

1 · Tuning lithium-6 and weighing its halo dimer (confidence builder)

The broad ⁶Li resonance has $B_0=832.2$ G, $\Delta=-262.3$ G, and $a_{\mathrm{bg}}=-1405\,a_0$ . Find the scattering length at $B=700$ G and the binding energy of the halo dimer there. Use $a_0=5.292\times10^{-11}$ m, $m=9.99\times10^{-27}$ kg, $\hbar=1.055\times10^{-34}$ J·s.

Step 1. Evaluate equation (1) at $B=700$ G. The offset is $B-B_0=-132.2$ G, so $\Delta/(B-B_0)=(-262.3)/(-132.2)=1.984$ , and $a=a_{\mathrm{bg}}(1-1.984)=(-1405)(-0.984)=\mathbf{+1383\,a_0}$ . Large and positive — the molecular side, below the pole, where the second figure places the green dot.

Step 2. Convert to SI: $a=1383\times5.292\times10^{-11}\,\mathrm{m}=7.32\times10^{-8}$ m. Since $a\gg r_0\approx30\,a_0$ (the van der Waals range of Li), formula (3) applies, with $a^2=5.36\times10^{-15}\ \mathrm{m}^2$ and $m a^2=5.35\times10^{-41}$ kg·m².

Step 3. Binding energy: $E_b=\hbar^2/(m a^2)=(1.055\times10^{-34})^2/(5.35\times10^{-41})=2.08\times10^{-28}$ J, i.e. $E_b/h=\mathbf{314\ kHz}$ and $E_b/k_B=\mathbf{15\ \mu K}$ . This is the molecule a field sweep would associate at 700 G — bound by a sub-microkelvin energy yet over 7 nm across, of size $\langle r\rangle=a/2\approx690\,a_0$ . At 700 G, $a=+1383\,a_0$ and $E_b\approx h\times314$ kHz $\approx k_B\times15\ \mu$ K.

2 · The unitarity-limited cross section of a microkelvin gas

A ⁶Li gas at $T=1\ \mu$ K is tuned to resonance, $a\to\infty$ . Find the collision wavevector, the unitarity-limited cross section, and the $|a|$ beyond which the gas is effectively unitary. Use $\mu=m/2=5.00\times10^{-27}$ kg, $k_B=1.381\times10^{-23}$ J/K.

Step 1. Estimate the relative collision wavevector from $E\sim k_B T=\hbar^2 k^2/2\mu$ , so $k=\sqrt{2\mu k_B T}/\hbar$ . Numerically $2\mu k_B T=1.38\times10^{-55}$ , whose root is $3.71\times10^{-28}$ , giving $k=3.71\times10^{-28}/1.055\times10^{-34}=\mathbf{3.5\times10^{6}\ m^{-1}}$ .

Step 2. The unitarity cross section is $\sigma=4\pi/k^2=4\pi/(3.5\times10^{6})^2=\mathbf{1.0\times10^{-12}\ m^2}$ — the largest elastic cross section the gas can have at this temperature, and independent of $a$ once $k|a|\gg1$ .

Step 3. The crossover happens at $k|a|\approx1$ , i.e. $|a|\approx1/k=2.84\times10^{-7}$ m $=\mathbf{5.4\times10^{3}\,a_0}$ . So any scattering length beyond about $5000\,a_0$ — whether $10^4$ or $10^6\,a_0$ — gives the same cross section, because the atoms already scatter with unit probability; the last stretch toward the pole buys no extra scattering, only extra loss. At $1\ \mu$ K, $\sigma_{\text{unitary}}\approx1.0\times10^{-12}\ \mathrm{m}^2$ and the gas is effectively unitary once $|a|\gtrsim5000\,a_0$ .

Physics of the portrait

The idea that a resonance can be built from a bound state hiding in a channel the particles cannot enter did not begin with cold atoms — it began in the nucleus, and carries the name of the theorist who formulated it most cleanly.

Origins · 1 minHerman Feshbach and how this was discovered

Herman Feshbach

1917–2000 · MIT nuclear theorist whose closed-channel resonance concept migrated into AMO physics

Herman Feshbach was a nuclear theorist at MIT, and the resonances that bear his name were first a story about neutrons hitting nuclei. In his 1958 unified theory of nuclear reactions he showed how to split a collision into "open" channels a projectile can escape through and "closed" channels it cannot, and how a bound state in a closed channel produces a sharp resonance in the open-channel cross section. The mathematics — projecting the closed channel out to leave an energy-dependent effective interaction with a pole — is exactly Step 1 of the first derivation above. Ugo Fano had described the same discrete–continuum coupling independently in atomic autoionization, which is why spectroscopists speak of "Fano resonances" for the asymmetric line shapes it produces. What made the concept explode in atomic physics was the realization, worked out by Verhaar's group and by Tiesinga and collaborators in the early 1990s, that in ultracold alkali collisions the closed channel is a molecular bound state and the knob is a magnetic field, because the channels differ in Zeeman energy. When Ketterle's group saw the predicted divergence in a sodium condensate in 1998, Feshbach's nuclear bookkeeping became the master control tool of a new field.

History

1958

— Herman Feshbach formulates a unified theory of nuclear reactions using closed-channel resonances

1961

— Ugo Fano casts the same discrete–continuum coupling as an asymmetric resonance line shape in atomic autoionization

1976

— Stwalley predicts magnetically tunable resonances in collisions of spin-polarized hydrogen

1993

— Tiesinga, Verhaar, and Stoof predict field-tunable Feshbach resonances in ultracold alkali gases

1998

— Inouye and Ketterle's group observes a Feshbach resonance in a sodium condensate

2003

— molecular Bose–Einstein condensates are created by associating Feshbach dimers

2006

— the Efimov effect is observed near a cesium Feshbach resonance

Going further

Texts. Chin, Grimm, Julienne & Tiesinga, Feshbach Resonances in Ultracold Gases, Rev. Mod. Phys. 82, 1225 (2010) — the definitive review · Pethick & Smith, Bose–Einstein Condensation in Dilute Gases, Ch. 5 · Foot, Atomic Physics, §10 (cold collisions) · Demtröder, Laser Spectroscopy (cold-atom collisions). The broad ⁶Li resonance parameters used throughout ( $B_0=832.2$ G, $\Delta=-262.3$ G, $a_{\mathrm{bg}}=-1405\,a_0$ ) are the accepted experimental values. Key results to keep in mind: the pole/zero structure of $a(B)$ ; the halo-dimer binding energy $E_b=\hbar^2/(m a^2)$ from matching an exponential tail with $\kappa=1/a$ ; the unitarity cross section $4\pi/k^2$ ( $8\pi/k^2$ for identical bosons); and the Efimov scaling ratio $e^{\pi/s_0}\approx22.7$ . The worked-example results are $a(700\,\mathrm{G})=+1383\,a_0$ , $E_b\approx h\times314$ kHz, and $\sigma_{\text{unitary}}\approx1.0\times10^{-12}\ \mathrm{m}^2$ at 1 μK.

Builds on: Magnetic Trapping of Neutral Atoms · Evaporative Cooling · Bose–Einstein Condensation in Dilute Gases. Leads to: Degenerate Fermi Gases and the BEC–BCS Crossover · Bose–Einstein Condensation in Dilute Gases.

Ask the tutor: Ask me to trace a magnetic-field sweep across a real resonance and show, step by step, how the scattering length, the halo-dimer binding energy, and the three-body loss all evolve together.

What is the difference between an open and a closed channel?

A channel is a specific pair of internal (hyperfine and Zeeman) states the two atoms occupy, each with a definite asymptotic energy when they are far apart. A channel is open if its asymptote lies below the collision energy — the atoms can separate to infinity in those states. It is closed if its asymptote lies above the collision energy — energy conservation forbids the atoms from flying apart in it, so at long range it is classically forbidden. Crucially, a closed channel can still hold bound molecular states in its well, and it is one of those, tuned onto the open threshold, that produces the resonance.

Why does the resonance have a zero as well as a pole?

The two-parameter form $a=a_{\mathrm{bg}}(1-\Delta/(B-B_0))$ carries both a pole at $B_0$ and a zero at $B_0+\Delta$ , both physical. The pole is where the closed-channel molecule sits at threshold and the resonant contribution overwhelms everything. The zero is where the resonant contribution exactly cancels the background $a_{\mathrm{bg}}$ , leaving no net s-wave interaction. Experimentally the zero is a gift: parking at $B_0+\Delta$ switches interactions off, giving a nearly ideal gas invaluable for atom interferometry and optical clocks, where collisional shifts are a systematic error to eliminate.

Is a Feshbach molecule a "real" molecule?

On the $a>0$ side, yes — a genuine bound state, but an extreme one. The halo dimer has binding energy $E_b=\hbar^2/(m a^2)$ , near resonance a fraction of a microkelvin, and size $\langle r\rangle=a/2$ reaching hundreds of nanometres, far larger than an ordinary molecule. The atoms spend most of their time in the classically forbidden region, so it is almost all "halo" and almost no bond. It is real enough to be counted, imaged, and Bose-condensed, yet so weakly bound that a small change of field dissolves it into free atoms. On the $a<0$ side there is no molecule at all — only a virtual state.

Why can't I just make interactions as strong as I like by sitting on the pole?

Because the elastic scattering you want and the inelastic loss you don't scale differently. The elastic cross section saturates at $4\pi/k^2$ once $k|a|\gg1$ , so pushing closer to the pole buys no extra elastic scattering — the atoms already interact as strongly as the collision energy permits. Meanwhile three-body recombination keeps climbing as $L_3\propto a^4$ , so the closer you sit to $B_0$ the faster three atoms bundle into a dimer plus a fast atom, heating and emptying the trap. Real experiments therefore work near resonance, balancing strong interactions against tolerable loss.

How does sweeping the field turn atoms into molecules?

The trick is adiabaticity. As the field ramps toward $B_0$ from the $a<0$ side, the molecular level crosses the open threshold, and a free atom pair and the bound molecule form an avoided crossing. If the ramp is slow compared with the coupling, the system follows the lower adiabatic branch — starting as two free atoms and ending as one Feshbach molecule, with efficiency set by ramp speed and local pair density. Those halo dimers are only the first step; an optical transfer (STIRAP) then carries them to the rovibrational ground state.

Do Feshbach resonances only occur in s-wave collisions?

No, but s-wave resonances dominate the ultracold regime. At microkelvin temperatures higher partial waves ( $p$ -wave, $d$ -wave) are suppressed by centrifugal barriers, so they matter mainly for identical fermions, which antisymmetry forbids from colliding in s-wave. Magnetically tunable $p$ -wave resonances have been observed, but they are narrower, more loss-prone, and less universal, because the centrifugal barrier ties them more tightly to short-range physics. For the great majority of experiments the relevant object is the broad s-wave resonance analysed here.