Magnetically Tuned Scattering in Ultracold Gases

Ultracold atomic gases provide a pristine platform for exploring quantum many-body physics. By cooling dilute vapors of alkali atoms to temperatures on the order of microkelvin or below, experimenters can reach regimes where the de Broglie wavelengths of particles overlap and quantum statistics dominate the dynamics. At such low energies, the details of interatomic potentials become irrelevant and the interactions between atoms can be parameterized by a single quantity: the s-wave scattering length $a$. The sign and magnitude of this parameter dictate whether particles effectively attract, repel, or do not interact. One of the most powerful tools for controlling $a$ is the Feshbach resonance. By tuning an external magnetic field, experimentalists adjust the relative energy between an open scattering channel and a closed molecular channel, allowing the two to resonantly couple. Near the resonance, the scattering length diverges, enabling access to both strongly attractive and strongly repulsive regimes. This calculator implements the simplest Feshbach resonance model to evaluate the scattering length $a$ and the corresponding zero-energy cross-section from a set of experimentally relevant parameters.

Within the single-resonance approximation, the dependence of the scattering length on magnetic field $B$ is expressed as

$a\left(B\right)=a{\mathrm{bg}}_{}(1-\frac{\Delta }{B-B{0}_{}})$

where $a{\mathrm{bg}}_{}$ is the background scattering length far from resonance, $\Delta$ is the width of the resonance, and $B{0}_{}$ is the resonance position. All three quantities are typically measured in Gauss and Bohr radii, as reflected in the calculator inputs. In physical units, the scattering length determines the low-energy cross-section via $\sigma =4\pi a{2}^{}$, implying that even moderate changes in $a$ can drastically modify collision rates. The ability to tune $a$ across zero by passing through the divergence underlies many landmark experiments, including the production of molecular Bose-Einstein condensates, the exploration of the BEC–BCS crossover, and studies of Efimov few-body states.

Understanding the behavior of the scattering length near resonance is critical for designing experiments. As the magnetic field approaches $B{0}_{}$, the denominator in the fractional term shrinks, causing $a$ to diverge to ±∞ depending on the sign of $\Delta$. On one side of the resonance, the scattering length is large and positive, supporting a weakly bound molecular state with binding energy approximated by

$E{b}_{}=\frac{{ħ}^2}{2\mu a^2}$

where $\mu$ is the reduced mass of the atom pair. On the other side, $a$ becomes large and negative, signifying effectively attractive interactions without a true bound state. The cross-section formula ensures that the collision probability always remains positive, even when $a$ is negative, because it depends on $a^2$. The resonance width determines how rapidly the scattering length varies with magnetic field; broad resonances (large $\Delta$) allow for smooth tuning with minimal sensitivity to magnetic noise, whereas narrow resonances demand precise field control but can offer specialized interaction features such as strong effective range corrections.

The table below lists several commonly used resonances as reference points. The parameters are illustrative and may vary slightly across experiments.

Species	abg (a0)	Δ (G)	B0 (G)
¹⁹⁵Rb	100	300	1007.4
⁴⁰K	174	7.8	202.1
³⁹K-⁴±K mixture	40	1.2	350.0

When using the calculator, enter the background scattering length in units of the Bohr radius $a{0}_{}$ (approximately 0.0529 nm). The magnetic field inputs are given in Gauss because most experimental coils are calibrated in this unit. The script converts $a{\mathrm{bg}}_{}$ to meters using the Bohr radius and performs the calculation entirely in SI units. The resulting scattering length is displayed both in Bohr radii and nanometers for convenience, along with the cross-section in square meters. By comparing the cross-section to typical values, users can gauge the likelihood of two-body collisions. For example, a scattering length of 1000 a₀ yields a cross-section on the order of 10⁻¹² m², greatly enhancing thermalization rates in a gas.

Beyond two-body physics, tuning the scattering length has profound consequences for many-body phenomena. In Bose gases, a positive $a$ sets the strength of the mean-field interaction energy $(4\pi {ħ}^{2}a/m)$ that enters the Gross-Pitaevskii equation. A large negative $a$ can drive attractive gases toward collapse, as observed in lithium condensates. In Fermi gases, the sign change of $a$ across the resonance connects the Bose-Einstein condensation of molecules on the positive side to Bardeen-Cooper-Schrieffer pairing on the negative side. Precisely at the resonance, where $a$ diverges, the gas exhibits universal unitary behavior independent of microscopic details, enabling tests of strongly interacting quantum theories.

From a practical standpoint, experiments rarely operate exactly at the pole because the divergence in cross-section can lead to enhanced three-body recombination and particle loss. Instead, researchers choose magnetic fields that realize specific finite values of $a$. The calculator therefore becomes an indispensable planning tool: by plugging in the desired $a$ and inverting the resonance equation, one can determine the required magnetic field setting. Additionally, precision measurement of resonance parameters, such as $\Delta$ and $B{0}_{}$, is itself a subject of ongoing research, as higher accuracy enables more reliable tuning and comparison with theoretical predictions that include coupling to higher partial waves or many-body shifts.

The Feshbach resonance concept also generalizes beyond magnetic tuning. Optical Feshbach resonances use laser light to couple scattering atoms to excited molecular states, allowing rapid, spatially patterned control of interactions, though at the cost of light-induced losses. Likewise, confinement-induced resonances in low-dimensional traps modify scattering properties purely through geometry. The underlying theme across these variations is the ability to manipulate a single effective parameter—the scattering length—to engineer exotic quantum states. As quantum simulation and quantum information processing with cold atoms advance, precise and dynamic control over $a$ will remain central, making intuitive tools for calculating its field dependence all the more valuable.

Finally, it is worth emphasizing the limitations of the simple resonance formula used here. Real systems exhibit additional physics such as effective range corrections, coupling to multiple closed channels, and magnetic field-dependent background scattering. For broad resonances, these effects are often negligible, but for narrow resonances they can significantly alter the relationship between $a$ and $B$. Nevertheless, the simple model captures the leading-order behavior in most situations and provides a solid starting point for deeper analysis. Users are encouraged to consult the experimental literature for the most accurate parameters relevant to their atomic species and to treat the calculator as an educational resource rather than a definitive reference.