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research34 min studyLesson 485 in Physics

Feshbach resonances and interaction control

Connect open–closed-channel coupling to the signed field dependence of scattering length and to the universal shallow-dimer binding law.

Before you begin
On this page

The question for today

What you’ll understand

  • Locate the pole and zero crossing of a declared scattering-length parametrization.
  • Distinguish the sign of low-energy scattering from the full microscopic potential.
  • Use reduced mass and range conditions in the universal binding energy.

Ultracold collisions can be tuned by shifting a molecular level relative to the threshold of two incoming atoms. Coupling between the incoming channel and that molecular channel creates a Feshbach resonance. Magnetic field is often the control because the two configurations have different magnetic moments. The resulting scattering length can become large, pass through a pole, and change sign. This tunability is powerful only when field calibration, resonance conventions, collision energy, and inelastic loss are kept explicit.

The width is a signed parameter

For an isolated magnetic resonance, use a background scattering length, a pole field, and a signed width to specify the usual zero-energy parametrization. The pole occurs at the stated resonance field, while the zero crossing is displaced by the signed width. Some references rewrite the same relation using a different sign convention for the width. Copying the width without its defining formula can therefore reverse which side is predicted to have positive scattering length.

The bare molecular threshold crossing need not coincide with the dressed scattering pole because channel coupling shifts the resonance. Finite temperature, field noise, and loss can broaden or move an observed feature relative to a zero-energy theoretical parameter. A maximum loss signal is not automatically an exact measurement of the pole. Binding spectroscopy, elastic thermalization, and independently calibrated magnetic fields provide complementary constraints on the scattering model.

a(B)=abg(1−ΔB−B0),a(B0+Δ)=0a(B)=a_{\mathrm{bg}}\left(1-\frac\Delta{B-B_0}\right),\qquad a(B_0+\Delta)=0
a and abg are in metres or a declared common length unit. B, B₀, and signed Δ must use the same field unit. The formula describes a single isolated zero-energy resonance, neglecting overlapping resonances and significant background variation.↔ Scroll wide equations horizontally.

Scattering sign is an effective low-energy statement

At sufficiently low relative momentum, s-wave scattering can be organized by scattering length and effective range. A positive scattering length gives a positive contact coupling in the weak Bose-gas mean-field description, while a negative value gives an attractive contact contribution. These statements do not mean the entire interatomic potential is repulsive or attractive at every separation. Real atomic potentials can support many deeply bound molecular levels on either side of the resonance.

The effective-range term becomes important for narrow resonances or higher collision energy. Large scattering length alone does not establish a universal zero-range gas if effective range is also large. At finite momentum the elastic scattering amplitude remains bounded by the outgoing-wave term, so the collision cross section does not literally diverge at the zero-energy pole. Distinguish tuning a low-energy parameter from producing infinite physical scattering or automatically establishing many-body equilibrium.

f(k)=1−a−1+rek2/2−ik,g=4πℏ2am(equal-mass bosons)f(k)=\frac1{-a^{-1}+r_ek^2/2-ik},\qquad g=\frac{4\pi\hbar^2a}{m}\quad\text{(equal-mass bosons)}
k is relative wavenumber in m⁻¹, re an effective range in metres, f a scattering amplitude in metres, and g a three-dimensional contact coupling in J m³. The truncated expansion requires low enough collision energy and a suitable short-range s-wave regime.↔ Scroll wide equations horizontally.

A shallow dimer has a range of validity

When a is positive and much larger than the interaction and resonance range scales, a shallow bound state's size is set by a. Its binding energy is positive by convention, with the molecular level lying that amount below the free-atom threshold. The reduced mass belongs in this two-body formula. For equal atoms it is half the individual atomic mass, removing the explicit factor two that sometimes causes errors when comparing formulas.

Negative scattering length does not support this universal free-space shallow dimer, but deeper bound states or confinement-induced molecules can still exist. Sweeping through a resonance can produce molecules or excite a gas, with outcomes depending on ramp rate and channel physics. Three-body recombination and other loss channels can restrict the useful interaction range. A static a(B) curve specifies equilibrium two-body scattering; it does not predict molecule conversion, heating, or quench dynamics on its own.

Eb=ℏ22μa2,μ=m1m2m1+m2,Eb=ℏ2ma2(m1=m2=m)E_b=\frac{\hbar^2}{2\mu a^2},\qquad \mu=\frac{m_1m_2}{m_1+m_2},\qquad E_b=\frac{\hbar^2}{ma^2}\quad(m_1=m_2=m)
Eb is positive binding energy in joules, μ and masses are in kilograms, and a > 0 must greatly exceed both relevant interaction-range and resonance-range scales. The bound molecular energy relative to threshold is −Eb. This is a universal near-threshold limit, not every molecular level.↔ Scroll wide equations horizontally.

Work it through

Read both sides of a toy resonance

An illustrative isolated resonance has abg = 1000a₀, B₀ = 200 G, and Δ = +10 G, using the displayed convention. Find scattering lengths at 195 G and 205 G and the zero-crossing field. Here a₀ is a common length unit, not the pole label.

  1. At 195 G, B − B₀ = −5 G. The bracket is 1−10/(−5)=31-10/(-5)=3, so a = 3000a₀.

  2. At 205 G the bracket is 1−10/5=−11-10/5=-1, so a = −1000a₀. Crossing the pole changes the sign in this interval.

  3. The zero crossing is B₀ + Δ = 210 G. The positive side can support the universal shallow dimer only when its large-a range conditions also hold.

The result

The values are +3000a₀ at 195 G, −1000a₀ at 205 G, and zero at 210 G. These are toy parameters rather than a claim about a particular isotope.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

A regular zero and a pole are different objects

Reset and set BOTH Coefficient a to 1 and Position x to 0; predict y = 0 with finite slope. Then Reset and set BOTH a to 1 and x to 1; predict y = 1 and slope 2. This smooth quadratic has no pole; it illustrates a regular zero only and cannot represent a(B), a Feshbach channel, a divergent scattering length, or molecule association.

A regular zero and a pole are different objectsThis lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Current readings: y at chosen x: 0.608; Tangent slope · 2ax: 1.35; Signed area · ax³/3: 0.182. All parameters are adjustable with the labeled controls below.−2.4−1.201.22.4−9−4.504.59xy(x, ax²)
y = ax²TangentSigned integral
y at chosen x
0.608
Tangent slope · 2ax
1.35
Signed area · ax³/3
0.182
t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA zero can be smooth, while a resonance pole requires a different functional structure and interpretation.

Pause & reason

Check your understanding

For equal atoms of mass m, what is the universal shallow-dimer binding energy at large positive a?

Your turn

Try it yourself

Equal atoms satisfy ℏ²/m = 8 × 10⁻⁴⁴ J m². In the universal large-positive-a regime, find Eb for a = 200 nm. If a is halved while remaining well above the relevant range scales, how does Eb change?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Convert nanometres to metres before squaring, then use the inverse-square dependence.

Compare with the solution

At a=2×10−7 ma=2\times10^{-7}\,\mathrm m, the squared length is 4×10−14 m24\times10^{-14}\,\mathrm{m^2}. Hence Eb=(8×10−44)/(4×10−14)=2×10−30 JE_b=(8\times10^{-44})/(4\times10^{-14})=2\times10^{-30}\,\mathrm J. Halving a quadruples Eb to 8 × 10⁻³⁰ J, provided universality remains valid. The molecular level is at minus this energy relative to threshold; Eb itself is defined positive.

Keep the idea

Take it with you

  • A resonance width has meaning only together with its signed scattering-length formula.
  • Large scattering length must be compared with collision momentum and effective range.
  • The universal shallow-dimer law requires positive large a and the two-body reduced mass.

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