Two-Channel Resonance Models

A two-channel model represents a resonant pair by an explicit molecular field coupled to two open-channel atoms. The model keeps the energy dependence that a single leading contact coupling removes. After matching, detuning fixes the scattering length and the atom–molecule coupling fixes a resonance-width scale, allowing broad and narrow many-body regimes to be distinguished quantitatively.

Required background. Effective Range, Shallow Poles, and Universality Windows defines range corrections, and Hubbard–Stratonovich and Collective Fields explains auxiliary and dynamical pair fields.

Atom–molecule field theory

For equal-mass fermions, a minimal Galilean-invariant model is

$$\begin{aligned} \mathcal L={}&\sum_{\sigma} \psi_\sigma^\dagger \left(i\partial_t+\frac{\nabla^2}{2m}\right)\psi_\sigma\\ &+\phi^\dagger \left(i\partial_t+\frac{\nabla^2}{4m}-\nu_0\right)\phi -g_0\left(\phi^\dagger\psi_\downarrow\psi_\uparrow+\text{h.c.}\right). \end{aligned}$$

The molecular kinetic term makes $\phi$ dynamical rather than a purely algebraic Hubbard–Stratonovich field. Its bare detuning $\nu_0$ and coupling $g_0$ are regulator dependent. Dressing the molecule by an atom pair gives

$$D^{-1}(E,\mathbf P) =D_0^{-1}(E,\mathbf P)-g_0^2\Pi_{\mathrm{pair}}(E,\mathbf P),$$

and the two-body amplitude is proportional to $-g_0^2D$. Matching its low-energy denominator to $-a^{-1}+r_ek^2/2-ik$ removes the cutoff dependence.

Resonance position and width

Near an isolated magnetic Feshbach resonance, the measured scattering length is conventionally parameterized as

$$a(B)=a_{\rm bg}\left(1-\frac{\Delta B}{B-B_0}\right),$$

where $a_{\rm bg}$ is the background scattering length, $B_0$ the pole position, and $\Delta B$ the width in magnetic field. This empirical width is not itself the many-body energy-width parameter; it combines with the magnetic-moment difference and background scattering to define a length $R^*$.

In a common low-energy convention for a narrow resonance,

$$k\cot\delta(k)=-\frac1a-R^*k^2+\cdots, \qquad r_e\simeq-2R^*.$$

Background range corrections modify the last relation. The mapping between $g_0$ and $R^*$ depends on field normalization and regulator, so a model should report the matched amplitude rather than a bare $g_0$ alone. The standard resonance parameters and their experimental meaning are reviewed in Chin et al. 2010, §§II.C–II.D.

Broad and narrow many-body limits

At a many-body momentum $Q$ such as $k_F$ or $\lambda_T^{-1}$:

$$QR^*\ll1 \quad\text{broad resonance}, \qquad QR^*\gtrsim1 \quad\text{width dynamics resolved}.$$

In the broad limit the molecular field can be integrated out to a leading contact interaction over the target energy window. In the narrow limit its propagating character is leading order, and a one-channel scattering-length theory is incomplete. Narrowness can sometimes provide a perturbative parameter through the small atom–molecule coupling, but only after density, temperature, and detuning scalings are declared.

The closed-channel fraction is an operator expectation value such as $2\langle\phi^\dagger\phi\rangle/N$ in a chosen normalized model. It is not identical to the pole residue of an arbitrary auxiliary field, because field redefinitions redistribute open- and closed-channel components. Comparison with experiment requires the physical probe matrix element used to calibrate that fraction.

Medium effects and double counting

In matter, the molecular self-energy uses medium-dressed pair propagators or occupation factors. Vacuum matching must remain fixed while the medium contribution is added. If a background contact interaction is retained alongside $\phi$, its diagrams must be organized so that the same scattering process is not counted both through molecule exchange and the background ladder.

Common pitfalls

Equating $\Delta B$ with $R^*$. Field width becomes an energy or length only after including $a_{\rm bg}$ and the differential magnetic moment.

Calling an auxiliary-field residue observable. Field normalization is conventional; probe-calibrated closed-channel content is physical.

Integrating out a narrow molecule. If $QR^*$ is not small, the induced interaction is strongly energy dependent and the one-channel reduction loses leading physics.

Exercises

Integrate out the molecule at tree level

Neglect the molecular derivatives relative to $|\nu_0|$ and eliminate $\phi$.

Solution

Its equation of motion gives $\phi\simeq-(g_0/\nu_0)\psi_\downarrow\psi_\uparrow$ in the displayed sign convention. Substitution generates a contact interaction proportional to $g_0^2/\nu_0$. The reduction fails when collision energies or momenta resolve the omitted molecular derivatives.

Classify a resonance

A gas has $k_F^{-1}=500\,\mathrm{nm}$ and $R^*=20\,\mathrm{nm}$. Is width physics leading for ground-state properties at momentum $k_F$?

Solution

$k_FR^*=20/500=0.04$. Width corrections are parametrically small at the few-percent scale, subject to background range and observable-dependent enhancements. This supports a broad-resonance treatment at modest accuracy, not an exact one-channel description.

Continue

Efimov Physics and the Three-Body Parameter shows why resonant bosons require another input. From Few-Body Inputs to Many-Body Predictions propagates width uncertainty. Cooper Instability and Pairing uses the pair field in a medium.

References

• Chin, Cheng, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. “Feshbach Resonances in Ultracold Gases.” Reviews of Modern Physics 82 (2010): 1225–1286. DOI.

Further reading

• Gurarie, Victor, and Leo Radzihovsky. “Resonantly Paired Fermionic Superfluids.” Annals of Physics 322 (2007): 2–119. DOI.
• Timmermans, Eddy, Paolo Tommasini, R. Côté, M. Hussein, and Arthur Kerman. “Feshbach Resonances in Atomic Bose–Einstein Condensates.” Physics Reports 315 (1999): 199–230. DOI.