Dimensional Crossover and Confinement-Induced Resonances

Transverse confinement changes scattering even when the microscopic interaction is unchanged. Virtual excitation of closed transverse modes modifies the open-channel boundary condition and can drive an effective one-dimensional coupling through a pole. This confinement-induced resonance is a matching effect between three-dimensional short-distance scattering and a discrete geometric spectrum.

Required background. Effective Range, Shallow Poles, and Universality Windows supplies the three-dimensional amplitude and its range domain.

Helpful background. Fundamental Solutions and Green Operators supplies the resolvent construction used to sum transverse modes.

Scattering in a harmonic waveguide

Take two equal-mass particles in an isotropic transverse harmonic trap of frequency $\omega_\perp$ and free longitudinal coordinate $z$ . Define the relative-coordinate transverse length

a_\perp=\sqrt{\frac{2}{m\omega_\perp}}.

Assume the interaction range $R\ll a_\perp$ and collision energy below the first excited transverse threshold. Matching the three-dimensional Bethe–Peierls boundary condition to the confined Green function gives

g_{1\mathrm D} =\frac{4a}{ma_\perp^2} \frac{1}{1-\mathcal C a/a_\perp}, \qquad \mathcal C=-\zeta\!\left(\frac12\right)\simeq1.4603,

in this transverse-length convention. The denominator vanishes at

\frac{a}{a_\perp}=\frac{1}{\mathcal C},

even though the three-dimensional scattering length is finite. This is the confinement-induced resonance found by Olshanii 1998.

Origin of the denominator

The relative Green function at contact separates into the open transverse ground state and closed excited modes,

G_{\mathrm{conf}}(E;0,0) =G_{0,\rm open}(E;0,0) +\sum_{n>0}G_{n,\rm closed}(E;0,0).

The ultraviolet divergence is the same as in free three-dimensional scattering and is removed by matching to $a$ . The finite difference between the confined and free Green functions yields $\mathcal C/a_\perp$ . Repeated open-channel scattering therefore contains the inverse combination $a^{-1}-\mathcal C/a_\perp$ , producing the pole in $g_{1\mathrm D}$ .

This derivation shows why simply averaging the three-dimensional potential over the transverse ground state is incomplete: that projection omits virtual closed modes, precisely the contribution responsible for the resonance.

Effective one-dimensional amplitude

Define the one-dimensional scattering length through

g_{1\mathrm D}=-\frac{2}{m a_{1\mathrm D}}.

With the displayed conventions,

a_{1\mathrm D} =-\frac{a_\perp^2}{2a} \left(1-\mathcal C\frac{a}{a_\perp}\right).

Near the resonance, energy dependence from the transverse resolvent becomes important. A constant $g_{1\mathrm D}$ is reliable only when longitudinal collision energy, temperature, chemical potential, and interaction shifts remain well below the transverse gap and when range corrections such as $R/a_\perp$ are controlled.

Crossover rather than abrupt dimensionality

A gas is kinematically one dimensional when excited transverse populations are negligible, but virtual transverse modes still renormalize its interactions. As $T$ , $\mu$ , or collision energy approaches $\omega_\perp$ , real occupation of excited modes creates a multichannel problem. Trap anisotropy splits thresholds and can generate multiple resonant structures. Finite effective range and narrow Feshbach-resonance physics add further energy dependence.

Common pitfalls

Projecting before renormalizing. Ground-state averaging misses the virtual-mode sum and its confinement-induced pole.

Mixing oscillator-length conventions. Factors of $\sqrt2$ move between $a_\perp$ , $\mathcal C$ , and the prefactor. State the relative or single-particle definition.

Calling the system one dimensional solely because $T<\omega_\perp$ . Chemical potential, interaction energy, collision energy, and nonequilibrium excitation must also lie below the gap.

Exercises

Locate the resonance

At what value of $a$ does the effective coupling diverge, and what happens to $a_{1\mathrm D}$ there?

Solution

The denominator vanishes at $a=a_\perp/\mathcal C$ . The effective one-dimensional scattering length then crosses zero. The divergence is generated by a confined two-body state reaching the open-channel threshold.

Recover weak projection

Expand $g_{1\mathrm D}$ for $|a|\ll a_\perp$ .