Effective Potential for the Second Virial Coefficient at Feshbach Resonance

In a recent paper entitled “High temperature expansion applied to fermions near Feshbach resonance”, (Phys. Rev. Lett. 92 160404 (2004)), Ho and Mueller have demonstrated a remarkable similarity between its high and low temerature properties at resonance. The quantum second virial coefficient plays a crucial role in their analysis, and has a universal value at resonance. In this paper, we explore the connection between the quantum and classical second virial coefficients, and show that near a Feshbach resonance an exact mapping from the quantum to classical form is possible. This gives rise to a scale-independent inverse square effective potential for the classical virial coefficient. It is suggested that this may be tested by measuring the isothermal compressibility of the gas on the repulsive side of the resonance.. PACS: 03,75.-b, 03.75.Ss Typeset using REVTEX 1 Great advances have been made in the study of ultracold trapped atoms in recent times. The interaction between the atoms is normally weak in a dilute gas, but may be enhanced drastically by making use of the so-called Feshbach resonance [1]. This may be achieved by applying a magnetic field to tune the energy of the Zeeman level of the trapped atoms to one of the molecular resonances. At a zero-energy resonance, the scattering phase shift is π/2, and the cross section reaches the unitary limit. It is argued that the properties of a gas at Feshbach resonance have a universal character [2], irrespective of the type of gas in the trap. In this connection, Ho and Mueller [3] recently considered the interaction energy of a twocomponent Fermi gas near a Feshbach resonance. They compared the available experimental data of the ultracold gas in the vicinity of a Feshbach resonance with the predictions in the Boltzmann regime using the quantum second virial coefficient. Surprisingly, it was found that the high temperature expression, when used at micro-Kelvin temperatures, explains the data reasonably well. In the present work, we examine the use of the quantal second virial coefficient in more detail, and its connection to its classical counterpart at the Feshbach resonance. We find that the quantal expression may be mapped onto the classical formula exactly in the vicinity of the Feshbach resonance using an inverse square interaction in the configuration space. To appreciate this in more detail, note that the virial expansion of a (one-component) gas (classical as well as quantal) for the pressure P at a temperature T is given by [4] P nτ = 1 + a2(nλ ) + a3(nλ ) + .... , (1) where a2, a3... etc are the dimensionless second, third..virial coefficients, τ = kBT , n = N/V is the number density of particles, and λ = √ 2πh̄/mτ is the thermal de Broglie wavelength. The free energy F = E−TS may be easily obtained by integrating the above P with respect to the volume V (since P = −(∂F/∂V )τ ), and hence also the energy E = −τ [∂(F/τ)/∂τ ]. After subtracting out the energy of the perfect gas part (classical or quantal), one obtains the virial series for the interaction energy : Eint V = 3 2 nτ [ (nλ){a2 − 2 3 τ da2 dτ }+ (nλ){a3 − 2 3 τ da3 dτ 1 2 }+ ...(nλ){aj − 2 3 τ daj dτ 1 j − 1}+ ... ] (2) This expression is valid for a classical as well as a quantum gas, provided the appropriate classical or quantal virial coefficients are used. Whereas for a classical gas the virial coefficients may be expressed as integrals involving the interaction potential, for the quantum problem a solution of the j-body problem is needed to obtain aj . Henceforth we shall denote the classical virial coefficients by Aj to differentiate from their quantum counterparts, to be still denoted as aj. For an interesting example where classical and quantum results are very different, consider a hard sphere classical Boltzmann gas. For a hard-sphere diameter rc, A2 = 2π 3 ( rc λ ), and the higher order Aj’s may be expressed as Aj = κ(A2), where κ is a constant. Substituting this in Eq.(2), we see that the interaction energy of the classical hard sphere gas vanishes identically in each power of (nλ). This, of course, is not the case for a bosonic or fermionic quantum gas at low temperatures. We note that for a classical gas, (nλ) << 1, and the series (2) may be terminated after the first order term in most cases: 2