Elastic collision rates of spin-polarized fermions in two dimensions

Abstract

We study the p-wave elastic collision rates in a two-dimensional spin-polarized ultracold Fermi gas in the presence of a p-wave Feshbach resonance. We derive the analytical relation of the elastic collision rate coefficient in the close vicinity of resonance when the effective range is dominant. The elastic collision rate is enhanced by an exponential scaling of $e^{-q_{\textrm{r}}^{2}/q_{\textrm{T}}^{2}}$ towards the resonance. Here, $q_{\textrm{r}}$ is the resonant momentum and $q_{\textrm{T}}$ is the thermal momentum. An analogous expression derived for the case of three dimensions successfully explains the thermalization rates measurement in the recent experiment (Phys Rev A 88:012710, 2013). In the zero-range limit where the effective range is negligible, the elastic collision rate coefficient is proportional to temperature $T^2$ and scattering area $A_{\textrm{p}}^2$. Using these elastic collision rates, we studied the energy transfer from high to low velocity through p-wave collision. We also discuss the collisional stability in the presence of three-body losses in the background scattering limit. Our results may provide insight into the dynamics of the two-dimensional spin-polarized ultracold Fermi gas.

Appendix A: Elastic collision rates in three dimensions

The scattering amplitude for p-wave interaction between two fermions with relative wave vector k in three dimension is given by

$$\begin{aligned} f(k)=\frac{-k^2}{1/V_{p}+k_e k^2+i k^3}. \end{aligned}$$

(A1)

Here, $V_{p}$ is scattering volume and $k_e > 0$ is the effective range. The p-wave S-matrix element is given by $S(k)= \exp$ $(2 i \delta (k))$. The elastic rate constant is $K=v \sigma (k)$, where $\sigma (k)=3 \pi \left| 1-S(k)\right| ^{2} /k^{2}$ is the p-wave elastic scattering cross-section, and $v=2 \hbar k / m$ is the relative velocity. As a result, the elastic rate coefficient becomes

$$\begin{aligned} K= {\frac{24\pi \hbar }{m}}{\frac{k^5}{(1/V_{p}+k_e k^2)^2 + k^6}}. \end{aligned}$$

(A2)

Very close to the resonance when $V_{p} \rightarrow \infty$, the largest contribution comes from the momenta of resonant bound state $k_r= 1 /\sqrt{k_e|V_{p}|}$. In the close vicinity of resonant regime, $k_{T} \gg k_r$, where $k_{T}=\sqrt{3\,m k_BT/2\hbar ^2}$ is the thermal momentum. As a result, only a small fraction of relative momenta contributes to the collision process. Following the procedure similar to the two-dimensional case in the main text, we find the expression for the elastic collision rate coefficient

$$\begin{aligned} \langle K\rangle ^n=\frac{96{\pi }^{3/2}\hbar }{m k_e}{\left( \frac{k_r}{k_{T}}\right) ^3} e^{-\left( k^2_{r}/k^2_{T} \right) }. \end{aligned}$$

(A3)

Thermalization rates can be obtained from the above equation as

$$\begin{aligned} \Gamma _{th}= \langle n \rangle \times \langle K\rangle ^{n}/ \alpha . \end{aligned}$$

(A4)

The mean density for a three-dimensional harmonically trapped Fermi gas at temperature T in the Boltzmann regime is given by $\left\langle n \right\rangle ={\frac{1}{48}}{\left( \frac{mk_B}{{\hbar ^2\pi }}\right) ^{3/2}} {\frac{T_F^3}{T^{3/2}}}$ [35]. The dashed curves in Fig. 4 show the fitted thermalization rates $\Gamma$ in comparison to the experimental data from Ref. [34] for four different sets of temperatures. During the fitting we kept all scattering parameters fixed and kept the mean density as the only free parameter. The expression A4 successfully reproduces the experimental results in the narrow range where interaction is sufficiently strong. The mean density obtained from fitting differs from the measured density of approximately $50 \%$ due to uncertainty in the estimation of atom numbers in the trap and as well as trap conditions such as trapping frequencies.

Fig. 4

Thermalization rates close to p-wave Feshbach resonance at $B-B_0=159.17(5)G$ from publication of Nakasuji et al. [34] for four different sets of temperatures. The dashed curves show the result obtained from Eq. A4 with all fixed scattering parameters, which successfully reproduces the experimental results in a narrow range closer to the resonance

At sufficiently far away from the resonance, interaction is weak ($V_{p} \rightarrow 0$) and $k_{T} \ll k_r$. In this regime, elastic collision rates can be approximated as [35]

$$\begin{aligned} \Gamma ^f= \left\langle n \right\rangle \times \frac{288\sqrt{\pi }V_{p}^2 m^{3/2}}{{\hbar }^4} { \left( k_BT \right) }^{\frac{5}{2}}. \end{aligned}$$

(A5)

The ratio of elastic scattering rate for an atom with velocity $v$ compared to average scattering rate $\Gamma ^{f}$ [35]:

$$\begin{aligned} {\tilde{\Gamma }} =\frac{\int _0^\infty \int _0^{\pi } \left( {\tilde{u}}^2+u^2-2{\tilde{u}}u \cos \theta \right) ^{5/2} \sin \theta \textrm{d}\theta u^2 e^{-u^2} \textrm{d}u}{24 \sqrt{3}} \end{aligned}$$

(A6)

At zero velocity one can substitute $v=0$ (${\tilde{u}}=0$) in Eq. A6 and in Eq. 13, which results the ratio ${\tilde{\gamma }}/{\tilde{\Gamma }}=\sqrt{2}$.