Many-body dynamics of $p$-wave Feshbach-molecule production: A mean-field approach

We study the mean-field dynamics of $p$-wave Feshbach-molecule production in an ultracold gas of Fermi atoms in the same internal state. We derive a separable potential to describe the low-energy scattering properties of such atoms, and use this potential to solve the mean-field dynamics during a magnetic-field sweep. Initially, on the negative scattering length side of a Feshbach resonance the gas is described by the BCS theory. We adapt the method by Szymańska et al. [Phys. Rev. Lett. 94, 170402 (2005)] to $p$-wave interacting Fermi gases and model the conversion dynamics of the gas into a Bose-Einstein condensate of molecules on the other side of the resonance under the influence of a linearly varying magnetic field. We have analyzed the dependence of the molecule production efficiency on the density of the gas, temperature, initial value of the magnetic field, and magnetic-field ramp speed. Our results show that in this approximation molecule production by a linear magnetic-field sweep is highly dependent on the initial state.

Figure 1

Variation of the $p$-wave binding energy with the magnetic field for the $m_1=0$ resonance in ${}^{40}\text{K}$. The solid blue line is the solution to Eq. (A3). The dashed red line is the low-energy expansion of Eq. (7). The black solid line shows the variation of the scattering volume (right axis) in the vicinity of the resonance.

Figure 2

$p$-wave elastic-scattering cross section for ${}^{40}$K colliding in the $|9/2,-7/2\rangle$ channel as a function of collision energy at three different magnetic-field strengths. The solid blue line corresponds to the pseudopotential model. The red crosses are from a coupled-channel calculation.

Figure 3

Variation of the parameter ${\Delta }_m$ with magnetic field for the $p$-wave resonance in ${}^{40}\text{K}$ for a density of ${10}^{13}$ cm${}^{-3}$ and a temperature of 70 nK. The solid green line is the value of the $m=0$ resonance and the dashed blue line is for the $\left|m\right|=1$ resonance. There is no significant difference between the results obtained when coupling between the components is included and when the coupling is excluded.

Figure 4

Values of the $p$-wave chemical potential for the resonance in ${}^{40}\text{K}$ for a density of ${10}^{13}$ cm${}^{-3}$ and a temperature of 70 nK. The dashed blue line is the $|m_1|=1$ resonance and the solid green line is the $m_1=0$ resonance. The values for coupled resonances and separated resonances are indistinguishable. The positions of the resonances are marked by the vertical lines.

Figure 5

Variation of the molecular production efficiency in ${}^{40}\text{K}$ as the initial value of the magnetic field is varied. The different lines represent different densities. Higher densities give a larger production efficiency. The temperature is 70 nK and the sweep rate is held constant at $\dot{B}=60$ G/ms.

Figure 6

Variation of the molecular production in ${}^{40}\text{K}$ efficiency as the atomic density is varied. The sweep rate is held constant at $\dot{B}=60$ G/ms. The lines represent temperatures of 70 nK (solid blue), 100 nK (dot-dashed green), and 200 nK (dashed red).

Figure 7

Variation of the molecular production efficiency in ${}^{40}\text{K}$ as the ramp speed is varied for a temperature of 70 nK. The blue, dot-dashed curve refers to an atomic density of ${10}^{15}$ cm${}^{-3}$ and an initial magnetic field of 199 G. The green, dashed curve refers to an atomic density of ${10}^{15}$ cm${}^{-3}$ and an initial magnetic field of 200 G. The red, solid curve refers to an atomic density of $4\times {10}^{14}$ cm${}^{-3}$ and an initial magnetic field of 200 G.

Figure 8

Difference in the molecule production efficiency from a sweep of 10 G/ms and a sweep of 500 G/ms as a function of density. Here, $n_{\dot{B}}$ is the number of molecules over the number of atoms after a sweep at a speed equal to $\dot{B}$ in G/ms for a temperature of 70 nK. This shows how many molecules are actually produced during the dynamics. The dashed curve refers to an initial magnetic-field position of $B_i=199.3$ G, while the lower curve refers to $B_i=199.2$ G. It can be seen there is an optimum density at which to produce molecules from the dynamics.

Figure 9

Evolution of the molecule production efficiency after an infinitely fast sweep of the magnetic field across the 198.85 G resonance in ${}^{40}$K. The initial magnetic field is 199 G, just above the resonance. The different lines correspond to differing final magnetic fields of 198.5 G (solid, green line), 197.5 G (dashed, blue line), and 196.5 G (dot-dashed, red line). $n\left(t\right)$ is the density of molecules as a function of time where $n\left(0\right)$ is the density of molecules directly after the magnetic-field variation.

Figure 10

Evolution of the molecule production efficiency after an infinitely fast sweep of the magnetic field across the 198.85 G resonance in ${}^{40}\text{K}$. The final field is held constant at 197.5 G. The different lines represent different initial magnetic fields of 199 G (solid, blue line), 199.5 G (dashed, green line), 200 G (dot-dashed, red line), and 200.5 G (dotted, black line). Other symbols are as defined in Fig. 9.

Figure 11

Variation of the quantity $\left|\Delta \left(t\right)\right|/{\Delta }_{\text{eq}}$ with time for final magnetic fields of 197 G (top panel) and 198 G (bottom panel). The different lines correspond to different initial magnetic-field positions of 198.2 G (top, red), 200.2 G (middle, green), and 201.2 G (bottom, blue).