Optimization of collisional Feshbach cooling of an ultracold nondegenerate gas

We optimize a collision-induced cooling process for ultracold atoms in the nondegenerate regime. It makes use of a Feshbach resonance, instead of rf radiation in evaporative cooling, to selectively expel hot atoms from a trap. Using functional minimization we analytically show that for the optimal cooling process the resonance energy must be tuned such that it linearly follows the temperature. Here, optimal cooling is defined as maximizing the phase-space density after a fixed cooling duration. The analytical results are confirmed by numerical Monte Carlo simulations. In order to simulate more realistic experimental conditions, we show that background losses do not change our conclusions, while additional nonresonant two-body losses make a lower initial resonance energy with nonlinear dependence on temperature preferable.

Figure 1

Schematic depiction of the Feshbach cooling process for a $p$-wave resonance. The figure shows the potentials of three scattering channels as a function of atomic separation. Scattering starts in open channel 1. The shaded band indicates the thermal distribution of the atoms. A semibound state with energy $E_{\mathrm{res}}$ exists in the closed channel and only atoms colliding with relative energy $E_{\mathrm{r}}$ close to $E_{\mathrm{res}}$ scatter significantly. The colliding atoms return to channel 1 with rate $\Gamma \left(E_{\mathrm{r}}\right)$ or go to open channel 2 with energy-independent rate ${\Gamma }_0$. The first process leads to thermalization of the sample while the second leads to selective loss of hot atoms.

Figure 2

Comparison of the cooling mechanism using Feshbach resonances (a) and evaporative cooling in a magnetic trap (b). We sketch an ensemble of atoms in a harmonic trap (orange line) with the position along the $x$ axis and the relative energy $E_{\mathrm{r}}$ and total energy $E$ of the atoms along the $y$ axis in (a) and (b), respectively. Atom loss, indicated by black arrows, occurs in the green shaded regions.

Figure 3

(a) Final phase-space density as a function of ${\eta }_0$ for several dimensionless cooling durations ${\tau }_f$. For all ${\tau }_f$ the optimal cooling procedure occurs for ${\eta }_0=9/2$. (b) Dimensionless density, temperature, and phase-space density as well as scaled resonance position as a function of dimensionless time $\tau$ for the optimal cooling path $E_{\text{res}}=(9/2)k_{B}T$. For both panels the quantities on the vertical axis have been scaled with respect to their initial values.

Figure 4

Three cooling paths as a function of $\tau$ (blue, green, and black) as obtained during the Monte Carlo procedure. The simulation assumes a piecewise-constant function for $E_{\mathrm{res}}\left(\tau \right)$. The red curve shows the optimal cooling path as obtained by functional minimization. For each cooling path we give the normalized final phase-space density to the right of the figure.

Figure 5

(a) Dimensionless inverse temperature as a function of ${\eta }_0$ for several dimensionless cooling times ${\tau }_f$. For each ${\tau }_f$ a marker indicates the optimal ${\eta }_0$. (b) Initial value ${\eta }_0$ maximizing the final value of the function $\mathcal{B}$ as a function of cooling duration ${\tau }_f$. Different curves correspond to different choices of the parameters $a$ and $b$, as indicated in the legend. The markers on the curve for $\mathcal{B}={\tilde{T}}^{-1}$ correspond to those in (a).

Figure 6

(a) Final phase-space density, including atom losses due to collisions with background atoms or molecules with rate ${\gamma }_{\text{bg}}=0.2{\mathrm{s}}^{-1}$, as a function of the initial value of $\eta$ for several dimensionless cooling durations ${\tau }_f$. For all ${\tau }_f$ optimal cooling occurs for ${\eta }_0=9/2$. (b) Dimensionless density, temperature, and phase-space density as well as scaled resonance position as a function of dimensionless time $\tau$ for the optimal cooling path $E_{\mathrm{res}}=(9/2)k_{B}T$ and ${\gamma }_{\text{bg}}=0.2{\mathrm{s}}^{-1}$. For both panels the quantities on the vertical axis have been scaled with respect to their initial values. The value of ${\gamma }_{\mathrm{bg}}$ is a typical experimental value and the cooling procedure has noticeable loss due to collisions with background atoms and molecules, while still showing significant cooling.

Figure 7

(a) Final phase-space density, including effects due to nonresonant collisions with $n_0{\gamma }_2=0.2{\mathrm{s}}^{-1}$ as shown in Eq. (21), as a function of ${\eta }_0$ for several dimensionless cooling durations ${\tau }_f$. In order to highlight the differences between the curves we have normalized the phase-space density with respect to the maximal value ${\varpi }_{\max }$ for each ${\tau }_f$. Optimal cooling occurs at smaller values of ${\eta }_0$ for longer ${\tau }_f$. (b) Time dependence of $\eta \left(\tau \right)$ for several dimensionless cooling durations ${\tau }_f$ and $n_0{\gamma }_2=0.2{\mathrm{s}}^{-1}$ with ${\eta }_0$ chosen such that the phase-space density is maximized. Each curve is only plotted up to the corresponding ${\tau }_f$.