Optimizing the catching of atoms or molecules in two-dimensional traps

Single-photon cooling is a recently introduced method to cool atoms and molecules for which standard methods might not be applicable. We numerically examine this method in a two-dimensional wedge trap as well as in a two-dimensional harmonic trap. An element of the method is a small optical box with “diodic” walls which moves slowly through the external potential and catches atoms irreversibly. We show that the cooling efficiency of the method can be improved by optimizing the trajectory of this box.

Figure 1

Schematic representation of the process for a single atom: snapshots for different, increasing times (from left to right); the passing direction of the diode is from left to right.

Figure 2

Schematic representation of the two different two-dimensional traps considered: (a) wedge trap, (b) harmonic trap; the red-filled squares indicate the “box.”

Figure 3

Wedge trap with the box at rest at $x_B=0$. (a) Fraction $F$ of trapped atoms versus box positions $y_B$: $\alpha ={30}^{\circ },w_B/l=0.1$ (solid blue line); $\alpha ={45}^{\circ },w_B/l=0.1$ (dashed red line); $\alpha ={60}^{\circ },w_B/l=0.1$ (dotted green line); $\alpha ={30}^{\circ },w_B/l=0.35$ (thick solid blue line); $\alpha ={45}^{\circ },w_B/l=0.35$ (thick dashed red line); $\alpha ={60}^{\circ },w_B/l=0.35$ (thick dotted green line). (b) Optimal position $y_{op}$ of the box at rest for different wedge angles (blue, red, respectively, green plus signs connected with lines), the black dots correspond to the straight line $y_{op}=w_B(1+\tan \alpha )/\tan \alpha$; see text for more details. $t_f/\tau =20$, $E_B/E_i=0.1$.

Figure 4

Wedge trap. Box moving with the linear trajectory Eq. (12). Fraction $F$ of trapped atoms versus box velocities: (a) wedge angle $\alpha ={30}^{\circ }$, $y_{op}/l=0.7$; (b) wedge angle $\alpha ={45}^{\circ }$, $y_{op}/l=0.6$; (c) wedge angle $\alpha ={60}^{\circ }$, $y_{op}/l=0.75$. $t_f/\tau =20$, $E_B/E_i=0.1$, $w_B/l=0.35$. The square symbols mark the maximal fraction for a box at rest; the dots mark the maximal fraction for a linear moving box.

Figure 5

Schematic representation of the box trajectories Eqs. (16) and (17); the initial position of the box is shown (see text for further details).

Figure 6

Wedge trap, $\alpha ={30}^{\circ }$. Fraction $F$ of trapped atoms versus box half widths $w_B$: (a) $t_f/\tau =20$, (b) $t_f/\tau =40$. Different box trajectories are as follows: box at rest (plus signs connected by thick blue line), box moving with trajectory Eq. (16) (crosses connected by red dashed line), box moving with analytical trajectory Eq. (17) (dots connected by thick green dotted line), box moving with wriggle trajectory Eq. (18) (circles connected by black solid line). $v/\nu =0.13$, $y_{W0}/l=2.0$, ${\omega }_W\tau =0.25$, $E_B/E_i=0.1$.

Figure 7

Wedge trap, $\alpha ={45}^{\circ }$. Fraction $F$ of trapped atoms versus box half widths $w_B$: (a) $t_f/\tau =20$, (b) $t_f/\tau =40$. $v/\nu =0.11$, $y_{W0}/l=1.5$, ${\omega }_W\tau =0.2$; see Fig. 6 for more details.

Figure 8

Wedge trap, $\alpha ={60}^{\circ }$. Fraction $F$ of trapped atoms versus box half widths $w_B$: (a) $t_f/\tau =20$, (b) $t_f/\tau =40$. $v/\nu =0.11$, $y_{W0}/l=1.5$, ${\omega }_W\tau =0.2$; see Fig. 6 for more details.

Figure 9

Harmonic trap, data versus different box half widths $w_B$. (a) Fraction $F$ of trapped atoms: box at rest with optimized $y_{op}$ (plus signs connected by thick blue line), linear moving box with optimized $v_{B,x}$ and optimized $y_c$ (crosses connected by red solid line), linear moving box with trajectory Eq. (25) (dots connected by thick green dotted line, on top of red line), box moving with a helix trajectory (circles connected by black solid line). (b) Optimized box parameter $y_{op}$, respectively, $y_c$ (crossing point of $y$ axis): for a box at rest (plus signs connected by thick blue line), for a linearly moving box (crosses connected by red dashed line), approximation Eq. (23) (black dashed line). (c) Optimized box velocity $v_{B,x}$: for a linearly moving box (crosses connected by red dashed line), approximation Eq. (24) (black dashed line). $E_B/E_i=0.1,t_f/\tau =60$.

Figure 10

Harmonic trap, box moving with the linear trajectory Eq. (22). Fraction $F$ for different positions $y_c$ and box velocities $v_{B,x}$. $v_{B,y}=0$, $w_B/l=0.2$; the square symbol marks the maximal fraction for a box at rest, the dot marks the maximal fraction for a linear moving box. $E_B/E_i=0.1,t_f/\tau =60$.

Figure 11

Harmonic trap, fraction $F$ versus box threshold energies $E_B$: box at rest (plus signs connected by thick blue line), box moving with the linear trajectory Eq. (25) (dots connected by thick green dotted line), box moving with a helix trajectory (circles connected by black solid line). (a) Box half width $w_B/l=0.2$, (b) box half width $w_B/l=0.35$. $t_f/\tau =60$.