Trap-depth determination from residual gas collisions

We present a method for determining the depth of an atomic or molecular trap of any type. This method relies on a measurement of the trap loss rate induced by collisions with background gas particles. Given a fixed gas composition, the loss rate uniquely determines the trap depth. Because of the “soft” long-range nature of the van der Waals interaction, these collisions transfer kinetic energy to trapped particles across a broad range of energy scales, from room temperature to the microkelvin energy scale. The resulting loss rate therefore exhibits a significant variation over an enormous range of trap depths, making this technique a powerful diagnostic with a large dynamic range. We present trap depth measurements of a Rb magneto-optical trap using this method and a different technique that relies on measurements of loss rates during optical excitation of colliding atoms to a repulsive molecular state. The main advantage of the method presented here is its large dynamic range and applicability to traps of any type requiring only knowledge of the background gas density and the interaction potential between the trapped and background gas particles.

Figure 1

(a) Normalized ${}^{87}\mathit{Rb}$ MOT atom number as a function of time fit to Eq. (18). (b) Normalized ${}^{87}\mathit{Rb}$ atom number in a quadrupole magnetic trap as a function of time fit to the decay law given in Eq. (19).

Figure 2

Total loss rate of trapped ${}^{87}\mathit{Rb}$ versus room-temperature argon gas density. The results are fit to a line and the slope provides the value of the velocity averaged collision rate ${\langle \sigma v\rangle }_{\mathit{Rb},\mathit{Ar}}$. The MOT (open squares) exhibits the smallest loss cross section due to a much larger effective trap depth ($\sim$2 K) than the MT (filled circles) where the $|F=1,m_F=-1\rangle$ state was confined with a trap depth of 3 mK. The vertical error bars, most of which are smaller than the symbols, represent the statistical uncertainty in the loss rate based on fits similar to those shown in Fig. 1. The range of argon gas density here corresponds to a pressure range from $1.8\times {10}^{-8}$ to $6\times {10}^{-8}$ torr.

Figure 3

The experimentally measured (squares and open circles) and theoretically computed (solid circles) loss-rate constant ${\langle \sigma v\rangle }_{\mathit{Rb},\mathit{Ar}}$ versus trap depth for trapped ${}^{87}\mathit{Rb}$ atoms and room-temperature ${}^{40}\mathit{Ar}$ atoms. The line is only a guide for the eye. The data above 100 mK (squares) were obtained with a MOT while the data below 10 mK (open circles) were obtained with a quadrupole magnetic trap and are reproduced from [34]. Due to the difference in axial and radial magnetic field gradients, the MT depth limited by the vacuum cell walls is anisotropic. The resulting range of trap depths is indicated by the horizontal error bars on the MT data. The MOT depth was experimentally verified using an independent technique described in the text.

Figure 4

$J=\frac{N_{\mathit{ss}}^0}{N_{\mathit{ss}}}-1$ as a function of the duty factor $d$ for three different catalysis laser detunings ${\Delta }_1=27.3$ GHz (solid triangles), ${\Delta }_2=16.2$ GHz (open squares), and ${\Delta }_3=5.7$ GHz (solid circles) with a MOT pump detuning of $-5$ MHz and pump intensity of $2.7$ mW/cm${}^2$. $\frac{N_{\mathit{ss}}^0}{N_{\mathit{ss}}}$ is the ratio of the steady-state number with the catalysis laser off to the number with the laser on. The dotted lines are guides for the eye, while the solid lines are linear fits to the data excluding the points at large $d$ where the variation is nonlinear. The slope of this ratio versus $d$ is equal to $\frac{{\beta }_{\mathit{cl}}{n}_{\mathit{ss}}}{\left(\Gamma +\beta n_{\mathit{ss}}\right)}$ and is proportional to the catalysis laser-induced loss rate, ${\beta }_{\mathit{cl}}$.

Figure 5

The quantity $\frac{{\beta }_{\mathit{cl}}{n}_{\mathit{ss}}}{\Gamma +\beta n_{\mathit{ss}}}$, proportional to the photoassociation induced loss rate, measured as a function of the catalysis laser detuning, $\Delta$. These data correspond to a MOT with a cooling laser detuning of $-5$ MHz and a total pump laser intensity of $2.7$ mW/cm${}^2$ (corresponding to a total power of 1.4 mW for the MOT). The peak loss rate occurred at 27(5) GHz and indicates that the effective trap depth was 0.64(0.12) K.

Figure 6

The open squares are the measured trap loss coefficients plotted at the trap depths deduced by the catalysis laser technique. The solid curve is the model trap loss-rate coefficient as a function of trap depth. The agreement between the data and the model indicate that the trap depth can be deduced from a measurement of the loss-rate coefficient.

Figure 7

The measured MOT trap depths using the trap loss method, $U_{\langle \sigma v\rangle }$, versus the depth determined by the catalysis laser method, $U_{\mathit{cl}}$. The line indicates the locus of values corresponding to hypothetically perfect agreement. As is evident, the two techniques yield the same results within measurement error.

Figure 8

Catalysis laser data for MOTs with pump laser settings of $-5$ MHz 2.7 mW/cm${}^2$ (circles), $-10$ MHz 2.7 mW/cm${}^2$ (squares), and $-12$ MHz 9.6 mW/cm ${}^2$ (triangles), corresponding to measured trap depths of 0.64 (0.12) K, 1.03 (0.12) K, and 1.99 (0.18) K, respectively. As the trap depth increases, and/or the photoassisted two-body losses induced by the MOT pump laser increase, the less sensitive is the catalysis laser trap depth measurement.