Multipartite model of evaporative cooling in optical dipole traps

We propose and study a model of forced evaporation of atomic clouds in crossed-beam optical dipole traps that explicitly includes the growth of a population in the “wings” of the trap and its subsequent impact on dimple temperature and density. It has long been surmised that a large wing population is an impediment to the efficient production of Bose-Einstein condensates in crossed-beam traps. Understanding the effect of the wings is particularly important for $\lambda =1.06\mu \mathrm{m}$ traps, for which a large ratio of Rayleigh range to beam waist results in wings that are large in volume and extend far from the dimple. Key ingredients to our model's realism are (1) our explicit treatment of the nonthermal, time-dependent energy distribution of wing atoms in the full anharmonic potential and (2) our accurate estimations of transition rates among dimple, wing, and free-atom populations, obtained with Monte Carlo simulations of atomic trajectories. We apply our model to trap configurations in which neither, one, or both of the wing potentials are made unbound by applying a “tipping” gradient. We find that forced evaporation in a trap with two bound wing potentials produces a large wing population which can collisionally heat the dimple so strongly as to preclude reaching quantum degeneracy. Evaporation in a trap with one unbound wing, such as that made by crossing one vertical beam and one horizontal beam, also leads to a persistent wing population which dramatically degrades the evaporation process. However, a trap with both wings tilted so as to be just unbound enjoys a nearly complete recovery of efficient evaporation. By introducing to our physical model an ad hoc, tunable escape channel for wing atoms, we study the effect of partially filled wings, finding that a wing population caused by single-beam potentials can drastically slow down evaporative cooling and increase the sensitivity to the choice of $\eta$.

Figure 1

Absorption images of a cloud of evaporatively cooled ${}^{87}\mathrm{Rb}$ in our apparatus, showing the formation of wings after 1 $\mu \mathrm{s}$ of forced evaporation at $\eta =9.5$ in a $\lambda =1.06\mu \mathrm{m}$ crossed-beam dipole trap.

Figure 2

Block diagram of our multipartite model of a crossed-beam optical trap. Symbols are described in the text. Arrows depict rate constants for population-changing processes. For clarity, only arrows connecting to the $i\mathrm{th}$ wing bin are shown. Dashed arrows depict mechanical work done on atoms due to changes to the trapping beam intensities. Rates shown as solid black arrows are taken from the model of Ref. [13]. Colors correspond to those of Fig. 11.

Figure 3

One-dimensional profiles through the total potential of a crossed-beam dipole trap, along a single beam direction. In this work we simulate evaporation in flat traps (black) and critically tilted traps [red (light gray)]. Trap parameters are given in the text. We do not present results for a strongly tilted trap [blue (dark gray)], as we find it to be overkill with regard to eliminating the wing population, while its extremely asymmetric evaporation lips make it difficult to compare it with the flat trap.

Figure 4

Example atomic trajectories in the total potential of a doubly critically tilted crossed-beam dipole trap. The “downhill” (i.e., unbounded) wings are shown on the bottom of the plot. Red crosses locate the positions in the potential which fix the location of the fiducial ellipsoid (black dashed curve) that demarcates the dimple volume. The three example atoms are launched in different directions from the center of the trap, each with energy equal to $0.6U_0$. The cyan curve is a “dimple-bound” trajectory because it has a turning point inside the ellipsoid. The orange curve is a “wing-bound” trajectory because it has a turning point outside the ellipsoid. The magenta curve is a “free” trajectory because it is accelerating away from the trap center and cannot return.

Figure 5

Monte Carlo calculations of dimple-to-wing ($d\to w$) and dimple-to-free ($d\to f$) transition rates in (a) flat traps and (b) tipped traps. Points are probability densities vs energy for the outcome of a collision between dimple atoms to kick one colliding partner into a wing-bound trajectory ($\alpha =w$, blue squares) or to a free trajectory ($\alpha =f$, red points). The probability densities are calculated with the Monte Carlo simulation described in Sec. 2d and are referred to the left axis. The solid curves show the transition probabilities (points) convolved with the Boltzmann distribution (thin black line) for $\eta =10$ traps and are referred to the right axis. (Note that the solid curves must be recomputed for every $\eta$.) The shaded area is the total probability for collisions between dimple atoms to eject a dimple atom from the trap entirely (area under the red dashed curve) or into the wing population (area under the blue solid curve); for the latter, each vertical bar represents the probability of an individual wing energy bin. These results show that, for this flat trap, 94% thermalizing collisions which kick dimple atoms out of the dimple send them into the wings, rather than ejecting them from the trap entirely. In stark contrast, for the critically tipped trap, 60% of the atoms kicked from the dimple by elastic collision with other dimple atoms promptly escape to infinity through the downhill wing.

Figure 6

Scaled $t_{\mathrm{apogee}}$ vs scaled energy for dimple atoms launched on new trajectories by collisions with other dimple atoms in a flat trap. Orbital energies corresponding to the bin energies ${\varepsilon }_i$ ($i=1,\cdots ,15$) are indicated. For $E/U_0>0.5$, $t_{\mathrm{apogee}}$ increases dramatically because atoms have enough energy to enter the wing part of the trapping potential. The large $t_{\mathrm{apogee}}$ for wing atoms occupying the last few energy bins greatly suppresses their back-action on the dimple. Note that these results are specific to given trap parameters (beam waist, crossing angle, etc.). However, as remarked in the text, the scaled $t_{\mathrm{apogee}}$ is independent of trap depth, so a single set of calculations is valid for the entire forced evaporation sequence. The inset shows the same data plotted on a semilog scale.

Figure 7

Monte Carlo calculations of wing-energy-bin residency times $t_{\mathrm{res},i}$. (a) $\dot{U_0}/U_0$ vs time for a hypothetical forced evaporation sequence in a flat trap. In (b)–(d), each point corresponds to one atom spending some amount of time in the energy range corresponding to a particular wing energy bin, at three representative epochs in the evaporation sequence of [circled points in (a)]. The distribution of $t_{\mathrm{res},i}$ in each energy bin reflects both the nonadiabaticity and the dependence on initial conditions. To fully implement our model, hundreds of these data sets are generated for ${\dot{U}}_0/U_0$ between 0.01 and 1000.

Figure 8

Optimizing forced evaporation at constant $\eta$ in a flat trap. Shown are the time to condensation $t_c$ (solid symbols, left axis) and number in the dimple at $t_c$ (open symbols, right axis) for $\eta$ between 6 and 9.5. For a sample in which background lifetime is ignored $(\tau =\infty$) evaporation at $\eta >9.5$ does not yield a condensate in less than 100 s. Circles represent calculations of forced evaporation for a somewhat more realistic case where the background gas collision rate corresponds to a background collision-limited lifetime of $\tau =60$ s. Lines are guides to the eye.

Figure 9

Phase-space density vs time for evaporation in a flat trap at $\eta$ of 9.4 (solid line) and 9.5 (dashed line). The curve for $\eta =9.5$ displays a local maximum just below threshold and does not recover forward progress until after $\sim 30$ s of evaporation.

Figure 10

Time to condensation $t_c$ for flat traps (blue circles), singly tipped traps (gold squares), and doubly tipped traps (red triangles). In these calculations, there is no loss term due to background gas collisions.

Figure 11

Evaporation calculations for (a1) and (a2) the flat trap, (b1) and (b2) the singly tipped trap, and (c1) and (c2) the doubly tipped trap. (top) Rates of energy flow scaled by the total trap energy in the dimple ${\dot{E}}_{d\to w}/E_d$ (blue dashed),${\dot{E}}_{d\to f}/E_d$ (gold solid), and ${\dot{E}}_{w\to d}/E_d$ (red thick). (bottom) Stacked area plots representing the wing population vs time. Each shaded band in the stack represents the population in one wing bin, from $i=1$ (bottom band) to $i=15$ (top band). Note how the wing populations in the flat and singly tipped traps reach (quasi-)steady states, whereas only in the doubly tipped case is the wing population steadily depleted as the dimple evaporation progresses towards condensation.

Figure 12

Time to condensation $t_c$ vs $\eta$ for evaporation in flat [blue (dark gray) points] and doubly tipped [red (light gray) points] traps for various choices of (a) beam waist, (b) initial temperature, and (c) initial number. The lines are guides to the eye. Note that in (c) there are no points associated with the lowest number evaporating in the flat trap, as this set of conditions did not yield a condensate in 100 s of simulation time.

Figure 13

The influence of wing population revealed by adding to our physical model a fictitious, tunable loss rate for wing atoms. (a) Time to condensate $t_c$ vs $\eta$ for $\kappa =(0,0.2,0.4,0.6,0.8,1)$. The red curve ($\kappa =0$) is identical to a true flat trap. The blue curve is a trap with the same physical (i.e., flat) potential but with the wing population eliminated by setting $\kappa =1$. The potential of a large wing population to degrade efficient dimple evaporation is revealed in the contraction of the range of viable $\eta$'s for the larger steady-state wing populations. (b) Time evolution of the wing population, at optimal values of $\eta$, for the same choices of $\kappa$ as in (a).