Universal Loss Dynamics in a Unitary Bose Gas

The low-temperature unitary Bose gas is a fundamental paradigm in few-body and many-body physics, attracting wide theoretical and experimental interest. Here, we present experiments performed with unitary ${}^{133}\mathrm{Cs}$ and ${}^7\mathrm{Li}$ atoms in two different setups, which enable quantitative comparison of the three-body recombination rate in the low-temperature domain. We develop a theoretical model that describes the dynamic competition between two-body evaporation and three-body recombination in a harmonically trapped unitary atomic gas above the condensation temperature. We identify a universal “magic” trap depth where, within some parameter range, evaporative cooling is balanced by recombination heating and the gas temperature stays constant. Our model is developed for the usual three-dimensional evaporation regime as well as the two-dimensional evaporation case, and it fully supports our experimental findings. Combined ${}^{133}\mathrm{Cs}$ and ${}^7\mathrm{Li}$ experimental data allow investigations of loss dynamics over 2 orders of magnitude in temperature and 4 orders of magnitude in three-body loss rate. We confirm the $1/T^2$ temperature universality law. In particular, we measure, for the first time, the Efimov inelasticity parameter ${\eta }_{*}=0.098(7)$ for the 47.8-G $d$-wave Feshbach resonance in ${}^{133}\mathrm{Cs}$. Our result supports the universal loss dynamics of trapped unitary Bose gases up to a single parameter ${\eta }_{*}$.

Popular Summary

Strongly interacting, dilute Bose gases are one of the most fundamental systems in few- and many-body physics. Naively, the equation of state of a Bose gas is governed by three parameters only: density, temperature, and the strength of two-body interactions. In the 1970s, researchers showed that processes involving more than two particles come into play in the limit of strong two-body interactions. In particular, scientists predicted the existence of an infinite set of trimer (i.e., three-body) states. In 2006, a group from the University of Innsbruck observed signatures of these states by measuring precisely the gas three-body decay as a function of two-body interaction strength. In the low-temperature regime, the two-body interactions can be tuned to the maximum value allowed by quantum mechanics, called unitarity. Here, we study both experimentally and theoretically the unitary regime and the related universal dynamics of the trapped gas.

We perform our measurements on two atomic species with very different mass: cesium-133 and lithium-7. We employ two different trapping potentials in two different laboratories. We derive the equations providing the time evolution of the gas density and temperature in these two systems resulting from the competition among three-body losses, heating, and trap evaporation at temperatures of several hundred nanokelvin. These losses result from the binding energy of the trimer being larger than the trap depth. Using setups spanning 2 orders of magnitude in temperature and 4 orders of magnitude in three-body loss rate, our results reveal that recombination heating can be balanced by evaporative cooling, rendering the gas temperature constant. We recover excellent agreement between theory and experiment, which demonstrates the universality of the observed physics.

We expect that our findings will yield a deeper understanding of the universal quantum dynamics of strongly interacting Bose gases and pave the way to observing log-periodic loss modulations resulting from the universal trimer states.

Figure 1

Evolution of the unitary ${}^{133}\mathrm{Cs}$ gas in (a) absolute and (b) relative numbers (points). The solid lines are fits of the data using the theory presented in Sec. 3, and the fitted initial relative trap depth ${\eta }_{\mathrm{in}}=U/k_{\mathrm{B}}{T}_{\mathrm{in}}$ is given in the legend. The condition for $(\mathrm{d}T/\mathrm{d}N){|}_{t=0}$ is expected for ${\eta }_{\mathrm{in}}\approx {\eta }_{\mathrm{m}}\approx 8.2$, very close to the measured data for ${\eta }_{\mathrm{in}}=7.4$ [green lines in (a) and (b)]. (c) Evolution of the unitary ${}^7\mathrm{Li}$ gas. The solid lines are fits of the data using our 2D evaporation model and the fitted initial relative trap depth ${\eta }_{\mathrm{in}}$ as in (b). In 2D evaporation, ${\eta }_{\mathrm{in}}\approx {\eta }_{\mathrm{m}}^{\mathrm{eff}}={\eta }_{\mathrm{m}}+1=8.5$ is required to meet the $(\mathrm{d}T/\mathrm{d}N){|}_{t=0}$ condition and is found to be in good agreement with the measured value 8.5 [green line in (c)]; see text. All error bars represent 1 standard deviation.

Figure 2

“Magic” ${\eta }_{\mathrm{m}}$ as a function of the dimensionless parameter $\alpha =N(\hslash \overline{\omega }/k_{B}T{)}^3(1-e^{-4{\eta }_{*}})$ (solid line). The blue solid circles correspond to the results obtained for ${}^{133}\mathrm{Cs}$ in Fig. 1 with ${\eta }_{*}=0.098$. The red solid squares correspond to the ${}^7\mathrm{Li}$ data of Fig. 1 with ${\eta }_{*}=0.21$ [28]. Error bars are statistical.

Figure 3

The magnitude of three-body loss rate at unitarity for ${}^7\mathrm{Li}$ (red) and ${}^{133}\mathrm{Cs}$ (blue), with the respective standard deviations (shaded areas). On the horizontal axis, masses are scaled to the hydrogen atom mass $m_{\mathrm{H}}$. The dashed line represents the unitary limit [Eq. (3) with ${\eta }_{*}\to \infty$]. For comparison, we show predictions of universal theory [15] with ${\eta }_{*}=0.21$ for ${}^7\mathrm{Li}$ [28] and ${\eta }_{*}=0.098(7)$ for ${}^{133}\mathrm{Cs}$ as solid lines (see text). The data confirm the universality of the $L_3\propto T^{-2}$ law.

Figure 4

(a) Schematic drawing of the hybrid trap. It consists of three intersecting lasers, and a magnetic-field gradient in the $z$ direction (vertical) created by a pair of coils. The light sheet beam (blue) confines dominantly along the vertical direction. The additional round beams (red) stabilize the horizontal confinement. (b) Trap shape along the vertical ($z$) direction. It is composed of two Gaussians and a linear contribution from the tilt [see Eq. (a1)]. Relative dimensions are to scale. Also indicated are the contributions of the tilt only (dashed line) and the round beams (dotted line). (c) Trap frequency measurements (dots) and fits (line) as a function of the tilt of the trap. The frequencies are normalized with respect to the zero-tilt frequency ${\omega }_{z,0}=2\pi \times 140\mathrm{Hz}$. We normalize the tilt from our knowledge of the critical ${\gamma }_{\mathrm{c}}$, where the trap opens (see text).

Figure 5

(a) Relative ${}^{133}\mathrm{Cs}$ trap-depth results from the fits to our data (dots) and theoretical model [Eqs. (a1) and (a3), line]. The trap depth is normalized with respect to the trap depth $U_{\mathrm{LS}}\approx 11\mu \mathrm{K}$ given by the Gaussian light sheet only. The shaded area corresponds to the region where the horizontal confinement beams significantly contribute to the trap depth. (b) Absolute ${}^7\mathrm{Li}$ trap-depth results from the fits to our data (dots). The solid line indicates theoretical knowledge of our trap [Eqs. (a1) and (a3), line], with $w_{\mathrm{R}}=38(1)\mu \mathrm{m}$. The shaded area accounts for the combined uncertainty of $w_{\mathrm{R}}$ and $P_{\text{trap}}$.