Loss features in ultracold ${}^{162}\mathrm{Dy}$ gases: Two- versus three-body processes

Dipolar gases like erbium and dysprosium have a dense spectrum of resonant loss features associated with their strong anisotropic interaction potential. These resonances display various behaviors with density and temperature, implying diverse microscopic properties. Here we quantitatively investigate the low-field ($B<6\mathrm{G}$) loss features in ultracold thermal samples of ${}^{162}\mathrm{Dy}$. The atoms are spin polarized in their lowest internal state so that pure two-body losses due to spin relaxation are forbidden. However, our analysis reveals that some resonances lead to a two-body-like decay law, while others show the expected three-body decay. We present microscopic one-step and two-step models for these losses, investigate their temperature dependence, and detect a feature compatible with a $d$-wave Fano-Feshbach resonance that has not been observed before. We also report the variation of the scattering length around these resonances, inferred from the time-of-flight expansion of the condensate.

Figure 1

Normalized atom number as a function of magnetic field for a thermal sample with temperature $T=190\mathrm{n}\mathrm{K}$ (top panel in blue) and $T=2.4\mu \mathrm{K}$ (bottom panel in red). The red vertical lines indicate the 11 loss features for which we characterize the density and temperature dependence.

Figure 2

Resonant dimer model. A pair of atoms, with relative motion of energy $E$ and angular momentum $\ell$, can resonantly form a dimer state of energy $E_d\approx E$, which then decays at a rate ${\mathrm{\Gamma }}_d$. In the case of a narrow Fano-Feshbach resonance [22], this process leads to a sharp energy feature described by the Lorentzian of Eq. (6).

Figure 3

Atom loss dynamics for varying total atom number $N_a$ at a magnetic field $B=5.130\mathrm{G}$. The averaged temperature is constant and equal to $T\approx 2.0\left(1\right)\mu \mathrm{K}$. Panels (a) and (b) show the atom number and temperature evolution for the case of an initial atom number $N_0=2.3\times {10}^5$. Panels (c) and (d) show the atom number and temperature evolution for $N_0=1.4\times {10}^5$. (e) Variation of the initial loss rate, $\beta =-{\dot{N}}_a/N_a$, with the density ${\overline{n}}_a\propto {\overline{N}}_a/{\overline{T}}^{3/2}$. The solid line is the fitting function $\beta \propto {\overline{n}}_a^{\gamma }$ with $\gamma =0.92\left(10\right)$.

Figure 4

Atom losses as a function of temperature $T$ for a fixed atom number $N_a\approx 1.5\times {10}^5$, for the loss feature $B=5.130\mathrm{G}$. (a) and (b) Atom number and temperature evolution, respectively, for a thermal cloud with temperature $\overline{T}=1.1\left(1\right)\mu \mathrm{K}$. (c) and (d) Atom number and temperature evolution for $\overline{T}=2.0\left(1\right)\mu \mathrm{K}$. (e) Variation of the initial loss rate, $\beta =-\dot{N}/N$, with temperature. The solid line is the fitting function $\beta \propto T^{\alpha }$, with $\alpha =-2.2\left(3\right)$.

Figure 5

Summary of density and temperature dependence for 11 loss features between 0 and $6\mathrm{G}$. (a) Determination of two- vs three-body dominated loss features. A value of $\gamma$ compatible with 1 indicates a two-body dominated loss feature, while a value of $\gamma =2$ indicates a three-body dominated feature. These values are represented by dashed horizontal lines. Loss features characterized by $s(d$)-wave two-body dominated loss processes correspond to blue circles (squares), while red circles correspond to three-body processes. The green circles indicate loss features in a transitional regime where it is not possible to determine a two- or three-body loss rate. (b) Temperature dependence of the loss rate $\beta \left(T\right)\propto T^{\alpha }$. (c) Strength of the different loss features. From the results shown in (a) and (b) we derive the two-body and three-body loss rates for a nominal density of $\overline{n}=1\times {10}^{20}{\text{m}}^{-3}$ and temperature $\overline{T}=1\mu \mathrm{K}$. (d) Temperature dependence of the two- and three-body loss coefficients ${\chi }_2=\alpha +3/2$ (blue) and ${\chi }_3=\alpha +3$ (red). The 11 loss features studied in this article are marked by vertical bars.

Figure 6

Variation of the scattering length with magnetic field, in the vicinity of six loss features. Central magnetic field: (a) $B=2.655\mathrm{G}$, (b) $B=3.138\mathrm{G}$, (c) $B=3.881\mathrm{G}$, (d) $B=4.620\mathrm{G}$, (e) $B=5.130\mathrm{G}$, and (f) $B=5.561\mathrm{G}$.

Figure 7

(a) $1/N\left(t\right)$ (blue) and $1/N^2\left(t\right)$ (red) as a function of time $t/t_0$ for $A\left(0\right)=1$ and $b=100$. (b) $1/N\left(t\right)$ (blue) and $1/N^2\left(t\right)$ (red) as a function of time $t/t_0$ for $A\left(0\right)=0.01$ and $b=0.01$.

Figure 8

Schematic representation of the experimental setup. The atoms are optically collimated at the output of the oven (transverse cooling) and then decelerated in a spin-flip Zeeman slower. The atoms are then confined in a magneto-optical trap (MOT) that comprises five laser beams and a magnetic field gradient. We compress the MOT and capture approximately $4\times {10}^6$ atoms into a crossed dipole trap (CDT) made up of two beams (ODT1 and ODT2) with a relative angle of ${144}^{\circ }$.

Figure 9

Schematic representation of the experimental sequence used to produce a degenerate gas of dysprosium. (1) MOT loading stage. (2) Compressed MOT. (3) First stage of evaporative cooling. (4) Second stage of evaporative cooling. (5) Plain evaporation in the CDT to purify the quantum gas. (ToF) Time-of-flight expansion and absorption imaging acquisition.

Figure 10

Efficiency of evaporative cooling. (a) Temperature as a function of time in log-linear scale. (b) Atom number as a function of time in log-linear scale. (c) Phase-space density ($\mathcal{D}$) as a function of atom number, $N_a$, in log-log scale. We fit our data with $\mathcal{D}\propto N_a^{-\vartheta }$ and retrieve $\vartheta \approx 4.0$. The gray region indicates the points for which a non-negligible condensed fraction is already present and the estimation of $\mathcal{D}$ is no longer quantitative. The horizontal dashed line corresponds to the noninteracting prediction for the emergence of a BEC at $\mathcal{D}\approx 2.612$.

Figure 11

Scattering length in the vicinity of the $B_0=5.130\mathrm{G}$ resonance (vertical red line). Expansion of a BEC in the $x$–$y$ plane after 30-ms time-of-flight, (a) far from the Feshbach resonance and near the resonance with $B<B_0$ (b) or $B>B_0$ (c). We interpret the strong expansion of the BEC for $B<B_0$ resulting from a large scattering length, while for $B>B_0$ we observe a dense cloud compatible with the formation of quantum droplets.