Universal Scaling Laws in the Dynamics of a Homogeneous Unitary Bose Gas

We study the dynamics of an initially degenerate homogeneous Bose gas after an interaction quench to the unitary regime at a magnetic Feshbach resonance. As the cloud decays and heats, it exhibits a crossover from degenerate- to thermal-gas behavior, both of which are characterized by universal scaling laws linking the particle-loss rate to the total atom number $N$. In the degenerate and thermal regimes, the per-particle loss rate is $\propto N^{2/3}$ and $N^{26/9}$, respectively. The crossover occurs at a universal kinetic energy per particle and at a universal time after the quench, in units of energy and time set by the gas density. By slowly sweeping the magnetic field away from the resonance and creating a mixture of atoms and molecules, we also map out the dynamics of correlations in the unitary gas, which display a universal temporal scaling with the gas density, and reach a steady state while the gas is still degenerate.

Figure 1

Measurement of atom-loss and correlation dynamics. (a) A BEC is quenched to unitarity, and the cloud is held there for a variable time $t_{\mathrm{hold}}$ and then ramped away from unitarity at a variable rate $R=-dB/dt$. An infinitely fast ramp-out (solid line) would project the resonantly interacting cloud onto free-atom states, while a finite-rate ramp-out (dashed line) creates a mixture of atoms and molecules. (b) Open symbols: The observed atom number versus $1/R$, for $N_0=98\times {10}^3$ and $t_{\mathrm{hold}}=80\mu \mathrm{s}$. Solid line: An exponential fit, which gives an exponential constant of $2.2(3)\mu \mathrm{s}/\mathrm{G}$. Solid symbols: $N_{\mathrm{obs}}$ if the molecules are dissociated by a second magnetic-field pulse to resonance. (c) Evolution of $N_{\mathrm{obs}}$ with $t_{\mathrm{hold}}$ for our fastest ramp-out ($0.3\mu \mathrm{s}/\mathrm{G}$) and a much slower one ($6\mu \mathrm{s}/\mathrm{G}$). The fast ramp-out data show the on-resonance atom loss, and the difference between the two curves reveals the correlation dynamics in the unitary gas.

Figure 2

Growth of the kinetic energy, for $N_0=214\times {10}^3$. (a) Absorption images taken for various $t_{\mathrm{hold}}$ and after 12 ms of TOF expansion at weak interactions. (b) Kinetic energy per particle, $E$, versus $t_{\mathrm{hold}}$. Note that the interaction energy per particle after the ramp-out is $<20\mathrm{nK}$. The solid line shows the $2/13$ power law predicted for a thermal gas at long $t_{\mathrm{hold}}$, where $E\propto T$.

Figure 3

Atom-loss scaling laws, for $N_0=214\times {10}^3$. We observe both the $\gamma =2/3$ and the $\gamma =26/9$ law, predicted (respectively) for a degenerate and a thermal unitary Bose gas. The crossover between the two regimes occurs at a well-defined atom number.

Figure 4

Universal crossover from a degenerate to a thermal unitary gas. (a) The same $2/3$ law, $\dot{N}/N=-0.28/t_n$ (solid line), describes degenerate gases with different $N_0$. The dashed lines show the $\gamma =26/9$ scaling. (b) For all $N_0$ the crossover occurs at almost the same $N/N_0$ and $t_{\mathrm{hold}}/t_{n_0}$; averaging gives $N_c=0.43(4)N_0$ and $t_c=4.0(4)t_{n_0}$. (c) Plotting the dimensionless loss-rate $\mathrm{\Gamma }$, defined in Eq. (3), versus $E/E_n$ yields a single universal curve, with the crossover at $E_c=1.7(2)E_n$. The solid line is $\mathrm{\Gamma }=0.28$, and the dashed one shows the expected $\mathrm{\Gamma }\propto (E_n/E{)}^2$.

Figure 5

Correlation dynamics. (a) $\mathrm{\Delta }N$ versus $t_{\mathrm{hold}}$ for different $N_0$ and the same slow ramp-out, $1/R=6\mu \mathrm{s}/\mathrm{G}$. (b) $\mathrm{\Delta }N/N$ versus the reduced time $t_{\mathrm{hold}}/t_{n_0}$, for different $N_0$ and $1/R$; see the legend in (a). The shaded region marks the crossover to the thermal regime at $t_c=4.0(4)t_{n_0}$. Inset: Zoom-in on the degenerate regime shows that the correlations reach a (quasi)steady state well before $t_c$.