Long-lived nonthermal states realized by atom losses in one-dimensional quasicondensates

We investigate the cooling produced by a loss process nonselective in energy on a one-dimensional (1D) Bose gas with repulsive contact interactions in the quasicondensate regime. By performing nonlinear classical-field calculations for a homogeneous system, we show that the gas reaches a nonthermal state where different modes have acquired different temperatures. After losses have been turned off, this state is robust with respect to the nonlinear dynamics, described by the Gross-Pitaevskii equation. We argue that the integrability of the Gross-Pitaevskii equation is linked to the existence of such long-lived nonthermal states and illustrate this by showing that such states are not supported within a nonintegrable model of two coupled 1D gases of different masses. We go beyond a classical-field analysis, taking into account the quantum noise introduced by the discreteness of losses, and show that the nonthermal state is still produced and its nonthermal character is even enhanced. Finally, we extend the discussion to gases trapped in a harmonic potential and present experimental observations of a long-lived nonthermal state within a trapped 1D quasicondensate following an atom-loss process.

Figure 1

Time evolution of the mode energy during the loss process ($\mathrm{\Gamma }t<1$), and subsequent dynamics after the loss rate is set to zero, for modes of wave vectors $k{\xi }_i=0.5$ (red, lowest curve), 2.5 (black), and 6 (green, highest curve). The dashed and dotted lines are the expected behavior for phonons ($e^{-3\mathrm{\Gamma }t/2}$) and high-energy excitations ($e^{-\mathrm{\Gamma }t}$), respectively. Here $\chi =T_i/\left({\rho }_i\sqrt{\hslash {\rho }_{i}g/m}\right)=0.05$ and the loss rate is $\mathrm{\Gamma }=2\times {10}^{-3}{\rho }_{i}g/\hslash$.

Figure 2

Time evolution of the mean energy of modes of wave vectors $k=0.3\sqrt{mg{\rho }_i}/\hslash$ (solid lines) and $k=6.0\sqrt{mg{\rho }_i}/\hslash$ (dashed lines), where ${\rho }_i$ is the initial density of each species, in the coupled (fat blue lines, $\tilde{g}=0.4g$) and uncoupled (thin red lines, $\tilde{g}=0$) cases. The loss process is turned off at time $t=2/\mathrm{\Gamma }$ and shown is the subsequent evolution of the isolated system. The modes in the two uncoupled gases retain their respective energies after dissipation has been turned off to form a long-lived nonthermal state as above, while the modes in the coupled system relax towards an equipartition of energy. This highlights the role of integrability in the establishment of the long-lived nonthermal configuration. Here results are obtained by averaging over 10 samples, and for each sample we average the mode energy over a $k$ interval of $0.2\sqrt{mg{\rho }_i}/\hslash$. The parameters of the simulation are $T_i/\left({\rho }_i\sqrt{\hslash g{\rho }_i/m}\right)=0.04$, $m^{\prime }/m=3$, and $\hslash \mathrm{\Gamma }/\left({\rho }_{i}g\right)=4\times {10}^{-3}$.

Figure 3

Temperature of each mode, obtained from the stochastic Gross-Pitaevskii equation, as a function of the wave vector $k$ of the mode, for different evolution times: $\mathrm{\Gamma }t=0$ (dotted green line), $\mathrm{\Gamma }t=2.5$ (dashed blue line), and $\mathrm{\Gamma }t=5.3$ (solid fat red line). The initial temperature is $k_B{T}_i/\left({\rho }_{i}g\right)=3$. As atoms are lost, the gas is driven out of equilibrium and $T_k$ acquire a $k$ dependence. For phononic particles, we observe that $T_k$ tends towards $g{\rho }_0\left(t\right)$, as expected from the linearized approach. The smooth red solid line is the asymptotic result of Eq. (8), valid at long times for excitations in the particle regime, computed for $\mathrm{\Gamma }t=5.3$. Parameters of the simulation are $k_B{T}_i/\left(\hslash {\rho }_i\sqrt{g{\rho }_i/m}\right)=3\times {10}^{-3}$, $\hslash \mathrm{\Gamma }/\left(g{\rho }_i\right)=2\times {10}^{-3}$, and $mg/\left({\hslash }^2{\rho }_i\right)={10}^{-6}$.

Figure 4

Results of the two thermometries we performed on the experiment. (a) Atom number fluctuations. Fitting the central region of the cloud, i.e., the high atom-number part, (red line) gives a first temperature $T_{fl}=80$ nK. However, the expected profile from this temperature [also the red curve on (b)] lies well below the experimental data. A second temperature $T_{pr}=140$ nK is found by fitting the profile with the equation of state (solid green line). By plotting the expected atom number fluctuations from $T_{pr}$ on (a), the prediction is in good agreement with the small atom number region. That is, the center of the cloud appears to be at a different temperature than the edges. The close to 50% discrepancy is well beyond the uncertainty we have on the temperature measurements, which is around 10%. (c) Data showing that this temperature difference is stationary over time: we observed a long-lived nonthermal state.