Spontaneous Solitons in the Thermal Equilibrium of a Quasi-1D Bose Gas

We show that solitons occur generically in the thermal equilibrium state of a weakly interacting elongated Bose gas, without the need for external forcing or perturbations. This reveals a major new quality to the experimentally widespread quasicondensate state, usually thought of as primarily phase-fluctuating. Thermal solitons are seen in uniform 1D, trapped 1D, and elongated 3D gases, appearing as shallow solitons at low quasicondensate temperatures, becoming widespread and deep as temperature rises. This behavior can be understood via thermal occupation of the type II excitations in the Lieb-Liniger model of a uniform 1D gas. Furthermore, we find that the quasicondensate phase includes very appreciable density fluctuations while leaving phase fluctuations largely unaltered from the standard picture derived from a density-fluctuation-free treatment.

Figure 1

Density (left) and phase (right) as a function of time of a 1D Bose gas at equilibrium for a single realization. Here $\gamma =0.02$ ($N={10}^3$), and the temperatures are $T={10}^4=\mu /2$ (top), $7\times {10}^4$ (middle), and ${10}^5$ (bottom) in units of ${\hslash }^2/m{L}^2{k}_B$. The superimposed red line in the upper left panel corresponds to travel at the speed of sound.

Figure 2

Comparison of the spectra of Lieb type II excitations (solid black line) and dips in the thermal state for three different temperatures: $T={10}^4$ (medium red dots), $4\times {10}^4$ (light green dots), and ${10}^5$ (dark blue dots). Here $\gamma =0.02$ ($N={10}^3$).

Figure 3

Solitons in trapped clouds. The left panels show the time evolution of a trapped 1D condensate with parameters: $N=1000$, 1D interaction strength $g=0.31$ in units of $\hslash {\omega }_z(\hslash /m{\omega }_z{)}^{1/2}$, and trap frequency ${\omega }_z=2\pi \times 10\mathrm{Hz}$. The temperatures are $k_{B}T=15\hslash {\omega }_z$ (upper frame) and $k_{B}T=260\hslash {\omega }_z$ (lower frame). The right panel shows the evolution of the central density in a three-dimensional elongated trapped cloud (horizontal direction). Here, the trap frequencies are ${\omega }_z=2\pi \times 10\mathrm{Hz}$ and ${\omega }_{\perp }=2\pi \times 1000\mathrm{Hz}$, with again $N=1000$, and the (3D) interaction strength $g_{3\mathrm{D}}=0.213$ in units of $\hslash {\omega }_z(\hslash /m{\omega }_z{)}^{3/2}$. The temperature is $80\hslash {\omega }_z/k_B$.

Figure 4

Phase and thermal density fluctuations in the center of the trapped 1D gas as a function of temperature. Parameters are as in Fig. 3 except for varying $T$ (in units of $\hslash {\omega }_z/k_B$). Purple circles show $\delta n/⟨n⟩$ in the center of the trap, orange squares the ratio of the phase coherence length $l_{\varphi }$ to the cloud width $W$. The latter is taken to be the distance between points at which the density has fallen to 10% of the central value. Vertical gray bars indicate temperatures $T_{\varphi }$ and an estimate of the location of the quasicondensate–decoherent-quantum-gas crossover at $\tau =\sqrt{\gamma }$ [25]. Yang-Yang predictions (black solid line) are for the average density in the center ($|z|\le 2$) of the cloud. Some statistical uncertainty is visible.

Figure 5

Number of solitons per micrometer on the edges of the cloud, per snapshot, versus temperature. There are ${10}^4$ ${}^{87}\mathrm{Rb}$ atoms in a 1D trap with ${\omega }_z=2\pi \times 1.9\mathrm{Hz}$. The 1D coupling assumes a Gaussian transverse profile corresponding to ${\omega }_{\perp }=2\pi \times 128\mathrm{Hz}$. Counting was over $7\mu \mathrm{m}$ intervals on either side of the cloud centered around the position where the average density is $10\mu {\mathrm{m}}^{-1}$ (12% of peak). Dips whose central density falls to below 10% of the mean density at their trap position were counted.