Fig. 2: Roton softening by thermal fluctuations.

Raising the temperature of a dipolar quantum fluid can induce a pronounced roton-maxon spectrum of its collective excitations, as shown in a for an infinitely elongated condensate along the z-axis [see Fig. 1c]. Heating the fluid tends to lower the energy of the roton minimum and eventually softens the roton excitation as the temperature increases. This effect can be traced back to the density dependence of the energy correction caused by fluctuations, shown in b. While quantum fluctuations yield an energy H_qu (dashed line) that increases with a rising condensate density ρ = ∣ψ∣², the contribution H_th from thermal fluctuations decreases (solid lines). The thermal energy correction H_th(r), therefore, acts as a focusing nonlinearity that supports the formation of regular density modulations. This is illustrated in c, d, where we show the axial density ρ_∣∣(z) = ∫dxdyρ(r) along with the axial potential \({\bar{H}}_{{{{{{{{\rm{th}}}}}}}}}={\rho }_{||}^{-1}\int{{{{{{{\rm{d}}}}}}}}x{{{{{{{\rm{d}}}}}}}}y\rho ({{{{{{{\bf{r}}}}}}}}){\bar{H}}_{{{{{{{{\rm{th}}}}}}}}}({{{{{{{\bf{r}}}}}}}})\), respectively. The calculations are performed for a/a_d = 0.7 and μ = ε_d.