High-Temperature Nonequilibrium Bose Condensation Induced by a Hot Needle

We investigate theoretically a one-dimensional ideal Bose gas that is driven into a steady state far from equilibrium via the coupling to two heat baths: a global bath of temperature $T$ and a “hot needle,” a bath of temperature $T_h\gg T$ with localized coupling to the system. Remarkably, this system features a crossover to finite-size Bose condensation at temperatures $T$ that are orders of magnitude larger than the equilibrium condensation temperature. This counterintuitive effect is explained by a suppression of long-wavelength excitations resulting from the competition between both baths. Moreover, for sufficiently large needle temperatures ground-state condensation is superseded by condensation into an excited state, which is favored by its weaker coupling to the hot needle. Our results suggest a general strategy for the preparation of quantum degenerate nonequilibrium steady states with unconventional properties and at large temperatures.

Figure 1

(a) Tight-binding chain with $N=nM$ bosons on $M$ sites and tunneling parameter $J$, coupled with strengths $\gamma$ and ${\gamma }_h$ to a global bath of temperature $T$ and to a hot needle of temperature $T_h$ at site $\ell$, respectively. [(b) and (c)] Condensate fraction $N_c/N$ indicated by green (blue) shading for ground-state (excited-state) condensation in the mode $k_c\approx 0$ ($k_c\approx \pi /\ell$) versus $T$ and $T_h$ or $M$; for $\ell =5$, ${\gamma }_h=0.5\gamma$, $n=3$, and (b) $M=500$ or (c) $T_h=60J$. Estimated temperature $T_c^{\mathrm{n}\mathrm{e}}$ for one-dimensional-like (three-dimensional-like) Bose condensation plotted as red dotted (solid) lines. Blue-white dotted lines give equilibrium condensation temperature $T_c^{\mathrm{eq}}$ (for ${\gamma }_h=0$). The black dashed line gives estimated needle temperature $T_h^{\mathrm{s},1}$, where excited-state condensation sets in. Black arrows indicate the parameters where (b) and (c) coincide.

Figure 2

[(a) and (b)] Mean occupations $⟨{\widehat{n}}_k⟩$ for the parameters of Fig. 1 with $T_h=60J$ and $M=500$, (a) versus $k$ [for $T=0.29J$, black dot in Fig. 1] and (b) versus $T$. In (a) the crosses indicate the condensate occupation; the gray (black) dashed line in the main panel shows thermal distributions for the temperature $T$ ($T_h$); the inset shows $⟨{\widehat{n}}_k⟩$ for small $k$. [(c) and (d)] Like (a) and (b), but for the equilibrium situation with ${\gamma }_h/\gamma =0$. Occupations obey the Bose-Einstein distribution. (e) Heat current $I$ from the needle through the system into the global bath versus $T$ for the nonequilibrium steady state [red line, parameters like in (b)]; the blue line shows the trivial equilibrium value $I=0$.

Figure 3

Mode occupations (d) versus needle temperature $T_h$ for $T=2J$, $\ell =7$, $M=500$, $n=3$, and ${\gamma }_h/\gamma =1$. With increasing $T_h$ the system passes from a state without Bose condensate through a sequence of states with a condensate in the states $k_0\approx 0$ (green line), $k_1\approx \pi /\ell$ (light blue line), and $k_2\approx 2\pi /\ell$ (violet line), before condensation breaks eventually down again. [(a)–(c)] Momentum distribution for states with condensates in three different modes.