Figure 1

The phase diagram of a homogeneous upper branch Bose gas with fixed density $n$: At the blue curve that separates the upper and lower branch, the energy density undergoes a discontinuous jump as shown in Fig. 2. The purple dashed curve represents a state with $\kappa =0$. In the “unstable” region, the number equation for chemical potential does not have a solution.Reuse & Permissions

Figure 2

The upper panel is the energy density across resonance at $T=4E_F=9.2T_c$: rescaled by energy $E_o(T)$ of a noninteracting system at the same temperature. The jump represents a transition from the upper to the lower branch. Even at this high temperature, interaction energy and the jump are substantial fractions of the total energy. The lower panel shows $a_c(q)$ as a function of $q$ for fixed $n$. The jump is due to the sudden change in $\mu$ as the system switches branches.Reuse & Permissions

Figure 3

Phase diagram in a trap at fixed temperature $T$ and trap frequency $\omega$: ${\mu }^a$ represents the state of compressibility $\kappa =0$. ${\mu }^{(b)}$ separates the equilibrium branch with the unstable region. The latter has $\kappa <0$. Each point on this diagram denotes a density profile of the Bose gas, with $\mu$ being the chemical potential at the trap center. The density profile can be generated by an upward vertical trajectory using LDA, (see text). The three horizontal lines with arrows are trajectories of a Bose gas across resonance into the atomic side at fixed $\mu$. The termination point (denoted by a black dot) on each $\mu$-trajectory indicates the critical scattering length $a_{*}$ for that $\mu$. For $a_s<a_{*}$ ($a_s>a_{*}$), the density profile is stable (unstable). Hence, the density profiles $(a)$ with chemical ${\mu }_1$ is stable, whereas the densities $(b)$ and $(c)$, with chemical potential ${\mu }_3$ are unstable. For $T=1\mu \mathrm{K}$, and $\omega =2\pi (250\mathrm{Hz})$, we find $(-\lambda /a_s{)}^{*}=0.14$, 0.18, and 0.21 for the trajectories $-(\mu /T{)}_1=3$, $-(\mu /T{)}_2=2.5$ and $-(\mu /T{)}_3=2$, respectively. The corresponding particle numbers are $(N_1,N_2,N_3)=(2.1,3.4,5.2)\times {10}^4$. All these systems satisfy the condition to be in the low-recombination regime, Eq. (1) or equivalently Eq. (10), as $⟨{\gamma }_3{⟩}_{\mathrm{ave}}/\omega =0.14$, 0.35, and 0.96, respectively, and $N_1$, $N_2$, $N_3<N^{*}$, where $N^{*}=6.5\times {10}^4$ for this temperature and trap frequency.Reuse & Permissions

Figure 4

The density profiles of an upper branch Bose gas in a trap: Fig. (a), (b), and (c) correspond to the densities (a), (b), and (c) in Fig. 3. Density (a) has an upper branch core and an outer shell of equilibrium state. The dashed purple curve is the density of the equilibrium branch, Eq. (6), at the same $T$, $a_s$, $\omega$, and $N$. The densities (b) and (c) are unstable as they contain regions where $dn/d\mu <0$. For density (a), the width of the unstable region becomes less than interparticle spacing and is therefore nonexistent.Reuse & Permissions