Mixture of two ultra cold bosonic atoms confined in a ring: stability and persistent currents

In this article we investigate the stability of quantized yrast (QY) states in a mixture of two distinguishable equal mass bosonic atoms, $A$ and $B$, confined in a ring. We focus our investigation in the study of the energetic stability since the Bloch analysis and the Bogoliubov theory establish that only energetically stable QY states are capable of sustain a persistent current. Based on physical considerations the stability is studied in two different two-dimensional planes. One is when we are studying the stability of a single QY state which is realized in the $U_{AB}\times U$ plane spanned by the inter and intraspecies interaction strengths with fixed values of angular momentum per particle $l$ and population imbalance $f$ equal to the labels of the QY state. We found that the energetic phase boundary is the positive branch of a hyperbola and the energetically stable domain the internal region of this positive branch. The other is when we are studying the stability at a fixed dynamics which is realized in the $l\times f$ plane spanned by $l$ and $f$ with fixed values of the interaction strengths. The QY states are introduced when we postulate a correspondence between points in sector of the $l\times f$ plane of physical significance (SPS), defined by $-\infty< l<\infty$ and $-1\leq f\leq 1$, and QY states. The stability diagram in the SPS is determined by the overlap of the stability diagram in all $l\times f$ plane and the SPS. We found that there are critical values of $f$ and $l$. $f_\mathrm{crit}(l)$ gives the size of the window of energetic stability in the sense that for a given $l$ only QY states with $0\leq f<f_\mathrm{crit}(l)$ are energetically stable. On the other hand, $l_\mathrm{crit}$ states that there is none energetically stable QY state with $l>l_\mathrm{crit}$.