Quenching small quantum gases: Genesis of the orthogonality catastrophe

We study the dynamics of two strongly interacting bosons with an additional impurity atom trapped in a harmonic potential. Using exact numerical diagonalization we are able to fully explore the dynamical evolution when the interaction between the two distinct species is suddenly switched on (quenched). We examine the behavior of the densities, the entanglement, the Loschmidt echo, and the spectral function for a large range of interspecies interactions and find that even in such small systems evidence of Anderson's orthogonality catastrophe can be witnessed.

Figure 1

Evolution of the densities after a quench in $g_{\mathrm{AB}}$. (a) and (b) show the densities for components A and B, respectively, for $g_{\mathrm{AB}}=0.05g_{\mathrm{A}}$. (c) and (d) show the densities for $g_{\mathrm{AB}}=0.25g_{\mathrm{A}}$. (e) and (f) show the densities for $g_{\mathrm{AB}}=0.5g_{\mathrm{A}}$. In all cases, $g_{\mathrm{A}}=25a_{\mathrm{ho}}\hslash \omega$.

Figure 2

(a) vNE between A and B as a function of time when $g_{\mathrm{A}}=25a_{\mathrm{ho}}\hslash \omega$ and $g_{\mathrm{AB}}=(0.05,0.5,1)g_{\mathrm{A}}$ for the black (lowest), blue (middle), and red (top) lines, respectively. (b) vNE of the reduced state of species A, red (lighter) line and species B, black (darker) line as a function of $g_{\mathrm{AB}}/g_{\mathrm{A}}$.

Figure 3

Natural orbital occupations ${\lambda }^0$ (upper three thin lines) and ${\lambda }^1$ (lower three thick lines) for (a) species A and (b) species B. Each different color (shade) corresponds to the same color convention as in Fig. 2, i.e., $g_{\mathrm{AB}}=(0.05,0.5,1)g_{\mathrm{A}}$ for black, blue, and red lines, respectively, and $g_{\mathrm{A}}=25a_{\mathrm{ho}}\hslash \omega$.

Figure 4

(a) and (b) Time evolution of the LE of the individual species for A and B when $g_{\mathrm{AB}}=0.05g_{\mathrm{A}}$ (topmost black line), $0.5g_{\mathrm{A}}$ (darker blue line), and $g_{\mathrm{A}}$ (lighter red line). (c) Time evolution of the LE for the total system versus time for the whole range of $g_{\mathrm{AB}}\in [0,g_{\mathrm{A}}]$. (d) same as (c) for three particular values $g_{\mathrm{AB}}=0.25g_{\mathrm{A}}$ (topmost black line), $0.5g_{\mathrm{A}}$ (middle blue line), and $g_{\mathrm{A}}$ (bottom red line). The inset shows a zoom over the first period. In all panels $g_{\mathrm{A}}=25a_{\mathrm{ho}}\hslash \omega$.

Figure 5

Spectral function $A\left(\omega \right)$ as a function of $g_{\mathrm{AB}}/g_{\mathrm{A}}$. The quasiparticle peak moves from $\omega =0$ to more negative finite frequencies as $g_{\mathrm{AB}}$ is increased until $g_{\mathrm{AB}}/g_{\mathrm{A}}⪆0.4$, after which the dynamics of the system does not drastically change. Also visible at more negative finite $\omega$ is a cusp which is a consequence of the oscillating behavior of species B as seen in Fig. 4. Cuts of the spectral function are shown for $g_{\mathrm{AB}}/g_{\mathrm{A}}=0.1,0.3,1$ and $g_{\mathrm{A}}=25a_{\mathrm{ho}}\hslash \omega$.