Orthogonality catastrophe as a consequence of qubit embedding in an ultracold Fermi gas

We investigate the behavior of a two-level atom coupled to a one-dimensional, ultracold Fermi gas. The sudden switching on of the scattering between the two entities leads to the loss of any coherence in the initial state of the impurity and we show that the exact dynamics of this process is strongly influenced by the effect of the orthogonality catastrophe within the gas. We highlight the relationship between the Loschmidt echo and the retarded Green's function—typically used to formulate the dynamical theory of the catastrophe—and demonstrate that the effect is reflected in the impurity dynamics. We show that the expected nonexponential decay of the spectral function can be observed using Ramsey interferometry on the two-level atom and comment on finite temperature effects.

Figure 1

(a) Time-dependent von Neumann entropy as a function of the particle number for $\kappa =200$ in units of $l_0/\left(\hslash \omega \right)$. The inset shows a time slice at $t=\pi /2$, in units of the inverse trapping frequency, ${\omega }^{-1}$. (b) Time-dependent von Neumann entropy as a function of the particle number for $\kappa =50$.

Figure 2

Spectral function. (a) The spectral function $A\left(\omega \right)$ for $N=100$ and $\kappa =100$ in units of $l_0/\left(\hslash \omega \right)$. The continuous blue curve is the envelop of the otherwise discrete spectrum. The asymmetry is clearly visible. (b) Comparison of the spectrum (black line) with an exponential (red line) and a power-law [continuous blue (gray) line] fit.

Figure 3

The effect of finite temperature for a system of $N=7$ particles is shown with $\kappa =5$ in units of $l_0/\left(\hslash \omega \right)$. The suppression given by the exponential coefficient appearing in Eq. (9) is evident as the spectrum for $T\ne 0$ has an amplitude lower than the one for $T=0$.

Figure 4

(a) Loschmidt echo $L\left(t\right)$ as a function of the particle number $N$ for $\kappa =200$ in units of $l_0/\left(\hslash \omega \right)$. (b) Behavior of $L\left(t\right)$ at time $t=\pi /2$ for $\kappa =25$, 50, and 100. The time axis is in units of the inverse trap frequency ${\omega }^{-1}$.