Loschmidt Echo and the Many-Body Orthogonality Catastrophe in a Qubit-Coupled Luttinger Liquid

We investigate the many-body generalization of the orthogonality catastrophe by studying the generalized Loschmidt echo of Luttinger liquids (LLs) after a global change of interaction. It decays exponentially with system size and exhibits universal behavior: the steady state exponent after quenching back and forth $n$ times between 2 LLs (bang-bang protocol) is $2n$ times bigger than that of the adiabatic overlap and depends only on the initial and final LL parameters. These are corroborated numerically by matrix-product state based methods of the $XXZ$ Heisenberg model. An experimental setup consisting of a hybrid system containing cold atoms and a flux qubit coupled to a Feshbach resonance is proposed to measure the Loschmidt echo using rf spectroscopy or Ramsey interferometry.

Figure 1

Schematics of the experimental setup. The black segments on the SQUID denote the Josephson junctions, and the arrows stand for the total magnetic field. By changing the eigenstate of the flux qubit, the total flux and hence the total magnetic field changes, controlling the Feshbach resonance in the cold atomic LL.

Figure 2

(a) The exponent of the generalized Loschmidt echo is shown for the SQ $n=1$ (circles), double quench $n=2$ (triangles), and adiabatic time evolution (squares) for the $XXZ$ Heisenberg model in the steady state, starting from the $XX$ point and ending up at a finite $J_z$, obtained numerically from MPS based methods (when convergence was reached). The solid lines are the analytical results from Eqs. (10, 13). The inset shows the ratio of the SQ and adiabatic exponents (diamonds) and the $n=2$ and 1 exponents (circles) from numerics, which agrees well with the theoretically expected value of 2 in the long time limit. (b) Typical numerical results (solid lines) of the LE of the $XXZ$ model are shown for a SQ from the $XX$ point to $J_z=0.5J$ and $-0.3J$; the dashed curve is the analytical expression using Eqs. (9, 11). (c) The scaling of the numerical data expected from Eq. (12) for short times for a SQ is visualized from $J_z=0$ to $0.1J$ to $0.6J$ with $0.1J$ steps from top to bottom in arbitrary units.