Loschmidt echo for quantum metrology

We propose a versatile Loschmidt echo protocol to detect and quantify multiparticle entanglement. It allows us to extract the quantum Fisher information for arbitrary pure states, and finds direct application in quantum metrology. In particular, the protocol applies to states that are generally difficult to characterize, as non-Gaussian states, and states that are not symmetric under particle exchange. We focus on atomic systems, including trapped ions, polar molecules, and Rydberg atoms, where entanglement is generated dynamically via long-range interaction, and show that the protocol is stable against experimental detection errors.

Figure 1

Loschmidt echo protocol applied to the OAT model. (a) Snapshot of the Husimi distribution. Left panel: A spin-polarized state is prepared at the north pole of the Bloch sphere. Central panel: Interaction is switched on for a time $t_1$ (transformation ${\widehat{U}}_{1,\mathrm{OAT}}$). The state is then rotated around the $y$ axis of an angle $\theta \left[{\widehat{R}}_y\left(\theta \right)\right]$. Right panel: Interaction is switched on again for a time $t_2$ (transformation ${\widehat{U}}_{2,\mathrm{OAT}}$) such that ${\widehat{U}}_{1,\mathrm{OAT}}{\widehat{U}}_{2,\mathrm{OAT}}=\mathbb{1}$. In these plots $\theta /\pi =0.01$ and $\tau /\pi =0.05$. (b) Probability $P_0\left(\theta \right)$ (color scale) as a function of time and phase shift. (c) and (d) Cuts of (b) showing $P_0\left(\theta \right)$ (solid line) as a function of $\theta$ for $\tau /\pi =0.05$ (c) and $\tau /\pi =0.01$ (d). The dashed line is the Taylor expansion as in Eq. (1). Here $N=100$.

Figure 2

Phase sensitivity (normalized to the standard quantum limit) calculated as spin squeezing (thin solid and dashed-dotted blue lines) and inverse QFI (thick solid and dashed red lines). Solid lines refer to a square lattice with $N=100$ particles with $L=10a_{\text{latt}}$ and $R_c=8a_{\text{latt}}$, red dashed and blue dashed-dotted lines to the OAT model with $N=100$ particles. (a) Nonoptimized case. (b) Optimized case, where the state is rotated by a suitable angle before applying $R_y\left(\theta \right)$. The inset shows the comparison of Eq. (2) for the optimized (solid) and nonoptimized (dot-dashed) dynamics for the QFI at short times.

Figure 3

(a) Comparison of phase sensitivities obtained via the inverse QFI (red dots) and spin squeezing (blue dots) as a function of the number of particles in a 2D square optical lattice with $R_c=5a_{\text{latt}}$. Here the sensitivity is optimized with respect of evolution time and rotation direction. Solid lines are a guide for the eye. The dashed green line is the Heisenberg limit $\left(\mathrm{\Delta }\theta \right)/{\left(\mathrm{\Delta }{\theta }_{\mathrm{SQL}}\right)}^2=1/N$ achieved when the soft-core radius is larger than the maximum interparticle distance. Red dashed-dotted (blue dotted) is the limiting value of the inverse QFI $F_Q^{\infty }$ (spin squeezing ${\xi }_{\infty }^2$) for an infinite system. (b) Minimum value of the inverse QFI and squeezing for an infinite two-dimensional system with varying soft-core radius. For the power-law scaling we find: ${\xi }_{\mathrm{R}}^2\propto {(Rc/a)}^{-1.52}$ and $N/F_Q\propto {(Rc/a)}^{-1.94}$. (c) Time $V_0{t}_{\text{min}}$ where the minimum of the squeezing and the quantum Fisher is obtained as a function of the soft-core radius.

Figure 4

Implementation of Loschmidt echo with Rydberg dressed atoms in lattices. Upper panel: Level scheme with two lower energy levels that form an effective spin-$1/2$ system. Spin-up is coupled to a highly excited Rydberg state which displays repulsive (attractive) interactions in the first (second) part of the protocol. Lower panel: Loschmidt echo protocol is implemented by two consecutive spin echoes with a global spin rotation in the middle by an angle $\theta$. Each spin echo guarantees that inhomogeneous laser detunings decouple from the collective spin dynamics.