Effect of Energy Extensivity on the Performance of Open Quantum Interferometers

Abstract

Studying the performance of a quantum interferometer coupled to an external environment is a problem of conceptual and practical importance. If we consider a quantum interferometer featuring Heisenberg-limited sensitivity, then a typical result is that introducing coupling with the environment degrades the sensitivity to the shot-noise limit. Here we argue that this result crucially depends on whether the interferometer-environment coupling term is subject (or not) to the so-called Kac rescaling that restores extensivity, i.e., whether the coupling Hamiltonian is extensive or not. We present results of the Lindblad equation in the presence and absence of Kac rescaling of the coupling constant. Our results show that for a linear coupling and a harmonic model of the environment, often used in modeling of a quantum interferometer coupled with an environment, the Heisenberg-limited sensitivity may be restored after the Kac rescaling. This result points out the need and the importance to characterize the (model of the) environment of the interferometer at hand.

Figures (2)

• Figure 1: Effect of rescaling $\gamma$. Figure shows the CRLB of sensitivity $\Delta \delta$ and its scaling with the number of particles $N$ for an open quantum interferometer governed by the Lindblad equantion with a particle conserving Lindblad operator $L=(a+ b)/\sqrt{2}$. Three different initial states are considered in the simulation, and the solid, dash-dotted, dotted lines correspond to $\gamma_1=0$, $\gamma_2=0.5$, $\gamma_3 N=1$ respectively. We see that in the NOON and TF cases scaling behavior is restored to Heisenberg-limited scaling by rescaling of the noise parameter.
• Figure 2: CRLB of the sensitivity $\Delta \delta$ and its scaling with the number of particles $N$ is plotted for an open quantum system governed by the Lindblad equation with a particle conserving Lindblad operator $L=b^\dagger a$. We consider three different initial states in the simulation, and the solid, dash-dotted, dashed and dotted lines correspond to $\gamma_1=0$, $\gamma_2 N=1$, $\gamma_3 N^2=2$ and $\gamma_4=0.5$, respectively. We show that in the NOON and $|N,0\rangle$ cases their scaling behavior is restored by rescaling with $N$ of the noise parameter, whereas for the TF state, rescaling with $N$ results in an improved sensitivity although the Heisenberg-limited scaling is not restored. When rescaling with $N^2$, the $\gamma=0$ behavior is always restored.