Temperature-enhanced quantum sensing for the cutoff frequency of Ohmic environments

TL;DR

The paper analyzes a dephasing qubit as a probe to estimate the cutoff frequency of Ohmic environments, quantified by the QSNR $\mathcal{Q}$. It shows that, at zero temperature, $\mathcal{Q}$ peaks at an optimal time and can reach a universal maximum $\mathcal{Q}_{\text{max}}=0.648$ under a short-time condition; the maximum is derived from a transcendental equation with $\gamma_{\text{opt}}\approx0.8$. At high temperature, the QSNR saturates to $\mathcal{Q}_{\text{sat}}=\mathcal{Q}_{\text{max}}/4\approx0.162$, and temperature can substantially enhance sensing for small coupling $η$ by compressing the optimal measurement window, effectively making temperature a resource for improved frequency estimation in Ohmic environments.

Abstract

We investigate the quantum sensing performance of a dephasing qubit as a probe in Ohmic environments, characterized by the coupling strength $η$, the Ohmicity parameter $s$, and the cutoff frequency $ω_c$ to be estimated. The performance is quantified by the dimensionless quantum signal-to-noise ratio $\mathcal{Q}$. We show that the evolution of $\mathcal{Q}$ with the scaled time $ω_c t$ is independent of $ω_c$, and peaks at an optimal time $t_{\text{opt}}$, yielding optimal sensitivity $\mathcal{Q}_{\text{opt}}$. We analyze how $\mathcal{Q}_{\text{opt}}$ depends on $η$, $s$ and the temperature $T$. Our results demonstrate that, for any Ohmic environment, provided that $ω_c t_{\text{opt}} \ll 1$, $\mathcal{Q}_{\text{opt}}$ always reaches the upper bound: $\mathcal{Q}_{\text{max}} = 0.648$ at zero temperature, and consistently attains $\mathcal{Q}_{\text{max}}/4$ at high temperatures. Remarkably, we find that increasing the scaled temperature $T/ω_c$ can enhance $\mathcal{Q}_{\text{opt}}$ by nearly two orders of magnitude compared to its zero-temperature counterpart for certain Ohmic environments. Our work reveals that temperature can serve as a resource to enhance sensing precision, as it accelerates the encoding of the cutoff frequency information into the probe state, thereby enabling optimal measurement within a short time window.

Figures (6)

• Figure 1: (color online) The QSNR versus dimensionless time $\omega_{c}t$ for various $\eta$: (a) small values ($\eta=0.1,0.5,1$); (b) large values ($\eta=200,300,400$), with $s=1$ throughout.
• Figure 2: (color online) Dependence of (a) the optimal QSNR and (b) the corresponding optimal measurement time on the coupling strength $\eta$, with the Ohmicity parameter fixed at $s=0.5,1,2$.
• Figure 3: (color online) Dependence of the optimal QSNR on the Ohmicity parameter $s$, with the coupling strength fixed at $\eta=0.01,0.1,1$. Inset: Variation of the Euler gamma function with $s$.
• Figure 4: (color online) Dependence of the QSNR on the scaled time $\omega_{c}t$ at various scaled temperatures: (a) $T/\omega_{c}=0,1,10,100$; (b) $T/\omega_{c}=200,300,400$, with fixed parameters $s=1$ and $\eta=0.1$.
• Figure 5: (color online) The optimal QSNR (top) and its corresponding optimal measurement time (bottom) as functions of the scaled temperature $T/\omega_{c}$: (left) for $\eta=0.1$ , varying $s=0.5,1,2$; (right) for $s=1$, varying $\eta=0.05,0.2,0.5$.