Quantum probes for the cutoff frequency of Ohmic environments

Quantum probing consists of suitably exploiting a simple, small, and controllable quantum system to characterize a larger and more complex system. Here, we address the estimation of the cutoff frequency of the Ohmic spectral density of a harmonic reservoir by quantum probes. To this aim, we address the use of single-qubit and two-qubit systems and different kinds of coupling with the bath of oscillators. We assess the estimation precision by the quantum Fisher information of the sole quantum probe as well as the corresponding quantum signal-to-noise ratio. We prove that, for most of the values of the Ohmicity parameter, a simple probe such as a single qubit is already optimal for the precise estimation of the cutoff frequency. Indeed for those values, upon considering a two-qubit probe either in a Bell or in separable state, we do not find improvement to the estimation precision. However, we also showed that there exist few conditions where employing two qubits in a Bell state interacting with a common bath is more suitable for precisely estimating the cutoff frequency.

Figure 1

Quantum Fisher information $H_{\text{max}}$ and optimal time $t^{\text{opt}}$ (inset) as a function of ${\omega }_c$ for different values of the parameter $s$ (in the legend), in the single-qubit case.

Figure 2

Dependency of the QSNR $R$ on the parameter $s$ for the single-qubit case. In the inset we report the behavior of the coefficient $G$ as a function of $s$.

Figure 3

Dependency of the QSNR $R$ on the parameter $s$, in the case of two qubits interacting with identical independent baths. In the inset we compare the QFI for product (solid red line) and Bell (dashed black line) states as a function of time for ${\omega }_c=0.8$.

Figure 4

Top: Fisher information as a function of time for two fixed values of the parameter $s$ in the cases of two qubits in independent environments in product (solid black line) and Bell (dashed red line) states and in a common environment prepared in a product (dotted green line) or Bell (dot-dashed blue line) state for ${\omega }_c=1$. Bottom: Comparison between the QSNR as a function of the Ohmicity $s$, obtained from the optimized quantum Fisher information, for four different initial conditions of the qubit: two qubits initially in a separable (black dots) or an entangled (red squares) state in independent reservoirs and two qubits in a common environment initialized in a product (green diamonds) or entangled (blue triangles) state.

Figure 5

Left: Ratio of the optimized QFI for Werner states in independent and common baths, as a function of the parameter $p$, for three different values of $s$. Right: Behavior of the optimal QFI for two qubits in a Werner state interacting in a common environment for ${\omega }_c=0.8$.