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.
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.
It is often the case that the environment of a quantum system may be described as a bath of oscillators with an ohmic density of states. In turn, the precise characterization of these classes of environments is a crucial tool to engineer decoherence or to tailor quantum information protocols. Recently, the use of quantum probes in characterizing ohmic environments at zero-temperature has been discussed, showing that a single qubit provides precise estimation of the cutoff frequency. On the other hand, thermal noise often spoil quantum probing schemes, and for this reason we here extend the analysis to a complex system at thermal equilibrium. In particular, we discuss the interplay between thermal fluctuations and time evolution in determining the precision attainable by quantum probes. Our results show that the presence of thermal fluctuations degrades the precision for low values of the cutoff frequency, i.e., values of the order ω c ≲ T (in natural units). For larger values of ω c , decoherence is mostly due to the structure of environment, rather than thermal fluctuations, such that quantum probing by a single qubit is still an effective estimation procedure.
Abstract The cutoff frequency is one of the crucial parameters that characterize the environment. In this paper, we estimate the cutoff frequency of the Ohmic spectral density by applying the π-pulse sequences (both equidistant and optimized) to a quantum probe coupled to a bosonic environment. To demonstrate the precision of cutoff frequency estimation, we theoretically derive the quantum Fisher information (QFI) and quantum signal-to-noise ratio (QSNR) across sub-Ohmic, Ohmic, and super-Ohmic environments, and investigate their behaviors through numerical examples. The results indicate that, compared to the equidistant π-pulse sequence, the optimized π-pulse sequence significantly shortens the time to reach maximum QFI while enhancing the precision of cutoff frequency estimation, particularly in deep sub-Ohmic and deep super-Ohmic environments.
Abstract Quantum probing is the art of exploiting simple quantum systems interacting with a complex environment to extract precise information about some environmental parameters, e.g. the temperature of the environment or its spectral density. Here we analyze the performance of a single-qubit probe in characterizing Ohmic bosonic environments at thermal equilibrium. In particular, we analyze the effects of tuning the interaction Hamiltonian between the probe and the environment, going beyond the traditional paradigm of pure dephasing. In the weak-coupling and short-time regime, we address the dynamics of the probe analytically, whereas numerical simulations are employed in the strong coupling and long-time regime. We then evaluate the quantum Fisher information for the estimation of the cutoff frequency and the temperature of the environment. Our results provide clear evidence that pure dephasing is not optimal, unless we focus attention to short times. In particular, we found several working regimes where the presence of a transverse interaction improves the maximum attainable precision, i.e. it increases the quantum Fisher information. We also explore the role of the initial state of the probe and of the probe characteristic frequency in determining the estimation precision, thus providing quantitative guidelines to design optimized detection to characterize bosonic environments at the quantum level.
AbstractQuantum probing is the art of exploiting simple quantum systems interacting with a complex environment to extract precise information about some environmental parameters, e.g. the temperature of the environment or its spectral density. Here we analyze the performance of a single-qubit probe in characterizing Ohmic bosonic environments at thermal equilibrium. In particular, we analyze the effects of tuning the interaction Hamiltonian between the probe and the environment, going beyond the traditional paradigm of pure dephasing. In the weak-coupling and short-time regime, we address the dynamics of the probe analytically, whereas numerical simulations are employed in the strong coupling and long-time regime. We then evaluate the quantum Fisher information for the estimation of the cutoff frequency and the temperature of the environment. Our results provide clear evidence that pure dephasing is not optimal, unless we focus attention to short times. In particular, we found several working regimes where the presence of a transverse interaction improves the maximum attainable precision, i.e. it increases the quantum Fisher information. We also explore the role of the initial state of the probe and of the probe characteristic frequency in determining the estimation precision, thus providing quantitative guidelines to design optimized detection to characterize bosonic environments at the quantum level.
We investigate the effect of the initial probe-environment correlations in estimating the cutoff frequency which is one of the crucial parameters characterizing the environment of a quantum system. The two-qubit probe, referred to as a probe constructed by two qubits, is prepared in the product state and allowed to interact with the environment. Under the initial probe-environment correlations, we derive the analytic quantum Fisher information (QFI) and the corresponding quantum signal-to-noise ratio (QSNR), which quantify the estimation accuracy of the cutoff frequency. By comparing the QFI and QSNR with the initial probe-environment correlations to those without correlations, it shows that the initial probe-environment correlations can be used to improve the accuracy of estimating the cutoff frequency in the estimation scheme of two-qubit probes.
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 ωc to be estimated. The performance is quantified by the dimensionless quantum signal-to-noise ratio Q. We show that the evolution of Q with the scaled time ωct is independent of ωc, and peaks at an optimal time topt, yielding optimal sensitivity Qopt. We analyze how Qopt depends on η, s and the temperature T. Our results demonstrate that, for any Ohmic environment, provided that ωctopt≪1, Qopt always reaches the upper bound: Qmax=0.648 at zero temperature, and consistently attains Qmax/4 at high temperatures. Remarkably, we find that increasing the scaled temperature T/ωc can enhance Qopt 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.