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Journal of Physics B Atomic Molecular and Optical Physics

Precision metrology of weak measurement with a thermal state pointer

Gang Li, Libo Chen, Tao Wang, Zhi-Hui He, He‐Shan Song

Published Apr 1, 2019• 2 citations
Publisher

Abstract

Abstract Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with a thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal quantum Fisher information (QFI) after the postselection for an accepted event (conversely, the postselection for a rejected event) can attain a level of thermal fluctuation, without surpassing the total QFI, and further, that QFI which increases with increasing temperature can constantly improve measurement precision. These results are much better than those of weak measurement with a pure state (i.e. Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using a thermal state pointer, we obtain in the phase space the accepted postselection and the postselected measurement both achieving the Heisenberg limit of quantum metrology, and showing that weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical-enhanced scaling of N 2 . Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of light because generating these states with larger uncertainty is limited under current technology, but thermal states with larger uncertainly are very easy to achieve with increasing temperature in nature, such as for optical systems and optomechanical systems.

DOI
10.1088/1361-6455/ab1500

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Abstract Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with a thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal quantum Fisher information (QFI) after the postselection for an accepted event (conversely, the postselection for a rejected event) can attain a level of thermal fluctuation, without surpassing the total QFI, and further, that QFI which increases with increasing temperature can constantly improve measurement precision. These results are much better than those of weak measurement with a pure state (i.e. Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using a thermal state pointer, we obtain in the phase space the accepted postselection and the postselected measurement both achieving the Heisenberg limit of quantum metrology, and showing that weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical-enhanced scaling of N 2 . Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of light because generating these states with larger uncertainty is limited under current technology, but thermal states with larger uncertainly are very easy to achieve with increasing temperature in nature, such as for optical systems and optomechanical systems.
Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal QFI after successful postselection can attain the level of thermal fluctuations, without surpassing total QFI, and that QFI which increases with increasing temperature can constantly improve the measurement precision. These results are much better than that of weak measurement with pure state (i.e., Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using thermal state pointer, we obtain in the phase space successful postselection and postselected measurements both achieve the Heisenberg limit of quantum metrology, and show weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical enhanced scaling of N^2. Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of the light because generating these states with more large uncertainty are limited under the current technology, but thermal states with more large uncertainly are very easy to achieve with increasing temperature in nature, regardless of thermal states of the light or the matter.
Abstract Most studies for postselected weak measurement focus on using the pure Gaussian state as a pointer, which can only give an amplification limit reaching the level of the ground state fluctuation. When the pointer is initialized in a thermal state, we find that the amplification limit after the postselection can reach the level of thermal fluctuation, indicating that the amplification effect achieving the level of thermal fluctuation is also increased with the temperature growth, and also that the amplification mechanism is different from that with pure Gaussian-state pointer. To illustrate these results, we propose two schemes to implement room temperature amplification of the mechanical oscillator’s displacement caused by a single photon in the optomechanical system. The two schemes both enhance the mechanical oscillator’s original displacement by nearly seven orders of magnitude, attaining sensitivity to displacements of ∼0.26 nm. Such amplification effect can be used to observe the impact of a single photon on a room temperature mechanical oscillator, which is hard to detect in traditional measurement.
In most studies on post-selected weak measurement, the zero-mean Gaussian state (i.e., pure state) is initialized as a pointer state. Recently, several theoretical studies give the limits that using fisher information metric, weak measurement amplification strategy can not outperform the standard metrology strategy. However, we show that the limits, especially for the imaginary part of weak value, is no longer tenable if thermal state is initialized as a pointer state. By adjusting the temperature, we can improve accuracy of the measurement precision.
Most studies for postselected weak measurement focus on using pure Gaussian state as a pointer, which can only give an amplification limit reaching the ground state fluctuation. In this paper we use the thermal state pointer. We find that its amplification limit can reach thermal fluctuations and we also give the amplification mechanism different from the one with pure Gaussian state pointer. To illustrate the results, we propose two schemes to implement room temperature amplification of the mechanical oscillator's displacement caused by one photon in optomechanical system. The two schemes can both enhance the original displacement by nearly seven orders of magnitude, attaining sensitivity to displacements of ~ 0.26 nm. Such amplification effect can be used to observe the impact of a single photon on a room temperature mechanical oscillator which is impossible to detect in traditional measurement.
Postselected weak measurement is a useful protocol for amplifying weak physical effects. However, there has recently been controversy over whether it gives any advantage in precision. While it is now clear that retaining failed postselections can yield more Fisher information than discarding them, the advantage of postselection measurement itself still remains to be clarified. In this Letter, we address this problem by studying two widely used estimation strategies: averaging measurement results, and maximum likelihood estimation, respectively. For the first strategy, we find a surprising result that squeezed coherent states of the pointer can give postselected weak measurements a higher signal-to-noise ratio than standard ones while all standard coherent states cannot, which suggests that raising the precision of weak measurements by postselection calls for the presence of "nonclassicality" in the pointer states. For the second strategy, we show that the quantum Fisher information of postselected weak measurements is generally larger than that of standard weak measurements, even without using the failed postselection events, but the gap can be closed with a proper choice of system state.
Weak measurement has ignited extensive research and discussions in the field of quantum metrology due to its sufficient potential in quantum measurement applications. Recent studies demonstrate that weak measurement protocols, when integrated with quantum resources, can achieve Heisenberg scaling while maintaining technical noise advantages. However, scenarios involving dephasing noise have not been addressed, as such noise induces rapid degradation of Heisenberg scaling to the standard quantum limit. Here, we incorporate phase fluctuations, a typical dephasing noise, into three Heisenberg-scaling systems. We find that postselection invariably reduces the robustness of quantum systems. Compared to coherent pointer states, squeezed coherent pointer states can exhibit superior resilience against phase fluctuations through appropriate squeezing-parameter optimization. In addition, in entanglement-assisted systems, we can optimize the postselected angle to compensate for Fisher information loss induced by the postselection process. Our findings reveal that weak measurement exhibits inherent limitations in dephasing noise metrology compared to conventional measurement.