Abstract
The energy-time uncertainty relation puts a fundamental limit on the precision of lidars for the estimation of range and velocity. The precision in the estimation of the range (through the time of arrival) and the velocity (through Doppler frequency shifts) of a target are inversely related to each other and are dictated by the bandwidth of the signal. Here, we use the theoretical toolbox of multiparameter quantum metrology to determine the ultimate precision of the simultaneous estimation of range and velocity. We consider the case of a single target as well as a pair of closely separated targets. In the latter case, we focus on the relative position and velocity. We show that the tradeoff between the estimation precision of position and velocity is relaxed for entangled probe states and is completely lifted in the limit of perfect photon time-frequency correlations. In the regime where the two targets are close to each other, the relative position and velocity can be estimated nearly optimally and jointly, even without entanglement, using the measurements determined by the symmetric logarithmic derivatives.
- Received 21 November 2020
- Revised 19 April 2021
- Accepted 21 June 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.030303
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Traditional lidar systems are bandwidth limited. This means that probe signals that are sharp in time will necessarily be broad in frequency and vice versa. These uncertainties limit the precision for ranging and velocity measurements. In addition, their ability to measure the separation between two close targets deteriorates if the signals start to overlap (the equivalent to Rayleigh’s criterion in imaging). Here, we show that Rayleigh’s curse in the time and frequency domain can be avoided by a coherent detection technique and we derive such a linear-optical measurement. In this work, the ultimate precision limits are obtained for simultaneously estimating the range and velocity of a target, as well as the range and velocity separation of a pair of targets by using lidar systems. These are deduced from the time of arrival and frequency of a pulse. The energy-time uncertainty relation puts a fundamental limit on the precision of lidars for the estimation of time and frequency. It is shown that this tradeoff between the estimation precision of time and frequency is relaxed for entangled probe states and is completely lifted in the limit of perfect entanglement. This result will significantly boost the research on superresolution for positioning weakly reflecting targets and will find a wide range of applications over a broad spectrum of frequencies, including the realistic implementation of superresolution lidar systems with limited entanglement in the probe beam.
