Relative phase distribution and the precision of optical phase sensing in quantum metrology

The corpuscular nature of light imposes limits for the maximum possible precision in the estimation of a phase difference introduced between two optical modes in an interferometer. This quantum behavior generates fluctuations in the photon counts by detectors at the interferometer exits, the so-called shot noise, which disturbs the phase estimation. With classical light having an average number of photons N̄, the uncertainty in the phase sensing scales with 1/N̄. But with quantum light sources, involving entanglement or light squeezing, this uncertainty may scale with 1/N̄ in some cases, such that a much greater precision can be achieved with a large value for N̄ [1], [2], [3].

The field of photonic quantum metrology has been dealing with this issue, and many recent experimental advances were reached [2], [3], [4]. The combination of entanglement, multiple samplings of the phase shift, and adaptive measurement have been use to optimize a phase shift estimation [5]. Coherent measurements involving entangling operations can be useful for phase estimation in the presence of noise [6]. Notable advances in multiple phases estimation have been reported [7], [8], [9], [10], [11]. The use of detection schemes with photon-number-resolution allows the realization of quantum sensing protocols without pos-selection, such as scalable protocols for quantum-enhanced optical phase estimation [12] and distributed quantum sensing [13]. Recently it was also shown how the insensitivity of Hong–Ou–Mandel two-photon interference [14] to phase fluctuations can be used to reduce the phase noise in the measurement of a mirror tilting angle [15] in relation to a similar classical procedure [16], increasing the precision. Quantum metrology with cavities and resonators are also interesting possibilities [17], [18]. But perhaps the most prominent application of photonic quantum metrology so far has been the improvement of the sensitivity of gravitational wave detectors [19], [20], [21], [22]. It is worth mentioning that many other physical systems, besides quantum light, are used in the broader area of quantum metrology [23], [24], [25].

Here we associate the precision limit in the estimation of a small phase difference introduced between two optical modes to the phase difference distribution of the initial two-mode quantum light state used for this purpose. For many pure quantum states useful in quantum metrology, we compute the relative phase distribution P(ϕ) introduced by Luis and Sánchez-Soto (LSS) [26]. We then compute the Fisher information based on this probability distribution and show that the result is very close to the quantum Fisher information for the treated states. The average difference between these quantities for the tested quantum states is smaller than 0.1%, a difference compatible with the numerical precision of our calculations. Since the quantum Fisher information is associated to the maximum possible precision in the phase estimation according to the rules of quantum mechanics [2], [3], [24], [27], [28], our results demonstrate the relevance of the LSS relative phase distribution in the field of quantum metrology. If the introduced method can be extended to treat mixed states, the LSS relative phase distribution could be a valuable tool to estimate the maximum precision in phase sensing for realistic situations involving mixed states subjected to decoherence processes, since this is usually a difficult task with the use of the quantum Fisher information [29].

We consider in this work NOON states [30], phase states [31], states produced with the incidence of a twin-Fock state [32] and with a correlated Fock state [33], [34] at the interferometer inputs, and with the incidence of a squeezed state in one interferometer input and a coherent state in the other [35], besides the “classical” situation of a Fock state sent at one of the interferometer inputs. Our results give some insights for the fundamental reason behind the improved sensitivity of the phase estimation by using quantum light sources.

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