Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state

Abstract

Phase precision in optimal 2-channel quantum interferometry is studied in the limit of large photon number N 1, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the losses or the number of photons increase. We further show that fixing the loss paramater determines a scale for quantum metrology -a crossover value of the photon number Nc beyond which the supra-classical precision is progressively lost. For large losses the optimal state also has a different structure from those considered previously. 42.50.St,06.20.Dk It has been recognized that using quantum states of light may increase the resolution of interferometric measurements [1][2][3]. Particular states of N photons achieve the Heisenberg limit of phase resolution for standard error on the phase estimate ∆ϕ = 1/N , an improvement over the classical (or shotnoise) limit ∆ϕ = 1/ √ N that is obtainable when N photons enter the interferometer one at a time. These bounds are derived by an application of the Cramer-Rao inequality [2] for the standard error of an unbiased estimator, ∆ϕ ≥ (νF) −1/2 , where F is the quantum Fisher information (QFI) [4] and ν is the number of repeated independent trials. Assuming any instrument is composed of three components: quantum input state, dynamics and measurement; the functional F depends only on the first two -it assumes an optimal measurement choice. For pure states in a single mode F/4 = ∆ 2n ≡ n 2 − n 2 (wheren is the number operator) and a familiar uncertainty relation is recovered: ∆n∆ϕ ≥ 1/2. Thus, for a lossless two-mode interferometer QFI and precision are greatest for the maximum variance state, or 'NOON state'; it saturates the Heisenberg limit. Unfortunately, it is also highly susceptible to noise, especially dissipation [5].To mitigate this problem various two-component states were proposed [6][7][8], where the loss of a number of photons in the first mode does not destroy the superposition. The precision performance under dissipation of various Gaussian states, e.g. squeezed, coherent and thermal states, has also been considered recently [9]. In all cases, the precision was found to be supra-classical for certain range of losses and N .In the lossy case the pure input state of two oscillator modes maximizing QFImust balance supra-classical precision against robustness to photon loss. In this notation the NOON state has two non-zero components, φ 0 = φ N = 1/ √ 2. For a lossy interferometer light propagates in each arm as a damped harmonic oscillator, with frequencies ω (1) , ω (2) and dissipation γ (1) , γ (2) . Equivalently, losses can be introduced by beam-splitters in each mode with reflectivity R (1,2) = 1 − exp{−γ (1,2) t}. Those lost photons siphoned out of the modes are then traced over.In the simpler case of losses in only one of the two modes, R (1) = R > 0, R (2) = 0, as might occur when that mode is directed through a partially transparent test sample, the state |φ decays into a mixtureρ = k ...

Advances in quantum metrology

“…Is it then really possible to outperform classical strategies in practical phase sensing [100]? Surprisingly, it has been shown that asymptotically it is possible to do so only by a constant factor [101,102]: for any nonzero loss, for sufficiently high number of photons N the scaling of the optimal phase sensing is proportional to the scaling of the shot noise ∝ N −1/2 . While this means that quantum approaches are useful in highly controlled environments [102] (such as for gravitational wave detection [1]), they only allow for very small enhancements in free-space target acquisition [102].…”

“…Surprisingly, it has been shown that asymptotically it is possible to do so only by a constant factor [101,102]: for any nonzero loss, for sufficiently high number of photons N the scaling of the optimal phase sensing is proportional to the scaling of the shot noise ∝ N −1/2 . While this means that quantum approaches are useful in highly controlled environments [102] (such as for gravitational wave detection [1]), they only allow for very small enhancements in free-space target acquisition [102]. Nonetheless, the shot noise can be beaten [103] and the optimal states to do so in the presence of loss have been calculated numerically using various optimization techniques for fixed number of input photons [104,105] and for photon-number detection [106].…”

Towards practical quantum metrology with photon counting

“…2 Moreover, loss will be present in any practical scenario, including absorbance in measured samples and non-unit efficiency detectors. Consequently, revised scaling laws of precision with photon flux have been derived, 17 along with optimised superposition states of fixed photon number, numerically for small photon number 18,19 and analytically for large. 17 Here, we demonstrate the underpinning principles of a practical -loss-tolerant-scheme for sub-shot-noise interferometry, illustrated in Figure 1.…”

“…Consequently, revised scaling laws of precision with photon flux have been derived, 17 along with optimised superposition states of fixed photon number, numerically for small photon number 18,19 and analytically for large. 17 Here, we demonstrate the underpinning principles of a practical -loss-tolerant-scheme for sub-shot-noise interferometry, illustrated in Figure 1. This scheme is designed to use the full fourmode multi-photon state naturally occurring in a non-linear optical process known as type-II spontaneous parametric downconversion (SPDC), which generates a coherent superposition of correlated photon-number states.…”

Heisenberg scaling with weak measurement: a quantum state discrimination point of view

“…This behavior has been argued to be generic to all Heisenberg scaling schemes (see Refs. [20][21][22][23]), so it is not surprising that we also find this behavior here as well. We also showed that the von Neumann measurement interaction also has the phase accumulation effect, provided we prepared the meter states in momentum eigenstates.…”