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Quantum Limited Parameter Estimation with Pulse Shaped Frequency Combs

Valérian Thiel, Pu Jian, Jonathan Roslund, Nicolas Treps, Claude Fabre

Published Jan 1, 2014• 0 citations
Publisher

Abstract

Combining a multi-color homodyne detection scheme with techniques from quantum optics, we show the feasibility of shot-noise limited measurements of a medium’s parameters, while also being able to compensate for their fluctuations.
DOI
10.1364/cleo_si.2014.sf1i.6

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Summary form only given. Optical techniques are widely used in many areas of science and technology to make accurate measurements. The precision of these measurements suffers from limits due to the unavoidable quantum fluctuations of light. When the light is in the quasi-classical coherent state, this limit is called ”standard quantum limit” and scales as 1/√N with N the number of photons. While better scaling can be achieved with exotic quantum states, coherent light is experimentally interesting as we are able to produce it with large N. We consider light as a probe for the measurement of a parameter p that affects the medium or the length through which the light propagates. We derived the ultimate limit of sensitivity, the so-called quantum Cramér-Rao bound, for the measurement of p [1]: for u(ω, p) the normalized mean field mode of the light, the minimal fluctuation of p that can be measured is given in Fig. 1. We also showed that this ultimate limit of sensitivity can be reached experimentally in a homodyne detection scheme, where the intense local oscillator is taken in the mode wp o ∂u ∂p, called detection mode associated with p. This measurement scheme can be applied to space-time positioning experiments such as ranging or clock synchronization [2]. The basic method for space-time positioning relies on the exchange of regular light pulses between two emitters. For coherent broadband light pulses, such as the ones delivered by a mode-locked femtosecond laser, of central frequency ω0 and of spectral width Δω, the quantum limits of the measurements of the delay t are given by δtphase = 2√Nω0 1 for interferometric measurements and δttof = 2√NΔω 1 for time-of-flight measurements. However, these two methods do not reach the quantum Cramer-Rao bound; this ultimate limit is achieved by a homodyne detection with local oscillator w(ω) o iωu(ω), and is given by 1 δtmin = (1) 2√N . Δω2 + ω2 0 For usual characteristics of a Titane:Sapphire femtosecond laser and 1 ms integration time, δtmin = 10-23s. We demonstrate this limit experimentally by using pulse shaping techniques to shape the local oscillator in the detection mode. We also study the influence of the intensity and phase noise of a mode-locked femtosecond laser on this limit. This technique can also be applied in the case of ranging in a dispersive medium such as air. In this case, dispersive effects due to environmental parameters such as pressure, temperature or humidity affect the precision of measurements. Usually, this problem is tacked by measuring separately the environmental parameters or using several wavelengths, which can be difficult and require post-processing. We show that using the same technique as previously shown, the parasitic effects due to the medium in a distance measurement can be canceled by shaping properly the local oscillator [3]: we choose for the local oscillator a mode that is orthogonal to the environmental effects, so the homodyne detection is insensitive to them, at the expense of a lower sensitivity. This scheme is an all-optical and real-time measurement of distance in a dispersive medium.
Accurate time-delay measurement has become a crucial issue for space positioning experiments which require increasing precision over large distances. The femtosecond frequency comb has been found to be a good source for its advantages of a large nonambiguity range and coherence length up to tens of kilometers or more. A scheme combining homodyne detection and frequency combs was proposed [B. Lamine et al., Phys. Rev. Lett. 101, 123601 (2008)] that leads to a timing sensitivity reaching the shot-noise limit; with a squeezed frequency comb as the input field, the measured timing sensitivity can be further improved to the level of the sub-shot-noise limit. Based on this scheme and a squeezed frequency comb that was measured to have a phase squeezing of 1.5 dB at an analyzing frequency of 2 MHz, a narrow-band interferometric timing measurement is implemented. The minimum measurable timing variation modulation at 2 MHz is reduced from a shot-noise-limited value of $(2.8\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}$ s to $(2.4\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}$ s.