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arXiv:quant-ph/0512151 · Optics Letters

Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam

V. Delaubert, Nicolas Treps, C. C. Harb, Ping Koy Lam, Hans‐A. Bachor

Published Apr 28, 2006• 45 citations
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Abstract

We consider the problem of measurement of optical transverse profile parameters and their conjugate variable. Using multimode analysis, we introduce the concept of detection noise modes. For Gaussian beams, displacement and tilt are a pair of transverse-profile conjugate variables. We experimentally demonstrate the optimal encoding and detection of these variables with a spatial homodyning scheme. Using higher-order spatial mode squeezing, we show the sub-shot-noise measurements for the displacement and tilt of a Gaussian beam.

DOI
10.1364/ol.31.001537

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We consider the problem of measurement of optical transverse profile parameters and their conjugate variable. Using multimode analysis, we introduce the concept of detection noise modes. For Gaussian beams, displacement and tilt are a pair of transverse-profile conjugate variables. We experimentally demonstrate the optimal encoding and detection of these variables with a spatial homodyning scheme. Using higher-order spatial mode squeezing, we show the sub-shot-noise measurements for the displacement and tilt of a Gaussian beam.
Spatial multi-mode quantum optics, also termed « quantum imaging », allows a study of the detection of any physical parameter within an optical image. The parameter considered here can for instance be the transverse displacement or the tilt of a laser beam, or even the bit sequence of an optical disc encoded onto the beam. Such measurements are ultimately limited by the quantum nature of optical images. We show that the use of non classical beams allows a reduction of the measurement fluctuations below the latter fundamental limit. Using this theoretical result, we have performed optimal measurements of laser beam nano-displacement, as well as entanglement between this quantity and its conjugate variable : beam tilt. The implementation of these experiments has led us to demonstrate all the key elements required for quantum imaging experiments. We have also investigated the possibility of using multi-mode light as parallel communication channels for quantum information.
We present a detailed description of small displacement and tilt measurements of a Gaussian beam using split detectors and TEM10{\mathrm{TEM}}_{10}TEM10​ homodyne detectors. Theoretical analysis and an experimental demonstration of measurements of these two conjugate variables are given. A comparison between the experimental efficiency of each scheme proves that the standard split detection is only 64% efficient relative to the TEM10{\mathrm{TEM}}_{10}TEM10​ homodyne detection, which is optimal for beam displacement and tilt. We also demonstrate experimentally that squeezed light in the appropriate spatial modes allows measurements beyond the quantum noise limit for both types of detectors. Finally, we explain how to choose the detection scheme best adapted to a given application.
Estimation of an optical beam's transverse displacement is a canonical imaging problem fundamental to numerous optical imaging and sensing tasks. Quantum enhancements to the measurement precision in this problem have been studied extensively. However, previous studies have neither accounted for diffraction loss in full generality, nor have they addressed how to jointly optimize the spatial mode and the balance between squeezing and coherent amplitude. Here we show that, in the small-displacement limit, the seemingly intractable infinite-spatial-mode problem can be reduced to a compact three-mode interaction framework. We quantify the improvement afforded by an optimized single-spatial-mode Gaussian-state probe over the optimal classical laser probe, and show that a two-spatial-mode homodyne receiver is asymptotically optimal for the former in the limit of high probe energy. Our findings reveal a strategy for identifying quantum-optimal probes in the presence of generic multimode linear probe-target interaction and photon loss.
We derive the quantum noise limit for the optical beam displacement of a TEM00 mode. Using a multimodal analysis, we show that the conventional split detection scheme for measuring beam displacement is non-optimal with 80% efficiency. We propose a new displacement measurement scheme that is optimal for small beam displacement. This scheme utilises a homodyne detection setup that has a TEM10 mode local oscillator. We show that although the quantum noise limit to displacement measurement can be surpassed using squeezed light in appropriate spatial modes for both schemes, the TEM10 homodyning scheme out-performs split detection for all values of squeezing.
Estimation of an optical beam’s transverse displacement is a canonical imaging problem fundamental to numerous optical imaging and sensing tasks. Quantum enhancements to the measurement precision in this problem have been studied extensively. However, previous studies have neither accounted for diffraction loss in full generality, nor have they addressed how to jointly optimize the spatial mode and the balance between squeezing and coherent amplitude. Here we show that, in the small-displacement limit, the seemingly intractable infinite-spatial-mode problem can be reduced to a compact three-mode interaction framework. We quantify the improvement afforded by an optimized single-spatial-mode Gaussian-state probe over the optimal classical laser probe, and show that a two-spatial-mode homodyne receiver is asymptotically optimal for the former in the limit of high probe energy. Our findings reveal a strategy for identifying quantum-optimal probes in the presence of generic multimode linear probe-target interaction and photon loss.
The application of quantum optics to spatial measurements with continuous-wave laser beams is discussed. In particular, we summarize the work on measuring small displacements and tilt. We present the required theoretical multi-mode model and discuss how it applies to split detectors and spatial homodyne detectors. We show how spatial measurements can be improved using squeezed light. Finally, we present a demonstration of improved spatial homodyne measurements using squeezed light in a higher order spatial mode.