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Physical Review Letters

Quantum State Tomography with Array Detectors

M. Beck

Published Jun 19, 2000• 48 citations
Publisher

Abstract

I propose a method for measuring the quantum state of an optical field that occupies a mode having a complicated spatial structure. The technique uses array detectors and a single, plane-wave local oscillator beam. The advantage of using array detectors is that the local oscillator is not mode matched to the field being measured, yet the deleterious effects of this mismatch on the effective detection efficiency are greatly reduced compared to using single detectors. Indeed, when the spatial mode of the signal field is describable by a real function, the effective mode-matching efficiency is unity.

DOI
10.1103/physrevlett.84.5748

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We have simultaneously measured the quantum states of two different spatial modes of the same optical beam by performing quantum-state tomography with an array detector. Both modes are well described by coherent states, but the projection of the signal onto the local oscillator mode contains a mean of 0.09 photons, while a more complicated mode has a mean of 4.3 photons. This demonstrates that for this particular mode the effective detection efficiency when using array detection is over 40 times greater than when using single detectors.

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I propose a method for measuring the quantum state of an optical field that occupies a mode having a complicated spatial structure. The technique uses array detectors and a single, plane-wave local oscillator beam. The advantage of using array detectors is that the local oscillator is not mode matched to the field being measured, yet the deleterious effects of this mismatch on the effective detection efficiency are greatly reduced compared to using single detectors. Indeed, when the spatial mode of the signal field is describable by a real function, the effective mode-matching efficiency is unity.
We have simultaneously measured the quantum states of two different spatial modes of the same optical beam by performing quantum-state tomography with an array detector. Both modes are well described by coherent states, but the projection of the signal onto the local oscillator mode contains a mean of 0.09 photons, while a more complicated mode has a mean of 4.3 photons. This demonstrates that for this particular mode the effective detection efficiency when using array detection is over 40 times greater than when using single detectors.
We demonstrate that it is possible to use the balanced homodyning with array detectors to measure the quantum state of a correlated two-mode signal field. We show the applicability of the method to fields with complex mode functions, thus generalizing the work of Beck [Phys. Rev. Lett. 84, 5748 (2000)] in several important ways. We further establish that, under suitable conditions, array detector measurements from one of the two ouputs are sufficient to determine the quantum state of signal. We show the power of the method by reconstructing a truncated Perelomov state, which exhibits complicated structure in the joint probability density for the quadratures.
We introduce a method for quantum-state tomography for a pair of optical modes, which we call generalized rotations in phase space. Rather than using a dual-mode local oscillator field for implementing balanced-homodyne detection, as in previous methods, a single-mode local oscillator field can be used in our method, greatly simplifying certain aspects of experiments. The signal field is put through a series of general transformations before entering the detection system where quantum statistics are measured.
We demonstrate that it is possible to choose an optimal signal mode for state reconstruction when performing quantum-state tomography with array detectors. The mode optimization is done during the data analysis (i.e., after all the data have been collected.) We develop theoretically a procedure for finding the mode that satisfies a criterion which is quadratic in field operators; as examples we explicitly show how to maximize the average photon number, or the amount of quadrature squeezing. We experimentally demonstrate the technique by finding the mode which maximizes the average photon number for coherent-state signal beams occupying both linear and sinusoidal modes.
We measure the joint QQQ function of a multispatial-mode field using a charge-coupled device array detector in an unbalanced heterodyne configuration. The intensity pattern formed by interference between a weak signal field and a strong local oscillator is resolved using Fourier analysis and used to reconstruct quadrature amplitude statistics for 22 spatial modes simultaneously. The local oscillator and signal propagate at an angle of 12mrad12\phantom{\rule{0.28em}{0ex}}\mathrm{mrad}12mrad, thus shifting the classical noise to modes that do not overlap with the signal. In this configuration, balanced detection is not necessary.
Transformation and detection of photons in higher-order spatial modes usually requires complicated holographic techniques. Detectors based on spatial holograms suffer from non-idealities and should be carefully calibrated. We report a novel method for analyzing the quality of projective measurements in spatial mode basis inspired by quantum detector tomography. It allows us to calibrate the detector response using only gaussian beams. We experimentally investigate the inherent inaccuracy of the existing methods of mode transformation and provide a full statistical reconstruction of the POVM (positive operator valued measure) elements for holographic spatial mode detectors.