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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Quantum state inference from photocount statistics: one-probe reconstruction and reconstruction checking the presence or absence of photons

Dmitri S. Mogilevtsev, Z. Hradil, Jan Peřina

Published Apr 1, 1998• 8 citations
Publisher

Abstract

We demonstrate the possibility of reconstructing the density matrix of a single-mode signal state of a radiation field by measuring the photocount statistics in a simple two-detector measurement scheme. We show that the reconstruction can be done by measuring either the complete photocount distribution for identical probe states or the probability of zero counts on both detectors for different probe states.

DOI
10.1088/1355-5111/10/2/008

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We demonstrate the possibility of reconstructing the density matrix of a single-mode signal state of a radiation field by measuring the photocount statistics in a simple two-detector measurement scheme. We show that the reconstruction can be done by measuring either the complete photocount distribution for identical probe states or the probability of zero counts on both detectors for different probe states.
We present various schemes for measuring the quantum state of a single mode of the electromagnetic field. These involve measuring the photon statistics for the mode before and after an interaction with either one or two two-level atoms. The photon statistics conditioned on the final state of the atoms, for two choices of the initial set of atomic states, along with the initial photon statistics, may be used to calculate the complete quantum state in a simple manner. Alternatively, when one atom is used, two unconditioned sets of photon statistics, each after interaction with a single atom in different initial states, along with the initial photon statistics may be used to calculate the initial state in a simple manner. When the cavity is allowed to interact with just one atom, only pure cavity states which do not contain zeros in the photon distribution may be reconstructed. When two atoms are used we may reconstruct pure states which do not contain adjacent zeros in the photon distribution. Coherent states and number states are among those that may be measured with one-atom interaction, and squeezed states and ‚Schrödinger cats‘ are among those that may be measured with a two-atom interaction.
Any single on-off photocounter, which can only detect the presence or absence of photons without discriminating their number, is not capable of identifying nonclassical nature of light. This limitation arises because any photocounting statistics obtained with such a detector can be easily reproduced with coherent states of a light mode. We show that a simple modification of an on-off detector -- introducing controlled attenuation as a tunable setting -- enables such detectors to reveal nonclassical properties of radiation fields.
Reconstruction of photon statistics of optical states provides fundamental information on the nature of any optical field and helps with various relevant applications. Nevertheless, no detector that can reliably discriminate the number of incident photons is available. On the other hand, the alternative of reconstructing the density matrix by quantum tomography leads to various technical difficulties that are particularly severe in the pulsed regime (where mode matching between a signal and local oscillator is very challenging). Even if on/off detectors, as usual avalanche photodiodes operating in Geiger mode, seem useless as photo-counters, it was recently shown how reconstruction of photon statistics is possible by considering a variable quantum efficiency. Here we present experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams in a scheme based on on/off avalanche photodetection assisted by maximum-likelihood estimation. Reconstructions of the distribution for both semiclassical and quantum states of light (as single photon, coherent, pseudothermal, and multithermal states) are reported for single-mode and multimode beams. The stability and good accuracy obtained in the reconstruction of these states clearly demonstrate the interesting potentialities of this simple technique.
An original approach has been developed to extract information on the photon distribution function recorded by the photocount method in the presence of interference. Photon statistics reconstruction has been studied in the presence of two types of interference: dark counts and background radiation. The calculation has been performed using a characteristic function. Correctly reconstructing photon statistics and appropriate experimental conditions have been discussed. The obtained results can be used in measuring photon statistics of low-photon radiation sources and non-classical light fields.