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Physical Review A

One-probe reconstruction of a quantum state

Dmitri S. Mogilevtsev

Published Mar 1, 1998• 2 citations
Publisher

Abstract

In this work we demonstrate a possibility to reconstruct a density matrix of a single-mode signal state of radiation field using a two-detector measurement scheme with fixed position of beam splitter and identical probe states for both ideal and nonideal detectors. It is also shown that by using different probe states in this measurement scheme it is possible to make the reconstruction measuring only the probability of no photon registration on both nonideal detectors.

DOI
10.1103/physreva.57.2146

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In this work we demonstrate a possibility to reconstruct a density matrix of a single-mode signal state of radiation field using a two-detector measurement scheme with fixed position of beam splitter and identical probe states for both ideal and nonideal detectors. It is also shown that by using different probe states in this measurement scheme it is possible to make the reconstruction measuring only the probability of no photon registration on both nonideal detectors.
We suggest a method for measuring the photon number distribution and the density matrix of a single-mode radiation field. The setup consists of a chain of active ring cavities, each fed by a strong coherent probe and coupled to the signal through cross-Kerr phase modulation. Each cavity is set at resonance by a different Fock component of the signal, so that the detection probability at the cavities output is proportional to the signal photon statistics. The off-diagonal elements of the density matrix can be evaluated by measuring the photon statistics of a set of displaced signal states. The suggested setup allows a reliable state measurement even with low quantum efficiency at detectors.
We propose a scheme to reconstruct the state of a two-mode Bose-Einstein condensate, with a given total number of atoms, using an atom interferometer that requires beam splitter, phase shift, and nonideal atom counting operations. The density matrix in the number-state basis can be computed directly from the probabilities of different counts for various phase shifts between the original modes, unless the beam splitter is exactly balanced. Simulated noisy data from a two-mode coherent state are produced and the state is reconstructed, for 49 atoms. The error can be estimated from the singular values of the transformation matrix between state and probability data.
Any measurement scheme involving interference of quantum states of the electromagnetic field necessarily mixes information about the spatiotemporal structure of these fields and quantum states in the recorded data. We show that in this case, a trade-off is possible between extracting information about the quantum states and the structure of the underlying fields, with the modal overlap being either a goal or a convenient tool of the reconstruction. We show that varying quantum states in a controlled way allows one to infer temporal profiles of modes. Vice versa, for the known quantum state of the probe and controlled variable overlap, one can infer the quantum state of the signal. We demonstrate this trade-off by performing an experiment using the simplest on–off detection in an unbalanced weak homodyning scheme. For the single-mode case, we demonstrate experimentally inference of the overlap and a few-photon signal state. Moreover, we show theoretically that the same single-detector scheme is sufficient even for arbitrary multi-mode fields.