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Physical Review A

Quantum theory for continuous photodetection processes

Masahito Ueda, Nobuyuki Imoto, Tetsuo Ogawa

Published Apr 1, 1990• 95 citations

Abstract

We develop a quantum theory for continuous photodetection processes that describes nonunitary time development of the field under continuous measurement of photon number. Exact expressions are obtained for time evolutions of the photon-field density operator, average and variance of the photon number, and the Fano factor. These are applied to typical quantum states, i.e., number, coherent, thermal, and squeezed states. The continuous photodetection process is made up of two elementary processes in terms of the referring measurement process, that is, one-count and no-count processes. Just after the one-count process in which a photodetector registers one photoelectron, the average photon number 〈n(t)〉 of the remaining field is shown to increase for super-Poissonian states (e.g., thermal state) and decrease for sub-Poissonian states (e.g., number state); for the Poissonian state (e.g., coherent light), 〈n(t)〉 does not change. During the no-count process in which the photodetector registers no photoelectrons, on the other hand, 〈n(t)〉 decreases in time for all states except the number state. The physical origins for these results are clarified from the viewpoint of nonunitary state reduction by continuous measurement of photon number. Furthermore, we introduce a nonreferring measurement process in which the detector registers photocounts, but we discard all readout information. We discuss the difference in the way the photon field evolves in this process compared to the referring measurement process.

DOI: 10.1103/physreva.41.3891

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We develop a quantum theory for continuous photodetection processes that describes nonunitary time development of the field under continuous measurement of photon number. Exact expressions are obtained for time evolutions of the photon-field density operator, average and variance of the photon number, and the Fano factor. These are applied to typical quantum states, i.e., number, coherent, thermal, and squeezed states. The continuous photodetection process is made up of two elementary processes in terms of the referring measurement process, that is, one-count and no-count processes. Just after the one-count process in which a photodetector registers one photoelectron, the average photon number 〈n(t)〉 of the remaining field is shown to increase for super-Poissonian states (e.g., thermal state) and decrease for sub-Poissonian states (e.g., number state); for the Poissonian state (e.g., coherent light), 〈n(t)〉 does not change. During the no-count process in which the photodetector registers no photoelectrons, on the other hand, 〈n(t)〉 decreases in time for all states except the number state. The physical origins for these results are clarified from the viewpoint of nonunitary state reduction by continuous measurement of photon number. Furthermore, we introduce a nonreferring measurement process in which the detector registers photocounts, but we discard all readout information. We discuss the difference in the way the photon field evolves in this process compared to the referring measurement process.

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A concept 'probability-density functional' combined with the path integral technique is shown to provide a new framework for quantum photodetection theory. It replaces the conventional picture of unitary evolution of the total density operator followed by von Neumann's projection postulate for the case in which the effects of nonunitary state reduction and continuous measurements must be taken into account. This paper shows that an expression of the probability-density functional which incorporates both these effects provides complete quantum-statistical information about photoelectron statistics. Simple procedures are given for calculating the ensemble average and generating the functional of a quantity distributed in time.

The detection of cavity radiation with the detector placed outside the cavity is studied. Each leaked photon has a certain probability of propagating away without being detected. It is viewed as a continuous quantum measurement in which the density matrix is continuously revised according to the readout of the detector. By analyzing in terms of the more elementary no-leak and one-leak-undetected processes, the superoperator for the no-count process is derived, which leads to a pseudo-master-equation governing the time evolution of unnormalized photon-number distribution.

This paper proposes a microscopic model for the continuous measurement of photon numbers where the continuous measurement is simulated by a sequence of infinitesimal processes. Each process is composed of a unitary evolution for the coupled system, which consists of the field and the measuring apparatus, followed by a projection according to the readout of the measuring apparatus. The proposed model consists of two-level atoms that are coupled one by one with an optical cavity field via the electric-dipole interaction. Each atomic level is then read out after the interaction, causing a nonunitary state reduction of the field. In particular, it is shown that this microscopic model leads to the Srinivas-Davies model that has been postulated for the photodetection process.