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Physical Review A

Continuous quantum-nondemolition measurement of photon number

Masahito Ueda, Nobuyuki Imoto, Hiroshi Nagaoka, Tetsuo Ogawa

Published Sep 1, 1992• 56 citations
Publisher

Abstract

This paper presents a general theory for a continuous quantum-nondemolition measurement of photon number. This theory treats a time-distributed measurement as a sequence of measurements in which at most one photon can be detected in an infinitesimal time, and shows that the average number of photons remaining in the measured field increases when a photon is detected and decreases when no photon is detected. The state of the measured system evolves nonunitarily and reduces continuously to a nunber state whose eigenvalue is uniquely determined by the average rate of photodetection and whose probability distribution coincides with the initial photon-number distribution. Applying the general theory to typical quantum states---coherent, thermal, and squeezed states---shows that the continuous-state reduction towards a number state depends strongly on the initial photon statistics. Despite the nonunitarity of state evolution, an initially pure state keeps its purity: the initial density operator becomes diagonalized only if the readout information is discarded.

DOI
10.1103/physreva.46.2859

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This paper presents a general theory for a continuous quantum-nondemolition measurement of photon number. This theory treats a time-distributed measurement as a sequence of measurements in which at most one photon can be detected in an infinitesimal time, and shows that the average number of photons remaining in the measured field increases when a photon is detected and decreases when no photon is detected. The state of the measured system evolves nonunitarily and reduces continuously to a nunber state whose eigenvalue is uniquely determined by the average rate of photodetection and whose probability distribution coincides with the initial photon-number distribution. Applying the general theory to typical quantum states---coherent, thermal, and squeezed states---shows that the continuous-state reduction towards a number state depends strongly on the initial photon statistics. Despite the nonunitarity of state evolution, an initially pure state keeps its purity: the initial density operator becomes diagonalized only if the readout information is discarded.
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.
Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths.
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.
We propose a quantum nondemolition measurement of the photon-number distribution for a two-mode cavity field. In the scheme two sequences of two-level atoms interact dispersively with the respective cavity modes and resonantly with two classical fields, and then are detected continually. The field finally reduces to a two-mode Fock state. The probability of collapsing to a given Fock state is determined by the initial photon-number distribution. The scheme can be easily generalized to a field with N modes.