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Physical Review A

Continuous measurement of photon number with double registration of counts

A. Lukš, Vlasta Peřinová, J. Křepelka

Published Aug 10, 2000• 1 citation
Publisher

Abstract

The theory of continuous measurement has been applied to photon counting by means of an atomic beam crossing a cavity and impinging on a field-ionization detector behind. An interesting feature of the proposed measurement consists in an incoherent frequency down-conversion coupling of the measured mode to another mode, which, on the assumption that the latter is initially in the vacuum state, results in a second registration of the counts.

DOI
10.1103/physreva.62.033806

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The theory of continuous measurement has been applied to photon counting by means of an atomic beam crossing a cavity and impinging on a field-ionization detector behind. An interesting feature of the proposed measurement consists in an incoherent frequency down-conversion coupling of the measured mode to another mode, which, on the assumption that the latter is initially in the vacuum state, results in a second registration of the counts.
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 an experimental scheme to probe the form of one-count operation used in the theory of continuous photodetection in cavities. Two main steps are: 1) an absorption of a single photon by an atom passing through a high-Q cavity containing electromagnetic field in a thermal or coherent state, 2) a subsequent measurement of the photon statistics in the new field state arising after the photon absorption. Then comparing the probabilities of finding 0 and 1 photons in the initial and final states of the field, one can make conclusions on the form of the one-count operation. This method can be readily applied in the microwave cavity QED with present technology.
The counting statistics give insight into the properties of quantum states of light and other quantum states of matter such as ultracold atoms or electrons. The theoretical description of photon counting was derived in the 1960s and was extended to massive particles more recently. Typically, the interaction between each particle and the detector is assumed to be limited to short time intervals, and the probability of counting particles in one interval is independent of the measurements in previous intervals. There has been some effort to describe particle counting as a continuous measurement, where the detector and the field to be counted interact continuously. However, the formalism based on continuous measurements does not provide a formula applicable to general time- and space-dependent fields. In our work, we derive a fully time- and space-dependent description of the counting process for linear quantum many-body systems, taking into account the back-action of the detector on the field. We apply our formalism to an expanding Bose-Einstein condensate of ultracold atoms, and show that it describes the process correctly, whereas the standard approach gives unphysical results in some limits. The example illustrates that, in certain situations, the back-action of the detector cannot be neglected and has to be included in the description.
We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.