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Optical Engineering

Role of detectors and their proper calibration in inter-relation between classical and quantum optics

Andrei Yu. Khrennikov

Published Jun 11, 2012• 5 citations
Publisher

Abstract

We describe quantum theory by using a complex valued random signal, which is (coherently) averaged over a time window, and a photon click is defined as the energy of this averaged signal surpassing a detector threshold. The random signal is modeled as white noise but grows over time s (proportional to ). Eventually this random signal will lead to a detector "click" defined by a detector threshold condition, at which point the random signal is reset. Further we proceed by analysing correlations and finally by modeling the quantum state of an entangled pair by a complex valued correlation function.

DOI
10.1117/1.oe.51.6.069001

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We describe quantum theory by using a complex valued random signal, which is (coherently) averaged over a time window, and a photon click is defined as the energy of this averaged signal surpassing a detector threshold. The random signal is modeled as white noise but grows over time s (proportional to ). Eventually this random signal will lead to a detector "click" defined by a detector threshold condition, at which point the random signal is reset. Further we proceed by analysing correlations and finally by modeling the quantum state of an entangled pair by a complex valued correlation function.
In this chapter a measurement model for PCSFT will be presented [201]. We explore the basic idea of PCSFT: a quantum state (e.g., a pure state) is simply a label for an ensemble of classical fields- classical random field. We model the process of interaction of a detector with a random field. By using another terminology, we can speak about classical random signalsa and, in particular, about switching to time averages (see Section 3.1.5). The model developed in this chapter is of the phenomenological type. It is based on (heuristically justified) the coupling between the energy distribution of a random field and the probability distribution of clicks of a detector. This model will be concreted in Chapter 9 by considering the detectors of the threshold type. In the later model we shall be able to describe the process of the interaction of a random signal with a threshold-type detector.
All previous chapters of this book were devoted to the development of the theory of classical (prequantum) random fields reproducing quantum averages (including correlations for entangled systems) as averages of quadratic forms of fields. Since the basic model of prequantum fields is the Gaussian one, these fields are continuous. Hence, in the same way as in the classical signal theory, averages are calculated for variables with a continuous range of values. This model, PCSFT, is really a prequantum model. In accordance with the Bohr’s viewpoint, we consider QM as an operational formalism describing (predicting) results of measurements on microsystems. QM cannot describe intrinsic physical processes in the microworld, but onlymeasurements performed bymacroscopic classical devices. In contrast, PCSFT describes intrinsic processes in the microworld. However, it does not describe results of measurements on the level of individual events, clicks of detectors. As we pointed in Section 1.1.8, in principle one might be satisfied with creation of a prequantum model which reproduces only quantum probabilistic predictions, i.e., without establishing a direct connectionwith theory of measurement on the level of individual events. This approach would match with views of Schro¨dinger and the Bild concept in general. However, historical development of QM demonstrated that the Bild concept was not attractive for the majority of physicists. Since the experimental verification is considered as the basic counterpart of any physical theory, Bild-like prequantum theories are considered as metaphysical. Measurement theory connecting PCSFT with experiment on the level of individual events was developed in my paper [211] (see also Refs. [209, 210, 212, 213]). In this paper I elaborated a scheme of discrete measurements of classical random signals which reproduces the basic rule of QM, the Born’s rule. This is the scheme of threshold-type detection: such a detector clicks after it has “eaten” a special portion of energy
The very old problem of the statistical content of quantum mechanics (QM) is studied in a novel framework. The Born's rule (one of the basic postulates of QM) is derived from theory of classical random signals. We present a measurement scheme which transforms continuous signals into discrete clicks and reproduces the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. Creation of such a detection model provides a possibility to represent the basic quantum phenomena as classical field phenomena completed by measurements with threshold type and properly calibrated detectors. Finally, the wave-particle duality has been resolved in favor of the purely wave description. It is well known that coincidence probability plays an important role in the debate on the possibility to reduce quantum theory (especially quantum optics) to classical field theory. Our model matches well the predictions of quantum theory and experiment. It violates the prediction of the classical field model which does not take into account the presence of threshold type detectors with proper calibration.
This paper is a contribution to the project "emergent quantum mechanics" unifying a variety of attempts to treat quantum mechanics (QMs) as emergent from other theories pretending on finer descriptions of quantum phenomena. More concretely it is about an attempt to model detection probabilities predicted by QM for single photon states by using classical random fields interacting with detectors of the threshold type. Continuous field model, prequantum classical statistical field theory (PCSFT), was developed in recent years and its predictions about probabilities and correlations match well with QM. The main problem is to develop the corresponding measurement theory which would describe the transition from continuous fields to discrete events, "clicks of detectors". Some success was achieved and the click-probabilities for quantum observables can be derived from PCSFT by modeling interaction of fields with the threshold type detectors. However, already for the coefficient of second-order coherence g2(0) calculations are too complicated and only an estimation of g2(0) was obtained. In this paper, we present results of numerical simulation based on PCSFT and modeling of interaction with threshold type detectors. The "prequantum random field" interacting with a detector is modeled as the Brownian motion in the space of classical fields (Wiener process in complex Hilbert space). Simulation for g2(0) shows that this coefficient approaches zero with increase of the number of detections.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.