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International Journal of Quantum Information

Emerging quantum mechanics: Coefficient of second-order coherence from classical random fields interacting with threshold type detectors

Andrei Yu. Khrennikov

Published Nov 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749915600072

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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.
In this paper we present results of numerical simulation based on Prequantum Classical Statistical Field Theory (PCSFT), a model with hidden variables of the field-type reproducing probabilistic predictions of quantum mechanics (QM). PCSFT is combined with measurement theory based on detectors of the threshold type. The latter describes discrete events corresponding to the continuous fields model, PCSFT. Numerical modeling demonstrated that the classical Brownian motion (the Wiener process valued in complex Hilbert space) producing clicks when approaching the detection threshold gives probabilities of detection predcited by the formalism QM (as well as PCSFT). This numerical result is important, since the transition from PCSFT to the threshold detection has a complex mathematical structure (in the framework of classical random processes) and it was modeled only approximately. We also perform numerical simulation for the PCSFT-value of the coefficient of second order coherence. Our result matches well with the prediction of quantum theory. Thus, opposite to semiclassical theory, PCSFT cannot be rejected as a consequence of measurements of g(2)(0).g^{(2)}(0).g(2)(0). Finally, we analyze the output of the recent experiment performed in NIST questioning the validity of some predictions of PCSFT.
The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.
The very old problem of the statistical content of quantum mechanics is studied in a novel framework. We show that quantum mechanics (QM) can be considered as a measurement model of prequantum classical statistical field theory (PCSFT): a pure field theory reproducing quantum correlations. We present a measurement scheme for classical signals which reproduces the basic rule of QM, the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. The size of the detection threshold is determined by the mean energy of a prequantum signal. QM is interpreted as theory of measurements of classical random signals with threshold type detectors which are calibrated in a proper way. The essence of this paper is the creation of a rigorous mathematical model of threshold detection of classical (ergodic) stochastic processes.
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.
We show that quantum probabilities for photon detection can be reproduced by a model in which classical random fields interact with detectors of the threshold type. This approach is applied to the old problem of distinguishing classical and quantum light sources with the aid of the coefficient of second-order coherence g(2)(0) (the problem of "existence of photon"). In our classical random field model, we obtain an estimate of this coefficient implying that it becomes strictly less than one for sufficiently small value of parameter [Formula: see text], where [Formula: see text] is the average energy of pulses (photons) emitted by a source and [Formula: see text] is the detection threshold. This prediction can in principle be tested experimentally. Thus in the presented model experimental technicalities (such as e.g. the detection thresholds) are lifted to the level of the fundamental entities of theory.