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arXiv:1112.5591 · Journal of Physics Conference Series

Classical signal model reproducing quantum probabilities for single and coincidence detections

Andrei Yu. Khrennikov, Börje Nilsson, Sven Nordebo

Published May 10, 2012• 14 citations
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Abstract

We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 [22] played a crucial role in rejection of (semi-)classical field models in favour of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favour of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient g (2) (0), is zero (for one photon states), but in (semi-)classical models g (2) (0) ≥ 1. In TSD the coefficient g (2) (0) decreases as 1/ ε 2 d , where ε d > 0 is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient g (2) (0) essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient g (2) (0) on the detection threshold. Structurally our model has some similarity with the prequantum model of Grossing et al. Subquantum stochasticity is composed of the two counterparts: a stationary process in the space of internal degrees of freedom and the random walk type motion describing the temporal dynamics.

DOI
10.1088/1742-6596/361/1/012030

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We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 [22] played a crucial role in rejection of (semi-)classical field models in favour of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favour of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient g (2) (0), is zero (for one photon states), but in (semi-)classical models g (2) (0) ≥ 1. In TSD the coefficient g (2) (0) decreases as 1/ ε 2 d , where ε d > 0 is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient g (2) (0) essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient g (2) (0) on the detection threshold. Structurally our model has some similarity with the prequantum model of Grossing et al. Subquantum stochasticity is composed of the two counterparts: a stationary process in the space of internal degrees of freedom and the random walk type motion describing the temporal dynamics.
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 present a purely wave model (based on classical random field) which reproduces quantum probabilities (given by the fundamental law of quantum mechanics, Born's rule) including probabilities for joint detection of a pair of quantum observables (e.g., spin or polarization projections). The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. The presence of the background field (vacuum fluctuations) is also the key-element of our prequantum model. It is of the classical signal type and the methods of classical signal theory (including statistical radiophysics) are used for its development. We stress that our prequantum model is not objective, i.e., the values of observables (clicks of detectors) cannot be assigned in advance, i.e., before measurement. Hence, the dilemma, nonobjectivity or nonlocality, is resolved in favor of nonobjectivity (our model is local of the classical field type). In particular, we reproduce the probabilities for the EPR-experiment for photon polarization and, hence, violate CHSH inequality for classical random signals (measured by the threshold type and properly calibrated detectors acting in the presence of the background field).
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.
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.
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.
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.