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Journal of Applied Mathematics and Physics

Vacuum Correlations, Detector Efficiency Fluctuations and Classical Dynamic Violations of CH-Inequality

Gianpaolo Bei

Published Jan 1, 2023• 2 citations
Publisher

Abstract

We study in this paper the possible influence of vacuum fluctuations on photo detection and its background noise in Bell tests. We analyze its consequences on the standard statistical analysis of data showing that it is not fulfilled anymore the conventional hypothesis of a Poisson like probability density distribution of single photodetection events. We assume that vacuum fluctuations are due to real and measurable fluctuating fields, as recently confirmed experimentally, and that their non null correlations outside the light cone contribute to photon coincidence rates making them time dependent. We introduce a generalized Bell like correlation function which contains a new term due to supposed vacuum induced photon counting events. We deduce then a generalization of CH-inequality which takes in account the effect of these vacuum electric fields on detector efficiency. We predict an apparatus temperature fluctuations during photon detection which we suggest could be observed by looking for colored noise thermal emission of the photodetectors, generalizing the standard white noise prediction of C.S.L. models on wave function collapse postulate. We discuss an experimental test of this prediction, based on the idea of inducing a thermal wave on the whole quantum detectors, aimed to observe time dependent deviations from standard stationary statistical predictions of Quantum Mechanics.

DOI
10.4236/jamp.2023.116109

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We study in this paper the possible influence of vacuum fluctuations on photo detection and its background noise in Bell tests. We analyze its consequences on the standard statistical analysis of data showing that it is not fulfilled anymore the conventional hypothesis of a Poisson like probability density distribution of single photodetection events. We assume that vacuum fluctuations are due to real and measurable fluctuating fields, as recently confirmed experimentally, and that their non null correlations outside the light cone contribute to photon coincidence rates making them time dependent. We introduce a generalized Bell like correlation function which contains a new term due to supposed vacuum induced photon counting events. We deduce then a generalization of CH-inequality which takes in account the effect of these vacuum electric fields on detector efficiency. We predict an apparatus temperature fluctuations during photon detection which we suggest could be observed by looking for colored noise thermal emission of the photodetectors, generalizing the standard white noise prediction of C.S.L. models on wave function collapse postulate. We discuss an experimental test of this prediction, based on the idea of inducing a thermal wave on the whole quantum detectors, aimed to observe time dependent deviations from standard stationary statistical predictions of Quantum Mechanics.
We extend a previous model of the author which generalizes Bell local hidden variable models to the case of entangled photon pairs assuming that the standard Bell correlation functions depend on a hidden vacuum index. We deduce a generalization of Bell theorem assuming that classical observables are not dichotomic and that photon pair emission and detection is not a stationary stochastic process. We derive a photon imperfect polarization correlation functions due to rotational invariance breaking induced by hidden vacuum spin currents. We implement formally this approach deducing a generalization of C.H.S.H. inequalities which asymptotically converges to the standard one and which might be competitive with standard quantum mechanics predictions. We suggest to test this inequalities conceiving new E.P.R.-Bell like tests with time dependent detector efficiency and photon flux. Finally, we suggest to apply these generalized inequalities to the correlation functions of entangled classical spinning waves realized recently with metamaterials.
Stochastic optics may be considered as simply a local realistic interpretation of quantum optics and, in this sense, it is a first step in the reinterpretation of the whole of quantum theory. However, as it is not possible to interpret all the details of quantum theory in a local realistic manner, as shown by Bell's theorem, minor changes are introduced in the formalism with the consequence that the new theory makes different predictions in some special cases. In stochastic optics, the quantum-operator formalism is simply considered a formal way of dealing with stochastic fields. In particular, the quantum zero point is taken as a real random electromagnetic radiation filling the whole of space. This radiation noise has the same nature as light signals, the only difference being the greater intensity of the latter. We assume that photon detectors have an intensity threshold just above the level of the noise, thus detecting only signals. Transmission of radiation through polarizers follows Malus's law, but the interplay of signal and noise leads quite naturally to the prediction that the detection probability of some signals is enhanced, which is known to be a necessary condition for the violation of the empirically tested Bell inequalities. In our view, correlated photon pairs are pairs of light signals supercorrelated in polarization, in the sense that, as well as the signal, the accompanying noise is also correlated. Thus stochastic optics allows predictions for the empirical correlations very close, but not identical, to the quantum ones. The theory is applied to the analysis of all experiments designed to test the Bell inequalities by measuring polarization correlations of photon pairs. The predictions agree with quantum optics and experiments within statistical errors, except for the Holt-Pipkin experiment. In this case, the experimental results agree with stochastic optical predictions within two standard deviations while violating quantum optics by four.
We investigate the role played by quantum operator ordering in the correlations that characterize two-photon polarization Bell measurements. The Clauser-Horne-Shimony-Holt (CHSH) criterion is investigated in the normal ordering imposed by the photodetection theory and in the symmetric ordering that constitutes the standard prescription for building Hermitian operators from products of noncommuting observables. The two approaches are obtained in a single theoretical framework, where operator ordering is directly associated with the representation used for the density matrix. Moreover, this discussion can be recast in terms of the contribution given by the vacuum fluctuations to the detected signals. We also envisage possible detection schemes sensitive to these fluctuations with recent technological developments.
We show that various experimental configurations using correlated photon-number states are predicted to violate many forms of Bell's inequality. This test is potentially one of macroscopic quantum mechanics in that violations are predicted for situations where large numbers of photons are detected at a single detector at a time. The configurations also allow higher-spin tests of quantum mechanics. We indicate how the violations of Bell inequalities might be achieved with both parametric amplification and parametric oscillation. The effect of photodetection inefficiencies is analyzed. In order to investigate the potential for Bell-inequality tests in the high-intensity regime of recent experiments where detection efficiencies may be very high, we introduce a simple model for the effect of electronic noise that introduces a poor resolution of the photon number detected.
Loophole-free Bell tests for quantum nonlocality and long-distance secure communication require photodetection efficiencies beyond a threshold ${\ensuremath{\eta}}_{\mathrm{crit}}$ that depends on the Bell inequality and the noise affecting the entangled state received by the distant parties. Most calculations of ${\ensuremath{\eta}}_{\mathrm{crit}}$ assume that the noise is random and can be modeled as white noise. However, most sources suffer from colored noise. Indeed, since entangled states are usually created as a superposition of two possible deexcitation paths, a partial distinguishability between the two processes leads to the appearance of colored noise in the generated state. Recently, there was a proposal for a loophole-free Bell test [A. Cabello and F. Sciarrino, Phys. Rev. X 2, 021010 (2012)], where a specific colored noise appears as a consequence of the precertification of the photon's presence through single-photon spontaneous parametric down-conversion. Here we obtain ${\ensuremath{\eta}}_{\mathrm{crit}}$, the optimal quantum states, and the local settings for a loophole-free Bell test as a function of the amount of colored noise. We consider three bipartite Bell inequalities with nnn dichotomic settings: Clauser-Horne-Shimony-Holt (n=2n=2n=2), I3322{I}_{3322}I3322​ (n=3n=3n=3), and A5{A}_{5}A5​ (n=4n=4n=4), both for the case of symmetric efficiencies, corresponding to photon-photon Bell tests, and for the totally asymmetric case, corresponding to atom-photon Bell tests. Remarkably, in all these cases, ${\ensuremath{\eta}}_{\mathrm{crit}}$ is robust against the colored noise. The present analysis can find application in any test of Bell inequalities in which the dominant noise is of the colored type.
In this paper we derive a general approach using correlation functions for interpreting the results of measurements on two-particle entangled states. In our analysis we explicitly take into account the detector efficiency. We show that all Bell-type inequalities have bounds depending on both the losses in the apparatus and the supplementary assumptions about the statistical distribution of the losses themselves. Moreover, our approach enables us to include in the causal and local theory the same assumptions as used in the quantum mechanical analysis of an experiment, and to deduce the inequality which has to be satisfied by any local realistic model incorporating these assumptions. We show that, although the most recent experiments on Bell's inequality can be considered to be a remarkable step in the direction of testing quantum mechanics predictions, the loophole of the low quantum efficiency of the detector is not yet solved. In fact, using the quantum mechanical assumption of random nondetection in the causal and local approach, we deduce a generalized Bell-type inequality which includes no-count events and whose upper bound can be violated by the quantum mechanical predictions only if the detector quantum efficiency is >0.811.