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arXiv:quant-ph/0003131 · Physical Review A

Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator

Damian T. Pope, P. D. Drummond, William J. Munro

Published Sep 18, 2000• 13 citations
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Abstract

Intracavity and external third order correlations in the damped nondegenerate parametric oscillator are calculated for quantum mechanics and stochastic electrodynamics (SED), a semiclassical theory. The two theories yield greatly different results, with the correlations of quantum mechanics being cubic in the system's nonlinear coupling constant and those of SED being linear in the same constant. In particular, differences between the two theories are present in at least a mesoscopic regime. They also exist when realistic damping is included. Such differences illustrate distinctions between quantum mechanics and a hidden variable theory for continuous variables.

DOI
10.1103/physreva.62.042108

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Intracavity and external third order correlations in the damped nondegenerate parametric oscillator are calculated for quantum mechanics and stochastic electrodynamics (SED), a semiclassical theory. The two theories yield greatly different results, with the correlations of quantum mechanics being cubic in the system's nonlinear coupling constant and those of SED being linear in the same constant. In particular, differences between the two theories are present in at least a mesoscopic regime. They also exist when realistic damping is included. Such differences illustrate distinctions between quantum mechanics and a hidden variable theory for continuous variables.
Semiclassical theories similar to stochastic electrodynamics are widely used in optics. The distinguishing feature of such theories is that the quantum uncertainty is represented by random statistical fluctuations. They can successfully predict some quantum-mechanical phenomena; for example, the squeezing of the quantum uncertainty in the parametric oscillator. However, since such theories are not equivalent to quantum mechanics, they will not always be useful. Complex number representations can be used to exactly model the quantum uncertainty, but care has to be taken that approximations do not reduce the description to a hidden variable one. This paper helps show the limitations of ``semiclassical theories,'' and helps show where a true quantum-mechanical treatment needs to be used. Third-order correlations are a test that provides a clear distinction between quantum and hidden variable theories in a way analogous to that provided by the ``all or nothing'' Greenberger-Horne-Zeilinger test of local hidden variable theories. \textcopyright{} 1996 The American Physical Society.
I propose that quantum mechanics is a stochastic theory and quantum phenomena derive from the existence of real vacuum stochastic fields filling space. I revisit stochastic electrodynamics (SED), a theory that studies classical systems of electrically charged particles immersed in an electromagnetic (zeropoint) radiation field with spectral density proportional to the cube of the frequency, Planck's constant appearing as the parameter fixing the scale. Asides from briefly reviewing known results, I make a detailed comparison between SED and quantum mechanics. Both theories make the same predictions when the stochastic equations of motion are of first order in Planck constant, but not in general. I propose that SED provides a clue for a realistic interpretation of quantum theory.
We develop the quantum theory of a driven, damped intracavity parametric oscillator in the case of nondegenerate signal and idler modes. An analysis of the results of different types of local-oscillator measurements on the outgoing modes is presented. The frequency-domain squeezing is calculated below threshold including the effect of detunings and nonequal decay rates. We also calculate a nonclassical correlation in the quadrature amplitudes of the output beams. The existence of an Einstein-Podolsky-Rosen paradox is demonstrated through violations of the inferred Heisenberg uncertainty principle at causally unrelated locations in the output field. This provides a way of maintaining the commutation relations of the measured operators in the exact form envisaged originally by Einstein, Podolsky, and Rosen.
We contrast a number of nonclassical features of quantum optics and quantum information science to those in either one or more local hidden variables theories (LHVTs). In quantum optics, we contrast certain third-order transient quantum correlations for the damped nondegenerate parametric oscillator to their analogues in the LHVT stochastic electrodynamics. We also demonstrate, in a novel manner, the occurrence of a recently formulated theoretical process called quantum state exchange in a particular model. This is a process by which we can transfer a degree of freedom from an electromagnetic field to one or more trapped atom(s). Finally, we show that a certain entangled state involving two groups of trapped ions located in two spatially separated ion traps can exhibit correlations beyond the scope of all LHVTs by virtue of violating a novel Bell inequality. Turning to quantum information science, we show that the entangled state mentioned above contains a certain type of multipartite entanglement, a resource which typically manifests correlations that cannot be simulated using a LHVT. We then derive a lower bound on the amount of this entanglement in the state using a novel entanglement measure. Next, we show that, given a certain assumption, instances of the Bennett et al. teleportation protocol for which the fidelity of the teleportation process is greater than approximately 0.902 cannot be modelled by any LHVT. Finally, we contrast the computational power of the quantum circuits model to that of two particular classical wave computing models, showing that one reason why the quantum circuits model is more powerful than these models is that, in this model, we compute using binary representation instead of a unary one.
We calculate the triple quadrature correlations in non-degenerate quantum parametric amplifiers and oscillators. Firstly, at low signal intensities, a highly non-classical triple correlation can emerge in a parametric amplifier. This cannot be represented using the positive Wigner function approximation, sometimes called the semiclassical method. For this reason, measurements of triple quadrature correlations are a very strong test of stochastic electrodynamics and other hidden variable theories. As the corresponding local oscillator measurements involve a macroscopic particle flux at the detector, a high detector efficiency is possible. We show that triple correlations also feature in the critical fluctuations of the non-degenerate quantum parametric oscillator at threshold. This result allows us to asymptotically calculate the critical squeezing produced for large system size.
The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed). Important commonalities and complementarities between the two theories are identified, notwithstanding their dissimilar origins and approaches. Further, the dynamical equation of sqm is completed with the radiation terms that are an integral element in sed. The central problem of the transition to the quantum dynamics is addressed, pointing to the key role of diffusion in the emergence of quantization.