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arXiv:1701.00353 · AIP Advances

How long single-photon detectors stay in quantum superpositions during detection according to the Diósi–Penrose criterion

Garrelt Quandt-Wiese

Published Mar 1, 2026• 1 citation
arXivHTMLPDF

Abstract

For special single-photon detectors that are isolated from their environment during detection (so-called indirect detectors), it is investigated how long they stay in a superposition of a photon-detected and a no-photon-detected state according to the Diósi–Penrose criterion for wavefunction collapse. To suppress interactions with the environment during detection, the avalanche photodiodes of the indirect detectors are biased using plate capacitors rather than conventional voltage sources, and the detection outcome is read out a sufficient time after the superposition in the detector has reduced. For the analysis, the Diósi–Penrose criterion is applied to solids in quantum superpositions that are slightly displaced relative to each other or have slightly different expansions in the superposed states, where both the parameter-free Diósi–Penrose model and Diósi’s version, in which the microscopic mass distribution is spatially averaged, are discussed. It is shown that indirect single-photon detectors can be constructed in such a way that they remain in superposition for seconds. It is proposed to use indirect detectors for the generation of mirror superpositions with the help of piezo-actuators, where the superposed mirror can have a displacement of about 50 Å for approximately half a microsecond. Even though the superposed mirror states generated in this way are decoherent superpositions (improper mixtures) and therefore cannot be detected by conventional methods, their generation opens new perspectives for probing wavefunction collapse.

DOI
10.1063/5.0293981

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For special single-photon detectors that are isolated from their environment during detection (so-called indirect detectors), it is investigated how long they stay in a superposition of a photon-detected and a no-photon-detected state according to the Diósi–Penrose criterion for wavefunction collapse. To suppress interactions with the environment during detection, the avalanche photodiodes of the indirect detectors are biased using plate capacitors rather than conventional voltage sources, and the detection outcome is read out a sufficient time after the superposition in the detector has reduced. For the analysis, the Diósi–Penrose criterion is applied to solids in quantum superpositions that are slightly displaced relative to each other or have slightly different expansions in the superposed states, where both the parameter-free Diósi–Penrose model and Diósi’s version, in which the microscopic mass distribution is spatially averaged, are discussed. It is shown that indirect single-photon detectors can be constructed in such a way that they remain in superposition for seconds. It is proposed to use indirect detectors for the generation of mirror superpositions with the help of piezo-actuators, where the superposed mirror can have a displacement of about 50 Å for approximately half a microsecond. Even though the superposed mirror states generated in this way are decoherent superpositions (improper mixtures) and therefore cannot be detected by conventional methods, their generation opens new perspectives for probing wavefunction collapse.
A formulary for the application of the Diosi-Penrose criterion to solids in quantum superpositions is developed, which takes the solid's microscopic mass distribution (resulting from its nuclei) and its macroscopic shape into account, where the solid's states can differ by slightly different positions or extensions of the solid. For small displacements, smaller than the spatial variation of the solid's nuclei, the characteristic energy of the Diosi-Penrose criterion is mainly determined by the mass distribution of the nuclei. For large displacements, much larger than the solid's lattice constant, the solid can be idealised as a continuum, and the characteristic energy depends on the solid's shape and the direction of the displacement. In Diosi's approach, in which the mass density operator has to be smeared, the solid's microscopic mass distribution plays no role. The results are applied to a special single-photon avalanche photodiode detector, which interacts as little as possible with its environment. This is realised by disconnecting the detector from the measurement devices when it is in a superposition, and by biasing the photodiode by a plate capacitor, which is charged shortly before the photon's arrival. For a suitable choice of components, the detector can stay in a superposition for seconds; its lifetime can be shortened to microseconds with the help of a piezoactuator, which displaces a mass in the case of photon detection.
An experiment for checking the Dynamical Spacetime approach to wavefunction collapse is proposed. The Dynamical Spacetime approach predicts deviations from Born's rule, when a solid evolves into a three-state superposition, and when the displacement between the superposed states is, at the reduction point in time, significantly larger than the spatial variation of the solids nuclei, being typically on the order of a tenth of an Angstroem. The solid is brought into the three-state superposition by splitting a photon into three beams and by detecting it in each beam by avalanche photodiodes, which displace the solid at different distances with the help of a piezoactuator. The challenge of the experiment is the precise prediction of the setup's reduction point in time to ensure a sufficient separation between the states at this point in time. This is addressed by avoiding interactions of the setup with the environment during superposition, and by a precise calculation of the setup's reduction point in time with the help of a formulary for the Diosi-Penrose criterion for solids in quantum superpositions. Since the measurement of reduction probabilities is not disturbed by state decoherence, the experiment can be performed at room temperature. The quantitative analysis demonstrates that the predicted increase of the reduction probability of one state by a factor of 1.5 with respect to Born's rule can be measured by a few hundred statistically significant measurements.
A novel approach for measuring wavefunction collapse is proposed which, unlike interferometric measurements, is not affected by decoherence. A mirror of a Michelson interferometer is transferred into a quantum superposition, where the decay of the mirror superposition is measured by the fact that it affects the probability of detecting photons in the interferometer. The experiment can be realised with currently available technology for photon generation and detection and is suitable for testing collapse models, which is shown by calculating the expected outcomes for the gravity-based models of Diósi and Penrose.
A new approach to wavefunction collapse is prepared by an analysis of semiclassical gravity. The fact that, in semiclassical gravity, superposed states must share a common classical spacetime geometry, even if they prefer (according to general relativity) differently curved spacetimes, leads to energy increases of the states, when their mass distributions are different. If one interprets these energy increases divided by Planck's constant as decay rates of the states, one obtains the lifetimes of superpositions according to the Diosi-Penrose criterion and reduction probabilities according to Born's rule. The derivation of Born's rule for two-state superpositions can be adapted to the typical quantum mechanical experiments with the help of a common property of these experiments. It is that they lead to never more than two different mass distributions at one location referring e.g. to the cases that a particle "is", or "is not", detected at the location. From the characteristic energy of the Diosi-Penrose criterion, an action is constructed whose relativistic generalisation becomes obvious by a decomposition of the Einstein-Hilbert action to the superposed states. In Part 2, semiclassical gravity is enhanced to the so-called Dynamical Spacetime approach to wavefunction collapse, which leads to a physical mechanism for collapse.
Abstract The possibility that gravity plays a role in the collapse of the quantum wave function has been considered in the literature, and it is of relevance not only because it would provide a solution to the measurement problem in quantum theory, but also because it would give a new and unexpected twist to the search for a unified theory of quantum and gravitational phenomena, possibly overcoming the current impasse. The Diósi–Penrose model is the most popular incarnation of this idea. It predicts a progressive breakdown of quantum superpositions when the mass of the system increases; as such, it is susceptible to experimental verification. Current experiments set a lower bound R 0 ≳ 4 Å for the free parameter of the model, excluding some versions of it. In this work we search for an upper bound, coming from the request that the collapse is effective enough to guarantee classicality at the macroscopic scale: we find out that not all macroscopic systems collapse effectively. If one relaxes this request, a reasonable (although to some degree arbitrary) bound is found to be: R 0 ≲ 10 6 Å. This will serve to better direct future experiments to further test the model.