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Quantum Gravitational Decoherence from Fluctuating Minimal Length And ARTICLE https://doi.org/10.1038/s41467-021-24711-7 OPEN Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale ✉ ✉ Luciano Petruzziello 1,2 & Fabrizio Illuminati 1,2 Schemes of gravitationally induced decoherence are being actively investigated as possible mechanisms for the quantum-to-classical transition. Here, we introduce a decoherence 1234567890():,; process due to quantum gravity effects. We assume a foamy quantum spacetime with a fluctuating minimal length coinciding on average with the Planck scale. Considering deformed canonical commutation relations with a fluctuating deformation parameter, we derive a Lindblad master equation that yields localization in energy space and decoherence times consistent with the currently available observational evidence. Compared to other schemes of gravitational decoherence, we find that the decoherence rate predicted by our model is extremal, being minimal in the deep quantum regime below the Planck scale and maximal in the mesoscopic regime beyond it. We discuss possible experimental tests of our model based on cavity optomechanics setups with ultracold massive molecular oscillators and we provide preliminary estimates on the values of the physical parameters needed for actual laboratory implementations. 1 Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Fisciano, (SA), Italy. 2 INFN, Sezione di Napoli, Gruppo collegato di Salerno, Fisciano, ✉ (SA), Italy. email: [email protected]; fi[email protected] NATURE COMMUNICATIONS | (2021) 12:4449 | https://doi.org/10.1038/s41467-021-24711-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24711-7 ollowing early pioneering studies1–4, the investigation of the gravitational effects are deemed to be comparably important. Fquantum-to-classical transition via the mechanism of Although a complete and consistent theory of quantum gravity is decoherence has become a very active area of research, both still lacking, all the different current candidates including string experimentally and theoretically, playing an increasingly central theory, loop quantum gravity, noncommutative geometry and role in the research area on the foundations of quantum doubly special relativity, predict the existence of a minimal length mechanics (QM) and the appearance of a classical world at the at the Planck scale. An immediate consequence of this common macroscopic scale, as one may gather, e.g., from the many aspect is the breakdown of the Heisenberg uncertainty principle excellent existing reviews on the subject (i.e., see for instance (HUP), whose most famous formulation conveys that spatial refs. 5–13 and references therein). In broad terms, decoherence resolution can be made arbitrarily small with a proper energetic appears to be due to the inevitable interaction and the ensuing probe. Therefore, as firstly derived in the framework of string creation of entanglement between a given quantum system and theory46 and strongly supported by thought experiments on the environment in which it is embedded. Indeed, no quantum micro black holes afterwards47, at the Planck scale the HUP must system can truly be regarded as isolated, and the entanglement be superseded by a generalized uncertainty principle (GUP), shared with the environment significantly affects the outcome of whose minimal one-dimensional expression reads local measurements, even in the circumstance in which “classical” _ ΔP2 disturbances (such as dissipation and noise) can be neglected. ΔXΔP ≥ 1 þ β ‘2 ; ð1Þ Decoherence has been corroborated over the years by several 2 p _2 experimental observations, among which it is worth mentioning 14,15 β ℓ its first detection obtained via cavity QED , as well as the where denotes the so-called deformation parameter and p is various laboratory tests involving superconductive devices16, the Planck length. Starting from the above basic picture, the GUP trapped ions17 and matter-wave interferometers18. has been discussed in the most disparate settings, from QM and – Apart from the domains of condensed matter and atomic quantum field theory48 54 to noncommutative geometry and – physics, decoherence has been studied in more exotic and black hole physics55 60. extreme contexts, ranging from particle physics to cosmological In this work, we introduce a model accounting for a universal large-scale structures. For instance, in the framework of particle quantum-gravitational decoherence process. The model assumes physics, decoherence has been investigated at the quantum field deformed canonical commutation relations (DCCRs) leading to theory level in the regime of nontrivial interactions, both at zero the deformed uncertainty relation (1) that accounts for the pre- and finite temperature6,19,20, and including lepton mixing and sence of a minimal length scale. Furthermore, the model assumes oscillations21. These studies and related ones have led to the the minimal length scale to be a fluctuating quantity, induced by a opportunity of predicting new physics phenomenology, especially conjectured foamy character of quantum spacetime, whose in connection with entanglement degradation in non-inertial average magnitude is fixed at the Planck length scale. As a con- frames and in the presence of strong gravitational fields22. sequence, the deformation parameter β that enters in the mod- In principle, as in the above examples, the main features of ified canonical momentum operator and in the DCCRs is each decoherence mechanism depend significantly on the type of promoted to a random quantity as well. As it is customary in the interaction with the environment. On the other hand, in an analysis of signatures of quantum gravitational effects at the attempt to explain wave function collapse and reduction to classical macroscopic scale61, in the following we treat the con- pointer states in general terms, Ghirardi, Rimini and Weber tribution associated to the GUP as a small perturbation. We then (GRW) proposed the existence of a universal decoherence show how to derive a master equation of the mechanism and spontaneous localization with the introduction of Lindblad–Gorini–Kossakowski–Sudarshan form62,63 for the just two model-dependent parameters, namely the collapse rate averaged quantum density matrix out of the corresponding β- and the localization distance23 (for important developments in deformed stochastic Schrödinger equation for the quantum state GRW-like models, we refer the reader to refs. 24–30). Inspired by vector, and we study the physical consequences of such open the original GRW suggestion, Diosi31 and Penrose32 singled out quantum state dynamics. In light of the apparent experimental classical Einstein gravity as the possible overarching interaction confutation of the Diosi–Penrose model, our proposal may be responsible for the loss of coherence in GRW theory, thus regarded as a possible alternative universal decoherence refining the previous intuition and fixing the free parameters in mechanism, provided that a set of reasonable assumptions holds. the GRW scheme of the quantum-to-classical transition. Since the At the same time, a laboratory test of our model and its predic- innovative Diosi–Penrose insight, much effort has been devoted tions would also yield an indirect evidence for the quantum to experimentally test the gravitational realization of the GRW nature of the gravitational interaction. Along these lines, building mechanism. In this regard, the attention has been mainly focused on a universal decoherence measure introduced in ref. 34,we on gravitational time dilation33, optomechanics34 and quantum investigate a scheme for laboratory tests involving a cavity clocks35. Very recently, a crucial experiment has demonstrated optomechanical system equipped with heavy molecular struc- that the parameter-free Diosi–Penrose model based on classical tures. We then provide numerical estimates and show that such a Einstein gravity fails to account for the correct rates of radiation setup can probe our predictions with a high degree of accuracy. emission due to quantum state reduction, thus ruling it out as a feasible candidate for the implementation of the GRW deco- Results herence mechanism36. Deformed commutation relations, generalized uncertainty The experimental falsification of the Diosi–Penrose version of principle and modified Schrödinger dynamics. In order to GRW theory can be considered an important milestone as well as define a consistent quantum mechanical theory in Hilbert space a new starting point. As a matter of fact, among the various ideas from Eq. (1), one can start from the β-deformed canonical that revolve around the concept of gravitational decoherence (i.e., commutation relation, as shown in ref. 64. In compliance with the – see for example refs. 37 44; for a comprehensive review and a above discussions, we can then consider the deformation of the more detailed list of references, see e.g., ref. 45), a significant commutator (and, in turn, the modification of the uncertainty portion deals with the high-energy regime in which quantum and relations) as the signature of the quantum gravitational 2 NATURE COMMUNICATIONS | (2021) 12:4449 | https://doi.org/10.1038/s41467-021-24711-7 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-24711-7 ARTICLE environment in which the system under examination is embed- ψ in the form ded. Notice that this kind of approach departs from refs. 42 and43
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