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arXiv:1908.00835 · Physical review. D/Physical review. D.

Entanglement production in the dynamical Casimir effect at parametric resonance

Ivan Romualdo, Lucas Hackl, Nelson Yokomizo

Published Sep 26, 2019• 27 citations
arXivHTMLPDF

Abstract

The particles produced from the vacuum in the dynamical Casimir effect are highly entangled. In order to quantify the correlations generated by the process of vacuum decay induced by moving mirrors, we study the entanglement evolution in the dynamical Casimir effect by computing the time-dependent R\'enyi and von Neumann entanglement entropy analytically in arbitrary dimensions. We consider the system at parametric resonance, where the effect is enhanced. We find that, in ($1+1$) dimensions, the entropies grow logarithmically for large times, ${S}_{A}(\ensuremath{\tau})\ensuremath{\sim}\frac{1}{2}\mathrm{log}(\ensuremath{\tau})$, while in higher dimensions ($n+1$) the growth is linear, ${S}_{A}(t)\ensuremath{\sim}\ensuremath{\lambda}\ensuremath{\tau}$, where $\ensuremath{\lambda}$ can be identified with the Lyapunov exponent of a classical instability in the system. In ($1+1$) dimensions, strong interactions among field modes prevent the parametric resonance from manifesting as a Lyapunov instability, leading to a sublinear entropy growth associated with a constant rate of particle production in the resonant mode. Interestingly, the logarithmic growth comes with a prefactor of $1/2$ which cannot occur in time-periodic systems with finitely many degrees of freedom and is thus a special property of bosonic field theories.

DOI
10.1103/physrevd.100.065022

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The particles produced from the vacuum in the dynamical Casimir effect are highly entangled. In order to quantify the correlations generated by the process of vacuum decay induced by moving mirrors, we study the entanglement evolution in the dynamical Casimir effect by computing the time-dependent R'enyi and von Neumann entanglement entropy analytically in arbitrary dimensions. We consider the system at parametric resonance, where the effect is enhanced. We find that, in (1+11+11+1) dimensions, the entropies grow logarithmically for large times, SA(\ensuremathτ)\ensuremath∼12log(\ensuremathτ){S}_{A}(\ensuremath{\tau})\ensuremath{\sim}\frac{1}{2}\mathrm{log}(\ensuremath{\tau})SA​(\ensuremathτ)\ensuremath∼21​log(\ensuremathτ), while in higher dimensions (n+1n+1n+1) the growth is linear, SA(t)\ensuremath∼\ensuremathλ\ensuremathτ{S}_{A}(t)\ensuremath{\sim}\ensuremath{\lambda}\ensuremath{\tau}SA​(t)\ensuremath∼\ensuremathλ\ensuremathτ, where \ensuremathλ\ensuremath{\lambda}\ensuremathλ can be identified with the Lyapunov exponent of a classical instability in the system. In (1+11+11+1) dimensions, strong interactions among field modes prevent the parametric resonance from manifesting as a Lyapunov instability, leading to a sublinear entropy growth associated with a constant rate of particle production in the resonant mode. Interestingly, the logarithmic growth comes with a prefactor of 1/21/21/2 which cannot occur in time-periodic systems with finitely many degrees of freedom and is thus a special property of bosonic field theories.
Abstract The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate h KS given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. The derivation takes into account the case of time-dependent Hamiltonians with Floquet instabilities. We show that the entanglement entropy S A of a Gaussian state grows linearly for large times in unstable systems, with a rate Λ A ≤ h KS determined by the Lyapunov exponents and the choice of the subsystem A. We apply our results to the analysis of entanglement production in unstable quadratic potentials and due to periodic quantum quenches in many-body quantum systems. Our results are relevant for quantum field theory, for which we present three applications: a scalar field in a symmetry-breaking potential, parametric resonance during post-inflationary reheating and cosmological perturbations during inflation. Finally, we conjecture that the same rate Λ A appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.
In the leading paradigm of cosmology, most particles are created during a reheating period following inflation. Part of this creation proceeds via parametric resonance [1], a process referred to as preheating. Since inflation essentially dilutes any pre-existing particles, particle production proceeds from fields in their vacuum state, a phenomenon of purely quantum origin similar to a dynamical Casimir effect (DCE). While the predicted quantum origin seems hard to test in the cosmological context, experiments involving genuine quantum fields evolving under the same dynamics have been realised. Ref. [2] reports on such an experiment in an elongated cigar-shaped Bose-Einstein condensate (BEC). The BEC is periodically modulated and acts as a classical field exciting its own quantum perturbations (phonons), created in pairs of opposite momenta ±k, with occupation number n_k and correlation amplitude c_k. Entanglement of the generated pairs is known to be a signature of pair production seeded by vacuum fluctuations [3] but is also fragile. Indeed, in [2] the observed pair correlation c_k was much weaker than expected and far from the regime c_k > n_k indicating entanglement. In a series of publications, the theoretical description behind the experiment was re-analysed in increasing detail to account for the absence of entanglement.First, in [3] weak dissipation of the excitations was phenomenologically introduced and shown to be able to prevent the generation of entanglement. A fully nonlinear numerical study of the modulated gas based on the Truncated Wigner Approximation (TWA) was then conducted in [4]. It confirmed that phonon-phonon interactions can induce an effective dissipation acting against the parametric resonance.In [5], we identified the dominant interaction channel for small occupation numbers in 1D quasi-BEC as Beliaev-Landau scatterings between the resonant mode and the thermal population. We analytically derived its rate and checked it against TWA simulations.Finally, in [6], whose results I will present in the talk, we demonstrate that the phenomenological model of [3], when supplemented by the rate derived in [5], accurately reproduces the particle and correlation generation seen in TWA simulations akin to that of [4].In this talk, I will first emphasise the subtle interplay between growth and dissipation, leading to distinct particle generation regimes. Then, informed by these findings, I will briefly discuss different optimal experimental strategies to achieve observation of entanglement.[1] D. Figueroa and F. Torrentí, JCAP02(2017)001 (2016)[2] J.-C. Jaskula et al., Phys. Rev. Lett. 109, 220401 (2012)[3] X. Busch, R. Parentani and S. Robertson, Phys. Rev. A 89, 063606 (2014)[4] S. Robertson, F. Michel and R. Parentani, Phys. Rev. D 98, 056003 (2018)[5] A. Micheli and S. Robertson, Phys. Rev. B 106, 214528 (2022)[6] A. Micheli and S. Robertson, Comptes Rendus Physique, Online first (2024), pp. 1-34. doi: 10.5802/crphys.250
Quantum entanglement, a defining feature of quantum theory, is generated dynamically by the Schwinger effect—the production of particle-antiparticle pairs from a strong electric field. In this work, we go beyond traditional particle-number analyses to study the quantum entanglement gen- erated in this process. We partition the system into momentum modes and calculate entanglement entropy and logarithmic negativity as functions of time. For a single Sauter pulse, entanglement exhibits a monotonic growth that closely tracks the pair creation process. Strikingly, a train of alternating pulses induces coherent interference, leading to a cumulative enhancement of entan- glement and revealing a clear interplay between successive creation events. Most significantly, we show that the temporal delay (𝑇) between pulses acts as a tunable parameter, controlling the level of entanglement via constructive or destructive interference. This tunability provides a distinct sig- nature of vacuum memory, establishing pair production as a platform for entanglement-controlled strong-field physics.
Several nontrivial phenomena emerge when a quantum field is subjected to dynamical perturbations, with prominent examples including the Hawking and Unruh effects, as well as the dynamical Casimir effect. In this work, we compute the number of particles produced via the dynamical Casimir effect in an ideal cavity, where one of the mirrors is allowed to move under the influence of a gravitational wave. Assuming an oscillatory mirror motion and a plane gravitational wave, we identify the resonance conditions that lead to an exponential increase in the number of created particles through parametric amplification.
We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models.
Establishing quantum gravity theory remains one of the major challenges in modern physics, as the lack of experimental evidence makes it difficult to explore. In response to this challenge, proposals to test quantum entanglement induced by Newtonian gravity in table-top experiments have attracted significant attention as a potentially feasible approach far below the Planck energy scale. In this work, we propose a scheme to amplify gravity-induced entanglement between two masses using parametric resonance. Specifically, we consider two parametrically resonant oscillators interacting through Newtonian gravity, each governed by the Mathieu equation. We analyzed the logarithmic negativity between two oscillators and investigate the effects of random force noise and linear damping. As a result, we find an exponential growth of gravity-induced entanglement between the oscillators, which reflects the dynamical instability of parametric resonant systems.