We probe the dynamics of a modified form of the Schrodinger-Newton system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of "collapse" (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrodinger ground state energies match the Dirac ones with an error of ~ 10%. At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, m ~ 3.3 Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the Schrodinger-Newton system. Unlike the Schrodinger-Newton dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.
We show that optomechanical systems can test the Schrödinger-Newton equation of gravitational quantum mechanics due to Yang et al. This equation is motivated by semiclassical gravity, a widely used theory of interacting gravitational and quantum fields. From the many-body Schrödinger-Newton equation follows an approximate equation for the center-of-mass dynamics of macroscopic objects. This predicts a distinctive double-peaked signature in the output optical quadrature power spectral density of certain optomechanical systems. Since the Schrödinger-Newton equation lacks free parameters, these w…
We study the Schrodinger-Newton system of equations with the addition of gravitational field energy sourcing - such additional nonlinearity is to be expected from a theory of gravity (like general relativity), and its appearance in this simplified scalar setting (one of Einstein's precursors to general relativity) leads to significant changes in the spectrum of the self-gravitating theory. Using an iterative technique, we compare the mass dependence of the ground state energies of both Schrodinger-Newton and the new, self-sourced system and find that they are dramatically different. The Bohr m…
The scope of this paper is twofold. First, we derive rigorously a low-velocity and Galilei-covariant limit of the gravitoelectromagnetic (GEM) equations. Subsequently, these reduced GEM equations are coupled to the Schrödinger equation with gravitoelectric and gravitomagnetic potentials. The resulting extended Schrödinger-Newton equations constitute a minimal model where the three fundamental constants of nature ($G$, $\hbar$, and $c$) appear naturally. We show that the relativistic correction coming from the gravitomagnetic potential scales as the ratio of the mass of the system to the Planck…
In this paper we show that the Schrödinger-Newton equation for spherically symmetric gravitational fields can be derived in a WKB-like expansion in 1/c from the Einstein-Klein-Gordon and Einstein-Dirac system.
The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was i…
Using numerical techniques, we study the collapse of a scalar field configuration in the Newtonian limit of the spherically symmetric Einstein--Klein--Gordon (EKG) system, which results in the so called Schrödinger--Newton (SN) set of equations. We present the numerical code developed to evolve the SN system and topics related, like equilibrium configurations and boundary conditions. Also, we analyze the evolution of different initial configurations and the physical quantities associated to them. In particular, we readdress the issue of the gravitational cooling mechanism for Newtonian systems…
The Schrödinger-Newton equation is a proposed model to explain the localization of macroscopic particles by suppressing quantum dispersion with the particle's own gravitational attraction. On cosmic scales, however, dark energy also acts repulsively, as witnessed by the accelerating rate of universal expansion. Here, we introduce the effects of dark energy in the form of a cosmological constant $Λ$, that drives the late-time acceleration of the Universe, into the Schrödinger-Newton approach. We then ask in which regime dark energy dominates both canonical quantum diffusion and gravitational se…
In this note we show that Newton-Schrödinger Equations (NSEs) [arXiv:1210.0457 and references therein] do not follow from general relativity (GR) and quantum field theory (QFT) by way of two considerations: 1) Taking the nonrelativistic limit of the semiclassical Einstein equation, the central equation of relativistic semiclassical gravity, a fully covariant theory based on GR+QFT with self-consistent backreaction of quantum matter on the spacetime dynamics; 2) Working out a model [see C. Anastopoulos and B. L. Hu, Class. Quant. Grav. 30, 165007 (2013), arXiv:1305.5231] with a matter scalar fi…
Entanglement generation in a two-body Schrödinger--Newton model
Qilimanjaro Quantum Tech（Qilimanjaro量子科技公司）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO光子科学研究所，巴塞罗那科学技术研究院）；Astronomical Observatory, Faculty of Physics, Astronomy and Applied Computer Science Jagiellonian University（雅盖隆大学物理、天文学与应用科学学院天文台）；Doctoral School of Exact and Natural Sciences, Jagiellonian University（雅盖隆大学精确与自然科学学院博士项目）；Institute of Theoretical Physics, Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University（雅盖隆大学物理、天文学与应用科学学院理论物理研究所）
The Schrödinger-Newton [SN] equation describes the effect of self-gravity on the evolution of a quantum system, and it has been proposed that gravitationally induced decoherence drives the system to one of the stationary solutions of the SN equation. However, the equation by itself lacks a decoherence mechanism, because it does not possess any stochastic feature. In the present work we derive a stochastic modification of the Schrödinger-Newton equation, starting from the Einstein-Langevin equation in the theory of stochastic semiclassical gravity. We specialize this equation to the case of a s…
We study time evolution of Schrodinger-Newton system using the self-consistent Crank-Nicolson method to understand the dynamical characteristics of nonlinear systems. Compactifying the radial coordinate by a new one, which brings the spatial infinity to a finite value, we are able to impose the boundary condition at infinity allowing for a numerically exact treatment of the Schrodinger-Newton equation. We study patterns of gravitational cooling starting from exponentially localized initial states. When the gravitational attraction is strong enough, we find that a small-sized oscillatory solito…
The Schrödinger-Newton equation has gained attention in the recent past as a nonlinear modification of the Schrödinger equation due to a gravitational self-interaction. Such a modification is expected from a fundamentally semi-classical theory of gravity, and can therefore be considered a test case for the necessity of the quantisation of the gravitational field. Here we provide a thorough study of the effects of the Schrödinger-Newton equation for a micron-sized sphere trapped in a harmonic oscillator potential. We discuss both the effect on the energy eigenstates and the dynamical behaviour …
In this paper we extend the WKB-like `non-relativistic' expansion of the minimally coupled Klein--Gordon equation after Kiefer and Singh [1], Lämmerzahl [2] and Giulini and Großardt [3] to arbitrary order in $c^{-1}$, leading to Schrödinger equations describing a quantum particle in a general gravitational field, and compare the results with canonical quantisation of a free particle in curved spacetime, following Wajima et al. [4]. Furthermore, using a more operator-algebraic approach, the Klein--Gordon equation and the canonical quantisation method are shown to lead to the same results for so…
It has been suggested that the nonlinear Schrödinger-Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the Møller-Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than $\sim 1.14(\hbar^2/(Gσ))^{1/3}$, a wave packet of width $σ$ partially "collapses" to a groundstate solution found by Moroz, Penrose, and Tod, with excess probability dispersing awa…
In this letter, we will analyse the deformation of a semi-classical gravitational system from minimal measurable length scale. In the semi-classical approximation, the gravitational field will be analysed as a classical field, and the matter fields will be treated quantum mechanically. Thus, using this approximation, this system will be represented by a deformation of Schrödinger-Newton equation by the generalised uncertainty principle (GUP). We will analyse the effects of this GUP deformed Schrödinger-Newton equation on the behaviour of such a semi-classical gravitational system. As the quant…
The Schrödinger-Newton model describes self-gravitating quantum particles, and it is often cited to explain the gravitational collapse of the wave function and the localization of macroscopic objects. However, this model is completely nonrelativistic. Thus, in order to study whether the relativistic effects may spoil the properties of this system, we derive a modification of the Schrödinger-Newton equation by considering certain relativistic corrections up to the first post-Newtonian order. The construction of the model begins by considering the Hamiltonian of a relativistic particle propagati…
We discuss aspects of gravitational modifications of Schrodinger dynamics proposed by Diosi and Penrose. We consider first the Diosi-Penrose criterion for gravitationally induced state vector reduction, and compute the reduction time expected for a superposition of a uniform density cubical solid in two positions displaced by a small fraction of the cube side. We show that the predicted effect is much smaller than would be observable in the proposed Marshall et al. mirror experiment. We then consider the ``Schrodinger -Newton'' equation for an N-particle system. We show that in the independent…
We consider the modification of a single particle Schrödinger equation by the inclusion of an additional gravitational self-potential term which follows from the prescription that the' mass-density'that enters this term is given by $m |ψ(\vec {r},t)|^2$, where $ψ(\vec {r}, t)$ is the wavefunction and $m$ is the mass of the particle. This leads to a nonlinear equation, the ' Newton Schrodinger' equation, which has been found to possess stationary self-bound solutions, whose energy can be determined exactly using an asymptotic method. We find that such a particle strongly violates superposition …
We consider in a pedagogical fashion alterations to Newtonian gravity due to the postulate that all energy corresponds to active gravitational mass when applied to the self-energy of the gravitational field. We show why a simple addition of ${1\over c^2}$ times the gravitational field energy to the matter density in Newton's field equation is inconsistent. A consistent prescription is shown and discussed. The connection to general relativity is pointed out.
We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of the Schrodinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the "active" gravitational masses are always positive, and approximately equal to inertial masses and to the mass $m$ of the quanta of Dirac field up to the corrections of order $α^2$. Here $α=(m/M_{pl})^2$ is the gravitational a…
In this letter, we develop a Newtonian approach to the collapse of galaxy fluctuations of scalar field dark matter under initial conditions inferred from simple assumptions. The full relativistic system, the so called Einstein-Klein-Gordon, is reduced to the Schrödinger-Newton one in the weak field limit. The scaling symmetries of the SN equations are exploited to track the non-linear collapse of single scalar matter fluctuations. The results can be applied to both real and complex scalar fields.
