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Modeling and experimental observation of mixing and violent relaxation in an analogue gravitational system

Lovisetto, Martino

Published May 10, 2022• 0 citations
Publisher

Abstract

Equations of the Schrödinger-Newton type appear naturally in different domains of research. In cosmology, they are used to model the formation of large scale structures in the universe, while, in astrophysics, they are employed as a semi-classical description for the evolution of elementary particles of dark matter. The Schrödinger-Newton equation is also used in optics, to describe the propagation of a beam of light through a medium with a thermo-optical nonlinearity. All of these systems, despite being very different from each other, are characterized by long range interactions. Long range interacting systems, relax generically to out-of-equilibrium quasi-stationary states, with significant astrophysical examples such as galaxies and globular clusters. These states cannot be described by standard statistical mechanics, but are formed through a process of very different nature, called violent relaxation. However, astrophysical time-scales are so large that it is not possible to directly observe the relaxation dynamics. In this thesis, we develop a table-top experimental model that captures the dynamics of long range interacting systems and allows to directly observe violent relaxation mechanism, leading to the formation of a table-top optical analogue of a galaxy. The experiment allows to control a range of parameters, including the nonlocal (gravitational) interaction strength and the strength of quantum gravity coupling ħ/m, thus providing an effective test-bed for gravitational models that cannot otherwise be directly studied in experimental settings. In addition, a central part of the thesis is devoted to the numerical study of the Schrödinger-Newton equation. Here, we present a new efficient numerical method which can be employed not only to solve the Schrödinger-Newton equation, but any Schrödinger-like equations, with different kind of interactions. The numerical method developed, which exploits the freedom provided by the gauge condition of the potential, is an improvement of the integrating factor technique. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. Moreover, we make an extensive comparison of the new method developed, together with other popular integrators, commonly use to numerically solve this kind of equations. We focus in particular on splitting algorithms and methods belonging to the integrating factor family. Comparisons are done in one and two spatial dimensions, with different boundary conditions, both for the Schrödinger--Newton equation and the non-linear Schrödinger equation. We conclude that for the short range potential of the non-linear Schrödinger equation, integrating factor methods perform better than the Split-Step algorithm, while for the long range potential of the Schrödinger--Newton equation it depends on the particular system considered.

DOI
10.70675/74a81d40zf8faz4956z8d36z3f5dbf5acdbc

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Equations of the Schrödinger-Newton type appear naturally in different domains of research. In cosmology, they are used to model the formation of large scale structures in the universe, while, in astrophysics, they are employed as a semi-classical description for the evolution of elementary particles of dark matter. The Schrödinger-Newton equation is also used in optics, to describe the propagation of a beam of light through a medium with a thermo-optical nonlinearity. All of these systems, despite being very different from each other, are characterized by long range interactions. Long range interacting systems, relax generically to out-of-equilibrium quasi-stationary states, with significant astrophysical examples such as galaxies and globular clusters. These states cannot be described by standard statistical mechanics, but are formed through a process of very different nature, called violent relaxation. However, astrophysical time-scales are so large that it is not possible to directly observe the relaxation dynamics. In this thesis, we develop a table-top experimental model that captures the dynamics of long range interacting systems and allows to directly observe violent relaxation mechanism, leading to the formation of a table-top optical analogue of a galaxy. The experiment allows to control a range of parameters, including the nonlocal (gravitational) interaction strength and the strength of quantum gravity coupling ħ/m, thus providing an effective test-bed for gravitational models that cannot otherwise be directly studied in experimental settings. In addition, a central part of the thesis is devoted to the numerical study of the Schrödinger-Newton equation. Here, we present a new efficient numerical method which can be employed not only to solve the Schrödinger-Newton equation, but any Schrödinger-like equations, with different kind of interactions. The numerical method developed, which exploits the freedom provided by the gauge condition of the potential, is an improvement of the integrating factor technique. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. Moreover, we make an extensive comparison of the new method developed, together with other popular integrators, commonly use to numerically solve this kind of equations. We focus in particular on splitting algorithms and methods belonging to the integrating factor family. Comparisons are done in one and two spatial dimensions, with different boundary conditions, both for the Schrödinger--Newton equation and the non-linear Schrödinger equation. We conclude that for the short range potential of the non-linear Schrödinger equation, integrating factor methods perform better than the Split-Step algorithm, while for the long range potential of the Schrödinger--Newton equation it depends on the particular system considered.
Cosmological simulations describing the evolution of density perturbations of a self-gravitating collisionless dark matter (DM) fluid in an expanding background provide a powerful tool to follow the formation of cosmic structures over wide dynamic ranges. The most widely adopted approach, based on the NNN-body discretization of the collisionless Vlasov-Poisson (VP) equations, is hampered by an unfavorable scaling when simulating the wide range of scales needed to cover at the same time the formation of single galaxies and of the largest cosmic structures. On the other hand, the dynamics described by the VP equations is limited by the rapid increase of the number of resolution elements (grid points and/or particles) which is required to simulate an ever growing range of scales. Recent studies showed an interesting mapping of the six-dimensional +++ 1 (6D+1\text{6D}+16D+1) VP problem into a more amenable 3D+1\text{3D}+13D+1 nonlinear Schr\"odinger-Poisson (SP) problem for simulating the evolution of DM perturbations. This opens up the possibility of improving the scaling of time propagation simulations using quantum computing. In this paper, we introduce a quantum algorithm for simulating the Schr\"odinger-Poisson (SP) equation by adapting a variational real-time evolution approach to a self-consistent, nonlinear, problem. To achieve this, we designed a novel set of quantum circuits that establish connections between the solution of the original Poisson equation and the solution of the corresponding time-dependent Schr\"odinger equation. We also analyzed how nonlinearity impacts the variance of observables. Furthermore, we explored how the spatial resolution behaves as the SP dynamics approaches the classical limit ($\ensuremath{\hbar}/m\ensuremath{\rightarrow}0$) and discovered an empirical logarithmic relationship between the required number of qubits and the scale of the SP equation ($\ensuremath{\hbar}/m$). This entire approach holds the potential to serve as an efficient alternative for solving the Vlasov-Poisson (VP) equation by means of classical algorithms.
The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of ∣ψ∣2|\psi|^2∣ψ∣2, where ψ\psiψ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the (n+1)(n+1)(n+1)th state has nnn zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for nnn quadruples of complex eigenvalues for the (n+1)(n+1)(n+1)th state, where a quadruple consists of {λ,λˉ,−λ,−λˉ}\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}{λ,λˉ,−λ,−λˉ}. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general WWW- and general GHZGHZGHZ-state, coupled to a global spin-1/21/21/2 bath or several independent spin-1/21/21/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.
One of the grand open problems of modern physics is the unification of Einstein’s general relativity and quantum mechanics. This challenge has been approached by the greatest physicists but there is still no complete theory and experimental evidence remains out of reach. The Newton–Schrödinger equation (NSE) offers insight into this puzzle, as it describes a quantum wavefunction under self-gravity dynamics. Having been studied theoretically for decades, optical experiment of this nonlinear model was demonstrated only in 2015. Although the NSE can be generalized to post-Newtonian gravity approximating general relativity, all experiments simulating nonlinear gravity so far have been limited to Newtonian gravity. Here we present experimental emulation of post-Newtonian dynamics by probing a new physical regime of nonlinearity that mimics larger masses in gravity. We find soliton solutions of the post-Newtonian–Schrödinger equations, distinct from their Newtonian counterparts and demonstrate them experimentally. We observe rich beam evolution requiring previously unconsidered nonlinear terms, thereby opening up new experimental capabilities for simulating wavefunction dynamics in the settings of general relativity. An optical emulation experiment of a quantum wave function dynamics under self-induced post Newtonian gravitational potential is presented, including observation of solitons of this system and rich beam evolution due to high-order nonlinear effects.
Abstract The Schrödinger–Newton (SN) model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the SN interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the SN model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.
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