We present high-performance and high-accuracy numerical simulations of quantum turbulence modelled by the Gross–Pitaevskii equation for the time-evolution of the macroscopic wave function of the ...system. The hydrodynamic analogue of this model is a flow in which the viscosity is absent and all rotational flow is carried by quantized vortices with identical topological line-structure and circulation. Numerical simulations start from an initial state containing a large number of quantized vortices and follow the chaotic vortex interactions leading to a vortex-tangle turbulent state. The Gross–Pitaevskii equation is solved using a parallel (MPI-OpenMP) code based on a pseudo-spectral spatial discretization and second order splitting for the time integration. We define four quantum-turbulence simulation cases based on different methods used to generate initial states: the first two are based on the hydrodynamic analogy with classical Taylor–Green and Arnold–Beltrami–Childress vortex flows, while the other two methods use a direct manipulation of the wave function by generating a smoothed random phase field, or seeding random vortex-ring pairs. The dynamics of the turbulent field corresponding to each case is analysed in detail by presenting statistical properties (spectra and structure functions) of main quantities of interest (energy, helicity, etc.). Some general features of quantum turbulence are identified, despite the variety of initial states. Numerical and physical parameters of each case are presented in detail by defining corresponding benchmarks that could be used to validate or calibrate new Gross–Pitaevskii codes. The efficiency of the parallel computation for a reference case is also reported.
The Lieb-Robinson bound asserts the existence of a maximal propagation speed for the quantum dynamics of lattice spin systems. Such general bounds are not available for most bosonic lattice gases due ...to their unbounded local interactions. Here we establish for the first time a general ballistic upper bound on macroscopic particle transport in the paradigmatic Bose-Hubbard model. The bound is the first to cover a broad class of initial states with positive density including Mott states, which resolves a longstanding open problem. It applies to Bose-Hubbard-type models on any lattice with not too long-ranged hopping. The proof is rigorous and rests on controlling the time evolution of a new kind of adiabatic spacetime localization observable via iterative differential inequalities.
In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb-Thirring constant when the eigenvalues of a ...Schr\"odinger operator $-\Delta+V(x)$ are raised to the power $\kappa$ is never given by the one-bound state case when $\kappa>\max(0,2-d/2)$ in space dimension $d\geq1$. When in addition $\kappa\geq1$ we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo-Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb-Thirring inequality, in the same spirit as in Part I of this work (arXiv:2002.04963). In a different but related direction, we also show that the cubic nonlinear Schr\"odinger equation admits no orthonormal ground state in 1D, for more than one function.
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