FACt: FORTRAN toolbox for calculating fluctuations in atomic condensates

Abstract

We develop a FORTRAN code to compute fluctuations in atomic condensates (FACt) by solving the Bogoliubov-de Gennes (BdG) equations for two component Bose-Einstein condensate (TBEC) in quasi two dimensions. The BdG equations are recast as matrix equations and solved self consistently. The code is suitable for handling quantum fluctuations as well as thermal fluctuations at temperatures below the critical point of Bose-Einstein condensation. The code is versatile, and the ground state density profile and low energy excitation modes obtained from the code can be easily adapted to compute different properties of TBECs -ground state energy, overlap integral, quasi particle amplitudes of BdG spectrum, dispersion relation and structure factor and other related experimental observables.External routines/libraries: ARPACK Subprograms used: Journal reference of previous version:* Nature of problem: Compute the ground state density profile, ground state energy and chemical potential for individual species, evaluate the quasiparticle mode energies and corresponding amplitudes which can capture the transformation of the modes against the change of the parameters (intraspecies interaction, interspecies interaction, anisotropy parameter etc.) using Hartree-Fock Bogoliubov theory with the Popov approximation Calculate the overlap integral, dispersion relation and structure factor. Solution method: In the first step, the pair of coupled Gross-Pitaevskii (GP) equations are solved using split step Crank-Nicolson method to compute the condensate density.To solve BdG equations, as a basic input the first N b harmonic oscillator eigenstates are chosen as a basis to generate the BdG matrix with dimension of 4(N b + 1)(N b + 1) × 4(N b + 1)(N b + 1). Since the matrix size rapidly increases with N b , ARPACK routines are used to diagonalise the BdG matrix efficiently. To compute the fluctuation and noncondensate density, a set of the low energy quasiparticle amplitudes above a threshold value of the Bose factor are considered. The equations are then solved iteratively till the condensate, and non-condensate densities converge to predefined accuracies. To accelerate the convergence we use the method of successive under-relaxation (SUR).Restrictions: For a large system size, if the harmonic oscillator basis size is also taken to be large, the dimension of the BDG matrix becomes huge. It may take several days to compute the low energy modes at finite temperature and this package may be computationally expensive.Additional comments:After successful computation of this package, one should obtain the equilibrium density profiles for TBEC, low energy Bogoliubov modes and the corresponding quasiparticle amplitudes. In addition, one can calculate the dispersion relation, structure factor, overlap integral, correlation function, etc. using this package with minimal modifications. In the theory section of the manuscript, we have provided the expressions to compute the above quantities numerically.Running time: ∼ 10 minutes for the sample case. For ...

Splitting of singly and doubly quantized composite vortices in two-component Bose-Einstein condensates

“…To go beyond the linear regime of the Bogoliubov stability analysis, we simulate the full two-dimensional dynamics of the unstable composite vortices by directly evolving the timedependent GP equations (1) in time. The time evolution is performed with a split-step Crank-Nicolson method [78,79] adapted for two-component BECs [80]. To obtain a convenient initial state, we first propagate Eqs.…”

Quantum fluctuation in the segregation phase of two-component weakly interacting Bose–Einstein condensates

“…Studying a miscible BECs based on the time-dependent Hartree-Fock-Bogoliubov theory, the author of [23] found that quantum fluctuation is proportional to the square power of the atomic interaction strengths. The numerical solving of the Bogoliubovde Gennes equations of the quantum fluctuation in BECs was reported in [24]. However, the influence of the atomic interaction on the quantum fluctuation in the immiscible phase was not mentioned.…”

A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

“…To the best of our knowledge, FACt (Fluctuations in Atomic Condensates, Roy et al, 2020) is the only publicly accessible code for the BdG problem. It considers thermal excitations of the BECs at non-zero temperatures by solving the BdG equation in two component BECs with a pseudo-spectral method.…”