DOI: 10.5281/zenodo.16110534
title: Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism data

description: Primary data and numerical codes of "Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism", arXiv:2506.21670 [cond-mat.quant-gas].Codes are written in the Julia programming language.[Structure]1. "codes" folder contains numerical codes written in Julia. (a) "codes/SPGPE" contains codes to solve the stochastic projected Gross-Pitaevskii equation (SPGPE) in parallel (HPC) and only save the data at the equilibration time, with a bash script and the example input parameter file.

(b) "codes/vortex_detect" contains codes to detect vortices, by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(c) "codes/nv_teq_KZM_test" contains codes to test the Kibble-Zurek mechanism (average number of vortices and the average equilibration time), by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(d) "codes/save_u_only_at_Tq_tf" contains codes to solve the SPGPE in parallel (HPC) and only save the data at the end of the quench (at t = Tq) and at the end of the time, with a bash script and the example input parameter file. 

(e) "codes/condensate_fraction_calc" contains codes to calculate the condensate fraction, by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(f) "codes/Extended_self_similarity_SP_struct_calc" contains codes to calculate the velocity structure function and test the extended self-similarity, by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(g) "codes/QuantumTurbulence" contains codes to calculate the compressible/incompressible kinetic energy spectra, by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(h) "codes/Ei_Ec_energy_calculation" contains codes to calculate the compressible/incompressible kinetic energy, by processing the SPGPE data in parallel (HPC), with a bash script and the example input parameter files.

(i) "codes/vortex_spacing_cluster" contains codes to check the existence of the vortex cluster, by processing the SPGPE data and vortex position data (obtained by vortex_detect code) in parallel (HPC), with a bash script and the example input parameter files.

(j) "codes/enstrophy_calc" contains codes to calculate the time evolution of the enstrophy, by processing the SPGPE data, with a bash script and the example input parameter files.

2. "results" folder contains numerical data and figures used in the arXiv:2506.21670 [cond-mat.quant-gas].(a) "results/u_data" contains the Fourier transform of the c-field wavefunction below the cutoff energy, and the norm of the c-field wavefunction as a function of time (Fig. 1 (a) and (b) in the main text, and Fig. 1 in the Supplementary Material), at equilibration time for different quench times. At T = 1, we calculated 1000 different noise realizations for each quench time. At T = 10^{-6}, we calculated 2000 different noise realizations for each quench time.

(b) "results/vortex_positions" contains positions of vortices (at equilibration time), histogram of the number of vortices (at equilibration time), and histogram of the equilibration time for different quench times.Data at T = 1 (1000 different noise realizations for each quench time) are in "T_1_data". Data at T = 10^{-6} (2000 different noise realizations for each quench time) are in "T_1e-6_data". 

(c) "results/nv_teq_KZM_test" contains the raw data and figure for Fig. 1 (c) in the main text (testing the Kibble-Zurek mechanism).

(d) "results/QuantumTurbulence_E_data" contains the raw data of compressible/incompressible kinetic energy spectra (Figs. 2 and 3 in the main text, Figs. 5 and 6 in the Supplementary Material).Data at T = 1 (1000 different noise realizations for each quench time) are in "T_1_data". Data at T = 10^{-6} (2000 different noise realizations for each quench time) are in "T_1e-6_data". 

(e) "results/u_data_at_Tq_and_t_tot" contains the Fourier transform of the c-field wavefunction below the cutoff energy, at T = 1, at the end of the quench (t = Tq) and at the end of the time (100 different noise realizations for each quench time).

(f) "results/condensate_fraction" contains the raw data of the condensate fraction (Fig. 2 in the Supplementary Material).

(g) "results/Extended_self_similarity" contains the raw data on testing the extended self-similarity (Fig. 3 in the Supplementary Material).

(h) "results/Ei_Ec_energy_data" contains the raw data on compressible/incompressible kinetic energy (Figs. 4 and 6 in the Supplementary Material).Data at T = 1 (1000 different noise realizations for each quench time) are in "T_1". Data at T = 10^{-6} (2000 different noise realizations for each quench time) are in "T_1e-6". 

(i) "results/u_time_evolve" contains the Fourier transform of the c-field wavefunction below the cutoff energy, at T = 1, for some selected quench times (Tq = 30, 60, 350), saving for every time interval (used in Fig. 7 in the Supplementary Material).

(j) "results/enstrophy" contains the raw data on the time evolution of the enstrophy, from the "u_time_evolve" data (Fig. 8 in the Supplementary Material).

(k) "results/vortex_spacing_and_cluster" contains the raw data to check the existence of the vortex cluster, from the "u_time_evolve" data. At most, there is only 1 vortex cluster, with only 2 vortices forming 1 vortex cluster.

(l) "results/mean_vortex_distance" contains the raw data on the mean nearest intervortex distance (used for the guidelines in Fig. 2 in the main text, and Fig. 3 in the Supplementary Material).

(m) "results/QuantumTurbulence_E_quadrature" contains the data and the figure showing that one needs smaller step size in space in order to reduce numerical errors at large momentum k (Fig. 9 in the Supplementary Material).