SPHERICAL BOSE-BOSE MIXTURE W/ LHY

Intro:

• Overview: These codes are to solve the Bose-Bose mixture, with the Lee-Huang-Yang (LHY) correction, in a spherically symmetric geometry (i.e. the density of the mixture is a function of radius only).

• Time-stepping scheme: The code uses a 4th-order Runge-Kutta Scheme.

• Boundary conditions: The code has options for boundary conditions in the boundary.py file though the only current options are: 1. Neumann boundary conditions at both ends; 2. Neumann boundary condition at centre of the box with density set to zero at the far end of the box.

Code Structure:

Main files and directories

• The main body of the code is within the 'main' directory.
• mix_sph_lhy/json_init is a directory which contains two files: default.config_dens_lck.json and default.config_dens_ulck.json. It is best to copy these into the sph_mix_lhy directory and rename them config_dens_lck.json and config_dens_ulck.json. These are the configuration files for each simulation and will be described below
• mix_sph_lhy.py is the file which reads in the simulation data and runs the codes in main
• running.py is the file which runs mix_sph_lhy.py and saves most of the data and files that each simulation produces

HPC workflow files

These are files that are designed to be ran on HPC clusters (using the SLURM queueing system):

• drop_run.sh is the shell scipt to run a single simulation via SLURM
• drop_array.sh is the shell script to run a job array of simulations via SLURM
• generator_imbal.sh is the shell script to generate config files for a job array of simulations for various atom number imbalances
• generator_size.sh is the shell script to generate config files for a job array of simulations for various droplet sizes

Processing files

The directory mix_sph_lhy/process contains files for processing output of simulations, particularly for real time breathing mode simulations:

• imbal_breath_fitting.py is a file which reads in multiple simulations (from a job array) and fits the oscillation of the central density of a droplet and outputs values of the breathing mode frequency and breathing mode decay rate
• breath_fit.sh is the shell script to run the imbal_breath_fitting.py file on the SLURM queueing system
• process_ulck.py is a file which generates movies and other output for an individual simulation

Describing a mixture:

A Bose-Bose mixture is here defined by:

• a11 - intraspecies scattering length for component 1
• a12 - interspecies scattering length
• a22 - intraspecies scattering length for component 2
• m1 - atomic mass of component 1
• m2 - atomic mass of component 2

We also require an atom number for the system and this is where the first separation occurs.

As mention the mixture can be density-locked (DENS_LCK = 1) or density-unlocked (DENS_LCK = 0). When in the density-locked regime, you need only specify the total number of atoms, N. However, when in the density-unlocked regime, you must specify the atom number for both components (i.e. N1 and N2).

One other important way to specify your system is the trapping potential. In the spherically symmetric case we are somewhat limited so the main choices are: 1. No potential; 2. Harmonic potential. The only subtlty is that in the density-locked case you can only specify harmonic trap frequency (OMEGA) but in the density-unlocked case you can impose different trapping potentials on each component (via OMEGA1 and OMEGA2).

Configuration for a simulation:

There are two possibly simulation config files: 1) config_dens_lck.json - for a density-locked mixture; 2) config_dens_ulck.json - for density-unlocked mixture. As mentioned before, copy the default versions of these files from the 'sph_mix_lhy/json_init' directory and rename them to remove the default. prefix. These config files give you some flexibility in what to look at with these simulations

Density-locked:

• The above experimental parameters for defining the mixture
• dr - the grid spacing
• Nr - the number of grid points (hence this also changes the box size, Lr = Nr*dr)
• BC_TYPE - choice of boundary conditions
  • 0 - Neumann boundary conditions at both r = 0 and r = Lr
  • 1 - Neumann boundary conditions at r = 0 and Dirichlet at r = Lr
• DENS_LCK - is by default set to 1 in the config_dens_lck.json file
  • 0 - Unlocked mixture
  • 1 - Locked mixture
• DT_COEF - the size of the time-step set by dt = DT_COEFx*dr^2)
• IM_T_STEPS - the number of imaginary time steps
• RE_T_STEPS - the number of real time steps
• T_SAVE - the number of steps between data saving
• GAUSS_SIGMA - the width of the initial condition
• OMEGA - the trapping potential strength
• INIT_TYPE - the form of the initial wavefunction guess
  • 'GAUSS' - Gaussian wavefunction profile
  • 'S_GAUSS' - Super-Gaussian wavefunction profile
• BREATH - sets whether to trigger a breathing mode for real time
  • 0 - No triggering of breathing mode
  • 1 - Trigger breathing mode using phase imprint exp(i\lambda r^2)
• ABSORB_BC - sets whether to turn on absorbing boundary conditions
  • 0 - Do not apply absorbing boundary conditions
  • 1 - Apply absorbing boundary conditions
• ABS_HEIGHT - sets height of tanh(r) function for absorbing boundary conditions
• ABS_SLOPE - sets the slope of tanh(r) function for absorbing boundary conditions
• ABS_POS - sets the position of tanh(r) function for absorbing boundary conditions

Density-unlocked:

As above but with some slight differences, such as:

• OMEGA1,OMEGA2 - two trapping potentials, one for each component
• INIT_TYPE1,INIT_TYPE2 - two initial guess functions for the wavefunction (for an imbalanced mixture it is often favourable to set the minority component to 'S_GAUSS' and the majority component to 'NON_ZERO_TAIL'
  • 'GAUSS' - Gaussian wavefunction profile
  • 'S_GAUSS' - Super-Gaussian wavefunction profile
  • 'NON_ZERO_TAIL' - Super-Gaussian + a small constant density mimicing the unbound density of an imbalanced mixture
• BREATH1,BREATH2 - two switches for triggering a breathing mode in real time
  • 0 - No triggering of breathing mode in the ith - component
  • 1 - Trigger breathing mode using phase imprint exp(i\lambda r^2) in the ith - component
• ABS_COMP - this flag allows for choosing with component to apply absorbing boundary conditions to
  • 'BOTH' - apply absorbing boundary conditions to both components
  • 'FIRST' - apply absorbing boundary conditions to component 1
  • 'SECOND' - apply absorbing boundary conditions to component 2
• T_DEPEN_POT - (soon to be removed) this switch allows for varying the absorbing boundary conditions in real time
  • 0 - Do not vary absorbing boundary conditions in real time
  • 1 - Turn off absorbing boundary conditions at some point in real time (this is vague as this feature is due to be removed)

Simulating mixture on a single process (local machine)

** Before running any simulations, ensure that you have created a sph_mix_lhy/data directory as this is where the data directories will go for each simulation **

In order to run this a single process on a single process, the following command is needed: python3 running.py -wp <path_to_data_dir> -rp <sim_config_file> where <sim_config_file> is either config_dens_lck.json or config_dens_ulck.json. This then creates a directory sph_mix_lhy/data/<path-to-data-dir> where the various outputs of the simulation will be.

One of the most important outputs of the simulation is the sim.out file which, during the simulation, will give some information regarding the completion percentage of the simulation and various other outputs such as chemical potential, energy and central density values.

Processing (real time) simulation data

Simple processing of a density-unlocked simulation (density-locked to come in due course) can be computed via the sph_mix_lhy/process/process_ulck.py file. This processing is ran by: python3 process_ulck.py -rp <path-to-data-dir> which will generate some movies and output of the real time simulation.

Simulating mixture on a single process (HPC cluster with SLURM queueing system)

To run a single simulation on a single process of a cluster (using SLURM) can be done via the sph_mix_lhy/drop_run.sh shell script. This is done via the command: sbatch drop_run.sh <path_to_data_dir> <sim_config_file> which will again save the simulation output to the directory sph_mix_lhy/data/<path_to_data_dir>.

Simulating job array mixture on HPC cluster with SLURM queueing system

These jobs are quasi-parallel, namely that this does not parallelise a single simulation, but for a given array of input data, will run a job array using SLURM. This is useful when wishing to explore a parameter space and want many simulations to run for different input parameters.

1. Generate simulation configs:

   • Exploring atom number imbalance requires use of sph_mix_lhy/generator_imbal.sh using the command: source generator_imbal.sh <start> <step> <end> where <start>,etc. specify a for loop of the various atom number e.g. source generator_imbal.sh 100 25 200 will generate 5 config_dens_ulck?.json files with N1 = 100, 125, 150, 175, 200.
   • Exploring droplet sizes is as above but now with the sph_mix_lhy/generator_size.sh shell script. Here, this shell script is varying the value of a12 but the minus sign is built into the shell script so e.g. source generator_size.sh 4 2 8 will generate 3 config_dens_ulck.json with a12 = -4, -6, -8.

2. From here, the config_dens_ulck?.json files can then be ran in a job array using the sph_mix_lhy/drop_array.sh script which can be ran using: sbatch --array 1-<num_of_sims> drop_array.sh <path_to_data_dir> which will then generate <num_of_sims> data files in the sph_mix_lhy/data directories.

Processing (real time) simulation data from job array

Processing of a density-unlocked job array set of simulations (density-locked to come in due course) can be computed via the sph_mix_lhy/process/imbal_breath_fitting.py file. This processing is ran by the shell script sph_mix_lhy/breath_fit.sh: sbatch breath_fit.sh <path-to-data-dir>_job <num_of_sims> <type_of_breathing_mode_sim> which will fit the real time breathing mode oscillations. The <type_of_breathing_mode_sims> flag takes values:

• 'IMBAL' - for looking at different atom number imbalances
• 'SIZE' - for looking at different droplet sizes which must be specified to let the fitting code know what to output.