Example 4: Raman Detuning Gradient Ground State

Starts with the Thomas-Fermi solution. Configures a uniform Raman coupling and a linear gradient in the Raman detuning. Propagates in imaginary time, before reaching the ground state of this configuration. The detuning gradient separates the two components vertically, and the line where they interfere is a row of vortices.

Physical Parameters

Atom number

\(\quad N_{\rm at} = 10,000\)

Atomic mass, Rubidium-87

\(\quad m = 1.4442 \times 10^{-25}~[\rm kg]\)

Trap frequencies

\(\quad (\omega_x, \omega_y, \omega_z) = 2 \pi \times (50, 50, 2000)~[{\rm Hz}]\)

\(\quad (\omega_x, \omega_y, \omega_z) = \omega_x \times (1, \gamma, \eta) = (1, 1, 40)~[\omega_x]\)

Harmonic oscillator length, x-axis

\(\quad a_x = \sqrt{\hbar / m \omega_x} = 1.525~[{\mu\rm m}]\)

3D scattering length, Rubidium-87

\(\quad a = 5.313~[{\rm nm}]\)

\(\quad a_{\rm sc} = a / a_x = 0.00348~[a_x]\)

Scattering 2D scale

\(\quad g_{\rm sc}^{2\rm D} = \sqrt{8\pi\eta}~a_{\rm sc} = 0.1105~[\omega_x a_x^2]\)

Scattering coupling

\(\quad (g_{\rm uu}, g_{\rm dd}, g_{\rm ud}) = g_{\rm sc}^{2 \rm D} \times (1, 0.995, 0.995)~[\omega_x a_x^2]\)

Chemical potential

\(\quad \mu = \sqrt{4 N_{\rm at} a_{\rm sc} \gamma \sqrt{\eta / 2 \pi}} = 18.754~[\omega_x]\)

Thomas-Fermi radius

\(\quad R_{\rm TF} = \sqrt{2 \mu} = 6.124~[a_x]\)

Initial population fractions

\(\quad (p_0, p_1) = (0.5, 0.5)\)

Raman wavelength

\(\quad \lambda_L = 790.1~[{\rm nm}]\)

Numerical Parameters

Number of grid points

\(\quad (N_x, N_y) = (256, 256)\)

r-grid half-size

\(\quad (x^{\rm max}, y^{\rm max}) = (16, 16)~[a_x]\)

r-grid spacing

\(\quad (\Delta x, \Delta y) = (0.125, 0.125)~[a_x]\)

k-grid half-size

\(\quad (k_x^{\rm max}, k_y^{\rm max}) = \pi / (\Delta x, \Delta y)\)

\(\quad (k_x^{\rm max}, k_y^{\rm max}) = (25.133, 25.133)~[a_x^{-1}]\)

k-grid spacing

\(\quad (\Delta k_x, \Delta k_y) = \pi / (x^{\rm max}, y^{\rm max})\)

\(\quad (\Delta k_x, \Delta k_y) = (0.1963, 0.1963)~[a_x^{-1}]\)

Time scale

\(\quad \tau_0 = 1 / \omega_x = 0.00318~[{\rm s/rad}]\)

\(\quad \tau_0 = 1~[\omega_x^{-1}]\)

Time step duration, imaginary

\(\quad \Delta \tau_{\rm im} = 1 / 50~[-i \tau_0]\)

Number of time steps, imaginary

\(\quad N_{\rm im} = 1000\)

import os
import sys
sys.path.insert(0, os.path.abspath('../..'))  # Adds project root to the PATH

import numpy as np

from spinor_gpe.pspinor import pspinor as spin

# 1. SETUP

DATA_PATH = 'examples/Trial_008'  # Default data path is in the /data/ folder

FREQ = 50
W = 2*np.pi*FREQ
Y_SCALE = 1
Z_SCALE = 40.0

ATOM_NUM = 1e4
OMEG = {'x': W, 'y': Y_SCALE * W, 'z': Z_SCALE * W}
G_SC = {'uu': 1, 'dd': 0.995, 'ud': 0.995}

ps = spin.PSpinor(DATA_PATH, overwrite=True,  # Initialize PSpinor object
                  atom_num=ATOM_NUM,
                  omeg=OMEG,
                  g_sc=G_SC,
                  pop_frac=(0.5, 0.5),
                  r_sizes=(16, 16),
                  mesh_points=(256, 256))

ps.coupling_setup(wavel=804e-9, kin_shift=True)

# Shifts the k-space density momentum peaks by `kshift_val` [`kL_recoil` units]
ps.shift_momentum(scale=0.6, frac=(0.5, 0.5))
ps.coupling_uniform(5 * ps.EL_recoil)
ps.detuning_grad(-12)

# Selects the form of the coupling operator in the rotated reference frame
ps.rot_coupling = True

ZOOM = 2  # Zooms the momentum-space density plots by a constant factor

ps.plot_spins(rscale=ps.rad_tf, kscale=ps.kL_recoil, zoom=ZOOM)


# 2. RUN (Imaginary-time)

DT = 1/50
N_STEPS = 1000
DEVICE = 'cuda'
ps.rand_seed = 99999

res, prop = ps.imaginary(DT, N_STEPS, DEVICE, is_sampling=True, n_samples=50)

# 3. ANALYZE

res.plot_spins(rscale=ps.rad_tf, kscale=ps.kL_recoil, zoom=ZOOM)
res.plot_total(kscale=ps.kL_recoil, zoom=ZOOM)
res.plot_pops()
res.make_movie(rscale=ps.rad_tf, kscale=ps.kL_recoil, play=True, zoom=ZOOM,
               norm_type='half')
print(f'\nFinal energy: {res.eng_final[0]} [hbar * omeg]')