Hyperfine Structure: Sech Pulse 2π, q=1 — Self-Induced Transparency¶
\(^{87}\mathrm{Rb}\) driven on the \(5S_{1/2} F=1 \rightarrow 5p_{1/2} F=1\) transition.
[1]:
import numpy as np
[2]:
from maxwellbloch import hyperfine
Rb87_5s12_F1 = hyperfine.LevelF(I=1.5, J=0.5, F=1)
Rb87_5s12_F2 = hyperfine.LevelF(I=1.5, J=0.5, F=2) # Needed for decay
Rb87_5p12_F1 = hyperfine.LevelF(I=1.5, J=0.5, F=1)
atom1e = hyperfine.Atom1e(element='Rb', isotope='87')
atom1e.add_F_level(Rb87_5s12_F1)
atom1e.add_F_level(Rb87_5s12_F2)
atom1e.add_F_level(Rb87_5p12_F1)
[3]:
NUM_STATES = atom1e.get_num_mF_levels()
print(NUM_STATES)
11
[4]:
ENERGIES = atom1e.get_energies()
print(ENERGIES)
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
[5]:
# Tune to be on resonance with the F1 -> F1 transition
DETUNING = 0
print(DETUNING)
0
[6]:
FIELD_CHANNELS = atom1e.get_coupled_levels(F_level_idxs_a=(0,), F_level_idxs_b=(2,))
print(FIELD_CHANNELS)
[[0, 8], [0, 9], [0, 10], [1, 8], [1, 9], [1, 10], [2, 8], [2, 9], [2, 10]]
[7]:
q = 1 # Field polarisation
FIELD_FACTORS = atom1e.get_clebsch_hf_factors(F_level_idxs_a=(0,), F_level_idxs_b=(2,), q=q)
print(FIELD_FACTORS)
[ 0. 0. 0. 0.28867513 -0. -0.
0. 0.28867513 0. ]
[8]:
strength_factor = np.sum(FIELD_FACTORS**2)
print(strength_factor)
0.16666666666666657
1/6 is the strength factor S_11
[9]:
hf_factor = np.max(FIELD_FACTORS)
print(hf_factor)
0.2886751345948128
[10]:
DECAY_CHANNELS = atom1e.get_coupled_levels(F_level_idxs_a=(0,1), F_level_idxs_b=(2,))
print(DECAY_CHANNELS)
[[0, 8], [0, 9], [0, 10], [1, 8], [1, 9], [1, 10], [2, 8], [2, 9], [2, 10], [3, 8], [3, 9], [3, 10], [4, 8], [4, 9], [4, 10], [5, 8], [5, 9], [5, 10], [6, 8], [6, 9], [6, 10], [7, 8], [7, 9], [7, 10]]
[11]:
DECAY_FACTORS = atom1e.get_decay_factors(F_level_idxs_a=(0, 1), F_level_idxs_b=(2,))
print(DECAY_FACTORS)
[ 0.28867513 -0.28867513 0. 0.28867513 -0. -0.28867513
0. 0.28867513 -0.28867513 0.70710678 0. 0.
0.5 0.5 -0. 0.28867513 0.57735027 0.28867513
-0. 0.5 0.5 0. 0. 0.70710678]
[12]:
INITIAL_STATE = (
[1.0/3.0]*3 + # s12_F1
[0.0/5.0]*5 + # s12_F2
[0.0]*3) # p12_F1
print(INITIAL_STATE)
[0.3333333333333333, 0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
[13]:
sech_fwhm_conv = 1./2.6339157938
WIDTH = 1.0*sech_fwhm_conv # [τ]
print('WIDTH', WIDTH)
n = 2.0 # For a pulse area of nπ
AMPL = n/WIDTH/(2*np.pi) # Pulse amplitude [2π Γ]
AMPL *= 1/hf_factor
print('ampl', AMPL)
WIDTH 0.3796628587572578
ampl 2.9043077704595337
[14]:
mb_solve_json = """
{{
"atom": {{
"decays": [
{{
"channels": {decay_channels},
"rate": 0.0,
"factors": {decay_factors}
}}
],
"energies": {energies},
"fields": [
{{
"coupled_levels": {field_channels},
"factors": {field_factors},
"detuning": {detuning},
"detuning_positive": true,
"label": "probe",
"rabi_freq": 1.0,
"rabi_freq_t_args": {{
"ampl": {ampl},
"centre": 0.0,
"width": {width}
}},
"rabi_freq_t_func": "sech"
}}
],
"num_states": {num_states},
"initial_state": {initial_state}
}},
"t_min": -2.0,
"t_max": 10.0,
"t_steps": 100,
"z_min": -0.5,
"z_max": 1.5,
"z_steps": 100,
"z_steps_inner": 1,
"num_density_z_func": "square",
"num_density_z_args": {{
"on": 0.0,
"off": 1.0,
"ampl": 1.0
}},
"interaction_strengths": [
5.0e2
],
"velocity_classes": null,
"method": "mesolve",
"opts": {{
"method": "bdf",
"atol": 1e-8,
"rtol": 1e-6,
"nsteps": 1e2
}},
"savefile": "mbs-Rb87_5s12_5p12_F11_q1-sech-2pi"
}}
""".format(num_states=NUM_STATES, energies=ENERGIES, initial_state=INITIAL_STATE,
detuning=DETUNING, field_channels=FIELD_CHANNELS, field_factors=list(FIELD_FACTORS),
decay_channels=DECAY_CHANNELS, decay_factors=list(DECAY_FACTORS),
ampl=AMPL, width=WIDTH)
[15]:
from maxwellbloch import mb_solve
mbs = mb_solve.MBSolve().from_json_str(mb_solve_json)
[16]:
%time Omegas_zt, states_zt = mbs.mbsolve(recalc=True, pbar_chunk_size=10)
10.0%. Run time: 2.68s. Est. time left: 00:00:00:24
20.0%. Run time: 5.68s. Est. time left: 00:00:00:22
30.0%. Run time: 8.65s. Est. time left: 00:00:00:20
40.0%. Run time: 11.89s. Est. time left: 00:00:00:17
50.0%. Run time: 15.18s. Est. time left: 00:00:00:15
60.0%. Run time: 18.68s. Est. time left: 00:00:00:12
70.0%. Run time: 22.33s. Est. time left: 00:00:00:09
80.0%. Run time: 26.16s. Est. time left: 00:00:00:06
90.0%. Run time: 30.09s. Est. time left: 00:00:00:03
Total run time: 34.04s
Saving MBSolve to mbs-Rb87_5s12_5p12_F11_q1-sech-2pi.qu
CPU times: user 34 s, sys: 65.3 ms, total: 34 s
Wall time: 34.4 s
[17]:
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
sns.set_style('darkgrid')
import numpy as np
[18]:
fig = plt.figure(1, figsize=(16, 6))
ax = fig.add_subplot(111)
# cmap_range = np.linspace(0.0, 12, 11)
cf = ax.contourf(mbs.tlist, mbs.zlist,
np.abs(mbs.Omegas_zt[0]/(2*np.pi)),
# cmap_range,
cmap=plt.cm.Blues
)
ax.set_title('Rabi Frequency ($\Gamma / 2\pi $)')
ax.set_xlabel('Time ($1/\Gamma$)')
ax.set_ylabel('Distance ($L$)')
ax.grid(alpha=0.5)
ax.set_axisbelow(False)
for y in [0.0, 1.0]:
ax.axhline(y, c='grey', lw=1.0, ls='dotted')
plt.colorbar(cf);
[19]:
fig = plt.figure(1, figsize=(16, 6))
ax = fig.add_subplot(111)
# cmap_range = np.linspace(0.0, 1.0e-3, 11)
cf = ax.contourf(mbs.tlist, mbs.zlist,
np.abs(mbs.populations_field(0, upper=False)),
# cmap_range,
cmap=plt.cm.Reds
)
ax.set_title('Rabi Frequency ($\Gamma / 2\pi $)')
ax.set_xlabel('Time ($1/\Gamma$)')
ax.set_ylabel('Distance ($L$)')
for y in [0.0, 1.0]:
ax.axhline(y, c='grey', lw=1.0, ls='dotted')
plt.colorbar(cf);
[20]:
fig = plt.figure(1, figsize=(16, 6))
ax = fig.add_subplot(111)
# cmap_range = np.linspace(0.0, 1.0e-3, 11)
cf = ax.contourf(mbs.tlist, mbs.zlist,
np.imag(mbs.coherences_field(0)),
# cmap_range,
cmap=plt.cm.Greens
)
ax.set_title('Rabi Frequency ($\Gamma / 2\pi $)')
ax.set_xlabel('Time ($1/\Gamma$)')
ax.set_ylabel('Distance ($L$)')
for y in [0.0, 1.0]:
ax.axhline(y, c='grey', lw=1.0, ls='dotted')
plt.colorbar(cf);
