This page was generated from the Jupyter notebook
spectrum_with_overlaps.ipynb.
MQDT Spectrum with overlaps of OSQDT states and Tunable MQDT models
This notebook compares the OSQDT channel spectra with the full MQDT spectrum and illustrates how the levels evolve as the tunable MQDT coupling factor increases from 0 to 1. Levels are plotted against the effective principal quantum number \(\nu\), and the colored segments of each full MQDT level show its summed squared overlaps with the different OSQDT channel groups.
The plots, including the spectra obtained by tuning the coupling, are intended only as didactic examples to illustrate channel mixing. Only the full MQDT results should be used for calculations.
[1]:
from __future__ import annotations
# %matplotlib widget
import logging
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.collections import LineCollection
from utils import build_segments
logging.basicConfig(level=logging.INFO)
logging.getLogger("matplotlib").setLevel(logging.WARNING)
logging.getLogger("rydstate").setLevel(logging.WARNING)
[2]:
from typing import TYPE_CHECKING
import rydstate
if TYPE_CHECKING:
from rydstate.rydberg_state import RydbergState
[ ]:
species = "Yb171"
l_r = 0
f_tot = 0.5
nu_range = (0, 150)
basis_osqdt = rydstate.basis.BasisOSQDT(species, nu=nu_range, l_r=(l_r, l_r + 1), f_tot=(f_tot, f_tot))
basis_mqdt = rydstate.BasisMQDT(species, nu=nu_range, l_r=(l_r, l_r), f_tot=(f_tot, f_tot))
[ ]:
coupling_factors = np.linspace(0, 1, 10)
bases_tunable_mqdt = [
rydstate.basis.BasisTunableMQDT(species, nu=nu_range, l_r=(l_r, l_r), f_tot=(f_tot, f_tot), coupling_factor=c)
for c in coupling_factors
]
[5]:
basis_dict: dict[str, rydstate.basis.BasisBase[RydbergState]] = {}
basis_dict["fc=0"] = (
basis_osqdt.shallow_copy().filter_states("l_c", 0).filter_states("l_r", l_r).filter_states("f_c", 0)
)
basis_dict["fc=1"] = (
basis_osqdt.shallow_copy().filter_states("l_c", 0).filter_states("l_r", l_r).filter_states("f_c", 1)
)
basis_dict["lc=1 jc=1/2"] = (
basis_osqdt.shallow_copy().filter_states("l_r", 1).filter_states("l_c", 1).filter_states("j_c", 1 / 2)
)
basis_dict["lc=1 jc=3/2"] = (
basis_osqdt.shallow_copy().filter_states("l_r", 1).filter_states("l_c", 1).filter_states("j_c", 3 / 2)
)
for key in ["4f13 5d 6snl a", "4f13 5d 6snl b", "4f13 5d 6snl c"]:
basis_dict[key] = basis_osqdt.shallow_copy().filter_states("parity", +1).filter_states_label(key)
basis_dict["full MQDT"] = basis_mqdt
for key, basis in basis_dict.items():
print(f"{key}: {len(basis.states)} states")
for i, state in enumerate(basis.states):
if i >= 3:
print(" ...\n")
break
print(f" {state}")
fc=0: 155 states
|Yb171:(6s_1/2,7s_1/2),f_c=0,F=1/2⟩
|Yb171:(6s_1/2,8s_1/2),f_c=0,F=1/2⟩
|Yb171:(6s_1/2,9s_1/2),f_c=0,F=1/2⟩
...
fc=1: 148 states
|Yb171:(6s_1/2,7s_1/2),f_c=1,F=1/2⟩
|Yb171:(6s_1/2,8s_1/2),f_c=1,F=1/2⟩
|Yb171:(6s_1/2,9s_1/2),f_c=1,F=1/2⟩
...
lc=1 jc=1/2: 2 states
|Yb171:(6p_1/2,6p_1/2),f_c=0,F=1/2⟩
|Yb171:(6p_1/2,6p_1/2),f_c=1,F=1/2⟩
lc=1 jc=3/2: 2 states
|Yb171:(6p_3/2,6p_3/2),f_c=1,F=1/2⟩
|Yb171:(6p_3/2,6p_3/2),f_c=2,F=1/2⟩
4f13 5d 6snl a: 1 states
|Yb171:nu=1.8,4f13 5d 6snl a,F=1/2⟩
4f13 5d 6snl b: 1 states
|Yb171:nu=1.7,4f13 5d 6snl b,F=1/2⟩
4f13 5d 6snl c: 0 states
full MQDT: 307 states
0.87*|Yb171:(6s_1/2,[2.5]s_1/2),f_c=0,F=1/2⟩ + 3.7e-06*|Yb171:nu=1.5,4f13 5d 6snl a,F=1/2⟩ - 5.7e-06*|Yb171:(6p_3/2,[1.5]p_3/2),f_c=1,F=1/2⟩ + 7.3e-06*|Yb171:(6p_3/2,[1.5]p_3/2),f_c=2,F=1/2⟩ + 1.5e-06*|Yb171:nu=1.5,4f13 5d 6snl b,F=1/2⟩ - 4.6e-06*|Yb171:(6p_1/2,[1.6]p_1/2),f_c=0,F=1/2⟩ + 7.9e-06*|Yb171:(6p_1/2,[1.6]p_1/2),f_c=1,F=1/2⟩ + 4.9e-06*|Yb171:nu=1.5,4f13 5d 6snl c,F=1/2⟩ + 0.5*|Yb171:(6s_1/2,[2.5]s_1/2),f_c=1,F=1/2⟩
-0.5*|Yb171:(6s_1/2,[2.6]s_1/2),f_c=0,F=1/2⟩ - 0.035*|Yb171:nu=1.5,4f13 5d 6snl a,F=1/2⟩ + 0.052*|Yb171:(6p_3/2,[1.5]p_3/2),f_c=1,F=1/2⟩ - 0.067*|Yb171:(6p_3/2,[1.5]p_3/2),f_c=2,F=1/2⟩ - 0.013*|Yb171:nu=1.5,4f13 5d 6snl b,F=1/2⟩ + 0.044*|Yb171:(6p_1/2,[1.6]p_1/2),f_c=0,F=1/2⟩ - 0.076*|Yb171:(6p_1/2,[1.6]p_1/2),f_c=1,F=1/2⟩ - 0.044*|Yb171:nu=1.5,4f13 5d 6snl c,F=1/2⟩ + 0.86*|Yb171:(6s_1/2,[2.6]s_1/2),f_c=1,F=1/2⟩
0.87*|Yb171:(6s_1/2,[3.5]s_1/2),f_c=0,F=1/2⟩ - 1.6e-05*|Yb171:nu=1.6,4f13 5d 6snl a,F=1/2⟩ + 9.4e-06*|Yb171:(6p_3/2,[1.7]p_3/2),f_c=1,F=1/2⟩ - 1.2e-05*|Yb171:(6p_3/2,[1.7]p_3/2),f_c=2,F=1/2⟩ - 1.4e-06*|Yb171:nu=1.6,4f13 5d 6snl b,F=1/2⟩ + 4.5e-05*|Yb171:(6p_1/2,[1.7]p_1/2),f_c=0,F=1/2⟩ - 7.7e-05*|Yb171:(6p_1/2,[1.7]p_1/2),f_c=1,F=1/2⟩ - 1.7e-05*|Yb171:nu=1.6,4f13 5d 6snl c,F=1/2⟩ + 0.5*|Yb171:(6s_1/2,[3.5]s_1/2),f_c=1,F=1/2⟩
...
[6]:
overlaps_dict: dict[str, list[float]] = {}
for key, basis in basis_dict.items():
if isinstance(basis, rydstate.basis.BasisMQDT):
continue
overlaps = basis_mqdt.calc_reduced_overlaps(basis)
overlaps_dict[key] = np.sum(overlaps**2, axis=1).tolist()
[ ]:
x_loc = {
# S_05
"fc=0": 0,
"fc=1": 1,
"lc=1 jc=1/2": 2,
"lc=1 jc=3/2": 3,
# general
"4f13 5d 6snl a": 4,
"4f13 5d 6snl b": 4,
"4f13 5d 6snl c": 4,
# mqdt
"tunable MQDT": 5,
"full MQDT": 6,
}
fig, axs = plt.subplots(1, 2, figsize=(14, 6))
keys = list(basis_dict.keys())
keys_mo_mqdt = [key for key in keys if "MQDT" not in key]
tunable_nus = [basis.calc_exp_qn("nu") for basis in bases_tunable_mqdt]
tunable_segments = [
(
np.array([coupling_factors[i] for i, j in seg]) * 0.8 + x_loc["tunable MQDT"] - 0.4,
[tunable_nus[i][j] for i, j in seg],
)
for seg in build_segments(tunable_nus)
]
# None means autoscale, i.e. the full nu range; only the zoomed panels get labels
ylims = [None, (0, 10)]
for ax, ylim in zip(axs, ylims, strict=True):
if ylim is not None:
ax.set_ylim(*ylim)
# collect all level markers in one LineCollection per axis, drawing them one by one is slow
lines: list[list[tuple[float, float]]] = []
colors: list[str] = []
# OQDT states
for i, key in enumerate(keys_mo_mqdt):
x = x_loc.get(key)
for state in basis_dict[key].states:
if np.isinf(state.nu):
continue
lines.append([(x - 0.4, state.nu), (x + 0.4, state.nu)])
colors.append(f"C{i}")
if ylim is None or not ylim[0] <= state.nu <= ylim[1]:
continue
label = rf"$n={state.n}$, $\nu={state.nu:.3f}$"
if not np.isclose(state.nu, state.nui):
label += rf", $\nu_{{i}}={state.nui:.3f}$"
ax.text(x, state.nu, label, ha="center", va="bottom", fontsize=6, clip_on=True)
# MQDT states
for ids, state in enumerate(basis_mqdt.states):
if np.isinf(state.nu):
continue
overlaps = {key: overlaps_dict[key][ids] for key in keys_mo_mqdt}
x0 = x_loc["full MQDT"] - 0.4
for i, key in enumerate(keys_mo_mqdt):
overlap = overlaps[key]
x1 = x0 + overlap * 0.8
lines.append([(x0, state.nu), (x1, state.nu)])
colors.append(f"C{i}")
x0 = x1
if ylim is None or not ylim[0] <= state.nu <= ylim[1]:
continue
ax.text(
x_loc["full MQDT"], state.nu, rf"$\nu={state.nu:.3f}$", ha="center", va="bottom", fontsize=6, clip_on=True
)
ax.add_collection(LineCollection(lines, colors=colors, linewidths=0.75))
ax.autoscale_view()
ylim_ = ax.get_ylim()
ax.plot([x_loc["full MQDT"] - 0.4, x_loc["full MQDT"] - 0.4], ylim_, color="0.5", lw=0.75, ls="--")
ax.plot([x_loc["full MQDT"] + 0.4, x_loc["full MQDT"] + 0.4], ylim_, color="0.5", lw=0.75, ls="--")
fig.suptitle(f"States for {species} {l_r=}, {f_tot=}")
xticklabels = {}
for key, x in x_loc.items():
if x not in xticklabels:
xticklabels[x] = key
else:
xticklabels[x] += f"\n{key}"
xticklabels[x_loc["tunable MQDT"]] += "\n" + r"coupling_factor $0 \to 1$"
ax.set_xticks(list(xticklabels.keys()))
ax.set_xticklabels(list(xticklabels.values()))
ax.set_ylabel(r"$\nu$")
ax.set_xlim(-0.5, max(x_loc.values()) + 0.5)
for xs, ys in tunable_segments:
ax.plot(xs, ys, color="blue", marker=".", markersize=2, lw=0.75)
plt.tight_layout()
plt.show()
