# -*- coding: utf-8 -*-
from scipy.integrate import odeint
from lmfit import minimize, Parameters, report_fit
import matplotlib.pyplot as plt
import numpy as np
import sys
from arc._database import UsedModulesARC
"""
**Contributors:**
getPopulationLifetime - written by Alessandro Greco,
Dipartimento di Fisica *E. Fermi*, Università di Pisa,
Largo Bruno Pontecorvo 3, 56127 Pisa, Italy (alessandrogreco08 at gmail dot com),
the simulations have been compared with experimental data [#greco2019]_
"""
[docs]def getPopulationLifetime(
atom,
n,
l,
j,
temperature=0,
includeLevelsUpTo=0,
period=1,
plotting=1,
thresholdState=False,
detailedOutput=False,
):
r"""
Calculates lifetime of atomic **population** taking into account
redistribution of population to other states under spontaneous and
black body induced transitions.
It simulates the time evolution of a system in which all the states,
from the fundamental one to the highest state which you want to include,
are taken into account.
The orbital angular momenta taken into account are only S,P,D,F.
This function is based on getStateLifetime but it takes into account
the re-population processess due to BBR-induced transitions.
For this reason lifetimes of Rydberg states are slightly longer
than those returned by getStateLifetime up to 5-10%.
This function creates a .txt file, plots the time evolution of the
population of the Rydberg states and yields the lifetime values by using
the fitting method from Ref. [#fit]_ .
**Contributed by:** Alessandro Greco (alessandrogreco08 at gmail dot com),
Dipartimento di Fisica *E. Fermi*, Università di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy.
The simulations have been compared with experimental data [#greco2019]_ .
**Please cite as:** `original ARC paper`_ and paper introducing
extension [#greco2019]_
.. _`original ARC paper`:
https://doi.org/10.1016/j.cpc.2017.06.015
References:
.. [#fit] https://people.duke.edu/~ccc14/sta-663/CalibratingODEs.html
.. [#greco2019] M. Archimi, C. Simonelli, L. Di Virgilio, A. Greco,
M. Ceccanti, E. Arimondo, D. Ciampini, I. I. Ryabtsev, I. I. Beterov,
and O. Morsch, *Phys. Rev. A* **100**, 030501(R) (2019)
https://doi.org/10.1103/PhysRevA.100.030501
**Some definitions:**
What are the **ensemble**, the **support**, the **ground**?
According to https://arxiv.org/abs/1907.01254
The sum of the populations of every state which is detected as Rydberg state
(above the threshold state which must be set\) is called **ensemble**
The sum of the populations of every state which is detected as Rydberg state,
without the target state, is called **support**
The sum of the populations of every state which cannot be detected as Rydberg state
(under the threshold state which must be set) is called **ground**
**gammaTargetSpont** is the rate which describes transitions
from Target State towards all the levels under the threshold
state, i.e. Ground State
**gammaTargetBBR** is the rate which describes transitions towards all
the levels above the threshold state, i.e. Support State
**gammaSupporSpont** is the rate which describes transitions from the
Support State towards all the levels under the threshold state,
i.e. Ground State
**gammaSupportBBR** is the rate which describes transitions from
upport State towards all the levels above the threshold state,
i.e. Target State)
Args:
n (int): principal quantum number of the state whose population lifetime
we are calculating, it's called the *Target* state and its color is
green in the plot.
l (int): orbital angular momentum number of the state whose population lifetime
we are calculating, it's called the *Target* state and its color is
green in the plot.
j (float): total angular momentum of the state whose population lifetime
we are calculating, it's called the *Target* state and its color is
green in the plot.
temperature (float): Temperature at which the atom environment
is, measured in K. If this parameter is non-zero, user has
to specify transitions up to which state (due to black-body
decay) should be included in calculation.
includeLevelsUpTo (int): At non zero temperatures,
this specifies maximum principal quantum number of the state
to which black-body induced transitions will be included.
Minimal value of the parameter in that case is =`n+1
period: Specifies the period that you want to consider for
the time evolution, in microseconds.
plotting (int): optional. It is set to 1 by default. The options are
(see also image at the bottom of documentation):
**plotting=0** no plot;
**plotting=1** plots the population of the target (n,l,j) state
with its fit and it yields the value of the target lifetime
in microseconds;
**plotting=2** plots the whole system (Ensemble, Support, Target),
no fit;
**plotting=3** plots the whole system (Ensemble, Support, Target)
and it fits the Ensemble and Target curves, it yields the values
of the Ensemble lifetime and Target lifetime in microseconds;
**plotting=4** it plots the whole system (Ensemble, Support, Target) +
the Ground (which is the complementary of the ensemble).
It considers the whole system like a three-level model (Ground
State, Support State, Target State) and yields four transition
rates.
thresholdState (int): optional. It specifies the principal quantum
number n of the lowest state (it's referred to S state!) which is
detectable by your experimental apparatus, it directly modifies
the *Ensemble* and the *Support* (whose colors are red and blue
respectively in the plot). It is necessary to define a threshold
state if plotting = 2, 3 or 4 has been selected. It is not necessary
to define a threshold state if plotting = 0 or 1 has been selected.
detailedOutput=True: optional. It writes a .txt file with the time
evolution of all the states. It is set to false by default.
(The first column is the time, the other are the population of all
the states. The order is time, nS, nP0.5, nP1.5, nD1.5, nD2.5,
nF2.5, nF3.5, and n is ordered from the lowest state to the highest one.
For example: time, 4S, 5S ,6S ,ecc... includeLevelsUpToS, 4P0.5,
5P0.5, 6P0.5, ecc... includeLevelsUpToP0.5, 4P1.5, 5P1.5, 6P1.5, ecc...)
Returns:
Plots and a .txt file.
**plotting = 0,1** create a .txt file with two coloumns
(time \t target population);
**plotting = 2,3,4** create a .txt file with four coloumns
(time \t ensemble population \t support population \t target population)
Example:
>>> from arc import *
>>> from arc.advanced.population_lifetime import getPopulationLifetime
>>> atom = Rubidium()
>>> getPopulationLifetime(atom, 10, 1, 1.5, temperature =300,
includeLevelsUpTo=15, detailedOutput=True, plotting=1)
"""
UsedModulesARC.advanced_getPopulationTime = True
if l > 3:
print("Error: this function takes into account only S, P, D, F states.")
return
if plotting > 4:
print("Error: plotting must be equal to 0, 1, 2, 3 or 4.")
return
if ((not thresholdState) and (plotting > 1)) or (thresholdState):
print(
"Error: you need to specify the principal quantum number of the "
"thresholdState if you use plotting=2, 3 or 4."
)
return
if (plotting == 0) or (plotting == 1):
thresholdState = False
import time
start = time.time()
# What state do you want to excite?
STATE = n
L = l
J = j
# Which states do you want to consider for the BBR width?
if includeLevelsUpTo - STATE < 0:
raise ValueError("Error: includeLevelsUpTo must be >= n")
WidthBBR = includeLevelsUpTo - STATE
# What is the temperature?
if temperature == 0:
raise ValueError(
"Error: if you don't want BBR-induced transition, use getStateLifetime"
)
TEMP_BBR = temperature
# What is the critical state for the ionization?
if thresholdState - STATE >= 0:
raise ValueError("Error: thresholdState must be < n")
CState = thresholdState
# It creates the references for the ensemble population
cutoffs = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 0, 0.5)
)
cutoffp05 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 1, 0.5)
)
cutoffp15 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 1, 1.5)
)
cutoffd15 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 2, 1.5)
)
cutoffd25 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 2, 2.5)
)
cutofff25 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 3, 2.5)
)
cutofff35 = int(
atom.getQuantumDefect(STATE, 0, 0.5)
- atom.getQuantumDefect(STATE, 3, 3.5)
)
# Total time of the dynamics
totaltime = period * 1e-6
# Parts of gammamax that you take for time step
partg = 2.0
#########################################################
# It takes into account of the extra levels
extraL = atom.extraLevels[1][:]
# It creates the references for the matrix
riftot = (STATE + WidthBBR - extraL[0] + 1) * 7
rifs = ((STATE + WidthBBR - extraL[0] + 1) * 0) - extraL[0]
rifp05 = ((STATE + WidthBBR - extraL[0] + 1) * 1) - extraL[0]
rifp15 = ((STATE + WidthBBR - extraL[0] + 1) * 2) - extraL[0]
rifd15 = ((STATE + WidthBBR - extraL[0] + 1) * 3) - extraL[0]
rifd25 = ((STATE + WidthBBR - extraL[0] + 1) * 4) - extraL[0]
riff25 = ((STATE + WidthBBR - extraL[0] + 1) * 5) - extraL[0]
riff35 = ((STATE + WidthBBR - extraL[0] + 1) * 6) - extraL[0]
# It creates the matrix of the rates
c = np.zeros(shape=(riftot, riftot))
print("Creating the rates matrix:")
for pqn in range(extraL[0], STATE + WidthBBR + 1):
for fpqn in range(extraL[0], STATE + WidthBBR + 1):
# rate from s
c[pqn + rifs, fpqn + rifp05] = atom.getTransitionRate(
pqn, 0, 0.5, fpqn, 1, 0.5, TEMP_BBR
) # rate s -> p0.5
c[pqn + rifs, fpqn + rifp15] = atom.getTransitionRate(
pqn, 0, 0.5, fpqn, 1, 1.5, TEMP_BBR
) # rate s -> p1.5
# rate from p0.5
c[pqn + rifp05, fpqn + rifs] = atom.getTransitionRate(
pqn, 1, 0.5, fpqn, 0, 0.5, TEMP_BBR
) # rate p0.5 -> s
c[pqn + rifp05, fpqn + rifd15] = atom.getTransitionRate(
pqn, 1, 0.5, fpqn, 2, 1.5, TEMP_BBR
) # rate p0.5 -> d1.5
# rate from p1.5
c[pqn + rifp15, fpqn + rifs] = atom.getTransitionRate(
pqn, 1, 1.5, fpqn, 0, 0.5, TEMP_BBR
) # rate p1.5 -> s
c[pqn + rifp15, fpqn + rifd15] = atom.getTransitionRate(
pqn, 1, 1.5, fpqn, 2, 1.5, TEMP_BBR
) # rate p1.5 -> d1.5
c[pqn + rifp15, fpqn + rifd25] = atom.getTransitionRate(
pqn, 1, 1.5, fpqn, 2, 2.5, TEMP_BBR
) # rate p1.5 -> d2.5
# rate from d1.5
c[pqn + rifd15, fpqn + rifp05] = atom.getTransitionRate(
pqn, 2, 1.5, fpqn, 1, 0.5, TEMP_BBR
) # rate d1.5 -> p0.5
c[pqn + rifd15, fpqn + rifp15] = atom.getTransitionRate(
pqn, 2, 1.5, fpqn, 1, 1.5, TEMP_BBR
) # rate d1.5 -> p1.5
c[pqn + rifd15, fpqn + riff25] = atom.getTransitionRate(
pqn, 2, 1.5, fpqn, 3, 2.5, TEMP_BBR
) # rate d1.5 -> f2.5
# rate from d2.5
c[pqn + rifd25, fpqn + rifp15] = atom.getTransitionRate(
pqn, 2, 2.5, fpqn, 1, 1.5, TEMP_BBR
) # rate d2.5 -> p1.5
c[pqn + rifd25, fpqn + riff25] = atom.getTransitionRate(
pqn, 2, 2.5, fpqn, 3, 2.5, TEMP_BBR
) # rate d2.5 -> f2.5
c[pqn + rifd25, fpqn + riff35] = atom.getTransitionRate(
pqn, 2, 2.5, fpqn, 3, 3.5, TEMP_BBR
) # rate d2.5 -> f3.5
# rate from f2.5
c[pqn + riff25, fpqn + rifd15] = atom.getTransitionRate(
pqn, 3, 2.5, fpqn, 2, 1.5, TEMP_BBR
) # rate f2.5 -> d1.5
c[pqn + riff25, fpqn + rifd25] = atom.getTransitionRate(
pqn, 3, 2.5, fpqn, 2, 2.5, TEMP_BBR
) # rate f2.5 -> d2.5
# rate from f3.5
c[pqn + riff35, fpqn + rifd25] = atom.getTransitionRate(
pqn, 3, 3.5, fpqn, 2, 2.5, TEMP_BBR
) # rate f3.5 -> d2.5
print(pqn, end=" ")
# It deletes all the gammas for states under the ground state which are not the extra levels
if extraL[1] > 2:
c[extraL[0] + rifd15, :] = 0
c[:, extraL[0] + rifd15] = 0
c[extraL[0] + rifd25, :] = 0
c[:, extraL[0] + rifd25] = 0
if extraL[1] > 3:
c[extraL[0] + riff25, :] = 0
c[:, extraL[0] + riff25] = 0
c[extraL[0] + riff35, :] = 0
c[:, extraL[0] + riff35] = 0
c[extraL[0] + rifs, :] = 0
c[:, extraL[0] + rifs] = 0
c[extraL[0] + rifp05, :] = 0
c[:, extraL[0] + rifp05] = 0
c[extraL[0] + rifp15, :] = 0
c[:, extraL[0] + rifp15] = 0
# It finds the maximum rate in the matrix
gammamax = c.max() # is from the 5P1.5 towards the 5S0.5
# It defines Dtmin
Dtmin = round(1 / (partg * gammamax), 9)
print("\n", Dtmin)
#########################################################
# It inizialites the population and the auxiliry population vectors
pop = np.zeros(shape=(1, riftot))
popaus = np.zeros(shape=(1, riftot))
# It inizializes the reference for the population vector
if L == 0:
rifinitial = rifs
if L == 1:
if J == 0.5:
rifinitial = rifp05
if J == 1.5:
rifinitial = rifp15
if L == 2:
if J == 1.5:
rifinitial = rifd15
if J == 2.5:
rifinitial = rifd25
if L == 3:
if J == 2.5:
rifinitial = riff25
if J == 3.5:
rifinitial = riff35
pop[0, (rifinitial + STATE)] = 1
#########################################################
# It inizializes the time and the time step
t = 0.0
Dt = 0.0
#########################################################
# References for the name of the .txt file
if L == 0:
StrL = "S"
elif L == 1:
StrL = "P"
elif L == 2:
StrL = "D"
elif L == 3:
StrL = "F"
if J == 0.5:
StrJ = "05"
elif J == 1.5:
StrJ = "15"
elif J == 2.5:
StrJ = "25"
elif J == 3.5:
StrJ = "35"
# It creates the file for the three curves
with open("Lifetime" + str(STATE) + StrL + StrJ + ".txt", "w") as fi:
fi.writelines("")
if detailedOutput:
# It creates the file for the all states
with open(
"Lifetime" + str(STATE) + StrL + StrJ + "All.txt", "w"
) as fiall:
fiall.writelines("")
#########################################################
# It creates four lists to quickly write the results to the file
ListTime = []
if thresholdState:
ListRed = []
ListBlue = []
ListGreen = []
# The core of the program starts
while t < (totaltime):
if detailedOutput:
ListStates = []
ListStates.append(t * 1e6)
for a in range(0, riftot):
popaus[0, a] = 0.0
for b in range(0, riftot):
popaus[0, a] += -c[a, b] * pop[0, a] + c[b, a] * pop[0, b]
popaus[0, a] = popaus[0, a] * Dt
pop += popaus
if t == 0:
Dt = Dtmin
if detailedOutput:
ListStates.extend(pop[0, :])
with open(
"Lifetime" + str(STATE) + StrL + StrJ + "All.txt", "a"
) as fall:
fall.writelines(
"%.5f \t" % (ListStates[ind])
for ind in range(0, len(ListStates))
)
fall.writelines("\n")
ListTime.append(t * 1e6)
if thresholdState:
popall = 0.0
for k in range(0, riftot):
if (
(CState + rifs - cutoffs <= k < rifp05 + extraL[0])
or (CState + rifp05 - cutoffp05 <= k < rifp15 + extraL[0])
or (CState + rifp15 - cutoffp15 <= k < rifd15 + extraL[0])
or (CState + rifd15 - cutoffd15 <= k < rifd25 + extraL[0])
or (CState + rifd25 - cutoffd25 <= k < riff25 + extraL[0])
or (CState + riff25 - cutofff25 <= k < riff35 + extraL[0])
or (CState + riff35 - cutofff35 <= k < riftot)
):
# above the threshold state
popall += pop[0, k]
ListRed.append(popall)
ListBlue.append(popall - pop[0, (rifinitial + STATE)])
ListGreen.append(pop[0, (rifinitial + STATE)])
sys.stdout.write("\rProgress: %d%%" % ((t / totaltime) * 100))
sys.stdout.flush()
t = t + Dt
if not thresholdState:
with open("Lifetime" + str(STATE) + StrL + StrJ + ".txt", "a") as f:
f.writelines(
"%.4f \t %.5f \n" % (ListTime[index], ListGreen[index])
for index in range(0, len(ListTime))
)
else:
with open("Lifetime" + str(STATE) + StrL + StrJ + ".txt", "a") as f:
f.writelines(
"%.4f \t %.5f \t %.5f \t %.5f \n"
% (
ListTime[index],
ListRed[index],
ListBlue[index],
ListGreen[index],
)
for index in range(0, len(ListTime))
)
#########################################################
if plotting == 1:
def f(xs, t, ps):
"""Lotka-Volterra predator-prey model."""
try:
gammaTarget = ps["gammaTarget"].value
except Exception:
gammaTarget = ps
x, y = xs
return [-gammaTarget * x, -gammaTarget * y]
def g(t, x0, ps):
"""
Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
"""
x = odeint(f, x0, t, args=(ps,))
return x
def residual(ps, ts, data):
x0 = ps["x0"].value, ps["y0"].value
model = g(ts, x0, ps)
return (model - data).ravel()
t = np.array(ListTime)
x0 = np.array([0, 0])
data = np.zeros(shape=(len(t), 2))
data[:, 0] = np.array(ListGreen)
data[:, 1] = np.array(ListGreen)
# set parameters incluing bounds
params = Parameters()
params.add("x0", value=1, vary=False)
params.add("y0", value=1, vary=False)
params.add("gammaTarget", value=0.01, min=0, max=1)
# fit model and find predicted values
result = minimize(residual, params, args=(t, data), method="leastsq")
final = data + result.residual.reshape(data.shape)
LifetimeTarget = 1.0 / (result.params["gammaTarget"].value)
# Grafico
fig, axes = plt.subplots(1, 1, figsize=(10, 6))
axes.plot(t, data[:, 0], "g*", label=r"Target")
axes.plot(t, final[:, 0], "k-", linewidth=2, label=r"Fit Target")
axes.set_ylim(0, max(ListGreen))
axes.set_xlim(0, max(ListTime))
axes.legend(loc=0, fontsize=12)
axes.set_ylabel("Number of Rydberg atoms", fontsize=12)
axes.set_xlabel(r"Time, $\mu s$", fontsize=12)
axes.grid()
plt.legend
plt.show()
# display fitted statistics
print("\n")
report_fit(result)
print("\n")
print("Lifetime Target: %.6f us" % (LifetimeTarget))
if plotting == 2:
# Make the plot of the three curves
fig, axes = plt.subplots(1, 1, figsize=(10, 6))
axes.plot(ListTime, ListRed, "r.", label=r"Ensemble")
axes.plot(ListTime, ListBlue, "b.", label=r"Other")
axes.plot(ListTime, ListGreen, "g.", label=r"Target")
axes.set_ylim(0, 1)
axes.set_xlim(0, ListTime[-1])
axes.legend(loc=0, fontsize=12)
axes.set_ylabel("Number of Rydberg atoms", fontsize=12)
axes.set_xlabel(r"Time [$\mu s$]", fontsize=12)
axes.grid()
plt.legend
plt.show()
if plotting == 3:
def f(xs, t, ps):
"""Lotka-Volterra predator-prey model."""
try:
gammaEnsemble = ps["gammaEnsemble"].value
gammaTarget = ps["gammaTarget"].value
except Exception:
gammaEnsemble, gammaTarget = ps
x, y = xs
return [-gammaEnsemble * x, -gammaTarget * y]
def g(t, x0, ps): # noqa: F821, F811
"""
Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
"""
x = odeint(f, x0, t, args=(ps,))
return x
def residual(ps, ts, data):
x0 = ps["x0"].value, ps["y0"].value
model = g(ts, x0, ps)
return (model - data).ravel()
t = np.array(ListTime)
x0 = np.array([0, 0]) # noqa: F841
dataAll = np.zeros(shape=(len(t), 3))
dataAll[:, 0] = np.array(ListRed)
dataAll[:, 1] = np.array(ListBlue)
dataAll[:, 2] = np.array(ListGreen)
data = np.zeros(shape=(len(t), 2))
data[:, 0] = dataAll[:, 0]
data[:, 1] = dataAll[:, 2]
# set parameters incluing bounds
params = Parameters()
params.add("x0", value=max(ListRed), vary=False)
params.add("y0", value=max(ListGreen), vary=False)
params.add("gammaEnsemble", value=0.005, min=0.0, max=1.0)
params.add("gammaTarget", value=0.01, min=0.0, max=1.0)
# fit model and find predicted values
result = minimize(residual, params, args=(t, data), method="leastsq")
final = data + result.residual.reshape(data.shape)
LifetimeEnsemble = 1.0 / (result.params["gammaEnsemble"].value)
LifetimeTarget = 1.0 / (result.params["gammaTarget"].value)
# Grafico
fig, axes = plt.subplots(1, 1, figsize=(10, 6))
axes.plot(t, dataAll[:, 0], "r*", label=r"Ensemble")
axes.plot(t, dataAll[:, 1], "b*", label=r"Support")
axes.plot(t, dataAll[:, 2], "g*", label=r"Target")
axes.plot(t, final[:, 0], "k-", linewidth=2, label=r"Fit Ensemble")
axes.plot(t, final[:, 1], "k-", linewidth=2, label=r"Fit Target")
axes.set_ylim(0, max(ListRed))
axes.set_xlim(0, max(ListTime))
axes.legend(loc=0, fontsize=12)
axes.set_ylabel("Number of Rydberg atoms", fontsize=12)
axes.set_xlabel(r"Time, $\mu s$", fontsize=12)
axes.grid()
plt.legend
plt.show()
# display fitted statistics
print("\n")
report_fit(result)
print("\n")
print(
"Lifetime Ensemble: %.6f us \nLifetime Target: %.6f us"
% (LifetimeEnsemble, LifetimeTarget)
)
if plotting == 4:
def f(xs, t, ps):
"""Lotka-Volterra predator-prey model."""
try:
gammaTargetSpont = ps["gammaTargetSpont"].value
gammaTargetBBR = ps["gammaTargetBBR"].value
gammaSupportSpont = ps["gammaSupportSpont"].value
gammaSupportBBR = ps["gammaSupportBBR"].value
except Exception:
(
gammaTargetSpont,
gammaTargetBBR,
gammaSupportSpont,
gammaSupportBBR,
) = ps
x, y, z = xs
return [
+gammaTargetSpont * z + gammaSupportSpont * y,
-gammaSupportSpont * y
- gammaSupportBBR * y
+ gammaTargetBBR * z,
-gammaTargetSpont * z
- gammaTargetBBR * z
+ gammaSupportBBR * y,
]
def g(t, x0, ps): # noqa: F821, F811
"""
Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
"""
x = odeint(f, x0, t, args=(ps,))
return x
def residual(ps, ts, data):
x0 = ps["x0"].value, ps["y0"].value, ps["z0"].value
model = g(ts, x0, ps)
return (model - data).ravel()
ListRedAus = np.zeros(shape=(len(ListRed)))
for i in range(0, len(ListRed)):
ListRedAus[i] = max(ListRed) - ListRed[i]
t = np.array(ListTime)
data = np.zeros(shape=(len(t), 3))
data[:, 0] = np.array(ListRedAus)
data[:, 1] = np.array(ListBlue)
data[:, 2] = np.array(ListGreen)
# set parameters incluing bounds
params = Parameters()
params.add("x0", value=0, vary=False)
params.add("y0", value=0, vary=False)
params.add("z0", value=max(ListGreen), vary=False)
params.add("gammaTargetSpont", value=0.02, min=0.0, max=1.0)
params.add("gammaTargetBBR", value=0.02, min=0.0, max=1.0)
params.add("gammaSupportSpont", value=0.02, min=0.0, max=1.0)
params.add("gammaSupportBBR", value=0.001, min=0.0, max=1.0)
# fit model and find predicted values
result = minimize(residual, params, args=(t, data), method="leastsq")
final = data + result.residual.reshape(data.shape)
# Grafico
fig, axes = plt.subplots(1, 1, figsize=(10, 6))
axes.plot(t, data[:, 0], "m*", label=r"Ground")
axes.plot(t, ListRed, "r*", label=r"Ensemble")
axes.plot(t, data[:, 1], "b*", label=r"Support")
axes.plot(t, data[:, 2], "g*", label=r"Target")
axes.plot(t, final[:, 0], "k-", linewidth=2, label=r"Fit Ground")
axes.plot(t, final[:, 1], "k-", linewidth=2, label=r"Fit Support")
axes.plot(t, final[:, 2], "k-", linewidth=2, label=r"Fit Target")
axes.set_ylim(0, max(ListRed))
axes.set_xlim(0, max(ListTime))
axes.legend(loc=0, fontsize=12)
axes.set_ylabel("Number of Rydberg atoms", fontsize=12)
axes.set_xlabel(r"Time, $\mu s$", fontsize=12)
axes.grid()
plt.legend()
plt.show()
print("\n")
# display fitted statistics
report_fit(result)
# It returns the time elapsed
print("\nIt took", time.time() - start, "seconds.")
return