# Imports and Simulation Parameters

```python
import numpy as np
import math
import cmath
import scipy
import scipy.integrate
import sys

import matplotlib
import matplotlib.pyplot as plt
%matplotlib inline

hbar = 1.0 / (2.0 * np.pi)

ZERO_TOLERANCE = 10**-6
```

    /Users/joseph/anaconda/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
      warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

```python
MAX_VIBRATIONAL_STATES = 1500

STARTING_GROUND_STATES = 9
STARTING_EXCITED_STATES = 9

time_scale_set = 100  #will divide the highest energy to give us the time step
low_frequency_cycles = 20.0 #will multiply the frequency of the lowest frequency mode to get the max time
```

```python
#See if a factorial_Storage dictionary exists already and if not, create one
try:
    a = factorial_storage
except:
    factorial_storage = {}
```

# Defining Parameters of the System

```python
energy_g = 0
energy_gamma = .1

energy_e = 5
energy_epsilon = .31

Huang_Rhys_Parameter = .80
S = Huang_Rhys_Parameter

#bookkeeping
overlap_storage = {}
electronic_energy_gap = energy_e + .5*energy_epsilon - (energy_g  + .5 * energy_gamma)
min_energy = energy_g + energy_gamma * .5

```

```python
mu_0 = 1.0
```

If we set the central frequency of a pulse at the 0->0 transition, and we decide we care about the ratio of the 0->1 transition to the 0->0 transition and set that to be $\tau$ then the desired pulse width will be
\begin{align}
    \sigma &= \sqrt{-\frac{2 \ln (\tau)}{\omega_{\epsilon}^2}}
\end{align}

```python
def blank_wavefunction(number_ground_states, number_excited_states):
    return np.zeros((number_ground_states + number_excited_states))

def perturbing_function(time):
#     stdev = 30000.0 * dt #very specific to 0->0 transition!
    stdev = 3000.0 * dt #clearly has a small amount of amplitude on 0->1 transition
    center = 6 * stdev
    return np.cos(electronic_energy_gap*(time - center) / hbar)*np.exp( - (time - center)**2 / (2 * stdev**2)) / stdev

def time_function_handle_from_tau(tau_proportion):
    stdev = np.sqrt( -2.0 * np.log(tau_proportion) / (energy_epsilon/hbar)**2)
    center = 6 * stdev
    return center, stdev, lambda t: np.cos(electronic_energy_gap*(t - center) / hbar)*np.exp( - (t - center)**2 / (2 * stdev**2)) / stdev

def perturbing_function_define_tau(time, tau_proportion):
    center, stdev, f = time_function_handle_from_tau(tau_proportion)
    return f(time)
```

```python

```

```python

```

# Defining Useful functions

$ O_{m}^{n} = \left(-1\right)^{n} \sqrt{\frac{e^{-S}S^{m+n}}{m!n!}} \sum_{j=0}^{\min \left( m,n \right)} \frac{m!n!}{j!(m-j)!(n-j)!}(-1)^j S^{-j} $

```python
def factorial(i):
    if i in factorial_storage:
        return factorial_storage[i]
    if i <= 1:
        return 1.0
    else:
        out =  factorial(i - 1) * i
        factorial_storage[i] = out
        return out

def ndarray_factorial(i_array):
    return np.array([factorial(i) for i in i_array])
```

```python
def overlap_function(ground_quantum_number, excited_quantum_number):
    m = ground_quantum_number
    n = excited_quantum_number
    if (m,n) in overlap_storage:
        return overlap_storage[(m,n)]
    output = (-1)**n
    output *= math.sqrt(math.exp(-S) * S**(m + n) /(factorial(m) * factorial(n)) )
    j_indeces = np.array(range(0, min(m,n) + 1))
    j_summation = factorial(m) * factorial(n) * np.power(-1.0, j_indeces) * np.power(S, -j_indeces)
    j_summation = j_summation / (ndarray_factorial(j_indeces) * ndarray_factorial( m - j_indeces) * ndarray_factorial(n - j_indeces) )
    output *= np.sum(j_summation)
    overlap_storage[(m,n)] = output
    return output
```

# Solving the Differential Equation

\begin{align*}
	\left(\frac{d G_a(t)}{dt} + \frac{i}{\hbar}\Omega_{(a)}\right) &=-E(t)\frac{i}{\hbar}  \sum_{b} E_b(t) \mu_{a}^{b}\\
	\left(\frac{d E_b(t)}{dt} + \frac{i}{\hbar} \Omega^{(b)} \right) &=-E(t)\frac{i}{\hbar} \sum_{a} G_a(t) \mu_{a}^{b}
\end{align*}
Or in a more compact form:
\begin{align*}
	\frac{d}{dt}\begin{bmatrix}
		G_a(t) \\
		E_b(t)
	\end{bmatrix}
	= -\frac{i}{\hbar}
	\begin{bmatrix}
		\Omega_{(a)} & E(t) \mu_{a}^{b} \\
		E(t) \mu_{a}^{b} & \Omega^{b}
	\end{bmatrix}
	\cdot
	\begin{bmatrix}
		G_a(t) \\
		E_b(t)
	\end{bmatrix}
\end{align*}

```python
def ode_diagonal_matrix(number_ground_states, number_excited_states):
    #Define the Matrix on the RHS of the above equation
    ODE_DIAGONAL_MATRIX = np.zeros((number_ground_states + number_excited_states, number_ground_states + number_excited_states), dtype=np.complex)
    #set the diagonals
    for ground_i in range(number_ground_states):
        ODE_DIAGONAL_MATRIX[ground_i, ground_i] = -1.0j * (energy_g + energy_gamma * (ground_i + .5)) / hbar

    for excited_i in range(number_excited_states):
        excited_index = excited_i + number_ground_states #the offset since the excited states comes later
        ODE_DIAGONAL_MATRIX[excited_index, excited_index] = -1.0j * (energy_e + energy_epsilon * (excited_i + .5)) / hbar

    return ODE_DIAGONAL_MATRIX

#now for the off-diagonals
def mu_matrix(c, number_ground_states, number_excited_states):
    MU_MATRIX = np.zeros((number_ground_states, number_excited_states), dtype = np.complex)
    for ground_a in range(number_ground_states):
        for excited_b in range(number_excited_states):
            new_mu_entry = overlap_function(ground_a, excited_b)
            if ground_a >0:
                new_mu_entry += c * math.sqrt(ground_a) * overlap_function(ground_a - 1, excited_b)

            new_mu_entry += c * math.sqrt(ground_a+1) * overlap_function(ground_a + 1, excited_b)

            MU_MATRIX[ground_a, excited_b] = new_mu_entry
    return MU_MATRIX

def ode_off_diagonal_matrix(c_value, number_ground_states, number_excited_states):
    output  = np.zeros((number_ground_states + number_excited_states, number_ground_states + number_excited_states), dtype=np.complex)
    MU_MATRIX = mu_matrix(c_value, number_ground_states, number_excited_states)
    output[0:number_ground_states, number_ground_states:] = -1.0j * mu_0 * MU_MATRIX  / hbar
    output[number_ground_states:, 0:number_ground_states] = -1.0j * mu_0 * MU_MATRIX.T  / hbar
    return output

def IR_transition_dipoles(number_ground_states, number_excited_states):
    "outputs matrices to calculate ground and excited state IR emission spectra.  Can be combined for total"
    output_g = np.zeros((number_ground_states, number_excited_states), dtype = np.complex)
    output_e = np.zeros((number_ground_states, number_excited_states), dtype = np.complex)

    for ground_a in range(number_ground_states):
        try:
            output_g[ground_a, ground_a + 1] = 1.0
            output_g[ground_a + 1, ground_a] = 1.0
        except:
            pass

        try:
            output_g[ground_a, ground_a - 1] = 1.0
            output_g[ground_a - 1, ground_a] = 1.0
        except:
            pass
    return output
```

\begin{align*}
	\mu(x) &= \mu_0 \left(1 + \lambda x \right) \\
	&= \mu_0 \left(1 + c\left(a + a^{\dagger} \right) \right) \\
	\mu_{a}^{b} &= \mu_0\left(O_{a}^{b}  + c\left(\sqrt{a}O_{a-1}^{b} + \sqrt{a+1}O_{a+1}^{b}\right) \right)
\end{align*}

```python

```

```python
class VibrationalStateOverFlowException(Exception):
    def __init__(self):
        pass
```

```python

```

```python
def propagate_amplitude_to_end_of_perturbation(c_value, ratio_01_00, starting_ground_states=STARTING_GROUND_STATES, starting_excited_states=STARTING_EXCITED_STATES):
    center_time, stdev, time_function = time_function_handle_from_tau(ratio_01_00)
    ending_time = center_time + 8.0 * stdev

    number_ground_states = starting_ground_states
    number_excited_states = starting_excited_states

    while number_excited_states + number_ground_states < MAX_VIBRATIONAL_STATES:

        #define time scales
        max_energy = energy_e + energy_epsilon * (.5 + number_excited_states)
        dt = 1.0 / (time_scale_set *  max_energy)

        ODE_DIAGONAL = ode_diagonal_matrix(number_ground_states, number_excited_states)
        ODE_OFF_DIAGONAL = ode_off_diagonal_matrix(c_value, number_ground_states, number_excited_states)

        def ODE_integrable_function(time, coefficient_vector):
            ODE_TOTAL_MATRIX = ODE_OFF_DIAGONAL * time_function(time) + ODE_DIAGONAL
            return np.dot(ODE_TOTAL_MATRIX, coefficient_vector)

        #define the starting wavefuntion
        initial_conditions = blank_wavefunction(number_ground_states, number_excited_states)
        initial_conditions[0] = 1

        #create ode solver
        current_time = 0.0
        ode_solver = scipy.integrate.complex_ode(ODE_integrable_function)
        ode_solver.set_initial_value(initial_conditions, current_time)

        #Run it
        results = []
        try:  #this block catches an overflow into the highest ground or excited vibrational state
            while current_time < ending_time:
#                 print(current_time, ZERO_TOLERANCE)
                #update time, perform solution
                current_time = ode_solver.t+dt
                new_result = ode_solver.integrate(current_time)
                results.append(new_result)
                #make sure solver was successful
                if not ode_solver.successful():
                    raise Exception("ODE Solve Failed!")
                #make sure that there hasn't been substantial leakage to the highest excited states
                re_start_calculation = False
                if abs(new_result[number_ground_states - 1])**2 >= ZERO_TOLERANCE:
                    number_ground_states +=1
                    print("Increasing Number of Ground vibrational states to %i " % number_ground_states)
#                     print("check this:", np.abs(new_result)**2, number_ground_states, abs(new_result[number_ground_states])**2)
#                     raise Exception()
                    re_start_calculation = True
                if abs(new_result[-1])**2 >= ZERO_TOLERANCE:
                    number_excited_states +=1
                    print("Increasing Number of excited vibrational states to %i " % number_excited_states)
                    re_start_calculation = True
                if re_start_calculation:
                    raise VibrationalStateOverFlowException()
        except VibrationalStateOverFlowException:
            #Move on and re-start the calculation
            continue

        #Finish calculating
        results = np.array(results)
        return results, number_ground_states, number_excited_states
    raise Exception("NEEDED TOO MANY VIBRATIONAL STATES!  RE-RUN WITH DIFFERENT PARAMETERS!")
```

```python
def get_average_quantum_number_time_series(c_value, ratio_01_00, starting_ground_states=STARTING_GROUND_STATES, starting_excited_states=STARTING_EXCITED_STATES):
    results, number_ground_states, number_excited_states = propagate_amplitude_to_end_of_perturbation(c_value, ratio_01_00, starting_ground_states, starting_excited_states)
    probabilities = np.abs(results)**2
    #calculate the average_vibrational_quantum_number series
    average_ground_quantum_number = probabilities[:,0:number_ground_states].dot(np.array(range(number_ground_states)) )
    average_excited_quantum_number = probabilities[:,number_ground_states:].dot(np.array(range(number_excited_states)))
    return average_ground_quantum_number, average_excited_quantum_number, results, number_ground_states, number_excited_states
```

```python
def IR_emission_spectrum_after_excitation(c_value, ratio_01_00, starting_ground_states=STARTING_GROUND_STATES, starting_excited_states=STARTING_EXCITED_STATES):
    center_time, stdev, time_function = time_function_handle_from_tau(ratio_01_00)
    ending_time = center_time + 8.0 * stdev
    ending_time +=  low_frequency_cycles * hbar/min_energy

    number_ground_states = starting_ground_states
    number_excited_states = starting_excited_states

    while number_excited_states + number_ground_states < MAX_VIBRATIONAL_STATES:

        #define time scales
        max_energy = energy_e + energy_epsilon * (.5 + number_excited_states)
        dt = 1.0 / (time_scale_set *  max_energy)

        ODE_DIAGONAL = ode_diagonal_matrix(number_ground_states, number_excited_states)
        ODE_OFF_DIAGONAL = ode_off_diagonal_matrix(c_value, number_ground_states, number_excited_states)

        def ODE_integrable_function(time, coefficient_vector):
            ODE_TOTAL_MATRIX = ODE_OFF_DIAGONAL * time_function(time) + ODE_DIAGONAL
            return np.dot(ODE_TOTAL_MATRIX, coefficient_vector)

        #define the starting wavefuntion
        initial_conditions = blank_wavefunction(number_ground_states, number_excited_states)
        initial_conditions[0] = 1

        #create ode solver
        current_time = 0.0
        ode_solver = scipy.integrate.complex_ode(ODE_integrable_function)
        ode_solver.set_initial_value(initial_conditions, current_time)

        #Run it
        results = []
        try:  #this block catches an overflow into the highest ground or excited vibrational state
            while current_time < ending_time:
#                 print(current_time, ZERO_TOLERANCE)
                #update time, perform solution
                current_time = ode_solver.t+dt
                new_result = ode_solver.integrate(current_time)
                results.append(new_result)
                #make sure solver was successful
                if not ode_solver.successful():
                    raise Exception("ODE Solve Failed!")
                #make sure that there hasn't been substantial leakage to the highest excited states
                re_start_calculation = False
                if abs(new_result[number_ground_states - 1])**2 >= ZERO_TOLERANCE:
                    number_ground_states +=1
                    print("Increasing Number of Ground vibrational states to %i " % number_ground_states)
#                     print("check this:", np.abs(new_result)**2, number_ground_states, abs(new_result[number_ground_states])**2)
#                     raise Exception()
                    re_start_calculation = True
                if abs(new_result[-1])**2 >= ZERO_TOLERANCE:
                    number_excited_states +=1
                    print("Increasing Number of excited vibrational states to %i " % number_excited_states)
                    re_start_calculation = True
                if re_start_calculation:
                    raise VibrationalStateOverFlowException()
        except VibrationalStateOverFlowException:
            #Move on and re-start the calculation
            continue

        #Finish calculating
        results = np.array(results)
        return results, number_ground_states, number_excited_states
    raise Exception("NEEDED TOO MANY VIBRATIONAL STATES!  RE-RUN WITH DIFFERENT PARAMETERS!"))

low_frequency_cycles
```

```python
c_values = np.logspace(-3, np.log10(.9), 31)
tau_values = np.logspace(-4, np.log10(.9), 30)

heating_results_ground = np.zeros((c_values.shape[0], tau_values.shape[0]))
heating_results_excited = np.zeros((c_values.shape[0], tau_values.shape[0]))

n_g = STARTING_GROUND_STATES
n_e = STARTING_EXCITED_STATES

dict_cTau_mapsTo_NgNe = {}
for i_c, c in enumerate(c_values):
    # as we increase in both tau and
    for i_tau, tau in enumerate(tau_values):
        print(c, tau)
        #make a good guess for the new needed number of simulated states
        try:
            last_c = c_values[i_c - 1]
        except:
            last_c = None
        try:
            last_tau = tau_values[i_tau - 1]
        except:
            last_tau = None

        if last_tau is not None and (c, last_tau) in dict_cTau_mapsTo_NgNe:
            n_g_candidate1, n_e_candidate1 = dict_cTau_mapsTo_NgNe[(c, last_tau)]
#             n_g_candidate1 += 1
#             n_e_candidate1 += 1
        else:
            n_g_candidate1, n_e_candidate1 = n_g, n_e
        if last_c is not None and (last_c, tau) in dict_cTau_mapsTo_NgNe:
            n_g_candidate2, n_e_candidate2 = dict_cTau_mapsTo_NgNe[(last_c, tau)]
#             n_g_candidate2 += 1
#             n_e_candidate2 += 1
        else:
            n_g_candidate2, n_e_candidate2 = n_g, n_e

        n_g = max([n_g_candidate1, n_g_candidate2])
        n_e = max([n_e_candidate1, n_e_candidate2])
        sys.stdout.flush()
        sys.stdout.write("Calculating c=%f, tau=%f at n_g = %i and n_e=%i..." %(c, tau, n_g, n_e))
        n_bar_g, n_bar_e, results, num_g, num_e = get_average_quantum_number_time_series(c,
                                                                                tau,
                                                                                starting_ground_states = n_g,
                                                                                starting_excited_states = n_e)
        dict_cTau_mapsTo_NgNe[(c, tau)] = (num_g, num_e)

        heating_results_ground[i_c, i_tau] = n_bar_g[-1]
        heating_results_excited[i_c, i_tau] = n_bar_e[-1]

        n_g = num_g
        n_e = num_e

```

    (0.001, 0.0001)
    Calculating c=0.001000, tau=0.000100 at n_g = 9 and n_e=9...(0.001, 0.00013688415834558614)
    Calculating c=0.001000, tau=0.000137 at n_g = 9 and n_e=9...(0.001, 0.00018737272805979496)
    Calculating c=0.001000, tau=0.000187 at n_g = 9 and n_e=9...(0.001, 0.00025648358177381397)
    Calculating c=0.001000, tau=0.000256 at n_g = 9 and n_e=9...(0.001, 0.00035108539220569837)
    Calculating c=0.001000, tau=0.000351 at n_g = 9 and n_e=9...(0.001, 0.00048058028419507029)
    Calculating c=0.001000, tau=0.000481 at n_g = 9 and n_e=9...(0.001, 0.00065783827719524712)
    Calculating c=0.001000, tau=0.000658 at n_g = 9 and n_e=9...(0.001, 0.00090047638901381789)
    Calculating c=0.001000, tau=0.000900 at n_g = 9 and n_e=9...(0.001, 0.0012326095262022907)
    Calculating c=0.001000, tau=0.001233 at n_g = 9 and n_e=9...(0.001, 0.0016872471756295224)
    Calculating c=0.001000, tau=0.001687 at n_g = 9 and n_e=9...(0.001, 0.0023095740955701451)
    Calculating c=0.001000, tau=0.002310 at n_g = 9 and n_e=9...(0.001, 0.0031614410620888727)
    Calculating c=0.001000, tau=0.003161 at n_g = 9 and n_e=9...(0.001, 0.0043275119894321127)
    Calculating c=0.001000, tau=0.004328 at n_g = 9 and n_e=9...(0.001, 0.0059236783640384772)
    Calculating c=0.001000, tau=0.005924 at n_g = 9 and n_e=9...(0.001, 0.0081085772717136469)
    Calculating c=0.001000, tau=0.008109 at n_g = 9 and n_e=9...(0.001, 0.011099357752186716)
    Calculating c=0.001000, tau=0.011099 at n_g = 9 and n_e=9...(0.001, 0.015193262440846353)
    Calculating c=0.001000, tau=0.015193 at n_g = 9 and n_e=9...(0.001, 0.020797169417388586)
    Calculating c=0.001000, tau=0.020797 at n_g = 9 and n_e=9...(0.001, 0.028468030316698006)
    Calculating c=0.001000, tau=0.028468 at n_g = 9 and n_e=9...(0.001, 0.038968223696578319)
    Calculating c=0.001000, tau=0.038968 at n_g = 9 and n_e=9...(0.001, 0.053341325029286485)
    Calculating c=0.001000, tau=0.053341 at n_g = 9 and n_e=9...(0.001, 0.073015823816722281)
    Calculating c=0.001000, tau=0.073016 at n_g = 9 and n_e=9...(0.001, 0.099947095890616205)
    Calculating c=0.001000, tau=0.099947 at n_g = 9 and n_e=9...(0.001, 0.13681174100072591)
    Calculating c=0.001000, tau=0.136812 at n_g = 9 and n_e=9...Increasing Number of Ground vibrational states to 10
    (0.001, 0.18727360018678682)
    Calculating c=0.001000, tau=0.187274 at n_g = 10 and n_e=9...(0.001, 0.25634789141916114)
    Calculating c=0.001000, tau=0.256348 at n_g = 10 and n_e=9...(0.001, 0.35089965360577574)
    Calculating c=0.001000, tau=0.350900 at n_g = 10 and n_e=9...(0.001, 0.48032603747584279)
    Calculating c=0.001000, tau=0.480326 at n_g = 10 and n_e=9...(0.001, 0.657490253713512)
    Calculating c=0.001000, tau=0.657490 at n_g = 10 and n_e=9...Increasing Number of Ground vibrational states to 11
    (0.001, 0.90000000000000002)
    Calculating c=0.001000, tau=0.900000 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0001)
    Calculating c=0.001255, tau=0.000100 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00013688415834558614)
    Calculating c=0.001255, tau=0.000137 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00018737272805979496)
    Calculating c=0.001255, tau=0.000187 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00025648358177381397)
    Calculating c=0.001255, tau=0.000256 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00035108539220569837)
    Calculating c=0.001255, tau=0.000351 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00048058028419507029)
    Calculating c=0.001255, tau=0.000481 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00065783827719524712)
    Calculating c=0.001255, tau=0.000658 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.00090047638901381789)
    Calculating c=0.001255, tau=0.000900 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0012326095262022907)
    Calculating c=0.001255, tau=0.001233 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0016872471756295224)
    Calculating c=0.001255, tau=0.001687 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0023095740955701451)
    Calculating c=0.001255, tau=0.002310 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0031614410620888727)
    Calculating c=0.001255, tau=0.003161 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0043275119894321127)
    Calculating c=0.001255, tau=0.004328 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0059236783640384772)
    Calculating c=0.001255, tau=0.005924 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.0081085772717136469)
    Calculating c=0.001255, tau=0.008109 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.011099357752186716)
    Calculating c=0.001255, tau=0.011099 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.015193262440846353)
    Calculating c=0.001255, tau=0.015193 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.020797169417388586)
    Calculating c=0.001255, tau=0.020797 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.028468030316698006)
    Calculating c=0.001255, tau=0.028468 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.038968223696578319)
    Calculating c=0.001255, tau=0.038968 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.053341325029286485)
    Calculating c=0.001255, tau=0.053341 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.073015823816722281)
    Calculating c=0.001255, tau=0.073016 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.099947095890616205)
    Calculating c=0.001255, tau=0.099947 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.13681174100072591)
    Calculating c=0.001255, tau=0.136812 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.18727360018678682)
    Calculating c=0.001255, tau=0.187274 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.25634789141916114)
    Calculating c=0.001255, tau=0.256348 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.35089965360577574)
    Calculating c=0.001255, tau=0.350900 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.48032603747584279)
    Calculating c=0.001255, tau=0.480326 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.657490253713512)
    Calculating c=0.001255, tau=0.657490 at n_g = 11 and n_e=9...(0.0012545117989873635, 0.90000000000000002)
    Calculating c=0.001255, tau=0.900000 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0001)
    Calculating c=0.001574, tau=0.000100 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00013688415834558614)
    Calculating c=0.001574, tau=0.000137 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00018737272805979496)
    Calculating c=0.001574, tau=0.000187 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00025648358177381397)
    Calculating c=0.001574, tau=0.000256 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00035108539220569837)
    Calculating c=0.001574, tau=0.000351 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00048058028419507029)
    Calculating c=0.001574, tau=0.000481 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00065783827719524712)
    Calculating c=0.001574, tau=0.000658 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.00090047638901381789)
    Calculating c=0.001574, tau=0.000900 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0012326095262022907)
    Calculating c=0.001574, tau=0.001233 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0016872471756295224)
    Calculating c=0.001574, tau=0.001687 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0023095740955701451)
    Calculating c=0.001574, tau=0.002310 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0031614410620888727)
    Calculating c=0.001574, tau=0.003161 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0043275119894321127)
    Calculating c=0.001574, tau=0.004328 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0059236783640384772)
    Calculating c=0.001574, tau=0.005924 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.0081085772717136469)
    Calculating c=0.001574, tau=0.008109 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.011099357752186716)
    Calculating c=0.001574, tau=0.011099 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.015193262440846353)
    Calculating c=0.001574, tau=0.015193 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.020797169417388586)
    Calculating c=0.001574, tau=0.020797 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.028468030316698006)
    Calculating c=0.001574, tau=0.028468 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.038968223696578319)
    Calculating c=0.001574, tau=0.038968 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.053341325029286485)
    Calculating c=0.001574, tau=0.053341 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.073015823816722281)
    Calculating c=0.001574, tau=0.073016 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.099947095890616205)
    Calculating c=0.001574, tau=0.099947 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.13681174100072591)
    Calculating c=0.001574, tau=0.136812 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.18727360018678682)
    Calculating c=0.001574, tau=0.187274 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.25634789141916114)
    Calculating c=0.001574, tau=0.256348 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.35089965360577574)
    Calculating c=0.001574, tau=0.350900 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.48032603747584279)
    Calculating c=0.001574, tau=0.480326 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.657490253713512)
    Calculating c=0.001574, tau=0.657490 at n_g = 11 and n_e=9...(0.0015737998537985109, 0.90000000000000002)
    Calculating c=0.001574, tau=0.900000 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0001)
    Calculating c=0.001974, tau=0.000100 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00013688415834558614)
    Calculating c=0.001974, tau=0.000137 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00018737272805979496)
    Calculating c=0.001974, tau=0.000187 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00025648358177381397)
    Calculating c=0.001974, tau=0.000256 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00035108539220569837)
    Calculating c=0.001974, tau=0.000351 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00048058028419507029)
    Calculating c=0.001974, tau=0.000481 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00065783827719524712)
    Calculating c=0.001974, tau=0.000658 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.00090047638901381789)
    Calculating c=0.001974, tau=0.000900 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0012326095262022907)
    Calculating c=0.001974, tau=0.001233 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0016872471756295224)
    Calculating c=0.001974, tau=0.001687 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0023095740955701451)
    Calculating c=0.001974, tau=0.002310 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0031614410620888727)
    Calculating c=0.001974, tau=0.003161 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0043275119894321127)
    Calculating c=0.001974, tau=0.004328 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0059236783640384772)
    Calculating c=0.001974, tau=0.005924 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.0081085772717136469)
    Calculating c=0.001974, tau=0.008109 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.011099357752186716)
    Calculating c=0.001974, tau=0.011099 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.015193262440846353)
    Calculating c=0.001974, tau=0.015193 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.020797169417388586)
    Calculating c=0.001974, tau=0.020797 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.028468030316698006)
    Calculating c=0.001974, tau=0.028468 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.038968223696578319)
    Calculating c=0.001974, tau=0.038968 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.053341325029286485)
    Calculating c=0.001974, tau=0.053341 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.073015823816722281)
    Calculating c=0.001974, tau=0.073016 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.099947095890616205)
    Calculating c=0.001974, tau=0.099947 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.13681174100072591)
    Calculating c=0.001974, tau=0.136812 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.18727360018678682)
    Calculating c=0.001974, tau=0.187274 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.25634789141916114)
    Calculating c=0.001974, tau=0.256348 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.35089965360577574)
    Calculating c=0.001974, tau=0.350900 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.48032603747584279)
    Calculating c=0.001974, tau=0.480326 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.657490253713512)
    Calculating c=0.001974, tau=0.657490 at n_g = 11 and n_e=9...(0.0019743504858348196, 0.90000000000000002)
    Calculating c=0.001974, tau=0.900000 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0001)
    Calculating c=0.002477, tau=0.000100 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00013688415834558614)
    Calculating c=0.002477, tau=0.000137 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00018737272805979496)
    Calculating c=0.002477, tau=0.000187 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00025648358177381397)
    Calculating c=0.002477, tau=0.000256 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00035108539220569837)
    Calculating c=0.002477, tau=0.000351 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00048058028419507029)
    Calculating c=0.002477, tau=0.000481 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00065783827719524712)
    Calculating c=0.002477, tau=0.000658 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.00090047638901381789)
    Calculating c=0.002477, tau=0.000900 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0012326095262022907)
    Calculating c=0.002477, tau=0.001233 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0016872471756295224)
    Calculating c=0.002477, tau=0.001687 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0023095740955701451)
    Calculating c=0.002477, tau=0.002310 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0031614410620888727)
    Calculating c=0.002477, tau=0.003161 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0043275119894321127)
    Calculating c=0.002477, tau=0.004328 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0059236783640384772)
    Calculating c=0.002477, tau=0.005924 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.0081085772717136469)
    Calculating c=0.002477, tau=0.008109 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.011099357752186716)
    Calculating c=0.002477, tau=0.011099 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.015193262440846353)
    Calculating c=0.002477, tau=0.015193 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.020797169417388586)
    Calculating c=0.002477, tau=0.020797 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.028468030316698006)
    Calculating c=0.002477, tau=0.028468 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.038968223696578319)
    Calculating c=0.002477, tau=0.038968 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.053341325029286485)
    Calculating c=0.002477, tau=0.053341 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.073015823816722281)
    Calculating c=0.002477, tau=0.073016 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.099947095890616205)
    Calculating c=0.002477, tau=0.099947 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.13681174100072591)
    Calculating c=0.002477, tau=0.136812 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.18727360018678682)
    Calculating c=0.002477, tau=0.187274 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.25634789141916114)
    Calculating c=0.002477, tau=0.256348 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.35089965360577574)
    Calculating c=0.002477, tau=0.350900 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.48032603747584279)
    Calculating c=0.002477, tau=0.480326 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.657490253713512)
    Calculating c=0.002477, tau=0.657490 at n_g = 11 and n_e=9...(0.0024768459798162144, 0.90000000000000002)
    Calculating c=0.002477, tau=0.900000 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0001)
    Calculating c=0.003107, tau=0.000100 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00013688415834558614)
    Calculating c=0.003107, tau=0.000137 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00018737272805979496)
    Calculating c=0.003107, tau=0.000187 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00025648358177381397)
    Calculating c=0.003107, tau=0.000256 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00035108539220569837)
    Calculating c=0.003107, tau=0.000351 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00048058028419507029)
    Calculating c=0.003107, tau=0.000481 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00065783827719524712)
    Calculating c=0.003107, tau=0.000658 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.00090047638901381789)
    Calculating c=0.003107, tau=0.000900 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0012326095262022907)
    Calculating c=0.003107, tau=0.001233 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0016872471756295224)
    Calculating c=0.003107, tau=0.001687 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0023095740955701451)
    Calculating c=0.003107, tau=0.002310 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0031614410620888727)
    Calculating c=0.003107, tau=0.003161 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0043275119894321127)
    Calculating c=0.003107, tau=0.004328 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0059236783640384772)
    Calculating c=0.003107, tau=0.005924 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.0081085772717136469)
    Calculating c=0.003107, tau=0.008109 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.011099357752186716)
    Calculating c=0.003107, tau=0.011099 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.015193262440846353)
    Calculating c=0.003107, tau=0.015193 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.020797169417388586)
    Calculating c=0.003107, tau=0.020797 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.028468030316698006)
    Calculating c=0.003107, tau=0.028468 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.038968223696578319)
    Calculating c=0.003107, tau=0.038968 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.053341325029286485)
    Calculating c=0.003107, tau=0.053341 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.073015823816722281)
    Calculating c=0.003107, tau=0.073016 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.099947095890616205)
    Calculating c=0.003107, tau=0.099947 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.13681174100072591)
    Calculating c=0.003107, tau=0.136812 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.18727360018678682)
    Calculating c=0.003107, tau=0.187274 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.25634789141916114)
    Calculating c=0.003107, tau=0.256348 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.35089965360577574)
    Calculating c=0.003107, tau=0.350900 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.48032603747584279)
    Calculating c=0.003107, tau=0.480326 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.657490253713512)
    Calculating c=0.003107, tau=0.657490 at n_g = 11 and n_e=9...(0.0031072325059538584, 0.90000000000000002)
    Calculating c=0.003107, tau=0.900000 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0001)
    Calculating c=0.003898, tau=0.000100 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00013688415834558614)
    Calculating c=0.003898, tau=0.000137 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00018737272805979496)
    Calculating c=0.003898, tau=0.000187 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00025648358177381397)
    Calculating c=0.003898, tau=0.000256 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00035108539220569837)
    Calculating c=0.003898, tau=0.000351 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00048058028419507029)
    Calculating c=0.003898, tau=0.000481 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00065783827719524712)
    Calculating c=0.003898, tau=0.000658 at n_g = 11 and n_e=9...(0.003898059840916188, 0.00090047638901381789)
    Calculating c=0.003898, tau=0.000900 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0012326095262022907)
    Calculating c=0.003898, tau=0.001233 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0016872471756295224)
    Calculating c=0.003898, tau=0.001687 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0023095740955701451)
    Calculating c=0.003898, tau=0.002310 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0031614410620888727)
    Calculating c=0.003898, tau=0.003161 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0043275119894321127)
    Calculating c=0.003898, tau=0.004328 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0059236783640384772)
    Calculating c=0.003898, tau=0.005924 at n_g = 11 and n_e=9...(0.003898059840916188, 0.0081085772717136469)
    Calculating c=0.003898, tau=0.008109 at n_g = 11 and n_e=9...(0.003898059840916188, 0.011099357752186716)
    Calculating c=0.003898, tau=0.011099 at n_g = 11 and n_e=9...(0.003898059840916188, 0.015193262440846353)
    Calculating c=0.003898, tau=0.015193 at n_g = 11 and n_e=9...(0.003898059840916188, 0.020797169417388586)
    Calculating c=0.003898, tau=0.020797 at n_g = 11 and n_e=9...(0.003898059840916188, 0.028468030316698006)
    Calculating c=0.003898, tau=0.028468 at n_g = 11 and n_e=9...(0.003898059840916188, 0.038968223696578319)
    Calculating c=0.003898, tau=0.038968 at n_g = 11 and n_e=9...(0.003898059840916188, 0.053341325029286485)
    Calculating c=0.003898, tau=0.053341 at n_g = 11 and n_e=9...(0.003898059840916188, 0.073015823816722281)
    Calculating c=0.003898, tau=0.073016 at n_g = 11 and n_e=9...(0.003898059840916188, 0.099947095890616205)
    Calculating c=0.003898, tau=0.099947 at n_g = 11 and n_e=9...(0.003898059840916188, 0.13681174100072591)
    Calculating c=0.003898, tau=0.136812 at n_g = 11 and n_e=9...(0.003898059840916188, 0.18727360018678682)
    Calculating c=0.003898, tau=0.187274 at n_g = 11 and n_e=9...(0.003898059840916188, 0.25634789141916114)
    Calculating c=0.003898, tau=0.256348 at n_g = 11 and n_e=9...(0.003898059840916188, 0.35089965360577574)
    Calculating c=0.003898, tau=0.350900 at n_g = 11 and n_e=9...(0.003898059840916188, 0.48032603747584279)
    Calculating c=0.003898, tau=0.480326 at n_g = 11 and n_e=9...(0.003898059840916188, 0.657490253713512)
    Calculating c=0.003898, tau=0.657490 at n_g = 11 and n_e=9...(0.003898059840916188, 0.90000000000000002)
    Calculating c=0.003898, tau=0.900000 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0001)
    Calculating c=0.004890, tau=0.000100 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00013688415834558614)
    Calculating c=0.004890, tau=0.000137 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00018737272805979496)
    Calculating c=0.004890, tau=0.000187 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00025648358177381397)
    Calculating c=0.004890, tau=0.000256 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00035108539220569837)
    Calculating c=0.004890, tau=0.000351 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00048058028419507029)
    Calculating c=0.004890, tau=0.000481 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00065783827719524712)
    Calculating c=0.004890, tau=0.000658 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.00090047638901381789)
    Calculating c=0.004890, tau=0.000900 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0012326095262022907)
    Calculating c=0.004890, tau=0.001233 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0016872471756295224)
    Calculating c=0.004890, tau=0.001687 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0023095740955701451)
    Calculating c=0.004890, tau=0.002310 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0031614410620888727)
    Calculating c=0.004890, tau=0.003161 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0043275119894321127)
    Calculating c=0.004890, tau=0.004328 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0059236783640384772)
    Calculating c=0.004890, tau=0.005924 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.0081085772717136469)
    Calculating c=0.004890, tau=0.008109 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.011099357752186716)
    Calculating c=0.004890, tau=0.011099 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.015193262440846353)
    Calculating c=0.004890, tau=0.015193 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.020797169417388586)
    Calculating c=0.004890, tau=0.020797 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.028468030316698006)
    Calculating c=0.004890, tau=0.028468 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.038968223696578319)
    Calculating c=0.004890, tau=0.038968 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.053341325029286485)
    Calculating c=0.004890, tau=0.053341 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.073015823816722281)
    Calculating c=0.004890, tau=0.073016 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.099947095890616205)
    Calculating c=0.004890, tau=0.099947 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.13681174100072591)
    Calculating c=0.004890, tau=0.136812 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.18727360018678682)
    Calculating c=0.004890, tau=0.187274 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.25634789141916114)
    Calculating c=0.004890, tau=0.256348 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.35089965360577574)
    Calculating c=0.004890, tau=0.350900 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.48032603747584279)
    Calculating c=0.004890, tau=0.480326 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.657490253713512)
    Calculating c=0.004890, tau=0.657490 at n_g = 11 and n_e=9...(0.0048901620635881634, 0.90000000000000002)
    Calculating c=0.004890, tau=0.900000 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0001)
    Calculating c=0.006135, tau=0.000100 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00013688415834558614)
    Calculating c=0.006135, tau=0.000137 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00018737272805979496)
    Calculating c=0.006135, tau=0.000187 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00025648358177381397)
    Calculating c=0.006135, tau=0.000256 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00035108539220569837)
    Calculating c=0.006135, tau=0.000351 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00048058028419507029)
    Calculating c=0.006135, tau=0.000481 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00065783827719524712)
    Calculating c=0.006135, tau=0.000658 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.00090047638901381789)
    Calculating c=0.006135, tau=0.000900 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0012326095262022907)
    Calculating c=0.006135, tau=0.001233 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0016872471756295224)
    Calculating c=0.006135, tau=0.001687 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0023095740955701451)
    Calculating c=0.006135, tau=0.002310 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0031614410620888727)
    Calculating c=0.006135, tau=0.003161 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0043275119894321127)
    Calculating c=0.006135, tau=0.004328 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0059236783640384772)
    Calculating c=0.006135, tau=0.005924 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.0081085772717136469)
    Calculating c=0.006135, tau=0.008109 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.011099357752186716)
    Calculating c=0.006135, tau=0.011099 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.015193262440846353)
    Calculating c=0.006135, tau=0.015193 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.020797169417388586)
    Calculating c=0.006135, tau=0.020797 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.028468030316698006)
    Calculating c=0.006135, tau=0.028468 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.038968223696578319)
    Calculating c=0.006135, tau=0.038968 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.053341325029286485)
    Calculating c=0.006135, tau=0.053341 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.073015823816722281)
    Calculating c=0.006135, tau=0.073016 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.099947095890616205)
    Calculating c=0.006135, tau=0.099947 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.13681174100072591)
    Calculating c=0.006135, tau=0.136812 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.18727360018678682)
    Calculating c=0.006135, tau=0.187274 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.25634789141916114)
    Calculating c=0.006135, tau=0.256348 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.35089965360577574)
    Calculating c=0.006135, tau=0.350900 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.48032603747584279)
    Calculating c=0.006135, tau=0.480326 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.657490253713512)
    Calculating c=0.006135, tau=0.657490 at n_g = 11 and n_e=9...(0.0061347660077317439, 0.90000000000000002)
    Calculating c=0.006135, tau=0.900000 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0001)
    Calculating c=0.007696, tau=0.000100 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00013688415834558614)
    Calculating c=0.007696, tau=0.000137 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00018737272805979496)
    Calculating c=0.007696, tau=0.000187 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00025648358177381397)
    Calculating c=0.007696, tau=0.000256 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00035108539220569837)
    Calculating c=0.007696, tau=0.000351 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00048058028419507029)
    Calculating c=0.007696, tau=0.000481 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00065783827719524712)
    Calculating c=0.007696, tau=0.000658 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.00090047638901381789)
    Calculating c=0.007696, tau=0.000900 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0012326095262022907)
    Calculating c=0.007696, tau=0.001233 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0016872471756295224)
    Calculating c=0.007696, tau=0.001687 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0023095740955701451)
    Calculating c=0.007696, tau=0.002310 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0031614410620888727)
    Calculating c=0.007696, tau=0.003161 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0043275119894321127)
    Calculating c=0.007696, tau=0.004328 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0059236783640384772)
    Calculating c=0.007696, tau=0.005924 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.0081085772717136469)
    Calculating c=0.007696, tau=0.008109 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.011099357752186716)
    Calculating c=0.007696, tau=0.011099 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.015193262440846353)
    Calculating c=0.007696, tau=0.015193 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.020797169417388586)
    Calculating c=0.007696, tau=0.020797 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.028468030316698006)
    Calculating c=0.007696, tau=0.028468 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.038968223696578319)
    Calculating c=0.007696, tau=0.038968 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.053341325029286485)
    Calculating c=0.007696, tau=0.053341 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.073015823816722281)
    Calculating c=0.007696, tau=0.073016 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.099947095890616205)
    Calculating c=0.007696, tau=0.099947 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.13681174100072591)
    Calculating c=0.007696, tau=0.136812 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.18727360018678682)
    Calculating c=0.007696, tau=0.187274 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.25634789141916114)
    Calculating c=0.007696, tau=0.256348 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.35089965360577574)
    Calculating c=0.007696, tau=0.350900 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.48032603747584279)
    Calculating c=0.007696, tau=0.480326 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.657490253713512)
    Calculating c=0.007696, tau=0.657490 at n_g = 11 and n_e=9...(0.0076961363407260759, 0.90000000000000002)
    Calculating c=0.007696, tau=0.900000 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0001)
    Calculating c=0.009655, tau=0.000100 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00013688415834558614)
    Calculating c=0.009655, tau=0.000137 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00018737272805979496)
    Calculating c=0.009655, tau=0.000187 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00025648358177381397)
    Calculating c=0.009655, tau=0.000256 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00035108539220569837)
    Calculating c=0.009655, tau=0.000351 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00048058028419507029)
    Calculating c=0.009655, tau=0.000481 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00065783827719524712)
    Calculating c=0.009655, tau=0.000658 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.00090047638901381789)
    Calculating c=0.009655, tau=0.000900 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0012326095262022907)
    Calculating c=0.009655, tau=0.001233 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0016872471756295224)
    Calculating c=0.009655, tau=0.001687 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0023095740955701451)
    Calculating c=0.009655, tau=0.002310 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0031614410620888727)
    Calculating c=0.009655, tau=0.003161 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0043275119894321127)
    Calculating c=0.009655, tau=0.004328 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0059236783640384772)
    Calculating c=0.009655, tau=0.005924 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.0081085772717136469)
    Calculating c=0.009655, tau=0.008109 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.011099357752186716)
    Calculating c=0.009655, tau=0.011099 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.015193262440846353)
    Calculating c=0.009655, tau=0.015193 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.020797169417388586)
    Calculating c=0.009655, tau=0.020797 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.028468030316698006)
    Calculating c=0.009655, tau=0.028468 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.038968223696578319)
    Calculating c=0.009655, tau=0.038968 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.053341325029286485)
    Calculating c=0.009655, tau=0.053341 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.073015823816722281)
    Calculating c=0.009655, tau=0.073016 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.099947095890616205)
    Calculating c=0.009655, tau=0.099947 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.13681174100072591)
    Calculating c=0.009655, tau=0.136812 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.18727360018678682)
    Calculating c=0.009655, tau=0.187274 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.25634789141916114)
    Calculating c=0.009655, tau=0.256348 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.35089965360577574)
    Calculating c=0.009655, tau=0.350900 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.48032603747584279)
    Calculating c=0.009655, tau=0.480326 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.657490253713512)
    Calculating c=0.009655, tau=0.657490 at n_g = 11 and n_e=9...(0.0096548938460562939, 0.90000000000000002)
    Calculating c=0.009655, tau=0.900000 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0001)
    Calculating c=0.012112, tau=0.000100 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00013688415834558614)
    Calculating c=0.012112, tau=0.000137 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00018737272805979496)
    Calculating c=0.012112, tau=0.000187 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00025648358177381397)
    Calculating c=0.012112, tau=0.000256 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00035108539220569837)
    Calculating c=0.012112, tau=0.000351 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00048058028419507029)
    Calculating c=0.012112, tau=0.000481 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00065783827719524712)
    Calculating c=0.012112, tau=0.000658 at n_g = 11 and n_e=9...(0.012112178247848111, 0.00090047638901381789)
    Calculating c=0.012112, tau=0.000900 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0012326095262022907)
    Calculating c=0.012112, tau=0.001233 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0016872471756295224)
    Calculating c=0.012112, tau=0.001687 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0023095740955701451)
    Calculating c=0.012112, tau=0.002310 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0031614410620888727)
    Calculating c=0.012112, tau=0.003161 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0043275119894321127)
    Calculating c=0.012112, tau=0.004328 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0059236783640384772)
    Calculating c=0.012112, tau=0.005924 at n_g = 11 and n_e=9...(0.012112178247848111, 0.0081085772717136469)
    Calculating c=0.012112, tau=0.008109 at n_g = 11 and n_e=9...(0.012112178247848111, 0.011099357752186716)
    Calculating c=0.012112, tau=0.011099 at n_g = 11 and n_e=9...(0.012112178247848111, 0.015193262440846353)
    Calculating c=0.012112, tau=0.015193 at n_g = 11 and n_e=9...(0.012112178247848111, 0.020797169417388586)
    Calculating c=0.012112, tau=0.020797 at n_g = 11 and n_e=9...(0.012112178247848111, 0.028468030316698006)
    Calculating c=0.012112, tau=0.028468 at n_g = 11 and n_e=9...(0.012112178247848111, 0.038968223696578319)
    Calculating c=0.012112, tau=0.038968 at n_g = 11 and n_e=9...(0.012112178247848111, 0.053341325029286485)
    Calculating c=0.012112, tau=0.053341 at n_g = 11 and n_e=9...(0.012112178247848111, 0.073015823816722281)
    Calculating c=0.012112, tau=0.073016 at n_g = 11 and n_e=9...(0.012112178247848111, 0.099947095890616205)
    Calculating c=0.012112, tau=0.099947 at n_g = 11 and n_e=9...(0.012112178247848111, 0.13681174100072591)
    Calculating c=0.012112, tau=0.136812 at n_g = 11 and n_e=9...(0.012112178247848111, 0.18727360018678682)
    Calculating c=0.012112, tau=0.187274 at n_g = 11 and n_e=9...(0.012112178247848111, 0.25634789141916114)
    Calculating c=0.012112, tau=0.256348 at n_g = 11 and n_e=9...(0.012112178247848111, 0.35089965360577574)
    Calculating c=0.012112, tau=0.350900 at n_g = 11 and n_e=9...(0.012112178247848111, 0.48032603747584279)
    Calculating c=0.012112, tau=0.480326 at n_g = 11 and n_e=9...(0.012112178247848111, 0.657490253713512)
    Calculating c=0.012112, tau=0.657490 at n_g = 11 and n_e=9...(0.012112178247848111, 0.90000000000000002)
    Calculating c=0.012112, tau=0.900000 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0001)
    Calculating c=0.015195, tau=0.000100 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00013688415834558614)
    Calculating c=0.015195, tau=0.000137 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00018737272805979496)
    Calculating c=0.015195, tau=0.000187 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00025648358177381397)
    Calculating c=0.015195, tau=0.000256 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00035108539220569837)
    Calculating c=0.015195, tau=0.000351 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00048058028419507029)
    Calculating c=0.015195, tau=0.000481 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00065783827719524712)
    Calculating c=0.015195, tau=0.000658 at n_g = 11 and n_e=9...(0.015194870523363546, 0.00090047638901381789)
    Calculating c=0.015195, tau=0.000900 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0012326095262022907)
    Calculating c=0.015195, tau=0.001233 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0016872471756295224)
    Calculating c=0.015195, tau=0.001687 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0023095740955701451)
    Calculating c=0.015195, tau=0.002310 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0031614410620888727)
    Calculating c=0.015195, tau=0.003161 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0043275119894321127)
    Calculating c=0.015195, tau=0.004328 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0059236783640384772)
    Calculating c=0.015195, tau=0.005924 at n_g = 11 and n_e=9...(0.015194870523363546, 0.0081085772717136469)
    Calculating c=0.015195, tau=0.008109 at n_g = 11 and n_e=9...(0.015194870523363546, 0.011099357752186716)
    Calculating c=0.015195, tau=0.011099 at n_g = 11 and n_e=9...(0.015194870523363546, 0.015193262440846353)
    Calculating c=0.015195, tau=0.015193 at n_g = 11 and n_e=9...(0.015194870523363546, 0.020797169417388586)
    Calculating c=0.015195, tau=0.020797 at n_g = 11 and n_e=9...(0.015194870523363546, 0.028468030316698006)
    Calculating c=0.015195, tau=0.028468 at n_g = 11 and n_e=9...(0.015194870523363546, 0.038968223696578319)
    Calculating c=0.015195, tau=0.038968 at n_g = 11 and n_e=9...(0.015194870523363546, 0.053341325029286485)
    Calculating c=0.015195, tau=0.053341 at n_g = 11 and n_e=9...(0.015194870523363546, 0.073015823816722281)
    Calculating c=0.015195, tau=0.073016 at n_g = 11 and n_e=9...(0.015194870523363546, 0.099947095890616205)
    Calculating c=0.015195, tau=0.099947 at n_g = 11 and n_e=9...(0.015194870523363546, 0.13681174100072591)
    Calculating c=0.015195, tau=0.136812 at n_g = 11 and n_e=9...(0.015194870523363546, 0.18727360018678682)
    Calculating c=0.015195, tau=0.187274 at n_g = 11 and n_e=9...(0.015194870523363546, 0.25634789141916114)
    Calculating c=0.015195, tau=0.256348 at n_g = 11 and n_e=9...(0.015194870523363546, 0.35089965360577574)
    Calculating c=0.015195, tau=0.350900 at n_g = 11 and n_e=9...(0.015194870523363546, 0.48032603747584279)
    Calculating c=0.015195, tau=0.480326 at n_g = 11 and n_e=9...(0.015194870523363546, 0.657490253713512)
    Calculating c=0.015195, tau=0.657490 at n_g = 11 and n_e=9...(0.015194870523363546, 0.90000000000000002)
    Calculating c=0.015195, tau=0.900000 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0001)
    Calculating c=0.019062, tau=0.000100 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00013688415834558614)
    Calculating c=0.019062, tau=0.000137 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00018737272805979496)
    Calculating c=0.019062, tau=0.000187 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00025648358177381397)
    Calculating c=0.019062, tau=0.000256 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00035108539220569837)
    Calculating c=0.019062, tau=0.000351 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00048058028419507029)
    Calculating c=0.019062, tau=0.000481 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00065783827719524712)
    Calculating c=0.019062, tau=0.000658 at n_g = 11 and n_e=9...(0.019062144355644864, 0.00090047638901381789)
    Calculating c=0.019062, tau=0.000900 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0012326095262022907)
    Calculating c=0.019062, tau=0.001233 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0016872471756295224)
    Calculating c=0.019062, tau=0.001687 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0023095740955701451)
    Calculating c=0.019062, tau=0.002310 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0031614410620888727)
    Calculating c=0.019062, tau=0.003161 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0043275119894321127)
    Calculating c=0.019062, tau=0.004328 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0059236783640384772)
    Calculating c=0.019062, tau=0.005924 at n_g = 11 and n_e=9...(0.019062144355644864, 0.0081085772717136469)
    Calculating c=0.019062, tau=0.008109 at n_g = 11 and n_e=9...(0.019062144355644864, 0.011099357752186716)
    Calculating c=0.019062, tau=0.011099 at n_g = 11 and n_e=9...(0.019062144355644864, 0.015193262440846353)
    Calculating c=0.019062, tau=0.015193 at n_g = 11 and n_e=9...(0.019062144355644864, 0.020797169417388586)
    Calculating c=0.019062, tau=0.020797 at n_g = 11 and n_e=9...(0.019062144355644864, 0.028468030316698006)
    Calculating c=0.019062, tau=0.028468 at n_g = 11 and n_e=9...(0.019062144355644864, 0.038968223696578319)
    Calculating c=0.019062, tau=0.038968 at n_g = 11 and n_e=9...(0.019062144355644864, 0.053341325029286485)
    Calculating c=0.019062, tau=0.053341 at n_g = 11 and n_e=9...(0.019062144355644864, 0.073015823816722281)
    Calculating c=0.019062, tau=0.073016 at n_g = 11 and n_e=9...(0.019062144355644864, 0.099947095890616205)
    Calculating c=0.019062, tau=0.099947 at n_g = 11 and n_e=9...(0.019062144355644864, 0.13681174100072591)
    Calculating c=0.019062, tau=0.136812 at n_g = 11 and n_e=9...(0.019062144355644864, 0.18727360018678682)
    Calculating c=0.019062, tau=0.187274 at n_g = 11 and n_e=9...(0.019062144355644864, 0.25634789141916114)
    Calculating c=0.019062, tau=0.256348 at n_g = 11 and n_e=9...(0.019062144355644864, 0.35089965360577574)
    Calculating c=0.019062, tau=0.350900 at n_g = 11 and n_e=9...(0.019062144355644864, 0.48032603747584279)
    Calculating c=0.019062, tau=0.480326 at n_g = 11 and n_e=9...(0.019062144355644864, 0.657490253713512)
    Calculating c=0.019062, tau=0.657490 at n_g = 11 and n_e=9...(0.019062144355644864, 0.90000000000000002)
    Calculating c=0.019062, tau=0.900000 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0001)
    Calculating c=0.023914, tau=0.000100 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00013688415834558614)
    Calculating c=0.023914, tau=0.000137 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00018737272805979496)
    Calculating c=0.023914, tau=0.000187 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00025648358177381397)
    Calculating c=0.023914, tau=0.000256 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00035108539220569837)
    Calculating c=0.023914, tau=0.000351 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00048058028419507029)
    Calculating c=0.023914, tau=0.000481 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00065783827719524712)
    Calculating c=0.023914, tau=0.000658 at n_g = 11 and n_e=9...(0.023913685008156854, 0.00090047638901381789)
    Calculating c=0.023914, tau=0.000900 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0012326095262022907)
    Calculating c=0.023914, tau=0.001233 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0016872471756295224)
    Calculating c=0.023914, tau=0.001687 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0023095740955701451)
    Calculating c=0.023914, tau=0.002310 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0031614410620888727)
    Calculating c=0.023914, tau=0.003161 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0043275119894321127)
    Calculating c=0.023914, tau=0.004328 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0059236783640384772)
    Calculating c=0.023914, tau=0.005924 at n_g = 11 and n_e=9...(0.023913685008156854, 0.0081085772717136469)
    Calculating c=0.023914, tau=0.008109 at n_g = 11 and n_e=9...(0.023913685008156854, 0.011099357752186716)
    Calculating c=0.023914, tau=0.011099 at n_g = 11 and n_e=9...(0.023913685008156854, 0.015193262440846353)
    Calculating c=0.023914, tau=0.015193 at n_g = 11 and n_e=9...(0.023913685008156854, 0.020797169417388586)
    Calculating c=0.023914, tau=0.020797 at n_g = 11 and n_e=9...(0.023913685008156854, 0.028468030316698006)
    Calculating c=0.023914, tau=0.028468 at n_g = 11 and n_e=9...(0.023913685008156854, 0.038968223696578319)
    Calculating c=0.023914, tau=0.038968 at n_g = 11 and n_e=9...(0.023913685008156854, 0.053341325029286485)
    Calculating c=0.023914, tau=0.053341 at n_g = 11 and n_e=9...(0.023913685008156854, 0.073015823816722281)
    Calculating c=0.023914, tau=0.073016 at n_g = 11 and n_e=9...(0.023913685008156854, 0.099947095890616205)
    Calculating c=0.023914, tau=0.099947 at n_g = 11 and n_e=9...(0.023913685008156854, 0.13681174100072591)
    Calculating c=0.023914, tau=0.136812 at n_g = 11 and n_e=9...(0.023913685008156854, 0.18727360018678682)
    Calculating c=0.023914, tau=0.187274 at n_g = 11 and n_e=9...(0.023913685008156854, 0.25634789141916114)
    Calculating c=0.023914, tau=0.256348 at n_g = 11 and n_e=9...(0.023913685008156854, 0.35089965360577574)
    Calculating c=0.023914, tau=0.350900 at n_g = 11 and n_e=9...(0.023913685008156854, 0.48032603747584279)
    Calculating c=0.023914, tau=0.480326 at n_g = 11 and n_e=9...(0.023913685008156854, 0.657490253713512)
    Calculating c=0.023914, tau=0.657490 at n_g = 11 and n_e=9...(0.023913685008156854, 0.90000000000000002)
    Calculating c=0.023914, tau=0.900000 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0001)
    Calculating c=0.030000, tau=0.000100 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00013688415834558614)
    Calculating c=0.030000, tau=0.000137 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00018737272805979496)
    Calculating c=0.030000, tau=0.000187 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00025648358177381397)
    Calculating c=0.030000, tau=0.000256 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00035108539220569837)
    Calculating c=0.030000, tau=0.000351 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00048058028419507029)
    Calculating c=0.030000, tau=0.000481 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00065783827719524712)
    Calculating c=0.030000, tau=0.000658 at n_g = 11 and n_e=9...(0.029999999999999995, 0.00090047638901381789)
    Calculating c=0.030000, tau=0.000900 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0012326095262022907)
    Calculating c=0.030000, tau=0.001233 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0016872471756295224)
    Calculating c=0.030000, tau=0.001687 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0023095740955701451)
    Calculating c=0.030000, tau=0.002310 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0031614410620888727)
    Calculating c=0.030000, tau=0.003161 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0043275119894321127)
    Calculating c=0.030000, tau=0.004328 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0059236783640384772)
    Calculating c=0.030000, tau=0.005924 at n_g = 11 and n_e=9...(0.029999999999999995, 0.0081085772717136469)
    Calculating c=0.030000, tau=0.008109 at n_g = 11 and n_e=9...(0.029999999999999995, 0.011099357752186716)
    Calculating c=0.030000, tau=0.011099 at n_g = 11 and n_e=9...(0.029999999999999995, 0.015193262440846353)
    Calculating c=0.030000, tau=0.015193 at n_g = 11 and n_e=9...(0.029999999999999995, 0.020797169417388586)
    Calculating c=0.030000, tau=0.020797 at n_g = 11 and n_e=9...(0.029999999999999995, 0.028468030316698006)
    Calculating c=0.030000, tau=0.028468 at n_g = 11 and n_e=9...(0.029999999999999995, 0.038968223696578319)
    Calculating c=0.030000, tau=0.038968 at n_g = 11 and n_e=9...(0.029999999999999995, 0.053341325029286485)
    Calculating c=0.030000, tau=0.053341 at n_g = 11 and n_e=9...(0.029999999999999995, 0.073015823816722281)
    Calculating c=0.030000, tau=0.073016 at n_g = 11 and n_e=9...(0.029999999999999995, 0.099947095890616205)
    Calculating c=0.030000, tau=0.099947 at n_g = 11 and n_e=9...(0.029999999999999995, 0.13681174100072591)
    Calculating c=0.030000, tau=0.136812 at n_g = 11 and n_e=9...(0.029999999999999995, 0.18727360018678682)
    Calculating c=0.030000, tau=0.187274 at n_g = 11 and n_e=9...(0.029999999999999995, 0.25634789141916114)
    Calculating c=0.030000, tau=0.256348 at n_g = 11 and n_e=9...(0.029999999999999995, 0.35089965360577574)
    Calculating c=0.030000, tau=0.350900 at n_g = 11 and n_e=9...(0.029999999999999995, 0.48032603747584279)
    Calculating c=0.030000, tau=0.480326 at n_g = 11 and n_e=9...(0.029999999999999995, 0.657490253713512)
    Calculating c=0.030000, tau=0.657490 at n_g = 11 and n_e=9...(0.029999999999999995, 0.90000000000000002)
    Calculating c=0.030000, tau=0.900000 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0001)
    Calculating c=0.037635, tau=0.000100 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00013688415834558614)
    Calculating c=0.037635, tau=0.000137 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00018737272805979496)
    Calculating c=0.037635, tau=0.000187 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00025648358177381397)
    Calculating c=0.037635, tau=0.000256 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00035108539220569837)
    Calculating c=0.037635, tau=0.000351 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00048058028419507029)
    Calculating c=0.037635, tau=0.000481 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00065783827719524712)
    Calculating c=0.037635, tau=0.000658 at n_g = 11 and n_e=9...(0.037635353969620901, 0.00090047638901381789)
    Calculating c=0.037635, tau=0.000900 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0012326095262022907)
    Calculating c=0.037635, tau=0.001233 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0016872471756295224)
    Calculating c=0.037635, tau=0.001687 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0023095740955701451)
    Calculating c=0.037635, tau=0.002310 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0031614410620888727)
    Calculating c=0.037635, tau=0.003161 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0043275119894321127)
    Calculating c=0.037635, tau=0.004328 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0059236783640384772)
    Calculating c=0.037635, tau=0.005924 at n_g = 11 and n_e=9...(0.037635353969620901, 0.0081085772717136469)
    Calculating c=0.037635, tau=0.008109 at n_g = 11 and n_e=9...(0.037635353969620901, 0.011099357752186716)
    Calculating c=0.037635, tau=0.011099 at n_g = 11 and n_e=9...(0.037635353969620901, 0.015193262440846353)
    Calculating c=0.037635, tau=0.015193 at n_g = 11 and n_e=9...(0.037635353969620901, 0.020797169417388586)
    Calculating c=0.037635, tau=0.020797 at n_g = 11 and n_e=9...(0.037635353969620901, 0.028468030316698006)
    Calculating c=0.037635, tau=0.028468 at n_g = 11 and n_e=9...(0.037635353969620901, 0.038968223696578319)
    Calculating c=0.037635, tau=0.038968 at n_g = 11 and n_e=9...(0.037635353969620901, 0.053341325029286485)
    Calculating c=0.037635, tau=0.053341 at n_g = 11 and n_e=9...(0.037635353969620901, 0.073015823816722281)
    Calculating c=0.037635, tau=0.073016 at n_g = 11 and n_e=9...(0.037635353969620901, 0.099947095890616205)
    Calculating c=0.037635, tau=0.099947 at n_g = 11 and n_e=9...(0.037635353969620901, 0.13681174100072591)
    Calculating c=0.037635, tau=0.136812 at n_g = 11 and n_e=9...(0.037635353969620901, 0.18727360018678682)
    Calculating c=0.037635, tau=0.187274 at n_g = 11 and n_e=9...(0.037635353969620901, 0.25634789141916114)
    Calculating c=0.037635, tau=0.256348 at n_g = 11 and n_e=9...(0.037635353969620901, 0.35089965360577574)
    Calculating c=0.037635, tau=0.350900 at n_g = 11 and n_e=9...(0.037635353969620901, 0.48032603747584279)
    Calculating c=0.037635, tau=0.480326 at n_g = 11 and n_e=9...(0.037635353969620901, 0.657490253713512)
    Calculating c=0.037635, tau=0.657490 at n_g = 11 and n_e=9...(0.037635353969620901, 0.90000000000000002)
    Calculating c=0.037635, tau=0.900000 at n_g = 11 and n_e=9...(0.047213995613955326, 0.0001)
    Calculating c=0.047214, tau=0.000100 at n_g = 11 and n_e=9...(0.047213995613955326, 0.00013688415834558614)
    Calculating c=0.047214, tau=0.000137 at n_g = 11 and n_e=9...(0.047213995613955326, 0.00018737272805979496)
    Calculating c=0.047214, tau=0.000187 at n_g = 11 and n_e=9...

```python
n_g, n_e
```

```python

```

```python

```

```python
plt.figure()
plt.title("Ground State Heating")
plt.contourf(np.log10(tau_values),np.log10(c_values),  heating_results_ground, 100)
plt.colorbar()
plt.ylabel(r"$c$ log scale")
plt.xlabel(r"$\tau$ log scale")
plt.savefig("ground_state_heating.png")

plt.figure()
plt.title("Excited State Heating")
plt.contourf(np.log10(tau_values),np.log10(c_values),  heating_results_excited, 100)
plt.colorbar()
plt.ylabel(r"$c$ log scale")
plt.xlabel(r"$\tau$ log scale")
plt.savefig("excited_state_heating.png")
```

```python
n_bar_g, n_bar_e, results, number_ground_states = get_average_quantum_number_time_series(.3, .8)
```

```python
plt.figure()
plt.plot(n_bar_g, label="g")
plt.plot(n_bar_e, label="e")
plt.legend(loc=0)
plt.figure("ground")
plt.title("Ground State Populations")
plt.figure("excited")
plt.title("Excited State Populations")
# plt.semilogy(np.abs(time_function) /np.max(np.abs(time_function)))
for index in range(results.shape[1]):
    if index < number_ground_states:
        plt.figure("ground")
        plt.semilogy(abs(results[:, index])**2, label=index)
    else:
        plt.figure("excited")
        plt.semilogy(abs(results[:, index])**2, label=index - number_ground_states)

plt.figure("ground")
plt.legend(loc=0)

plt.figure("excited")
plt.legend(loc=0)
```

```python

```

```python

```

```python

```

```python

```

```python

```