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master-thesis/python/richard_hops/energy_flow_thermal.org
Valentin Boettcher 3461403cb1
detailed run
2021-11-02 20:13:03 +01:00

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#+PROPERTY: header-args :session finite_temp :kernel python :pandoc t :async yes
* Setup
** Jupyter
#+begin_src jupyter-python
%load_ext autoreload
%autoreload 2
%load_ext jupyter_spaces
#+end_src
#+RESULTS:
: The autoreload extension is already loaded. To reload it, use:
: %reload_ext autoreload
: The jupyter_spaces extension is already loaded. To reload it, use:
: %reload_ext jupyter_spaces
** Matplotlib
#+begin_src jupyter-python
import matplotlib
import matplotlib.pyplot as plt
#matplotlib.use("TkCairo", force=True)
%gui tk
%matplotlib inline
plt.style.use('ggplot')
#+end_src
#+RESULTS:
** Richard (old) HOPS
#+begin_src jupyter-python
import hierarchyLib
import hierarchyData
import numpy as np
from stocproc.stocproc import StocProc_FFT
import bcf
from dataclasses import dataclass, field
import scipy
import scipy.misc
import scipy.signal
import pickle
from scipy.special import gamma as gamma_func
from scipy.optimize import curve_fit
#+end_src
#+RESULTS:
** Auxiliary Definitions
#+begin_src jupyter-python
σ1 = np.matrix([[0,1],[1,0]])
σ2 = np.matrix([[0,-1j],[1j,0]])
σ3 = np.matrix([[1,0],[0,-1]])
#+end_src
#+RESULTS:
* Model Setup
Basic parameters.
#+begin_src jupyter-python
class params:
T = 2
t_max = 15
t_steps = int(t_max * 1/.05)
k_max = 10
N = 4000
seed = 100
dim = 2
H_s = σ3 + np.eye(dim)
L = σ2 #1 / 2 * (σ1 - 1j * σ2)
ψ_0 = np.array([0, 1])
s = 1
num_exp_t = 4
wc = 1
with open("good_fit_data_abs_brute_force", "rb") as f:
good_fit_data_abs = pickle.load(f)
alpha = 0.8
# _, g_tilde, w_tilde = good_fit_data_abs[(numExpFit, s)]
# g_tilde = np.array(g_tilde)
# w_tilde = np.array(w_tilde)
# g = 1 / np.pi * gamma_func(s + 1) * wc ** (s + 1) * np.asarray(g_tilde)
# w = wc * np.asarray(w_tilde)
bcf_scale = np.pi / 2 * alpha * wc ** (1 - s)
#+end_src
#+RESULTS:
** BCF and Thermal BCF
#+begin_src jupyter-python
@dataclass
class CauchyBCF:
δ: float
wc: float
def I(self, ω):
return np.sqrt(self.δ) / (self.δ + (ω - self.wc) ** 2 / self.δ)
def __call__(self, τ):
return np.sqrt(self.δ) * np.exp(-1j * self.wc * τ - np.abs(τ) * self.δ)
def __bfkey__(self):
return self.δ, self.wc
α = bcf.OBCF(s=params.s, eta=1, gamma=params.wc)
I = bcf.OSD(s=params.s, eta=1, gamma=params.wc)
#+end_src
#+RESULTS:
*** Fit
We now fit a sum of exponentials against the BCF.
#+begin_src jupyter-python
from lmfit import minimize, Parameters
def α_apprx(τ, g, w):
return np.sum(
g[np.newaxis, :] * np.exp(-w[np.newaxis, :] * (τ[:, np.newaxis])), axis=1
)
def _fit():
def residual(fit_params, x, data):
resid = 0
w = np.array([fit_params[f"w{i}"] for i in range(params.num_exp_t)]) + 1j * np.array(
[fit_params[f"wi{i}"] for i in range(params.num_exp_t)]
)
g = np.array([fit_params[f"g{i}"] for i in range(params.num_exp_t)]) + 1j * np.array(
[fit_params[f"gi{i}"] for i in range(params.num_exp_t)]
)
resid = data - α_apprx(x, g, w)
return resid.view(float)
fit_params = Parameters()
for i in range(params.num_exp_t):
fit_params.add(f"g{i}", value=.1)
fit_params.add(f"gi{i}", value=.1)
fit_params.add(f"w{i}", value=.1)
fit_params.add(f"wi{i}", value=.1)
ts = np.linspace(0, params.t_max, 1000)
out = minimize(residual, fit_params, args=(ts, α(ts)))
w = np.array([out.params[f"w{i}"] for i in range(params.num_exp_t)]) + 1j * np.array(
[out.params[f"wi{i}"] for i in range(params.num_exp_t)]
)
g = np.array([out.params[f"g{i}"] for i in range(params.num_exp_t)]) + 1j * np.array(
[out.params[f"gi{i}"] for i in range(params.num_exp_t)]
)
return w, g
w, g = _fit()
#+end_src
#+RESULTS:
*** Plot
Let's look a the result.
#+begin_src jupyter-python
class bcfplt:
t = np.linspace(0, params.t_max, 1000)
ω = np.linspace(params.wc - 10, params.wc + 10, 1000)
fig, axs = plt.subplots(2)
axs[0].plot(t, np.real(α(t)))
axs[0].plot(t, np.imag(α(t)))
axs[0].plot(t, np.real(α_apprx(t, g, w)))
axs[0].plot(t, np.imag(α_apprx(t, g, w)))
axs[1].plot(ω, I(ω).real)
axs[1].plot(ω, I(ω).imag)
#+end_src
#+RESULTS:
[[file:./.ob-jupyter/9f05a1fbf06920c271f0667db664ce2972415437.png]]
Seems ok for now.
** Hops setup
#+begin_src jupyter-python
HierachyParam = hierarchyData.HiP(
k_max=params.k_max,
# g_scale=None,
# sample_method='random',
seed=params.seed,
nonlinear=True,
normalized=False,
# terminator=False,
result_type=hierarchyData.RESULT_TYPE_ALL,
# accum_only=None,
# rand_skip=None
)
#+end_src
#+RESULTS:
Integration.
#+begin_src jupyter-python
IntegrationParam = hierarchyData.IntP(
t_max=params.t_max,
t_steps=params.t_steps,
# integrator_name='zvode',
# atol=1e-8,
# rtol=1e-8,
# order=5,
# nsteps=5000,
# method='bdf',
# t_steps_skip=1
)
#+end_src
#+RESULTS:
And now the system.
#+begin_src jupyter-python
SystemParam = hierarchyData.SysP(
H_sys=params.H_s,
L=params.L,
psi0=params.ψ_0, # excited qubit
g=np.array(g),
w=np.array(w),
H_dynamic=[],
bcf_scale=params.bcf_scale, # some coupling strength (scaling of the fit parameters 'g_i')
gw_hash=None, # this is used to load g,w from some database
len_gw=len(g),
)
#+end_src
#+RESULTS:
The quantum noise.
#+begin_src jupyter-python
Eta = StocProc_FFT(
I,
params.t_max,
α,
negative_frequencies=False,
seed=params.seed,
intgr_tol=1e-3,
intpl_tol=1e-3,
scale=params.bcf_scale,
)
#+end_src
#+RESULTS:
#+begin_example
stocproc.stocproc - INFO - non neg freq only
stocproc.method_ft - INFO - get_dt_for_accurate_interpolation, please wait ...
stocproc.method_ft - INFO - acc interp N 33 dt 9.38e-01 -> diff 2.83e-01
stocproc.method_ft - INFO - acc interp N 65 dt 4.69e-01 -> diff 8.53e-02
stocproc.method_ft - INFO - acc interp N 129 dt 2.34e-01 -> diff 1.76e-02
stocproc.method_ft - INFO - acc interp N 257 dt 1.17e-01 -> diff 3.92e-03
stocproc.method_ft - INFO - acc interp N 513 dt 5.86e-02 -> diff 9.52e-04
stocproc.method_ft - INFO - requires dt < 5.859e-02
stocproc.method_ft - INFO - get_N_a_b_for_accurate_fourier_integral, please wait ...
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 32 yields: interval [0.00e+00,6.47e+00] diff 9.83e-03
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [0.00e+00,9.12e+00] diff 8.12e-03
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [0.00e+00,6.47e+00] diff 1.11e-02
stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 32 yields: interval [0.00e+00,1.17e+01] diff 1.32e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [0.00e+00,9.12e+00] diff 1.22e-03
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [0.00e+00,6.47e+00] diff 1.14e-02
stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level
stocproc.method_ft - INFO - J_w_min:1.00e-05 N 32 yields: interval [0.00e+00,1.42e+01] diff 1.94e-02
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [0.00e+00,1.17e+01] diff 2.57e-03
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [0.00e+00,9.12e+00] diff 8.98e-04
stocproc.method_ft - INFO - return, cause tol of 0.001 was reached
stocproc.method_ft - INFO - requires dx < 7.123e-02
stocproc.stocproc - INFO - Fourier Integral Boundaries: [0.000e+00, 1.251e+02]
stocproc.stocproc - INFO - Number of Nodes : 2048
stocproc.stocproc - INFO - yields dx : 6.107e-02
stocproc.stocproc - INFO - yields dt : 5.023e-02
stocproc.stocproc - INFO - yields t_max : 1.028e+02
#+end_example
The sample trajectories are smooth.
#+begin_src jupyter-python
%%space plot
ts = np.linspace(0, params.t_max, 1000)
Eta.new_process()
plt.plot(ts, Eta(ts).real)
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f708dab9f40> |
[[file:./.ob-jupyter/10385a90753d7683d5740b88622cc4274de9c86a.png]]
:END:
And now the thermal noise.
#+begin_src jupyter-python
EtaTherm = StocProc_FFT(
spectral_density=bcf.OFTDens(s=params.s, eta=1, gamma=params.wc, beta=1 / params.T),
t_max=params.t_max,
alpha=bcf.OFTCorr(s=params.s, eta=1, gamma=params.wc, beta=1 / params.T),
intgr_tol=1e-3,
intpl_tol=1e-3,
seed=params.seed,
negative_frequencies=False,
scale=params.bcf_scale,
)
#+end_src
#+RESULTS:
#+begin_example
stocproc.method_ft - INFO - acc interp N 129 dt 2.34e-01 -> diff 3.20e-03
stocproc.method_ft - INFO - acc interp N 257 dt 1.17e-01 -> diff 7.67e-04
stocproc.method_ft - INFO - requires dt < 1.172e-01
stocproc.method_ft - INFO - get_N_a_b_for_accurate_fourier_integral, please wait ...
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 32 yields: interval [0.00e+00,4.10e+00] diff 2.00e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [0.00e+00,5.82e+00] diff 4.75e-02
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [0.00e+00,4.10e+00] diff 7.88e-03
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 32 yields: interval [0.00e+00,7.50e+00] diff 8.51e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [0.00e+00,5.82e+00] diff 1.01e-02
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [0.00e+00,4.10e+00] diff 7.56e-03
stocproc.method_ft - INFO - J_w_min:1.00e-05 N 32 yields: interval [0.00e+00,9.16e+00] diff 1.04e-01
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [0.00e+00,7.50e+00] diff 1.78e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [0.00e+00,5.82e+00] diff 2.38e-03
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 256 yields: interval [0.00e+00,4.10e+00] diff 7.48e-03
stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level
stocproc.method_ft - INFO - J_w_min:1.00e-06 N 32 yields: interval [0.00e+00,1.08e+01] diff 1.22e-01
stocproc.method_ft - INFO - J_w_min:1.00e-05 N 64 yields: interval [0.00e+00,9.16e+00] diff 2.81e-02
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 128 yields: interval [0.00e+00,7.50e+00] diff 4.17e-03
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 256 yields: interval [0.00e+00,5.82e+00] diff 7.86e-04
stocproc.method_ft - INFO - return, cause tol of 0.001 was reached
stocproc.method_ft - INFO - requires dx < 2.272e-02
stocproc.stocproc - INFO - Fourier Integral Boundaries: [0.000e+00, 7.064e+01]
stocproc.stocproc - INFO - Number of Nodes : 4096
stocproc.stocproc - INFO - yields dx : 1.725e-02
stocproc.stocproc - INFO - yields dt : 8.895e-02
stocproc.stocproc - INFO - yields t_max : 3.642e+02
stocproc.stocproc - INFO - non neg freq only
stocproc.method_ft - INFO - get_dt_for_accurate_interpolation, please wait ...
stocproc.method_ft - INFO - acc interp N 33 dt 9.38e-01 -> diff 6.53e-02
stocproc.method_ft - INFO - acc interp N 65 dt 4.69e-01 -> diff 1.47e-02
stocproc.method_ft - INFO - acc interp N 129 dt 2.34e-01 -> diff 3.20e-03
stocproc.method_ft - INFO - acc interp N 257 dt 1.17e-01 -> diff 7.67e-04
stocproc.method_ft - INFO - requires dt < 1.172e-01
stocproc.method_ft - INFO - get_N_a_b_for_accurate_fourier_integral, please wait ...
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 32 yields: interval [0.00e+00,4.10e+00] diff 2.00e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [0.00e+00,5.82e+00] diff 4.75e-02
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [0.00e+00,4.10e+00] diff 7.88e-03
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 32 yields: interval [0.00e+00,7.50e+00] diff 8.51e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [0.00e+00,5.82e+00] diff 1.01e-02
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [0.00e+00,4.10e+00] diff 7.56e-03
stocproc.method_ft - INFO - J_w_min:1.00e-05 N 32 yields: interval [0.00e+00,9.16e+00] diff 1.04e-01
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [0.00e+00,7.50e+00] diff 1.78e-02
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [0.00e+00,5.82e+00] diff 2.38e-03
stocproc.method_ft - INFO - J_w_min:1.00e-02 N 256 yields: interval [0.00e+00,4.10e+00] diff 7.48e-03
stocproc.method_ft - INFO - increasing N while shrinking the interval does lower the error -> try next level
stocproc.method_ft - INFO - J_w_min:1.00e-06 N 32 yields: interval [0.00e+00,1.08e+01] diff 1.22e-01
stocproc.method_ft - INFO - J_w_min:1.00e-05 N 64 yields: interval [0.00e+00,9.16e+00] diff 2.81e-02
stocproc.method_ft - INFO - J_w_min:1.00e-04 N 128 yields: interval [0.00e+00,7.50e+00] diff 4.17e-03
stocproc.method_ft - INFO - J_w_min:1.00e-03 N 256 yields: interval [0.00e+00,5.82e+00] diff 7.86e-04
stocproc.method_ft - INFO - return, cause tol of 0.001 was reached
stocproc.method_ft - INFO - requires dx < 2.272e-02
stocproc.stocproc - INFO - Fourier Integral Boundaries: [0.000e+00, 7.064e+01]
stocproc.stocproc - INFO - Number of Nodes : 4096
stocproc.stocproc - INFO - yields dx : 1.725e-02
stocproc.stocproc - INFO - yields dt : 8.895e-02
stocproc.stocproc - INFO - yields t_max : 3.642e+02
#+end_example
The sample trajectories are smooth too.
#+begin_src jupyter-python
%%space plot
ts = np.linspace(0, params.t_max, 1000)
EtaTherm.new_process()
plt.plot(ts, EtaTherm(ts).real)
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f70a2159100> |
[[file:./.ob-jupyter/20d0cdb5d62012a02c6ac14bc39a3b9fe59968ce.png]]
| <matplotlib.lines.Line2D | at | 0x7f708d47b430> |
[[file:./.ob-jupyter/6d082076097694cb45a42d4608ed411592bfcd3e.png]]
:END:
* Actual Hops
Generate the key for binary caching.
#+begin_src jupyter-python
hi_key = hierarchyData.HIMetaKey_type(
HiP=HierachyParam,
IntP=IntegrationParam,
SysP=SystemParam,
Eta=Eta,
EtaTherm=EtaTherm,
)
#+end_src
#+RESULTS:
Initialize Hierarchy.
#+begin_src jupyter-python
myHierarchy = hierarchyLib.HI(hi_key, number_of_samples=params.N, desc="calculate the heat flow")
#+end_src
#+RESULTS:
: /home/hiro/Documents/Projects/UNI/master/masterarb/python/richard_hops/hierarchyLib.py:1058: UserWarning: sum_k_max is not implemented! DO SO BEFORE NEXT USAGE (use simplex).HierarchyParametersType does not yet know about sum_k_max
: warnings.warn(
: init Hi class, use 2002 equation
Run the integration.
#+begin_src jupyter-python
myHierarchy.integrate_simple(data_path="data", data_name="energy_flow_therm_new_again_less.data")
#+end_src
#+RESULTS:
#+begin_example
(10, 300, 1001)
samples :0.0%
integration :0.0%
samples :0.1%
integration :0.0%
samples :0.1%
integration :23.7%
samples :0.2%
integration :22.3%
samples :0.3%
integration :49.7%
samples :0.4%
integration :68.0%
samples :0.4%
integration :96.7%
samples :0.5%
integration :19.7%
samples :0.6%
integration :39.3%
samples :0.7%
integration :58.0%
samples :0.8%
integration :88.0%
samples :0.9%
integration :20.7%
samples :0.9%
integration :44.0%
samples :1.0%
integration :64.0%
samples :1.1%
integration :83.3%
samples :1.2%
integration :5.0%
samples :1.2%
integration :86.3%
samples :1.2%
integration :4.3%
samples :1.3%
integration :99.7%
samples :1.3%
integration :95.7%
samples :1.4%
integration :20.3%
samples :1.5%
integration :79.7%
samples :1.6%
integration :12.0%
samples :1.7%
integration :31.7%
samples :1.7%
integration :62.0%
samples :1.8%
integration :88.0%
samples :1.9%
integration :99.7%
samples :2.0%
integration :25.3%
samples :2.1%
integration :54.0%
samples :2.1%
integration :80.3%
samples :2.2%
integration :4.0%
samples :2.3%
integration :21.3%
samples :2.4%
integration :23.3%
samples :2.5%
integration :45.7%
samples :2.5%
integration :40.0%
samples :2.6%
integration :54.7%
samples :2.6%
integration :75.3%
samples :2.7%
integration :16.3%
samples :2.8%
integration :45.0%
samples :2.8%
integration :99.7%
samples :2.9%
integration :94.7%
samples :2.9%
integration :99.0%
samples :3.0%
integration :44.7%
samples :3.1%
integration :54.7%
samples :3.1%
integration :48.3%
samples :3.2%
integration :50.0%
samples :3.3%
integration :47.0%
samples :3.3%
integration :67.7%
samples :3.4%
integration :75.0%
samples :3.5%
integration :19.7%
samples :3.5%
integration :25.3%
samples :3.6%
integration :33.3%
samples :3.7%
integration :57.3%
samples :3.8%
integration :47.0%
samples :3.8%
integration :99.7%
samples :3.9%
integration :35.7%
samples :4.0%
integration :42.7%
samples :4.0%
integration :17.7%
samples :4.1%
integration :24.3%
samples :4.2%
integration :99.7%
samples :4.2%
integration :66.7%
samples :4.3%
integration :78.0%
samples :4.3%
integration :80.7%
samples :4.4%
integration :87.0%
samples :4.5%
integration :76.0%
samples :4.6%
integration :19.3%
samples :4.7%
integration :0.0%
samples :4.7%
integration :51.3%
samples :4.8%
integration :67.3%
samples :4.9%
integration :75.7%
samples :4.9%
integration :67.3%
samples :5.0%
integration :86.3%
samples :5.0%
integration :99.7%
samples :5.1%
integration :17.0%
samples :5.1%
integration :24.3%
samples :5.2%
integration :84.7%
samples :5.3%
integration :14.7%
samples :5.3%
integration :99.7%
samples :5.4%
integration :79.0%
samples :5.5%
integration :82.0%
samples :5.6%
integration :9.3%
samples :5.7%
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samples :5.7%
integration :99.7%
samples :5.7%
integration :48.0%
samples :5.8%
integration :65.0%
samples :5.9%
integration :3.3%
samples :5.9%
integration :23.3%
samples :6.0%
integration :99.7%
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#+end_example
Get the samples.
#+BEGIN_SRC jupyter-python
# to access the data the 'hi_key' is used to find the data in the hdf5 file
class int_result:
with hierarchyData.HIMetaData(
hid_name="energy_flow_therm_new_again_less.data", hid_path="data"
) as metaData:
with metaData.get_HIData(hi_key, read_only=True) as data:
smp = data.get_samples()
print("{} samples found in database".format(smp))
τ = data.get_time()
rho_τ = data.get_rho_t()
#s_proc = np.array(data.stoc_proc)
#states = np.array(data.aux_states).copy()
ψ_1 = np.array(data.aux_states[:, :, 0 : params.num_exp_t * params.dim])
ψ_0 = np.array(data.stoc_traj)
y = np.array(data.y)
#η = np.array(data.stoc_proc)
temp_y = np.array(data.temp_y)
#+end_src
#+RESULTS:
: 4000 samples found in database
Calculate energy.
#+begin_src jupyter-python
%matplotlib inline
energy = np.einsum("ijk,kj", int_result.rho_τ,params.H_s).real
plt.plot(int_result.τ, energy)
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f70abf84a30> |
[[file:./.ob-jupyter/8e6031964f047808246e7cc787bff3a3133b953e.png]]
:END:
* Energy Flow
:PROPERTIES:
:ID: 9ce93da8-d323-40ec-96a2-42ba184dc963
:END:
#+begin_src jupyter-python
int_result.ψ_1.shape
#+end_src
#+RESULTS:
| 5120 | 300 | 8 |
Let's look at the norm.
#+begin_src jupyter-python
plt.plot(int_result.τ, (int_result.ψ_0[0].conj() * int_result.ψ_0[0]).sum(axis=1).real)
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f70a9ac3f10> |
[[file:./.ob-jupyter/46197d5b229e031e905aea1a0f5b517e977f711d.png]]
:END:
And try to calculate the energy flow.
#+begin_src jupyter-python
def flow_for_traj(ψ_0, ψ_1, temp_y):
a = np.array((params.L @ ψ_0.T).T)
EtaTherm.new_process(temp_y)
η_dot = scipy.misc.derivative(EtaTherm, int_result.τ, dx=1e-3, order=5)
ψ_1 = (-w * g * params.bcf_scale)[None, :, None] * ψ_1.reshape(
params.t_steps, params.num_exp_t, params.dim
)
# return np.array(np.sum(ψ_0.conj() * ψ_0, axis=1)).flatten().real
j_0 = np.array(
2
,* (
1j
,* (np.sum(a.conj()[:, None, :] * ψ_1, axis=(1, 2)))
/ np.sum(ψ_0.conj() * ψ_0, axis=1)
).real
).flatten()
j_therm = np.array(
2
,* (
1j
,* (np.sum(a.conj() * ψ_0, axis=1)) * η_dot
/ np.sum(ψ_0.conj() * ψ_0, axis=1)
).real
).flatten()
return j_0, j_therm
#+end_src
#+RESULTS:
Now we calculate the average over all trajectories.
#+begin_src jupyter-python
class Flow:
j_0 = np.zeros_like(int_result.τ)
j_therm = np.zeros_like(int_result.τ)
for i in range(0, params.N):
dj, dj_therm = flow_for_traj(
int_result.ψ_0[i], int_result.ψ_1[i], int_result.temp_y[i]
)
j_0 += dj
j_therm += dj_therm
j_0 /= params.N
j_therm /= params.N
j = j_0 + j_therm
#+end_src
#+RESULTS:
And plot it :).
#+begin_src jupyter-python
%matplotlib inline
plt.plot(int_result.τ, Flow.j_0, label=r"$j_0$")
plt.plot(int_result.τ, Flow.j_therm, label=r"$j_\mathrm{therm}$")
plt.plot(int_result.τ, Flow.j, label=r"$j$")
plt.legend()
#+end_src
#+RESULTS:
:RESULTS:
: <matplotlib.legend.Legend at 0x7f708c3a6880>
[[file:./.ob-jupyter/5a475f8087beb6abe79b48efe888fe5c4e4e182d.png]]
:END:
Let's calculate the integrated energy.
#+begin_src jupyter-python
E_t = np.array(
[0]
+ [
scipy.integrate.simpson(Flow.j[0:n], int_result.τ[0:n])
for n in range(1, len(int_result.τ))
]
)
E_t[-1]
#+end_src
#+RESULTS:
: 0.18349268980598232
With this we can retrieve the energy of the interaction Hamiltonian.
#+begin_src jupyter-python
E_I = - energy - E_t
#+end_src
#+RESULTS:
#+begin_src jupyter-python
%%space plot
#plt.plot(τ, j, label="$J$", linestyle='--')
plt.plot(int_result.τ, E_t, label=r"$\langle H_{\mathrm{B}}\rangle$")
plt.plot(int_result.τ, E_I, label=r"$\langle H_{\mathrm{I}}\rangle$")
plt.plot(int_result.τ, energy, label=r"$\langle H_{\mathrm{S}}\rangle$")
plt.xlabel("τ")
plt.legend()
plt.show()
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f709f479d00> |
| <matplotlib.lines.Line2D | at | 0x7f709f481130> |
| <matplotlib.lines.Line2D | at | 0x7f709f4814f0> |
: Text(0.5, 0, 'τ')
: <matplotlib.legend.Legend at 0x7f709f479f40>
[[file:./.ob-jupyter/070f131d17f8426b384f26f16434edde81064452.png]]
:END:
* System + Interaction Energy
#+begin_src jupyter-python
def h_si_for_traj(ψ_0, ψ_1, temp_y):
a = np.array((params.L @ ψ_0.T).T)
b = np.array((params.H_s @ ψ_0.T).T)
ψ_1 = (g*params.bcf_scale)[None, :, None] * ψ_1.reshape(
params.t_steps, params.num_exp_t, params.dim
)
EtaTherm.new_process(temp_y)
E_i = np.array(
2
,* (
-1j
,* np.sum(
a.conj()[:, None, :]
,* ψ_1,
axis=(1, 2),
)
).real
).flatten()
E_i += np.array(
2
,* (
EtaTherm(int_result.τ)
,* np.sum(
a.conj()
,* ψ_0,
axis=1,
)
).real
).flatten()
E_s = np.array(np.sum(b.conj() * ψ_0, axis=1)).flatten().real
return (E_i + E_s) / np.sum(ψ_0.conj() * ψ_0, axis=1).real
#+end_src
#+RESULTS:
#+begin_src jupyter-python
e_si = np.zeros_like(int_result.τ)
for i in range(0, params.N):
e_si += h_si_for_traj(int_result.ψ_0[i], int_result.ψ_1[i], int_result.temp_y[i])
e_si /= params.N
#+end_src
#+RESULTS:
Doesn't work out.
#+begin_src jupyter-python
plt.plot(int_result.τ, e_si -energy, label=r"direct")
plt.plot(int_result.τ, E_I)
plt.legend()
#+end_src
#+RESULTS:
:RESULTS:
: <matplotlib.legend.Legend at 0x7f709f3f1f10>
[[file:./.ob-jupyter/5d121c7f9496bd6f69931ad6252f2ae57660bd48.png]]
:END:
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