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master-thesis/python/richard_hops/energy_flow_nonlinear.org
Valentin Boettcher 97ac28637d
updates
2021-11-02 14:39:09 +01:00

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Setup

Jupyter 

  %load_ext autoreload
  %autoreload 2
  %load_ext jupyter_spaces
The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

Matplotlib 

  import matplotlib
  import matplotlib.pyplot as plt

  #matplotlib.use("TkCairo", force=True)
  %gui tk
  %matplotlib inline
  plt.style.use('ggplot')

Richard (old) HOPS 

  import hierarchyLib
  import hierarchyData
  import numpy as np

  from stocproc.stocproc import StocProc_FFT
  import bcf
  from dataclasses import dataclass
  import scipy
  import scipy.misc
  import scipy.signal

Auxiliary Definitions 

  σ1 = np.matrix([[0,1],[1,0]])
  σ2 = np.matrix([[0,-1j],[1j,0]])
  σ3 = np.matrix([[1,0],[0,-1]])

Model Setup

Basic parameters.

  γ = 3 # coupling ratio
  ω_c = 2  # center of spect. dens
  δ = 2  # breadth BCF
  t_max = 10
  t_steps = 4000
  k_max = 10
  seed = 100
  H_s = σ3 + np.eye(2)
  L =  σ2 * γ # 1 / 2 * (σ1 - 1j * σ2) * γ
  ψ_0 = np.array([1, 0])
  W = ω_c * 1j + δ  # exponent BCF
  N = 1000

BCF 

  @dataclass
  class CauchyBCF:
      δ: float
      ω_c: float

      def I(self, ω):
          return np.sqrt(self.δ) / (self.δ + (ω - self.ω_c) ** 2 / self.δ)

      def __call__(self, τ):
          return np.sqrt(self.δ) * np.exp(-1j * self.ω_c * τ - np.abs(τ) * self.δ)

      def __bfkey__(self):
          return self.δ, self.ω_c

  α = CauchyBCF(δ, ω_c)

Plot 

  %%space plot
  t = np.linspace(0, t_max, 1000)
  ω = np.linspace(ω_c - 10, ω_c + 10, 1000)
  fig, axs = plt.subplots(2)
  axs[0].plot(t, np.real(α(t)))
  axs[0].plot(t, np.imag(α(t)))
  axs[1].plot(ω, α.I(ω))
<matplotlib.lines.Line2D at 0x7fa1394d1400>
<matplotlib.lines.Line2D at 0x7fa1394d1640>
<matplotlib.lines.Line2D at 0x7fa1394d1a60>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/cba6b2fb1d87a1f5c3a8e444c4b8851586bd49ef.png

Hops setup 

  HierachyParam = hierarchyData.HiP(
      k_max=k_max,
      # g_scale=None,
      # sample_method='random',
      seed=seed,
      nonlinear=True,
      normalized=False,
      # terminator=False,
      result_type=hierarchyData.RESULT_TYPE_ALL,
      # accum_only=None,
      # rand_skip=None
  )

Integration.

  IntegrationParam = hierarchyData.IntP(
      t_max=t_max,
      t_steps=t_steps,
      # integrator_name='zvode',
      # atol=1e-8,
      # rtol=1e-8,
      # order=5,
      # nsteps=5000,
      # method='bdf',
      # t_steps_skip=1
  )

And now the system.

  SystemParam = hierarchyData.SysP(
      H_sys=H_s,
      L=L,
      psi0=ψ_0,  # excited qubit
      g=np.array([np.sqrt(δ)]),
      w=np.array([W]),
      H_dynamic=[],
      bcf_scale=1,  # 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=1,
  )

The quantum noise.

  Eta = StocProc_FFT(
      α.I,
      t_max,
      α,
      negative_frequencies=True,
      seed=seed,
      intgr_tol=1e-2,
      intpl_tol=1e-2,
      scale=1,
  )
  stocproc.stocproc - INFO - use neg freq
  stocproc.method_ft - INFO - get_dt_for_accurate_interpolation, please wait ...
  stocproc.method_ft - INFO - acc interp N 33 dt 1.44e-01 -> diff 7.57e-03
  stocproc.method_ft - INFO - requires dt < 1.439e-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 [-1.47e+01,1.87e+01] diff 3.37e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [-5.11e+01,5.51e+01] diff 6.70e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [-1.47e+01,1.87e+01] diff 3.37e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 32 yields: interval [-1.66e+02,1.70e+02] diff 2.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [-5.11e+01,5.51e+01] diff 1.11e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [-1.47e+01,1.87e+01] diff 3.37e-01
  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 [-5.30e+02,5.34e+02] diff 3.68e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [-1.66e+02,1.70e+02] diff 1.34e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [-5.11e+01,5.51e+01] diff 1.06e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-02 N 256 yields: interval [-1.47e+01,1.87e+01] diff 3.37e-01
  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 [-1.68e+03,1.68e+03] diff 4.19e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 64 yields: interval [-5.30e+02,5.34e+02] diff 3.04e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 128 yields: interval [-1.66e+02,1.70e+02] diff 4.07e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 256 yields: interval [-5.11e+01,5.51e+01] diff 1.06e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-02 N 512 yields: interval [-1.47e+01,1.87e+01] diff 3.37e-01
  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-07 N 32 yields: interval [-5.32e+03,5.32e+03] diff 4.36e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 64 yields: interval [-1.68e+03,1.68e+03] diff 3.94e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 128 yields: interval [-5.30e+02,5.34e+02] diff 2.09e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 256 yields: interval [-1.66e+02,1.70e+02] diff 3.72e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 512 yields: interval [-5.11e+01,5.51e+01] diff 1.06e-01
  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-08 N 32 yields: interval [-1.68e+04,1.68e+04] diff 4.42e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 64 yields: interval [-5.32e+03,5.32e+03] diff 4.28e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 128 yields: interval [-1.68e+03,1.68e+03] diff 3.50e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 256 yields: interval [-5.30e+02,5.34e+02] diff 9.79e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 512 yields: interval [-1.66e+02,1.70e+02] diff 3.36e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 1024 yields: interval [-5.11e+01,5.51e+01] diff 1.06e-01
  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-09 N 32 yields: interval [-5.32e+04,5.32e+04] diff 4.43e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-08 N 64 yields: interval [-1.68e+04,1.68e+04] diff 4.39e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 128 yields: interval [-5.32e+03,5.32e+03] diff 4.12e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 256 yields: interval [-1.68e+03,1.68e+03] diff 2.75e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 512 yields: interval [-5.30e+02,5.34e+02] diff 2.16e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 1024 yields: interval [-1.66e+02,1.70e+02] diff 3.36e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-03 N 2048 yields: interval [-5.11e+01,5.51e+01] diff 1.06e-01
  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-10 N 32 yields: interval [-1.68e+05,1.68e+05] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-09 N 64 yields: interval [-5.32e+04,5.32e+04] diff 4.43e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-08 N 128 yields: interval [-1.68e+04,1.68e+04] diff 4.34e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 256 yields: interval [-5.32e+03,5.32e+03] diff 3.82e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 512 yields: interval [-1.68e+03,1.68e+03] diff 1.71e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 1024 yields: interval [-5.30e+02,5.34e+02] diff 1.07e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 2048 yields: interval [-1.66e+02,1.70e+02] diff 3.36e-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-11 N 32 yields: interval [-5.32e+05,5.32e+05] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-10 N 64 yields: interval [-1.68e+05,1.68e+05] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-09 N 128 yields: interval [-5.32e+04,5.32e+04] diff 4.41e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-08 N 256 yields: interval [-1.68e+04,1.68e+04] diff 4.24e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 512 yields: interval [-5.32e+03,5.32e+03] diff 3.28e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 1024 yields: interval [-1.68e+03,1.68e+03] diff 6.56e-01
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 2048 yields: interval [-5.30e+02,5.34e+02] diff 1.06e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 4096 yields: interval [-1.66e+02,1.70e+02] diff 3.36e-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-12 N 32 yields: interval [-1.68e+06,1.68e+06] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-11 N 64 yields: interval [-5.32e+05,5.32e+05] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-10 N 128 yields: interval [-1.68e+05,1.68e+05] diff 4.43e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-09 N 256 yields: interval [-5.32e+04,5.32e+04] diff 4.38e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-08 N 512 yields: interval [-1.68e+04,1.68e+04] diff 4.04e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 1024 yields: interval [-5.32e+03,5.32e+03] diff 2.43e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 2048 yields: interval [-1.68e+03,1.68e+03] diff 9.69e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-05 N 4096 yields: interval [-5.30e+02,5.34e+02] diff 1.06e-02
  stocproc.method_ft - INFO - J_w_min:1.00e-04 N 8192 yields: interval [-1.66e+02,1.70e+02] diff 3.36e-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-13 N 32 yields: interval [-5.32e+06,5.32e+06] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-12 N 64 yields: interval [-1.68e+06,1.68e+06] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-11 N 128 yields: interval [-5.32e+05,5.32e+05] diff 4.44e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-10 N 256 yields: interval [-1.68e+05,1.68e+05] diff 4.42e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-09 N 512 yields: interval [-5.32e+04,5.32e+04] diff 4.31e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-08 N 1024 yields: interval [-1.68e+04,1.68e+04] diff 3.67e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-07 N 2048 yields: interval [-5.32e+03,5.32e+03] diff 1.33e+00
  stocproc.method_ft - INFO - J_w_min:1.00e-06 N 4096 yields: interval [-1.68e+03,1.68e+03] diff 3.37e-03
  stocproc.method_ft - INFO - return, cause tol of 0.01 was reached
  stocproc.method_ft - INFO - requires dx < 8.212e-01
  stocproc.method_ft - INFO - increase N to match dt_new*(N-1) < t_max
  stocproc.stocproc - INFO - Fourier Integral Boundaries: [-1.680e+03, 1.684e+03]
  stocproc.stocproc - INFO - Number of Nodes            : 8192
  stocproc.stocproc - INFO - yields dx                  : 4.106e-01
  stocproc.stocproc - INFO - yields dt                  : 1.868e-03
  stocproc.stocproc - INFO - yields t_max               : 1.530e+01

Actual Hops

Generate the key for binary caching.

  hi_key = hierarchyData.HIMetaKey_type(
      HiP=HierachyParam,
      IntP=IntegrationParam,
      SysP=SystemParam,
      Eta=Eta,
      EtaTherm=None,
  )

Initialize Hierarchy.

  myHierarchy = hierarchyLib.HI(hi_key, number_of_samples=N, desc="run a test case")
init Hi class, use 22 equation
/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(

Run the integration.

  myHierarchy.integrate_simple(data_path="data", data_name="energy_flow_nl_2.data")
  (10, 4000, 11)
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  integration :17.9%
  samples     :99.5%
  integration :38.2%
  samples     :99.5%
  integration :57.9%
  samples     :99.5%
  integration :77.8%
  samples     :99.5%
  integration :97.8%
  samples     :99.6%
  integration :18.8%
  samples     :99.6%
  integration :38.6%
  samples     :99.6%
  integration :58.7%
  samples     :99.6%
  integration :78.6%
  samples     :99.6%
  integration :98.6%
  samples     :99.7%
  integration :20.2%
  samples     :99.7%
  integration :40.4%
  samples     :99.7%
  integration :60.6%
  samples     :99.7%
  integration :80.8%
  samples     :99.8%
  integration :1.5%
  samples     :99.8%
  integration :22.6%
  samples     :99.8%
  integration :42.7%
  samples     :99.8%
  integration :62.7%
  samples     :99.8%
  integration :82.9%
  samples     :99.9%
  integration :3.8%
  samples     :99.9%
  integration :24.5%
  samples     :99.9%
  integration :44.6%
  samples     :99.9%
  integration :65.0%
  samples     :99.9%
  integration :84.8%
  samples     : 100%
  integration :0.0%
  

Get the samples.

  # to access the data the 'hi_key' is used to find the data in the hdf5 file
  with hierarchyData.HIMetaData(hid_name="energy_flow_nl_2.data", hid_path=".") 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)[:, :, 0:2]
          ψ_0 = np.array(data.stoc_traj)
          y = np.array(data.y)
          η = np.array(data.stoc_proc)
3 samples found in database

Calculate energy.

  %matplotlib inline
  energy = np.array([np.trace(ρ @ H_s).real for ρ in rho_τ])
  plt.plot(τ, energy)
<matplotlib.lines.Line2D at 0x7fa179f70ac0>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/bfe332f67cff9b40ce48b8bb5f61e681ad2e8733.png

  %%space plot
  plt.plot(τ, np.trace(rho_τ.T).real)
<matplotlib.lines.Line2D at 0x7fa179ee0df0>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/895fca546bda1a90b476d05fdcbd3d8502fd2f8a.png

Energy Flow 

  ψ_1.shape
10 4000 2

Let's look at the norm.

  plt.plot(τ, (ψ_0[0].conj() * ψ_0[0]).sum(axis=1).real)
<matplotlib.lines.Line2D at 0x7fa179e439d0>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/1356cc6776127b29a8f0bbe253101d83e6dad8d7.png

And try to calculate the energy flow.

  def flow_for_traj(ψ_0, ψ_1):
      a = np.array((L @ ψ_0.T).T)
      #return np.array(np.sum(ψ_0.conj() * ψ_0, axis=1)).flatten().real
      return np.array(2 * (1j * -W * np.sum(a.conj() * ψ_1, axis=1)/np.sum(ψ_0.conj() * ψ_0, axis=1)).real).flatten()


  def flow_for_traj_alt(ψ_0, y):
      Eta.new_process(y)
      eta_dot = scipy.misc.derivative(Eta, τ, dx=1e-8)
      a = np.array((L @ ψ_0.T).T)

      return -(
          2j * eta_dot.conj() * np.array((np.sum(ψ_0.conj() * a, axis=1))).flatten()
      ).real

Now we calculate the average over all trajectories.

  j = np.zeros_like(τ)
  for i in range(0, N):
      j += flow_for_traj(ψ_0[i], ψ_1[i])
  j /= N
<ipython-input-20-0e4c3c950b41>:4: RuntimeWarning: invalid value encountered in true_divide
  return np.array(2 * (1j * -W * np.sum(a.conj() * ψ_1, axis=1)/np.sum(ψ_0.conj() * ψ_0, axis=1)).real).flatten()


IndexErrorTraceback (most recent call last)
<ipython-input-21-b9081128ed40> in <module>
      1 j = np.zeros_like(τ)
      2 for i in range(0, N):
----> 3     j += flow_for_traj(ψ_0[i], ψ_1[i])
      4 j /= N

IndexError: index 10 is out of bounds for axis 0 with size 10

And do the same with the alternative implementation.

  ja = np.zeros_like(τ)
  for i in range(0, N):
      ja += flow_for_traj_alt(ψ_0[i], y[i])
  ja /= N

IndexErrorTraceback (most recent call last)
<ipython-input-22-5dc6ccf09941> in <module>
      1 ja = np.zeros_like(τ)
      2 for i in range(0, N):
----> 3     ja += flow_for_traj_alt(ψ_0[i], y[i])
      4 ja /= N

IndexError: index 10 is out of bounds for axis 0 with size 10

And plot it :)

  %matplotlib inline
  plt.plot(τ, j)
  #plt.plot(τ, ja)
  plt.show()

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/218e7fb38922b30990d9af124bdd5670bcd91914.png

Let's calculate the integrated energy.

  E_t = np.array([0] + [scipy.integrate.simpson(j[0:n], τ[0:n]) for n in range(1, len(τ))])
  E_t[-1]
nan

With this we can retrieve the energy of the interaction Hamiltonian.

  E_I = 2 - energy - E_t
  %%space plot
  plt.rcParams['figure.figsize'] = [15, 10]
  #plt.plot(τ, j, label="$J$", linestyle='--')
  plt.plot(τ, E_t, label=r"$\langle H_{\mathrm{B}}\rangle$")
  plt.plot(τ, E_I, label=r"$\langle H_{\mathrm{I}}\rangle$")
  plt.plot(τ, energy, label=r"$\langle H_{\mathrm{S}}\rangle$")

  plt.xlabel("τ")
  plt.legend()
  plt.show()
<matplotlib.lines.Line2D at 0x7fa179ca4130>
<matplotlib.lines.Line2D at 0x7fa179ca4520>
<matplotlib.lines.Line2D at 0x7fa179ca48e0> 
Text(0.5, 0, 'τ')
<matplotlib.legend.Legend at 0x7fa179ca4310>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/82103f483195bce4ebc6e8850cd69aa22ae5e90d.png

System + Interaction Energy 

  def h_si_for_traj(ψ_0, ψ_1):
      a = np.array((L @ ψ_0.T).T)
      b = np.array((H_s @ ψ_0.T).T)
      E_i = np.array(2 * (-1j * np.sum(a.conj() * ψ_1, 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

  def h_si_for_traj_alt(ψ_0, y):
      Eta.new_process(y)

      a = np.array((L.conj().T @ ψ_0.T).T)
      b = np.array((H_s @ ψ_0.T).T)
      E_i = np.array(2 * (Eta(τ) * 1j * 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
  e_si = np.zeros_like(τ)
  for i in range(0, N):
      e_si += h_si_for_traj(ψ_0[i], ψ_1[i])
  e_si /= N
<ipython-input-27-a81ea186bce4>:7: RuntimeWarning: invalid value encountered in true_divide
  return (E_i + E_s)/np.sum(ψ_0.conj() * ψ_0, axis=1).real


IndexErrorTraceback (most recent call last)
<ipython-input-28-b6519289ec80> in <module>
      1 e_si = np.zeros_like(τ)
      2 for i in range(0, N):
----> 3     e_si += h_si_for_traj(ψ_0[i], ψ_1[i])
      4 e_si /= N

IndexError: index 10 is out of bounds for axis 0 with size 10

Not too bad…

  plt.plot(τ, e_si)
  plt.plot(τ, E_I + energy)
<matplotlib.lines.Line2D at 0x7fa179b5e4f0>

/hiro/master-thesis/media/commit/97ac28637dd0bb30acc8ecdce80535c563b44b42/python/richard_hops/.ob-jupyter/2dc622d07d507b3b2fa068687a21150c630c9ead.png