master-thesis/python/graveyard/richard_hops/energy_flow_nonlinear.org

• Setup
  • Jupyter
  • Matplotlib
  • Richard (old) HOPS
  • Auxiliary Definitions
• Model Setup
  • BCF
    • Plot
  • Hops setup
• Actual Hops
• Energy Flow
• System + Interaction Energy

Setup

Jupyter

  %load_ext autoreload
  %autoreload 2
  %load_ext jupyter_spaces

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 = 4
  t_steps = 4000
  k_max = 3
  seed = 100
  g = np.sqrt(δ)
  H_s = σ3 + np.eye(2)
  L =  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))

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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([g]),
      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.stocproc - INFO - Fourier Integral Boundaries: [-1.680e+03, 1.684e+03]
  stocproc.stocproc - INFO - Number of Nodes            : 4096
  stocproc.stocproc - INFO - yields dx                  : 8.212e-01
  stocproc.stocproc - INFO - yields dt                  : 1.868e-03
  stocproc.stocproc - INFO - yields t_max               : 7.649e+00

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 8 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")

samples     :0.0%
integration :0.0%
samples     :49.0%
integration :0.0%
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="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)[:, :, 0:2]
          ψ_0 = np.array(data.stoc_traj)
          y = np.array(data.y)
          η = np.array(data.stoc_proc)

1000 samples found in database

Calculate energy.

  %matplotlib inline
  import qutip
  energy = np.array([np.trace(ρ @ H_s).real for ρ in rho_τ])
  plt.plot(τ, energy)

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  %%space plot
  plt.plot(τ, np.trace(rho_τ.T).real)

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Energy Flow

  ψ_1.shape

1280

4000

2

Let's look at the norm.

  plt.plot(τ, (ψ_0[0].conj() * ψ_0[0]).sum(axis=1).real)

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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(np.sqrt(δ) * 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

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

And plot it :)

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

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]

1.992784078082371

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()

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Text(0.5, 0, 'τ')
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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

Not too bad…

  plt.plot(τ, e_si)
  plt.plot(τ, E_I + energy)

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