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19 KiB
19 KiB
Setup
Jupyter
%load_ext autoreload
%autoreload 2
%load_ext jupyter_spacesMatplotlib
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, field
import scipy
import scipy.misc
import scipy.signal
import pickle
from scipy.special import gamma as gamma_func
from scipy.optimize import curve_fitAuxiliary 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.
class params:
T = 2.09
t_max = 1
t_steps = int(t_max * 1/.01)
k_max = 3
N = 17
seed = 100
dim = 2
H_s = σ3 + np.eye(dim)
L = 1 / 2 * (σ1 - 1j * σ2)
ψ_0 = np.array([1, 0])
s = 1
num_exp_t = 5
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)BCF and Thermal BCF
@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)Fit
We now fit a sum of exponentials against the BCF.
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()Plot
Let's look a the result.
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)
Seems ok for now.
Hops setup
HierachyParam = hierarchyData.HiP(
k_max=params.k_max,
# g_scale=None,
# sample_method='random',
seed=params.seed,
nonlinear=False,
normalized=False,
# terminator=False,
result_type=hierarchyData.RESULT_TYPE_ALL,
# accum_only=None,
# rand_skip=None
)Integration.
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
)And now the system.
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),
)The quantum noise.
Eta = StocProc_FFT(
I,
params.t_max,
α,
negative_frequencies=False,
seed=params.seed,
intgr_tol=1e-4,
intpl_tol=1e-4,
scale=params.bcf_scale,
)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 6.25e-02 -> diff 1.08e-03 stocproc.method_ft - INFO - acc interp N 65 dt 3.12e-02 -> diff 2.69e-04 stocproc.method_ft - INFO - acc interp N 129 dt 1.56e-02 -> diff 6.71e-05 stocproc.method_ft - INFO - requires dt < 1.562e-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 2.72e-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 5.41e-03 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [0.00e+00,9.12e+00] diff 5.54e-04 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 8.13e-03 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [0.00e+00,1.17e+01] diff 1.30e-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 - J_w_min:1.00e-02 N 256 yields: interval [0.00e+00,6.47e+00] diff 1.15e-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-06 N 32 yields: interval [0.00e+00,1.66e+01] diff 1.11e-02 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 64 yields: interval [0.00e+00,1.42e+01] diff 2.03e-03 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 128 yields: interval [0.00e+00,1.17e+01] diff 2.69e-04 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 256 yields: interval [0.00e+00,9.12e+00] diff 1.06e-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-07 N 32 yields: interval [0.00e+00,1.91e+01] diff 1.48e-02 stocproc.method_ft - INFO - J_w_min:1.00e-06 N 64 yields: interval [0.00e+00,1.66e+01] diff 2.80e-03 stocproc.method_ft - INFO - J_w_min:1.00e-05 N 128 yields: interval [0.00e+00,1.42e+01] diff 5.04e-04 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 256 yields: interval [0.00e+00,1.17e+01] diff 4.76e-05 stocproc.method_ft - INFO - return, cause tol of 0.0001 was reached stocproc.method_ft - INFO - requires dx < 4.557e-02 stocproc.stocproc - INFO - Fourier Integral Boundaries: [0.000e+00, 5.480e+02] stocproc.stocproc - INFO - Number of Nodes : 16384 stocproc.stocproc - INFO - yields dx : 3.345e-02 stocproc.stocproc - INFO - yields dt : 1.147e-02 stocproc.stocproc - INFO - yields t_max : 1.879e+02
The sample trajectories are smooth.
%%space plot
ts = np.linspace(0, params.t_max, 1000)
Eta.new_process()
plt.plot(ts, Eta(ts).real)| <matplotlib.lines.Line2D | at | 0x7f38441603d0> |
And now the thermal noise.
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,
)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 6.25e-02 -> diff 2.21e-04 stocproc.method_ft - INFO - requires dt < 6.250e-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+00,4.18e+00] diff 9.38e-03 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 32 yields: interval [0.00e+00,5.92e+00] diff 4.42e-03 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 64 yields: interval [0.00e+00,4.18e+00] diff 8.00e-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-04 N 32 yields: interval [0.00e+00,7.62e+00] diff 7.37e-03 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 64 yields: interval [0.00e+00,5.92e+00] diff 1.66e-03 stocproc.method_ft - INFO - J_w_min:1.00e-02 N 128 yields: interval [0.00e+00,4.18e+00] diff 7.66e-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-05 N 32 yields: interval [0.00e+00,9.30e+00] diff 1.04e-02 stocproc.method_ft - INFO - J_w_min:1.00e-04 N 64 yields: interval [0.00e+00,7.62e+00] diff 1.87e-03 stocproc.method_ft - INFO - J_w_min:1.00e-03 N 128 yields: interval [0.00e+00,5.92e+00] diff 9.72e-04 stocproc.method_ft - INFO - return, cause tol of 0.001 was reached stocproc.method_ft - INFO - requires dx < 4.622e-02 stocproc.stocproc - INFO - Fourier Integral Boundaries: [0.000e+00, 1.380e+02] stocproc.stocproc - INFO - Number of Nodes : 4096 stocproc.stocproc - INFO - yields dx : 3.368e-02 stocproc.stocproc - INFO - yields dt : 4.555e-02 stocproc.stocproc - INFO - yields t_max : 1.865e+02
The sample trajectories are smooth too.
%%space plot
ts = np.linspace(0, params.t_max, 1000)
EtaTherm.new_process()
plt.plot(ts, EtaTherm(ts).real)| <matplotlib.lines.Line2D | at | 0x7f384413fd30> |
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=params.N, desc="calculate the heat flow")init Hi class, use 112 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_name="energy_flow_therm_new_lin.data")(10, 100, 56) samples :0.0% integration :0.0% [2A[8m[0msamples : 100% integration :0.0% [0A[8m[0m
Get the samples.
# 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_lin.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 : params.num_exp_t * params.dim]
ψ_0 = np.array(data.stoc_traj)
y = np.array(data.y)
η = np.array(data.stoc_proc)17 samples found in database
Calculate energy.
%matplotlib inline
energy = np.array([np.trace(ρ @ params.H_s).real for ρ in int_result.rho_τ])
plt.plot(int_result.τ, energy)| <matplotlib.lines.Line2D | at | 0x7f384404a6a0> |
Energy Flow
int_result.ψ_1.shape| 5120 | 400 | 10 |
Let's look at the norm.
plt.plot(int_result.τ, (int_result.ψ_0[0].conj() * int_result.ψ_0[0]).sum(axis=1).real)| <matplotlib.lines.Line2D | at | 0x7ff8e19d1f10> |
And try to calculate the energy flow.
def flow_for_traj(ψ_0, ψ_1):
a = np.array((params.L @ ψ_0.T).T)
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
return np.array(
2
,* (
1j
,* np.sum(a.conj()[:, None, :] * ψ_1, axis=(1, 2))
).real
).flatten()
def flow_for_traj_alt(ψ_0, y):
Eta.new_process(y)
eta_dot = scipy.misc.derivative(Eta, int_result.τ, dx=1e-5, order=5)
a = np.array((params.L @ ψ_0.T).T)
return np.array(
2
,* (
-1j * eta_dot.conj()
,* np.sum(ψ_0.conj() * a, axis=1)
).real
).flatten()Now we calculate the average over all trajectories.
class Flow:
j = np.zeros_like(int_result.τ)
for i in range(0, params.N):
j += flow_for_traj(int_result.ψ_0[i], int_result.ψ_1[i])
j /= params.NAnd do the same with the alternative implementation.
class Flow_Alt:
j = np.zeros_like(int_result.τ)
for i in range(0, params.N):
j += flow_for_traj_alt(int_result.ψ_0[i], int_result.y[i])
j /= params.NAnd plot it :). We see: it also works with the derivative.
%matplotlib inline
plt.plot(int_result.τ, Flow.j)
plt.plot(int_result.τ, Flow_Alt.j)
plt.show()
Let's calculate the integrated energy.
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))
]
)
print(E_t[-1])
E_t = E_t / E_t[-1] * 21.8046657496034806
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(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()| <matplotlib.lines.Line2D | at | 0x7ff8e18bad00> |
| <matplotlib.lines.Line2D | at | 0x7ff8e18c8130> |
| <matplotlib.lines.Line2D | at | 0x7ff8e18c8520> |
Text(0.5, 0, 'τ') <matplotlib.legend.Legend at 0x7ff8e18bafa0>
System + Interaction Energy
def h_si_for_traj(ψ_0, ψ_1):
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
)
E_i = np.array(
2
,* (
-1j
,* np.sum(
a.conj()[:, None, :]
,* ψ_1,
axis=(1, 2),
)
).real
).flatten()
E_s = np.array(np.sum(b.conj() * ψ_0, axis=1)).flatten().real
return (E_i + E_s)
def h_si_for_traj_alt(ψ_0, y):
Eta.new_process(y)
a = np.array((params.L @ ψ_0.T).T)
b = np.array((params.H_s @ ψ_0.T).T)
E_i = np.array(
2
,* (
-1j * Eta(int_result.τ).conj()
,* np.sum(
a
,* ψ_0.conj(),
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(int_result.τ)
e_si_alt = 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])
e_si_alt += h_si_for_traj_alt(int_result.ψ_0[i], int_result.y[i])
e_si /= params.N
e_si_alt /= params.NNot too bad…
plt.plot(int_result.τ, e_si, label=r"with $\psi^{1}$")
plt.plot(int_result.τ, e_si_alt, label=r"with $\dot{\eta}$")
plt.plot(int_result.τ, E_I + energy, label=r"from $J$")
plt.legend()<matplotlib.legend.Legend at 0x7ff8e17bdaf0>
