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https://github.com/vale981/HOPSFlow-Paper
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6.9 KiB
6.9 KiB
In here, we'll try to find out how long the cycle has to be, to ensure complete thermalization.
Boilerplate
import figsaver as fs
import plot_utils as pu
from hiro_models.one_qubit_model import StocProcTolerances
from hiro_models.otto_cycle import OttoEngine
import hiro_models.model_auxiliary as aux
import numpy as np
import qutip as qt
import utilities as ut
import stocproc
import matplotlib.pyplot as plt
import otto_utilities as ot
import ray
ray.shutdown()
#ray.init(address='auto')
ray.init()
from hops.util.logging_setup import logging_setup
import logging
logging_setup(logging.INFO)
plt.rcParams['figure.figsize'] = (12,4)Cycle
This model reflects the currently "best-working" configuration. The only parameter we'll vary is the cycle time θ. We'll be interested in whether the system thermalizes.
Therefore we need a way to keep the coupling switching speed constant.
def timings(τ_c, τ_i):
τ_th = (1 - 2 * τ_c) / 2
τ_i_on = τ_th - 2 * τ_i
timings_H = (0, τ_c, τ_c + τ_th, 2 * τ_c + τ_th)
timings_L_hot = (τ_c, τ_c + τ_i, τ_c + τ_i + τ_i_on, τ_c + 2 * τ_i + τ_i_on)
timings_L_cold = tuple(time + timings_H[2] for time in timings_L_hot)
return timings_H, (timings_L_cold, timings_L_hot)Now we define the prototype. The numeric accuracy is jacked down, as we don't need precision. We only use two cycles because of the long cycle time. With δ=.4 θ=70 seemed ok. So let's try with .6 and 50 ok. 1 and 40 is ok too. Let's try 1 and 45. Seems OK, but let's ease off on the coupling strength and try .8.
def make_cycle(θ):
(p_H, p_L) = timings(3. / θ, 3. / θ)
return OttoEngine(
δ=[.8, .8],
ω_c=[2, 2],
ψ_0=qt.basis([2], [1]),
description=f"Classic Cycle",
k_max=4,
bcf_terms=[4] * 2,
truncation_scheme="simplex",
driving_process_tolerances=[StocProcTolerances(1e-3, 1e-3)] * 2,
thermal_process_tolerances=[StocProcTolerances(1e-3, 1e-3)] * 2,
T=[0.5, 4],
therm_methods=["tanhsinh", "tanhsinh"],
Δ=1,
num_cycles=2,
Θ=θ,
dt=0.001,
timings_H=p_H,
timings_L=p_L,
streaming_mode=True,
shift_to_resonance=(False, False),
L_shift=(0, 0),
) ot.plot_cycle(make_cycle(100))| <Figure | size | 1200x400 | with | 1 | Axes> | <AxesSubplot: | xlabel= | $\tau$ | ylabel= | Operator Norm | > |
Cursory Scanning
We can now test the model to find wether it allows enough time for complete thermalization. We'll start with a really long cycle.
We tried 100, now we try 60. 60 is a bit too short. So let's try 70
long_cycle = make_cycle(45) ot.integrate_online(long_cycle, 100_000)We can also import from taurus.
from hiro_models.model_auxiliary import import_results
import_results(
other_data_path="./taurus/.data",
other_results_path="./taurus/results",
interactive=False,
models_to_import=[long_cycle],
force=True,
)Now we look at the system energy.
f, a, *_ = pu.plot_with_σ(long_cycle.t, long_cycle.system_energy())
a.set_xlim(0, long_cycle.Θ)| 0.0 | 45.0 |
ot.plot_energy(long_cycle)| <Figure | size | 1200x400 | with | 1 | Axes> | <AxesSubplot: | xlabel= | $\tau$ | ylabel= | Energy | > |
pu.plot_with_σ(long_cycle.t, long_cycle.bath_energy_flow().sum_baths())| <Figure | size | 1200x400 | with | 1 | Axes> | <AxesSubplot: | > | ((<matplotlib.lines.Line2D at 0x7fb4f3b63bb0>) <matplotlib.collections.PolyCollection at 0x7fb4f3b73d00>) |
We would like to know how far away from the thermal state the system is.
def thermal_state(Ω, T):
ρ = np.array([[np.exp(-Ω/T), 0], [0, 1]])
ρ /= np.sum(np.diag(ρ))
return ρ import hops.util.utilities
from hopsflow.util import EnsembleValue
with aux.get_data(long_cycle) as data:
trace_dist_c = hops.util.utilities.trace_distance(data, relative_to=thermal_state(long_cycle.T[0], long_cycle.energy_gaps[0]))
trace_dist_h = hops.util.utilities.trace_distance(data, relative_to=thermal_state(long_cycle.T[1], long_cycle.energy_gaps[1]))
f, a = plt.subplots()
pu.plot_with_σ(long_cycle.t, EnsembleValue(trace_dist_c), ax=a, label=r"$||\rho(\tau)-\rho_c||$")
pu.plot_with_σ(long_cycle.t, EnsembleValue(trace_dist_h), ax=a, label=r"$||\rho(\tau)-\rho_h||$")
a.plot(long_cycle.t, (long_cycle.H(long_cycle.t)[:, 0, 0] - 1)/2, label="H Modulation")
a.set_xlabel(r"$\tau$")
#a.set_xlim(155)
a.legend()
fs.export_fig("thermalization")/nix/store/vkzza81mzwyk5br1c6cm67g48xycvmvl-python3-3.9.15-env/lib/python3.9/site-packages/matplotlib/cbook/__init__.py:1369: ComplexWarning: Casting complex values to real discards the imaginary part return np.asarray(x, float) /nix/store/vkzza81mzwyk5br1c6cm67g48xycvmvl-python3-3.9.15-env/lib/python3.9/site-packages/matplotlib/axes/_axes.py:5340: ComplexWarning: Casting complex values to real discards the imaginary part pts[0] = start /nix/store/vkzza81mzwyk5br1c6cm67g48xycvmvl-python3-3.9.15-env/lib/python3.9/site-packages/matplotlib/axes/_axes.py:5341: ComplexWarning: Casting complex values to real discards the imaginary part pts[N + 1] = end /nix/store/vkzza81mzwyk5br1c6cm67g48xycvmvl-python3-3.9.15-env/lib/python3.9/site-packages/matplotlib/axes/_axes.py:5344: ComplexWarning: Casting complex values to real discards the imaginary part pts[1:N+1, 1] = dep1slice /nix/store/vkzza81mzwyk5br1c6cm67g48xycvmvl-python3-3.9.15-env/lib/python3.9/site-packages/matplotlib/axes/_axes.py:5346: ComplexWarning: Casting complex values to real discards the imaginary part pts[N+2:, 1] = dep2slice[::-1]
