def generate_pt(temp):
start_time = 0.0
end_time = 10.0
correlations = tempo.PowerLawSD(alpha=alpha,
zeta=3,
cutoff=omega_cutoff,
cutoff_type='gaussian',
temperature=temp)
bath = tempo.Bath(0.5 * tempo.operators.sigma("z"), correlations)
dt = 0.1 # 0.01
dkmax = 20 # 200
epsrel = 1.0e-6 # 1.0e-7
tempo_parameters = tempo.TempoParameters(dt=dt, dkmax=dkmax, epsrel=epsrel)
pt_tempo_parameters = tempo.PtTempoParameters(dt=dt,
dkmax=dkmax,
epsrel=epsrel)
pt = tempo.pt_tempo_compute(bath=bath,
start_time=start_time,
end_time=end_time,
parameters=pt_tempo_parameters,
progress_type='bar')
pt.export("details_pt_tempo_{}K.processTensor".format(temp),
overwrite=True)
def test_tensor_network_tempo_backend_non_diag(backend):
Omega = 1.0
omega_cutoff = 5.0
alpha = 0.3
sx = tempo.operators.sigma("x")
sy = tempo.operators.sigma("y")
sz = tempo.operators.sigma("z")
bases = [{"sys_op":sx, "coupling_op":sz, \
"init_state":tempo.operators.spin_dm("y+")},
{"sys_op":sy, "coupling_op":sx, \
"init_state":tempo.operators.spin_dm("z+")},
{"sys_op":sz, "coupling_op":sy, \
"init_state":tempo.operators.spin_dm("x+")}]
results = []
for i, base in enumerate(bases):
system = tempo.System(0.5 * base["sys_op"])
correlations = tempo.PowerLawSD(alpha=alpha,
zeta=1,
cutoff=omega_cutoff,
cutoff_type='exponential',
max_correlation_time=8.0)
bath = tempo.Bath(0.5 * base["coupling_op"], correlations)
tempo_parameters = tempo.TempoParameters(dt=0.1,
dkmax=30,
epsrel=10**(-5))
dynamics = tempo.tempo_compute(system=system,
bath=bath,
initial_state=base["init_state"],
start_time=0.0,
end_time=1.0,
parameters=tempo_parameters,
backend=backend)
_, s_x = dynamics.expectations(0.5 * tempo.operators.sigma("x"),
real=True)
_, s_y = dynamics.expectations(0.5 * tempo.operators.sigma("y"),
real=True)
_, s_z = dynamics.expectations(0.5 * tempo.operators.sigma("z"),
real=True)
if i == 0:
results.append(np.array([s_x, s_y, s_z]))
elif i == 1:
results.append(np.array([s_y, s_z, s_x]))
elif i == 2:
results.append(np.array([s_z, s_x, s_y]))
assert np.allclose(results[0], results[1], atol=tempo_parameters.epsrel)
assert np.allclose(results[0], results[2], atol=tempo_parameters.epsrel)
def test_tempo_dynamics_reference():
system = tempo.System(0.5 * tempo.operators.sigma("x"))
correlations = tempo.PowerLawSD(alpha=0.1,
zeta=1,
cutoff=1.0,
cutoff_type='exponential',
max_correlation_time=0.5)
bath = tempo.Bath(0.5 * tempo.operators.sigma("z"), correlations)
tempo_parameters = tempo.TempoParameters(dt=0.1, dkmax=10, epsrel=10**(-4))
tempo_A = tempo.Tempo(system=system,
bath=bath,
parameters=tempo_parameters,
initial_state=tempo.operators.spin_dm("up"),
start_time=0.0)
dynamics_1 = tempo_A.compute(end_time=0.2)
t_1, sz_1 = dynamics_1.expectations(tempo.operators.sigma("z"))
tempo_A.compute(end_time=0.4)
dynamics_2 = tempo_A.get_dynamics()
t_2, sz_2 = dynamics_2.expectations(tempo.operators.sigma("z"))
assert dynamics_1 == dynamics_2
assert len(t_2) > len(t_1)
plt.legend()
# ### B.3: Create time dependent system object
# In[6]:
def hamiltonian_t(t):
return detuning(t) / 2.0 * tempo.operators.sigma("z") + gaussian_shape(
t, area=np.pi / 2.0, tau=0.245) / 2.0 * tempo.operators.sigma("x")
system = tempo.TimeDependentSystem(hamiltonian_t)
correlations = tempo.PowerLawSD(alpha=alpha,
zeta=3,
cutoff=omega_cutoff,
cutoff_type='gaussian',
max_correlation_time=5.0,
temperature=temperature)
bath = tempo.Bath(tempo.operators.sigma("z") / 2.0, correlations)
# ### B.4: TEMPO computation
# With all physical objects defined, we are now ready to compute the dynamics of the quantum dot using TEMPO (using quite rough convergence parameters):
# In[7]:
tempo_parameters = tempo.TempoParameters(dt=0.05, dkmax=40, epsrel=10**(-5))
tempo_sys = tempo.Tempo(system=system,
bath=bath,
initial_state=initial_state,
# exact result:
def exact_result(t):
x = (t*cutoff_E)**2
phi = 2 * alpha_E * (1 + (x-1)/(x+1)**2)
y_plus = np.exp(-phi)
dm = (1 - y_plus) * tempo.operators.spin_dm("mixed") \
+ y_plus * tempo.operators.spin_dm("y+")
return dm
rho_E = exact_result(t_end_E)
correlations_E = tempo.PowerLawSD(alpha=alpha_E,
zeta=3.0,
cutoff=cutoff_E,
cutoff_type="exponential",
temperature=temperature_E,
name="superohmic",
)
bath_E = tempo.Bath(coupling_operator_E,
correlations_E,
name="superohmic phonon bath")
system_E = tempo.System(h_sys_E)
# -----------------------------------------------------------------------------
@pytest.mark.parametrize('backend,backend_config',
[("tensor-network", {"backend":"numpy"}),])
def test_tensor_network_tempo_backend_E(backend, backend_config):
tempo_params_E = tempo.TempoParameters(dt=0.4,
dkmax=5,
"""
import pytest
import time_evolving_mpo as tempo
TEMP_FILE = "tests/data/temp.processTensor"
# -- prepare a process tensor -------------------------------------------------
system = tempo.System(tempo.operators.sigma("x"))
initial_state = tempo.operators.spin_dm("z+")
correlations = tempo.PowerLawSD(alpha=0.3,
zeta=1.0,
cutoff=5.0,
cutoff_type="exponential",
temperature=0.2,
name="ohmic")
bath = tempo.Bath(0.5*tempo.operators.sigma("z"),
correlations,
name="phonon bath")
tempo_params = tempo.PtTempoParameters(dt=0.1,
dkmax=5,
epsrel=10**(-5))
pt = tempo.pt_tempo_compute(bath,
start_time=0.0,
end_time=1.0,
parameters=tempo_params)
pt.export(TEMP_FILE, overwrite=True)
del pt
# -------------------------------------------------
# ## Example A - The Spin Boson Model
#
# As a first example let's try to reconstruct one of the lines in figure 2a of [Strathearn2018] ([Nat. Comm. 9, 3322 (2018)](https://doi.org/10.1038/s41467-018-05617-3) / [arXiv:1711.09641v3](https://arxiv.org/abs/1711.09641)). In this example we compute the time evolution of a spin which is strongly coupled to an ohmic bath (spin-boson model). Before we go through this step by step below, let's have a brief look at the script that will do the job - just to have an idea where we are going:
# In[3]:
Omega = 1.0
omega_cutoff = 5.0
alpha = 0.3
system = tempo.System(0.5 * Omega * tempo.operators.sigma("x"))
correlations = tempo.PowerLawSD(alpha=alpha,
zeta=1,
cutoff=omega_cutoff,
cutoff_type='exponential',
max_correlation_time=8.0)
bath = tempo.Bath(0.5 * tempo.operators.sigma("z"), correlations)
tempo_parameters = tempo.TempoParameters(dt=0.1, dkmax=30, epsrel=10**(-4))
dynamics = tempo.tempo_compute(system=system,
bath=bath,
initial_state=tempo.operators.spin_dm("up"),
start_time=0.0,
end_time=15.0,
parameters=tempo_parameters)
t, s_z = dynamics.expectations(0.5 * tempo.operators.sigma("z"), real=True)
plt.plot(t, s_z, label=r'$\alpha=0.3$')
plt.xlabel(r'$t\,\Omega$')
alpha_C = 0.3
cutoff_C = 5.0
temperature_C = 0.0
# end time
t_end_C = 1.0
# result obtained with release code (made hermitian, dkmax=10):
rho_C = np.array(
[[ 0.12576653+0.j ,-0.11739956-0.14312036j, 0.12211454-0.05963583j],
[-0.11739956+0.14312036j, 0.61315893+0.j ,-0.06636825+0.26917271j],
[ 0.12211454+0.05963583j,-0.06636825-0.26917271j, 0.26107455+0.j ]])
correlations_C = tempo.PowerLawSD(alpha=alpha_C,
zeta=1.0,
cutoff=cutoff_C,
cutoff_type="exponential",
temperature=temperature_C,
name="ohmic")
bath_C = tempo.Bath(coupling_operator_C,
correlations_C,
name="phonon bath")
system_C = tempo.System(h_sys_C,
gammas=[gamma_C_1, gamma_C_2],
lindblad_operators=[lindblad_operators_C_1,
lindblad_operators_C_2])
# -----------------------------------------------------------------------------
@pytest.mark.parametrize('backend',["tensor-network"])
def test_tensor_network_tempo_backend_C(backend):