def test_pure_dephasing():
"""
pure_dephasing: Tests the pure_dephasing integrand.
"""
coup_strength, bath_broad, bath_freq = 0.08, 0.4, 1.0
# Test only at short times
tlist = np.linspace(0, 20, 100)
# Set qubit frequency to 0 to see only the dynamics due to the interaction.
wq = 0
# Zero temperature case
beta = np.inf
ck1, vk1 = nonmatsubara_exponents(coup_strength, bath_broad, bath_freq,
beta)
# ck2, vk2 = matsubara_exponents(lam, gamma, w0, beta, N_exp)
mats_data_zero = matsubara_zero_analytical(coup_strength, bath_broad,
bath_freq, tlist)
ck20, vk20 = biexp_fit(tlist, mats_data_zero)
ck = np.concatenate([ck1, ck20])
vk = np.concatenate([vk1, vk20])
pd_analytical = pure_dephasing_evolution(tlist, coup_strength, bath_broad,
bath_freq, beta, wq)
pd_numerical_fitting = pure_dephasing_evolution_analytical(
tlist, wq, ck, vk)
residue = np.abs(pd_analytical - pd_numerical_fitting)
assert_(np.max(residue) < 1e-3)
def test_heom():
"""
heom: Tests the HEOM method.
"""
Q = sigmax()
wq = 1.0
lam, gamma, w0 = 0.2, 0.05, 1.0
tlist = np.linspace(0, 200, 1000)
Nc = 9
# zero temperature case
beta = np.inf
Hsys = 0.5 * wq * sigmaz()
initial_ket = basis(2, 1)
rho0 = initial_ket * initial_ket.dag()
omega = np.sqrt(w0 ** 2 - (gamma / 2.0) ** 2)
a = omega + 1j * gamma / 2.0
lam_coeff = lam ** 2 / (2 * (omega))
options = Options(nsteps=1500, store_states=True, atol=1e-12, rtol=1e-12)
ck1, vk1 = nonmatsubara_exponents(lam, gamma, w0, beta)
mats_data_zero = matsubara_zero_analytical(lam, gamma, w0, tlist)
ck20, vk20 = biexp_fit(tlist, mats_data_zero)
hsolver = HeomUB(
Hsys,
Q,
lam_coeff,
np.concatenate([ck1, ck20]),
np.concatenate([-vk1, -vk20]),
ncut=Nc,
)
output = hsolver.solve(rho0, tlist)
result = np.real(expect(output.states, sigmaz()))
# Ignore Matsubara
hsolver2 = HeomUB(Hsys, Q, lam_coeff, ck1, -vk1, ncut=Nc)
output2 = hsolver2.solve(rho0, tlist)
result2 = np.real(expect(output2.states, sigmaz()))
steady_state_error = np.abs(result[-1] - result2[-1])
assert_(steady_state_error > 0.0)
wq = 1. delta = 0. coup_strength, bath_broad, bath_freq = 0.2, 0.05, 1. tlist = np.linspace(0, 200, 1000) Nc = 9 Hsys = 0.5 * wq * sigmaz() + 0.5 * delta * sigmax() initial_ket = basis(2, 1) rho0 = initial_ket*initial_ket.dag() omega = np.sqrt(bath_freq**2 - (bath_broad/2.)**2) options = Options(nsteps=1500, store_states=True, atol=1e-12, rtol=1e-12) # zero temperature case, renormalized coupling strength beta = np.inf lam_coeff = coup_strength**2/(2*(omega)) ck1, vk1 = nonmatsubara_exponents(coup_strength, bath_broad, bath_freq, beta) # Ignore Matsubara hsolver_no_matsubara = HeomUB(Hsys, Q, lam_coeff, ck1, -vk1, ncut=Nc) output_no_matsubara = hsolver_no_matsubara.solve(rho0, tlist, options) heom_result_no_matsubara = (np.real(expect(output_no_matsubara.states, sigmaz())) + 1)/2 # Add zero temperature Matsubara coefficients mats_data_zero = matsubara_zero_analytical(coup_strength, bath_broad, bath_freq, tlist) ck20, vk20 = biexp_fit(tlist, mats_data_zero) hsolver = HeomUB(Hsys, Q, lam_coeff, np.concatenate([ck1, ck20]), np.concatenate([-vk1, -vk20]), ncut=Nc) output = hsolver.solve(rho0, tlist, options) heom_result_with_matsubara = (np.real(expect(output.states, sigmaz())) + 1)/2
def test_exponents():
"""
correlation: Tests the Matsubara and non Matsubara exponents.
"""
lam, gamma, w0 = 0.2, 0.05, 1.0
tlist = np.linspace(0, 100, 1000)
# Finite temperature cases
beta = 0.1
N_exp = 200
ck_nonmats = [0.190164 - 0.004997j, 0.21017 + 0.004997j]
vk_nonmats = [-0.025 + 0.999687j, -0.025 - 0.999687j]
ck1, vk1 = nonmatsubara_exponents(lam, gamma, w0, beta)
assert_array_almost_equal(ck1, ck_nonmats)
assert_array_almost_equal(vk1, vk_nonmats)
ck2, vk2 = matsubara_exponents(lam, gamma, w0, beta, N_exp)
corr_fit = sum_of_exponentials(np.concatenate([ck1, ck2]),
np.concatenate([vk1, vk2]), tlist)
corr = sum_of_exponentials(ck_nonmats, vk_nonmats, tlist)
max_residue = np.max(np.abs(corr_fit - corr))
max_amplitude = np.max(np.abs(corr))
assert_(max_residue < max_amplitude / 1e5)
# Lower temperature
beta = 1.0
N_exp = 100
ck_nonmats = [0.011636 - 0.00046j, 0.031643 + 0.00046j]
vk_nonmats = [-0.025 + 0.999687j, -0.025 - 0.999687j]
ck1, vk1 = nonmatsubara_exponents(lam, gamma, w0, beta)
assert_array_almost_equal(ck1, ck_nonmats)
assert_array_almost_equal(vk1, vk_nonmats)
ck2, vk2 = matsubara_exponents(lam, gamma, w0, beta, N_exp)
corr_fit = sum_of_exponentials(np.concatenate([ck1, ck2]),
np.concatenate([vk1, vk2]), tlist)
corr = sum_of_exponentials(ck_nonmats, vk_nonmats, tlist)
max_residue = np.max(np.abs(corr_fit - corr))
max_amplitude = np.max(np.abs(corr))
assert_(max_residue < max_amplitude / 1e3)
# Zero temperature case
beta = np.inf
N_exp = 100
ck_nonmats = [0.0, 0.020006]
vk_nonmats = [-0.025 + 0.999687j, -0.025 - 0.999687j]
ck1, vk1 = nonmatsubara_exponents(lam, gamma, w0, beta)
assert_array_almost_equal(ck1, ck_nonmats)
assert_array_almost_equal(vk1, vk_nonmats)
ck2, vk2 = matsubara_exponents(lam, gamma, w0, beta, N_exp)
mats_data_zero = matsubara_zero_analytical(lam, gamma, w0, tlist)
ck_mats_zero = [-0.000208, -0.000107]
vk_mats_zero = [-1.613416, -0.329604]
ck20, vk20 = biexp_fit(tlist, mats_data_zero)
assert_array_almost_equal(ck20, ck_mats_zero)
assert_array_almost_equal(vk20, vk_mats_zero)