def floquet_master_equation_rates(f_modes_0,
f_energies,
c_op,
H,
T,
args,
J_cb,
w_th,
kmax=5,
f_modes_table_t=None):
"""
Calculate the rates and matrix elements for the Floquet-Markov master
equation.
Parameters
----------
f_modes_0 : list of :class:`qutip.qobj` (kets)
A list of initial Floquet modes.
f_energies : array
The Floquet energies.
c_op : :class:`qutip.qobj`
The collapse operators describing the dissipation.
H : :class:`qutip.qobj`
System Hamiltonian, time-dependent with period `T`.
T : float
The period of the time-dependence of the hamiltonian.
args : dictionary
Dictionary with variables required to evaluate H.
J_cb : callback functions
A callback function that computes the noise power spectrum, as
a function of frequency, associated with the collapse operator `c_op`.
w_th : float
The temperature in units of frequency.
k_max : int
The truncation of the number of sidebands (default 5).
f_modes_table_t : nested list of :class:`qutip.qobj` (kets)
A lookup-table of Floquet modes at times precalculated by
:func:`qutip.floquet.floquet_modes_table` (optional).
Returns
-------
output : list
A list (Delta, X, Gamma, A) containing the matrices Delta, X, Gamma
and A used in the construction of the Floquet-Markov master equation.
"""
N = len(f_energies)
M = 2 * kmax + 1
omega = (2 * pi) / T
Delta = np.zeros((N, N, M))
X = np.zeros((N, N, M), dtype=complex)
Gamma = np.zeros((N, N, M))
A = np.zeros((N, N))
nT = 100
dT = T / nT
tlist = np.arange(dT, T + dT / 2, dT)
if f_modes_table_t is None:
f_modes_table_t = floquet_modes_table(f_modes_0, f_energies,
np.linspace(0, T, nT + 1), H, T,
args)
for t in tlist:
# TODO: repeated invocations of floquet_modes_t is
# inefficient... make a and b outer loops and use the mesolve
# instead of the propagator.
# f_modes_t = floquet_modes_t(f_modes_0, f_energies, t, H, T, args)
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
for a in range(N):
for b in range(N):
k_idx = 0
for k in range(-kmax, kmax + 1, 1):
X[a, b, k_idx] += (dT / T) * exp(-1j * k * omega * t) * \
(f_modes_t[a].dag() * c_op * f_modes_t[b])[0, 0]
k_idx += 1
Heaviside = lambda x: ((np.sign(x) + 1) / 2.0)
for a in range(N):
for b in range(N):
k_idx = 0
for k in range(-kmax, kmax + 1, 1):
Delta[a, b, k_idx] = f_energies[a] - f_energies[b] + k * omega
Gamma[a, b, k_idx] = 2 * pi * Heaviside(Delta[a, b, k_idx]) * \
J_cb(Delta[a, b, k_idx]) * abs(X[a, b, k_idx]) ** 2
k_idx += 1
for a in range(N):
for b in range(N):
for k in range(-kmax, kmax + 1, 1):
k1_idx = k + kmax
k2_idx = -k + kmax
A[a, b] += Gamma[a, b, k1_idx] + \
n_thermal(abs(Delta[a, b, k1_idx]), w_th) * \
(Gamma[a, b, k1_idx] + Gamma[b, a, k2_idx])
return Delta, X, Gamma, A
def floquet_master_equation_rates(f_modes_0, f_energies, c_op, H, T,
args, J_cb, w_th, kmax=5,
f_modes_table_t=None):
"""
Calculate the rates and matrix elements for the Floquet-Markov master
equation.
Parameters
----------
f_modes_0 : list of :class:`qutip.qobj` (kets)
A list of initial Floquet modes.
f_energies : array
The Floquet energies.
c_op : :class:`qutip.qobj`
The collapse operators describing the dissipation.
H : :class:`qutip.qobj`
System Hamiltonian, time-dependent with period `T`.
T : float
The period of the time-dependence of the hamiltonian.
args : dictionary
Dictionary with variables required to evaluate H.
J_cb : callback functions
A callback function that computes the noise power spectrum, as
a function of frequency, associated with the collapse operator `c_op`.
w_th : float
The temperature in units of frequency.
k_max : int
The truncation of the number of sidebands (default 5).
f_modes_table_t : nested list of :class:`qutip.qobj` (kets)
A lookup-table of Floquet modes at times precalculated by
:func:`qutip.floquet.floquet_modes_table` (optional).
Returns
-------
output : list
A list (Delta, X, Gamma, A) containing the matrices Delta, X, Gamma
and A used in the construction of the Floquet-Markov master equation.
"""
N = len(f_energies)
M = 2 * kmax + 1
omega = (2 * pi) / T
Delta = np.zeros((N, N, M))
X = np.zeros((N, N, M), dtype=complex)
Gamma = np.zeros((N, N, M))
A = np.zeros((N, N))
nT = 100
dT = T / nT
tlist = np.arange(dT, T + dT / 2, dT)
if f_modes_table_t is None:
f_modes_table_t = floquet_modes_table(f_modes_0, f_energies,
np.linspace(0, T, nT + 1), H, T,
args)
for t in tlist:
# TODO: repeated invocations of floquet_modes_t is
# inefficient... make a and b outer loops and use the mesolve
# instead of the propagator.
# f_modes_t = floquet_modes_t(f_modes_0, f_energies, t, H, T, args)
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
for a in range(N):
for b in range(N):
k_idx = 0
for k in range(-kmax, kmax + 1, 1):
X[a, b, k_idx] += (dT / T) * exp(-1j * k * omega * t) * \
(f_modes_t[a].dag() * c_op * f_modes_t[b])[0, 0]
k_idx += 1
Heaviside = lambda x: ((np.sign(x) + 1) / 2.0)
for a in range(N):
for b in range(N):
k_idx = 0
for k in range(-kmax, kmax + 1, 1):
Delta[a, b, k_idx] = f_energies[a] - f_energies[b] + k * omega
Gamma[a, b, k_idx] = 2 * pi * Heaviside(Delta[a, b, k_idx]) * \
J_cb(Delta[a, b, k_idx]) * abs(X[a, b, k_idx]) ** 2
k_idx += 1
for a in range(N):
for b in range(N):
for k in range(-kmax, kmax + 1, 1):
k1_idx = k + kmax
k2_idx = -k + kmax
A[a, b] += Gamma[a, b, k1_idx] + \
n_thermal(abs(Delta[a, b, k1_idx]), w_th) * \
(Gamma[a, b, k1_idx] + Gamma[b, a, k2_idx])
return Delta, X, Gamma, A