__all__ = ['spectrum', 'spectrum_correlation_fft']
import numpy as np
import scipy.fftpack
from .steadystate import steadystate
from ..core import liouvillian, spre, expect
from ..core import data as _data
from qutip.settings import settings
[docs]def spectrum(H, wlist, c_ops, a_op, b_op, solver="es"):
r"""
Calculate the spectrum of the correlation function
:math:`\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>`,
i.e., the Fourier transform of the correlation function:
.. math::
S(\omega) = \int_{-\infty}^{\infty}
\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>
e^{-i\omega\tau} d\tau.
using the solver indicated by the ``solver`` parameter. Note: this spectrum
is only defined for stationary statistics (uses steady state rho0)
Parameters
----------
H : :class:`.qobj`
system Hamiltonian.
wlist : array_like
List of frequencies for :math:`\omega`.
c_ops : list
List of collapse operators.
a_op : :class:`.Qobj`
Operator A.
b_op : :class:`.Qobj`
Operator B.
solver : str, {'es', 'pi', 'solve'}, default: 'es'
Choice of solver, ``es`` for exponential series and
``pi`` for psuedo-inverse, ``solve`` for generic solver.
Returns
-------
spectrum : array
An array with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
if not H.issuper:
L = liouvillian(H, c_ops)
else:
L = H + sum([lindblad_dissipator(c) for c in c_ops])
if solver == "es":
return _spectrum_es(L, wlist, a_op, b_op)
elif solver in ["pi", "solve"]:
return _spectrum_pi(L, wlist, a_op, b_op, use_pinv=solver=="pi")
raise ValueError(
f"Unrecognized choice of solver {solver} (use 'es', 'pi' or 'solve')."
)
[docs]def spectrum_correlation_fft(tlist, y, inverse=False):
"""
Calculate the power spectrum corresponding to a two-time correlation
function using FFT.
Parameters
----------
tlist : array_like
list/array of times :math:`t` which the correlation function is given.
y : array_like
list/array of correlations corresponding to time delays :math:`t`.
inverse: bool, default: False
boolean parameter for using a positive exponent in the Fourier
Transform instead. Default is False.
Returns
-------
w, S : tuple
Returns an array of angular frequencies 'w' and the corresponding
two-sided power spectrum 'S(w)'.
"""
tlist = np.asarray(tlist)
N = tlist.shape[0]
dt = tlist[1] - tlist[0]
if not np.allclose(np.diff(tlist), dt * np.ones(N - 1, dtype=float)):
raise ValueError('tlist must be equally spaced for FFT.')
F = (N * scipy.fftpack.ifft(y)) if inverse else scipy.fftpack.fft(y)
# calculate the frequencies for the components in F
f = scipy.fftpack.fftfreq(N, dt)
# re-order frequencies from most negative to most positive (centre on 0)
idx = np.array([], dtype='int')
idx = np.append(idx, np.where(f < 0.0))
idx = np.append(idx, np.where(f >= 0.0))
return 2 * np.pi * f[idx], 2 * dt * np.real(F[idx])
def _spectrum_es(L, wlist, a_op, b_op):
r"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\tau)B(0)\right>`.
"""
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steadystate(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
states, rates = _diagonal_evolution(L, b_op * rho0)
# correlation
ampls = [_data.expect(a_op.data, state) for state in states]
# make covariance
ampls += [-a_op_ss * b_op_ss]
rates += [0]
# Tidy up similar rates.
order = np.argsort(rates)
clean_rates = []
clean_ampls = []
prev_rate = np.nan
for idx in order:
if np.abs(rates[idx] - prev_rate) < settings.core["atol"]:
clean_ampls[-1] += ampls[idx]
else:
clean_rates.append(rates[idx])
clean_ampls.append(ampls[idx])
prev_rate = rates[idx]
# Remove 0 amplitude
rates, ampls = zip(*[
(rate, ampl)
for rate, ampl in zip(clean_rates, clean_ampls)
if np.abs(ampl) > settings.core["atol"]
])
ampls, rates = np.array(ampls), np.array(rates)
LW = np.subtract.outer(1j * np.array(wlist), rates).T
return (ampls @ (2 / LW)).real
#
# pseudo-inverse solvers
def _spectrum_pi(L, wlist, a_op, b_op, use_pinv=False):
r"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\tau)B(0)\right>`.
"""
dtype = type(L.data)
rho_ss = steadystate(L)
tr_mat = _data.identity[dtype](rho_ss.shape[0])
tr_vec = _data.column_stack(tr_mat).transpose()
rho = _data.column_stack(rho_ss.data)
A = L.data
ket = spre(b_op).data @ rho
bra = tr_vec @ spre(a_op).data
I = _data.identity[dtype](L.shape[0])
P = _data.kron(rho, tr_vec)
Q = I - P
spectrum = np.zeros(len(wlist))
for idx, w in enumerate(wlist):
if use_pinv and np.abs(w) > settings.core["atol"]:
# At w == 0., "L - iw" is singular
MMR = _data.inv(-1.0j * w * I + A)
else:
MMR = Q @ _data.solve(-1.0j * w * I + A, Q)
spectrum[idx] = -2 * _data.inner_op(bra, MMR, ket).real
return spectrum
def _diagonal_evolution(L, rho0, sparse=False):
if rho0.norm() < settings.core["atol"]:
return [_data.zeros["CSR"](*rho0.shape)], [0]
if isinstance(L.data, _data.CSR) and not sparse:
L = L.to(_data.Dense)
evals, evecs = _data.eigs(L.data)
size = rho0.shape[0] * rho0.shape[1]
r0 = _data.column_stack(rho0.data)
v0 = _data.solve(evecs, r0)
vv = evecs @ _data.diag(v0.to_array().flatten(), [0])
states = []
rates = []
for ket, rate in zip(_data.split_columns(vv), evals):
if _data.norm.l2(ket) < settings.core["atol"]:
continue
states.append(_data.column_unstack(ket, rho0.shape[0]))
rates.append(rate)
return states, rates