__all__ = ['correlation_2op_1t', 'correlation_2op_2t', 'correlation_3op_1t',
'correlation_3op_2t', 'coherence_function_g1',
'coherence_function_g2', 'spectrum', 'spectrum_correlation_fft',
'correlation_ss', 'correlation', 'correlation_4op_1t',
'correlation_4op_2t', 'spectrum_ss', 'spectrum_pi']
from re import sub
from warnings import warn
import warnings
import types
import numpy as np
import scipy.fftpack
from qutip.eseries import esval, esspec
from qutip.essolve import ode2es
from qutip.expect import expect
from qutip.mesolve import mesolve
from qutip.mcsolve import mcsolve
from qutip.operators import qeye
from qutip.qobj import Qobj, isket, issuper
from qutip.qobjevo import QobjEvo
from qutip.rhs_generate import rhs_clear, _td_wrap_array_str
from qutip.cy.utilities import _cython_build_cleanup
from qutip.settings import debug
from qutip.solver import Options, config
from qutip.steadystate import steadystate
from qutip.states import ket2dm
from qutip.superoperator import liouvillian, spre, mat2vec
from qutip.tensor import tensor
if debug:
import inspect
# -----------------------------------------------------------------------------
# PUBLIC API
# -----------------------------------------------------------------------------
# low level correlation
[docs]def correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the two-operator two-time correlation function:
:math:`\left<A(t+\tau)B(t)\right>`
along one time axis using the quantum regression theorem and the evolution
solver indicated by the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool {False, True}
If `True`, calculate :math:`\left<A(t)B(t+\tau)\right>` instead of
:math:`\left<A(t+\tau)B(t)\right>`.
solver : str {'me', 'mc', 'es'}
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
Solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_vec : ndarray
An array of correlation values for the times specified by `taulist`.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
if debug:
print(inspect.stack()[0][3])
if reverse:
A_op = a_op
B_op = b_op
C_op = 1
else:
A_op = 1
B_op = a_op
C_op = b_op
return _correlation_2t(H, state0, [0], taulist, c_ops, A_op, B_op, C_op,
solver=solver, args=args, options=options)[0]
[docs]def correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the two-operator two-time correlation function:
:math:`\left<A(t+\tau)B(t)\right>`
along two time axes using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\rho_0` or state vector
:math:`\psi_0`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
tlist : array_like
list of times for :math:`t`. tlist must be positive and contain the
element `0`. When taking steady-steady correlations only one tlist
value is necessary, i.e. when :math:`t \rightarrow \infty`; here
tlist is automatically set, ignoring user input.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool {False, True}
If `True`, calculate :math:`\left<A(t)B(t+\tau)\right>` instead of
:math:`\left<A(t+\tau)B(t)\right>`.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_mat : ndarray
An 2-dimensional array (matrix) of correlation values for the times
specified by `tlist` (first index) and `taulist` (second index). If
`tlist` is `None`, then a 1-dimensional array of correlation values
is returned instead.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
if debug:
print(inspect.stack()[0][3])
if tlist is None:
return correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op,
solver=solver, reverse=reverse, args=args,
options=options)
else:
if reverse:
A_op = a_op
B_op = b_op
C_op = 1
else:
A_op = 1
B_op = a_op
C_op = b_op
return _correlation_2t(H, state0, tlist, taulist,
c_ops, A_op, B_op, C_op,
solver=solver, args=args, options=options)
[docs]def correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op,
solver="me", args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the three-operator two-time correlation function:
:math:`\left<A(t)B(t+\tau)C(t)\right>`
along one time axis using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Note: it is not possibly to calculate a physically meaningful correlation
of this form where :math:`\tau<0`.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
rho0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_vec : array
An array of correlation values for the times specified by `taulist`
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
if debug:
print(inspect.stack()[0][3])
return _correlation_2t(H, state0, [0], taulist, c_ops, a_op, b_op, c_op,
solver=solver, args=args, options=options)[0]
[docs]def correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
solver="me", args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the three-operator two-time correlation function:
:math:`\left<A(t)B(t+\tau)C(t)\right>`
along two time axes using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Note: it is not possibly to calculate a physically meaningful correlation
of this form where :math:`\tau<0`.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
rho0 : Qobj
Initial state density matrix :math:`\rho_0` or state vector
:math:`\psi_0`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
tlist : array_like
list of times for :math:`t`. tlist must be positive and contain the
element `0`. When taking steady-steady correlations only one tlist
value is necessary, i.e. when :math:`t \rightarrow \infty`; here
tlist is automatically set, ignoring user input.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_mat : array
An 2-dimensional array (matrix) of correlation values for the times
specified by `tlist` (first index) and `taulist` (second index). If
`tlist` is `None`, then a 1-dimensional array of correlation values
is returned instead.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
if debug:
print(inspect.stack()[0][3])
if tlist is None:
return correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op,
solver=solver, args=args, options=options)
else:
return _correlation_2t(H, state0, tlist, taulist,
c_ops, a_op, b_op, c_op,
solver=solver, args=args, options=options)
# high level correlation
[docs]def coherence_function_g1(H, state0, taulist, c_ops, a_op, solver="me",
args={}, options=Options(ntraj=[20, 100])):
r"""
Calculate the normalized first-order quantum coherence function:
.. math::
g^{(1)}(\tau) =
\frac{\langle A^\dagger(\tau)A(0)\rangle}
{\sqrt{\langle A^\dagger(\tau)A(\tau)\rangle
\langle A^\dagger(0)A(0)\rangle}}
using the quantum regression theorem and the evolution solver indicated by
the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
solver : str
choice of solver (`me` for master-equation and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
g1, G1 : tuple
The normalized and unnormalized second-order coherence function.
"""
# first calculate the photon number
if state0 is None:
state0 = steadystate(H, c_ops)
n = np.array([expect(state0, a_op.dag() * a_op)])
else:
n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op],
options=options).expect[0]
# calculate the correlation function G1 and normalize with n to obtain g1
G1 = correlation_2op_1t(H, state0, taulist, c_ops, a_op.dag(), a_op,
solver=solver, args=args, options=options)
g1 = G1 / np.sqrt(n[0] * n)
return g1, G1
[docs]def coherence_function_g2(H, state0, taulist, c_ops, a_op, solver="me", args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the normalized second-order quantum coherence function:
.. math::
g^{(2)}(\tau) =
\frac{\langle A^\dagger(0)A^\dagger(\tau)A(\tau)A(0)\rangle}
{\langle A^\dagger(\tau)A(\tau)\rangle
\langle A^\dagger(0)A(0)\rangle}
using the quantum regression theorem and the evolution solver indicated by
the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
args : dict
Dictionary of arguments to be passed to solver.
solver : str
choice of solver (`me` for master-equation and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
g2, G2 : tuple
The normalized and unnormalized second-order coherence function.
"""
# first calculate the photon number
if state0 is None:
state0 = steadystate(H, c_ops)
n = np.array([expect(state0, a_op.dag() * a_op)])
else:
n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op], args=args).expect[0]
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_3op_1t(H, state0, taulist, c_ops,
a_op.dag(), a_op.dag()*a_op, a_op,
solver=solver, args=args, options=options)
g2 = G2 / (n[0] * n)
return g2, G2
# spectrum
[docs]def spectrum(H, wlist, c_ops, a_op, b_op, solver="es", use_pinv=False):
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:`qutip.qobj`
system Hamiltonian.
wlist : array_like
list of frequencies for :math:`\omega`.
c_ops : list
list of collapse operators.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
solver : str
choice of solver (`es` for exponential series and
`pi` for psuedo-inverse).
use_pinv : bool
For use with the `pi` solver: if `True` use numpy's pinv method,
otherwise use a generic solver.
Returns
-------
spectrum : array
An array with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
if debug:
print(inspect.stack()[0][3])
if solver == "es":
return _spectrum_es(H, wlist, c_ops, a_op, b_op)
elif solver == "pi":
return _spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv)
else:
raise ValueError("Unrecognized choice of solver" +
"%s (use es or pi)." % solver)
[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: boolean
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)'.
"""
if debug:
print(inspect.stack()[0][3])
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 Exception('tlist must be equally spaced for FFT.')
if inverse:
F = N * scipy.fftpack.ifft(y)
else:
F = 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])
# -----------------------------------------------------------------------------
# LEGACY API
# -----------------------------------------------------------------------------
# low level correlation
[docs]def correlation_ss(H, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the two-operator two-time correlation function:
.. math::
\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>
along one time axis (given steady-state initial conditions) using the
quantum regression theorem and the evolution solver indicated by the
`solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : *bool*
If `True`, calculate
:math:`\lim_{t \to \infty} \left<A(t)B(t+\tau)\right>` instead of
:math:`\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>`.
solver : str
choice of solver (`me` for master-equation and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_vec : array
An array of correlation values for the times specified by `tlist`.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
warn("correlation_ss() now legacy, please use correlation_2op_1t() with" +
"initial state as None", FutureWarning)
if debug:
print(inspect.stack()[0][3])
return correlation_2op_1t(H, None, taulist, c_ops, a_op, b_op,
solver=solver, reverse=reverse, args=args,
options=options)
[docs]def correlation(H, state0, tlist, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the two-operator two-time correlation function:
:math:`\left<A(t+\tau)B(t)\right>`
along two time axes using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
tlist : array_like
list of times for :math:`t`. tlist must be positive and contain the
element `0`. When taking steady-steady correlations only one tlist
value is necessary, i.e. when :math:`t \rightarrow \infty`; here
tlist is automatically set, ignoring user input.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : *bool*
If `True`, calculate :math:`\left<A(t)B(t+\tau)\right>` instead of
:math:`\left<A(t+\tau)B(t)\right>`.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_mat : array
An 2-dimensional array (matrix) of correlation values for the times
specified by `tlist` (first index) and `taulist` (second index). If
`tlist` is `None`, then a 1-dimensional array of correlation values
is returned instead.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
warn("correlation() now legacy, please use correlation_2op_2t()",
FutureWarning)
if debug:
print(inspect.stack()[0][3])
return correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op,
solver=solver, reverse=reverse, args=args,
options=options)
[docs]def correlation_4op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, d_op,
solver="me", args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the four-operator two-time correlation function:
:math:`\left<A(t)B(t+\tau)C(t+\tau)D(t)\right>`
along one time axis using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Note: it is not possibly to calculate a physically meaningful correlation
of this form where :math:`\tau<0`.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
rho0 : Qobj
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
d_op : Qobj
operator D.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_vec : array
An array of correlation values for the times specified by `taulist`.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
.. note:: Deprecated in QuTiP 3.1
Use correlation_3op_1t() instead.
"""
warn("correlation_4op_1t() now legacy, please use correlation_3op_1t()",
FutureWarning)
warn("the reverse argument has been removed as it did not contain any" +
"new physical information", DeprecationWarning)
if debug:
print(inspect.stack()[0][3])
return correlation_3op_1t(H, state0, taulist, c_ops,
a_op, b_op * c_op, d_op,
solver=solver, args=args, options=options)
[docs]def correlation_4op_2t(H, state0, tlist, taulist, c_ops,
a_op, b_op, c_op, d_op, solver="me", args={},
options=Options(ntraj=[20, 100])):
r"""
Calculate the four-operator two-time correlation function:
:math:`\left<A(t)B(t+\tau)C(t+\tau)D(t)\right>`
along two time axes using the quantum regression theorem and the
evolution solver indicated by the `solver` parameter.
Note: it is not possibly to calculate a physically meaningful correlation
of this form where :math:`\tau<0`.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
rho0 : Qobj
Initial state density matrix :math:`\rho_0` or state vector
:math:`\psi_0`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
tlist : array_like
list of times for :math:`t`. tlist must be positive and contain the
element `0`. When taking steady-steady correlations only one tlist
value is necessary, i.e. when :math:`t \rightarrow \infty`; here
tlist is automatically set, ignoring user input.
taulist : array_like
list of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
d_op : Qobj
operator D.
solver : str
choice of solver (`me` for master-equation, `mc` for Monte Carlo, and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
corr_mat : array
An 2-dimensional array (matrix) of correlation values for the times
specified by `tlist` (first index) and `taulist` (second index). If
`tlist` is `None`, then a 1-dimensional array of correlation values
is returned instead.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
warn("correlation_4op_2t() now legacy, please use correlation_3op_2t()",
FutureWarning)
warn("the reverse argument has been removed as it did not contain any" +
"new physical information", DeprecationWarning)
if debug:
print(inspect.stack()[0][3])
return correlation_3op_2t(H, state0, tlist, taulist, c_ops,
a_op, b_op * c_op, d_op,
solver=solver, args=args, options=options)
# spectrum
[docs]def spectrum_ss(H, wlist, c_ops, a_op, b_op):
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 an eseries based solver Note: this spectrum is only defined for
stationary statistics (uses steady state rho0).
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian.
wlist : array_like
list of frequencies for :math:`\omega`.
c_ops : *list* of :class:`qutip.qobj`
list of collapse operators.
a_op : :class:`qutip.qobj`
operator A.
b_op : :class:`qutip.qobj`
operator B.
use_pinv : *bool*
If `True` use numpy's `pinv` method, otherwise use a generic solver.
Returns
-------
spectrum : array
An array with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
warn("spectrum_ss() now legacy, please use spectrum()", FutureWarning)
return spectrum(H, wlist, c_ops, a_op, b_op, solver="es")
[docs]def spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False):
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 a psuedo-inverse method. Note: this spectrum is only defined for
stationary statistics (uses steady state rho0)
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian.
wlist : array_like
list of frequencies for :math:`\omega`.
c_ops : *list* of :class:`qutip.qobj`
list of collapse operators.
a_op : :class:`qutip.qobj`
operator A.
b_op : :class:`qutip.qobj`
operator B.
use_pinv : *bool*
If `True` use numpy's pinv method, otherwise use a generic solver.
Returns
-------
spectrum : array
An array with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
warn("spectrum_pi() now legacy, please use spectrum()", FutureWarning)
return spectrum(H, wlist, c_ops, a_op, b_op,
solver="pi", use_pinv=use_pinv)
# -----------------------------------------------------------------------------
# PRIVATE SOLVER METHODS
# -----------------------------------------------------------------------------
# master 2t correlation solver
def _correlation_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
solver="me", args={}, options=Options()):
"""
Internal function for calling solvers in order to calculate the
three-operator two-time correlation function:
<A(t)B(t+tau)C(t)>
"""
# Note: the current form of the correlator is sufficient for all possible
# two-time correlations (incuding those with 2ops vs 3). Ex: to compute a
# correlation of the form <A(t+tau)B(t)>: a_op = identity, b_op = A,
# and c_op = B.
if debug:
print(inspect.stack()[0][3])
if min(tlist) != 0:
raise TypeError("tlist must be positive and contain the element 0.")
if min(taulist) != 0:
raise TypeError("taulist must be positive and contain the element 0.")
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
H, c_ops, args = _td_wrap_array_str(H, c_ops, args, tlist)
if solver == "me":
return _correlation_me_2t(H, state0, tlist, taulist,
c_ops, a_op, b_op, c_op,
args=args, options=options)
elif solver == "mc":
return _correlation_mc_2t(H, state0, tlist, taulist,
c_ops, a_op, b_op, c_op,
args=args, options=options)
elif solver == "es":
return _correlation_es_2t(H, state0, tlist, taulist,
c_ops, a_op, b_op, c_op)
else:
raise ValueError("Unrecognized choice of solver" +
"%s (use me, mc, or es)." % solver)
# master equation solvers
def _correlation_me_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
args={}, options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a master equation solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
rho_t = mesolve(H, rho0, tlist, c_ops, [],
args=args, options=options).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
taulist = np.asarray(taulist)
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
for t_idx, rho in enumerate(rho_t):
corr_mat[t_idx, :] = mesolve(
H, c_op * rho * a_op, taulist + tlist[t_idx], c_ops,
[b_op], args=args, options=options
).expect[0]
if t_idx == 1:
options.rhs_reuse = True
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
return corr_mat
# exponential series solvers
def _correlation_es_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using an exponential series solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
# The user-facing eseries and ode2es are deprecated from QuTiP 4.6, pending
# removal in QuTiP 5.0, however they are being maintained for internal use
# in correlation, so should not raise warnings to the user not matter what
# their settings.
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=DeprecationWarning)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, c_op * rho_t * a_op)
corr_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
return corr_mat
def _spectrum_es(H, wlist, c_ops, a_op, b_op):
r"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\tau)B(0)\right>`.
"""
if debug:
print(inspect.stack()[0][3])
# construct the Liouvillian
L = liouvillian(H, c_ops)
# 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)
# The user-facing eseries and ode2es are deprecated from QuTiP 4.6, pending
# removal in QuTiP 5.0, however they are being maintained for internal use
# in spectrum, so should not raise warnings to the user not matter what
# their settings.
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=DeprecationWarning)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covariance
cov_es = corr_es - a_op_ss * b_op_ss
# tidy up covariance (to combine, e.g., zero-frequency components that
# cancel)
cov_es.tidyup()
# spectrum
return esspec(cov_es, wlist)
# Monte Carlo solvers
def _correlation_mc_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
args={}, options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a Monte Carlo solver.
"""
if not c_ops:
raise TypeError("If no collapse operators are required, use the `me`" +
"or `es` solvers")
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
raise NotImplementedError("steady state not implemented for " +
"mc solver, please use `es` or `me`")
elif not isket(state0):
raise TypeError("state0 must be a state vector.")
psi0 = state0
if debug:
print(inspect.stack()[0][3])
psi_t_mat = mcsolve(
H, psi0, tlist, c_ops, [],
args=args, ntraj=options.ntraj[0], options=options, progress_bar=None
).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
tlist = np.asarray(tlist)
taulist = np.asarray(taulist)
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
# calculation of <A(t)B(t+tau)C(t)> from only knowledge of psi0 requires
# averaging over both t and tau
for t_idx in range(np.size(tlist)):
for trial_idx in range(options.ntraj[0]):
if isinstance(a_op, Qobj) and isinstance(c_op, Qobj):
if a_op.dag() == c_op:
# A shortcut here, requires only 1/4 the trials
chi_0 = (options.mc_corr_eps + c_op) * \
psi_t_mat[trial_idx, t_idx]
# evolve these states and calculate expectation value of B
c_tau = chi_0.norm()**2 * mcsolve(
H, chi_0/chi_0.norm(), taulist + tlist[t_idx], c_ops,
[b_op],
args=args, ntraj=options.ntraj[1], options=options,
progress_bar=None
).expect[0]
# final correlation vector computed by combining the
# averages
corr_mat[t_idx, :] += c_tau/options.ntraj[1]
else:
# otherwise, need four trial wavefunctions
# (Ad+C)*psi_t, (Ad+iC)*psi_t, (Ad-C)*psi_t, (Ad-iC)*psi_t
if isinstance(a_op, Qobj):
a_op_dag = a_op.dag()
else:
# assume this is a number, ex. i.e. a_op = 1
# if this is not correct, the over-loaded addition
# operation will raise errors
a_op_dag = a_op
chi_0 = [(options.mc_corr_eps + a_op_dag +
np.exp(1j*x*np.pi/2)*c_op) *
psi_t_mat[trial_idx, t_idx]
for x in range(4)]
# evolve these states and calculate expectation value of B
c_tau = [
chi.norm()**2 * mcsolve(
H, chi/chi.norm(), taulist + tlist[t_idx], c_ops,
[b_op],
args=args, ntraj=options.ntraj[1], options=options,
progress_bar=None
).expect[0]
for chi in chi_0
]
# final correlation vector computed by combining the averages
corr_mat_add = np.asarray(
1.0 / (4*options.ntraj[0]) *
(c_tau[0] - c_tau[2] - 1j*c_tau[1] + 1j*c_tau[3]),
dtype=corr_mat.dtype
)
corr_mat[t_idx, :] += corr_mat_add
if t_idx == 1:
options.rhs_reuse = True
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
return corr_mat
# pseudo-inverse solvers
def _spectrum_pi(H, wlist, c_ops, 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>`.
"""
L = H if issuper(H) else liouvillian(H, c_ops)
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
N = np.prod(L.dims[0][0])
A = L.full()
b = spre(b_op).full()
a = spre(a_op).full()
tr_vec = np.transpose(mat2vec(tr_mat.full()))
rho_ss = steadystate(L)
rho = np.transpose(mat2vec(rho_ss.full()))
I = np.identity(N * N)
P = np.kron(np.transpose(rho), tr_vec)
Q = I - P
spectrum = np.zeros(len(wlist))
for idx, w in enumerate(wlist):
if use_pinv:
MMR = np.linalg.pinv(-1.0j * w * I + A)
else:
MMR = np.dot(Q, np.linalg.solve(-1.0j * w * I + A, Q))
s = np.dot(tr_vec,
np.dot(a, np.dot(MMR, np.dot(b, np.transpose(rho)))))
spectrum[idx] = -2 * np.real(s[0, 0])
return spectrum