__all__ = [
'correlation_3op',
'correlation_2op_1t', 'correlation_2op_2t', 'correlation_3op_1t',
'correlation_3op_2t', 'coherence_function_g1', 'coherence_function_g2'
]
import numpy as np
import scipy.fftpack
from ..core import (
Qobj, QobjEvo, liouvillian, spre, unstack_columns, stack_columns,
tensor, expect, qeye_like, isket
)
from .mesolve import MESolver
from .mcsolve import MCSolver
from .brmesolve import BRSolver
from .heom.bofin_solvers import HEOMSolver
from .steadystate import steadystate
from ..ui.progressbar import progress_bars
# -----------------------------------------------------------------------------
# PUBLIC API
# -----------------------------------------------------------------------------
# low level correlation
[docs]def correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args=None,
options=None):
r"""
Calculate the two-operator one-time correlation function:
:math:`\left<A(\tau)B(0)\right>`
along one time axis using the quantum regression theorem and the evolution
solver indicated by the `solver` parameter.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of `me`.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list of {:obj:`.Qobj`, :obj:`.QobjEvo`}
List of collapse operators
a_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator A.
b_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator B.
reverse : bool, default: False
If ``True``, calculate :math:`\left<A(t)B(t+\tau)\right>` instead of
:math:`\left<A(t+\tau)B(t)\right>`.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to `me` with
``options={"method": "diag"}``.
options : dict, optional
Options for the solver.
Returns
-------
corr_vec : ndarray
An array of correlation values for the times specified by ``taulist``.
See Also
--------
:func:`correlation_3op` :
Similar function supporting various solver types.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
solver = _make_solver(H, c_ops, args, options, solver)
if reverse:
A_op, B_op, C_op = a_op, b_op, 1
else:
A_op, B_op, C_op = 1, a_op, b_op
if state0 is None:
state0 = steadystate(H, c_ops)
return correlation_3op(solver, state0, [0], taulist, A_op, B_op, C_op)[0]
[docs]def correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op,
solver="me", reverse=False, args=None,
options=None):
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 : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of `me`.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
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`.
If ``tlist`` is ``None``, ``tlist=[0]`` is assumed.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list of {:obj:`.Qobj`, :obj:`.QobjEvo`}
List of collapse operators
a_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator A.
b_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator B.
reverse : bool, default: False
If ``True``, calculate :math:`\left<A(t)B(t+\tau)\right>` instead of
:math:`\left<A(t+\tau)B(t)\right>`.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to `me` with
``options={"method": "diag"}``.
options : dict, optional
Options for the solver.
Returns
-------
corr_mat : ndarray
An 2-dimensional array (matrix) of correlation values for the times
specified by ``tlist`` (first index) and ``taulist`` (second index).
See Also
--------
:func:`correlation_3op` :
Similar function supporting various solver types.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
solver = _make_solver(H, c_ops, args, options, solver)
if tlist is None:
tlist = [0]
if state0 is None:
state0 = steadystate(H, c_ops)
if reverse:
A_op, B_op, C_op = a_op, b_op, 1
else:
A_op, B_op, C_op = 1, a_op, b_op
return correlation_3op(solver, state0, tlist, taulist, A_op, B_op, C_op)
[docs]def correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op,
solver="me", args=None, options=None):
r"""
Calculate the three-operator two-time correlation function:
:math:`\left<A(0)B(\tau)C(0)\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 : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of ``me``.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list of {:obj:`.Qobj`, :obj:`.QobjEvo`}
List of collapse operators
a_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator A.
b_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator B.
c_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator C.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to `me` with
``options={"method": "diag"}``.
options : dict, optional
Options for the solver.
Returns
-------
corr_vec : array
An array of correlation values for the times specified by ``taulist``.
See Also
--------
:func:`correlation_3op` :
Similar function supporting various solver types.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
solver = _make_solver(H, c_ops, args, options, solver)
if state0 is None:
state0 = steadystate(H, c_ops)
return correlation_3op(solver, state0, [0], taulist, a_op, b_op, c_op)[0]
[docs]def correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
solver="me", args=None, options=None):
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 : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of ``me``.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
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`.
If ``tlist`` is ``None``, ``tlist=[0]`` is assumed.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list of {:obj:`.Qobj`, :obj:`.QobjEvo`}
List of collapse operators
a_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator A.
b_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator B.
c_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator C.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to `me` with
``options={"method": "diag"}``.
options : dict, optional
Options for the solver. Only used with ``me`` solver.
Returns
-------
corr_mat : array
An 2-dimensional array (matrix) of correlation values for the times
specified by ``tlist`` (first index) and ``taulist`` (second index).
See Also
--------
:func:`correlation_3op` :
Similar function supporting various solver types.
References
----------
See, Gardiner, Quantum Noise, Section 5.2.
"""
solver = _make_solver(H, c_ops, args, options, solver)
if tlist is None:
tlist = [0]
if state0 is None:
state0 = steadystate(H, c_ops)
return correlation_3op(solver, state0, tlist, taulist, a_op, b_op, c_op)
# high level correlation
[docs]def coherence_function_g1(
H, state0, taulist, c_ops, a_op, solver="me", args=None, options=None
):
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 : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of ``me``.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
c_ops : list of {:obj:`.Qobj`, :obj:`.QobjEvo`}
List of collapse operators
a_op : :obj:`.Qobj`, :obj:`.QobjEvo`
Operator A.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to `me` with
``options={"method": "diag"}``.
args : dict, optional
dictionary of parameters for time-dependent Hamiltonians
options : dict, optional
Options for the solver.
Returns
-------
g1, G1 : tuple
The normalized and unnormalized second-order coherence function.
"""
solver = _make_solver(H, c_ops, args, options, solver)
# 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 = solver.run(state0, taulist, e_ops=[a_op.dag() * a_op]).expect[0]
# calculate the correlation function G1 and normalize with n to obtain g1
G1 = correlation_3op(solver, state0, [0], taulist, None, a_op.dag(), a_op)[0]
g1 = G1 / np.sqrt(n[0] * np.array(n))[0]
return g1, G1
[docs]def coherence_function_g2(H, state0, taulist, c_ops, a_op, solver="me",
args=None, options=None):
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 : :obj:`.Qobj`, :obj:`.QobjEvo`
System Hamiltonian, may be time-dependent for solver choice of ``me``.
state0 : :obj:`.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
if ``c_ops`` are provided and the Hamiltonian is constant.
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``.
a_op : :obj:`.Qobj`
Operator A.
args : dict, optional
Dictionary of arguments to be passed to solver.
solver : str {'me', 'es'}, default: 'me'
Choice of solver, ``me`` for master-equation, and ``es`` for
exponential series. ``es`` is equivalent to ``me`` with
``options={"method": "diag"}``.
options : dict, optional
Options for the solver.
Returns
-------
g2, G2 : tuple
The normalized and unnormalized second-order coherence function.
"""
solver = _make_solver(H, c_ops, args, options, solver)
# 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 = solver.run(state0, taulist, e_ops=[a_op.dag() * a_op]).expect[0]
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_3op(solver, state0, [0], taulist,
a_op.dag(), a_op.dag() * a_op, a_op)[0]
g2 = G2 / (n[0] * np.array(n))
return g2, G2
def _make_solver(H, c_ops, args, options, solver):
H = QobjEvo(H, args=args)
c_ops = [QobjEvo(c_op, args=args) for c_op in c_ops]
if solver == "me":
solver_instance = MESolver(H, c_ops, options=options)
elif solver == "es":
options = {"method": "diag"}
solver_instance = MESolver(H, c_ops, options=options)
elif solver == "mc":
raise ValueError("MC solver for correlation has been removed")
return solver_instance
[docs]def correlation_3op(solver, state0, tlist, taulist, A=None, B=None, C=None):
r"""
Calculate the three-operator two-time correlation function:
:math:`\left<A(t)B(t+\tau)C(t)\right>`.
from a open system :class:`.Solver`.
Note: it is not possible to calculate a physically meaningful correlation
where :math:`\tau<0`.
Parameters
----------
solver : :class:`.MESolver`, :class:`.BRSolver`
Qutip solver for an open system.
state0 : :obj:`.Qobj`
Initial state density matrix :math:`\rho(t_0)` or state vector
:math:`\psi(t_0)`.
tlist : array_like
List of times for :math:`t`. tlist must be positive and contain the
element `0`.
taulist : array_like
List of times for :math:`\tau`. taulist must be positive and contain
the element `0`.
A, B, C : :class:`.Qobj`, :class:`.QobjEvo`, optional, default=None
Operators ``A``, ``B``, ``C`` from the equation ``<A(t)B(t+\tau)C(t)>``
in the Schrodinger picture. They do not need to be all provided. For
exemple, if ``A`` is not provided, ``<B(t+\tau)C(t)>`` is computed.
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.
"""
taulist = np.asarray(taulist)
if isket(state0):
state0 = state0.proj()
A = QobjEvo(qeye_like(state0) if A in [None, 1] else A)
B = QobjEvo(qeye_like(state0) if B in [None, 1] else B)
C = QobjEvo(qeye_like(state0) if C in [None, 1] else C)
if isinstance(solver, (MESolver, BRSolver)):
out = _correlation_3op_dm(solver, state0, tlist, taulist, A, B, C)
elif isinstance(solver, MCSolver):
raise TypeError("Monte Carlo support for correlation was removed. "
"Please, tell us on GitHub issues if you need it!")
elif isinstance(solver, HEOMSolver):
raise TypeError("HEOM is not supported by correlation. "
"Please, tell us on GitHub issues if you need it!")
else:
raise TypeError("Only solvers able to evolve density matrices"
" are supported.")
return out
def _correlation_3op_dm(solver, state0, tlist, taulist, A, B, C):
old_opt = solver.options.copy()
try:
# We don't want to modify the solver
# TODO: Solver could have a ``with`` or ``copy``.
solver.options["normalize_output"] = False
solver.options["progress_bar"] = False
progress_bar = progress_bars[old_opt['progress_bar']](
len(taulist) + 1, **old_opt['progress_kwargs']
)
rho_t = solver.run(state0, tlist).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
progress_bar.update()
for t_idx, rho in enumerate(rho_t):
t = tlist[t_idx]
corr_mat[t_idx, :] = solver.run(
C(t) @ rho @ A(t),
taulist + t,
e_ops=B
).expect[0]
progress_bar.update()
progress_bar.finished()
finally:
solver.options = old_opt
return corr_mat