"""
This module provides utilities for describing baths when using the
HEOM (hierarchy equations of motion) to model system-bath interactions.
See the ``qutip.nonmarkov.bofin_solvers`` module for the associated solver.
The implementation is derived from the BoFiN library (see
https://github.com/tehruhn/bofin) which was itself derived from an earlier
implementation in QuTiP itself.
"""
import enum
import numpy as np
from scipy.linalg import eigvalsh
from qutip.core import data as _data
from qutip.core.qobj import Qobj
from qutip.core.superoperator import spre, spost
__all__ = [
"BathExponent",
"Bath",
"BosonicBath",
"DrudeLorentzBath",
"DrudeLorentzPadeBath",
"UnderDampedBath",
"FermionicBath",
"LorentzianBath",
"LorentzianPadeBath",
]
def _isequal(Q1, Q2, tol):
""" Return true if Q1 and Q2 are equal to within the given tolerance. """
return _data.iszero(_data.sub(Q1.data, Q2.data), tol=tol)
[docs]class BathExponent:
"""
Represents a single exponent (naively, an excitation mode) within the
decomposition of the correlation functions of a bath.
Parameters
----------
type : {"R", "I", "RI", "+", "-"} or BathExponent.ExponentType
The type of bath exponent.
"R" and "I" are bosonic bath exponents that appear in the real and
imaginary parts of the correlation expansion.
"RI" is combined bosonic bath exponent that appears in both the real
and imaginary parts of the correlation expansion. The combined exponent
has a single ``vk``. The ``ck`` is the coefficient in the real
expansion and ``ck2`` is the coefficient in the imaginary expansion.
"+" and "-" are fermionic bath exponents. These fermionic bath
exponents must specify ``sigma_bar_k_offset`` which specifies
the amount to add to ``k`` (the exponent index within the bath of this
exponent) to determine the ``k`` of the corresponding exponent with
the opposite sign (i.e. "-" or "+").
dim : int or None
The dimension (i.e. maximum number of excitations for this exponent).
Usually ``2`` for fermionic exponents or ``None`` (i.e. unlimited) for
bosonic exponents.
Q : Qobj
The coupling operator for this excitation mode.
vk : complex
The frequency of the exponent of the excitation term.
ck : complex
The coefficient of the excitation term.
ck2 : optional, complex
For exponents of type "RI" this is the coefficient of the term in the
imaginary expansion (and ``ck`` is the coefficient in the real
expansion).
sigma_bar_k_offset : optional, int
For exponents of type "+" this gives the offset (within the list of
exponents within the bath) of the corresponding "-" bath exponent.
For exponents of type "-" it gives the offset of the corresponding
"+" exponent.
tag : optional, str, tuple or any other object
A label for the exponent (often the name of the bath). It
defaults to None.
Attributes
----------
fermionic : bool
True if the type of the exponent is a Fermionic type (i.e. either
"+" or "-") and False otherwise.
All of the parameters are also available as attributes.
"""
types = enum.Enum("ExponentType", ["R", "I", "RI", "+", "-"])
def _check_ck2(self, type, ck2):
if type == self.types["RI"]:
if ck2 is None:
raise ValueError("RI bath exponents require ck2")
else:
if ck2 is not None:
raise ValueError(
"Second co-efficient (ck2) should only be specified for RI"
" bath exponents"
)
def _check_sigma_bar_k_offset(self, type, offset):
if type in (self.types["+"], self.types["-"]):
if offset is None:
raise ValueError(
"+ and - bath exponents require sigma_bar_k_offset"
)
else:
if offset is not None:
raise ValueError(
"Offset of sigma bar (sigma_bar_k_offset) should only be"
" specified for + and - bath exponents"
)
def _type_is_fermionic(self, type):
return type in (self.types["+"], self.types["-"])
def __init__(
self, type, dim, Q, ck, vk, ck2=None,
sigma_bar_k_offset=None, tag=None,
):
if not isinstance(type, self.types):
type = self.types[type]
self._check_ck2(type, ck2)
self._check_sigma_bar_k_offset(type, sigma_bar_k_offset)
self.type = type
self.fermionic = self._type_is_fermionic(type)
self.dim = dim
self.Q = Q
self.ck = ck
self.vk = vk
self.ck2 = ck2
self.sigma_bar_k_offset = sigma_bar_k_offset
self.tag = tag
def __repr__(self):
dims = getattr(self.Q, "dims", None)
return (
f"<{self.__class__.__name__} type={self.type.name}"
f" dim={self.dim!r}"
f" Q.dims={dims!r}"
f" ck={self.ck!r} vk={self.vk!r} ck2={self.ck2!r}"
f" sigma_bar_k_offset={self.sigma_bar_k_offset!r}"
f" fermionic={self.fermionic!r}"
f" tag={self.tag!r}>"
)
[docs]class Bath:
"""
Represents a list of bath expansion exponents.
Parameters
----------
exponents : list of BathExponent
The exponents of the correlation function describing the bath.
Attributes
----------
All of the parameters are available as attributes.
"""
def __init__(self, exponents):
self.exponents = exponents
[docs]class BosonicBath(Bath):
"""
A helper class for constructing a bosonic bath from the expansion
coefficients and frequencies for the real and imaginary parts of
the bath correlation function.
If the correlation functions ``C(t)`` is split into real and imaginary
parts::
C(t) = C_real(t) + i * C_imag(t)
then::
C_real(t) = sum(ck_real * exp(- vk_real * t))
C_imag(t) = sum(ck_imag * exp(- vk_imag * t))
Defines the coefficients ``ck`` and the frequencies ``vk``.
Note that the ``ck`` and ``vk`` may be complex, even through ``C_real(t)``
and ``C_imag(t)`` (i.e. the sum) is real.
Parameters
----------
Q : Qobj
The coupling operator for the bath.
ck_real : list of complex
The coefficients of the expansion terms for the real part of the
correlation function. The corresponding frequencies are passed as
vk_real.
vk_real : list of complex
The frequencies (exponents) of the expansion terms for the real part of
the correlation function. The corresponding ceofficients are passed as
ck_real.
ck_imag : list of complex
The coefficients of the expansion terms in the imaginary part of the
correlation function. The corresponding frequencies are passed as
vk_imag.
vk_imag : list of complex
The frequencies (exponents) of the expansion terms for the imaginary
part of the correlation function. The corresponding ceofficients are
passed as ck_imag.
combine : bool, default True
Whether to combine exponents with the same frequency (and coupling
operator). See :meth:`combine` for details.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def _check_cks_and_vks(self, ck_real, vk_real, ck_imag, vk_imag):
if len(ck_real) != len(vk_real) or len(ck_imag) != len(vk_imag):
raise ValueError(
"The bath exponent lists ck_real and vk_real, and ck_imag and"
" vk_imag must be the same length."
)
def _check_coup_op(self, Q):
if not isinstance(Q, Qobj):
raise ValueError("The coupling operator Q must be a Qobj.")
def __init__(
self, Q, ck_real, vk_real, ck_imag, vk_imag, combine=True,
tag=None,
):
self._check_cks_and_vks(ck_real, vk_real, ck_imag, vk_imag)
self._check_coup_op(Q)
exponents = []
exponents.extend(
BathExponent("R", None, Q, ck, vk, tag=tag)
for ck, vk in zip(ck_real, vk_real)
)
exponents.extend(
BathExponent("I", None, Q, ck, vk, tag=tag)
for ck, vk in zip(ck_imag, vk_imag)
)
if combine:
exponents = self.combine(exponents)
super().__init__(exponents)
[docs] @classmethod
def combine(cls, exponents, rtol=1e-5, atol=1e-7):
"""
Group bosonic exponents with the same frequency and return a
single exponent for each frequency present.
Exponents with the same frequency are only combined if they share the
same coupling operator ``.Q``.
Note that combined exponents take their tag from the first
exponent in the group being combined (i.e. the one that occurs first
in the given exponents list).
Parameters
----------
exponents : list of BathExponent
The list of exponents to combine.
rtol : float, default 1e-5
The relative tolerance to use to when comparing frequencies and
coupling operators.
atol : float, default 1e-7
The absolute tolerance to use to when comparing frequencies and
coupling operators.
Returns
-------
list of BathExponent
The new reduced list of exponents.
"""
groups = []
remaining = exponents[:]
while remaining:
e1 = remaining.pop(0)
group = [e1]
for e2 in remaining[:]:
if (
np.isclose(e1.vk, e2.vk, rtol=rtol, atol=atol) and
_isequal(e1.Q, e2.Q, tol=atol)
):
group.append(e2)
remaining.remove(e2)
groups.append(group)
new_exponents = []
for combine in groups:
exp1 = combine[0]
if (exp1.type != exp1.types.RI) and all(
exp2.type == exp1.type for exp2 in combine
):
# the group is either type I or R
ck = sum(exp.ck for exp in combine)
new_exponents.append(BathExponent(
exp1.type, None, exp1.Q, ck, exp1.vk, tag=exp1.tag,
))
else:
# the group includes both type I and R exponents
ck_R = (
sum(exp.ck for exp in combine if exp.type == exp.types.R) +
sum(exp.ck for exp in combine if exp.type == exp.types.RI)
)
ck_I = (
sum(exp.ck for exp in combine if exp.type == exp.types.I) +
sum(exp.ck2 for exp in combine if exp.type == exp.types.RI)
)
new_exponents.append(BathExponent(
"RI", None, exp1.Q, ck_R, exp1.vk, ck2=ck_I,
tag=exp1.tag,
))
return new_exponents
[docs]class DrudeLorentzBath(BosonicBath):
"""
A helper class for constructing a Drude-Lorentz bosonic bath from the
bath parameters (see parameters below).
Parameters
----------
Q : Qobj
Operator describing the coupling between system and bath.
lam : float
Coupling strength.
gamma : float
Bath spectral density cutoff frequency.
T : float
Bath temperature.
Nk : int
Number of exponential terms used to approximate the bath correlation
functions.
combine : bool, default True
Whether to combine exponents with the same frequency (and coupling
operator). See :meth:`BosonicBath.combine` for details.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def __init__(
self, Q, lam, gamma, T, Nk, combine=True, tag=None,
):
ck_real, vk_real, ck_imag, vk_imag = self._matsubara_params(
lam=lam,
gamma=gamma,
T=T,
Nk=Nk,
)
super().__init__(
Q, ck_real, vk_real, ck_imag, vk_imag, combine=combine, tag=tag,
)
self._dl_terminator = _DrudeLorentzTerminator(
Q=Q, lam=lam, gamma=gamma, T=T,
)
[docs] def terminator(self):
"""
Return the Matsubara terminator for the bath and the calculated
approximation discrepancy.
Returns
-------
delta: float
The approximation discrepancy. That is, the difference between the
true correlation function of the Drude-Lorentz bath and the sum of
the ``Nk`` exponential terms is approximately ``2 * delta *
dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.
terminator : Qobj
The Matsubara terminator -- i.e. a liouvillian term representing
the contribution to the system-bath dynamics of all exponential
expansion terms beyond ``Nk``. It should be used by adding it to
the system liouvillian (i.e. ``liouvillian(H_sys)``).
"""
delta, L = self._dl_terminator.terminator(self.exponents)
return delta, L
def _matsubara_params(self, lam, gamma, T, Nk):
""" Calculate the Matsubara coefficents and frequencies. """
ck_real = [lam * gamma / np.tan(gamma / (2 * T))]
ck_real.extend([
(8 * lam * gamma * T * np.pi * k * T /
((2 * np.pi * k * T)**2 - gamma**2))
for k in range(1, Nk + 1)
])
vk_real = [gamma]
vk_real.extend([2 * np.pi * k * T for k in range(1, Nk + 1)])
ck_imag = [lam * gamma * (-1.0)]
vk_imag = [gamma]
return ck_real, vk_real, ck_imag, vk_imag
[docs]class DrudeLorentzPadeBath(BosonicBath):
"""
A helper class for constructing a Padé expansion for a Drude-Lorentz
bosonic bath from the bath parameters (see parameters below).
A Padé approximant is a sum-over-poles expansion (
see https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).
The application of the Padé method to spectrum decompoisitions is described
in "Padé spectrum decompositions of quantum distribution functions and
optimal hierarchical equations of motion construction for quantum open
systems" [1].
The implementation here follows the approach in the paper.
[1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466
This is an alternative to the :class:`DrudeLorentzBath` which constructs
a simpler exponential expansion.
Parameters
----------
Q : Qobj
Operator describing the coupling between system and bath.
lam : float
Coupling strength.
gamma : float
Bath spectral density cutoff frequency.
T : float
Bath temperature.
Nk : int
Number of Padé exponentials terms used to approximate the bath
correlation functions.
combine : bool, default True
Whether to combine exponents with the same frequency (and coupling
operator). See :meth:`BosonicBath.combine` for details.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def __init__(
self, Q, lam, gamma, T, Nk, combine=True, tag=None
):
eta_p, gamma_p = self._corr(lam=lam, gamma=gamma, T=T, Nk=Nk)
ck_real = [np.real(eta) for eta in eta_p]
vk_real = [gam for gam in gamma_p]
# There is only one term in the expansion of the imaginary part of the
# Drude-Lorentz correlation function.
ck_imag = [np.imag(eta_p[0])]
vk_imag = [gamma_p[0]]
super().__init__(
Q, ck_real, vk_real, ck_imag, vk_imag, combine=combine, tag=tag,
)
self._dl_terminator = _DrudeLorentzTerminator(
Q=Q, lam=lam, gamma=gamma, T=T,
)
[docs] def terminator(self):
"""
Return the Padé terminator for the bath and the calculated
approximation discrepancy.
Returns
-------
delta: float
The approximation discrepancy. That is, the difference between the
true correlation function of the Drude-Lorentz bath and the sum of
the ``Nk`` exponential terms is approximately ``2 * delta *
dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.
terminator : Qobj
The Padé terminator -- i.e. a liouvillian term representing
the contribution to the system-bath dynamics of all exponential
expansion terms beyond ``Nk``. It should be used by adding it to
the system liouvillian (i.e. ``liouvillian(H_sys)``).
"""
delta, L = self._dl_terminator.terminator(self.exponents)
return delta, L
def _corr(self, lam, gamma, T, Nk):
beta = 1. / T
kappa, epsilon = self._kappa_epsilon(Nk)
eta_p = [lam * gamma * (self._cot(gamma * beta / 2.0) - 1.0j)]
gamma_p = [gamma]
for ll in range(1, Nk + 1):
eta_p.append(
(kappa[ll] / beta) * 4 * lam * gamma * (epsilon[ll] / beta)
/ ((epsilon[ll]**2 / beta**2) - gamma**2)
)
gamma_p.append(epsilon[ll] / beta)
return eta_p, gamma_p
def _cot(self, x):
return 1. / np.tan(x)
def _kappa_epsilon(self, Nk):
eps = self._calc_eps(Nk)
chi = self._calc_chi(Nk)
kappa = [0]
prefactor = 0.5 * Nk * (2 * (Nk + 1) + 1)
for j in range(Nk):
term = prefactor
for k in range(Nk - 1):
term *= (
(chi[k]**2 - eps[j]**2) /
(eps[k]**2 - eps[j]**2 + self._delta(j, k))
)
for k in [Nk - 1]:
term /= (eps[k]**2 - eps[j]**2 + self._delta(j, k))
kappa.append(term)
epsilon = [0] + eps
return kappa, epsilon
def _delta(self, i, j):
return 1.0 if i == j else 0.0
def _calc_eps(self, Nk):
alpha = np.diag([
1. / np.sqrt((2 * k + 5) * (2 * k + 3))
for k in range(2 * Nk - 1)
], k=1)
alpha += alpha.transpose()
evals = eigvalsh(alpha)
eps = [-2. / val for val in evals[0: Nk]]
return eps
def _calc_chi(self, Nk):
alpha_p = np.diag([
1. / np.sqrt((2 * k + 7) * (2 * k + 5))
for k in range(2 * Nk - 2)
], k=1)
alpha_p += alpha_p.transpose()
evals = eigvalsh(alpha_p)
chi = [-2. / val for val in evals[0: Nk - 1]]
return chi
class _DrudeLorentzTerminator:
""" A class for calculating the terminator of a Drude-Lorentz bath
expansion.
"""
def __init__(self, Q, lam, gamma, T):
self.Q = Q
self.lam = lam
self.gamma = gamma
self.T = T
def terminator(self, exponents):
""" Calculate the terminator for a Drude-Lorentz bath. """
Q = self.Q
lam = self.lam
gamma = self.gamma
beta = 1 / self.T
delta = 2 * lam / (beta * gamma) - 1j * lam
for exp in exponents:
if exp.type == BathExponent.types["R"]:
delta -= exp.ck / exp.vk
elif exp.type == BathExponent.types["RI"]:
delta -= (exp.ck + 1j * exp.ck2) / exp.vk
else:
delta -= 1j * exp.ck / exp.vk
op = -2*spre(Q)*spost(Q.dag()) + spre(Q.dag()*Q) + spost(Q.dag()*Q)
L_bnd = -delta * op
return delta, L_bnd
[docs]class UnderDampedBath(BosonicBath):
"""
A helper class for constructing an under-damped bosonic bath from the
bath parameters (see parameters below).
Parameters
----------
Q : Qobj
Operator describing the coupling between system and bath.
lam : float
Coupling strength.
gamma : float
Bath spectral density cutoff frequency.
w0 : float
Bath spectral density resonance frequency.
T : float
Bath temperature.
Nk : int
Number of exponential terms used to approximate the bath correlation
functions.
combine : bool, default True
Whether to combine exponents with the same frequency (and coupling
operator). See :meth:`BosonicBath.combine` for details.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def __init__(
self, Q, lam, gamma, w0, T, Nk, combine=True, tag=None,
):
ck_real, vk_real, ck_imag, vk_imag = self._matsubara_params(
lam=lam,
gamma=gamma,
w0=w0,
T=T,
Nk=Nk,
)
super().__init__(
Q, ck_real, vk_real, ck_imag, vk_imag, combine=combine, tag=tag,
)
def _matsubara_params(self, lam, gamma, w0, T, Nk):
""" Calculate the Matsubara coefficents and frequencies. """
beta = 1/T
Om = np.sqrt(w0**2 - (gamma/2)**2)
Gamma = gamma/2.
ck_real = ([
(lam**2 / (4 * Om))
* (1 / np.tanh(beta * (Om + 1.0j * Gamma) / 2)),
(lam**2 / (4*Om))
* (1 / np.tanh(beta * (Om - 1.0j * Gamma) / 2)),
])
ck_real.extend([
(-2 * lam**2 * gamma / beta) * (2 * np.pi * k / beta)
/ (
((Om + 1.0j * Gamma)**2 + (2 * np.pi * k/beta)**2)
* ((Om - 1.0j * Gamma)**2 + (2 * np.pi * k / beta)**2)
)
for k in range(1, Nk + 1)
])
vk_real = [-1.0j * Om + Gamma, 1.0j * Om + Gamma]
vk_real.extend([
2 * np.pi * k * T
for k in range(1, Nk + 1)
])
ck_imag = [
1.0j * lam**2 / (4 * Om),
-1.0j * lam**2 / (4 * Om),
]
vk_imag = [-1.0j * Om + Gamma, 1.0j * Om + Gamma]
return ck_real, vk_real, ck_imag, vk_imag
[docs]class FermionicBath(Bath):
"""
A helper class for constructing a fermionic bath from the expansion
coefficients and frequencies for the ``+`` and ``-`` modes of
the bath correlation function.
There must be the same number of ``+`` and ``-`` modes and their
coefficients must be specified in the same order so that ``ck_plus[i],
vk_plus[i]`` are the plus coefficient and frequency corresponding
to the minus mode ``ck_minus[i], vk_minus[i]``.
In the fermionic case the order in which excitations are created or
destroyed is important, resulting in two different correlation functions
labelled ``C_plus(t)`` and ``C_plus(t)``::
C_plus(t) = sum(ck_plus * exp(- vk_plus * t))
C_minus(t) = sum(ck_minus * exp(- vk_minus * t))
where the expansions above define the coeffiients ``ck`` and the
frequencies ``vk``.
Parameters
----------
Q : Qobj
The coupling operator for the bath.
ck_plus : list of complex
The coefficients of the expansion terms for the ``+`` part of the
correlation function. The corresponding frequencies are passed as
vk_plus.
vk_plus : list of complex
The frequencies (exponents) of the expansion terms for the ``+`` part
of the correlation function. The corresponding ceofficients are passed
as ck_plus.
ck_minus : list of complex
The coefficients of the expansion terms for the ``-`` part of the
correlation function. The corresponding frequencies are passed as
vk_minus.
vk_minus : list of complex
The frequencies (exponents) of the expansion terms for the ``-`` part
of the correlation function. The corresponding ceofficients are passed
as ck_minus.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def _check_cks_and_vks(self, ck_plus, vk_plus, ck_minus, vk_minus):
if len(ck_plus) != len(vk_plus) or len(ck_minus) != len(vk_minus):
raise ValueError(
"The bath exponent lists ck_plus and vk_plus, and ck_minus and"
" vk_minus must be the same length."
)
if len(ck_plus) != len(ck_minus):
raise ValueError(
"The must be the same number of plus and minus exponents"
" in the bath, and elements of plus and minus arrays"
" should be arranged so that ck_plus[i] is the plus mode"
" corresponding to ck_minus[i]."
)
def _check_coup_op(self, Q):
if not isinstance(Q, Qobj):
raise ValueError("The coupling operator Q must be a Qobj.")
def __init__(self, Q, ck_plus, vk_plus, ck_minus, vk_minus, tag=None):
self._check_cks_and_vks(ck_plus, vk_plus, ck_minus, vk_minus)
self._check_coup_op(Q)
exponents = []
for ckp, vkp, ckm, vkm in zip(ck_plus, vk_plus, ck_minus, vk_minus):
exponents.append(BathExponent(
"+", 2, Q, ckp, vkp, sigma_bar_k_offset=1, tag=tag,
))
exponents.append(BathExponent(
"-", 2, Q, ckm, vkm, sigma_bar_k_offset=-1, tag=tag,
))
super().__init__(exponents)
[docs]class LorentzianBath(FermionicBath):
"""
A helper class for constructing a Lorentzian fermionic bath from the
bath parameters (see parameters below).
.. note::
This Matsubara expansion used in this bath converges very slowly
and ``Nk > 20`` may be required to get good convergence. The
Padé expansion used by :class:`LorentzianPadeBath` converges much
more quickly.
Parameters
----------
Q : Qobj
Operator describing the coupling between system and bath.
gamma : float
The coupling strength between the system and the bath.
w : float
The width of the environment.
mu : float
The chemical potential of the bath.
T : float
Bath temperature.
Nk : int
Number of exponential terms used to approximate the bath correlation
functions.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def __init__(self, Q, gamma, w, mu, T, Nk, tag=None):
ck_plus, vk_plus = self._corr(gamma, w, mu, T, Nk, sigma=1.0)
ck_minus, vk_minus = self._corr(gamma, w, mu, T, Nk, sigma=-1.0)
super().__init__(
Q, ck_plus, vk_plus, ck_minus, vk_minus, tag=tag,
)
def _corr(self, gamma, w, mu, T, Nk, sigma):
beta = 1. / T
kappa = [0.]
kappa.extend([1. for _ in range(1, Nk + 1)])
epsilon = [0]
epsilon.extend([(2 * ll - 1) * np.pi for ll in range(1, Nk + 1)])
def f(x):
return 1 / (np.exp(x) + 1)
eta_list = [0.5 * gamma * w * f(1.0j * beta * w)]
gamma_list = [w - sigma * 1.0j * mu]
for ll in range(1, Nk + 1):
eta_list.append(
-1.0j * (kappa[ll] / beta) * gamma * w**2 /
(-(epsilon[ll]**2 / beta**2) + w**2)
)
gamma_list.append(epsilon[ll] / beta - sigma * 1.0j * mu)
return eta_list, gamma_list
[docs]class LorentzianPadeBath(FermionicBath):
"""
A helper class for constructing a Padé expansion for Lorentzian fermionic
bath from the bath parameters (see parameters below).
A Padé approximant is a sum-over-poles expansion (
see https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).
The application of the Padé method to spectrum decompoisitions is described
in "Padé spectrum decompositions of quantum distribution functions and
optimal hierarchical equations of motion construction for quantum open
systems" [1].
The implementation here follows the approach in the paper.
[1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466
This is an alternative to the :class:`LorentzianBath` which constructs
a simpler exponential expansion that converges much more slowly in
this particular case.
Parameters
----------
Q : Qobj
Operator describing the coupling between system and bath.
gamma : float
The coupling strength between the system and the bath.
w : float
The width of the environment.
mu : float
The chemical potential of the bath.
T : float
Bath temperature.
Nk : int
Number of exponential terms used to approximate the bath correlation
functions.
tag : optional, str, tuple or any other object
A label for the bath exponents (for example, the name of the
bath). It defaults to None but can be set to help identify which
bath an exponent is from.
"""
def __init__(self, Q, gamma, w, mu, T, Nk, tag=None):
ck_plus, vk_plus = self._corr(gamma, w, mu, T, Nk, sigma=1.0)
ck_minus, vk_minus = self._corr(gamma, w, mu, T, Nk, sigma=-1.0)
super().__init__(
Q, ck_plus, vk_plus, ck_minus, vk_minus, tag=tag,
)
def _corr(self, gamma, w, mu, T, Nk, sigma):
beta = 1. / T
kappa, epsilon = self._kappa_epsilon(Nk)
def f_approx(x):
f = 0.5
for ll in range(1, Nk + 1):
f = f - 2 * kappa[ll] * x / (x**2 + epsilon[ll]**2)
return f
eta_list = [0.5 * gamma * w * f_approx(1.0j * beta * w)]
gamma_list = [w - sigma * 1.0j * mu]
for ll in range(1, Nk + 1):
eta_list.append(
-1.0j * (kappa[ll] / beta) * gamma * w**2
/ (-(epsilon[ll]**2 / beta**2) + w**2)
)
gamma_list.append(epsilon[ll] / beta - sigma * 1.0j * mu)
return eta_list, gamma_list
def _kappa_epsilon(self, Nk):
eps = self._calc_eps(Nk)
chi = self._calc_chi(Nk)
kappa = [0]
prefactor = 0.5 * Nk * (2 * (Nk + 1) - 1)
for j in range(Nk):
term = prefactor
for k in range(Nk - 1):
term *= (
(chi[k]**2 - eps[j]**2) /
(eps[k]**2 - eps[j]**2 + self._delta(j, k))
)
for k in [Nk - 1]:
term /= (eps[k]**2 - eps[j]**2 + self._delta(j, k))
kappa.append(term)
epsilon = [0] + eps
return kappa, epsilon
def _delta(self, i, j):
return 1.0 if i == j else 0.0
def _calc_eps(self, Nk):
alpha = np.diag([
1. / np.sqrt((2 * k + 3) * (2 * k + 1))
for k in range(2 * Nk - 1)
], k=1)
alpha += alpha.transpose()
evals = eigvalsh(alpha)
eps = [-2. / val for val in evals[0: Nk]]
return eps
def _calc_chi(self, Nk):
alpha_p = np.diag([
1. / np.sqrt((2 * k + 5) * (2 * k + 3))
for k in range(2 * Nk - 2)
], k=1)
alpha_p += alpha_p.transpose()
evals = eigvalsh(alpha_p)
chi = [-2. / val for val in evals[0: Nk - 1]]
return chi