"""
Classes that describe environments of open quantum systems
"""
# Required for Sphinx to follow autodoc_type_aliases
from __future__ import annotations
__all__ = ['BosonicEnvironment',
'DrudeLorentzEnvironment',
'UnderDampedEnvironment',
'OhmicEnvironment',
'ExponentialBosonicEnvironment',
'FermionicEnvironment',
'LorentzianEnvironment',
'ExponentialFermionicEnvironment',
'CFExponent',
'system_terminator']
import abc
import enum
from time import time
from typing import Any, Callable, Literal, Sequence, overload, Union
import warnings
import numpy as np
from numpy.typing import ArrayLike
from scipy.linalg import eigvalsh
from scipy.interpolate import CubicSpline
try:
from mpmath import mp
_mpmath_available = True
except ModuleNotFoundError:
_mpmath_available = False
from ..utilities import (n_thermal, iterated_fit, aaa,
prony_methods, espira1, espira2)
from .superoperator import spre, spost
from .qobj import Qobj
[docs]
class BosonicEnvironment(abc.ABC):
"""
The bosonic environment of an open quantum system. It is characterized by
its spectral density and temperature or, equivalently, its power spectrum
or its two-time auto-correlation function.
Use one of the classmethods :meth:`from_spectral_density`,
:meth:`from_power_spectrum` or :meth:`from_correlation_function` to
construct an environment manually from one of these characteristic
functions, or use a predefined sub-class such as the
:class:`DrudeLorentzEnvironment`, the :class:`UnderDampedEnvironment` or
the :class:`OhmicEnvironment`.
Bosonic environments offer various ways to approximate the environment with
a multi-exponential correlation function, which can be used for example in
the HEOM solver. The approximated environment is represented as a
:class:`ExponentialBosonicEnvironment`.
All bosonic environments can be approximated by various fitting methods,
see the description of :meth:`approximate`.
Subclasses may offer additional approximation methods such as the
analytical Matsubara or Pade expansions.
Parameters
----------
T : optional, float
The temperature of this environment.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(self, T: float = None, tag: Any = None):
self.T = T
self.tag = tag
[docs]
@abc.abstractmethod
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
The spectral density of this environment. For negative frequencies,
a value of zero will be returned. See the Users Guide on
:ref:`bosonic environments <bosonic environments guide>` for specifics
on the definitions used by QuTiP.
If no analytical expression for the spectral density is known, it will
be derived from the power spectrum. In this case, the temperature of
this environment must be specified.
If no analytical expression for the power spectrum is known either, it
will be derived from the correlation function via a fast fourier
transform.
Parameters
----------
w : array_like or float
The frequencies at which to evaluate the spectral density.
"""
...
[docs]
@abc.abstractmethod
def correlation_function(
self, t: float | ArrayLike, *, eps: float = 1e-10
) -> (float | ArrayLike):
"""
The two-time auto-correlation function of this environment. See the
Users Guide on :ref:`bosonic environments <bosonic environments guide>`
for specifics on the definitions used by QuTiP.
If no analytical expression for the correlation function is known, it
will be derived from the power spectrum via a fast fourier transform.
If no analytical expression for the power spectrum is known either, it
will be derived from the spectral density. In this case, the
temperature of this environment must be specified.
Parameters
----------
t : array_like or float
The times at which to evaluate the correlation function.
eps : optional, float
Used in case the power spectrum is derived from the spectral
density; see the documentation of
:meth:`BosonicEnvironment.power_spectrum`.
"""
...
[docs]
@abc.abstractmethod
def power_spectrum(
self, w: float | ArrayLike, *, eps: float = 1e-10
) -> (float | ArrayLike):
"""
The power spectrum of this environment. See the Users Guide on
:ref:`bosonic environments <bosonic environments guide>` for specifics
on the definitions used by QuTiP.
If no analytical expression for the power spectrum is known, it will
be derived from the spectral density. In this case, the temperature of
this environment must be specified.
If no analytical expression for the spectral density is known either,
the power spectrum will instead be derived from the correlation
function via a fast fourier transform.
Parameters
----------
w : array_like or float
The frequencies at which to evaluate the power spectrum.
eps : optional, float
To derive the zero-frequency power spectrum from the spectral
density, the spectral density must be differentiated numerically.
In that case, this parameter is used as the finite difference in
the numerical differentiation.
"""
...
# --- user-defined environment creation
[docs]
@classmethod
def from_correlation_function(
cls,
C: Callable[[float], complex] | ArrayLike,
tlist: ArrayLike = None,
tMax: float = None,
*,
T: float = None,
tag: Any = None,
args: dict[str, Any] = None,
) -> BosonicEnvironment:
r"""
Constructs a bosonic environment from the provided correlation
function. The provided function will only be used for times
:math:`t \geq 0`. At times :math:`t < 0`, the symmetry relation
:math:`C(-t) = C(t)^\ast` is enforced.
Parameters
----------
C : callable or array_like
The correlation function. Can be provided as a Python function or
as an array. When using a function, the signature should be
``C(t: array_like, **args) -> array_like``
where ``t`` is time and ``args`` is a dict containing the
other parameters of the function.
tlist : optional, array_like
The times where the correlation function is sampled (if it is
provided as an array).
tMax : optional, float
Specifies that the correlation function is essentially zero outside
the interval [-tMax, tMax]. Used for numerical integration
purposes.
T : optional, float
Environment temperature. (The spectral density of this environment
can only be calculated from the correlation function if a
temperature is provided.)
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
args : optional, dict
Extra arguments for the correlation function ``C``.
"""
return _BosonicEnvironment_fromCF(C, tlist, tMax, T, tag, args)
[docs]
@classmethod
def from_power_spectrum(
cls,
S: Callable[[float], float] | ArrayLike,
wlist: ArrayLike = None,
wMax: float = None,
*,
T: float = None,
tag: Any = None,
args: dict[str, Any] = None,
) -> BosonicEnvironment:
r"""
Constructs a bosonic environment with the provided power spectrum.
Parameters
----------
S : callable or array_like
The power spectrum. Can be provided as a Python function or
as an array. When using a function, the signature should be
``S(w: array_like, **args) -> array_like``
where ``w`` is the frequency and ``args`` is a dict containing the
other parameters of the function.
wlist : optional, array_like
The frequencies where the power spectrum is sampled (if it is
provided as an array).
wMax : optional, float
Specifies that the power spectrum is essentially zero outside the
interval [-wMax, wMax]. Used for numerical integration purposes.
T : optional, float
Environment temperature. (The spectral density of this environment
can only be calculated from the powr spectrum if a temperature is
provided.)
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
args : optional, dict
Extra arguments for the power spectrum ``S``.
"""
return _BosonicEnvironment_fromPS(S, wlist, wMax, T, tag, args)
[docs]
@classmethod
def from_spectral_density(
cls,
J: Callable[[float], float] | ArrayLike,
wlist: ArrayLike = None,
wMax: float = None,
*,
T: float = None,
tag: Any = None,
args: dict[str, Any] = None,
) -> BosonicEnvironment:
r"""
Constructs a bosonic environment with the provided spectral density.
The provided function will only be used for frequencies
:math:`\omega > 0`. At frequencies :math:`\omega \leq 0`, the spectral
density is zero according to the definition used by QuTiP. See the
Users Guide on :ref:`bosonic environments <bosonic environments guide>`
for a note on spectral densities with support at negative frequencies.
Parameters
----------
J : callable or array_like
The spectral density. Can be provided as a Python function or
as an array. When using a function, the signature should be
``J(w: array_like, **args) -> array_like``
where ``w`` is the frequency and ``args`` is a tuple containing the
other parameters of the function.
wlist : optional, array_like
The frequencies where the spectral density is sampled (if it is
provided as an array).
wMax : optional, float
Specifies that the spectral density is essentially zero outside the
interval [-wMax, wMax]. Used for numerical integration purposes.
T : optional, float
Environment temperature. (The correlation function and the power
spectrum of this environment can only be calculated from the
spectral density if a temperature is provided.)
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
args : optional, dict
Extra arguments for the spectral density ``J``.
"""
return _BosonicEnvironment_fromSD(J, wlist, wMax, T, tag, args)
# --- spectral density, power spectrum, correlation function conversions
def _ps_from_sd(self, w, eps, derivative=None):
# derivative: value of J'(0)
if self.T is None:
raise ValueError(
"The temperature must be specified for this operation.")
w = np.asarray(w, dtype=float)
if self.T == 0:
return 2 * np.heaviside(w, 0) * self.spectral_density(w)
# at zero frequency, we do numerical differentiation
# S(0) = 2 J'(0) / beta
zero_mask = (w == 0)
nonzero_mask = np.invert(zero_mask)
S = np.zeros_like(w)
if derivative is None:
S[zero_mask] = 2 * self.T * self.spectral_density(eps) / eps
else:
S[zero_mask] = 2 * self.T * derivative
S[nonzero_mask] = (
2 * np.sign(w[nonzero_mask])
* self.spectral_density(np.abs(w[nonzero_mask]))
* (n_thermal(w[nonzero_mask], self.T) + 1)
)
return S.item() if w.ndim == 0 else S
def _sd_from_ps(self, w):
w = np.asarray(w, dtype=float)
J = np.zeros_like(w)
positive_mask = (w > 0)
power_spectrum = self.power_spectrum(w[positive_mask])
if self.T is None:
raise ValueError(
"The temperature must be specified for this operation.")
J[positive_mask] = (
power_spectrum / 2 / (n_thermal(w[positive_mask], self.T) + 1)
)
return J.item() if w.ndim == 0 else J
def _ps_from_cf(self, w, tMax):
w = np.asarray(w, dtype=float)
if w.ndim == 0:
wMax = np.abs(w)
elif len(w) == 0:
return np.array([])
else:
wMax = max(np.abs(w[0]), np.abs(w[-1]))
mirrored_result = _fft(self.correlation_function, wMax, tMax=tMax)
result = np.real(mirrored_result(-w))
return result.item() if w.ndim == 0 else result
def _cf_from_ps(self, t, wMax, **ps_kwargs):
t = np.asarray(t, dtype=float)
if t.ndim == 0:
tMax = np.abs(t)
elif len(t) == 0:
return np.array([])
else:
tMax = max(np.abs(t[0]), np.abs(t[-1]))
result_fct = _fft(lambda w: self.power_spectrum(w, **ps_kwargs),
tMax, tMax=wMax)
result = result_fct(t) / (2 * np.pi)
return result.item() if t.ndim == 0 else result
# --- fitting
@overload
def approximate(
self,
method: Literal['cf'],
tlist: ArrayLike,
target_rmse: float = 2e-5,
Nr_max: int = 10,
Ni_max: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
full_ansatz: bool = False,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['ps'],
wlist: ArrayLike,
target_rmse: float = 5e-6,
Nmax: int = 5,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal["sd"],
wlist: ArrayLike,
Nk: int = 1,
target_rmse: float = 5e-6,
Nmax: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['aaa'],
wlist: ArrayLike,
tol: float = 1e-13,
Nmax: int = 10,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
tlist: ArrayLike,
separate: bool = False,
Nr: int = 3,
Ni: int = 3,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@property
def _approximation_methods(self):
return {
"cf": (self._approx_by_cf_fit, "Correlation Function NLSQ"),
"ps": (self._approx_by_ps_fit, "Power Spectrum NLSQ"),
"sd": (self._approx_by_sd_fit, "Spectral Density NLSQ"),
"aaa": (self._approx_by_aaa, "Power spectrum AAA"),
"prony": (self._approx_by_prony, "Correlation Function Prony"),
"esprit": (self._approx_by_prony, "Correlation Function ESPRIT"),
"espira-i": (self._approx_by_prony,
"Correlation Function ESPIRA-I"),
"espira-ii": (self._approx_by_prony,
"Correlation Function ESPIRA-II"),
}
[docs]
def approximate(self, method: str, *args, **kwargs):
"""
Generates a multi-exponential approximation of this environment.
The available methods are ``"cf"``, ``"ps"``, ``"sd"``, ``"aaa"``,
``"prony"``, ``"esprit"``, ``"espira-I"`` and
``"espira-II"``. The methods and the parameters required per method are
documented in the :ref:`Users Guide<environment approximations api>`.
"""
dispatch = self._approximation_methods
if method.lower() not in dispatch:
error_string = (f"Unsupported method: {method}."
" The available methods are:\n")
for key in dispatch:
error_string += f" - {dispatch[key][1]} ({key})\n"
error_string += ("If unsure what fitting method to use, you should"
" probably use ESPIRA-I, or Power Spectrum NLSQ."
" For more information about the fitting methods,"
" see the Users Guide.")
raise ValueError(error_string)
func = dispatch[method.lower()][0]
return func(method, *args, **kwargs)
def _approx_by_cf_fit(
self, method, tlist, target_rmse=2e-5, Nr_max=10, Ni_max=10,
guess=None, lower=None, upper=None, sigma=None, maxfev=None,
full_ansatz=False, combine=True, tag=None
):
# Process arguments
if tag is None and self.tag is not None:
tag = (self.tag, f"{method.upper()} Fit")
if full_ansatz:
num_params = 4
else:
num_params = 3
if target_rmse is None:
target_rmse = 0
Nr_min, Ni_min = Nr_max, Ni_max
else:
Nr_min, Ni_min = 1, 1
clist = self.correlation_function(tlist)
if guess is None and lower is None and upper is None:
guess_re, lower_re, upper_re = _default_guess_cfreal(
tlist, np.real(clist), full_ansatz)
guess_im, lower_im, upper_im = _default_guess_cfimag(
np.imag(clist), full_ansatz)
else:
guess_re, lower_re, upper_re = guess, lower, upper
guess_im, lower_im, upper_im = guess, lower, upper
# Fit real part
start_real = time()
rmse_real, params_real = iterated_fit(
_cf_real_fit_model, num_params, tlist, np.real(clist), target_rmse,
Nr_min, Nr_max, guess=guess_re, lower=lower_re, upper=upper_re,
sigma=sigma, maxfev=maxfev
)
end_real = time()
fit_time_real = end_real - start_real
# Fit imaginary part
start_imag = time()
rmse_imag, params_imag = iterated_fit(
_cf_imag_fit_model, num_params, tlist, np.imag(clist), target_rmse,
Ni_min, Ni_max, guess=guess_im, lower=lower_im, upper=upper_im,
sigma=sigma, maxfev=maxfev
)
end_imag = time()
fit_time_imag = end_imag - start_imag
# Generate summary
Nr = len(params_real)
Ni = len(params_imag)
full_summary = _cf_fit_summary(
params_real, params_imag, fit_time_real, fit_time_imag,
Nr, Ni, rmse_real, rmse_imag, n=num_params
)
fit_info = {"Nr": Nr, "Ni": Ni, "fit_time_real": fit_time_real,
"fit_time_imag": fit_time_imag, "rmse_real": rmse_real,
"rmse_imag": rmse_imag, "params_real": params_real,
"params_imag": params_imag, "summary": full_summary}
# Finally, generate environment and return
ckAR = []
vkAR = []
for term in params_real:
if full_ansatz:
a, b, c, d = term
else:
a, b, c = term
d = 0
ckAR.extend([(a + 1j * d) / 2, (a - 1j * d) / 2])
vkAR.extend([-b - 1j * c, -b + 1j * c])
ckAI = []
vkAI = []
for term in params_imag:
if full_ansatz:
a, b, c, d = term
else:
a, b, c = term
d = 0
ckAI.extend([-1j * (a + 1j * d) / 2, 1j * (a - 1j * d) / 2])
vkAI.extend([-b - 1j * c, -b + 1j * c])
approx_env = ExponentialBosonicEnvironment(
ckAR, vkAR, ckAI, vkAI, combine=combine, T=self.T, tag=tag)
return approx_env, fit_info
def _approx_by_sd_fit(
self, method, wlist, Nk=1, target_rmse=5e-6, Nmax=10,
guess=None, lower=None, upper=None, sigma=None, maxfev=None,
combine=True, tag=None
):
# Process arguments
if tag is None and self.tag is not None:
tag = (self.tag, f"{method.upper()} Fit")
if target_rmse is None:
target_rmse = 0
Nmin = Nmax
else:
Nmin = 1
jlist = self.spectral_density(wlist)
if guess is None and lower is None and upper is None:
guess, lower, upper = _default_guess_sd(wlist, jlist)
# Fit
start = time()
rmse, params = iterated_fit(
_sd_fit_model, 3, wlist, jlist, target_rmse, Nmin, Nmax,
guess=guess, lower=lower, upper=upper, sigma=sigma, maxfev=maxfev
)
end = time()
fit_time = end - start
# Generate summary
N = len(params)
summary = _fit_summary(
fit_time, rmse, N, "the spectral density", params
)
fit_info = {
"N": N, "Nk": Nk, "fit_time": fit_time, "rmse": rmse,
"params": params, "summary": summary}
ckAR, vkAR, ckAI, vkAI = [], [], [], []
# Finally, generate environment and return
for a, b, c in params:
lam = np.sqrt(a + 0j)
gamma = 2 * b
w0 = np.sqrt(c**2 + b**2)
env = UnderDampedEnvironment(self.T, lam, gamma, w0)
coeffs = env._matsubara_params(Nk)
ckAR.extend(coeffs[0])
vkAR.extend(coeffs[1])
ckAI.extend(coeffs[2])
vkAI.extend(coeffs[3])
approx_env = ExponentialBosonicEnvironment(
ckAR, vkAR, ckAI, vkAI, combine=combine, T=self.T, tag=tag)
return approx_env, fit_info
def _approx_by_ps_fit(
self, method, wlist, target_rmse=5e-6, Nmax=5,
guess=None, lower=None, upper=None, sigma=None, maxfev=None,
combine=True, tag=None
):
# Process arguments
if tag is None and self.tag is not None:
tag = (self.tag, f"{method.upper()} Fit")
if target_rmse is None:
target_rmse = 0
Nmin = Nmax
else:
Nmin = 1
jlist = self.power_spectrum(wlist)
if guess is None and lower is None and upper is None:
guess, lower, upper = _default_guess_ps(wlist, jlist)
# Fit
start = time()
rmse, params = iterated_fit(
_ps_fit_model, 4, wlist, jlist, target_rmse, Nmin, Nmax,
guess=guess, lower=lower, upper=upper, sigma=sigma, maxfev=maxfev
)
end = time()
fit_time = end - start
# Generate summary
N = len(params)
summary = _fit_summary(
fit_time, rmse, N, "the power spectrum", params,
columns=['a', 'b', 'c', 'd']
)
fit_info = {
"N": N, "fit_time": fit_time, "rmse": rmse,
"params": params, "summary": summary}
ck, vk = [], []
# Finally, generate environment and return
for a, b, c, d in params:
ck.append(a+1j*b)
vk.append(c+1j*d)
ck = np.array(ck)
vk = np.array(vk)
ckAR = np.concatenate((ck/2, ck.conj()/2))
ckAI = np.concatenate((-1j*ck/2, 1j*ck.conj()/2))
vkAR = np.concatenate((vk, vk.conj()))
approx_env = ExponentialBosonicEnvironment(
ckAR, vkAR, ckAI, vkAR, T=self.T, combine=combine, tag=tag)
return approx_env, fit_info
def _approx_by_aaa(
self, method, wlist, tol=1e-13, Nmax=10, combine=True, tag=None
):
if tag is None and self.tag is not None:
tag = (self.tag, f"{method.upper()} Fit")
start = time()
# The *2 is there because half the poles will be filtered out
result = aaa(self.power_spectrum, wlist,
tol=tol,
max_iter=Nmax * 2)
end = time()
pol = result['poles']
res = result['residues']
mask = np.imag(pol) < 0
new_pols, new_res = pol[mask], res[mask]
vk = 1j * new_pols
ck = -1j * new_res
# Create complex conjugates for both vk and ck
ckAR = np.concatenate((ck/2, ck.conj()/2))
ckAI = np.concatenate((-1j*ck/2, 1j*ck.conj()/2))
vkAR = np.concatenate((vk, vk.conj()))
cls = ExponentialBosonicEnvironment(
ck_real=ckAR, vk_real=vkAR, ck_imag=ckAI,
vk_imag=vkAR, T=self.T, combine=combine, tag=tag)
# Generate summary
N = len(vk)
fit_time = end - start
params = [(ck.real[i], ck.imag[i], vk[i].real, vk[i].imag)
for i in range(len(ck))]
summary = _fit_summary(
fit_time, result['rmse'], N, "the power spectrum", params,
columns=['ckr', 'cki', 'vkr', 'vki']
)
fitinfo = {
"N": N, "fit_time": fit_time, "rmse": result['rmse'],
"params": params, "summary": summary}
return cls, fitinfo
def _approx_by_prony(
self, method, tlist, separate=False, Nr=3, Ni=None,
combine=True, tag=None
):
if not separate and Ni is not None:
raise ValueError("The number of imaginary exponents (Ni) cannot be"
" specified if real and imaginary parts are fit"
" together (separate=False).")
if Ni is None:
Ni = 3 # default value
if tag is None and self.tag is not None:
tag = (self.tag, f"{method.upper()} Fit")
def prony(x, n):
return prony_methods(method, x, n)
def phase_to_exponent(phases):
return -((len(tlist) - 1) / tlist[-1]) * (
np.log(np.abs(phases)) + 1j * np.angle(phases)
)
methods = {"prony": prony, "esprit": prony,
"espira-I": espira1, "espira-II": espira2}
if separate:
start_real = time()
rmse_real, params_real = methods[method](
self.correlation_function(tlist).real, Nr)
end_real = time()
start_imag = time()
rmse_imag, params_imag = methods[method](
self.correlation_function(tlist).imag, Ni)
end_imag = time()
ckAR, phases = params_real.T
ckAI, phases2 = params_imag.T
vkAR = phase_to_exponent(phases)
vkAI = phase_to_exponent(phases2)
cls = ExponentialBosonicEnvironment(
ck_real=ckAR, vk_real=vkAR, ck_imag=ckAI,
vk_imag=vkAI, T=self.T, combine=combine, tag=tag)
params_real = [(ckAR[i].real, ckAR[i].imag, vkAR[i].real,
vkAR[i].imag) for i in range(len(ckAR))]
params_imag = [(ckAI[i].real, ckAI[i].imag, vkAI[i].real,
vkAI[i].imag) for i in range(len(ckAI))]
fit_time_real = end_real-start_real
fit_time_imag = end_imag-start_imag
full_summary = _cf_fit_summary(
params_real, params_imag, fit_time_real, fit_time_imag,
Nr, Ni, rmse_real, rmse_imag, n=4)
fit_info = {
"Nr": Nr, "Ni": Ni, "fit_time_real": fit_time_real,
"fit_time_imag": fit_time_imag, "rmse_real": rmse_real,
"rmse_imag": rmse_imag, "params_real": params_real,
"params_imag": params_imag, "summary": full_summary,
}
else:
start_real = time()
rmse_real, params_real = methods[method](
self.correlation_function(tlist), Nr)
end_real = time()
amp, phases = params_real.T
ck = amp
vk = phase_to_exponent(phases)
# Create complex conjugates for both vk and ck
ckAR = np.concatenate((ck/2, ck.conj()/2))
ckAI = np.concatenate((-1j*ck/2, 1j*ck.conj()/2))
vkAR = np.concatenate((vk, vk.conj()))
cls = ExponentialBosonicEnvironment(
ck_real=ckAR, vk_real=vkAR, ck_imag=ckAI,
vk_imag=vkAR, T=self.T, combine=combine, tag=tag)
params_real = [(ck[i].real, ck[i].imag, vk[i].real,
vk[i].imag) for i in range(len(amp))]
fit_time_real = end_real-start_real
full_summary = _fit_summary(fit_time_real, rmse_real, Nr,
"Correlation Function", params_real,
columns=['ckr', 'cki', 'vkr', 'vki'])
fit_info = {
"N": Nr, "fit_time": fit_time_real, "rmse": rmse_real,
"params": params_real, "summary": full_summary,
}
return cls, fit_info
def approx_by_cf_fit(self, *args, **kwargs):
# TODO remove by 5.3
warnings.warn('The API has changed. Please use approximate("cf", ...)'
' instead of approx_by_cf_fit(...).', FutureWarning)
return self.approximate("cf", *args, **kwargs)
def approx_by_sd_fit(self, *args, **kwargs):
# TODO remove by 5.3
warnings.warn('The API has changed. Please use approximate("sd", ...)'
' instead of approx_by_sd_fit(...).', FutureWarning)
return self.approximate("sd", *args, **kwargs)
class _BosonicEnvironment_fromCF(BosonicEnvironment):
def __init__(self, C, tlist, tMax, T, tag, args):
super().__init__(T, tag)
self._cf = _complex_interpolation(
C, tlist, 'correlation function', args)
if tlist is not None:
self.tMax = max(np.abs(tlist[0]), np.abs(tlist[-1]))
else:
self.tMax = tMax
def correlation_function(self, t, **kwargs):
t = np.asarray(t, dtype=float)
result = np.zeros_like(t, dtype=complex)
positive_mask = (t >= 0)
non_positive_mask = np.invert(positive_mask)
result[positive_mask] = self._cf(t[positive_mask])
result[non_positive_mask] = np.conj(
self._cf(-t[non_positive_mask])
)
return result.item() if t.ndim == 0 else result
def spectral_density(self, w):
return self._sd_from_ps(w)
def power_spectrum(self, w, **kwargs):
if self.tMax is None:
raise ValueError('The support of the correlation function (tMax) '
'must be specified for this operation.')
return self._ps_from_cf(w, self.tMax)
class _BosonicEnvironment_fromPS(BosonicEnvironment):
def __init__(self, S, wlist, wMax, T, tag, args):
super().__init__(T, tag)
self._ps = _real_interpolation(S, wlist, 'power spectrum', args)
if wlist is not None:
self.wMax = max(np.abs(wlist[0]), np.abs(wlist[-1]))
else:
self.wMax = wMax
def correlation_function(self, t, **kwargs):
if self.wMax is None:
raise ValueError('The support of the power spectrum (wMax) '
'must be specified for this operation.')
return self._cf_from_ps(t, self.wMax)
def spectral_density(self, w):
return self._sd_from_ps(w)
def power_spectrum(self, w, **kwargs):
w = np.asarray(w, dtype=float)
ps = self._ps(w)
return ps.item() if w.ndim == 0 else self._ps(w)
class _BosonicEnvironment_fromSD(BosonicEnvironment):
def __init__(self, J, wlist, wMax, T, tag, args):
super().__init__(T, tag)
self._sd = _real_interpolation(J, wlist, 'spectral density', args)
if wlist is not None:
self.wMax = max(np.abs(wlist[0]), np.abs(wlist[-1]))
else:
self.wMax = wMax
def correlation_function(self, t, *, eps=1e-10):
if self.T is None:
raise ValueError('The temperature must be specified for this '
'operation.')
if self.wMax is None:
raise ValueError('The support of the spectral density (wMax) '
'must be specified for this operation.')
return self._cf_from_ps(t, self.wMax, eps=eps)
def spectral_density(self, w):
w = np.asarray(w, dtype=float)
result = np.zeros_like(w)
positive_mask = (w > 0)
result[positive_mask] = self._sd(w[positive_mask])
return result.item() if w.ndim == 0 else result
def power_spectrum(self, w, *, eps=1e-10):
return self._ps_from_sd(w, eps)
[docs]
class DrudeLorentzEnvironment(BosonicEnvironment):
r"""
Describes a Drude-Lorentz bosonic environment with the spectral density
.. math::
J(\omega) = \frac{2 \lambda \gamma \omega}{\gamma^{2}+\omega^{2}}
(see Eq. 15 in [BoFiN23]_).
Parameters
----------
T : float
Environment temperature.
lam : float
Coupling strength.
gamma : float
Spectral density cutoff frequency.
Nk : optional, int, defaults to 10
The number of Pade exponents to be used for the calculation of the
correlation function.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(
self, T: float, lam: float, gamma: float, *,
Nk: int = 10, tag: Any = None
):
super().__init__(T, tag)
self.lam = lam
self.gamma = gamma
self.Nk = Nk
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
Calculates the Drude-Lorentz spectral density.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
w = np.asarray(w, dtype=float)
result = np.zeros_like(w)
positive_mask = (w > 0)
w_mask = w[positive_mask]
result[positive_mask] = (
2 * self.lam * self.gamma * w_mask / (self.gamma**2 + w_mask**2)
)
return result.item() if w.ndim == 0 else result
[docs]
def correlation_function(
self, t: float | ArrayLike, Nk: int = None, **kwargs
) -> (float | ArrayLike):
"""
Calculates the two-time auto-correlation function of the Drude-Lorentz
environment. The calculation is performed by summing a large number of
exponents of the Pade expansion.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
Nk : int, optional
The number of exponents to use. If not provided, then the value
that was provided when the class was instantiated is used.
"""
if self.T == 0:
raise ValueError("The Drude-Lorentz correlation function diverges "
"at zero temperature.")
t = np.asarray(t, dtype=float)
abs_t = np.abs(t)
Nk = Nk or self.Nk
ck_real, vk_real, ck_imag, vk_imag = self._pade_params(Nk)
def C(c, v):
return np.sum([ck * np.exp(-np.asarray(vk * abs_t))
for ck, vk in zip(c, v)], axis=0)
result = C(ck_real, vk_real) + 1j * C(ck_imag, vk_imag)
result = np.asarray(result, dtype=complex)
result[t < 0] = np.conj(result[t < 0])
return result.item() if t.ndim == 0 else result
[docs]
def power_spectrum(
self, w: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Calculates the power spectrum of the Drude-Lorentz environment.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
sd_derivative = 2 * self.lam / self.gamma
return self._ps_from_sd(w, None, sd_derivative)
# --- approximation methods
@overload
def approximate(
self,
method: Literal['matsubara', 'pade'],
Nk: int,
combine: bool = True,
compute_delta: Literal[False] = False,
tag: Any = None
) -> ExponentialBosonicEnvironment: ...
@overload
def approximate(
self,
method: Literal['matsubara', 'pade'],
Nk: int,
combine: bool = True,
compute_delta: Literal[True] = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, float]: ...
# region overloads from parent class
# unfortunately, @overload definitions must be duplicated in subclasses
@overload
def approximate(
self,
method: Literal['cf'],
tlist: ArrayLike,
target_rmse: float = 2e-5,
Nr_max: int = 10,
Ni_max: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
full_ansatz: bool = False,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['ps'],
wlist: ArrayLike,
target_rmse: float = 5e-6,
Nmax: int = 5,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal["sd"],
wlist: ArrayLike,
Nk: int = 1,
target_rmse: float = 5e-6,
Nmax: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['aaa'],
wlist: ArrayLike,
tol: float = 1e-13,
Nmax: int = 10,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
tlist: ArrayLike,
separate: bool = False,
Nr: int = 3,
Ni: int = 3,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
# endregion
@property
def _approximation_methods(self):
return {
**super()._approximation_methods,
"matsubara": (self._approx_by_matsubara, "Matsubara Truncation"),
"pade": (self._approx_by_pade, "Pade Truncation")
}
[docs]
def approximate(self, method: str, *args, **kwargs):
"""
Generates a multi-exponential approximation of this environment.
The available methods are ``"matsubara"``, ``"pade"``, ``"cf"``,
``"ps"``, ``"sd"``, ``"aaa"``, ``"prony"``, ``"esprit"``,
``"espira-I"`` and ``"espira-II"``. The methods and the parameters
required per method are documented in the
:ref:`Users Guide<environment approximations api>`.
"""
return super().approximate(method, *args, **kwargs)
def _approx_by_matsubara(
self, method, Nk, combine=True, compute_delta=False, tag=None
):
if self.T == 0:
raise ValueError("The Drude-Lorentz correlation function diverges "
"at zero temperature.")
if tag is None and self.tag is not None:
tag = (self.tag, "Matsubara Truncation")
lists = self._matsubara_params(Nk)
approx_env = ExponentialBosonicEnvironment(
*lists, T=self.T, combine=combine, tag=tag)
if not compute_delta:
return approx_env
delta = 2 * self.lam * self.T / self.gamma - 1j * self.lam
for exp in approx_env.exponents:
delta -= exp.coefficient / exp.exponent
return approx_env, delta
def _approx_by_pade(
self, method, Nk, combine=True, compute_delta=False, tag=None
):
if self.T == 0:
raise ValueError("The Drude-Lorentz correlation function diverges "
"at zero temperature.")
if tag is None and self.tag is not None:
tag = (self.tag, "Pade Truncation")
ck_real, vk_real, ck_imag, vk_imag = self._pade_params(Nk)
approx_env = ExponentialBosonicEnvironment(
ck_real, vk_real, ck_imag, vk_imag,
T=self.T, combine=combine, tag=tag
)
if not compute_delta:
return approx_env
delta = 2 * self.lam * self.T / self.gamma - 1j * self.lam
for exp in approx_env.exponents:
delta -= exp.coefficient / exp.exponent
return approx_env, delta
def _pade_params(self, Nk):
ck_real, vk_real = self._corr(Nk)
# There is only one term in the expansion of the imaginary part of the
# Drude-Lorentz correlation function.
ck_imag = [-self.lam * self.gamma]
vk_imag = [self.gamma]
return ck_real, vk_real, ck_imag, vk_imag
def _matsubara_params(self, Nk):
""" Calculate the Matsubara coefficients and frequencies. """
ck_real = [self.lam * self.gamma / np.tan(self.gamma / (2 * self.T))]
ck_real.extend([
(8 * self.lam * self.gamma * self.T * np.pi * k * self.T /
((2 * np.pi * k * self.T)**2 - self.gamma**2))
for k in range(1, Nk + 1)
])
vk_real = [self.gamma]
vk_real.extend([2 * np.pi * k * self.T for k in range(1, Nk + 1)])
ck_imag = [-self.lam * self.gamma]
vk_imag = [self.gamma]
return ck_real, vk_real, ck_imag, vk_imag
# --- Pade approx calculation ---
def _corr(self, Nk):
kappa, epsilon = self._kappa_epsilon(Nk)
eta_p = [self.lam * self.gamma *
self._cot(self.gamma / (2 * self.T))]
gamma_p = [self.gamma]
for ll in range(1, Nk + 1):
eta_p.append(
(kappa[ll] * self.T) * 4 * self.lam *
self.gamma * (epsilon[ll] * self.T)
/ ((epsilon[ll]**2 * self.T**2) - self.gamma**2)
)
gamma_p.append(epsilon[ll] * self.T)
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
def approx_by_matsubara(self, *args, **kwargs):
# TODO remove by 5.3
warnings.warn(
'The API has changed. Please use approximate("matsubara", ...)'
' instead of approx_by_matsubara(...).', FutureWarning)
return self.approximate("matsubara", *args, **kwargs)
def approx_by_pade(self, *args, **kwargs):
# TODO remove by 5.3
warnings.warn(
'The API has changed. Please use approximate("pade", ...)'
' instead of approx_by_pade(...).', FutureWarning)
return self.approximate("pade", *args, **kwargs)
[docs]
class UnderDampedEnvironment(BosonicEnvironment):
r"""
Describes an underdamped environment with the spectral density
.. math::
J(\omega) = \frac{\lambda^{2} \Gamma \omega}{(\omega_0^{2}-
\omega^{2})^{2}+ \Gamma^{2} \omega^{2}}
(see Eq. 16 in [BoFiN23]_).
Parameters
----------
T : float
Environment temperature.
lam : float
Coupling strength.
gamma : float
Spectral density cutoff frequency.
w0 : float
Spectral density resonance frequency.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(
self, T: float, lam: float, gamma: float, w0: float, *, tag: Any = None
):
super().__init__(T, tag)
self.lam = lam
self.gamma = gamma
self.w0 = w0
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
Calculates the underdamped spectral density.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
w = np.asarray(w, dtype=float)
result = np.zeros_like(w)
positive_mask = (w > 0)
w_mask = w[positive_mask]
result[positive_mask] = (
self.lam**2 * self.gamma * w_mask / (
(w_mask**2 - self.w0**2)**2 + (self.gamma * w_mask)**2
)
)
return result.item() if w.ndim == 0 else result
[docs]
def power_spectrum(
self, w: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Calculates the power spectrum of the underdamped environment.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
sd_derivative = self.lam**2 * self.gamma / self.w0**4
return self._ps_from_sd(w, None, sd_derivative)
[docs]
def correlation_function(
self, t: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Calculates the two-time auto-correlation function of the underdamped
environment.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
"""
# we need an wMax so that spectral density is zero for w>wMax, guess:
wMax = self.w0 + 25 * self.gamma
return self._cf_from_ps(t, wMax)
# --- approximation methods
@overload
def approximate(
self,
method: Literal['matsubara'],
Nk: int,
combine: bool = True,
compute_delta: Literal[False] = False,
tag: Any = None
) -> ExponentialBosonicEnvironment: ...
@overload
def approximate(
self,
method: Literal['matsubara'],
Nk: int,
combine: bool = True,
compute_delta: Literal[True] = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, float]: ...
# region overloads from parent class
# unfortunately, @overload definitions must be duplicated in subclasses
@overload
def approximate(
self,
method: Literal['cf'],
tlist: ArrayLike,
target_rmse: float = 2e-5,
Nr_max: int = 10,
Ni_max: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
full_ansatz: bool = False,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['ps'],
wlist: ArrayLike,
target_rmse: float = 5e-6,
Nmax: int = 5,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal["sd"],
wlist: ArrayLike,
Nk: int = 1,
target_rmse: float = 5e-6,
Nmax: int = 10,
guess: list[float] = None,
lower: list[float] = None,
upper: list[float] = None,
sigma: float | ArrayLike = None,
maxfev: int = None,
combine: bool = True,
tag: Any = None,
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['aaa'],
wlist: ArrayLike,
tol: float = 1e-13,
Nmax: int = 10,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
@overload
def approximate(
self,
method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
tlist: ArrayLike,
separate: bool = False,
Nr: int = 3,
Ni: int = 3,
combine: bool = True,
tag: Any = None
) -> tuple[ExponentialBosonicEnvironment, dict[str, Any]]:
...
# endregion
@property
def _approximation_methods(self):
return {
**super()._approximation_methods,
"matsubara": (self._approx_by_matsubara, "Matsubara Truncation")
}
[docs]
def approximate(self, method: str, *args, **kwargs):
"""
Generates a multi-exponential approximation of this environment.
The available methods are ``"matsubara"``, ``"cf"``,
``"ps"``, ``"sd"``, ``"aaa"``, ``"prony"``, ``"esprit"``,
``"espira-I"`` and ``"espira-II"``. The methods and the parameters
required per method are documented in the
:ref:`Users Guide<environment approximations api>`.
"""
return super().approximate(method, *args, **kwargs)
def _approx_by_matsubara(
self, method, Nk, combine=True, compute_delta=False, tag=None
):
if tag is None and self.tag is not None:
tag = (self.tag, "Matsubara Truncation")
lists = self._matsubara_params(Nk)
approx_env = ExponentialBosonicEnvironment(
*lists, T=self.T, combine=combine, tag=tag)
if not compute_delta:
return approx_env
delta = self.gamma * self.lam**2 * self.T / self.w0**4
for exp in approx_env.exponents:
delta -= np.real(exp.coefficient / exp.exponent)
return approx_env, delta
def _matsubara_params(self, Nk):
""" Calculate the Matsubara coefficients and frequencies. """
if Nk > 0 and self.T == 0:
warnings.warn("The Matsubara expansion cannot be performed at "
"zero temperature. Use other approaches such as "
"fitting the correlation function.")
Nk = 0
Om = np.sqrt(self.w0**2 - (self.gamma / 2)**2)
Gamma = self.gamma / 2
z = np.inf if self.T == 0 else (Om + 1j * Gamma) / (2 * self.T)
# we set the argument of the hyperbolic tangent to infinity if T=0
ck_real = ([
(self.lam**2 / (4 * Om)) * (1 / np.tanh(z)),
(self.lam**2 / (4 * Om)) * (1 / np.tanh(np.conjugate(z))),
])
ck_real.extend([
(-2 * self.lam**2 * self.gamma * self.T) * (2 * np.pi * k * self.T)
/ (
((Om + 1j * Gamma)**2 + (2 * np.pi * k * self.T)**2)
* ((Om - 1j * Gamma)**2 + (2 * np.pi * k * self.T)**2)
)
for k in range(1, Nk + 1)
])
vk_real = [-1j * Om + Gamma, 1j * Om + Gamma]
vk_real.extend([
2 * np.pi * k * self.T
for k in range(1, Nk + 1)
])
ck_imag = [
1j * self.lam**2 / (4 * Om),
-1j * self.lam**2 / (4 * Om),
]
vk_imag = [-1j * Om + Gamma, 1j * Om + Gamma]
return ck_real, vk_real, ck_imag, vk_imag
def approx_by_matsubara(self, *args, **kwargs):
# TODO remove by 5.3
warnings.warn(
'The API has changed. Please use approximate("matsubara", ...)'
' instead of approx_by_matsubara(...).', FutureWarning)
return self.approximate("matsubara", *args, **kwargs)
[docs]
class OhmicEnvironment(BosonicEnvironment):
r"""
Describes Ohmic environments as well as sub- or super-Ohmic environments
(depending on the choice of the parameter `s`). The spectral density is
.. math::
J(\omega)
= \alpha \frac{\omega^s}{\omega_c^{s-1}} e^{-\omega / \omega_c} .
This class requires the `mpmath` module to be installed.
Parameters
----------
T : float
Temperature of the environment.
alpha : float
Coupling strength.
wc : float
Cutoff parameter.
s : float
Power of omega in the spectral density.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(
self, T: float, alpha: float, wc: float, s: float, *, tag: Any = None
):
super().__init__(T, tag)
self.alpha = alpha
self.wc = wc
self.s = s
if _mpmath_available is False:
warnings.warn(
"The mpmath module is required for some operations on "
"Ohmic environments, but it is not installed.")
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
r"""
Calculates the spectral density of the Ohmic environment.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
w = np.asarray(w, dtype=float)
result = np.zeros_like(w)
positive_mask = (w > 0)
w_mask = w[positive_mask]
result[positive_mask] = (
self.alpha * w_mask ** self.s
/ (self.wc ** (self.s - 1))
* np.exp(-np.abs(w_mask) / self.wc)
)
return result.item() if w.ndim == 0 else result
[docs]
def power_spectrum(
self, w: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Calculates the power spectrum of the Ohmic environment.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
if self.s > 1:
sd_derivative = 0
elif self.s == 1:
sd_derivative = self.alpha
else:
sd_derivative = np.inf
return self._ps_from_sd(w, None, sd_derivative)
[docs]
def correlation_function(
self, t: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
r"""
Calculates the correlation function of an Ohmic environment using the
formula
.. math::
C(t)= \frac{1}{\pi} \alpha w_{c}^{1-s} \beta^{-(s+1)} \Gamma(s+1)
\left[ \zeta\left(s+1,\frac{1+\beta w_{c} -i w_{c} t}{\beta w_{c}}
\right) +\zeta\left(s+1,\frac{1+ i w_{c} t}{\beta w_{c}}\right)
\right] ,
where :math:`\Gamma` is the gamma function, and :math:`\zeta` the
Riemann zeta function.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
"""
t = np.asarray(t, dtype=float)
t_was_array = t.ndim > 0
if not t_was_array:
t = np.array([t], dtype=float)
if self.T != 0:
corr = (self.alpha * self.wc ** (1 - self.s) / np.pi
* mp.gamma(self.s + 1) * self.T ** (self.s + 1))
z1_u = ((1 + self.wc / self.T - 1j * self.wc * t)
/ (self.wc / self.T))
z2_u = (1 + 1j * self.wc * t) / (self.wc / self.T)
result = np.asarray(
[corr * (mp.zeta(self.s + 1, u1) + mp.zeta(self.s + 1, u2))
for u1, u2 in zip(z1_u, z2_u)],
dtype=np.cdouble
)
else:
corr = (self.alpha * self.wc**2 / np.pi
* mp.gamma(self.s + 1)
* (1 + 1j * self.wc * t) ** (-self.s - 1))
result = np.asarray(corr, dtype=np.cdouble)
if t_was_array:
return result
return result[0]
[docs]
class CFExponent:
"""
Represents a single exponent (naively, an excitation mode) within an
exponential decomposition of the correlation function of a environment.
Parameters
----------
type : {"R", "I", "RI", "+", "-"} or one of `CFExponent.types`
The type of exponent.
"R" and "I" are bosonic exponents that appear in the real and
imaginary parts of the correlation expansion, respectively.
"RI" is a combined bosonic 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 exponents.
ck : complex
The coefficient of the excitation term.
vk : complex
The frequency of the exponent 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).
tag : optional, str, tuple or any other object
A label for the exponent (often the name of the environment). 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.
coefficient : complex
The coefficient of this excitation term in the total correlation
function (including real and imaginary part).
exponent : complex
The frequency of the exponent of the excitation term. (Alias for `vk`.)
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 exponents require ck2")
else:
if ck2 is not None:
raise ValueError(
"Second co-efficient (ck2) should only be specified for"
" RI exponents"
)
def _type_is_fermionic(self, type):
return type in (self.types["+"], self.types["-"])
def __init__(
self, type: str | CFExponent.ExponentType,
ck: complex, vk: complex, ck2: complex = None, tag: Any = None
):
if not isinstance(type, self.types):
type = self.types[type]
self._check_ck2(type, ck2)
self.type = type
self.ck = ck
self.vk = vk
self.ck2 = ck2
self.tag = tag
self.fermionic = self._type_is_fermionic(type)
[docs]
def rescale(self, alpha: float) -> CFExponent:
"""Rescale the coefficient of the exponent by a factor of alpha."""
ck_new = self.ck * alpha
if self.type == self.types["RI"]:
ck2_new = self.ck2 * alpha
else:
ck2_new = None
return CFExponent(
type=self.type, ck=ck_new, vk=self.vk, ck2=ck2_new, tag=self.tag
)
def __repr__(self):
return (
f"<{self.__class__.__name__} type={self.type.name}"
f" ck={self.ck!r} vk={self.vk!r} ck2={self.ck2!r}"
f" fermionic={self.fermionic!r}"
f" tag={self.tag!r}>"
)
@property
def coefficient(self) -> complex:
coeff = 0
if self.type != self.types['I']:
coeff += self.ck
else:
coeff += 1j * self.ck
if self.type == self.types['RI']:
coeff += 1j * self.ck2
return coeff
@property
def exponent(self) -> complex:
return self.vk
def _can_combine(self, other, rtol, atol):
if type(self) is not type(other):
return False
if self.fermionic or other.fermionic:
return False
if not np.isclose(self.vk, other.vk, rtol=rtol, atol=atol):
return False
return True
def _combine(self, other, **init_kwargs):
# Assumes can combine was checked
cls = type(self)
if self.type == other.type and self.type != self.types['RI']:
# Both R or both I
return cls(type=self.type, ck=(self.ck + other.ck),
vk=self.vk, tag=self.tag, **init_kwargs)
# Result will be RI
real_part_coefficient = 0
imag_part_coefficient = 0
for exp in [self, other]:
if exp.type == self.types['RI'] or exp.type == self.types['R']:
real_part_coefficient += exp.ck
if exp.type == self.types['I']:
imag_part_coefficient += exp.ck
if exp.type == self.types['RI']:
imag_part_coefficient += exp.ck2
return cls(type=self.types['RI'], ck=real_part_coefficient, vk=self.vk,
ck2=imag_part_coefficient, tag=self.tag, **init_kwargs)
[docs]
class ExponentialBosonicEnvironment(BosonicEnvironment):
"""
Bosonic environment that is specified through an exponential decomposition
of its correlation function. The list of coefficients and exponents in
the decomposition may either be passed through the four lists `ck_real`,
`vk_real`, `ck_imag`, `vk_imag`, or as a list of bosonic
:class:`CFExponent` objects.
Parameters
----------
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 coefficients 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 coefficients are
passed as ck_imag.
exponents : list of :class:`CFExponent`
The expansion coefficients and exponents of both the real and the
imaginary parts of the correlation function as :class:`CFExponent`
objects.
combine : bool, default True
Whether to combine exponents with the same frequency. See
:meth:`combine` for details.
T: optional, float
The temperature of the environment.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
_make_exponent = CFExponent
def _check_cks_and_vks(self, ck_real, vk_real, ck_imag, vk_imag):
# all None: returns False
# all provided and lengths match: returns True
# otherwise: raises ValueError
lists = [ck_real, vk_real, ck_imag, vk_imag]
if all(x is None for x in lists):
return False
if any(x is None for x in lists):
raise ValueError(
"If any of the exponent lists ck_real, vk_real, ck_imag, "
"vk_imag is provided, all must be provided."
)
if len(ck_real) != len(vk_real) or len(ck_imag) != len(vk_imag):
raise ValueError(
"The exponent lists ck_real and vk_real, and ck_imag and "
"vk_imag must be the same length."
)
return True
def __init__(
self,
ck_real: ArrayLike = None, vk_real: ArrayLike = None,
ck_imag: ArrayLike = None, vk_imag: ArrayLike = None,
*,
exponents: Sequence[CFExponent] = None,
combine: bool = True, T: float = None, tag: Any = None
):
super().__init__(T, tag)
lists_provided = self._check_cks_and_vks(
ck_real, vk_real, ck_imag, vk_imag)
if exponents is None and not lists_provided:
raise ValueError(
"Either the parameter `exponents` or the parameters "
"`ck_real`, `vk_real`, `ck_imag`, `vk_imag` must be provided."
)
if exponents is not None and any(exp.fermionic for exp in exponents):
raise ValueError(
"Fermionic exponent passed to exponential bosonic environment."
)
exponents = exponents or []
if lists_provided:
exponents.extend(self._make_exponent("R", ck, vk, tag=tag)
for ck, vk in zip(ck_real, vk_real))
exponents.extend(self._make_exponent("I", ck, vk, tag=tag)
for ck, vk in zip(ck_imag, vk_imag))
if combine:
exponents = self.combine(exponents)
self.exponents = exponents
[docs]
@classmethod
def combine(
cls, exponents: Sequence[CFExponent],
rtol: float = 1e-5, atol: float = 1e-7
) -> Sequence[CFExponent]:
"""
Group bosonic exponents with the same frequency and return a
single exponent for each frequency present.
Parameters
----------
exponents : list of :class:`CFExponent`
The list of exponents to combine.
rtol : float, default 1e-5
The relative tolerance to use to when comparing frequencies.
atol : float, default 1e-7
The absolute tolerance to use to when comparing frequencies.
Returns
-------
list of :class:`CFExponent`
The new reduced list of exponents.
"""
remaining = exponents[:]
new_exponents = []
while remaining:
new_exponent = remaining.pop(0)
for other_exp in remaining[:]:
if new_exponent._can_combine(other_exp, rtol, atol):
new_exponent = new_exponent._combine(other_exp)
remaining.remove(other_exp)
new_exponents.append(new_exponent)
return new_exponents
[docs]
def correlation_function(
self, t: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Computes the correlation function represented by this exponential
decomposition.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
"""
t = np.asarray(t, dtype=float)
corr = np.zeros_like(t, dtype=complex)
for exp in self.exponents:
corr += exp.coefficient * np.exp(-exp.exponent * np.abs(t))
corr[t < 0] = np.conj(corr[t < 0])
return corr.item() if t.ndim == 0 else corr
[docs]
def power_spectrum(
self, w: float | ArrayLike, **kwargs
) -> (float | ArrayLike):
"""
Calculates the power spectrum corresponding to the multi-exponential
correlation function.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
w = np.asarray(w, dtype=float)
S = np.zeros_like(w)
for exp in self.exponents:
S += 2 * np.real(
exp.coefficient / (exp.exponent - 1j * w)
)
return S.item() if w.ndim == 0 else S
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
Calculates the spectral density corresponding to the multi-exponential
correlation function.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
return self._sd_from_ps(w)
[docs]
def rescale(
self, alpha: float, tag: Any = None
) -> ExponentialBosonicEnvironment:
"""
Returns a new environment where all exponents are scaled by the factor
alpha. This corresponds to changing the coupling constant. The spectral
density, the correlation function and the power spectrum are all scaled
by the same factor.
Parameters
----------
alpha : float
Rescaling factor.
tag : optional, str, tuple or any other object
An identifier (name) for the rescaled environment.
Returns
-------
ExponentialBosonicEnvironment
The new environment with the rescaled exponents.
"""
tag = tag or self.tag
return ExponentialBosonicEnvironment(
exponents=[exp.rescale(alpha) for exp in self.exponents],
combine=False, T=self.T, tag=tag
)
[docs]
def system_terminator(Q: Qobj, delta: float) -> Qobj:
"""
Constructs the terminator for a given approximation discrepancy.
Parameters
----------
Q : :class:`Qobj`
The system coupling operator.
delta : float
The approximation discrepancy of approximating an environment with a
finite number of exponentials, see for example the description of the
:ref:`Matsubara approximation<matsubara approximations api>`.
Returns
-------
terminator : :class:`Qobj`
A superoperator acting on the system Hilbert space. Liouvillian term
representing the contribution to the system-environment dynamics of all
neglected expansion terms. It should be used by adding it to the system
Liouvillian (i.e. ``liouvillian(H_sys)``).
"""
op = 2 * spre(Q) * spost(Q.dag()) - spre(Q.dag() * Q) - spost(Q.dag() * Q)
return delta * op
# --- utility functions ---
def _real_interpolation(fun, xlist, name, args=None):
args = args or {}
if callable(fun):
return lambda w: fun(w, **args)
else:
if xlist is None or len(xlist) != len(fun):
raise ValueError("A list of x-values with the same length must be "
f"provided for the discretized function ({name})")
return CubicSpline(xlist, fun)
def _complex_interpolation(fun, xlist, name, args=None):
args = args or {}
if callable(fun):
return lambda t: fun(t, **args)
else:
real_interp = _real_interpolation(np.real(fun), xlist, name)
imag_interp = _real_interpolation(np.imag(fun), xlist, name)
return lambda x: real_interp(x) + 1j * imag_interp(x)
def _fft(f, wMax, tMax):
r"""
Calculates the Fast Fourier transform of the given function. We calculate
Fourier transformations via FFT because numerical integration is often
noisy in the scenarios we are interested in.
Given a (mathematical) function `f(t)`, this function approximates its
Fourier transform
.. math::
g(\omega) = \int_{-\infty}^\infty dt\, e^{-i\omega t}\, f(t) .
The function f is sampled on the interval `[-tMax, tMax]`. The sampling
discretization is chosen as `dt = pi / (4*wMax)` (Shannon-Nyquist + some
leeway). However, `dt` is always chosen small enough to have at least 500
samples on the interval `[-tMax, tMax]`.
Parameters
----------
wMax: float
Maximum frequency of interest
tMax: float
Support of the function f (i.e., f(t) is essentially zero for
`|t| > tMax`).
Returns
-------
The fourier transform of the provided function as an interpolated function.
"""
# Code adapted from https://stackoverflow.com/a/24077914
numSamples = int(
max(500, np.ceil(2 * tMax * 4 * wMax / np.pi + 1))
)
t, dt = np.linspace(-tMax, tMax, numSamples, retstep=True)
f_values = f(t)
# Compute Fourier transform by numpy's FFT function
g = np.fft.fft(f_values)
# frequency normalization factor is 2 * np.pi / dt
w = np.fft.fftfreq(numSamples) * 2 * np.pi / dt
# In order to get a discretisation of the continuous Fourier transform
# we need to multiply g by a phase factor
g *= dt * np.exp(1j * w * tMax)
return _complex_interpolation(
np.fft.fftshift(g), np.fft.fftshift(w), 'FFT'
)
def _cf_real_fit_model(tlist, a, b, c, d=0):
return np.real((a + 1j * d) * np.exp((b + 1j * c) * np.abs(tlist)))
def _cf_imag_fit_model(tlist, a, b, c, d=0):
return np.sign(tlist) * np.imag(
(a + 1j * d) * np.exp((b + 1j * c) * np.abs(tlist))
)
def _default_guess_cfreal(tlist, clist, full_ansatz):
corr_abs = np.abs(clist)
corr_max = np.max(corr_abs)
tc = 2 / np.max(tlist)
# Checks if constant array, and assigns zero
if (clist == clist[0]).all():
if full_ansatz:
return [[0] * 4] * 3
return [[0] * 3] * 3
if full_ansatz:
lower = [-100 * corr_max, -np.inf, -np.inf, -100 * corr_max]
guess = [corr_max, -100 * corr_max, 0, 0]
upper = [100 * corr_max, 0, np.inf, 100 * corr_max]
else:
lower = [-20 * corr_max, -np.inf, 0]
guess = [corr_max, -tc, 0]
upper = [20 * corr_max, 0.1, np.inf]
return guess, lower, upper
def _default_guess_cfimag(clist, full_ansatz):
corr_max = np.max(np.abs(clist))
# Checks if constant array, and assigns zero
if (clist == clist[0]).all():
if full_ansatz:
return [[0] * 4] * 3
return [[0] * 3] * 3
if full_ansatz:
lower = [-100 * corr_max, -np.inf, -np.inf, -100 * corr_max]
guess = [0, -10 * corr_max, 0, 0]
upper = [100 * corr_max, 0, np.inf, 100 * corr_max]
else:
lower = [-20 * corr_max, -np.inf, 0]
guess = [-corr_max, -10 * corr_max, 1]
upper = [10 * corr_max, 0, np.inf]
return guess, lower, upper
def _sd_fit_model(wlist, a, b, c):
return (
2 * a * b * wlist / ((wlist + c)**2 + b**2) / ((wlist - c)**2 + b**2)
)
def _default_guess_sd(wlist, jlist):
sd_abs = np.abs(jlist)
sd_max = np.max(sd_abs)
wc = wlist[np.argmax(sd_abs)]
if sd_max == 0:
return [0] * 3
lower = [-100 * sd_max, 0.1 * wc, 0.1 * wc]
guess = [sd_max, wc, wc]
upper = [100 * sd_max, 100 * wc, 100 * wc]
return guess, lower, upper
def _ps_fit_model(wlist, a, b, c, d):
return (
2 * (a*c + b*(d-wlist)) / ((wlist - d)**2 + c**2)
)
def _default_guess_ps(wlist, jlist):
sd_abs = np.abs(jlist)
sd_max = np.max(sd_abs)
wc = np.abs(wlist[np.argmin(sd_abs)])
if sd_max == 0:
return [0] * 4
lower = [-1.5 * sd_max, -1.5 * sd_max, 0.01 * wc, -np.pi]
guess = [sd_max, sd_max, wc, -np.pi/2]
upper = [1.5 * sd_max, 1.5 * sd_max, 10 * wc, np.pi]
return guess, lower, upper
def _fit_summary(time, rmse, N, label, params,
columns=['a', 'b', 'c']):
# Generates summary of fit by nonlinear least squares
if len(columns) == 3:
summary = (f"Result of fitting {label} "
f"with {N} terms: \n \n {'Parameters': <10}|"
f"{columns[0]: ^10}|{columns[1]: ^10}|{columns[2]: >5} \n ")
for k in range(N):
summary += (
f"{k + 1: <10}|{params[k][0]: ^10.2e}|{params[k][1]:^10.2e}|"
f"{params[k][2]:>5.2e}\n ")
elif len(columns) == 4:
summary = (
f"Result of fitting {label} "
f"with {N} terms: \n \n {'Parameters': <10}|"
f"{columns[0]: ^10}|{columns[1]: ^10}|{columns[2]: ^10}"
f"|{columns[3]: >5} \n ")
for k in range(N):
summary += (
f"{k + 1: <10}|{params[k][0]: ^10.2e}|{params[k][1]:^10.2e}"
f"|{params[k][2]:^10.2e}|{params[k][3]:>5.2e}\n ")
else:
raise ValueError("Unsupported number of columns")
summary += (f"\nA RMSE of {rmse: .2e}"
f" was obtained for the {label}.\n")
summary += f"The current fit took {time: 2f} seconds."
return summary
def _cf_fit_summary(
params_real, params_imag, fit_time_real, fit_time_imag, Nr, Ni,
rmse_real, rmse_imag, n=3
):
# Generate nicely formatted summary with two columns for CF fit
columns = ["ckr", "vkr", "vki"]
if n == 4:
columns.append("cki")
summary_real = _fit_summary(
fit_time_real, rmse_real, Nr,
"the real part of\nthe correlation function",
params_real, columns=columns
)
summary_imag = _fit_summary(
fit_time_imag, rmse_imag, Ni,
"the imaginary part\nof the correlation function",
params_imag, columns=columns
)
full_summary = "Correlation function fit:\n\n"
lines_real = summary_real.splitlines()
lines_imag = summary_imag.splitlines()
max_lines = max(len(lines_real), len(lines_imag))
# Fill the shorter string with blank lines
lines_real = (
lines_real[:-1]
+ (max_lines - len(lines_real)) * [""] + [lines_real[-1]]
)
lines_imag = (
lines_imag[:-1]
+ (max_lines - len(lines_imag)) * [""] + [lines_imag[-1]]
)
# Find the maximum line length in each column
max_length1 = max(len(line) for line in lines_real)
max_length2 = max(len(line) for line in lines_imag)
# Print the strings side by side with a vertical bar separator
for line1, line2 in zip(lines_real, lines_imag):
formatted_line1 = f"{line1:<{max_length1}} |"
formatted_line2 = f"{line2:<{max_length2}}"
full_summary += formatted_line1 + formatted_line2 + "\n"
return full_summary
# --- fermionic environments ---
[docs]
class FermionicEnvironment(abc.ABC):
r"""
The fermionic environment of an open quantum system. It is characterized by
its spectral density, temperature and chemical potential or, equivalently,
by its power spectra or its two-time auto-correlation functions.
This class is included as a counterpart to :class:`BosonicEnvironment`, but
it currently does not support all features that the bosonic environment
does. In particular, fermionic environments cannot be constructed from
manually specified spectral densities, power spectra or correlation
functions. The only types of fermionic environment implemented at this time
are Lorentzian environments (:class:`LorentzianEnvironment`) and
environments with multi-exponential correlation functions
(:class:`ExponentialFermionicEnvironment`).
Parameters
----------
T : optional, float
The temperature of this environment.
mu : optional, float
The chemical potential of this environment.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(self, T: float = None, mu: float = None, tag: Any = None):
self.T = T
self.mu = mu
self.tag = tag
[docs]
@abc.abstractmethod
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
r"""
The spectral density of this environment. See the Users Guide on
:ref:`fermionic environments <fermionic environments guide>` for
specifics on the definitions used by QuTiP.
Parameters
----------
w : array_like or float
The frequencies at which to evaluate the spectral density.
"""
...
[docs]
@abc.abstractmethod
def correlation_function_plus(
self, t: float | ArrayLike
) -> (float | ArrayLike):
r"""
The "+"-branch of the auto-correlation function of this environment.
See the Users Guide on
:ref:`fermionic environments <fermionic environments guide>` for
specifics on the definitions used by QuTiP.
Parameters
----------
t : array_like or float
The times at which to evaluate the correlation function.
"""
...
[docs]
@abc.abstractmethod
def correlation_function_minus(
self, t: float | ArrayLike
) -> (float | ArrayLike):
r"""
The "-"-branch of the auto-correlation function of this environment.
See the Users Guide on
:ref:`fermionic environments <fermionic environments guide>` for
specifics on the definitions used by QuTiP.
Parameters
----------
t : array_like or float
The times at which to evaluate the correlation function.
"""
...
[docs]
@abc.abstractmethod
def power_spectrum_plus(self, w: float | ArrayLike) -> (float | ArrayLike):
r"""
The "+"-branch of the power spectrum of this environment. See the Users
Guide on :ref:`fermionic environments <fermionic environments guide>`
for specifics on the definitions used by QuTiP.
Parameters
----------
w : array_like or float
The frequencies at which to evaluate the power spectrum.
"""
...
[docs]
@abc.abstractmethod
def power_spectrum_minus(
self, w: float | ArrayLike
) -> (float | ArrayLike):
r"""
The "-"-branch of the power spectrum of this environment. See the Users
Guide on :ref:`fermionic environments <fermionic environments guide>`
for specifics on the definitions used by QuTiP.
Parameters
----------
w : array_like or float
The frequencies at which to evaluate the power spectrum.
"""
...
# --- user-defined environment creation
[docs]
@classmethod
def from_correlation_functions(cls, **kwargs) -> FermionicEnvironment:
r"""
User-defined fermionic environments are currently not implemented.
"""
raise NotImplementedError("User-defined fermionic environments are "
"currently not implemented.")
[docs]
@classmethod
def from_power_spectra(cls, **kwargs) -> FermionicEnvironment:
r"""
User-defined fermionic environments are currently not implemented.
"""
raise NotImplementedError("User-defined fermionic environments are "
"currently not implemented.")
@classmethod
def from_spectral_density(cls, **kwargs) -> FermionicEnvironment:
r"""
User-defined fermionic environments are currently not implemented.
"""
raise NotImplementedError("User-defined fermionic environments are "
"currently not implemented.")
[docs]
class LorentzianEnvironment(FermionicEnvironment):
r"""
Describes a Lorentzian fermionic environment with the spectral density
.. math::
J(\omega) = \frac{\gamma W^2}{(\omega - \omega_0)^2 + W^2}.
(see Eq. 46 in [BoFiN23]_).
Parameters
----------
T : float
Environment temperature.
mu : float
Environment chemical potential.
gamma : float
Coupling strength.
W : float
The spectral width of the environment.
omega0 : optional, float (default equal to ``mu``)
The resonance frequency of the environment.
Nk : optional, int, defaults to 10
The number of Pade exponents to be used for the calculation of the
correlation functions.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def __init__(
self, T: float, mu: float, gamma: float, W: float,
omega0: float = None, *, Nk: int = 10, tag: Any = None
):
super().__init__(T, mu, tag)
self.gamma = gamma
self.W = W
self.Nk = Nk
if omega0 is None:
self.omega0 = mu
else:
self.omega0 = omega0
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
Calculates the Lorentzian spectral density.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
w = np.asarray(w, dtype=float)
return self.gamma * self.W**2 / ((w - self.omega0)**2 + self.W**2)
[docs]
def correlation_function_plus(
self, t: float | ArrayLike, Nk: int = None
) -> (float | ArrayLike):
r"""
Calculates the "+"-branch of the two-time auto-correlation function of
the Lorentzian environment. The calculation is performed by summing a
large number of exponents of the Pade expansion.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
Nk : int, optional
The number of exponents to use. If not provided, then the value
that was provided when the class was instantiated is used.
"""
Nk = Nk or self.Nk
return self._correlation_function(t, Nk, 1)
[docs]
def correlation_function_minus(
self, t: float | ArrayLike, Nk: int = None
) -> (float | ArrayLike):
r"""
Calculates the "-"-branch of the two-time auto-correlation function of
the Lorentzian environment. The calculation is performed by summing a
large number of exponents of the Pade expansion.
Parameters
----------
t : array_like or float
The time at which to evaluate the correlation function.
Nk : int, optional
The number of exponents to use. If not provided, then the value
that was provided when the class was instantiated is used.
"""
Nk = Nk or self.Nk
return self._correlation_function(t, Nk, -1)
def _correlation_function(self, t, Nk, sigma):
if self.T == 0:
raise NotImplementedError(
"Calculation of zero-temperature Lorentzian correlation "
"functions is not implemented yet.")
t = np.asarray(t, dtype=float)
abs_t = np.abs(t)
c, v = self._corr(Nk, sigma)
result = np.sum([ck * np.exp(-np.asarray(vk * abs_t))
for ck, vk in zip(c, v)], axis=0)
result = np.asarray(result, dtype=complex)
result[t < 0] = np.conj(result[t < 0])
return result.item() if t.ndim == 0 else result
[docs]
def power_spectrum_plus(self, w: float | ArrayLike) -> (float | ArrayLike):
r"""
Calculates the "+"-branch of the power spectrum of the Lorentzian
environment.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
return self.spectral_density(w) / (np.exp((w - self.mu) / self.T) + 1)
[docs]
def power_spectrum_minus(
self, w: float | ArrayLike
) -> (float | ArrayLike):
r"""
Calculates the "-"-branch of the power spectrum of the Lorentzian
environment.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
return self.spectral_density(w) / (np.exp((self.mu - w) / self.T) + 1)
[docs]
def approx_by_matsubara(
self, Nk: int, tag: Any = None
) -> ExponentialFermionicEnvironment:
"""
Generates an approximation to this environment by truncating its
Matsubara expansion.
Parameters
----------
Nk : int
Number of Matsubara terms to include. In total, the "+" and "-"
correlation function branches will include `Nk+1` terms each.
tag : optional, str, tuple or any other object
An identifier (name) for the approximated environment. If not
provided, a tag will be generated from the tag of this environment.
Returns
-------
The approximated environment with multi-exponential correlation
function.
"""
if self.T == 0:
raise NotImplementedError(
"Calculation of zero-temperature Lorentzian correlation "
"functions is not implemented yet.")
if tag is None and self.tag is not None:
tag = (self.tag, "Matsubara Truncation")
ck_plus, vk_plus = self._matsubara_params(Nk, 1)
ck_minus, vk_minus = self._matsubara_params(Nk, -1)
return ExponentialFermionicEnvironment(
ck_plus, vk_plus, ck_minus, vk_minus, T=self.T, mu=self.mu,
tag=tag
)
[docs]
def approx_by_pade(
self, Nk: int, tag: Any = None
) -> ExponentialFermionicEnvironment:
"""
Generates an approximation to this environment by truncating its
Pade expansion.
Parameters
----------
Nk : int
Number of Pade terms to include. In total, the "+" and "-"
correlation function branches will include `Nk+1` terms each.
tag : optional, str, tuple or any other object
An identifier (name) for the approximated environment. If not
provided, a tag will be generated from the tag of this environment.
Returns
-------
The approximated environment with multi-exponential correlation
function.
"""
if self.T == 0:
raise NotImplementedError(
"Calculation of zero-temperature Lorentzian correlation "
"functions is not implemented yet.")
if tag is None and self.tag is not None:
tag = (self.tag, "Pade Truncation")
ck_plus, vk_plus = self._corr(Nk, sigma=1)
ck_minus, vk_minus = self._corr(Nk, sigma=-1)
return ExponentialFermionicEnvironment(
ck_plus, vk_plus, ck_minus, vk_minus, T=self.T, mu=self.mu, tag=tag
)
def _matsubara_params(self, Nk, sigma):
""" Calculate the Matsubara coefficients and frequencies. """
def f(x):
return 1 / (np.exp(x / self.T) + 1)
coeff_list = [(
self.W * self.gamma / 2 *
f(sigma * (self.omega0 - self.mu) + 1j * self.W)
)]
exp_list = [self.W - sigma * 1j * self.omega0]
xk_list = [(2 * k - 1) * np.pi * self.T for k in range(1, Nk + 1)]
for xk in xk_list:
coeff_list.append(
1j * self.gamma * self.W**2 * self.T /
((sigma * xk - 1j * self.mu + 1j * self.omega0)**2 - self.W**2)
)
exp_list.append(
xk - sigma * 1j * self.mu
)
return coeff_list, exp_list
# --- Pade approx calculation ---
def _corr(self, Nk, sigma):
beta = 1 / self.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 * self.gamma * self.W *
f_approx(beta * sigma * (self.omega0 - self.mu)
+ beta * 1j * self.W))]
gamma_list = [self.W - sigma * 1.0j * self.omega0]
for ll in range(1, Nk + 1):
eta_list.append(
-1.0j * (kappa[ll] / beta) * self.gamma * self.W**2
/ ((self.mu - self.omega0 + sigma * 1j * epsilon[ll] / beta)**2
+ self.W**2)
)
gamma_list.append(epsilon[ll] / beta - sigma * 1.0j * self.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
[docs]
class ExponentialFermionicEnvironment(FermionicEnvironment):
"""
Fermionic environment that is specified through an exponential
decomposition of its correlation function. The list of coefficients and
exponents in the decomposition may either be passed through the four lists
`ck_plus`, `vk_plus`, `ck_minus`, `vk_minus`, or as a list of fermionic
:class:`CFExponent` objects.
Alternative constructors :meth:`from_plus_exponents` and
:meth:`from_minus_exponents` are available to compute the "-" exponents
automatically from the "+" ones, or vice versa.
Parameters
----------
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 coefficients 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 coefficients are passed
as ck_minus.
exponents : list of :class:`CFExponent`
The expansion coefficients and exponents of both parts of the
correlation function as :class:`CFExponent` objects.
T: optional, float
The temperature of the environment.
mu: optional, float
The chemical potential of the environment.
tag : optional, str, tuple or any other object
An identifier (name) for this environment.
"""
def _check_cks_and_vks(self, ck_plus, vk_plus, ck_minus, vk_minus):
# all None: returns False
# all provided and lengths match: returns True
# otherwise: raises ValueError
lists = [ck_plus, vk_plus, ck_minus, vk_minus]
if all(x is None for x in lists):
return False
if any(x is None for x in lists):
raise ValueError(
"If any of the exponent lists ck_plus, vk_plus, ck_minus, "
"vk_minus is provided, all must be provided."
)
if len(ck_plus) != len(vk_plus) or len(ck_minus) != len(vk_minus):
raise ValueError(
"The exponent lists ck_plus and vk_plus, and ck_minus and "
"vk_minus must be the same length."
)
return True
def __init__(
self,
ck_plus: ArrayLike = None, vk_plus: ArrayLike = None,
ck_minus: ArrayLike = None, vk_minus: ArrayLike = None,
*,
exponents: Sequence[CFExponent] = None,
T: float = None, mu: float = None, tag: Any = None
):
super().__init__(T, mu, tag)
lists_provided = self._check_cks_and_vks(
ck_plus, vk_plus, ck_minus, vk_minus)
if exponents is None and not lists_provided:
raise ValueError(
"Either the parameter `exponents` or the parameters "
"`ck_plus`, `vk_plus`, `ck_minus`, `vk_minus` must be "
"provided."
)
if (exponents is not None and
not all(exp.fermionic for exp in exponents)):
raise ValueError(
"Bosonic exponent passed to exponential fermionic environment."
)
self.exponents = exponents or []
if lists_provided:
self.exponents.extend(CFExponent("+", ck, vk, tag=tag)
for ck, vk in zip(ck_plus, vk_plus))
self.exponents.extend(CFExponent("-", ck, vk, tag=tag)
for ck, vk in zip(ck_minus, vk_minus))
[docs]
def spectral_density(self, w: float | ArrayLike) -> (float | ArrayLike):
"""
Computes the spectral density corresponding to the multi-exponential
correlation function.
Parameters
----------
w : array_like or float
Energy of the mode.
"""
return self.power_spectrum_minus(w) + self.power_spectrum_plus(w)
[docs]
def correlation_function_plus(
self, t: float | ArrayLike
) -> (float | ArrayLike):
r"""
Computes the "+"-branch of the correlation function represented by this
exponential decomposition.
Parameters
----------
t : array_like or float
The times at which to evaluate the correlation function.
"""
return self._cf(t, CFExponent.types['+'])
[docs]
def correlation_function_minus(
self, t: float | ArrayLike
) -> (float | ArrayLike):
r"""
Computes the "-"-branch of the correlation function represented by this
exponential decomposition.
Parameters
----------
t : array_like or float
The times at which to evaluate the correlation function.
"""
return self._cf(t, CFExponent.types['-'])
def _cf(self, t, type):
t = np.asarray(t, dtype=float)
corr = np.zeros_like(t, dtype=complex)
for exp in self.exponents:
if exp.type == type:
corr += exp.coefficient * np.exp(-exp.exponent * np.abs(t))
corr[t < 0] = np.conj(corr[t < 0])
return corr.item() if t.ndim == 0 else corr
[docs]
def power_spectrum_plus(
self, w: float | ArrayLike
) -> (float | ArrayLike):
r"""
Calculates the "+"-branch of the power spectrum corresponding to the
multi-exponential correlation function.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
return self._ps(w, CFExponent.types['+'], 1)
[docs]
def power_spectrum_minus(
self, w: float | ArrayLike
) -> (float | ArrayLike):
r"""
Calculates the "-"-branch of the power spectrum corresponding to the
multi-exponential correlation function.
Parameters
----------
w : array_like or float
The frequency at which to evaluate the power spectrum.
"""
return self._ps(w, CFExponent.types['-'], -1)
def _ps(self, w, type, sigma):
w = np.asarray(w, dtype=float)
S = np.zeros_like(w)
for exp in self.exponents:
if exp.type == type:
S += 2 * np.real(
exp.coefficient / (exp.exponent + sigma * 1j * w)
)
return S.item() if w.ndim == 0 else S
[docs]
def rescale(
self, alpha: float, tag: Any = None
) -> ExponentialFermionicEnvironment:
"""
Returns a new environment where all exponents are scaled by the factor
alpha. This corresponds to changing the coupling constant. The spectral
density, the correlation function and the power spectrum are all scaled
by the same factor.
Parameters
----------
alpha : float
Rescaling factor.
tag : optional, str, tuple or any other object
An identifier (name) for the rescaled environment.
Returns
-------
ExponentialBosonicEnvironment
The new environment with the rescaled exponents.
"""
tag = tag or self.tag
return ExponentialFermionicEnvironment(
exponents=[exp.rescale(alpha) for exp in self.exponents],
T=self.T, mu=self.mu, tag=tag
)
Environment = Union[BosonicEnvironment, FermionicEnvironment]