# Source code for qutip.core.environment

    """
    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
    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)
    from .superoperator import spre, spost
    from .qobj import Qobj

    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 directly fitting their
        correlation function with a multi-exponential ansatz
        (:meth:`approx_by_cf_fit`) or by fitting their spectral density with a sum
        of Lorentzians (:meth:`approx_by_sd_fit`), which correspond to
        underdamped environments with known multi-exponential decompositions.
        Subclasses may offer additional approximation methods such as
        :meth:`DrudeLorentzEnvironment.approx_by_matsubara` or
        :meth:`DrudeLorentzEnvironment.approx_by_pade` in the case of a
        Drude-Lorentz environment.

        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

        @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.
            """

            ...

        @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`.
            """

            ...

        @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

        @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)

        @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)

        @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 ``S``.
            """
            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

        def approx_by_cf_fit(
            self,
            tlist: ArrayLike,
            target_rsme: 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]]:
            r"""
            Generates an approximation to this environment by fitting its
            correlation function with a multi-exponential ansatz. The number of
            exponents is determined iteratively based on reducing the normalized
            root mean squared error below a given threshold.

            Specifically, the real and imaginary parts are fit by the following
            model functions:

            .. math::
                \operatorname{Re}[C(t)] = \sum_{k=1}^{N_r} \operatorname{Re}\Bigl[
                    (a_k + \mathrm i d_k) \mathrm e^{(b_k + \mathrm i c_k) t}\Bigl]
                    ,
                \\
                \operatorname{Im}[C(t)] = \sum_{k=1}^{N_i} \operatorname{Im}\Bigl[
                    (a'_k + \mathrm i d'_k) \mathrm e^{(b'_k + \mathrm i c'_k) t}
                    \Bigr].

            If the parameter `full_ansatz` is `False`, :math:`d_k` and :math:`d'_k`
            are set to zero and the model functions simplify to

            .. math::
                \operatorname{Re}[C(t)] = \sum_{k=1}^{N_r}
                    a_k  e^{b_k  t} \cos(c_{k} t)
                    ,
                \\
                \operatorname{Im}[C(t)] = \sum_{k=1}^{N_i}
                    a'_k  e^{b'_k  t} \sin(c'_{k} t) .

            The simplified version offers faster fits, however it fails for
            anomalous spectral densities with
            :math:`\operatorname{Im}[C(0)] \neq 0` as :math:`\sin(0) = 0`.

            Parameters
            ----------
            tlist : array_like
                The time range on which to perform the fit.
            target_rmse : optional, float
                Desired normalized root mean squared error (default `2e-5`). Can be
                set to `None` to perform only one fit using the maximum number of
                modes (`Nr_max`, `Ni_max`).
            Nr_max : optional, int
                The maximum number of modes to use for the fit of the real part
                (default 10).
            Ni_max : optional, int
                The maximum number of modes to use for the fit of the imaginary
                part (default 10).
            guess : optional, list of float
                Initial guesses for the parameters :math:`a_k`, :math:`b_k`, etc.
                The same initial guesses are used for all values of k, and for
                the real and imaginary parts. If `full_ansatz` is True, `guess` is
                a list of size 4, otherwise, it is a list of size 3.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            lower : optional, list of float
                Lower bounds for the parameters :math:`a_k`, :math:`b_k`, etc.
                The same lower bounds are used for all values of k, and for
                the real and imaginary parts. If `full_ansatz` is True, `lower` is
                a list of size 4, otherwise, it is a list of size 3.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            upper : optional, list of float
                Upper bounds for the parameters :math:`a_k`, :math:`b_k`, etc.
                The same upper bounds are used for all values of k, and for
                the real and imaginary parts. If `full_ansatz` is True, `upper` is
                a list of size 4, otherwise, it is a list of size 3.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            sigma : optional, float or list of float
                Adds an uncertainty to the correlation function of the environment,
                i.e., adds a leeway to the fit. This parameter is useful to adjust
                if the correlation function is very small in parts of the time
                range. For more details, see the documentation of
                ``scipy.optimize.curve_fit``.
            maxfev : optional, int
                Number of times the parameters of the fit are allowed to vary
                during the optimization (per fit).
            full_ansatz : optional, bool (default False)
                If this is set to False, the parameters :math:`d_k` are all set to
                zero. The full ansatz, including :math:`d_k`, usually leads to
                significantly slower fits, and some manual tuning of the `guesses`,
                `lower` and `upper` is usually needed. On the other hand, the full
                ansatz can lead to better fits with fewer exponents, especially
                for anomalous spectral densities with
                :math:`\operatorname{Im}[C(0)] \neq 0` for which the simplified
                ansatz will always give :math:`\operatorname{Im}[C(0)] = 0`.
                When using the full ansatz with default values for the guesses and
                bounds, if the fit takes too long, we recommend choosing guesses
                and bounds manually.
            combine : optional, bool (default True)
                Whether to combine exponents with the same frequency. See
                :meth:`combine <.ExponentialBosonicEnvironment.combine>` for
                details.
            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
            -------
            approx_env : :class:`ExponentialBosonicEnvironment`
                The approximated environment with multi-exponential correlation
                function.
            fit_info : dictionary
                A dictionary containing the following information about the fit.

                "Nr"
                    The number of terms used to fit the real part of the
                    correlation function.
                "Ni"
                    The number of terms used to fit the imaginary part of the
                    correlation function.
                "fit_time_real"
                    The time the fit of the real part of the correlation function
                    took in seconds.
                "fit_time_imag"
                    The time the fit of the imaginary part of the correlation
                    function took in seconds.
                "rmse_real"
                    Normalized mean squared error obtained in the fit of the real
                    part of the correlation function.
                "rmse_imag"
                    Normalized mean squared error obtained in the fit of the
                    imaginary part of the correlation function.
                "params_real"
                    The fitted parameters (array of shape Nx3 or Nx4) for the real
                    part of the correlation function.
                "params_imag"
                    The fitted parameters (array of shape Nx3 or Nx4) for the
                    imaginary part of the correlation function.
                "summary"
                    A string that summarizes the information about the fit.
            """

            # Process arguments
            if tag is None and self.tag is not None:
                tag = (self.tag, "CF Fit")

            if full_ansatz:
                num_params = 4
            else:
                num_params = 3

            if target_rsme is None:
                target_rsme = 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_rsme,
                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_rsme,
                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,
            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]]:
            r"""
            Generates an approximation to this environment by fitting its spectral
            density with a sum of underdamped terms. Each underdamped term
            effectively acts like an underdamped environment. We use the known
            exponential decomposition of the underdamped environment, keeping `Nk`
            Matsubara terms for each. The number of underdamped terms is determined
            iteratively based on reducing the normalized root mean squared error
            below a given threshold.

            Specifically, the spectral density is fit by the following model
            function:

            .. math::
                J(\omega) = \sum_{k=1}^{N} \frac{2 a_k b_k \omega}{\left(\left(
                    \omega + c_k \right)^2 + b_k^2 \right) \left(\left(
                    \omega - c_k \right)^2 + b_k^2 \right)}

            Parameters
            ----------
            wlist : array_like
                The frequency range on which to perform the fit.
            Nk : optional, int
                The number of Matsubara terms to keep in each mode (default 1).
            target_rmse : optional, float
                Desired normalized root mean squared error (default `5e-6`). Can be
                set to `None` to perform only one fit using the maximum number of
                modes (`Nmax`).
            Nmax : optional, int
                The maximum number of modes to use for the fit (default 10).
            guess : optional, list of float
                Initial guesses for the parameters :math:`a_k`, :math:`b_k` and
                :math:`c_k`. The same initial guesses are used for all values of
                k.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            lower : optional, list of float
                Lower bounds for the parameters :math:`a_k`, :math:`b_k` and
                :math:`c_k`. The same lower bounds are used for all values of
                k.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            upper : optional, list of float
                Upper bounds for the parameters :math:`a_k`, :math:`b_k` and
                :math:`c_k`. The same upper bounds are used for all values of
                k.
                If none of `guess`, `lower` and `upper` are provided, these
                parameters will be chosen automatically.
            sigma : optional, float or list of float
                Adds an uncertainty to the spectral density of the environment,
                i.e., adds a leeway to the fit. This parameter is useful to adjust
                if the spectral density is very small in parts of the frequency
                range. For more details, see the documentation of
                ``scipy.optimize.curve_fit``.
            maxfev : optional, int
                Number of times the parameters of the fit are allowed to vary
                during the optimization (per fit).
            combine : optional, bool (default True)
                Whether to combine exponents with the same frequency. See
                :meth:`combine <.ExponentialBosonicEnvironment.combine>` for
                details.
            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
            -------
            approx_env : :class:`ExponentialBosonicEnvironment`
                The approximated environment with multi-exponential correlation
                function.
            fit_info : dictionary
                A dictionary containing the following information about the fit.

                "N"
                    The number of underdamped terms used in the fit.
                "Nk"
                    The number of Matsubara modes included per underdamped term.
                "fit_time"
                    The time the fit took in seconds.
                "rmse"
                    Normalized mean squared error obtained in the fit.
                "params"
                    The fitted parameters (array of shape Nx3).
                "summary"
                    A string that summarizes the information about the fit.
            """

            # Process arguments
            if tag is None and self.tag is not None:
                tag = (self.tag, "SD 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

    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)

    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

        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

        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

        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)

        @overload
        def approx_by_matsubara(
            self, Nk: int, combine: bool = ...,
            compute_delta: Literal[False] = False, tag: Any = ...
        ) -> ExponentialBosonicEnvironment: ...

        @overload
        def approx_by_matsubara(
            self, Nk: int, combine: bool = ...,
            compute_delta: Literal[True] = True, tag: Any = ...
        ) -> tuple[ExponentialBosonicEnvironment, float]: ...

        def approx_by_matsubara(
            self, Nk, combine=True, compute_delta=False, tag=None
        ):
            """
            Generates an approximation to this environment by truncating its
            Matsubara expansion.

            Parameters
            ----------
            Nk : int
                Number of Matsubara terms to include. In total, the real part of
                the correlation function will include `Nk+1` terms and the
                imaginary part `1` term.

            combine : bool, default `True`
                Whether to combine exponents with the same frequency.

            compute_delta : bool, default `False`
                Whether to compute and return the approximation discrepancy
                (see below).

            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
            -------
            approx_env : :class:`ExponentialBosonicEnvironment`
                The approximated environment with multi-exponential correlation
                function.

            delta : float, optional
                The approximation discrepancy. That is, the difference between the
                true correlation function of the Drude-Lorentz environment and the
                sum of the ``Nk`` exponential terms is approximately ``2 * delta *
                dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.
                It can be used to create a "terminator" term to add to the system
                dynamics to take this discrepancy into account, see
                :func:`.system_terminator`.
            """

            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

        @overload
        def approx_by_pade(
            self, Nk: int, combine: bool = ...,
            compute_delta: Literal[False] = False, tag: Any = ...
        ) -> ExponentialBosonicEnvironment: ...

        @overload
        def approx_by_pade(
            self, Nk: int, combine: bool = ...,
            compute_delta: Literal[True] = True, tag: Any = ...
        ) -> tuple[ExponentialBosonicEnvironment, float]: ...

        def approx_by_pade(
            self, Nk, combine=True, compute_delta=False, tag=None
        ):
            """
            Generates an approximation to this environment by truncating its
            Pade expansion.

            Parameters
            ----------
            Nk : int
                Number of Pade terms to include. In total, the real part of
                the correlation function will include `Nk+1` terms and the
                imaginary part `1` term.

            combine : bool, default `True`
                Whether to combine exponents with the same frequency.

            compute_delta : bool, default `False`
                Whether to compute and return the approximation discrepancy
                (see below).

            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
            -------
            approx_env : :class:`ExponentialBosonicEnvironment`
                The approximated environment with multi-exponential correlation
                function.

            delta : float, optional
                The approximation discrepancy. That is, the difference between the
                true correlation function of the Drude-Lorentz environment and the
                sum of the ``Nk`` exponential terms is approximately ``2 * delta *
                dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.
                It can be used to create a "terminator" term to add to the system
                dynamics to take this discrepancy into account, see
                :func:`.system_terminator`.
            """

            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):
            eta_p, gamma_p = self._corr(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]]

            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)) - 1.0j)]
            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

    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

        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

        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)

        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)

        def approx_by_matsubara(
            self, Nk: int, combine: bool = True, tag: Any = None
        ) -> ExponentialBosonicEnvironment:
            """
            Generates an approximation to this environment by truncating its
            Matsubara expansion.

            Parameters
            ----------
            Nk : int
                Number of Matsubara terms to include. In total, the real part of
                the correlation function will include `Nk+2` terms and the
                imaginary part `2` terms.

            combine : bool, default `True`
                Whether to combine exponents with the same frequency.

            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
            -------
            :class:`ExponentialBosonicEnvironment`
                The approximated environment with multi-exponential correlation
                function.
            """

            if tag is None and self.tag is not None:
                tag = (self.tag, "Matsubara Truncation")

            lists = self._matsubara_params(Nk)
            result = ExponentialBosonicEnvironment(
                *lists, T=self.T, combine=combine, tag=tag)
            return result

        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

    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.")

        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

        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)

        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**(self.s + 1) / 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]

    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)

        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)

    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

        @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

        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

        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

        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)

    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
            :meth:`.DrudeLorentzEnvironment.approx_by_matsubara`.

        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 _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 normalized 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 = ["a", "b", "c"]
        if n == 4:
            columns.append("d")
        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 ---

    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

        @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.
            """

            ...

        @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.
            """

            ...

        @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.
            """

            ...

        @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.
            """

            ...

        @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

        @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.")

        @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.")

    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

        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)

        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)

        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

        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)

        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)

        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
            )

        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

    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))

        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)

        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['+'])

        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

        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)

        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