# Source code for qutip.solver.heom.bofin_baths

    """
    This module provides utilities for describing baths when using the HEOM
    (hierarchy equations of motion) to model system-bath interactions. See the
    ``qutip.nonmarkov.bofin_solvers`` module for the associated solver.

    The implementation is derived from the BoFiN library (see
    https://github.com/tehruhn/bofin) which was itself derived from an earlier
    implementation in QuTiP itself.

    The "bath" classes in this module are closely related to the "environment"
    classes in the `qutip.core.environment` module. The bath classes were
    implemented first, specifically for the HEOM solver. The environment classes,
    added later provide additional functionality and are designed to work also with
    other QuTiP solvers.

    The bath classes are kept partly for backwards compatibility, partly to use as
    a "shortcut" when one only wants to use the HEOM solver.

    Note that this module also contains the `BathExponent` class, which is used by
    the HEOM solver internally and in the result object describing its output.
    """

    from qutip.core import data as _data
    from qutip.core import environment
    from qutip.core.qobj import Qobj

    __all__ = [
        "BathExponent",
        "Bath",
        "BosonicBath",
        "DrudeLorentzBath",
        "DrudeLorentzPadeBath",
        "UnderDampedBath",
        "FermionicBath",
        "LorentzianBath",
        "LorentzianPadeBath",
    ]

    def _isequal(Q1, Q2, tol):
        """ Return true if Q1 and Q2 are equal to within the given tolerance. """
        return _data.iszero(_data.sub(Q1.data, Q2.data), tol=tol)

    class BathExponent(environment.CFExponent):
        """
        Represents a single exponent (naively, an excitation mode) within the
        decomposition of the correlation functions of a bath.

        This class extends the
        :class:`CFExponent <qutip.core.environment.CFExponent>` of the
        environment-module for use with the HEOM solver. In addition to the
        exponent itself, the `BathExponent` keeps track of the corresponding system
        coupling operator ``Q``, as well as the parameters ``dim`` and
        ``sigma_bar_k_offset``.

        Parameters
        ----------
        type : {"R", "I", "RI", "+", "-"} or ``CFExponent.ExponentType``
            The type of bath exponent.

            "R" and "I" are bosonic bath exponents that appear in the real and
            imaginary parts of the correlation expansion.

            "RI" is combined bosonic bath exponent that appears in both the real
            and imaginary parts of the correlation expansion. The combined exponent
            has a single ``vk``. The ``ck`` is the coefficient in the real
            expansion and ``ck2`` is the coefficient in the imaginary expansion.

            "+" and "-" are fermionic bath exponents. These fermionic bath
            exponents must specify ``sigma_bar_k_offset`` which specifies
            the amount to add to ``k`` (the exponent index within the bath of this
            exponent) to determine the ``k`` of the corresponding exponent with
            the opposite sign (i.e. "-" or "+").

        dim : int or None
            The dimension (i.e. maximum number of excitations for this exponent).
            Usually ``2`` for fermionic exponents or ``None`` (i.e. unlimited) for
            bosonic exponents.

        Q : Qobj
            The coupling operator for this excitation mode.

        vk : complex
            The frequency of the exponent of the excitation term.

        ck : complex
            The coefficient of the excitation term.

        ck2 : optional, complex
            For exponents of type "RI" this is the coefficient of the term in the
            imaginary expansion (and ``ck`` is the coefficient in the real
            expansion).

        sigma_bar_k_offset : optional, int
            For exponents of type "+" this gives the offset (within the list of
            exponents within the bath) of the corresponding "-" bath exponent.
            For exponents of type "-" it gives the offset of the corresponding
            "+" exponent.

        tag : optional, str, tuple or any other object
            A label for the exponent (often the name of the bath). It
            defaults to None.

        Attributes
        ----------
        fermionic : bool
            True if the type of the exponent is a Fermionic type (i.e. either
            "+" or "-") and False otherwise.

        All of the parameters are also available as attributes.
        """
        types = environment.CFExponent.types

        def _check_sigma_bar_k_offset(self, type, offset):
            if type in (self.types["+"], self.types["-"]):
                if offset is None:
                    raise ValueError(
                        "+ and - type exponents require sigma_bar_k_offset"
                    )
            else:
                if offset is not None:
                    raise ValueError(
                        "Offset of sigma bar (sigma_bar_k_offset) should only be"
                        " specified for + and - type exponents"
                    )

        def __init__(
                self, type, dim, Q, ck, vk, ck2=None,
                sigma_bar_k_offset=None, tag=None,
        ):
            super().__init__(type, ck, vk, ck2, tag)

            self.dim = dim
            self.Q = Q

            self._check_sigma_bar_k_offset(self.type, sigma_bar_k_offset)
            self.sigma_bar_k_offset = sigma_bar_k_offset

        def __repr__(self):
            dims = getattr(self.Q, "dims", None)
            return (
                f"<{self.__class__.__name__} type={self.type.name}"
                f" dim={self.dim!r}"
                f" Q.dims={dims!r}"
                f" ck={self.ck!r} vk={self.vk!r} ck2={self.ck2!r}"
                f" sigma_bar_k_offset={self.sigma_bar_k_offset!r}"
                f" fermionic={self.fermionic!r}"
                f" tag={self.tag!r}>"
            )

        def rescale(self, alpha):
            """Rescale the coefficient of the exponent by a factor of alpha."""
            ck_new = self.ck * alpha
            if self.type == self.types["RI"]:
                ck2_new = self.ck2 * alpha
            else:
                ck2_new = None
            return BathExponent(
                self.type, self.dim, self.Q, ck_new, self.vk, ck2=ck2_new,
                sigma_bar_k_offset=self.sigma_bar_k_offset, tag=self.tag
            )

        def _can_combine(self, other, rtol, atol):
            if not super()._can_combine(other, rtol, atol):
                return False
            if not _isequal(self.Q, other.Q, tol=atol):
                return False
            return True

        def _combine(self, other):
            # Assumes can combine was checked
            return super()._combine(other, dim=None, Q=self.Q)

    class Bath:
        """
        Represents a list of bath expansion exponents.

        Parameters
        ----------
        exponents : list of :class:`.BathExponent`
            The exponents of the correlation function describing the bath.
        """

        def __init__(self, exponents):
            self.exponents = exponents

    class BosonicBath(environment.ExponentialBosonicEnvironment):
        """
        A helper class for constructing a bosonic bath from the expansion
        coefficients and frequencies for the real and imaginary parts of
        the bath correlation function.

        If the correlation functions ``C(t)`` is split into real and imaginary
        parts::

            C(t) = C_real(t) + i * C_imag(t)

        then::

            C_real(t) = sum(ck_real * exp(- vk_real * t))
            C_imag(t) = sum(ck_imag * exp(- vk_imag * t))

        Defines the coefficients ``ck`` and the frequencies ``vk``.

        Note that the ``ck`` and ``vk`` may be complex, even through ``C_real(t)``
        and ``C_imag(t)`` (i.e. the sum) is real.

        Parameters
        ----------
        Q : Qobj
            The coupling operator for the bath.

        ck_real : list of complex
            The coefficients of the expansion terms for the real part of the
            correlation function. The corresponding frequencies are passed as
            vk_real.

        vk_real : list of complex
            The frequencies (exponents) of the expansion terms for the real part of
            the correlation function. The corresponding ceofficients are passed as
            ck_real.

        ck_imag : list of complex
            The coefficients of the expansion terms in the imaginary part of the
            correlation function. The corresponding frequencies are passed as
            vk_imag.

        vk_imag : list of complex
            The frequencies (exponents) of the expansion terms for the imaginary
            part of the correlation function. The corresponding ceofficients are
            passed as ck_imag.

        combine : bool, default True
            Whether to combine exponents with the same frequency (and coupling
            operator). See :meth:`.ExponentialBosonicEnvironment.combine` for
            details.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. This class is an extended version
        of the :class:`.ExponentialBosonicEnvironment`, adding the parameter ``Q``,
        which is not included in the newer "environment" API.
        """

        def _make_exponent(self, type, ck, vk, ck2=None, tag=None):
            return BathExponent(type, None, self._Q, ck, vk, ck2, tag=tag)

        def _check_coup_op(self, Q):
            if not isinstance(Q, Qobj):
                raise ValueError("The coupling operator Q must be a Qobj.")

        def __init__(
            self, Q, ck_real, vk_real, ck_imag, vk_imag, combine=True, tag=None
        ):
            self._check_coup_op(Q)
            self._Q = Q

            super().__init__(ck_real, vk_real, ck_imag, vk_imag,
                             combine=combine, tag=tag)

        @classmethod
        def from_environment(cls, env, Q, dim=None):
            """
            Converts from the "environment" API to the "bath" API. A `BosonicBath`
            combines the information from an `ExponentialBosonicEnvironment` and a
            coupling operator.

            Parameters
            ----------
            env : :class:`.ExponentialBosonicEnvironment`
                The bath.
            Q : Qobj
                The coupling operator for the bath.
            dim : optional, int or ``None`` (default ``None``)
                The maximum number of excitations for each exponent. Usually
                ``None`` (i.e. unlimited).
            """
            bath_exponents = []
            for exponent in env.exponents:
                new_exponent = BathExponent(
                    exponent.type, dim, Q, exponent.ck, exponent.vk,
                    exponent.ck2, None, env.tag
                )
                bath_exponents.append(new_exponent)

            result = cls(Q, [], [], [], [], tag=env.tag)
            result.exponents = bath_exponents
            return result

    class DrudeLorentzBath(BosonicBath):
        """
        A helper class for constructing a Drude-Lorentz bosonic bath from the
        bath parameters (see parameters below).

        Parameters
        ----------
        Q : Qobj
            Operator describing the coupling between system and bath.

        lam : float
            Coupling strength.

        gamma : float
            Bath spectral density cutoff frequency.

        T : float
            Bath temperature.

        Nk : int
            Number of exponential terms used to approximate the bath correlation
            functions.

        combine : bool, default True
            Whether to combine exponents with the same frequency (and coupling
            operator). See :meth:`.ExponentialBosonicEnvironment.combine` for
            details.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. Creating a `DrudeLorentzBath` is
        equivalent to creating a :class:`.DrudeLorentzEnvironment`, performing a
        :meth:`Matsubara <.DrudeLorentzEnvironment.approximate>`
        approximation, and finally bundling the result together with the coupling
        operator ``Q`` for convenient use with the HEOM solver.
        """

        def __new__(
            mcs, Q, lam, gamma, T, Nk, combine=True, tag=None,
        ):
            # Basically this makes `DrudeLorentzBath` a function
            # (Q, lam, ...) -> BosonicBath
            # but it is made to look like a class because it was a class in the
            # initial bofin release
            env = environment.DrudeLorentzEnvironment(T, lam, gamma)
            matsubara_approx, delta = env.approximate(
                "matsubara", Nk=Nk, combine=combine, compute_delta=True, tag=tag
            )

            result = BosonicBath.from_environment(matsubara_approx, Q)
            result.terminator = lambda: (
                delta, environment.system_terminator(Q, delta)
            )
            return result

        def terminator(self):
            """
            Return the Matsubara terminator for the bath and the calculated
            approximation discrepancy.

            Returns
            -------
            delta: float

                The approximation discrepancy. That is, the difference between the
                true correlation function of the Drude-Lorentz bath and the sum of
                the ``Nk`` exponential terms is approximately ``2 * delta *
                dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.

            terminator : Qobj

                The Padé terminator -- i.e. a liouvillian term representing
                the contribution to the system-bath dynamics of all exponential
                expansion terms beyond ``Nk``. It should be used by adding it to
                the system liouvillian (i.e. ``liouvillian(H_sys)``).
            """
            # This is only here to keep the API doc
            ...

    class DrudeLorentzPadeBath(BosonicBath):
        """
        A helper class for constructing a Padé expansion for a Drude-Lorentz
        bosonic bath from the bath parameters (see parameters below).

        A Padé approximant is a sum-over-poles expansion (see
        https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).

        The application of the Padé method to spectrum decompoisitions is described
        in "Padé spectrum decompositions of quantum distribution functions and
        optimal hierarchical equations of motion construction for quantum open
        systems" [1].

        The implementation here follows the approach in the paper.

        [1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466

        This is an alternative to the :class:`DrudeLorentzBath` which constructs
        a simpler exponential expansion.

        Parameters
        ----------
        Q : Qobj
            Operator describing the coupling between system and bath.

        lam : float
            Coupling strength.

        gamma : float
            Bath spectral density cutoff frequency.

        T : float
            Bath temperature.

        Nk : int
            Number of Padé exponentials terms used to approximate the bath
            correlation functions.

        combine : bool, default True
            Whether to combine exponents with the same frequency (and coupling
            operator). See :meth:`.ExponentialBosonicEnvironment.combine` for
            details.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. Creating a `DrudeLorentzPadeBath`
        is equivalent to creating a :class:`.DrudeLorentzEnvironment`, performing a
        :meth:`Pade <.DrudeLorentzEnvironment.approximate>` approximation, and
        finally bundling the result together with the coupling operator ``Q`` for
        convenient use with the HEOM solver.
        """

        def __new__(
            mcs, Q, lam, gamma, T, Nk, combine=True, tag=None,
        ):
            # See DrudeLorentzBath comment
            env = environment.DrudeLorentzEnvironment(T, lam, gamma)
            pade_approx, delta = env.approximate(
                "pade", Nk=Nk, combine=combine, compute_delta=True, tag=tag
            )

            result = BosonicBath.from_environment(pade_approx, Q)
            result.terminator = lambda: (
                delta, environment.system_terminator(Q, delta)
            )
            return result

        def terminator(self):
            """
            Return the Padé terminator for the bath and the calculated
            approximation discrepancy.

            Returns
            -------
            delta: float

                The approximation discrepancy. That is, the difference between the
                true correlation function of the Drude-Lorentz bath and the sum of
                the ``Nk`` exponential terms is approximately ``2 * delta *
                dirac(t)``, where ``dirac(t)`` denotes the Dirac delta function.

            terminator : Qobj

                The Padé terminator -- i.e. a liouvillian term representing
                the contribution to the system-bath dynamics of all exponential
                expansion terms beyond ``Nk``. It should be used by adding it to
                the system liouvillian (i.e. ``liouvillian(H_sys)``).
            """
            # This is only here to keep the API doc
            ...

    class UnderDampedBath(BosonicBath):
        """
        A helper class for constructing an under-damped bosonic bath from the
        bath parameters (see parameters below).

        Parameters
        ----------
        Q : Qobj
            Operator describing the coupling between system and bath.

        lam : float
            Coupling strength.

        gamma : float
            Bath spectral density cutoff frequency.

        w0 : float
            Bath spectral density resonance frequency.

        T : float
            Bath temperature.

        Nk : int
            Number of exponential terms used to approximate the bath correlation
            functions.

        combine : bool, default True
            Whether to combine exponents with the same frequency (and coupling
            operator). See :meth:`.ExponentialBosonicEnvironment.combine` for
            details.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. Creating an `UnderDampedBath` is
        equivalent to creating an :class:`.UnderDampedEnvironment`, performing a
        :meth:`Matsubara <.UnderDampedEnvironment.approximate>`
        approximation, and finally bundling the result together with the coupling
        operator ``Q`` for convenient use with the HEOM solver.
        """

        def __new__(
            mcs, Q, lam, gamma, w0, T, Nk, combine=True, tag=None,
        ):
            # See DrudeLorentzBath comment
            env = environment.UnderDampedEnvironment(T, lam, gamma, w0)
            matsubara_approx = env.approximate(
                "matsubara", Nk=Nk, combine=combine, tag=tag
            )
            return BosonicBath.from_environment(matsubara_approx, Q)

    class FermionicBath(environment.ExponentialFermionicEnvironment):
        """
        A helper class for constructing a fermionic bath from the expansion
        coefficients and frequencies for the ``+`` and ``-`` modes of
        the bath correlation function.

        There must be the same number of ``+`` and ``-`` modes and their
        coefficients must be specified in the same order so that ``ck_plus[i],
        vk_plus[i]`` are the plus coefficient and frequency corresponding
        to the minus mode ``ck_minus[i], vk_minus[i]``.

        In the fermionic case the order in which excitations are created or
        destroyed is important, resulting in two different correlation functions
        labelled ``C_plus(t)`` and ``C_plus(t)``::

            C_plus(t) = sum(ck_plus * exp(- vk_plus * t))
            C_minus(t) = sum(ck_minus * exp(- vk_minus * t))

        where the expansions above define the coeffiients ``ck`` and the
        frequencies ``vk``.

        Parameters
        ----------
        Q : Qobj
            The coupling operator for the bath.

        ck_plus : list of complex
            The coefficients of the expansion terms for the ``+`` part of the
            correlation function. The corresponding frequencies are passed as
            vk_plus.

        vk_plus : list of complex
            The frequencies (exponents) of the expansion terms for the ``+`` part
            of the correlation function. The corresponding ceofficients are passed
            as ck_plus.

        ck_minus : list of complex
            The coefficients of the expansion terms for the ``-`` part of the
            correlation function. The corresponding frequencies are passed as
            vk_minus.

        vk_minus : list of complex
            The frequencies (exponents) of the expansion terms for the ``-`` part
            of the correlation function. The corresponding ceofficients are passed
            as ck_minus.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. This class is an extended version
        of the :class:`.ExponentialFermionicEnvironment`, adding the parameter
        ``Q``, which is not included in the newer "environment" API.
        """

        def _check_cks_and_vks(self, ck_plus, vk_plus, ck_minus, vk_minus):
            lists_provided = super()._check_cks_and_vks(
                ck_plus, vk_plus, ck_minus, vk_minus)
            if lists_provided and len(ck_plus) != len(ck_minus):
                raise ValueError(
                    "The must be the same number of plus and minus exponents"
                    " in the bath, and elements of plus and minus arrays"
                    " should be arranged so that ck_plus[i] is the plus mode"
                    " corresponding to ck_minus[i]."
                )
            return lists_provided

        def _check_coup_op(self, Q):
            if not isinstance(Q, Qobj):
                raise ValueError("The coupling operator Q must be a Qobj.")

        def __init__(self, Q, ck_plus, vk_plus, ck_minus, vk_minus, tag=None):
            self._check_cks_and_vks(ck_plus, vk_plus, ck_minus, vk_minus)
            self._check_coup_op(Q)

            exponents = []
            for ckp, vkp, ckm, vkm in zip(ck_plus, vk_plus, ck_minus, vk_minus):
                exponents.append(BathExponent(
                    "+", 2, Q, ckp, vkp, sigma_bar_k_offset=1, tag=tag,
                ))
                exponents.append(BathExponent(
                    "-", 2, Q, ckm, vkm, sigma_bar_k_offset=-1, tag=tag,
                ))
            super().__init__(exponents=exponents, tag=tag)

        @classmethod
        def from_environment(cls, env, Q, dim=2):
            """
            Converts from the "environment" API to the "bath" API. A
            `FermionicBath` combines the information from an
            `ExponentialFermionicEnvironment` and a coupling operator.

            Note that the "environment" API does not require fermionic exponents to
            come in pairs. This method will add additional exponents with
            zero coefficient in order to achieve the same amount of ``+`` and ``-``
            exponents.

            Parameters
            ----------
            env : :class:`.ExponentialFermionicEnvironment`
                The bath.
            Q : Qobj
                The coupling operator for the bath.
            dim : optional, int or ``None`` (default ``2``)
                The maximum number of excitations for each exponent. Usually ``2``.
            """

            # make same amount of plus and minus exponents by adding zeros
            plus_exponents = []
            minus_exponents = []
            for exponent in env.exponents:
                if exponent.type == BathExponent.types['+']:
                    plus_exponents.append(exponent)
                else:
                    minus_exponents.append(exponent)
            while len(plus_exponents) > len(minus_exponents):
                minus_exponents.append(environment.CFExponent('-', 0, 0))
            while len(minus_exponents) > len(plus_exponents):
                plus_exponents.append(environment.CFExponent('+', 0, 0))

            bath_exponents = []
            for expp, expm in zip(plus_exponents, minus_exponents):
                bath_exponents.append(BathExponent(
                    '+', dim, Q, expp.ck, expp.vk, None, 1, env.tag
                ))
                bath_exponents.append(BathExponent(
                    '-', dim, Q, expm.ck, expm.vk, None, -1, env.tag
                ))

            result = cls(Q, [], [], [], [], tag=env.tag)
            result.exponents = bath_exponents
            return result

    class LorentzianBath(FermionicBath):
        """
        A helper class for constructing a Lorentzian fermionic bath from the
        bath parameters (see parameters below).

        .. note::

            The Matsubara expansion used in this bath converges very slowly
            and ``Nk > 20`` may be required to get good convergence. The
            Padé expansion used by :class:`LorentzianPadeBath` converges much
            more quickly.

        Parameters
        ----------
        Q : Qobj
            Operator describing the coupling between system and bath.

        gamma : float
            The coupling strength between the system and the bath.

        w : float
            The width of the environment.

        mu : float
            The chemical potential of the bath.

        T : float
            Bath temperature.

        Nk : int
            Number of exponential terms used to approximate the bath correlation
            functions.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. Creating a `LorentzianBath` is
        equivalent to creating a :class:`.LorentzianEnvironment`, performing a
        :meth:`Matsubara <.LorentzianEnvironment.approx_by_matsubara>`
        approximation, and finally bundling the result together with the coupling
        operator ``Q`` for convenient use with the HEOM solver.
        """

        def __new__(self, Q, gamma, w, mu, T, Nk, tag=None):
            # See DrudeLorentzBath comment
            env = environment.LorentzianEnvironment(T, mu, gamma, w)
            mats_approx = env.approx_by_matsubara(Nk, tag)
            return FermionicBath.from_environment(mats_approx, Q)

    class LorentzianPadeBath(FermionicBath):
        """
        A helper class for constructing a Padé expansion for Lorentzian fermionic
        bath from the bath parameters (see parameters below).

        A Padé approximant is a sum-over-poles expansion (
        see https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).

        The application of the Padé method to spectrum decompoisitions is described
        in "Padé spectrum decompositions of quantum distribution functions and
        optimal hierarchical equations of motion construction for quantum open
        systems" [1].

        The implementation here follows the approach in the paper.

        [1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466

        This is an alternative to the :class:`LorentzianBath` which constructs
        a simpler exponential expansion that converges much more slowly in
        this particular case.

        Parameters
        ----------
        Q : Qobj
            Operator describing the coupling between system and bath.

        gamma : float
            The coupling strength between the system and the bath.

        w : float
            The width of the environment.

        mu : float
            The chemical potential of the bath.

        T : float
            Bath temperature.

        Nk : int
            Number of exponential terms used to approximate the bath correlation
            functions.

        tag : optional, str, tuple or any other object
            A label for the bath exponents (for example, the name of the
            bath). It defaults to None but can be set to help identify which
            bath an exponent is from.

        Notes
        -----
        This class is part of the "bath" API, which is now mirrored by the newer
        "environment" API. The bath classes are kept in QuTiP for reasons of
        backwards compatibility and convenience. Creating a `LorentzianPadeBath` is
        equivalent to creating a :class:`.LorentzianEnvironment`, performing a
        :meth:`Pade <.LorentzianEnvironment.approx_by_pade>` approximation, and
        finally bundling the result together with the coupling operator ``Q`` for
        convenient use with the HEOM solver.
        """

        def __new__(self, Q, gamma, w, mu, T, Nk, tag=None):
            # See DrudeLorentzBath comment
            env = environment.LorentzianEnvironment(T, mu, gamma, w)
            mats_approx = env.approx_by_pade(Nk, tag)
            return FermionicBath.from_environment(mats_approx, Q)