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API reference

BosonicBath

Source code in nmm/utils/baths.py 
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class BosonicBath:
    def __init__(self, T):
        self.T = T

    def bose(self, ν):
        r"""
        It computes the Bose-Einstein distribution

        $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

        Parameters:
        ----------
        ν: float
            The mode at which to compute the thermal population

        Returns:
        -------
        float
            The thermal population of mode ν
        """
        if self.T == 0:
            return 0
        if np.isclose(ν, 0).all():
            return 0
        return np.exp(-ν / self.T) / (1-np.exp(-ν / self.T))

    def spectral_density(self, w):
        return None

    def correlation_function(self, t):
        return None

    def power_spectrum(self, w):
        return 2*(self.bose(w)+1)*self.spectral_density(w)

bose(ν)

It computes the Bose-Einstein distribution

\[ n(\omega)=\frac{1}{e^{\beta \omega}-1} \]
Parameters:

ν: float The mode at which to compute the thermal population

Returns:

float The thermal population of mode ν

Source code in nmm/utils/baths.py 
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def bose(self, ν):
    r"""
    It computes the Bose-Einstein distribution

    $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

    Parameters:
    ----------
    ν: float
        The mode at which to compute the thermal population

    Returns:
    -------
    float
        The thermal population of mode ν
    """
    if self.T == 0:
        return 0
    if np.isclose(ν, 0).all():
        return 0
    return np.exp(-ν / self.T) / (1-np.exp(-ν / self.T))

OhmicBath

Bases: BosonicBath

Source code in nmm/utils/baths.py 
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class OhmicBath(BosonicBath):
    def __init__(self, T, coupling, cutoff):
        super().__init__(T)
        self.coupling = coupling
        self.cutoff = cutoff
        self.label = "ohmic"

    def spectral_density(self, w):
        r"""
        It describes the spectral density of an Ohmic spectral density given by

        $$ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} $$

        Parameters
        ----------
        """
        return self.coupling*w*np.exp(-abs(w)/self.cutoff)

    def correlation_function(self, t):
        return None

spectral_density(w)

It describes the spectral density of an Ohmic spectral density given by

\[ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} \]
Source code in nmm/utils/baths.py 
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def spectral_density(self, w):
    r"""
    It describes the spectral density of an Ohmic spectral density given by

    $$ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} $$

    Parameters
    ----------
    """
    return self.coupling*w*np.exp(-abs(w)/self.cutoff)

baths

BosonicBath

Source code in nmm/utils/baths.py 
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class BosonicBath:
    def __init__(self, T):
        self.T = T

    def bose(self, ν):
        r"""
        It computes the Bose-Einstein distribution

        $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

        Parameters:
        ----------
        ν: float
            The mode at which to compute the thermal population

        Returns:
        -------
        float
            The thermal population of mode ν
        """
        if self.T == 0:
            return 0
        if np.isclose(ν, 0).all():
            return 0
        return np.exp(-ν / self.T) / (1-np.exp(-ν / self.T))

    def spectral_density(self, w):
        return None

    def correlation_function(self, t):
        return None

    def power_spectrum(self, w):
        return 2*(self.bose(w)+1)*self.spectral_density(w)

bose(ν)

It computes the Bose-Einstein distribution

\[ n(\omega)=\frac{1}{e^{\beta \omega}-1} \]
Parameters:

ν: float The mode at which to compute the thermal population

Returns:

float The thermal population of mode ν

Source code in nmm/utils/baths.py 
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def bose(self, ν):
    r"""
    It computes the Bose-Einstein distribution

    $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

    Parameters:
    ----------
    ν: float
        The mode at which to compute the thermal population

    Returns:
    -------
    float
        The thermal population of mode ν
    """
    if self.T == 0:
        return 0
    if np.isclose(ν, 0).all():
        return 0
    return np.exp(-ν / self.T) / (1-np.exp(-ν / self.T))

OhmicBath

Bases: BosonicBath

Source code in nmm/utils/baths.py 
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class OhmicBath(BosonicBath):
    def __init__(self, T, coupling, cutoff):
        super().__init__(T)
        self.coupling = coupling
        self.cutoff = cutoff
        self.label = "ohmic"

    def spectral_density(self, w):
        r"""
        It describes the spectral density of an Ohmic spectral density given by

        $$ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} $$

        Parameters
        ----------
        """
        return self.coupling*w*np.exp(-abs(w)/self.cutoff)

    def correlation_function(self, t):
        return None

spectral_density(w)

It describes the spectral density of an Ohmic spectral density given by

\[ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} \]
Source code in nmm/utils/baths.py 
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def spectral_density(self, w):
    r"""
    It describes the spectral density of an Ohmic spectral density given by

    $$ J(\omega)= \alpha \omega e^{-\frac{|\omega|}{\omega_{c}}} $$

    Parameters
    ----------
    """
    return self.coupling*w*np.exp(-abs(w)/self.cutoff)

cumulant

csolve

Source code in nmm/cumulant/cumulant.py 
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class csolve:
    def __init__(self, Hsys, t, baths, Qs, eps=1e-4, cython=False, limit=50,
                 matsubara=True,ls=False):
        self.Hsys = Hsys
        self.t = t
        self.eps = eps
        self.limit = limit
        self.dtype = Hsys.dtype
        self.ls=ls

        if isinstance(Hsys, qutip_Qobj):
            self._qutip = True
        else:
            self._qutip = False
        if cython:
            self.baths = [bath_csolve(b.T, eps, b.coupling, b.cutoff, b.label)
                          for b in baths]
        else:
            self.baths = baths
        self.Qs = Qs
        self.cython = cython
        self.matsubara = matsubara

    def _tree_flatten(self):
        children = (self.Hsys, self.t, self.eps,
                    self.limit, self.baths, self.dtype)
        aux_data = {}
        return (children, aux_data)

    @classmethod
    def _tree_unflatten(cls, aux_data, children):
        return cls(*children, **aux_data)
    def bose(self,w,bath):
        r"""
        It computes the Bose-Einstein distribution

        $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

        Parameters:
        ----------
        nu: float
            The mode at which to compute the thermal population

        Returns:
        -------
        float
            The thermal population of mode nu
        """
        if bath.T == 0:
            return 0
        if np.isclose(w, 0).all():
            return 0
        return np.exp(-w / bath.T) / (1-np.exp(-w / bath.T))

    def gamma_fa(self, bath, w, w1, t):
        r"""
        It describes the decay rates for the Filtered Approximation of the
        cumulant equation

        $$\gamma(\omega,\omega^\prime,t)= 2\pi t e^{i \frac{\omega^\prime
        -\omega}{2}t}\mathrm{sinc} \left(\frac{\omega^\prime-\omega}{2}t\right)
         \left(J(\omega^\prime) (n(\omega^\prime)+1)J(\omega) (n(\omega)+1)
         \right)^{\frac{1}{2}}$$

        Parameters
        ----------

        w : float or numpy.ndarray

        w1 : float or numpy.ndarray

        t : float or numpy.ndarray

        Returns
        -------
        float or numpy.ndarray
            It returns a value or array describing the decay between the levels
            with energies w and w1 at time t

        """
        var = (2 * t * np.exp(1j * (w1 - w) * t / 2)
               * np.sinc((w1 - w) * t / (2 * np.pi))
               * np.sqrt(bath.spectral_density(w1) * (self.bose(w1,bath) + 1))
               * np.sqrt(bath.spectral_density(w) * (self.bose(w,bath) + 1)))
        return var

    def _gamma_(self, nu, bath, w, w1, t):
        r"""
        It describes the Integrand of the decay rates of the cumulant equation
        for bosonic baths

        $$\Gamma(w,w',t)=\int_{0}^{t} dt_1 \int_{0}^{t} dt_2
        e^{i (w t_1 - w' t_2)} \mathcal{C}(t_{1},t_{2})$$

        Parameters:
        ----------

        w: float or numpy.ndarray

        w1: float or numpy.ndarray

        t: float or numpy.ndarray

        Returns:
        --------
        float or numpy.ndarray
            It returns a value or array describing the decay between the levels
            with energies w and w1 at time t

        """

        self._mul = 1/np.pi
        var = (
            np.exp(1j * (w - w1) / 2 * t)
            * bath.spectral_density(nu)
            * (np.sinc((w - nu) / (2 * np.pi) * t)
               * np.sinc((w1 - nu) / (2 * np.pi) * t))
            * (self.bose(nu,bath) + 1)
        )
        var += (
            np.exp(1j * (w - w1) / 2 * t)
            * bath.spectral_density(nu)
            * (np.sinc((w + nu) / (2 * np.pi) * t)
               * np.sinc((w1 + nu) / (2 * np.pi) * t))
            * self.bose(nu,bath)
        )

        var = var*self._mul

        return var

    def gamma_gen(self, bath, w, w1, t, approximated=False):
        r"""
        It describes the the decay rates of the cumulant equation
        for bosonic baths

        $$\Gamma(\omega,\omega',t) = t^{2}\int_{0}^{\infty} d\omega 
        e^{i\frac{\omega-\omega'}{2} t} J(\omega) \left[ (n(\omega)+1) 
        sinc\left(\frac{(\omega-\omega)t}{2}\right)
        sinc\left(\frac{(\omega'-\omega)t}{2}\right)+ n(\omega) 
        sinc\left(\frac{(\omega+\omega)t}{2}\right) 
        sinc\left(\frac{(\omega'+\omega)t}{2}\right)   \right]$$

        Parameters
        ----------

        w : float or numpy.ndarray
        w1 : float or numpy.ndarray
        t : float or numpy.ndarray

        Returns
        -------
        float or numpy.ndarray
            It returns a value or array describing the decay between the levels
            with energies w and w1 at time t

        """
        if isinstance(t, type(jnp.array([2]))):
            t = np.array(t.tolist())
        if isinstance(t, list):
            t = np.array(t)
        if approximated:
            return self.gamma_fa(bath, w, w1, t)
        if self.matsubara:
            if w == w1:
                return self.decayww(bath, w, t)
            else:
                return self.decayww2(bath, w, w1, t)
        if self.cython:
            return bath.gamma(np.real(w), np.real(w1), t, limit=self.limit)

        else:
            integrals = quad_vec(
                self._gamma_,
                0,
                np.inf,
                args=(bath, w, w1, t),
                epsabs=self.eps,
                epsrel=self.eps,
                quadrature="gk21"
            )[0]
            return t*t*integrals

    def sparsify(self, vectors, tol=10):
        dims = vectors[0].dims
        new = []
        for vector in vectors:
            top = np.max(np.abs(vector.full()))
            vector = (vector/top).full().round(tol)*top
            vector = qutip_Qobj(vector).to("CSR")
            vector.dims = dims

            new.append(vector)
        return new

    def jump_operators(self, Q):
        evals, all_state = self.Hsys.eigenstates()
        N = len(all_state)
        collapse_list = []
        ws = []
        for j in range(N):
            for k in range(j + 1, N):
                Deltajk = evals[k] - evals[j]
                ws.append(Deltajk)
                collapse_list.append(
                    (
                        all_state[j]
                        * all_state[j].dag()
                        * Q
                        * all_state[k]
                        * all_state[k].dag()
                    )
                )  # emission
                ws.append(-Deltajk)
                collapse_list.append(
                    (
                        all_state[k]
                        * all_state[k].dag()
                        * Q
                        * all_state[j]
                        * all_state[j].dag()
                    )
                )  # absorption
        collapse_list.append(Q - sum(collapse_list))  # Dephasing
        ws.append(0)
        output = defaultdict(list)
        for k, key in enumerate(ws):
            output[jnp.round(key, 12).item()].append(collapse_list[k])
        eldict = {x: sum(y) for x, y in output.items()}
        dictrem = {}
        empty = 0*self.Hsys
        for keys, values in eldict.items():
            if not (values == empty):
                if isinstance(values,qutip_Qobj):
                    dictrem[keys] = values.to("CSR")
                else:
                    dictrem[keys] = values
        return dictrem

    def decays(self, combinations, bath, approximated):
        rates = {}
        done = []
        for i in tqdm(combinations, desc='Calculating Integrals ...',
                      dynamic_ncols=True):
            done.append(i)
            j = (i[1], i[0])
            if (j in done) & (i != j):
                rates[i] = np.conjugate(rates[j])
            else:
                rates[i] = self.gamma_gen(bath, i[0], i[1], self.t,
                                          approximated)
        return rates

    def matrix_form(self, jumps, combinations):
        matrixform = {}
        lsform={}
        for i in tqdm(
                combinations, desc='Calculating time independent matrices...',
                dynamic_ncols=True):
            ada=jumps[i[0]].dag() * jumps[i[1]]
            matrixform[i] = (
                spre(jumps[i[1]]) * spost(jumps[i[0]].dag()) - 1 *
                (0.5 *
                 (spre(ada) +spost(ada))))
            lsform[i]= -1j*(spre(ada)-spost(ada))

        return matrixform,lsform

    def generator(self, approximated=False):
        generators = []
        for Q, bath in zip(self.Qs, self.baths):
            jumps = self.jump_operators(Q)
            ws = list(jumps.keys())
            combinations = list(itertools.product(ws, ws))
            rates = self.decays(combinations, bath, approximated)
            matrices,lsform = self.matrix_form(jumps, combinations)
            if self.ls is False:
                superop = sum(
                    (rates[i] * np.array(matrices[i])
                    for i in tqdm(
                        combinations,
                        desc="Calculating time dependent generators")))
            else:
                LS= self.LS(combinations,bath,self.t)            
                superop = sum(
                    (LS[i]*np.array(lsform[i])+rates[i] * np.array(matrices[i])
                    for i in tqdm(
                        combinations,
                        desc="Calculating time dependent generators")))
            generators.extend(superop)
            del superop
        self.generators = self._reformat(generators)

    def _reformat(self, generators):
        if len(generators) == len(self.t):
            return generators
        else:
            one_list_for_each_bath = [
                generators
                [i * len(self.t): (i + 1) * len(self.t)]
                for i in range(
                    0, int(
                        len(generators) / len(self.t)))]
            composed = list(map(sum, zip(*one_list_for_each_bath)))
            return composed

    def evolution(self, rho0, approximated=False):
        r"""
        This function computes the evolution of the state $\rho(0)$

        Parameters
        ----------

        rho0 : numpy.ndarray or qutip.Qobj
            The initial state of the quantum system under consideration.

        approximated : bool
            When False the full cumulant equation/refined weak coupling is
            computed, when True the Filtered Approximation (FA is computed),
            this greatly reduces computational time, at the expense of
            diminishing accuracy particularly for the populations of the system
            at early times.

        Returns
        -------
        list
            a list containing all of the density matrices, at all timesteps of
            the evolution
        """
        self.generator(approximated)
        states = [
            (i).expm()(rho0)
            for k,i in tqdm(
                enumerate(self.generators),
                desc='Computing Exponential of Generators . . . .')]  # this counts time incorrectly
        return states
    def _decayww(self,bath, w, t):
        cks=np.array([i.coefficient for i in bath.exponents])
        vks=np.array([i.exponent for i in bath.exponents])
        result=[]
        for i in range(len(cks)):
            term1 =(vks[i]*t-1j*w*t-1)+np.exp(-(vks[i]-1j*w)*t)
            term1=term1*cks[i]/(vks[i]-1j*w)**2
            result.append(term1)
        return 2*np.real(sum(result))

    def _decayww2(self,bath ,w, w1, t):
        cks=np.array([i.coefficient for i in bath.exponents])
        vks=np.array([i.exponent for i in bath.exponents])
        result=[]
        for i in range(len(cks)):
            a=(vks[i]-1j*w1)
            b=(vks[i]-1j*w)
            term1=cks[i]*np.exp(-b*t)/(a*b)
            term2=np.conjugate(cks[i])*np.exp(-np.conjugate(a)*t)/(np.conjugate(a)*np.conjugate(b))
            term3=cks[i]*((1/b)-(np.exp(1j*(w-w1)*t)/a))
            term4=np.conjugate(cks[i])*((1/np.conjugate(a))-(np.exp(1j*(w-w1)*t)/np.conjugate(b)))
            actual=term1+term2+(1j*(term3+term4)/(w-w1))
            result.append(actual)
        return sum(result)


    def decayww2(self, bath, w, w1, t):
        t_array = np.asarray(t)
        result = self._decayww2(bath, w,w1, t_array)
        zero_indices = np.where(t_array == 0)
        result[zero_indices] = 0
        return result

    def decayww(self, bath, w, t):
        t_array = np.asarray(t)
        result = self._decayww(bath, w, t_array)
        zero_indices = np.where(t_array == 0)
        result[zero_indices] = 0
        return result

    def _LS(self, bath, w,w1, t):
        if w!=w1:
            cks=np.array([i.coefficient for i in bath.exponents])
            vks=np.array([i.exponent for i in bath.exponents])
            mask = np.imag(cks) >= 0
            cks=cks[mask]
            vks=vks[mask]
            result=[]
            for i in range(len(cks)):
                a=(vks[i]-1j*w1)
                b=(vks[i]-1j*w)
                term1=cks[i]*np.exp(-b*t)/(a*b)
                term2=np.conjugate(cks[i])*np.exp(-np.conjugate(a)*t)/(np.conjugate(a)*np.conjugate(b))
                term3=cks[i]*((1/b)-(np.exp(1j*(w-w1)*t)/a))
                term4=np.conjugate(cks[i])*((1/np.conjugate(a))-(np.exp(1j*(w-w1)*t)/np.conjugate(b)))
                actual=term1-term2+(1j*(term3-term4)/(w-w1))
                result.append(actual)
            return sum(result)/2j
        else:
            cks=np.array([i.coefficient for i in bath.exponents])
            vks=np.array([i.exponent for i in bath.exponents])
            result=[]
            for i in range(len(cks)):
                term1 =(vks[i]*t-1j*w*t-1)+np.exp(-(vks[i]-1j*w)*t)
                term1=term1*cks[i]/(vks[i]-1j*w)**2
                result.append(term1)
            return np.imag(sum(result))/2
    def LS(self, combinations, bath, t):
        rates = {}
        done = []
        for i in combinations:
            done.append(i)
            j = (i[1], i[0])
            if (j in done) & (i != j):
                rates[i] = np.conjugate(rates[j])
            else:
                rates[i] = self._LS(bath, i[0], i[1], t)
        return rates

bose(w, bath)

It computes the Bose-Einstein distribution

\[ n(\omega)=\frac{1}{e^{\beta \omega}-1} \]
Parameters:

nu: float The mode at which to compute the thermal population

Returns:

float The thermal population of mode nu

Source code in nmm/cumulant/cumulant.py 
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def bose(self,w,bath):
    r"""
    It computes the Bose-Einstein distribution

    $$ n(\omega)=\frac{1}{e^{\beta \omega}-1} $$

    Parameters:
    ----------
    nu: float
        The mode at which to compute the thermal population

    Returns:
    -------
    float
        The thermal population of mode nu
    """
    if bath.T == 0:
        return 0
    if np.isclose(w, 0).all():
        return 0
    return np.exp(-w / bath.T) / (1-np.exp(-w / bath.T))

evolution(rho0, approximated=False)

This function computes the evolution of the state \(\rho(0)\)

Parameters:

Name Type Description Default
rho0 ndarray or Qobj

The initial state of the quantum system under consideration.

 required
approximated bool

When False the full cumulant equation/refined weak coupling is computed, when True the Filtered Approximation (FA is computed), this greatly reduces computational time, at the expense of diminishing accuracy particularly for the populations of the system at early times.

 False

Returns:

Type Description
list

a list containing all of the density matrices, at all timesteps of the evolution
Source code in nmm/cumulant/cumulant.py 
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def evolution(self, rho0, approximated=False):
    r"""
    This function computes the evolution of the state $\rho(0)$

    Parameters
    ----------

    rho0 : numpy.ndarray or qutip.Qobj
        The initial state of the quantum system under consideration.

    approximated : bool
        When False the full cumulant equation/refined weak coupling is
        computed, when True the Filtered Approximation (FA is computed),
        this greatly reduces computational time, at the expense of
        diminishing accuracy particularly for the populations of the system
        at early times.

    Returns
    -------
    list
        a list containing all of the density matrices, at all timesteps of
        the evolution
    """
    self.generator(approximated)
    states = [
        (i).expm()(rho0)
        for k,i in tqdm(
            enumerate(self.generators),
            desc='Computing Exponential of Generators . . . .')]  # this counts time incorrectly
    return states

gamma_fa(bath, w, w1, t)

It describes the decay rates for the Filtered Approximation of the cumulant equation

\[\gamma(\omega,\omega^\prime,t)= 2\pi t e^{i \frac{\omega^\prime -\omega}{2}t}\mathrm{sinc} \left(\frac{\omega^\prime-\omega}{2}t\right) \left(J(\omega^\prime) (n(\omega^\prime)+1)J(\omega) (n(\omega)+1) \right)^{\frac{1}{2}}\]

Parameters:

Name Type Description Default
w float or ndarray
required
w1 float or ndarray
required
t float or ndarray
required

Returns:

Type Description
float or ndarray

It returns a value or array describing the decay between the levels with energies w and w1 at time t
Source code in nmm/cumulant/cumulant.py 
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def gamma_fa(self, bath, w, w1, t):
    r"""
    It describes the decay rates for the Filtered Approximation of the
    cumulant equation

    $$\gamma(\omega,\omega^\prime,t)= 2\pi t e^{i \frac{\omega^\prime
    -\omega}{2}t}\mathrm{sinc} \left(\frac{\omega^\prime-\omega}{2}t\right)
     \left(J(\omega^\prime) (n(\omega^\prime)+1)J(\omega) (n(\omega)+1)
     \right)^{\frac{1}{2}}$$

    Parameters
    ----------

    w : float or numpy.ndarray

    w1 : float or numpy.ndarray

    t : float or numpy.ndarray

    Returns
    -------
    float or numpy.ndarray
        It returns a value or array describing the decay between the levels
        with energies w and w1 at time t

    """
    var = (2 * t * np.exp(1j * (w1 - w) * t / 2)
           * np.sinc((w1 - w) * t / (2 * np.pi))
           * np.sqrt(bath.spectral_density(w1) * (self.bose(w1,bath) + 1))
           * np.sqrt(bath.spectral_density(w) * (self.bose(w,bath) + 1)))
    return var

gamma_gen(bath, w, w1, t, approximated=False)

It describes the the decay rates of the cumulant equation for bosonic baths

\[\Gamma(\omega,\omega',t) = t^{2}\int_{0}^{\infty} d\omega e^{i\frac{\omega-\omega'}{2} t} J(\omega) \left[ (n(\omega)+1) sinc\left(\frac{(\omega-\omega)t}{2}\right) sinc\left(\frac{(\omega'-\omega)t}{2}\right)+ n(\omega) sinc\left(\frac{(\omega+\omega)t}{2}\right) sinc\left(\frac{(\omega'+\omega)t}{2}\right) \right]\]

Parameters:

Name Type Description Default
w float or ndarray
required
w1 float or ndarray
required
t float or ndarray
required

Returns:

Type Description
float or ndarray

It returns a value or array describing the decay between the levels with energies w and w1 at time t
Source code in nmm/cumulant/cumulant.py 
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def gamma_gen(self, bath, w, w1, t, approximated=False):
    r"""
    It describes the the decay rates of the cumulant equation
    for bosonic baths

    $$\Gamma(\omega,\omega',t) = t^{2}\int_{0}^{\infty} d\omega 
    e^{i\frac{\omega-\omega'}{2} t} J(\omega) \left[ (n(\omega)+1) 
    sinc\left(\frac{(\omega-\omega)t}{2}\right)
    sinc\left(\frac{(\omega'-\omega)t}{2}\right)+ n(\omega) 
    sinc\left(\frac{(\omega+\omega)t}{2}\right) 
    sinc\left(\frac{(\omega'+\omega)t}{2}\right)   \right]$$

    Parameters
    ----------

    w : float or numpy.ndarray
    w1 : float or numpy.ndarray
    t : float or numpy.ndarray

    Returns
    -------
    float or numpy.ndarray
        It returns a value or array describing the decay between the levels
        with energies w and w1 at time t

    """
    if isinstance(t, type(jnp.array([2]))):
        t = np.array(t.tolist())
    if isinstance(t, list):
        t = np.array(t)
    if approximated:
        return self.gamma_fa(bath, w, w1, t)
    if self.matsubara:
        if w == w1:
            return self.decayww(bath, w, t)
        else:
            return self.decayww2(bath, w, w1, t)
    if self.cython:
        return bath.gamma(np.real(w), np.real(w1), t, limit=self.limit)

    else:
        integrals = quad_vec(
            self._gamma_,
            0,
            np.inf,
            args=(bath, w, w1, t),
            epsabs=self.eps,
            epsrel=self.eps,
            quadrature="gk21"
        )[0]
        return t*t*integrals