hopsflow/gaussflow.py

import numpy as np
import itertools
from dataclasses import dataclass, field
from typing import Callable, Optional
import numpy.typing as npt
from . import util
from numpy.polynomial import Polynomial
from .util import BCF, expand_t
import scipy


@dataclass
class SystemParams:
    r"""A parameter object to hold information about the physical
    system and the global parameters for the algorithm.

    :param Ω: the system hamiltonian energy scale, :math:`H_s =
              \Omega (p^2 + q^2)`
    :param η: the coupling strength (in essence a prefactor to the BCF)
    :param α_0: the zero temperature BCF


    :attr t_max: the maximum simulation time, will be copied from :attr:`α`
    :attr W: the exponential factors in the BCF expansion
    :attr G: the pre-factors in the BCF expansion
    """

    Ω: float
    η: float
    α_0: BCF

    t_max: float = field(init=False)

    # BCF coefficients
    W: np.ndarray = field(init=False)
    G: np.ndarray = field(init=False)

    root_tol: float = 1e-7
    """
    The relative tolerance of the roots.

    The roots are being calculated with the numpy polynomial module
    and then refined with newtons method.  This parameter is passed as
    ``rtol`` to :any:`scipy.optimize.newton`.
    """

    def __post_init__(self):
        self.t_max = self.α_0.t_max

        self.W = self.α_0.exponents
        self.G = self.α_0.factors.copy() * self.η


def construct_polynomials(sys: SystemParams) -> tuple[Polynomial, Polynomial]:
    r"""
    Construct the polynomials required to find the coefficients
    and exponents of the solution.

    :param sys: a parameter object with the system parameters
    :returns: a list of polynomials

        - :math:`f_0(z) = \prod_k (z-z_k) (z-z_k^{\ast})`
        - :math:`p(z) = p_1(z) + \sum_n q_n(z)` where
            - :math:`p_1(z) = (z^2 + \Omega^2)\prod_k (z-z_k)(z-z_k^{\ast})`
            - :math:`q_n(z)=\Omega f_n(z) \prod_{k\neq n}(z-z_k) (z-z_k^{\ast})`

    """
    # we begin by calculating all the constants we'll need
    φ = -np.angle(sys.G)
    G_abs = np.abs(sys.G)
    γ, δ = sys.W.real, sys.W.imag
    s, c = np.sin(φ), np.cos(φ)

    # the roots of the denominator of the laplace transform of α
    roots_z = -sys.W

    # now the polynomials
    f_0 = util.poly_real(
        Polynomial.fromroots(np.concatenate((roots_z, roots_z.conj())))
    )
    p_1 = Polynomial([sys.Ω ** 2, 0, 1]) * f_0

    q = [
        -G_c
        * sys.Ω
        * Polynomial((δ_c * c_c + γ_c * s_c, s_c))
        * (
            util.poly_real(
                Polynomial.fromroots(
                    np.concatenate(
                        (
                            util.except_element(roots_z, i),
                            util.except_element(roots_z, i).conj(),
                        )
                    )
                )
            )
            if len(roots_z) > 1
            else 1  # corner case
        )
        for G_c, γ_c, δ_c, s_c, c_c, i in zip(G_abs, γ, δ, s, c, range(len(c)))
    ]

    p = p_1 + sum(q)

    return f_0, p


def calculate_coefficients(sys: SystemParams) -> tuple[np.ndarray, np.ndarray]:
    r"""
    Calculates the coefficients required to construct the
    propagator matrix :math:`G`.

    :param sys: a parameter object with the system parameters
    :returns: the exponents and the residuals which play a role as pre-factors
    """

    f_0, p = construct_polynomials(sys)

    master_roots = p.roots()

    if np.unique(master_roots).size != master_roots.size:
        raise RuntimeError(
            """The roots of the polynomial are not unique.
You can try to alter the number of terms in the expansion of the BCF."""
        )

    # improving the accuracy of the roots seems to be essential
    p_prime = p.deriv()
    p_prime_2 = p_prime.deriv()
    for i, root in enumerate(master_roots):
        res = scipy.optimize.newton(
            p,
            root,
            p_prime,
            rtol=sys.root_tol,
            fprime2=p_prime_2,
            maxiter=100000,  # we can allow that, there aren't that many roots
        )
        master_roots[i] = res

    resiquals = f_0(master_roots) / p.deriv()(master_roots)
    return master_roots, resiquals


class Propagator:
    """The propagator matrix :math:`G` for the system.

    You can get inidividual matrix elements as functions of time
    through indexing this object.

    Calling it with a time argument returns the whole matrix as
    array of shape ``(time, 2, 2)``.

    :param sys: a parameter object with the system parameters
    """

    def __init__(self, sys: SystemParams):
        self.params = sys
        self._roots, self._res = calculate_coefficients(sys)
        self._roots_exp = np.expand_dims(self._roots, axis=1)
        self._res_exp = np.expand_dims(self._res, axis=1)
        self._Ω = sys.Ω

        self._res_times_roots = self._res_exp * self._roots_exp
        self._res_times_roots_squared = self._res_times_roots * self._roots_exp
        self._elements = np.array([[self.el_11, self.el_12], [self.el_21, self.el_22]])
        pass

    @expand_t
    def el_11(self, t):
        return (self._res_times_roots * np.exp(t * self._roots_exp)).real.sum(axis=0)

    @expand_t
    def el_12(self, t):
        return self._Ω * (self._res_exp * np.exp(t * self._roots_exp)).real.sum(axis=0)

    @expand_t
    def el_21(self, t):
        return (self._res_times_roots_squared * np.exp(t * self._roots_exp)).real.sum(
            axis=0
        ) / self._Ω

    def el_22(self, t):
        return self.el_11(t)

    def __getitem__(self, indices) -> Callable[[npt.ArrayLike], np.ndarray]:
        """Get an individual matrix elment as function."""
        return self._elements[indices]

    @expand_t
    def __call__(self, t) -> np.ndarray:
        """Get the propagator as array of shape ``(time, 2, 2)``."""

        g_12 = self._res_exp * np.exp(t * self._roots_exp)
        diag = self._roots_exp * g_12
        g_21 = self._roots_exp * diag / self._Ω

        return (
            np.array([[diag, g_12 * self._Ω], [g_21, diag]])
            .real.sum(axis=2)
            .swapaxes(0, 2)
            .swapaxes(1, 2)
        )

    def inv(self, t) -> np.ndarray:
        return np.linalg.inv(self.__call__(t))


class Flow:
    r"""
    A collection of methods to calculate the time derivative
    of the bath energy expectation value :math:`\frac{d}{dt}\langle H_B\rangle`
    which can be retrieved as a function of time by calling this object.

    :param system: the system parameters, see :any:`SystemParams`
    :param α: the (finite temperature) BCF
    :param α_0_dot: the zero temperature BCF time derivative
    :param n: the excitation number of the initial state of the system :math:`|n\rangle`

    """

    def __init__(self, system: SystemParams, α: BCF, α_0_dot: BCF, n: int):
        #: the BCF
        self.α = α

        #: the zero temperature BCF
        self.α_0 = system.α_0

        #: the exponential factors in the BCF derivative
        #: expansion :math:`\dot{\alpha}_0=\sum_k P_k e^{-L_k \cdot t}`
        self.L = α_0_dot.exponents

        #: the pre-factors in the BCF derivative expansion
        self.P = α_0_dot.factors.copy() * system.η

        self.C, self.B = calculate_coefficients(system)

        #: the pre-factors of the exponential sum of :math:`A=G_{11}=\sum_k A_k e^{-C_k \cdot t}`
        self.A = self.B * self.C

        #: the exponents of the exponential sum of :math:`B=G_{12}=\sum_k B_k e^{-C_k \cdot t}`
        #:
        #:   - note the minus sign
        self.C = -self.C  # mind the extra -

        #: the pre-factors of the exponential sum of ``B``
        self.B = system.Ω * self.B

        #: the exponential factors in the BCF
        #: expansion :math:`\alpha=\sum_k G_k e^{-W_k \cdot t}`
        self.W = α.exponents

        #: the pre-factors factors in the BCF expansion
        self.G = α.factors.copy() * system.η

        #: the exponential factors in the zero temperature BCF
        #: expansion :math:`\alpha=\sum_k U_k e^{-Q_k \cdot t}`
        self.Q = system.α_0.exponents

        #: the pre-factors factors in the zero temperature BCF expansion
        self.U = system.α_0.factors.copy() * system.η

        #: the expectation value :math:`\langle q(0)^2\rangle`
        self.q_s_0 = 1 + 2 * n

        #: the expectation value :math:`\langle q(0)^2\rangle`
        self.p_s_0 = 1 + 2 * n

        #: the expectation value :math:`\langle q(0)p(0)\rangle`
        self.qp_0 = 1j

        #: the propagator matrix :math:`G(t)`, see :any:`Propagator`
        self.prop = Propagator(system)

        #: the coefficient matrix :math:`\Gamma^1`
        self.Γ1 = (self.B[:, None] * self.P[None, :]) / (
            self.L[None, :] - self.C[:, None]
        )

        #: the coefficient matrix :math:`\Gamma^2`
        self.Γ2 = (self.B[:, None] * self.G[None, :]) / (
            self.C[:, None] - self.W[None, :]
        )

        #: the coefficient matrix :math:`\Gamma^3`
        self.Γ3 = (self.B[:, None] * self.G.conj()[None, :]) / (
            self.C[:, None] + self.W.conj()[None, :]
        )

        #: the coefficient matrix :math:`\Gamma^A`
        self.ΓA = (self.A[:, None] * self.P[None, :]) / (
            self.L[None, :] - self.C[:, None]
        )

        #: the system parameters
        self.system = system

    def A_conv(self, t: npt.ArrayLike):
        r"""The integral :math:`\int_0^t A(s)\dot{\alpha}_0(t-s)ds`."""
        result = np.zeros_like(t, dtype="complex128")

        for (n, m) in itertools.product(range(len(self.A)), range(len(self.P))):
            result += self.ΓA[n, m] * (np.exp(-self.C[n] * t) - np.exp(-self.L[m] * t))

        return result

    def B_conv(self, t: npt.ArrayLike):
        r"""The integral :math:`\int_0^t B(s)\dot{\alpha}_0(t-s)ds`."""
        result = np.zeros_like(t, dtype="complex128")
        for (n, m) in itertools.product(range(len(self.B)), range(len(self.P))):
            result += self.Γ1[n, m] * (np.exp(-self.C[n] * t) - np.exp(-self.L[m] * t))

        return result

    def flow_nontherm(self, t: npt.ArrayLike):
        r"""The part of the flow that **does not** involve :math:`\alpha`."""
        a, b = self.prop[0, 0](t), self.prop[0, 1](t)
        ac, bc = self.A_conv(t), self.B_conv(t)

        return (
            -1
            / 2
            * (
                self.q_s_0 * a * ac
                + self.p_s_0 * b * bc
                + self.qp_0 * a * bc
                + self.qp_0.conjugate() * b * ac
            ).imag
        )

    def flow_therm_zero(self, t: npt.ArrayLike):
        r"""The part of the flow that **does** involve :math:`\alpha`."""
        t = np.asarray(t)
        result = np.zeros_like(t, dtype="float")

        for (m, k, n, l) in itertools.product(
            range(len(self.B)),
            range(len(self.P)),
            range(len(self.B)),
            range(len(self.G)),
        ):
            g_1_2 = (
                self.Γ1[m, k]
                * self.Γ2[n, l]
                * (
                    (1 - np.exp(-(self.C[m] + self.W[l]) * t)) / (self.C[m] + self.W[l])
                    + (np.exp(-(self.C[m] + self.C[n]) * t) - 1)
                    / (self.C[m] + self.C[n])
                    + (np.exp(-(self.L[k] + self.W[l]) * t) - 1)
                    / (self.L[k] + self.W[l])
                    + (1 - np.exp(-(self.L[k] + self.C[n]) * t))
                    / (self.L[k] + self.C[n])
                )
            ).imag

            g_1_3 = (
                self.Γ1[m, k]
                * self.Γ3[n, l]
                * (
                    (1 - np.exp(-(self.C[m] + self.C[n]) * t)) / (self.C[m] + self.C[n])
                    - (1 - np.exp(-(self.L[k] + self.C[n]) * t))
                    / (self.L[k] + self.C[n])
                    - (
                        np.exp(-(self.C[n] + self.W[l].conj()) * t)
                        - np.exp(-(self.C[m] + self.C[n]) * t)
                    )
                    / (self.C[m] - self.W[l].conj())
                    + (
                        np.exp(-(self.C[n] + self.W[l].conj()) * t)
                        - np.exp(-(self.L[k] + self.C[n]) * t)
                    )
                    / (self.L[k] - self.W[l].conj())
                )
            ).imag

            result += -1 / 2 * (g_1_2 + g_1_3)

        return result

    def flow_therm_nonzero(self, t: npt.ArrayLike):
        r"""The part of the flow that **does** involve :math:`\alpha-\alpha_0`."""
        t = np.asarray(t)
        result = np.zeros_like(t, dtype="float")

        for (n, m) in itertools.product(
            range(len(self.A)),
            range(len(self.G)),
        ):
            result += (
                self.A[n]
                * self.G[m]
                / (self.C[n] + self.W[m])
                * (1 - np.exp(-(self.C[n] + self.W[m]) * t))
            ).real

        for (n, m) in itertools.product(
            range(len(self.A)),
            range(len(self.U)),
        ):
            result -= (
                self.A[n]
                * self.U[m]
                / (self.C[n] + self.Q[m])
                * (1 - np.exp(-(self.C[n] + self.Q[m]) * t))
            ).real

        return (
            1
            / 2
            * (
                -self.system.Ω * result
                + self.prop.el_12(t)
                * (self.α.approx(t) - self.α_0.approx(t))
                * self.system.η
            )
        ).real

    def __call__(self, t: npt.ArrayLike) -> np.ndarray:
        r"""
        The flow. Time derivative of the bath energy
        expectation value :math:`\frac{d}{dt}\langle H_B\rangle`.
        """

        return (
            self.flow_nontherm(t) + self.flow_therm_zero(t) + self.flow_therm_nonzero(t)
        )