Source code for arc.calculations_atom_single

# -*- coding: utf-8 -*-

"""
    This module provides calculations of single-atom properties.

    Included calculations are Stark maps, level plot visualisations,
    lifetimes and radiative decays.

"""

from __future__ import print_function

from .alkali_atom_functions import (
    printStateString,
    _EFieldCoupling,
    printStateLetter,
    printStateStringLatex,
    formatNumberSI,
)
import datetime
import matplotlib
from matplotlib.colors import LinearSegmentedColormap
from math import sqrt
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
import numpy as np
import warnings

from scipy.constants import physical_constants, pi, epsilon_0, hbar
from scipy.constants import c as C_c
from scipy.constants import h as C_h
from scipy.constants import e as C_e
from scipy.constants import m_e as C_m_e
from scipy.optimize import curve_fit
from scipy import interpolate

# for matrices
from numpy.linalg import eigh

import scipy.sparse as sp
from scipy.sparse import csr_matrix
from scipy.special import sph_harm

import sys

from arc._database import UsedModulesARC
from arc.divalent_atom_functions import DivalentAtom
from arc.wigner import Wigner6j, CG


if sys.version_info > (2,):
    xrange = range


def Ylm(l, m, theta, phi):
    return sph_harm(m, l, phi, theta)


__all__ = [
    "Ylm",
    "Wavefunction",
    "StarkMap",
    "LevelPlot",
    "AtomSurfaceVdW",
    "OpticalLattice1D",
    "DynamicPolarizability",
    "StarkBasisGenerator",
    "ShirleyMethod",
    "RWAStarkShift",
]


[docs] class Wavefunction: r""" Calculates and plots electron wavefunctions. For an example see `wavefunction plotting example snippet`_. .. _`wavefunction plotting example snippet`: ./ARC_3_0_introduction.html#Wavefunction-calculations-for-Alkali-atom-Rydberg-states Args: atom: atom type considered (for example :obj:`Rubidum87()`) basisStates (array): array of states in fine basis that contribute\ to the state whose wavefunction is requested. :math:`[[n_1, \ell_1, j_1, m_{j1}], ...]` For efficient calculation **do not** pass all the possible basis states, but just the once that have significant contribution to the reqested state. coefficients (array): array `[c1, ...]` of complex coefficents :math:`c_i = \langle \psi_i |\psi\rangle` corresponding to decomposition of required state :math:`|\psi\rangle` on basis states :math:`|\psi_i \rangle` . """ def __init__(self, atom, basisStates, coefficients): # n, l, j, mj UsedModulesARC.arc3_0_methods = True self.atom = atom if ( len(basisStates) == 0 or len(basisStates[0]) != 4 or len(basisStates) != len(coefficients) ): raise ValueError( "basisStates should be defined as array of" "states in fine basis [[n1, l1, j1, mj1], ... ]" "contributing to the required states " "(do not use unecessarily whole basis) " "while coefficients corresponding to decomposition " "of requested state on these basis state " "should be given as" "separete array [c1, ...]" ) self.basisStates = basisStates self.coef = coefficients self.basisWavefunctions = [] for state in self.basisStates: n = state[0] l = state[1] j = state[2] # calculate radial wavefunction step = 0.001 r, rWavefunc = atom.radialWavefunction( l, 0.5, j, self.atom.getEnergy(n, l, j) / 27.211, self.atom.alphaC ** (1 / 3.0), 2.0 * n * (n + 15.0), step, ) suma = np.trapz(rWavefunc**2, x=r) rWavefunc = rWavefunc / (sqrt(suma)) self.basisWavefunctions.append( interpolate.interp1d( r, rWavefunc, bounds_error=False, fill_value=(0, 0) ) )
[docs] def getRtimesPsiSpherical(self, theta, phi, r): r""" Calculates list of :math:`r \cdot \psi_{m_s} (\theta, \phi, r)` At point defined by spherical coordinates, returns list of :math:`r \cdot \psi_{m_s} (\theta, \phi, r)` wavefunction values for different electron spin projection values :math:`m_s`. Coordinates are defined relative to atomic core. Args: theta (float): polar angle (angle between :math:`z` axis and vector pointing towards selected point) (in units of radians). phi (float): azimuthal angle (angle between :math:`x` axis and projection at :math:`x-y` plane of vector pointing towards selected point) (in units of radians). r (float): distance between coordinate origin and selected point. (in atomic units of Bohr radius :math:`a_0`) Returns: list of complex values corresponding to :math:`\psi_{m_s} (\theta, \phi, r)` for different spin states :math:`m_s` contributing to the state in **decreasing** order of :math:`m_s`. For example, for :obj:`arc.AlkaliAtom` returns :math:`r \cdot \psi_{m_s=+1/2} (\theta, \phi, r)` and :math:`r \cdot \psi_{m_s=-1/2} (\theta, \phi, r) `. )` """ wfElectronP = 0 + 0j # electron spin +1/2 wfElectronM = 0 + 0j # electron spin -1/2 for i, state in enumerate(self.basisStates): l = state[1] j = state[2] mj = state[3] if abs(mj - 0.5) - 0.1 < l: wfElectronP += ( CG(l, mj - 0.5, 0.5, +0.5, j, mj) * Ylm(l, mj - 0.5, theta, phi) * self.basisWavefunctions[i](r) * self.coef[i] ) if abs(mj + 0.5) - 0.1 < l: wfElectronM += ( CG(l, mj + 0.5, 0.5, -0.5, j, mj) * Ylm(l, mj + 0.5, theta, phi) * self.basisWavefunctions[i](r) * self.coef[i] ) return wfElectronP, wfElectronM
[docs] def getRtimesPsi(self, x, y, z): r""" Calculates list of :math:`r \cdot \psi_{m_s} (x, y, z)` At a point defined by Cartesian coordinates returns list of :math:`r \cdot \psi_{m_s} (x, y, z)` wavefunction values for different electron spin projection values :math:`m_s`. Args: x (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) y (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) z (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) Returns: list of complex values corresponding to :math:`r \cdot \psi_{m_s} (\theta, \phi, r)` for different spin states :math:`m_s` contributing to the state in **decreasing** order of :math:`m_s`. For example, for :obj:`arc.AlkaliAtom` returns :math:`r \cdot \psi_{m_s=+1/2} (\theta, \phi, r)` and :math:`r \cdot \psi_{m_s=-1/2} (\theta, \phi, r)` . )`, where :math:`r=\sqrt{x^2+y^2+z^2}`. """ theta = np.arctan2((x**2 + y**2) ** 0.5, z) phi = np.arctan2(y, x) r = np.sqrt(x**2 + y**2 + z**2) return self.getRtimesPsiSpherical(theta, phi, r)
[docs] def getPsi(self, x, y, z): r""" Calculates list of :math:`\psi_{m_s} (x,y,z)` At point define by Cartesian coordinates returns list of :math:`\psi_{m_s} (x,y,z)` wavefunction values corresponding to different electron spin projection values :math:`m_s`. Args: x (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) y (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) z (float): Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius :math:`a_0`) Returns: list of complex values corresponding to :math:`\psi_{m_s} (\theta, \phi, r)` for different spin states :math:`m_s` contributing to the state in **decreasing** order of :math:`m_s`. For example, for :obj:`arc.AlkaliAtom` returns :math:`\psi_{m_s=+1/2} (\theta, \phi, r)` and :math:`\psi_{m_s=-1/2} (\theta, \phi, r)` . )`. """ r = np.sqrt(x * x + y * y + z * z) return self.getRtimesPsi(x, y, z) / r
[docs] def getRtimesPsiSquaredInPlane( self, plane="x-z", pointsPerAxis=150, axisLength=None, units="atomic" ): r""" Calculates :math:`|r \cdot \psi|^2` on a mesh in a given plane. Args: plane (str): optiona, set's calculation plane to `'x-y'` or `'x-z'`. Default value `'x-y'` pointsPerAxis (int): optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of :math:`150 \times 150 = 22500` points. axisLength (float): optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is largw enough to fit the whole wavefunction (in atomic units of Bohr radius :math:`a_0`). units (str): optional, units of length in which calculated mesh will be **returned** (note that `axisLength` is on the other hand always in atomi units.). Supported values are `'atomic'` or `'nm'`. Default value `'atomic'` . Returns: meshCoordinate1, meshCoordinate2 and :math:`|r \cdot \psi|^2 = \sum_{m_s} |r \cdot \psi_{m_s}|^2`, where sum is over possible electron spin projection values :math:`m_s`. """ if axisLength is None: nMax = 1 for state in self.basisStates: nMax = max(nMax, state[0]) axisLength = 2.0 * 2.0 * nMax * (nMax + 15.0) coord1 = np.linspace(-axisLength / 2.0, axisLength / 2.0, pointsPerAxis) coord2 = np.linspace(-axisLength / 2.0, axisLength / 2.0, pointsPerAxis) meshCoord1, meshCoord2 = np.meshgrid(coord1, coord2) coord = [] if plane == "x-z": coord = [meshCoord1, 0, meshCoord2] elif plane == "x-y": coord = [meshCoord1, meshCoord2, 0] else: raise ValueError("Only 'x-y' and 'x-z' planes are supported.") wfP, wfM = self.getRtimesPsi(*coord) # change units if units == "nm": scale = physical_constants["Bohr radius"][0] * 1e9 meshCoord1 *= scale meshCoord2 *= scale wfP /= scale wfM /= scale elif units == "atomic": pass else: raise ValueError( "Only 'atomic' (a_0) and 'nm' are recognised" "as possible units. Received: %s" % units ) f = np.power(np.abs(wfP), 2) + np.power(np.abs(wfM), 2) return meshCoord1, meshCoord2, f
[docs] def plot2D( self, plane="x-z", pointsPerAxis=150, axisLength=None, units="atomic", colorbar=True, labels=True, ): r""" 2D colour plot of :math:`|r \cdot \psi|^2` wavefunction in a requested plane. Args: plane (str): optiona, set's calculation plane to `'x-y'` or `'x-z'`. Default value `'x-y'` pointsPerAxis (int): optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of :math:`150 \times 150 = 22500` points. axisLength (float): optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is large enough to fit the whole wavefunction (in atomic units of Bohr radius :math:`a_0`). units (str): optional, units of length in which calculated mesh will be **returned** (note that `axisLength` is on the other hand always in atomi units.). Supported values are `'atomic'` or `'nm'`. Default value `'atomic'` . colorbar (bool): optional, determens if the colour bar scale of should be shown. Default value is `True`. labels (bool): optional, determines if the labels on the axis of the plot should be shown. Default value is `True`. Returns: :obj:`matplotlib.pyplot.figure` object with a requested plot. Use `show()` method to see figure. """ x, y, f = self.getRtimesPsiSquaredInPlane( plane=plane, pointsPerAxis=pointsPerAxis, axisLength=axisLength, units=units, ) fig = plt.figure(figsize=(6, 4)) ax = fig.add_subplot(1, 1, 1) cp = ax.pcolor(x, y, f, vmin=0, vmax=f.max(), cmap="viridis") if labels: if units == "atomic": unitLabel = r"$a_0$" else: unitLabel = "nm" if plane == "x-y": plt.xlabel(r"$x$ (%s)" % unitLabel) plt.ylabel(r"$y$ (%s)" % unitLabel) elif plane == "x-z": plt.xlabel(r"$x$ (%s)" % unitLabel) plt.ylabel(r"$z$ (%s)" % unitLabel) else: raise ValueError( "Only 'atomic' (a_0) and 'nm' are recognised" "as possible units. Received: %s" % units ) ax.set_aspect("equal", "box") if colorbar: cb = fig.colorbar(cp) cb.set_label( r"$|r\cdot\psi(x,y,z)|^2$" ) # NOTE: change label if plotting Imaginart part! return fig
# return figure
[docs] def plot3D( self, plane="x-z", pointsPerAxis=150, axisLength=None, units="atomic", labels=True, ): r""" 3D colour surface plot of :math:`|r \cdot \psi|^2` wavefunction in a requested plane. Args: plane (str): optiona, set's calculation plane to `'x-y'` or `'x-z'`. Default value `'x-y'` pointsPerAxis (int): optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of :math:`150 \times 150 = 22500` points. axisLength (float): optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is large enough to fit the whole wavefunction (in atomic units of Bohr radius :math:`a_0`). units (str): optional, units of length in which calculated mesh will be **returned** (note that `axisLength` is on the other hand always in atomi units.). Supported values are `'atomic'` or `'nm'`. Default value `'atomic'` . labels (bool): optional, determines if the labels on the axis of the plot should be shown. Default value is `True`. Returns: :obj:`matplotlib.pyplot.figure` object with a requested plot. Use `show()` method to see figure. """ x, y, f = self.getRtimesPsiSquaredInPlane( plane=plane, pointsPerAxis=pointsPerAxis, axisLength=axisLength, units=units, ) fig = plt.figure(figsize=(6, 4)) ax = fig.add_subplot(projection="3d") ax.view_init(40, -35) # Plot the surface. ax.plot_surface( x, y, f, cmap="Reds", vmin=0, vmax=f.max(), linewidth=0, antialiased=False, rstride=1, cstride=1, ) ax.plot_wireframe( x, y, f, rstride=10, cstride=10, alpha=0.05, color="k" ) if labels: if units == "atomic": unitLabel = r"$a_0$" else: unitLabel = "nm" if plane == "x-y": plt.xlabel(r"$x$ (%s)" % unitLabel) plt.ylabel(r"$y$ (%s)" % unitLabel) elif plane == "x-z": plt.xlabel(r"$x$ (%s)" % unitLabel) plt.ylabel(r"$z$ (%s)" % unitLabel) else: raise ValueError( "Only 'atomic' (a_0) and 'nm' are recognised" "as possible units. Received: %s" % units ) plt.xlim(x.min(), x.max()) plt.ylim(y.min(), y.max()) return fig
[docs] class StarkMap: """ Calculates Stark maps for single atom in a field This initializes calculation for the atom of a given type. For details of calculation see Zimmerman [1]_. For a quick working example see `Stark map example snippet`_. Args: atom (:obj:`arc.alkali_atom_functions.AlkaliAtom` or :obj:`arc.divalent_atom_functions.DivalentAtom`): ={ :obj:`arc.alkali_atom_data.Lithium6`, :obj:`arc.alkali_atom_data.Lithium7`, :obj:`arc.alkali_atom_data.Sodium`, :obj:`arc.alkali_atom_data.Potassium39`, :obj:`arc.alkali_atom_data.Potassium40`, :obj:`arc.alkali_atom_data.Potassium41`, :obj:`arc.alkali_atom_data.Rubidium85`, :obj:`arc.alkali_atom_data.Rubidium87`, :obj:`arc.alkali_atom_data.Caesium`, :obj:`arc.divalent_atom_data.Strontium88`, :obj:`arc.divalent_atom_data.Calcium40` :obj:`arc.divalent_atom_data.Ytterbium174` } Select the alkali metal for energy level diagram calculation Examples: State :math:`28~S_{1/2}~|m_j|=0.5` polarizability calculation >>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> calc.diagonalise(np.linspace(00.,6000,600)) >>> print("%.5f MHz cm^2 / V^2 " % calc.getPolarizability()) 0.76705 MHz cm^2 / V^2 Stark map calculation >>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> calc.diagonalise(np.linspace(00.,60000,600)) >>> calc.plotLevelDiagram() >>> calc.showPlot() << matplotlib plot will open containing a Stark map >> Examples: **Advanced interfacing of Stark map calculations (StarkMap class)** Here we show one easy way to obtain the Stark matrix (from diagonal :obj:`mat1` and off-diagonal part :obj:`mat2` ) and basis states (stored in :obj:`basisStates` ), if this middle-product of the calculation is needed for some code build on top of the existing ARC package. >>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> # Now we have matrix and basis states, that we can used in our own code >>> # Let's say we want Stark map at electric field of 0.2 V/m >>> eField = 0.2 # V/m >>> # We can easily extract Stark matrix >>> # as diagonal matrix (state detunings) >>> # + off-diagonal matrix (propotional to electric field) >>> matrix = calc.mat1+calc.mat2*eField >>> # and the basis states as array [ [n,l,j,mj] , ...] >>> basisStates = calc.basisStates >>> # you can do your own calculation now... References: .. [1] M. L. Zimmerman et.al, PRA **20**:2251 (1979) https://doi.org/10.1103/PhysRevA.20.2251 .. _`Stark map example snippet`: ./Rydberg_atoms_a_primer_notebook.html#Rydberg-Atom-Stark-Shifts """ def __init__(self, atom): self.atom = atom self.basisStates = [] """ List of basis states for calculation in the form [ [n,l,j,mj], ...]. Calculated by :obj:`defineBasis` . """ self.mat1 = [] """ diagonal elements of Stark-matrix (detuning of states) calculated by :obj:`defineBasis` in the basis :obj:`basisStates`. """ self.mat2 = [] """ off-diagonal elements of Stark-matrix divided by electric field value. To get off diagonal elemements multiply this matrix with electric field value. Full Stark matrix is obtained as `fullStarkMatrix` = :obj:`mat1` + :obj:`mat2` *`eField`. Calculated by :obj:`defineBasis` in the basis :obj:`basisStates`. """ self.indexOfCoupledState = [] """ Index of coupled state (initial state passed to :obj:`defineBasis`) in :obj:`basisStates` list of basis states """ # finding energy levels self.eFieldList = [] """ Saves electric field (in units of V/m) for which energy levels are calculated See also: :obj:`y`, :obj:`highlight`, :obj:`diagonalise` """ self.y = [] # eigenValues """ `y[i]` is an array of eigenValues corresponding to the energies of the atom states at the electric field `eFieldList[i]`. For example `y[i][j]` is energy of the `j` eigenvalue (energy of the state) measured in cm :math:`{}^{-1}` relative to the ionization threshold. See also: :obj:`eFieldList`, :obj:`highlight`, :obj:`diagonalise` """ self.highlight = ( [] ) # contribution of initial state there (overlap |<original state | given state>|^2) """ `highlight[i]` is an array of values measuring highlighted feature in the eigenstates at electric field intensity `eFieldList[i]`. E.g. `highlight[i][j]` measures highlighted feature of the state with energy `y[i][j]` at electric field `eFieldList[i]`. What will be highlighted feature is defined in the call of :obj:`diagonalise` (see that part of documentation for details). See also: :obj:`eFieldList`, :obj:`y`, :obj:`diagonalise` """ #: pointer towards matplotlib figure after :obj:`plotLevelDiagram` #: is called to create figure self.fig = 0 #: pointer towards matplotlib figure axis after :obj:`plotLevelDiagram` #: is called to create figure self.ax = 0 # values used for fitting polarizability, and fit self.fitX = [] self.fitY = [] self.fittedCurveY = [] self.drivingFromState = [0, 0, 0, 0, 0] self.maxCoupling = 0.0 # STARK memoization self.eFieldCouplingSaved = False #: spin manifold in which we are working #: default value of 0.5 is correct for Alkaline Atoms. Otherwise it has #: to be specified when calling `defineBasis` as `s=0` or `s=1` for #: singlet and triplet states respectively self.s = 0.5 def _eFieldCouplingDivE(self, n1, l1, j1, mj1, n2, l2, j2, mj2, s=0.5): # eFied coupling devided with E (witout actuall multiplication to getE) # delta(mj1,mj2') delta(l1,l2+-1) if (abs(mj1 - mj2) > 0.1) or (abs(l1 - l2) != 1): return 0 # matrix element result = ( self.atom.getRadialMatrixElement(n1, l1, j1, n2, l2, j2, s=s) * physical_constants["Bohr radius"][0] * C_e ) sumPart = self.eFieldCouplingSaved.getAngular( l1, j1, mj1, l2, j2, mj2, s=s ) return result * sumPart def _eFieldCoupling(self, n1, l1, j1, mj1, n2, l2, j2, mj2, eField, s=0.5): return ( self._eFieldCouplingDivE(n1, l1, j1, mj1, n2, l2, j2, mj2, s=s) * eField )
[docs] def defineBasis( self, n, l, j, mj, nMin, nMax, maxL, Bz=0, progressOutput=False, debugOutput=False, s=0.5, ): """ Initializes basis of states around state of interest Defines basis of states for further calculation. :math:`n,l,j,m_j` specify state whose neighbourhood and polarizability we want to explore. Other parameters specify basis of calculations. This method stores basis in :obj:`basisStates`, while corresponding interaction matrix is stored in two parts. First part is diagonal electric-field independent part stored in :obj:`mat1`, while the second part :obj:`mat2` corresponds to off-diagonal elements that are propotional to electric field. Overall interaction matrix for electric field `eField` can be then obtained as `fullStarkMatrix` = :obj:`mat1` + :obj:`mat2` *`eField` Args: n (int): principal quantum number of the state l (int): angular orbital momentum of the state j (flaot): total angular momentum of the state mj (float): projection of total angular momentum of the state nMin (int): *minimal* principal quantum number of the states to be included in the basis for calculation nMax (int): *maximal* principal quantum number of the states to be included in the basis for calculation maxL (int): *maximal* value of orbital angular momentum for the states to be included in the basis for calculation Bz (float): optional, magnetic field directed along z-axis in units of Tesla. Calculation will be correct only for weak magnetic fields, where paramagnetic term is much stronger then diamagnetic term. Diamagnetic term is neglected. progressOutput (:obj:`bool`, optional): if True prints the progress of calculation; Set to false by default. debugOutput (:obj:`bool`, optional): if True prints additional information usefull for debuging. Set to false by default. s (float): optional. Total spin angular momentum for the state. Default value of 0.5 is correct for Alkaline Atoms, but value **has to** be specified explicitly for divalent atoms (e.g. `s=0` or `s=1` for singlet and triplet states, that have total spin angular momenutum equal to 0 or 1 respectively). """ global wignerPrecal wignerPrecal = True self.eFieldCouplingSaved = _EFieldCoupling() states = [] # save calculation details START self.n = n self.l = l self.j = j self.mj = mj self.nMin = nMin self.nMax = nMax self.maxL = maxL self.Bz = Bz self.s = s # save calculation details END for tn in xrange(nMin, nMax): for tl in xrange(min(maxL + 1, tn)): for tj in np.linspace(tl - s, tl + s, round(2 * s + 1)): if (abs(mj) - 0.1 <= tj) and ( tn >= self.atom.groundStateN or [tn, tl, tj] in self.atom.extraLevels ): states.append([tn, tl, tj, mj]) dimension = len(states) if progressOutput: print("Found ", dimension, " states.") if debugOutput: print(states) indexOfCoupledState = 0 index = 0 for st in states: if ( (st[0] == n) and (abs(st[1] - l) < 0.1) and (abs(st[2] - j) < 0.1) and (abs(st[3] - mj) < 0.1) ): indexOfCoupledState = index index += 1 if debugOutput: print("Index of initial state") print(indexOfCoupledState) print("Initial state = ") print(states[indexOfCoupledState]) self.mat1 = np.zeros((dimension, dimension), dtype=np.double) self.mat2 = np.zeros((dimension, dimension), dtype=np.double) self.basisStates = states self.indexOfCoupledState = indexOfCoupledState if progressOutput: print("Generating matrix...") progress = 0.0 for ii in xrange(dimension): if progressOutput: progress += (dimension - ii) * 2 - 1 sys.stdout.write( "\r%d%%" % (float(progress) / float(dimension**2) * 100) ) sys.stdout.flush() # add diagonal element self.mat1[ii][ii] = ( self.atom.getEnergy( states[ii][0], states[ii][1], states[ii][2], s=self.s ) * C_e / C_h * 1e-9 + self.atom.getZeemanEnergyShift( states[ii][1], states[ii][2], states[ii][3], self.Bz, s=self.s, ) / C_h * 1.0e-9 ) # add off-diagonal element for jj in xrange(ii + 1, dimension): coupling = ( self._eFieldCouplingDivE( states[ii][0], states[ii][1], states[ii][2], mj, states[jj][0], states[jj][1], states[jj][2], mj, s=self.s, ) * 1.0e-9 / C_h ) self.mat2[jj][ii] = coupling self.mat2[ii][jj] = coupling if progressOutput: print("\n") if debugOutput: print(self.mat1 + self.mat2) print(self.mat2[0]) self.atom.updateDipoleMatrixElementsFile() self.eFieldCouplingSaved._closeDatabase() self.eFieldCouplingSaved = False return 0
[docs] def diagonalise( self, eFieldList, drivingFromState=[0, 0, 0, 0, 0], progressOutput=False, debugOutput=False, upTo=4, totalContributionMax=0.95, ): """ Finds atom eigenstates in a given electric field Eigenstates are calculated for a list of given electric fields. To extract polarizability of the originaly stated state see :obj:`getPolarizability` method. Results are saved in :obj:`eFieldList`, :obj:`y` and :obj:`highlight`. Args: eFieldList (array): array of electric field strength (in V/m) for which we want to know energy eigenstates progressOutput (:obj:`bool`, optional): if True prints the progress of calculation; Set to false by default. debugOutput (:obj:`bool`, optional): if True prints additional information usefull for debuging. Set to false by default. upTo ('int', optional): Number of top contributing bases states to be saved into composition attribute; Set to 4 by default. To keep all contributing states, set upTo = -1. totalContributionMax ('float', optional): Ceiling for contribution to the wavefunction from basis states included in composition attribute. Composition will contain a list of [coefficient, state index] pairs for top contributing unperturbed basis states until the number of states reaches upTo or their total contribution reaches totalContributionMax, whichever comes first. totalContributionMax is ignored if upTo = -1. """ # if we are driving from some state # ========= FIND LASER COUPLINGS (START) ======= coupling = [] dimension = len(self.basisStates) self.maxCoupling = 0.0 self.drivingFromState = drivingFromState if self.drivingFromState[0] != 0: if progressOutput: print("Finding driving field coupling...") # get first what was the state we are calculating coupling with state1 = drivingFromState n1 = round(state1[0]) l1 = round(state1[1]) j1 = state1[2] m1 = state1[3] q = state1[4] for i in xrange(dimension): thisCoupling = 0.0 if progressOutput: sys.stdout.write( "\r%d%%" % (i / float(dimension - 1) * 100.0) ) sys.stdout.flush() if ( (round(abs(self.basisStates[i][1] - l1)) == 1) and (round(abs(self.basisStates[i][2] - j1)) <= 1) and (round(abs(self.basisStates[i][3] - m1 - q)) == 0) ): state2 = self.basisStates[i] n2 = round(state2[0]) l2 = round(state2[1]) j2 = state2[2] m2 = state2[3] if debugOutput: print( n1, " ", l1, " ", j1, " ", m1, " < - ", q, " - >", n2, " ", l2, " ", j2, " ", m2, "\n", ) dme = self.atom.getDipoleMatrixElement( n1, l1, j1, m1, n2, l2, j2, m2, q, s=self.s ) thisCoupling += dme thisCoupling = abs(thisCoupling) ** 2 if thisCoupling > self.maxCoupling: self.maxCoupling = thisCoupling if (thisCoupling > 0.00000001) and debugOutput: print("coupling = ", thisCoupling) coupling.append(thisCoupling) if progressOutput: print("\n") if self.maxCoupling < 0.00000001: raise Exception( "State that you specified in drivingFromState, for a " + "given laser polarization, is uncoupled from the specified Stark " + "manifold. If you just want to see the specified Stark manifold " + "remove driveFromState optional argument from call of function " + "diagonalise. Or specify state and driving that is coupled " + "to a given manifold to see coupling strengths." ) # ========= FIND LASER COUPLINGS (END) ======= indexOfCoupledState = self.indexOfCoupledState self.eFieldList = eFieldList self.y = [] self.highlight = [] self.composition = [] if progressOutput: print("Finding eigenvectors...") progress = 0.0 for eField in eFieldList: if progressOutput: progress += 1.0 sys.stdout.write( "\r%d%%" % (float(progress) / float(len(eFieldList)) * 100) ) sys.stdout.flush() m = self.mat1 + self.mat2 * eField ev, egvector = eigh(m) self.y.append(ev) if drivingFromState[0] < 0.1: sh = [] comp = [] for i in xrange(len(ev)): sh.append(abs(egvector[indexOfCoupledState, i]) ** 2) comp.append( self._stateComposition2( egvector[:, i], upTo=upTo, totalContributionMax=totalContributionMax, ) ) self.highlight.append(sh) self.composition.append(comp) else: sh = [] comp = [] for i in xrange(len(ev)): sumCoupledStates = 0.0 for j in xrange(dimension): sumCoupledStates += abs( coupling[j] / self.maxCoupling ) * abs(egvector[j, i] ** 2) comp.append( self._stateComposition2( egvector[:, i], upTo=upTo, totalContributionMax=totalContributionMax, ) ) sh.append(sumCoupledStates) self.highlight.append(sh) self.composition.append(comp) if progressOutput: print("\n") return
[docs] def exportData(self, fileBase, exportFormat="csv"): """ Exports StarkMap calculation data. Only supported format (selected by default) is .csv in a human-readable form with a header that saves details of calculation. Function saves three files: 1) `filebase` _eField.csv; 2) `filebase` _energyLevels 3) `filebase` _highlight For more details on the format, see header of the saved files. Args: filebase (string): filebase for the names of the saved files without format extension. Add as a prefix a directory path if necessary (e.g. saving outside the current working directory) exportFormat (string): optional. Format of the exported file. Currently only .csv is supported but this can be extended in the future. """ fmt = "on %Y-%m-%d @ %H:%M:%S" ts = datetime.datetime.now().strftime(fmt) commonHeader = "Export from Alkali Rydberg Calculator (ARC) %s.\n" % ts commonHeader += "\n *** Stark Map for %s %s m_j = %d/2. ***\n\n" % ( self.atom.elementName, printStateString(self.n, self.l, self.j), round(2.0 * self.mj), ) commonHeader += ( " - Included states - principal quantum number (n) range [%d-%d].\n" % (self.nMin, self.nMax) ) commonHeader += ( " - Included states with orbital momentum (l) in range [%d,%d] (i.e. %s-%s).\n" % (0, self.maxL, printStateLetter(0), printStateLetter(self.maxL)) ) commonHeader += ( " - Calculated in manifold where total spin angular momentum is s = %.1d\n" % (self.s) ) if self.drivingFromState[0] < 0.1: commonHeader += ( " - State highlighting based on the relative contribution \n" + " of the original state in the eigenstates obtained by diagonalization." ) else: commonHeader += ( " - State highlighting based on the relative driving strength \n" + " to a given energy eigenstate (energy level) from state\n" + " %s m_j =%d/2 with polarization q=%d.\n" % ( printStateString(*self.drivingFromState[0:3]), round(2.0 * self.drivingFromState[3]), self.drivingFromState[4], ) ) if exportFormat == "csv": print("Exporting StarkMap calculation results as .csv ...") commonHeader += " - Export consists of three (3) files:\n" commonHeader += " 1) %s,\n" % ( fileBase + "_eField." + exportFormat ) commonHeader += " 2) %s,\n" % ( fileBase + "_energyLevels." + exportFormat ) commonHeader += " 3) %s.\n\n" % ( fileBase + "_highlight." + exportFormat ) filename = fileBase + "_eField." + exportFormat np.savetxt( filename, self.eFieldList, fmt="%.18e", delimiter=", ", newline="\n", header=(commonHeader + " - - - eField (V/m) - - -"), comments="# ", ) print(" Electric field values (V/m) saved in %s" % filename) filename = fileBase + "_energyLevels." + exportFormat headerDetails = " NOTE : Each row corresponds to eigenstates for a single specified electric field" np.savetxt( filename, self.y, fmt="%.18e", delimiter=", ", newline="\n", header=( commonHeader + " - - - Energy (GHz) - - -\n" + headerDetails ), comments="# ", ) print( " Lists of energies (in GHz relative to ionisation) saved in %s" % filename ) filename = fileBase + "_highlight." + exportFormat np.savetxt( filename, self.highlight, fmt="%.18e", delimiter=", ", newline="\n", header=( commonHeader + " - - - Highlight value (rel.units) - - -\n" + headerDetails ), comments="# ", ) print(" Highlight values saved in %s" % filename) print("... data export finished!") else: raise ValueError("Unsupported export format (.%s)." % format)
[docs] def plotLevelDiagram( self, units="cm", highlightState=True, progressOutput=False, debugOutput=False, highlightColour="red", addToExistingPlot=False, ): r""" Makes a plot of a stark map of energy levels To save this plot, see :obj:`savePlot`. To print this plot see :obj:`showPlot`. Pointers (handles) towards matplotlib figure and axis used are saved in :obj:`fig` and :obj:`ax` variables respectively. Args: units (:obj:`char`,optional): possible values {'*cm*','GHz','eV'}; [case insensitive] if the string contains 'cm' (default) Stark diagram will be plotted in energy units cm :math:`{}^{-1}`; if value is 'GHz', Stark diagram will be plotted as energy :math:`/h` in units of GHz; if the value is 'eV', Stark diagram will be plotted as energy in units eV. highlightState (:obj:`bool`, optional): False by default. If True, scatter plot colour map will map in red amount of original state for the given eigenState progressOutput (:obj:`bool`, optional): if True prints the progress of calculation; Set to False by default. debugOutput (:obj:`bool`, optional): if True prints additional information usefull for debuging. Set to False by default. addToExistingPlot (:obj:`bool`, optional): if True adds points to existing old plot. Note that then interactive plotting doesn't work. False by default. """ rvb = LinearSegmentedColormap.from_list( "mymap", ["0.9", highlightColour, "black"] ) # for back-compatibilirt with versions <= 3.0.11 # where units were chosen as integer 1 or 2 if not isinstance(units, str): units = ["ev", "ghz", "cm"][units - 1] if units.lower() == "ev": self.units = "eV" self.scaleFactor = 1e9 * C_h / C_e Elabel = "" elif units.lower() == "ghz": self.units = "GHz" self.scaleFactor = 1 Elabel = "/h" elif "cm" in units.lower(): self.units = "cm$^{-1}$" self.scaleFactor = 1e9 / (C_c * 100) Elabel = "/(h c)" self.addToExistingPlot = addToExistingPlot if progressOutput: print("plotting...") originalState = self.basisStates[self.indexOfCoupledState] n = originalState[0] l = originalState[1] j = originalState[2] existingPlot = False if self.fig == 0 or not addToExistingPlot: if self.fig != 0: plt.close() self.fig, self.ax = plt.subplots(1, 1, figsize=(11.0, 5)) else: existingPlot = True eFieldList = [] y = [] yState = [] for br in xrange(len(self.y)): for i in xrange(len(self.y[br])): eFieldList.append(self.eFieldList[br]) y.append(self.y[br][i]) yState.append(self.highlight[br][i]) yState = np.array(yState) sortOrder = yState.argsort(kind="heapsort") eFieldList = np.array(eFieldList) y = np.array(y) eFieldList = eFieldList[sortOrder] y = y[sortOrder] yState = yState[sortOrder] if not highlightState: self.ax.scatter( eFieldList / 100.0, y * self.scaleFactor, s=1, color="k", picker=5, ) else: cm = rvb cNorm = matplotlib.colors.Normalize(vmin=0.0, vmax=1.0) self.ax.scatter( eFieldList / 100, y * self.scaleFactor, c=yState, s=5, norm=cNorm, cmap=cm, lw=0, picker=5, ) if not existingPlot: cax = self.fig.add_axes([0.91, 0.1, 0.02, 0.8]) cb = matplotlib.colorbar.ColorbarBase(cax, cmap=cm, norm=cNorm) if self.drivingFromState[0] < 0.1: cb.set_label( r"$|\langle %s | \mu \rangle |^2$" % printStateStringLatex(n, l, j, s=self.s) ) else: cb.set_label(r"$( \Omega_\mu | \Omega )^2$") self.ax.set_xlabel("Electric field (V/cm)") eV2GHz = C_e / C_h * 1e-9 halfY = 300 # GHz, half Y range upperY = ( self.atom.getEnergy(n, l, j, s=self.s) * eV2GHz + halfY ) * self.scaleFactor lowerY = ( self.atom.getEnergy(n, l, j, s=self.s) * eV2GHz - halfY ) * self.scaleFactor self.ax.set_ylabel(r"State energy, $E%s$ (%s)" % (Elabel, self.units)) self.ax.set_ylim(lowerY, upperY) ## self.ax.set_xlim(min(eFieldList) / 100.0, max(eFieldList) / 100.0) return 0
[docs] def savePlot(self, filename="StarkMap.pdf"): """ Saves plot made by :obj:`plotLevelDiagram` Args: filename (:obj:`str`, optional): file location where the plot should be saved """ if self.fig != 0: self.fig.savefig(filename, bbox_inches="tight") else: print("Error while saving a plot: nothing is plotted yet") return 0
[docs] def showPlot(self, interactive=True): """ Shows plot made by :obj:`plotLevelDiagram` """ if self.fig != 0: if interactive: if self.addToExistingPlot: print( "NOTE: Interactive plotting doesn't work with" " addToExistingPlot option set to True" "\nPlease turn off this option in plotLevelDiagram.\n" ) else: self.ax.set_title("Click on state to see state composition") self.clickedPoint = 0 self.fig.canvas.draw() self.fig.canvas.mpl_connect("pick_event", self._onPick) plt.show() else: print("Error while showing a plot: nothing is plotted yet") return 0
def _onPick(self, event): if isinstance(event.artist, matplotlib.collections.PathCollection): scaleFactor = self.scaleFactor x = event.mouseevent.xdata * 100.0 y = event.mouseevent.ydata / scaleFactor i = np.searchsorted(self.eFieldList, x) if i == len(self.eFieldList): i -= 1 if (i > 0) and ( abs(self.eFieldList[i - 1] - x) < abs(self.eFieldList[i] - x) ): i -= 1 j = 0 for jj in xrange(len(self.y[i])): if abs(self.y[i][jj] - y) < abs(self.y[i][j] - y): j = jj # now choose the most higlighted state in this area distance = abs(self.y[i][j] - y) * 1.5 for jj in xrange(len(self.y[i])): if abs(self.y[i][jj] - y) < distance and ( abs(self.highlight[i][jj]) > abs(self.highlight[i][j]) ): j = jj if self.clickedPoint != 0: self.clickedPoint.remove() (self.clickedPoint,) = self.ax.plot( [self.eFieldList[i] / 100.0], [self.y[i][j] * scaleFactor], "bs", linewidth=0, zorder=3, ) self.ax.set_title( ("[%s] = " % self.atom.elementName) + self._stateComposition(self.composition[i][j]) + (" Colourbar value = %.2f" % self.highlight[i][j]), fontsize=11, ) event.canvas.draw() def _stateComposition(self, stateVector): i = 0 totalContribution = 0 value = "$" while (i < len(stateVector)) and (totalContribution < 0.95): if i != 0 and stateVector[i][0] > 0: value += "+" value = ( value + ("%.2f" % stateVector[i][0]) + self._addState(*self.basisStates[stateVector[i][1]]) ) totalContribution += abs(stateVector[i][0]) ** 2 i += 1 if totalContribution < 0.999: value += "+\\ldots" return value + "$" def _stateComposition2( self, stateVector, upTo=300, totalContributionMax=0.999 ): contribution = np.absolute(stateVector) order = np.argsort(contribution, kind="heapsort") index = -1 totalContribution = 0 mainStates = [] # [state Value, state index] if upTo == -1: for index in range(len(order)): i = order[-index - 1] mainStates.append([stateVector[i], i]) else: while (index > -upTo) and ( totalContribution < totalContributionMax ): i = order[index] mainStates.append([stateVector[i], i]) totalContribution += contribution[i] ** 2 index -= 1 return mainStates def _addState(self, n1, l1, j1, mj1): if abs(self.s - 0.5) < 0.1: # we have Alkali Atoms return "|%s m_j=%d/2\\rangle" % ( printStateStringLatex(n1, l1, j1), round(2 * mj1), ) else: # we have singlets or triplets states of divalent atoms return "|%s m_j=%d\\rangle" % ( printStateStringLatex(n1, l1, j1, s=self.s), round(mj1), )
[docs] def getPolarizability( self, maxField=1.0e10, showPlot=False, debugOutput=False, minStateContribution=0.0, ): r""" Returns the polarizability of the state (set during the initalization process). Fits offset of the energy level of the state to :math:`\frac{1}{2} \alpha_0 E^2`, where :math:`E` is the applied static electric field, and returns fitted value :math:`\alpha_0` Parameters: maxField (:obj:`float`, optional): maximum field (in V/m) to be used for fitting the polarizability. By default, max field is very large, so it will use eigenvalues calculated in the whole range. showPlot (:obj:`bool`, optional): shows plot of calculated eigenValues of the given state (dots), and the fit (solid line) for extracting polarizability debugOutput (:obj:`bool`, optional): if True prints additional information usefull for debuging. Set to false by default. Returns: float: scalar polarizability in units of MHz cm :math:`^2` / V \ :math:`^2` """ if self.drivingFromState[0] != 0: raise Exception( "Program can only find Polarizability of the original " + "state if you highlight original state. You can do so by NOT " + "specifying drivingFromState in diagonalise function." ) eFieldList = self.eFieldList yState = self.highlight y = self.y originalState = self.basisStates[self.indexOfCoupledState] n = originalState[0] l = originalState[1] j = originalState[2] energyOfOriginalState = ( self.atom.getEnergy(n, l, j, s=self.s) * C_e / C_h * 1e-9 ) # in GHz if debugOutput: print("finding original state for each electric field value") stopFitIndex = 0 while ( stopFitIndex < len(eFieldList) - 1 and eFieldList[stopFitIndex] < maxField ): stopFitIndex += 1 xOriginalState = [] yOriginalState = [] for ii in xrange(stopFitIndex): maxPortion = 0.0 yval = 0.0 jj = 0 for jj in xrange(len(y[ii])): if yState[ii][jj] > maxPortion: maxPortion = yState[ii][jj] yval = y[ii][jj] # measure state energy relative to the original state if minStateContribution < maxPortion: xOriginalState.append(eFieldList[ii]) yOriginalState.append(yval - energyOfOriginalState) xOriginalState = np.array(xOriginalState) / 100.0 # converts to V/cm yOriginalState = np.array(yOriginalState) # in GHz # in GHz uppery = 5.0 lowery = -5.0 if debugOutput: print("found ", len(xOriginalState)) if showPlot: self.fig, self.ax = plt.subplots(1, 1, figsize=(6.5, 3)) self.ax.scatter(xOriginalState, yOriginalState, s=2, color="k") self.ax.set_xlabel("E field (V/cm)") self.ax.set_ylim(lowery, uppery) self.ax.set_ylabel(r"Energy/$h$ (GHz)") self.ax.set_xlim(xOriginalState[0], xOriginalState[-1]) def polarizabilityFit(eField, offset, alpha): return offset - 0.5 * alpha * eField**2 try: popt, pcov = curve_fit( polarizabilityFit, xOriginalState, yOriginalState, [0, 0] ) except Exception as ex: print(ex) print( "\nERROR: fitting energy levels for extracting polarizability\ of the state failed. Please check the range of electric \ fields where you are trying to fit polarizability and ensure\ that there is only one state with continuous energy change\ that has dominant contribution of the initial state.\n\n" ) return 0 if debugOutput: print( "Scalar polarizability = ", popt[1] * 1.0e3, " MHz cm^2 / V^2 " ) y_fit = [] for val in xOriginalState: y_fit.append(polarizabilityFit(val, popt[0], popt[1])) y_fit = np.array(y_fit) if showPlot: self.ax.plot(xOriginalState, y_fit, "r--") self.ax.legend( ("fitted model function", "calculated energy level"), loc=1, fontsize=10, ) self.ax.set_ylim(min(yOriginalState), max(yOriginalState)) plt.show() self.fitX = xOriginalState self.fitY = yOriginalState self.fittedCurveY = y_fit return popt[1] * 1.0e3 # returned value is in MHz cm^2 / V^2
[docs] def getState( self, state, electricField, minN, maxN, maxL, accountForAmplitude=0.95, debugOutput=False, ): r""" Returns basis states and coefficients that make up for a given electric field the eigenstate with largest contribution of the original state. Args: state (array): target basis state in format :math:`[n,\ell,j,m_j]` corresponding to the state whose composition we want to track as we apply the electric field electricField (float): applied DC electric field in units of V/m. minN (int): minimal principal quantum number to be taken for calculation of the Stark mixing maxN (int): maximal principal quantum nunber to be take for calculation of the Start mixing maxL (int): maximal orbital angular momentum of states that should be taken in calculation of the Stark mixing accountForAmplitude (float): optinal, relative amplitude of state that should be reached with the subset of the eigen states returned. The returned eigen states will be sorted in the declining relative contribution to the final eigen state, and once total accounted amplitude of the state reaches 0.95, further output of additional small contribution of the other basis states to the final states will be supressed. Default value of 0.95 will force output until basis state accounts for 95\% of the state amplitude. debugOutput (bool): optional, prints additional debug information if True. Default False. Returns: **array of states** in format [[n1, l1, j1, mj1], ...] and **array of complex coefficients** in format [c1, c2, ...] corresponding the projections of the eigenstate (thas has largest contribution of the original state in the given electric field) on the basis states, and **energy** of the found state in (eV) """ self.defineBasis( state[0], state[1], state[2], state[3], minN, maxN, maxL ) m = self.mat1 + self.mat2 * electricField ev, egvector = eigh(m) # find which state in the electric field has strongest contribution # of the requested state? maxOverlap = 0 eigenvectorIndex = 0 for i in range(len(ev)): if abs(egvector[self.indexOfCoupledState, i]) ** 2 > maxOverlap: maxOverlap = abs(egvector[self.indexOfCoupledState, i]) ** 2 eigenvectorIndex = i energy = ev[eigenvectorIndex] * 1e9 * C_h / C_e if debugOutput: print("Max overlap = %.3f" % maxOverlap) print( "Eigen energy (state index %d) = %.2f eV" % (eigenvectorIndex, energy) ) contributions = egvector[:, eigenvectorIndex] sortedContributions = np.argsort(abs(contributions)) if debugOutput: print("Maximum contributions to this state") for i in range(4): index = sortedContributions[-i - 1] print(contributions[index]) print(self.basisStates[index]) print("===========\n") i = 0 coef = [] contributingStates = [] while accountForAmplitude > 0 and i < len(self.basisStates): index = sortedContributions[-i - 1] coef.append(contributions[index]) accountForAmplitude -= abs(coef[-1]) ** 2 contributingStates.append(self.basisStates[index]) i += 1 return contributingStates, coef, energy
# ================= Level plots, decays, cascades etc =======================
[docs] class LevelPlot: """ Single atom level plots and decays (a Grotrian diagram, or term diagram) For an example see `Rydberg energy levels example snippet`_. .. _`Rydberg energy levels example snippet`: ./Rydberg_atoms_a_primer_notebook.html#Rydberg-Atom-Energy-Levels Args: atom (:obj:`arc.alkali_atom_functions.AlkaliAtom` or :obj:`arc.divalent_atom_functions.DivalentAtom`): ={ :obj:`arc.alkali_atom_data.Lithium6`, :obj:`arc.alkali_atom_data.Lithium7`, :obj:`arc.alkali_atom_data.Sodium`, :obj:`arc.alkali_atom_data.Potassium39`, :obj:`arc.alkali_atom_data.Potassium40`, :obj:`arc.alkali_atom_data.Potassium41`, :obj:`arc.alkali_atom_data.Rubidium85`, :obj:`arc.alkali_atom_data.Rubidium87`, :obj:`arc.alkali_atom_data.Caesium`, :obj:`arc.divalent_atom_data.Strontium88`, :obj:`arc.divalent_atom_data.Calcium40` :obj:`arc.divalent_atom_data.Ytterbium174` } Alkali atom type whose levels we want to examine """ def __init__(self, atomType): self.atom = atomType self.nFrom = 0 self.nTo = 0 self.lFrom = 0 self.lTo = 0 self.sList = [] self.listX = [] self.listY = [] # list of energies self.levelLabel = [] self.fig = 0 self.ax = 0 self.width = 0.2 self.state1 = [0, 0, 0] self.state2 = [0, -1, 0] self.transitionMatrix = [] self.populations = [] self.transitionMatrixWavelength3 = [] # characterization of the graph self.spectraX = [] self.spectraY = [] self.spectraLine = []
[docs] def makeLevels(self, nFrom, nTo, lFrom, lTo, sList=[0.5]): """ Constructs energy level diagram in a given range Args: nFrom (int): minimal principal quantum number of the states we are interested in nTo (int): maximal principal quantum number of the states we are interested in lFrom (int): minimal orbital angular momentum of the states we are interested in lTo (int): maximal orbital angular momentum of the states we are interested in sList (float): optional, spin angular momentum. Default value of [0.5] corresponds to Alkali atoms. For Alkaline Earth it has to be specified. For divalent atoms one can plot either one spin state by setting for example `sList=[0]``, or both spin states `sList=[0,1]`` """ if ( issubclass(type(self.atom), DivalentAtom) and abs(sList[0] - 0.5) < 0.1 ): raise ValueError( "For divalent atoms requested spin state(s) have " "to be explicitly specified e.g. sList=[0] or " "sList=[0,1]" ) # save local copy of the space restrictions self.nFrom = nFrom self.nTo = nTo self.lFrom = lFrom self.lTo = lTo self.sList = sList # find all the levels within this space restrictions xPositionOffset = 0 for s in sList: n = max(self.nFrom, self.atom.groundStateN) while n <= nTo: l = lFrom if l == 0 and s == 1 and n == self.atom.groundStateN: # for ground state S state, there is only singlet l += 1 while l <= min(lTo, n - 1): for j in np.linspace(l - s, l + s, round(2 * s + 1)): if j > -0.1: self.listX.append(l - lFrom + xPositionOffset) self.listY.append(self.atom.getEnergy(n, l, j, s=s)) self.levelLabel.append([n, l, j, s]) l = l + 1 n += 1 # if user requested principal quantum nuber below theself.listX_l.append(l) # ground state principal quantum number # add those L states that are higher in energy then the ground state for state in self.atom.extraLevels: if ( state[1] <= lTo and state[0] >= self.nFrom and (len(state) == 3 or state[3] == s) ): # last line means: either is Alkali, when we don't need to # check the spin, or it's divalent, when we do need to check # the spin self.listX.append(state[1] - lFrom + xPositionOffset) self.listY.append( self.atom.getEnergy(state[0], state[1], state[2], s=s) ) self.levelLabel.append([state[0], state[1], state[2], s]) xPositionOffset += lTo + 1 - lFrom
def makeTransitionMatrix( self, environmentTemperature=0.0, printDecays=True ): self.transitionMatrix = [] for i in xrange(len(self.levelLabel)): state1 = self.levelLabel[i] transitionVector = [] # decay of the stay decay = 0.0 for state2 in self.levelLabel: dipoleAllowed = (abs(state1[1] - state2[1]) == 1) and ( abs(state1[2] - state2[2]) <= 1.01 ) if dipoleAllowed: # decay to this state rate = self.atom.getTransitionRate( state2[0], state2[1], state2[2], state1[0], state1[1], state1[2], temperature=environmentTemperature, ) transitionVector.append(rate) # decay from this state rate = self.atom.getTransitionRate( state1[0], state1[1], state1[2], state2[0], state2[1], state2[2], temperature=environmentTemperature, ) decay = decay - rate else: transitionVector.append(0.0) transitionVector[i] = decay if printDecays: print("Decay time of ") printStateString(state1[0], state1[1], state1[2]) if decay < -1e-20: print("\t is\t", -1.0e9 / decay, " ns") self.transitionMatrix.append(transitionVector) np.array(self.transitionMatrix) self.transitionMatrix = np.transpose(self.transitionMatrix) def drawSpectra(self): self.fig, self.ax = plt.subplots(1, 1, figsize=(16, 5)) lineWavelength = [] lineStrength = [] lineName = [] i = 0 while i < len(self.levelLabel): j = 0 while j < len(self.levelLabel): if i != j: wavelength = self.atom.getTransitionWavelength( self.levelLabel[i][0], self.levelLabel[i][1], self.levelLabel[i][2], self.levelLabel[j][0], self.levelLabel[j][1], self.levelLabel[j][2], ) intensity = self.atom.getTransitionRate( self.levelLabel[i][0], self.levelLabel[i][1], self.levelLabel[i][2], self.levelLabel[j][0], self.levelLabel[j][1], self.levelLabel[j][2], ) lineWavelength.append(abs(wavelength) * 1.0e9) lineStrength.append(abs(intensity)) lineName.append( printStateString( self.levelLabel[i][0], self.levelLabel[i][1], self.levelLabel[i][2], ) + " -> " + printStateString( self.levelLabel[j][0], self.levelLabel[j][1], self.levelLabel[j][2], ) ) j = j + 1 i = i + 1 self.spectraX = np.copy(lineWavelength) self.spectraY = np.copy(lineStrength) self.spectraLine = np.copy(lineName) def drawSpectraConvoluted( self, lowerWavelength, higherWavelength, points, gamma ): wavelengths = np.linspace(lowerWavelength, higherWavelength, points) spectra = np.zeros(points) i = 0 while i < len(wavelengths): value = 0 j = 0 while j < len(self.spectraX): value = value + self.spectraY[j] * gamma / ( (self.spectraX[j] - wavelengths[i]) ** 2 + gamma**2 ) j = j + 1 spectra[i] = value i = i + 1 self.ax.plot(wavelengths, spectra, "g-") def showSpectra(self, saveInFile="", showTransitionPoints=True): if showTransitionPoints: self.ax.plot(self.spectraX, self.spectraY, "ro", picker=5) self.ax.set_xlabel("Wavelength (nm)") self.ax.set_ylabel("Intensity (arb.un)") self.fig.subplots_adjust(right=0.95, left=0.1) # self.ax.set_xlim(300,600) self.fig.canvas.mpl_connect("pick_event", self.onpick3) if saveInFile != "": self.fig.savefig(saveInFile) plt.show()
[docs] def drawLevels(self, units="eV"): r""" Draws a level diagram plot Arg: units (:obj:`char`,optional): possible values {'eV','*cm*','GHz'}; [case insensitive] if the value is 'eV' (default), Stark diagram will be plotted as energy in units eV; if the string contains 'cm' Stark diagram will be plotted in energy units cm :math:`{}^{-1}`; if value is 'GHz', Stark diagram will be plotted as energy :math:`/h` in units of GHz; """ self.fig, self.ax = plt.subplots(1, 1, figsize=(9.0, 11.5)) if units.lower() == "ev": self.scaleFactor = 1 self.units = "eV" elif units.lower() == "ghz": self.scaleFactor = C_e / C_h * 1e-9 self.units = "GHz" elif "cm" in units.lower(): self.scaleFactor = C_e / (C_h * C_c * 100) self.units = "cm$^{-1}$" i = 0 while i < len(self.listX): self.ax.plot( [self.listX[i] - self.width, self.listX[i] + self.width], [ self.listY[i] * self.scaleFactor, self.listY[i] * self.scaleFactor, ], "b-", picker=True, ) if i < len(self.populations) and (self.populations[i] > 1e-3): self.ax.plot( [self.listX[i]], [self.listY[i] * self.scaleFactor], "ro", alpha=self.populations[i], ) i = i + 1 # Y AXIS self.listX = np.array(self.listX) self.ax.set_ylabel("Energy (%s)" % self.units) self.ax.set_xlim(-0.5 + np.min(self.listX), np.max(self.listX) + 0.5) # X AXIS majorLocator = MultipleLocator(1) self.ax.xaxis.set_major_locator(majorLocator) tickNames = [] for s in self.sList: sNumber = round(2 * s + 1) for l in xrange(self.lFrom, self.lTo + 1): tickNames.append("$^%d %s$" % (sNumber, printStateLetter(l))) tickNum = len(tickNames) self.fig.canvas.draw() self.ax.set_xticks(np.arange(tickNum)) self.ax.set_xticklabels(tickNames) self.ax.set_xlim(-0.5 + np.min(self.listX), np.max(self.listX) + 0.5) # TITLE self.ax.set_title( "%s: $n \\in [%d,%d]$" % (self.atom.elementName, self.nFrom, self.nTo) )
[docs] def showPlot(self): """ Shows a level diagram plot """ self.fig.canvas.mpl_connect("pick_event", self.onpick2) self.state1[0] = -1 # initialise for picking plt.show()
def findState(self, x, y): y /= self.scaleFactor distance = 100000000.0 state = [0, 0, 0] i = 0 while i < len(self.listX): dx = self.listX[i] - x dy = self.listY[i] - y dist = sqrt(dx * dx + dy * dy) if dist < distance: distance = dist state = self.levelLabel[i] i = i + 1 return state def findStateNo(self, state): # returns no of the given state in the basis i = 0 while i < len(self.levelLabel): if ( (self.levelLabel[i][0] == state[0]) and (self.levelLabel[i][1] == state[1]) and (abs(self.levelLabel[i][2] - state[2]) < 0.01) ): return i i = i + 1 print("Error: requested state ") print(state) print("could not be found!") return -1 def findLine(self, x, y): distance = 1.0e40 line = "" i = 0 while i < len(self.spectraLine): dx = self.spectraX[i] - x dy = self.spectraY[i] - y dist = sqrt(dx * dx + dy * dy) if dist < distance: distance = dist line = self.spectraLine[i] i = i + 1 return line def onpick2(self, event): if isinstance(event.artist, matplotlib.lines.Line2D): thisline = event.artist xdata = thisline.get_xdata() ydata = thisline.get_ydata() state = self.findState((xdata[0] + xdata[0]) / 2.0, ydata[0]) if self.state1[0] == -1: if state[1] != self.state2[1] or state[0] != self.state2[0]: self.state1 = state self.ax.set_title( r"$%s \rightarrow$ " % ( printStateStringLatex( state[0], state[1], state[2], s=state[3] ) ) ) self.state2 = [-1, -1, -1] else: title = "" if (state[0] != self.state1[0]) or (state[1] != self.state1[1]): title = r"$ %s \rightarrow %s $ " % ( printStateStringLatex( self.state1[0], self.state1[1], self.state1[2], s=self.state1[3], ), printStateStringLatex( state[0], state[1], state[2], s=state[3] ), ) transitionEnergy = ( self.atom.getTransitionFrequency( self.state1[0], self.state1[1], self.state1[2], state[0], state[1], state[2], s=self.state1[3], s2=state[3], ) * C_h / C_e ) # in eV title = title + ( " %sm (%s%s)" % ( formatNumberSI( self.atom.getTransitionWavelength( self.state1[0], self.state1[1], self.state1[2], state[0], state[1], state[2], s=self.state1[3], s2=state[3], ) ), formatNumberSI(transitionEnergy * self.scaleFactor), self.units, ) ) self.ax.set_title(title) self.state1 = [-1, 0, 0] self.state2 = state event.canvas.draw() def onpick3(self, event): if isinstance(event.artist, matplotlib.lines.Line2D): thisline = event.artist xdata = thisline.get_xdata() ydata = thisline.get_ydata() ind = event.ind print(ind[0]) line = self.findLine(xdata[ind][0], ydata[ind][0]) self.ax.set_title(line) event.canvas.draw()
[docs] class AtomSurfaceVdW: r""" Calculates atom-surface Van der Waals interaction. Energy of atom state :math:`|i\rangle` at distance :math:`z` from the surface of material is offseted in energy by :math:`V_{\rm VdW}` at small distances :math:`z\ll\rm{min}(\lambda_{i,j})` , where :math:`\lambda_{i,j}` are the wavelengths from atom state :math:`|i \rangle` to all strongly-coupled states :math:`j` , due to (unretarded) atom-surface interaction, also called Van der Waals interaction. The interaction potential can be expressed as :math:`V_{\rm VdW} = - \frac{C_3}{z^3}` This class calculates :math:`C_3` for individual states :math:`|i\rangle`. See example `atom-surface calculation snippet`_. .. _`atom-surface calculation snippet`: ./ARC_3_0_introduction.html#Atom-surface-van-der-Waals-interactions-(C3-calculation) Args: atom (:obj:`AlkaliAtom` or :obj:`DivalentAtom`): specified Alkali or Alkaline Earth atom whose interaction with surface we want to explore material (from :obj:`arc.materials`): specified surface material Note: To find frequecy shift of a transition :math:`|\rm a \rangle\rightarrow |\rm b \rangle`, one needs to calculate difference in :math:`C_3` coefficients obtained for the two states :math:`|\rm a\rangle` and :math:`|\rm b\rangle` respectively. See example TODO (TO-DO) """ def __init__(self, atom, surfaceMaterial=None): UsedModulesARC.arc3_0_methods = True self.atom = atom if surfaceMaterial is None: print( "NOTE: No surface material specified. " "Assuming perfect mirror." ) self.surfaceMaterial = surfaceMaterial
[docs] def getC3contribution(self, n1, l1, j1, n2, l2, j2, s=0.5): r""" Contribution to :math:`C_3` of :math:`|n_1, \ell_1, j_1\rangle` state due to dipole coupling to :math:`|n_2, \ell_2, j_2\rangle` state. Calculates :math:`\frac{1}{4\pi\varepsilon_0}\ \frac{ n(\omega_{\rm ab})^2 - 1}{ n(\omega_{\rm ab})^2 + 1}\ \frac{ \left| \langle a| D_x | b \rangle \right|^2 \ + \left| \langle a | D_y | b \rangle \right|^2 + \ 2 \cdot \left|\langle a |D_z| b \rangle \right|^2}{16}` where :math:`|{\rm a}\rangle \equiv |n_1, \ell_1, j_1\rangle` , :math:`|{\rm b}\rangle \equiv |n_2, \ell_2, j_2\rangle`, :math:`\mathbf{D} \equiv e \cdot \mathbf{r} \ \equiv \hat{x} D_x + \hat{y} D_y\ + \hat{z} D_z` is atomic dipole operator and :math:`n(\omega_{\rm ab})` is refractive index of the considered surface at transition frequency :math:`\omega_{\rm ab}` . Args: n1 (int): principal quantum number of state 1 l1 (int): orbital angular momentum of state 1 j1 (float): total angular momentum of state 1 n2 (int): principal quantum number od state 2 l2 (int): orbital angular momentum of state 2 j2 (float): total angular momentum of state 2 s (float): optional, spin angular momentum of states. Default value of 0.5 is correct for AlkaliAtoms. For DivalentAtom it has to be explicitly stated Returns: float, float, float: contribution to VdW coefficient :math:`C_3` ,\ estimated error :math:`\delta C_3` \ (in units of :math:`{\rm J}\cdot{\rm m}^3`), and refractive \ index :math:`n` of the surface material for the given \ transition. Warning: This is just contribution of one transition to the level shift of a particular state. To calculate total level shift, check :obj:`AtomSurfaceVdW.getStateC3` """ result = 0.0 error = 0.0 hasLiteratureValue, dme, info = self.atom.getLiteratureDME( n1, l1, j1, n2, l2, j2, s=0.5 ) if hasLiteratureValue: dme_reduced_J = self.atom.getReducedMatrixElementJ( n1, l1, j1, n2, l2, j2, s=0.5 ) relativeError = abs(info[1] / dme_reduced_J) else: relativeError = ( 0.05 # 5 percent for calculated values (note: estimate only!) ) # sum over mj1 for mj1 in np.linspace(-j1, j1, round(2 * j1 + 1)): # calculate sum_mj2 |<j1,mj1|Dx|j2,mj2>|^2 + |<j1,mj1|Dy|j2,mj2>|^2 + 2* |<j1,mj1|Dz|j2,mj2>|^2 # which is equal to (check!) |<j1,mj1|D+|j2,mj2>|^2 + |<j1,mj1|D-|j2,mj2>|^2 + 2* |<j1,mj1|Dz|j2,mj2>|^2 for mj2 in np.linspace(-j2, j2, round(2 * j2 + 1)): for q in [-1, +1]: result += ( abs( self.atom.getDipoleMatrixElement( n1, l1, j1, mj1, n2, l2, j2, mj2, q, s=s ) * C_e * physical_constants["Bohr radius"][0] ) ** 2 ) error += ( 2 * abs( self.atom.getDipoleMatrixElement( n1, l1, j1, mj1, n2, l2, j2, mj2, q, s=s ) * C_e * physical_constants["Bohr radius"][0] ) ** 2 * relativeError ) # for q = 0 q = 0 result += ( 2 * abs( self.atom.getDipoleMatrixElement( n1, l1, j1, mj1, n2, l2, j2, mj2, q ) * C_e * physical_constants["Bohr radius"][0] ) ** 2 ) error += ( 2 * abs( self.atom.getDipoleMatrixElement( n1, l1, j1, mj1, n2, l2, j2, mj2, q ) * C_e * physical_constants["Bohr radius"][0] ) ** 2 * relativeError ) materialFactor = 1.0 n = 10000 # effectively infinite refractive index would correspond to perfect # reflector (perfect mirror) if self.surfaceMaterial is not None: wavelength = ( np.abs( self.atom.getTransitionWavelength( n1, l1, j1, n2, l2, j2, s=s, s2=s ) ) * 1e6 ) # in mum n = self.surfaceMaterial.getN(vacuumWavelength=wavelength) materialFactor = (n**2 - 1.0) / (n**2 + 1.0) # include factor of 16 result = result / (2 * j1 + 1) / 16 error = error / (2 * j1 + 1) / 16 C3 = materialFactor * 1 / (4.0 * pi * epsilon_0) * result error = materialFactor * 1 / (4.0 * pi * epsilon_0) * error return C3, error, n # C3 and error in units of J m^3
[docs] def getStateC3(self, n, l, j, coupledStatesList, s=0.5, debugOutput=False): r""" Van der Waals atom-surface interaction coefficient for a given state (:math:`C_3` in units of :math:`\mathrm{J}\cdot\mathrm{m}^3` ) Args: n (int): principal quantum number of the state l (int): orbital angular momentum of the state j (int): total angular momentum of state coupledStatesList (array): array of states that are strongly dipole-coupled to the initial state, whose contribution to :math:`C_3` will be take into account. Format `[[n1,l1,j1],...]` s (float, optional): total spin angular momentum for the considered state. Default value of 0.5 is correct for `AlkaliAtoms`, but it has to be explicitly specifiied for `DivalentAtom`. debugOutput (bool, optional): prints additional output information, False by default. Returns: float, float: :math:`C_3` (in units of :math:`{\rm J}\cdot {\rm m}^3` ), estimated error :math:`\delta C_3` """ if debugOutput: print( "%s ->\tC3 contr. (kHz mum^3) \tlambda (mum)\tn" % (printStateString(n, l, j, s=s)) ) totalShift = 0 sumSqError = 0 for state in coupledStatesList: c3, err, refIndex = self.getC3contribution( n, l, j, state[0], state[1], state[2], s=s ) if debugOutput: print( "-> %s\t%.3f +- %.3f \t%.3f\t\t%.3f\n" % ( printStateString(state[0], state[1], state[2], s=s), c3 / C_h * (1e6) ** 3 * 1e-3, err / C_h * (1e6) ** 3 * 1e-3, self.atom.getTransitionWavelength( n, l, j, state[0], state[1], state[2], s=s, s2=s ) * 1e6, refIndex, ) ) totalShift += c3 sumSqError += err**2 error = np.sqrt(sumSqError) if debugOutput: print( "= = = = = = \tTotal shift of %s\t= %.3f+-%.4f kHz mum^3\n" % ( printStateString(n, l, j, s=s), totalShift / C_h * (1e6) ** 3 * 1e-3, error / C_h * (1e6) ** 3 * 1e-3, ) ) return totalShift, error # in J m^3
[docs] class OpticalLattice1D: r""" Atom properties in optical lattices in 1D. See example `optical lattice calculations snippet`_. .. _`optical lattice calculations snippet`: ./ARC_3_0_introduction.html#Optical-lattice-calculations-(Bloch-bands,-Wannier-states...) Args: atom: one of AlkaliAtom or DivalentAtom trapWavenegth (float): wavelength of trapping laser light (in units of m) """ energy = [] """ energy of states obtained by :obj:`OpticalLattice1D.diagonalise` method in format `[[energies for quasimomentum1 ], [energies for quasimomentum2 ], ...]` """ quasimomentum = [] """ list of quzimomentum for which the energies of states was calculated by :obj:`OpticalLattice1D.diagonalise` method in format `[quasimomentum1, quasimomentum2, ...]` """ savedBlochBand = [] """ list of saved eigen energy state compositions for each of the Calculated quasimomentums for the selected index of the Bloch band in :obj:`OpticalLattice1D.diagonalise` method in format `[[eigen state decomposition for quasimomentum 1], [eigen state decomposition for quasimomentum 2], ...]` """ trapPotentialDepth = 0 """ save slattice trap potential depth for which calculation :obj:`OpticalLattice1D.diagonalise` was done """ def __init__(self, atom, trapWavenegth): UsedModulesARC.arc3_0_methods = True self.atom = atom self.trapWavenegth = trapWavenegth
[docs] def getRecoilEnergy(self): """ Recoil energy for atoms in given optical lattice Returns: float: recoil energy in units of J """ latticeConstant = self.trapWavenegth / 2 Er = C_h**2 / (8 * self.atom.mass * latticeConstant**2) return Er
[docs] def getTrappingFrequency(self, trapPotentialDepth): """ Atom's trapping frequecy for given trapth depth Args: trapPotentialDepth (float): lattice depth (in units of J) Returns: float: trapping frequency (in Hz) """ Er = self.getRecoilEnergy() return 2.0 * Er / hbar * np.sqrt(trapPotentialDepth / Er)
def _BlochFunction(self, x, stateVector, q, k=1.0): r""" Bloch wavefunctions Args: x (x): position (in units \2 pi/k, for default value of laser wavevector unit k=1, one full wavelength is 2\pi) stateVector: eigen vector obtained by diagonalisation of interaction Hamiltonian in a subspace given by the selected quasimomentum q (float): quasimomentum (in units of driving laser k) k (float): driving laser wavevector, define units for momentum and distance; if k==1 (default value), reciprocal lattice momentum is 2, and the full range of quasimomentum is from -1 to +1; one full wavelength is the 2\pi. Retruns: float: """ index = len(stateVector) // 2 + 2 # Align Bloch functions in phase angle = np.angle(stateVector[index]) sign = np.exp(-1j * angle) temp = 0 + 0j for l in np.arange(-self.lLimit, self.lLimit + 1, 1): temp += ( sign * stateVector[l + self.lLimit] * np.exp(1.0j * (2.0 * k * l + q) * x) ) return temp
[docs] def BlochWavefunction( self, trapPotentialDepth, quasimomentum, blochBandIndex ): r""" Bloch wavefunction as a **function** of 1D coordinate. Paraeters: trapPotentialDepth (float): (in units of recoil energy :obj:`OpticalLattice1D.getRecoilEnergy`) quasimomentum (float): (in units of 2 \pi / :obj:`OpticalLattice1D.trapWavenegth`; note that reciprocal lattice momentum in this units is 2, and that full range of quasimomentum is from -1 to +1) Returns: Bloch wavefunction as a **function** of coordinate (see call example below) Example: Returns Bloch wavefunction. Use as following:: trapPotentialDepth = 40 # units of recoil energy quasimomentum = 0 blochBandIndex = 0 # Bloch band lowest in energy is 0 wf = lattice.BlochWavefunction(trapPotentialDepth, quasimomentum, blochBandIndex) wf(x) # returns complex number corresponding to value of Bloch # wavefunction at point x (cooridnate given in units of # 1/k where k = 2 \pi / trapWavenegth ) # by default k=1, so one full wavelength is 2\pi """ temp1 = self.energy temp2 = self.quasimomentum temp3 = self.savedBlochBand self.diagonalise( trapPotentialDepth, [quasimomentum], saveBandIndex=blochBandIndex ) state = np.copy(self.savedBlochBand[0]) self.energy = temp1 self.quasimomenutm = temp2 self.savedBlochBand = temp3 return lambda x: self._BlochFunction(x, state, quasimomentum)
[docs] def defineBasis(self, lLimit=35): """ Define basis for Bloch band calculations Bloch states are calculated suming up all relevant states with momenta in range `[-lLimit * 4 * pi /trapWavenegth, +lLimit * 4 * pi /trapWavenegth]` Note that factor of 4 occurs since potential lattice period is twice the `trapWavelength` for standing wave. Args: lLimit (integer): Optional, defines maximal momentum to be taken for calculation of Bloch States as `lLimit * 4 * pi / trapWavenegth` . By default set to 35. """ self.lLimit = lLimit
def _getLatticeHamiltonian(self, q, Vlat): """ Lattice Hamiltonian Args: q (float): Vlat (float): lLimit (int): """ # assemble Hamiltonian hConstructor = [[], [], []] # [[values],[columnIndex],[rowIndex]] for l in np.arange(-self.lLimit, self.lLimit + 1, 1): # basis index exp(2*l*k*x) state has index lLimit+l column = self.lLimit + l if l - 1 >= -self.lLimit: hConstructor[0].append(-Vlat / 4.0) hConstructor[1].append(column) hConstructor[2].append(column - 1) if l + 1 <= self.lLimit: hConstructor[0].append(-Vlat / 4.0) hConstructor[1].append(column) hConstructor[2].append(column + 1) # diagonal term # with global energy offset (- Vlat / 2.) factored out hConstructor[0].append((2.0 * l + q) ** 2 + Vlat / 2.0) hConstructor[1].append(column) hConstructor[2].append(column) dimension = 2 * self.lLimit + 1 hamiltonianQ = csr_matrix( (hConstructor[0], (hConstructor[1], hConstructor[2])), shape=(dimension, dimension), ) return hamiltonianQ
[docs] def diagonalise( self, trapPotentialDepth, quasimomentumList, saveBandIndex=None ): r""" Calculates energy levels (Bloch bands) for given `quasimomentumList` Energy levels and their quasimomentum are saved in internal variables `energy` and `quasimomentum`. Energies are saved in units of recoil energy, and quasimomentum in units of The optional parameter `saveBandIndex` specifies index of the Bloch band for which eigenvectrors should be saved. If provided, eigenvectors for each value `quasimomentumList[i]` are saved in `savedBlochBand[i]`. Args: latticePotential (float): lattice depth formed by the standing wave of laser, with wavelength specified during initialisation of the lattice (in units of recoil energy). quasimomentumList (array): array of quasimomentum values for which energy levels will be calculated (in units of :math:`\hbar \cdot k`, where :math:`k` is trapping laser wavevector; since reciprocal lattice has twice the trapping laser wavevector due to standing wave, full range of quasimomentum is from -1 to +1) saveBandIndex (int): optional, default None. If provided, specifies for which Bloch band should the eignevectors be also saved. `saveBlochBand=0` corresponds to lowest energy band. """ self.energy = [] self.quasimomentum = quasimomentumList self.savedBlochBand = [] self.trapPotentialDepth = trapPotentialDepth for q in quasimomentumList: hamiltonianQ = self._getLatticeHamiltonian(q, trapPotentialDepth) ev, egvector = np.linalg.eig(hamiltonianQ.todense()) egvector = np.transpose(np.array(egvector)) orderInEnergy = np.argsort(ev) ev = ev[orderInEnergy] egvector = egvector[orderInEnergy] self.energy.append(ev) if saveBandIndex is not None: self.savedBlochBand.append(egvector[saveBandIndex])
[docs] def plotLevelDiagram(self): """ Plots energy level diagram (Bloch bands). Based on diagonalisation of the lattice potential, plots descrete eigen energy spectra obtained for each value of the quasimomentum used in :obj:`OpticalLattice1D.diagonalise` method. Returns: matploltib figure with a Bloch bands """ f = plt.figure(figsize=(6, 10)) ax = f.add_subplot(1, 1, 1) for i, energyLevels in enumerate(self.energy): ax.plot( [self.quasimomentum[i]] * len(energyLevels), energyLevels, ".", color="0.8", ) ax.set_xlabel(r"Quasimomentum, $q$ $(\hbar k)$") ax.set_ylabel(r"State energy, E ($E_{\rm r}$)") ax.set_ylim(-0.2, 50) ax.set_xlim(-1, 1) return f
[docs] def getWannierFunction(self, x, latticeIndex=0, k=1): r""" Gives value at cooridnate x of a Wannier function localized at given lattice index. Args: x (float): spatial coordinate (in units of :math:`2\pi/k` ; for default value of laser drivng wavevecto :math:`k=1` , one trappinWavelength is :math:`2\pi` ). Coordinate origin is at `latticeIndex=0` . latticeIndex (int): optional, lattice index at which the Wannier function is localised. By defualt 0. k (float): optional; laser driving wavevector, defines unit of length. Default value is 1, making one trapping laser wavelenth equal to :math:`2\pi` """ value = 0 localizedAt = 2.0 * pi / k * latticeIndex / 2.0 # last division by 2 is because lattice period is # 2 x smaleler then wavelenth of the driving laser for i in range(len(self.quasimomentum)): q = self.quasimomentum[i] value += np.exp(-1j * q * localizedAt) * self._BlochFunction( x, self.savedBlochBand[i], q, k=k ) return value
[docs] class DynamicPolarizability: """ Calculations of magic wavelengths and dynamic polarizability (scalar and tensor). Args: atom: alkali or alkaline element of choice n (int): principal quantum number of the selected stated l (int): orbital angular momentum of the selected state j (float): total angular momentum of selected state s (float): optional, spin state of the atom. Default value of 0.5 is correct for Alkali atoms, but it has to be explicitly specified for DivalentAtom. """ def __init__(self, atom, n, l, j, s=0.5): UsedModulesARC.arc3_0_methods = True self.atom = atom self.n = n self.l = l self.j = j self.s = s
[docs] def defineBasis(self, nMin, nMax): """ Defines basis for calculation of dynamic polarizability Args: nMin (int): minimal principal quantum number of states to be taken into account for calculation nMax (int): maxi,al principal quantum number of states to be taken into account for calculation """ self.nMin = nMin self.nMax = nMax self.basis = [] self.lifetimes = [] for n1 in np.arange( max(self.nMin, self.atom.groundStateN), self.nMax + 1 ): lmin = self.l - 1 if lmin < -0.1: lmin = self.l + 1 for l1 in range(lmin, min(self.l + 2, n1)): j1 = l1 - self.s if j1 < 0.1: j1 += 1 while j1 <= l1 + self.s + 0.1: if self.__isDipoleCoupled( self.n, self.l, self.j, n1, l1, j1 ): # print([n1, l1, j1, self.s]) self.basis.append([n1, l1, j1, self.s]) j1 += 1 for state in self.atom.extraLevels: if ( len(state) == 3 or abs(state[3] - self.s) < 0.1 ) and self.__isDipoleCoupled( self.n, self.l, self.j, state[0], state[1], state[2] ): self.basis.append(state)
def __isDipoleCoupled(self, n1, l1, j1, n2, l2, j2, s=0.5): if ( not ( abs(l1 - l2) != 1 and ( ( abs(j1 - 0.5) < 0.1 and abs(j2 - 0.5) < 0.1 ) # j = 1/2 and j'=1/2 forbidden or ( abs(j1) < 0.1 and abs(j2 - 1) < 0.1 ) # j = 0 and j'=1 forbidden or ( abs(j1 - 1) < 0.1 and abs(j2) < 0.1 ) # j = 1 and j'=0 forbidden ) ) and not (abs(j1) < 0.1 and abs(j2) < 0.1) # j = 0 and j'=0 forbiden and not ( abs(l1) < 0.1 and abs(l2) < 0.1 ) # l = 0 and l' = 0 is forbiden ): dl = abs(l1 - l2) dj = abs(j1 - j2) if dl == 1 and (dj < 1.1): return True else: return False return False
[docs] def getPolarizability( self, driveWavelength, units="SI", accountForStateLifetime=False, mj=None, ): r""" Calculates of scalar, vector, tensor, core and pondermotive polarizability, and returns state corresponding to the closest transition resonance. Note that pondermotive polarisability is calculated as :math:`\alpha_P = e^2 / (2 m_e \omega^2)`, i.e. assumes that the definition of the energy shift in field :math:`E` is :math:`\frac{1}{2}\alpha_P E^2`. For more datils check the preprint `arXiv:2007.12016`_ that introduced the update. .. _`arXiv:2007.12016`: https://arxiv.org/abs/2007.12016 Args: driveWavelength (float): wavelength of driving field (in units of m) units (string): optional, 'SI' or 'a.u.' (equivalently 'au'), switches between SI units for returned result (:math:`Hz V^{-2} m^2` ) and atomic units (":math:`a_0^3` "). Defaul 'SI' accountForStateLifetime (bool): optional, should we account for finite transition linewidths caused by finite state lifetimes. By default False. Returns: scalar, vector, and tensor, polarizabilities of the state specified, as well as the core, and ponderomotive polarizabilities of the atom, followed by the atomic state whose resonance is closest in energy. Returned units depend on `units` parameter (default SI). """ if accountForStateLifetime and len(self.lifetimes) == 0: for state in self.basis: self.lifetimes.append( self.atom.getStateLifetime( state[0], state[1], state[2], s=self.s ) ) driveEnergy = C_c / driveWavelength * C_h initialLevelEnergy = ( self.atom.getEnergy(self.n, self.l, self.j, s=self.s) * C_e ) # prefactor for vector polarisability prefactor1 = 1.0 / ((self.j + 1) * (2 * self.j + 1)) # prefactor for tensor polarisability prefactor2 = ( 6 * self.j * (2 * self.j - 1) / (6 * (self.j + 1) * (2 * self.j + 1) * (2 * self.j + 3)) ) ** 0.5 alpha0 = 0.0 alpha1 = 0.0 alpha2 = 0.0 closestState = [] closestEnergy = -1 targetStateLifetime = self.atom.getStateLifetime( self.n, self.l, self.j, s=self.s ) for i, state in enumerate(self.basis): n1 = state[0] l1 = state[1] j1 = state[2] if (mj is None) or (abs(mj) < j1 + 0.1): if abs(j1 - self.j) < 1.1 and ( abs(l1 - self.l) > 0.5 and abs(l1 - self.l) < 1.1 ): coupledLevelEnergy = ( self.atom.getEnergy(n1, l1, j1, s=self.s) * C_e ) diffEnergy = abs( (coupledLevelEnergy - initialLevelEnergy) ** 2 - driveEnergy**2 ) if (diffEnergy < closestEnergy) or (closestEnergy < 0): closestEnergy = diffEnergy closestState = state if diffEnergy < 1e-65: # print("For given frequency we are in exact resonance with state %s" % printStateString(n1,l1,j1,s=s)) return None, None, None, None, None, state # common factors if accountForStateLifetime: transitionLinewidth = ( 1 / self.lifetimes[i] + 1 / targetStateLifetime ) * C_h else: transitionLinewidth = 0.0 # transitionEnergy transitionEnergy = coupledLevelEnergy - initialLevelEnergy d = ( self.atom.getReducedMatrixElementJ( self.n, self.l, self.j, n1, l1, j1, s=self.s ) ** 2 * (C_e * physical_constants["Bohr radius"][0]) ** 2 * transitionEnergy * ( transitionEnergy**2 - driveEnergy**2 + transitionLinewidth**2 / 4 ) / ( ( transitionEnergy**2 - driveEnergy**2 + transitionLinewidth**2 / 4 ) ** 2 + transitionLinewidth**2 * driveEnergy**2 ) ) alpha0 += d # vector polarsizavility alpha1 += ( (-1) * (self.j * (self.j + 1) + 2 - j1 * (j1 + 1)) * self.atom.getReducedMatrixElementJ( self.n, self.l, self.j, n1, l1, j1, s=self.s ) ** 2 * (C_e * physical_constants["Bohr radius"][0]) ** 2 * driveEnergy * ( transitionEnergy**2 - driveEnergy**2 - transitionLinewidth**2 / 4 ) / ( ( transitionEnergy**2 - driveEnergy**2 + transitionLinewidth**2 / 4 ) ** 2 + transitionLinewidth**2 * driveEnergy**2 ) ) # tensor polarizability vanishes for j=1/2 and j=0 states # because Wigner6j is then zero if self.j > 0.6: alpha2 += ( (-1) ** (self.j + j1 + 1) * self.atom.getReducedMatrixElementJ( self.n, self.l, self.j, n1, l1, j1, s=self.s ) ** 2 * (C_e * physical_constants["Bohr radius"][0]) ** 2 * Wigner6j(self.j, 1, j1, 1, self.j, 2) * (coupledLevelEnergy - initialLevelEnergy) / ( (coupledLevelEnergy - initialLevelEnergy) ** 2 - driveEnergy**2 ) ) alpha0 = 2.0 * alpha0 / (3.0 * (2.0 * self.j + 1.0)) alpha0 = alpha0 / C_h # Hz m^2 / V^2 alpha1 = prefactor1 * alpha1 / C_h alpha2 = -4 * prefactor2 * alpha2 / C_h # core polarizability -> assumes static polarisability alphaC = self.atom.alphaC * 2.48832e-8 # convert to Hz m^2 / V^2 # podermotive shift driveOmega = 2 * np.pi / driveWavelength * C_c alphaP = C_e**2 / (2 * C_m_e * driveOmega**2 * C_h) if units == "SI": return ( alpha0, alpha1, alpha2, alphaC, alphaP, closestState, ) # in Hz m^2 / V^2 elif units == "a.u." or units == "au": return ( alpha0 / 2.48832e-8, alpha1 / 2.48832e-8, alpha2 / 2.48832e-8, alphaC / 2.48832e-8, alphaP / 2.48832e-8, closestState, ) else: raise ValueError( "Only 'SI' and 'a.u' (atomic units) are recognised" " as 'units' parameter. Entered value '%s' is" " not recognised." % units )
[docs] def plotPolarizability( self, wavelengthList, mj=None, addToPlotAxis=None, line="b-", units="SI", addCorePolarisability=True, addPondermotivePolarisability=False, accountForStateLifetime=False, debugOutput=False, ): r""" Plots of polarisability for a range of wavelengths. Can be combined for different states to allow finding magic wavelengths for pairs of states. Currently supports only driving with linearly polarised light. See example `magic wavelength snippet`_. .. _`magic wavelength snippet`: ../ARC_3_0_introduction.html#Calculations-of-dynamic-polarisability-and-magic-wavelengths-for-optical-traps Parameters: wavelengthList (array): wavelengths for which we want to calculate polarisability (in units of m). mj (float): optional, `mj` projection of the total angular momenutum for the states for which we are calculating polarisability. By default it's `+j`. line (string): optional, line style short definition to be passed to matplotlib when plotting calculated polarisabilities units (string): optional, 'SI' or 'a.u.' (equivalently 'au'), switches between SI units for returned result (:math:`Hz V^-2 m^2` ) and atomic units (":math:`a_0^3` "). Deafault 'SI'. addCorePolarisability (bool): optional, should ionic core polarisability be taken into account. By default True. addPondermotivePolarisability (bool): optional, should pondermotive polarisability (also called free-electron polarisability) be added to the total polarisability. Default is False. It assumes that there is no significant variation of trapping field intensity over the range of the electric cloud. If this condition is not satisfied, one has to calculate total shift as average over the electron wavefunction. accountForStateLifetime (bool): optional, should we account for finite transition linewidths caused by finite state lifetimes. By default False. debugOutput (bool): optonal. Print additional output on resonances Default value False. """ pFinal = [] wFinal = [] p = [] w = [] resonances = [] if mj is None: mj = self.j if self.j > 0.5 + 0.1: tensorPrefactor = (3 * mj**2 - self.j * (self.j + 1)) / ( self.j * (2 * self.j - 1) ) else: tensorPrefactor = 0 for wavelength in wavelengthList: ( scalarP, vectorP, tensorP, coreP, pondermotiveP, state, ) = self.getPolarizability( wavelength, accountForStateLifetime=accountForStateLifetime, units=units, mj=mj, ) if scalarP is not None: # we are not hitting directly the resonance totalP = scalarP + tensorPrefactor * tensorP if addCorePolarisability: totalP += coreP if addPondermotivePolarisability: # Subtract pondermotive contribution since the sign convention # is opposite to that of the dynamical polarizability. totalP -= pondermotiveP if ( (len(p) > 0) and p[-1] * totalP < 0 and (len(p) > 2 and (p[-2] - p[-1]) * totalP > 0) ): pFinal.append(p) wFinal.append(w) p = [] w = [] resonances.append(wavelength) if debugOutput: print( r"Resonance: %.2f nm %s" % ( wavelength * 1e9, printStateString( state[0], state[1], state[2], s=self.s ), ) ) p.append(totalP) w.append(wavelength) pFinal.append(p) wFinal.append(w) if addToPlotAxis is None: fig = plt.figure() ax = fig.add_subplot(1, 1, 1) else: ax = addToPlotAxis for i in range(len(wFinal)): ax.plot(np.array(wFinal[i]) * 1e9, pFinal[i], line, zorder=1) ax.set_xlabel(r"Driving field wavelength (nm)") if units == "SI": ax.set_ylabel(r"Polarizability (Hz/V$^2$ m$^2$)") else: ax.set_ylabel(r"Polarizability (a.u.)") for resonance in resonances: ax.axvline( x=resonance * 1e9, linestyle=":", color="0.5", zorder=0 ) return ax
[docs] class StarkBasisGenerator: """ Base class for determining the basis of the Rydberg manifold and associated properties. Defines logic for determining the basis of states to include in a calculation and obtains the energy levels and dipole moments to build the Hamiltonian from the provided ARC atom. This class should be inherited from to create a specific calculation. Args: atom (:obj:`arc.alkali_atom_functions.AlkaliAtom` or :obj:`arc.divalent_atom_functions.DivalentAtom`): ={ :obj:`arc.alkali_atom_data.Lithium6`, :obj:`arc.alkali_atom_data.Lithium7`, :obj:`arc.alkali_atom_data.Sodium`, :obj:`arc.alkali_atom_data.Potassium39`, :obj:`arc.alkali_atom_data.Potassium40`, :obj:`arc.alkali_atom_data.Potassium41`, :obj:`arc.alkali_atom_data.Rubidium85`, :obj:`arc.alkali_atom_data.Rubidium87`, :obj:`arc.alkali_atom_data.Caesium`, :obj:`arc.divalent_atom_data.Strontium88`, :obj:`arc.divalent_atom_data.Calcium40` :obj:`arc.divalent_atom_data.Ytterbium174` } Select the alkali metal for energy level diagram calculation """ def __init__(self, atom): UsedModulesARC.ac_stark = True self.atom = atom """ Instance of an ARC atom to perform calculations of the energy levels and coupling strengths. """ # basis definitions self.basisStates = [] """ List of basis states for calculation in the form [ [n,l,j,mj], ...]. Calculated by :obj:`defineBasis` . """ self.indexOfCoupledState = None """ Index of coupled state (initial state passed to :obj:`defineBasis`) in :obj:`basisStates` list of basis states """ self.targetState = [] """ Target state. Found by :obj:`basisStates`[:obj:`indexOfCoupledState`]. """ self.bareEnergies = [] """ `bareEnergies` is list of energies corresponding to :obj:`basisStates`. It is calculated in :obj:`defineBasis` in the basis of :obj:`basisStates` in units of GHz. """ self.targetEnergy = None """ `targetEnergy` stores the energy of the target state (initial state passed to :obj:`defineBasis`) """ self.n = None """ Stores the principle quantum number of the target state """ self.l = None """ Stores the orbital quantum number of the target state """ self.j = None """ Stores the total angular momentum number of the target state """ self.mj = None """ Stores the projection of the total angular moment of the target state """ self.s = None """ Stores the total spin angular momentum of the target state """ self.nMin = None """ Stores the minimum n to consider for the basis """ self.nMax = None """ Stores the maximum n to consider for the basis """ self.maxL = None """ Stores the max L to consider for the basis """ self.Bz = None """ Stores the applied magnetic field used to Zeeman shift states in the basis """ self.q = None """ Stores polarization of electric field for determining dipole coupled states. """ # hamiltonian components self.H = [] """ Diagonal elements of Stark-matrix. Not to be confused with :obj:`H0` for the Time-Independant Formulation of the Floquet Hamiltonian. Given in units of GHz. """ self.V = [] """ Off-diagonal elements of Stark-matrix divided by electric field value. To get off diagonal elemements multiply this matrix with electric field value. Full DC Stark matrix is obtained as ``fullStarkMatrix`` = np.diag(:obj:`bareEnergies`) + :obj:`V` * ``eField``. Calculated by :obj:`defineBasis` in the basis :obj:`basisStates` in units of GHz/(V/m). """ # STARK memoization self.eFieldCouplingSaved = False def _eFieldCouplingDivE(self, n1, l1, j1, mj1, n2, l2, j2, mj2, s=0.5): # eFied coupling devided with E (witout actuall multiplication to getE) # delta(mj1,mj2') delta(l1,l2+-1) if (abs(mj1 - mj2) > 0.1) or (abs(l1 - l2) != 1): return 0 # matrix element result = ( self.atom.getRadialMatrixElement(n1, l1, j1, n2, l2, j2, s=s) * physical_constants["Bohr radius"][0] * C_e ) sumPart = self.eFieldCouplingSaved.getAngular( l1, j1, mj1, l2, j2, mj2, s=s ) return result * sumPart def _eFieldCoupling(self, n1, l1, j1, mj1, n2, l2, j2, mj2, eField, s=0.5): return ( self._eFieldCouplingDivE(n1, l1, j1, mj1, n2, l2, j2, mj2, s=s) * eField ) def _onePhotonCoupling(self, ns, ls, js, mjs, nt, lt, jt, mjt, q, s=0.5): """ Tests if state s can be dipole coupled with a single photon to target state t. Given ss==st, true only for Delta-l==+-1 and (Delta-l==Delta-j or Delta-j=0 and j=l+s for either state) transitions. Args: ns (int): principle quantum number of potentially coupled state ls (int): orbital quantum number of potentially coupled state js (float): total angular quantum number of potentially coupled state mjs (float): projection of total angular momentum of potentially coupled state nt (int): principle quantum number of target state lt (int): orbital quantum number of target state jt (float): total angular quantum number of target state mjt (float): projection of total angular momentum of target state q (int): polarization of coupling field, must be -1,0,1 s (float, optional): total spin angular momentum quantum number. Defaults to 1/2, appropriate for alkali atoms. Returns: bool: True if transition is electric dipole allowed via a single photon """ # ignore the target state if (ns == nt) and (ls == lt) and (js == jt) and (mjs == mjt): return False # transitions that change l by 1 elif (abs(ls - lt) == 1) and (mjs - mjt == q): if ls - lt == js - jt: return True elif (js == jt) and ((js == ls + s) or (jt == lt + s)): return True else: return False else: return False def _twoPhotonCoupling(self, ns, ls, js, mjs, nt, lt, jt, mjt, q, s=0.5): """ Tests if states can be dipole coupled with two photons. Args: ns (int): principle quantum number of potentially coupled state ls (int): angular quantum number of potentially coupled state js (float): total angular quantum number of potentially coupled state mjs (float): projection of total angular momentum of potentially coupled state nt (int): principle quantum number of target state lt (int): angular quantum number of target state jt (float): total angular quantum number of target state mjt (float): projection of total angular momentum of target state q (int): polarization of coupling light, must be -1,0,1 s (float, optional): total spin angular momentum quantum number. Defaults to 1/2, appropriate for alkali atoms. Returns: bool: True if two photon coupling between states """ # ignore target state if (ns == nt) and (ls == lt) and (js == jt) and (mjs == mjt): return False # transitions that change l by 2 elif ( (abs(ls - lt) == 2) and (ls - lt == js - jt) and ((mjs - mjt) / 2 == q) ): return True # transitions that don't change l elif ((ls - lt) == 0) and (js == jt) and ((mjs - mjt) / 2 == q): return True else: return False
[docs] def defineBasis( self, n, l, j, mj, q, nMin, nMax, maxL, Bz=0, edN=0, progressOutput=False, debugOutput=False, s=0.5, ): """ Initializes basis of states around state of interest Defines basis of states for further calculation. :math:`n,l,j,m_j` specify target state whose neighbourhood and shifts we want to explore. Other parameters specify breadth of basis. This method stores basis in :obj:`basisStates`, then calculates the interaction Hamiltonian of the system. Args: n (int): principal quantum number of the state l (int): angular orbital momentum of the state j (flaot): total angular momentum of the state mj (float): projection of total angular momentum of the state q (int): polarization of coupling field is spherical basis. Must be -1, 0, or 1: corresponding to sigma-, pi, sigma+ nMin (int): *minimal* principal quantum number of the states to be included in the basis for calculation nMax (int): *maximal* principal quantum number of the states to be included in the basis for calculation maxL (int): *maximal* value of orbital angular momentum for the states to be included in the basis for calculation Bz (float, optional): magnetic field directed along z-axis in units of Tesla. Calculation will be correct only for weak magnetic fields, where paramagnetic term is much stronger then diamagnetic term. Diamagnetic term is neglected. edN (int, optional): Limits the basis to electric dipole transitions of the provided photon number. Default of 0 means include all states. Setting to 1 means only include single-photon dipole-allowed transitions. Setting to 2 means include up to 2 photon transitions. Higher numbers not supported. progressOutput (:obj:`bool`, optional): if True prints the progress of calculation; Set to false by default. debugOutput (:obj:`bool`, optional): if True prints additional information usefull for debuging. Set to false by default. s (float, optional): Total spin angular momentum for the state. Default value of 0.5 is correct for Alkaline Atoms, but value **has to** be specified explicitly for divalent atoms (e.g. `s=0` or `s=1` for singlet and triplet states, that have total spin angular momenutum equal to 0 or 1 respectively). """ # save calculation details START self.n = n self.l = l self.j = j self.mj = mj self.q = q if edN in [0, 1, 2]: self.edN = edN else: raise ValueError("EN must be 0, 1, or 2") self.nMin = nMin self.nMax = nMax self.maxL = maxL self.Bz = Bz self.s = s # save calculation details END self._findBasisStates(progressOutput, debugOutput) self._buildHamiltonian(progressOutput, debugOutput)
def _findBasisStates(self, progressOutput=False, debugOutput=False): """ Creates the list of basis states we want to include. Details about calculation are taken from class attributes. Results saved to class attributes are: :obj:`basisStates`, :obj:`indexOfCoupledState`, and :obj:`targetState`. Args: progressOutput (bool, optional): Whether to print calculation progress. debugOutput (bool, optional): Whether to print debug information. """ states = [] n = self.n l = self.l j = self.j mj = self.mj q = self.q s = self.s edN = self.edN nMin = self.nMin nMax = self.nMax maxL = self.maxL # track where target state is inserted in this list indexOfCoupledState = 0 index = 0 for tn in range(nMin, nMax): for tl in range(min(maxL + 1, tn)): for tj in np.linspace(tl - s, tl + s, round(2 * s + 1)): # ensure we add the target state if (n == tn) and (l == tl) and (j == tj): states.append([tn, tl, tj, mj]) indexOfCoupledState = index # adding all manifold states elif ( (edN == 0) and (abs(mj) + q - 0.1 <= tj) and ( tn >= self.atom.groundStateN or [tn, tl, tj] in self.atom.extraLevels ) ): states.append([tn, tl, tj, mj + q]) index += 1 # add states that are electric dipole allowed elif (edN == 1 or edN == 2) and self._onePhotonCoupling( n, l, j, mj, tn, tl, tj, mj + q, q, s ): states.append([tn, tl, tj, mj + q]) index += 1 # add states that are electric dipole allowed via 2-photon transition elif edN == 2 and self._twoPhotonCoupling( n, l, j, mj, tn, tl, tj, mj + 2 * q, q, s ): states.append([tn, tl, tj, mj + 2 * q]) index += 1 dimension = len(states) if progressOutput: print("Found ", dimension, " states.") if debugOutput: print(states) print("Index of initial state") print(indexOfCoupledState) print("Initial state = ") print(states[indexOfCoupledState]) # save info about states self.basisStates = states self.indexOfCoupledState = indexOfCoupledState self.targetState = states[indexOfCoupledState] def _buildHamiltonian(self, progressOutput=False, debugOutput=False): """ Creates the base matrices needed to produce the Floquet Hamiltonians. Details about calculation are taken from class attributes. Matrices correspond to two parts: field dependent and independent. Results saved to class attributes are: :obj:`bareEnergies`, :obj:`H`, and :obj:`V`. Args: progressOutput (bool, optional): Whether to print calculation progress. debugOutput (bool, optional): Whether to print debug information. """ global wignerPrecal wignerPrecal = True self.eFieldCouplingSaved = _EFieldCoupling() dimension = len(self.basisStates) states = self.basisStates indexOfCoupledState = self.indexOfCoupledState self.bareEnergies = np.zeros((dimension), dtype=np.double) self.V = np.zeros((dimension, dimension), dtype=np.double) if progressOutput: print("Generating matrix...") progress = 0.0 for ii in range(dimension): if progressOutput: progress += (dimension - ii) * 2 - 1 print(f"{progress / dimension**2:.0%}", end="\r") # add diagonal element self.bareEnergies[ii] = ( self.atom.getEnergy( states[ii][0], states[ii][1], states[ii][2], s=self.s ) * C_e / C_h + self.atom.getZeemanEnergyShift( states[ii][1], states[ii][2], states[ii][3], self.Bz, s=self.s, ) / C_h ) # add off-diagonal element for jj in range(ii + 1, dimension): coupling = ( 0.5 * self._eFieldCouplingDivE( states[ii][0], states[ii][1], states[ii][2], self.mj, states[jj][0], states[jj][1], states[jj][2], self.mj, s=self.s, ) / C_h ) self.V[jj][ii] = coupling self.V[ii][jj] = coupling self.H = np.diag(self.bareEnergies) if progressOutput: print("\nEnergies and Couplings Generated") if debugOutput: print(np.diag(self.bareEnergies) + self.V) # save info about target state self.targetEnergy = self.bareEnergies[indexOfCoupledState] if debugOutput: print("Target State:", self.targetState, self.targetEnergy) self.atom.updateDipoleMatrixElementsFile() self.eFieldCouplingSaved._closeDatabase() self.eFieldCouplingSaved = False
[docs] class ShirleyMethod(StarkBasisGenerator): """ Calculates Stark Maps for a single atom in a single oscillating field Uses Shirley's Time Independent Floquet Hamiltonian Method [1]_. More detail can be found in the review of Semiclassical Floquet Theories by Chu [2]_ and its application in Meyer et al [3]_. For examples demonstrating basic usage see `Shirley Method Examples`_. Args: atom (:obj:`arc.alkali_atom_functions.AlkaliAtom` or :obj:`arc.divalent_atom_functions.DivalentAtom`): ={ :obj:`arc.alkali_atom_data.Lithium6`, :obj:`arc.alkali_atom_data.Lithium7`, :obj:`arc.alkali_atom_data.Sodium`, :obj:`arc.alkali_atom_data.Potassium39`, :obj:`arc.alkali_atom_data.Potassium40`, :obj:`arc.alkali_atom_data.Potassium41`, :obj:`arc.alkali_atom_data.Rubidium85`, :obj:`arc.alkali_atom_data.Rubidium87`, :obj:`arc.alkali_atom_data.Caesium`, :obj:`arc.divalent_atom_data.Strontium88`, :obj:`arc.divalent_atom_data.Calcium40` :obj:`arc.divalent_atom_data.Ytterbium174` } Select the alkali metal for energy level diagram calculation Examples: AC Stark Map calculation >>> from arc import Rubidium85, ShirleyMethod >>> calc = ShirleyMethod(Rubidium85()) >>> calc.defineBasis(56, 2, 2.5, 0.5, 0, 45, 70, 10) >>> calc.defineShirleyHamiltonian(fn=1) >>> calc.diagonalise(0.01, np.linspace(1.0e9, 40e9, 402)) >>> print(calc.targetShifts.shape) (402,) References: .. [1] J. H. Shirley, Physical Review **138**, B979 (1965) https://link.aps.org/doi/10.1103/PhysRev.138.B979 .. [2] Shih-I Chu, "Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes", in Adv. At. Mol. Phys., vol. 21 (1985) http://www.sciencedirect.com/science/article/pii/S0065219908601438 .. [3] D. H. Meyer, Z. A. Castillo, K. C. Cox, P. D. Kunz, J. Phys. B: At. Mol. Opt. Phys., **53**, 034001 (2020) https://doi.org/10.1088/1361-6455/ab6051 .. _`Shirley Method Examples`: ./AC_Stark_primer.html#Shirley's-Time-Independent-Floquet-Hamiltonian """ def __init__(self, atom): UsedModulesARC.ac_stark = True super().__init__(atom) # Shirley Floquet Hamiltonian components self.fn = None """ Saves rank of Floquet Hamiltonian expansion. Only fn+1 photon processes are accurately accounted for in the diagonalisation. """ self.H0 = [] """ diagonal elements of Floquet-matrix (detuning of states) calculated by :obj:`defineShirleyHamiltonian` with units GHz relative to ionization energy. It is a 'csr' sparse matrix. """ self.B = [] """ off-diagonal elements of Floquet Hamiltonian. Get final matrix by multiplying by the electric field amplitude in V/m. Calculated by :obj:`defineShirleyHamiltonian`. """ self.dT = [] """ diagonal prefactors of frequency elements of Floquet Hamiltonian. To get diagonal elements multiply this matrix diagonal by electric field frequency. Calculated by :obj:`defineShirleyHamiltonian` and is unitless. Multiplying frequency should be in GHz. """ # calculation inputs self.eFields = None """ Saves electric field (in units of V/m) for which energy levels vs frequency are calculated See also: :obj:`diagonalise` """ self.freqs = None """ Saves frequency (in units of Hz) for which energy levels vs electric field are calculated See also: :obj:`diagonalise` """ # calculation outputs self.eigs = [] """ Array of eigenValues corresponding to the energies of the atom states for the electric field `eField` at the frequency `freq`. In units of Hz. """ self.eigVectors = [] """ Array of eigenvectors corresponding to the eigenValues of the solve. """ self.transProbs = [] """ Probability to transition from the target state to another state in the basis. """ self.targetShifts = [] """ This is the shift of the target state relative to the zero field energy for an applied field of :obj:`eField` and :obj:`freq`. Given in units of Hz. """
[docs] def defineShirleyHamiltonian(self, fn, debugOutput=False): """ Create the Shirley time-independent Floquet Hamiltonian. Uses :obj:`~StarkBasisGenerator.bareEnergies` and :obj:`~StarkBasisGenerator.V` from :class:`StarkBasisGenerator` to build. Matrix is stored in three parts. First part is diagonal electric-field independent part stored in :obj:`H0`, while the second part :obj:`B` corresponds to off-diagonal elements that are propotional to electric field amplitude. The third part is the diagonal Floquet expansion proportional to electric field frequency. Overall interaction matrix for electric field `eField` and `freq` can be then obtained from A B blocks ``A`` = :obj:`H0` + :obj:`dT` * ``freq`` and ``B`` = :obj:`B` * ``eField``. These matrices are saved as sparse CSR to facilitate calculations and minimize memory footprint. Args: fn (int): rank of Floquet Hamiltonian expansion. Only fn+1 multi-photon processes are accurately accounted for. """ self.fn = fn if not fn >= 1: raise ValueError( "Floquet expansion must be greater than 1." + " Rank of 0 is equivalent to rotating wave approximation" + " solution and is not covered by this method." ) dimension = len(self.bareEnergies) # create the sparse building blocks for the Floquet Hamiltonian # ensure everything is converted to csr format for efficient math self.H0 = sp.diags(np.tile(self.bareEnergies, 2 * fn + 1)).tocsr() self.dT = sp.block_diag( [ sp.diags([i], 0, shape=(dimension, dimension)) for i in range(-fn, fn + 1, 1) ], dtype=np.double, ).tocsr() self.B = sp.bmat( [ [ self.V if abs(i - j) == 1 else None for i in range(-fn, fn + 1, 1) ] for j in range(-fn, fn + 1, 1) ], dtype=np.double, ).tocsr() if debugOutput: print(self.H0.shape, self.dT.shape, self.B.shape) print(self.H0[(0, 0)], self.dT[(0, 0)], self.B[(0, 0)])
[docs] def diagonalise( self, eFields, freqs, progressOutput=False, debugOutput=False ): """ Finds atom eigenstates versus electric field and driving frequency Eigenstates are calculated for the outer product `eFields` and `freqs`. Inputs are saved in class attributes :obj:`eFields`, :obj:`freqs`. Resulting sorted eigenvalues, eigenvectors, transition probabilities, and target state shifts are saved in the class attributes :obj:`eigs`, :obj:`eigVectors`, :obj:`transProbs` and :obj:`targetShifts`. Function automatically produces the outer product space of the inputs. For example, if `eFields` has two elements and `freqs` and 10, the output shifts will have a shape of `(2,10)`. If one of the inputs is a single value, that dimension is squeezed out. Args: eFields (float or sequence of floats): electric field strengths (in V/m) for which we want to know energy eigenstates freqs (float or sequence of floats): driving frequency (in Hz) for which we want to know energy eigenstates progressOutput (bool, optional): if True prints the progress of calculation; Set to false by default. debugOutput (bool, optional): if True prints additional information usefull for debuging. Set to false by default. """ # get basic info about solve structure from class dim0 = len(self.basisStates) targetEnergy = self.targetEnergy # ensure inputs are numpy arrays, if scalars, 0d-arrays self.eFields = np.array(eFields, ndmin=1) self.freqs = np.array(freqs, ndmin=1) # pre-allocation of results array eig = np.zeros( (*self.eFields.shape, *self.freqs.shape, dim0 * (2 * self.fn + 1)), dtype=np.double, ) eigVec = np.zeros( ( *self.eFields.shape, *self.freqs.shape, dim0 * (2 * self.fn + 1), dim0 * (2 * self.fn + 1), ), dtype=np.complex128, ) transProbs = np.zeros( (*self.eFields.shape, *self.freqs.shape, dim0), dtype=np.double ) targetShifts = np.zeros( (*self.eFields.shape, *self.freqs.shape), dtype=np.double ) if progressOutput: print("Finding eigenvectors...") # create numpy iterator object it = np.nditer( [self.eFields, self.freqs], flags=["multi_index"], op_flags=[["readonly"], ["readonly"]], op_axes=[ list(range(self.eFields.ndim)) + [-1] * self.freqs.ndim, [-1] * self.eFields.ndim + list(range(self.freqs.ndim)), ], ) with it: for field, freq in it: if progressOutput: print(f"{(it.iterindex + 1) / it.itersize:.0%}", end="\r") # define the Shirley Hamiltonian for this combo of field and frequency Hf = self.H0 + self.dT * freq + self.B * field # convert Hf to dense array to get all eigenvectors ev, egvector = eigh(Hf.toarray()) # save the eigenvalues and eigenvectors eig[it.multi_index] = ev eigVec[it.multi_index] = egvector # get transition probabilities from target state to other basis states # index of first basis state in k=0 block diagonal refInd = self.fn * dim0 # index of target state in basis tarInd = self.indexOfCoupledState + refInd transProbs[it.multi_index] = np.array( [ np.sum( [ np.abs( np.conj(egvector[refInd + k * dim0 + i]) * egvector[tarInd] ) ** 2 for k in range(-self.fn, self.fn + 1, 1) ] ) for i in range(0, dim0, 1) ] ) # get the target shift by finding the max overlap with the target state evInd = np.argmax( np.abs(egvector[tarInd].conj() * egvector[tarInd]) ** 2 ) if np.count_nonzero(ev == ev[evInd]) > 1: warnings.warn( "Multiple states have same overlap with target. Only saving first one." ) targetShifts[it.multi_index] = targetEnergy - ev[evInd] if debugOutput: print( f"E field {field:.5f} V/m, Freq {freq * 1e-9:.3f} GHz" ) print( f"Eigenvalue with largest overlap of target state {evInd}: {ev[evInd] * 1e-9:.3f} GHz" ) print(f"Shift: {(targetEnergy - ev[evInd]) * 1e-9:.3e} GHz") print(f"Eigenstate: {egvector[evInd]}") # squeeze out unused dimensions corresponding to single element inputs self.eigs = eig.squeeze() self.eigVectors = eigVec.squeeze() self.transProbs = transProbs.squeeze() self.targetShifts = targetShifts.squeeze()
[docs] class RWAStarkShift(StarkBasisGenerator): """ Approximately calculates Stark Maps for a single atom in a single oscillating field Assumes the rotating wave approximation applies independently for the field interaction with all possible dipole transitions. Approximation is generally reasonable for weak driving fields such that no more than a single resonance contributes significantly to the overall Stark shift. When field is far-detuned from all transitions, error tends to a factor of 2. For an example of usage and comparison to other methods see `RWAStarkShift Example`_. Args: atom (:obj:`AlkaliAtom`): ={ :obj:`arc.alkali_atom_data.Lithium6`, :obj:`arc.alkali_atom_data.Lithium7`, :obj:`arc.alkali_atom_data.Sodium`, :obj:`arc.alkali_atom_data.Potassium39`, :obj:`arc.alkali_atom_data.Potassium40`, :obj:`arc.alkali_atom_data.Potassium41`, :obj:`arc.alkali_atom_data.Rubidium85`, :obj:`arc.alkali_atom_data.Rubidium87`, :obj:`arc.alkali_atom_data.Caesium` } Select the alkali metal for energy level diagram calculation Examples: Approximate AC Stark Map calculation >>> from arc import Rubidium85, RWAStarkShift >>> calc = RWAStarkShift(Rubidium85()) >>> calc.defineBasis(56, 2, 2.5, 0.5, 0, 45, 70, 10) >>> calc.findDipoleCoupledStates() >>> calc.makeRWA(0.01, np.linspace(1.0e9, 40e9, 402)) >>> print(calc.starkShifts.shape) (402,) .. _`RWAStarkShift Example`: ./AC_Stark_primer.html#RWAStarkShift:-Approximating-AC-Stark-Map-Calculations """ def __init__(self, atom): UsedModulesARC.ac_stark = True super().__init__(atom) self.dipoleCoupledStates = [] """ List of basis states that are dipole coupled to the target state. This is a subset of :obj:`~StarkBasisGenerator.basisStates`. """ self.dipoleCoupledFreqs = [] """ Transition frequencies in Hz between :obj:`~StarkBasisGenerator.targetState` and :obj:`dipoleCoupledStates`. """ self.starkShifts = [] """ Saves results of :obj:`makeRWA` caclulations. """
[docs] def findDipoleCoupledStates(self, debugOutput=False): r""" Finds the states in :obj:`basisStates` that directly couple to :obj:`targetState` via single photon electric dipole transitions. Saves the states and their detunings relative to :obj:`targetState` to :obj:`dipoleCoupledStates` and :obj:`dipoleCoupledFreqs`. Args: q (int): laser polarization (-1,0,1 corresponds to :math:`\sigma^-` :math:`\pi` and :math:`\sigma^+` respectively) """ coupledStates = [] coupledFreqs = [] for i, st in enumerate(self.basisStates): if self._onePhotonCoupling( self.n, self.l, self.j, self.mj, st[0], st[1], st[2], self.mj + self.q, self.q, self.s, ): coupledStates.append(st) coupledFreqs.append(self.targetEnergy - self.bareEnergies[i]) self.dipoleCoupledStates = coupledStates self.dipoleCoupledFreqs = np.array(coupledFreqs) if debugOutput: print(f"Found {len(coupledStates):d} dipole coupled states") print( f"Nearest dipole coupled state is detuned by: {np.abs(self.dipoleCoupledFreqs).min() * 1e-9:.3f} GHz" )
def _getRabiFrequency2_broadcast( self, n1, l1, j1, mj1, n2, l2, j2, q, electricFieldAmplitude, s=0.5 ): eFields = np.array(electricFieldAmplitude, ndmin=1) rabis = np.array( [ self.atom.getRabiFrequency2( n1, l1, j1, mj1, n2, l2, j2, q, eField, s ) for eField in eFields ] ) return rabis
[docs] def makeRWA(self, efields, freqs, maxRes=0.0, zip_inputs=False): """ Calculates the total Rotating-Wave Approximation AC stark shift Interaction is between :obj:`targetState` with each :obj:`dipoleCoupledStates` ``[i]``. Resulting shifts are saved in Hz to :obj:`starkShifts` . Function automatically produces the outer product space of the inputs. For example, if `eFields` has two elements and `freqs` and 10, the output shifts will have a shape of `(2,10)`. If one of the inputs is a single value, that dimension is squeezed out. :obj:`findDipoleCoupledStates` must be run fist. Args: efields (float or sequence of floats): electric field amplitude in V/m freqs (float or sequence of floats): electric field frequency in Hz maxRes (float, optional): only include dipole transitions with frequences less than this. Specified in Hz. zip_inputs (bool, optional): Causes the calculation to zip the inputs instead of an outer product. Inputs must be of equal shape when `True`. Default is `False`. """ # ensure inputs are numpy arrays, even if single values eFields = np.array(efields, ndmin=1) Freqs = np.array(freqs, ndmin=1) if zip_inputs: if freqs.shape != eFields.shape: raise ValueError("Zipped inputs must have same shape") delta_slice = np.s_[:] Omega_slice = np.s_[:] starkShift = np.zeros(Freqs.shape, dtype=np.double) else: delta_slice = np.s_[np.newaxis, :] Omega_slice = np.s_[:, np.newaxis] starkShift = np.zeros( (*eFields.shape, *Freqs.shape), dtype=np.double ) if maxRes != 0.0: inds = np.where( (self.dipoleCoupledFreqs > -maxRes) & (self.dipoleCoupledFreqs < maxRes) ) states = [self.dipoleCoupledStates[i] for i in inds[0]] else: states = self.dipoleCoupledStates print( f"Calculating RWA Stark Shift approximation with {len(states):d} levels" ) for st in states: Omega = ( self._getRabiFrequency2_broadcast( *self.targetState, *st[:-1], self.q, eFields ) / 2 / np.pi ) trans = self.atom.getTransitionFrequency( *self.targetState[:-1], *st[:-1] ) if trans > 0.0: delta = -(trans - freqs) else: delta = -(trans + freqs) starkShiftplus = 0.5 * ( delta[delta_slice] + np.sqrt(delta[delta_slice] ** 2 + Omega[Omega_slice] ** 2) ) starkShiftminus = 0.5 * ( delta[delta_slice] - np.sqrt(delta[delta_slice] ** 2 + Omega[Omega_slice] ** 2) ) starkShift += np.where(delta < 0.0, starkShiftplus, starkShiftminus) self.starkShifts = starkShift.squeeze()