Density Matrices notes + demo (ePSproc + PEMtk dev.)

30/08/21, 24/07/24

In this notebook density matrices are considered & numerical routines demonstrated.

24/07/24: Updated notes and added density matrix from spherical tensor example.

Density Matrices

The general density operator, for a mixture of indepent states \(|\psi_{n} \rangle\), can be defined as (Eqn. 2.8, Blum [1]):

\begin{equation} \hat{\rho}=\sum_{n}W_{n}|\psi_{n}\rangle\langle\psi_{n}| \end{equation}

Where \(W_{n}\) defines the (statistical) weighting of each state \(\psi_{n}\) in the mixture.

For a given basis set, \(|\phi_{m}\rangle\), the states can be expanded and the matrix elements of \(\boldsymbol{\rho}\) defined (Eqns. 2.9 - 2.11, Blum [1]): \begin{equation} | \psi_{n} \rangle = \sum_{m'} a_{m'}^{(n)}| \phi_{m'}\rangle \end{equation}

\begin{equation} \hat{\rho}=\sum_{n}\sum_{mm'}W_{n}a_{m'}^{(n)}a_{m}^{(n)*}|\phi_{m'}\rangle\langle\phi_{m}| \end{equation}

And the matrix elements - the density matrix - given explicitly as:

\begin{equation} \boldsymbol{\rho}_{i,j}=\langle\phi_{i}|\hat{\rho}|\phi_{j}\rangle=\sum_{n}W_{n}a_{i}^{(n)}a_{j}^{(n)*} \end{equation}

For all pairs of basis states \((i,j)\). This defines the density matrix in the \(\{|\phi_n\rangle\}\) representation (basis space).

The expectation value of an operator \(\hat{Q}\) can then be found as (Eqn. 2.18, Blum [1]):

:nbsphinx-math:`begin{eqnarray} langle hat{Q}rangle & = & sum_{n}sum_{mm’}W_{n}a_{m’}^{(n)}a_{m}^{(n)*}langlephi_{m}|hat{Q}|phi_{m’}rangle\

& = & sum_{mm’}langlephi_{m’}|hat{rho}|phi_{m}ranglelanglephi_{m}|hat{Q}|phi_{m’}rangle\ & = & mathrm{tr}(rho Q)

end{eqnarray}`

For basic examples, in terms of \(|J,M\rangle\) states and alignment, see the Quantum Metrology with Photoelectrons notes.

Here, we’ll look at applying the formalism to photoionization, using ePSproc for the numerics…

[1] Blum, K. (2012). Density Matrix Theory and Applications (3rd Editio, Vol. 64). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-20561-3

Setup

[1]:

import sys
import os
from pathlib import Path
import numpy as np
# import epsproc as ep
import xarray as xr

import matplotlib.pyplot as plt

from datetime import datetime as dt
timeString = dt.now()

import epsproc as ep

# Multijob class dev code
from epsproc.classes.multiJob import ePSmultiJob

OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.

* sparse not found, sparse matrix forms not available.
* natsort not found, some sorting functions not available.

* Setting plotter defaults with epsproc.basicPlotters.setPlotters(). Run directly to modify, or change options in local env.

* Set Holoviews with bokeh.
* pyevtk not found, VTK export not available.

[2]:

epDemoDataPath = Path(ep.__path__[0]).parent/'data'/'photoionization'/'n2_multiorb'

Load test data

[3]:

# Class dev code
from epsproc.classes.multiJob import ePSmultiJob
from epsproc.classes.base import ePSbase

# Instantiate class object.
# Minimally this needs just the dataPath, if verbose = 1 is set then some useful output will also be printed.
data = ePSbase(epDemoDataPath, verbose = 1)

# ScanFiles() - this will look for data files on the path provided, and read from them.
data.scanFiles()


*** Job orb6 details
Key: orb6
Dir /home/jovyan/code-share/github-share/ePSproc/data/photoionization/n2_multiorb, 1 file(s).
{   'batch': 'ePS n2, batch n2_1pu_0.1-50.1eV, orbital A2',
    'event': ' N2 A-state (1piu-1)',
    'orbE': -17.09691397835426,
    'orbLabel': '1piu-1'}

*** Job orb5 details
Key: orb5
Dir /home/jovyan/code-share/github-share/ePSproc/data/photoionization/n2_multiorb, 1 file(s).
{   'batch': 'ePS n2, batch n2_3sg_0.1-50.1eV, orbital A2',
    'event': ' N2 X-state (3sg-1)',
    'orbE': -17.34181645456815,
    'orbLabel': '3sg-1'}

Versions

[32]:

import scooby
scooby.Report(additional=['epsproc', 'holoviews', 'hvplot', 'xarray', 'matplotlib', 'bokeh'])

[32]:

Thu Jul 25 09:44:53 2024 EDT
OS	Linux	CPU(s)	64	Machine	x86_64	Architecture	64bit
RAM	62.8 GiB	Environment	Jupyter	File system	btrfs
Python 3.10.11 \| packaged by conda-forge \| (main, May 10 2023, 18:58:44) [GCC 11.3.0]
epsproc	1.3.2-dev	holoviews	1.16.2	hvplot	0.8.4	xarray	2022.3.0
matplotlib	3.5.3	bokeh	3.1.1	numpy	1.23.5	scipy	1.10.1
IPython	8.13.2	scooby	0.7.2

[33]:

# Check current Git commit for local ePSproc version
from pathlib import Path
!git -C {Path(ep.__file__).parent} branch
!git -C {Path(ep.__file__).parent} log --format="%H" -n 1

* 3d-AFPAD-dev
  dev
  master
ba3d6780d95999e518caf85e08e80e7cbc68b6b2

[34]:

# Check current remote commits
!git ls-remote --heads git://github.com/phockett/ePSproc

fatal: unable to connect to github.com:
github.com[0: 140.82.114.4]: errno=Connection timed out

[35]:

!hostname

8fdb9101787f

[36]:

!conda env list

# conda environments:
#
base                  *  /opt/conda

[ ]:

[ ]:

Density Matrices notes + demo (ePSproc + PEMtk dev.)