MultidimensionalSpectroscopy.jl <!-- omit in toc -->

Simulate coherent multidimensional spectroscopy signals from quantum mechanical models.

Note that examples are currently not working, due to transition from CMDS.jl to MultidimensionalSpectroscopy.jl

Working examples:

• 01_coupled_dimer
• 02_displaced_harmonic_oscillator

Table of contents <!-- omit in toc -->

<a name="examplesTOC"></a>

• Introduction
• Installation
• Function overview
  • Available functions:
• How to
• Examples
  • The coupled dimer
  • Displaced (harmonic) oscillators
  • Light-matter coupled systems
  • 2D spectrum of J-aggregates
• Useful stuff
  • Evolution of density matrix
  • Disentangling GSB, SE and ESA contributions
  • Disentangling rephasing and non-rephasing signals
  • Convolution with laser spectrum

Introduction

The code relies primarily on qojulia/QuantumOptics.jl, which is well described here. Further helpful examples and functionalities are found the Python project QuTiP.

The module MultidimensionalSpectroscopy.jl contains the necessary functions to calculate 2D spectra from QM models and will be described below. examples/ shows example scenarios.

Installation

MultidimensionalSpectroscopy.jl requires the Julia language and qojulia/QuantumOptics.jl, which can be installed from via the standard sources:

• Julia

• QOJulia

Clone the repo ... and include the module via using MultidimensionalSpectroscopy.

Function overview

Type ?<function> into the REPL to access the documentation for a certain function.

Available functions:

• create_colormap: creates a blue-white-green-red colormap with zero values being white (scheme="bright") or dark (scheme="dark")

• zeropad: zeropadding of time domain data before Fourier transformation into spectral domain

• interpt: interpolate time vector after zeropadding

• make2Dspectra: invokes cmds.correlations to calculate different signals

• correlations: calculate time evolution and interaction with laser pulses

• view_dm_evo: quick way to visualize evolution of density matrix

• save_2d: saves 2D plots to folder fn_base

• plot2d: plots 2D data in out2d

• plot_timeTrace: plots time traces from T evolution of 2D spectrum

• crop2d: crops 2D data to reduce size

• tri: select upper or lower triangular matrix, used to simplify pathways

• create_subspace: create subspace (ground state -> single excitation sector, ground state -> double excitation sector, ...) Hamiltonian and operators

• absorptionSpectrum: shortcut to calculate and plot absorption spectrum

• plot2d_comps: plot components (GSB, SE, ESA) of 2D spectrum

• vib_analysis: plot rephasing and non-rephasing parts of GSB, SE and ESA 2D spectra -> beating maps

• plot_levels: diagonalise Hamiltonian and plot level structure

How to

Set up your QM model of interest!

Example (see displaced harmonic oscillator, as well as documentation of QuantumOptics.jl):

b_tls = NLevelBasis(2)  # Hilbert-space of system                   Basis: {|ui⟩}
b_vib = FockBasis(5)    # Hilbert-space of oscillator               Basis: {|vi⟩}
b     = b_tls ⊗ b_vib  # combined basis

j21 = transition(b_tls,2,1) # |e⟩⟨g|
j12 = dagger(j21)           # |g⟩⟨e|
at  = create(b_vib)         # ...
a   = dagger(at)
D   = displace(b_vib,d)     # diplacement operator with the displacement d

H = ...

Use plot_levels() to get an overview of the energy level structure (Eigenstates) of the full systems and its components.

plot_levels(H,0) # 0: shift in x-direction (change to display multiple energy diagrams next to each other)

Result:

<a name="ordering"></a> Operators can be ordered by increasing energy (diagonal elements of Hamiltonian) by

idx = sortperm(real(diag((H).data)))
H.data = H.data[idx,idx]
...

<!--- and grouped into excitation sectors (0, 1, 2, ..., N excitations) by -->

This is required to convert between eigen- and site basis (or coupled- and uncoupled basis) when working with a subspace of the full Hilbert space, by using the transform operator (transf_op) from the output of create_subspace() (see below):

H, transf_op, P, L, rho0, ... = create_subspace(H,"bi", L, rho0, ...)
op_site = transf_op * op  * transf_op'

If the dimension of the basis is too large, create_subspace() can be used to project all necessary operators into the ground + single excitation subspace (e.g. for linear spectra), or the ground + single + double excitation subspace (e.g. for 2D spectra). NOTE that creating a subspace requires ordering the operators into excitation sectors

H, transf_op, P, L, ... = create_subspace([H],"bi", L, ...)

... replaces as maybe operators is desired.

To calculate 2D spectra first initialize the output array

spectra2d = Array{out2d}(undef, length(T))

where T = [0., 5., 10., ...] is a vector containing population/evolution time steps.

Next call make2dspectra() in a for loop

for i = 1:length(T)
    spectra2d[i] = make2Dspectra(tlist,rho0,H,F,μ12,μ23,T[i],"lindblad";debug=false,zp=zp)
end

with tlist being the coherence/detection time steps, rho0 the equilibrium/ground state density matrix, H the system Hamiltonian, F the jump operator (Lindblad operator) or the Redfield tensor, μ12 the transition dipole operator between the ground and single excited states, and μ23 the transition dipole operator between the single and double excited states. T[i] is the current population time. The option "lindblad" or "redfield" selects which ... to use. debug=true activates the debugging output and zp is the zero padding value of 10^zp.

Using multithreading, several population time steps can be evaluated simultaneously:

Threads.@threads for i = 1:length(T)
    spectra2d[i] = make2Dspectra(...)
end

Make sure to disable all output plots within make2Dspectra() when using multithreading, as these might crash the execution.

JLD2 can be used to conveniently save the spectra2d structure (does not work with cloud drives, such as pCloud). Remember to round2d() the data to save disk space.

@save "C:\\path\\to\\data\\file.jld2" out2d

Data can then be load as

@load "C:\\path\\to\\data\\file.jld2" out2d

However, the data format is not compatible with other software. save2d() saves ASCII files for real (.re) and imaginary (.im) parts.

...

You can create a slider to flip through 2D spectra (work in progress):

using Blink, Interactive

mp = @manipulate for i in slider(1:length(spectra2d))
          clf(); plot2d(spectra2d[i].ω,spectra2d[i].full2d)
          end

w = Window(); 
body!(w, mp);

Examples

The following examples (scripts) are available:

<a name="coupledDimer"></a>

The coupled dimer

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>> link

<a name="DO"></a>

Displaced (harmonic) oscillators

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>> link

<!--- <a name="vibrationalCoherences"></a> --> <!--- <a name="FCFmorse"></a> -->

<a name="jaynesCummings"></a>

Light-matter coupled systems

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>> link

<!--- <a name="ensembleDisorder"></a> ### Ensemble of two-level systems with disorder [back to TOC](#examplesTOC) In order to study the effect of disorder on the 2D signal [examples\ensemble_of_TLSs_w_disorder.jl](examples\ensemble_of_TLSs_w_disorder.jl) creates a composite Hamiltonian of ``num_of_TLSs`` two-level systems, whose energies are distributed by the function used to create ``disorder``. For a Gaussian distribution of energies and 5 ``num_of_TLSs = 5`` the energy diagram looks as follows:  and the relevant single excited manifold  This system leads to the following correlation function and absorption spectrum with the Lindblad-operator ....  ... and the system dynamics: