# Encoder–Decoder Neural Networks in Interpretation of X-ray Spectra

Jalmari Passilahti  Department of Physics and Astronomy, University of Turku,
20014 Turun yliopisto, Finland<EMAIL_ADDRESS>Anton Vladyka  Department of
Physics and Astronomy, University of Turku, 20014 Turun yliopisto, Finland
Johannes Niskanen<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Turku, 20014 Turun yliopisto, Finland

###### Abstract

Encoder–decoder neural networks (EDNN) condense information most relevant to
the output of the feedforward network to activation values at a bottleneck
layer. We study the use of this architecture in emulation and interpretation
of simulated X-ray spectroscopic data with the aim to identify key structural
characteristics for the spectra, previously studied using emulator-based
component analysis (ECA). We find an EDNN to outperform ECA in covered target
variable variance, but also discover complications in interpreting the latent
variables in physical terms. As a compromise of the benefits of these two
approaches, we develop a network where the linear projection of ECA is used,
thus maintaining the beneficial characteristics of vector expansion from the
latent variables for their interpretation. These results underline the
necessity of information recovery after its condensation and identification of
decisive structural degrees for the output spectra for a justified
interpretation.

##  I Introduction

X-ray spectroscopic methods are sensitive to the local atomic structure of
matter, and are therefore used for characterization [1, 2, 3, 4, 5]. In
systems with significant structural variation, that of the spectra of
individual structures can also be expected. The relationship between a
structure and a spectrum is, however, far from trivial due to the origin of
the spectral effects in quantum physics of the electronic-nuclear system.
Moreover, this relationship may be indicative of only certain structural
characteristics for a given spectroscopic method [6]. Thus interpretation of
spectra requires analysis based on wide sampling, enabled quite recently by
growth of computational resources. Whereas measurement of spectra for
individual structures in liquid systems seems unfeasible, their computation is
possible; statistical simulation of the ensemble average corresponds to the
experiment [7].

To aid these studies machine learning (ML) may be utilized. Models such as
neural networks (NN) are known for their ability to make predictions for
complicated functions with little computational effort, provided that
sufficient training data is available. While still notable, training with
standard model selection requires less computational effort than simulations
for X-ray spectra. We have recently used these benefits of NNs in an emulator-
based component analysis (ECA), a member of project-pursuit algorithms, to
identify the structural characteristics a given spectrum is reflective of [6,
8, 9, 10].

An encoder–decoder neural network (EDNN), illustrated in Figure 1, is a
pipeline of the encoder part, a narrow hidden layer (bottleneck), and the
decoder part. If accurate emulation is achieved by such a feed-forward
network, knowledge most relevant for spectrum prediction is necessarily
condensed in the activation values of the bottleneck neurons. This ability of
EDNNs to compress information into few values in the latent space is widely
used in computer vision [11], for feature extraction [12] or denoising [13,
14]. The aforementioned information condensing takes place intrinsically to
training and model selection of ML, and thus knowledge discovery would only
require interpretation of the activation values output by the encoder part,
passed to the decoder part as an input. As intriguing the potential for
automatized information compression by an EDNN sounds for interpretation of
X-ray spectra, as relevant it is to find out whether this can be done in
practice – and at which cost.

Figure 1: The principle of spectra prediction with encoder-decoder neural
network (EDNN). Successful emulation of output for given input requires
essential latent information to be condensed into the activation values of the
neurons at the bottleneck, which may be only a few for reasonably accurate
emulation of the structure–spectrum relationship.

We benchmark the use of EDNN in the analysis of structural dependence of X-ray
spectra using data of the previously studied H2O molecule [6] and amorphous
germanium dioxide [8]. To achieve a meaningful comparison, we closely follow
the workflow of these prior analyses. We find that the more flexible EDNN
approach captures more target variable variance than the ECA approach of the
same number of latent variables. However, finding a reasonable approximate
inverse solution for the EDNN architecture is highly untrivial, whereas the
ECA allows this owing to inherent linear-only operations. As a compromise we
develop an architecture to utilize the strengths of both approaches: a linear
projection encoder (such as ECA), a narrow layer of neurons for latent
variables, and a freely adjusting decoder. In this neural-network component
analysis (NNCA) approach the basis vectors of ECA are optimized jointly with
the training of the subsequent neural network, in the spirit of EDNN, but the
constrained architecture allows for stable approximate solution of the inverse
problem.

##  II Methods

In this work we use two previously studied simulated datasets. The first one
consists of 10 000 snapshots from AIMD trajectories for H2O molecule and
respective O K-edge X-ray photoelectron spectra (XPS), X-ray emission spectra
(XES) and X-ray absorption spectra (XAS) [6, 15], openly available in [16].
The second dataset consists of 13 896 structures and respective Ge K$\beta$
XES of amorphous GeO2 simulated at 11 different pressures ranging from 0 to
120 GPa. Each data point consists of a local Ge-centered structure based on
AIMD simulations [17], and the corresponding calculated spectrum [18, 8].

Successful ML of X-ray spectra requires input data to be presented in a format
different to that of the raw data, i.e. feature engineering [10]. We encoded
the geometry of the H2O molecule simply in a form of H–O–H angle and the
lengths of O–H bonds. For amorphous GeO2 we followed Ref. 8 by describing the
local structure as Coulomb matrices [19]. The output features may require
engineering as well; whereas for the H2O system spectra tabulated on a grid in
energy could be used, the limited amount of data necessitated encoding the
spectral information of GeO2 into 4 statistical moments (mean, standard
deviation, skewness and excess kurtosis) for K$\beta^{\prime\prime}$ and
K$\beta_{2}$ emission lines as done in Ref. 8. We trained the models using 80
% (train set) of the data and evaluated the final performance with the
remaining 20 % (test set), unseen to the model. Furthermore, we z-score
standardized the structural features for the train set to have zero mean and
unit variance. For spectra of the H2O molecule, we used the mean and variance
over all channels in the standardization as in Ref. 6. The spectral moments of
GeO2 were individually z-score standardized.

We carried out randomized-grid-search model selection using one and two latent
variables (bottleneck widths) for both systems. This procedure used 5-fold
cross validation, and the best model was trained with the full train set
afterwards. In each case, the depth and the width of the network were varied
together with the learning rate and the strength of the L2-regularization. We
used the rectified linear unit (ReLU) activation function [20] for these
networks. In total we tested over 500 000 hyperparameter combinations: 120 000
for H2O (20 000 per model type), and 400 000 for the GeO2 (200 000 per model
type). We used Python[21] package scikit-learn[22] for these tasks. The model
selection search spaces are provided in Supplementary Information.

As a general measure for the performance of models we use the R2 score
(generalized covered target variance):

$\displaystyle\mathrm{R}^{2}=1-\frac{\mathrm{tr\left(\mathbf{\tilde{S}}^{T}\mathbf{\tilde{S}}\right)}}{\mathrm{tr\left(\mathbf{S}^{T}\mathbf{S}\right)}},$
(1)

where matrix $\mathbf{S}$ contains the true target features (spectra or
related measures) as row vectors and $\mathbf{\tilde{S}}$ is the difference
between true and predicted spectral output:

$\displaystyle\mathbf{\tilde{S}}=\mathbf{S}-\mathbf{S}^{(\mathrm{pred})}.$ (2)

The R2 metric takes values in the range $\left]-\infty,1\right]$, where 1
corresponds to perfect match, and 0 corresponds to the performance of constant
prediction of the mean value. The R2 score is a relative measure with an
interpretable range of values independent of unit or absolute scale. Analysis
of changes with respect to a baseline spectrum, expected for the chemical
entity, is a meaningful task for interpretation of a spectrum, fulfilled by
mean-subtracted data and R2. Last, since structure–spectrum relationship is
nonlinear, a small spectral feature may be indicative of a physically
interesting structure or a physical mechanism. These points motivate the use
of R2, but we note that any other metric could be used if desired.
Additionally, this choice allows for direct comparison to the earlier works by
Niskanen et al. [6] and Vladyka et al. [8].

We benchmark EDNN against ECA [6]. This projection-pursuit-spirited analysis
utilizes a pre-trained emulator $\mathbf{S}_{\mathrm{emu}}$, such as a NN, to
identify relevant input variations in terms of the variations of the output.
The method carries out a decomposition in the input space, in which the basis
vectors $\mathbf{v}_{j}$ are optimized one at a time to maximize the
prediction performance score for rank $k$ of the projection

$\mathbf{x}_{i}^{(k)}=\sum_{j=1}^{k}\underbrace{(\mathbf{v}_{j}\cdot\mathbf{x}_{i})}_{=:t_{j}}\mathbf{v}_{j}.$
(3)

It is inherently assumed that feature-wise z-score-standardized input data
$\mathbf{x}_{i}$ is used. Here we use the test dataset to maximize the R2
score for $\mathbf{S}_{\mathrm{emu}}(\mathbf{x}_{i}^{(k)})$ and the known
spectra $\mathbf{S}_{i}$. In this iterative procedure the quickness of the ML
emulator $\mathbf{S}_{\mathrm{emu}}$ is crucial.

As motivated and discussed in section III.3, we implemented an EDNN variant
(NNCA, Neural Network Component Analysis), where the encoder consisted of a
single hidden layer with orthonormality constraints for the basis vectors,
without bias and without activation function. This corresponds to the
projection step of Equation (3) for $t_{j}$ and allows fitting the ECA basis
vectors $\mathbf{v}_{j}$ during the training of the network. In this manner,
the model selection phase of ML is also leveraged for ECA. We implemented the
aforementioned network with ReLU activation function using PyTorch[23] (v
2.2.1). We ran exhaustive network architecture model selection for the decoder
part using the same search spaces that were used for the emulators of the ECA
implementations [6, 8] which are provided in the Supplementary Information.

##  III Results and discussion

Figure 2 shows the mean spectra of the studied data. The grey areas represent
the variation in the dataset and in each case the spectra show noticeable
structural dependency. Furthermore, in panel (d) a pressure dependency of the
GeO2 Ge K$\beta$ XES spectrum is clearly visible. Conveniently, the H2O system
has only a few inputs and many outputs while GeO2 has the opposite. This
contrast may help unveil interesting insights about our solution and the
problem in general.

Figure 2: Studied spectra for (a-c) H2O and (d) GeO2. Black line depicts the
mean and grey areas $\pm\sigma$ cut to zero. For GeO2 colored lines represent
mean spectra for different pressures and the intervals containing the studied
peaks are marked with vertical dashed lines.

###  III.1 H2O molecule

The performance of a dimensionality reduction is measured by the spectral
variance that it allows to cover, for which we use the R2 score. These numbers
are presented for ECA and EDNN in Table 1. The table also contains values for
NNCA, discussed later in Section III.3. A clear improvement for test data is
seen with the more flexible EDNN, when the bottleneck width equals the
corresponding ECA rank. With one-component model for XES, a score improvement
of 0.12 units is achieved in comparison to ECA, and the spectral dependence
can be completely encoded into two latent variables by using either approach.
For XAS, 0.10 units better performance than that of ECA is achieved by one-
component EDNN, and the two-component EDNN leaves only a minor 0.02-unit
fraction out of the complete spectral behavior. For XPS, a single degree of
freedom suffices to describe the whole variation. Using two components,
virtually no relevant structural information is lost in the bottleneck in any
of the cases.

Table 1: Generalized covered spectral variance (R2 score) for H2O EDNN, NNCA and ECA with 1 and 2 bottleneck neurons or ECA-components $k$. Notable improvement is achieved owing to nonlinearity of the encoder. Same train and test splits as in [6] were used with all models. | $k$ | EDNN | NNCA | ECA [6]
---|---|---|---|---
XES | 1 | 0.86 | 0.79 | 0.74
| 2 | 1.00 | 1.00 | 1.00
XAS | 1 | 0.85 | 0.76 | 0.75
| 2 | 0.98 | 0.92 | 0.91
XPS | 1 | 1.00 | 0.99 | 0.99
| 2 | 1.00 | 1.00 | 1.00
Figure 3: Isosurface plots a)-c) of the one-component EDNN model activation
values $t_{1}$ compared to d)-f) isosurfaces of metric $M$ defined in equation
(4) a polynomial model[6] for H2O spectra. Drawn following Ref. 6.

The structural simplicity of the molecular H2O system allows for inspection of
performance by visualization. As an metric for evaluation of the final model
for H2O we use the spectral deviation from that at the centre of the training
set

$\displaystyle
M(\mathbf{x}):=\frac{\|\mathbf{S}_{\mathrm{emu}}(\mathbf{x})-\mathbf{S}_{\mathrm{emu}}(\mathbf{x}_{\mathrm{cen}})\|_{2}}{\|\mathbf{S}_{\mathrm{emu}}(\mathbf{x}_{\mathrm{cen}})\|_{2}},$
(4)

where $\mathbf{x}_{\mathrm{cen}}$ is located at origin when using standardized
data. Isosurface plots for EDNN activation value for the one-component model
are compared to isosurfaces of the metric with the polynomial emulator [6] in
Figure 3, which shows a good match. We conclude that the bottleneck activation
captures the content of the difference metric $M$ and spectral change in the
structural space. We note that some of the XES spectrum isosurfaces are close
to each other, indicating structural regions of high spectral sensitivity, in
addition to identification of this spectrally dominant direction in the
structural space. Furthermore, the $t$ score isosurfaces would be planes for
ECA due to the linear formalism, and therefore best one-component ($k=1$) ECA
performance is obtained for XPS. The deviation of EDNN from this planar
behaviour is owing to the flexibility of the respective encoder.

For interpretation of spectra, or latent variables derived thereof, a
structural correspondent might be searched for. While the EDNN succeeds in the
forward problem with a bottleneck, finding an inverse for the network proves
problematic. We trace this to non-bijective nature of the structure–spectrum
problem, manifested here by ill-conditioned weight matrices in the network,
which will amplify errors when propagation to the inverse direction is carried
out. In our try to overcome these problems, we carried out a separate model
selection with a bijective activation function and an architecture with as
many square weight matrices as possible. The details and results for the H2O
molecule are documented in Supplementary Information, and we will return to
this topic with the GeO2.

###  III.2 Amorphous GeO2

An ECA analysis of K$\beta$ X-ray emission spectrum (XES) of amorphous GeO2
has been previously carried out by Vladyka et al. [8]. In the work prediction
of full spectra was not achieved but instead the authors focused on eight
statistical moments for two spectral lines as target features. The generalized
covered variance (R2 score) for the features with the EDNN and ECA are given
in Table 2, where EDNN is again seen to achieve greater scores than ECA.
Notably, the score of the one-component EDNN surpasses that of the
corresponding two-component ECA model. Thus, more significant portion of the
information affecting the spectral moments can be condensed even into a single
latent variable using EDNN architecture.

Table 2:  Generalized covered spectral variance (R2 score) for GeO2 EDNN, NNCA and ECA with one and two bottleneck neurons/ECA-components $k$. Again significant improvement is observed owing to nonlinearity of the encoder of the EDNN. Same train and test splits as in [8] were used with all models. | $k$ | EDNN | NNCA | ECA [8]
---|---|---|---|---
XES moments | 1 | 0.84 | 0.84 | 0.77
| 2 | 0.90 | 0.88 | 0.83

As shown by Vladyka et al. ECA provides an approximate solution to the
spectrum-to-structure inverse problem [8]. In the work, latent variables
$\mathbf{t}$ for ranks $k=1,2$ could be reconstructed from spectral moments,
after which general trends in the Coulomb matrix and in the interactomic
distances from the active Ge site could be reconstructed. Our trials using a
bijective activation function Leaky ReLU [24] for EDNN were in general
unsuccesful in reconstructing the bottleneck activation values. As with the
H2O molecule, we applied the specific architecture of maximal number of square
weight matrices for this specific task of inverse (see Supplementary
Information). The data from the GeO2 and H2O systems reveal a trend: the
larger matrices, whether in encoder or decoder, cause error when an inverse is
attempted. In addition the step from encoder to the bottleneck is always
problematic.

###  III.3 NNCA: ECA implemented during training

For ECA the problem of finding an inverse may not be as severe as for EDNN:
while reconstruction of the $\mathbf{t}$ scores may be susceptible to non-
bijective behavior, at least structural interpretation of these scores is
straightforward by application of Equation (3) providing an approximate
structural data point [8]. Even though in some cases the activation values of
the neurons at the bottleneck could be reconstructable, the EDNN would then
require approximate structural reconstruction to achieve the same. We combined
the automatized optimization for information condensing by the EDNN to the
interpretability brought by ECA into neural network component analysis (NNCA),
illustrated in Figure 4. In this network architecture, the first layer carries
out the projection step of ECA to latent variables (without the expansion of
Eq. (3)). As the basis vectors $\\{\mathbf{v}_{j}\\}$ then are optimized
during training, and as the whole process is subject to model selection, more
covered spectral variance than for ECA is expected. Moreover, NNCA using
built-in orthonormality may give better performance for $k>1$ components than
rank-by-rank proceeding ECA, owing to collaborative adapdation of the vectors
subject to optimization. Last, latent variables condensed in the bottleneck
can be converted back to structural space by simple vector expansion. As a
downside of this approach, the hierarchy of the ECA basis vectors in terms of
covered target variance is lost.

Figure 4: The principle of implementing ECA as a simple encoder in the NNCA
network with $k$=1. A broader bottleneck implies corresponding orthonormal set
of vectors $\\{\mathbf{v}_{j}\\}_{j=1}^{k}$ for the projection of the input
vector $\mathbf{x}$. The matrix operation after input has zero bias followed
by linear activation.

Tables 1 and 2 indicate the performance of NNCA to expectedly lie in between
those of EDNN and ECA, as NNCA is a special case of EDNN and ECA a special
case of NNCA. Notably the NNCA manages to match the performance of EDNN with
one-component for the spectral moments of the Ge K$\beta$ XES of GeO2. This
limited comparison hints that the performance of NNCA is closer to EDNN when
the system is more complex particularly when the structural descriptor is more
complex.

We studied structural reconstruction by NNCA following closely the earlier ECA
procedure of Vladyka and co-workers [8]. In the cited work reasonable, but not
complete, pressure-wise reconstruction of ensemble-mean interatomic distances
from the active Ge site for Ge K$\beta$ XES of amorphous GeO2 was achieved
from the eight spectral moments. Motivated by earlier observations of unstable
approximate inverse, even due to small discrepancies, in the NNCA procedure
the activation values $\mathbf{t}$ of the bottleneck are optimized by least-
squares fitting for the target variables, after which the expansion of
Equation (3) is taken. This expansion corresponds to (pseudo-)inverting the
first matrix layer between input $\mathbf{x}$ and scores $\mathbf{t}$,
illustrated for the NN of Figure 4.

Figure 5: Mean-structural-parameter-based distances in each pressure from the
central Ge atom (a) Reconstructed from the fitted NNCA components for both one
and two-component models. Dashed lines correspond to the two-component model.
(b) Known mean distances calculated from the atomic coordinates of the test
dataset. Drawn following Ref. 8

Figure 5 depicts the results for each pressure in the dataset together with
the known values. The rapid fall of Ge–O interatomic distances up to 20 GPa
coincides with the rapid decrease of 4-fold Ge–O coordination in the amorphous
GeO2 [18]. However, the changes related to interatomic Ge–Ge distances are not
reconstructed by NNCA. When compared to reconstruction by ECA [8] the NNCA
outperforms in regard to Ge–O distances, but leaves out the effect in Ge–Ge
distances. Moreover, the one-component model gives the more accurate
structural reconstruction out of the two. The applied expansion method
predicts the mean value for spectrally irrelevant parameters due to z-score
standardization of the input. Thus a criterion probably exists for a component
of a basis vector (and the respective input feature) to be deemed indecisive
in expansion of Eq. (3). We leave the investigation of this condition for
future, because the study requires data on more systems than currently
available to us.

##  IV Conclusions

Encoder–decoder neural networks (EDNN) with a bottleneck of $k$ neurons can
cover more spectral variance than an emulator-based component analysis (ECA)
decomposition of the same rank. This happens owing to the flexibility of the
encoder and its related adaptivity to nonlinear behaviour in the
multidimensional input space. However, the improved covered target variance
does not come without a cost: interpretation of the latent degrees of freedom
behind spectral sensitivity, i.e. the activation values at the bottleneck,
becomes problematic in itself.

These problems of EDNN could be cured by implementation of the projection of
input to latent variables by basis vectors such as in ECA. When this new NNCA
approach was implemented as a part of the neural network training and model
selection, we observed expected rise in the covered variance to that of ECA.
The approach helped to recover the coordination change in amorphous GeO2 at
elevated pressures, in a qualitative agreement with previous studies. However,
this improved performance implied the loss of the hierarchy present of the
basis vectors, which could in principle be fixed by posterior unitary
transformation of the vectors.

Problems in finding inverse functions in the EDNN are indicative of the loss
of structural information upon spectrum formation. However this information
may be available from the structural simulation data behind the spectrum
calculations, which may leak into interpretation of spectra. Due to this risk
of overinterpretation of spectra by information leakage, investigation of
reconstructable information is an essential aspect of interpreting X-ray
spectra with the help of simulations. Indeed, a realistic interpretation of
spectra can only comment on structural characteristics that have an effect on
them.

###### Acknowledgements.

Academy of Finland is acknowledged for funding via project 331234. The authors
acknowledge CSC – IT Center for Science, Finland, and the FGCI - Finnish Grid
and Cloud Infrastructure for computational resources. The authors thank Mr.
E.A. Eronen for discussions.

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## Supplementary information

### Model Selections

The parameter grids for the randomized searches carried out for EDNN models
are presented in Table 3.

Table 3: Model selection search spaces for H2O and GeO2 EDNN models. Here lr is the learning rate, $\alpha$ the strength of L2-regularization and $n_{hidden,enc/dec}$ the number of hidden layers in encoder and decoder parts of the model. The width of each layer was independently varied in the search. | lr | $\alpha$ | $n_{hidden,enc}$ | $n_{hidden,dec}$ | Width of hidden layers
---|---|---|---|---|---
H2O | $10^{-4},10^{-3}$ | $10^{-9},10^{-8},...10^{-4}$ | 2, 3, 4 | 2, 3, 4 | 64, 128, 256
GeO2 | $10^{-4},10^{-3}$ | $10^{-9},10^{-8},...10^{-1}$ | 2, 3, 4 | 2, 3, 4 | 8, 32, 64, 128, 256

The parameter grids for the exhaustive searches carried out for EDNN models
with maximal number of square weight matrices and LReLU activation, aiming for
easist inverse problem, are presented in Table 4. The condition for square
weight matrices implies the layer widths of the input size for the encoder,
and the layer widths of the output size for the decoder.

Table 4: Model selection search spaces for H2O and GeO2 EDNN inverse optimized models. Here lr is the learning rate, $\alpha$ the strength of L2-regularization and $n_{hidden,enc/dec}$ the number of hidden layers in encoder and decoder parts of the model. | lr | $\alpha$ | $n_{hidden,enc}$ | $n_{hidden,dec}$ | Encoder width | Decoder width
---|---|---|---|---|---|---
H2O | $10^{-3}$ | $10^{-4},...10^{-1}$ | 1, 2, 3, 4, 5 | 1, 2, 3, 4, 5 | 3 | 100
GeO2 | $10^{-3}$ | $10^{-4},...10^{-1}$ | 1, 2, 3, 4, 5 | 1, 2, 3, 4, 5 | 153 | 8

The parameter grids for the exhaustive searches carried out for NNCA models
are presented in Table 5.

Table 5: Model selection search spaces for H2O and GeO2 NNCA models. Here lr is the learning rate, $\alpha$ the strength of L2-regularization and $n_{hidden,dec}$ the number of hidden layers in the decoder part of the model. The width of each layer was independently varied in the search. Decoder search spaces are identical with ECA emulator search spaces apart from the learning rate [6, 8]. | lr | $\alpha$ | $n_{hidden,dec}$ | Width of hidden layers
---|---|---|---|---
H2O | $5\cdot 10^{-3}$ | $10^{-12},...10^{4}$ | 2, 3, 4, 5 | 5, 10, 50, 100, 200, 500
GeO2 | $5\cdot 10^{-3}$ | $10^{-10},10^{-8},...10^{2}$ | 2, 3 | 64, 128

### Inverse studies for GeO2

Layer-wise maximum absolute errors (MAE) for EDNN activation values using
amorphous GeO2 are shown in figure 6. The discussed EDNN architecture contains
maximal number of square weight matrices and LReLU activation aiming for
easist inverse problem. We find that direct inverse transform (without
fitting) of the decoder part of the model is possible from predicted values,
when we composed the model from suitable parts. However, this reconstruction
from known test set values, or values with added noise, diverged. This
indicates high instability in regard to the exact value of
$\mathbf{\tilde{m}}$. We chose the weight matrices to be square, and thus to
have as many equally-sized hidden layers as possible. The latter can not be
fulfilled for the encoder, unless the dimension of input does not already
equal to the width of the bottleneck. For a complicated system, this is not
the case and non-bijective behaviour guaranteed.

The study of error build-up, presented in Figure 6, indicates this effort was
futile. Panel a) of the figure shows the maximum absolute error made to the
activation values, when approached from the final predicted spectral moments.
The reconstruction error originates from the bottleneck and the large weight
matrices of the encoder, whereas selection of the decoder to match the
dimensions of the output introduces rather small deviations – when the exact
predicted value for $\mathbf{\tilde{m}}$ is started from. A complementary view
is obtained when this inverse is taken from the correct values of the forwards
pass, shown in Figure 6 b). Whereas the encoder causes a rapid rise, the
decoder introduces no further effects. Similar problems were observed for the
simpler H2O system, where this analysis shows the error to build up solely in
the decoder and the bottleneck.

Figure 6: (a) Layer-wise maximum absolute error during inverse for the
spectral moments of K$\beta$ XES of amorphous GeO2 with maximal number of
square weight matrices and LReLU activation. (b) Maximum absolute error at the
input layer after forward pass to a given layer and approximate inverse from
there. The relative threshold $th$ for inclusion of small singular values
(with respect to the largest singular value) was varied for the Moore–Penrose
pseudo-inverse. The default method applies numerically ”exact” inverse and
uses pseudo-inverse with $th=10^{-15}$ only when necessary. Dashed gray line
denotes the location of the bottleneck ($k=1$) layer.

### Inverse studies for H2O

Layer-wise maximum absolute errors (MAE) for EDNN activation values using H2O
molecule are shown in figure 7. The discussed EDNN architecture contains
maximal number of square weight matrices and LReLU activation aiming for
easist inverse problem. From the figure error is seen to originate from the
bottleneck and subsequent layers. Similar to GeO2, the bottleneck and the
associated ill-conditioned weight matrices produce error Figures. 7(b,d,f). On
the contrary to GeO2 the error also builds up through the decoder (Figure
7(a,c,e)) whereas the inverse of the encoder behaves accurately. Investigation
of the decoder weight matrices for H2O models shows a wide spread in the order
of magnitude of their singular values, which indicates these matrices to be
ill-conditioned and sensitive to small perturbations.

Figure 7: (a,c,e): Layer-wise maximum absolute error (MAE) during inverse for
the three spectra of H2O with maximal number of square weight matrices and
LReLU activation. (b,d,f): Maximum absolute error at the input layer after
forward pass to a given layer and approximate inverse from there. The relative
threshold $th$ for inclusion of small singular values (with respect to the
largest singular value) was varied for the Moore–Penrose pseudo-inverse. The
default method applies numerically ”exact” inverse and uses pseudo-inverse
with $th=10^{-15}$ only when necessary. Dashed gray line denotes the location
of the bottleneck ($k=1$) layer.

###  GeO2 component vectors

Figure 8 depicts the NNCA basis vectors, compared to the ECA basis vectors
from Ref. 8, in the standardized feature space reordered in the shape of the
according Coulomb matrix. For NNCA, interatomic distances to the active site
have largest absolute values, which indicates strongets spectral sensitivity.
Noteworthily correlations between non-active sites plays a role in the
spectral moment outcome, especially for ECA. Differences between the two can
be observed; e.g. ECA shows more sensitivity towards Ge distances from the
active site. As a result the distance reconstruction using NNCA shows flatter
curves for Ge atoms.

Figure 8: NNCA vectors of the one and two-component GeO2 models compared to
ECA basis vectors from Ref. [8]. Drawn following Ref. [8].


# Resonance fluorescence in $\Lambda$, $V$ and $\Xi$ \- type three-level
configurations

Surajit Sen†,‡,1, Tushar Kanti Dey‡,2 and Bimalendu Deb§,3 † Department of
Physics, Guru Charan College, Silchar 788004, India  ‡ Centre of Advanced
Studies and Innovation Lab (CASILab)
18/27 Kali Mohan Road, Silchar 788003, India  § Indian Association for the
Cultivation of Science, Jadavpur, Kolkata 700032, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We theoretically study the resonance fluorescence spectra of the lambda
($\Lambda$), vee ($V$) and cascade ($\Xi$) type three-level configurations. It
is shown that each system with two detuning frequencies can be modelled using
the $SU(3)$ symmetry group to derive a generalized optical Bloch equation. For
each configuration, this equation is solved to calculate the two-time
correlation function by invoking the quantum regression theorem. The
incoherent part of the power spectra gives the characteristic multi-peak
fluorescence profiles which are different for different configurations. We
also discuss how the dressed-state structure of such system can explain the
origin of quintuplet profile of the fluorescent spectrum.

Keywords: Resonance Fluorescence, Three-level System, SU(3) Group, Optical
Bloch Equation, Power Spectrum

##  1 Introduction

The behavior of two-, three-, or multi-level systems when subjected to an
intense electromagnetic field gives rise to numerous intriguing spectral
features that may find a wide range of applications in quantum optics and
quantum metrology. Historically, a two-level system driven by a strong
resonant field was first investigated by Mollow who showed the existence of
two symmetrical sidebands in the both sides of the central Rayleigh peak [1,
2]. In the three-peak spectrum, popularly known as Mollow triplet, the
sidebands are located at a shift proportional to the Rabi frequency of the
system with their intensity equal to one-third that of the central peak. This
phenomenon is known as Resonance Fluorescence (RF) which arises due to the
transitions between the dressed states of the effective system formed under
the action of an intense external field [3]. The predecessors of this light-
matter interaction model are, Wigner-Weisskopf model of the spontaneous
emission [4] which leads to Lorentzian spectra, Autler-Townes (AT) effect,
where a doublet spectral feature is observed if a two-level system is
resonantly driven by a strong oscillatory electromagnetic field [5] and the
Fano model [6] which is characterized by an asymmetric spectrum resulting due
to the presence of a continuum state among the states involved. The
experimental observation of the anti-bunching of photons in the intensity-
intensity correlation is an important outcome of the strongly coupled light-
matter quantum dynamics [7, 8].

$E_{3},\ket{3}$$\Delta_{13}^{\Lambda}$$E_{2},\ket{2}$$\Delta_{23}^{\Lambda}$$E_{1},\ket{1}$$V_{\pm}$$T_{\pm}$

(a) $\Lambda$ Configuration

$E_{3},\ket{3}$$\Delta_{13}^{V}$$E_{2},\ket{2}$$\Delta_{12}^{V}$$E_{1},\ket{1}$$V_{\pm}$$U_{\pm}$

(b) $V$ Configuration

$E_{3},\ket{3}$$\Delta_{23}^{\Xi}$$E_{2},\ket{2}$$\Delta_{12}^{\Xi}$$E_{1},\ket{3}$$T_{\pm}$$U_{\pm}$

(c) $\Xi$ Configuration

Figure 1: a) The $\Lambda$ configuration is characterized by the transitions
$|1\rangle$ $\longleftrightarrow$ $|3\rangle$ ($V_{\pm}$) and $|3\rangle$
$\longleftrightarrow$ $|2\rangle$ ($T_{\pm}$) with detuning frequencies
$\Delta_{13}^{\Lambda}$ and $\Delta_{23}^{\Lambda}$; b) the $V$ configuration
is characterized by the transitions $|3\rangle$ $\longleftrightarrow$
$|1\rangle$ ($V_{\pm}$) and $|2\rangle$ $\longleftrightarrow$ $|1\rangle$
($U_{\pm}$) with the detuning frequencies $\Delta_{13}^{V}$ and
$\Delta_{12}^{V}$.; c) the $\Xi$ configuration with transitions $|1\rangle$
$\longleftrightarrow$ $|2\rangle$ ($U_{\pm}$) and $|2\rangle$
$\longleftrightarrow$ $|3\rangle$ ($T_{\pm}$) with detuning frequencies
$\Delta_{23}^{\Xi}$ and $\Delta_{12}^{\Xi}$. The energy $E_{i}$ ($i=1,2,3$) of
the uncoupled atomic states $\ket{i}$ are arranged as $E_{1}<E_{2}<E_{3}$.

The generalization of the two-level system to the three-level one is quite
nontrivial due to three distinct categories of transitions known as, the
lambda ($\Lambda$), vee ($V$) and cascade ($\Xi$) type configurations shown in
Fig.1. In the recent past the three-level system has drawn considerable
attention because it exhibits wide range of quantum-optical phenomena such as
two photon coherence [9], double resonance process [10], three-level super-
radiance [11], resonance Raman scattering [12], population trapping [2], tri-
level echoes [13], STIRAP [14], quantum jump [15], quantum zeno effect [16],
three-level Fano effect [17], electromagnetically induced transparency (EIT)
[18, 19, 20] etc. In context with the resonance fluorescence, it is revealed
that Mollow-like sideband is not limited to the two-level systems alone, but
is also displayed by the multilevel system, including three-level systems
driven by two intense driving fields [21, 22, 23].

In the eighties, the RF in a two-level system gained prominence due to its
inherent significance, particularly with the introduction of damping induced
by the squeezed vacuum [24, 25]. Since then, the RF in three-level system is
being addressed in context with the squeezed vacuum using the $V$ [26, 27, 28]
and $\Xi$ type of three-level configurations were studied [29]. Apart from
that, it was further investigated by considering the emission spectrum of two-
photon resonant excitation using the $\Xi$ model [30], in the nano-particle
system [31] and in trapped condition using $\Lambda$ configuration [32]. In
the realm of three-level configurations, the topology of each configuration is
fundamentally different from one another due to their distinct
characteristics. Therefore, despite the aforesaid works, it is crucial to
adopt a comprehensive approach to study the RF for all three-level
configurations particularly to decipher the distinct structure of the dressed
states of each system.

Operationally, a three-level configuration consists of two coupled two-level
systems where two dipole transitions are involved shown in Fig.1. Such a
system with multiple Lindblad terms takes into account the effects of
spontaneous emission and dephasing caused by the vacuum fluctuations in two
distinct transition pathways. Consequently, the solution to such a model
becomes quite complex to handle theoretically. Recently, the SU(3) group has
been shown to have the potential to unveil a wide range of quantum optical
phenomena involving three-level systems [33, 34, 35, 20, 17]. It is therefore
interesting to explore the RF of all three-level configurations using this
group theoretic method and give a comprehensive comparison of this quantum
phenomenon. Our method provides an elegant approach to solve a set of Optical
Bloch equations (OBE) and subsequently give the correlation function for
fluctuations of the atomic variables around the steady state.

The remaining sections of the paper are organized as follows: In Section II,
using the Gell-Mann matrices of $SU(3)$ group, we have revisited the
construction of the dissipative $\Lambda$, $V$ and $\Xi$ configurations where
the dissipation is taken care of by the Lindblad term. Then in Section-III,
the OBE of each configuration is derived. The two-time correlation function
using the quantum regression theorem is presented in Section-IV. The power
spectra of all configurations are compared and the structure of the dressed
states of three-level system is enunciated in Section V to understand the the
quintuplet spectrum of resonance fluorescence. In the final Section, we
highlight the main results of the paper and discuss the outlook.

##  2 The Models

The dissipative process in the $\Lambda,V$ and $\Xi$ type three-level
configurations can be described by the Liouville equation with Lindblad term
($A=\Lambda,V,\Xi$),

$\displaystyle\frac{{d\rho^{A}}}{{dt}}=-i\left[{H^{A},\rho^{A}}\right]+\mathfrak{L}_{D}^{A},$
(1)

where the density matrix is given by

$\displaystyle\rho^{A}=\left(\begin{array}[]{ccc}{\rho_{33}}&{\rho_{32}}&{\rho_{31}}\cr{\rho_{23}}&{\rho_{22}}&{\rho_{21}}\cr{\rho_{13}}&{\rho_{12}}&{\rho_{11}}\cr\end{array}\right)$
(5)

written in the the atomic basis states are $|1\rangle=(0,0,1)^{T}$,
$|2\rangle=(0,1,0)^{T}$ and $|3\rangle=(1,0,0)^{T}$ with
$\mathfrak{L}_{D}^{A}$ as the Lindblad term. Using Gellmann matrices of the
$SU(3)$ representation, the Hamiltonian and Lindblad terms of these
configurations are given by [34, 35],

$\displaystyle
H^{\Lambda}=\left(\begin{array}[]{ccc}\frac{1}{3}\big{(}\Delta_{13}^{\Lambda}+\Delta_{23}^{\Lambda}\big{)}&g_{23}&g_{13}\\\
g_{23}&\frac{1}{3}\big{(}\Delta_{13}^{\Lambda}-2\Delta_{23}^{\Lambda}\big{)}&0\\\
g_{13}&0&-\frac{1}{3}\big{(}2\Delta_{13}^{\Lambda}-\Delta_{23}^{\Lambda}\big{)},\end{array}\right),$
(6d) $\displaystyle\mathfrak{L}_{D}^{\Lambda}$
$\displaystyle=\Gamma_{31}^{\Lambda}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+})$
(6e)
$\displaystyle+\Gamma_{32}^{\Lambda}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}),$

for the $\Lambda$ system with the detuning offsets,
$\Delta_{13}^{\Lambda}=2\omega_{31}+\omega_{32}-\Omega_{13}$ and
$\Delta_{23}^{\Lambda}=\omega_{31}+2\omega_{32}-\Omega_{23}$,

$\displaystyle
H^{V}=\left(\begin{array}[]{ccc}\frac{1}{3}\big{(}2\Delta_{13}^{V}-\Delta_{12}^{V}\big{)}&0&g_{13}\\\
0&\frac{1}{3}\big{(}2\Delta_{12}^{V}-\Delta_{13}^{V}\big{)}&g_{12}\\\
g_{13}&g_{12}&-\frac{1}{3}\big{(}\Delta_{12}^{V}+\Delta_{13}^{V}\big{)},\\\
\end{array}\right)$ (6gd) $\displaystyle\mathfrak{L}_{D}^{V}$
$\displaystyle=\Gamma_{31}^{V}(V_{-}\rho^{V}V_{+}-\frac{1}{2}V_{+}V_{-}\rho^{V}-\frac{1}{2}\rho^{V}V_{+}V_{-})$
(6ge)
$\displaystyle+\Gamma_{21}^{V}(U_{-}\rho^{V}U_{+}-\frac{1}{2}U_{+}U_{-}\rho^{V}-\frac{1}{2}\rho^{V}U_{+}U_{-}),$

for the $V$ system with
$\Delta_{12}^{V}=\omega_{31}+2\omega_{21}-\Omega_{12}$,
$\Delta_{13}^{V}=2\omega_{31}+\omega_{21}-\Omega_{13}$ and

$\displaystyle
H^{\Xi}=\left(\begin{array}[]{ccc}\frac{1}{3}\big{(}\Delta_{12}^{\Xi}+2\Delta_{23}^{\Xi}\big{)}&g_{23}&0\\\
g_{23}&\frac{1}{3}\big{(}\Delta_{12}^{\Xi}-\Delta_{23}^{\Xi}\big{)}&g_{12}\\\
0&g_{12}&-\frac{1}{3}\big{(}2\Delta_{12}^{\Xi}+\Delta_{23}^{\Xi}\big{)}\end{array}\right),$
(6ghd) $\displaystyle\mathfrak{L}_{D}^{\Xi}$
$\displaystyle=\Gamma_{32}^{\Xi}(T_{-}\rho^{\Xi}T_{+}-\frac{1}{2}T_{+}T_{-}\rho^{\Xi}-\frac{1}{2}\rho^{\Xi}T_{+}T_{-})$
(6ghe)
$\displaystyle+\Gamma_{21}^{\Xi}(U_{-}\rho^{\Xi}U_{+}-\frac{1}{2}U_{+}U_{-}\rho^{\Xi}-\frac{1}{2}\rho^{\Xi}U_{+}U_{-}),$

for the $\Xi$ system with detuning
$\Delta_{23}^{\Xi}=2\omega_{21}+\omega_{32}-\Omega_{23}$ and
$\Delta_{12}^{\Xi}=2\omega_{32}-2\omega_{21}-\Omega_{12}$. Here, $g_{ij}$
($i,j=1,2,3$) is the system-field coupling parameter with the laser field with
frequency $\Omega_{ij}$, $\Gamma_{ji}$ be the decay constant for the
transition pathway $j\rightarrow i$ ($j>i$) and
$\omega_{ij}=\omega_{i}-\omega_{j}$ be the difference between transition
frequencies of $\ket{i}$-th and $\ket{j}$-th levels with $\hbar\omega_{1}$
($=E_{1}$), $\hbar\omega_{2}$ ($=E_{2}$) and $\hbar\omega_{3}$ ($=E_{3}$) be
the absolute energies of three levels shown in Fig.1. The SU(3) shift vectors
appearing in Equations (6d-6ghe) are given by [36],

$\displaystyle\eqalign{T_{\pm}&=\frac{1}{2}(\lambda_{1}\pm i\lambda_{2}),\quad
V_{\pm}=\frac{1}{2}(\lambda_{4}\pm i\lambda_{5}),\quad
U_{\pm}=\frac{1}{2}(\lambda_{6}\pm i\lambda_{7})\cr T_{3}&=\lambda_{3},\quad
V_{3}=\frac{1}{2}(\sqrt{8}\lambda_{8}+\lambda_{3}),\quad
U_{3}=\frac{1}{2}(\sqrt{8}\lambda_{8}-\lambda_{3}).}$ (6ghk)

where the Gell-Mann matrices are given by,

$\displaystyle\eqalign{\lambda_{0}=&\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\\\
\end{array}\right),\quad\lambda_{1}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\\\
\end{array}\right),\quad\lambda_{2}=\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&0\\\ 0&0&0\\\
\end{array}\right),\cr\lambda_{3}=&\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\\\
\end{array}\right),\quad\lambda_{4}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\\\
\end{array}\right),\quad\lambda_{5}=\left(\begin{array}[]{ccc}0&0&-i\\\
0&0&0\\\ i&0&0\\\
\end{array}\right),\cr\lambda_{6}=&\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&1&0\\\
\end{array}\right),\quad\lambda_{7}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\ 0&i&0\\\
\end{array}\right),\quad\lambda_{8}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\\\ \end{array}\right),}$ (6ghap)

which are normalized as
$\lambda_{l}\lambda_{m}=\delta_{lm}+d_{lmn}\lambda_{n}+f_{lmp}\lambda_{p}$
with $d_{lmn}$ and $f_{lmp}$ ($l,m,n,p=1,2,\dots,8$) being the completely
symmetric and completely antisymmetric structure constants. It is worth
mentioning here that aforesaid time-independent Hamiltonian appearing in
Equations (6d,6gd,6ghd) can be obtained by using the transformation [34, 35],

$\displaystyle\eqalign{H^{A}={U^{A}}^{\dagger}(t)H^{A}(t)U^{A}(t)-i\bigg{[}\frac{d{U^{A}}(t)}{dt}\bigg{]}{U^{A}}^{{\dagger}}(t),}$
(6ghar)

where the unitary operators are given by,

$\displaystyle
U^{\Lambda}(t)=\exp[-\frac{i}{3}\big{(}(2\Omega_{13}-\Omega_{23})V_{3}+(2\Omega_{23}-\Omega_{13})T_{3}\big{)}\big{]}t,$
(6ghasa) $\displaystyle
U^{V}(t)=\exp[-\frac{i}{3}\big{(}(2\Omega_{12}-\Omega_{13})U_{3}+(\Omega_{13}-2\Omega_{23})V_{3}\big{)}\big{]}t,$
(6ghasb) $\displaystyle
U^{\Xi}(t)=\exp[-\frac{i}{3}\big{(}(2\Omega_{12}+\Omega_{23})U_{3}+(\Omega_{12}+2\Omega_{23})T_{3}\big{)}\big{]}t,$
(6ghasc)

with the time-dependent Hamiltonians of three configurations,

$\displaystyle\qquad
H^{\Lambda}(t)=\omega_{31}V_{3}+\omega_{32}T_{3}+g_{13}V_{+}\exp(-i\Omega_{13}t)+g_{23}T_{+}\exp(-i\Omega_{23}t)+h.c.,$
(6ghasata) $\displaystyle\qquad
H^{V}(t)=\omega_{31}V_{3}+\omega_{21}U_{3}+g_{13}V_{+}\exp(-i\Omega_{13}t)+g_{12}U_{+}\exp(-i\Omega_{12}t)+h.c.,$
(6ghasatb) $\displaystyle\qquad
H^{\Xi}(t)=\omega_{32}T_{3}+\omega_{21}U_{3}+g_{12}U_{+}\exp(-i\Omega_{12}t)+g_{23}T_{+}\exp(-i\Omega_{23}t)+h.c..$
(6ghasatc)

Having knowledge about the model Hamiltonian of all three configurations and
the Lindblad term which characterizes the dissipation in such system, we
proceed to derive the Optical Bloch Equations.

##  3 Bloch Equation for Three-level Configuration

The Bloch vector $\mathbf{S}^{A}_{\mathbb{P}_{i}}(t)$ of a generic three-level
configuration is defined as,

$\displaystyle{\mathbf{S}}^{A}_{\mathbb{P}_{i}}(t)=Tr[\rho^{A}(t){\mathbb{P}_{i}}],$
(6ghasatau)

where $\mathbb{P}_{i}$ ($T_{+,-,3},V_{+,-,3},U_{+,-,3}$) is the $SU(3)$ shift
operators. Using the algebra of the shift vectors it is easy to see that,

$\displaystyle\frac{d\mathbf{S}^{A}_{\mathbb{P}_{i}}(t)}{dt}=Tr\big{[}\frac{d\rho^{A}(t)}{dt}{\mathbb{P}_{i}}\big{]},$
(6ghasatav)
$\displaystyle=-i\big{\langle}\big{[}H^{A},\rho^{A}\big{]}{\mathbb{P}_{i}}\big{\rangle}+\big{\langle}\mathfrak{L}_{D}^{A}{\mathbb{P}_{i}}\big{\rangle},$
(6ghasataw)

where Equation (1) is used along with
$\big{\langle}{\mathcal{\hat{O}}}\big{\rangle}^{A}=\Tr[{\mathcal{\hat{O}}}\rho^{A}]$.
Using the Lindblad terms in Equations (6e, 6ge, 6ghe), Equation (6ghasatav)
for three configurations are given by,

$\displaystyle\frac{d\mathbf{S}^{\Lambda}_{\mathbb{P}_{i}}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda}){\mathbb{P}_{i}}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda}){\mathbb{P}_{i}}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda}){\mathbb{P}_{i}}\big{]},$
(6ghasataxa)
$\displaystyle\frac{d\mathbf{S}^{V}_{\mathbb{P}_{i}}(t)}{dt}=-i\Tr[(H^{V}\rho^{V}-\rho^{V}H^{V}){\mathbb{P}_{i}}\big{]}+\Gamma_{31}Tr\big{[}(V_{-}\rho^{\Lambda}V_{+}-\frac{1}{2}\rho^{\Lambda}V_{+}V_{-}-\frac{1}{2}V_{+}V_{-}\rho^{\Lambda}\big{)}{\mathbb{P}_{i}}]$
$\displaystyle+\Gamma_{21}Tr\big{[}(U_{-}\rho^{V}U_{+}-\frac{1}{2}\rho^{V}U_{+}U_{-}-\frac{1}{2}U_{+}U_{-}\rho^{V}){\mathbb{P}_{i}}\big{]},$
(6ghasataxb)
$\displaystyle\frac{d\mathbf{S}^{\Xi}_{\mathbb{P}_{i}}(t)}{dt}=-i\Tr[(H^{\Xi}\rho^{\Xi}-\rho^{\Xi}H^{\Xi}){\mathbb{P}_{i}}\big{]}+\Gamma_{21}Tr\big{[}(U_{-}\rho^{\Xi}U_{+}-\frac{1}{2}\rho^{\Xi}U_{+}U_{-}-\frac{1}{2}U_{+}U_{-}\rho^{\Xi}){\mathbb{P}_{i}}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{-}\rho^{\Xi}T_{+}-\frac{1}{2}\rho^{\Xi}T_{+}T_{-}-\frac{1}{2}T_{+}T_{-}\rho^{\Xi}){\mathbb{P}_{i}}\big{]},$
(6ghasataxc)

Thus we note that only a pair of SU(3) projection vectors are involved for a
specific configuration.

To derive the OBE of the $\Lambda$ configuration, we first substitute the
shift vector $\mathbb{P}_{i}=V_{\pm},V_{3},T_{\pm},T_{3},U_{\pm},U_{3}$ in
Equation (6ghasatax) and obtain,

$\displaystyle\frac{dS_{T_{+}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})T_{+}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})T_{+}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})T_{+}\big{]},$
(6ghasataxaya)
$\displaystyle\frac{dS_{T_{-}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})T_{-}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})T_{-}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})T_{-}\big{]},$
(6ghasataxayb)
$\displaystyle\frac{dS_{T_{3}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})T_{3}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda}\big{)}T_{3}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+})T_{3}\big{]},$
(6ghasataxayc)
$\displaystyle\frac{dS_{V_{+}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})V_{+}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})V_{+}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})V_{+}\big{]},$
(6ghasataxayd)
$\displaystyle\frac{dS_{V_{-}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})V_{-}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})V_{-}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})V_{-}\big{]},$
(6ghasataxaye)
$\displaystyle\frac{dS_{V_{3}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})V_{3}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})V_{3}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})V_{3}\big{]},$
(6ghasataxayf)
$\displaystyle\frac{dS_{U_{+}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})U_{+}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})U_{+}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{(}T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})U_{+}\big{]},$
(6ghasataxayg)
$\displaystyle\frac{dS_{U_{-}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})U_{-}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{-}V_{+}-\frac{1}{2}V_{-}V_{+}\rho^{\Lambda})U_{-}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{-}T_{+}-\frac{1}{2}T_{-}T_{+}\rho^{\Lambda})U_{-}\big{]},$
(6ghasataxayh)
$\displaystyle\frac{dS_{U_{3}}^{\Lambda}(t)}{dt}=-i\Tr[(H^{\Lambda}\rho^{\Lambda}-\rho^{\Lambda}H^{\Lambda})U_{3}\big{]}+\Gamma_{31}Tr\big{[}(V_{+}\rho^{\Lambda}V_{-}-\frac{1}{2}\rho^{\Lambda}V_{+}V_{-}-\frac{1}{2}V_{+}V_{-}\rho^{\Lambda})U_{3}\big{]}$
$\displaystyle+\Gamma_{32}Tr\big{[}(T_{+}\rho^{\Lambda}T_{-}-\frac{1}{2}\rho^{\Lambda}T_{+}T_{-}-\frac{1}{2}T_{+}T_{-}\rho^{\Lambda})U_{3}\big{]},$
(6ghasataxayi)

Equation (6ghasatau) supplemented by the normalization condition
$\Tr\rho^{\Lambda}(t)=1$ gives the elements of the density matrix in terms of
the Bloch vectors,

$\eqalign{\rho_{33}^{\Lambda}&=\frac{1}{3}(1+S_{T_{3}}^{\Lambda}+S_{V_{3}}^{\Lambda}),\quad\rho_{32}^{\Lambda}=S_{T_{-}^{\Lambda}},\quad\rho_{31}^{\Lambda}=S_{V_{-}}^{\Lambda},\\\
\rho_{23}^{\Lambda}&=S_{T_{+}}^{\Lambda},\quad\rho_{22}^{\Lambda}=\frac{1}{3}(1-S_{T_{3}}^{\Lambda}+S_{U_{3}}^{\Lambda}),\quad\rho_{21}=S_{U_{-}}^{\Lambda},\\\
\rho_{13}^{\Lambda}&=S_{V_{+}}^{\Lambda},\quad\rho_{12}^{\Lambda}=S_{U_{+}}^{\Lambda},\quad\rho_{11}^{\Lambda}=\frac{1}{3}(1-S_{U_{3}}^{\Lambda}-S_{V_{3}}^{\Lambda}).}$
(6ghasataxayaz)

Finally substituting the elements of the density matrix from Equation
(6ghasataxayaz) into (6ghasataxay-6ghasataxay), we obtain the desired OBE of
the $\Lambda$ configuration,

$\displaystyle\frac{d\mathbf{S^{\Lambda}_{\mathbb{P}_{i}}}(t)}{dt}=M^{\Lambda}\mathbf{S^{\Lambda}_{\mathbb{P}_{i}}}+\mathbf{B^{\Lambda}},$
(6ghasataxayba)

where the Bloch matrix $M^{\Lambda}$ and the inhomogeneous term
$\mathbf{B^{\Lambda}}$ are given in Appendix. Proceeding similar way, the OBE
for the $V$ and $\Xi$ type configurations are obtained as shown in Appendix.

##  4 Power Spectrum

The evaluation of the power spectrum requires the information about the two-
time correlation function of the fluctuation about the steady state. This is
usually done by using the quantum regression theorem. In the long time limit
(${t\rightarrow\infty}$) when the system attains the steady state we have,

$\displaystyle\frac{d\langle\mathbf{S^{A}_{\mathbb{P}_{i}}(t)\rangle}}{dt}\bigg{|}_{t\rightarrow\infty}=0=M^{\Lambda}\mathbf{\langle
S_{{\mathbb{P}_{i}}}^{A}(\infty)\rangle}_{s}+\mathbf{B^{A}}.$ (6ghasataxaybb)

Now introducing the fluctuation of the atomic variable around the steady
state, namely,
$\langle\delta{\mathbf{S^{A}_{\mathbb{P}_{i}}}(t)}\rangle=\mathbf{S^{A}_{\mathbb{P}_{i}}}(t)-\langle\mathbf{S^{A}_{\mathbb{P}_{i}}(\infty)\rangle}_{s}\equiv\langle\langle\mathbb{P}_{i}(t)\rangle\rangle$,
the in-homogeneous term $\mathbf{B^{A}}$ can be eliminated,

$\displaystyle\frac{d\langle\langle\mathbb{P}_{i}(t)\rangle\rangle}{dt}=M^{A}\langle\langle\mathbb{P}_{i}(t)\rangle\rangle.$
(6ghasataxaybc)

According to the quantum regression theorem [37], the two-time correlation
function $\mathbf{K}^{A}_{\mathbb{P}_{i}\mathbb{P}_{j}}(\tau)$ is given by

$\displaystyle\frac{d\mathbf{K}^{A}_{\mathbb{P}_{i}\mathbb{P}_{j}}(\tau)}{d\tau}=M^{A}\mathbf{K}^{A}_{\mathbb{P}_{i}\mathbb{P}_{j}}(\tau),$
(6ghasataxaybd)

where the incoherent part of the correlation function is given by,

$\displaystyle\mathbf{K}^{\Lambda}_{\mathbb{P}_{i}\mathbb{P}_{j}}(\tau)=\lim_{t\rightarrow\infty}\langle{\delta\mathbf{S^{\Lambda}_{\mathbb{P}_{i}}}}(t+\tau)\mathbf{\delta
S^{\Lambda}_{\mathbb{P}_{j}}(t)}\rangle\equiv\langle\langle\mathbb{P}_{i}(\tau)\mathbb{P}_{j}(0)\rangle\rangle.$
(6ghasataxaybe)

with the two-time correlation function depends upon the difference of time
$\tau$.

In the case of the $\Lambda$ configuration ($A=\Lambda$), in order to
calculate the power spectrum for the $\langle\langle
V_{+}(\tau)V_{-}(0)\rangle\rangle$ in transition pathway ${3}\rightarrow{1}$,
we need to solve the equation,

$\displaystyle\frac{d\mathbf{K}^{\Lambda}_{V_{+}\mathbb{P}_{j}}(\tau)}{d\tau}=M^{\Lambda}\mathbf{K}^{\Lambda}_{V_{+}\mathbb{P}_{j}}(\tau),$
(6ghasataxaybfa) where the correlation vector
$\mathbf{K}^{\Lambda}_{V_{+}\mathbb{P}_{j}}(\tau)$ is given by,
$\displaystyle\mathbf{K}^{\Lambda}_{V_{+}\mathbb{P}_{j}}(\tau)=$
$\displaystyle\big{(}\langle\langle
V_{+}(\tau)T_{+}(0)\rangle\rangle,\langle\langle
V_{+}(\tau)T_{-}(0)\rangle\rangle,\langle\langle
V_{+}(\tau)T_{3}(0)\rangle\rangle,$ $\displaystyle\langle\langle
V_{+}(\tau)V_{+}(0)\rangle\rangle,\langle\langle
V_{+}(t)V_{-}(0)\rangle\rangle,\langle\langle
V_{+}(\tau)V_{3}(0)\rangle\rangle,$ $\displaystyle\langle\langle
V_{+}(\tau)U_{+}(0)\rangle\rangle,\langle\langle
V_{+}(\tau)U_{-}(0)\rangle\rangle,\langle\langle
V_{+}(\tau)U_{3}(0)\rangle\rangle\big{)}^{T},$ To evaluate the correlation
function, we first obtain the initial conditions using GellMann algebra,
$\displaystyle
K_{V_{+}T_{+}}^{A}(0)=0,K_{V_{+}T_{-}}^{A}(0)=0,K_{V_{+}T_{3}}^{A}(0)=0,K_{V_{+}V_{+}}^{A}(0)=0,$
$\displaystyle
K_{V_{+}V_{-}}^{\Lambda}(0)=\frac{1}{3}(1+\langle\langle{T_{3}}\rangle\rangle_{s}+\langle\langle{V_{3}}\rangle\rangle_{s}),\quad
K_{V_{+}V_{3}}^{\Lambda}(0)=-\langle\langle{V_{+}}\rangle\rangle_{s},$
(6ghasataxaybfc) $\displaystyle K_{V_{+}U_{+}}^{\Lambda}(0)=0,\quad
K_{V_{+}U_{-}}^{\Lambda}(0)=\langle\langle{T_{+}}\rangle\rangle_{s},\quad
K_{V_{+}U_{3}}^{\Lambda}(0)=-\langle\langle{V_{+}}\rangle\rangle_{s}$

where $\langle\langle\mathbb{P}_{i}\rangle\rangle_{s}$ is obtained from
Equation (6ghasataxaybc) under the steady state condition.

Proceeding in a similar way, to find the correlation vector
$\mathbf{K}^{\Lambda}_{T_{+}\mathbb{P}_{j}}(\tau)$ and hence $\langle\langle
T_{+}(\tau)T_{-}(0)\rangle\rangle$, we solve,

$\displaystyle\frac{d\mathbf{K}^{\Lambda}_{T_{+}\mathbb{P}_{j}}(\tau)}{d\tau}=M^{\Lambda}\mathbf{K}^{\Lambda}_{T_{+}\mathbb{P}_{j}}(\tau),$
(6ghasataxaybfbga) where the correlation vector is given by
$\displaystyle\mathbf{K}^{\Lambda}_{T_{+}\mathbb{P}_{j}}(\tau)=$
$\displaystyle\big{(}\langle\langle T_{+}(\tau)T_{+}(0)\rangle,\langle\langle
T_{+}(\tau)T_{-}(0)\rangle\rangle,\langle\langle
T_{+}(\tau)T_{3}(0)\rangle\rangle,$ $\displaystyle\langle\langle
T_{+}(\tau)V_{+}(0)\rangle,\langle\langle
T_{+}(\tau)V_{-}(0)\rangle\rangle,\langle\langle
T_{+}(\tau)V_{3}(0)\rangle\rangle,$ $\displaystyle\langle\langle
T_{+}(\tau)U_{+}(0)\rangle\rangle,\langle\langle
T_{+}(\tau)U_{-}(0)\rangle\rangle,\langle\langle
T_{+}(\tau)U_{3}(0)\rangle\rangle\big{)}^{T},$ with the initial conditions,
$\displaystyle
K_{T_{+}T_{+}}^{A}(0)=0,K_{T_{+}T_{-}}^{A}(0)=\frac{1}{3}(1+\langle\langle{T_{3}}\rangle\rangle_{s}+\langle\langle{V_{3}}\rangle\rangle_{s}),$
$\displaystyle
K_{T_{+}T_{3}}^{A}(0)=-\langle\langle{T_{+}}\rangle\rangle_{s},K_{T_{+}V_{+}}^{A}(0)=0,K_{T_{+}V_{-}}^{\Lambda}(0)=0,K_{T_{+}V_{3}}^{\Lambda}(0)=0,$
(6ghasataxaybfbgc) $\displaystyle
K_{T_{+}U_{+}}^{\Lambda}(0)=\langle\langle{V_{+}}\rangle\rangle_{s},\quad
K_{T_{+}U_{-}}^{\Lambda}(0)=0,\quad
K_{T_{+}U_{3}}^{\Lambda}(0)=\langle\langle{T_{+}}\rangle\rangle_{s}$

Finally having knowledge about the correlation function, we proceed to
evaluate the power spectrum of the three-level configuration given by [27,
38],

$\displaystyle
S_{ij}^{A}(\omega)=Re\bigg{\\{}\int_{0}^{\infty}d\tau\mathbf{K}^{A}_{P_{i}P_{j}}(\tau)e^{-i(\omega-\Omega_{jk})\tau}\bigg{\\}}.$
(6ghasataxaybfbgbh)

where $\omega$ be the frequency of the fluorescent light emanating from the
three-level system corresponding to the transition
$\ket{i}\leftrightarrow\ket{j}$. To compute the incoherent spectrum, we solve
the evolution equations of the correlation function
$\mathbf{K}^{A}_{P_{i}P_{j}}(\tau)$ using Laplace transformation with the
initial conditions discussed above. Taking two detuning offsets at resonant
frequencies, $\Delta_{ij}^{A}\approx 0$ and $\Delta_{jk}^{A}\approx 0$, the
incoherent spectrum can be given by the algabraic equation [38, 27],

$\displaystyle S_{ij}^{A}(\omega)\approx
Re\bigg{\\{}\mathbf{K}^{A}_{P_{i}P_{j}}(s)\bigg{|}_{s=-i(\omega-\Omega_{ij}^{A})}^{\Delta_{ij}=0,\Delta_{jk}=0}\bigg{\\}}.$
(6ghasataxaybfbgbi)

with the Laplacian parameter given by $s\approx-i(\omega-\Omega_{ij}^{A})$.

For example, in the $\Lambda$ configuration, we have two distinct transition
pathways $|1\rangle\leftrightarrow|3\rangle$ and
$|2\rangle\leftrightarrow|3\rangle$ which correspond to the correlation
function $\langle\langle V_{+}(\tau)V_{-}(0)\rangle\rangle$ and
$\langle\langle T_{+}(\tau)T_{-}(0)\rangle\rangle$. Then for the pathway
$|1\rangle\leftrightarrow|3\rangle$, the power spectrum is given by,

$\displaystyle S_{13}^{\Lambda}(\omega)\approx
Re\bigg{\\{}\mathbf{K}^{\Lambda}_{V_{+}V_{-}}(s)\bigg{|}_{s=-i(\omega-\Omega_{13}^{\Lambda})}^{\Delta_{13}=0,\Delta_{23}=0}\bigg{\\}},$
(6ghasataxaybfbgbja) and for the transition $\ket{2}\leftrightarrow\ket{3}$ we
obtain, $\displaystyle S_{23}^{\Lambda}(\omega)\approx
Re\bigg{\\{}\mathbf{K}^{\Lambda}_{T_{+}T_{-}}(s)\bigg{|}_{s=-i(\omega-\Omega_{23}^{\Lambda})}^{\Delta_{13}=0,\Delta_{23}=0}\bigg{\\}},$
(6ghasataxaybfbgbjb)

respectively.

The evaluation of the power spectrum for the $V$ and $\Xi$ configurations is
similar where the pairs of correlation functions,
$\\{\expectationvalue{\expectationvalue{V_{+}(\tau)V_{-}(0)}},\expectationvalue{\expectationvalue{U_{+}(\tau)U_{-}(0)}}\\}$
and
$\\{\expectationvalue{\expectationvalue{U_{+}(\tau)U_{-}(0)}},\expectationvalue{\expectationvalue{T_{+}(\tau)T_{-}(0)}}\\}$
are relevant.

##  5 Results and Discussions

We now turn our attention to the resonance fluorescence profiles of different
configurations. To get the desired power spectrum of the $\Lambda$
configuration for two distinct transition pathways, we have to solve Equations
(6ghasataxaybfa) and (6ghasataxaybfbga), subject to the steady state
conditions given by Equations (6ghasataxaybf) and (6ghasataxaybfbg). It is
convenient to work with the dimensionless parameters,
$\tilde{\Delta}_{ij}=\Delta_{ij}/\Gamma$,
$\tilde{\Gamma}_{ij}=\Gamma_{ij}/\Gamma$, $\tilde{g}_{ij}=g_{ij}/\Gamma$,
where the scaling factor $\Gamma$ is different for different configurations
(See figures 2-7). As mentioned earlier, we solve the correlation function
using Laplace transformation method while taking the resonant detuning to be
$\Delta_{13}^{\Lambda}\approx 0$ and $\Delta_{23}^{\Lambda}\approx 0$ and
then, retrace back to frequency domain with the Laplace variables
$s=-i(\omega-\Omega_{13})$ or $s=-i(\omega-\Omega_{23})$ [27, 38].

The power spectra of $\Lambda$ configuration are plotted in Fig.2 and 3 for
various combination of input parameters. It is noteworthy that in the strong
coupling regime with $g_{23}>1$ and $g_{13}>1$, we observe distinctive
quintuplet spectrum as depicted in Fig.2(a,b) and Fig.3(a,b) for the
transition pathways, $\ket{3}\rightarrow\ket{1}$ and
$\ket{3}\rightarrow\ket{2}$, respectively. In contrast, when the laser
coupling in one transition is weak and that in the other transition is strong,
then we observe Mollow-like triplet in the RF of the latter transition as
shown in Figs. 2(d) and 3(c), while a doublet in the RF profile depicted in
Figs.2(c) and 3(d). The appearance of a Mollow-like triplet structure clearly
indicates that the system effectively reduces to a strongly driven two-level
atom when the laser-coupling to the other transition pathway for which the RF
is not observed goes to zero. The emergence of a doublet structure is a
signature of the RF counterpart where we have familiar Autler-Townes
splitting. A strong laser-coupling in $\ket{3}\rightarrow\ket{2}$
($\ket{3}\rightarrow\ket{1}$) transition creates two excited dressed states
the signature of which is then manifested as a doublet in the RF in weakly
coupled $\ket{3}\rightarrow\ket{1}$ ($\ket{3}\rightarrow\ket{2}$) transition.

In a similar way, we can proceed to discuss the spectral features of the
$V$-type configuration as shown in Figs. 4 and 5 which exhibit distinct shapes
of the fluorescent spectra. The distinct shapes of the fluorescent spectra
contrast to the $\Lambda$ configurations is clearly evident. The RF profiles
of the $\Xi$ -type configuration are displayed in Fig.6 and 7. Once again,
with strong couplings for both transition pathways we observe the quintet
resonance profile. However, in the weak coupling regime for one transition and
strong-coupling for other one, we observe a Mollow-like triplet or a doublet
profile only for the transition $\ket{2}\leftrightarrow\ket{1}$. In contrast,
for the $\ket{3}\leftrightarrow\ket{2}$ transition, we witness five-peak
spectra with different shapes.

At this point, it is important to emphasize that the magnitude of some
specific profiles in the $V$ and $\Xi$ configurations which are quite small
compare to other profiles as revealed in Fig.4(a,c), 5(b,d) for the $V$
configuration and Fig.6c, 7(c,d) for the $\Xi$ configuration. In essence, the
unique spectral signatures of the resonance fluorescence serve as another key
identifying feature to differentiate each three-level configuration from the
other two. Among various profiles, appearance of the quintuplet resonance
profile for different type of three-level configurations is in agreement with
the results of previous theoretical investigations [21, 22, 23, 27], however,
in different parameter regimes, the emergence of the Mollow-like triplet and
Autler-Townes-like doublet is noteworthy.

$\ket{D(9,m-1,n})$$\ket{D(8,m-1,n)}$$\ket{D(7,m-1,n)}$$\ket{D(6,m-1,n+1)}$$\ket{D(5,m-1,n+1)}$$\ket{D(4,m-1,n+1)}$$\ket{D(3,m,n)}$$\ket{D(2,m,n)}$$\ket{D(1,m,n)}$$T_{91}$$T_{92}$$T_{93}$$T_{81}$$T_{82}$$T_{83}$$T_{71}$$T_{72}$$T_{73}$$T_{94}$$T_{95}$$T_{96}$$T_{84}$$T_{85}$$T_{86}$$T_{74}$$T_{75}$$T_{76}$$|1\rangle\leftrightarrow|{3}\rangle$$|2\rangle\leftrightarrow|{3}\rangle$
$\mathcal{E}(m,n)$ $\mathcal{E}(m-1,n+1)$ $\mathcal{E}(m-1,n)$

Figure 2: *

Figure 8: Three energy manifolds of the dressed states of the $\Lambda$
configuration with all allowed transitions. For transition pathway
$|3\rangle\rightarrow|{1}\rangle$, (i.e., from energy manifold
$\mathcal{E}(m-1,n)$ to $\mathcal{E}(m,n)$), the nonvanishing transition
amplitude is proportional to the coupling parameter $g_{13}$, while for the
route $|2\rangle\rightarrow|{1}\rangle$, (i.e., from the energy manifold
$\mathcal{E}(m-1,n)$ to $\mathcal{E}(m-1,n+1)$), the amplitude is proportional
to $g_{23}$.

Table 1: Allowed transitions in pathway $\ket{3}\rightarrow\ket{1}$ Sl. No. |  Transition in dressed states | Order of Peak
---|---|---
Transition I |  $\ket{D(9,m-1,n)}$ $\Longrightarrow$ $\ket{D(1,m,n)}$ | 2nd oder
Transition II |  $\ket{D(7,m-1,n)}$ $\Longrightarrow$ $\ket{D(2,m,n)}$ | 1st order
|  $\ket{D(8,m,n)}$ $\Longrightarrow$ $\ket{D(3,m,n)}$ |
Transition III |  $\ket{D(7,m-1,n)}$ $\Longrightarrow$ $\ket{D(1,m,n)}$ | (Central Peak)
|  $\ket{D(8,m-1,n)}$ $\Longrightarrow$ $\ket{D(2,m,n)}$ | Zero-th order
|  $\ket{D(9,m-1,n)}$ $\Longrightarrow$ $\ket{D(3,m,n)}$ |
Transition IV |  $\ket{D(9,m-1,n)}$ $\Longrightarrow$ $\ket{D(2,m,n)}$ | 1st order
|  $\ket{D(8,m-1,n)}$ $\Longrightarrow$ $\ket{D(1,m,n)}$ |
Transition V |  $\ket{D(7,m-1,n)}$ $\Longrightarrow$ $\ket{D(3,m,n)}$ | 2nd order

Finally for completeness, we discuss the dressed states of the three-level
configuration which is indispensable part to understand the origin of
quintuplet fluorescence spectrum and other related phenomena. It is well known
that in presence of strong laser field, the dressed states $\ket{D(i,m,n)}$ is
a coherent superposition of three atom-field bare states, namely,

$\displaystyle\\{\ket{-,m,n},\quad\ket{0,m-1,n+1}\quad\ket{+,m-1,n}\\},$
(6ghasataxaybfbgbjbk)

where $\\{\ket{-},\ket{0},\ket{+}\\}$ represent the atomic states with $m$,
$n$ be the photon numbers of the bi-chromatic fields. The first triplet of the
dressed states for the $\Lambda$ configuration, which constitutes the energy
manifold $\mathcal{E}({m,n})$, is given by [35],

$\displaystyle\begin{bmatrix}\ket{D(1,m,n)}\\\ \ket{D(2,m,n)}\\\
\ket{D(3,m,n)}\end{bmatrix}=\mathcal{R}_{m,n}\begin{bmatrix}\ket{+,m-1,n}\\\
\ket{0,m-1,n+1}\\\ \ket{-,m,n}\end{bmatrix}.\;$ (6ghasataxaybfbgbjbl)

where $\mathcal{R}_{m,n}(\theta,\phi,\psi)$ be the Euler matrix with mixing
angles $\\{\theta,\phi,\psi\\}$ which mixes the bare states. This triplet
provides the basis of constructing the tower of remaining dressed states with
judicious choice of the photon numbers of the bare states. Fig.8 shows the
first three manifolds which precisely preserves the topology of the $\Lambda$
configuration having two well-defined transition pathways, namely,
$\ket{3}\rightarrow\ket{1}$ (solid lines) and $\ket{3}\rightarrow\ket{2}$
(dotted lines). The dipole transition amplitudes between $j$ to $i$-th dressed
states of either of these pathways are given by the dipole matrix element,

$\displaystyle d_{ij}=\bra{D(j,m,n)}\hat{\mathcal{O}}\ket{D(i,m,n)},$
(6ghasataxaybfbgbjbm)

where $\hat{\mathcal{O}}$ be the interaction operator described by the
Hamiltonian (6d) proportional to the dipole coupling strength $g_{13}$ or
$g_{23}$ depending upon the route. The exact evaluation of the transition
amplitudes requires the knowledge of the Euler angles for different
configurations which is beyond the scope of the paper.

To understand the quintuplet structure of the $\Lambda$ configuration of
different order, in Table-I we have displayed all possible spontaneous
transitions from the dressed states of manifold $\mathcal{E}(m-1,n)$ to those
in $\mathcal{E}(m,n)$ at zero detuning, namely $\tilde{\Delta}_{23}=0$ and
$\tilde{\Delta}_{13}=0$. It is worth noting here that corresponding to zero-th
order central peak we have three transitions, then a pair of of sidebands with
diminishing intensity with two and then single transitions, respectively.
Similarly we have spontaneous transitions from the energy manifold
$\mathcal{E}(m-1,n)$ to $\mathcal{E}(m-1,n+1)$ which has same structure (Table
not shown). The treatment for the $V$ and $\Xi$ configurations of the three-
level system is similar.

##  6 Conclusion

The quantum interference in the $\Lambda$, $V$ and $\Xi$ \- type of three-
level configuration in presence of intense bi-chromatic resonant laser field
exhibits several non-trivial features of this strong coupling phenomena. In
this paper we have developed a comprehensive method to study the resonance
fluorescence spectrum of different three-level configurations. In contrast to
other available methods [21, 23, 27], our approach rooted in the SU(3) based
group theoretical technique which offers a succinct derivation of the optical
Bloch equation, presented in a concise format. When combined with the quantum
regression theorem, this derivation enables the calculation of the
fluorescence profiles of all configurations across different parameter
regimes. Finally to explain the origin of quintuplet spectra, we have touched
upon the phenomenological description of the dressed state of the $\Lambda$
system. The emergence of the heralded photons steaming from the transitions of
the dressed states of various three-level configurations is another important
aspect which requires further exploration of the structural aspects of the
dressed states. In the parlance of significant development of the atom-field
interaction, further study of the dressed states of the three-level system may
unravel uncharted phenomena of quantum information processing.

##  Appendix A Bloch matrices of Lambda, Vee and Cascade configurations

In Equation (6ghasataxayba), the Bloch matrix $M^{\Lambda}$ of the $\Lambda$
configuration is found to be,
$M^{\Lambda}=$

$\begin{pmatrix}D_{11}^{\Lambda}&0&-\frac{2ig_{23}}{3}&0&0&-\frac{ig_{23}}{3}&0&ig_{13}&\frac{ig_{23}}{3}\\\
0&D_{22}^{\Lambda}&\frac{2ig_{23}}{3}&0&0&\frac{ig_{23}}{3}&-ig_{13}&0&-\frac{ig_{23}}{3}\\\
-2ig_{23}&2ig_{23}&D_{33}^{\Lambda}&-ig_{13}&ig_{13}&-\frac{\Gamma_{31}}{3}&0&0&\frac{1}{3}\left(2\Gamma_{32}-\Gamma_{31}\right)\\\
0&0&-\frac{ig_{13}}{3}&D_{44}^{\Lambda}&0&-\frac{2ig_{13}}{3}&ig_{23}&0&-\frac{ig_{13}}{3}\\\
0&0&\frac{ig_{13}}{3}&0&D_{55}^{\Lambda}&\frac{2ig_{13}}{3}&0&-ig_{23}&\frac{ig_{13}}{3}\\\
-ig_{23}&ig_{23}&-\frac{\Gamma_{32}}{3}&-2ig_{13}&2ig_{13}&D_{66}^{\Lambda}&0&0&\frac{1}{3}\left(\Gamma_{32}-2\Gamma_{31}\right)\\\
0&-ig_{13}&0&ig_{23}&0&0&D_{77}^{\Lambda}&0&0\\\
ig_{13}&0&0&0&-ig_{23}&0&0&D_{88}^{\Lambda}&0\\\
ig_{23}&-ig_{23}&\frac{\Gamma_{32}}{3}&-ig_{13}&ig_{13}&-\frac{\Gamma_{31}}{3}&0&0&D_{99}^{\Lambda}\end{pmatrix}$
(6ghasataxaybfbgbjbn)

where the diagonal terms are given by,

$\displaystyle D_{11}^{\Lambda}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{32}}{2}+i\Delta_{23},\quad
D_{22}^{\Lambda}=-\frac{\Gamma_{32}}{2}-i\Delta_{23},\quad
D_{33}^{\Lambda}=-\frac{2\Gamma_{32}}{3},$ $\displaystyle D_{44}^{\Lambda}$
$\displaystyle=$ $\displaystyle-\frac{\Gamma_{31}}{2}+i\Delta_{13},\quad
D_{55}^{\Lambda}=-\frac{\Gamma_{31}}{2}-i\Delta_{13},\quad
D_{66}^{\Lambda}=-\frac{2\Gamma_{31}}{3}$ $\displaystyle D_{77}^{\Lambda}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\Gamma_{31}-\Gamma_{32}+2i\left(\Delta_{13}-\Delta_{23}\right)\right),$
(6ghasataxaybfbgbjbo) $\displaystyle D_{88}^{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\Gamma_{31}-\Gamma_{32}-2i\left(\Delta_{13}-\Delta_{23}\right)\right),\quad
D_{99}^{\Lambda}=\frac{1}{3}\left(-\Gamma_{31}-\Gamma_{32}\right)$

with the inhomogeneous term,

$\mathbf{B}^{\Lambda}=\big{[}0,0,\frac{1}{3}(\Gamma_{31}+2\Gamma_{32}),0,0,\frac{1}{3}(2\Gamma_{31}+\Gamma_{32}),0,0,\frac{1}{3}(\Gamma_{31}-\Gamma_{32})\big{]}^{T}.$
(6ghasataxaybfbgbjbp)

For the $V$ configuration, the components of the density matrix in terms of
Bloch vectors are given by,

$\displaystyle\eqalign{\rho_{33}^{V}&=\frac{1}{3}(1+S_{T_{3}}^{V}+S_{V_{3}}^{V}),\quad\rho_{32}^{V}=S_{T_{-}}^{V},\quad\rho_{31}^{V}=S_{V_{-}}^{V},\\\
\rho_{23}^{V}&=S_{T_{+}}^{V},\quad\rho_{22}^{V}=\frac{1}{3}(1-S_{T_{3}}^{V}+S_{U_{3}}^{V}),\quad\rho_{21}=S_{U_{-}}^{V},\\\
\rho_{13}^{V}&=S_{V_{+}}^{V},\quad\rho_{12}^{V}=S_{U_{+}}^{V},\quad\rho_{11}^{V}=\frac{1}{3}(1-S_{U_{3}}^{V}-S_{V_{3}}^{V}).}$
(6ghasataxaybfbgbjbt)

Substituting Equation (6ghasataxaybfbgbjbt) into (6ghasatax), the OBE of the
$V$ configuration is given by,

$\displaystyle\frac{d\mathbf{S^{V}_{\mathbb{P}_{i}}}(t)}{dt}=M^{V}\mathbf{S^{V}_{\mathbb{P}_{i}}}+\mathbf{B^{V}},$
(6ghasataxaybfbgbjbu)

where the Bloch matrix $M^{V}$ is given by,
$M^{V}=$

$\begin{pmatrix}D_{11}^{V}&0&0&-ig_{12}&0&0&0&ig_{13}&0\\\
0&D_{22}^{V}&0&0&ig_{12}&0&-ig_{13}&0&0\\\
0&0&D_{33}^{V}&-ig_{13}&ig_{13}&-\frac{\Gamma_{31}}{3}&ig_{12}&-ig_{12}&\frac{\Gamma_{21}}{3}\\\
-ig_{12}&0&-\frac{ig_{13}}{3}&D_{44}^{V}&0&-\frac{2ig_{13}}{3}&0&0&-\frac{ig_{13}}{3}\\\
0&ig_{12}&\frac{ig_{13}}{3}&0&D_{55}^{V}&\frac{2ig_{13}}{3}&0&0&\frac{ig_{13}}{3}\\\
0&0&\frac{1}{3}\left(\Gamma_{21}-2\Gamma_{31}\right)&-2ig_{13}&2ig_{13}&D_{66}^{V}&-ig_{12}&ig_{12}&-\frac{\Gamma_{21}}{3}\\\
0&-ig_{13}&\frac{ig_{12}}{3}&0&0&-\frac{ig_{12}}{3}&D_{77}^{V}&0&-\frac{2ig_{12}}{3}\\\
ig_{13}&0&-\frac{ig_{12}}{3}&0&0&\frac{ig_{12}}{3}&0&D_{88}^{V}&\frac{2ig_{12}}{3}\\\
0&0&\frac{1}{3}\left(2\Gamma_{21}-\Gamma_{31}\right)&-ig_{13}&ig_{13}&-\frac{\Gamma_{31}}{3}&-2ig_{12}&2ig_{12}&D_{99}^{V}\end{pmatrix}$
(6ghasataxaybfbgbjbv)

where the diagonal terms are given by

$\displaystyle D_{11}^{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\Gamma_{21}-\Gamma_{31}-2i\left(\Delta_{12}-\Delta_{13}\right)\right),$
$\displaystyle D_{22}^{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\Gamma_{21}-\Gamma_{31}+2i\left(\Delta_{12}-\Delta_{13}\right)\right),\quad
D_{33}=\frac{1}{3}\left(-\Gamma_{21}-\Gamma_{31}\right),$ $\displaystyle
D_{44}^{V}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{31}}{2}+i\Delta_{13},\quad
D_{55}^{V}=-\frac{\Gamma_{31}}{2}-i\Delta_{13},\quad
D_{66}^{V}=-\frac{2\Gamma_{31}}{3},$ (6ghasataxaybfbgbjbw) $\displaystyle\quad
D_{77}^{V}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{21}}{2}+i\Delta_{12},\quad
D_{88}^{V}=-\frac{\Gamma_{21}}{2}-i\Delta_{12},\quad
D_{99}^{V}=-\frac{2\Gamma_{21}}{3}$

with the corresponding inhomogeneous term,

$B^{V}=\big{[}0,0,\frac{1}{3}(\Gamma_{21}-\Gamma_{31}),0,0,-\frac{1}{3}(\Gamma_{21}+2\Gamma_{31}),0,0,-\frac{1}{3}(2\Gamma_{21}+\Gamma_{31})\big{]}^{T},$
(6ghasataxaybfbgbjbx)

Finally for the $\Xi$ configuration, the components of the density matrix in
terms of Bloch vectors are given by,

$\displaystyle\eqalign{\rho_{33}^{\Xi}&=\frac{1}{3}(1+S_{T_{3}}^{\Xi}+S_{V_{3}}^{\Xi}),\quad\rho_{32}^{\Xi}=S_{T_{-}}^{\Xi},\quad\rho_{31}^{\Xi}=S_{V_{-}}^{\Xi},\\\
\rho_{23}^{\Xi}&=S_{T_{+}}^{\Xi},\quad\rho_{22}^{\Xi}=\frac{1}{3}(1-S_{T_{3}}^{\Xi}+S_{U_{3}}^{\Xi}),\quad\rho_{21}^{\Xi}=S_{U_{-}}^{\Xi},\\\
\rho_{13}^{\Xi}&=S_{V_{+}}^{\Xi},\quad\rho_{12}^{\Xi}=S_{U_{+}}^{\Xi},\quad\rho_{11}^{\Xi}=\frac{1}{3}(1-S_{U_{3}}^{\Xi}-S_{V_{3}}^{\Xi}).}$
(6ghasataxaybfbgbjcb)

Substituting (6ghasataxaybfbgbjcb) into (6ghasatax) the OBE of the $\Xi$
configuration is given by,

$\displaystyle\frac{d\mathbf{S^{\Xi}_{\mathbb{P}_{i}}}(t)}{dt}=M^{\Xi}\mathbf{S^{\Xi}_{\mathbb{P}_{i}}}+\mathbf{B^{\Xi}},$
(6ghasataxaybfbgbjcc)

with the Bloch matrix given by, $M^{\Xi}=$

$\begin{pmatrix}D_{11}^{\Xi}&0&-\frac{2ig_{23}}{3}&-ig_{12}&0&-\frac{ig_{23}}{3}&0&0&\frac{ig_{23}}{3}\\\
0&D_{22}^{\Xi}&\frac{2ig_{23}}{3}&0&ig_{12}&\frac{ig_{23}}{3}&0&0&-\frac{ig_{23}}{3}\\\
-2ig_{23}&2ig_{23}&D_{33}^{\Xi}&0&0&-\frac{2\Gamma_{32}}{3}&ig_{12}&-ig_{12}&\frac{\Gamma_{21}}{3}\\\
-ig_{12}&0&0&D_{44}^{\Xi}&0&0&ig_{23}&0&0\\\
0&ig_{12}&0&0&D_{55}^{\Xi}&0&0&-ig_{23}&0\\\
-ig_{23}&ig_{23}&\frac{1}{3}\left(\Gamma_{21}-\Gamma_{32}\right)&0&0&D_{66}^{\Xi}&-ig_{12}&ig_{12}&-\frac{\Gamma_{21}}{3}\\\
0&0&\frac{ig_{12}}{3}&ig_{23}&0&-\frac{ig_{12}}{3}&D_{77}^{\Xi}&0&-\frac{2ig_{12}}{3}\\\
0&0&-\frac{ig_{12}}{3}&0&-ig_{23}&\frac{ig_{12}}{3}&0&D_{88}^{\Xi}&\frac{2ig_{12}}{3}\\\
ig_{23}&-ig_{23}&\frac{1}{3}\left(2\Gamma_{21}+\Gamma_{32}\right)&0&0&\frac{\Gamma_{32}}{3}&-2ig_{12}&2ig_{12}&D_{99}^{\Xi}\end{pmatrix}$
(6ghasataxaybfbgbjcd)

where the diagonal terms are given by

$\displaystyle D_{11}^{\Xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\Gamma_{21}-\Gamma_{32}+2i\Delta_{23}\right),\quad
D_{22}^{\Xi}=\frac{1}{2}\left(-\Gamma_{21}-\Gamma_{32}-2i\Delta_{23}\right),$
$\displaystyle D_{33}^{\Xi}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(-\Gamma_{21}-2\Gamma_{32}\right),\quad
D_{44}^{\Xi}=-\frac{\Gamma_{32}}{2}+i\left(\Delta_{12}+\Delta_{23}\right),$
$\displaystyle D_{55}^{\Xi}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{32}}{2}-i\left(\Delta_{12}+\Delta_{23}\right),\quad
D_{66}^{\Xi}=-\frac{\Gamma_{32}}{3},$ (6ghasataxaybfbgbjce) $\displaystyle
D_{77}^{\Xi}$ $\displaystyle=$
$\displaystyle-\frac{\text{$\Gamma$21}}{2}+i\Delta_{12},\quad
D_{88}^{\Xi}=-\frac{\Gamma_{21}}{2}-i\Delta_{12},\quad
D_{99}^{\Xi}=-\frac{2\Gamma_{21}}{3}$

with corresponding inhomogeneous term,

$\mathbf{B}^{\Xi}=\big{[}0,0,\frac{1}{3}(\Gamma_{21}-2\Gamma_{32}),0,0,-\frac{1}{3}(\Gamma_{21}+\Gamma_{32}),0,0,\frac{1}{3}(\Gamma_{32}-2\Gamma_{21})\big{]}^{T}.$
(6ghasataxaybfbgbjcf)

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## Figure captions:

Figure 3: *

Figure 2: Power spectrum of $\Lambda$ configuration for
$\ket{3}\rightarrow\ket{1}$ ($\langle\langle
V_{+}(\tau)V_{-}(0)\rangle\rangle$ correlation) transition (Scaled with
$\Gamma_{31}^{\Lambda}$): Plot of $S_{13}^{\Lambda}(\omega)$ versus
$(\omega-\Omega_{13}^{\Lambda})/\Gamma_{31}^{\Lambda}$ with following
parameters: a) $\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=10$,
$\tilde{g}_{13}=7$ (Red), b) $\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=7$,
$\tilde{g}_{13}=10$ (Blue), c) $\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=5$,
$\tilde{g}_{13}=.1$ (Green), d) $\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=.1$,
$\tilde{g}_{13}=5$ (Black)

Figure 4: *

Figure 3: Power spectrum of $\Lambda$ configuration for
$\ket{3}\rightarrow\ket{2}$ ($\langle\langle
T_{+}(\tau)T_{-}(0)\rangle\rangle$ correlation) transition (Scaled with
$\Gamma_{32}^{\Lambda}$): Plot of $S_{13}^{\Lambda}(\omega)$ versus
$(\omega-\Omega_{13}^{\Lambda})/\Gamma_{32}^{\Lambda}$ with following
parameters: a) $\tilde{\Gamma}_{31}=.5$, $\tilde{g}_{23}=10$,
$\tilde{g}_{13}=7$ (Red), b) $\tilde{\Gamma}_{31}=.5$, $\tilde{g}_{23}=7$,
$\tilde{g}_{13}=10$ (Blue), c) $\tilde{\Gamma}_{31}=.5$, $\tilde{g}_{23}=5$,
$\tilde{g}_{13}=.1$ (Green), d) $\tilde{\Gamma}_{31}=.5$, $\tilde{g}_{23}=.1$,
$\tilde{g}_{13}=5$ (Black)

Figure 5: *

Figure 4: Power spectrum of $V$ configuration for $\ket{3}\rightarrow\ket{1}$
($\langle\langle V_{+}(\tau)V_{-}(0)\rangle\rangle$ correlation) transition
(Scaled with $\Gamma_{31}^{V}$): Plot of $S_{13}^{V}(\omega)$ versus
$(\omega-\Omega_{13}^{V})/\Gamma_{31}^{V}$ with following parameters: a)
$\tilde{\Gamma}_{21}=8$, $\tilde{g}_{12}=20$, $\tilde{g}_{13}=7$ (Red), b)
$\tilde{\Gamma}_{21}=8$, $\tilde{g}_{12}=7$, $\tilde{g}_{13}=20$ (Blue), c)
$\tilde{\Gamma}_{21}=8$, $\tilde{g}_{12}=7$, $\tilde{g}_{13}=.1$ (Green), d)
$\tilde{\Gamma}_{21}=8$, $\tilde{g}_{12}=.1$, $\tilde{g}_{13}=7$ (Black).

Figure 6: *

Figure 5: Power spectrum of $V$ configuration for $\ket{2}\rightarrow\ket{1}$
($\langle\langle U_{+}(\tau)U_{-}(0)\rangle\rangle$ correlation) transition
(Scaled with $\Gamma_{21}^{V}$): Plot of $S_{12}^{V}(\omega)$ versus
$(\omega-\Omega_{12}^{V})/\Gamma_{21}^{V}$ with following parameters: a)
$\tilde{\Gamma}_{31}=8$, $\tilde{g}_{12}=20$, $\tilde{g}_{13}=7$ (Red), b)
$\tilde{\Gamma}_{31}=8$, $\tilde{g}_{12}=7$, $\tilde{g}_{13}=20$ (Blue), c)
$\tilde{\Gamma}_{31}=8$, $\tilde{g}_{12}=7$, $\tilde{g}_{13}=.1$ (Green), d)
$\tilde{\Gamma}_{31}=8$, $\tilde{g}_{12}=.1$, $\tilde{g}_{13}=7$ (Black)

Figure 7: *

Figure 6: Power spectrum of $\Xi$ configuration for
$\ket{2}\rightarrow\ket{1}$ ($\langle\langle
U_{+}(\tau)U_{-}(0)\rangle\rangle$ correlation) transition (Scaled with
$\Gamma_{21}^{\Xi}$): Plot of $S_{12}^{\Xi}(\omega)$ versus
$(\omega-\Omega_{12}^{\Xi})/\Gamma_{21}^{\Xi}$ with following parameters: a)
$\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=10$, $\tilde{g}_{12}=7$ (Red), b)
$\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=7$, $\tilde{g}_{12}=10$ (Blue), c)
$\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=7$, $\tilde{g}_{12}=.1$ (Green), d)
$\tilde{\Gamma}_{32}=.5$, $\tilde{g}_{23}=.1$, $\tilde{g}_{12}=7$ (Black)

Figure 8: *

Figure 7: Power spectrum of $\Xi$ configuration for
$\ket{3}\rightarrow\ket{2}$ ($\langle\langle
T_{+}(\tau)T_{-}(0)\rangle\rangle$ correlation) transition (Scaled with
$\Gamma_{32}^{\Xi}$): Plot of $S_{23}^{\Xi}(\omega)$ versus
$(\omega-\Omega_{23}^{\Xi})/\Gamma_{32}^{\Xi}$ with following parameters: a)
$\tilde{\Gamma}_{21}=.5$, $\tilde{g}_{23}=10$, $\tilde{g}_{12}=7$ (Red), b)
$\tilde{\Gamma}_{21}=.5$, $\tilde{g}_{23}=7$, $\tilde{g}_{12}=10$ (Blue), c)
$\tilde{\Gamma}_{21}=.5$, $\tilde{g}_{23}=7$, $\tilde{g}_{12}=.1$ (Green), d)
$\tilde{\Gamma}_{21}=.5$, $\tilde{g}_{23}=.1$, $\tilde{g}_{12}=7$ (Black)