Benchmarking Semantic Communications for Image Transmission Over MIMO Interference Channels

Yanhu Wang, Shuaishuai Guo, Senior Member, IEEE,  Anming Dong, and Hui Zhao

Yanhu Wang, and Shuaishuai Guo are with School of Control Science and Engineering, and also with Shandong Key Laboratory of Wireless Communication Technologies, Shandong University, Jinan, 250061, China (e-mail<EMAIL_ADDRESS>shuaishuaiguo@sdu.edu.cn).

Anming Dong is with the School of Computer Science and Technology, Qilu University of Technology,
Jinan, 250353, China (e-mail: anmingdong@qlu.edu.cn).

Hui Zhao is with the Communication Systems Department, EURECOM,
06410 Sophia Antipolis, France (e-mail: hui.zhao@eurecom.fr).

Semantic communications offer promising prospects for enhancing data transmission efficiency.
However, existing schemes have predominantly concentrated on point-to-point transmissions. In this paper, we aim to investigate the validity of this claim in interference scenarios compared to baseline approaches. Specifically, our focus is on general multiple-input multiple-output (MIMO) interference channels, where we propose an interference-robust semantic communication (IRSC) scheme.
This scheme involves the development of transceivers based on neural networks (NNs), which integrate channel state information (CSI) either solely at the receiver or at both transmitter and receiver ends.
Moreover, we establish a composite loss function for training IRSC transceivers, along with a dynamic mechanism for updating the weights of various components in the loss function to enhance system fairness among users.  Experimental results demonstrate that the proposed IRSC scheme effectively learns to mitigate interference and outperforms baseline approaches, particularly in low signal-to-noise (SNR) regimes.

Semantic communication, MIMO, interference channel, deep learning.

§ INTRODUCTION

Recent times have witnessed a resurgence of interest in semantic communications<cit.>, propelled by the expanding realm of intelligent applications such as augmented reality/virtual reality (AR/VR)<cit.>.
These advanced applications impose stringent requirements on communication services, prompting the exploration of novel theories and technologies in wireless networks.
Semantic communication, as a new paradigm, aims to convey the meaning of data rather than focusing solely on the precise transmission of individual symbols<cit.>.
This approach holds significant potential to reduce data traffic, thereby greatly improving communication efficiency and effectively supporting various intelligent applications.

With advancements in deep learning (DL), particularly in natural language processing (NLP) and computer vision (CV), significant research efforts have been dedicated to exploring the domain of semantic communications<cit.>.
Neural networks (NNs) are employed in <cit.> to construct semantic encoder/decoder models, which are subsequently trained on large-scale datasets to acquire the ability to extract and interpret semantic information.
These NNs-based semantic communications can handle various types of data, including images<cit.>, speech<cit.> and text<cit.>.
When compared to conventional communication systems based on source-channel separation, they demonstrate superior performance at the same transmission rate.
However, most existing NNs-based works have focused on simplistic single-user channel models, such as additive white Gaussian noise (AWGN) channels or fading channels, which do not adequately address interference issues.
Interference is a common challenge in wireless communications, particularly for users located at the edge of cell networks.
Thus, the question of how to effectively integrate semantic communication in environments with interference remains unanswered.

Introducing unexpected interference to semantic communications may lead to erroneous data recovery.
Real-time monitoring of the communication environment and dynamically adjusting transmission power can reduce interference.
However, such adjustments may compromise communication efficiency and reliability.
To tackle these challenges, we propose a NNs-based multiple-antenna transceiver design to handle interference.
Specifically, we introduce an interference-robust semantic communication (IRSC) design over multiple-input multiple-output (MIMO) interference channels.
The IRSC scheme utilizes NNs for transceiver design and integrates channel state information (CSI).
We explore two variations in our design: one with CSI provided only at the receiver end and another with CSI available at both transmitter and receiver ends.
The transceivers are trained using a composite loss function. Additionally, we develop a dynamic mechanism to adjust the weights of various components in the loss function, thereby enhancing system fairness among users. The results demonstrate the effectiveness of the proposed IRSC scheme in mitigating interference. Compared to benchmark schemes, our approach performs well in low SNR regimes.

§ SYSTEM MODEL

As depicted in Fig. <ref>, we consider a system comprising $K$ transmitter-receiver pairs, all of which operate over a shared physical channel. Without loss of generality, we presume that each transmitter is equipped with $N_t$ antennas, and each receiver with $N_r$ antennas.
In this setup, transmitters send semantic information to their corresponding receivers. However, each receiver can also accidentally receive semantic information from other transmitters, causing interference between users, as illustrated by the dashed lines in Fig. <ref>.

Semantic communication system model over MIMO interference channels.
The direct link is depicted by a solid line, and the interfering link is depicted by a dashed line.

§.§ Transmitter

Let $\mathbf{S}_k \in \mathbb{R}^n$ represent the image input of user $k$, where $n$ denotes the dimension of the input images and $k=1,\cdots,K$.
As shown in Fig. <ref>, the transmitter consists
of semantic encoder and joint source-channel (JSC) encoder.
The semantic encoder extracts semantic information from $\mathbf{S}_k$ and then the JSC encoder maps them to symbols $\widetilde{\mathbf{X}}_k \in \mathbb{C}^{N_tN_B}$ to ensure reliable transmission over the channel.
Here, $N_tN_B$ denotes the number of complex-valued symbols.
This encoding process can be represented as follows:

\begin{equation}
\label{enc1}
\widetilde{\mathbf{X}}_k = f_{\boldsymbol{\mathit{\beta}}_k}(f_{\boldsymbol{\mathit{\alpha}}_k}(\mathbf{S}_k)),
\end{equation}

where $f_{\boldsymbol{\mathit{\alpha}}_k}(\cdot)$ and $f_{\boldsymbol{\mathit{\beta}}_k}(\cdot)$ represent the semantic encoder function and JSC encoder function for the transmitter $k$, parameterized by $\boldsymbol{\mathit{\alpha}}_k$ and $\boldsymbol{\mathit{\beta}}_k$ respectively.
The symbols $\widetilde{\mathbf{X}}_k$ are normalized and reshaped before transmission according to

\begin{equation}
\label{ap}
\mathbf{X}_k = {\rm Reshape} \left(\sqrt{PN_tN_B}\frac{\widetilde{\mathbf{X}}_k}{\left \|\widetilde{\mathbf{X}}_k \right\|_2} \right),
\end{equation}

where $P$ is the average transmit power constraint;
$\rm{Reshape}(\cdot)$ is a function that changes the shape of input; and
$\mathbf{X}_k \in \mathbb{C}^{N_t\times N_B}$ is the signal matrix transmitted by transmitter $k$.

§.§ Communication Model

The received symbol matrix at the downlink user $k$ can be given as

\begin{equation}
\label{chn}
\mathbf{Y}_k = \mathbf{H}_{kk} \mathbf{X}_k
+ \sum_{j=1, j\neq k}^K \mathbf{H}_{kj} \mathbf{X}_j + \mathbf{N}_k,
\end{equation}

where $\mathbf{Y}_k \in \mathbb{C}^{N_r\times N_B}$.
The downlink channel $\mathbf{H}$ is flat Rayleigh fading,
and $\mathbf{H}_{kj} \in \mathbb{C}^{N_r\times N_t} $ stands for the channel matrix from the $j$-th transmitter to the $k$-th receiver.
$\mathbf{N}_k\in\mathbb{C}^{N_r\times N_B}$ is the noise matrices with each entry obeying an independent identical zero-mean  Gaussian distribution with variance $\sigma^2$.
In (<ref>), the first term on the right side of the equation is the expected signal, the second term is the interference between users, and the last term is the additive Gaussian noise.

§.§ Receiver

Similar to the transmitter, the receiver also comprises two parts, namely a JSC decoder and a semantic decoder.
JSC decoder is used to recover the transmitted symbols, and the semantic decoder is used to recover the transmitted images.
The decoding process at the receiver $k$ can be written as

\begin{equation}
\label{dec1}
\hat{\mathbf{S}}_k = f_{\bm{\theta}_k}(f_{\bm{\gamma}_k}(\mathbf{Y}_k)),
\end{equation}

where $\hat{\mathbf{S}}_k \in \mathbb{R}^n$ is the target image of user $k$;
$f_{\bm{\gamma}_k}(\cdot)$ and $f_{\bm{\theta}_k}(\cdot)$ are the JSC decoder function and semantic decoder function for the receiver $k$, parameterized by $\bm{\gamma}_k$ and $\bm{\theta}_k$ respectively.

§.§ Problem Description

The interference channels considered in this work may disrupt the transmission of semantic information and distort the intended meaning of data, thus affecting the overall efficiency and reliability of the semantic communication system.
We aim to maximize resistance to interference channels for accurate image transmission.
This problem can be formulated as

\begin{equation}
\begin{split}
\label{P1}
(\bm{\alpha}_k^*, \bm{\beta}_k^*, \bm{\gamma}_k^*, \bm{\theta}_k^*) &= \underset{\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k} {\rm argmin} \mathcal{L}_{MSE}(\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k)
\\&=\underset{\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k} {\rm argmin}  \frac{1}{M} \sum^M_i (\mathbf{S}_{k,i}-\hat{\mathbf{S}}_{k,i})^2,
\end{split}
\end{equation}

where the function $\mathcal{L}_{MSE}(\cdot)$ represents MSE between  $\mathbf{S}_k$ and $\hat{\mathbf{S}}_k$.
$M$ denotes the number of samples.

The proposed neural network structure for IRSC.

§ INTERFERENCE ROBUST SEMANTIC COMMUNICATION

To address problem (<ref>), we design an interference-robust semantic communication system, named IRSC.
Specifically, we first propose to integrate CSI at the receiver only or at both the transmitter and receiver
ends, allowing IRSC to learn to adapt to channel interference.
Subsequently, we provide a detailed description of the model design.
Finally, we introduce the design of the loss function and the training strategy.

§.§ CSI Integration for Enhancing Semantic Communications

CSI plays a critical role in characterizing communication channels,
encompassing aspects such as fading and interference.
Having knowledge of CSI is imperative for optimizing the performance of communication systems.
In our work, we propose the integration of CSI into the IRSC system.
Specifically, we consider two different design options.
In the first case, where CSI is available at each receiver, we provide CSI, denoted as $\mathbf{H}$, along with the received signal $\mathbf{Y}_k$ to the JSC decoder.
$\mathbf{H}$ is initially transformed into a real-valued vector of length $2K^2 N_t N_r$ and then concatenated with $\mathbf{Y}_k$, which is similarly converted into a real-valued vector.
In the second scenario, where CSI is available at both the transmitter and receiver, the usage of $\mathbf{H}$ at the receiver follows the previously described approach.
At the transmitter, $\mathbf{H}$, which is converted into a real-valued vector with a length of $2K^2 N_t N_r$, is input to the JSC encoder together with the extracted semantic information to learn the features adapted to the interference channel.

§.§ Model Design

As shown in Fig. <ref>,

the semantic encoder first learns semantic information from the inputs $\mathbf{S}_k$ and outputs the learned features.
It is noteworthy that the proposed IRSC is a universal framework, and the NN structure of the semantic encoder is not fixed.
For devices with limited computational and memory resources, we use a fully connected layer as the semantic encoder network to meet resource constraints.
Assuming abundant computational and memory resources, the semantic encoder can be composed of ResNet<cit.>.
After passing through the semantic encoder network, we determine whether CSI is available at the transmitter.
If available, the CSI is fed to the JSC encoder along with the semantic information.
The first layer of the JSC encoder is a reshape layer that concatenates the semantic information with CSI and performs dimension transformation.
The second layer is a fully connected layer where the dimensions are reduced to the length of semantic information.
The third layer is also a fully connected layer, mapping the fused features to channel input symbols $\widetilde{\mathbf{X}}_k$.
If CSI is not available, the JSC encoder directly maps the semantic information to $\widetilde{\mathbf{X}}_k$, and in this case, the JSC encoder consists of only a single fully connected layer.
After that, $\widetilde{\mathbf{X}}_k$ is normalized and reshaped to $\mathbf{X}_k$.

In order to be able to jointly optimize the communication system in Fig. <ref> in an end-to-end manner, we model the interference channels as untrainable layers and incorporate them into the entire neural network architecture.
After the channel layer, $\mathbf{X}_k$ is converted to $\mathbf{Y}_k$.
$\mathbf{Y}_k$ and CSI are jointly input into the JSC decoder,
and its network structure is the same as the network structure of JSC encoder when CSI is available at the transmitter, which is also composed of three layers.
The semantic decoder network reverses the operations performed by the semantic encoder network and generates an estimate $\hat{\mathbf{S}}_k$ of the original image based on the output of the JSC decoder.
All the transmitters and receivers in IRSC are trained jointly, the loss is calculated at the end of the receiver and backpropagated to the transmitter, and the training parameters are updated simultaneously.

§.§ Loss Function Design

To jointly optimize the entire IRSC neural network,

we use the weighted combination of (<ref>) as the loss function, which can be written as

\begin{equation}
\begin{split}
\label{P2}
\mathcal{L}_{total} &= \underset{\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k} {\rm argmin}
\bigg\{ \sum^K_k  \omega_k \mathcal{L}_{MSE}(\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k) \bigg\}
\\&=\underset{\bm{\alpha}_k, \bm{\beta}_k, \bm{\gamma}_k, \bm{\theta}_k} {\rm argmin}  \bigg\{ \omega_1 \cdot \frac{1}{M} \sum^M_i (\mathbf{S}_{1,i}-\hat{\mathbf{S}}_{1,i})^2 + \cdots \\
& \quad +\omega_K \cdot \frac{1}{M} \sum^M_i (\mathbf{S}_{K,i}- \hat{\mathbf{S}}_{K,i})^2 \bigg\},
\end{split}
\end{equation}

where $\omega_k$ represents the weight assigned to the loss for user $k$.
In order to ensure equitable system performance among users, we implement a dynamic updating mechanism for $\omega_k$.
Specifically, the setting of $\omega_k$ is determined by the proportion of the corresponding user's contribution to the total loss in the previous iteration, which is given by

\begin{equation}
\label{xs}
\omega_{k}^{(t+1)} = \frac{\mathcal{L}_{MSE}(\bm{\alpha}_k^{(t)}, \bm{\beta}_k^{(t)}, \bm{\gamma}_k^{(t)}, \bm{\theta}_k^{(t)})}{\sum_k^K \mathcal{L}_{MSE}(\bm{\alpha}_k^{(t)}, \bm{\beta}_k^{(t)}, \bm{\gamma}_k^{(t)}, \bm{\theta}_k^{(t)})},
\end{equation}

where $t$ denotes the index of the current training epoch.
Initially, all weights are set to $1/K$.

IRSC Training Algorithm.

[1]
Initialize parameters, $\bm{\alpha}_k^{(0)}, \bm{\beta}_k^{(0)}, \bm{\gamma}_k^{(0)}, \bm{\theta}_k^{(0)}$,

Input: A batch of images $\mathbf{S}_k$, channel $\mathbf{H}$, epochs $T$.
while $t=1$ to $T$ do
if CSI is only available at the receiver then
$\widetilde{\mathbf{X}}_k$ $\leftarrow$ $f_{\boldsymbol{\mathit{\beta}}_k}(f_{\boldsymbol{\mathit{\alpha}}_k}(\mathbf{S}_k))$.
else CSI is available at both transmitter and receiver then
$\widetilde{\mathbf{X}}_k$ $\leftarrow$ $f_{\boldsymbol{\mathit{\beta}}_k}(f_{\boldsymbol{\mathit{\alpha}}_k}(\mathbf{S}_k), \mathbf{H})$.
end if
Power normalization.
Transmit $\mathbf{X}_k$ over MIMO interference channel.
$\hat{\mathbf{S}}_k \leftarrow f_{\bm{\theta}_k}(f_{\bm{\gamma}_k}(\mathbf{Y}_k, \mathbf{H}))$.
Compute loss $\mathcal{L}_{total}$ via (<ref>).
Update parameters simultaneously via SGD.
$t\leftarrow t+1$.
end while
Output: $f_{\bm{\alpha}_k}(\cdot)$, $f_{\bm{\beta}_k}(\cdot)$, $f_{\bm{\gamma}_k}(\cdot)$, $f_{\bm{\theta}_k}(\cdot)$.

§.§ Training Strategy

The goal of the aforementioned IRSC is to train an end-to-end model for transmitting images, particularly by leveraging CSI to enhance suitability for MIMO interference channels.
According to Fig. <ref>, the training algorithm for IRSC is outlined in Algorithm <ref>.
Initially, the NN parameters are initialized.
After extracting semantic information by the semantic encoder, the obtained semantic information is encoded, transmitted, and decoded alongside the CSI.
Subsequently, compute the loss and iteratively update the parameters using stochastic gradient descent (SGD).
Training persists until meeting termination criteria, such as reaching the maximum iteration count or observing no further reduction in the loss.

Complexity analysis:
The computational complexity of the proposed IRSC depends on the neural network structure.
In the subsequent experiments, IRSC is implemented using fully connected layers.
The computational complexity of the fully connected layer is determined by the matrix multiplication operations.
Assuming the input feature dimension is $D_{\text{in}}$, the output feature dimension is $D_{\text{out}}$, and the batch size is $M$, the computational complexity of a single layer is $\mathcal{O}(M \times D_{\text{in}} \times D_{\text{out}})$.
Assuming there are $L$ layers of fully connected layer, the total computational complexity is $\mathcal{O}\left(\sum\limits_{l=1}\limits^L M \times D_{\text{in}}^{(l)}\times D_{\text{out}}^{(l)}\right)$.

§ EXPERIMENTS AND DISCUSSIONS

In this section, several experiments are
provided to investigate the performance of the proposed IRSC scheme.

§.§ Dataset and Parameter Setting

We utilize the MNIST<cit.> dataset and the Fashion-MNIST<cit.> dataset for evaluation, as they are datasets commonly used in the field of machine learning and computer vision.
The MNIST dataset consists of handwritten digits from “0” to “9”, while the Fashion-MNIST dataset contains $10$ different clothing categories.
They both contain $70,000$ grayscale images with a training set of $60,000$ examples and a test set of $10,000$ examples.
In the experiment, we set $K$ to $2$, indicating consideration for a two-user system.
Each user's transmitter and receiver are equipped with $2$ antennas respectively.
The neural network structures of the transmitter and receiver are listed in Table <ref>.
Throughout the training process, we employ the Adam<cit.> optimizer with
learning rate $0.001$, betas of $0.9$ and $0.98$, batch size of $128$, and epoch of $200$.

§.§ Evaluation Metric and Performance Baselines

We use Structural Similarity (SSIM) as the performance metric<cit.>, which measures the similarity of two images:

\begin{equation}
\label{ssim}
{\rm SSIM}(\mathbf{s},\hat{\mathbf{s}}) = \frac{(2\mu_s\mu_{\hat{s}}+C_1)(2\sigma_{s\hat{s}}+C_2)}{(\mu_s^2 +\mu_{\hat{s}}^2 +C_1)(\sigma_s^2 +\sigma_{\hat{s}}^2 +C_2)},
\end{equation}

where $\mu_s$ and $\sigma_s$ are the mean and the standard deviation for image ${\rm s}$;  $\sigma_{s\hat{s}}$ is the covariance of ${\rm s}$ and ${\rm \hat{s}}$;
$C_1$ and $C_2$ are constants to stabilize the division.
The SSIM index ranges from $-1$ to $1$.
A higher SSIM value suggests that the two images being compared are more similar.

Neural Network Structure of The
Transmitter and Receiver

Layer                                                                                Output
Transmitter     c]@c@Fully-connected Layer + ReLU
Reshape + Fully-connected Layer + ReLU
Fully-connected Layer     64
Receiver        c]@c@Fully-connected Layer + ReLU
Fully-connected Layer +Tanh                                        784

To fully verify the effectiveness of the proposed IRSC, we consider the following baselines:

Interference-Free scheme:
The semantic information is transmitted to the receiver without noise and interference, which serves as the upper bound.

Semi-Conventional scheme:
Before semantic information is transmitted by the transmitter $k$, it undergoes preprocessing by the transmitting precoder.
At the receiving end, the signal received by the receiving antenna is first passed through a receiving filter, and then fed into the JSC decoder and semantic decoder for image recovery.
The signal of receiver $k$ after filtering can be written as

\begin{equation}
\label{semi}
\mathbf{Y}_k = \mathbf{U}_{k}^{\rm H} \mathbf{H}_{kk} \mathbf{V}_{k} \mathbf{X}_k
+ \sum_{j=1, j\neq k}^K \mathbf{U}_{k}^{\rm H} \mathbf{H}_{kj} \mathbf{V}_{j}\mathbf{X}_j + \mathbf{U}_{k}^{\rm H} \mathbf{N}_k,
\end{equation}

where $\mathbf{V}_{k} \in \mathbb{C}^{N_t\times N_r}$ and $\mathbf{U}_{k} \in \mathbb{C}^{N_r\times N_t}$ represent the precoding matrix and the receiving filtering matrix of user $k$, respectively.
For the design of $\mathbf{V}_k$ and $\mathbf{U}_k$, we adopt the method proposed in <cit.>.

CSI-Free scheme:
The neural network structure of this scheme is the same as that of IRSC, and also adopts (<ref>) as the loss function.
The difference is that CSI is not integrated into the transmitter and receiver.
The purpose of this scheme is to verify the validity of the method proposed in Section III-A.

§.§ Performance of IRSC

In the first experiment, the two users are assumed to have different message sets for transmission, where user $1$ aims to complete the MNNIST image transmission, while user $2$ aims to complete the Fashion-MNNIST image transmission.
Fig. <ref> plots the performance of the proposed IRSC schemes and baselines over MIMO interference channels for different test SNR values.
We can observe that, whether it is user $1$ or user $2$, both the proposed IRSC CSIR (integrating CSI into receiver networks) scheme and IRSC CSITR (integrating CSI into transmitter and receiver networks) scheme outperform the CSI-Free scheme.
This indicates that the proposed schemes can effectively mitigate the impact of MIMO interference channels.
In the high SNR regimes, the Semi-Conventional scheme approaches the upper bound more closely.
However, this scheme requires the estimation of CSI at the receiver and then feeding it back to the transmitter.
In contrast, the proposed IRSC CSIR scheme does not require feedback CSI to the transmitter, and we also observe that it exhibits enhanced competitiveness in the low SNR regimes.
To illustrate more intuitively, the example reconstruction image with SNR = $0$ dB is shown in Fig. <ref>. Comparing IRSC CSIR with IRSC CSITR, we find they are achieving almost the same performance, implying that IRSC with CSI only at the receiver can handle the interference well.

Performance comparison of IRSC with other baseline schemes on the MNIST dataset and Fashion-MNIST dataset over MIMO interference channels.

Comparison of reconstructed images produced by various schemes in the low SNR regime: An Example at SNR = $0$dB.

Next, we discuss the effect of different user numbers on the performance of the proposed IRSC schemes.
Assuming that all users are aiming to complete MNIST image transmission, we average the SSIM of all users as the performance of the algorithm.
The simulation results are shown in the Fig. <ref>.
It can be seen that in the case of fewer users, the proposed IRSC schemes can deal with interference more effectively, and they are easier to achieve good SSIM.
As the number of users increases, the interference problem becomes more complex, so SSIM of the proposed IRSC schemes decreases.
We also note that IRSC CSITR performs almost the same as IRSC CSIR.
Since IRSC CSIR does not require CSI feedback to the transmitter, it is preferred when handling interference.

§ CONCLUSION

In this paper, we evaluated the performance gains of semantic communication systems in interference scenarios.
Specifically, we proposed an interference-robust semantic communication scheme, named IRSC, over MIMO interference channels.
The IRSC scheme uses NNs for transceiver design and integrates CSI.
In particular, we have considered two design options: IRSC CSIR, where CSI is input solely at the receiver end, and IRSC CSITR, where CSI is input at both transmitter and receiver ends.
We trained the transceivers by establishing a composite loss function and implemented a dynamic mechanism to enhance system fairness among users.
Experiments show that the proposed IRSC CSIR and IRSC CSITR can effectively mitigate interference and significantly outperform the baselines in the low SNR regime. Additionally, the IRSC CSIR does not require CSI feedback.

SSIM comparisons versus the number of users over MIMO interference channels with test SNR of $0$dB and $15$dB.


C. Morisset et al. Abundance determination in PNe

Planetary Nebulae: a Universal Toolbox in the Era of Precision Astrophysics
2023 10.1017/xxxxx 384

Proceedings IAU Symposium

# Abundance determination in PNe: How to deal with large chemical
inhomogeneities

C. Morisset1,2 J. Garcia-Rojas3,4 V. Gomez-Llanos3,4 & H. Monteiro5 1Instituto
de Astronomía (IA), Universidad Nacional Autónoma de México, Apdo. postal 106,
C.P. 22800 Ensenada, Baja California, México
2 Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av.
Universidad s/n, 62210 Cuernavaca, Mor., México
3Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
4Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain
5Instituto de Física e Química, Universidade Federal de Itajubá, Av. BPS
1303-Pinheirinho, 37500-903, Itajubá, Brazil

###### Abstract

Abundance determinations in planetary nebulae (PNe) are crucial for
understanding stellar evolution and the chemical evolution of the host galaxy.

We discuss the complications involved when the presence of a metal-rich phase
is suspected in the nebula. We demonstrate that the presence of a cold region
emitting mainly metal recombination lines necessitates a detailed treatment to
obtain an accurate assessment of the enrichment of this cold gas phase.

###### keywords:

(stars:) binaries: general, ISM: abundances,(ISM:) planetary nebulae: general

††editors: O. De Marco, A. Zijlstra, R. Szczerba, eds.

##  1 Introduction

For more than 80 years, the abundance discrepancy (AD) has been a challenge in
the field of abundance determination in ionized nebulae. This discrepancy is
the fact that the ionic abundances relative to hydrogen (X+i/H+) obtained from
observations of weak recombination lines (RLs) of ions heavier than helium are
systematically higher than those obtained from much brighter collisionally
excited lines (CELs) corresponding to the same ion (Wyse, 1942). This
discrepancy is generally parameterized through the abundance discrepancy
factor (ADF, Tsamis et al., 2003), which is the ratio between RLs and CELs
abundances determined for a given ion (i. e. ADF = [X+i/H+]RLs/[X+i/H+]CELs).
Recently, it has been found that a scenario in which the presence of
temperature fluctuations (Peimbert, 1967) in the zones where the gas is most
ionized, possibly produced due to the stellar feedback effect, may be
responsible for the AD in H ii regions (Méndez-Delgado et al., 2023). However,
this scenario does not appear to work in the case of planetary nebulae (PNe;
Méndez-Delgado & García-Rojas, 2023, these proceedings).

The problem is further complicated when considering that in some PNe, the AD
is much greater (ADF $\geq$ 10) than that measured in the vast majority of
ionized nebulae (ADF $\sim$ 2–4). In these cases, a binary star that has gone
through a common-envelope phase is usually present. It has been shown that a
large part of the AD is caused by two plasma components with very different
physical conditions. The first component has temperatures similar to those of
typical photoionized nebulae with a spectrum mainly composed of RLs of H and
He and CELs of heavy elements. The second component is much colder and emits
almost no CELs, but its spectrum of RLs of heavy elements is very strong
(Corradi et al., 2015; Wesson et al., 2018). The presence of this cold
component significantly affects the determination of the physical conditions
of the gas and the determination of the chemical abundances in both components
(Gómez-Llanos et al., 2020; García-Rojas et al., 2022).

Gómez-Llanos & Morisset (2020) developed a grid of photoionization models that
combine cold metal-rich clumps embedded in a normal metallicity region and
found that the real contrast between chemical abundances relative to H in both
components (the Abundance Contrast Factor, ACF) could be different from the
observationally computed ADF. These results call for a review of the
methodology for computing ion abundances in this type of objects.

##  2 Using a correct temperature for H$\beta$

In the following, we use the same formalism and equations as in Gómez-Llanos
et al. (2024, in prep.).

The classical way of determining ionic abundances from emission line is using
the following relations:

$I_{\lambda}=\int_{V}n_{\rm e}.n_{X^{i}}.\epsilon_{\lambda}(T_{\rm e},n_{\rm
e})dv\simeq\frac{X^{i}}{H^{+}}.n_{\rm e}^{2}.\epsilon_{\lambda}(T_{\rm
e},n_{\rm e}).V,$ (1)

where $n_{\rm e}$ and $n_{X^{i}}$ are, respectively, the densities of
electrons and of the ions under consideration, ${\epsilon_{\lambda}(T_{\rm
e},n_{\rm e})}$ is the line emissivity in erg/s.cm3, $X^{i}/H^{+}$ is the
ionic abundance to be determined, $T_{\rm e}$ is the electron temperature (but
see below). The integral is carried out over the volume $V$ of the nebula. In
the case of H$\beta$ line this relation is:

$I_{\beta}=\int_{V}n_{\rm e}.n_{H^{+}}.\epsilon_{\beta}(T_{\rm e},n_{\rm
e})dv\simeq n_{\rm e}^{2}.\epsilon_{\beta}(T_{\rm e},n_{\rm e}).V,$ (2)

neglecting the contribution of elements other than H to free electrons.
Combining both relations leads to:

$\frac{X^{i}}{H^{+}}=\frac{I_{\lambda}}{I_{\beta}}.\frac{\epsilon_{\beta}(T_{\rm
e},n_{\rm e})}{\epsilon_{\lambda}(T_{\rm e},n_{\rm e})},$ (3)

where ${I_{\lambda}}/{I_{\beta}}$ is the line ratio involving the $X^{i}$ ion.
Here, we find the first potential problem: the temperature and density used in
both emissivities are not necessarily the same. It is very common in detailed
analysis to consider two or even tree zones in the considered nebula, the
difference between the zones being described by the main ions it contains. For
example, a two-zone description could have a low ionization zone where the
temperature is determined from [N ii] $\lambda$5755/$\lambda$6584, while the
high ionization zone is described by the [O iii] $\lambda$4363/$\lambda$5007
temperature diagnostic. The important issue here is the temperature (and
density in a less important way) used in the H$\beta$ emissivity: as the
H$\beta$ line is the same in both zones, its emissivity should not depend on
where the metal line is coming from. The emissivity of the H$\beta$ line must
be obtained from a pair of $T_{\rm e},N_{\rm e}$ representative of the whole
$H^{+}$ region, which encloses both zones.

As an illustration, we consider a nebula where the low ionization zone has a
temperature of 12,000 K, while the high ionization zone is at 9,000 K. The H i
line can be considered at 10,000 K. The classical way to determine abundances,
using the same electron temperature for the metal line and for H$\beta$ would
underestimate the $N^{+}/H^{+}$ ionic abundance in the low ionization zone
(using [N ii] $\lambda$6584) by 15%, while it would overpredict the
$O^{++}/H^{+}$ ionic abundance in the high ionization zone (using [O iii]
$\lambda$5007) by 9%.

The PyNeb `getIonabunce` method gives the user the opportunity to specify the
electron temperature for the line under consideration, as well as the
temperature for the H$\beta$ line used to normalize the observation:

    O3.getIonAbundance(int_ratio=350, tem=9000, den=100., wave=5007,
                       Hbeta=100, tem_HI=10000)

##  3 Cold regions disguised as hot ones

In the case of a metal-rich cold phase, the dominant channel to produce
emission lines such as [N ii] $\lambda$5755 or [O iii] $\lambda$4363 is not
longer the radiative decay following a collisional excitation of N+ or O++
ions, but rather the recombination of N++ to N+ or O3+ to O++, respectively.

This phenomenon has already been reported in observations (e.g. Jones et al.,
2016; García-Rojas et al., 2016, 2022) and in theoretical studies (e.g. Gómez-
Llanos & Morisset, 2020). Failing to consider the contribution of the
recombination to the auroral lines leads to significantly inaccurate
estimations of the electron temperature and subsequently the chemical
abundances.

##  4 Detailed abundance determinations for two-components plasma

In the case of two different phases of gas mixed in the line of sight, the
previous relations lead to the following.

$\begin{split}I_{\lambda}=&I_{\lambda}^{w}+I_{\lambda}^{c}\\\
=&[\frac{X^{i}}{H^{+}}]^{w}\cdot(n_{\rm
e}^{w})^{2}\cdot\epsilon_{\lambda}(T_{\rm e}^{w},n_{\rm e}^{w})\cdot V^{w}+\\\
&[\frac{X^{i}}{H^{+}}]^{c}\cdot(n_{\rm
e}^{c})^{2}\cdot\epsilon_{\lambda}(T_{\rm e}^{c},n_{\rm e}^{c})\cdot V^{c}\\\
=&[\frac{X^{i}}{H^{+}}]^{w}\cdot[N_{\rm e}.n_{\rm
e}]^{w}\cdot\epsilon_{\lambda}(T_{\rm e}^{w},n_{\rm e}^{w})+\\\
&[\frac{X^{i}}{H^{+}}]^{c}\cdot[N_{\rm e}.n_{\rm
e}]^{c}\cdot\epsilon_{\lambda}(T_{\rm e}^{c},n_{\rm e}^{c}),\end{split}$ (4)

the case of H$\beta$ being:

$\begin{split}I_{\beta}=&I_{\beta}^{w}+I_{\beta}^{c}\\\ =&(n_{\rm
e}^{w})^{2}\cdot\epsilon_{\beta}(T_{\rm e}^{w},n_{\rm e}^{w}).V^{w}+(n_{\rm
e}^{c})^{2}\cdot\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm e}^{c}).V^{c}\\\
=&[N_{\rm e}.n_{\rm e}]^{w}\cdot\epsilon_{\beta}(T_{\rm e}^{w},n_{\rm
e}^{w})+[N_{\rm e}.n_{\rm e}]^{c}\cdot\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm
e}^{c})\\\ =&(1-\omega)\cdot I_{\beta}+\omega\cdot I_{\beta},\end{split}$ (5)

where the $w$ and $c$ superscripts are used for the warm and cold regions,
respectively, and $\omega$ is defined as the weight of the cold region in the
$I_{\beta}$ emission: $\omega=I_{\beta}^{c}/I_{\beta}$. The total number of
electrons in each region is $N_{\rm e}=n_{\rm e}.V$.

Combining the 2 previous equations, we obtain:

$\begin{split}\frac{I_{\lambda}}{I_{\beta}}=&[\frac{X^{i}}{H^{+}}]^{w}\cdot(1-\omega)\cdot\frac{\epsilon_{\lambda}(T_{\rm
e}^{w},n_{\rm e}^{w})}{\epsilon_{\beta}(T_{\rm e}^{w},n_{\rm e}^{w})}+\\\
&[\frac{X^{i}}{H^{+}}]^{c}\cdot\omega\cdot\frac{\epsilon_{\lambda}(T_{\rm
e}^{c},n_{\rm e}^{c})}{\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm
e}^{c})}.\end{split}$ (6)

When a line is mainly emitted through collisional excitation followed by
radiative transitions, the ratio of its emissivities
$\epsilon_{\lambda}/\epsilon_{\beta}$ vanishes at low electron temperature,
and the line is produced exclusively by the warm region. On the other hand,
when a line originates from recombination followed by radiative transitions,
the ratio of emissivities $\epsilon_{\lambda}/\epsilon_{\beta}$ remains almost
independent of the electron temperature. The contribution of the warm (cold)
region to the observed line intensity depends primarily on $1-\omega$
($\omega$ respectively), as well as on the ionic abundance of the emitting
ion.

Using the findings of bi-abundance models built by Gómez-Llanos & Morisset
(2020) and depicted in their Fig. 18, in the following we explore the
hypothesis that [O iii] and O ii come from exclusive regions (warm and cold,
respectively). The ionic abundances for each region are determined from
different lines using the following equations:

$[\frac{X^{i}}{H^{+}}]^{w}=\frac{I_{\lambda}}{(1-\omega)\cdot
I_{\beta}}\cdot\frac{\epsilon_{\beta}(T_{\rm e}^{w},n_{\rm
e}^{w})}{\epsilon_{\lambda}(T_{\rm e}^{w},n_{\rm e}^{w})},$ (7)

when $I_{\lambda}^{c}=0$, and:

$[\frac{X^{i}}{H^{+}}]^{c}=\frac{I_{\lambda}}{\omega\cdot
I_{\beta}}\cdot\frac{\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm
e}^{c})}{\epsilon_{\lambda}(T_{\rm e}^{c},n_{\rm e}^{c})},$ (8)

when $I_{\lambda}^{w}=0$.

It is important to note here that the $T_{\rm e}$ and $n_{\rm e}$ values used
to compute the H$\beta$ emissivity should correspond to the specific region
under consideration (warm or cold). Refer to Sec. 2 for the case of the warm
region.

The ionic abundances derived from both collisionally excited lines and
recombination lines using the generic eq. 3 are both underestimated by a
factor close to $1/(1-\omega)$ and $1/\omega$ respectively. If the effect of
$\omega$ is not taken into account, the ratio between the two determinations
is the classical ADF. Thus, ACF is linked to ADF by a factor of
$(1-\omega)/\omega$, which can be notable. Hence, there is a clear necessity
to determine $\omega$ for each spaxel in the images to attain reliable
abundances.

The determination of the weight of the cold region $\omega$ is a challenging
task. It can be computed by considering that the hydrogen emission arises from
both regions, and that the Balmer or Paschen jump normalized to an H i line is
indicative of the emission contribution from each region. This ratio is
commonly employed to determine the so-called Balmer or Paschen temperature. In
the case of two regions with very different temperatures, the derived
temperature from the Balmer or Paschen jump may not accurately reflect the
actual temperature of any plasma component within the observed object.

The theoretical Paschen jump and H-lines are computed as the sum of
contributions from both the cold and the warm regions:

$\begin{split}PJ\left(\omega,T_{\rm e}^{w},T_{\rm e}^{c},n_{\rm e}^{w},n_{\rm
e}^{c}\right)=&[N_{\rm e}.n_{\rm e}]^{w}\cdot C_{8100}(T_{\rm e}^{w},n_{\rm
e}^{w})+[N_{\rm e}.n_{\rm e}]^{c}\cdot C_{8100}(T_{\rm e}^{c},n_{\rm
e}^{c})-\\\ &[N_{\rm e}.n_{\rm e}]^{w}\cdot C_{8400}(T_{\rm e}^{w},n_{\rm
e}^{w})-[N_{\rm e}.n_{\rm e}]^{c}\cdot C_{8400}(T_{\rm e}^{c},n_{\rm
e}^{c})\\\ =&\left[(1-\omega)\cdot\frac{I_{\beta}}{\epsilon_{\beta}(T_{\rm
e}^{w},n_{\rm e}^{w})}\cdot(C_{8100}(T_{\rm e}^{w},n_{\rm
e}^{w})-C_{8400}(T_{\rm e}^{w},n_{\rm e}^{w}))\right]+\\\
&\left[\omega\cdot\frac{I_{\beta}}{\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm
e}^{c})}\cdot(C_{8100}(T_{\rm e}^{c},n_{\rm e}^{c})-C_{8400}(T_{\rm
e}^{c},n_{\rm e}^{c}))\right]\end{split}$ (9)

$\begin{split}I_{9229}\left(\omega,T_{\rm e}^{w},T_{\rm e}^{c},n_{\rm
e}^{w},n_{\rm
e}^{c}\right)=&(1-\omega)\cdot\frac{I_{\beta}}{\epsilon_{\beta}(T_{\rm
e}^{w},n_{\rm e}^{w})}\cdot\epsilon_{9229}(T_{\rm e}^{w},n_{\rm e}^{w})+\\\
&\omega\cdot\frac{I_{\beta}}{\epsilon_{\beta}(T_{\rm e}^{c},n_{\rm
e}^{c})}\cdot\epsilon_{9229}(T_{\rm e}^{c},n_{\rm e}^{c})\end{split}$ (10)

In the first equation, we omit explicitly stating the dependencies of the
continuum on He+/H+ and He2+/H+, but they are considered.

The normalized Paschen jump PJ/I9229 is computed using PyNeb (Luridiana et
al., 2015) across a grid of $T_{\rm e}^{w}$, $T_{\rm e}^{c}$, and $\omega$,
and constant values for $n_{\rm e}^{w}$, $n_{\rm e}^{c}$, He+/H+, and He2+/H+.
These numerical values must be adapted for each object. As an illustration,
the values of PJ/I9229 obtained for $T_{\rm e}^{w}$ between 6,000 and 14,000
K, and $\omega$ between 0 and 0.25, and fixed values of $T_{\rm e}^{c}$ =
3,000 K, $n_{\rm e}^{w}$ = $n_{\rm e}^{c}$ = 1,000 cm-3, He+/H+ = 0.1, and
He2+/H+ = 0.005 are used in Fig. 1 to draw as colored hexagons the variations
of $\omega$ relative to PJ/I9229 and $T_{\rm e}^{w}$. Lines are also showing
the effect of changing $T_{\rm e}^{c}$ to 1,000, 2,000, 3,000 (the same value
used for the colored hexagons), and 4,000 K for the case $\omega$ = 0.1.

A numerical algorithm (which can be an Artificial Neuron Network) is then used
to interpolate the inverse problem to predict $\omega$ from T${}_{\rm e}^{w}$,
T${}_{\rm e}^{c}$, and PJ/I9229.

In the case of NGC 6153, Gómez-Llanos et al. (2024, in prep.) generate a
spatially resolved map of $\omega$ and use it to compute the ACF of O2+.

Figure 1: Colored hexagons show the variation of $\omega$ when $T_{\rm e}^{w}$
varies from 6,000 to 14,000 K and the normalized Paschen jump PJ/I9229 varies
from 0.010 to 0.017. Fixed values are set to: $T_{\rm e}^{c}$ = 3,000 K,
$n_{\rm e}^{w}$ = $n_{\rm e}^{c}$ = 1,000 cm-3, He+/H+ = 0.1, and He2+/H+ =
0.005. Lines of different styles show the values of PJ/I9229 for different
values of $T_{\rm e}^{c}$ and a fixed value of $\omega$ = 0.1.

##  5 Conclusions

As highlighted by Richer et al. (2022) regarding the PN NGC 6153, “reality is
complicated” when determining the chemical abundances in PNe, particularly in
those with high ADFs. It is essential to pay close attention to details when
attempting to obtain accurate chemical abundances in these objects, as has
been shown in this proceeding.

CM acknowledges funding from PAPIIT/UNAM grants IN101220 and IG101223. JG-R
and VG-LL acknowledge funding from the Canarian Agency for Research,
Innovation and Information Society (ACIISI), of the Canary Islands Government,
and the European Regional Development Fund (ERDF), under grant with reference
ProID2021010074. JG-R also acknowledges the support of the Spanish Ministry of
Science and Innovation (MICINN) through the Spanish State Research Agency,
under the Severo Ochoa Centers of Excellence Program 2020-2023
(CEX2019-000920-S).

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# Impact of non-Markovian evolution on characterizations of quantum
thermodynamics

Devvrat Tiwari<EMAIL_ADDRESS>Subhashish Banerjee<EMAIL_ADDRESS>Indian Institute of Technology Jodhpur-342030, India

###### Abstract

Here we study the impact of non-Markovian evolution on prominent
characteristics of quantum thermodynamics, such as ergotropy and power. These
are benchmarked by the behavior of the quantum speed limit time. We make use
of both geometric-based, particularly quantum Fisher and Wigner-Yanase
information metric, and physical properties based-measures, particularly
relative purity measure and relative entropy of coherence measure, to compute
the quantum speed limit time. A simple non-Markovian model of a qubit in a
bosonic bath exhibiting non-Markovian amplitude damping evolution is
considered, which, from the quantum thermodynamic perspective with finite
initial ergotropy, can be envisaged as a quantum battery. To this end, we
explore the connections between the physical properties-based measures of
quantum speed limit time and the coherent component of ergotropy. The non-
Markovian evolution is shown to impact the recharging process of the quantum
battery. Further, a connection between the discharging-charging cycle of the
quantum battery and the geometric measures of quantum speed limit time is
observed.

Quantum speed limit, ergotropy, non-Markovian evolution

##  I Introduction

A realistic quantum system is subjected to the influence of the environment.
The dynamics of the system is altered by this, and as a consequence,
information is lost from the system to the external environment. A paradigm
for examining how the ambient environment affects a quantum system is provided
by the theory of open quantum systems (OQS) [1, 2, 3]. Ideas pertaining to
open quantum systems are applicable to a number of scenarios [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. A Markovian approximation, which
implies that the environment instantly recovers from its contact with the
system, can be used to describe the evolution of an OQS in many situations.
This results in a constant movement of information from the system to the
environment. However, research has being pushed into domains beyond the
Markovian evolution due to increasing technical and technological
advancements. In many of these situations, a clean division between system and
environment time scales cannot be anticipated, leading to non-Markovian
behavior [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

One of the cornerstones of quantum physics is Heisenberg’s uncertainty
principle. The uncertainty principle for position and momentum demonstrates
that it is impossible to accurately measure both position and momentum at the
same time; however, the meaning of the energy-time uncertainty relation in
this statement is not immediately clear. The energy-time uncertainty principle
($\Delta t\geq\hbar/\Delta E$) is a statement about the intrinsic time in
which a quantum system develops, as demonstrated by Mandelstam and Tamm (MT)
in [37]. Using the Fubini-Study metric on the space of quantum pure states,
the concept of the quantum speed limit (QSL) time, or the speed at which a
quantum system can evolve, was first presented in [38]. Margolus and Levitin
(ML) [39] gave a different QSL time depending on the mean energy. Combining
the MT and ML bounds results in a tighter QSL time for unitary dynamics,
limited to orthogonal pure states. QSL time alludes to the shortest period of
time needed for a quantum system to evolve from one state to another. It plays
a crucial role in the development of quantum technologies such as quantum
computing and quantum communication. Over the years, many measures to quantify
the QSL time have emerged. Here we use the geometric based measures,
particularly Fisher and Wigner-Yanase (WY) information metrics-based measures,
as well as the inherent dynamics-based measures, particularly relative entropy
and entropy of coherence-based measures to compute the QSL time. The design
and execution of quantum information processing algorithms are significantly
impacted by the expansion of the QSL time to open quantum systems [40, 41, 42,
43]. Due to its applicability to other technical areas, this has been a
subject of considerable recent research [44, 45, 46, 47, 48, 49, 50].

The fundamental laws of thermodynamic equilibrium and nonequilibrium in the
quantum regime are the subject of quantum thermodynamics [51, 52, 53, 54]. The
emerging field of quantum thermodynamics has grown rapidly over the last
decade. The main objective of quantum thermodynamics is to extend classical
thermodynamics to incorporate quantum effects and tiny ensemble sizes. This is
facilitated by the rapid experimental control of quantum systems and the
engineering of small environments. The extraction of maximal work from a
system is an old problem in thermodynamics. It was shown [55, 56] that in
quantum systems, this can be quantified by the ergotropy of the system. The
ergotropy has been established as an important quantity in the emerging field
of quantum thermodynamics [57, 58, 59, 60, 61] and has recently been measured
experimentally [62, 63].

Quantum batteries are quantum-mechanical energy storage devices [64, 52]. The
role of quantum effects on the issue of energy storage has been extensively
studied in the last years [65, 66, 67, 68, 69, 70, 71]. The problem of quantum
battery’s charging and discharging has been studied in an open quantum system
setting [72] along with the impact of non-Markovian evolution [73, 74]. In
recent years, research on quantum batteries has gained much attention, and
various methods have been employed to realize a practical quantum battery [75,
76]. To this end, we aim to study a simple OQS model exhibiting non-Markovian
behavior, viz., a non-Markovian amplitude damping (NMAD) evolution. We exploit
characterizers of quantum thermodynamics, particularly ergotropy, along with
instantaneous and average powers, to study this model from the perspective of
a quantum battery. This is benchmarked by the QSL time of the evolution of the
system, where we use geometric and inherent dynamics-based measures of QSL
time.

This paper is organized as follows. In Sec. II, we discuss the preliminaries
used throughout the paper, including the discussion of QSL time, ergotropy,
instantaneous and average powers, and an OQS model with NMAD evolution.
Section III discusses a direct connection between the physical properties-
based measure of QSL time and the coherent component of the ergotropy. We
study the connection of geometrical based-measures of QSL time with ergotropy,
instantaneous and average powers, and impact of non-Markovian evolution over
the discharging-charging process of quantum batteries in Sec. IV. This is
followed by conclusions.

##  II Preliminaries

This section briefly discusses different bounds on the quantum speed limit
time, ergotropy followed by instantaneous and average powers, and a single
qubit model with non-Markovian evolution modeled by the non-Markovian
amplitude damping (NMAD) master equation.

###  II.1 Quantum speed limit (QSL) time

Here we discuss four forms of the quantum speed limit (QSL) time. The first
two are related to the geometric QSL time, where we use quantum Fisher
information and Wigner Yanase (WY) information metrics for the geodesic
distance between the initial and the final state at time $t$. The other two
measures depend on the inherent dynamics of the system, one of which is a
definition of QSL time using the relative purity measure of a quantum state
between initial and final states, and the other is a QSL time based on the
coherence of the initial and final states.

####  II.1.1 Using quantum Fisher information metric

Mandelstam and Tamm (MT) and Margolus and Levitin (ML)-type bounds on speed
limit time [37, 39] are estimated by using the geometric approach, using the
Bures angle, to quantify the closeness between the initial and final states.
This approach was used to provide a bound for the initial pure state
$\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$, on the quantum speed limit time
$(\tau_{QSL})$ [40] as

$\tau_{QSL}=\max\Bigg{\\{}\frac{1}{\Lambda^{\textrm{op}}_{\tau}},\frac{1}{\Lambda^{\textrm{tr}}_{\tau}},\frac{1}{\Lambda^{\textrm{hs}}_{\tau}}\Bigg{\\}}\sin^{2}[\mathcal{B}],$
(1)

where $\mathcal{B}(\rho_{0},\rho_{t})=\arccos\left(\left\\{{\rm
Tr}\left[\sqrt{\sqrt{\rho_{0}}\rho_{t}\sqrt{\rho_{0}}}\right]\right\\}^{2}\right)$,
and

$\Lambda^{\textrm{op,tr,hs}}_{\tau}=\frac{1}{\tau}\int^{\tau}_{0}dt||\mathcal{L}(\rho_{t})||_{\textrm{op,tr,hs}}.$
(2)

The three norms $||\cdot||_{\text{op}}$, $||\cdot||_{\text{tr}}$, and
$||\cdot||_{\text{hs}}$ are the operator, trace, and Hilbert-Schmidt norms,
respectively. $\mathcal{L}$ is the Liouvillian superoperator acting on $\rho$.
From the norm inequalities, it can be shown that the operator norm of the
generator provides a tighter bound on the quantum speed limit time.

####  II.1.2 Using Wigner-Yanase information metric

Here we make use of the Wigner-Yanase information metric given by

$\mathcal{B}(\rho_{0},\rho_{t})=\arccos\left({\rm
Tr}\left[\sqrt{\rho_{0}}\sqrt{\rho_{t}}\right]\right),$ (3)

in the expression of the QSL time given in Eq. (1). The QSL time obtained
using this metric is $\tilde{\tau}_{QSL}$. In [43], it was shown that using
this metric, one can get an upper bound on the QSL time in the case of mixed
states.

####  II.1.3 QSL time using relative purity measure

A bound on the QSL time for the open quantum systems can be given based on the
relative purity measure given in [41]. The bound to the required time of
evolution, in this case, is given as

$\tau\geq\tau^{\prime}_{QSL}=\frac{4\theta^{2}{\rm
Tr}(\rho_{0}^{2})}{\pi^{2}\overline{\sqrt{{\rm
Tr}\left[\left(\mathcal{L}^{\dagger}\rho_{0}\right)^{2}\right]}}},$ (4)

where $\theta=\cos^{-1}[f(t)]$. $f(t)={\rm Tr}(\rho_{\tau}\rho_{0})/{\rm
Tr}(\rho_{0}^{2})$ is the relative purity measure, where $\rho_{0}$ is the
initial state and $\rho_{\tau}$ is the state evolved to time $t=\tau$. In the
denominator, we have $\bar{(\cdot)}=\tau^{-1}\int_{0}^{\tau}(\cdot)dt$.
Further, we discuss another quantifier of the coherence of a quantum state
given by $l_{1}$ norm of coherence [77], defined as

$\mathcal{C}_{l_{1}}(\rho)=\sum_{i,j,i\neq j}|\rho_{i,j}|,$ (5)

where $\rho_{i,j}$ are the off-diagonal elements of the density matrix of the
state. The QSL time defined in Eq. (4) can now be written using the $l_{1}$
norm of the coherence $\mathcal{C}_{l_{1}}$ as [47]

$\tau^{\prime}_{QSL}=\frac{4\sqrt{2}\cos^{-1}[f(t)]^{2}{\rm
Tr}(\rho_{0}^{2})}{\frac{\pi^{2}}{\tau}\int_{0}^{\tau}\left|\frac{\dot{p}_{t}}{p_{t}^{2}}\mathcal{C}_{l_{1}}(\rho_{t})\right|dt},$
(6)

where $p_{t}$ is the decoherence function that describes the dynamical
evolution of the state.

####  II.1.4 QSL time for the coherence

There are several widely known (basis dependent) quantum coherence measures,
such as the relative entropy of coherence, the $l_{1}$ norm of coherence, the
geometric coherence, and the robustness of coherence. The relative entropy of
coherence is used because of its operational meaning as distillable coherence.
In addition, it is also easier to work and compute compared to some other
measures of coherence. For a given state $\rho$, the relative entropy of
coherence $C(\rho)$ deﬁned as

$C(\rho)=S(\rho^{D})-S(\rho),$ (7)

where $\rho^{D}=\sum_{i}\bra{i}\rho\ket{i}|i\rangle\langle i|$ is the density
operator that is diagonal in the reference basis, obtained by dephasing off-
diagonal elements of $\rho$, and $S(\rho)$ is the von Neumann entropy of the
state $\rho$.

For an arbitrary quantum dynamics of a ﬁnite-dimensional quantum system
describable as the time-evolution of its state, the minimum time needed for
the state $\rho_{\tau}$ to attain coherence $C(\rho_{\tau})$, starting with
the initial coherence $C(\rho_{0})$, is lower bounded by quantum speed limit
time for coherence ($\tau_{CSL}$) given by [50]

$\tau\geq\tau_{CSL}=\frac{|C(\rho_{\tau})-C(\rho_{0})|}{\Lambda^{\text{rms},D}_{\tau}\overline{||\ln\rho_{t}^{D}||^{2}_{HS}}+\Lambda^{\text{rms}}_{\tau}\overline{||\ln\rho_{t}||^{2}_{HS}}},$
(8)

where
$\Lambda^{\text{rms},D}_{\tau}=\sqrt{\frac{1}{\tau}\int_{0}^{\tau}||\mathcal{L}_{t}(\rho_{t}^{D})||^{2}_{HS}dt}$,
$\Lambda^{\text{rms}}_{\tau}=\sqrt{\frac{1}{\tau}\int_{0}^{\tau}||\mathcal{L}_{t}(\rho_{t})||^{2}_{HS}dt}$,
$\overline{||\ln\rho_{t}^{D}||^{2}_{HS}}=\sqrt{\frac{1}{\tau}\int_{0}^{\tau}||\ln\rho_{t}^{D}||^{2}_{HS}dt}$,
and
$\overline{||\ln\rho_{t}||^{2}_{HS}}=\sqrt{\frac{1}{\tau}\int_{0}^{\tau}||\ln\rho_{t}||^{2}_{HS}dt}$.
$||\cdot||_{HS}$ is the Hilbert-Schmidt norm, and $\mathcal{L}_{t}$ is the
Liouvillian superoperator acting on $\rho$.

###  II.2 Ergotropy, instantaneous and average power

Here we define the ergotropy, followed by instantaneous and average powers.
The categorization of ergotropy as a sum of incoherent and coherence
ergotropies is also defined.

####  II.2.1 Ergotropy

Ergotropy refers to the maximum amount of work that can be extracted from a
quantum system through a cyclic unitary transformation of the initial state.
We assume that we are given a quantum state $\rho$ with its internal
Hamiltonian $H$ such that they have the following spectral decomposition:

$\rho=\sum_{i}r_{i}\ket{r_{i}}\bra{r_{i}},$ (9)

and

$H=\sum_{i}\epsilon_{i}\ket{\epsilon_{i}}\bra{\epsilon_{i}},$ (10)

where ordering of the eigenvalues for $\rho$ and $H$ is in the decreasing,
$r_{1}\geq r_{2}\geq....,$ and increasing,
$\epsilon_{1}\leq\epsilon_{2}\leq...$, order, respectively. Since unitary
dynamics is considered, any decrease in the internal energy of the system at
hand, with respect to its self-Hamiltonian $H$, will be extracted as work.
Thus, in order to find the ergotropy, one aims to minimize the internal energy
of the final state

$\mathcal{W}={\rm Tr}(\rho H)-\min\\{{\rm Tr}(U\rho U^{\dagger}H)\\},$ (11)

where the minimization is performed over all possible unitaries.

It can be shown that the final state $\rho_{f}=U\rho U^{\dagger}$ that
achieves this minimum has the form
$\rho_{f}=\sum_{j}r_{j}\ket{\epsilon_{j}}\bra{\epsilon_{j}}$. This is known as
the passive state, i.e., no work can be extracted from this state. A unitary
operator which performs such a transformation is
$U=\sum_{j}\ket{\epsilon_{j}}\bra{r_{j}}$. Now we can rewrite Eq. (11) as

$\mathcal{W}=\sum_{j,i}r_{j}\epsilon_{i}(|\bra{r_{j}}\epsilon_{i}\rangle|^{2}-\delta_{ij}).$
(12)

It was recently recognized [60, 61] that quantum ergotropy can be separated
into two different (coherent and incoherent) contributions

$\mathcal{W}=\mathcal{W}_{i}+\mathcal{W}_{c}.$ (13)

The incoherent ergotropy $\mathcal{W}_{i}$ denotes the maximal work that can
be extracted from $\rho$ without changing its coherence, which is defined as

$\mathcal{W}_{i}(\rho)={\rm Tr}\left\\{\left(\rho-\sigma\right)H\right\\},$
(14)

where $\sigma$ is the coherence invariant state of $\rho$, which has property

${\rm Tr}\left\\{\sigma H\right\\}=\min_{\mathcal{U}\in\mathcal{U}^{(i)}}{\rm
Tr}\left\\{\mathcal{U}\rho\mathcal{U}^{\dagger}H\right\\},$ (15)

where $\mathcal{U}^{(i)}$ is the set of unitary operations without changing
the coherence of $\rho$.

The coherent ergotropy $\mathcal{W}_{c}$ is the work that is exclusively
stored in the coherence. This can be quantified by the relative entropy of
coherence defined in Eq. (7). The expression for the coherent ergotropy was
derived in [60] and can be written in terms of classical relative entropy as

$\beta\mathcal{W}_{c}(\rho)=C(\rho)+S\left(\mathcal{E}(\sigma)||\rho^{eq}\right)-D\left(\rho||\rho^{eq}\right),$
(16)

where $S\left(\mathcal{E}(\sigma)||\rho^{eq}\right)={\rm
Tr}\left\\{\sigma\ln(\sigma)\right\\}-{\rm
Tr}\left\\{\mathcal{E}(\sigma)\ln(\rho^{eq})\right\\}$ is the quantum relative
entropy and
$D\left(\rho||\rho^{eq}\right)=\sum_{i}p_{i}\ln\left(\frac{p_{i}}{s_{i}}\right)$
is the classical relative entropy; $p_{i}$ and $s_{i}$ are the eigenvalues of
the state $\rho$ and $\rho^{eq}$, respectively. $\mathcal{E(\sigma)}$ is the
pure dephasing operation ($\sum_{i}\bra{i}\sigma\ket{i}|i\rangle\langle i|$)
on the coherence invariant state $\sigma$ defined in Eq. (15). The state
$\rho^{eq}$ is the Gibbs state

$\rho^{eq}=\frac{\exp(-\beta H)}{Z}\,\,\,\,\text{with}\,\,\,\,Z={\rm
Tr}\left\\{\exp(-\beta H)\right\\},$ (17)

and $\beta=1/k_{B}T$, with $T$ being the temperature. In the case of a single
qubit having energy eigenvalues $0$ and $\omega_{0}$, the coherent part of the
ergotropy can be directly expressed in terms of the purity of the state
$F(\rho)={\rm Tr}(\rho^{2})$, and in terms of $l_{1}$ norm of the coherence
defined in Eq. (5) as

$\mathcal{W}_{c}(\rho)=\frac{\omega_{0}}{2}\left[\sqrt{2F(\rho)-1}+\sqrt{2F(\rho)-1-C^{2}_{l_{1}}}\right].$
(18)

Interestingly, it can be observed that, in the case considered here, if we
replace the positive sign in between the two square roots in right hand side
of the above equation, the left-hand side becomes the ergotropy of the system.

####  II.2.2 Average and instantaneous power

The instantaneous charging power is defined by available work in the battery
as

$\mathcal{P}(t)=\lim_{\Delta t\rightarrow 0}\frac{\mathcal{W}(t+\Delta
t)-\mathcal{W}(t)}{\Delta t}=\frac{d\mathcal{W}}{dt}.$ (19)

It is also possible to define the average power-to-energy transfer given by

$\mathcal{P}_{av}=\frac{\mathcal{W}(t)-\mathcal{W}(t_{0})}{t-t_{0}},$ (20)

where $t-t_{0}$ refers to the charging time of the battery.

###  II.3 The model

We consider an example of the decay of a two-state atom into a bosonic
reservoir [21]. The general form of the total Hamiltonian is given as

$H=H_{S}\otimes\mathbb{I}_{B}+\mathbb{I}_{S}\otimes H_{B}+H_{I},$ (21)

where $H_{S}$ is the system’s Hamiltonian, $H_{B}$ is the Hamiltonian of the
reservoir, and interaction between the system and the reservoir is given by
the Hamiltonian $H_{I}$. The form of system Hamiltonian is

$H_{S}=\omega_{0}\sigma_{+}\sigma_{-},$ (22)

where $\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=|g\rangle\langle e|$
are the atomic raising and lowering operators, respectively, with
$\ket{g}(\ket{e})$ denoting the ground (excited) state. The environment is
represented by a reservoir of harmonic oscillators given as

$H_{B}=\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k},$ (23)

where $a_{k}$ and $a_{k}^{\dagger}$ are the bosonic creation and annihilation
operators, satisfying the commutation relation
$[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}$. The form of
interaction Hamiltonian can be given as

$H_{I}=\sum_{k}\left(g_{k}\sigma_{+}\otimes a_{k}+g^{*}_{k}\sigma_{-}\otimes
a_{k}^{\dagger}\right),$ (24)

where $g_{k}$ is the coupling constant. In this case, the total number of
excitations in the system
$N=\sigma_{+}\sigma_{-}+\sum_{k}a_{k}^{\dagger}a_{k}$ is a conserved quantity
due to the rotating wave approximation. The dynamical map of the evolution of
the reduced state of the system was derived in [78] and is given by the
quantum master equation

$\frac{d}{dt}\rho_{s}(t)=\frac{-i}{2}S(t)\left[\sigma_{+}\sigma_{-},\rho_{s}\right]+\gamma(t)\left[\sigma_{-}\rho_{s}\sigma_{+}-\frac{1}{2}\left\\{\sigma_{+}\sigma_{-},\rho_{s}\right\\}\right],$
(25)

where $\gamma(t)=-2\Re\left(\frac{\dot{G}(t)}{G(t)}\right)$, and
$S(t)=-2\Im\left(\frac{\dot{G}(t)}{G(t)}\right)$. $\Re$ and $\Im$ represents
real and imaginary parts of the quantity inside brackets, respectively. For a
Lorentzian spectral density of the environment in resonance with the
transition frequency of the qubit, the expression for the function $G(t)$ can
be given as

$G(t)=e^{-\lambda
t/2}\left[\cosh\left(\frac{lt}{2}\right)+\frac{\lambda}{l}\sinh\left(\frac{lt}{2}\right)\right],$
(26)

where $l=\sqrt{\lambda^{2}-2\gamma_{0}\lambda}$. Here $\gamma_{0}$ describes
the strength of the system-environment coupling, and $\lambda$ is the spectral
width related to the environment. The quantities $S(t)$ and $\gamma(t)$ in Eq.
(25) provide the time-dependent frequency shift and decay rates, respectively.
Further, the quantity $-2\left(\frac{\dot{G}(t)}{G(t)}\right)$ can be written
as

$-2\frac{\dot{G}(t)}{G(t)}=2\left(\frac{\gamma_{0}}{\sqrt{1-\frac{2\gamma_{0}}{\lambda}\coth\left(\frac{1}{2}\lambda
t\sqrt{1-\frac{2\gamma_{0}}{\lambda}}\right)+1}}\right).$ (27)

In the limit $\lambda<2\gamma_{0}$, the decay rate becomes negative for
certain intervals giving rise to non-Markovian evolution, referred to as the
non-Markovian amplitude damping (NMAD) evolution. In the limit
$\lambda>2\gamma_{0}$, the dynamics become time-dependent Markovian. It can
also be noted that for $\lambda\gg\gamma_{0}$, the decay rate
$\gamma(t)=\gamma_{0}$, that is, it becomes time-independent, and the
evolution corresponds to the standard AD channel.

We can envisage the system as a quantum battery in the model studied here. The
battery discharges when the ergotropy of the system dissipates to the
environment. We start with a state with finite ergotropy, and the ergotropy
vanishes due to interaction with the environment during the evolution of the
system. However, due to the system’s P-indivisible non-Markovian evolution
[35], the battery recharges again as the ergotropy revives. This is
highlighted in the subsequent sections.

Figure 1: Comparison between QSL time obtained using Fisher information metric
($\tau_{QSL}$) WY information metric ($\tilde{\tau}_{QSL}$), and relative
purity measure ($\tau^{\prime}_{QSL}$) under the evolution of state through
NMAD master equation. The parameters are: $\omega_{0}=1$, $\lambda=0.5$,
$\gamma_{0}=10$, and $T=1$.

In Fig. 1, we compare the QSL time obtained using the Fisher information
metric ($\tau_{QSL}$), Wigner-Yanase (WY) information metric
($\tilde{\tau}_{QSL}$), and the relative purity measure
($\tau^{\prime}_{QSL}$). The evolution of the quantum state is through by the
NMAD master equation defined in Eq. (25). The initial state, in this case, and
subsequently is taken to be $\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$, where
$\ket{\psi_{0}}=\frac{\sqrt{3}}{2}\ket{0}+\frac{1}{2}\ket{1}$, with $\ket{0}$
and $\ket{1}$ being the ground and excited state of the system, respectively.
It can be observed that qualitatively the QSL time obtained using WY and
Fisher information metrics are similar, with WY information giving a tighter
bound on the QSL time. This is consistent with the comparison of both metrics
in [43].

We will now see the connection between various forms of the quantum speed
limit, particularly QSL time for coherence and using relative purity measure,
with the ergotropy, with the dynamics being generated by the NMAD model.

##  III Connection of the quantum speed limits with the coherent ergotropy

This section first discusses the connection between the QSL for coherence and
coherent ergotropy. Further, we connect the QSL time obtained using the
relative purity measure and the coherent ergotropy.

###  III.1 Connection between the QSL for coherence and the coherent ergotropy

We can rewrite Eq. (16) for the relative entropy of coherence in terms of
coherent ergotropy as

$C(\rho)=\beta\mathcal{W}_{c}(\rho)-S\left(\mathcal{E}(\sigma)||\rho^{eq}\right)+D\left(\rho||\rho^{eq}\right).$
(28)

Plugging the above equation in Eq. (8) for the state $\rho_{t}$ at time
$t=\tau$ and at time $t=0$, we get a relation between the QSL time for
coherence and the coherent ergotropy.

$\tau_{CSL}=\frac{|\beta\mathcal{W}_{c}(\rho_{\tau})-S\left(\mathcal{E}(\sigma_{\tau})||\rho^{eq}\right)+D\left(\rho_{\tau}||\rho^{eq}\right)-C(\rho_{0})|}{\Lambda^{\text{rms},D}_{\tau}\overline{||\ln\rho_{t}^{D}||^{2}_{HS}}+\Lambda^{\text{rms}}_{\tau}\overline{||\ln\rho_{t}||^{2}_{HS}}},$
(29)

where
$C(\rho_{0})=\beta\mathcal{W}_{c}(\rho_{0})-S\left(\mathcal{E}(\sigma_{0})||\rho^{eq}\right)+D\left(\rho_{0}||\rho^{eq}\right)$.

Figure 2: Variation of the scaled QSL time for coherence $10\tau_{CSL}$,
ergotropy $\mathcal{W}$, coherent ergotropy $\mathcal{W}_{C}$, instantaneous
and average powers, $\mathcal{P}(t)$ and $\mathcal{P}_{av}$, respectively. The
state is evolved through the NMAD master equation. The parameters are chosen
to be $\omega_{0}=1$, $\lambda=0.5$, $\gamma_{0}=10$, and $T=1$.

In Fig. [2], we can see the variation of the QSL time for coherence
($\tau_{CSL}$) and the characterizers of quantum thermodynamics, particularly
ergotropy ($\mathcal{W}$), coherent ergotropy ($\mathcal{W}_{C}$),
instantaneous ($\mathcal{P}(t)$) and average ($\mathcal{P}_{av}$) powers. The
region in the ergotropy curve denoted by the up triangle shows the battery’s
charging process, and the down triangle shows the discharging process in the
given cycle from time $\tau=1$ to 3. During the charging process, the
instantaneous power is positive and negative while the battery is discharging.
Further, in this case, we observe that the maxima and minima of the
$\tau_{CSL}$ match that of the coherent ergotropy and contrast with the
erogtropy’s maxima and minima. This shows that the speed of evolution of the
coherence of the state during the discharging process is getting slower. This
speed hits its minimum when the ergotropy is zero. However, during the
charging process, when ergotropy again becomes non-zero, the speed of
evolution of the coherence increases and reaches its maximum when the battery
is fully charged.

###  III.2 QSL time for relative purity measure and coherent ergotropy

The relation between coherent ergotropy $\mathcal{W}_{c}$ and the $l_{1}$ norm
of coherence $\mathcal{C}_{l_{1}}$ given in Eq. (18) can be rewritten as

$\mathcal{C}_{l_{1}}(\rho)=\sqrt{2F(\rho)-1-\left(\frac{2\mathcal{W}_{c}(\rho)}{\omega_{0}}-\sqrt{2F(\rho)-1}\right)^{2}}.$
(30)

The above equation can be directly substituted in Eq. (6), to get a relation
between the coherent ergotropy $\mathcal{W}_{c}$ and the QSL time using
relative purity measure $\tau^{\prime}_{QSL}$ as

$\tau^{\prime}_{QSL}=\frac{4\sqrt{2}\cos^{-1}[f(t)]^{2}{\rm
Tr}(\rho_{0}^{2})}{\frac{\pi^{2}}{\tau}\int_{0}^{\tau}\left|\frac{\dot{p}_{t}}{p_{t}^{2}}\sqrt{2F(\rho_{t})-1-\left(\frac{2\mathcal{W}_{c}(\rho_{t})}{\omega_{0}}-\sqrt{2F(\rho_{t})-1}\right)^{2}}\right|dt},$
(31)

where the decoherence function $p_{t}$, in the case of the NMAD evolution,
given in Eq. (25) is the function $G(t)$ of Eq. (26).

Figure 3: Variation of QSL time with relative purity measure
($\tau^{\prime}_{QSL}$), ergotropy ($\mathcal{W}$), coherent ergotropy
($\mathcal{W}_{c}$), instantaneous ($\mathcal{P}(t)$) and average
($\mathcal{P}_{av}$) powers, with evolution time $\tau$. The evolution is
through the NMAD master equation. The parameters are taken to be
$\omega_{0}=1$, $\lambda=0.5$, $\gamma_{0}=10$, and $T=1$.

Figure 3 depicts the connection between the QSL time using relative purity
measure ($\tau^{\prime}_{QSL}$) with ergotropy ($\mathcal{W}$), coherent
ergotropy ($\mathcal{W}_{c}$), instantaneous ($\mathcal{P}(t)$), average
($\mathcal{P}_{av}$) powers . It can be observed that, in this case, the peaks
and valleys of $\tau^{\prime}_{QSL}$ are in contrast with the peaks and
valleys of instantaneous power. This indicates that changes in the
$\tau^{\prime}_{QSL}$ pick up the maximal rate of charging or discharging of
the battery, with a maximum in $\tau^{\prime}_{QSL}$ corresponding to the
maximum discharging rate of the battery and a minimum in $\tau^{\prime}_{QSL}$
corresponds to the maximum charging rate, in a given charging and discharging
cycle.

##  IV Quantum speed limit and Ergotropy in non-Markovian evolution

Now we discuss the impact of the non-Markovian evolution on the charging and
discharging process of the quantum battery together with its impact on the QSL
time, particularly those obtained using geometric measures. In Fig. 4, the
evolution of the system under Markovian evolution is depicted.

Figure 4: Variation of QSL time using Fisher information metric
($\tau_{QSL}$), ergotropy ($\mathcal{W}$), instantaneous ($\mathcal{P}(t)$)
and average ($\mathcal{P}_{av}$) powers, with time. The evolution of the state
is through the Markovian AD channel. The parameters are taken to be
$\omega_{0}=1$, $T=1$, $\lambda=0.5$, $\gamma_{0}=0.1$.

Here we observe that the system (qubit) keeps on discharging, but due to the
Markovian evolution, there is no revival in the ergotropy or charging of the
battery. This brings out the fact that in the present qubit-bath setup, the
non-Markovian nature of the evolution plays a crucial role in highlighting the
role of the system as a battery. Further, the QSL time obtained using the
Fisher information metric ($\tau_{QSL}$) is also monotonically increasing;
that is, the speed of evolution of the system keeps slowing down with time.

Figure 5: Variation of QSL time using Fisher information metric
($\tau_{QSL}$), ergotropy ($\mathcal{W}$), and instantaneous
($\mathcal{P}(t)$) and average ($\mathcal{P}_{av}$) powers, with time. The
evolution of the state is through the NMAD channel. The parameters are taken
to be $\omega_{0}=1$, $T=1$, $\lambda=0.5$, $\gamma_{0}=10$.

Figures 5 and 6 show the variation of the QSL time obtained using geometric
measures (using Fisher and Wigner-Yanase information metrics), ergotropy,
instantaneous and average powers, with time.

Figure 6: Variation of QSL time using Wigner-Yanase information metric
($\tilde{\tau}_{QSL}$), ergotropy ($\mathcal{W}$), and instantaneous
($\mathcal{P}(t)$) and average ($\mathcal{P}_{av}$) powers, with time. The
evolution of the state is through the NMAD channel. The parameters are taken
to be $\omega_{0}=1$, $T=1$, $\lambda=0.5$, $\gamma_{0}=10$.

Qualitatively the QSL time obtained using Fisher ($\tau_{QSL}$) and WY
($\tilde{\tau}_{QSL}$) information metrics show a similar pattern. In both
cases, we observe that either of the peaks or valleys of the $\tau_{QSL}$ and
$\tilde{\tau}_{QSL}$ occurs exactly at the points when a cycle of complete
discharging and recharging to a maximum value complete. This coincides with
the points where the average power is calculated. The revivals in the
ergotropy (or recharging of the battery) are completely due to the non-
Markovian nature of the system in both figures. This also brings out a
difference between the QSL time obtained using inherent dynamics of the
system, particularly QSL time for coherence ($\tau_{CSL}$ in Fig. 2) and using
relative purity measure ($\tau^{\prime}_{QSL}$ in Fig. 3), and the QSL time
obtained using geometric measures in Figs. 5 and 6. The peaks and valleys of
the $\tau_{CSL}$ and $\tau^{\prime}_{QSL}$ specify the discharging and
charging processes, being exact for $\tau_{CSL}$. However, the peaks and
valleys of $\tau_{QSL}$ and $\tilde{\tau}_{QSL}$ specify the completion of a
discharging-charging cycle. This thus brings out a difference between the
geometric and physical properties-based measures of QSL time from the
perspective of discharging-charging processes. It can also be seen that the
speed of evolution first decreases and then increases, oscillating around a
fixed point later. This again benchmarks the non-Markovian nature of the
system.

##  V Conclusions

Here we have studied the impact of non-Markovian evolution on the quantities
characterizing quantum thermodynamics, particularly ergotropy and its
components, and instantaneous and average powers. We explored the connection
of these quantities with the quantum speed limit (QSL) time of evolution. The
characterization of the QSL time was done using both geometric and physical
properties-based measures. Fisher and Wigner-Yanase information metrics were
used for the geometric based measures ($\tau_{QSL}$ and $\tilde{\tau}_{QSL}$,
respectively), and relative purity and relative entropy of coherence were used
for the physical properties-based measure ($\tau_{CSL}$ and
$\tau^{\prime}_{QSL}$, respectively) of QSL time. We considered the evolution
of a single qubit system interacting with the bosonic bath through the non-
Markovian amplitude damping (NMAD) master equation. Having an initial finite
ergotropy, we proposed that the system could be envisaged as a quantum battery
discharging into the bosonic environment.

The coherent and incoherent components of erogtropy were discussed, and direct
connections between the coherent ergotropy and QSL time using relative purity
and relative entropy of coherence measures were explored. Further, it was
observed that QSL time obtained using physical properties-based measures
identified the discharging and charging process of the quantum battery. This
was different from the geometric based measures that brought out the
connection between the completion of the discharging-charging cycle and the
change in the speed of evolution of the system. The revivals in the ergotropy
of the system (or recharging of the battery) only occurred in the non-
Markovian limit. It was observed that in the Markovian limit, the battery only
discharged, and the speed of evolution was monotonically decreasing (QSL time
was monotonically increasing). Therefore, the non-Markovian evolution is
crucial when modeling the system as a quantum battery.

## Acknowledgements

SB acknowledges support from the Interdisciplinary Cyber-Physical Systems
(ICPS) programme of the Department of Science and Technology (DST), India,
Grant No.: DST/ICPS/QuST/Theme-1/2019/13. SB also acknowledges support from
the Interdisciplinary Research Platform (IDRP) on Quantum Information and
Computation (QIC) at IIT Jodhpur.

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