# Constructive Interference in Linear Precoding Systems: Power Allocation and
User Selection

Dimitrios Christopoulos1, Symeon Chatzinotas1, Ioannis Krikidis2 and Björn
Ottersten1  1SnT - securityandtrust.lu, University of Luxembourg
e-mail: {dimitrios.christopoulos, symeon.chatzinotas<EMAIL_ADDRESS>2Department of Electrical and Computer Engineering, University of Cyprus
e-mail<EMAIL_ADDRESS>
###### Abstract

The exploitation of interference in a constructive manner has recently been
proposed for the downlink of multiuser, multi-antenna transmitters. This novel
linear precoding technique, herein referred to as constructive interference
zero forcing (CIZF) precoding, has exhibited substantial gains over
conventional approaches; the concept is to cancel, on a symbol-by-symbol
basis, only the interfering users that do not add to the intended signal
power. In this paper, the power allocation problem towards maximizing the
performance of a CIZF system with respect to some metric (throughput or
fairness) is investigated. What is more, it is shown that the performance of
the novel precoding scheme can be further boosted by choosing some of the
constructive multiuser interference terms in the precoder design. Finally,
motivated by the significant effect of user selection on conventional, zero
forcing (ZF) precoding, the problem of user selection for the novel precoding
method is tackled. A new iterative, low complexity algorithm for user
selection in CIZF is developed. Simulation results are provided to display the
gains of the algorithm compared to known user selection approaches.

##  I Introduction

The capacity of the multiple input multiple output (MIMO) broadcast channel
(BC) can be reached by non-linear precoding methods, namely dirty paper coding
(DPC)[1]. However, linear precoding methods, like zero forcing (ZF) precoding,
can still attain the channel capacity in a multiuser environment [2, 3, 4],
while proven more realistic in terms of practical implementation. Linear
precoding techniques, especially ZF, have been extensively investigated in [5,
3] and the references therein. In these cases, ZF precoding constitutes a
simple precoder design solution. By inverting the channel, multiuser
interferences are cancelled and the precoding design problem is reduced to a
power allocation problem over new equivalent channels; hence a simple concave
optimization problem[6] needs to be solved. To maximize the throughput (sum-
rate, SR), the well known water-filling solution can be straightforwardly
applied [7]. To maximize the minimum offered rate (i.e. the fairness problem),
the problem is still convex and thus solvable[5]. The key assumption of all
the above considerations however is the assumption of Gaussian signaling.

The concept of constructive interference linear precoding, initially proposed
in [8] for code division multiple access (CDMA) systems and then extended to
apply for MIMO communications in [9], is based on the multiuser interference
cancellation concept of channel inversion. An example of the concept is
described in Fig. 1. The novelty of this precoder design lies in considering
practical constellations and allowing users that add up to the intended user’s
signal power to interfere. This is referred to as constructive interference
(CI) and it can be exploited by acknowledging each users’ channel and
modulated signal. The problem of power allocation in constructive interference
zero forcing (CIZF) precoding techniques has not been studied in existing
literature. Existing works on this topic only assumed CIZF precoding with
equal power allocation for all users[9].

Figure 1: The constructive interference (CI) concept over binary phase shift
keying (BPSK) modulations: The $k$-th user’s transmit symbol is $s_{k}=+1$.
The $i$-th user’s symbol, $s_{i}$, multiplied by the cross-correlation between
the $k$-th and the $j$-th user’s channels, $\rho_{kj}$ (see Sec. II-B), is a
vector that when added to $s_{k}$ will move the resulting vector further away
from the decision threshold (0 for BPSK). Subsequently, not cancelling this
user will benefit the $k$-th user. On the other hand, the $j$-th user is still
interfering thus needs to be cancelled by the precoding design.

Another very important aspect of linear precoding is the user selection
problem investigated in [3, 6]. ZF performance is increased when user channels
are orthogonal to each other. Under the assumption of large random user sets,
the probability of orthogonal users increases and with that the complexity of
the user selection problem. Nevertheless, simple suboptimal algorithms in the
existing literature provide substantial gains with affordable complexity.
Based on existing methods, Yoo et al [2] proposed a low complexity, iterative
user selection algorithm that allows ZF to achieve the performance of non-
linear precoding[10] as the number of available for selection users grows to
infinity.

The contribution of the present paper is twofold. Firstly, the effect of power
allocation on CIZF precoding transmitters is investigated; a design parameter
that has not been examined in existing literature. Secondly, motivated by the
fact that user selection can optimize the ZF performance, the problem of user
selection in CIZF systems is defined and solved by a low complexity algorithm.
This algorithm achieves substantial gains and approaches the performance of
the optimal user selection, as derived by full space search.

The rest of the present paper is structured as follows. The considered system
model is described with detail in Section II, where the concepts of
conventional ZF and novel CIZF are also described. Section III explores the
effects of power allocation in CI based linear precoding methods with the
support of simulation results. In Section IV, the user selection problem is
defined and solved via a novel heuristic algorithm. Conclusions are drawn in
Section V.

Notation: Throughout the paper, $\left(\cdot\right)^{\dagger}$,
$\mathfrak{Re}(\cdot)$ and $||\cdot||,$ denote the conjugate transpose, the
real part of complex elements and the Euclidean norm operations, respectively,
while $[\cdot]_{ij}$ denotes the $i,j$-th element of a matrix. The element-
wise matrix product is denoted by $\circ$. Bold face lower case characters
denote column vectors and upper case denote matrices while the operator
diag($\mathbf{x}$) produces a diagonal square matrix composed of the elements
of $\mathbf{x}$. An identity matrix of size $n$ is denoted by
$\mathbf{I}_{n}$. Upper case calligraphic characters denote sets. The
operation $\mathcal{A-B}$ is the relative complement of $\mathcal{B}$ in
$\mathcal{A}$, while $|\mathcal{A}|$, denotes the cardinality of a set.

##  II System Model

A multi-user (MU) multiple input single output (MISO) network consisting of
one transmitter with $N_{t}$ antennas and $K\geq N_{t}$ single antenna
receivers, is considered. At each time, the transmitter serves exactly
$K=N_{t}$ users which are selected either randomly or based on a selection
scheme as described in Sec. IV. The received signal at the $k$-th user can be
expressed as

$y_{k}=\mathbf{h}^{{\dagger}}_{k}\mathbf{x}+n_{k},$ (1)

where $\mathbf{h}_{k}$ is an $N_{t}\times 1$ vector composed of the channel
coefficients between the $k$-th user and the $N_{t}$ antennas of the source,
$\mathbf{x}$ is an $N_{t}\times 1$ vector of transmitted symbols and $n_{k}$
is the independent identically distributed (i.i.d) zero mean Additive White
Gaussian Noise (AWGN) measured at the $k$-th user’s receive antenna. The noise
is assumed normalized, thus $\mathcal{E}\left\\{|n_{k}|^{2}\right\\}=1$. In
matrix form, this MU MISO BC reads as

$\displaystyle\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n},$ (2)

where $\mathbf{y}=[y_{1},y_{2},\ldots,y_{N_{t}}]^{\dagger}$,
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{N_{t}}]^{\dagger}$, $\mathbf{H}$ is the
$N_{t}\times N_{t}$ square matrix that contains the user complex vector
channels, i.e.
$\mathbf{H}=\left[\mathbf{h}_{1},\mathbf{h}_{2}\dots,\mathbf{h}_{N_{t}}\right]^{\dagger}$
and $\mathbf{n}=[n_{1},n_{2},\ldots,n_{N_{t}}]^{{\dagger}}$. The transmitter
linearly precodes the information symbols:

$\displaystyle\mathbf{x}=\mathbf{W}\mathbf{P}^{1/2}\mathbf{s},$ (3)

where the $N_{t}\times N_{t}$ matrix $\mathbf{W}$ is the precoding matrix,
$s_{k}$ denotes the symbol for the $k$-th destination and $\mathbf{s}=[s_{1},\
s_{2},\ldots,s_{K}]^{\dagger}$ with
$\mathbb{E}[\mathbf{s}\mathbf{s}^{\dagger}]=\mathbf{I}$ and
$\mathbf{P}^{1/2}=\operatorname{diag}([\sqrt{p_{1}},\sqrt{p_{2}}\ldots\sqrt{p_{k}}])$
is a diagonal $K\times K$ matrix composed of the transmit powers allocated to
the $k$ users111The notion of transmit power allocated to a user is explained
in [3].. For shortness and since the results can be straightforwardly
generalized for higher order constellations, in this study only real valued
signals will be assumed (binary phase shift keying-BPSK modulation), hence

$\displaystyle{s_{i}}=\pm 1,\ i=1\ldots N_{t}.$ (4)

###  II-A Zero Forcing beamforming

Transmit beamforming is a multiuser precoding technique that separates user
data streams in different parallel beamforming directions[3]. A linear
precoding technique with reasonable computational complexity that still
achieves full spatial multiplexing and multiuser diversity gains, is ZF
precoding [11, 12, 3]. The ability of ZF to fully cancel out multiuser
interference makes it useful in the high Signal to Noise Ratio
($\mathrm{SNR}$) regime. However, it performs far from the optimal in the
noise limited regime. In addition, it can only simultaneously serve as many
single antenna users as the number of transmit antennas. A common solution for
the ZF precoding matrix is the pseudo-inverse of the $K\times N_{t}$ channel
matrix. Under a total power constraint, the pseudo-inverse is the optimal
solution (rather than any generalized inverse) in terms of maximum SR and
maximum fairness [5]. The precoding matrix can be expressed as

$\displaystyle\mathbf{W}=\mathbf{H}^{\dagger}\mathbf{R}^{-1}\mathbf{T},$ (5)

where we define the matrix $\mathbf{R}$ as

$\displaystyle\mathbf{R}=\mathbf{HH}^{\dagger}.$ (6)

The matrix $\mathbf{T}$ has been introduced by [9] to model the CI scheme as
explained in the next subsection. The general model described by (5), for

$\displaystyle\mathbf{T}=\mathbf{I}\circ\mathbf{R,}$ (7)

where the non zero elements of $\mathbf{T}$ are
$\tau_{kk}=\sum\mathbf{h}_{k}^{\dagger}\mathbf{h}_{k}$, will yield the
conventional ZF design. The complete cancellation of interferences in this
case, will reduce the $k$-th users received signal to

$\displaystyle
y_{k}=\mathbf{h}^{{\dagger}}_{k}\sqrt{p_{k}}\mathbf{w}_{k}s_{k}+n_{k},$ (8)

where $\mathbf{w}_{k}$ is the $k$-th column of the total precoding matrix.
Assuming uniform power allocation across users, as in [9], the total power
constraint over the transmit antennas $P_{tot}$ will yield[5]:

$\displaystyle\mathcal{E}\\{||\mathbf{x}||^{2}\\}=\operatorname{Tr}\\{\mathbf{xx}^{\dagger}\\}\leq
P_{tot}.$ (9)

From (3) and (9), the transmit power allocated to the $k$-th user becomes

$\displaystyle\sqrt{p}_{k}=\sqrt{\frac{P_{tot}}{\operatorname{Tr}\\{\mathbf{T}^{\dagger}\mathbf{R}^{-1}\mathbf{T}\\}}},\
\forall\ k.$ (10)

By examining (10), it is clear that the transmit power allocated to the $k$-th
user is a function of the precoder design. In precoding, channel inversion
i.e. projecting the actual channels on orthogonal dimensions, leads to the
reduction of each users effective channel. Subsequently, since the precoders
are not normalized, the sum of the individual powers allocated to each user
($\sqrt{p_{k}}$) is not equal to the sum of powers transmitted (9). The notion
of individual user consumption can be introduced to better explain this power
loss due to the precoding. Finally, the $k$-th user $\mathrm{SINR}$ for the ZF
precoding will read as

$\mathrm{SINR}_{k}^{\text{ZF}}=|\tau_{kk}|^{2}p_{k}.$ (11)

###  II-B Constructive Interference Zero Forcing beamforming

The CIZF scheme, introduced in [9], allows the so-called constructive
multiuser interference (cross-interference) to be added to the useful signal
at each receiver. In general, given the full channel state information
available at the transmitter and acknowledging the signal constellation, the
CIZF scheme does not suppress the part of the cross-interference that is
constructive and thus increases the power of the useful signal. A simple
example of this concept is explained in Fig. 1.

As discussed with detail in [9], the symbol to symbol multiuser interference
results from the $i,j$-th element of the matrix $\mathbf{R}$ :
$\rho_{ij}=\sum_{n=1}^{N_{t}}h_{in}\cdot({h_{jn}}^{\dagger})$. In the CI
scenario, the received signal of (8) will become

$\displaystyle y_{k}=\mathbf{\tau}_{kk}\sqrt{p_{k}}s_{k}+\sum_{j\neq
k}\text{CI}_{kj}+n_{k}$ (12)

where $\text{CI}_{kj}=\tau_{ki}\sqrt{p_{j}}s_{j}$ denotes the constructive
cross-interference from the $j$-th data flow ($j$-th user) to the $k$-th user.
Subsequently, the $k$-th user’s signal to interference plus noise ratio
($\mathrm{SINR}$) will read as

$\mathrm{SINR}_{k}=\sum^{K}_{j=1}|\tau_{kj}|^{2}p_{j}.$ (13)

Let us define as
$\mathbf{G}=\text{diag}(\mathbf{s})\cdot\mathfrak{Re}(\mathbf{R})\cdot\text{diag}(\mathbf{s}),$
which yields

$\displaystyle\mathbf{G}=\begin{pmatrix}s_{1}\mathfrak{Re}(\rho_{11})s_{1}&\dots&&\\\
\vdots&\ddots&&\\\ &&s_{k}\mathfrak{Re}(\rho_{kj})s_{j}&\\\ &&&\\\
\end{pmatrix}.$ (14)

In order to indicate the cross-interference as CI, the signal constellation
needs to be accounted for. Subsequently, the terms that position the received
signal into the decision region of the transmitted symbols are beneficial and
thus not cancelled by the precoding design. For the simple case of BPSK
modulation, the cross-interference generated by the $j$-th data flow to the
$k$-th destination, is considered to be constructive when

$\displaystyle s_{k}\mathfrak{Re}(\rho_{kj})s_{j}>0,$ (15)

which can be expressed as $\mathbf{G}_{kj}>0.$ Thus the CIZF precoder is
deduced from

$\displaystyle\tau_{kk}=\rho_{kk}$ (16)
$\displaystyle\tau_{ki}=\begin{cases}\rho_{ki},&\text{If}\
[\mathbf{G}]_{kj}>0\\\ 0,&\text{elsewhere.}\end{cases}$

Therefore, the precoding matrix is computed on a symbol-by-symbol basis.

##  III Power Allocation in Linear Precoding

The impact of power allocation (PA) on the CIZF has not been addressed in
existing literature on this topic [8, 9]. Therein, the problem was simplified
by a uniform power allocation assumption, as defined in (10). In general, PA
is performed to the end of maximizing some performance metric. The performance
metrics commonly addressed in literature involve either the total throughput
performance (i.e. max SR criterion) or the $\mathrm{SINR}$ level of the worst
user (i.e. max fairness criterion). Another important parameter in linear
precoding is the type of constraints that will be assumed. Usually, a total
sum power constraint simplifies the analysis and provides better results since
the available power is freely allocated across antennas. Herein, two objective
functions of the achievable user rates that ensure maximum fairness
(availability) and maximum SR (throughput), are considered. More specifically,
the optimization problem reads as

$\displaystyle\max_{\mathbf{P}\geq\mathbf{0}}f(\mathbf{P})$ (17) s.t.
$\displaystyle\sum_{i=1}^{N_{t}}\sum_{j=1}^{K}|\omega_{ij}|^{2}p_{j}\leq
P_{tot}$

where $\omega_{ij}$ is the $i,j$-th element of $\mathbf{W}$ and the objective
function $f$ is given by [5]:

$\displaystyle
f(\mathbf{P})=\begin{cases}\sum_{k}\log_{2}(1+\mathrm{SINR}_{k}),&\text{\
Throughput}\\\ \min_{k}\mathrm{SINR}_{k},&\text{\ Fairness}\end{cases}$ (18)

where $\mathrm{SINR}_{k}$ is given by (13) and $1\leq k\leq N_{t}$. Based on
(18), the objective function is concave in $\mathbf{P}\geq 0$, for both
scenarios and therefore the optimization problem is a simple concave
maximization with one linear constraint. It is worth noting that for the case
of the maximum throughput, the optimization problem can be solved using the
water-filling solution. The problem of allocating the power to the end of
maximising some system performance metric is discussed in the following
Section (III-A).

###  III-A Power Allocation

An appropriate PA scheme distributes the total available power to the data
flows in a way that maximizes an objective function of the achievable rates.
In the case of conventional ZF precoding, the max throughput PA problem, under
a total power constraint, reads as in (17) with objective function

$\displaystyle
f(\mathbf{P})=\sum_{k}\log_{2}\left(1+\mathrm{SINR}_{k}^{\text{ZF}}\right),$
(19)

where the $\mathrm{SINR}_{k}^{\text{ZF}}$ is given by (11).

In order to be able to easily show the impact of PA on CIZF and maintain
concavity for the formulated optimization problems, we assume that the
equivalent CIZF channel (with the modulation-based CI) refers to Gaussian
inputs; this assumption allows to approximate the channel capacity of the
system with the simple $\log$-based Shannon expressions. It should be
clarified here, that the optimal power allocation problem for linear precoders
under the constraint of finite input alphabets is a highly complex problem.
The most recent attempt to solve it can be found in [13] where a heuristic
optimization algorithm is developed. However, in the present paper, a
preliminary study to exhibit the impact of PA on CIZF is performed, hence the
strictly optimal solution is beyond the scope of the present work.

####  III-A1 Simulation Results

The effect of power allocation in the CIZF precoding design is plotted in Fig.
2. The power allocation problem (18) has been solved using the CVX tool in
MATLAB [14]. Simulations where carried for $100$ channel instances, and for
$K=N_{t}=4$. In Fig. 2, the gain from CIZF with uniform power allocation, as
proposed in [9], over the conventional ZF is evident by comparing the dashed
lines. The novel result, depicted in Fig. 2, is that power allocation further
boosts the CIZF gain (continuous lines). More specifically, for the
conventional ZF scheme power allocation introduces approximately $1$ dB of
gain over the uniform allocation. However, when PA is applied in CIZF, then
more than $2$ dB gain can be gleaned. It is therefore concluded that power
allocation over the CIZF precoding scheme is an important aspect that
introduces significant gains. Finally, in the same figure, the realistic
region where the results apply is defined by a dashed line. This restriction
comes from the acknowledgement of BPSK modulation as a practical constellation
choice. The alleviation of this restriction via adaptive modulation methods is
part of the future extensions of this work.

Figure 2: Per user spectral efficiency of conventional ZF precoding under
uniform and optimized PA, compared to the efficiency of the novel CIZF
precoding under equivalent PA assumptions. PA optimization for max system
throughput. The modulation constrained threshold for BPSK is also plotted.

###  III-B Power Constrained Transmission

In the present section, the existence of redundant CI interference terms is
discussed. As can be seen from (10) and (13), the consideration of non-zero,
off diagonal elements in the precoding matrices, i.e. CI terms, has a double
impact on each user’s $\mathrm{SINR}$ and thus the sum system capacity; from
one hand, it increases the expression $\sum_{i}|\tau_{ki}|^{2}p_{k}$ by adding
more positive terms in the summation, but on the other hand, it changes the
power allocated to each user in (9), (10). Therefore, a CI term in the CIZF is
not always beneficial for the system performance. This partially constructive
interference zero forcing (P-CIZF) scheme examines the tradeoff between the
positive and the negative impact of an CI term by searching all combinations
and selecting the most beneficial set of CI terms. The P-CIZF scheme can be
formulated as

$\displaystyle\max_{\\{\mathbf{p},\mathbf{T}^{(\mathcal{S})}\\}}f(\mathbf{p},\mathbf{T}^{(s)})$
(20) s.t.
$\displaystyle\mathbf{T}^{(\mathcal{S})}\subseteq\mathbf{T}^{(\mathcal{S}_{tot})},$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{K}|\omega_{ij}|^{2}p_{j}\leq P_{tot}.$
(21)

where

$\displaystyle\mathcal{S}\subseteq\mathcal{S}_{tot}=2^{m},$

with $m=\left|[\mathbf{G}]_{ki}>0\right|$, the number of CI terms. The above
definition means that the Partially-CIZF scheme searches all the possible
combinations ($2^{m}$) of the CI terms and holds the one that maximizes the
objective function considered. The optimization problem (20), will be solved
under uniform (10) and optimal (21) power allocation considerations.
Intuitively, the second consideration provides more flexibility in the design
but finding the strictly beneficial terms while at the same time optimizing
the PA is a highly complex procedure; for every possible combination of the CI
terms, a new convex optimization PA problem is solved.

As the purpose of this work is to demonstrate this interesting trade-off, more
practical implementation of the P-CIZF scheme are beyond the scope of this
paper.

####  III-B1 Simulation Results

In Fig. 3 the improvements of P-CIZF over the CIZF scheme are depicted. The
performance is evaluated under uniform and optimized power allocation.
Starting with the CIZF scheme under uniform power allocation, in Fig. 3,
finding the strictly beneficial CI terms provides some gains. In each point of
the figure, the percentage of the CI terms kept in the P-CIZF scheme over the
total CI terms of the CIZF scheme is also depicted. Focusing on the dotted
curves, in the uniform power allocation case, it is apparent that maintaining
approximately $88\%$ of the total number of CI terms, provides small gains.
These results exhibit the optimality of this approach, while analytical proof
of the optimality is part of future work. The continuous curves in Fig. 3
correspond to an optimal (with respect to maximizing the total SR) power
allocation assumption in P-CIZF precoding and again some gains are gleaned;
thus the strict optimality of this approach is exhibited. Since the degrees of
freedom in the precoder design are increased, less CI terms are maintained
(approximately $65\%$).

Intuitively, the above results can be explained by the fact that allowing only
strictly beneficial terms in the precoder reduces power consumption, thus
allowing for more power to be allocated to the users without exceeding the
total power constraint imposed on the transmit antennas. It should be noted
here, however, that the effects of this approach on the minimum supported rate
(fairness) are not examined in the present work. Finally, higher gains of this
approach could be gleaned over larger user sets, but by exponentially
increasing the iterations of the searching algorithm. The investigation of
such scenarios is part of future work.

Figure 3: Evaluation of the P-CIZF compared to fully CIZF under equal and
optimal power allocation. The percentages of the constructive terms maintained
in the P-CIZF scheme, for each $\mathrm{SNR}$ point are also presented.

##  IV User selection

In this section, the user selection problem is formulated and a novel
selection technique based on the exploitation of CI is proposed. Considering
the decoupled nature between the user selection and the PA problem, as proven
in [3], the performance of user selection is not affected by PA. However, in
accordance with the previous results and more importantly to maximise the
overall performance of the assumed system, PA is optimized independently after
the user selection process.

###  IV-A User Selection Methods

####  IV-A1 Orthogonal user selection

ZF beamforming has the potential of approaching the optimal channel capacity,
otherwise only achievable by DPC, when all the users are perfectly orthogonal
to each other, as proven by [3]. The same authors provided a heuristic low
complexity user selection algorithm, namely the semi-orthogonal user selection
($\mathrm{SUS}$) algorithm, which was proven to select the optimal users, as
the number of available users approaches infinity. However, it has never been
applied in the CIZF framework. Due to lack of space, the reader is referred
to[3, 2] for more details on this algorithm. It should be mentioned here, that
the orthogonal user selection does not take into account the CI during the
selection procedure. However, to maintain fairness in the study, after the
selection, any CI terms that exist are maintained.

####  IV-A2  Optimal Constructive Interference User Selection

The investigated precoding scheme strongly depends on the transmitted signal
(user constellation) and therefore the above conventional user selection
schemes become unsuitable. A CIZF-based user selection metric should also
consider the transmitted symbols, since the precoding matrix is defined via
the CI and affects the final performance. In order to attain an upper bound
for any CIZF-based user selection policy, all the possible combinations of
users $\mathcal{Q}$ are examined and out of them, the combination
$\mathcal{U}$ that maximizes the system throughput is chosen:

$\displaystyle\mathcal{U}=\arg_{\mathcal{U}}\max_{\mathcal{U}\in\mathcal{Q}}\sum_{m}\log_{2}\left(1+\mathrm{SINR}_{m}\right),$
(22)

where $\mathrm{SINR}_{m}$ is given by (13) under a uniform power allocation
assumption, i.e. (10). It should be stressed that this method relies on
exhaustive search over all possible combinations of users. As a result, a
searching algorithm requires $\binom{K}{N_{t}}=K!/(K-N_{t})!$ iterations in
order to decide about the optimal combination at each transmission.
Considering that user selection methods perform better as the number of users
increases[3], i.e. as $K>>N_{t}$, then the optimal solution becomes difficult
to compute. In the scope of this work, a simpler heuristic algorithm is
presented hereafter.

####  IV-A3 Semi parallel user selection

Inspired by the concept of user orthogonality, the purpose of this selection
method is dual: users with CI need to be selected and furthermore these users
need to be aligned (rather than orthogonal) so that the aggregate beneficial
receive power is increased. Following this concept the semi-parallel user
selection ($\mathrm{SPUS}$) algorithm, provided in pseudo-code in Alg. 1, has
been developed. An analytic description of the algorithm follows.

Initially, the algorithm accepts as input the CI matrix $\mathbf{G}$. The
first step is to choose the user with the larger diagonal element. For this
user, the cross-correlation elements with all other users are stored in the
buffer vector $\mathbf{c}_{(i)}$ where $i$ is the iteration counter. Also the
sets $\mathcal{S},\mathcal{T}$ that include the available and the selected
users, are initialized and updated in every iteration. Cross-correlation is
the inner product of the vector channels of two users and represents the level
of orthogonality between the users. In this scenario, the purpose is to have
as little orthogonality as possible so as to increase the received CI. In each
of the $M$ iterations, (Step 2) the user with the strongest element in
$\mathbf{c}_{(i)}$ is chosen. Then this user’s corresponding cross-
correlations with all the users is added in the buffer matrix. By adding the
cross-correlation of each selected user in the buffer matrix, a metric for the
subspace of the previously selected users is created, since the aim of the
algorithm is to find the most parallel users to the subspace spanned by the
ones already selected. This is achieved by choosing the strongest element of
the $\mathbf{c}$ vector in each iteration. Since in the previous steps no
guarantee exists that a selected user has only CI towards the selected ones,
in Step 3, the residual non-CI terms are removed from the precoding matrix.
The developed heuristic, iterative algorithm runs for exactly $K$ iterations.

#### Simulation Results

A comparison in terms of maximum SR performance of the algorithms described in
the previous section is presented in Fig. 4, where the performance of these
algorithms was studied with optimized PA to maximize the total throughput. In
this figure, the gain of the optimal user selection algorithm compared to
existing approaches is clear. The best algorithm for ZF precoding, i.e.
$\mathrm{SUS}$ [3], performs better than assuming no selection; however,
approximately $6$ dB loss is expected over the optimal CIZF selection. This
observation emanates the need for better user selection algorithms, when
exploiting the benefits of CI. Accounting also for the complexity of an
exhaustive search selection algorithm, as discussed in the previous section, a
less complex heuristic algorithm is further necessitated. In this direction,
$\mathrm{SPUS}$ has been developed. In Fig. 4, the close to optimal
performance of the developed algorithm is clear; the proposed technique
performs less than 1 dB away from the optimal selection.

Furthermore, the performance of the developed algorithm under fairness
maximization PA is examined in Fig. 5. Results indicate, that when the
$\mathrm{SPUS}$ algorithm is combined with max fairness PA optimization, the
performance degradation with respect to the optimal selection policy, is
relatively small. Therefore, the proposed algorithm, developed for maximising
the total throughput of the system, does not severely compromise the fairness
of the system. By comparing Fig. 4 and 5, it is also noted that the the
minimum user rates are not far from the average rate. This result indicates
that the variance between the user rates is kept in reasonable levels when
user selection is combined with PA to optimize the total SR. It is therefore
concluded that fairness is not severely compromised in user selection
scenarios, even when PA optimization is performed to maximize to total system
SR.

In Fig. 6 the performance of the discussed algorithms is studied with respect
the size of the available user set. The beneficial effect of the increasing
number of users is clear for all algorithms. What is more, for relatively
small user pools the performance of the algorithms is beginning to saturate
thus indicating that the main gains are gleaned for finite numbers of users,
that are in line with the dimensions of practical operating multiuser systems.

Figure 4: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a total
pool of $K=12$ users. PA has been optimized to maximise the total system
throughput.
Figure 5: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a total
pool of $K=12$ users. PA has been optimized to maximise minimum user rate.
Figure 6: Performance of user selection algorithms with respect to the total
available transmit power. Results for $N_{t}=4$ users selected out of a
variable pool of users, for a fixed total transmit power of $15$ dB.

##  V Conclusions and future work

The concept of constructive interference in linear precoding systems has been
examined under the framework of power optimization for the maximization of the
system performance. Results indicate that an excess of 2dB gain can be gleaned
by optimizing the power allocation with the aim of increasing the total system
throughput in constructive interference precoding systems. Moreover, as it has
been shown, the individual user power consumption of these schemes can be
further reduced, thus leading to some gains. Finally, the user selection
problem has been tackled for the novel type of precoding and a heuristic, low
complexity, iterative algorithm with close to optimal performance has been
proposed.

Future extensions of this work include the investigation of constructive
interference amongst users with different constellations in an adaptive
modulation environment where the limitations induced by these practical
constellations are alleviated.

## Acknowledgment

This work was partially supported by the National Research Fund, Luxembourg
under the project “$CO^{2}SAT:$ Cooperative & Cognitive Architectures for
Satellite Networks’.

Semi-Parallel User Selection (SPUS) algorithm

Output: $\mathbf{G}_{out}$

Input:
$\mathbf{G}=\text{diag}(\mathbf{s})\cdot\mathfrak{Re}(R)\cdot\text{diag}(\mathbf{s}),$

_Step 1: Initialization_
$\pi_{(0)}=\arg\max||\mathbf{h}_{k}||=\arg\max[\mathbf{G}]_{kk}$,

$\forall k=1,...M:\mathbf{c}_{(0)}=\mathbf{G}(\pi_{(0)},k)$,
$\mathcal{S}_{(0)}=\pi_{(0)}$

$\mathcal{T}_{(0)}=\\{1,\dots K\\}-\\{\pi_{(0)}\\}\ $ set of unprocessed
users.

for  _$i=1\to M$ _ do

_Step 2: Selection_ $\pi_{(i)}=\arg\max\mathbf{c}_{(i-1)},\text{ Provided that
}\pi_{(i)}\in\ \mathcal{T}_{(i-1)}$;

$\forall k=1,...M:\
\mathbf{c}_{(i)}=\mathbf{G}(\pi_{(i-1)},k)+\mathbf{G}(\pi_{(i)},k);$

$\mathcal{T}_{(i)}=\mathcal{T}_{(i-1)}-\\{\pi_{(i)}\\};$

$\mathcal{S}_{(i)}=\mathcal{S}_{(i-1)}+\\{\pi_{(i)}\\};$

end for

_Step 3: Output_

$\mathbf{G}_{out}=\mathbf{G}(\mathcal{S}_{(M)})$;

for  _$m=1\to M$_ do

for  _$l=1\to M$_ do

if  _$\mathbf{G}(m,k) <0$ _ then

$\mathbf{G}_{out}(m,k)=0$

end if

end for

end for

Algorithm 1 Semi-Parallel User Selection Algorithm (SPUS)

## References

  * [1] M. Costa, “Writing on dirty paper,” _IEEE Trans. Inf. Theory_ , vol. 29, no. 3, pp. 439–441, 1983.
  * [2] T. Yoo and A. Goldsmith, “Optimality of zero-forcing beamforming with multiuser diversity,” in _IEEE Int. Conf. on Commun. (ICC)_ , vol. 1, May 2005, pp. 542–546.
  * [3] ——, “On the optimality of multi-antenna broadcast scheduling using zero-forcing beamforming,” _IEEE J. Select. Areas Commun._ , vol. 24, Mar. 2006.
  * [4] B. L. Ng, J. Evans, S. Hanly, and D. Aktas, “Distributed downlink beamforming with cooperative base stations,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 12, pp. 5491 –5499, dec. 2008.
  * [5] A. Wiesel, Y. C. Eldar, and S. Shamai, “Zero forcing precoding and generalized inverses,” _IEEE Trans. Signal Process._ , vol. 56, no. 9, pp. 4409–4418, Sept. 2008.
  * [6] G. Dimic and N. Sidiropoulos, “On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm,” _IEEE Trans. Signal Process._ , vol. 53, no. 10, pp. 3857 – 3868, Oct. 2005.
  * [7] T. Yoo and A. Goldsmith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 5, pp. 2203–2214, May 2006.
  * [8] C. Masouros and E. Alsusa, “A novel transmitter-based selective-precoding technique for DS/CDMA systems,” _IEEE Signal Process. Lett._ , vol. 14, no. 9, pp. 637 –640, Sep. 2007.
  * [9] ——, “Dynamic linear precoding for the exploitation of known interference in MIMO broadcast systems,” _IEEE Trans. Wireless Commun._ , vol. 8, no. 3, pp. 1396 –1404, Mar. 2009.
  * [10] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 9, pp. 3936–3964, 2006.
  * [11] H. Viswanathan, S. Venkatesan, and H. Huang, “Downlink capacity evaluation of cellular networks with known-interference cancellation,” _IEEE J. Select. Areas Commun._ , vol. 21, no. 5, pp. 802–811, 2003.
  * [12] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 49, no. 7, pp. 1691–1706, July 2003.
  * [13] Y. Wu, M. Wang, C. Xiao, Z. Ding, and X. Gao, “Linear precoding for MIMO broadcast channels with finite-alphabet constraints,” _IEEE Trans. Wireless Commun._ , vol. 11, no. 8, pp. 2906–2920, Aug. 2012.
  * [14] S. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge Univ. Press, 2004.


You have access to:

i) assignment context (provided below and delineated with <assignment context>
) and

ii) feedback criteria (provided below and delineated with <feedback criteria>)

Here is the assignment context:

<assignment context>

We have the following schema declaration:

\\#\\#\\#

Account [ username, firstName, middleName, lastName, email, dateOfBirth,
registrationDate, subscriptionTier] Standard [ username ]

Premium [ username ]

PremiumFriend [ username, friendUsername ]

Playlist [ username, playlistName ]

PlaylistProduct [ username, playlistName, code ]

Watches [ username, code, timestampStopped ]

Product [ code, synopsis, title ]

ProductSubtitleOptions [ code, subtitleLanguage ]

ProductAudioOptions [ code, audioLanguage ]

ProductTags [ code, tag ]

Movie [ code, dateOfRelease, runtime, sequelCode]

TVShow [ code ]

Episode [ code, seasonNumber, episodeNumber, dateOfRelease, runtime ] Acts [
code, id ]

ActsRoles [ code, id, roleName ]

CastMember [ id, nationality, firstName, middleName, lastName ]

Standard.username references Account.username

Premium.username references Account.username

PremiumFriend.username references Premium.username
PremiumFriend.friendUsername references Premium.username

Playlist.username references Premium.username

PlaylistProduct.{username, playlistName} references Playlist.{username,
playlistName} PlaylistProduct.code references Product.code

Watches.username references Account.username Watches.code references
Product.code ProductSubtitleOptions.code references Product.code
ProductAudioOptions.code references Product.code ProductTags.code references
Product.code Movie.code references Product.code Movie.sequelCode references
Movie.code TVShow.code references Product.code

Episode.code references TVShow.code Acts.code references Product.code

Acts.id references CastMember.id ActsRoles.{code, id} references Acts.{code,
id}

\\#\\#\\#

</assignment context>

<feedback criteria>

Constructive feedback

Empathetic feedback

Feedback being detailed and actionable

Feedback encouraging self-reflection and independence

</feedback criteria>\\\

##  Appendix B Feedback Evaluation Prompt

Listing 3: Prompt template used for feedback evaluation.

⬇

Imagine that you are a feedback expert who can evaluate a given feedback based
on how well it is aligned with different criteria.

1. Wait for the user input.

User will provide you with the feedback to be provided to students.

2. Your role is to evaluate the feedback against the set of criteria.

Process the criteria and their descriptions which are specified to you below:

(1). Constructive Feedback:

Feedback that highlights areas for improvement while offering specific
suggestions and positive reinforcement.

Such feedback avoids criticism without guidance, and focuses on fostering a
growth mindset.

(2). Empathetic Feedback:

Such feedback demonstrates understanding and sensitivity in the judgements.

It acknowledges student’s efforts and challenges, and usew language that
encourages motivation rather than discouragement.

(3). Detailed and Actionable Feedback:

Such feedback is specific in the comments and points out particular strengths
and weaknesses.

It offer actionable advice by breaking down larger tasks into manageable
steps, facilitating the student’s ability to implement changes effectively.

(4). Encouraging Self-reflection and Independence:

Such feedback encourages students to think critically about their work and to
become more self-directed in their learning.

It can include questions or suggestions that prompt students to reflect and
think more deeply.

3. Be as strict as possible when conducting evaluations against criteria.

4. For each criterion provide a score from 0 to 10, where 0 means that the criterion is not followed at all and 10 means that the criterion is followed perfectly.

5. IMPORTANT: for each criteria give *a verbatim example* from the user-provided feedback.

These examples would be used to justify your evaluation.

Format output as a JSON object with the following structure:

‘{

constructive\_feedback: … , constructive\_feedback-justification: …,

empathetic\_feedback: … , empathetic\_feedback-justification: … ,

detailed\_and\_actionable\_feedback: … , detailed\_and\_actionable\_feedback-
justification: … ,

encouraging\_self\_reflection\_and\_independence: …,
encouraging\_self\_reflection\_and\_independence-justification: …

}‘\\\

##  Appendix C Assignment Context: Question and Solution

Table 2: Tasks descriptions used im the assignement and a reference solution.  |  |
---|---|---
Task |  Question context |  Sample solution
Task 1 |  Find all accounts where the owner is older than 21 years old. (Note: This need to be correct at the time the query is run) |  SELECT * FROM Account A WHERE YEAR(now()) - YEAR(A.dateOfBirth) >= 21
Task 2 |  Find all playlists that do not contain a product that aired after January 1st, 2012 |  SELECT playlistName FROM Playlist WHERE playlistName NOT IN( SELECT playlistName FROM PlaylistProduct WHERE PlaylistProduct.code IN( SELECT code FROM Episode WHERE dateOfRelease > ’2012-1-1’ UNION SELECT code FROM Movie WHERE dateOfRelease > ’2012-1-1’) )
Task 3 |  Find all movies that are the third move in their franchise (i.e. at least the sequel of a sequel) |  SELECT * FROM Movie A WHERE EXISTS (SELECT * FROM Movie B WHERE A.Code = B.sequelCode AND EXISTS (SELECT * FROM Movie C WHERE B.code = C.sequelCode AND NOT EXISTS( SELECT * FROM Movie D WHERE D.sequelCode = C.code)))
Task 4 |  Find all playlists which contain a movie that the account owner has not watched |  SELECT plp.playlistName FROM PlaylistProduct plp WHERE plp.code NOT IN (SELECT plp1.code FROM PlaylistProduct plp1,Watches w, Movie M WHERE plp1.username = w.username AND plp1.code=M.code)
Task 5 |  Find which account(s) have watched at least all the products that ‘Idris Elba’ has been cast in |  SELECT * FROM Account A WHERE NOT EXISTS(SELECT * FROM CastMember C,Acts A1 WHERE C.firstName = ’Idris’ AND C.lastName = ’Elba’ AND A1.id = C.id AND C.id NOT IN (SELECT C.id FROM Watches W WHERE A1.code = W.code AND W.username = A.username))
Table 2: Tasks descriptions used im the assignement and a reference solution.
(continued)


# Self-induced Transparency in a Semiconductor Quantum Dot medium at ultra-
cold temperatures

Samit Kumar Hazra<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Guwahati, Guwahati 781039, Assam, India  P. K. Pathak
<EMAIL_ADDRESS>Indian Institute of Technology Mandi, Mandi 175001,
Himachal Pradesh, India  Tarak Nath Dey<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam,
India

###### Abstract

We investigate the feasibility of minimum absorption and minimum broadening of
pulse propagation in an inhomogeneously broadened semiconductor quantum dot
medium. The phonon interaction is inevitable in studying any semiconductor
quantum dot system. We have used the polaron transformation technique to deal
with quantum dot phonon interaction in solving system dynamics. We demonstrate
that a short pulse can propagate inside the medium with minimal absorption and
broadening in pulse shape. The stable pulse area becomes slightly higher than
the prediction of the pulse area theorem and is also dependent on the
environment temperature. The change in the final pulse shape is explained very
well by numerically solving the propagation equation supported by the
susceptibility of the medium. Our system also exhibits the pulse breakup
phenomena for higher input pulse areas. Therefore, the considered scheme can
have important applications in quantum communication, quantum information, and
mode-locking with the advantage of scalability and controllability.

††preprint: APS/123-QED

##  I INTRODUCTION

In Self-induced transparency (SIT), an optical pulse propagates resonantly
through the two-level absorbing medium without any loss and distortion. This
pioneering work was carried out by McCall and Hahn [1, 2]. SIT originates from
the generated coherence of a strongly coupled light-medium interaction.
Therefore for observing SIT, the incident pulse should be short compared to
the various relaxation times present in the system, such that the coherence
will not vanish during the pulse propagation. Further, the pulse should also
be strong enough to excite the atom from the ground state. One of the best
theoretical estimations of the input pulse was reported in the “area theorem”
[2]. This theorem dictates that a 2$\pi$ secant pulse can propagate through
the medium without any loss and distortion in the pulse shape. In general, for
an initial pulse area $\theta_{0}$ obeying the condition
$(n+1)\pi>\theta_{0}>n\pi$, evolves the area towards $(n+1)\pi$ or $n\pi$
depending on whether $n$ is odd or even. Therefore input pulse with a larger
area of $2n\pi$, breaks up into $n$ number of $2\pi$ pulses with different
propagation velocities. These effects have been observed experimentally in
atomic rubidium medium by Slusher and Gibbs [3]. In particular, they have
found excellent agreement between numerical simulations and experimental
results. These fundamental properties of the SIT were investigated several
times, both theoretically and experimentally [4, 5, 6].

However, in atomic medium, the preparation and trapping of atomic gas required
a vast and sophisticated setup. Moreover, due to the gaseous nature of the
medium, the different velocity of the atom shows Dopler broadening in output
result. For the last two decades, solid-state semiconductor mediums have
emerged as a potential candidate for optical applications, particularly for
scalable on-chip quantum technology. Earlier, the resonant coherent pulse
propagation in bulk and quantum-well semiconductors behaves differently
compared to a two-level atomic medium. The discrepancy mentioned above occurs
due to the many-body Coulomb interaction of the different momentum states
present in a bulk medium[7, 8, 9]. This problem has been overcome in three-
dimensionally confined excitons in quantum dots (QD’s). The quantum dots can
easily be engineered to get the desired transition frequency to avoid the
problem of laser availability. The scalability and fabrication technology make
the semiconductor QDs suitable for modern quantum optics experiments. There
have been some interesting theoretical proposals about the possibility of
observing SIT in self-organized InGaAs QDs[10]. Excitonic transition in InGaAs
QDs have large transition dipole moments and long dephasing time in the range
of nanoseconds at cryogenic temperatures [11] and are, therefore a promising
candidate for SIT.

Though the QD medium is a potential candidate for observing SIT, it has a few
drawbacks also. All the QDs inside the medium are not identical, so an
inhomogeneous level broadening is always present in the system. In
semiconductors, longitudinal acoustic phonon interaction is vital because of
the environment temperature. Interactions between phonon and exciton lead to
dephasing in coupled dynamics of exciton-photon interaction[12, 13]. Several
theoretical models and experiments have recently explained SIT in the
semiconductor QD medium [14, 15, 16]. Few of them consider the effect of the
phonon environment on the system dynamics in the context of group velocity
dispersion[17]. Another recent experimental work showed the SIT mode-locking
and area theorem for semiconductor QD medium, and rubidium atom [18, 19].

In this paper, we discuss the possibility of SIT in a semiconductor QD medium
incorporating the effect of phonon bath in our model. We utilize the recently
developed polaron transformed master equation keeping all orders of exciton-
phonon interaction [20, 21, 22]. Our model’s pulse propagation dynamics depend
on system and bath parameters. Hence, the propagation dynamics become more
transparent by knowing both the system and the bath’s contribution.The
motivation behind this work is to find long-distance optical communication
without loss of generality in an array of QD. Due to strong confinement of
electron hole pairs, QDs have discrete energy levels thus QD arrays mimic
atomic medium with the added advantage of scalability and controllability with
advanced semiconductor technology. It is also possible to create QD fibers
which can be used for quantum communication channels [23, 24]. Motivated by
this work, we theoretically investigate the self-induced transparency effect
in a semiconductor QD medium.

Our paper is organized as follows. Sec. I contains a brief introduction of the
SIT in a QD medium and its application. In Sec. II, we present our considered
model system along with the theoretical formalism of the polaron master
equation. In Sec. III we discuss the result after numerically solving the
relevant system equations. Finally, we draw a conclusion in Sec. IV.

##  II MODEL SYSTEM

The phonon contribution to QD dynamics at low temperature is mandatory. We
assume the propagation of an optical pulse along the $z$-direction.
Accordingly, we define the electric field of the incident optical pulse as

$\vec{E}(z,t)=\hat{e}\mathcal{E}(z,t)e^{i(kz-\omega_{L}t)}+c.c,$ (1)

where $\mathcal{E}(z,t)$ is the slowly varying envelope of the field. The bulk
QD medium comprises multiple alternating InGaAs/GaAs QD deposition layers.
Every QD inside the medium strongly interacts with the electric field due to
the significant dipole moment. Since all the QD inside the medium is not
identical, the exciton energy of the different QD will vary depending on the
dot size. The $l^{th}$ type QD can be modeled as a two-level system with
exciton state $|1\rangle_{l}$, and ground state $|2\rangle_{l}$ with energy
gap $\hbar\omega_{l}$ by taking the proper choice of biexciton binding energy
and polarisation as shown in the Fig.1. The raising and lowering operator for
the $l^{th}$ type QD can be written as
$\sigma^{+}_{l}=|1\left(\omega_{l}\right)\rangle_{l}\langle
2\left(\omega_{l}\right)|_{l}$ and
$\sigma^{-}_{l}=|2\left(\omega_{l}\right)\rangle_{l}\langle
1\left(\omega_{l}\right)|_{l}$.

In case of semiconductor QD’s, the optical properties get modified due to the
lattice mode of vibration i.e., the acoustic phonon. Hence, QD exciton
transition coupled to an acoustic phonon bath model mimics the desired
interaction. The phonon bath consists of a large number of closely spaced
harmonic oscillator modes. Therefore, we introduce the annihilation and
creation operators associated with $k^{th}$ phonon mode having frequency
$\omega_{k}$ as $b_{k}$ and $b_{k}^{\dagger}$. The mode frequency can be
expressed as $\omega_{k}=c_{s}k$ where $k$ and $c_{s}$ are the wave vector and
velocity of sound. The Hamiltonian for the described model system after making
dipole and rotating wave approximation is given by

Figure 1: A Schematic diagram of the QD level system with ground state
$|2\rangle$ and exciton state $|1\rangle$ driven by the optical pulse with
effective coupling $\langle B\rangle\Omega$(blue line). The spontaneous decay
from the exciton state to the ground state is shown using a curly red line.
The parallel violet lines represent the phonon modes interacting with the
exciton state. The red and blue dashed lines represent the phonon-induced
decay and pumping rate respectively.

$\begin{split}H&=\sum\limits_{l}\Bigr{[}-\hbar\delta_{l}\sigma^{+}_{l}\sigma^{-}_{l}+\frac{1}{2}\hbar\Bigl{(}\Omega(z,t)\sigma^{+}_{l}+\Omega^{*}(z,t)\sigma^{-}_{l}\Bigr{)}\\\
&+\hbar\sigma^{+}_{l}\sigma^{-}_{l}\sum\limits_{k}\lambda_{k}\left(b_{k}+b_{k}^{{\dagger}}\right)\Bigr{]}+\hbar\sum\limits_{k}\omega_{k}b_{k}^{{\dagger}}b_{k},\end{split}$
(2)

where $\lambda_{k}$ is the exciton phonon mode coupling constant and
$\Omega(z,t)=-2\vec{d}_{12}\cdot\hat{e}\mathcal{E}(z,t)/\hbar$ is the Rabi
frequency with transition dipole moment vector $\vec{d}_{12}$. The detuning of
the optical field with QD transition is defined as
$\delta_{l}=\omega_{L}-\omega_{l}$.

We notice that the Hamiltonian contains an infinite sum over phonon modes.
Keeping all order of exciton phonon interaction, we made a transformation in
the polaron frame. The transformation rule for modified Hamiltonian is given
by $H^{{}^{\prime}}=e^{P}He^{-P}$ where the operator
$P=\sum_{l}\sigma^{+}_{l}\sigma^{-}_{l}\sum_{k}\lambda_{k}(b_{k}^{\dagger}-b_{k})/\omega_{k}$.
This transformation also helps us to separate the system Hamiltonian from the
total Hamiltonian which is our primary interest. The transformed Hamiltonian
is divided into system, bath and interaction part, which can be decomposed as
$H^{\prime}=H_{s}+H_{b}+H_{I}$, where

$\displaystyle H_{s}$ $\displaystyle=$
$\displaystyle\sum\limits_{l}-\hbar\Delta_{l}\sigma^{+}_{l}\sigma^{-}_{l}+\langle
B\rangle X_{l}^{g},$ (3) $\displaystyle H_{b}$ $\displaystyle=$
$\displaystyle\hbar\sum_{k}\omega_{k}b_{k}^{{\dagger}}b_{k},$ (4)
$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum\limits_{l}\xi_{g}X_{l}^{g}+\xi_{u}X_{l}^{u},$ (5)

and $\Delta_{l}$ is the redefined detuning by considering the polaron shift
$\sum_{k}\lambda_{k}^{2}/\omega_{k}$. The definition of phonon-modified system
operators is given by

$\displaystyle X_{l}^{g}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\left(\Omega(z,t)\sigma^{+}_{l}+\Omega^{*}(z,t)\sigma^{-}_{l}\right),$
(6) $\displaystyle X_{l}^{u}$ $\displaystyle=$
$\displaystyle\frac{i\hbar}{2}\left(\Omega(z,t)\sigma^{+}_{l}-\Omega^{*}(z,t)\sigma^{-}_{l}\right).$
(7)

The phonon bath fluctuation operators are

$\displaystyle\xi_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(B_{+}+B_{-}-2\langle B\rangle\right),$ (8)
$\displaystyle\xi_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left(B_{+}-B_{-}\right),$ (9)

where $B_{+}$ and $B_{-}$ are the coherent-state phonon displacement
operators. Explicitly, the phonon displacement operators in terms of the
phonon mode operators can be written as

$B_{\pm}=\exp\left[{\pm\sum_{k}\frac{\lambda_{k}}{\omega_{k}}\left(b_{k}^{\dagger}-b_{k}\right)}\right].$
(10)

From this expression, it is clear that the exponential of the phonon operator
takes care of all the higher order phonon processes. Therefore, the phonon
displacement operator averaged over all closely spaced phonon modes at a
temperature T, obeys the relation $\langle B_{+}\rangle=\langle
B_{-}\rangle=\langle B\rangle$ where

$\langle
B\rangle=\text{exp}\left[-\frac{1}{2}\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega^{2}}\coth\left(\frac{\hbar\omega}{2K_{B}T}\right)\right],$
(11)

and $K_{B}$ is the Boltzmann constant. The phonon spectral density function
$J(\omega)=\alpha_{p}\omega^{3}\exp[-\omega^{2}/2\omega_{b}^{2}]$ describes
longitudinal acoustic(LA) phonon coupling via a deformation potential [25] for
QD system, where the parameters $\alpha_{p}$ and $\omega_{b}$ are the
electron-phonon coupling and cutoff frequency, respectively.

Next we use the master equation(ME) approach to solve the polaron-transformed
system Hamiltonian dynamics by considering the phonon bath as a perturbation.
The Born-Markov approximation can be performed with respect to the polaron-
transformed perturbation in the case of nonlinear excitation. Hence, the
density matrix equation for the reduced system under Born-Markov approximation
can be written as

$\dot{\rho}=\frac{1}{i\hbar}[H_{s},\rho]+\sum\limits_{l}\Bigr{(}{\cal
L}_{ph}\rho+\frac{\gamma}{2}{\cal
L}[\sigma^{-}_{l}]\rho+\frac{\gamma_{d}}{2}{\cal
L}[\sigma^{+}_{l}\sigma^{-}_{l}]\rho\Bigr{)},$ (12)

where $\gamma$ is the spontaneous decay rate of the exciton state. The
spontaneous decay originates from the quantum fluctuations of the vacuum
state. Similarly, for thermal fluctuation, we have adopted the final
Lindbladian form of the dephasing interaction model described by a simple
stochastic Hamiltonian[26]. Therefore, we incorporate the pure-dephasing
process phenomenologically in ME with a decay rate $\gamma_{d}$. This
additional dephasing term explains the broadening of the zero-phonon line
(ZPL) in QD with increasing temperatures [27, 28]. The Lindblad superoperator
$\cal L$ is expressed as ${\cal L}[\cal O]\rho=\text{2}{\cal
O}\rho{\cal}O^{\dagger}-{\cal O}^{\dagger}{\cal O}\rho-\rho\cal
O^{\dagger}\cal O$, under the operation of $\cal O$ operator. The term ${\cal
L}_{ph}$ represents the effect of phonon bath on the system dynamics.
Therefore the explicit form of ${\cal L}_{ph}\rho$ in terms of previously
defined system operators can be expressed as

$\displaystyle{\cal L}_{ph}\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{\hbar^{2}}\int_{0}^{\infty}d\tau\sum_{j=g,u}G_{j}(\tau)[X_{l}^{j}(z,t),X_{l}^{j}(z,t,\tau)\rho(t)]~{}$
(13) $\displaystyle+$ $\displaystyle H.c.,$

where
$X_{l}^{j}(z,t,\tau)=e^{-iH_{s}\tau/\hbar}X_{l}^{j}(z,t)e^{iH_{s}\tau/\hbar}$,
and the polaron Green’s functions are $G_{g}(\tau)=\langle
B\rangle^{2}\\{\cosh\left[\phi(\tau)\right]-1\\}$ and $G_{u}(\tau)=\langle
B\rangle^{2}\sinh[\phi(\tau)]$. The phonon Green’s functions depend on phonon
correlation function given below

$\phi(\tau)=\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega^{2}}\left[\coth\left(\frac{\hbar\omega}{2K_{B}T}\right)\cos(\omega\tau)-i\sin(\omega\tau)\right].$
(14)

The polaron ME formalism is not generally valid for arbitrary excitation
strength and exciton phonon coupling. The validity of polaron ME is stated as
[20]

$\left(\frac{\Omega}{\omega_{b}}\right)^{2}\left(1-\langle
B\rangle^{4}\right)\ll 1.$ (15)

It is clear from the above equation that, at low temperatures $\langle
B\rangle\approx 1$ and $\Omega/\omega_{b}<1$ fulfil the above criteria. Hence,
we restrict our calculation in the weak field regime satisfying
$\Omega/\omega_{b}<1$ at a low phonon bath temperature.

The full polaron ME (12) contains multiple commutator brackets and complex
operator exponents, which require involved numerical treatment for studying
time dynamics. We make some simplifications of the full ME by using various
useful identities. These reduce ME into a simple analytical form with decay
rates corresponding to the various phonon-induced processes. Though we have
not made any approximation, simplified ME scales down the numerical
computation efforts and gives better insight into the physical process. By
expanding all the commutators in Eq.(12) and rearranging using fermion
operator identities, we get the simplified ME as

$\begin{split}\dot{\rho}&=\frac{1}{i\hbar}[H_{s},\rho]+\sum\limits_{l}\Bigr{(}\frac{\gamma}{2}{\cal
L}[\sigma^{-}_{l}]\rho+\frac{\gamma_{d}}{2}{\cal
L}[\sigma^{+}_{l}\sigma^{-}_{l}]\rho\\\
&+\frac{\Gamma^{\sigma^{+}}_{l}}{2}{\cal
L}[\sigma^{+}_{l}]\rho+\frac{\Gamma^{\sigma^{-}}_{l}}{2}{\cal
L}[\sigma^{-}_{l}]\rho-\Gamma^{cd}_{l}(\sigma^{+}_{l}\rho\sigma^{+}_{l}+\sigma^{-}_{l}\rho\sigma^{-}_{l})\\\
&-i\Gamma^{sd}_{l}(\sigma^{+}_{l}\rho\sigma^{+}_{l}-\sigma^{-}_{l}\rho\sigma^{-}_{l})+i\Delta^{\sigma^{+}\sigma^{-}}_{l}[\sigma^{+}_{l}\sigma^{-}_{l},\rho]\\\
&-[i\Gamma^{gu+}_{l}(\sigma^{+}_{l}\sigma^{-}_{l}\rho\sigma^{+}_{l}+\sigma^{-}_{l}\rho-\sigma^{+}_{l}\sigma^{-}_{l}\rho\sigma^{-}_{l})+H.c.]\\\
&-[\Gamma^{gu-}_{l}(\sigma^{+}_{l}\sigma^{-}_{l}\rho\sigma^{+}_{l}-\sigma^{-}_{l}\rho+\sigma^{+}_{l}\sigma^{-}_{l}\rho\sigma^{-}_{l})+H.c.]\Bigr{)}.\end{split}$
(16)

The phonon-induced decay rates are given by

$\displaystyle\Gamma^{\sigma^{+}/\sigma^{-}}_{l}$
$\displaystyle=\frac{\Omega_{R}(z,t)^{2}}{2}\int_{0}^{\infty}\Bigg{(}\operatorname{Re}\bigg{\\{}(\cosh(\phi(\tau))-1)f(z,t,\tau)+\sinh(\phi(\tau))\cos(\eta(z,t)\tau)\bigg{\\}}$
$\displaystyle\mp\operatorname{Im}\left\\{(e^{\phi(\tau)}-1)\frac{\Delta_{l}\sin(\eta(z,t)\tau)}{\eta(z,t)}\right\\}\Bigg{)}\,d\tau,$
(17) $\displaystyle\Gamma^{\mathrm{cd}}_{l}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\operatorname{Re}\bigg{\\{}\Omega_{S}(z,t)\sinh(\phi(\tau))\cos(\eta(z,t)\tau)-\Omega_{S}(z,t)(\cosh(\phi(\tau))-1)f(z,t,\tau)$
$\displaystyle+\Omega_{T}(z,t)(e^{-\phi(\tau)}-1)\frac{\Delta_{l}\sin(\eta(z,t)\tau)}{\eta(z,t)}\bigg{\\}}d\tau,$
(18) $\displaystyle\Gamma^{\mathrm{sd}}_{l}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\operatorname{Re}\bigg{\\{}\Omega_{T}(z,t)\sinh(\phi(\tau))\cos(\eta(z,t)\tau)-\Omega_{T}(z,t)(\cosh(\phi(\tau))-1)f(z,t,\tau)$
$\displaystyle-\Omega_{S}(z,t)(e^{-\phi(\tau)}-1)\frac{\Delta_{l}\sin(\eta(z,t)\tau)}{\eta(z,t)}\bigg{\\}}d\tau,$
(19) $\displaystyle\Delta^{\sigma^{+}\sigma^{-}}_{l}$
$\displaystyle=\frac{\Omega_{R}(z,t)^{2}}{2}\int_{0}^{\infty}\operatorname{Re}\bigg{\\{}(e^{\phi(\tau)}-1)\frac{\Delta_{l}\sin(\eta(z,t)\tau)}{\eta(z,t)}\bigg{\\}}d\tau,$
(20) $\displaystyle\Gamma^{\mathrm{gu+}}_{l}$
$\displaystyle=\frac{\Omega_{R}(z,t)^{2}}{2}\int_{0}^{\infty}\bigg{\\{}(\cosh(\phi(\tau))-1)\operatorname{Im}[\langle
B\rangle\Omega]h(z,t,\tau)+\sinh(\phi(\tau))\frac{\operatorname{Re}[\langle
B\rangle\Omega]\sin(\eta(z,t)\tau)}{\eta(z,t)}\bigg{\\}}\,d\tau,$ (21)
$\displaystyle\Gamma^{\mathrm{gu-}}_{l}$
$\displaystyle=\frac{\Omega_{R}(z,t)^{2}}{2}\int_{0}^{\infty}\bigg{\\{}(\cosh(\phi(\tau))-1)\operatorname{Re}[\langle
B\rangle\Omega]h(z,t,\tau)-\sinh(\phi(\tau))\frac{\operatorname{Im}[\langle
B\rangle\Omega]\sin(\eta(z,t)\tau)}{\eta(z,t)}\bigg{\\}}\,d\tau,\ $ (22)

where
$f(z,t,\tau)=(\Delta_{l}^{2}\cos(\eta(z,t)\tau)+\Omega_{R}(z,t)^{2})/\eta(z,t)^{2}$,
$h(z,t,\tau)=\Delta_{l}(1-\cos(\eta(z,t)\tau))/\eta^{2}(z,t)$ and
$\eta(z,t)=\sqrt{\Omega_{R}(z,t)^{2}+\Delta_{l}^{2}}$ with the polaron-shifted
Rabi frequency, $\Omega_{R}(z,t)=\langle B\rangle|\Omega(z,t)|$,
$\Omega_{S}(z,t)=\operatorname{Re}[\langle
B\rangle\Omega(z,t)]^{2}-\operatorname{Im}[\langle B\rangle\Omega(z,t)]^{2}$,
$\Omega_{T}(z,t)=2\operatorname{Re}[\langle
B\rangle\Omega(z,t)]\operatorname{Im}[\langle B\rangle\Omega(z,t)]$.

Next, we use Maxwell wave equation to describe the propagation dynamics of the
electromagnetic field inside the QD medium

$\bigg{(}\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\bigg{)}\vec{E}(z,t)=\mu_{0}\frac{\partial^{2}}{\partial
t^{2}}\vec{P}(z,t)$ (23)

where $\mu_{0}$ is the permeability of free space. The induced polarisation
$\vec{P}(z,t)$ originates from the alignment of the medium dipole in the
presence of an applied field. Therefore it depends on the coherence term of
the density matrix equation. For $l^{th}$ QD, the coherence term of the
density matrix equation can be written as $\rho_{12}(\Delta_{l},z,t)=\langle
1(\omega_{l})|_{l}\rho(z,t)|2(\omega_{l})\rangle_{l}$. The medium consists of
a large number of QD with continuous frequency distribution centered at
$\omega_{c}$. Therefore we can safely replace the summation with integration
by redefining the discrete variable $\Delta_{l}$ to a continuous variable
$\Delta$. The induced macroscopic polarisation can be written in terms of the
density matrix element as

$\vec{P}(z,t)=N\int_{-\infty}^{\infty}\left(\vec{d}_{12}\rho_{12}(\Delta,z,t)e^{i(kz-\omega_{L}t)}+c.c.\right)g(\Delta)d\Delta,$
(24)

where $N$ is the QD volume number density. The inhomogeneous level broadening
function in the frequency domain is defined by $g(\Delta)$. In our
calculation, the form of $g(\Delta)$ is

$g(\Delta)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(\Delta-\Delta_{c})^{2}}{2\sigma^{2}}},$
(25)

where the standard deviation is $\sigma$. The detuning between the applied
field and the QD’s central frequency is represented by $\Delta_{c}$. By
applying slowly varying envelope approximation, one can cast inhomogeneous
second order partial differential Eq.(23) to first order differential equation
as

$\bigg{(}\frac{\partial}{\partial z}+\frac{1}{c}\frac{\partial}{\partial
t}\bigg{)}\Omega(z,t)=i\hskip
1.0pt\eta\int_{-\infty}^{\infty}\rho_{12}(\Delta,z,t)g(\Delta)d\Delta,$ (26)

where the coupling constant $\eta$ is defined by

$\eta=-3N\lambda^{2}\gamma/4\pi$ (27)

and $\lambda$ is the carrier wavelength of the QD transition. The self
consistent solution of Eq.(16) and (26) with proper initial conditions can
display the spatiotemporal evolution of the field inside the medium. Moreover
the analytical solution of the coupled partial differential equation is known
only for some special conditions, hence we adopted numerical integration of
Eq.(16) and (26) to depict the results. For numerical computation, a useful
frame transformation $\tau=t-z/c$ and $\zeta=z$ is needed which removes the
explicit time variable from Eq.(26), which now only depends on the one
variable $\zeta$.

##  III NUMERICAL RESULT

###  III.1 Phonon-induced scattering rates

First we discuss various decay rates for the QD system with experimentally
available parameter regions [29, 30]. The medium comprises InGaAs/GaAs QDs
with volume density $N=5\times 10^{20}\text{m}^{-3}$ and a length of 1 mm. The
central QD excitation energy is $\hbar\omega_{c}$ =1.3 eV with a Gaussian
spectral distribution having FWHM of 23.5 meV. The QD is driven by the optical
pulse at $\zeta$ = 0 with a hyperbolic secant profile

$|\Omega(0,\tau)|=\Omega_{0}\hskip
1.0pt\text{sech}\left(\frac{\tau-\tau_{c}}{\tau_{0}}\right)$ (28)

where $\tau_{0}$, and $\tau_{c}$ defines the width, and center of the pulse,
respectively. For numerical computation, the amplitude and width of the pulse
are taken to be $\Omega_{0}$ = 0.2 meV and $\tau_{0}$ = 6.373 ps. The phonon
bath temperature T = 4.2 K gives $\langle B\rangle=0.95$. Other parameters are
$\alpha_{p}=0.03\ \text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$. The system under
consideration has a relaxation rate $\gamma=\gamma_{d}=2\ \mu\text{eV}$(2 ns).
In order to normalize all the system parameters to a dimensionless quantity we
have chosen normalization frequency to be $\gamma_{n}$ = 1 rad/ps.

In Fig.(2), the color bar represents the variation of various phonon-induced
scattering rates as a function of detuning and time, both at normalised units
along the $x$\- and $y$-axis respectively. In the QD system, various phonon
processes are connected with exciton transitions. In the case of ground state
to exciton transition, phonon absorption occurs while in the opposite process,
phonon emission occurs.

Figure 2: The variation of phonon-induced scattering rates with detuning and
time of a QD at $\zeta$ = 0 for the applied secant pulse in Eq.(28). a)
Phonon-induced pumping rate $\Gamma^{\sigma^{+}}$[Eq.(17)] b) Phonon-induced
decay rate $\Gamma^{\sigma^{-}}$[Eq.(17)] c) Phonon induced dephasing
$\Gamma^{\mathrm{cd}}$[Eq.(18)] d) Phonon induced detuning
$\Delta^{\sigma^{+}\sigma^{-}}$[Eq.(20)] for peak Rabi frequency $\Omega_{0}$
= 0.2 meV, pulse width $\tau_{0}$ = 6.373 ps and pulse center
$\gamma_{n}\tau_{c}=40$. The phonon bath temperature T = 4.2 K corresponds to
$\langle B\rangle=0.95$ with spectral density function parameters
$\alpha_{p}=0.03\ \text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$.

Now we discuss the physical process associated with the phonon scattering
rates $\Gamma^{\sigma^{+}}$ and $\Gamma^{\sigma^{-}}$. For positive detuning,
the applied field frequency is larger than the QD transition frequency.
Subsequently a phonon generates with $\Delta$ frequency in order to make a
resonant QD transition. These emitted phonons develop an incoherent excitation
in the system referred by the $\Gamma^{\sigma^{+}}$. Oppositely for negative
detuning, the applied field frequency is smaller than the QD transition
frequency, and a resonant QD transition is possible only when some phonon of
frequency $\Delta$ will be absorbed from the bath. With this mechanism, QD
exciton to ground state decay enhances the radiation which is represented by
the $\Gamma^{\sigma^{-}}$. This low-temperature asymmetry is clearly visible
in Fig.2(a) and 2(b). At higher temperatures, this asymmetry gets destroyed,
and both rates overlap and are centered at $\Delta$ = 0. Fig.2(c) shows the
variation of $\Gamma^{cd}$ which is only present in the off-diagonal density
matrix element and responsible for the additional dephasing in the system
dynamics. The additional detuning $\Delta^{\sigma^{+}\sigma^{-}}$ from the
simplified master equation plotted in Fig.2(d), shows a very tiny value
compared to the system detuning $\Delta$. We also notice that the sign of
$\Delta^{\sigma^{+}\sigma^{-}}$ changes according to the system detuning
$\Delta$. It is important to keep in mind that we display the variation along
the $y$-axis around $\gamma_{n}\tau$ =40, which is the centre of the pulse
with the secant profile.

###  III.2 Pulse area theorem

It is well know from Beer’s law, that a weak pulse gets absorbed inside the
medium due to the presence of opacity at the resonance condition. However,
McCall and Hahn showed that some specific envelope pulse shape remains intact
for a long distance without absorption, even at resonance[1, 2]. Inspired of
this phenomena, we have taken into account of a time-varying pulse whose
envelope shape is stated in the Eq.(28). The area $\Theta(z)$ enclosed by its
hyperbolic envelope shape is defined as

$\Theta(z)=\int_{-\infty}^{+\infty}\Omega(z,t^{{}^{\prime}})dt^{{}^{\prime}}.$
(29)

By formally integrating Eq.(26) over time and detuning, one can find the
spatial variation of the pulse area closely followed by the McCall and Hahn
work. The evolution of the pulse area $\Theta(z)$ during its propagation in a
two-level absorbing QD medium is given by

$\frac{d\Theta(z)}{dz}=-\frac{\alpha}{2}\sin\Theta(z)$ (30)

where $\alpha$ is the optical extinction per unit length. The optical
extinction depends on the various system parameters as $\alpha=2\pi\eta g(0)$.
The solution of the Eq.(30) is

$\tan\frac{\Theta(z)}{2}=\tan\frac{\Theta(0)}{2}e^{-\alpha z/2},$ (31)

where $\Theta(0)$ is the pulse area at $z$ =0. It is clear from the above
expression that $\Theta(z)=2n\pi$ is the stable solution, whereas
$\Theta(z)=(2n+1)\pi$ is an unstable one. The pulse area of the given envelope
as stated in Eq.(28) is $\Theta(0)=\pi\Omega_{0}\tau_{0}$. Thus, the envelope
with amplitude $\Omega_{0}=2/\tau_{0}$ gives 2$\pi$ area pulse. This envelope
shape remains preserve for the long propagation distance even though it
interacts resonantly with the medium.

Figure 3: Evolution of the pulse area($\Theta$) as a function of propagation
distance $\zeta$ started with $2\pi$ sech-type pulse for different
temperatures. The applied pulse has a width of $\tau_{0}$ = 6.373 ps and
centered at $\gamma_{n}\tau_{c}=40$. The system under consideration without
phonon bath(black) and with phonon bath maintaining temperature T = 4.2K(red),
10K(blue), 20K(green) with electron phonon coupling $\alpha_{p}=0.03\
\text{ps}^{2}$ and cut off frequency $\omega_{b}=1\ \text{meV}$.The central QD
detuning $\Delta_{c}$ = 0 with spontaneous decay and the pure dephasing rate
$\gamma=\gamma_{d}=2\ \mu\text{eV}$(2 ns).The optical extinction per unit
length $\alpha$ = 10 mm-1. The inset figure shows the stability of the pulse
area higher than $2\pi$ for different phonon bath temperatures.

Fig.(3) exhibits the variation of pulse area with the propagation distance
inside the QD medium. It is evident from this figure that the propagation
dynamics of 2$\pi$ area pulse through the medium of length $L$ has negligible
loss in pulse area. In the absence of phonon(black line) interaction, the
system behaves identical to the atomic system and hence follows $\Theta\approx
2\pi(1-\tau_{0}/T_{2}^{{}^{\prime}})$ reported earlier by McCall and Hahn [2].
The loss in pulse area comes from the finite lifetime $T_{2}^{{}^{\prime}}$ of
the QD which is inversely proportional to $\gamma_{d}$. Ideally, the pulse
will retain initial pulse area for an arbitrary distance in absence of decay
and decoherence. However, in presence of phonon contribution, we have noticed
the pulse area gets enhanced by a small amount. The amount of raise in the
pulse area linearly depends on the bath temperature as indicated in Fig.(3).
This effect can be explained by carefully examing the definition of an
effective Rabi frequency $\Omega_{R}(z,t)=\langle B\rangle|\Omega(z,t)|$ where
$\langle B\rangle$ is dependent on the bath temperatures. The inset of Fig.(3)
illustrate the convergence of the pulse area shifted from the $2\pi$ value at
different temperatures.

Figure 4: The real(black) and imaginary(red) part of the coherence $\rho_{12}$
of a single QD at different times for a $2\pi$ sech-type short pulse with
pulse center at $\gamma_{n}\tau_{c}$ = 40 as a function of detuning. The pulse
has a width $\tau_{0}$ = 6.373 ps. Corresponding phonon bath parameters are T
= 4.2K, $\alpha_{p}=0.03\ \text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$.
Considered QD relaxation rates are $\gamma=\gamma_{d}=2\ \mu\text{eV}$(2 ns).

To explain the behavior of Fig.(3), we study the absorption and dispersion
properties of the medium as a function of detuning at various time intervals
of the pulse. Fig.(4) delineates the physical process behind the dispersion
and absorption. We assume all the population in the ground state, before the
leading edge of the pulse reaches the medium. The peak of incident pulse
enters inside the medium at $\gamma_{n}\tau_{c}$ = 40. It is clear from
Fig.4(a) that most of the leading edge pulse energy gets absorbed by the
ground state population and the population goes to the excited state. Hence
the medium shows maximum absorption at $\gamma_{n}\tau$ = 30, hence
elucidating the absorption phenomenon at resonance. Simultaneously, the nature
of the dispersion curve is anomalous as previously reported [31]. The
anomalous dispersion accompanied fast velocity is completely prohibited due to
huge absorption at the resonance condition. The medium becomes saturated as
the centre of the pulse enters the medium; consequently, the medium turns less
absorbent to the pulse. Nonetheless, a tiny absorption peak still exists at
the resonance condition due to the presence of various decay processes of the
medium as indicated by Fig.4(b). Therefore, the excited state gets populated
during the passage of the leading edge pulse. This population can leave the
excited state and return to the ground by stimulated emission in the presence
of the trailing edge of the pulse. As a results, a gain can be experienced by
the incident pulse at $\gamma_{n}\tau$ = 50 as revealed in Fig.4(c). From
these three panels, we can conclude that the leading edge of the pulse gets
absorbed by the medium, while the tailing edge of the pulse experiences gain.
Towards the trailing end of the pulse, the dispersive nature of the medium
changes from anomalous to normal, as shown in Fig.4(d). The positive slope of
the dispersion curve lead to slow group velocity that started at
$\gamma_{n}\tau$ = 60 shown in Fig.4(d). Fig.4(d) to Fig.4(f) indicate that
the optical pulse regeneration process is completed due to the medium-assisted
gain; hence, the pulse shape remains preserved. This is the explanation of the
underpinning mechanism behind SIT.

Figure 5: The variation of excited state population with input pulse area at
resonance condition $\Delta_{c}$ = 0. The system and bath parameters are
$\tau_{0}$ = 6.373 ps, $\gamma_{n}\tau_{c}$ = 40, T = 4.2K, $\alpha_{p}=0.03\
\text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$, $\gamma=\gamma_{d}=2\
\mu\text{eV}$(2 ns).

The claim of the above physical mechanism can be supported by studying
population dynamics at the excited state. For this purpose, we have plotted
the excited state population as a function of the pulse area in Fig.(5). A
noticeable population redistribution among the levels is feasible within few
widths of incident pulse wherein intensity is appreciable. As soon as the
pulse intensity diminishes at the trailing end, spontaneous emission takes
care of depletion of the excited state population. This leads to vanishing
population at the excited state after a sufficiently long time from the pulse
centre. As a consequence, it is crucial to decide the observation time of the
QD population. Hence, we display the exciton population at just the end of the
pulse $\gamma_{n}\tau=60$, to capture the outcome of the pulse. It is clear
from Fig.(5) that the excited state population shows a decaying Rabi
oscillation kind of behaviour. It is also confirmed that the population never
fully transferred to the excited state or fully returned to the ground state
for any pulse area, indicating to non-constant phonon induced decay and gain
process involved in the system. The decaying features of the local population
maximum can be justified by the examining the photon and phonon induced decay
rates. The various phonon decay rates are given in Eq.(17)-(22) where
increasing incident pulse amplitude $\Omega(z,t)$ results in the enhancement
of these decay rates. This field amplitude dependent phonon decay together
with constant photon decay can explain the gradual decay of the population
local maxima. On the contrary, the dip of local minima increases due to the
presence of phonon induced gain processes $\Gamma^{\sigma+}$ as suggested in
Eq.(17). The local maximum and minimum of the exciton population are located
respectively near odd and even integer multiples of $\pi$ pulse area. The
maxima signifies the pulse absorption by the medium, resulting in population
inversion. Similarly, minima manifests the transparency of the medium. Thus,
the leading edge of the pulse excite the population whereas the tailing edge
assists in stimulated emission leaving the population in the ground state of
the medium. It is evident that only even integer multiples of $\pi$ pulse can
propagate through the medium without absorption that is consistent with the
pulse area theorem. That the local maxima and minima of exciton population
never match exactly with the integer value can be figured out later by
investigating pulse propagation dynamics. Previously, we found the stable
pulse area is higher than $2\pi$ as shown in Fig.(3) which also agrees with
the above observation. Therefore, the analysis of coherence and population
ensures us that SIT phenomena can be accomplished in the QD medium.

Figure 6: The Rabi frequency normalized with the input peak value is plotted
against retarded time at different propagation distances inside the medium at
resonance condition $\Delta_{c}$ = 0. The input pulse has the following
parameters $\Theta(0)=2\pi$, $\tau_{0}$ = 6.373 ps, $\gamma_{n}\tau_{c}$ = 40.
The chosen QD inhomogeneous level broadening in normalized units
$\sigma/\gamma_{n}=15$. Other parameters are T = 4.2K, $\alpha_{p}=0.03\
\text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$, $\gamma=\gamma_{d}=2\
\mu\text{eV}$(2 ns).

###  III.3 Self Induced Transparency

A homogenous QD medium with length $1$ mm is taken into account for studying
spatio-temporal evolution of hyperbolic secant optical pulse. To achieve a
stable pulse propagation, we have chosen the initial pulse area to be 2$\pi$.
Fig.6 confirms the area theorem by showing a stable optical pulse propagation
for a longer distance. However, the pulse shape at larger distances has
noticed some distortion and absorption. Fig.6 also indicates that the pulse’s
peak value gradually decreases by increasing the propagation distance. This
suggests a finite absorption in the QD medium that prohibited complete
transparency in the system. In particular, the statement agrees well with the
small absorption peak at resonance in the absorption profile shown in Fig.4b.

Figure 7: The Rabi frequency normalized with the individual peak value is
plotted against retarded time at different propagation distances inside the
medium at resonance condition $\Delta_{c}$ = 0. All the other parameters are
the same as Fig.(6).

Figure 7 displays the individually normalized pulse for different propagation
distances. Inspection says that the input pulse experiences delay and a little
broadening during the propagation through the medium. The sole reason behind
the pulse broadening is the dispersive nature of the system. In the frequency
domain, a temporal pulse can be treated as a linear superposition of many
travelling plane waves with different frequencies. These individual frequency
waves gather different phases and move with varying velocities during the
pulse propagation in a dispersive medium. Therefore the pulse gets broader as
the leading part(low frequency) moves faster, and the tailing end(high
frequency) goes slower. In the QD system, the pure dephasing rate is also
responsible for this broadening as it destroys the coherence. From Fig.7, a
distinct peak shift is observed while optical pulse propagating through the
medium. This peak shift arises because of normal dispersive medium that
induced slow group velocity of the optical pulse inside the medium. We adopt
the analytical expression of time delay in the ideal case by considering
$\sigma\gg 1/\tau_{0}$ reported earlier[32]. The analytical expression for
time delay found to be $\gamma_{n}\tau_{d}=\alpha L\gamma_{n}\tau_{0}/4$. Here
the absorption coefficient $\alpha$ is approximately 10 mm-1 calculated from
the chosen parameters. Therefore the calculated analytical time delay
$\gamma_{n}\tau_{d}\approx$ 15 shows excellent agreement with the numerical
result. The inhomogeneous level broadening $\sigma$ plays an important role in
pulse propagation dynamics. In our calculation, we are in the regime where the
pulse width is greater than the inhomogeneous broadening time
$\sigma\tau_{0}\gg 1$. Therefore, the higher spread of the QD parameter
$\sigma$ leads to fewer QD resonantly interacting with the propagating pulse.
This results in a negligibly small change in pulse shape. Alternatively, the
effective QD density becomes less, indicating the lower value of the optical
extinction parameter $\alpha$. Henceforth a lower time delay is expected in
the final output pulse due to its presence in the righthand term of Eq.(26).
In Fig.(8), the final output pulse shape variation is presented for the three
different QD spreads. The pulse delay decreases with an increasing QD
broadening $\sigma$. On the other hand, the pulse peak value decreases with
the lower QD spreads. This observation matches our previous prediction that
higher $\sigma$ produce a lower pulse delay in the medium. Also, more resonant
QD absorb more energy from the pulse, resulting in a lesser peak value in the
final pulse shape. Hence spread of the QD is also a determining factor for the
shape and delay of the output pulse.

Figure 8: The normalized Rabi frequency displayed with retarded time after
passing the medium for three different QD broadening $\sigma$. All the other
parameters are the same as Fig.(6).

Recalling the pulse area theorem again, we observe that the pulse area is
almost constant throughout the propagation near $2\pi$. The result is
consistent because as the pulse amplitude decreases, the pulse width
increases, maintaining the constant area under the curve. Therefore an
absorbing QD medium can exhibit the SIT phenomena at low temperatures.

###  III.4 Phonon bath parameter dependence on SIT

In the simplified master equation (16), various phonon-induced scattering
rates depend on both the system and bath parameters. Hence it is crucial to
study the effect of phonon bath on the SIT dynamics. The phonon contribution
comes to the picture in two ways; one from the reduced Rabi frequency, which
depends on the $\langle B\rangle$ and the other is the phonon-induced
scattering rates connected with the phonon spectral density function.

Figure 9: The plot of Rabi frequency envelope with time at a propagation
distance $\zeta\eta/\gamma_{n}$ = 50 for different phonon bath temperatures at
resonance condition $\Delta_{c}$ = 0. The common parameters are
$\Theta(0)=2\pi$, $\tau_{0}$ = 6.373 ps, $\gamma_{n}\tau_{c}$ = 40,
$\alpha_{p}=0.03\ \text{ps}^{2}$, $\omega_{b}=1\ \text{meV}$,
$\gamma=\gamma_{d}=2\ \mu\text{eV}$(2 ns). The figure display four different
configurations, system without a phonon bath (black) and with a phonon bath at
a temperature T = 4.2K(red), 10K(blue), 20K(green).

Therefore increasing phonon bath temperatures reduces the value of $\langle
B\rangle$ and $\hbar\omega/2K_{b}T$ present in the expression of $\phi(\tau)$
given in the Eq.(14). Consequently, effective coupling between QD and applied
field gets reduced, but the phonon-induced decay rates get enhanced. From
Fig.(9), we notice that the final pulse shape experiences more deformation for
higher temperatures. The peak of the output pulse is also very much reduced
for the higher temperature T =20 K. Therefore, the bath temperature should be
minimised to see the SIT in the QD medium.

Figure 10: The Rabi frequency envelope with time at a propagation distance
$\zeta\eta/\gamma_{n}$ = 50 for different electron-phonon coupling strength
$\alpha_{p}$ at resonance condition $\Delta_{c}$ = 0. All the parameters are
same as Fig.(9) except T = 4.2K and various electron-phonon coupling
$\alpha_{p}=0.03\ \text{ps}^{2}$(red), 0.06 $\text{ps}^{2}$(blue), 0.12
$\text{ps}^{2}$(green).

Another controlling factor of the SIT is the interaction strength between the
QD and the phonon bath. So the increment of system-bath coupling leads to the
reduction of the coherence in the system. This statement is understandable by
looking at the phonon correlation function shown in Eq.(14). Thus the final
pulse shape for the equal propagation distances is significantly modified by
the electron-phonon coupling constant, as shown in Fig.(10). Therefore we also
have to ensure that the QD bath interacts weakly to get SIT phenomena in the
QD medium.

###  III.5 Higher pulse area and pulse breakup

Finally, we discuss the behaviour of a pulse propagating through the absorbing
QD medium with a higher pulse area than $2\pi$. Therefore we consider the next
stable pulse area solution 4$\pi$ for further investigation. The numerical
result of the pulse propagation in both space and time is shown in Fig.(11).
Unlike the 2$\pi$ pulse case, here, the initial pulse breaks into two pulses
as it travels through the medium. This phenomenon is also well explained by
the pulse area theorem where $2n\pi$ pulse is split into $n$ number of 2$\pi$
pulses. Surprisingly, the initial pulse breakup into two pulses is not
identical in shape. One pulse gets sharper, and the other gets broader in the
time domain and adjusts the peak value such that the area under the curve is
2$\pi$. The broader pulse component shows a prominent time delay, whereas the
sharper pulse component propagates with a tiny time delay. As a result, total
pulse area is constant throughout the propagation distance near 4$\pi$.

Figure 11: The propagation dynamics of a 4$\pi$ area pulse in an absorbing QD
medium as a function of both space and time at resonance condition
$\Delta_{c}=0$. All other parameters are same as Fig.(6).

##  IV CONCLUSIONS

We have investigated the SIT phenomena in an inhomogeneously broadened
semiconductor QD medium. In our model, we have included the effect of phonon
in the total Hamiltonian to describe the modified optical properties of QD in
the presence of a thermal environment. We then adopted the polaron ME
formalism to analytically derive the simplified ME with various phonon-induced
decay rates. These phonon-induced scattering rates are plotted against
detuning and time, which verify the presence of low-temperature asymmetry of
phonon-induced pumping and decay in our system. We solve numerically the
density matrix equation and Maxwell equation selfconsistently with suitable
parameters. We observe that stable pulse propagation is possible in the QD
medium with pulse area slightly higher than 2$\pi$, depending on the phonon
bath temperature. The physical mechanism of the SIT is clearly understood by
analyzing the absorption and dispersion of the medium. The leading edge of the
pulse gets absorbed by the medium, whereas the tailing edge of the pulse
experience gain, hence the pulse shape remains intact and propagate through
medium with short length. However, for longer propagation distances, we find
that even though the pulse propagation through the medium is possible, the
propagating pulse gets absorbed and broadened. The final pulse shape is
preserved on exiting the medium. Increasing the phonon bath temperature and
coupling produce more deformation in the final pulse shape, as it destroys the
coherence in the system. Finally, we explore the propagation of a 4$\pi$ pulse
in the QD medium, which shows prominent pulse breakup phenomena reported
earlier in the literature. Therefore our investigation ensures that a short
pulse can propagate through the considered QD medium with a tiny change in
shape. Hence, this work may have potential applications in quantum
communication, quantum information, and mode-locking.

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