# Environment-induced entanglement generation for two qubits in the presence
of qubit-qubit interaction noise

Muhammad Abdullah Mutahar  School of Science & Engineering, Lahore University
of Management Sciences (LUMS),
Opposite Sector U, D.H.A, Lahore 54792, Pakistan  Adam Zaman Chaudhry
<EMAIL_ADDRESS>School of Science & Engineering, Lahore University of
Management Sciences (LUMS),
Opposite Sector U, D.H.A, Lahore 54792, Pakistan

###### Abstract

Using an exactly solvable pure dephasing model, we show how entanglement
between qubits can be generated via the interaction with a common environment
and concurrent application of suitable control pulses. The control pulses are
able to effectively remove the detrimental effect of the environment while
preserving the indirect interaction between the qubits, thereby leading to the
generation of near-perfect entanglement. Furthermore, we also investigate the
entanglement dynamics if the qubits are directly interacting; this interaction
may even contain a noise term. The present of this additional noise leads to
an additional decoherence term. This decoherence term cannot be removed by
applying the pulses at the same time to both qubits. Rather, we show that by
introducing a time delay between the two pulse sequences, near-perfect
entanglement can still be generated via the interaction with the common
environment.

††preprint: APS/123-QED

##  I  Introduction

Entanglement is a fundamental concept in quantum mechanics, and one of the key
resources in quantum information processing [1, 2, 3, 4]. It finds many
practical applications in quantum computation and communication such as
quantum cryptography [5, 6] and superdense coding [7]. Unfortunately, quantum
resources such as entanglement and coherence are very delicate - realistic
quantum systems inevitably interact with their surrounding environments,
deteriorating these resources very quickly as a result of the decoherence
process [8, 9, 10]. Consequently, the environment has largely been considered
very problematic for quantum technologies. However, with the advent of
‘reservoir engineering’, it has been shown that the system-environment
interaction can instead be used to generate useful quantum resources [11, 12,
13, 14, 15, 16]. This is carried out by tuning the system-environment coupling
and properties of the environment. These techniques have been used
experimentally to generate superposition states in superconducting circuits
[17], atomic ensembles [18], and trapped ions [19, 20].

Of key interest to us in this work is the fact that two independent qubits
interacting with a common environment can become entangled [21, 22, 23]. This
environment-induced entanglement has several applications, including, but not
limited to, quantum control of two-dimensional quantum systems [24],
entanglement of two-mode squeezed states [25, 26], and entanglement of charged
qubits [27]. Therefore, it is worthwhile to optimize and enhance this
environment-induced entanglement. Attention has been directed towards
optimising the properties of the environment to enhance the induced
entanglement dynamics [28]. On the other hand, considerable work has been done
to mitigate the detrimental influence of the environment. Inspired by the
technique of dynamical decoupling that uses suitable control fields to remove
the effect of the environment [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42], in this paper we investigate the effect of pulse sequences on the
entanglement dynamics of two qubits interacting with a common environment. Our
hope is that the applied control pulses would be such that the decoherence of
the two qubits is highly mitigated while largely leaving the entanglement
generating influence of the environment untouched.

We start by considering a system of two independent qubits, initially in a
pure product state, interacting with a common environment. This environment is
modeled as a collection of harmonic oscillators. As the dephasing time is much
shorter than the relaxation time scale, we restrict our model to the pure
dephasing case. After working out the exact two-qubit dynamics in the presence
of pulses, we show that near-perfect entanglement can be generated via the
application of pulses. We then further modify our system to include a direct
noisy interaction between the two qubits. We show that, as a result of the
noise in the interaction between the qubits, an additional decoherence term
emerges. Interestingly, to mitigate the influence of this additional term on
the entanglement, the pulses cannot be applied at the same time to both
qubits. Rather, there must be a time delay. By incorporating such a time-
delay, we show how near-perfect entanglement can still be generated.

This paper is organised in the following way. In Sec. II, we introduce our
basic model and work out its dynamics. In Sec. III, we work out the
entanglement dynamics to show the near-perfect generation of entanglement with
the application of pulses. Then, in Sec. IV, we modify our Hamiltonian to
accommodate for the qubits interacting directly with each other. This
interaction term contains a classical stochastic variable depicting a noisy
interaction. We conclude our findings in Sec. V, with some details deferred to
the appendix.

##  II The model and its dynamics

We begin with two qubits interacting with a common bosonic environment. The
total system-environment Hamiltonian is then given by (we use $\hbar=1$
throughout)

$\displaystyle H$ $\displaystyle=H_{S}+H_{B}+H_{I}+H_{C}$ (1)
$\displaystyle=\frac{\omega_{0}}{2}\left(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}\right)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$
$\displaystyle\hskip
11.38109pt+\left(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}\right)\sum_{k}\left(g_{k}^{*}b_{k}+g_{k}b_{k}^{\dagger}\right)$
$\displaystyle\hskip 11.38109pt+H_{C}^{(1)}+H_{C}^{(2)}.$

Here, the superscripts $(1)$ and $(2)$ refer to the first qubit and the second
qubit, $\omega_{0}$ is the spacing between the energy states of a two-level
system (assumed to be the same for both qubits), $\sigma_{z}$ is the standard
Pauli matrix, while $H_{B}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$ represents
the common bosonic environment composed of a collection of harmonic
oscillators. The third term describes the qubit-environment interaction for
the qubits, while the last terms, $H_{C}^{(1)}$ and $H_{C}^{(2)}$, represent
the control fields (pulses) applied to the first and the second qubit
respectively. We note that our system Hamiltonian is restricted to the pure-
dephasing case of the widely known spin-boson model for two qubits [43, 44].
This restriction is allowed if we consider the dynamics within a time scale
much shorter than the relaxation time-scale of the system.

To solve the dynamics, we first consider the free time-evolution operator
$U_{0}(t)=\text{exp}[-i(H_{S}+H_{B})t]$. The Hamiltonian in the interaction
picture is then defined as

$\displaystyle H_{I}(t)$
$\displaystyle=U_{0}^{\dagger}(t)H_{I}U_{0}(t)+U_{0}^{\dagger}(t)H_{C}^{(1)}U_{0}(t)$
(2) $\displaystyle\hskip 11.38109pt+U_{0}^{\dagger}(t)H_{C}^{(2)}U_{0}(t)$
$\displaystyle=\left(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}\right)\sum_{k}\left(g_{k}^{*}b_{k}e^{-i\omega_{k}t}+g_{k}b_{k}^{\dagger}e^{i\omega_{k}t}\right)$
$\displaystyle\hskip 11.38109pt+H_{C}^{(1)}(t)+H_{C}^{(2)}(t).$

It is useful to move to the toggling frame of the pulses applied on the
qubits. Assuming that $H_{C}^{(1)}(t)$ and $H_{C}^{(2)}(t)$ generate a series
of instantaneous $\pi$ rotations of the qubits around an axis orthogonal to
the $z$-axis at times $t_{k}$, where $0\leq k\leq n+1$ with $n$ being the
number of pulses applied and $t_{0}=0,t_{n+1}=t$, the interaction Hamiltonian
in the toggling frame of the pulses takes the form

$\displaystyle H_{T}(t)$
$\displaystyle=\left(s^{(1)}(t)\sigma_{z}^{(1)}+Z^{(2)}(t)\sigma_{z}^{(2)}\right)$
(3) $\displaystyle\hskip
11.38109pt\times\sum_{k}\left(g_{k}^{*}b_{k}e^{-i\omega_{k}t}+g_{k}b_{k}^{\dagger}e^{i\omega_{k}t}\right),$

where

$\displaystyle Z^{(1)}(t)=\int_{0}^{t}s^{(1)}(t^{\prime})dt^{\prime},$
$\displaystyle Z^{(2)}(t)=\int_{0}^{t}s^{(2)}(t^{\prime})dt^{\prime}.$

As the role of the pulses can essentially be reduced to flipping the sign of
$\sigma_{z}$, we have attached a switching function $s(t)$ above for both the
qubits, and the superscript is there to remind us that the applied pulse
sequence need not be the same for both qubits. The switching function is

$s(t^{\prime})=\sum_{k=0}^{n}(-1)^{k}\Theta\left(t_{k+1}-t^{\prime}\right)\Theta\left(t^{\prime}-t_{k}\right),$
(4)

where $\Theta(t)$ is the Heaviside step function. We further denote the times
at which pulses are applied $t_{k}$ as $t_{k}=\delta_{k}t$ with
$0\leq\delta_{k}\leq 1$, where $k=1,\ldots,n$. Next, we move on to evaluating
the unitary time evolution operator corresponding to $H_{T}(t)$. This is found
to be

$\displaystyle U_{I}(t)$
$\displaystyle=\text{exp}\left[\sum_{i=1}^{2}A_{i}\right].$ (5)

Here

$\displaystyle A_{1}(t)=$
$\displaystyle-i\sigma_{z}^{(1)}\sum_{k}\left[\phi^{*(1)}_{k}(t)g_{k}^{*}b_{k}+\phi^{(1)}_{k}(t)g_{k}b_{k}^{\dagger}\right]$
(6)
$\displaystyle-i\sigma_{z}^{(2)}\sum_{k}\left[\phi^{*(2)}_{k}(t)g_{k}^{*}b_{k}+\phi^{(2)}_{k}(t)g_{k}b_{k}^{\dagger}\right],$
$\displaystyle=$
$\displaystyle-i\sigma_{z}^{(1)}\chi_{1}(t)-i\sigma_{z}^{(2)}\chi_{2}(t),$

where

$\displaystyle\phi_{k}^{(1)}(t)$
$\displaystyle\equiv\int_{0}^{t}dt^{\prime}s^{(1)}(t^{\prime})e^{i\omega_{k}t^{\prime}},$
(7) $\displaystyle\phi_{k}^{(2)}(t)$
$\displaystyle\equiv\int_{0}^{t}dt^{\prime}s^{(2)}(t^{\prime})e^{i\omega_{k}t^{\prime}},$

and $\chi_{1}(t)$ and $\chi_{2}(t)$ are defined using Eq. (6). We work out
$A_{2}(t)$ to be

$\displaystyle A_{2}(t)$
$\displaystyle=iB_{1}(t)+iB_{2}(t)+iB_{3}(t)\sigma_{z}^{(1)}\sigma_{z}^{(2)}$
(8) $\displaystyle\hskip
11.38109pt+iB_{4}(t)\sigma_{z}^{(1)}\sigma_{z}^{(2)},$

with

$\displaystyle B_{1}(t)=$ $\displaystyle i\sum_{k}\left|g_{k}\right|^{2}$ (9)
$\displaystyle\times\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}s^{(1)}(t_{1})s^{(1)}(t_{2})\sin[\omega_{k}(t_{1}-t_{2})],$
$\displaystyle B_{2}(t)=$ $\displaystyle i\sum_{k}\left|g_{k}\right|^{2}$
$\displaystyle\times\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}s^{(2)}(t_{1})s^{(2)}(t_{2})\sin[\omega_{k}(t_{1}-t_{2})],$
$\displaystyle B_{3}(t)=$ $\displaystyle i\sum_{k}\left|g_{k}\right|^{2}$
$\displaystyle\times\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}s^{(1)}(t_{1})s^{(2)}(t_{2})\sin[\omega_{k}(t_{1}-t_{2})],$
$\displaystyle B_{4}(t)=$ $\displaystyle i\sum_{k}\left|g_{k}\right|^{2}$
$\displaystyle\times\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}s^{(2)}(t_{1})s^{(1)}(t_{2})\sin[\omega_{k}(t_{1}-t_{2})].$

The full unitary time evolution operator corresponding to the Hamiltonian in
the toggling frame of the control fields then becomes

$\displaystyle U(t)$
$\displaystyle=\text{exp}\left[\frac{-i\omega_{0}}{2}\left(Z^{(1)}(t)\sigma_{z}^{(1)}+Z^{(2)}(t)\sigma_{z}^{(2)}\right)\right]$
(10) $\displaystyle\hskip 11.38109pt\times\text{exp}\left[-iH_{B}t\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-i\sigma_{z}^{(1)}\chi_{1}(t)\right]\text{exp}\left[-i\sigma_{z}^{(2)}\chi_{2}(t)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[i\left(B_{1}(t)+B_{2}(t)\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[i\left(B_{3}(t)+B_{4}(t)\right)\sigma_{z}^{(1)}\sigma_{z}^{(2)}\right].$

We now proceed to compute the dynamics of the system in the density matrix
formalism, given any arbitrary system-environment initial state. For
simplicity we consider the initial state to be a product state of any qubit-
system state with the environment in a thermal bath state

$\displaystyle\rho(0)$ $\displaystyle=\rho_{S}(0)\otimes\rho_{B},$ (11)
$\displaystyle\rho_{B}$ $\displaystyle=\frac{\text{exp}\left[-\beta
H_{B}\right]}{Z_{B}},$ $\displaystyle Z_{B}$
$\displaystyle=\text{Tr}_{B}[e^{-\beta H_{B}}].$

Here, $\beta$ is the inverse temperature and $Z_{B}$ is the partition function
of the environment. $\text{Tr}_{B}[\hskip 5.69054pt]$ represents the trace
over the environment states. Given the initial system-environment state, we
can work out the state at a later time $t$ via
$\rho(t)=U(t)\rho(0)U^{\dagger}(t)$. We do so in the basis $\ket{k,l}$ given
by

$\displaystyle\sigma_{z}^{(1)}\ket{k,l}$ $\displaystyle=k\ket{k,l},$
$\displaystyle\sigma_{z}^{(2)}\ket{k,l}$ $\displaystyle=l\ket{k,l}.$

It is also useful to define

$\displaystyle\bra{k^{\prime},l^{\prime}}\rho_{S}(t)\ket{k,l}$
$\displaystyle=\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}=\text{Tr}_{S}[\rho_{S}(t)P_{kl,k^{\prime}l^{\prime}}],$
(12)

where $P_{kl,k^{\prime}l^{\prime}}=\ket{k,l}\bra{k^{\prime},l^{\prime}}$, and
$\rho_{S}(t)=\text{Tr}_{B}[\rho(t)]$ is the density matrix of the two qubits
only. Now, it is easy to show that
$\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}=\text{Tr}_{S+B}\left[U^{\dagger}(t)P_{kl,k^{\prime}l^{\prime}}U(t)\rho(0)\right]$,
where $\text{Tr}_{S+B}$ is the total trace over both system and environment.
Using the expression for the total time-evolution operator already worked out,
we find that

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime},l^{\prime};k,l}$ (13)
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{i\omega_{0}}{2}\left(k\hskip
1.42262ptZ^{(1)}(t)+l\hskip 1.42262ptZ^{(2)}(t)\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{-i\omega_{0}}{2}\left(k^{\prime}\hskip
1.42262ptZ^{(1)}(t)+l^{\prime}\hskip 1.42262ptZ^{(2)}(t)\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-iB(t)\left(kl-k^{\prime}l^{\prime}\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{Tr}_{B}\left[e^{iR_{kl,k^{\prime}l^{\prime}}}\rho_{B}\right].$

Here $B(t)=B_{3}(t)+B_{4}(t)$ and
$R_{kl,k^{\prime}l^{\prime}}=(k-k^{\prime})\chi_{1}(t)+(l-l^{\prime})\chi_{2}(t)$.
To further evaluate the trace in the last line above, we use the identity
$\text{Tr}_{B}[e^{A}\rho_{B}]=e^{\text{Tr}_{B}[A^{2}\rho_{B}]/2}$, leading to

$\text{Tr}_{B}\left[e^{iR_{kl,k^{\prime}l^{\prime}}}\rho_{B}\right]=\text{exp}\left[-\frac{1}{2}\text{Tr}_{B}\left[\left(R_{kl,k^{\prime}l^{\prime}}\right)^{2}\rho_{B}\right]\right],$
(14)

with

$\displaystyle\text{Tr}_{B}$
$\displaystyle\left[\left(R_{kl,k^{\prime}l^{\prime}}\right)^{2}\rho_{B}\right]=$
(15)
$\displaystyle(k-k^{\prime})^{2}\sum_{r}\left|\phi_{r}^{(1)}(t)\right|^{2}\left|g_{r}\right|^{2}\left(2n_{r}+1\right)$
$\displaystyle+$
$\displaystyle(l-l^{\prime})^{2}\sum_{r}\left|\phi_{r}^{(2)}(t)\right|^{2}\left|g_{r}\right|^{2}\left(2n_{r}+1\right)$
$\displaystyle+$ $\displaystyle
2(k-k^{\prime})(l-l^{\prime})\sum_{r}\phi_{r}^{*(1)}(t)\hskip
2.84526pt\phi_{r}^{(2)}(t)\left|g_{r}\right|^{2}\left(n_{r}+1\right)$
$\displaystyle+$ $\displaystyle
2(k-k^{\prime})(l-l^{\prime})\sum_{r}\phi_{r}^{(1)}(t)\hskip
2.84526pt\phi_{r}^{*(2)}(t)\left|g_{r}\right|^{2}n_{r}.$

Note that
$\text{Tr}_{B}\left[b_{r}^{\dagger}b_{r}\rho_{B}\right]=n_{r}=1/(e^{\beta\omega_{r}}-1)$,
which is Bose-Einstein distribution since the environment is in thermal
equilibrium.

Until this point we have kept the formalism arbitrary with respect to the
pulses. The purpose for keeping it arbitrary, as we would see later, is to use
this formalism to analyze the more complicated problem where a ‘noisy’ qubit-
qubit interaction is present as well. However, for our current purpose, we now
assume that both qubits are subjected to the same pulse sequence. This would
imply $s^{(1)}(t)=s^{(2)}(t)=s(t)$, $Z^{(1)}(t)=Z^{(2)}(t)=Z(t)$,
$\phi_{r}^{(1)}(t)=\phi_{r}^{(2)}(t)=\phi_{r}(t)$ and
$B_{3}(t)=B_{4}(t)=D(t)$. By this assumption, Eq. (15) becomes

$\text{Tr}_{B}\left[\left(R_{kl,k^{\prime}l^{\prime}}\right)^{2}\rho_{B}\right]=\\\
\left(k+l-k^{\prime}-l^{\prime}\right)^{2}\sum_{r}\left|\phi_{r}(t)\right|^{2}\left|g_{r}\right|^{2}\coth\left(\frac{\beta\omega_{r}}{2}\right).$

It is useful to further express our formalism in terms of filter functions
[45]. The action of a pulse sequence is captured by its filter function which
is essentially in terms of the Fourier transform of $s(t)$. The filter
function is defined as

$F(\omega t)=\frac{\omega^{2}}{2}\left|\int_{0}^{t}s(t^{\prime})e^{i\omega
t^{\prime}}dt^{\prime}\right|^{2}.$ (16)

At this point, the spectral density $J(\omega)$ is also introduced via the
substitution

$\sum_{r}4\left|g_{r}\right|^{2}\rightarrow\int_{0}^{\infty}d\omega
J(\omega)\delta\left(\omega_{r}-\omega\right).$ (17)

This replaces the discrete summation with a continuous spectrum of environment
frequencies $\omega$. The spectral density $J(\omega)$ is typically expressed
using a power law, i.e
$J(\omega)=g\frac{\omega^{s}}{\omega_{c}^{s-1}}G(\omega,\omega_{c})$, where
$g$ is the system-environment coupling strength, $s$ is the Ohmicity
parameter, and $G(\omega,\omega_{c})$ is the cutoff function with $\omega_{c}$
being the cutoff frequency. For the sake of simplicity, we choose to work in
the Ohmic ($s=1$) regime [46, 10]. Similarly, although different varieties of
cutoff functions are used in literature [10], for simplicity, we would stick
to the exponential cutoff function, that is,
$G(\omega,\omega_{c})=\text{exp}\left[-\omega/\omega_{c}\right]$.

In summary, putting everything together, we finally have

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime},l^{\prime};k,l}$ (18)
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{-i\omega_{0}Z(t)}{2}\left(k^{\prime}+l^{\prime}-k-l\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{iD(t)}{2}\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{\gamma(t)}{4}\left(k+l-k^{\prime}-l^{\prime}\right)^{2}\right],$

where

$\displaystyle D(t)$ $\displaystyle=\int_{0}^{\infty}d\omega
J(\omega)\int_{0}^{t}dt_{1}\hskip 2.84526pts(t_{1})$
$\displaystyle\times\int_{0}^{t_{1}}dt_{2}\hskip
2.84526pts(t_{2})\sin\left[\omega\left(t_{1}-t_{2}\right)\right],$
$\displaystyle\gamma(t)$ $\displaystyle=\int_{0}^{\infty}d\omega
J(\omega)\frac{F\left(\omega
t\right)}{\omega^{2}}\coth\left(\frac{\beta\omega}{2}\right).$

##  III Generating entanglement

As stated in the introduction, our goal is to analyze the entanglement
generation of two qubits under various pulse sequences and evaluate if the
pulse sequences can possibly enhance the entanglement generation of the two
qubits as compared to the qubits interacting with the common environment
without any pulses applied. Let us try to interpret Eq. (18). The first factor
containing
$\text{exp}\left[-i\omega_{0}Z(t)\left(k^{\prime}+l^{\prime}-k-l\right)/2\right]$
describes the free evolution of the qubits in the presence of the pulses. The
second factor, that is, containing
$\text{exp}\left[\frac{iD(t)}{2}\left(k^{\prime}l^{\prime}-kl\right)\right]$
describes the indirect interaction of the qubits, modulated by the pulses, due
to the common bosonic environment. It is this interaction that generates the
entanglement, so our goal would be to preserve this interaction as much as we
could. The third term, namely
$\left[-\frac{\gamma(t)}{4}\left(k+l-k^{\prime}-l^{\prime}\right)^{2}\right]$,
represents the decoherence factor $\gamma(t)$. We would try to minimize this
term as much as we can by applying control pulses with short pulse spacing -
this, of course, is the idea behind dynamical decoupling. Essentially, the
decoherence rate depends on the overlap between the spectral density and the
filter function; if pulses are applied rapidly, this overlap becomes very
small, leading to a very small decoherence rate. However, the indirect
interaction can still survive.

Let us then analyze the functions $D(t)$ and $\gamma(t)$ in more detail.
Starting with $D(t)$,

$\displaystyle D(t)=\int_{0}^{\infty}d\omega\hskip 2.84526ptJ(\omega)\hskip
2.84526ptd(\omega,t),$

where

$\displaystyle d(\omega,t)=\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\hskip
2.84526pts(t_{1})s(t_{2})\sin\left[\omega\left(t_{1}-t_{2}\right)\right].$

For $n$ pulses, we denote $d(\omega,t)$ by $d_{n}(\omega,t)$, which can be
simplified to [47]

$\displaystyle d_{n}(\omega,t)=\vartheta(\omega,t)+\nu(\omega,t)-t/\omega,$

with

$\displaystyle\vartheta(\omega,t)=$
$\displaystyle\frac{1}{\omega^{2}}\left[2\sum_{m=1}^{n}(-1)^{m}\sin(\omega
t_{m})+(-1)^{n+2}\sin(\omega t)\right],$ $\displaystyle\nu(\omega,t)=$
$\displaystyle\frac{2}{\omega^{2}}\Biggl{\\{}\sum_{m=1}^{n}\sum_{j=1}^{n}(-1)^{m+j}\left(\sin\left[\omega\left(t_{m+1}-t_{j}\right)\right]\right)$
$\displaystyle-\sin\left[\omega\left(t_{m}-t_{j}\right)\right]\Biggl{\\}}.$

For $\vartheta(\omega,t)$, if we assume a very large number of pulses
($n>>1$), we see that the first term effectively approaches zero since it is
rapidly switching sign, and, in the high pulse limit, $t_{m+1}$ approaches
$\approx t_{m}$. The same logic applies to $\nu(\omega,t)$. We are then left
with a very simple $d(\omega,t)$ in the high pulse limit. Coming towards
$\gamma(t)$, it is useful to first define
$F_{1}(\omega)=J(\omega)\coth(\beta\omega/2)$, and $F_{2}(\omega)=F(\omega
t_{0})/\omega^{2}$. We can then rewrite $\gamma(t)$ as

$\displaystyle\gamma(t_{0})=\int_{0}^{\infty}d\omega
F_{1}(\omega)F_{2}(\omega).$ (19)

It can be seen that the maximum contribution to the integral occurs where the
functions $F_{1}(\omega)$ and $F_{2}(\omega)$ overlap the most. The effect of
increasing the number of pulses $n$ can essentially be understood as the
shifting of the peak of the filter function and hence of $F_{2}(\omega)$
towards the side of increasing $\omega$. As $F_{1}(\omega)$ is an
exponentially decreasing function after $\omega=\omega_{c}$, the overlap of
both functions keeps on decreasing as we increase the number of pulses. This
also suggests in the high pulse limit ($n>>1$), $\gamma(t)\rightarrow 0$.
Therefore, we can expect to get negligible decoherence but significant
entanglement due to the indirect interaction encapsulated by $D(t)$.

Figure 1:  Concurrence $C(t)$ in the absence of pulses vs pulses applied
according to the PDD scheme for $n=256$. The solid, blue line represents the
concurrence in the absence of pulses, while the red, dashed line represents
the $C(t)$ with the applied pulses. We are using dimensionless units
throughout with $\hbar=1$, and we have used $\omega_{c}=20$, $\beta=1$, and
$g=0.1$.

We now numerically check our claim regarding the generation of entanglement.
First, to compute the dynamics of entanglement between the two qubits, we the
use concurrence $C(t)$ [48]. It is defined via the Hermitian matrix $V$ such
that

$\displaystyle V=\sqrt{\sqrt{\rho_{S}}\hskip
2.84526pt\widetilde{\rho_{S}}\hskip 2.84526pt\sqrt{\rho_{S}}},$

where

$\displaystyle\widetilde{\rho_{S}}=\left(\sigma_{y}\otimes\sigma_{y}\right)\rho_{S}^{*}\left(\sigma_{y}\otimes\sigma_{y}\right).$

Here, $\sigma_{y}$ is the standard Pauli matrix. $C(\rho_{S})$ is then
computed via

$C(\rho_{S})=\text{max}\left[0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\right].$
(20)

The $\lambda_{i}$ are the eigenvalues of $V$ in decreasing order. Now, in
order to see the dynamics of entanglement, we should start with the initial
system state to be a product state of the two qubits, and then we would
analyse if and how the environment generates entanglement in between those
qubits. Furthermore, the initial two-qubit state should not be part of a
decoherence-free subspace. We begin our analysis by choosing the initial
system state to be $\rho_{S}(t)=\ket{+,+}\bra{+,+}$ where
$\sigma_{x}\ket{+}=\ket{+}$. The system density matrix at time $t$ is then
given by

$\rho_{S}(t)=\frac{1}{4}\begin{pmatrix}1&e^{-i\omega_{0}Z(t)}e^{-\gamma(t)+iD(t)}&e^{-i\omega_{0}Z(t)}e^{-\gamma(t)+iD(t)}&e^{-4\gamma(t)-2i\omega_{0}Z(t)}\\\
e^{i\omega_{0}Z(t)}e^{-\gamma(t)-iD(t)}&1&1&e^{-i\omega_{0}Z(t)}e^{-\gamma(t)-iD(t)}\\\
e^{i\omega_{0}Z(t)}e^{-\gamma(t)-iD(t)}&1&1&e^{-i\omega_{0}Z(t)}e^{-\gamma(t)-iD(t)}\\\
e^{-4\gamma(t)+2i\omega_{0}Z(t)}&e^{i\omega_{0}Z(t)}e^{-\gamma(t)+iD(t)}&e^{i\omega_{0}Z(t)}e^{-\gamma(t)+iD(t)}&1\end{pmatrix}.$
(21)

We now investigate concurrence $C(t)$ against time $t$ by applying pulses
according to the periodic dynamical decoupling (PDD) scheme to both qubits,
and compare it with the evolution when no pulses are applied. Results are
illustrated in Fig. 1. By comparing the solid, blue line with the red, dashed
line, it is clear that near-perfect entanglement can be generated via the
application of the pulses. These results are in accordance with our prediction
that the effect of the decoherence factor is greatly nullified, while the
indirect interaction remains. Note, however, that after some time, the
decoherence becomes significant, leading to a decrease in the value of the
peak entanglement.

We next consider the performance of different pulse sequences in generating
entanglement. We numerically analyse $C(t)$ and compare the results for
different types of pulse sequences, namely, Periodic Dynamical Decoupling
(PDD) pulses, Carr-Purcell-Meiboom-Gill (CPMG) pulses[49, 50], and Uhrig’s
dynamical decoupling (UDD) pulses [51]. Results are shown in Fig. 2. It is
clear that while all three pulse sequences are able to generate entanglement,
the entanglement decays in a different manner for the different sequences.
This can be traced back to the fact that the mitigation of the decoherence
factor with different pulse sequences is different.

Figure 2:  Concurrence $C(t)$ for PDD, CPMG and UDD pulses. The dashed, red
line represents $C(t)$ for PDD pulses, the dotted, black line is for CPMG
pulses, and the blue, solid line represents $C(t)$ for UDD pulses. The number
of pulses applied is $n=256$. We use the same parameters as before, that is,
$\omega_{c}=20$, $\beta=1$, and $g=0.1$.

##  IV Introducing the direct qubit-qubit interaction

###  IV.1 The Model and Time Evolution

We now allow the two qubits to interact directly with each other. The system-
environment Hamiltonian now becomes

$\displaystyle H$ $\displaystyle=H_{S}+H_{B}+H_{I}+H_{C}$ (22)
$\displaystyle=\frac{\omega_{0}}{2}\left(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}\right)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$
$\displaystyle\hskip 11.38109pt+\lambda(t)\hskip
2.84526pt\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ $\displaystyle\hskip
11.38109pt+\left(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}\right)\sum_{k}\left(g_{k}^{*}b_{k}+g_{k}b_{k}^{\dagger}\right)$
$\displaystyle\hskip 11.38109pt+H_{C}^{(1)}+H_{C}^{(2)}$

where

$\displaystyle\lambda(t)=\lambda_{0}+\xi(t)$

Here, $\lambda(t)$ represents the direct qubit-qubit interaction, expressed as
the sum of a constant value $\lambda_{0}$ and a classical stochastic function
$\xi(t)$. In other words, we allow for the possibility of noise in the
interaction. For convenience, it is useful to express $\xi(t)$ in terms of a
coupling constant $v$ and a dimensionless variable $\eta(t)$, that is,
$\xi(t)=v\eta(t)$. We proceed as before and obtain the effective Hamiltonian
in the toggling frame as

$\displaystyle H_{T}(t)$ $\displaystyle=\lambda(t)\hskip
2.84526pts^{(1)}(t)\hskip 2.84526pts^{(2)}(t)\hskip
2.84526pt\sigma_{z}^{(1)}\sigma_{z}^{(2)}$ (23) $\displaystyle\hskip
11.38109pt+\left(s^{(1)}(t)\sigma_{z}^{(1)}+s^{(2)}(t)\sigma_{z}^{(2)}\right)$
$\displaystyle\hskip
11.38109pt\times\sum_{k}\left(g_{k}^{*}b_{k}e^{-i\omega_{k}t}+g_{k}b_{k}^{\dagger}e^{i\omega_{k}t}\right)$

The unitary time-evolution operator corresponding to this is $U_{I}(t)$.
Following our previous derivation, we now find that

$\displaystyle A_{1}(t)=$
$\displaystyle-i\sigma_{z}^{(1)}\chi_{1}(t)-i\sigma_{z}^{(2)}\chi_{2}(t)$ (24)
$\displaystyle-i\sigma_{z}^{(1)}\sigma_{z}^{(2)}\lambda_{0}h(t)-i\sigma_{z}^{(1)}\sigma_{z}^{(2)}\Gamma(t),$

where

$\displaystyle h(t)$ $\displaystyle=\int_{0}^{t}dt_{1}\hskip
2.84526pts^{(1)}(t_{1})\hskip 2.84526pts^{(2)}(t_{1}),$
$\displaystyle\Gamma(t)$ $\displaystyle=\int_{0}^{t}dt_{1}\hskip
2.84526pt\xi(t_{1})\hskip 2.84526pts^{(1)}(t_{1})\hskip
2.84526pts^{(2)}(t_{1}).$

Here $\chi_{1}(t)$ and $\chi_{2}(t)$ are given in Eq. (6). $A_{2}(t)$ turns
out to be same as Eq. (8) with $B_{i}(t)$ given in Eq. (9). With these
changes, it is clear that the system state is now given by

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime},l^{\prime};k,l}$ (25)
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{i\omega_{0}}{2}\left(k\hskip
1.42262ptZ^{(1)}(t)+l\hskip 1.42262ptZ^{(2)}(t)\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{-i\omega_{0}}{2}\left(k^{\prime}\hskip
1.42262ptZ^{(1)}(t)+l^{\prime}\hskip 1.42262ptZ^{(2)}(t)\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[iB(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-i\lambda_{0}h(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-i\Gamma(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{Tr}_{B}\left[e^{iR_{kl,k^{\prime}l^{\prime}}}\rho_{B}\right]$

Here, $B(t)=B_{3}(t)+B_{4}(t)$ and
$R_{kl,k^{\prime}l^{\prime}}=(k-k^{\prime})\chi_{1}(t)+(l-l^{\prime})\chi_{2}(t)$
as defined previously, and
$\text{Tr}_{B}[e^{iR_{kl,k^{\prime}l^{\prime}}}\rho_{B}]$ is evaluated in Eq.
(14) and Eq. (15).

We now clearly observe the effect of the direct interaction in the terms
$\text{exp}\left[-i\lambda_{0}h(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
and $\left[-i\Gamma(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$. The former
describes the effect of the constant interaction $\lambda_{0}$, while the
latter is the effect of the noise in the interaction. Let us focus on the
latter. Noticing the form of $\Gamma(t)$, it contains an integral over the
stochastic function $\xi(t)$. To proceed, we compute
$\Bigl{\langle}\text{exp}\left[-i\Gamma(t)\left(k^{\prime}l^{\prime}-kl\right)\right]\Bigr{\rangle}$,
where $\langle...\rangle$ corresponds to average over all possible noise
realizations. We assume that the noise $\xi(t)$ is Gaussian, and that the
noise is stationary. We then find that

$\displaystyle\biggl{\langle}e^{-i\Gamma(t)\left(k^{\prime}l^{\prime}-kl\right)}\biggr{\rangle}=e^{-\frac{1}{2}\left(k^{\prime}l^{\prime}-kl\right)^{2}\bigl{\langle}\Gamma(t)^{2}\bigl{\rangle}},$

with

$\displaystyle\biggl{\langle}\Gamma(t)^{2}\biggl{\rangle}\hskip 2.84526pt=$
$\displaystyle\hskip
2.84526pt\biggl{\langle}\int_{0}^{t}dt_{1}\xi(t_{1})s^{(1)}(t_{1})s^{(2)}(t_{1})$
$\displaystyle\times\int_{0}^{t}dt_{2}\xi(t_{2})s^{(1)}(t_{2})s^{(2)}(t_{2})\biggl{\rangle}$
$\displaystyle=$ $\displaystyle\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}\hskip
2.84526pts^{(1)}(t_{1})s^{(2)}(t_{1})s^{(1)}(t_{2})s^{(2)}(t_{2})$
$\displaystyle\times\biggl{\langle}\xi(t_{1})\xi(t_{2})\biggl{\rangle}$

Introducing the change of variables $\tau=t_{1}-t_{2}$,
$\bigl{\langle}\xi(t_{1})\xi(t_{2})\bigl{\rangle}=\bigl{\langle}\xi(\tau)\xi(0)\bigl{\rangle}$.
We further write this autocorrelation function as the Fourier transform over
the spectral density $S(\omega)$ of the noise, that is,
$\bigl{\langle}\xi(\tau)\xi(0)\bigl{\rangle}=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\hskip
2.84526pte^{-i\omega\tau}S(\omega)$. For simplicity, we introduce the combined
switching function $f_{c}(t)=s^{(1)}(t)s^{(2)}(t)$. This allows us to write
the Fourier transform of $\bigl{\langle}\Gamma(t)^{2}\bigl{\rangle}$ as

$\displaystyle\biggl{\langle}\Gamma(t)^{2}\biggl{\rangle}$
$\displaystyle=\hskip
2.84526pt\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\left|\int_{0}^{t}dt_{1}f_{c}(t_{1})e^{-i\omega
t_{1}}\right|^{2}S(\omega)$ (26) $\displaystyle=\hskip
2.84526pt\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega\frac{F_{c}(\omega
t)}{\omega^{2}}S(\omega)$ $\displaystyle=\hskip 2.84526pt\mu(t).$

Here $F_{c}(\omega t)$ is the filter function corresponding to $f_{c}(t)$.

The key point is that the presence of the noise in the interaction leads to
the decay of the off-diagonal terms of the system density matrix. In other
words, the noise in the interaction acts as an additional source of
decoherence on top of the effect of $\gamma(t)$. Therefore, our goal now would
be to eliminate the effect of both $\gamma(t)$ and $\mu(t)$, while preserving
the effect of $D(t)$ and $\lambda_{0}$. We begin by noticing the form of
$f_{c}(t)$. This is the product of the switching functions of the pulses
applied to both qubits. If the same pulse sequence is used for both qubits,
$f_{c}(t)=1$ regardless of the form of the pulse sequence because
$s(t)^{2}=1$. This means that $\mu(t)$ would be unchanged upon the application
of the pulses. Therefore, we investigate displacing the switching function of
the pulse sequence on the second qubit with respect to the first qubit, that
is, we consider $s^{(2)}(t)=s^{(1)}(t+\alpha)$, where $\alpha$ encapsulates
the time shift between the pulse sequences applied to the two qubits. An
important point to notice here is that when $\alpha=\Delta/2$ for switching
functions with constant pulse spacing $\Delta$,
$f_{c}(t)=s^{(1)}(t)s^{(1)}(t+\Delta/2)$ actually corresponds to a new
switching function with double the number of pulses $n$. That is, if
$s^{(1)}(t)$ is a PDD sequence with $n$ pulses, $f_{c}(t)$ would be a PDD
sequence with $2n$ pulses when $\alpha=\Delta/2$. This would make the
computation of $F_{c}(\omega t)$ much simpler in this case. However, for pulse
sequences with changing pulse spacing such as the UDD pulse sequence, we would
have to compute the filter function for $f_{c}(t)$ for arbitrary values of
$\alpha$ by Eq. (16).

We now note that, with $s^{(2)}(t)=s^{(1)}(t+\alpha)$, $\phi_{r}^{(2)}(t)$ is
simplified to (see the appendix for details)

$\phi_{r}^{(2)}(t)=e^{-i\omega_{r}\alpha}\phi_{r}^{(1)}(t).$ (27)

This greatly simplifies Eq. (15), which yields

$\displaystyle\text{Tr}_{B}$
$\displaystyle\left[\left(R_{kl,k^{\prime}l^{\prime}}\right)^{2}\rho_{B}\right]=\frac{1}{2}(k-k^{\prime})^{2}\gamma(t)+\frac{1}{2}(l-l^{\prime})^{2}\gamma(t)$
(28) $\displaystyle+$
$\displaystyle(k-k^{\prime})(l-l^{\prime})\Bigl{[}2p(t)+q(t)+ir(t)\Bigl{]},$

where

$\displaystyle\gamma(t)$ $\displaystyle=\int_{0}^{\infty}d\omega\hskip
2.84526ptJ(\omega)\frac{F(\omega t)}{\omega^{2}}\hskip
2.84526pt\text{coth}\left(\frac{\beta\omega}{2}\right),$ $\displaystyle p(t)$
$\displaystyle=\int_{0}^{\infty}d\omega\hskip 2.84526ptJ(\omega)\frac{F(\omega
t)}{\omega^{2}}\hskip
2.84526pt\text{cos}\left(\omega\alpha\right)\frac{1}{e^{\beta\omega}-1},$
$\displaystyle q(t)$ $\displaystyle=\int_{0}^{\infty}d\omega\hskip
2.84526ptJ(\omega)\frac{F(\omega t)}{\omega^{2}}\hskip
2.84526pt\text{cos}\left(\omega\alpha\right),$ $\displaystyle r(t)$
$\displaystyle=\int_{0}^{\infty}d\omega\hskip 2.84526ptJ(\omega)\frac{F(\omega
t)}{\omega^{2}}\hskip 2.84526pt\text{sin}\left(\omega\alpha\right).$

Here we have lifted the superscripts because everything is expressed in terms
of $F^{(1)}(\omega t)$ using Eq. (27). This results in the density matrix
$\rho_{S}(t)$ attaining the form

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime},l^{\prime};k,l}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime},l^{\prime};k,l}$ (29)
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[\frac{-i\omega_{0}Z(t)}{2}\left(k^{\prime}+l^{\prime}-k-l\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[iB(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-i\lambda_{0}h(t)\left(k^{\prime}l^{\prime}-kl\right)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{1}{2}\left(k^{\prime}l^{\prime}-kl\right)^{2}\mu(t)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{1}{4}(k^{\prime}-k)^{2}\gamma(t)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{1}{4}(l^{\prime}-l)^{2}\gamma(t)\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{1}{2}(k^{\prime}-k)(l^{\prime}-l)\left[2p(t)+q(t)\right]\right]$
$\displaystyle\hskip
11.38109pt\times\text{exp}\left[-\frac{i}{2}(k^{\prime}-k)(l^{\prime}-l)r(t)\right]$

To numerically study the dynamics of the two-qubit system, we need to specify
the spectral density of the noise, which we denote by $S(\omega)$. For
$1/f^{a}$ noise [52], this is given by

$S_{f}(\omega)=\frac{A_{0}^{1+a}}{\omega^{a}},$ (30)

where we consider $0.5\leq a\leq 1.5$. As before, we choose the initial system
state to be $\rho_{S}(t)=\ket{+,+}\bra{+,+}$. The system density matrix at
time $t$ in matrix form is then given by

$\rho_{S}(t)=\frac{1}{4}\begin{pmatrix}1&e^{-i\omega_{0}Z(t)+iM(t)}e^{-\gamma(t)-2\mu(t)}&e^{-i\omega_{0}t+iM(t)}e^{-\gamma(t)-2\mu(t)}&e^{-2i\omega_{0}Z(t)-2ir(t)}e^{-2\gamma(t)-2R(t)}\\\
e^{i\omega_{0}Z(t)-iM(t)}e^{-\gamma(t)-2\mu(t)}&1&e^{-2\gamma(t)+2R(t)}e^{2ir(t)}&e^{-i\omega_{0}Z(t)-iM(t)}e^{-\gamma(t)-2\mu(t)}\\\
e^{i\omega_{0}Z(t)-iM(t)}e^{-\gamma(t)-2\mu(t)}&e^{-2\gamma(t)+2R(t)}e^{2ir(t)}&1&e^{-i\omega_{0}Z(t)-iM(t)}e^{-\gamma(t)-2\mu(t)}\\\
e^{2i\omega_{0}Z(t)-2ir(t)}e^{-2\gamma(t)-2R(t)}&e^{i\omega_{0}Z(t)+iM(t)}e^{-\gamma(t)-2\mu(t)}&e^{i\omega_{0}Z(t)+iM(t)}e^{-\gamma(t)-2\mu(t)}&1\end{pmatrix}.$
(31)

where $M(t)=2B(t)-2\lambda_{0}h(t)$ , and $R(t)=2p(t)+q(t)$.

Figure 3:  Concurrence $C(t)$ for the noise spectral density $S_{f}(\omega)$
with no pulses and in the presence of PDD pulses. The blue, solid line
represents the concurrence with no pulses. The black, dot-dashed line is the
concurrence with PDD pulses ($n=371$) with zero time delay ($\alpha=0$). The
red, dashed line represents the concurrence of PDD pulse sequence for $n=371$
pulses with non-zero $\alpha$. The parameter used are $\alpha=0.0139$,
$\omega_{ir}=2\pi$, $A_{0}=10$, $\lambda_{0}=0.1$, $\omega_{c}=20$, $\beta=1$,
and $g=0.1$.

We now numerically investigate the generation of entanglement. In Fig. 3, we
present the entanglement dynamics of the two qubits with and without PDD
pulses. It is clear from the blue, solid curve that in the absence of any
applied pulses, the decoherence factors from both the environment and the
qubit-qubit interaction noise do not allow the environment-induced
entanglement to be generated properly. Moreover, if there is no time-shift
between the pulse sequences (that is, $\alpha=0$), the system is able to
overcome the decoherence factor of the environment, but the decoherence factor
of the interaction noise $\mu(t)$ still leads to significant decoherence (see
the black, dot-dashed curve). Finally, if we displace the switching function
of the second qubit, that is, $s^{(2)}(t)=s^{(1)}(t+\alpha)$, $\mu(t)$ can
also be minimised and near perfect entanglement can be generated. In Fig. 4,
we illustrate that CPMG pulses and UDD pulses also generate significant
entanglement when there is a time delay between the pulses applied to the two
qubits.

Figure 4:  Concurrence $C(t)$ with the noise spectral density $S_{f}(\omega)$
for PDD (black, short-dashed curve), CPMG (red, long-dashed curve), and UDD
pulses (blue, solid curve). The number of pulses $n=64$ is kept constant for
all pulse sequences. The parameters used are $\alpha=0.0139$,
$\omega_{ir}=2\pi$, $A_{0}=10$, $\lambda_{0}=0.1$, $\omega_{c}=20$, $\beta=1$,
and $g=0.1$.

##  V Conclusion

We started with a two-qubit system interacting with a common bosonic
environment and developed the formalism to compute the dynamics of the two
qubits without any direct interaction. The results showed that for this model,
applying control pulses to the two qubits does enhance the environment-
generated entanglement between them. We then proceeded to add a qubit-qubit
interaction term in the model Hamiltonian along with noise in this
interaction. Due to this noise, the qubits were shown to undergo additional
decoherence. To mitigate the effect of this additional term, it was shown that
the same pulse sequence cannot be applied to both qubits, that is, there must
be a time shift. If such a time shift is allowed, we showed that near perfect
entanglement can again be generated. Our results should be useful in the
context of reservoir engineering to generate entanglement between two qubits.

##  Appendix A Simplifying $\phi_{r}^{(2)}(t)$

Under the assumption taken in the paper that $s^{(2)}(t)=s^{(1)}(t+\alpha)$,
$\phi_{r}^{(2)}(t)$ can be simplified as follows:

$\displaystyle\phi_{r}^{(2)}(t)$
$\displaystyle=\int_{0}^{t}dt^{\prime}s^{(2)}(t^{\prime})e^{i\omega_{r}t^{\prime}}$
$\displaystyle=\int_{0}^{t}dt^{\prime}s^{(1)}(t^{\prime}+\alpha)e^{i\omega_{r}t^{\prime}}$
$\displaystyle=e^{-i\omega_{r}\alpha}e^{i\omega_{r}\alpha}\int_{0}^{t}dt^{\prime}s^{(1)}(t^{\prime}+\alpha)e^{i\omega_{r}t^{\prime}}$
$\displaystyle=e^{-i\omega_{r}\alpha}\int_{0}^{t}dt^{\prime}s^{(1)}(t^{\prime}+\alpha)e^{i\omega_{r}(t^{\prime}+\alpha)}.$

Now performing a change of variables $t^{\prime\prime}=t^{\prime}+\alpha$, we
get

$\displaystyle\phi_{r}^{(2)}(t)=$ $\displaystyle
e^{-i\omega_{r}\alpha}\int_{\alpha}^{t+\alpha}dt^{\prime\prime}s^{(1)}(t^{\prime\prime})e^{i\omega_{r}t^{\prime\prime}}$
$\displaystyle=$ $\displaystyle
e^{-i\omega_{r}\alpha}\Biggr{[}\int_{0}^{t+\alpha}dt^{\prime\prime}s^{(1)}(t^{\prime\prime})e^{i\omega_{r}t^{\prime\prime}}$
$\displaystyle-\int_{0}^{\alpha}dt^{\prime\prime}s^{(1)}(t^{\prime\prime})e^{i\omega_{r}t^{\prime\prime}}\Biggr{]}$
$\displaystyle=$ $\displaystyle
e^{-i\omega_{r}\alpha}\left[\phi_{r}^{(1)}(t+\alpha)-\phi_{r}^{(1)}(\alpha)\right].$

It can be shown that

$\displaystyle F(\omega
t)=\frac{1}{2}\left|\sum_{k=0}^{n}(-1)^{k}\left(e^{i\omega t_{k+1}}-e^{i\omega
t_{k}}\right)\right|^{2}.$

Equating this with our definition of $\phi_{r}^{(1)}(t)$ in Eq. (7) and using
Eq. (16), we get

$\displaystyle
F^{(1)}(\omega_{r}t)=\frac{\omega_{r}^{2}}{2}\left|\phi_{r}^{(1)}(t)\right|^{2}.$

This implies

$\displaystyle\phi_{r}^{(1)}(t)=\frac{1}{\omega_{r}}\sum_{k=0}^{n}(-1)^{k}\left(e^{i\omega_{r}t_{k+1}}-e^{i\omega_{r}t_{k}}\right).$

This leads to

$\displaystyle\phi_{r}^{(1)}(t+\alpha)-\phi_{r}^{(1)}(\alpha)$
$\displaystyle=$ $\displaystyle\frac{1}{\omega_{r}}\sum_{k=0}^{n}(-1)^{k}$
$\displaystyle\left(e^{i\omega_{r}(t)_{k+1}}-e^{i\omega_{r}(t)_{k}}\right)$
$\displaystyle=\phi_{r}^{(1)}(t).$

Therefore, $\phi_{r}^{(2)}(t)$ is simplified as:

$\displaystyle\phi_{r}^{(2)}(t)=e^{-i\omega_{r}\alpha}\phi_{r}^{(1)}(t).$

## References

  * Nielsen [2000] M. Nielsen, A & chuang, il quantum computation and information (2000).
  * Bennett [1992] C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Physical review letters 68, 3121 (1992).
  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Physical review letters 70, 1895 (1993).
  * Schumacher and Westmoreland [1998] B. Schumacher and M. D. Westmoreland, Quantum privacy and quantum coherence, Physical Review Letters 80, 5695 (1998).
  * Ekert [1992] A. K. Ekert, Quantum cryptography and bell’s theorem, in  _Quantum Measurements in Optics_ (Springer, 1992) pp. 413–418.
  * Deutsch  _et al._ [1996] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy amplification and the security of quantum cryptography over noisy channels, Physical review letters 77, 2818 (1996).
  * Bennett and Wiesner [1992] C. H. Bennett and S. J. Wiesner, Communication via one-and two-particle operators on einstein-podolsky-rosen states, Physical review letters 69, 2881 (1992).
  * Zurek [2003] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of modern physics 75, 715 (2003).
  * Schlosshauer [2007] M. A. Schlosshauer,  _Decoherence: and the quantum-to-classical transition_ (Springer Science & Business Media, 2007).
  * Breuer  _et al._ [2002] H.-P. Breuer, F. Petruccione, _et al._ ,  _The theory of open quantum systems_ (Oxford University Press on Demand, 2002).
  * Zagoskin  _et al._ [2006] A. Zagoskin, S. Ashhab, J. Johansson, and F. Nori, Quantum two-level systems in josephson junctions as naturally formed qubits, Physical review letters 97, 077001 (2006).
  * Neeley  _et al._ [2008] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’connell, H. Wang, A. N. Cleland, and J. M. Martinis, Process tomography of quantum memory in a josephson-phase qubit coupled to a two-level state, Nature Physics 4, 523 (2008).
  * Murch  _et al._ [2012] K. Murch, U. Vool, D. Zhou, S. Weber, S. Girvin, and I. Siddiqi, Cavity-assisted quantum bath engineering, Physical review letters 109, 183602 (2012).
  * Cirac  _et al._ [1993] J. Cirac, A. Parkins, R. Blatt, and P. Zoller, “dark”squeezed states of the motion of a trapped ion, Physical review letters 70, 556 (1993).
  * Poyatos  _et al._ [1996] J. Poyatos, J. I. Cirac, and P. Zoller, Quantum reservoir engineering with laser cooled trapped ions, Physical review letters 77, 4728 (1996).
  * Carvalho  _et al._ [2001] A. Carvalho, P. Milman, R. de Matos Filho, and L. Davidovich, Decoherence, pointer engineering and quantum state protection, in  _Modern Challenges in Quantum Optics_ (Springer, 2001) pp. 65–79.
  * Shankar  _et al._ [2013] S. Shankar, M. Hatridge, Z. Leghtas, K. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Autonomously stabilized entanglement between two superconducting quantum bits, Nature 504, 419 (2013).
  * Krauter  _et al._ [2011] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, Entanglement generated by dissipation and steady state entanglement of two macroscopic objects, Physical review letters 107, 080503 (2011).
  * Barreiro  _et al._ [2011] J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, An open-system quantum simulator with trapped ions, Nature 470, 486 (2011).
  * Lin  _et al._ [2013] Y. Lin, J. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. Sørensen, D. Leibfried, and D. J. Wineland, Dissipative production of a maximally entangled steady state of two quantum bits, Nature 504, 415 (2013).
  * Orszag and Hernandez [2010] M. Orszag and M. Hernandez, Coherence and entanglement in a two-qubit system, Advances in Optics and Photonics 2, 229 (2010).
  * Braun [2002] D. Braun, Creation of entanglement by interaction with a common heat bath, Phys. Rev. Lett. 89, 277901 (2002).
  * Dajka and Łuczka [2008] J. Dajka and J. Łuczka, Origination and survival of qudit-qudit entanglement in open systems, Phys. Rev. A 77, 062303 (2008).
  * Romano and D’Alessandro [2006] R. Romano and D. D’Alessandro, Environment-mediated control of a quantum system, Phys. Rev. Lett. 97, 080402 (2006).
  * An and Zhang [2007] J.-H. An and W.-M. Zhang, Non-markovian entanglement dynamics of noisy continuous-variable quantum channels, Phys. Rev. A 76, 042127 (2007).
  * Paz and Roncaglia [2009] J. P. Paz and A. J. Roncaglia, Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment, Phys. Rev. A 79, 032102 (2009).
  * Contreras-Pulido and Aguado [2008] L. D. Contreras-Pulido and R. Aguado, Entanglement between charge qubits induced by a common dissipative environment, Phys. Rev. B 77, 155420 (2008).
  * Chaudhry  _et al._ [2015] A. Z. Chaudhry, J. Gong, _et al._ , Optimization of the environment for generating entanglement and spin squeezing, Journal of Physics B: Atomic, Molecular and Optical Physics 48, 115505 (2015).
  * Addis  _et al._ [2014] C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, Coherence trapping and information backflow in dephasing qubits, Phys. Rev. A 89, 024101 (2014).
  * Viola and Lloyd [1998] L. Viola and S. Lloyd, Dynamical suppression of decoherence in two-state quantum systems, Phys. Rev. A 58, 2733 (1998).
  * Viola  _et al._ [1999] L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett. 82, 2417 (1999).
  * Viola  _et al._ [2000] L. Viola, E. Knill, and S. Lloyd, Dynamical generation of noiseless quantum subsystems, Phys. Rev. Lett. 85, 3520 (2000).
  * Wu and Lidar [2002] L.-A. Wu and D. A. Lidar, Creating decoherence-free subspaces using strong and fast pulses, Phys. Rev. Lett. 88, 207902 (2002).
  * Yang  _et al._ [2011] W. Yang, Z.-Y. Wang, and R.-B. Liu, Preserving qubit coherence by dynamical decoupling, Frontiers of Physics in China 6, 2 (2011).
  * Stollsteimer and Mahler [2001] M. Stollsteimer and G. Mahler, Suppression of arbitrary internal coupling in a quantum register, Phys. Rev. A 64, 052301 (2001).
  * Viola and Knill [2003] L. Viola and E. Knill, Robust dynamical decoupling of quantum systems with bounded controls, Phys. Rev. Lett. 90, 037901 (2003).
  * Wocjan [2006] P. Wocjan, Efficient decoupling schemes with bounded controls based on eulerian orthogonal arrays, Phys. Rev. A 73, 062317 (2006).
  * Gordon  _et al._ [2008] G. Gordon, G. Kurizki, and D. A. Lidar, Optimal dynamical decoherence control of a qubit, Phys. Rev. Lett. 101, 010403 (2008).
  * Wu  _et al._ [2009] L.-A. Wu, G. Kurizki, and P. Brumer, Master equation and control of an open quantum system with leakage, Phys. Rev. Lett. 102, 080405 (2009).
  * Uhrig [2007a] G. S. Uhrig, Keeping a quantum bit alive by optimized $\pi$-pulse sequences, Phys. Rev. Lett. 98, 100504 (2007a).
  * Chaudhry and Gong [2012a] A. Z. Chaudhry and J. Gong, Decoherence control: Universal protection of two-qubit states and two-qubit gates using continuous driving fields, Phys. Rev. A 85, 012315 (2012a).
  * Chaudhry and Gong [2012b] A. Z. Chaudhry and J. Gong, Protecting and enhancing spin squeezing via continuous dynamical decoupling, Phys. Rev. A 86, 012311 (2012b).
  * Palma  _et al._ [1996] G. M. Palma, K.-a. Suominen, and A. Ekert, Quantum computers and dissipation, Proc. R. Soc. London A 452, 567 (1996).
  * Reina  _et al._ [2002] J. H. Reina, L. Quiroga, and N. F. Johnson, Decoherence of quantum registers, Phys. Rev. A 65, 032326 (2002).
  * Cywiński  _et al._ [2008] Ł. Cywiński, R. M. Lutchyn, C. P. Nave, and S. D. Sarma, How to enhance dephasing time in superconducting qubits, Physical Review B 77, 174509 (2008).
  * Weiss [2012] U. Weiss,  _Quantum dissipative systems_ (World Scientific, 2012).
  * Tan  _et al._ [2014] Q.-S. Tan, Y. Huang, L.-M. Kuang, and X. Wang, Dephasing-assisted parameter estimation in the presence of dynamical decoupling, Phys. Rev. A 89, 063604 (2014).
  * Wootters [1998] W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Physical Review Letters 80, 2245 (1998).
  * Carr and Purcell [1954] H. Y. Carr and E. M. Purcell, Effects of diffusion on free precession in nuclear magnetic resonance experiments, Physical review 94, 630 (1954).
  * Meiboom and Gill [1958] S. Meiboom and D. Gill, Modified spin-echo method for measuring nuclear relaxation times, Review of scientific instruments 29, 688 (1958).
  * Uhrig [2007b] G. S. Uhrig, Keeping a quantum bit alive by optimized $\pi$-pulse sequences, Physical Review Letters 98, 100504 (2007b).
  * Paladino  _et al._ [2014] E. Paladino, Y. Galperin, G. Falci, and B. Altshuler, 1/f noise: Implications for solid-state quantum information, Reviews of Modern Physics 86, 361 (2014).