\section{Motivation}

Backaction in cavity optomechanics has shown to lead to novel physical phenomena including laser cooling, self-oscillation, and non-linear dynamics in systems that go from kilometer size interferometers to single or few trapped ions.~\cite{ReviewCOM} Briefly, a resonant photon field in a cavity exerts a force and induces a mechanical motion on the mirrors, which in turn leads to a delayed modification of the resonant condition of the trapped field. Such coupled dynamics can be exploited for a large variety of applications that span for example from gravitational wave detection~\cite{Ligo} to the study of quantum motion states in mesoscopic mechanical systems.~\cite{O'Connell,Teufel,Chan,Verhagen} How light exerts force on matter is at the center of this investigations. Photons can apply stress through radiation pressure, transferring impulse when reflected from the mirrors.~\cite{Cohadon}  Related mechanisms also deriving from the same fundamental interaction (Lorentz forces) are gradient forces (also exploited in optical tweezers)~\cite{Lin}, and electrostriction. The latter, linked to the material´s photoelasticity, has been shown to play a role that can be of the same magnitude as radiation pressure,~\cite{Rakich1,Rakich2} or even larger if optical resonances are exploited in direct bandgap materials as for example GaAs.~\cite{FainsteinPRL2013,Rozas_Polariton,Baker}

In the presence of radiation pressure forces, the energy $E$ of the photon is shifted by the Doppler effect by an amount of the order $(v/c)E$, where $v$ and $c$ are the mirror and light velocities, respectively. The mechanical energy transferred from the photon to the mirror is thus very small. Electrostriction leads to Raman-like processes, for which the transferred energy $\Delta E_R$ corresponds to the involved vibrational frequency. This leads to inelastic scattering side-bands. Again, tipically $\Delta E_R << E$. Contrastingly, if the photon is absorbed in the process of interaction, all its energy is transferred to the mirror. This fundamental difference has been used in cavity optomechanics to demonstrate strongly enhanced light-matter interactions based on photothermal forces.~\cite{MetzgerPRL2008,MetzgerPRB2008,Restrepo} In materials displaying optical resonances, the photons can be resonantly absorbed with the consequent transfer of electrons to excited states. Photoexcited carrier-mediated optomechanical interactions have been reported in semiconductor modulation-doped heterostructure-cantilever cavity-less hybrid systems. Efficient cavity-less optomechanical transduction involving opto-piezoelectric backaction from the bound photoexcited electron-hole pairs have been demonstrated in these systems, including self-feedback cooling, amplification of the thermomechanical motion, and control of the mechanical quality factor through carrier excitation.~\cite{17,18,19} The change of electronic landscape produced by photoexcited carriers also induces a stress in the structure through electron-phonon coupling mediated by deformation potential interaction. This stress can be identified as an optoelectronic force, and should have the same kind of temporal behavior (with different time-scales and details depending on the carrier dynamics) and amplified strength as observed for photothermal forces.~\cite{MetzgerPRL2008,MetzgerPRB2008,Restrepo} Recently optical cooling of mechanical modes of a GaAs nanomembrane forming part of an optical cavity was reported ~\cite{Usami} and its relation to optoelectronic stress via the deformation potential was analysed. Because of the very fast relaxation rate of excited carriers due to surface recombination in such nanometer size structures, it was concluded that thermal (and not optoelectronic) stress was the primary cause of cooling in that case. We will demonstrate here that the carrier dynamics can be fundamentally modified when the nanometer size GaAs layer is embedded in a monolithic microcavity, making optoelectronic forces the  main mechanism of interaction of light with vibrations in such semiconductor devices. The diffusion of the photoexcited carriers thus assumes a central role, a dynamics that can be engineered using embedded quantum wells (QWs).

Because of their optoelectronic properties, semiconductor GaAs/AlAs microcavities are interesting candidates for novel functionalities in cavity optomechanics. Perfect photon-phonon overlap, and access to electronically resonant coupling in addition to radiation pressure could lead to strong optomechanical interactions of photoelastic origin.~\cite{FainsteinPRL2013,Baker,JusserandPRL2015} The vibrational frequencies in these microresonators are determined by the vertical layering of the device (fabricated with the ultra-high quality of molecular beam epitaxy), and not by the lateral pattering (defined by the more limited performance of microfabrication techniques). This has allowed access to much higher frequencies for the optomechanical vibrational modes, in the 20-100~GHz range, without significant reduction of the mechanical Q-factors.~\cite{AnguianoPRL} In addition, these optomechanical resonators enable the conception of hybrid architectures involving artificial atoms (semiconductor excitons) coupled to the optical cavity mode, and thus combining the physics of cavity optomechanics with cavity quantum electrodynamics.~\cite{Rozas_Polariton,Restrepo2,Kyriienko}  Our motivation here is to search for evidence of optoelectronic forces in these devices, and for that purpose we study the light-sound coupling involving the resonator mechanical modes and with the optical cavity at resonance with the material exciton transition energy. We demonstrate that the main phonon generation mechanism using pulsed lasers close to resonance in these devices involves indeed the real excitation of carriers and the deformation potential mechanism.~\cite{Baker,RuelloUltrasonics} We show that in microcavities with ``bulk'' GaAs spacers (i.e. cavity spacers constituted by a thick $\lambda/2$ layer of GaAs) ultrafast carrier redistribution leads to an enhanced coupling to the more uniformly distributed fundamental 20~GHz cavity vibrational mode. An engineering of the structure taking into account this effect and using embedded quantum wells is used to limit the carrier diffusion and enhance the optomechanical coupling.  This opens the way to new opportunities to the field of optomechanics.

\section{Results}

We consider two {\em planar} microcavity structures, specifically a ``bulk'' GaAs and a multiple quantum well (MQW) resonator. The ``bulk'' GaAs microcavity is made of a $\lambda/2$ GaAs-spacer enclosed by ($\lambda/4,\lambda/4)$ Al$_{0.18}$Ga$_{0.82}$As /AlAs  DBRs, 20 pairs of layers on the bottom, 18 on top, grown on a GaAs substrate (a scheme of the structure is  presented in Fig.~\ref{Fig1}(b)).~\cite{Tredicucci,AFainsteinBulkGaAs}  The DBRs stop band is centered at the design wavelength of $817~\mathrm{nm}$.
As we have demonstrated previously, this structure performs as an optomechanical resonator that simultaneously confines photons and acoustic phonons of the same wavelength.~\cite{FainsteinPRL2013,Trigo,Anguiano,PSesin,AnguianoPRL} In the MQW microcavity the $\lambda/2$ spacer is constituted by six $14.5~\mathrm{nm}$ GaAs QWs separated by $6.1~\mathrm{nm}$ AlAs barriers. To further enhance the light-sound coupling the second and fourth $\lambda/2$ DBR alloy layers on each side are also replaced by three GaAs/AlAs QWs. The reason for this design will become clear below. The DBRs in this case are ($\lambda/4,\lambda/4)$ Al$_{0.10}$Ga$_{0.90}$As /AlAs  multilayers, 27 pairs on the bottom, 23 on top, grown again on a GaAs substrate. A scheme of this structure is displayed in Fig.~\ref{Fig1}(c). The number of DBR periods in both structures is designed to assure an optical Q-factor $Q \geq 10^4$ (cavity photon lifetime $\tau \sim 5$~ps).

\begin{figure}[!t]
    \begin{center}
    \includegraphics[trim = 0mm 0mm 0mm 20mm,clip,scale=0.45,angle=0]{Fig1.pdf}
    \end{center}
    \vspace{-0.8 cm}
\caption{(Color online) (a) Time resolved reflectance difference experimental set-up. PBS stands for polarizing beam splitter. (b) and (c) are schems of the GaAs ``bulk'' and ``MQW'' microcavities, respectively.\label{Fig1}}
\end{figure}
Reflection-type degenerate pump-probe experiments were performed with the laser wavelength tuned with the optical cavity mode (see the scheme in Fig.~\ref{Fig1}(a)).~\cite{Thomsen,Bartels}
Picosecond pulses ($\sim 1$~ps) from a mode-locked Ti:Sapphire laser, with repetition rate 80~MHz, were split into cross polarized pump (power 20~mW) and probe (1~mW) pulses.  Both pulses were focused onto superimposed $\sim 50$ $\mu$m-diameter spots. To couple the light to the microcavity the probe beam propagates close to  the sample normal and is tuned to the high derivative flank of the reflectivity dip, while the pump incidence angle is set for resonant condition precisely at the cavity mode.~\cite{Kimura_coherentcavity,KimuraTheory,Kimura_doubleresonance} The laser wavelength was also set so that the phonon generation and detection would be a few meV below resonance with the direct bandgap of the GaAs ``bulk'' cavity spacer ($E_\textrm{\tiny GaAs} \sim 1.425$~eV) or of the QWs ($E_\textrm{\tiny QW} \sim 1.526$~eV). To accomplish this resonant excitation the temperature was also used as a tuning parameter; the ``bulk'' cavity experiments were done at room temperature, while the MQW structure was studied at 80~K.  Light is thus coupled resonantly with the optical cavity mode and the semiconductor excitonic resonance. Acoustic phonons confined in the same space as the optical cavity mode are selectively generated within the resonator, and are detected through their modulation of the optical cavity mode frequency.\cite{FainsteinPRL2013}

These pump-probe ultrafast laser experiments are conceptually similar to the ring-down techniques recently exploited in the cavity optomechanics domain,~\cite{Usami} but more appropriate to the study of ultra-high frequency vibrations (GHz-THz range.) The pulsed laser phonon generation efficiency can be described as:~\cite{Flor2012}
\begin{equation}
g(\omega)\propto \int{\kappa(z)\eta_0(\omega, z)|F_p(z)|^2}dz.
\label{eq2}
\end{equation}
Here $\omega$ is the phonon frequency, $\eta_0$ describes the elastic strain eigenstates, $F_p(z)$ is the spatially dependent pump laser induced perturbation and $\kappa$ is an effective material-dependent generation parameter that considers different light-matter couplings. All parameters are implicitly assumed to depend on the laser wavelength. We will be interested here in the relative intensity of the vibrational modes, not in their absolute values so that the main physical ingredients are expressed in the functional form of Eq.~\eqref{eq2}. Equation~\eqref{eq2} reflects the spatial overlap of the strain eigenstates with the light-induced stress. The latter can be written (independently of the mechanism involved) as $\sigma_p(z,t)=\kappa(z)|F_p(z)|^2T(t)$.~\cite{Flor2012} Here $T(t)$ is the function describing the temporal evolution of the light-induced perturbation. Typically it is a delta-like function for radiation pressure and electrostriction forces, and a step-like function for the photothermal and optoelectronic mechanisms, broadened by the time-delay of the mechanism involved. The spatial distribution of the optical excitation $F_p(z)$ corresponds to the cavity confined electric field $E_c(z)$ for radiation pressure and electrostriction forces, but can be different from it for the other two mechanisms depending on the spatial distribution and dynamics of excited charges and laser-induced temperature variations.
As we argue next, this will be a way to identify the main optical force under play in the studied devices.

\begin{figure*}[!t]
    \begin{center}
    \includegraphics[trim = 0mm 0mm 0mm 10mm,clip,scale=0.6,angle=0]{Fig2-3.pdf}
    \end{center}
    \vspace{-0.8 cm}
\caption{(Color online) (a) and (c): Measured and calculated acoustic phonon spectra for excitation resonantly tuned to the optical cavity mode for the GaAs and QW cavities, respectively. The calculations consider a resonant generation only at the GaAs layers, either with a spatial pattern reproducing the squared modulus of the pump field (unrelaxed), or a flat generation that assumes a rapid distribution of carriers within the full width of the corresponding GaAs layer (relaxed). Excitation pattern for the two considered situations (unrelaxed or relaxed), and strain distribution associated to the first three confined acoustic modes corresponding to the GaAs (b) and MQW (d) cavity structures. The position $z=0$ marks the center of the cavity spacer. \label{Fig2}}
\end{figure*}

Figures~\ref{Fig2} a-b present the case of the ``bulk'' GaAs cavity. Panel (a) in the figure displays the experimental spectrum, compared with calculations assuming that $F_p(z)$ reproduces the instantaneous spatial distribution of the light intensity  ($|E_c(z)|^2$), or that it corresponds to a flat photoexcited carrier distribution within the GaAs-spacer (``relaxed''). That is, it assumes that within the time in which the pump laser-induced perturbation is effective (of the order of the half period of the vibrational frequencies involved), the photoexcited carrier spatial distribution relaxes extending their presence throughout the full width of the GaAs cavity spacer material.  Three peaks are clearly visible at $\sim 20$, $\sim 60$ and $\sim 100$~GHz, corresponding to the fundamental, third and fifth overtones of the z-polarized cavity confined breathing mechanical modes.~\cite{FainsteinPRL2013,AnguianoPRL} The associated spatial distribution of the strain fields $\eta_0(\omega,z)$, together with that of the light induced optoelectronic stress, are shown in panel (b) of the figure. $\kappa(z)$ is assumed to be non-zero {\em only} in GaAs. It reflects the physical ingredient that photoelastic (electrostrictive) coupling is resonantly enhanced in GaAs, and that photons are only absorbed in GaAs, all other materials being fully transparent at the involved wavelengths. The solid curves in $\sigma_\textrm{\tiny stress}$ thus represent the region where the light-induced stress is finite, either reflecting the excitation field (yellow), or the relaxed situation (red). It is clear in the experiment that the vibrational mode's amplitude decreases systematically with increasing frequency of the mode, something that according to the calculations is only compatible with the carriers having rapidly spread filling the full width of the cavity spacer. The explanation is straightforward considering the overlap integral given by Eq.~\eqref{eq2}, and the involved spatial distributions depicted in panel(b) Fig.~\ref{Fig2}. And it clearly excludes radiation pressure and electrostriction as the possible driving mechanisms.

Based on the above discussion it is also clear from the top curve in Fig.~\ref{Fig2}(a) that the relative weight of the higher frequency 60~GHz mode could be enhanced if the spatial distribution of the optical perturbation $F_p(z)$ could be forced to map-out the cavity field intensity $|E_c(z)|^2$. If, as argued, the main generation mechanism is indeed governed by optoelectronic forces, one could accomplished this task by artificially limiting the vertical diffusion of the photoexcited carriers. A natural way to do this is through an adequate  engineering of the cavity spacer, e.g. by introducing GaAs/AlAs MQWs.  This case is shown in Figs.~\ref{Fig2}(c-d). Again panel (c) shows the experiment and calculation of the coherent phonon spectral intensity. Panel (d) displays the spatial distribution of the photoexcited stress, and that of the strain related to the involved vibrational modes. In this case relaxing the carrier distribution to fill the full width of the QWs, or maintaining the exact laser excitation pattern, makes no observable difference in the calculated spectra. Interestingly, by tailoring the spatial distribution of the photoexcited carriers using quantum wells we are able to confirm the role of optoelectronic stress as the main optomechanical coupling, and furthermore we have pushed the main vibrational frequency of these optomechanical resonators from the fundamental mode at 20~GHz to the third overtone at 60~GHz. These frequencies are one order of magnitude larger than the record frequencies demonstrated in other cavity optomechanics approaches.~\cite{ReviewCOM} From Fig.~\ref{Fig2}(d) it also becomes clear why additional QWs were introduced in the design at the second and fourth DBR periods, and {\em not} at the first and third: because of a change of sign of the strain fields, the latter contribute to the overlap integral in Eq.~\eqref{eq2} with the sign reversed respect to the cavity spacer.

Having demonstrated the central role of optoelectronic forces in the generation of GHz phonons in semiconductor microcavities upon resonant pulsed excitation,  we address next the potentiality of this mechanism for the observation of laser cooling and optically induced self-oscillation. The fundamental concepts describing backaction dynamics in cavity optomechanics are grasped by the delayed force model which, for the optically modified vibrational damping rate $\Gamma_{\mathrm{eff}}$ gives~\cite{MetzgerPRB2008}
\begin{equation}
\Gamma_\textrm{\tiny eff}=\Gamma_\textrm{\tiny M}\left( 1+Q_\textrm{\tiny M}\frac{\Omega_\textrm{\tiny M} \tau}{1+\Omega^2_\textrm{\tiny M} \tau^2}\frac{\nabla{F}}{K} \right),
\label{eq1}
\end{equation}
with $Q_\textrm{\tiny \tiny M}$ and $\Gamma_\textrm{\tiny M}=\Omega_\textrm{\tiny M}/Q_\textrm{\tiny M}$  the unperturbed mechanical quality factor and damping, respectively. $\Omega_\textrm{\tiny M}$ and $K$ are respectively the unperturbed mechanical mode frequency
and stiffness, while $\nabla{F} = \partial F/\partial u|_{u_0}$ represents the change in the steady-state optical force for a small displacement $\delta u$ of the mechanical resonator around the equilibrium position. To lead to backaction optical forces need to respond with a delay to fluctuations that change the frequency of the optical mode. The simplest delay function to consider is an exponential function $h(t) = 1 - \textrm{exp} (t/\tau)$, with $\tau$ the corresponding time-delay. Typically $\tau$ would correspond to the cavity photon lifetime $\tau_\textrm{\tiny cav}$, but in general and particularly for photothermal and optoelectronic forces, it can be significantly longer than $\tau_\textrm{\tiny cav}$. $\Gamma_\textrm{\tiny eff}$ in the presence of dynamical backaction can thus increase if $\nabla{F} > 0$, leading to laser cooling, or decrease and eventually attain zero (self-oscillation) if $\nabla{F} < 0$. The magnitude of this effect is proportional to the gradient of the optical force, which is a function of the deposited optical energy and the involved optomechanical coupling mechanism (that is, how this deposited energy is translated into a mechanical deformation). It is also proportional to a delay tuning factor $f_D = \frac{\Omega_\textrm{\tiny M} \tau}{1+\Omega^2_\textrm{\tiny M} \tau^2}$. $f_D$  has a maximum for $\Omega_\textrm{\tiny M} \tau = 1$, which reflects the intuitive fact that the force fluctuations are more effective to induce vibrations if their time-constant is neither too short, nor too long, but tuned to the vibrational frequency so that $\tau \approx 1/\Omega_\textrm{\tiny M}$.

\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Mechanism & Opt. force (N) & $\textrm{ f}_{D} =\frac{\Omega_m \tau}{1+(\Omega_m \tau)^2}$  \\
\hline
Rad. Pressure &  $3.4 \times 10^{-14}$ & $\sim 0.44$ \\
\hline
Electrostriction &  $3 \times 10^{-14}-10^{-12}$ & $\sim0.44$ \\
\hline
Photothermal &  $5.4 \times 10^{-12}$  & $\sim 10^{-5}$ \\
\hline
Optoelectronic &  $6.3 \times 10^{-11}$ & $\sim 0.04$ \\
 \hline
\end{tabular}
\caption{\label{table} Optical force per trapped (or absorbed) photon, and the value of the delay tuning factor $f_D$ for the different optical forces present in the GaAs and MQW microcavities. The forces have been evaluated using finite-element methods and considering a cylindrical pillar of 2 microns diameter. For the case of electrostriction two values are given, corresponding to out of resonance and in resonance values of the photoelastic constant.~\cite{JusserandPRL2015}}
\end{table}
Table~\ref{table} presents the different factors intervening in Eq.~\eqref{eq1} for the studied DBR microcavities and considered mechanisms, namely, the corresponding magnitude of the involved optical forces and delay tuning factor $f_D$. The optical forces are given per photon (trapped in the cavity or absorbed depending on the mechanism). They have been evaluated using finite-element methods (see Appendix \ref{sec: appendix}),~\cite{Baker,MetzgerPRB2008} and considering a micro-pillar with 20 period DBRs of 2 micrometer diameter. As argued above, the main difference between photothermal and optoelectronic forces respect to radiation pressure and electrostriction, is the larger amount of deposited energy per trapped cavity photon. Similarly to what is observed in GaAs microdisks,~\cite{Baker} well below the gap (far from the excitonic resonances) both geometric (radiation-pressure) and photoelastic contributions to the optomechanical coupling factor have similar values.  Ultra-strong resonant enhancement of the photoelastic coupling has been experimentally demonstrated in bare GaAs/AlAs MQWs.~\cite{JusserandPRL2015} In Table~\ref{table} we provide the magnitude of the electrostrictive force per incident photon considering the two situations, far from resonance and at resonance as used in our experiments, assuming that the room temperature values of the photoelastic constant given in Ref.\,\onlinecite{JusserandPRL2015} are valid for a similar MQW embedded in a pillar microcavity.   Concerning the photothermal coupling, a quantitative evaluation of its relevance in semiconductor GaAs membranes and microdisks indicates that it can be significant.~\cite{Baker,Usami}  It is also significant in our microcavities, as evidenced in Table~\ref{table}.  What excludes it as a potentially relevant optomechanical force is its slow dynamics, which for the high frequency vibrations considered here leads to  delay tuning factors $f_D \sim 10^{-5}$ even for the fundamental breathing mode at 20~GHz and considering a relatively fast thermal relaxation $\tau$ of the order of a $\mu$s. The delay tuning factor is close to its maximum value 0.5 for the impulsive radiation pressure and electrostrictive mechanisms, considering a cavity photon lifetime of a few picoseconds as in our vertical microcavities. It is also reasonably well tuned for the optoelectronic forces and 200~ps recombination times as demonstrated for pillars of a few microns lateral size in Ref.\,\onlinecite{AnguianoPRL}. In fact, $f_D \sim 0.44$ is precisely the same value as attained for optimized cantilever resonators that have evidenced strong optical cooling and self-oscillation dynamics based on photothermal forces.~\cite{MetzgerPRL2008,MetzgerPRB2008,Restrepo} The factor $1/10$ decrease of $f_D$ with respect to radiation pressure and electrostriction is overly compensated by the much larger efficiency of optoelectronic forces. Note that the physical mechanism at the base of optoelectronic forces is the same as in photoelastic coupling, namely, deformation potential interaction. The qualitative difference is that a photon is scattered in the latter, while it is  absorbed with the subsequent creation of real electron-hole pairs in the former.

\section{Conclusions and Outlook}

In conclusion, we have shown that optoelectronic deformation potential interactions are at the origin of the optical forces acting for excitation close to the semiconductor gaps in GaAs/AlAs microcavities. The carriers dynamics following photoexcitation is determinant of the emitted coherent acoustic phonon spectrum, and can be tailored using quantum wells to push the vibrational optomechanic frequencies from 20 up to 60~GHz, an order of magnitude larger than the highest standards in cavity optomechanics.  The strong potentiality of optoelectronic forces for the demonstration of backaction dynamical effects in semiconductor microcavities was addressed. This could open the way to ultra-high frequency cavity optomechanics, and through it to quantum measurements and applications at higher temperatures than currently accessible.

\section*{Funding Information}
This work was partially supported by the ANPCyT Grants PICT 2012-1661 and 2013-2047, the Labex NanoSaclay,  and the international franco-argentinean laboratory LIFAN (CNRS-CONICET).


\section{Introduction}
\label{intro}

Entangled states are linear superpositions of quantum states that cannot be separated into product states of the individual systems. This property is long recognized as being a fully quantum-mechanical effect. Throughout the years entanglement have played a crucial role in many researches on the nature of quantum measurements, especially in the developments of quantum communications~\cite{1,haroche}.

Current investigations on atomic radiative processes have proved to be of paramount significance in quantum optics. Recent examples are connected with studies of radiative processes involving entanglement. For instance, in Ref.~\cite{yang} the authors investigate two entangled atoms coupled with an electromagnetic field in unbounded space. On the other hand, in Ref.~\cite{eberly} it was shown how vacuum fluctuations promote the entanglement decay of two qubits in a finite time. Ref.~\cite{yun} establishes a scheme to construct a highly controlled atom-field coupling in a cavity quantum electrodynamics system. Nonlinear optical processes in chains of ions in an entangled state were investigated in Ref.~\cite{aga}. More recently, in a series of papers the radiative processes of entangled atoms were investigated in different physical settings~\cite{pra15,pra16,jhep,prd}. We observe that radiative processes of maximally entangled states should be distinguished from the non-entangled states~\cite{rep}.

All realizations of entangled atoms must confront with the decoherence phenomenon that is present in open systems. In other words, every operations of entanglement must furnish a strong coherent coupling between the atoms in order to minimize as much as possible the effects of decoherence coming from the interaction with the environment. Complications related with the endeavors of isolating the atoms from the environment emerge as the prime barrier preventing practical applications of entangled states.

In this work we demonstrate a possible way to protect entanglement from a particular kind of decoherence, or at least to reduce the associated entanglement decay. Here we consider a system formed by two identical two-level atoms (which we call qubits) that are coupled individually with two spatially separated cavities. We suppose that in one of such cavities the structure of the light cones is altered due to the presence of disorder, consequently the eletromagnetic vacuum fluctuations in each cavity behave differently.

The interplay between disorder and quantum entanglement have been intensively studied in the recent literature. Let us briefly mention some of such important works. For instance, Anderson localization mechanism was reported in experiments involving entangled photons~\cite{disent1}. In a somewhat related investigation, the study of a disordered quantum Ising chain was explored in order to gather information on many-body localization transition~\cite{disent4}. In the context of quantum phase transitions, in Ref.~\cite{disent2} the entanglement entropy of random quantum critical points was investigated. The influence of disorder on quantum computing performance is studied in Ref.~\cite{disent5} in the context of spin-$1/2$ lattice models. With the objective of providing an efficient survey for experiments with quantum devices that could be appropriate for implementation of quantum computing, the authors in Ref.~\cite{disent3} analysed the entanglement dynamics of arrays of qubits in the presence of disorder. In all these papers the role of random fluctuations in quantum entanglement is uncovered in several different settings, each of which exploring a crucial aspect associated with disorder in the physical processes under consideration.

This paper is devoted to assess the impact of light-cone fluctuations on the nonlocal decoherence of initially entangled atoms. Here such fluctuations are treated classically and as a consequence their effects on the propagation of quantum fields are governed by random differential equations~\cite{pe13}. Indeed, the adoption of a Klein-Gordon equation with a random coefficient has been considered in a series of works~\cite{pe12,pe21,ellis,Bessa}. In these papers it has been shown that such a method suitably describes the effects of the random fluctuations in the propagation of signals associated with the quantum fields.

The aim of this work is to consider two initially entangled qubits and analyze the dynamics of their disentanglement when one of them is coupled with a disordered electromagnetic cavity. We show that nonlocal decoherence caused by vacuum fluctuations of the quantum field is attenuated in the presence of random fluctuations of the light cone. The organization of this paper is as follows. In Sec.~\ref{lcf} we derive the positive-frequency Wightman function of the electric field in the presence of light-cone fluctuations. We work in the Coulomb gauge. We employ a perturbative expansion in order to take into account the effects of random fluctuations. In Sec.~\ref{master} we discuss the quantum master equation which describes the time evolution of the reduced density matrix of the two-qubit system. Sec.~\ref{entanglement} is devoted to the presentation and examination of the main results of this paper. Conclusions are given in Sec.~\ref{conclude}. We use units such that $\hbar=c=k_{B}=1$. Summation over repeated indices is assumed, unless otherwise stated. All graphics were drawn using the Mathematica packages.

\section{Light-cone fluctuations in electrodynamics}
\label{lcf}

As discussed in the previous section, we take the following situation. One of the atoms is immersed in a disordered medium in such a way that the associated photon propagation is described by a random wave equation, being more specific these photons evolve with random velocities. The other atom is inside an ``empty" cavity (in the sense of not having a noise field). We consider that both atoms follow inertial world lines. We are working in the interaction picture and in a four-dimensional Minkowski space-time. Let us consider the generic Hamiltonian of the two-atom system as:
\begin{equation}
H_A(\tau) = \frac{\omega_0}{2}\,{\bf n}\cdot\biggl[\boldsymbol\sigma_{(1)}(\tau) + \boldsymbol\sigma_{(2)}(\tau)\biggr],
\label{ha-kerrbh}
\end{equation}
where $\boldsymbol\sigma_{(1)} = \boldsymbol\sigma\otimes\sigma_0$, $\boldsymbol\sigma_{(2)} = \sigma_0\otimes \boldsymbol\sigma$, $\sigma^0$ is the unit $2 \times 2$ unit matrix, $\sigma^j$, $j = 1, 2, 3$ are the Pauli matrices and
${\bf n} = (0,0,1)$ in Cartesian coordinates. Also, $\tau$ is the proper time of the atom, while $\omega_0$ represents the gap between the two energy eigenvalues.

The qubits are coupled with an electromagnetic field. In addition, we consider the case of atoms sufficiently far away from the walls of the cavities in such a way that their effects may be ignored. By exploiting the usual gauge invariance of the electromagnetic Lagrangian, we employ the Coulomb gauge. Using the usual decomposition of the electric and magnetic field operators in terms of creation and annihilation operators, the free electromagnetic field Hamiltonian can be written as
\begin{equation}
H_F(t) = \sum_{\lambda}\int \frac{d^3 k}{(2\pi)^3}\, \omega_{{\bf k}}\,a^{\dagger}_{{\bf k},\lambda}(t)a_{{\bf k},\lambda}(t),
\label{hf}
\end{equation}
where $\omega_{{\bf k}} = |{\bf k}|$ and $a^{\dagger}_{{\bf k},\lambda}, a_{{\bf k},\lambda}$ are the creation and annihilation operators of the electromagnetic field. We neglect the zero-point energy. The index $\lambda$ classifies different polarizations of the field: The orthonormal vectors ${\hat \varepsilon}_{\lambda}({\bf k})$, $\lambda = 1, 2$ correspond to the two polarization states of the photon. These obey known relations that can be found elsewhere~\cite{cohen}. We also take the electric field as divergenceless.

Let us discuss the interaction-picture Hamiltonian of interaction between the atoms and the electromagnetic field. Within the multipolar coupling scheme and in the dipole approximation one has that
\begin{equation}
H_{I}(\tau) = -{\bf P}_{(1)}(\tau)\cdot {\bf E}(x_1(\tau)) - {\bf P}_{(2)}(\tau)\cdot {\bf E}(x_2(\tau)),
\label{hi-kerr2}
\end{equation}
where ${\bf P}_{(k)}(\tau)$ is the atomic dipole moment operator. It can be written as
\begin{equation}
{\bf P}_{(k)}(\tau) = \sum_{\eta = \pm} e^{i\eta\omega_0 \tau} {\bf A}_{(k)}(\eta \omega_0),
\end{equation}
where ${\bf A}_{(k)}(-\omega_0) = {\bf p}_{(k)}\sigma_{-\,(k)}$ and ${\bf A}_{(k)}(\omega_0) = {\bf p}^{*}_{(k)}\sigma_{+\,(k)}$, ${\bf p}_{(k)}$ being the transition matrix element of the dipole operator of the atom $k$ in the usual basis formed by the eigenvectors of $\sigma^{3}_{(k)}$. In addition, $\sigma_{\pm\,(k)} = (1/2)(\sigma_{(k)}^{1} \pm i \sigma_{(k)}^{2})$.

Henceforth we consider that atom $2$ is immersed in a medium with light-cone fluctuations. We take as a disordered medium a heterogeneous, non-dissipative, non-magnetic dielectric medium whose dielectric constant $\epsilon({\bf x})/\epsilon_{0} = 1 + \chi({\bf x})$ is real and positive, and fluctuates randomly~\cite{ake}. Then a random wave equation for the vector potential follows from the Maxwell's equations. In other words, in the Coulomb gauge one has the following random wave equation in a four dimensional space-time:
\begin{equation}
\left[\bigl(1+\,\chi({\bf x})\bigr) \frac{\partial^{2}}{\partial\,t^{2}}
- \Delta\right] {\bf A}(t,{\bf x})=0,
\label{nami20}
\end{equation}
where as usual ${\bf E} = - \partial {\bf A}/\partial t$ and $\Delta$ is the three dimensional Laplacian. We have imposed that $c^2 = 1/(\epsilon_0\mu_0) = 1$. Note that in this medium $(\nabla \chi) \cdot {\bf E} = 0$. In addition, observe that $\chi({\bf x})$ works like a local refractive index, in that it perturbs locally the wave speed due to the replacement
$$
\frac{\partial^{2}}{\partial t^{2}}\rightarrow (1+\chi({\bf x}))
\frac{\partial^{2}}{\partial t^{2}}.
$$
For the random function $\chi({\bf x}\,)$ we will take a zero-mean Gaussian random
function:
\begin{equation}
\langle \chi({\bf x}\,) \rangle_{\chi} = 0,
\label{nami21}
\end{equation}
with a white-noise correlation function given by
\begin{equation}
\langle \chi({\bf x}\,)\chi({\bf x}\,') \rangle_{\chi} = \sigma^2\,\delta^{(3)}({\bf x}-{\bf x}\,'),
\label{nami22}
\end{equation}
where $\sigma^2$ gives the intensity of random fluctuations. The symbol $\langle\, ...\,\rangle_{\chi}$ denotes an average over
all possible realizations of this random variable. Note that we assume a time-independent random function for inertial observers. In addition, we impose that the fields inside and outside the random medium do not couple with each other, i.e., any correlation function containing products of such fields is zero.

Following Refs.~\cite{pe12,pe21}, in the weak-disorder limit the random Klein-Gordon equation of Eq.~(\ref{nami20}) can be solved using a perturbative expansion in the noise function. In this way, the Feynman Green's function can be written as
\begin{eqnarray}
\hspace{-0.25cm}
G_{ab}(x,x') &=& \langle 0_{\chi} |T[A_{a}(x)A_{b}(x')]| 0_{\chi}\rangle  =  G^{0}_{ab}(x-x')
\nonumber\\
&+&\, \sum_{n=1}^{\infty}\int dz_1 \,G^{0}_{ak}(x-z_1){\cal G}^{(n)}_{kb}(z_1,x'),
\label{a1}
\end{eqnarray}
where $|0_{\chi}\rangle$ is a modified vacuum state in the presence of disorder, $G^{0}_{ab}(x-x')$, $a, b = 1,2,3$, is the usual Feynman Green's function without random fluctuations and $T$ is the Dyson time-ordering symbol. The contributions from the light-cone fluctuations are represented by the following quantity
\begin{equation}
{\cal G}^{(n)}_{kb}(z_1,x') = (-1)^{n}\prod_{j=1}^n {\cal D}(z_j)
\int dz_{j+1} G^{0}_{m_{j}m_{j+1}}(z_j,z_{j+1})
\label{genericterm}
\end{equation}
with ${\cal D} (x)$ being the random differential operator
\begin{equation}
{\cal D} (x) ={\cal D}(t,{\bf r\,}) = - \chi({\bf r\,}) \frac{\partial^2}{\partial t^2}.
\label{L1-coord}
\end{equation}
In Eq.~(\ref{genericterm}), it is to be understood that $z_{n+1} = x'$, $m_{1} = k$, $m_{n+1} = b$ and that there is no integration in $z_{n+1}$. Details on similar derivations concerning a massless scalar field can be found in Ref.~\cite{pe21}.

In the Coulomb gauge, the free photon propagator is given by
\begin{eqnarray}
G^{0}_{ij}(x-x') &=& \langle 0 |T[A_{i}(x)A_{j}(x')]| 0\rangle
\nonumber\\
&=&\, \int \frac{d^{4}k}{(2\pi)^4}\,\frac{i}{k^2 + i\epsilon}\left(\delta_{ij} - \frac{k_{i}k_{j}}
{|{\bf k}|^2}\right)\,e^{-ik\cdot(x-x')}
\nonumber\\
\end{eqnarray}
where $|0\rangle$ is the Minkowski vacuum state, $d^{4}k = dk^{0}d{\bf k}$ and $k^2 = (k^{0})^2 - |{\bf k}|^2$. Hence the zeroth order Feynman Green's function for the electric field reads:
\begin{equation}
D^{0}_{ij}(x,x') = \int \frac{d^{4}k}{(2\pi)^4}\,\frac{i\,(k^{0})^2}{k^2 + i\epsilon}\left(\delta_{ij} - \frac{k_{i}k_{j}}
{|{\bf k}|^2}\right)\,e^{-ik\cdot(x-x')}.
\end{equation}
In what follows we will be interested in the positive-frequency Wightman function for the electric field. The zeroth order contribution for such a function is given by~\cite{cohen}
\begin{eqnarray}
\hspace{-0.07cm}
 D^{0,+}_{ij}(x,x') &=&
-\frac{1}{\pi^2}\frac{1}{\left[(\Delta t - i\epsilon)^2 - |\Delta{\bf x}|^2\right]^3}
\nonumber\\
&\times&\Bigl\{2(\Delta{\bf x})_i(\Delta{\bf x})_j - \delta_{ij}\left[|\Delta{\bf x}|^2 + (\Delta t - i\epsilon)^2\right]\Bigr\},
\nonumber\\
\end{eqnarray}
where $\Delta t = t - t'$ and $\Delta {\bf x} = {\bf x} - {\bf x}'$. For the case of a static atom the unperturbed Wightman function of the electric field is given by
\begin{equation}
D^{0,+}_{ij}(x(\tau), x(\tau')) = \frac{ \delta_{ij}}{\pi^2(\tau-\tau' - i\epsilon)^4}.
\label{free}
\end{equation}
Coming back to Eq.~(\ref{a1}), after performing the averages over the noise function, the Feynman propagator associated with the photon field can be written in the form of a Dyson equation:
\begin{equation}
\langle G_{ab} \rangle_{\chi} =  G^{0}_{ab} - G^{0}_{am}\Sigma_{mk} \, \langle G_{kb} \rangle_{\chi},
\label{p32}
\end{equation}
with $\Sigma$ being the self-energy. The form of $\Sigma$ is obtained from noise averaging up to the second-order contribution in the random function $\chi$ and can be written generically as
\begin{equation}
\langle \, G_{ab}(x,x') \rangle_{\chi}= G^{0}_{ab}(x-x')
+ \langle\bar G^{1}_{ab}(x,x')\rangle_{\chi}.
\label{gg}
\end{equation}
For $\langle\bar G^{1}_{ab}(x,x')\rangle_{\chi}$ one obtains
\begin{equation}
\hspace{-0.04cm}
\langle\bar G^{1}_{ab}(x,x')\rangle_{\chi} = \int\frac{d{\bf k}}{(2\pi)^3}
\int\frac{dk^{0}}{2\pi} \, e^{-ik\cdot(x-x')} \, \langle\bar G^{1}_{ab}(k^{0},{\bf k})\rangle_{\chi}
\label{FT-G1}
\end{equation}
where
\begin{widetext}
\begin{eqnarray}
\langle\bar G^{1}_{ab}(k^{0},{\bf k})\rangle_{\chi} &=&
- \left[\frac{i}{k^2 + i\epsilon}\left(\delta_{aj} - \frac{k_{a}k_{j}}{|{\bf k}|^2}\right)\right]
\Sigma_{jm}(k^{0})
\left[\frac{i(k^{0})^2}{k^2 + i\epsilon}\left(\delta_{mb} - \frac{k_{m}k_{b}}{|{\bf k}|^2}\right)\right],
\label{eu}
\end{eqnarray}
with
\begin{equation}
\Sigma_{jm}(k^{0}) = - \sigma^2 \,(k^{0})^2\,\int \frac{d^{3}p}{(2\pi)^3}\,\frac{i}{(k^{0})^2 - |{\bf p}|^2 + i\epsilon}
\left(\delta_{jm} - \frac{p_{j}p_{m}}{|{\bf p}|^2}\right) = \sigma^2\,\frac{(k^{0})^3}{3\pi}\,\delta_{jm}.
\label{Sigma}
\end{equation}
After inserting the above results in Eq.~(\ref{FT-G1}) one obtains:
\begin{equation}
\langle\bar G^{1}_{ab}(x,x')\rangle_{\chi} = \frac{\sigma^2}{3\pi}\,
\int\frac{d{\bf k}}{(2\pi)^3}\int\frac{dk^{0}}{2\pi} \, e^{-ik\cdot(x-x')} \,
\frac{(k^{0})^5}{(k^2 + i\epsilon)^2}\left(\delta_{ab} - \frac{k_{a}k_{b}}{|{\bf k}|^2}\right).
\end{equation}
\end{widetext}
The integral over $k^{0}$ can be evaluated within contour-integration methods. On the other hand, in order to present the corrections to the positive-frequency Wightman function, we only need to consider $t > t'$ and the pole at
$k^{0} = |{\bf k}| - i\epsilon$. The contour we use is the usual contour for the positive-frequency Wightman function, see page $22$ of Ref.~\cite{birrel}. One gets
\begin{eqnarray}
\hspace{-0.3cm}
\langle\bar G^{1,+}_{ab}(x,x')\rangle_{\chi} &=& \frac{\sigma^2}{12\pi}\,
\int\frac{d{\bf k}}{(2\pi)^3}|{\bf k}|^2  \Bigl(4 - i |{\bf k}|(t-t')\Bigr)
\nonumber\\
&\times&\left(\delta_{ab} - \frac{k_{a}k_{b}}{|{\bf k}|^2}\right)
e^{i{\bf k} \cdot ({\bf x} - {\bf x}') - i|{\bf k}|(t-t')}.
\label{wigh-A}
\end{eqnarray}
Hence the contribution to the positive-frequency Wightman function of the electric field is given by
\begin{eqnarray}
\hspace{-0.25cm}
\langle\bar D^{1,+}_{ab}(x,x')\rangle_{\chi} &=& \frac{\sigma^2}{12\pi}\,
\int\frac{d{\bf k}}{(2\pi)^3}|{\bf k}|^4  \Bigl(6 - i |{\bf k}|(t-t')\Bigr)
\nonumber\\
&\times& \left(\delta_{ab} - \frac{k_{a}k_{b}}{|{\bf k}|^2}\right)
e^{i{\bf k} \cdot ({\bf x} - {\bf x}') - i|{\bf k}|(t-t')}.
\label{wigh-E}
\end{eqnarray}
Evaluating the integrals one finds that
\begin{widetext}
\begin{eqnarray}
\langle\bar D^{1,+}_{ab}(x,x')\rangle_{\chi} &=& \frac{4 i \sigma^2}
{\pi^3 \left[(\Delta t-i \epsilon )^2-|\Delta {\bf x}|^2\right]^7}
\left\{
12(\Delta{\bf x})_a(\Delta{\bf x})_b
\Bigl[2(\Delta t-i \epsilon)\left(3 |\Delta {\bf x}|^4 + 4 |\Delta {\bf x}|^2 (\Delta t-i \epsilon )^2 -7 (\Delta t - i \epsilon )^4\right)
\right.
\nonumber\\
&+&\, \left. \Delta t\left(|\Delta {\bf x}|^4 + 18 |\Delta {\bf x}|^2 (\Delta t - i \epsilon )^2
+ 21 (\Delta t - i \epsilon )^4\right)\Bigr] \right.
\nonumber\\
&-&\, \left. \delta_{ab}
\Bigl[6(\Delta t-i \epsilon)\left(9 |\Delta {\bf x}|^6 + 17 |\Delta {\bf x}|^4 (\Delta t - i \epsilon )^2
-21 |\Delta {\bf x}|^2 (\Delta t - i \epsilon )^4 - 5 (\Delta t -i \epsilon )^6\right)
\right.
\nonumber\\
&+&\, \left. \Delta t\left(9|\Delta {\bf x}|^6 + 177 |\Delta {\bf x}|^4(\Delta t - i \epsilon )^2
+259  |\Delta {\bf x}|^2 (\Delta t-i \epsilon )^4 + 35 (\Delta t -i \epsilon )^6\right)\Bigr]\right\}.
\end{eqnarray}
\end{widetext}
For a static atom, one has that
\begin{equation}
\left\langle\bar D^{1,+}_{ij}(x(\tau), x(\tau'))\right\rangle_{\chi} = -\frac{20\, i \sigma^2\,\delta_{ij}}
{\pi^3 (\tau - \tau' - i \epsilon )^{7}}.
\label{disorder}
\end{equation}

\section{The master equation approach}
\label{master}

In this section we discuss the time evolution of the qubit system coupled with a quantized electromagnetic field within the master equation approach. A similar analysis is encountered in Ref.~\cite{yu}, in which the authors study entanglement dynamics of two uniformly accelerated atoms coupled with an electromagnetic field within master-equation techniques. We will consider the so-called Born approximation in which initial interaction-induced correlations between the subsystems are neglected. This is justified as long as the spectral width of the quantum field energies is much broader than the spectral width associated with the interaction between the subsystems. We also consider the relaxation time associated with the noise field to be much larger than the reciprocals of the spectral widths aforementioned. This requirement is necessary in order to qualify the disorder as being of the quenched type.

Taking into account the Born approximation aforecited, one can take an initial density operator of the whole system of qubits and fields in a factorized form: $\rho(0) = \rho_{A}(0) \otimes \rho_{F}$, where $\rho_{A}(0)$ is the initial density operator associated with the qubit system whereas $\rho_{F}$ is the analogous one for the quantum electromagnetic field. The latter is given by $\rho_{F} = |\Omega\rangle\langle \Omega|$, where $|\Omega\rangle = |0\rangle$ for the empty cavity and $|\Omega\rangle = |0_{\chi}\rangle$ for the disordered cavity. On the other hand, for the initial state $\rho_{A}(0)$ we will consider an entangled state whose specific form will be expounded in the due course.

The subdynamics describing the evolution of the qubits subsystem alone is obtained with a trace operation over the field degrees of freedom. The details are discussed in many works~\cite{edavies1,edavies2,lindblad,koss,cohen4,rep,ben1,ben2,ben3}. In the limit of weak coupling between the atoms and the field, and within a suitable Markovian approximation, the evolution of the reduced density matrix $\rho_{A}(\tau)$ can be written in the Kossakowski-Lindblad form
\begin{equation}
\frac{\partial \rho_{A}(\tau)}{\partial \tau} = -i[H_{\textrm{eff}}, \rho_{A}(\tau)] + {\cal L}[\rho_{A}(\tau)],
\label{kerr-master}
\end{equation}
where
\begin{eqnarray}
{\cal L}[\rho_{A}(\tau)] &=& \frac{1}{2}\sum_{a,b = 1}^{2}\,{\cal K}_{ij}^{(a)(b)}\,\left\{2\sigma_{(b)}^{j}\,
\rho_{A}(\tau)\sigma_{(a)}^{i} \right.
\nonumber\\
&-&\, \left. \{\sigma_{(a)}^{i}\,\sigma_{(b)}^{j},\rho_{A}(\tau)\}\right\},
\label{kerr-nonunitary}
\end{eqnarray}
and the effective Hamiltonian reads
\begin{equation}
H_{\textrm{eff}} = H_A - \frac{i}{2}\sum_{a,b=1}^{2}{\cal H}_{ij}^{(a)(b)}\,\sigma_{(a)}^{i}\sigma_{(b)}^{j},
\label{eff}
\end{equation}
with
\begin{equation}
{\cal H}_{ij}^{(a)(b)}(\omega_0) = d_{+}^{(a)(b)}\delta_{ij} - d_{+}^{(a)(b)}\,n_{i}n_{j} - i d_{-}^{(a)(b)}\,
\epsilon_{ijm}\,n^{m}.
\end{equation}
In the above equation $\epsilon_{ijk}$ is the usual Levi-Civita symbol. The Kossakowski matrix is given by
\begin{equation}
{\cal K}_{ij}^{(a)(b)}(\omega_0) = g_{+}^{(a)(b)}\delta_{ij} - g_{+}^{(a)(b)}\,n_{i}n_{j} - i g_{-}^{(a)(b)}\,
\epsilon_{ijm}\,n^{m}.
\label{koss}
\end{equation}
In the above expressions we have defined the quantities
\begin{eqnarray}
g_{\pm}^{(a)(b)} &=& \frac{1}{4}\Bigl[{\cal G}^{(a)(b)}(\omega_0) \pm {\cal G}^{(a)(b)}(-\omega_0)\Bigr]
\nonumber\\
d_{\pm}^{(a)(b)} &=& \frac{1}{4}\Bigl[{\cal D}^{(a)(b)}(\omega_0) \pm {\cal D}^{(a)(b)}(-\omega_0)\Bigr]
\end{eqnarray}
where (no summation over $a,b$)
\begin{eqnarray}
{\cal G}^{(a)(b)}(\omega_0) &=&  p^{i}_{(a)}p^{j\,*}_{(b)}{\cal G}^{(a)(b)}_{ij}(\omega_0)
\nonumber\\
{\cal D}^{(a)(b)}(\omega_0) &=& p^{i}_{(a)}p^{j\,*}_{(b)}{\cal D}^{(a)(b)}_{ij}(\omega_0)
\end{eqnarray}
In the above equations we have introduced the following Fourier and Hilbert transform of the electromagnetic field correlations
\begin{equation}
{\cal G}^{(a)(b)}_{ij}(\eta\omega_0) = \int_{-\infty}^{\infty}du\,e^{i\eta\omega_0 u}\,D^{(a)(b)}_{ij}(u),
\label{fourier}
\end{equation}
and
\begin{eqnarray}
{\cal D}^{(a)(b)}_{ij}(\eta\omega_0) &=&  \int_{-\infty}^{\infty}du\,e^{i\eta\omega_0 u}\,\mathrm{sgn}(u)\,D^{(a)(b)}_{ij}(u)
\nonumber\\
&=& \frac{\textrm{P}}{\pi i}\int_{-\infty}^{\infty}d\lambda\,\frac{{\cal G}^{(a)(b)}_{ij}(\lambda)}{\lambda - \eta\omega_0},
\label{hilbert}
\end{eqnarray}
where $\textrm{P}$ denotes the principal value and the correlation function
\begin{eqnarray}
D^{(a)(b)}_{ij}(\tau-\tau') &=& D_{ij}(x_a(\tau)-y_b(\tau'))
\nonumber\\
&=&\, \langle \Omega|E_{i}(x_a(\tau))E_{j}(y_b(\tau'))|\Omega\rangle.
\nonumber
\end{eqnarray}
In the above equations $\eta = \pm$. A suitable divergent (frequency-independent) term may need to be subtracted by a suitable renormalization procedure in order to make the contribution $H_{\textrm{eff}}$ well defined. This will be discussed below. A similar situation also appears in~\cite{ben2}. On the other hand, we recall that in the present situation the atoms are inside cavities whose fields do not interact with each other. This means that crossed terms in the Kossakowski matrix are absent in our analysis.

Let us evaluate the quantities ${\cal G}^{(a)(b)}(\omega_0)$ and ${\cal D}^{(a)(b)}(\omega_0)$ which enter in the above definitions of the effective Hamiltonian and the Kossakowski matrix, respectively Eqs.~(\ref{eff}) and~(\ref{koss}). For the qubit inside the empty cavity, the positive-frequency Wightman function is given by Eq.~(\ref{free}). Inserting such an expression in Eq.~(\ref{fourier}) one obtains
\begin{equation}
{\cal G}^{(1)(1)}_{kl}(\eta\omega_0) = \frac{\eta ^3 \omega _0^3}{3 \pi}\,\theta(\eta)\,\delta_{kl}.
\end{equation}
On the other hand, concerning the other qubit which is coupled with the disordered cavity, the Fourier transform of the positive-frequency Wightman function of the electric field can be written as
\begin{equation}
{\cal G}^{(2)(2)}_{kl}(\eta\omega_0) = {\cal G}^{(2)(2)\,0}_{kl}(\eta\omega_0)
+ \langle \bar {\cal G}^{(2)(2)\,1}_{kl}(\eta\omega_0) \rangle
\end{equation}
where ${\cal G}^{(2)(2)\,0}_{kl}(\eta\omega_0) = {\cal G}^{(1)(1)}_{kl}(\eta\omega_0)$ and
\begin{equation}
\langle \bar {\cal G}^{(2)(2)\,1}_{kl}(\eta\omega_0) \rangle = -\frac{\sigma ^2\eta^6\omega_0^6}{18 \pi ^2}\,\theta(\eta)\,\delta_{kl}.
\end{equation}
As discussed above one must have ${\cal G}^{(a)(b)}_{kl}(0) = 0$. In this way the explicit expression for the Kossakowski matrix reads
\begin{equation}
{\cal K}_{ij}^{(a)(b)}(\omega_0) = A^{(a)(b)}[\delta_{ij} - n_{i}n_{j} - i  \epsilon_{ijm}\,n^{m}],
\label{koss1}
\end{equation}
where
\begin{equation}
A^{(1)(1)} = \frac{\omega _0^3}{12 \pi}\Bigl(\delta_{ij}\,p^{i}_{(1)}p^{j\,*}_{(1)}\Bigr)
\label{a11}
\end{equation}
and
\begin{equation}
A^{(2)(2)} = \frac{\omega _0^3}{12 \pi}\left(1 - \frac{\sigma ^2\omega_0^3}{6 \pi}\right)
\Bigl(\delta_{ij}\,p^{i}_{(2)}p^{j\,*}_{(2)}\Bigr).
\label{a22}
\end{equation}
with ${\cal K}_{ij}^{(a)(b)} = 0$ for $a \neq b$. Concerning the Hilbert transform, one gets the following result for the empty cavity
\begin{equation}
{\cal D}_{kl}^{(1)(1)}(\eta\omega_0) = \frac{\delta_{kl}}{3 \pi}\frac{\textrm{P}}{\pi i}\int_{0}^{\infty}d\lambda\,\frac{\lambda^3}{\lambda - \eta\omega_0}.
\end{equation}
For the qubit inside the disordered cavity one finds that
\begin{equation}
{\cal D}_{kl}^{(2)(2)}(\eta\omega_0) = {\cal D}_{kl}^{(2)(2)\,0}(\eta\omega)
+ \langle \bar {\cal D}_{kl}^{(2)(2)\,1}(\eta\omega_0) \rangle,
\end{equation}
with ${\cal D}_{kl}^{(2)(2)\,0}(\eta\omega_0) = {\cal D}_{kl}^{(1)(1)}(\eta\omega_0)$ and
\begin{equation}
\langle \bar {\cal D}_{kl}^{(2)(2)\,1}(\eta\omega_0) \rangle = -\frac{\delta_{kl}\,\sigma^2}{18 \pi^2}
\frac{\textrm{P}}{\pi i}\int_{0}^{\infty}d\lambda\,\frac{\lambda^6}{\lambda - \eta\omega_0}.
\end{equation}
Hence using that $\sigma^{(a)}_{i}\sigma^{(a)}_{j} = \delta_{ij}\,{\bf 1}^{(a)} +  i \epsilon_{ijk}\sigma^{(a)}_{k}$, together with $\epsilon_{aij}\,\epsilon^{bij} = 2 \delta_{a}^{\,\,b}$, the effective Hamiltonian can be written as
\begin{equation}
H_{\textrm{eff}} = \frac{\,\,\,\Omega^{(1)}}{2}\,{\bf n}\cdot\boldsymbol\sigma^{(1)}
+ \frac{\,\,\,\Omega^{(2)}}{2}\,{\bf n}\cdot\boldsymbol\sigma^{(2)},
\end{equation}
where we have defined the renormalized frequencies $\Omega^{(a)}$, $a = 1, 2$, as (no summation over $a$)
\begin{equation}
\Omega^{(a)} = \omega_0 - \frac{i}{2}\,\Bigl[{\cal D}^{(a)(a)}(\omega_0) - {\cal D}^{(a)(a)}(-\omega_0)\Bigr].
\end{equation}
Note that the contributions ${\cal D}^{(a)(a)}(\eta\omega_0)$ produce the usual Lamb shift in the energy eigenvalues. As can be easily checked, the Lamb contribution turns out to be infinite, and its definition calls for the introduction of an adequate cutoff and a subsequent renormalization procedure. This is a well-known fact; the appearance of the divergences is related to the nonrelativistic treatment of the two-level atoms. It is mandatory the use of quantum field theory techniques for any sensible assessment of the energy shifts.

\section{Quantum entanglement in the presence of light-cone fluctuations}
\label{entanglement}

Now let us finally move on to the analysis of the contribution of light-cone fluctuations upon the quantum entanglement of inertial qubits in Minkowski spacetime. We find convenient to write the density matrix in the basis formed by the Dirac matrices~\cite{gamel}
\begin{equation}
\Sigma_{\mu\nu} = \sigma_{\mu}\otimes\sigma_{\nu}.
\end{equation}
In this way the reduced density matrix associated with the qubits subsystem can be written in a generic way as
\begin{equation}
\rho_{A}(\tau) = \frac{1}{4}\,\rho_{\mu\nu}(\tau)\Sigma_{\mu\nu},
\end{equation}
where $\rho_{\mu\nu}(\tau)$ is the Bloch matrix. With the normalization condition $\textrm{Tr}[\rho_{A}(\tau)] = 1$, and imposing that $\rho_{A}(\tau)$ should be Hermitian, one has that $\rho_{00} = 1$ and $\rho_{i0}(\tau), \rho_{0i}(\tau), \rho_{ij}(\tau)$ are real. Explicitly ($i,j = 1,2,3$):
\begin{eqnarray}
\rho_{A}(\tau) &=& \frac{1}{4}\left[\sigma^0\otimes\sigma^0 + \rho_{0i}(\tau)\,\sigma^{0}\otimes\sigma^{i}
+ \rho_{i0}(\tau)\,\sigma^{i}\otimes\sigma^{0} \right.
\nonumber\\
&+&\, \left. \rho_{ij}(\tau)\,\sigma^{i}\otimes\sigma^{j}\right]
\nonumber\\
&=&\,\frac{1}{4}\left[{\bf 1}_{4 \times 4} + \rho_{0i}(\tau)\,\sigma_{(2)}^{i}
+ \rho_{i0}(\tau)\,\sigma_{(1)}^{i} \right.
\nonumber\\
&+&\, \left. \rho_{ij}(\tau)\,\sigma_{(1)}^{i}\sigma_{(2)}^{j}\right].
\end{eqnarray}
In what follows we will ignore the Hamiltonian piece since it cannot give rise to entanglement phenomena, and focus attention on the study of the effects induced by the dissipative part ${\cal L}[\rho_{A}(\tau)]$. For the particular case in which ${\bf n} = (0,0,1)$, one gets the following evolution equations for the components of $\rho_{A}(\tau)$:
\begin{eqnarray}
\frac{\partial\rho_{0k}}{\partial \tau} &=& - 2A^{(2)(2)}\rho_{0k}(\tau),\,\,\, k \neq 3
\nonumber\\
\frac{\partial\rho_{03}}{\partial \tau} &=& -4A^{(2)(2)}\rho_{03}(\tau)  - 4 A^{(2)(2)}
\nonumber\\
\frac{\partial\rho_{k0}}{\partial \tau} &=& -2A^{(1)(1)}\rho_{k0}(\tau),\,\,\, k \neq 3
\nonumber\\
\frac{\partial\rho_{30}}{\partial \tau} &=& -4A^{(1)(1)}\rho_{30}(\tau) - 4 A^{(1)(1)}
\end{eqnarray}
and finally
\begin{widetext}
\begin{eqnarray}
\frac{\partial\rho_{ij}}{\partial \tau} &=& - 2(A^{(1)(1)} + A^{(2)(2)})\rho_{ij}(\tau),\,\,\, i,j \neq 3
\nonumber\\
\frac{\partial\rho_{i3}}{\partial \tau} &=& - 2(A^{(1)(1)} + 2A^{(2)(2)})\rho_{i3}(\tau)
 - 4 A^{(2)(2)} \rho_{i0},\,\,\,\, i =1,2
\nonumber\\
\frac{\partial\rho_{3j}}{\partial \tau} &=& - 2(2A^{(1)(1)} + A^{(2)(2)})\rho_{3j}(\tau)
- 4 A^{(1)(1)}\rho_{0j},\,\,\, j =1,2
\nonumber\\
\frac{\partial\rho_{33}}{\partial \tau} &=& - 4(A^{(1)(1)} + A^{(2)(2)})\rho_{33}(\tau)
- 4 A^{(1)(1)}\rho_{03}(\tau) - 4 A^{(2)(2)}\rho_{30}(\tau).
\end{eqnarray}
The solutions of such differential equations are given by
\begin{eqnarray}
\rho_{0i}(\tau) &=& \rho_{0i}(0)e^{-2A^{(2)(2)}\tau},\,\,\, i = 1,2
\nonumber\\
\rho_{03}(\tau) &=& -1+ \Bigl[\rho_{03}(0) + 1 \Bigr]e^{-4A^{(2)(2)}\tau}
\nonumber\\
\rho_{i0}(\tau) &=& \rho_{i0}(0)e^{-2A^{(1)(1)}\tau},\,\,\, i = 1,2
\nonumber\\
\rho_{30}(\tau) &=& - 1 + \Bigl[\rho_{30}(0) + 1 \Bigr]e^{-4A^{(1)(1)}\tau}
\nonumber\\
\rho_{ij}(\tau) &=& \rho_{ij}(0)e^{- 2(A^{(1)(1)} + A^{(2)(2)})\tau},\,\,\, i,j \neq 3
\nonumber\\
\rho_{i3}(\tau) &=& \left[\rho_{i3}(0) + \rho_{i0}(0)\left(1- e^{4 A^{(2)(2)} \tau}\right)\right]
e^{-2 \tau (A^{(1)(1)} + 2 A^{(2)(2)})},\,\,\,\, i =1,2
\nonumber\\
\rho_{3j}(\tau) &=& \left[\rho_{3j}(0) + \rho_{0j}(0)\left(1 -  e^{4 A^{(1)(1)} \tau}\right)\right]
e^{-2 \tau (2A^{(1)(1)} + A^{(2)(2)})},\,\,\, j =1,2
\nonumber\\
\rho_{33}(\tau) &=& e^{-4 (A^{(1)(1)} + A^{(2)(2)})\tau} \left\{1 + \rho_{03}(0) + \rho_{30}(0) + \rho_{33}(0) +
e^{4 (A^{(1)(1)} + A^{(2)(2)})\tau} \right.
\nonumber\\
&-&\, \left. \Bigl[\rho_{03}(0) + 1 \Bigr] e^{4 A^{(1)(1)} \tau} -  \Bigl[\rho_{30}(0) + 1 \Bigr] e^{4 A^{(2)(2)} \tau}\right\}.
\end{eqnarray}
\end{widetext}
We consider the situation in which the atoms are initially prepared in an entangled state. Being more specific, let us consider the case in which the initial density operator $\rho_{A}(0)$ is given by the $2$-qubit family of Werner states
\begin{equation}
\rho_{A}(0) = \kappa\,\rho^{-} + \left(\frac{1-\kappa}{4}\right)\sigma^0\otimes\sigma^0,
\end{equation}
where $\kappa$ is a real number and $\rho^{-}$ is the maximally entangled singlet state:
\begin{equation}
\rho^{-} = \frac{1}{4}\left[\sigma^0\otimes\sigma^0 - \delta_{ij}\,\sigma^{i}\otimes\sigma^{j}\right].
\end{equation}
For $\kappa > 1/3$ such a state is entangled.

In order to discuss the measure of the entanglement amount of the reduced density matrix $\rho_{A}(\tau)$,we resort to the concurrence ${\cal C}[\rho]$~\cite{conc1,conc2,conc3}, which is a monotonically increasing function of the entanglement of formation~\cite{conc4}. The concurrence takes values in the interval $[0,1]$, where zero is for separable states and one is for maximally entangled states, as the Bell states. In order to evaluate the concurrence of any $4\times4$ density matrix $\rho$ portraying the state of two qubits, one considers the square roots $\lambda$ of each of the four eigenvalues of the matrix $\rho(\sigma^{2}\otimes\sigma^{2})\rho^{{\cal T}}(\sigma^{2}\otimes\sigma^{2})$, where ${\cal T}$ denotes transposition. Then one considers such values in decreasing order: $\lambda_{1} \geq \lambda_{2} \geq \lambda_{3} \geq \lambda_{4}$. The concurrence of $\rho$ is defined to be ${\cal C}[\rho] = \textrm{max}\{\lambda_{1} - \lambda_{2} - \lambda_{3} - \lambda_{4},0\}$.

Inspection of the square roots $\lambda$ reveals that one has to impose certain conditions over the possible values of $\kappa$ in order for the concurrence to be a purely real number, for fixed values of $\omega_0$, $\sigma^2$ and polarizations of the atoms. The simplest of such conditions is to assume that $ \kappa \leq 1$. For $\kappa = 1$ ($\rho_{A}(0) = \rho^{-}$), the concurrence is given by the simple expression
\begin{equation}
{\cal C}[\rho_{A}(\tau)] = e^{-2 \tau (A^{(1)(1)} + A^{(2)(2)})}.
\label{concur}
\end{equation}
Taking into account Eqs.~(\ref{a11}) and~(\ref{a22}) it is easy to see that disorder contributes with a positive argument in the above exponential. As a consequence, the nonlocal decoherence can be strongly softened. In fact, assuming equal polarizations for the atoms, one notices that, for a fixed $\omega_0$, there is a critical value for the disorder strength $\sigma_{c}^{2}$ such that the initial entangled state behaves as a decoherence-free state:
\begin{equation}
\sigma_{c}^{2} = 12\pi/\omega_{0}^3.
\end{equation}
For $\sigma^{2} > \sigma_{c}^{2}$, ${\cal C}[\rho_{A}(\tau)]$ could be larger than one for $\tau > 0$ which is meaningless in the present case. Hence there must be a compromise between the atomic energy gap and the disorder strength so that the investigation of the entanglement by means of the concurrence yields physically acceptable results. A similar qualitative conclusion can be drawn for atoms with different polarizations.

It is possible to find an explicit expression for the concurrence, yet cumbersome, for a generic value of $\kappa$. For the sake of clarity we will not display such an expression here. In this case we will limit ourselves to a qualitative numerical description. The behavior of the concurrence as a function of the proper time $\tau$ of the atoms is depicted in Fig.~\ref{1} below. We choose the polarizations of the atoms such that
$$
\delta_{ij}\,p^{i}_{(1)}p^{j\,*}_{(1)} = \delta_{ij}\,p^{i}_{(2)}p^{j\,*}_{(2)} = 1.
$$
The conclusions are not qualitatively affected by considering an arbitrary configuration for the atomic polarizations. Observe the reduction in the entanglement decay as the intensity of disorder increases. Even though we do not observe a specific value for the disorder strength such that the concurrence remains constant for all $\tau$ as above, we do find a certain critical value for $\sigma^2$ from which the concurrence indicates creation of entanglement within a given proper time interval. In this case once again one must accept a sort of fine tuning between $\omega_0$ and $\sigma^2$ for a particular observational time interval so that the concurrence produces real values less than one.
\begin{figure}[htb]
\begin{center}
\includegraphics[height=88mm,width=88mm]{figure.pdf}
\caption{The concurrence ${\cal C}[\rho(\tau)]$ as a function of $\tau$ for different values of the disorder strength $\sigma^2$. We choose $\kappa = 0.8$ and $\omega_0 = 5$. We remark that all physical quantities are given in terms of the natural units associated with the transition between the eigenstates of the atoms, with wavelength $\lambda = 2\pi/\omega_0$.}
\label{1}
\end{center}
\end{figure}

\vspace{-7mm}
\section{Conclusions}
\label{conclude}

In this paper we studied how the entanglement dynamics between two-level atoms is affected by random fluctuations of the light cone. In our model the atoms were coupled individually to two spatially separated cavities, one of which is called a disordered cavity. We demonstrated that in certain situations the disorder may contribute to enhance the degree of entanglement. This implies in a reduction in the entanglement decay and in turn that qubit systems can decohere slower in the presence of disorder. When the initial state of the qubits subsystem is given by the maximally entangled singlet state, we found a critical value for the disorder strength such that one observes an almost complete inhibition of the decay due to destructive interference. This somewhat resembles the usual result that claims that, in atomic radiative processes, for small distances between the atoms the singlet Bell state is a decoherence-free state~\cite{rep}.

We believe that the approach investigated here for suppressing the effect of decoherence can be valid for other kinds of decoherence and for multipartite entangled states. In fact, we conjecture that in a strong disorder environment the mean life of entangled states can be enhanced in such a way as to nearly completely suppress nonlocal decoherence. Indeed, recently it was proposed that in a certain class of magnetic materials disorder generates quantum entanglement~\cite{savary}. Such outcomes corroborate our findings.

A possible generalization of the present work is to investigate whether the same results are attainable in a non-perturbative framework. In turn, an alternative route to investigate the relationship between radiative processes of two-level atoms and disorder models is provided by the use of ultra-cold atomic systems. Indeed, from a soluble model of quantum optics, a spin-glass behavior can be obtained~\cite{Sarang, Sachev, Rotondo1, Rotondo2}. This spin-boson model describes a phase transition from the fluorescent to superradiant phase~\cite{andreev}. On the other hand, a recent mathematical rigorous method for the evaluation of the average free energy of a disordered system was developed in Ref.~\cite{nb}. The basic quantity is a distributional zeta-function. Within such a framework it has been verified the presence of a spontaneous symmetry breaking mechanism in a disordered $d$-dimensional Euclidean $\lambda\varphi^{4}$ model~\cite{prdisorder}. In addition, in the low temperature regime the existence of $N$ instantons in such a model has been confirmed. It has been discussed in the literature the possibility of representing a system of $N$ real instantons as $N$ macroscopic structures by $N$ two-level systems~\cite{Leggett}. The distributional zeta-function method could be an appropriate approach in order to demonstrate the emergence of an effective model of a bosonic field interacting with a reservoir of $N$ identical two-level systems within a disordered model.  Such subjects are under investigation by the authors.

\vspace{-4mm}
\section*{Acknowledgements}

This paper was partially supported by Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol{\'o}gico (CNPq, Brazil).


\section{Introduction}\label{sec1}

A black hole has an event horizon through which any particle cannot escape from its gravity, even light. Classically, it implies that no energy or matter can reach an observer located outside of a black hole. However, in consideration of a quantum effect, a small portion of energy can be radiated to outside of the horizon in a black hole spacetime. Then, from the radiated energy, the temperature of the black hole is defined, and the black hole can be treated as a thermal system having this Hawking temperature\cite{Hawking:1974sw,Hawking:1976de}. Further, black holes have an irreducible mass, which is a property that increases in an irreversible process\cite{Christodoulou:1970wf,Bardeen:1970zz,Christodoulou:1972kt,Smarr:1972kt}. The irreducible mass is similar to the entropy in a thermal system, and based on this similarity, the entropy of a black hole is obtained from the irreducible mass. This is the Bekenstein-Hawking entropy of a black hole\cite{Bekenstein:1973ur,Bekenstein:1974ax} and is proportional to the area of the horizon. Using these two thermal properties, temperature and entropy, the laws of thermodynamics are constructed for the black hole as a thermal system.

An interesting conjecture has been applied to the horizon of a black hole. It is called the weak cosmic censorship conjecture in which the horizon of the black hole should cover its inside because a singularity of the black hole geometry is located at the center of the spacetime\cite{Penrose:1964wq,Penrose:1969pc}. This is an inevitable conjecture to save the causality of the spacetime from a naked singularity. Although the conjecture is generally suggested for black holes, its validity should be tested for each case because there is no general procedure to prove it. Moreover, the validity of the conjecture depends on the methods of investigation. For the Kerr black hole, the conjecture is valid under adding a particle\cite{Wald:1974ge}. However, the near-extremal Kerr black hole can be overspun beyond its extremal bound by a particle, so the conjecture is invalid\cite{Jacobson:2009kt}. To resolve this invalidity, in consideration of the self-force effect, the conjecture is known to be valid for the Kerr black hole\cite{Barausse:2010ka,Barausse:2011vx,Colleoni:2015ena}. The Reissner-Nordstr\"{o}m (RN) black hole was tested for validity of the conjecture with back-reaction effect\cite{Hubeny:1998ga,Isoyama:2011ea}. Furthermore, various investigations have been conducted on the conjecture for not only black holes in Einstein's gravity, but also anti-de Sitter (AdS), lower-dimensional, or higher-dimensional black holes\cite{Colleoni:2015afa,BouhmadiLopez:2010vc,Rocha:2011wp,Gao:2012ca,Rocha:2014gza,Rocha:2014jma,McInnes:2015vga,Hod:2016hqx,Natario:2016bay,Horowitz:2016ezu,Duztas:2016xfg,Gwak:2017icn,Sorce:2017dst}. From a thermodynamic point of view, the conjecture is quite consistently related to the laws of thermodynamics. If the entropy of the black hole increases as ensured by the second law for an irreversible process, the horizon can cover the inside of a black hole as the conjecture suggests. In addition, in the process, the variation of a black hole is consistent with the first law of thermodynamics under a particle absorption\cite{Gwak:2015fsa,Gwak:2016gwj}.

The thermal properties have an important role in AdS spacetime. The gravity theory in $D$-dimensional AdS spacetime is associated with the conformal field theory (CFT) defined on the boundary of the AdS spacetime. This is the AdS/CFT correspondence\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj,Aharony:1999ti}. Under this duality, the thermal properties of the AdS black hole are also found in the dual CFT, so that the CFT is given at finite temperature\cite{Witten:1998zw}. Currently, there are various applications of AdS/CFT duality. One of the representative applications is anti-de Sitter/quantum chromodynamics (AdS/QCD) correspondence\cite{Babington:2003vm,Kruczenski:2003uq,Sakai:2005yt,Erlich:2005qh}. Another is anti-de Sitter/condensed matter theory (AdS/CMT) correspondence\cite{Hartnoll:2008vx,Hartnoll:2008kx}. Because each solution of black holes is based on various gravity theories, its dual theory and physical interpretation depend on the black hole concerned. Further, the instability of black holes in perturbation or thermodynamics affects the states of the dual CFT. For example, the charged AdS black hole is mainly related to the AdS/CMT applications. In the (2+1)-dimensional charged AdS black hole, its dual theory is associated with the holographic superconductor\cite{Jensen:2010em,Andrade:2011sx,Chang:2014jna}. In addition, Fermi-Luttinger liquids is a model having resemblance to its dual theory\cite{Hung:2009qk,Davison:2013uha}.

The cosmological constant is a parameter which plays an important role in determining the asymptotic topology of a black hole spacetime. In the action of Einstein gravity, the cosmological constant is fixed, so it is set to a constant value at any time. Recently, various interesting studies have been conducted on the thermodynamics of black holes wherein the cosmological constant has been set to a dynamic variable and interpreted as a pressure. In fact, the cosmological constant as a dynamic variable was first considered a long time ago\cite{Teitelboim:1985dp,Brown:1988kg}. Furthermore, the pressure of the black hole spacetime is associated with the cosmological constant\cite{Caldarelli:1999xj, Padmanabhan:2002sha}, and its thermal conjugate is found to be a thermodynamic volume\cite{Dolan:2010ha,Cvetic:2010jb}. Under the cosmological constant as a dynamic variable, the mass of the black hole corresponds to the enthalpy of the black hole system\cite{Kastor:2009wy}. Owing to the pressure-volume contribution, the first law of thermodynamics is extended to have $PV$ term\cite{Dolan:2011xt}. This considerably affects the thermal phase of the black hole; various phenomena have been already investigated such as Van der Waals fluids, reentrant phase transitions, and holographic heat engines\cite{Kubiznak:2012wp,Johnson:2014yja,Karch:2015rpa,Kubiznak:2016qmn,Hennigar:2017apu,Wei:2017vqs}.

In this work, we prove that the variation of the $D$-dimensional charged AdS black hole including four dimensions follows the first law of thermodynamics considering the thermodynamic volume term via the charged particle absorption. Further, we investigate the second law of thermodynamics. Because particle absorption is an irreversible process, the entropy of the black hole should increase. It has been already proved that the satisfaction of the first law of thermodynamics is a necessary condition to ensure the second law of thermodynamics under particle absorption\cite{Gwak:2015fsa,Gwak:2016gwj} in the case of the non-thermodynamic volume term. Nevertheless, if the second law of thermodynamics is not valid under the absorption, it would be the first violation of the second law of thermodynamics under the correct first law of thermodynamics, which would only be seen in the case considering the pressure and volume term. The second law of thermodynamics plays an important role in physical processes such as the collision of black holes\cite{Gwak:2017zwm}. Considering the thermodynamic volume term, we assume the cosmological constant as a dynamic variable in the metric of the black hole. Under this assumption, we test the weak cosmic censorship conjecture by the charged particle absorption. Moreover, in the Einstein-Maxwell action, the cosmological constant is not a dynamic variable, so we cannot test the conjecture under the level of equations of motion. Thus, the particle absorption is almost the only method to investigate the conjecture from the variation of the black hole including the pressure and volume term.

This paper is organized as follows. In section~\ref{sec2}, the charged AdS black hole is introduced, and the laws of thermodynamics are presented along with the dynamic cosmological constant. In section~\ref{sec3}, we establish the first law of thermodynamics under the charged particle absorption. Further, the second law of thermodynamics is shown to be violated in specific cases. In section~\ref{sec4}, we describe the investigation of the weak cosmic censorship conjecture for the extremal and near-extremal black holes. In section~\ref{sec5}, we briefly summarize our results.

\section{Thermodynamic Volume in Charged AdS Black Hole}\label{sec2}

The Einstein-Maxwell action with the cosmological constant in the $D$-dimensional spacetime is given as
\begin{align}\label{eq:EMCaction}
S=-\frac{1}{16\pi}\int d^D x \sqrt{-g} \left(R-F_{\mu\nu}F^{\mu\nu}-2\Lambda\right).
\end{align}
where the spacetime dimensions are denoted as $D$ and include four dimensions. Maxwell field strength $F_{\mu\nu}$ and electric potential $A_\mu$ are$A_\mu$ are
\begin{align}
F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_{\mu}, \quad  A=-\frac{Q}{r^{D-3}}dt.
\end{align}
The equations of motion from Eq.\,(\ref{eq:EMCaction}) have a static solution for the charged AdS black hole. The metric of the black hole is in $D$-dimensional spacetime
\begin{align}\label{eq:metric01}
ds^2 = - f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega_{D-2},\quad f(r) = 1-\frac{2M}{r^{D-3}}+\frac{Q^2}{r^{2D-6}}+\frac{r^2}{\ell^2},
\end{align}
where the $D-2$-dimensional sphere is analytically denoted as
\begin{align}
d\Omega_{D-2}=\sum^{D-2}_{i=1}\left( \prod_{j=1}^{i} \sin^2\theta_{j-1}\right)d\theta_i^2,\quad \theta_0\equiv\frac{\pi}{2},\quad \theta_{D-2}\equiv\phi.
\end{align}
The metric components are determined in terms of mass and charge parameters $M$ and $Q$ with an AdS radius $\ell$ in Eq.\,(\ref{eq:metric01}). These parameters are proportional to the mass $M_\text{b}$, electric charge $Q_\text{b}$, and cosmological constant $\Lambda$\cite{Tian:2010gn}.
\begin{align}
M_\text{b}=\frac{(D-2)\Omega_{D-2}}{8\pi}M,\quad Q_\text{b}=\frac{(D-2)\Omega_{D-2}}{8\pi}Q,\quad \Lambda=-\frac{(D-1)(D-2)}{2\ell^2},
\end{align}
where we set $G=1$ and $\hbar=1$ in all dimensions for simplicity. The thermal properties can be defined on the horizon $r_\text{h}$ of the black hole. Hawking temperature, Bekenstein-Hawking entropy, and electric potential are given as
\begin{align}\label{eq:temperature02}
T_\text{h}=\frac{1}{2\pi \ell^2 }\left(r_\text{h}-\frac{(D-3)Q^2\ell^2}{r_\text{h}^{2D-5}}+\frac{(D-3)M\ell^2}{r_\text{h}^{D-2}}\right),\quad S_\text{h}=\frac{A_\text{h}}{4}=\frac{\Omega_{D-2}r_\text{h}^{D-2}}{4},\quad \Phi_\text{h}=\frac{Q}{r_\text{h}^{D-3}}.
\end{align}
Then, the thermodynamic laws can be constructed for the black hole. Recently, an interesting approach has been followed to treat the cosmological constant as a thermodynamic variable. From this point of view, the cosmological constant is not a fixed value. Its actual value can be obtained from the vacuum expectation value of the theory considered, so that it can vary under a perturbation\cite{Cvetic:2010jb}. Although the variation of the cosmological constant is not concrete in the Lagrangian theory, the cosmological constant as a thermodynamic variable represents quite consistent behaviors with other thermodynamic variables\cite{Dolan:2010ha,Dolan:2011xt}. In this extended thermodynamics, the cosmological constant plays the role of pressure $P$, and its conjugate variable is thermodynamic volume of the black hole $V_\text{b}$. The definitions of thermodynamic pressure and volume are in $D$-dimensional AdS spacetime\cite{Dolan:2013ft}
\begin{align}\label{eq:volume03}
P=-\frac{\Lambda}{8\pi}=\frac{(D-1)(D-2)}{16\pi \ell^2},\quad V_\text{b}=\frac{\Omega_{D-2}}{D-1}r_\text{h}^{D-1}.
\end{align}
When we consider the pressure term in the laws of thermodynamics, the key difference is that the mass is now enthalpy in the first law of thermodynamics\cite{Kastor:2009wy,Cvetic:2010jb}. Thus, the first law of thermodynamics determines the infinitesimal change of the mass of the black hole as\cite{Dolan:2012jh,Dolan:2013ft}
\begin{align}
dM_\text{b}=T_\text{h}dS_\text{h}+\Phi_\text{h}dQ+V_\text{h}dP,
\end{align}
where the $M_\text{b}$ plays a role as an enthalpy. The enthalpy is related to the internal energy of the black hole $U_\text{b}$ and the $PV_\text{b}$ term as
\begin{align}
M_\text{b}=U_\text{b}+PV_\text{b}.
\end{align}
Therefore, the variation of the mass causes rebalancing not only of the horizon and electric charge, but also the AdS radius in $PV_\text{b}$ term. In the following section, we will investigate the change in the black hole by the charged particle absorption when the AdS radius is assumed to be infinitesimally varied because of it.

\section{Thermodynamics with Pressure and Volume under Charged Particle Absorption}\label{sec3}

We assume that the charged AdS black hole is varied by absorbing a charged particle. When the black hole absorbs the particle, the conserved quantities of the particle can perturb both the mass and charge of the black hole, and the AdS radius is subordinated to these changes owing to the contribution of thermodynamic pressure and volume. To analyze this charged particle absorption, we will obtain the relation between conserved quantities of the particle, because the conserved quantities of the black hole are changed as much as those of the particle. Then, Hamiltonian of the charged particle under an electric potential $A_\mu$ is given as
\begin{align}
\mathcal{H}=\frac{1}{2}g^{\mu\nu}(p_\mu-q A_\mu)(p_\nu-q A_\nu),
\end{align}
of which Hamiltonian equations are separable under Hamilton-Jacobi action\cite{Carter}. The momentum $p_\mu$ is obtained in terms of a partial derivative of Hamilton-Jacobi action that
\begin{align}\label{eq:hjaction}
S=\frac{1}{2}m^2 \lambda -Et+L\phi+S_r(r)+\sum_{i=1}^{D-3}S_{\theta_i}(\theta_i),\quad p_\mu=\partial_\mu S.
\end{align}
The Hamilton-Jacobi action describes a particle having a mass $m^2$, and the affine parameter is $\lambda$. The conserved quantities $E$ and $L$ are assumed from the translation symmetries of the metric in Eq.\,(\ref{eq:metric01}). Owing to $D$-dimensional solution, the black hole includes a $D-2$-dimensional sphere $\Omega_{D-2}$. The angular momentum $L$ is defined as the conserved quantity from the translation symmetry of the last angle coordinate of $\Omega_{D-2}$. Thus, the summation in Eq.\,(\ref{eq:hjaction}) runs from $i=1$ to $D-3$. To solve Hamilton-Jacobi equations, we can use the inverse metric including $D-2$-dimensional sphere, so
\begin{align}
g^{\mu\nu}\partial_\mu\partial_\nu = -f(r)^{-1}(\partial_t)^2+f(r)(\partial_r)^2+r^{-2}\sum^{D-2}_{i=1}\left( \prod_{j=1}^{i} \sin^{-2}\theta_{j-1}\right)(\partial_{\theta_i})^2.
\end{align}
The Hamilton-Jacobi equation is
\begin{align}
-2\frac{\partial S}{\partial \lambda}=-m^2=&-f(r)^{-1}(-E-qA_t)^2+f(r)(\partial_r S_r(r))^2\\
&+r^{-2}\sum^{D-3}_{i=1}\left( \prod_{j=1}^{i} \sin^{-2}\theta_{j-1}\right)(\partial_{\theta_i}S_{\theta_i}(\theta_i))^2+r^{-2}\left(\prod_{j=1}^{D-2} \sin^{-2}\theta_{j-1}\right)(L)^2,\nonumber
\end{align}
which is divided by separate variables, $\mathcal{K}$ and $R_i$.
\begin{align}
\mathcal{K}=-m^2 r^2+\frac{r^2}{f(r)}(-E+\frac{Qq}{r^{D-3}})^2-r^2 f(r)(\partial_r S_r(r))^2,\quad R_i^2=(\partial_i S_{\theta_i}(\theta_i))^2 + \sin^2\theta_i R_{i+1}^2,
\end{align}
where two variables are defined that
\begin{align}
\mathcal{K}=R_1^2,\quad L=R_{D-2}.
\end{align}
Then, we can determine the entire equations of motion. The radial- and $\theta$-directional equations are sufficient to obtain the relation between energy and electric charge of the particle. The momenta of the particle are
\begin{align}\label{eq:eoms01}
p^r\equiv \frac{\partial r}{\partial \lambda}= f(r) \sqrt{-\frac{\mathcal{K}+m^2r^2}{r^2 f(r)}+\frac{1}{f(r)^2}\left(E-\frac{Qq}{r^{D-3}}\right)^2},\quad p^\theta\equiv \frac{\partial \theta}{\partial \lambda}=\frac{1}{r^2}\sqrt{\mathcal{K}-\sin^2\theta_1 R_2^2}.
\end{align}
We attempt to determine the variation of the black hole which absorbs a charged particle. The particle is supposed to be absorbed to the black hole when it passes through the outer horizon $r_\text{h}$, because the conserved quantities of the particle are not distinguishable anymore from those of the black hole at that moment by an observer outside of the horizon. By removing the separate variable $\mathcal{K}$ in Eq.\,(\ref{eq:eoms01}), we obtain the relation between conserved quantities and momenta for a given radial location $r$. Then, at the outer horizon $r_h$, conserved quantities of the particle are absorbed to those of the black hole. At the limit of the outer horizon, the energy relation between conserved quantities and momenta is obtained as
\begin{align}\label{eq:dispersion01}
E=\frac{Q}{r_\text{h}^{D-3}}q+|p^r|,
\end{align}
in which various dependencies between variables are reduced to this simple relation. A positive sign is required in front of the $|p^r|$ term. In the positive flow of time, the particle comes into the black hole. At this moment, the energy of the particle should be defined as a  positive value, so that the signs in front of $E$ and $|p^r|$ are the same and positive\cite{Christodoulou:1970wf,Christodoulou:1972kt}. Note that we consider the energy dependent on the electric potential term. However, the potential is independent of the flow of time and only related to the interaction between particle and black hole. Thus, the total value of energy under the sum of the potential is not important, and we simply choose the positive sign in front of $|p^r|$.

Absorbing the charged particle, the black hole is varied by the same quantity as that of the particle, assuming no loss of conserved quantities in this process. This is supported by the change of the black hole following the first law of thermodynamics. The charge of the particle $q$ is coincident to the change of the charge of the black hole $dQ_\text{b}$. The energy of the particle is only given as $q$ and $|p^r|$ at the horizon in Eq.\,(\ref{eq:dispersion01}), so we must find a thermodynamic potential of which the variables also change by $q$ and $|p^r|$. If we assume that the energy of the particle changes the internal energy of the black hole, the internal energy is given as $U_\text{b}(Q_\text{b},S_\text{b},V_\text{b})$, so that its variation will be given as $dQ_\text{b}$, $dS_\text{b}$, and $dV_\text{b}$. Fortunately, these variables will be denoted in terms of those of the particle. The energy and electric charge of the particle become
\begin{align}
E=dU_\text{b}=d(M_b-PV_\text{b}),\quad q=dQ_\text{b}.
\end{align}
Then, the energy relation in Eq.\,(\ref{eq:dispersion01}) is rewritten as
\begin{align}\label{eq:dispersion02}
dU_\text{b}=\frac{Q}{r_\text{h}^{D-3}}dQ_b+|p^r|.
\end{align}
Thus, the charged particle changes the black hole as much as $(dU_\text{b},dQ_\text{b})$, and the change of the black hole volume induces the change of its conjugate variable, pressure. Under the charged particle absorption, the changed variables are $(dM_\text{b},dQ_\text{b}, d\ell)$. The other variables depend on these. To rewrite Eq.\,(\ref{eq:dispersion02}) to the first law of thermodynamics, we need to find $dS_\text{h}$ changed by the absorption. Under the variation,
\begin{align}
dS_\text{h}=\frac{1}{4}(D-2)\Omega_{D-2}r_\text{h}^{D-3}dr_\text{h},
\end{align}
where the change of the outer horizon $dr_\text{h}$ should be rewritten as independent variables such as $(q,|p^r|)$ of the particle. The particle absorption varies the function $f(r)$, and its change is the reason of moved outer horizon $r_\text{h}+dr_\text{h}$. The infinitesimally moved outer horizon $dr_\text{h}$ satisfies
\begin{align}\label{eq:function05}
df_\text{h}=\frac{\partial f_\text{h}}{\partial M_\text{b}}dM_\text{b}+\frac{\partial f_\text{h}}{\partial Q_\text{b}}dQ_\text{b}+\frac{\partial f_\text{h}}{\partial \ell}d\ell+\frac{\partial f_\text{h}}{\partial r_\text{h}}dr_\text{h}=0,\quad f_\text{h}=f(M_\text{b},Q_\text{b},\ell,r_\text{h}),
\end{align}
with
\begin{align}
\frac{\partial f_\text{h}}{\partial M_\text{b}}&=-\frac{16\pi}{(D-2)\Omega_{D-2}r_\text{h}^{D-3}},\quad \frac{\partial f_\text{h}}{\partial Q_\text{b}}=\frac{16\pi Q}{(D-2)\Omega_{D-2}r_\text{h}^{2D-6}},\\
\frac{\partial f_\text{h}}{\partial \ell}&=-\frac{2 r_\text{h}^2}{\ell^3},\quad \frac{\partial f_\text{h}}{\partial r_\text{h}}=-\frac{(2D-6)Q^2}{r_\text{h}^{2D-5}}+\frac{2(D-3)M}{r_\text{h}^{D-2}}+\frac{2r_\text{h}}{\ell^2}.
\nonumber
\end{align}
In addition, the energy relation in Eq.\,(\ref{eq:dispersion02}) is rewritten in terms of the enthalpy
\begin{align}\label{eq:dispersion03}
dM_\text{b}-d(PV_\text{h})=\frac{8\pi Q_\text{b}}{(D-2)\Omega_{D-2}r^{D-3}}dQ_\text{b}+|p^r|.
\end{align}
By combining Eqs.\,(\ref{eq:function05}) and (\ref{eq:dispersion03}), we can remove the $d\ell$ term. Then, interestingly, except for the $|p^r|$ and $dr_\text{h}$ variables, the others are removed. The change of the outer horizon becomes
 \begin{align}\label{eq:variables02}
dr_\text{h}=\frac{16 \pi r_\text{h}^{4}\ell^2 |p^r|}{\Omega_{D-2}(D-2)(D-3)(r_\text{h}^{D+2}-2 M r_\text{h}^3\ell^2 +2r_\text{h}^{D}\ell^2)}.
\end{align}
Therefore, under the energy relation, the variations of entropy, and thermodynamic volume of the black hole is obtained as
\begin{align}\label{eq:variables01}
dS_\text{h}&=\frac{4 \pi r_\text{h}^{D+1}\ell^2 |p^r|}{(D-3)(r_\text{h}^{D+2}-2 M r_\text{h}^3\ell^2 +2r_\text{h}^{D}\ell^2)},\\
dV_\text{h}&=\frac{16 \pi r_\text{h}^{D+1}\ell^2 |p^r|}{(D-2)(D-3)(r_\text{h}^{D+2}-2 M r_\text{h}^3\ell^2 +2r_\text{h}^{D}\ell^2)}.\nonumber
\end{align}
Incorporating Eqs.\,(\ref{eq:temperature02}), (\ref{eq:volume03}), and (\ref{eq:variables01}), the energy relation of Eq.\,(\ref{eq:dispersion02}) becomes the first law of thermodynamics that
\begin{align}
dU_\text{h}=\Phi_\text{h}dQ_b + T_\text{h} dS_\text{h}-PdV_\text{h}.
\end{align}
Because the mass of the black hole is already defined to the enthalpy of the black hole, the internal energy can be exchanged with the enthalpy by the Legendre transformation, so
\begin{align}
dM_\text{b}=T_\text{h}dS_\text{h}+\Phi_\text{h}dQ+V_\text{h}dP.
\end{align}
Thus, we prove the coincidence between the variation of the $D$-dimensional charged black hole and the first law of thermodynamics under the charged particle absorption.

The second law of thermodynamics expects the increase of the entropy of the black hole in an irreversible process. As the charged particle absorption is an irreversible process, the entropy becomes greater than before the absorption. The validity of this statement is easily checked by the sign of $dS_\text{h}$ in Eq.\,(\ref{eq:variables01}). Moreover, we obtain the violation of the second law of thermodynamics in parameter ranges. Specifically, the denominator of $dS_\text{h}$ has a negative value for the extremal black hole,
\begin{align}\label{eq:extremaldS01}
r_\text{h}^{D+2}-2 M r_\text{h}^3\ell^2 +2r_\text{h}^{D}\ell^2=-\frac{(D-1)r_\text{h}^{D+2}}{(D-3)}<0,
\end{align}
which means that the entropy of the black hole decreases at least for the extremal case for all dimensions, $D\geq 4$.
\begin{figure}[h] \centering\subfigure[{$Q-M$ diagram for $D=4$.}] {\includegraphics[scale=0.49,keepaspectratio]{entropy4d.eps}} \quad \centering\subfigure[{$Q-M$ diagram for $D=5$.}] {\includegraphics[scale=0.49,keepaspectratio]{entropy5d.eps}} \quad \centering\subfigure[{$Q-M$ diagram for $D=6$.}] {\includegraphics[scale=0.49,keepaspectratio]{entropy6d.eps}}\\
\caption{{\small The scaled $dS_\text{h}$ in $Q-M$ diagrams with $\ell=1$.}}
\label{fig:f1a}
\end{figure}
Therefore, the second law of thermodynamics is violated under the consideration of $PV_\text{b}$ term for the charged particle absorption. This behavior is interesting and observable only with the $PV_\text{b}$ term in the first law of thermodynamics. Note that without the $PV$ term, the second law of thermodynamics is always ensured under satisfying the first law of thermodynamics\cite{Gwak:2015fsa}. Using $dS_\text{h}$ in Eq.\,(\ref{eq:variables01}), we investigate parameter ranges within which this violation occurs. The area of the violation depends on the spacetime dimensions. The parameter space $(Q,M)$ is limited due to the extremal condition for $D$-dimensional black holes in Fig.\,\ref{fig:f1a} in which negative maxima is scaled to have around $-5$.   As already shown in Eq.\,(\ref{eq:extremaldS01}), the decrease of the entropy appears in ranges close to the extremal black holes. Then, as the black hole electrically neutralizes more than the extremal ones, the entropy increases. The area of the decrease is obtained in higher dimensions as shown in Fig.\,\ref{fig:f1a} from (a) to (c). In Fig.\,\ref{fig:f1a} (a), (b), and (c), we show parameter spaces of higher-dimensional black holes from four to six dimensions. The range of the violation appears in extremal and near-extremal condition. This also implies that the entropy needs a correction term to resolve the violation when we consider thermodynamic pressure and volume.

\section{Weak Cosmic Censorship Conjecture with Pressure and Volume}\label{sec4}

In consideration of thermodynamic volume, the charged particle absorption can reproduce the first law of thermodynamics in terms of enthalpy. However, although the particle absorption is an irreversible process, we find the violation of the second law of thermodynamics in the process, and the entropy of the black hole decreases for extremal and near-extremal black holes. Without $PV_\text{b}$ term, the second law of thermodynamics is satisfied to validate the weak cosmic censorship conjecture under the absorption. Thus, owing to the violation of the second law with the $PV_\text{b}$ term, we can expect that the cosmic censorship is affected by the term\cite{Gwak:2016gwj}.

As the violation of the second law of thermodynamics occurs in extremal and near-extremal black holes, the changes of extremal and near-extremal black holes become very different from those of non-extremal ones. This change can be estimated from a behavior of the function $f(r)\equiv f(M_\text{b},Q_\text{b},\ell,r)$ in the metric of Eq.\,(\ref{eq:metric01}). The function $f(M_\text{b},Q_\text{b},\ell,r)$ of the black hole has only one minimum point $r_\text{min}$ which satisfies
\begin{align}\label{eq:cosmicExtremal}
f(M_\text{b},Q_\text{b},\ell,r)|_{r=r_\text{min}}\equiv f_\text{min}=\delta\leq 0,\quad \partial_{r} f(M_\text{b},Q_\text{b},\ell,r)|_{r=r_\text{min}}\equiv f'_\text{min}=0,
\end{align}
with
\begin{align}
 (\partial_{r})^2 f(M_\text{b},Q_\text{b},\ell,r)|_{r=r_\text{min}}>0.
\end{align}
The minimum value of the function $f(M_\text{b},Q_\text{b},\ell,r)$ is $\delta$, and $\delta=0$ for the extremal black hole. In addition, the inner and outer horizons are located around the minimum point. We explicitly denote variables $(M_\text{b},Q_\text{b},\ell)$ changed by the absorption. The conserved quantities of the particle infinitesimally change variables into $(M_\text{b}+dM_\text{b},Q_\text{b}+dQ_\text{b},\ell+d\ell)$. Then, owing to these changes, the locations of minimum point and outer horizon both are infinitesimally moved to $r_\text{min}\rightarrow r_\text{min}+dr_\text{min}$ and $r_h\rightarrow r_h+dr_h$. Under these changes, the configuration of the black hole after the absorption can be expected from a change of the minimum value of the function $df_\text{min}$. Then, the moved minimum point satisfies
\begin{align}
\partial_{r} f(M_\text{b}+dM_\text{b},Q_\text{b}+dQ_b,\ell+d\ell,r)|_{r=r_\text{min}+dr_\text{min}}=f'_\text{min}+df'_\text{min}=0,
\end{align}
which is in terms of variables with partial derivatives
\begin{align}
df'_\text{min}=\frac{\partial f'_\text{min}}{\partial M_\text{b}}dM_\text{b}+\frac{\partial f'_\text{min}}{\partial Q_\text{b}}dQ_\text{b}+\frac{\partial f'_\text{min}}{\partial \ell}d\ell+\frac{\partial f'_\text{min}}{\partial r_\text{min}}dr_\text{min}=0,
\end{align}
with
\begin{align}
\frac{\partial f'_\text{min}}{\partial M_\text{b}}=\frac{16(D-3)\pi}{(D-2)\Omega_{D-2}r_\text{min}^{D-2}},\quad \frac{\partial f'_\text{min}}{\partial Q_\text{b}}=-\frac{16(2D-6)\pi Q}{(D-2)\Omega_{D-2}r_\text{min}^{2D-5}},\quad \frac{\partial f'_\text{min}}{\partial \ell}=-\frac{4 r_\text{min}}{\ell^3}.
\end{align}
The value of the minimum at $r_\text{min}+dr_\text{min}$ becomes
\begin{align}
f(M_b+dM_b,Q_b+dQ_b,\ell+d\ell,r)|_{r=r_\text{min}+dr_\text{min}}&=f_\text{min}+df_\text{min}\\
&=\delta+\left(\frac{\partial f_\text{min}}{\partial M_b}dM_b+\frac{\partial f_\text{min}}{\partial Q_b}dQ_b+\frac{\partial f_\text{min}}{\partial \ell}d\ell\right),\nonumber
\end{align}
where we use $f'_\text{min}=0$ in Eq.\,(\ref{eq:cosmicExtremal}). Then, we can obtain $(dM_\text{b}, d\ell)$ in terms of the particle charges $(q,|p^r|)$ under the particle absorption. Owing to the location of the absorption, the outer horizon, the value of the minimum is obtained under $(r_\text{min},r_\text{h})$. However, this is too complex to analyze and write. Instead, we can impose the condition for the near-extremal black hole that
\begin{align}
\delta \rightarrow \delta_\epsilon,\quad r_\text{h}\rightarrow r_\text{min}+\epsilon.
\end{align}
The outer horizon of the near-extremal black hole is located slightly to the right of the minimum point, and the minimum value is a very small negative value. This is given as $|\delta_\epsilon|,\,\epsilon \ll 1$. For the near-extremal black hole, the moved minimum value is
\begin{align}
f_\text{min}+df_\text{min}=&\left(\delta_\epsilon+\frac{32\pi r_\text{min}^5 (-1-(D-2)r_\text{min}^{1-2D}(-Q^2 r_\text{min}^3+M r_\text{min}^D)\ell^2)|p^r|}{\Omega_{D-2}(D-3)(D-2)(r_\text{min}^{D+2}-2M r_\text{min}^3 \ell^2 + 2 r_\text{min}^D \ell^2)}\right)+\mathcal{O}(\epsilon),
\end{align}
where we skip to write the first order of $\epsilon$. To simplify this expansion, if we remove $Q^2$ term by using $f'_\text{min}=0$, then, this becomes
\begin{align}
f_\text{min}+df_\text{min}=\delta_\epsilon + \mathcal{O}(\epsilon^2).
\end{align}
The first order of $\epsilon$ is also removed, and the minimum value of the extremal black hole is
\begin{align}
f_\text{min}+df_\text{min}=0,\quad \delta_\epsilon=0,\quad \epsilon=0.
\end{align}
Therefore, extremal and near-extremal black holes stay at their minimum, so their phases cannot be changed, even if they are charged or discharged by the charged particle absorption. This result is quite different from that in cases of no $PV_\text{b}$ term where the extremal black holes are easily broken into a non-extremal one by the absorption. Because the extremal black hole does not change its minimum value, the varied extremal black hole can stay on the $\delta=0$ surface as shown in Fig.\,\ref{fig:f2b}\,(a).
\begin{figure}[h] \centering\subfigure[{$(Q,M,\ell)$ surface satisfying $\delta=0$.}] {\includegraphics[scale=0.5,keepaspectratio]{phase4d.eps}} \quad \centering\subfigure[{$Q-M$ lines in the $\ell=1$ slice for values of $\delta$.}] {\includegraphics[scale=0.8,keepaspectratio]{phase4db.eps}}\caption{{\small $(Q,M,\ell)$ surface and $Q-M$ diagram}} \label{fig:f2b}\end{figure}
Further, from Eq.\,(\ref{eq:extremaldS01}), to decrease the entropy, the extremal black hole should be contracted under the particle absorption in Fig.\,\ref{fig:f2b}\,(b). On the plane of $(Q,M,\ell)$, the phase of the extremal black hole moves in a three-dimensional direction, because the AdS radius is changed by the absorption. However, as shown in the $M-Q$ diagram of Fig.\,\ref{fig:f2b}\,(b), the extremal black hole stays on the extremal line. The near-extremal black hole also moves its own value of $\delta$.

\section{Summary}\label{sec5}

We investigated variations of the charged AdS black hole under the charged particle absorption by considering the pressure as a cosmological constant. It is known that the thermal conjugate of the pressure is the volume of the black hole inside of the horizon. However, the cosmological constant is not a variable in the action and equations of the motion, so the dynamical effect is not easy to predict with the pressure term. To elucidate the effect of the pressure and volume terms, we consider an infinitesimal variation of the black hole by a charged particle. This is almost the only way to demonstrate the effect of an external particle without using entire equations of the motion. Then, when the particle is absorbed into the black hole, the black hole varies by as much as the conserved quantities of the charged particle. The change of the black hole exactly corresponds to the first law of thermodynamics in terms of the enthalpy. However, the second law of thermodynamics is violated for the extremal and near-extremal black holes in which the entropy decreases under the absorption. It implies that, at least, the entropy of the black hole needs a correction which should not be proportional to the outer horizon. The violation of the second law of thermodynamics can be related to the weak cosmic censorship conjecture which is related to the stability of the horizon. The stability can be shown from the change of the minimum value of the function $f(M,Q,\ell,r)$ under the absorption. Interestingly, the variation of the minimum value is quite different from that in the case without the pressure term. The minimum value of the function $f(M,Q,\ell,r)$ is not changed for extremal and near-extremal black holes under the absorption. It implies that the extremal or near-extremal black holes still stay as they are after an absorption of the external particle. Thus, even if extremal or near-extremal black holes are charged or discharged by the absorption, they maintain their extremality or near-extremality. This result cannot be seen in the charged particle absorption without the thermodynamic pressure and volume term. In addition, owing to the maintenance of its minimum value, the extremal black hole cannot be overcharged in the process. This ensures the stability of the horizon under the charged particle absorption.

\vspace{10pt}
\newpage
{\bf Acknowledgments}

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT \& Future Planning (NRF-2015R1C1A1A02037523).