text:
\section{Introduction}
Study of matter-light interaction in semiconductor nano-structures like quantum dots and quantum well (QW) have led to a large variety of interesting phenomena like the Autler-Townes doublet \citep{aut} and vacuum Rabi splitting \citep{agg, tho, khi}. The quantum properties of light like antibunching, squeezing and bistability are also exhibited by quantum systems due to the synonymity between excitonic and atomic resonances \citep{kim, man, baa, mes}. The cavity quantum electrodynamics (QED) in semiconductor systems like quantum wells and quantum dots have potential application in opto-electronic devices \citep{shi, ele1, ele2}. The cavity QED in atomic systems exhibit antibunching whereas bunching is observed in the fluorescent spectrum of the QW \citep{ere, vya}.

The cavity quantum optomechanics is the latest field of research which has grabbed attention in a vast variety of systems ranging from gravitational wave detectors (LIGO project) \citep{cor, cor1}, nanomechanical cantilevers \citep{hoh, gig, arc, kle, fav, reg}, vibrating microtoroids \citep{carm, sch1}, membranes \citep{jdt} and ultracold atoms \citep{bre, mur, bha, bha1, tre}. In recent years, there has been a continuous growing interest in cooling of optomechanical systems to their quantum ground state \citep{vit1, vit2, vit3, tar2, son1, son2}. This field has a potential application in a large variety of sensitive measurements like detection of weak forces \citep{bra1, abr}, small masses \citep{jen} and small displacements \citep{lat}. In a pioneering work \citep{bra}, it has been predicted that the dynamics of back-action arises due to the radiation pressure force by the optical field on the moving mirror of optomechanical systems. The radiation pressure exerted by an optical field on the moving end mirror of the cavity gives rise to the coupling between the intensity of the intracavity light field and displacement of the mirror. This nonlinearity arises in the system due to change in the path of light in an intensity-dependent way. Such nonlinear effects have been studied for an optomechanical system consisting of a QW \citep{eyo}. The non-linearity analogous to Kerr nonlinearity can be achieved in an optomechanical system \citep{tar1}.

Two decades ago, Moore \citep{moo} discussed the quantization of electromagnetic field in a cavity with movable perfectly reflecting boundaries. Later, Dodonov and co-workers \citep{dodc, dodc1} generalized the Moore's theory so that the effects of time-varying refractive index of the medium inside the cavity were also included. The main interest in such kind of system is the creation of photons due to the nonadiabatic distortion of the electromagnetic vacuum state. It has been presumed that a moving boundary with nonuniform motion \citep{dew, ful} or a fast changes in the refractive index of the cavity medium \citep{yab, hiz} can generate real photons from the vacuum state. This phenomenon is nowadays termed as dynamical Casimir effect which was apparently introduced by Yablonovitch \citep{yab} and Schwinger \citep{sch}. Theoretically, an effective Hamiltonian was derived for this type of system consisting of radiation inside a cavity with moving boundary and a time-varying dielectric medium in the cavity \citep{law}. It was predicted that a nonadiabatic process characterized by two-photon process helps in creating photon pairs from the vacuum state. It was identified that this two-photon character of the optical field is related to the squeezing phenomenon \citep{sar, dod1}. Further, parametric process can be observed in an optomechanical system by periodically modulating the external driving field which is also efficient in squeezing phenomenon \citep{law2, mar2}.

Motivated by these interesting features in the field of non-stationary cavity QED, we propose a system containing a quantum well within an optomechanical cavity undergoing the periodic oscillations. We investigate this non-stationary system with classical mirror motion and quantized mirror motion. The cavity frequency is periodically modulated with time. We examined the fluorescent intensity of light emitted by exciton in QW and the average number of photons within the cavity. In the end, we have proposed a quantum system composed of a QW inside an optomechanical cavity with the fixed cavity frequency and the amplitude modulated external laser pump. In this case, we observe the amplification of phonons inside the cavity.

\section{Optomechanical Cavity with Classical Mirror Motion}

In this section, we introduce the basic model of the system with QW in an optomechanical cavity. The cavity has one mirror fixed and other mirror movable as shown in figure \ref{b}. The system investigated here is composed of a semiconductor QW in an optomechanical cavity driven by an external pump laser and interacting with a single-mode optical field. The optomechanical cavity has frequency $\omega_{c}$ which is driven by an external coherent light field with frequency  $\omega_{p}$.
\begin{figure}[h]
\hspace{-0.0cm}
\includegraphics [scale=0.8]{qwmodelfig2.eps}
\caption{(color online) Schematic figure of the system consisting of Quantum Well in an Optomechanical Cavity with one fixed mirror and other movable mirror attached to Cantilever.}
\label{b}
\end{figure}
The exciton-exciton scattering is ignored as the density of excitons is small which restricts the exciton-cavity mode to a linear regime. In this section, we treat the motion of the movable mirror as a classical harmonic oscillator. Also, the dynamics of this system is studied with time varying cavity frequency. In the present system, the one-dimensional single quantized cavity mode has time-dependent frequency $\omega_{c}\left(t\right)=\omega_{c}\left(1+\epsilon\sin\left(\Omega t\right)\right)$ with unperturbed frequency $\omega_{c}$. Here $\epsilon$ is the modulation amplitude and $\Omega$ is the frequency of the periodic modulation used for time varying cavity frequency. This particular form of time dependent cavity frequency is a result of the harmonic motion of the cavity mirror. The time-dependent Hamiltonian of the system in the rotating-wave and dipole approximation is given by \citep{eyo, law}

\begin{eqnarray}\label{one}
H_{I}=\hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}(t)a^{\dagger}a+\hbar g_{0}\left(a^{\dagger}b+b^{\dagger}a\right)+i\hbar\chi\left(t\right) \left(a^{\dagger 2}e^{-2i\omega_{p}t}-a^{2}e^{2i\omega_{p}t}\right)+i\hbar\epsilon_{p}\left(a^{\dagger}e^{-i\omega_{p}t}-a e^{i\omega_{p}t}\right).
\end{eqnarray}

The first term describes the free energy of exciton in the QW. Here $b(b^{\dagger})$ is the exciton annihilation (creation) operator obeying the commutation relation $\left(\left[b,b^{\dagger}\right]=1\right)$ and $\omega_{b}$ is the exciton frequency in the QW. The second term is the energy of the single optical mode with $a(a^{\dagger})$ being the lowering (raising) operator for the photon in the cavity which follows the commutation relation as $\left(\left[a,a^{\dagger}\right]=1\right)$. The third term depicts the interaction energy between the exciton and photon. Here $g_{0}$ is the coupling constant for the exciton-photon interaction energy. The fourth term is due to the two-photon process that arises because of the time-varying cavity field. This term has close connection with the Hamiltonian of the harmonic oscillator with a time-dependent frequency \citep{law}. Here $\chi(t)$ is the effective frequency which is arbitrary function of time. The last term in the Hamiltonian is the energy due to the external pump laser where $\epsilon_{p}$ is the amplitude of laser pump.

Now we will discuss in brief about the fourth term in the Hamiltonian. There are normally two types of nonadiabatic processes in a system composed of an optical cavity with one mirror fixed and other mirror movable as mentioned in ref. \citep{law}. First one is characterized by $a^{\dagger}a$ terms in the Hamiltonian. In this type of process, the total number of photons inside the cavity remains same. The second one is characterized by the terms $a^{\dagger 2}$ or ${a^{2}}$. In this process, photon pairs can be created from the vacuum state. This kind of process is known as two-photon process. This two-photon process is responsible for the fourth term in the Hamiltonian $\left(\ref{one}\right)$. It is related to the squeezing phenomenon \citep{plu}. Due to the fact that the two-photon character of the optical field helps in creating the photon pairs from the vacuum state, therefore, it is also used to study the Dynamical Casimir Effect in various systems \citep{dod5, dod2, dod3}.

The dynamics of the system is fully described when the fluctuation dissipation processes affecting the optical and excitonic modes are included into the system. The interaction of the system with external degrees of freedom brings the dissipation into the system. The leakage of photons through the mirrors damps the optomechanical cavity field. $\kappa$ is the decay constant of cavity light field.The spontaneous decay rate of exciton is $\gamma$. The full dynamics of the system can be described by the following set of quantum Langevin equations (QLE) by taking into account all the dissipative processes in the frame rotating at the frequency $\omega_{p}$:

\begin{equation}\label{two}
\dot{b}=-i\delta_{b}b-ig_{0}a-\gamma b+\sqrt{2\gamma}b_{in},
\end{equation}

\begin{equation}\label{three}
\dot{a}=-i\Delta a-i\omega_{c}\epsilon\sin\left(\Omega t \right)a-ig_{0}b+2\chi(t)a^{\dagger}+\epsilon_{p}-\kappa a+\sqrt{2\kappa}a_{in},
\end{equation}

where $\delta_{b}=\omega_{b}-\omega_{p}$ is the exciton-pump detuning and $\Delta=\omega_{c}-\omega_{p}$ is the cavity-pump detuning. In the above equations, $b_{in}$ and $a_{in}$ represent the input vacuum noise of exciton and cavity photons respectively. Their non zero correlation functions are given as follows \citep{eyo, eyo21}

\begin{equation}\label{four}
\left\langle b_{in}(t)b_{in}^{\dagger}(t')\right\rangle =\left\langle a_{in}(t)a_{in}^{\dagger}(t')\right\rangle =\delta\left(t-t'\right).
\end{equation}

In various studies, there is a relation between the functions $\omega_{c}(t)$ and $\chi(t)$ which is given as \citep{law}

\begin{equation}\label{fivea}
\chi(t)=\frac{1}{4\omega_{c}(t)}\frac{d\omega_{c}(t)}{dt}.
\end{equation}

The time modulation with small amplitude is considered for the realistic case i.e. we take $\vert\epsilon\vert\ll1$. Therefore from Eqn. \ref{fivea}, we obtain

\begin{equation}\label{fiveb}
\chi(t)\approx\frac{\epsilon \Omega}{4}\cos\left(\Omega t\right)\approx2\chi_{0}\cos\left(\Omega t \right),
\end{equation}

where $\chi_{0}=\left(\epsilon\Omega\right)/8$. We solve the coupled QLE [Eqs. \ref{two} and \ref{three}] numerically using MATHEMATICA 9.0 to study the fluorescent intensity of light emitted by the excitons in the QW. The dynamics of the system is also studied via the evolution of power $\left(\left\langle a^{\dagger}a\right\rangle\right)$. We have investigated the system under the strong modulation $\left(\chi_{0}\gtrsim g_{0}\right)$ and weak modulation $\left(\chi_{0}\ll g_{0}\right)$ and in two different limits i.e. in the bad cavity limit $(\kappa\gg\gamma)$ in the regime $\kappa>\epsilon$ and in the good cavity limit $(\kappa\ll\gamma)$ in the regime $\kappa\lesssim\epsilon$.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig1a.eps}& \includegraphics [scale=0.75] {fig1b.eps}\\
\includegraphics [scale=0.75]{fig1c.eps}& \includegraphics [scale=0.75] {fig1d.eps}\\
 \end{tabular}
\caption{(Color online) Plots of intracavity mean number of photons $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ inside an optomechanical cavity with classical mirror motion versus scaled time $\left(\gamma t\right)$ using time modulated cavity frequency at resonant frequency $\left(\Omega = \omega_{c}\right)$ for two modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line). The parameters used are $\delta_{b} = 2\gamma$, $\Delta = 4.712\gamma$, $\epsilon_{p} = 5\gamma$, $\gamma_{m} = 10^{-5}\gamma$, $n_{th} = 175$ and $\omega_{c} =1.36\gamma$ . Plots (a) and (b) show the variation of $A(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ and weak modulation $\left(g_{0} = 5\gamma\right)$ respectively in the bad cavity limit for $\kappa>\epsilon \left(\kappa = 1.5\gamma\right)$. Plots (c) and (d) show the variation of $A(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ and weak modulation $\left(g_{0} = 5\gamma\right)$ in the good cavity limit for $\kappa\lesssim\epsilon \left(\kappa = 0.1\gamma\right)$.}
\label{c}
\end{figure}

In figure \ref{c}, the intracavity photon number $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ is plotted with scaled time $\left(\gamma t\right)$ under the strong and weak modulation regimes for both $\kappa>\epsilon$ and $\kappa\lesssim\epsilon$ at resonant modulating frequency $\left(\Omega=\omega_{c}\right)$ for two different modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line). It is clearly seen from the plot that under the strong modulation, there is slight difference between the photon number for the two modulation amplitudes. For the case of weak modulation, there is not much deviation between the two plots. Under the strong modulation, periodic oscillations are observed with time. It shows an increase in the photon number inside the cavity due to the two-photon process. But for the weak modulation, the oscillations decay with time. This shows that the two-photon process is not dominant under the weak modulation. Initially, an increase in the intracavity photon number is noticed both for the weak and strong modulation. This is because initially excitons are in the excited state. They deexcite to the ground state by releasing photons in the cavity. In the case of strong modulation, the amplitude of initial peak is high as here photon number is amplified inside the cavity due to the two-photon process. Also, under the strong and weak modulations, we observe that increase in modulation amplitude enhances the average number of photons inside the cavity. Hence for a non-stationary system composed of a QW confined in an optomechanical cavity with classical mirror motion shows the periodic increase and decrease in photon number inside the cavity under the strong modulation but for the weak modulation, the photon number exhibits non-periodic damped oscillations.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig2a.eps}& \includegraphics [scale=0.75] {fig2b.eps}\\
\includegraphics [scale=0.75]{fig2c.eps}& \includegraphics [scale=0.75] {fig2d.eps}\\
 \end{tabular}
\caption{(Color online) Plots of intensity of fluorescent light $\left(B(t)=\langle b^{\dagger}b\rangle\right)$ versus scaled time $\left(\gamma t\right)$ inside an optomechanical cavity using time modulated cavity frequency at resonant frequency $\left(\Omega = \omega_{c}\right)$ for two modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line). Plots (a) and (b) show the variation of $B(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ and weak modulation $\left(g_{0} = 5\gamma\right)$ respectively in the bad cavity limit for $\kappa>\epsilon \left(\kappa = 1.5\gamma\right)$. Plots (c) and (d) show the variation of $B(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ and weak modulation $\left(g_{0} = 5\gamma\right)$ in the good cavity limit for $\kappa\lesssim\epsilon \left(\kappa = 0.1\gamma\right)$. The other parameters used are same as in figure \ref{c}}.
\label{d}
\end{figure}

Figure \ref{d} shows the fluorescent spectrum of light $\left(B(t)=\langle b^{\dagger}b \rangle\right)$ with scaled time $\left(\gamma t\right)$ at resonant modulating frequency $\left(\Omega=\omega_{c}\right)$ under the strong and weak modulation regimes for both $\kappa>\epsilon$ and $\kappa\lesssim\epsilon$ for two different modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line). The fluorescent light amplitude is proportional to the number of excitons. The figure depicts that the intensity of the fluorescent light increases as the modulation amplitude of time-modulated cavity frequency increases. However, the difference between the intensities is very small. Also, one can see that the intensity of fluorescent light decays with time. In general, the fluorescent spectrum exhibits nonperiodic damped oscillations. Moreover, there is an amplification in the mean number of excitons in the QW for the initial moment under strong modulation which gradually decreases and becomes stationary with time. However, under the weak modulation, the average number of excitons decreases initially, but gradually, the intracavity photons excite one or more excitons in the quantum well, leading to enhanced emission of fluorescence under the weak modulation. Furthermore, a saturation is observed in excitation of excitons for both strong and weak modulations. A system composed of a QW in an optical cavity with constant cavity frequency was considered \citep{eyo3}. It has been observed that the intensity of the fluorescent light emitted by excitons in the QW exhibits nonperiodic damped oscillations. Here, similar behaviour is observed in the oscillations of the fluorescent spectrum.

We have also analyzed the effect on the mean number of intracavity photons $\left(A(t)\right)$ and fluorescent spectrum of light $\left(B(t)\right)$ by turning off the external driving field $\left( \epsilon_{p}=0\right)$. It was noticed that the behaviour of the intracavity photons $\left(A(t)\right)$ and fluorescent spectrum of light $\left(B(t)\right)$ remain same but their magnitude decreases. It should be noted here that fluorescence photons is directly proportional to number of excitons in the cavity. Comparing figures of fluorescent spectrum of light $\left(B(t)\right)$ under strong modulation (see figs. \ref{d}(a) and (c)) and weak modulation (see figs. \ref{d}(b) and (d)), one can notice that initially there is a rise in the mean number of excitons under the strong modulation and a corresponding dip under weak modulation. This clearly shows that under the strong modulation the character of two-photon dominates which amplifies photons in the cavity leading to enhancement in the number of excitons initially. However, under the weak modulation, the effect of two-photon process is less as a result of which there is a decrease in the initial number of excitons in the cavity. Similar result was observed in a system with constant cavity frequency \citep{eyo3}. Therefore, by looking at the fluorescent spectrum of light at initial stage, one could easily detect the two-photon process. As time progresses, it is also observed that the steady state value of intracavity photon number $\left(A(t)\right)$ due to the two-photon process is more than the photon number due to the fluorescence $\left(B(t)\right)$ under the strong modulation. However, under the weak modulation, the photon number due to two-photon process $\left(A(t)\right)$ inside the cavity is less as compared to the intracavity photon number due to the process of fluorescence $\left(B(t)\right)$. This can be explained as under the strong modulation, the phenomenon of two-photon process dominates which amplifies photons in the cavity whereas under the weak modulation, the phenomenon of fluorescence dominates. This also shows the balance of energy between different degrees of freedom (optical mode, excitonic mode) of the system. One more thing noticeable here is that the peak intensities of intracavity photons, in general, are more in the good cavity limit.

In the next section, we will study the phenomenon of two-photon process in the same system by considering quantized motion of the movable mirror.

\section{Optomechanical Cavity with Quantized Mirror motion}

In this section, we investigate the same system consisting of QW with an optomechanical oscillator in a light field as shown in figure \ref{b}. In addition, here we consider the quantized mirror motion. Therefore, again due to the rapid motion of the mirror, the cavity frequency sinusoidally modulates with time. In this section, the movable mirror is treated as a quantum mechanical oscillator with frequency $\omega_{m}$. A force proportional to the photon number in the cavity acts on the resonator. This further modulates the various couplings between the different modes of the system. The Hamiltonian with an optomechanical resonator under rotating-wave and dipole approximation is given as

\begin{eqnarray}\label{six}\nonumber
H_{II} &=& \hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}\left(p^{2}+q^{2}\right)+\hbar g_{m}\left(t\right)a^{\dagger}aq+\hbar g_{0}\left(a^{\dagger}b+b^{\dagger}a\right)\nonumber \\ &+& \hbar g\left(t\right)\left(a^{\dagger}b+b^{\dagger}a\right)q+i\hbar\chi\left(t\right) \left(a^{\dagger 2}e^{-2i\omega_{p}t}-a^{2}e^{2i\omega_{p}t}\right)q+i\hbar\epsilon_{p}\left(a^{\dagger}e^{-i\omega_{p}t}-a e^{i\omega_{p}t}\right),
\end{eqnarray}

The derivation of the above Hamiltonian is given in Appendix. Here the third term represents the free energy of the mechanical oscillator. The dimensionless position and momentum operator of the movable mirror are represented by $q$ and $p$ respectively which obeys the commutation relation $\left(\left[q,p\right]=i\hbar\right)$. The fourth term gives the time-dependent interaction energy between the oscillator and cavity photons, where $g_{m}(t)=\omega_{c}\epsilon\sin\left(\Omega t\right)$ is the time-dependent coupling parameter. The sixth term depicts the three-body time-dependent interaction between exciton, cavity photon and mechanical mode with $g(t)=g_{0}\epsilon\sin\left(\Omega t\right)/2$ as time-dependent interaction parameter. A viscious force acts on the mechanical oscillator which damps the mechanical mode with damping rate $\gamma_{m}$. A Brownian stochastic force also effects the mechanical mode with zero-mean $\zeta$. This force is non-Markovian Gaussian noise \citep{abd, gio}. Taking into account all the dissipation processes, the QLE of the system with an optomechanical resonator in the rotating frame at frequency $\omega_{p}$ are given as:

\begin{equation}\label{sevena}
\dot{b}=-i\delta_{b}b-ig_{0}a-ig(t)aq-\gamma b+\sqrt{2\gamma}b_{in},
\end{equation}

\begin{equation}\label{sevenb}
\dot{a}=-i\Delta a-ig_{m}(t)aq-ig_{0}b-ig(t)bq+2\chi(t)a^{\dagger}q+\epsilon_{p}-\kappa a+\sqrt{2\kappa}a_{in},
\end{equation}

\begin{equation}\label{sevenc}
\dot{q}=\omega_{m}p,
\end{equation}

\begin{equation}\label{sevend}
\dot{p}=-\omega_{m}q-g_{m}(t)a^{\dagger}a-ig(t)\left(a^{\dagger}b+b^{\dagger}a\right)-i\chi(t)\left(a^{\dagger 2}-a^{2}\right)-\gamma_{m}p+\zeta(t),
\end{equation}

To a good approximation, the correlation function of the Brownian noise in the limit of a large mechanical quality factor i.e. $\omega_{m}/\gamma_{m} \gg 1$ can be given as \citep{eyo}

\begin{equation}\label{eight}
\left\langle\zeta(t)\zeta(t')\right\rangle\cong\gamma_{m}\left(2n_{th}+1\right) \delta\left(t-t'\right)
\end{equation}

where $n_{th}=\left[\exp\left(\hbar\omega_{m}/k_{B}T\right)-1\right]^{-1}$ gives the average number of thermal photons with $k_{B}$ representing the Boltzmann constant and $T$ is the temperature of the bath connected to the movable mirror. By using all the correlations of different noises [Eqs. \ref{four} and \ref{eight}] , the coupled differential equations [Eqs. \ref{sevena}-\ref{sevend}] are solved with the help of MATHEMATICA 9.0 to study the effect of time modulation in the system. Here also, we investigate this system under the same regimes as done in the previous section. Apart from the features noticed in the previous section for the photon number inside the cavity and the fluorescent spectrum of light, here we note some additional observations in the plots discussed below.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig3a.eps}& \includegraphics [scale=0.75] {fig3b.eps}\\
\includegraphics [scale=0.75]{fig3c.eps}& \includegraphics [scale=0.75] {fig3d.eps}\\
 \end{tabular}
\caption{(Color online) Plots (a) and (b) show the intracavity mean number of photons $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ and plots (c) and (d) show the corresponding fluorescent spectrum of light $\left(B(t)=\langle b^{\dagger}b\rangle\right)$ emitted by excitons in the QW  within an optomechanical cavity versus scaled time $\left(\gamma t\right)$ using time modulated cavity frequency for two modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line) in the bad cavity limit with $\kappa>\epsilon \left(\kappa = 1.5\gamma\right)$ at resonant modulating frequency $\left(\Omega=\omega_{m}\right)$. The parameters are $\delta_{b} = 2\gamma$, $\Delta = 4.712\gamma$, $\epsilon_{p} = 5\gamma$, $\gamma_{m} = 10^{-5}\gamma$, $n_{th} = 175$, $\omega_{c} =1.36\gamma$ and $\omega_{m} = 4.712\gamma$. Plots (a) and (c) show the variation of $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ and $\left(B(t)=\langle b^{\dagger}b\rangle\right)$ under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ and plots (b) and (d) show the same variation under the weak modulation $\left(g_{0} = 5\gamma\right)$.}
\label{e}
\end{figure}

In plot \ref{e}, we investigate the system consisting of QW confined within an optomechanical cavity with quantized mirror motion. Plots \ref{e}(a) and \ref{e}(b) illustrate the mean number of intracavity photons $\left(A(t)=\langle a^{\dagger}a\rangle\right)$  with scaled time $\left(\gamma t\right)$ under the strong modulation $(g_{0}=0.005\gamma)$ and weak modulation $(g_{0}=5\gamma)$ respectively in the bad cavity limit $(\kappa\gg\gamma)$ and for $\kappa>\epsilon$ $(\kappa=1.5\gamma)$ at resonant modulating frequency $(\Omega=\omega_{m})$ using two different values of modulation amplitude, $\epsilon=0.1$ (thick line) and $\epsilon=0.2$ (thin line). As time progresses, amplification is observed in the intracavity photon number due to the two-photon process. This additional amplification in photon number in the optomechanical cavity arises due to the external force applied by the movable mirror. In this case, there is an additional degree of freedom (mechanical mode) which exchanges energy with different degrees of freedom of the system. Also, the maximum amount of energy gets transferred between these degrees of freedom at resonant modulating frequency. This is because at resonant frequency, the modulation frequency is same as the natural frequency of the mirror. Furthermore, increase in the modulation amplitude enhances the amplification in the photon number inside the cavity. Also, under the weak modulation, the magnitude of amplification in the intracavity photons is less as compared to that in the strong modulation as the effect of two-photon process is less in the weak modulation. Plots \ref{e}(c) and \ref{e}(d) illustrate the corresponding intensity of fluorescent light emitted by excitons in the QW as a function of scaled time $\left(\gamma t\right)$ under the strong modulation $(g_{0}=0.005\gamma)$ and weak modulation $(g_{0}=5\gamma)$ respectively in the bad cavity limit $(\kappa\gg\gamma)$ and for $\kappa>\epsilon$ $(\kappa=1.5\gamma)$ at resonant frequency $(\Omega=\omega_{m})$ for two modulation amplitudes, $\epsilon=0.1$ (thick line) and $\epsilon=0.2$ (thin line). Plot \ref{e}(c) shows a very small amplification in the intensity of light emitted by excitons in the QW for higher modulation amplitude $(\epsilon=0.2)$. Plot \ref{e}(d) shows the significant amplification in the intensity of light emitted by excitons in the QW under the weak modulation. The increase in the modulation amplitude enhances the amplification in the intensity of light with time.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig4a.eps}& \includegraphics [scale=0.75] {fig4b.eps}\\
\includegraphics [scale=0.75]{fig4c.eps}& \includegraphics [scale=0.75] {fig4d.eps}\\
 \end{tabular}
\caption{(Color online) Plots of intracavity mean number of photons $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ inside an optomechanical cavity with scaled time $\left(\gamma t\right)$ using time modulated cavity frequency for two modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line) in the good cavity limit with $\kappa\lesssim\epsilon \left(\kappa = 0.1\gamma\right)$. Plots (a) and (b) show the variation of $A(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ for resonant $\left(\Omega = \omega_{m}\right)$ and off-resonant modulating frequency $\left(\Omega = 2\omega_{m}\right)$ respectively . Plots (c) and (d) show the variation of $A(t)$ with time under the weak modulation $\left(g_{0} = 5\gamma\right)$ for resonant $\left(\Omega = \omega_{m}\right)$ and off-resonant modulating frequency $\left(\Omega = 2\omega_{m}\right)$ respectively. The other parameters used are same as in figure \ref{e}.}
\label{f}
\end{figure}

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig5a.eps}& \includegraphics [scale=0.75] {fig5b.eps}\\
\includegraphics [scale=0.75]{fig5c.eps}& \includegraphics [scale=0.75] {fig5d.eps}\\
 \end{tabular}
\caption{(Color online) Plots of intensity of fluorescent light $\left(B(t)=\langle b^{\dagger}b\rangle\right)$ vs scaled time $\left(\gamma t\right)$ inside an optomechanical cavity using time modulated cavity frequency for two modulation amplitudes $\epsilon = 0.1$ (thick line) and $\epsilon = 0.2$ (thin line) in the good cavity limit with $\kappa\lesssim\epsilon \left(\kappa = 0.1\gamma\right)$. Plots (a) and (b) show the variation of $B(t)$ with time under the strong modulation $\left(g_{0} = 0.005\gamma\right)$ for resonant $\left(\Omega = \omega_{m}\right)$ and off-resonant modulating frequency $\left(\Omega = 2\omega_{m}\right)$ respectively . Plots (c) and (d) show the variation of $B(t)$ with time under the weak modulation $\left(g_{0} = 5\gamma\right)$ for resonant $\left(\Omega = \omega_{m}\right)$ and off-resonant modulating frequency $\left(\Omega = 2\omega_{m}\right)$ respectively. The other parameters used are same as in figure \ref{e}.}
\label{h}
\end{figure}

Figure \ref{f} shows the mean number of photons inside the optomechanical cavity with scaled time $\left(\gamma t\right)$ under strong and weak modulation in the good cavity limit $(\kappa\ll\gamma)$ and for $\kappa\lesssim\epsilon$ $(\kappa=0.1\gamma)$ using two different values of modulation amplitude, $\epsilon=0.1$ (thick line) and $\epsilon=0.2$ (thin line). Plot \ref{f}(a) illustrates the case of strong modulation $(g_{0}=0.005\gamma)$ at resonant modulating frequency $\left(\Omega=\omega_{m}\right)$. It depicts that as time progresses, the mean number of photons inside the optomechanical cavity firstly decreases and then increases (revival of oscillations). This clearly shows the character of two-photon process as there is amplification in the intracavity photons with time. Fig \ref{f}(b) demonstrates the off-resonant case $\left(\Omega=2\omega_{m}\right)$ under the strong modulation $(g_{0}=0.005\gamma)$. It clearly shows a wavepacket like behaviour with periodic increase and decrease in the intracavity photon number for both the modulation amplitudes. Moreover, here amplification is observed much faster for higher modulation amplitude $(\epsilon=0.2)$. Plots \ref{f}(c) and \ref{f}(d) show the case of weak modulation $(g_{0}=5\gamma)$ at resonant modulating frequency $\left(\Omega=\omega_{m}\right)$ and off-resonant frequency $\left(\Omega=2\omega_{m}\right)$ respectively. At resonant frequency, we are able to distinguish the plots corresponding to two modulation amplitudes (see fig. \ref{f}(c)) while at off-resonant frequency, we are unable to distinguish these plots (see fig. \ref{f}(d)).

Plot \ref{h} illustrates the intensity of fluorescent light emitted by the excitons in the QW with scaled time $(\gamma t)$ under the strong and weak modulation in the good cavity limit $(\kappa\ll\gamma)$ and for $\kappa\lesssim\epsilon$ $(\kappa=0.1\gamma)$ using two modulation amplitudes, $\epsilon=0.1$ (thick line) and $\epsilon=0.2$ (thin line). Figures \ref{h}(a) and \ref{h}(b) show the case of strong modulation $(g_{0}=0.005\gamma)$ for resonant $\left(\Omega=\omega_{m}\right)$ and off-resonant $\left(\Omega = 2\omega_{m}\right)$ frequency respectively. Figures \ref{h}(c) and \ref{h}(d) show the case of weak modulation $(g_{0}=5\gamma)$ for resonant $\left(\Omega=\omega_{m}\right)$ and off-resonant $\left(\Omega = 2\omega_{m}\right)$ frequency respectively. The features observed in this case, at resonant modulating frequency (see figs. \ref{h}(a) and \ref{h}(c)), are very similar to the one observed in the bad cavity limit (see figs. \ref{e}(c) and \ref{e}(d)). At off resonant frequency, here we observe wavepacket like behaviour with very small amplitude under the strong modulation (see fig. \ref{h}(b)). Also, least disparity is observed in the oscillations of the intensity of fluorescent light for the two values of modulation amplitudes under the weak modulation (see figure \ref{h}(d)).

In addition to all the above observations seen in the previous section where we considered classical motion of the mirror, here, we observe some more interesting features using the quantized motion of cavity mirror. Firstly, the amplification is clearly observed in intracavity photons in all the cases. This is due to the fact that the system now possesses an extra mechanical mode arising from the quantized mirror motion. So in this case, the coherent energy exchange takes place among three degrees of freedom. Moreover, the off-resonant case of intracavity photon number (see fig. \ref{f}(b)) in the good cavity limit also shows the amplification and wavepacket like behaviour under the strong modulation which is not observed in the weak modulation. This shows that to observe the amplification in the intracavity photons and the phenomenon of two-photon process effectively in a system composed of QW confined in optomechanical cavity, the motion of the cavity movable mirror should be treated quantum mechanically rather than classically.

In the next section, we investigate the phonon amplification for the system consisting of QW in an optomechanical cavity with constant cavity frequency and amplitude modulated external pump laser beam.

\section{Phonon Amplification}

In this section, we study the effect of periodic modulation of external laser pump on different modes of the system consisting of QW in an optomechanical cavity as shown in figure \ref{b}. Here, we consider fixed cavity frequency. Rewriting the Hamiltonian for this case as \citep{eyo,far, mar1, mar2, law2}

\begin{eqnarray}\label{nine}
H_{III}=\hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}\left(p^{2}+q^{2}\right)+\hbar g_{m}a^{\dagger}aq+\hbar g_{0}\left(a^{\dagger}b+b^{\dagger}a\right)+i\hbar\epsilon_{p}'(t)\left(a^{\dagger}e^{-i\omega_{p}t}-a e^{i\omega_{p}t}\right),
\end{eqnarray}

where \begin{eqnarray}
\epsilon_{p}'(t)=\epsilon_{p}\left(1+\eta\cos\left(\lambda t\right)\right).
\end{eqnarray}

Here $\eta$ represents the ampitude of modulation and $\lambda$ represents the frequency of modulation. In this case, the external driving field undergoes the oscillatory motion but the internal parameters of the system remains unchanged. Now, the QLE for the system in the frame rotating at frequency $\omega_{p}$ are given as follows:

\begin{equation}\label{tena}
\dot{b}=-i\delta_{b}b-ig_{0}a-\gamma b+\sqrt{2\gamma}b_{in},
\end{equation}

\begin{equation}\label{tenb}
\dot{a}=-i\Delta a-ig_{m}aq-ig_{0}b+\epsilon_{p}\left(1+\eta\cos\left(\lambda t\right)\right)-\kappa a+\sqrt{2\kappa}a_{in},
\end{equation}

\begin{equation}\label{tenc}
\dot{q}=\omega_{m}p,
\end{equation}

\begin{equation}\label{tend}
\dot{p}=-\omega_{m}q-g_{m}a^{\dagger}a-\gamma_{m}p+\zeta(t),
\end{equation}

By using the correlations given in Eqns. $\left(\ref{four}\right) $ and $\left( \ref{eight}\right)$, we have solved these coupled differential equations using MATHEMATICA 9.0. We have investigated the position dynamics of the mirror, the intracavity photon number and fluorescent spectrum of light with time for the present system. We have considered the strong and weak modulation of the amplitude modulated laser beam in both the limits of good cavity $(\kappa\ll\omega_{m})$ and bad cavity $(\kappa\gg\omega_{m})$.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig7a.eps}& \includegraphics [scale=0.75] {fig7b.eps}\\
\includegraphics [scale=0.75]{fig7c.eps}& \includegraphics [scale=0.75] {fig7d.eps}\\
 \end{tabular}
\caption{(Color online) Plots of mean position of mirror $\left\langle q(t)\right\rangle$ of an optomechanical cavity with scaled time $\left(\omega_{m} t\right)$ at resonant modulating frequency $\left(\lambda = \omega_{m}\right)$ for two different modulation amplitudes $\eta = 0.1$ (thick line) and $\eta=0.4$ (thin line) using an amplitude modulated external laser beam. Plots (a) and (b) show the variation of $\left\langle q(t)\right\rangle$ with time for weak modulation $\left(g_{0} = 1.061\omega_{m}\right)$ and strong modulation $\left(g_{0} = 0.005\omega_{m}\right)$ respectively in the good cavity limit with $\kappa\lesssim\epsilon \left(\kappa = 0.1\omega_{m}\right)$. Plots (c) and (d) show the variation of $\left\langle q(t)\right\rangle$ with time for weak modulation $\left(g_{0} = 1.061\omega_{m}\right)$ and strong modulation $\left(g_{0} = 0.005\omega_{m}\right)$ respectively in the bad cavity limit with $\kappa>\epsilon \left(\kappa = 1.5\omega_{m}\right)$. The various parameters used are $\delta_{b} = 0.459\omega_{m}$, $\Delta = \omega_{m}$, $\epsilon_{p} = 1.5\omega_{m}$, $\gamma_{m} = 10^{-5}\omega_{m}$, $n_{th} = 175$, $\gamma =0.212\omega_{m}$ and $g_{m} = 0.1\omega_{m}$.}
\label{i}
\end{figure}

Figure \ref{i} shows the position dynamics $\langle q(t)\rangle$ of the movable mirror with scaled time $\left(\omega_{m}t\right)$ at resonant modulating frequency $\left(\lambda=\omega_{m}\right)$ under strong and weak modulation for two different modulation amplitudes, $\eta=0.1$ (thick line) and $\eta=0.4$ (thin line). Oscillatory behaviour is observed in all the cases. Figures \ref{i}(a) and \ref{i}(b) represent the case of strong and weak modulations respectively in the good cavity limit $(\kappa\ll\omega_{m})$ and for $\kappa\lesssim\epsilon$ $(\kappa=0.1\omega_{m})$. The plots clearly show the amplification in the number of phonons with time. As the modulation amplitude increases the number of phonons in the cavity increases. Plot \ref{i}(a) shows huge enhancement in the phonons with time. Plot \ref{i}(b) demonstrates a slight increase in the phonon number initially. As time passes, the oscillations become steady. Plots \ref{i}(c) and \ref{i}(d) depict the position dynamics for strong and weak modulations respectively in the bad cavity limit $(\kappa\gg\omega_{m})$ and for $\kappa>\epsilon$ $(\kappa=1.5\gamma)$. As observed in the previous case, the phonon number increases with increase in modulation amplitude. In this case, large amplification is observed in the phonons inside the cavity for both the strong and weak modulations. However, the magnitude of amplification in the phonons is more in strong modulation than in the weak modulation.

\begin{figure}[h]
\hspace{-0.0cm}
\begin{tabular}{cc}
\includegraphics [scale=0.75]{fig8a.eps}& \includegraphics [scale=0.75] {fig8b.eps}\\
\includegraphics [scale=0.75]{fig8c.eps}& \includegraphics [scale=0.75] {fig8d.eps}\\
 \end{tabular}
\caption{(Color online) Plots (a) and (b) show the intracavity mean number of photons $\left(A(t)=\langle a^{\dagger}a\rangle\right)$ and plots (c) and (d) show the corresponding fluorescent spectrum of light $\left( B(t)=\langle b^{\dagger}b\rangle\right)$ emitted by excitons in the QW within an optomechanical cavity versus scaled time $\left(\omega_{m} t\right)$ for two modulation amplitudes $\eta = 0.1$ (thick line) and $\eta = 0.4$ (thin line) at resonant modulating frequency $\left(\lambda = \omega_{m}\right)$ in the bad cavity limit with $\kappa>\epsilon \left(\kappa = 1.5\omega_{m}\right)$ using an amplitude modulated external laser beam. The other parameters used are same as in figure \ref{i}.}
\label{j}
\end{figure}

Figures \ref{j}(a) and \ref{j}(b) show the intracavity photon number $(A(t))$ with scaled time $\left(\omega_{m}t\right)$ using amplitude modulated external laser pump beam at resonant modulating frequency $\lambda=\omega_{m}$ in the bad cavity limit $(\kappa\gg\omega_{m})$ and for $\kappa>\epsilon\left(\kappa = 1.5\omega_{m}\right)$ using two modulation amplitudes, $\eta=0.1$ (thick line) and $\eta=0.4$ (thin line). Figure \ref{j}(a) shows the case of weak modulation $\left(g_{0}=1.061\omega_{m}\right)$. It shows a rise in the photon number initially which eventually decreases with time. Periodic oscillations are observed gradually in the mean number of intracavity photons. Again, the amplitude of intracavity photon number increases with higher modulation amplitude. Figure \ref{j}(b) represents the case of strong modulation $\left(g_{0}=0.005\omega_{m}\right)$. Initial rise is observed in the photons inside the cavity which gradually decreases with time. This clearly illustrates the reverse of two-photon process. Plots \ref{j}(c) and \ref{j}(d) show the corresponding fluorescent spectrum of light $(B(t))$ with scaled time $\left(\omega_{m}t\right)$ at resonant modulating frequency $\lambda=\omega_{m}$ in the bad cavity limit $(\kappa\gg\omega_{m})$ and for $\kappa>\epsilon\left(\kappa = 1.5\omega_{m}\right)$ using two modulation amplitudes, $\eta=0.1$ (thick line) and $\eta=0.4$ (thin line). Figure \ref{j}(c) depicts the case of weak modulation $\left(g_{0}=1.061\omega_{m}\right)$. It shows periodic oscillations with an initial rise in the intensity of fluorescent light. This intensity increases with increase in the modulation amplitude. Figure \ref{j}(d) represents the case of strong modulation $\left(g_{0}=0.005\omega_{m}\right)$. This plot demonstrates that the fluorescent spectrum increases initially but gradually becomes steady with time. However, one can notice that there is slight amplification in the intensity of fluorescent light which increases with increase in modulation amplitude.

Comparing figures \ref{i}(d) and \ref{j}(b), one can clearly notice that there is enhancement in position dynamics of movable mirror and decrease in the intracavity photon number. This shows that the external pump force helps in amplifying phonons. Moreover, the intensity of fluorescent light (see fig. \ref{j}(d)) also shows slight amplification implying that the intracavity photons produced due to two-photon process are converted into fluorescence photons. This clearly illustrates the reverse of two-photon process.

In our calculations, the experimentally realizable parameters used for all the three systems are discussed as follows. The cavity field can have decay rate $\kappa = 2\pi \times 8.75$ kHz \citep{nag} ($2\pi \times 0.66$ MHz \citep{mur}). The frequency of the mechanical mode in an optomechanical system can be varied from $2\pi \times 100$ Hz \citep{gar}, $2\pi \times 10$ kHz \citep{hun}, to $2\pi \times 73.5$ MHz \citep{schl}. The corresponding damping rate for the optomechanical resonantor can thus be varied from $2\pi \times 10^{-3}$ Hz \citep{gar}, $2\pi \times 3.22$ Hz \citep{hun}, to $2\pi \times 1.3$ kHz \citep{schl}. Here we have also taken an appropriate regime for cooling the optomechanical system to the quantum ground state by considering the cavity-pump detuning to be $\Delta =\omega_{m}$ \citep{schl}. The other parameters used in the present calculations are taken from \citep{eyo, eyo3}.

\section{Conclusion}
In conclusion, we have proposed a scheme to study the periodic oscillations in a non-stationary system consisting of a QW inside the optomechanical cavity. We have analysed the system with classical cavity mirror motion and with quantized cavity mirror motion. Also, the cavity frequency is periodically modulated with time. The photon number inside the cavities is not steady but it changes with periodic increase and decrease. Amplification is observed in the intracavity photon number with quantized mirror motion. But the photon number inside the cavity with classical mirror motion show periodic oscillations. The fluorescent light emitted by the excitons in the QW has also been investigated. The QW in the optomechanical cavity produces non-periodic damped fluorescent intensity of light when the mirror motion is considered to be classical. However, the fluorescent spectrum of light emitted by excitons in the QW in an optomechanical cavity with quantized mirror motion, shows amplification under certain regimes. It is also observed that under the strong modulation, the phenomenon of two-photon process dominates whereas under the weak modulation, the phenomenon of fluorescence dominates. The system also shows a balance of energy between different degrees of freedom. The present system provides a good way to detect two-photon process at initial stage of fluorescence spectrum. Furthermore, we have also observed reverse of two-photon process for a non-stationary system composed of a QW confined in an optomechanical cavity with an amplitude modulated external laser pump and constant cavity frequency. Significant amplification is observed in the position dynamics of the mirror. The intracavity photons decreases with time. Thus, the modulation in the external laser beam helps in amplifying the phonons. The present scheme can be used as an optical switch with two-photon process as an external control parameter.

\section{Acknowledgements}

Sonam Mahajan acknowledges University of Delhi for the University Teaching Assistantship. A. Bhattacherjee and Neha Aggarwal acknowledge financial support from the Department of Science and Technology, New Delhi for financial assistance vide grant SR/S2/LOP-0034/2010.

\section{Appendix}

In this Appendix, we derive the time-dependent Hamiltoninan for optomechanical cavity with quantized motion of the movable mirror. As one knows that, the cavity frequency is given as \citep{dod5}

\begin{eqnarray}
\omega_{c}=\frac{C_{1}}{L},
\end{eqnarray}

where $C_{1}$ is constant and $L$ is the length of the cavity. Therefore, change in the cavity length modifies the cavity frequency. Hence, the time varying cavity frequency is given as

\begin{eqnarray}
\omega_{c}(x,t)=\frac{C_{1}}{L-x(t)}.
\end{eqnarray}

Since the perturbation in the cavity length is very small as compared to its original length. Therefore the time varying cavity frequency becomes

\begin{eqnarray}\label{1}
\omega_{c}(x,t)=\omega_{c}\left(1+\frac{x(t)}{L}\right).
\end{eqnarray}

Now one can take the time-modulated perturbation in cavity length as

\begin{eqnarray}
x(t)=x'\epsilon '\sin\left(\Omega t\right).
\end{eqnarray}

where $x'=(\Delta x)q$, $x'$ is the quantized perturbation in cavity length, $q$ is the dimensionless position operator of movable mirror, $\epsilon '$ is the modulation amplitude and $\Omega$ is the modulation frequency. Using the time-modulated perturbation of cavity length in equation (\ref{1}), we get

\begin{eqnarray}
\omega_{c}(x,t)=\omega_{c}\left(1+\frac{\left(\Delta x\right)q\epsilon '\sin\left(\Omega t \right)}{L}\right).
\end{eqnarray}

Now we take normalized modulation amplitude as $\epsilon=\left(\left(\Delta x\right)\epsilon '\right)/L$, therefore the time varying cavity frequency becomes

\begin{eqnarray}\label{2}
\omega_{c}(t)=\omega_{c}\left(1+q\epsilon\sin\left(\Omega t \right)\right).
\end{eqnarray}

The effective frequency also changes in this case as it is dependent on time varying cavity frequency (see eqn \ref{fivea}). Therefore, modified effective frequency for the optomechanical cavity becomes

\begin{eqnarray}\label{6}
\chi'(t)=\left(\epsilon'\Omega\cos\left(\Omega t\right)\right)/4=\chi(t)q.
\end{eqnarray}

The coupling constant for the photon-exciton coupling is given as \citep{dod5}

\begin{eqnarray}
g_{0}=\frac{C_{2}}{\sqrt{L}},
\end{eqnarray}

where, $C_{2}$ is a constant. Now the time varying coupling parameter becomes

\begin{eqnarray}
g_{0}(x,t)=\frac{C_{2}}{\sqrt{L-x(t)}}.
\end{eqnarray}

Under the small cavity length perturbation, the time varying coupling parameter becomes

\begin{eqnarray}
g_{0}(x,t)=\frac{C_{2}}{\sqrt{L}}\left(1+\frac{x(t)}{2L}\right).
\end{eqnarray}

Again, using the small quantized perturbation of the cavity length, we get

\begin{eqnarray}
g_{0}(x,t)=g_{0}\left(1+\frac{\left(\Delta x\right)q\epsilon '\sin\left(\Omega t \right)}{2L}\right).
\end{eqnarray}

Therefore, the time varying coupling parameter becomes

\begin{eqnarray}\label{3}
g_{0}(t)=g_{0}\left(1+\frac{q\epsilon\sin\left(\Omega t \right)}{2}\right).
\end{eqnarray}

Hence, the Hamitonian for the optomechanical cavity with quantized mirror motion under rotating-wave and dipole approximation can be written as

\begin{eqnarray}\label{4}\nonumber
H_{II} &=& \hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}(t)a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}\left(p^{2}+q^{2}\right)+\hbar g_{0}(t)\left(a^{\dagger}b+b^{\dagger}a\right)\nonumber \\ &+& i\hbar\chi'\left(t\right)\left(a^{\dagger 2}e^{-2i\omega_{p}t}-a^{2}e^{2i\omega_{p}t}\right)+i\hbar\epsilon_{p}\left(a^{\dagger}e^{-i\omega_{p}t}-a e^{i\omega_{p}t}\right).
\end{eqnarray}

In the above Hamiltonian, the free energy of the movable mirror is represented by $\frac{1}{2}\hbar\omega_{m}\left(p^{2}+q^{2}\right)$ where $\omega_{m}$ is the mechanical frequency and $p$ is the dimensionless momentum operator of the movable mirror. Now substituting the time varying cavity frequency (eqn. \ref{2}), time varying effective frequency (eqn. \ref{6}) and time varying coupling parameter (eqn. \ref{3}) in the above Hamiltonian (eqn. \ref{4}), the Hamiltonian becomes

\begin{eqnarray}\label{5}\nonumber
H_{II} &=& \hbar\omega_{b}b^{\dagger}b+\hbar\omega_{c}a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}\left(p^{2}+q^{2}\right)+\hbar g_{m}\left(t\right)a^{\dagger}aq+\hbar g_{0}\left(a^{\dagger}b+b^{\dagger}a\right)\nonumber \\ &+& \hbar g\left(t\right)\left(a^{\dagger}b+b^{\dagger}a\right)q+i\hbar\chi\left(t\right) \left(a^{\dagger 2}e^{-2i\omega_{p}t}-a^{2}e^{2i\omega_{p}t}\right)q+i\hbar\epsilon_{p}\left(a^{\dagger}e^{-i\omega_{p}t}-a e^{i\omega_{p}t}\right),
\end{eqnarray}

where $g_{m}(t)=\omega_{c}\epsilon\sin\left(\Omega t\right)$ and $g(t)=g_{0}\epsilon\sin\left(\Omega t\right)/2$ are the time-dependent coupling parameters.

redpajama_set_name: RedPajamaArXiv


COSHOCTON – Jack Clasen, a judge for America in Bloom, can sum up the difference in Coshocton from his visit six years ago to today in one word: confidence.
Official judges' comments and rating for the city in the national floral and landscape competition won't be released until the national symposium in September. However, Clasen and judge Susie Stratton had positive statements on their two-day visit during a closing reception Tuesday at Clary Gardens.
The pair led by representatives of Coshocton in Bloom visited many local sites such as Roscoe Village, the Coshocton County Courthouse, artPARK and new Coshocton Planetarium. Areas evaluated include flowers, landscaped areas, urban forestry, environmental efforts, celebrating heritage, community vitality and overall impression.
Judges named improved signage to direct people to attractions and having landscape guidelines for those constructing new homes or businesses as elements to improve upon in the future. Yet, much of what the judges saw demonstrates to them that Coshocton is a city on the rise.
Here are four main points the judges mentioned as positives for the community.
Floral Displays – In 2017, Coshocton earned four blooms out of five and won an award for best floral displays out of 40 communities, which surprised Clasen. He remembers the floral displays being good, but maybe not that good. Clasen and Stratton were impressed by the floral displays they saw around the Coshocton Court Square, along Main Street and across the city.
Community Vitality – A new category this year looked at hiking trails, biking trails, parks and other outdoor venues, which Coshocton has in Clary Gardens, Lake Park, the newly renovated Bancroft Park and other places. Clasen said community vitality was right up there with floral displays and heritage preservation for categories that Coshocton excelled in.
Roscoe Village – The recreation of a traditional canal era town with shops, restaurants, museums, historic buildings and more has long been a top local attraction. Stratton feels that can be leveraged more once the Ohio 16/541 bridge project is finished and Roscoe Village can have a stronger connection to downtown.
Engaging Residents – Clasen said he will sometimes visit a community and never meet any officials. Here he met with the mayor, service director and parks director who were thrilled to talk about their park plans and all Coshocton has to offer. People wanted to be involved with Coshocton in Bloom.
After a day, Clasen can usually tell if a municipality is going up, down or staying stagnant. If they are going down, he said it's because people aren't working together. If everyone is on the same page and moving in the same direction, he knows the community has a good future and he saw that in Coshocton.


Whenever you decide to move to another home, we take care of moving your Pinocchio -purchased furniture to your new residence. The service includes disassembling, wrapping, moving and re-assembling.
Taking measurements of your rooms and preparing the floor plans for you is part of our job. We offer you this service for living, dining and bedrooms. Designing your customized dressing room to fit to your personal needs is another service that we provide at Pinocchio.
Repairing your furniture at home is one of Pinocchio's services that ensure maintaining your furniture's pristine, brand-new condition. And if you need to refinish your furniture, it is not a problem. Pinocchio will collect your furniture and refinish it at Pinocchio's factory. Bored with your furniture's upholstery? You can also ask for a fabric change.
All services are available only in the local market.


text:
Лидия Матвеева (4 апреля 1925 — 10 ноября 1941) — юная героиня Великой Отечественной войны, замучена и повешена фашистами 10 ноября 1941 года.

Биография
Родилась 4 апреля 1925 года в городе Ленинграде. Проживала вместе с мамой и сестрой Женей. Училась в ленинградской средней школе №9 Смольнинского района по улице Калашниковская набережная. В летний период приезжала погостить к бабушке в деревню Иваново Рузского района. В 1941 году также находилась в Подмосковье, отдыхала вместе с двоюродной сестрой Екатериной.

Глубокой осенью 1941 года враг уже был на подступах к Москве. В деревне Иванове, где находилась Лида, уже были слышны орудийные залпы орудий, пулеметные и автоматная очереди. Немцы оккупировали деревню 26 октября. В начале ноября к населённому пункту подошли два советских танка. Их цель была разведка в тылу врага. Экипаж танковой бригады Иван Мороза и Алексея Сенцова совершали очередной рейд. На одной из улиц навстречу танкам выбежала девушка, Лида Матвеева, которая предупредила об опасности.

По некоторым данным сами фашисты заметили танки и девушку, по другим — её им сдал один из местных жителей. 10 ноября 1941 года немцы арестовали Лиду, жестоко избив, повесили на развесистой липе у сельсовета,  трое суток фашисты не разрешали снять тело. Отступая, бросили тело юной девушки в горящий дом.

Похоронили пятнадцатилетнюю девочку в больничном саду местные жители. После войны захоронение перенесли в сквер около клуба, где установили белый мраморный обелиск.

Память
 Имя юной героини Лиды Матвеевой было занесено в Книгу почёта Московской Областной пионерской организации им. В. И. Ленина.
 В центре деревне Иваново, на месте захоронения Лиды, установлен белый мраморный обелиск.

Примечания

Ссылки
 Герой Лида Матвеева
 Лидия Матвеева. Рузский краеведческий музей
 О Лидии Матвеевой

Дети и подростки — герои Великой Отечественной войны
Женщины в Великой Отечественной войне
Пионеры-герои

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text:
Businessmens
Earl Sweatshirt Net Worth 2019, Bio, Age, Height
Learning about celebrity net worth is sometimes hard, especially because we cannot know all the contracts and projects they have behind them. Every celebrity is different and we can only do an estimate amount of money they earn in general.
In today's text we are going to learn more about Earl Sweatshirt. This famous rapper made his money by doing what he does best, and that is creating good music. We are going to learn more about his early days and childhood, his career path development and his current personal life. If Earl Sweatshirt is your favorite rapper, then here is an opportunity to learn more about his life and career.
Earl Sweatshirt is better known to us as Thebe Neruda Kgositsile. He was born on February 24th 1994 in Los Angeles. Son of Cheryl Harris, professor of Laws at the California University, and of Keorapetse Kgositsile, a political agitator of South Africa jazz enthusiast and firm defender of the rights of blacks, the future rapper grew up in a very rich cultural environment and, however, unstructured, since his father left the family nucleus to return to Johannesburg when he was just six years old.
The truth is that Earl Sweatshirt did not have a role model adolescence. Soon he joined a group of kids obsessed with skateboarding, rap, the insane MTV donkeys and fast food, who called themselves Odd Future Wolf Gang Kill Them All. Beyond the clan's nihilistic hooliganism, several of them began to break through provocative home videos and perversely magnetic rhymes.
Her mother, when listening to a tape with 12 songs (mixtape Earl, of 2010) in which her son savors fantasies about raping girls, murders and drugs with a morbid and uncomfortable language, decided to send him to a juvenile boarding school in Samoa, an Island of the South Pacific Ocean.
That's when they begin to appear shadows in his biography, paranoid theories that suggest that the rapper has actually fallen into the networks of a sector remains in rehabilitation, while his Odd Future colleagues are no longer treated as a macabre joke the Yankee counterculture and they become a commercial phenomenon.
There aren't any more information about his early days, so we are going to continue on to his amazing career and see how he managed to reach the success he did.
Propelled by that infallible launch pad that is Odd Future and following in the footsteps of Frank Ocean, the young rapper Earl Sweatshirt has set out to give a good wiggle to contemporary hip hop. Earl is still a diamond that has yet to be polished, but it is true that it is becoming clearer what his sound references are (cosmic jazz via Sun Ra, that dark and dark rap with sharp and repetitive bases, the fumetas atmospheres) and his lyrical interests (introspective dialogue as a vehicle for reaffirmation of a different personality, sick thoughts, provocation, private jokes and references to trash culture).
He abandons the stridencies of his first installments to exhibit a much deeper and more varied profile in an effort to sublimate the darkness of urban music with a lyrical potential out of the ordinary.
After Earl's success, which depicts him as the star of the then emerging collective Odd Future led by Tyler The Creator among others, Kgositsile is forced into a period of forced removal from the scene at the behest of his mother. The guys at Complex tracked him down to Samora, in a school that more than a college is a military boot camp for the recovery of children "at risk".
The international success will soon arrive on his return to Los Angeles: a handful of lucky feats. They see him as the protagonist one of the highlights of the noble album of another member of the Odd Future, the Channel Ocean's Frank Ocean.
In 2013 Doris is released, a record that sees a bevy of important collaborations featuring and a production mainly in his hands. It is a generally minimal, sinister work, dominated by off-key accordions and keyboards in full Odd Future style and wisely measured in subtraction, to leave the field free for an alienated and scazzi-free slaker flow that knows how to be shrewd and technically impeccable, a rap on the border with songwriters, an original and convincing expression of self that finds, unanimous, the consent of the critics and a good success in terms of sales (debuts in 5th place in the top 200 of Billboard).
In 2015, in March, I don't like Shit comes out surprisingly, I Don't Go Outside: An Album By Earl Sweatshirt, an even more drained and personal work, we report in the review, with Earl Sweatshirt recluse to dialogue with his demons, to look their paranoia in the face, to sign the entire production (the only exception being Off Top with Left Brain).
An alcoholic and smoked disc that reduces feats. a Da $ H, Wiki, Na'Kel and Vince Staples and photographs, halfway between mockery and the disarming confession, a new and important step in a coherent and introspective artistic journey, slowed down in the flow compared to Doris, but not for this less effective in communicating disorders, deaths of love and an extremely vivid imagery, a prose with complex references that flows freely in the maze of memories.
Hardly a decade has passed since the Odd Future collective appeared in Los Angeles but it is enough time to say that it changed the rules of contemporary hip hop in the United States. Of those insane teenagers who recorded chunky versions of the videos of Jackass (his tricks included cocktails with psychotropic, blood and hangings) today only the ashes of the group remain. In the meantime, at least three of its members have become references of the current urban music: Tyler the Creator, Frank Ocean and Earl Sweatshirt.
The latter is perhaps the rarest bug in this gang of weirdos. During his beginnings in music at the age of 16 he became the martyr of the band when he was held in a reformatory in Samoa. When he returned home he delivered Doris, one of the most important records of 2013.
Then the features that he has polished for his third length were already present: the taste for free jazz and the dark bases, the oppressive thoughts (loneliness, abandonment, and marginalization) and his innate ability to write. "I think … I've wasted my whole depressed life, the only thing in my head was death," he says on the single Nowhere2go.
Some Rap Songs is much more than a few rap songs. At 24, Thebe Neruda Kgositsile (his real name, a tribute to Pablo Neruda) pays tribute to his family heritage: his father, who died earlier this year, was the South African poet and activist Bra Willie, while his uncle (who died a few days later) was the legend of the black jazz Hugh Masekela.
They both appear on this record, just like their mother, a university professor in California. And they do it in two ways: with their voices and rhythms and as characters in the lyrics. "My mother used to say that she saw my father in me, I answered that she did not offend me," she sings in Azucar, where she makes peace with that father figure who was always absent in her life.
Earl's net worth is estimated at around 4$ million. Most of the money comes from his music and songs, while some of it comes from his endorsements and other deals. Earl mad himself a household name on the rap scene and we can definitely expect to see more from him in the future. This will probably influence his net worth as well, so we can expect this number to go up.
When he was very young, he became an instant hero thanks to Earl, a rap rapture mixtape published in 2010 by the Odd Future collective, of which he was part at the time, Thebe Neruda Kgositsile, better known as Earl Sweatshirt, born in '94, born in Los Angeles by a law teacher and a South African poet, she is the protagonist of an exciting ascent to success based on a rigorous underground approach to hip hop, where the protagonist is a rap prose that rejects hooks and easy winks, for a witty flow arranged on a production essential, crumpled and fi, often entrusted to ramshackle accordions and jazz piano, coated bass never invasive and little else.
His style, immediately recognizable, finds comparisons with Eminem and Doom, for spelling and creativity in the associations of thoughts and words, while his career seems to be that of a natural talent that finds its own expressive path and follows it coherently without giving discounts to nobody.
Earl's father passed away recently at the age of 79. He was a very famous South African poet and Earl often mentioned his father in his lyrics. He mostly described him as a complicated man. Earl currently lives in Hollywood.
He has a history of drug abuse and was even sent to a Boarding school by his mother, which helped him a lot. As far as we know, he is currently sober and plans on staying that way. He even said that his habit of drug and alcohol abuse led him to isolation, and even caused him to lose precious people from his life.
Hopefully he will continue to make great music and inspire others to become artists. We can only enjoy his old songs until he releases something new and exciting.
Full name: Thebe Neruda Kgositsile aka Earl Sweatshirt
Date of birth: February 24th 1994
Birthplace: Illinois (Chicago, USA)
Profession: Producer, Songwriter, Rapper
Net Worth: 4$ million
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However, before you invest the time and equipment, read this explainer to see whether asic miner is really for you. We will focus primarily on Bitcoin (throughout, we'll use "Bitcoin" when referring to the network or the cryptocurrency as a concept, and "bitcoin" when we're referring to a quantity of individual tokens).
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