I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle ,\\ \langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ) ,\\ \langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume it is a standard procedure. I would appreciate it if you could point me to a reference in which these calculations are presented in more detail or if you could answer this thread by deriving one of the above EOMs explicitly.

I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \partial_{t}\langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle ,\\ \partial_{t}\langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ) ,\\ \partial_{t}\langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume it is a standard procedure. I would appreciate it if you could point me to a reference in which these calculations are presented in more detail or if you could answer this thread by deriving one of the above EOMs explicitly.

I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle ,\\ \langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ) ,\\ \langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume it is a standard procedure. I would appreciate it if you could point me to a reference in which these calculations are presented in more detail or if you could answer this thread by deriving one of the above EOMs explicitly.

I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle ,\\ \langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ) ,\\ \langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume it is a standard procedure. I would appreciate it if you could point me to a reference in which these calculations are presented in more detail or if you could answer this thread by deriving one of the above EOMs explicitly.

I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle , ,\\ \langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ), ,\\ \langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume the standard procedure. If you can point me to a reference in which these calculations are presented in more detail or can answer this thread by deriving one of the above EOMs, I would appreciate it.

I am trying to reproduce the calculations presented on page 4 in arXiv:1511.03347. The Hamiltonian (Eq. 2.4) is given by

$H= \hbar \omega (a^{\dagger} a + \frac{1}{2}) - B \sqrt{\frac{\hbar g}{2 \omega}} (a^{\dagger} + a) $,

where $a^{\dagger}(a)$ denotes creation~(annihilation) operator, $B$ and $ g$ are real constants and $\omega$ is time-dependent. The time-evolution of the density matrix ($\rho$) is governed by the Lindblad master equation given by

$\frac{d \rho}{dt}= {\cal L}\rho=-\frac{i }{\hbar} [H, \rho] + \gamma (n_{\omega}+1) \left[a \rho a^{\dagger} -\frac{1}{2} (a^{\dagger}a \rho + \rho a^{\dagger} a) \right] + \gamma n_{\omega} \left[a^{\dagger} \rho a -\frac{1}{2} (aa^{\dagger} \rho + \rho a a^{\dagger}) \right] $,

where $\gamma$ is the coupling constant and the Bose-Einstein distribution at temperature $T$ is given by $n_{\omega}=1/(e^{\hbar \omega /T}-1)$.

In Eqs.~(2.7-2.9), equation of motion for three expectation values, namely $\langle aa\rangle $, $\langle a^{\dagger}a\rangle $ and $\langle a\rangle $, are presented

$ \langle aa\rangle =-2 \left[ \frac{\gamma}{2} + i \omega \right] \langle aa\rangle + i \sqrt{\frac{2g}{\hbar \omega}} B \langle a\rangle ,\\ \langle a^{\dagger}a\rangle =-\gamma \langle a^{\dagger} a\rangle +\gamma n_{\omega} +i \sqrt{\frac{g}{2 \hbar \omega}} B (\langle a^{\dagger} \rangle -\langle a\rangle ) ,\\ \langle a\rangle =-\left[ \frac{\gamma}{2} + i \omega \right] \langle a \rangle + i \sqrt{\frac{g}{2 \hbar \omega}} B. $

Terms proportional to $\omega$ and $B$ in the above equations, originated from $[H, \rho]$, are easy to get. However, I am more concerned about dissipative terms proportional to $\gamma$.

I have tried to either work with ${\cal L}^{\dagger}$ or ${\cal L}$ as has been discussed in this and this threads. But in both cases, I ended up with higher-order correlation functions such as $\langle a \rho aaa^{\dagger}\rangle $ $\langle a^{\dagger} \rho aaa^{\dagger}\rangle $. As both terms with $\gamma$ in the master equation possess positive signs, even after using the commutation relations of operators, I couldn't simplify my equations to reproduce the above equations of motions. As I have encountered these relations in other papers, I assume it is a standard procedure. I would appreciate it if you could point me to a reference in which these calculations are presented in more detail or if you could answer this thread by deriving one of the above EOMs explicitly.