Optomechanical cavity-atom interaction through field coupling in a composed quantum system: a filtering approach

Abstract

In this paper, a composed quantum system is investigated which can potentially be used to manipulate the state of a qubit through measurement-based feedback (MBF) control. The composed quantum system consists of a nonlinear single-sided Fabry-P´erot optomechanical cavity cascaded with a 2-level atom and it is driven by a coherent quantum field. Due to the presence of the quantum measurement, the state of the atom needs to be estimated based on the measurement results. The effect of the optomechanical cavity on the state of the 2-level atom is disclosed through a quantum filtering approach. To this end, the SLH framework is utilized to analyze and model the considered composed quantum system. Then, filtering equations of the composed quantum system are derived. Numerical simulations are done for different initial conditions as well as different input quantum fields. In addition, the effects of atom and cavity parameters are studied and the result shows that choosing appropriate values for these parameters can change the behavior of the system in the desired way. The filter simulation results are validated by demonstrating that the average of the estimations for 50 ensembles follows the result from the master equation.

Appendix: Deriving SME equations

The procedure of deriving filter equations for ${\sigma }_{z}$ is presented in the following and equations for other operators can be derived similarly.

Substituting ${\sigma }_{z}$ into (8) gives:

$$\begin{array}{c}\mathrm{d}{\pi }_{t}\left({\sigma }_{z}\right)={\pi }_{t}(-i[{\sigma }_{z},{H}_{T}]+{\sum }_{j=1}^{n}({{L}_{j}}^{\dag }{\sigma }_{z}{L}_{j}-\frac{1}{2}{\sigma }_{z}{{L}_{j}}^{\dag }{L}_{j}-\frac{1}{2}{{L}_{j}}^{\dag }{L}_{j}{\sigma }_{z}))\mathrm{d}t+\\ \left({\pi }_{t}\left({\sigma }_{z}{L}_{1}+{{L}_{1}}^{\dag }{\sigma }_{z}\right)-{\pi }_{t}\left({L}_{1}+{{L}_{1}}^{\dag }\right){\pi }_{t}\left({\sigma }_{z}\right)\right)\mathrm{d}W\left(t\right)\end{array}$$

(15)

Pauli matrices of the 2-level atom can be written as ${\sigma }_{z}=|1\rangle \langle 1|-|0\rangle \langle 0|,$ ${\sigma }_{x}=|1\rangle \langle 0|+|0\rangle \langle 1|$,${\sigma }_{y}=i(|0\rangle \langle 1|-|1\rangle \langle 0|)$. Substituting these relations into $-i[{\sigma }_{z},H]$ and simplifying by doing some calculations using $\langle 1|0\rangle =\langle 0|1\rangle =0$ and $\langle 1|1\rangle =\langle 0|0\rangle =1$ gives:

$$\begin{aligned} & - i\left[ {\sigma _{z} ,H} \right] = \frac{{\sqrt {k\gamma } }}{2}\left( {\left( {i\sigma _{y} - \sigma _{x} } \right)a^{\dag } - a\left( {\sigma _{x} + i\sigma _{y} } \right)} \right) \nonumber \\ & \quad + \frac{{\sqrt \gamma }}{2}\left( {\left( {i\sigma _{y} - \sigma _{x} } \right)\alpha ^{{\text{*}}} - \alpha \left( {\sigma _{x} + i\sigma _{y} } \right)} \right) \end{aligned}$$

(16)

Substituting $L_{1} = \sqrt k a + {\raise0.7ex\hbox{${\sqrt \gamma }$} \!\mathord{\left/ {\vphantom {{\sqrt \gamma } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\sigma _{x} - {\raise0.7ex\hbox{${\sqrt \gamma }$} \!\mathord{\left/ {\vphantom {{\sqrt \gamma } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}i\sigma _{y} + \alpha$ into the drift part of (15) gives:

$${\sigma }_{z}{{L}_{1}}^{\dag }{L}_{1}-\frac{1}{2}{\sigma }_{z}{{L}_{1}}^{\dag }{L}_{1}-\frac{1}{2}{{L}_{1}}^{\dag }{L}_{1}{\sigma }_{z}=-\gamma \left(I+{\sigma }_{z}\right)+\frac{\sqrt{k\gamma }}{2}\left(\left(i{\sigma }_{y}-{\sigma }_{x}\right){a}^{\dag }-a\left({\sigma }_{x}+{i\sigma }_{y}\right)\right)+\frac{\sqrt{\gamma }}{2}\left(\left(i{\sigma }_{y}-{\sigma }_{x}\right){\alpha }^{*}-\alpha \left({\sigma }_{x}+{i\sigma }_{y}\right)\right)$$

(17)

For ${L}_{2}$ and ${L}_{3}$ we have:

$$\begin{array}{c}{\sigma }_{z}{{L}_{2}}^{\dag }{L}_{2}-\frac{1}{2}{\sigma }_{z}{{L}_{2}}^{\dag }{L}_{2}-\frac{1}{2}{{L}_{2}}^{\dag }{L}_{2}{\sigma }_{z}=0\\ {\sigma }_{z}{{L}_{3}}^{\dag }{L}_{3}-\frac{1}{2}{\sigma }_{z}{{L}_{3}}^{\dag }{L}_{3}-\frac{1}{2}{{L}_{3}}^{\dag }{L}_{3}{\sigma }_{z}=0\end{array}$$

(18)

Substituting ${\sigma }_{z}$ and ${L}_{1}$ from (7) to stochastic term of (15) and simplifying gives:

$$\begin{array}{c}{\pi }_{t}\left({\sigma }_{z}{L}_{1}+{{L}_{1}}^{\dag }{\sigma }_{z}\right)-{\pi }_{t}\left({L}_{1}+{{L}_{1}}^{\dag }\right){\pi }_{t}\left({\sigma }_{z}\right)={\pi }_{t}\left(-\sqrt{\gamma }{\sigma }_{x}+\sqrt{k}{\sigma }_{z}\left(a+{a}^{\dag }\right)+{\sigma }_{z}\left(\alpha +{\alpha }^{*}\right)\right)-\\ {\pi }_{t}\left(\sqrt{\gamma }{\sigma }_{x}+\sqrt{k}(a+{a}^{\dag })+(\alpha +{\alpha }^{*})\right){\pi }_{t}({\sigma }_{z})\end{array}$$

(19)

Aggregating (16)–(19) gives (9).