1. Introduction
The interaction between photons and phonons attracted considerable attention in experimental and theoretical quantum optics during the 1990s [1]. Under experimental conditions, optical setup cooling based on the radiation pressure force was studied as an optomechanical cavity [2]. In theoretical quantum optics, the optomechanical cavity has been investigated extensively because of its wide applications in quantum memory, laser stabilization performance, tunable optical filters and single-photon detectors, etc [3–5]. In optomechanical systems, the movable mirror interacts with electromagnetic radiation in an optical cavity via the radiation pressure, similar to a Fabry–Pèrot cavity with a movable end-mirror [6–13]. These systems have inevitable coupling with the environment and, therefore, the decay rates of the cavity and the mechanical oscillator can play an important role in the dynamics of the systems. In an atom-cavity optomechanical system, which is known as a hybrid optomechanical system, the subject can be evaluated by applying the Jaynes–Cummings model and its generalizations [14–17]. In addition, in this system, the decay rates of the atom are considered. Due to interactions between quantum systems and the environment, the study of optomechanical systems has attracted considerable attention in the literature [18–20]. In hybrid optomechanical systems, the exact calculation of some parameters such as the decay rate is difficult, either in principle due to non-observable quantities or the uncertainty principle of the observable quantities and experimental impediments. Fortunately, we can estimate the parameters using estimation theory [21].
Estimation theory investigates the estimation precision of physical quantities (such as temperature, phase and decay parameters) via different measurements [22]. It is useful for detecting gravitational waves [23–25], interacting systems in optical interferometry [26], quantum thermometry [27], etc. In estimation theory, the Cramèr–Rao bound provides one of the best precision methods for parameter estimation [28] and gives information about the mean-squared error of any unbiased estimator [29]. For every value of the optional parameter θ, the Cramèr–Rao inequality shows a complementarity relationship between the mean-squared error or variance Var(θ) and the precision F(θ) [28, 29]: 1 ) shows that the higher estimation precision can be obtained by the larger value of F(θ) of the unknown parameter θ. The F(θ) measures information about the parameter in the state and how accurately the state is used to estimate the parameter [30]. It shows the sensitivity of a quantum state to changes in particular parameters. Interestingly, the quantum Fisher information (QFI) can also distinguish chaotic from regular dynamics in the system. Indeed, the chaotic nature enhances the sensitivity of the system, making it important for parameter estimation [31].
$\begin{eqnarray}{\rm{Var}}(\theta )\,\geqslant \,\frac{1}{F(\theta )}.\end{eqnarray}$
Equation (According to quantum estimation theory, the lower bound of the QFI can be characterized by the quantum version of the Cramér–Rao inequality [32, 33]. Basically, the QFI is a quantum version of classical Fisher information, which was proposed by Ronald Aylmer Fisher as a concept in mathematical statistics [34–36]. It is used for all statistical fields in physical, biological, ecological, economic and even sociological studies [28]. Helstrom, Holevo and others extended Fisher information to a quantum mechanical formalism [32, 37, 38]. For a state with spectrum decomposition, $\hat{\rho }=\displaystyle {\sum }_{i}^{N}{p}_{i}| {\phi }_{i}\rangle \langle {\phi }_{i}| $ , where N is equal to the rank of the density matrix, {pi} and ∣φi⟩ are the full set of eigenvalues and eigenvectors of the density operator, respectively, and the support of a density matrix is defined as ${ \mathcal S }:= \{{p}_{i}\in \{{p}_{i}\}| {p}_{i}\ne 0\}$ . Therefore, the QFI for the estimation precision of parameter γ is given by [33, 39, 40]:
$\begin{eqnarray}\begin{array}{rcl}F(\gamma ) & = & \displaystyle \sum _{i,{p}_{i}\in { \mathcal S }}\frac{{({\partial }_{\gamma }{p}_{i})}^{2}}{{p}_{i}}+\displaystyle \sum _{i}4{p}_{i}\langle {\partial }_{\gamma }{\phi }_{i}| {\partial }_{\gamma }{\phi }_{i}\rangle \\ & & -\displaystyle \sum _{{p}_{i},{p}_{j}\in { \mathcal S }}\frac{8{p}_{i}{p}_{j}}{{p}_{i}+{p}_{j}}| \langle {\phi }_{i}| {\partial }_{\gamma }{\phi }_{j}\rangle {| }^{2}.\end{array}\end{eqnarray}$
Now, the QFI plays a crucial role in the field of quantum metrology [36, 41, 42], with many applications in quantum phase transitions [43, 44], quantum information geometry [45, 46], quantum information [47–49] and so on. A system consisting of a moving three-level atom interacting with a single-mode cavity field has been considered to study the atomic QFI [50]. The QFI has been investigated in a scheme of two atoms that interact with two non-degenerate modes, both in the absence and presence of intrinsic decoherence [51]. Additionally, the QFI has been studied in an atom-field system in the presence of negative binomial states and in a two-atom system [52, 53]. The QFI has been studied in a system consisting of two-level atoms interacting with a single-mode cavity in the presence of Kerr medium [54]. The QFI of a two-level atom has been investigated in anisotropic and isotropic photonic band-gap crystals [55]. The behavior of the QFI has been studied in a plasmonic waveguide at the nanoscale [56]. In various research studies, temperature [57, 58], the damping rate [40], frequency [59], phase [60], the coupling strength [61] and the detuning parameters [62] were estimated in various systems, such as a micromaser, optical interferometers, an optical cavity and an optomechanical cavity [63, 64].
Inspired by the aforementioned motivations, in this paper, we investigate the temporal behavior of the atomic QFI for the estimation of the decay rate of the mechanical oscillator in a hybrid optomechanical cavity. The system contains a ♢-type four-level atom and a mechanical resonator, which both interact with a single-mode cavity field. The effects of different sources of dissipation, i.e. cavity decay, loss of the mechanical oscillator, spontaneous emission and multi-photon transition in the atom-field interaction, are considered in the model. By introducing the total Hamiltonian for the considered quantum system and then obtaining the state vector of the whole system, we intend to examine the effects of the strength of optomechanical and atom-field coupling, the dissipation parameters and multi-photon transition on the dynamics of the atomic QFI.
The rest of the paper is organized as follows. In section 2 , we obtain the explicit form of the state vector of the hybrid optomechanical system. In section 3 , we study the numerical results of the QFI and the results are summarized in section 4 .
2. Description of the model
Let us consider a ♢-type four-level atom that interacts with a single-mode quantized radiation field with k-photon transition in cavity optomechanics, where one of its mirrors is movable (figure 1). The four-level atom is described by a quantum system with four energy levels ∣1⟩, ∣2⟩, ∣3⟩ and ∣4⟩, where the transitions ∣1⟩ → ∣2⟩, ∣1⟩ → ∣3⟩, ∣2⟩ → ∣4⟩ and ∣3⟩ → ∣4⟩ are permitted. These transitions are coupled to a single-mode quantized field with frequency Ω. Additionally, the cavity mode couples to the mechanical oscillator through the radiation pressure interaction.
Figure 1. A scheme for a dissipative hybrid optomechanical cavity containing spontaneous emission by the atom, cavity decay and losses of the cavity mirror, with multi-photon transition. Here, ωj, Ω and ωm (Γj, κ and γ) are the frequencies (decay rates) of the four-level atom, the cavity and the mechanical oscillator, respectively. |
The Hamiltonian of the above-outlined system can be described as (ℏ = 1): 3 ), correspond to the free Hamiltonian of the four-level atom, optical and mechanical modes, respectively. Also, the fourth, fifth and sixth terms denote the interaction between the single-mode quantized field and the four-level atom. In addition, the last term describes the interaction between the cavity mode and the mechanical oscillator via the radiation pressure. Here, we assume the conditions as kΩ = (ω1 − ωj) + ωm and also kΩ = (ωj − ω4) + ωm (j = 2, 3). It should be noted that this condition implies that ω2 = ω3. Basically, we consider the degeneracy in intermediate levels of the ♢-type four-level atom (specifically, levels 2 and 3). Therefore, the above Hamiltonian in the interaction picture can be written as follows:
$\begin{eqnarray}\begin{array}{rcl}\hat{H} & = & \displaystyle \sum _{j=1}^{4}{\omega }_{j}{\hat{\sigma }}_{jj}+{\rm{\Omega }}{\hat{a}}^{\dagger }\hat{a}+{\omega }_{m}{\hat{b}}^{\dagger }\hat{b}\\ & & +{\lambda }_{1}({\hat{a}}^{k}{\hat{\sigma }}_{12}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{21})+{\lambda }_{2}({\hat{a}}^{k}{\hat{\sigma }}_{13}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{31})\\ & & +{\lambda }_{3}({\hat{a}}^{k}{\hat{\sigma }}_{24}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{42})\\ & & +{\lambda }_{4}({\hat{a}}^{k}{\hat{\sigma }}_{34}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{43})-G{\hat{a}}^{\dagger }\hat{a}(\hat{b}+{\hat{b}}^{\dagger }),\end{array}\end{eqnarray}$
where ${\hat{\sigma }}_{ij}=| i\rangle \langle j| )$ (i, j = 1, 2, 3, 4) represents the atomic operators. Here, ωj, Ω and ωm are the frequencies of the four-level atom, the cavity and the mechanical oscillator, respectively; $\hat{a}$ ($\hat{b}$ ) and ${\hat{a}}^{\dagger }$ (${\hat{b}}^{\dagger }$ ) are the bosonic annihilation and creation operators of the optical mode (mechanical mode). The parameters λj (j = 1, 2, 3, 4) are related to the atom-field coupling constants, and G is the optomechanical coupling constant. It may be noted that the first, second and third terms of the Hamiltonian, equation ( $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{I}}} & = & {\lambda }_{1}({\hat{a}}^{k}{\hat{\sigma }}_{12}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{21}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{2}({\hat{a}}^{k}{\hat{\sigma }}_{13}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{31}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{3}({\hat{a}}^{k}{\hat{\sigma }}_{24}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{42}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{4}({\hat{a}}^{k}{\hat{\sigma }}_{34}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{43}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & -G{\hat{a}}^{\dagger }\hat{a}(\hat{b}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{b}}^{\dagger }{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t}).\end{array}\end{eqnarray}$
At this time, by performing some lengthy calculations, the effective Hamiltonian of the whole system can be obtained using the coarse-grained method [65, 66] as the following form (see appendix A for more details):
$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{eff}}} & = & -\frac{{G}^{2}}{{\omega }_{m}}{({\hat{a}}^{\dagger }\hat{a})}^{2}+\frac{k{\lambda }_{1}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{12}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{21})\\ & & +\frac{k{\lambda }_{2}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{13}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{31})\\ & & +\frac{k{\lambda }_{3}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{24}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{42})+\frac{k{\lambda }_{4}G}{{\omega }_{m}}\\ & & \times ({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{34}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{43})\\ & & +\frac{{\lambda }_{1}{\lambda }_{2}({\hat{a}}^{\dagger k}{\hat{a}}^{k})}{{\omega }_{m}}({\hat{\sigma }}_{21}{\hat{\sigma }}_{13}+{\hat{\sigma }}_{31}{\hat{\sigma }}_{12})\\ & & -\frac{{\lambda }_{3}{\lambda }_{4}({\hat{a}}^{k}{\hat{a}}^{\dagger k})}{{\omega }_{m}}({\hat{\sigma }}_{21}{\hat{\sigma }}_{13}+{\hat{\sigma }}_{31}{\hat{\sigma }}_{12})\\ & & +\frac{{\lambda }_{1}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{22}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{11})\\ & & +\frac{{\lambda }_{2}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{33}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{11})\\ & & +\frac{{\lambda }_{3}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{44}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{22})\\ & & +\frac{{\lambda }_{4}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{44}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{33}).\end{array}\end{eqnarray}$
It is noticeable that equation (5 ) exhibits the tripartite interaction between the four-level atom, cavity mode and the mechanical mode, even though there is no direct interaction between the vibrating mirror and the atom in the original form of the Hamiltonian in equation (3 ). Taking the dissipation effects into account, one would be able to rewrite the entire Hamiltonian of the system in the form [67, 68]: 6 ) and the initial state of the system, the wave function ∣$\Psi$(t)⟩ corresponding to the whole system at any time t is governed as:
$\begin{eqnarray}\begin{array}{rcl}{\hat{{ \mathcal H }}}_{{\rm{eff}}} & = & {\hat{H}}_{{\rm{eff}}}-\frac{{\rm{i}}}{2}(\kappa {\hat{a}}^{\dagger }\hat{a}+\gamma {\hat{b}}^{\dagger }\hat{b}\\ & & +({{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}){\hat{\sigma }}_{11}+{{\rm{\Gamma }}}_{3}{\hat{\sigma }}_{33}+{{\rm{\Gamma }}}_{4}{\hat{\sigma }}_{22}),\end{array}\end{eqnarray}$
where Γj, κ and γ are the decay rates of the four-level atom, the cavity mode and the mechanical oscillator, respectively. To obtain the dynamics of the considered system, we assume that the atom is initially in the exited state ∣1⟩ and that the cavity and the mechanical modes are in the Fock state ∣n⟩ and ∣m + 2⟩, respectively. Consequently, the initial state of the whole system has the form ∣$\Psi$(0)⟩ = ∣1, n, m + 2⟩. According to the effective Hamiltonian in equation ( $\begin{eqnarray}\begin{array}{rcl}| \psi (t)\rangle & = & {C}_{1}(t)| 1,n,m+2\rangle +{C}_{2}(t)| 2,n+k,m\\ & & +1\rangle +{C}_{3}(t)| 3,n+k,m+1\rangle \\ & & +{C}_{4}(t)| 4,n+2k,m\rangle ,\end{array}\end{eqnarray}$
where Ci(t)(i = 1, 2, 3, 4) are the unknown probability amplitudes. To find these amplitudes, by applying the Schrödinger equation, ${\rm{i}}| \dot{\psi }(t)\rangle ={\hat{{ \mathcal H }}}_{{\rm{eff}}}| \psi (t)\rangle $ , one may arrive at the following coupled differential equations: $\begin{eqnarray}{\rm{i}}\left(\begin{array}{c}{\dot{C}}_{1}(t)\\ {\dot{C}}_{2}(t)\\ {\dot{C}}_{3}(t)\\ {\dot{C}}_{4}(t)\end{array}\right)=\left(\begin{array}{cccc}{V}_{1} & {U}_{1} & {U}_{2} & 0\\ {U}_{1} & {V}_{2} & {U}_{3} & {U}_{4}\\ {U}_{2} & {U}_{3} & {V}_{3} & {U}_{5}\\ 0 & {U}_{4} & {U}_{5} & {V}_{4}\end{array}\right)\left(\begin{array}{c}{C}_{1}(t)\\ {C}_{2}(t)\\ {C}_{3}(t)\\ {C}_{4}(t)\end{array}\right),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{V}_{1} & = & \frac{-{G}^{2}{n}^{2}-{\lambda }_{1}^{2}\frac{(n+k)!}{n!}-{\lambda }_{2}^{2}\frac{(n+k)!}{n!}}{{\omega }_{m}}\\ & & -\frac{{\rm{i}}}{2}(\kappa n+\gamma (m+2)+{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{V}_{2} & = & \frac{-{G}^{2}{(n+k)}^{2}+{\lambda }_{2}^{2}\frac{(n+k)!}{n!}-{\lambda }_{4}^{2}\frac{(n+2k)!}{(n+k)!}}{{\omega }_{m}}\\ & & -\frac{{\rm{i}}}{2}(\kappa (n+k)+\gamma (m+1)+{{\rm{\Gamma }}}_{3}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{V}_{3} & = & \frac{-{G}^{2}{(n+k)}^{2}+{\lambda }_{1}^{2}\frac{(n+k)!}{n!}-{\lambda }_{3}^{2}\frac{(n+2k)!}{(n+k)!}}{{\omega }_{m}}\\ & & -\frac{{\rm{i}}}{2}(\kappa (n+k)+\gamma (m+1)+{{\rm{\Gamma }}}_{4}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{V}_{4} & = & \frac{-{G}^{2}{(n+2k)}^{2}+{\lambda }_{3}^{2}\frac{(n+2k)!}{(n+k)!}+{\lambda }_{4}^{2}\frac{(n+2k)!}{(n+k)!}}{{\omega }_{m}}\\ & & -\frac{{\rm{i}}}{2}(\kappa (n+2k)+\gamma m)\end{array}\end{eqnarray}$
$\begin{eqnarray}{U}_{1}=\frac{kG{\lambda }_{1}}{{\omega }_{m}}\sqrt{\frac{(n+k)!(m+2)}{n!}},\end{eqnarray}$
$\begin{eqnarray}{U}_{2}=\frac{kG{\lambda }_{2}}{{\omega }_{m}}\sqrt{\frac{(n+k)!(m+2)}{n!}},\end{eqnarray}$
$\begin{eqnarray}{U}_{3}=\frac{kG{\lambda }_{3}}{{\omega }_{m}}\sqrt{\frac{(n+2k)!(m+1)}{(n+k)!}},\end{eqnarray}$
$\begin{eqnarray}{U}_{4}=\frac{kG{\lambda }_{4}}{{\omega }_{m}}\sqrt{\frac{(n+2k)!(m+1)}{(n+k)!}},\end{eqnarray}$
$\begin{eqnarray}{U}_{5}=\frac{{\lambda }_{1}{\lambda }_{2}}{{\omega }_{m}}\frac{(n+k)!}{n!}-\frac{{\lambda }_{3}{\lambda }_{4}}{{\omega }_{m}}\frac{(n+2k)!}{(n+k)!}.\end{eqnarray}$
Under the initial conditions of the system, the coupled differential equation (8 ) can be solved exactly using the Laplace transform technique, and the probability amplitudes take the following expressions (see appendix B for details):
$\begin{eqnarray}\left(\begin{array}{c}{C}_{1}(t)\\ {C}_{2}(t)\\ {C}_{3}(t)\\ {C}_{4}(t)\end{array}\right)=\left(\begin{array}{cccc}{c}_{11} & {c}_{12} & {c}_{13} & {c}_{14}\\ {c}_{21} & {c}_{22} & {c}_{23} & {c}_{24}\\ {c}_{31} & {c}_{32} & {c}_{33} & {c}_{34}\\ {c}_{41} & {c}_{42} & {c}_{43} & {c}_{44}\end{array}\right)\left(\begin{array}{c}{{\rm{e}}}^{{y}_{1}t}\\ {{\rm{e}}}^{{y}_{2}t}\\ {{\rm{e}}}^{{y}_{3}t}\\ {{\rm{e}}}^{{y}_{4}t}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{c}_{1j}=\frac{{r}_{0}+{r}_{1}{y}_{j}+{r}_{2}{y}_{j}^{2}+{y}_{j}^{3}}{{b}_{j}},\end{eqnarray}$
$\begin{eqnarray}{c}_{2j}=\frac{{h}_{0}+{h}_{1}{y}_{j}+{h}_{2}{y}_{j}^{2}}{{b}_{j}},\end{eqnarray}$
$\begin{eqnarray}{c}_{3j}=\frac{{l}_{0}+{l}_{1}{y}_{j}+{l}_{2}{y}_{j}^{2}}{{b}_{j}},\end{eqnarray}$
$\begin{eqnarray}{c}_{4j}=\frac{{m}_{0}+{m}_{1}{y}_{j}}{{b}_{j}},\quad j=1,2,3,4,\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{l}{b}_{j}={y}_{jk}{y}_{jl}{y}_{jm},\quad {y}_{jk}={y}_{j}-{y}_{k},\\ j\ne k\ne l\ne m=1,2,3,4,\end{array}\end{eqnarray}$
$\begin{eqnarray}{r}_{0}={\rm{i}}({V}_{3}({U}_{4}^{2}-{V}_{2}{V}_{4})+{V}_{2}{U}_{3}^{2}+{V}_{4}{U}_{5}^{2}-2{U}_{3}{U}_{4}{U}_{5}),\end{eqnarray}$
$\begin{eqnarray}{r}_{1}={U}_{3}^{2}+{U}_{4}^{2}+{U}_{5}^{2}-{V}_{2}(V3+{V}_{4})-{V}_{3}{V}_{4},\end{eqnarray}$
$\begin{eqnarray}{r}_{2}={\rm{i}}({V}_{2}+{V}_{3}+{V}_{4}),\end{eqnarray}$
$\begin{eqnarray}{h}_{0}={\rm{i}}({U}_{2}({U}_{3}{U}_{4}-{V}_{4}{U}_{5})+{U}_{1}({V}_{3}{V}_{4}-{U}_{3}^{2})),\end{eqnarray}$
$\begin{eqnarray}{h}_{1}={U}_{1}({V}_{3}+{V}_{4})-{U}_{2}{U}_{5},\end{eqnarray}$
$\begin{eqnarray}{h}_{2}=-{\rm{i}}{U}_{1},\end{eqnarray}$
$\begin{eqnarray}{l}_{0}=-{\rm{i}}({U}_{2}({U}_{4}^{2}-{V}_{2}{V}_{4})+{U}_{1}({V}_{4}{U}_{5}-{U}_{3}{U}_{4})),\end{eqnarray}$
$\begin{eqnarray}{l}_{1}={U}_{2}({V}_{2}+{V}_{4})-{U}_{1}{U}_{5},\end{eqnarray}$
$\begin{eqnarray}{l}_{2}=-{\rm{i}}{U}_{2},\end{eqnarray}$
$\begin{eqnarray}{m}_{0}={\rm{i}}({U}_{1}({U}_{3}{U}_{5}-{V}_{3}{U}_{4})+{U}_{2}({U}_{4}{U}_{5}-{V}_{2}{U}_{3})),\end{eqnarray}$
$\begin{eqnarray}{m}_{1}=-({U}_{1}{U}_{4}+{U}_{2}{U}_{3}).\end{eqnarray}$
Also, yj satisfies the fourth-order algebraic equation
$\begin{eqnarray}{y}^{4}+{x}_{1}{y}^{3}+{x}_{2}{y}^{2}+{x}_{3}y+{x}_{4}=0,\end{eqnarray}$
where $\begin{eqnarray}{x}_{1}={\rm{i}}({V}_{1}+{V}_{2}+{V}_{3}+{V}_{4}),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{x}_{2} & = & {U}_{1}^{2}+{U}_{2}^{2}+{U}_{3}^{2}+{U}_{4}^{2}+{U}_{5}^{2}-{V}_{1}({V}_{2}+{V}_{3}+{V}_{4})\\ & & -{V}_{2}{V}_{3}-{V}_{4}({V}_{2}+{V}_{3}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{x}_{3}={\rm{i}}({U}_{1}^{2}({V}_{3}+{V}_{4})+{U}_{2}^{2}({V}_{2}+{V}_{4})+{U}_{3}^{2}({V}_{1}+{V}_{2})\\ \,+{U}_{4}^{2}({V}_{1}+{V}_{3})+{U}_{5}^{2}({V}_{1}+{V}_{4})-2{U}_{5}({U}_{1}{U}_{2}+{U}_{3}{U}_{4})\\ \,-{V}_{1}{V}_{2}({V}_{3}+{V}_{4})-{V}_{3}{V}_{4}({V}_{1}+{V}_{2})),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{x}_{4}={V}_{1}{V}_{2}{V}_{3}{V}_{4}-2{U}_{1}{U}_{2}{U}_{4}({U}_{3}-{U}_{5})\\ \,+{U}_{1}^{2}({U}_{3}^{2}-{V}_{3}{V}_{4})+{U}_{2}^{2}({U}_{4}^{2}-{V}_{2}{V}_{4})\\ \,-{V}_{1}({V}_{2}{U}_{3}^{2}+{V}_{3}{U}_{4}^{2}+{V}_{4}{U}_{5}^{2}-2{U}_{3}{U}_{4}{U}_{5}).\end{array}\end{eqnarray}$
The four roots of the quartic equation (13 ) are given by using MATHEMATICA in the following form:
$\begin{eqnarray}{y}_{1(2)}=-\frac{{x}_{1}}{4}-\frac{{z}_{1}}{2}\mp \frac{1}{2}\sqrt{{z}_{2}-\frac{{z}_{3}}{4{z}_{1}}},\end{eqnarray}$
$\begin{eqnarray}{y}_{3(4)}=-\frac{{x}_{1}}{4}-\frac{{z}_{1}}{2}\mp \frac{1}{2}\sqrt{{z}_{2}+\frac{{z}_{3}}{4{z}_{1}}},\end{eqnarray}$
where
$\begin{eqnarray}{z}_{1}=\sqrt{{w}_{2}+\frac{{w}_{1}}{3{v}_{2}}+\frac{{v}_{2}}{3}},\end{eqnarray}$
$\begin{eqnarray}{z}_{2}=(2{w}_{2}-\frac{{w}_{1}}{3{v}_{2}}-\frac{{v}_{2}}{3}),\end{eqnarray}$
$\begin{eqnarray}{z}_{3}=-8{x}_{3}+4{x}_{1}{x}_{2}-{x}_{1}^{3},\end{eqnarray}$
$\begin{eqnarray}{w}_{1}=12{x}_{4}+{x}_{2}^{2}-3{x}_{1}{x}_{3},\end{eqnarray}$
$\begin{eqnarray}{w}_{2}=\frac{-2{x}_{2}}{3}+\frac{{x}_{1}^{2}}{4},\end{eqnarray}$
$\begin{eqnarray}{v}_{1}=27{x}_{3}^{2}-72{x}_{2}{x}_{4}+2{x}_{2}^{3}-9{x}_{1}{x}_{2}{x}_{3}+27{x}_{1}^{2}{x}_{4},\end{eqnarray}$
$\begin{eqnarray}{v}_{2}={\left(\frac{{v}_{1}+\sqrt{{v}_{1}^{2}-4{w}_{1}^{3}}}{2}\right)}^{1/3}.\end{eqnarray}$
In conclusion, as is seen, the wave function ∣$\Psi$(t)⟩ introduced in equation (7 ) and, consequently, the density matrix of the whole system ρ(t) = ∣$\Psi$(t)⟩⟨$\Psi$(t)∣ are obtained. Then, the reduced density matrix of the atom can be obtained by tracing ρ(t) over the cavity and mechanical modes. Therefore, we are now able to study the QFI via equation (2 ).
3. Results and discussion
We studied the atomic QFI of the considered system in terms of the scaled time τ = ωmt to estimate the precision of the decay rate of the mechanical oscillator. Additionally, we investigated the effects of coupling constants and dissipation parameters on the QFI with k-photon transition (k = 1 and k = 2) in both the absence and presence of various dissipation sources. To achieve better alignment with the experimental results, in numerical calculations, we selected the realistic parameters scaled with ωm as, for instance, G = λj = λ = 0.78 (j = 1, 2, 3, 4), γ = 9.43 × 10−6, κ = 0.01 and Γj = Γ = 0.04 (j = 1, 2, 3, 4) [69, 70]. Also, In figures 2 and 3, the left and right plots represent the atomic QFI in the absence and presence of atomic and cavity field dissipations, respectively.
Figure 2. The time evaluation of the QFI versus the scaled time τ = ωmt under a single-photon process and γ = 9.43 × 10−6. Plots (a), (c), (e) and (g) correspond to the absence of the atomic and cavity field dissipations (κ = 0, Γ = 0) and plots (b), (d), (f) and (h) correspond to the presence of the atomic and cavity field dissipations (κ = 0.01, Γ = 0.04). Also, (a) and (b) G = λ = 0.78, n = 1, m = 1; (c) and (d) G = λ = 0.28, n = 1, m = 1; (e) and (f) G = λ = 0.78, n = 1, m = 0; (g) and (h) G = λ = 0.78, n = 0, m = 1. |
Figure 3. The time evaluation of the QFI versus the scaled time τ = ωmt under a two-photon process and γ = 9.43 × 10−6. Plots (a), (c), (e) and (g) correspond to the absence of the atomic and cavity field dissipations (κ = 0, Γ = 0), and plots (b), (d), (f) and (h) correspond to the presence of the atomic and cavity field dissipations (κ = 0.01, Γ = 0.04). Also, (a) and (b) G = λ = 0.78, n = 1, m = 1; (c) and (d) G = λ = 0.28, n = 1, m = 1; (e) and (f) G = λ = 0.78, n = 1, m = 0; (g) and (h) G = λ = 0.78, n = 0, m = 1. |
As shown in figure 2, the QFI oscillates between minimum and maximum values in the presence and absence of the dissipation effects. In the absence of atomic decay and cavity losses, the local maximum values of atomic QFI are increased as τ proceeds (figure 2(a)). The QFI is gradually increased over a long time, attributing to greater chaotic behavior in the quantum system. Indeed, the amplitude of oscillations increases as τ increases and becomes more chaotic compared with the initial state. This phenomenon can be understood via the concept of information.
In the presence of the atomic and cavity field dissipations, the maximum value of the QFI decreases (figure 2(b)). Consequently, the estimation precision of the decay rate of the mechanical oscillator diminishes with these dissipations, as dissipation plays a destructive role in estimating the accuracy of mechanical oscillator loss.
The effect of atom-field and optomechanical coupling strengths on the temporal behavior of the atomic QFI is investigated in figure 2(c) with G = λj = λ = 0.28 (j = 1, 2,3, 4). This figure shows a signification decrease in the number of fluctuations of the QFI. Consequently, it is found that the oscillation frequency of the QFI is deeply affected by the coupling constants. We investigate the temporal behavior of the QFI for different values of the phonon and photon numbers in figures 2(e) and 2(g). From these figures, we can see that decreasing the number of phonons and photons is helpful for high-precision mechanical oscillation decay rate estimation.
Figure 3 shows the atomic QFI of the considered system with two-photon transition, in the presence (magenta dashed line) and absence (green solid line) of the atomic and cavity field dissipations. Generally, it shows that the temporal behavior of the QFI for parameter variations with two-photon transition is qualitatively similar to the QFI with single-photon transition. Comparison of figures 2 and 3 demonstrates that there is a significant increase in the atomic QFI. Therefore, the accuracy of the estimation of the mechanical oscillator decay with the two-photon process is better than with the single-photon process under the same conditions.
Figure 4 shows the time evolution of QFI for different values of γ in the presence of the dissipation effects. It is demonstrated that the QFI increases as the value of the decay rate of the mechanical oscillator decreases. The QFI reaches its maximum value of 1568.93 at τ ≈ 243.25 for γ =9.43 × 10−7 under the single-photon transition in the presence of the dissipation effects. Therefore, the minimum possible mean-squared error is 6.37 × 10−4 from the quantum Cramèr–Rao inequality. In addition, the QFI reaches 11205.6 at τ ≈ 168.623 with a mean-squared error of 8.92 × 10−5 for the two-photon transition. This indicates, that the minimum possible mean-squared error decreases with the two-photon process. Therefore, comparison of the left and right plots of figure 4 shows that the accuracy of the estimation of the mechanical oscillator decay with the two-photon process is better than with the single-photon process. It is mentioned that, according to our numerical calculations (not shown here), the temporal behavior patterns of atomic QFI for all values of γ ≤ 9.43 × 10−7 would be similar to each other. Therefore, it is not possible to achieve more accuracy by further decreasing the decay rate of the mechanical oscillator.
Figure 4. The time evolution of the QFI for different values of the decay rate of the mechanical oscillator in the presence of the atomic and cavity field dissipations (κ = 0.01 and Γ = 0.04). The left and right plots correspond to single- and two-photon transitions (k = 1 and k = 2), respectively. Also, G = λ = 0.78 and n = m = 1. |
4. Conclusion
In this paper, we studied a hybrid optomechanical system comprising a single-mode cavity field coupled to a ♢-type four-level atom with k-photon transition. We investigated the dynamics of the QFI by varying the atom-field and optomechanical coupling strengths, the number of photons and phonons, and the dissipation parameters. We considered different sources of dissipation arising from the atom, cavity mode and mechanical mode. We observed that the local maximum values of atomic QFI increase as the scaled time τ progresses in the absence of dissipation. However, in the presence of dissipation, the atomic QFI gradually decreases and approaches zero after sufficient time, as the chaotic nature of the system diminishes and the state of the system stabilizes. The QFI oscillates between minimum and maximum values, and these oscillations gradually decrease with the inclusion of dissipation effects from the interaction subsystems. Overall, the estimation precision of the mechanical oscillation decay rate decreases in the presence of atomic and cavity field dissipations. Additionally, we examined the roles of the atom-field and optomechanical coupling constants on the time evolution of atomic QFI to estimate the precision of the mechanical oscillation decay rate. The results indicate that the estimation precision of the mechanical oscillation decay rate for a coupling parameter of 0.28 is comparable to that for a coupling parameter of 0.78, while the number of oscillations of the QFI for the coupling parameter 0.78 is greater than that for 0.28 with k-photon processes. By decreasing the number of phonons and photons, the maximum value of the QFI increases, the period of oscillation extends and the number of fluctuations of the QFI decreases with k-photon processes. In the two-photon transition, the maximum QFI is greater than that in the one-photon transition. Generally, the amount of atomic QFI can be appropriately controlled according to the key physical parameters of the considered system. This study could be expanded by considering different configurations of the four-level atom under various arrangements (e.g. ladder, N, V, Y, etc.). Such work will be reported in the near future.
Appendix A
Here, we want to demonstrate how the effective Hamiltonian of the whole system, which is introduced in equation (5 ), can be obtained. The effective Hamiltonian of the whole system can be obtained using the coarse-grained method [65, 66]. To this end, first suppose that an interaction Hamiltonian could be written as follows: A3 ) with equation (A1 ), one would be able to find the operators ${\hat{h}}_{i}(i=1,2,...5)$ and frequencies ${\omega }_{i}^{{\prime} }(i=1,2,...5)$ , associated with the Hamiltonian, equation (A3 ), as the following form: A2 ), we get to the following formula: A4 ) into equation (A2 ) and evaluating the commutators, we arrive at the effective Hamiltonian as follows:
$\begin{eqnarray}{\hat{H}}_{{\rm{I}}}(t)=\displaystyle \sum _{n=0}^{N}{\hat{h}}_{n}{{\rm{e}}}^{-{\rm{i}}{\omega }_{n}^{{\prime} }t}+{\hat{h}}_{n}^{\dagger }{{\rm{e}}}^{{\rm{i}}{\omega }_{n}^{{\prime} }t},\end{eqnarray}$
where N is the total number of different harmonic terms, which make up the interaction Hamiltonian, with oscillating frequency ${\omega }_{n}^{{\prime} }\gt 0$ . Therefore, the effective Hamiltonian reduces to: $\begin{eqnarray}{\hat{H}}_{{\rm{eff}}}(t)=\displaystyle \sum _{n,m=0}^{N}\frac{1}{\hslash {\bar{\omega }}_{mn}^{{\prime} }}[{\hat{h}}_{m}^{\dagger },{\hat{h}}_{n}]{{\rm{e}}}^{{\rm{i}}({\omega }_{m}^{{\prime} }-{\omega }_{n}^{{\prime} })t},\end{eqnarray}$
where ${\bar{\omega }}_{mn}^{{\prime} }$ is the harmonic average of ${\omega }_{m}^{{\prime} }$ and ${\omega }_{n}^{{\prime} }$ , defined as $\frac{1}{{\bar{\omega }}_{mn}^{{\prime} }}=\frac{1}{2}(\frac{1}{{\omega }_{m}^{{\prime} }}+\frac{1}{{\omega }_{n}^{{\prime} }})$ [14, 15, 71–74]. Now, we use the above equations and calculate the effective Hamiltonian for our considered model. In our system, the interaction Hamiltonian reads as (ℏ = 1): $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{I}}} & = & {\lambda }_{1}({\hat{a}}^{k}{\hat{\sigma }}_{12}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{21}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{2}({\hat{a}}^{k}{\hat{\sigma }}_{13}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{31}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{3}({\hat{a}}^{k}{\hat{\sigma }}_{24}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{42}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & +{\lambda }_{4}({\hat{a}}^{k}{\hat{\sigma }}_{34}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{a}}^{\dagger k}{\hat{\sigma }}_{43}{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})\\ & & -G{\hat{a}}^{\dagger }\hat{a}(\hat{b}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{b}}^{\dagger }{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t}).\end{array}\end{eqnarray}$
Comparing equation ( $\begin{eqnarray}\begin{array}{rcl}{\hat{h}}_{1} & = & {\lambda }_{1}{\hat{a}}^{k}{\hat{\sigma }}_{12},\quad \quad \quad \quad \quad \quad \quad {\hat{h}}_{2}={\lambda }_{2}{\hat{a}}^{k}{\hat{\sigma }}_{13},\\ {\hat{h}}_{3} & = & {\lambda }_{3}{\hat{a}}^{k}{\hat{\sigma }}_{24},\quad \quad \quad \quad \quad \quad \quad {\hat{h}}_{4}={\lambda }_{4}{\hat{a}}^{k}{\hat{\sigma }}_{34},\\ {\hat{h}}_{5} & = & -G{\hat{a}}^{\dagger }\hat{a}\hat{b},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega }_{i}^{{\prime} }={\omega }_{m},\quad i=1,2,\ldots ,5.\end{array}\end{eqnarray}$
Now, using the equation ( $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{eff}}} & = & \displaystyle \sum _{i=1}^{5}\frac{1}{{\bar{\omega }}_{i1}^{{\prime} }}[{\hat{h}}_{i}^{\dagger },{\hat{h}}_{1}]+\frac{1}{{\bar{\omega }}_{i2}^{{\prime} }}[{\hat{h}}_{i}^{\dagger },{\hat{h}}_{2}]+\frac{1}{{\bar{\omega }}_{i3}^{{\prime} }}[{\hat{h}}_{i}^{\dagger },{\hat{h}}_{3}]\\ & & +\frac{1}{{\bar{\omega }}_{i4}^{{\prime} }}[{\hat{h}}_{i}^{\dagger },{\hat{h}}_{4}]+\frac{1}{{\bar{\omega }}_{i5}^{{\prime} }}[{\hat{h}}_{i}^{\dagger },{\hat{h}}_{5}].\end{array}\end{eqnarray}$
By substituting the defined parameters ${\hat{h}}_{i}$ and ${\omega }_{i}^{{\prime} }$ (i = 1, 2, . . . 5) from equation ( $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{eff}}} & = & -\frac{{G}^{2}}{{\omega }_{m}}{({\hat{a}}^{\dagger }\hat{a})}^{2}+\frac{k{\lambda }_{1}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{12}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{21})\\ & & +\frac{k{\lambda }_{2}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{13}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{31})\\ & & +\frac{k{\lambda }_{3}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{24}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{42})\\ & & +\frac{k{\lambda }_{4}G}{{\omega }_{m}}({\hat{a}}^{k}{\hat{b}}^{\dagger }{\hat{\sigma }}_{34}+{\hat{a}}^{\dagger k}\hat{b}{\hat{\sigma }}_{43})\\ & & +\frac{{\lambda }_{1}{\lambda }_{2}({\hat{a}}^{\dagger k}{\hat{a}}^{k})}{{\omega }_{m}}({\hat{\sigma }}_{21}{\hat{\sigma }}_{13}+{\hat{\sigma }}_{31}{\hat{\sigma }}_{12})\\ & & -\frac{{\lambda }_{3}{\lambda }_{4}({\hat{a}}^{k}{\hat{a}}^{\dagger k})}{{\omega }_{m}}({\hat{\sigma }}_{21}{\hat{\sigma }}_{13}+{\hat{\sigma }}_{31}{\hat{\sigma }}_{12})\\ & & +\frac{{\lambda }_{1}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{22}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{11})\\ & & +\frac{{\lambda }_{2}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{33}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{11})\\ & & +\frac{{\lambda }_{3}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{44}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{22})\\ & & +\frac{{\lambda }_{4}^{2}}{{\omega }_{m}}({\hat{a}}^{\dagger k}{\hat{a}}^{k}{\hat{\sigma }}_{44}-{\hat{a}}^{k}{\hat{a}}^{\dagger k}{\hat{\sigma }}_{33}).\end{array}\end{eqnarray}$
Appendix B
In this appendix, we present the detailed derivation of equation (10 ). The coupled differential equation in equation (8 ) can be written as follows: 10 ).
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{\dot{C}}_{1}(t) & = & {V}_{1}{C}_{1}(t)+{U}_{1}{C}_{2}(t)+{U}_{2}{C}_{3}(t),\\ {\rm{i}}{\dot{C}}_{2}(t) & = & {U}_{1}{C}_{1}(t)+{V}_{2}{C}_{2}(t)+{U}_{3}{C}_{3}(t)+{U}_{4}{C}_{4}(t),\\ {\rm{i}}{\dot{C}}_{3}(t) & = & {U}_{2}{C}_{1}(t)+{U}_{3}{C}_{2}(t)+{V}_{3}{C}_{3}(t)+{U}_{5}{C}_{4}(t),\\ {\rm{i}}{\dot{C}}_{4}(t) & = & {U}_{4}{C}_{2}(t)+{U}_{5}{C}_{3}(t)+{V}_{4}{C}_{4}(t).\end{array}\end{eqnarray}$
By introducing the Laplace transform of probability amplitudes Ci(t), (i = 1, 2, 3, 4) as follows: $\begin{eqnarray}\tilde{{C}_{i}}(y)={\int }_{0}^{\infty }{C}_{i}(t){{\rm{e}}}^{-yt}{\rm{d}}t,\end{eqnarray}$
and by using the initial conditions of the system (∣$\Psi$(0)⟩ = ∣1, n, m + 2⟩), the probability amplitudes obey the following set of algebraic equations: $\begin{eqnarray}\begin{array}{rcl}{\rm{i}}(y\tilde{{C}_{1}}(y)-1) & = & {V}_{1}\tilde{{C}_{1}}(y)+{U}_{1}\tilde{{C}_{2}}(y)+{U}_{2}\tilde{{C}_{3}}(y),\\ {\rm{i}}y\tilde{{C}_{2}}(y) & = & {U}_{1}\tilde{{C}_{1}}(y)+{V}_{2}\tilde{{C}_{2}}(y)+{U}_{3}\tilde{{C}_{3}}(y)+{U}_{4}\tilde{{C}_{4}}(y),\\ {\rm{i}}y\tilde{{C}_{3}}(y) & = & {U}_{2}\tilde{{C}_{1}}(y)+{U}_{3}\tilde{{C}_{2}}(y)+{V}_{3}\tilde{{C}_{3}}(y)+{U}_{5}\tilde{{C}_{4}}(y),\\ {\rm{i}}y\tilde{{C}_{4}}(y) & = & {U}_{4}\tilde{{C}_{2}}(y)+{U}_{5}\tilde{{C}_{3}}(y)+{V}_{4}\tilde{{C}_{4}}(y).\end{array}\end{eqnarray}$
The solution of the algebraic equation results are: $\begin{eqnarray}\begin{array}{rcl}\tilde{{C}_{1}}(y) & = & \frac{{y}^{3}+{r}_{2}{y}^{2}+{r}_{1}y+{r}_{0}}{{y}^{4}+{x}_{1}{y}^{3}+{x}_{2}{y}^{2}+{x}_{3}y+{x}_{4}},\\ \tilde{{C}_{2}}(y) & = & \frac{{y}^{3}+{h}_{2}{y}^{2}+{h}_{1}y+{h}_{0}}{{y}^{4}+{x}_{1}{y}^{3}+{x}_{2}{y}^{2}+{x}_{3}y+{x}_{4}},\\ \tilde{{C}_{3}}(y) & = & \frac{{y}^{3}+{l}_{2}{y}^{2}+{l}_{1}y+{l}_{0}}{{y}^{4}+{x}_{1}{y}^{3}+{x}_{2}{y}^{2}+{x}_{3}y+{x}_{4}},\\ \tilde{{C}_{4}}(y) & = & \frac{{m}_{1}y+{m}_{0}}{{y}^{4}+{x}_{1}{y}^{3}+{x}_{2}{y}^{2}+{x}_{3}y+{x}_{4}}.\end{array}\end{eqnarray}$
Finally, by obtaining the inverse Laplace transform of the above equations, the probability amplitudes can be obtained in equation (Acknowledgments
One of the authors (HRB) would like to thank Dr M J Faghihi for his kind help duringthis work.