Decoherence dynamics of a charged particle within a non-demolition type interaction in non-commutative phase-space

In this work, both the momentum and coordinate space are assumed to be non-commutative. In the usual commutative phase-space, coordinates and momenta satisfy the following relations of commutation:

$$\begin{eqnarray}&&[{x}_{i},{x}_{j}]=0,[{p}_{i},{p}_{j}]=0,[{x}_{i},{p}_{j}]=i{\hslash }{\delta }_{{ij}}.\end{eqnarray} \tag{ 1 }$$

However, at a very tiny scale (or string scale), not only coordinate-momentum is non-commutative, but also coordinate-coordinate and momentum-momentum may not commute too. Therefore, the non-commutative phase-space in quantum mechanics is a Hilbert space where coordinate and momentum operators satisfy the following relations of commutation:

$$\begin{eqnarray}&&[{\hat{x}}_{i},{\hat{x}}_{j}]=i\theta {\epsilon }_{{ij}},[{\hat{p}}_{i},{\hat{p}}_{j}]=i\eta {\epsilon }_{{ij}},[{\hat{x}}_{j},{\hat{p}}_{j}]=i\check{{\hslash }}{\delta }_{{ij}},i,j=1,2,\end{eqnarray} \tag{ 2 }$$

where $\check{{\hslash }}$ is the effective Plank constant given by $\check{{\hslash }}={\hslash }\left(1+\displaystyle \frac{\theta \eta }{4{{\hslash }}^{2}}\right)$, ${\hat{x}}_{i(j)}$ and ${\hat{p}}_{i(j)}$ the coordinate and momentum operators in non-commutative phase-space, respectively. Here θ and η are constants and represent the non-commutative parameters for coordinates and momentum, respectively. Moreover, θ and η have the dimensions of (length)² and (momentum)², respectively. The mapping between the non-commutative phase-space (${\hat{x}}_{i}$ , ${\hat{p}}_{i}$) and the commutative ones (x_i , p_i ), is obtained using the Bopp-Shift linear transformation [32–35]:

$$\begin{eqnarray}&&\,\begin{array}{l}{\hat{x}}_{i}={x}_{i}-{\varepsilon }_{{ij}}\displaystyle \frac{\theta }{2{\hslash }}{p}_{j},\\ {\hat{p}}_{i}={p}_{i}+{\varepsilon }_{{ij}}\displaystyle \frac{\eta }{2{\hslash }}{x}_{j},\end{array}\ \ \ i,j=1,2.\end{eqnarray} \tag{ 3 }$$

Let's consider a quantum charged moving particle with mass m in the presence of an external magnetic field, and confined by a square potential, then its Hamiltonian can be given by [36, 37]:

$$\begin{eqnarray}&&{\hat{H}}_{S}=\displaystyle \frac{1}{2m}{\left(\hat{p}-\displaystyle \frac{e\hat{A}}{c}\right)}^{2}+V({\hat{x}}_{1},{\hat{x}}_{2}),\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&V({\hat{x}}_{1},{\hat{x}}_{2})=\displaystyle \frac{1}{2}m{\omega }_{0}^{2}({\hat{x}}_{1}^{2}+{\hat{x}}_{2}^{2})(1+\displaystyle \frac{\cos (4\phi )}{5}),\end{eqnarray} \tag{ 5 }$$

characterized by its confinement frequency ω₀ and the polar angle between the position vector and x₁, ϕ. Here, $({\hat{x}}_{1},{\hat{x}}_{2})$ and $({\hat{p}}_{1},{\hat{p}}_{2})$ are respectively the non-commutative coordinates and momentum. Considering the symetric gauge $\hat{A}=\tfrac{B}{2}(-{\hat{x}}_{2},{\hat{x}}_{1},0)$ and the Bopp-Shift transformation given by equation (3), the Hamiltonian can be rewritten as:

$$\begin{eqnarray}&&{\hat{H}}_{S}=\displaystyle \frac{1}{2M}({p}_{1}^{2}+{p}_{2}^{2})+\displaystyle \frac{K}{2}({x}_{1}^{2}+{x}_{2}^{2})-\lambda ({x}_{1}{p}_{2}-{x}_{2}{p}_{1}),\end{eqnarray} \tag{ 6 }$$

where $M=\displaystyle \frac{m}{\left[{\left(1+\tfrac{{\omega }_{c}}{2{\omega }_{\theta }}\right)}^{2}+{\left(\tfrac{{\omega }_{0}}{{\omega }_{\theta }}\right)}^{2}\left(1+\tfrac{\cos (4\phi )}{5}\right)\right]}$, $\lambda =\left[\left(1+\tfrac{{\omega }_{c}}{2{\omega }_{\theta }}\right)\left({\omega }_{\eta }+\tfrac{{\omega }_{c}}{2}\right)+\tfrac{{\omega }_{0}^{2}}{{\omega }_{\theta }}\left(1+\tfrac{\cos (4\phi )}{5}\right)\right]$,

${\rm{\Omega }}=\sqrt{\left[{\left(1+\tfrac{{\omega }_{c}}{2{\omega }_{\theta }}\right)}^{2}+{\left(\tfrac{{\omega }_{0}}{{\omega }_{\theta }}\right)}^{2}(1+\tfrac{\cos (4\phi )}{5})][{\omega }_{0}^{2}(1+\tfrac{{\cos }(4\phi )}{5})+{\left({\omega }_{\eta }+\tfrac{{\omega }_{c}}{2}\right)}^{2}\right]}$, ${\omega }_{\theta }=\tfrac{2{\hslash }}{m\theta }$, ${\omega }_{\eta }=\tfrac{\eta }{2m{\hslash }}$, ${\rm{\Omega }}=\sqrt{\tfrac{K}{M}}$, ${\omega }_{c}=\tfrac{{eB}}{{mc}}$ is the cyclotron frequency, ω_θ and ω_η are the non-commutative frequency due to non-commutativity effects. Ω depends on θ and η, and we can easily observe that when η = 0, the above Hamiltonian reduces to the case where only the space is non-commutative, while if in addition θ = 0, then its results to the usual 2-dimensional harmonic oscillator on commutative phase-space.

We note that the Hamiltonian (6) contains an additional term comparing to the usual harmonic oscillator Hamiltonian with commuting operators. This is because of the non-commutativity aspect of the phase-space, and can be assumed as the orbital angular momentum along the z-direction. In addition, the non-commutative system has an effective mass which coincides with the actual mass m for θ = 0. Further, in the commutative case (i.e. θ = 0, η = 0), and in the absence of magnetic field (i.e. B = 0), the last term in the Hamiltonian vanishes. In this situation, the system reduces to a simple two dimensional harmonic oscillator with energy ${E}_{{n}_{1}{n}_{2}}\,={\hslash }\omega ^{\prime} ({n}_{1}+{n}_{2}+1)$. Otherwise (i.e. θ ≠ 0, η ≠ 0 and B ≠ 0), the system is assumed to be an harmonic oscillator in a Hilbert phase-space with non-commutative structure, and presenting non-diagonal Hamiltonian due to the last term in equation (6). In order to diagonalize this Hamiltonian, let's introduce two new operators, a₁ and a₂ such that the position and momentum variables are rewritten as [38, 39]

$$\begin{eqnarray}\begin{array}{rcl}{x}_{1} & = & \sqrt{\displaystyle \frac{{\hslash }{\rm{\Omega }}}{4K}}({a}_{1}-{a}_{2}+{a}_{1}^{+}-{a}_{2}^{+}),\qquad {x}_{2}=i\sqrt{\displaystyle \frac{{\hslash }{\rm{\Omega }}}{4K}}(-{a}_{1}-{a}_{2}+{a}_{1}^{+}+{a}_{2}^{+}),\\ {p}_{1} & = & i\sqrt{\displaystyle \frac{M{\hslash }{\rm{\Omega }}}{4K}}({a}_{2}-{a}_{1}+{a}_{1}^{+}-{a}_{2}^{+}),\qquad {p}_{2}=-\sqrt{\displaystyle \frac{M{\hslash }{\rm{\Omega }}}{4K}}({a}_{1}+{a}_{2}+{a}_{1}^{+}+{a}_{2}^{+}),\end{array}\end{eqnarray} \tag{ 7 }$$

where a_i and ${a}_{j}^{+}$ satisfy the usual commutation relations $[{a}_{j}^{+},{a}_{i}]=-{\delta }_{{ij}}$, i, j = 1, 2. Considering equation (7), the Hamiltonian becomes

$$\begin{eqnarray}&&{H}_{S}={\hslash }{{\rm{\Omega }}}_{1}\left({a}_{1}^{+}{a}_{1}+\displaystyle \frac{1}{2}\right)+{\hslash }{{\rm{\Omega }}}_{2}\left({a}_{2}^{+}{a}_{2}+\displaystyle \frac{1}{2}\right),\end{eqnarray} \tag{ 8 }$$

with the frequencies

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{1}={\rm{\Omega }}+\lambda ,{{\rm{\Omega }}}_{2}={\rm{\Omega }}-\lambda .\end{eqnarray} \tag{ 9 }$$

It can be observed from the Hamiltonian (8) that, the non-commutativity structure of the phase space introduces anisotropy in the system, given that the frequencies are different for the x₁ − and x₂ − directions. In this case, the energy of the system strongly depends on the structure of the space and is given by:

$$\begin{eqnarray}&&{E}_{{n}_{1}{n}_{2}}={\hslash }{{\rm{\Omega }}}_{1}\left({n}_{1}+\displaystyle \frac{1}{2}\right)+{\hslash }{{\rm{\Omega }}}_{2}\left({n}_{2}+\displaystyle \frac{1}{2}\right),\end{eqnarray} \tag{ 10 }$$

where n₁ and n₂ are the quantum numbers (n₁, n₂ = 0, 1, 2, ⋯), Ω₁ and Ω₂ the frequencies defined by equation (9). We assume that the interaction system-environment leads only to decoherence, but not to energy dissipation [26, 31], which has been already introduced for a two-level atom in [40–42]. The total Hamiltonian of such system interacting with a bath of harmonic oscillators, via a QND type interaction [30, 31, 43] is defined by:

$$\begin{eqnarray}&&H={H}_{S}+{H}_{{SB}}+{H}_{B},\end{eqnarray} \tag{ 11 }$$

where H_S and H_B are respectively the Hamiltonian of non-commutative magneto-oscillator and that of the bath; H_SB the interaction Hamiltonian between the system and the bath. In this work, the bath is modeled as an infinite number of harmonic oscillators, considered as two independent heat bath in the x₁- and x₂-directions. Then, the bath Hamiltonian is given by:

$$\begin{eqnarray}&&{H}_{B}=\sum _{n=0}^{\infty }\sum _{j=1}^{2}{\hslash }{\omega }_{n,j}\left({b}_{n,j}^{+}{b}_{n,j}+\displaystyle \frac{1}{2}\right),\end{eqnarray} \tag{ 12 }$$

where ${\omega }_{n,j}=\sqrt{\left(1+{\left(\tfrac{{m}_{n}{\omega }_{n}\theta }{2{\hslash }}\right)}^{2})({\omega }_{n}^{2}+{\left(\tfrac{\eta }{2{m}_{n}{\hslash }}\right)}^{2}\right)}\pm (\tfrac{\eta }{2{m}_{n}{\hslash }}+\tfrac{{m}_{n}{\omega }_{n}^{2}\theta }{2{\hslash }})$ are the non-commutative frequencies of n^th oscillators in the x₁ and x₂ directions respectively. b_{n, j} and ${b}_{n,j}^{+}$ (j = 1, 2) are the annihilation and creation bosonic operators of the heat bath in the x₁ and x₂ directions respectively, and satisfy the commutation relations $[{b}_{n,j},{b}_{n,j}^{+}]={\delta }_{{nn}^{\prime} }{\delta }_{{jj}^{\prime} }$. Moreover, by considering the interaction part H_SB as QND coupling, one has [H_S , H_SB ] = 0. This means that in our fully quantified model, there is no energy transfer from the system to the environment. It implies that H_SB is a constant of motion, which can be generated by H_S and taken as some function of H_S as:

$$\begin{eqnarray}&&{H}_{{SB}}=\sum _{n=0}^{\infty }\sum _{j=1}^{2}{h}_{j}({g}_{n,j}{b}_{n,j}+{g}_{n,j}^{* }{b}_{n,j}^{+}),\end{eqnarray} \tag{ 13 }$$

with ${h}_{j}={\hslash }{{\rm{\Omega }}}_{j}\left({a}_{j}^{+}{a}_{j}+\tfrac{1}{2}\right)$.

To solve the dynamics of our model described by equations (8), (12) and (13), let us first consider the interaction picture transformation. That is, the Hamiltonian (13) becomes

$$\begin{eqnarray}&&{H}_{{SB}}(t)={e}^{{{iH}}_{0}t}{H}_{{SB}}{e}^{-{{iH}}_{0}t},\end{eqnarray} \tag{ 23 }$$

where H₀ = H_S + H_B is the free Hamiltonian in the absence of interaction. Then, considering equation (23) we have

$$\begin{eqnarray}&&{H}_{{SB}}(t)=\sum _{n=0}^{\infty }\sum _{j=1}^{2}{h}_{j}({g}_{n,j}^{* }{b}_{n,j}{e}^{-i{\omega }_{n,j}t}+{g}_{n,j}{b}_{n,j}^{+}{e}^{i{\omega }_{n,j}t}).\end{eqnarray} \tag{ 24 }$$

By defining ${U}_{0}(t)=\exp (-{{iH}}_{0}t)$, one has:

$$\begin{eqnarray}&&{U}_{{SB}}(t)={U}_{0}^{+}(t)U(t),\end{eqnarray} \tag{ 25 }$$

with U(t), the usual time evolution operator depicting a unitary evolution of the total system. The time-derivative of equation (25) shows that U_SB (t) verifies the following differential equation:

$$\begin{eqnarray}&&\displaystyle \frac{{{dU}}_{{SB}}(t)}{{dt}}=-{{iH}}_{{SB}}(t){U}_{{SB}}(t),\end{eqnarray} \tag{ 26 }$$

where its solution can be set as [44, 45]:

$$\begin{eqnarray}&&{U}_{{SB}}(t)=\exp \{\sum _{k=1}^{\infty }{C}_{k}(t)\},\end{eqnarray} \tag{ 27 }$$

Thus, the solution of equation (26) takes the following form (details can be found in appendix A):

$$\begin{eqnarray}&&{U}_{{SB}}(t)=\prod _{j=1}^{2}\exp [\sum _{n=0}^{\infty }{h}_{j}({b}_{n,j}^{+}{\alpha }_{n,j}-{b}_{n,j}{\alpha }_{n,j}^{* })-{{ih}}_{j}^{2}{{\rm{\Delta }}}_{j}].\end{eqnarray} \tag{ 28 }$$

Further, the exact unitary time evolution operator according to equation (25) is given by:

$$\begin{eqnarray}&&U(t)=\prod _{j=1}^{2}{e}^{-i({E}_{{m}_{j}^{{\prime} }}-{E}_{{n}_{j}^{{\prime} }})t}{e}^{-{{iH}}_{B}t}{U}_{{SB}}(t).\end{eqnarray} \tag{ 29 }$$

The dynamical behavior of a magneto-oscillator can be completely described by its density matrix state. So, by using the master equation approach, one is able to determine the time evolution of the density matrix and thereby the complete behavior of the system. In the presence of interactions between the system and the environment, it may not be possible to fully derive an exact master equation for the system only and consequently, we use the Born-Markov-approximation. This approach considers the reduced density matrix of the system in the limit of a weak interaction with dissipative environment. Thus, by considering the system-environment coupling Hamiltonian as follows:

$$\begin{eqnarray}&&{H}_{{SB}}(t)=\sum _{j=1}^{2}{h}_{j}\otimes {B}_{j}(t),\end{eqnarray} \tag{ 30 }$$

where the quantities h_j contain operators described in the Hilbert space of the system defined by ${h}_{j}={\hslash }{{\rm{\Omega }}}_{j}\left({a}_{j}^{+}{a}_{j}+\tfrac{1}{2}\right)$ and B_j (t) the operators in the environmental Hilbert space given by equation (16). Then, assuming a weak system-environment interaction, the total initial state defined by the product state ρ(0) = ρ_B (0)ρ_S (0), where ρ_B (0) and ρ_S (0) are the density matrix of the environment and the system respectively, one has [46, 47]:

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}\displaystyle \frac{d{\rho }_{S}(t)}{{dt}} & = & \displaystyle \frac{i}{{\hslash }}[{\rho }_{S},{H}_{S}]-\sum _{j=1}^{2}{\displaystyle \int }_{0}^{t}{dt}^{\prime} ([{h}_{j},{\tilde{h}}_{j}(t^{\prime} -t){\rho }_{S}(t)]{K}_{j}(t-t^{\prime} )\\ & & -[{h}_{j},{\rho }_{S}(t){\tilde{h}}_{j}(t^{\prime} -t)]{K}_{j}{\left(t-t^{\prime} \right)}^{* }),\end{array}\end{eqnarray} \tag{ 31 }$$

where ${\tilde{h}}_{j}(t,t^{\prime} )={U}_{S}(t,t^{\prime} ){h}_{j}{U}_{S}^{+}(t,t^{\prime} )$, with ${U}_{S}(t,t^{\prime} )$ a unitary operator describing the evolution of the system and ${K}_{j}(t-t^{\prime} )$ the bath correlation function given by equation (19). In addition, the physical meaning of the different terms of equation (31) are as follows: the first term on the right hand side introduces the coherent evolution of the system, while the remaining terms introduce the coupling with the bath. Considering equations (8) and (21), the master equation (31) becomes [30, 31]:

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}\displaystyle \frac{d{\rho }_{S}}{{dt}} & = & -i\sum _{j=1}^{2}{{\rm{\Omega }}}_{j}[{a}_{j}^{+}{a}_{j},{\rho }_{s}]+i{{\hslash }}^{2}\sum _{j=1}^{2}{{\rm{\Omega }}}_{j}^{2}{\dot{{\rm{\Delta }}}}_{j}[({\left({a}_{j}^{+}{a}_{j}\right)}^{2}+{a}_{j}^{+}{a}_{j}),{\rho }_{s}]\\ & & -{{\hslash }}^{2}\sum _{j=1}^{2}{{\rm{\Omega }}}_{j}^{2}{\dot{{\rm{\Gamma }}}}_{j}[{\left({a}_{j}^{+}{a}_{j}\right)}^{2}{\rho }_{s}-2{a}_{j}^{+}{a}_{j}{\rho }_{S}{a}_{j}^{+}{a}_{j}+{\rho }_{s}{\left({a}_{j}^{+}{a}_{j}\right)}^{2}],\end{array}\end{eqnarray} \tag{ 32 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{{\rm{\Delta }}}_{j}={\int }_{0}^{t}{\chi }_{j}(t-t^{\prime} ){dt}^{\prime} =\sum _{n=0}^{\infty }\displaystyle \frac{| {g}_{n,j}{| }^{2}}{{\omega }_{n,j}}(\cos ({\omega }_{n,j}t)-1),\end{eqnarray} \tag{ 33 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{{\rm{\Gamma }}}_{j}={\int }_{0}^{t}{\nu }_{j}(t-t^{\prime} ){dt}^{\prime} =\sum _{n=0}^{\infty }\displaystyle \frac{| {g}_{n,j}{| }^{2}\sin ({\omega }_{n,j}t)}{{\omega }_{n,j}}\coth \left(\displaystyle \frac{{\omega }_{n,j}}{2{K}_{B}T}\right).\end{eqnarray} \tag{ 34 }$$

The state of the system alone at anytime t is found by tracing over the reservoir degree of freedom as

$$\begin{eqnarray}&&{\rho }_{S}(t)={{tr}}_{B}(U(t)\rho (0){U}^{+}(t)),\end{eqnarray} \tag{ 35 }$$

with ρ(0) = ρ_S (0)ρ_B (0) the corresponding density operator of the total system (system +bath), ρ_S (0) and ρ_B (0) the density operators of the system and the bath respectively. Then the reduced density matrix elements in the eigenbasis are:

$$\begin{eqnarray}&&{\left[{\rho }_{S}(t)\right]}_{{m}_{j}^{{\prime} }{n}_{j}^{{\prime} }}={{tr}}_{B}[U(t)\rho (0){U}^{+}(t){T}_{{m}_{j}^{{\prime} }{n}_{j}^{{\prime} }}],\end{eqnarray} \tag{ 36 }$$

where ${T}_{{m}_{j}^{{\prime} }{n}_{j}^{{\prime} }}=| {m}_{j}^{{\prime} }{n}_{j}^{{\prime} }\rangle$ , $| {n}_{j}^{{\prime} }\rangle$ is the eigenstates of h_j with eigenvalues ${E}_{{n}_{j}^{{\prime} }}$. Therefore, we find that the exact solution of the reduced density matrix of equation (32) is given by (see appendix B for detailed calculations):

$$\begin{eqnarray}&&{\left[{\rho }_{S}(t)\right]}_{n^{\prime} m^{\prime} }={\left[{\rho }_{S}(0)\right]}_{n^{\prime} m^{\prime} }\prod _{j=1}^{2}{e}^{-i({E}_{{m}_{j}^{{\prime} }}-{E}_{{n}_{j}^{{\prime} }})t-i({E}_{{m}_{j}^{{\prime} }}^{2}-{E}_{{n}_{j}^{{\prime} }}^{2}){{\rm{\Delta }}}_{j}(t)}{e}^{-{{\rm{\Gamma }}}_{j}(t){\left({E}_{{m}_{j}^{{\prime} }}-{E}_{{n}_{j}^{{\prime} }}\right)}^{2}},\end{eqnarray} \tag{ 37 }$$

where

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)={\int }_{0}^{t}{ds}{\int }_{0}^{s}{\nu }_{j}(s-t^{\prime} ){dt}^{\prime} =\sum _{n=0}^{\infty }\displaystyle \frac{| {g}_{n,j}{| }^{2}}{{\omega }_{n,j}^{2}}(1-\cos ({\omega }_{n,j}t))\coth \left(\displaystyle \frac{{\omega }_{n,j}}{2{K}_{B}T}\right),\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{j}(t)={\int }_{0}^{t}{ds}{\int }_{0}^{s}{\chi }_{j}(s-t^{\prime} ){dt}^{\prime} =\sum _{n=0}^{\infty }\displaystyle \frac{| {g}_{n,j}{| }^{2}}{{\omega }_{n,j}^{2}}(\sin ({\omega }_{n,j}t)-{\omega }_{n,j}t),\end{eqnarray} \tag{ 39 }$$

with ${\prod }_{j=1}^{2}{e}^{-i({E}_{{m}_{j}^{{\prime} }}^{2}-{E}_{{n}_{j}^{{\prime} }}^{2}){{\rm{\Delta }}}_{j}(t)}$ describing the indirect atom-atom interaction due to the common bath. Comparing the master equation obtained here with that obtained in the case of quantum Brownian motion as in [34, 35, 43], it turns out that the term responsible of decoherence in the system is the derivative of Γ(t) with respect to time ($\dot{{\rm{\Gamma }}}(t)$), consequently, Γ(t) is responsible for coherence in the system.

The result in equation (37) is very interesting in the sense that, it demonstrates that the decoherence is mostly controlled by the heat bath parameters instead of those of the system. Moreover the non-commutative parameters and magnetic field content in the eigenvalues ${E}_{{n}_{j}^{{\prime} }}$ of the systems determine the rate while the function Γ_j (t) is controlled by the bath coupling type. Thus, the term

$$\begin{eqnarray}&&r(t)=\prod _{j=1}^{2}{e}^{-{{\rm{\Gamma }}}_{j}(t){\left({E}_{{m}_{j}^{{\prime} }}-{E}_{{n}_{j}^{{\prime} }}\right)}^{2}},\end{eqnarray} \tag{ 40 }$$

in equation (37) introduces the decoherence rate. On the other hand, one can realize that Γ_j (t) is a sum of positive terms, which means that, to get decoherence (i.e., for this sum to diverge as the time goes to infinity), we need continuum frequency values and a strong interaction with the bath modes at low frequencies. Therefore, let us define the environment function, namely the spectral density function in order to perform the sum over all environmental modes as ${J}_{j}(\omega )={\sum }_{n=0}^{\infty }{g}_{n,j}^{2}\delta (\omega -{\omega }_{n,j})$. This functions contains all relevant information of environment and the coupling of the system, apart from the temperature. Further, we assume that the baths are in the thermal equilibrium state with the same value of temperature T. Then, the spectral density is selected as [50, 51]:

$$\begin{eqnarray}&&{J}_{j}(\omega )=\displaystyle \frac{{G}_{0}}{\pi }{\omega }_{j}^{s}{e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}},\qquad j=1,2\end{eqnarray} \tag{ 41 }$$

where ${\omega }_{\mathrm{1,2}}=\sqrt{(1+{\alpha }_{\theta }^{2}{\omega }^{2})({\omega }^{2}+{\beta }_{\eta }^{2})}\pm ({\beta }_{\eta }+{\alpha }_{\theta }{\omega }^{2})$, (with ${\alpha }_{\theta }=\tfrac{m^{\prime} \theta }{2{\hslash }}$, ${\beta }_{\eta }=\tfrac{\eta }{2m^{\prime} {\hslash }}$), G₀ the coupling constant and Λ_c the cut-off frequency of the bath. Therefore, we have

$$\begin{eqnarray}&&\sum _{n=0}^{\infty }| {g}_{n,j}{| }^{2}f({\omega }_{n,j})\to {\int }_{0}^{+\infty }d\omega {J}_{j}(\omega ){f}_{j}(\omega ).\end{eqnarray} \tag{ 42 }$$

In this work, we focus on two special cases, which include the ohmic (s = 1) and super-ohmic (s = 3 ) spectral densities, respectively. These two cases describe different physical contexts.

5.1.1. (a) Case of ohmic spectral density

Let's thus first consider the ohmic case, and evaluate the quantities Δ_j (t) and Γ_j (t). Substituting equations (41) and (42) into equations (38) and (39), we obtain:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{j}(t)=\displaystyle \frac{{G}_{0}}{\pi }{\int }_{0}^{+\infty }\displaystyle \frac{(\sin ({\omega }_{j}t)-{\omega }_{j}t)}{{\omega }_{j}}{e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega ,\end{eqnarray} \tag{ 43 }$$

and

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)=\displaystyle \frac{{G}_{0}}{\pi }{\int }_{0}^{+\infty }\displaystyle \frac{(1-\cos ({\omega }_{j}t))}{{\omega }_{j}}\coth \left(\displaystyle \frac{{\omega }_{j}}{2{K}_{B}T}\right){e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega .\end{eqnarray} \tag{ 44 }$$

In the commutative limit (i.e. θ = η = 0), equations (43) and (44) reduce to:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}(t)={{\rm{\Delta }}}_{2}(t)={\rm{\Delta }}(t)=\displaystyle \frac{{G}_{0}}{\pi }\left[{\tan }^{-1}({{\rm{\Lambda }}}_{c}t)-{{\rm{\Lambda }}}_{c}t\right],\end{eqnarray} \tag{ 45 }$$

and

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}(t)={{\rm{\Gamma }}}_{2}(t)={\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}}{2\pi }\mathrm{ln}(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2}),\end{eqnarray} \tag{ 46 }$$

respectively, for zero temperature, which confirm the results obtained in [31, 52]. For low temperature (i.e. Λ_c ≫ K_B T), equation (44) reduces to [26, 40]:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}(t)={{\rm{\Gamma }}}_{2}(t)={\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}}{\pi }\left[\mathrm{ln}(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2})+2\mathrm{ln}\left[\displaystyle \frac{1}{\pi {K}_{B}{Tt}}\sinh (\pi {K}_{B}{Tt})\right]\right].\end{eqnarray} \tag{ 47 }$$

The first term arises from the quantum vacuum fluctuations while the second is due to thermal ones. The above expression reduces to ${\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}{\left({{\rm{\Lambda }}}_{c}t\right)}^{2}}{\pi }$ for $t\leqslant {{\rm{\Lambda }}}_{c}^{-1}$ (quiet regime), where the fluctuations are ineffective in the decoherece process. While it reduces to ${\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}\mathrm{ln}({{\rm{\Lambda }}}_{c}t)}{\pi }$ for ${{\rm{\Lambda }}}_{c}^{-1}\leqslant t\leqslant {\left({K}_{B}T\right)}^{-1}$. In this case, the main causes of coherence loss (decoherence) are the quantum fluctuations. However, one has ${\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}{K}_{B}{Tt}}{\pi }$ for ${\left({K}_{B}T\right)}^{-1}\leqslant t$ (thermal regime). In this case, the thermal fluctuations play the major role in eroding the system's coherence.

For high temperature, equation (44) reduces to:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)={{\rm{\Gamma }}}_{0}{\int }_{0}^{+\infty }\displaystyle \frac{(1-\cos ({\omega }_{j}t))}{{\omega }_{j}^{2}}{e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega .\end{eqnarray} \tag{ 48 }$$

Then, in commutative case (θ = η = 0), the previous equation becomes,

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}(t)={{\rm{\Gamma }}}_{2}(t)={\rm{\Gamma }}(t)={{\rm{\Gamma }}}_{0}\left[t{{\rm{\Lambda }}}_{c}{\tan }^{-1}({{\rm{\Lambda }}}_{c}t)-\mathrm{ln}(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2})\right],\end{eqnarray} \tag{ 49 }$$

with ${{\rm{\Gamma }}}_{0}=\tfrac{2{G}_{0}{K}_{B}T}{\pi }$, which coincides with those obtained in [31, 52].

5.1.2. (b) Case of super-ohmic spectral density

It is also interesting to consider the super-ohmic case (s = 3 in equation (41)). In this particular case, the quantity Γ_j (t) that leads decoherence can still be evaluated in three different temperature bands, which include zero-temperature, low-temperature and high-temperature.

• For zero-temperature, one has:
  $$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)=\displaystyle \frac{{G}_{0}}{\pi }{\int }_{0}^{+\infty }(1-\cos ({\omega }_{j}t)){\omega }_{j}{e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega ,\qquad j=1,2.\end{eqnarray} \tag{ 50 }$$
• For low temperature, one has:
  $$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)=\displaystyle \frac{{G}_{0}}{\pi }{\int }_{0}^{+\infty }(1-\cos ({\omega }_{j}t)){\omega }_{j}\coth \left(\displaystyle \frac{{\omega }_{j}}{2{K}_{B}T}\right){e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega ,\qquad j=1,2.\end{eqnarray} \tag{ 51 }$$
  In the commutative limit (i.e. θ = η = 0), equation (51) becomes [40]
  $$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{1}(t) & = & {{\rm{\Gamma }}}_{2}(t)={\rm{\Gamma }}(t)=\displaystyle \frac{{G}_{0}}{\pi }\left[{\left({K}_{B}T\right)}^{2}\left(2\zeta \left[2,\displaystyle \frac{{K}_{B}T}{{{\rm{\Lambda }}}_{c}}\right]-\zeta \left[2,\displaystyle \frac{{K}_{B}T(1+i{{\rm{\Lambda }}}_{c}t)}{{{\rm{\Lambda }}}_{c}}\right]-\zeta \left[2,\displaystyle \frac{{K}_{B}T(1-i{{\rm{\Lambda }}}_{c}t)}{{{\rm{\Lambda }}}_{c}}\right]\right)\right.\\ & & \left.+{{\rm{\Lambda }}}_{c}^{2}\left[\displaystyle \frac{1}{{\left(1+i{{\rm{\Lambda }}}_{c}t\right)}^{2}}+\displaystyle \frac{1}{{\left(1-i{{\rm{\Lambda }}}_{c}t\right)}^{2}}-2\right]\right],\end{array}\end{eqnarray} \tag{ 52 }$$
  where ζ is the generalized Riemann Zeta function.
• For high temperature, one has:
  $$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}(t)={{\rm{\Gamma }}}_{0}{\int }_{0}^{+\infty }(1-\cos ({\omega }_{j}t)){e}^{-\tfrac{{\omega }_{j}}{{{\rm{\Lambda }}}_{c}}}d\omega ,\qquad j=1,2,\end{eqnarray} \tag{ 53 }$$
  which reduces to
  $$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}(t)={{\rm{\Gamma }}}_{2}(t)=\displaystyle \frac{{{\rm{\Gamma }}}_{0}{{\rm{\Lambda }}}_{c}^{3}{t}^{2}}{1+{\left(t{{\rm{\Lambda }}}_{c}\right)}^{2}}\end{eqnarray} \tag{ 54 }$$
  in the commutative limit (i.e. θ = η = 0).

In the above expressions, we have as previously mentioned ${\omega }_{\mathrm{1,2}}=\sqrt{(1+{\alpha }_{\theta }^{2}{\omega }^{2})({\omega }^{2}+{\beta }_{\eta }^{2})}\pm ({\beta }_{\eta }+{\alpha }_{\theta }{\omega }^{2})$, (with ${\alpha }_{\theta }=\tfrac{m^{\prime} \theta }{2{\hslash }}$, ${\beta }_{\eta }=\tfrac{\eta }{2m^{\prime} {\hslash }}$), G₀ the coupling constant and ${{\rm{\Gamma }}}_{0}=\tfrac{2{G}_{0}{K}_{B}T}{\pi }$. These expressions are helpful in evaluating the decoherence factor given by equation (40).

It can be observed a rapid decay of the system's coherence in time, which vanishes after a very short given period of time when the system is considered in both commutative and non-commutative phase-space. This means that the suppression of coherence in a system in permanent interaction with a dissipative environment appears at a very short time scale. However, this decrease is more rapid when the system is found in commutative phase-space at low temperature, while at high temperature, the inverse phenomenon occurs. This behavior is observed when the system interact with an ohmic (figure 1) and super-ohmic (figure 3) reservoirs. The oscillatory behavior observed at low temperature traduces the revivals of coherence in the system due to non-commutativity effects. Similar behaviors are observed in figures 2 and 4 representing the decoherence-causing term with respect simultaneously to the time and the temperature (figures 2(a) and 4(a)) and with respect to the magnetic field and the cut-off frequency (figures 2(b) and 4(b)), respectively. As regards to figure 2(b), it can be observed an exponential decrease of this term with the magnetic field in the commutative case (the blue color curve), while it decreases very slowly to zero in the non-commutative system (the black and white color curves).

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In addition, from those figures, it can be identified no matter if the system is in non-commutative or commutative phase-space as well, three time regimes of decoherence: (i) the first time regime is when the decoherence factor tends to an initial value of one. This corresponds to the quiet regime where the fluctuations are ineffective in the system (decoherence process), (ii) the second phase is when the decoherence factor decreases, which corresponds to the thermal regime where the quantum vacuum fluctuations are the main cause of coherence loss and (iii) the third regime is when the decoherence factor vanishes, which also corresponds to the thermal regime, where thermal fluctuation play the crucial role in suppressing the system coherence. Similar results were found by Palma et al [40], when analyzing the possibility to control decoherence in systems interacting with different environment.

Moreover, from figures 2(b) and 4(b) it can be observed that in an intense magnetic field effects, the decay of coherence is more dominated by the thermal fluctuations of the reservoir, while, in low magnetic field effect, there is an intermediate regime where the vacuum fluctuation dominates, and which corresponds to the thermal regime. However, for a very tiny magnetic field effects, no decay is observed, thereby this regime represent the quiet regime. In order to provide further interpretation, we evaluate in the next section the linear entropy of the system.

The linear entropy is an important quantity which help measuring the degree of purity or mixing of quantum states (measure of coherence of a quantum open system) beside the Von Neumann entropy S [53, 54]. It is defined as:

$$\begin{eqnarray}&&S(t)=1-{tr}[{\rho }^{2}(t)],\end{eqnarray} \tag{ 55 }$$

where ρ(t) defines the system's density matrix. For pure state, we have ρ²(t) = ρ(t) and ${tr}[{\rho }^{2}(t)]=1$, and then S(t) = 0. For mixed state, we have ${tr}[{\rho }^{2}(t)]\lt 1$ and thus, 0 < S(t) < 1. It is important to mention that, the increase in linear entropy S(t) due to the system-environment interaction is closely related to the decoherence phenomenon (which implies loss of quantum coherence in the system), induced by the diffusion process [53]. We can define the measure of coherence as follows [30]:

$$\begin{eqnarray}&&C(t)={tr}[{\rho }^{2}(t)].\end{eqnarray} \tag{ 56 }$$

If we assume that the initial state is a coherent state, then the initial density matrix is ρ(0) = ∣α₁, α₂〉〈α₁, α₂∣, with ${\alpha }_{j}={\gamma }_{j}{e}^{i{\phi }_{j}}$. Thus, the initial density matrix elements are:

$$\begin{eqnarray}&&{\left[\rho (0)\right]}_{{n}_{1}^{{\prime} }{n}_{2}^{{\prime} }{m}_{1}^{{\prime} }{m}_{2}^{{\prime} }}=\langle {n}_{1}^{{\prime} },{n}_{2}^{{\prime} }| {\alpha }_{1},{\alpha }_{2}\rangle \langle {\alpha }_{1},{\alpha }_{2}| {m}_{1}^{{\prime} },{m}_{2}^{{\prime} }\rangle .\end{eqnarray} \tag{ 57 }$$

In addition, by expanding the coherent states in terms of states number, we get:

$$\begin{eqnarray}&&\langle {\alpha }_{1},{\alpha }_{2}| {n}_{1}^{{\prime} },{n}_{2}^{{\prime} }\rangle =\displaystyle \frac{| {\alpha }_{1}{| }^{{n}_{1}^{{\prime} }}| {\alpha }_{2}{| }^{{n}_{2}^{{\prime} }}}{\sqrt{{n}_{1}^{{\prime} }!{n}_{2}^{{\prime} }!}}{e}^{-(\tfrac{| {\alpha }_{1}^{2}| }{2}+\displaystyle \frac{| {\alpha }_{2}^{2}| }{2})}{e}^{-i({n}_{1}^{{\prime} }{\phi }_{1}+{n}_{2}^{{\prime} }{\phi }_{2})}.\end{eqnarray} \tag{ 58 }$$

Applying the results of equation (37) giving the elements of the density matrix, equation (56) becomes:

$$\begin{eqnarray}&&C(t)={ \mathcal F }({\alpha }_{1},{\alpha }_{2}){r}^{2}(t),\end{eqnarray} \tag{ 59 }$$

where

$$\begin{eqnarray}&&{ \mathcal F }({\alpha }_{1},{\alpha }_{2})={e}^{-({| {\alpha }_{1}| }^{2}+{| {\alpha }_{2}| }^{2})}\sum _{{n}_{1}^{{\prime} }{m}_{1}^{{\prime} }}^{\infty }\sum _{{n}_{2}^{{\prime} }{m}_{2}^{{\prime} }}^{\infty }\displaystyle \frac{| {\alpha }_{1}{| }^{2({n}_{1}^{{\prime} }+{m}_{1}^{{\prime} })}}{\sqrt{{n}_{1}^{{\prime} }!{m}_{1}^{{\prime} }!}}\times \displaystyle \frac{| {\alpha }_{2}{| }^{2({n}_{2}^{{\prime} }+{m}_{2}^{{\prime} })}}{\sqrt{{n}_{2}^{{\prime} }!{m}_{2}^{{\prime} }!}},\end{eqnarray} \tag{ 60 }$$

and r(t) defined by equation (40). Considering these results, the linear entropy S(t) can be derived as:

$$\begin{eqnarray}&&S(t)=1-C(t)=1-{ \mathcal F }({\alpha }_{1},{\alpha }_{2}){r}^{2}(t).\end{eqnarray} \tag{ 61 }$$

• Case of ohmic reservoir

In the case of ohmic reservoir (s = 1) and in the commutative limit (i.e. θ = η = 0), we obtain:

$$\begin{eqnarray}\begin{array}{rcl}C(t) & = & { \mathcal F }({\alpha }_{1},{\alpha }_{2})\exp \left[-2b[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}]\right.\\ & & \left.\times [t{{\rm{\Lambda }}}_{c}{\tan }^{-1}({{\rm{\Lambda }}}_{c}t)-\mathrm{ln}(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2})]\right],\end{array}\end{eqnarray} \tag{ 62 }$$

at zero temperature (T = 0), with $b=\displaystyle \frac{{G}_{0}{K}_{B}T}{\pi {{\rm{\Lambda }}}_{c}}$,

$$\begin{eqnarray}\begin{array}{rcl}C(t) & = & { \mathcal F }({\alpha }_{1},{\alpha }_{2})\exp \left[-2a\left[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}\right]\right.\\ & & \left.\times \left[\mathrm{ln}(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2})+2\mathrm{ln}[\displaystyle \frac{1}{\pi {K}_{B}{Tt}}\sinh (\pi {K}_{B}{Tt})]\right]\right],\end{array}\end{eqnarray} \tag{ 63 }$$

at low temperature (Λ_c ≫ K_B T), and

$$\begin{eqnarray}&&C(t)={ \mathcal F }({\alpha }_{1},{\alpha }_{2}){\left(1+{{\rm{\Lambda }}}_{c}^{2}{t}^{2}\right)}^{-a[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}]},\end{eqnarray} \tag{ 64 }$$

at high temperature, with $a=\tfrac{{G}_{0}}{\pi }$.

• Case of super-ohmic reservoir

In the case of super-ohmic reservoir (s = 3) and in the commutative limit (i.e. θ = η = 0), we obtain:

$$\begin{eqnarray}\begin{array}{rcl}C(t) & = & { \mathcal F }({\alpha }_{1},{\alpha }_{2})\exp \left[-4a{\left({K}_{B}T\right)}^{2}\left[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}\right]\zeta \left(2,\displaystyle \frac{{K}_{B}T}{{{\rm{\Lambda }}}_{c}}\right)\right]\\ & & \times \exp \left[2a{\left({K}_{B}T\right)}^{2}\left[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}\right]\zeta \left(2,\displaystyle \frac{{K}_{B}T(1+i{{\rm{\Lambda }}}_{c}t)}{{{\rm{\Lambda }}}_{c}}\right)\right]\\ & & \times \exp \left[2a{\left({K}_{B}T\right)}^{2}\left[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}\right]\zeta \left(2,\displaystyle \frac{{K}_{B}T(1-i{{\rm{\Lambda }}}_{c}t)}{{{\rm{\Lambda }}}_{c}}\right)\right]\\ & & \times \exp \left[-2a{{\rm{\Lambda }}}_{c}^{2}\left[{\left({E}_{{n}_{1}^{{\prime} }}-{E}_{{m}_{1}^{{\prime} }}\right)}^{2}+{\left({E}_{{n}_{2}^{{\prime} }}-{E}_{{m}_{2}^{{\prime} }}\right)}^{2}\right]\left[\displaystyle \frac{1}{{\left(1+i{{\rm{\Lambda }}}_{c}t\right)}^{2}}+\displaystyle \frac{1}{{\left(1-i{{\rm{\Lambda }}}_{c}t\right)}^{2}}-2\right]\right].\end{array}\end{eqnarray} \tag{ 65 }$$

Figures 5 and 7 depict the evolution of the linear entropy in time in the system of harmonic oscillator under the magnetic fields effects at low (figures 5(a) and 7(a)) and high (figures 5(b) and 7(b)) temperature, when the system interacts with an ohmic and super-ohmic reservoir respectively. One can easily observe that the non-commutativity effects present significant impact on the linear entropy. For the case of the system interacting with an ohmic reservoir (figure 5(b)), the linear entropy is enhanced with an increase in the non-commutative parameters, while in contrary the inverse phenomenon is observed for the case of super-ohmic reservoir (figure 7(b)). Therefore, increasing the degree of mixedless in the system state, induce loss of coherent dynamics. This result is in perfect agreement with the previous as regard to decoherence causing factor.

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In addition, it is important to notice that, the coherence time scale in which the system is coupled to an ohmic reservoir is much larger than that of the super-ohmic reservoir when the non-commutative effects are taken into consideration. However, in the commutative phase-space case, the coherence is better preserved for a larger period of time for the super-ohmic reservoir. Moreover, at a finite time, both linear entropy curves asymptotically increase in the case of non-commutative phase-space as well as in the commutative phase-space cases characterizing a complete decoherence state of the system.

Furthermore, plotting the linear entropy with respect simultaneously to the time and the temperature (figures 6(a) and 8(a)) and with respect to the magnetic field and the cut-off frequency (figures 6(b) and 8(b)), respectively leads to similar conclusion as previously mentioned in Figs 2 and 4. This implies that the decoherence process occurs very quickly in the commutative phase-space systems than in the non-commutative phase-space systems. We also realize from both figures that, in a lower magnetic field effects, the linear entropy goes to zero which characterizes a total coherent state of the system. However, for intense magnetic field effects, there is an intermediate zone, which characterize an abrupt appearance of decoherence process in the system, afterward it reaches a linear regime where the system is completely decoherent.

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