1. Introduction
The quantum correlation among distinct quantum systems (or entanglement) is a complex and powerful phenomenon in the framework of quantum mechanics, which nowadays is widely used in all branches of science [1–3]. The creation of entanglement due to the Lorentz invariance violation in high energy physics processes [4], the expansion of the Universe [5–10], the variation of entanglement due to the environmental interactions [11–13], the effect of observer acceleration on the entanglement degradation (the Unruh effect) [14–17], the quantum self-organization applied in physics [18] and biophysics [19] are some examples in this research area.
Based on the different mechanisms, entanglement of photons with other objects can be studied. The entangled atom-photon state can be formed during the spontaneous emission process by exciting an atom to a state which ideally has two decay channels [20–22]. Theoretical ideas of generations of entanglement between a quantum-dot spin and a photon have been proposed by several authors [23–28]. Recently, the problem of entanglement between a photon and a quantum dot has been experimentally studied [29]. The results of this investigation can play a major role in realizing a quantum node having the capability of interchanging information with flying photons and on-chip quantum logic, as it is necessary for quantum networks and quantum repeaters. A deterministic photon-photon entanglement protocol can be defined as a procedure when a single atom trapped in an optical cavity subsequently emits two single entangled photons [30–32]. A demonstration of entanglement between the rotational states of a 40CaH+ molecular ion and the internal states of a 40Ca+ atomic ion has been done in [33], where the quantum coherence of the Coulomb coupled motion between the atomic and molecular ions enables subsequent entangling manipulations.
Even though many theoretical works on the features of entanglement have been presented [34, 35], there are still relatively few experimental studies in this area [36, 37], because of many difficulties in measuring the quantum entanglement in laboratory and needing very high technological tools. In one successful experimental method of measuring the entanglement in bi-particle interaction, Saraga et al [38] observed the creation of Einstein−Podolsky−Rosen (EPR) pairs [39] during the 2-dimensional Coulomb scattering in an electron gas. They found that created entanglement is very sensitive to the potential energy function of a bi-electron system. Measuring of entanglement in terms of wave packet localization has been investigated by Fedorov et al [40], while Mishima et al [41] have found a suitable measure for entanglement during the scattering interactions which is now called entanglement fidelity (EF). The EF is now widely used because of its effectiveness in the realization of quantum entanglement and quantifying information processes [42, 43].
Several inter-particle interactions which are described by different functions of potential energies, produce different degrees of entanglement. By considering such descriptions, we can clearly understand that plasma is an ideal medium for the creation and measuring of some features of entanglement. It is because of the presence of high density charged particles. It is clear that in such media, collective effects caused by the long range electromagnetic interactions play an important role and result in the quantum correlations between particles [44]. Presenting a successful definition of an appropriate potential, which includes all important features of interactions, is the first and most essential step in studying quantum entanglement in plasmas. There are several theoretical and experimental investigations which provide more accurate knowledge about the nature of potential energy function in classical, semi-classical and quantum plasmas. The simplest description for interaction of charged particles in plasma is presented using the Debye–Huckel screened potential, which is applied in ideal plasmas where the energy of inter-particle interactions is smaller than the average kinetic energy of plasma constituents [45, 46].
The effective interaction potential between projectile electrons and dressed ions by considering the strong quantum recoil effects in quantum plasmas has been investigated in [47]. Shukla and Eliasson [48] have presented a practical definition for the interaction potential function in degenerate electron-ion plasmas, by considering the screening and electron exchange effects. The effective potential in electron-ion interaction for collisional quantum plasma has been presented in [49] by considering the standard Debye potential, as well as the effective Friedel far-field interaction term between particles. An acceptable form for interaction potential in strongly coupled semi-classical plasmas can be constructed by an effective pseudo-potential [50] interaction between different kinds of plasma particles based on the dielectric response function analysis. The inter-particle interaction potential for dressed electron-ion scattering in hot quantum plasmas [51] as well as in dense quantum plasmas with different initial conditions has been investigated [52–54]. Quantum mechanical effects based on the definition of effective potential functions are widely investigated in dense plasmas that appear in the core of giant planets [55], high energy laser-solid plasma interaction [56–60], ion accelerators [61], compact astrophysical objects and cosmological environments [62, 63], the construction of ultra-small electronic devices [64], metal nanostructures [65] and many other subjects.
The EF of various types of interaction potentials has been calculated [66–69] for a charged particles elastic scattering process. Falaye et al [43] have investigated the EF of elastic scattering at the presence of an external constant magnetic field. In this work, we have extended this problem by including the effects of electromagnetic fields of an applied laser beam in any arbitrary quantum plasma media. As it was expected, the laser beam fields effectively change the interaction potential in the system and cause important changes in the dynamics of entanglement in the system which can be described by the EF.
The paper is organized as follows: we summarize the bi-partite scattering process and obtained a stationary-state Schrodinger equation by considering the laser effect in section 2 . EF at the presence of a laser field is represented in section 3 . Effective potential for a hot quantum plasma is introduced in section 4 . We focus on deriving analytical relationships and evaluate the EF for hot quantum plasmas in section 5 . In section 6 , we have obtained some numerical discussions about effects of laser beams and other parameters on EF. Finally, we conclude the paper in section 7 .
2. Theory and calculations
We consider the bi-partite scattering of particles in the framework of non-relativistic quantum mechanics. The Total Hamiltonian of the system is constructed for two particles in the center of a mass system in the presence of linearly polarized intense laser beam fields radiation as follows [70]:
$\begin{eqnarray}\hat{H}=\displaystyle \frac{1}{2\mu }{\left(\,\,{\boldsymbol{p}}+e{\boldsymbol{A}}\left({\boldsymbol{r}},t\right)\right)}^{2}-e\phi \left({\boldsymbol{r}},t\right)+V\left(\,{\boldsymbol{r}}\right),\end{eqnarray}$
where $\phi \left({\boldsymbol{r}},t\right)$ and ${\bf{A}}\left({\boldsymbol{r}},t\right)$ are scalar and vector potentials of the laser beam respectively, which are invariant under the gauge transformation. μ and $V\left({\boldsymbol{r}}\right)$ are the reduced mass and the potential energy between the particles, respectively. Also, we consider spherically symmetric potential energy in our work.In order to derive the equation of motion for two particles interacted with a spherically symmetric potential under the influence of linearly polarized intense laser beam, we have to solve time-dependent Schrodinger wave equation: ${\rm{i}}{\hslash }\partial \psi \left({\boldsymbol{r}},t\right)$/$\partial t=\hat{H}\psi \left({\boldsymbol{r}},t\right)$ where considering Hamiltonian is given by the equation (1 ) leads to the following equation:
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\hslash }\displaystyle \frac{\partial }{\partial t}\psi \left({\boldsymbol{r}},t\right)=\left[-\displaystyle \frac{{{\hslash }}^{2}}{2\mu }{{\rm{\nabla }}}^{2}-{\rm{i}}{\hslash }\displaystyle \frac{e}{2\mu }\right.\left({\boldsymbol{A}}\left({\boldsymbol{r}},t\right).{\rm{\nabla }}+{\rm{\nabla }}.{\boldsymbol{A}}\left({\boldsymbol{r}},t\right)\right)\\ \left.+\displaystyle \frac{{e}^{2}}{2\mu }{\boldsymbol{A}}{\left({\boldsymbol{r}},t\right)}^{2}-e\phi \left({\boldsymbol{r}},t\right)+V\left({\boldsymbol{r}} \right)\right]\,\psi \left({\boldsymbol{r}},t\right).\end{array}\end{eqnarray}$
Taking into account the Coulomb gauge [71], where ${\rm{\nabla }}.{\boldsymbol{A}}\left({\boldsymbol{r}},t\right)=0$ (also $\phi \left({\boldsymbol{r}},t\right)=0$ in empty space), applying gauge transformations within the framework of dipole approximation, we can simplify the interaction term in equation (2 ). In this approximation, for an atom which is located at the position r0, the vector potential can be considered independent of space, such that ${\boldsymbol{A}}\left({\boldsymbol{r}},t\right)\approx {\boldsymbol{A}}\left(t\right)$. Moreover, term ${\boldsymbol{A}}{\left({\boldsymbol{r}},t\right)}^{2}$ appearing in equation (2 ) is noticeable only in media with extremely high field strength and can be eliminated by extracting a time-dependent phase factor from the wave function via [72]:3 ) in equation (2 ), we can write:4 ) into the Kramers–Henneberger (KH) accelerated frame [73, 74]. The KH frame is a reference frame in which a free electron moves at the influence of an applied laser beam field [75]. The wave functions in the laboratory and KH frames are related by a unitary transformation as [75]:4 )) is eliminated. More explicitly, by substituting equation (5 ) in equation (4 ) and multiplying both sides by U† from the left side, we get:6 ) is straightforward and using the well-known Campbell–Baker–Hausdorff formula [76] for the term ${U}^{\dagger }V\left({\boldsymbol{r}} \right)U$, we have:6 ) becomes:8 ) is a space-translated version of the time-dependent Schrodinger wave equation with incorporation of ${\boldsymbol{\xi }}\left(t\right)$ into the potential, in order to simulate the interaction of atomic systems with the laser beam fields. Now, by considering the steady field condition, the vector potential takes the form $A\left(t\right)={E}_{0}\,{\omega }^{-1}\;\cos \;\left(\omega t\right)$ with $\xi \left(t\right)={\xi }_{0}\;\sin \;\left(\omega t\right)$. Where ξ0 = eE0/μω2 is the amplitude of oscillation of a free electron in the laser beam electric field (which is called the laser-dressing parameter), E0 denotes the amplitude of the laser beam electric field and ω is the angular frequency of the laser beam. Now, we consider a pulse of the electric field with a steady amplitude where the wave function in the KH frame is represented by the following Floquet form [72]:9 ), (10 ) and (11 ) into equation (8 ) yields a set of coupled differential equations:
$\begin{eqnarray}{\rm{\Lambda }}\left({\boldsymbol{r}},t\right)=\exp \;\left[\displaystyle \frac{{\rm{i}}{e}^{2}}{2\mu {\hslash }}{\int }_{-\infty }^{t}\,{\boldsymbol{A}}{\left(t^{\prime} \right)}^{2}{\rm{d}}t^{\prime} \right]\,\psi \left({\boldsymbol{r}},t\right).\end{eqnarray}$
By replacing equation ( $\begin{eqnarray}{\rm{i}}\hslash \displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}{\rm{\Lambda }}\left({\boldsymbol{r}},t\right)=\left[-\displaystyle \frac{{\hslash }^{2}}{2\mu }{{\rm{\nabla }}}^{2}-{\rm{i}}\hslash \displaystyle \frac{e}{\mu }{\boldsymbol{A}}\left(\,t\right).{\rm{\nabla }}+V\left(\,{\boldsymbol{r}}\,\right)\right]{\rm{\Lambda }}\left({\boldsymbol{r}},t\right).\end{eqnarray}$
In order to find the state function of a system composed of two particles under an intense high-frequency laser field, we have to transform equation ( $\begin{eqnarray}{\rm{\Lambda }}\left({\boldsymbol{r}},t\right)=U{\rm{\Psi }}\left({\boldsymbol{r}},t\right),\end{eqnarray}$
where $U=\exp \;\left(-\tfrac{{\rm{i}}}{{\hslash }}{\boldsymbol{\xi }}\left(t\right).{\boldsymbol{p}}\,\right)$ is a local unitary transformation of the particle where ${\boldsymbol{\xi }}\left(t\right)=\tfrac{e}{\mu }{\int }_{}^{t}\,{\boldsymbol{A}}\left(t^{\prime} \right)\,{\rm{d}}t^{\prime} $ represents a shift to the accelerated frame of reference. It is indeed a semi-classical counterpart of the Block–Nordsieck transformation in the quantized field formalism, so that the coupling term ${\boldsymbol{A}}\left(t\right).{\rm{\nabla }}$ in the velocity gauge (i.e. equation ( $\begin{eqnarray}\begin{array}{l}{\rm{i}}{\hslash }{U}^{\dagger }\displaystyle \frac{\partial }{\partial t}U{\rm{\Psi }}\left({\boldsymbol{r}},t\right)={U}^{\dagger }\,\left[-\displaystyle \frac{{{\hslash }}^{2}}{2\mu }{{\rm{\nabla }}}^{2}\right.\\ \left.-{\rm{i}}{\hslash }\displaystyle \frac{e}{\mu }{\boldsymbol{A}}\left(t\right).{\rm{\nabla }}+V\left({\boldsymbol{r}} \right)\right]\,U{\rm{\Psi }}\left({\boldsymbol{r}},t\right).\end{array}\end{eqnarray}$
The calculation of the terms in equation ( $\begin{eqnarray}\begin{array}{l}{U}^{\dagger }V\left({\boldsymbol{r}} \right)U=\exp \;\left(\displaystyle \frac{{\rm{i}}}{{\hslash }}{\boldsymbol{\xi }}\left(t\right).{\boldsymbol{p}}\,\right)\,V\left({\boldsymbol{r}}\right)\;\exp \;\left(-\displaystyle \frac{{\rm{i}}}{{\hslash }}{\boldsymbol{\xi }}\left(t\right).{\boldsymbol{p}}\,\right)\\ =V\left({\boldsymbol{r}}\right)+\left[{\boldsymbol{\xi }}\left(t\right).{\rm{\nabla }},\,\right]V\left({\boldsymbol{r}}\right)+\displaystyle \frac{1}{2!}{\left[{\boldsymbol{\xi }}\left(t\right).{\rm{\nabla }}\right]}^{2}V\left({\boldsymbol{r}}\right)+...\\ =V\left({\boldsymbol{r}}+{\boldsymbol{\xi }}\left(t\right)\right),\end{array}\end{eqnarray}$
where ${\boldsymbol{\xi }}\left(t\right)$ represents the displacement of a free electron under the influence of the incident laser beam fields. Hence, equation ( $\begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial }{\partial t}{\rm{\Psi }}\left({\boldsymbol{r}},t\right)=\left[-\displaystyle \frac{{{\hslash }}^{2}}{2\mu }{{\rm{\nabla }}}^{2}+V\left({\boldsymbol{r}}+{\boldsymbol{\xi }}\left(t\right),\,\right)\right]\,{\rm{\Psi }}\left({\boldsymbol{r}},t\right).\end{eqnarray}$
Equation ( $\begin{eqnarray}{\rm{\Psi }}\left({\boldsymbol{r}},t\right)={{\rm{e}}}^{-\tfrac{{{iE}}_{\mathrm{KH}}}{{\hslash }}t}\displaystyle \sum _{n}\,{{\rm{\Psi }}}_{k}^{n}\left({\boldsymbol{r}}\right)\,{{\rm{e}}}^{-{\rm{i}}n\omega t},\end{eqnarray}$
where Floquet quasi-energy has been denoted by EKH. In this framework, the effect of the time-dependent incoming electric field is converted to the time dependence of the atomic potential. The potential in the KH frame can be expanded in the Fourier series as [77–79]: $\begin{eqnarray}V\left({\boldsymbol{r}}+{\boldsymbol{\xi }}\left(t\right)\right)=\displaystyle \sum _{m=-\infty }^{+\infty }\,{V}_{m}\left({\xi }_{0},r\right)\,{{\rm{e}}}^{-{\rm{i}}\,m\omega t},\end{eqnarray}$
where with some algebraic operations, the coefficients can be written in the form [80]: $\begin{eqnarray}{V}_{m}\left({\xi }_{0},r\right)=\displaystyle \frac{{{\rm{i}}}^{m}}{\pi }{\int }_{-1}^{+1}\,{V}_{m}\left(r+{\xi }_{0}\,\rho ,\,\right)\displaystyle \frac{{T}_{n}\left(\rho \right)}{\sqrt{1-{\rho }^{2}}}{\rm{d}}\rho ,\end{eqnarray}$
where we have taken the period as 2π/ω and introduced a new transformation of the form $\rho =\sin \;\left(\omega t\right)$ while ${T}_{n}\left(\rho \right)$ is Chebyshev polynomials. Substituting equations ( $\begin{eqnarray}\begin{array}{l}\left[-\displaystyle \frac{{{\hslash }}^{2}}{2\mu }{{\rm{\nabla }}}^{2}+{V}_{m}\left({\xi }_{0},r\right)-\left({E}_{\mathrm{KH}}+n{\hslash }\omega ,\,\right)\right]\,{{\rm{\Psi }}}_{k}^{m}\left({\boldsymbol{r}}\right)\\ =-\displaystyle \sum _{m=-\infty }^{+\infty }\,{V}_{n-m}{{\rm{\Psi }}}_{k}^{m}\left({\boldsymbol{r}}\right),\end{array}\end{eqnarray}$
that n ≠ m.If the coupling terms in the right-hand side of equation (12 ) could be neglected, then the time-dependent KH frame potential $V\left({\boldsymbol{r}}+{\boldsymbol{\xi }}\left(t\right)\right)$ can be substituted by ${V}_{0}\left({\xi }_{0},r\right)$, which is simply the time-averaged KH frame potential. On the other hand, considering the lowest order of approximation $\left(n=0\right)$ and high-frequency limit condition (which means Vm vanishes when m ≠ 0), then equation (12 ) would be written as:11 ) and considering the zeroth order Chebyshev relationship ${T}_{0}\left(\rho \right)=1$, ${V}_{0}\left({\xi }_{0},r\right)$ can be written as:13 ) can be strongly valid.
$\begin{eqnarray}\left[-\displaystyle \frac{{{\hslash }}^{2}}{2\mu }{{\rm{\nabla }}}^{2}+{V}_{0}\left({\xi }_{0},r\right)-{E}_{\mathrm{KH}}\,\right]\,{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right)=0,\end{eqnarray}$
according to equation ( $\begin{eqnarray}{V}_{0}\left({\xi }_{0},r\right)=\displaystyle \frac{1}{\pi }{\int }_{-1}^{+1}\,V\left(r+{\xi }_{0}\,\rho ,\,\right)\displaystyle \frac{{\rm{d}}\rho }{\sqrt{1-{\rho }^{2}}}.\end{eqnarray}$
It is worth mentioning that in [80], it is pointed out that the condition for having the relationship Vn≠0 = 0 is that the frequency of the incident laser beam is infinitely high. More importantly, in [81] it is shown that under the conditions $\omega \gg \left| {E}_{{\rm{KH}}},\,\right|$ and ${\xi }_{0}^{2}\,\omega \gg 1$, equation (Employing Ehlotzkys approximation [82]:14 ), we obtain:16 ) is the approximate expression to include the contribution of the laser beam fields in the potential. Now we can write the stationary-state Schrodinger equation by considering laser effect for potential $V\left(r\right)$ as follows:17 ) represents motion for the spherically confined two particles exposed to linearly polarized intense laser beam field radiation.
$\begin{eqnarray}V\left(r-{\xi }_{0}\,\rho \right)+V\left(r+{\xi }_{0}\,\rho \right)\approx V\left(r-{\xi }_{0}\right)+V\left(r+{\xi }_{0}\right),\end{eqnarray}$
and hence, by calculating the integral in equation ( $\begin{eqnarray}\begin{array}{l}{V}_{0}\left({\xi }_{0},r\right)\\ =\,\displaystyle \frac{1}{\pi }\\ {\times \displaystyle \int }_{-1}^{0}\,V\left(r+{\xi }_{0}\,\rho ,\,\right)\displaystyle \frac{{\rm{d}}\rho }{\sqrt{1-{\rho }^{2}}}+\displaystyle \frac{1}{\pi }{\displaystyle \int }_{0}^{1}\,V\left(r+{\xi }_{0}\,\rho ,\,\right)\displaystyle \frac{{\rm{d}}\rho }{\sqrt{1-{\rho }^{2}}}\\ =\,\displaystyle \frac{1}{2}\left(V\left(r-{\xi }_{0}\right)+V\left(r+{\xi }_{0}\right)\right),\end{array}\end{eqnarray}$
where ${\int }_{0}^{1}\,{\rm{d}}x/\sqrt{1-{x}^{2}}=\pi /2$. Equation ( $\begin{eqnarray}\left({{\rm{\nabla }}}^{2}+{k}^{2} \right){{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right)=\displaystyle \frac{\mu }{{{\hslash }}^{2}}\left[V\left(r+{\xi }_{0}\right)+V\left(r-{\xi }_{0}\,\right) \right]{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right),\end{eqnarray}$
where $k=\sqrt{2\mu {E}_{\mathrm{KH}}/{{\hslash }}^{2}}$ is the wave number. The equation (3. EF in the presence of laser beam fields
In order to give a quantitative measurement for the degree of entanglement, we introduce the EF. The EF shows the overlap between the maximally entangled state and the interesting wave function according to the following definition [41, 83]:
$\begin{eqnarray}{f}_{k}=\,{\left|\left\langle {\psi }_{e}\left({\boldsymbol{r}} \right) \right|\left.{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right) \right\rangle \right|}^{2},\end{eqnarray}$
where ${\psi }_{e}\left({\boldsymbol{r}}\right)$ is the maximally entangled state in the representation coordinate. It has been shown that the collisional EF for the scattering process can be represented by [41]: $\begin{eqnarray}{f}_{k}\propto {\left|\int {{\rm{d}}}^{3}{\boldsymbol{r}}{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}} \right)\right|}^{2},\end{eqnarray}$
which ${\left|\int {{\rm{d}}}^{3}{\boldsymbol{r}}{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}} \right)\right|}^{2}$ is the square of the absolute of the scattered wave function for a given interaction potential.The final state wave function ${{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right)$ satisfies the stationary-state Schrodinger equation containing the potential which characterizes the quantum collision processes and laser beam effect and we can represented by the partial wave expansion [84, 85] in the following form:17 ):
$\begin{eqnarray}{{\rm{\Psi }}}_{k}^{0}\left({\boldsymbol{r}}\right)=\displaystyle \sum _{l=0}^{\infty }\,{{\rm{i}}}^{l}\left(2l+1,\,\right){D}_{l,k}{R}_{l,k}\,(r)\,{P}_{l}\,(\cos \;\theta ),\end{eqnarray}$
where Dl,k is the expansion coefficient, i is the pure imaginary number and Rl,k(r) is the solution of the radial wave equation which is obtained from equation ( $\begin{eqnarray}\begin{array}{l}\left\{\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}r}\,\left({r}^{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}r}\right)-\displaystyle \frac{l\left(l+1\right)}{{r}^{2}}\right.\\ \left.-\displaystyle \frac{\mu }{{{\hslash }}^{2}}\left[V\left(r+{\xi }_{0}\right)+V\left(r-{\xi }_{0}\right)\right]+{k}^{2}\right\}\,{R}_{l,k}\,(r)=0,\end{array}\end{eqnarray}$
${P}_{l}\,(\cos \;\theta )$ is the well-known Legendre polynomial of order l, while l is the angular momentum quantum number. For a spherically symmetric potential, it has been shown that the radial wave equation Rl,k(r) and the expansion coefficient Dl,k are given by [41, 44]: $\begin{eqnarray}\begin{array}{l}{D}_{l,k}=\,{\left(2\pi \right)}^{-3/2}\\ \times \left\{1+\displaystyle \frac{{\rm{i}}\mu k}{{{\hslash }}^{2}}\right.{\left.{\int }_{0}^{\infty }\,{\rm{d}}{{rr}}^{2}{j}_{l}\,({kr})\,\left[V\left(r+{\xi }_{0}\right)+V\left(r-{\xi }_{0}\,\right) \right]{R}_{l,k}\,(r)\right\}}^{-1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{R}_{l,k}\,(r)={j}_{l}\,({kr})+\displaystyle \frac{\mu k}{{{\hslash }}^{2}}\,\left\{{n}_{l}\left({kr} \right){\displaystyle \int }_{0}^{r}\,{\rm{d}}r^{\prime} r{{\prime} }^{2}{j}_{l}\,({kr}^{\prime} )\,\left[V\left(r^{\prime} +{\xi }_{0}\right)+V\left(r^{\prime} -{\xi }_{0}\,\right) \right]{R}_{l,k}\,(r^{\prime} )\,\right.\\ \left.+{j}_{l}\,({kr})\,{\displaystyle \int }_{r}^{\infty }\,{\rm{d}}r^{\prime} r{{\prime} }^{2}{n}_{l}\,({kr}^{\prime} )\,\left[V\left(r^{\prime} +{\xi }_{0}\right)+V\left(r^{\prime} -{\xi }_{0}\,\right) \right]\right\}\,{R}_{l,k}\,(r^{\prime} ).\end{array}\end{eqnarray}$
The asymptotic form of the radial wave function can be achieved by the phase-shift δl such that ${R}_{l,k}\left(r\right)\,\propto {\left({kr}\right)}^{-1}\;\sin \;\left({kr}-\pi l/2+{\delta }_{l}\right)$.In low energy collisions, the main contribution in the scattering process is related to the partial s-wave scattering (l = 0). Therefore, the EF (i.e. fk) can be calculated through the expansion coefficient Dl,k and the radial wave equation Rl,k(r), as follows:24 ) reduces to the result of the Mishima et al's work [41] (their equation (18 )).
$\begin{eqnarray}{f}_{k}\propto \displaystyle \frac{{\left| {D}_{0,k},\,\right|}^{2}{\left|{\int }_{0}^{\infty }\,{\rm{d}}{{rr}}^{2}{R}_{0,k}\left(r,\,\right)\right|}^{2}}{1+{\left|\tfrac{\mu k}{{{\hslash }}^{2}}{\int }_{0}^{\infty }\,{{\rm{d}}{rr}}^{2}\left[V\left(r+{\xi }_{0}\right)+V\left(r-{\xi }_{0}\,\right) \right]{R}_{0,k}\left(r,\,\right)\right|}^{2}}.\end{eqnarray}$
The above equation can successfully explain most features of the collisional EF in the presence of the laser beam. As a cross check, one can find that in the limit ξ0 → 0 (when the influence of the laser field vanishes) the equation (4. Effective potential of hot quantum plasma
An analytical formulation for the effective interaction potential in hot quantum plasmas has been derived using the quantum approach, including the influence of the effective plasma screening effects due to the collective plasma oscillations. The picture of a dressed Debye interaction between charged particles in hot quantum plasmas usually finds a complicated situation if we consider effective screening effects. Using the effective screening potential, the dressed electron-ion interaction potential Veff(r, β, λD) in hot quantum plasmas can be written as [51, 66, 86]:
$\begin{eqnarray}\begin{array}{l}{V}_{\mathrm{eff}}\,(r,\beta ,{\lambda }_{D})\\ \,=-\displaystyle \frac{{k}_{q}\,{{Ze}}^{2}}{r}\displaystyle \frac{1}{4\sqrt{1-{\beta }^{2}}}\left[\left(4-\beta ,\,\right){{\rm{e}}}^{-r/{L}_{1}\left(\beta ,{\lambda }_{D} \right)}-2\left(1-\sqrt{1-{\beta }^{2}} \right){{\rm{e}}}^{-r/{L}_{2}\left(\beta ,{\lambda }_{D} \right)}\right],\end{array}\end{eqnarray}$
where Z is the charge number of the ion, kq = 1/4πε0 is the Coulomb constant, ε0 is the vacuum permittivity, β = ℏωp/kβT is the ratio of the plasmon energy ℏωP to the thermal energy kBT, ωP denotes the plasmon frequency, kB is the Boltzmann constant, T stands for the plasma temperature, L1 and L2 are the effective screening lengths which are defined as: $\begin{eqnarray}\begin{array}{l}{L}_{1}=\displaystyle \frac{{\lambda }_{D}}{\sqrt{2}}{\left(1+\sqrt{1-{\beta }^{2}}\right)}^{1/2},\\ \Space{0ex}{0.34em}{0ex}{L}_{2}=\displaystyle \frac{{\lambda }_{D}}{\sqrt{2}}{\left(1-\sqrt{1-{\beta }^{2}}\right)}^{1/2},\end{array}\end{eqnarray}$
while λD is the standard Debye radius. This potential is expected to be valid when the plasmon energy is smaller than the thermal energy, i.e., 0 ≤ β < 1. The effective potential Veff(r, β, λD) is a bright example for the renomarlization processes which can be applied to study the collective interactions in hot quantum plasmas. However, if the plasmon effects are neglected, the effective interaction potential Veff(r, β, λD) goes toward the classical Debye–Huckel potential: ${V}_{\mathrm{eff}}\to {V}_{\mathrm{DH}}=-\tfrac{{k}_{q}\,{{Ze}}^{2}}{r}{{\rm{e}}}^{-r/{\lambda }_{D}}$, since L1 → λD and L2 → 0 as β → 0.5. EF of hot quantum plasma in the presence of laser beam fields
The entanglement fidelity ratio (EFR) is a powerful measure for investigating the plasmon and plasma screening effects on the entanglement for the elastic collisions in hot quantum plasmas. The EFR ${R}_{F}={f}_{k}^{\mathrm{eff}}/{f}_{k}^{\mathrm{Coul}}$ is calculated by the ratio of the EF ${{f}_{k}}^{\mathrm{eff}}$ for the elastic electron-ion collision (using the effective interaction potential Veff(r, β, λD) in hot quantum plasmas) to the EF ${{f}_{k}}^{\mathrm{Coul}}$ for the elastic electron-ion collision using the pure Coulomb interaction, i.e. ${V}_{C}=-\tfrac{{k}_{q}\,{{Ze}}^{2}}{r}$. Thus, we have:
$\begin{eqnarray}{R}_{F}=\displaystyle \frac{1+{\left|-\tfrac{2{k}_{q}\,\mu {{kZe}}^{2}}{{{\hslash }}^{2}}{\int }_{0}^{\infty }\,{\rm{d}}{rr}\tfrac{\sin \,({kr})}{{kr}}\right|}^{2}\;}{1+{\left|\tfrac{\mu }{{{\hslash }}^{2}}{\int }_{0}^{\infty }\,{\rm{d}}{rr}\left[{V}_{{\rm{eff}}}\,(r+{\xi }_{0},\beta ,{\lambda }_{D})+{V}_{{\rm{eff}}}\,(r-{\xi }_{0},\beta ,{\lambda }_{D}) \right]\sin \,({kr})\,\right|}^{2}}.\end{eqnarray}$
After some straightforward mathematical operations, we arrive at the following relation for the EFR:28 ):28 ) reduces to the same one derived by Jung [66] for a similar system but in the absence of the laser beam fields.
$\begin{eqnarray}{R}_{F}\left(\bar{E},\beta ,{\bar{\lambda }}_{D},{\xi }_{0}\right)=\displaystyle \frac{1+\tfrac{4{{k}_{q}}^{2}{Z}^{2}{\mu }^{2}{e}^{4}}{{{\hslash }}^{4}{k}^{2}}}{1+\tfrac{4{{k}_{q}}^{2}{Z}^{2}{\mu }^{2}{e}^{4}}{{{\hslash }}^{4}{k}^{2}}{\left| {G}_{L},\,\right|}^{2}},\end{eqnarray}$
in which $\begin{eqnarray}\begin{array}{l}{G}_{L}=\displaystyle \frac{\left(4-\beta \right)}{8\sqrt{1-{\beta }^{2}}}\left[\left.\displaystyle \frac{2{k}^{2}\cosh \left({\xi }_{0}/{L}_{1}\right)}{1/{{L}_{1}}^{2}+{k}^{2}}+k{\xi }_{0}\mathrm{Im}\left\{{{\rm{e}}}^{{\rm{i}}k{\xi }_{0}}\,{\rm{Ei}}\left(1,-\left(1/{L}_{1}-{\rm{i}}k\right){\xi }_{0}\right)-{{\rm{e}}}^{-{\rm{i}}k{\xi }_{0}}\,{\rm{Ei}}\left(1,\left(1/{L}_{1}-{\rm{i}}k\right){\xi }_{0}\right)\right\}\right]\right.\\ -\displaystyle \frac{\left(1-\sqrt{1-{\beta }^{2}}\right)}{4\sqrt{1-{\beta }^{2}}}\left[\left.\displaystyle \frac{2{k}^{2}\cosh \left({\xi }_{0}/{L}_{2}\right)}{1/{{L}_{2}}^{2}+{k}^{2}}+k{\xi }_{0}\mathrm{Im}\left\{{{\rm{e}}}^{{\rm{i}}k{\xi }_{0}}\,{\rm{Ei}}\left(1,-\left(1/{L}_{2}-{\rm{i}}k\right){\xi }_{0}\right)-{{\rm{e}}}^{-{\rm{i}}k{\xi }_{0}}\,{\rm{Ei}}\left(1,\left(1/{L}_{2}-{\rm{i}}k\right){\xi }_{0}\right)\right\}\right]\right..\end{array}\end{eqnarray}$
We have used the following integration relation to calculate equation ( $\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{0}^{\infty }\,\displaystyle \frac{r}{r\pm \varepsilon }\,{{\rm{e}}}^{-\eta \left(r\pm \varepsilon ,\,\right)}\;\sin \;\left({kr} \right){\rm{d}}r=\mathrm{Im}\;\left\{{{\rm{e}}}^{\mp {\rm{i}}k\varepsilon }\right.\\ \times \left.\left[\displaystyle \frac{{\rm{\Gamma }}\left(1,\pm \left(\eta -{\rm{i}}k,\,\right)\varepsilon \right)}{\eta -{\rm{i}}k}\mp \varepsilon {\rm{Ei}}\left(1,\pm \left(\eta -{\rm{i}}k,\,\right)\varepsilon ,\,\right)\right]\,\right\},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}\left(a,x\right)={\displaystyle \int }_{x}^{\infty }\,{{\rm{e}}}^{-t}{t}^{a-1}{\rm{d}}t;\,\mathrm{Re}\;\left(a\right)\gt 0,\\ {\rm{Ei}}\left(1,x\right)={\displaystyle \int }_{x}^{\infty }\,\displaystyle \frac{{{\rm{e}}}^{-t}}{t}{\rm{d}}t.\end{array}\end{eqnarray}$
It may be noted that ${\rm{\Gamma }}\left(a,x\right)$ is called an incomplete Gamma function and ${\rm{Ei}}\left(1,x\right)$ is the exponential integral [87]. In the limit ξ0 → 0, equation (6. Numerical discussion
Analytical discussion on the behavior of the system is very difficult because of the complicated functionality of the physical parameters in the EFR. Therefore, we have set up several numerical calculations in order to understand different features of the system. All numerical calculations have been done using the following dimensionless variables:
$\begin{eqnarray}\bar{E}={{a}_{Z}}^{2}{k}^{2},{\bar{L}}_{1}=\displaystyle \frac{{L}_{1}}{{a}_{Z}},{\bar{L}}_{2}=\displaystyle \frac{{L}_{2}}{{a}_{Z}},{\bar{\lambda }}_{D}=\displaystyle \frac{{\lambda }_{D}}{{a}_{Z}},{\bar{\xi }}_{0}=\displaystyle \frac{{\xi }_{0}}{{a}_{Z}},\end{eqnarray}$
where aZ = ℏ2/kqμZe2 and for all cases, we get Z = 1. The amplitude of free electron oscillation in an applied electric field, ξ0 is an important parameter. It is clear that ξ0 is an increasing function of an applied electric field and inversely proportional to the laser beam angular frequency. Notice that even for high intensity lasers, this parameter is very less than 1 m. For example for a CO2 laser with an intensity of 1022W/cm2 with a wavelength of λ = 10 μm, this parameter is ξ0 = 0.0016 m (${\bar{\xi }}_{0}=3.03\times {10}^{7}$).In order to understand the effect of ${\bar{\xi }}_{0}$ on the EFR, we have plotted RF for several values of scaled collision energy $\bar{E}$. Figure 1 presents RF as a function of β (ratio of plasmon energy to the thermal energy) for different values of ${\bar{\xi }}_{0}$ and $\bar{E}$. Figure 1 shows that the general behavior of RF is the same for all values of scaled collision energy ($\bar{E}$) and for the free electron oscillation dimensionless amplitude (${\bar{\xi }}_{0}$) as well. The EFR rises to a maximum value and rapidly falls as β increases. The maximum value of the EFR increases when the scaled collision energy increases. This means that the entanglement between particles reduces, as the collision energy is increased. This result is in agreement with the findings of previous related works [42, 66, 68]. The presence of the laser beam fields in the system for the case when ξ0 = 0.005 m (${\bar{\xi }}_{0}=9.40\times {10}^{7}$), results in the increase in the EFR at a fixed temperature as one could expect, because oscillations of electrons under electromagnetic fields of the laser beam cause the electron to spend more time around the ion during interaction and it leads to more entanglement of the system. The increase in RF with the presence of the laser beam is more noticeable for small values of β.
Figure 1. The variations of RF as a function of the plasmon parameter β. |
To get a better view of the behavior of EFR, in figure 2, we have plotted RF as a function of the scaled collision energy $\bar{E}$ for different values of β and ${\bar{\xi }}_{0}$. The curves related to the ${\bar{\xi }}_{0}=0$ denote the scattering process in the absence of the laser beam which show less entanglement in comparison with the cases when we have applied the laser beam fields in the system. The laser beam fields affect on the system entanglement is more evident for high energies of projectile particles. Also, one can find from figure 2 that the entanglement in the system is destroyed as collisional energy $\bar{E}$ increases and this result was predictable since an increase in the incident particle energy leads to the decrease in the interaction time and consequently to the weakening of correlation between particles. Similar results can be found in [42, 66–69, 88] where the entanglement is studied without considering laser beam presence, i.e. for ${\bar{\xi }}_{0}=0$.
Figure 2. The variations of RF as a function of the scaled collision energy $\bar{E}$. |
Another important parameter is the scaled Debye length ${\bar{\lambda }}_{D}$. The effects of this parameter on the entanglement of the system can be explained using the figure 3.
Figure 3. The surface plot of RF as a function of the scaled Debye length ${\bar{\lambda }}_{D}$ and plasmon parameter β. |
Figure 3 shows that the EFR slightly decreases when Debye length increases. In distances smaller than the Debye length, long range effects are screened, therefore, it is expected that entanglement due to long range interactions decreases as the Debye length increases. On the other hand, the distance between entangled atoms are much smaller than the Debye length, thus the screening effect in the entanglement between plasma particles is small as we learn from figure 3. It is worth mentioning that the curve of case ${\bar{\xi }}_{0}=0$ is exactly in agreement with the findings of [66, 67].
Figure 4 provides a better view to understand the effects of the dimensionless amplitude of free electron oscillation (${\bar{\xi }}_{0}$) and the collisional energy $\bar{E}$ on the behavior of the system entanglement characterized through the parameter RF. It shows that the RF generally is an increasing function of ${\bar{\xi }}_{0}$ and for a fixed value of ${\bar{\xi }}_{0}$, the parameter RF decreases as collisional energy $\bar{E}$ increases. In [43], the effect of a constant magnetic field on the entanglement of an electron-electron interaction in strongly coupled semi-classical plasma is considered and it is shown that the presence of a magnetic field causes the enhancement of EF and here, in our system, the laser electric field acts in the same manner.
Figure 4. The variations of RF as a function of the dimensionless laser beam electric field amplitude ${\bar{\xi }}_{0}$. |
7. Conclusions
In this work, we have studied the effects of electromagnetic fields due to the applied laser beam on the effective potential, describing bi-partite systems during the entanglement of the scattering process. We have used the Kramers–Henneberger unitary transformation for solving the Schrodinger equation, in the quantized field formalism. In the resultant equation, wave function is expanded using the Fourier series by considering the Ehlotzkys approximation. The stationary-state Schrodinger equation containing laser beam effect has been calculated which represents motions of the spherically confined two particles, exposed to the linearly polarized intense laser beam fields radiation in the center of mass coordinate. The partial wave method is employed to obtain the EF containing a laser beam effect as well. Our calculations indicate that high intensity laser fields change the behavior of the entanglement, extensively. We have studied an example for using the effective potential energy function to understand the role of free electron oscillation and its relation to plasmon and quantum screening effects on the entanglement fidelity. Our results showed that the laser beam fields improve the entanglement of the charged particles scattering process in the hot quantum plasma. Inversely, an increase in the collisional energy of projectile particles leads to the decrease in the entanglement quality. These results should provide useful information on the laser-plasma interaction, quantum screening effect and the quantum information.