Abstract
The atomphoton entanglement using the LaguerreGaussian (LG) beams is studied in the closedloop threelevel Vtype quantum systems. We consider two schemes with neardegenerate and nondegenerate upper levels: in the first, the effect of the quantum interference due to the spontaneous emission is taken into account and in the second, a microwave plane wave is applied to the upper levels transition. It is shown that the atomphoton entanglement in both schemes depends on the intensity profile as well as the orbital angular momentum (OAM) of the applied fields so that the various spatially dependent entanglement patterns can be generated by LaguerreGaussian beams with different OAMs. However, due to the zero intensity,no entanglement appears in the center of the optical vortex beams. As a result, the entanglement between dressed atom and its spontaneous emissions in different points of the atomic vapor cell can be controlled by the OAM of the applied fields. Moreover, our numerical results show that the number of the local maximum degree of entanglement (DEM) peaks depends on the OAM of the applied fields. The degrees of freedom for OAM play a crucial role in spatially dependent atomphoton entanglement in such a way that it may possess broad applications in highdimensional quantum information processing and data storage.
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Introduction
Entanglement is generally a quantum mechanical phenomenon without any classical analogous which makes a correlation between the parts of a multipartite system^{1,2}. A quantum state of a bipartite entangled system cannot be described by a simple tensor product of quantum states of two subsystems^{3}. Not only does atomphoton entanglement play an outstanding role in quantum mechanics such as early measurement of Bell inequality violations^{4}, but it has also high potential applications in teleportation^{5}, cryptography^{6}, error correction^{7} and other quantum computation processes^{8}.
Generally speaking, the angular momentum carried by light can be distinguished by the spin angular momentum associated with circular polarization^{9} and the orbital angular momentum (OAM) associated with the spatial distribution of the wavefront. In 1992, Allen et al. showed that the LaguerreGaussian (LG) laser mode has a welldefined OAM, as \(l\hslash \) per photon, and proposed an experiment to observe the torque on cylindrical lenses arising from the reversal of the helicity of a LG mode^{10}. An additional motivation for recent studies of LG beams is as a route to obtain narrower electromagnetically induced transparency (EIT) spectrum than the conventional Gaussian one^{11,12,13}. The transmission of structured light has been measured using the phase profile in cold rubidium atoms and been shown that the EIT is spatially dependent for vortex light beams^{14}. Recently, Mahmoudi et al. has investigated the effect of LG intensity profile on the optical spectrum of multiphoton resonance phenomena. It has been found that the linewidth of the optical spectrum due to the multiphoton transition becomes narrower in the presence of a LG beam^{15}. They have also studied trap split in an atomic system interacting with femtosecond LG laser pulses^{16}. More recently, we have demonstrated that the LG beam can decrease the full width at the half maximum of the output probe intensity in the electromagnetically induced focusing^{17}.
It is well known that a LG light beam with an azimuthal phase dependence of \({e}^{il\varphi }\) carries quantized OAM^{18}. Due to the fact that the optical properties of the closedloop quantum systems, in multiphoton resonance condition, depend on the relative phase of applied fields^{19} the effect of the OAM of the LG beams can be seen in this system due to the azimuthal phase factor of LG beam via the multiphoton transitions.
Increasing the complexity of entangled systems is important in the various applications of the entanglement in quantum information tasks which can be essentially done by increasing the number of particles involved in the entanglement^{20} or the entanglement dimensionality of a system^{21}. The alternative method to increase the entanglement dimensionality of a system is to use the LG modes. Highdimensional entanglement has recently attracted increasing attention in both fundamental and applied research in quantum mechanics. The use of these states can enhance quantum communication schemes by increasing their channel capacity and offering improved robustness against sophisticated eavesdropping attacks. However, one of the methods for generating higher order entangled states is using the OAM basis^{22}. Recently, quantum logic gates also have been achieved experimentally^{23} and applied for generating high dimensional Bell States in this basis^{24}. Although there are a lot of scientific reports on high dimensional photon states in OAM basis, it seems a deep fundamental research is necessary for atomphoton interaction in OAM basis, too. Moreover, we believe that the simulated atomphoton system can be a good source for high dimensional entangled states.
In this paper, we study the effect of intensity profile and OAM of light beams on the quantum entanglement in the threelevel Vtype quantum systems. In the first scheme, we consider Vtype atomic system with two neardegenerate excited levels. In the presence of quantum coherence due to the spontaneous emission the transition of the electron between two excited states becomes possible and the Vtype atomic system switches to the closedloop configuration. In the second scheme, we consider two nondegenerate excited states and the quantum coherence between two upper levels is induced by applying a microwave field to the upper levels transition. Then, the optical properties of such systems depend on the OAM of the applied fields. We are interested in studying the entanglement of dressed atom and its spontaneous emission, using the LG fields with different OAMs. We use the von Neumann entropy as a measure of the degree of entanglement (DEM). It is shown that the DEM completely depends on the position of space points, so that the various points in the atomic vapor cell experience different DEM. Moreover, we find that the DEM is dependent on both magnitude and sign of the OAM of light beams. Then, the DEM can be controlled by the intensity profile, as well as the OAM of the applied fields. It is worth noting that the maximal value of DEM is obtained in the second scheme.
Theoretical Framework
In general, an entangled quantum system consisting of two subsystems, A and B, is described by the reduced density matrix. The state of the entangled system is not a simple tensor product of the subsystems reduced density matrices, \({\rho }_{AB}\ne {\rho }_{A}\otimes {\rho }_{B}\). In thermodynamics, entropy is a fundamental physical quantity describing the degree of randomness of the system. Thus, we can represent the entanglement adopting the reduced quantum entropy. Various measures such as the reduced entropy of entanglement^{25}, the relative entropy of entanglement^{7}, entanglement of formation^{26} and entanglement of distillation^{27} have been introduced for DEM. Here, we use the definition of the von Neumann reduced entropy of entanglement. The von Neumann entropy, as a helpful quantity for calculating the DEM between the subsystems, is given by
where \(\rho \) is the density matrix operator. It can be easily found that the von Neumann entropy vanishes for a bipartite system in pure state^{28}. In 1970, Araki and Lieb demonstrated that the subsystems entropies satisfy in the triangle inequality^{29}
at any time t. Here, \({S}_{AB}(t)\) is the total entropy of the composite system. Based on equation (2), for a atomfield system initially in a disentangled pure state, partial entropies of the field and the atom will be equal at all times after the interaction of two subsystems is switched on. Then, our information about any of the subsystems is an indication of the entanglement of the whole system. A decreasing partial entropy means that each subsystem evolves towards a pure quantum state, whereas in an initially pure system an increasing partial entropy drives the two components to lose their individuality and become entangled^{30}. So the DEM for atomfield system would be
where λ_{i} is the reduced density matrix operator eigenvalues. The maximum value of DEM for a Nlevel quantum system is given by lnN, in which the population is uniformly distributed in the dressed states of the system^{31,32}. Now, we are going to study the effect of applied LG fields on the atomphoton entanglement in two schemes of closedloop threelevel atomic systems.
ThreeLevel VType Atomic Systems
We assume an ensemble of threelevel Vtype atomic system with two neardegenerate excited states \(\mathrm{2}\rangle \) and \(\mathrm{3}\rangle \) and a ground state \(\mathrm{1}\rangle \), as shown in Fig. 1. As a realistic example, we consider the sodium \({D}_{2}\) line (\({\mathrm{3}}^{2}{S}_{\mathrm{1/2}},F=1\rangle \leftrightarrow {\mathrm{3}}^{2}{P}_{\mathrm{3/2}}\rangle \)); \(\mathrm{1}\rangle \), \(\mathrm{2}\rangle \), and \(\mathrm{3}\rangle \) correspond to sublevels \({\mathrm{3}}^{2}{S}_{\mathrm{1/2}},F=1\rangle \), \({\mathrm{3}}^{2}{P}_{\mathrm{3/2}},F^{\prime} =0\rangle \), and \({\mathrm{3}}^{2}{P}_{\mathrm{3/2}},F^{\prime} =1\rangle \), respectively. Here, the upper levels are not exactly degenerated, however, if the energy difference of the upper levels is less than the natural line width, it can be considered as the neardegenerate levels and both transitions interact with a common vacuum state. Here the energy difference of upper levels is 15.8 MHz, which is less than the natural line width of the transition \((\,\sim \,61.35\,MHz)\). The spontaneous emission rates from two neardegenerate excited states \(\mathrm{2}\rangle \) and \(\mathrm{3}\rangle \) to the ground state is denoted by \(2{\gamma }_{1}\) and \(2{\gamma }_{2}\), respectively. In such a system, the spontaneous emission passing through the indistinguishable paths lead to the quantum interference called as spontaneously generated coherence (SGC). The transition \(\mathrm{1}\rangle \leftrightarrow \mathrm{2}\rangle \) is driven by the right LG field with frequency \({\omega }_{R}\) and Rabi frequency \({{\rm{\Omega }}}_{R}={\overrightarrow{\mu }}_{12}\mathrm{.}{\overrightarrow{E}}_{R}/\hslash \), while the transition \(\mathrm{1}\rangle \leftrightarrow \mathrm{3}\rangle \) is excited by the left LG field with frequency \({\omega }_{L}\) and Rabi frequency \({{\rm{\Omega }}}_{L}={\overrightarrow{\mu }}_{13}.{\overrightarrow{E}}_{L}/\hslash \). Here \({\overrightarrow{E}}_{R(L)}\) denotes the amplitude of right (left) LG field. The parameter \({\mu }_{ij}\) represents the induced dipole moment of the \(i\rangle \leftrightarrow j\rangle \) transition.
The interaction Hamiltonian of the system under the electricdipole and rotatingwave approximations can be written as
where \({{\rm{\Delta }}}_{R}={\omega }_{R}{\omega }_{21}\) and \({{\rm{\Delta }}}_{L}={\omega }_{L}{\omega }_{31}\) are the applied fields frequency detunings with respect to the atomic transition frequencies. The parameter \({\omega }_{ij}\) is the central frequency of \(i\rangle \leftrightarrow j\rangle \) transition. Both applied fields are chosen as the LG fields and are given by
where \({E}_{{0}_{L}}\)(\({E}_{{0}_{R}}\)) and \({l}_{L}\)(\({l}_{R}\)) are the left(right) field amplitude and Orbital angular momentum, respectively. Gaussian and LG beam waists are denoted by \({w}_{G}\) and \({w}_{LG}\), respectively. Note that the beam waists of left and right applied fields are considered to be equal for enhancement the effect of their interaction with atomic system on the DEM. Using the von Neumann equation, the Bloch equations for such a system takes the form
where \({\rm{\Delta }}={{\rm{\Delta }}}_{R}{{\rm{\Delta }}}_{L}={\omega }_{R}{\omega }_{L}\). The parameter \(\eta \) in equation (6) stands for the SGC strength.
We now introduce the dressed states generated by applied fields which are useful for understanding the optical properties of the system. The physics of the entanglement can be explained via the population distribution of the dressed states. For the proposed model, the dressed states in the absence of spontaneous emissions are given by
The dressed states population can be obtained as
Now, we are interested in studying the DEM for Vtype atomic system by numerically solving equations (3) and (6). It is assumed that the multiphoton resonance condition is fulfilled. All frequency parameters are scaled by \({\gamma }_{1}\), which is in the order of MHz for the chosen atomic system. We consider different intensity profiles, i.e., Gaussian and LG modes for the applied fields and investigate steadystate behavior of the DEM in different points of atomic vapor cell. Figure 2 shows the DEM as a function of x for the applied Gaussian fields, (a) \({l}_{L}={l}_{R}=0\), first mode (b) \({l}_{L}={l}_{R}=1\) and second mode of applied LG fields (c) \({l}_{L}={l}_{R}=2\). Used parameters are \({\gamma }_{1}={\gamma }_{2}=\gamma \), \(\eta =0.99\), \({{\rm{\Omega }}}_{{0}_{L}}=7\gamma \), \({{\rm{\Omega }}}_{{0}_{R}}=9\gamma \), \({w}_{G}=1\,mm\), \({w}_{LG}=270\,\mu m\) and \({{\rm{\Delta }}}_{L}={{\rm{\Delta }}}_{R}=0\), multiphoton resonance condition. Red lines indicate the maximum of DEM positions in panels (a)–(c). In panel (a), it can be easily seen that the DEM changes from zero to a maximum value, for different values of x. The DEM vanishes where the beams’ intensity becomes zero. Panels (b) and (c) shows that the zero intensity in the center of LG profile implies a disentanglement in the center of LG modes. So, the spatially dependent behavior of DEM is related to the intensity profile of two applied fields. Note that the effect of OAMs’ sign on the DEM is not seen in one dimension calculation.
A physical meaning of entanglement lies on the dressed states conception. For explanation of DEM behavior in the presence of spontaneous emissions, we calculate the density matrix operator eigenvalues which describe the dressed states population in the presence of spontaneous emmision. The uniform population distribution of the dressed states induce the maximal entanglement to the atomic system. By numerically solving equation (6) and diagonalizing the density matrix, the dressed states population can be obtained as a function of x. We would like to investigate the dressed states population in different points of the atomic vapor cell. In the following, Fig. 3 shows the dressed states population versus x corresponding to Fig. 2. As seen in all panels of Fig. 3, the population is distributed only in two states around \(x=0\) and the DEM value reaches to \(ln2\), while a part of population transfers to the third state at the maximum point of DEM and the population is nearly uniformly distributed in the three dressed states, with respect to the population distribution at \(x=0\).
Note that in one dimension calculation, the effect of the OAM of light beams on the atomphoton entanglement does not take into account, so, we have to continue our calculations in two dimensions to include the contribution of the OAM of applied LG fields. According to equation (5), the OAM of light appears as a phase term, \({e}^{il\phi }\). It is expected that the optical properties of the Vtype atomic system depend on the OAM of light beam in closedloop configuration. In Fig. 4, we depict the DEM density plots as a function of x and \(y\) for different modes of applied fields with \({l}_{L}=\,3,\ldots ,3\) and \({l}_{R}=0,\ldots ,3\). Used parameters are \({\gamma }_{1}={\gamma }_{2}=\gamma \), \(\eta =0.99\), \({{\rm{\Omega }}}_{{0}_{L}}=7\gamma \), \({{\rm{\Omega }}}_{{0}_{R}}=9\gamma \), \({w}_{G}=1\,mm\), \({w}_{LG}=270\,\mu m\) and \({{\rm{\Delta }}}_{L}={{\rm{\Delta }}}_{R}=0\). There are \(28\) panels for different choices of LG modes of applied fields. For \({l}_{L}={l}_{R}=0\), the DEM changes from zero to its maximum value in a determined radius and decreases to nearly maximal value of DEM of twolevel system, \(ln2\). However, for equal OAM modes of two applied LG fields, \({l}_{L}={l}_{R}=l\), the DEM value has a central symmetry with a disentanglement point at the origin coordinates due to the zero intensity of applied fields. These results in the \(x\) direction are in good agreement with one dimensional results of Fig. 2. Here, we show that the DEM behavior completely depends on the OAM of applied fields in two dimensions. The local maximum DEM ring switches to \(2l\) maximum DEM regions when the rotation direction of wavefront is turned to counterclockwise, \({l}_{L}=\,{l}_{R}=l\). The obtained results are in good agreement with the presented petallike patterns in references^{33} and^{34} in which, it was shown that the superposition pattern of two LG beams with \(l\) of opposite sign has a symmetric structure with \(2l\) petals. The DEM profile is similar to the interference pattern of two applied LG fields. Moreover, we extend our calculations for different LG modes and it can be generally seen that the number of local maximum peaks equals \({l}_{L}{l}_{R}\) in all panels. So, the atomphoton entanglement pattern can be controlled by OAM of light beams.
Now, we are going to investigate the DEM behavior of the system beyond multiphoton resonance condition, \({\rm{\Delta }}\ne 0\). However, we drop the explicit time dependent phase factors in equation (6). Then, our results are valid only near multiphoton resonance condition. The right (left) field is considered as a planewave (LG field). In Fig. 5, we display the DEM density plot as a function of \(x\) and \(y\) in different points of the crosssection of atomic vapor cell for a Gaussian left field, \({l}_{L}\mathrm{=0}\). Used parameters for the applied fields are assumed to be \({{\rm{\Omega }}}_{R}=9\gamma \), \({{\rm{\Omega }}}_{{0}_{L}}=7\gamma \), \({w}_{G}=1\,mm\) and \({\rm{\Delta }}=\gamma \), beyond multiphoton resonance condition. The atomic parameters are \({\gamma }_{1}={\gamma }_{2}=\gamma \) and \(\eta =0.99\). It is shown that the DEM tends to its maximal value, \(ln3\), in a special radius which depends on the characteristics of the Gaussian field. The DEM density plots for the different modes of LG field are plotted versus \(x\) and \(y\), as shown in Fig. 6. The applied fields parameters are \({{\rm{\Omega }}}_{R}=9\gamma \), \({{\rm{\Omega }}}_{{0}_{L}}=7\gamma \), \({w}_{LG}=270\,\mu m\) and \({\rm{\Delta }}=\gamma \). The atomic properties are chosen to be same as in Fig. 5. Figure 6(a) shows the DEM variations for the first mode of LG field, \({l}_{L}=1\). The DEM behavior, specially around maximal DEM regions, is very similar to spiralshaped fringes due to the interference of first mode of LG light beam with a planewave^{18,34}. In Fig. 6(b) it is clearly seen that the DEM behavior for the negative OAM of the first mode of LG field is in good agreement with the rotation direction of spiral arms of LG field and planewave interferogram. We repeat our numerical calculations for positive and negative OAMs of the second and third modes of LG left field, \({l}_{L}=\mathrm{2,}\,\mathrm{2,}\,\mathrm{3,}\,3\) in panels (c)–(f), respectively. An investigation on Fig. 6(c–f) shows that the DEM patterns are similar to the interference pattern of different modes of LG beam with the planewave. Therefore, the spatially dependent DEM can be controlled either magnitude or sign of the OAM of the applied LG field.
ClosedLoop ThreeLevel VType Atomic Systems
In this section, we propose an ensemble of threelevel Vtype atomic system with two nondegenerate excited states and apply a planar microwave field, \({{\rm{\Omega }}}_{m}\), to the \(\mathrm{2}\rangle \leftrightarrow \mathrm{3}\rangle \) transition and two other fields shown in Fig. 7, which can be prepared in the Rb Rydberg atoms^{35}.
Using the von Neumann equation and the interaction Hamiltonian of the system, the Bloch equations for the system are obtained by
where \({\rm{\Delta }}={{\rm{\Delta }}}_{L}{{\rm{\Delta }}}_{R}{{\rm{\Delta }}}_{m}\).
Here, we are going to show the behavior of the DEM in a closedloop threelevel \(V\)type atomic system using equations (3) and (9). A planar microwave field is applied to the \(\mathrm{2}\rangle \leftrightarrow \mathrm{3}\rangle \) transition. It is assumed that the multiphoton resonance condition is fulfilled. Figure 8 presents the density plots of DEM versus \(x\) and \(y\) for different modes of applied fields with \({l}_{L}=\,\mathrm{3,}\,\mathrm{...,}\,3\) and \({l}_{R}=\mathrm{0,}\,\mathrm{...,}\,3\). Size of each density plot is \(2\,mm\,\times \,2\,mm\) in which horizontal and vertical axes are \(x\) and \(y\) axes, respectively. The microwave Rabi frequency, relaxation rate between two excited states and detuning of the microwave field frequency are considered to be \({{\rm{\Omega }}}_{m}=7\gamma \), \({\gamma }_{3}=2\gamma \) and \({{\rm{\Delta }}}_{m}=0\), respectively. Other used parameters are \({\gamma }_{1}={\gamma }_{2}=\gamma \), \({{\rm{\Omega }}}_{{0}_{L}}=7\gamma \), \({{\rm{\Omega }}}_{{0}_{R}}=9\gamma \), \({w}_{G}=1\,mm\), \({w}_{LG}=270\,\mu m\) and \({{\rm{\Delta }}}_{L}={{\rm{\Delta }}}_{R}=0\). It is shown that the maximal entanglement can be obtained in different regions of atomic vapor cell by applying a planar microwave field instead of the SGC effect. Two applied Gaussian fields, \({l}_{L}={l}_{R}=0\), create maximal value of DEM in a central circular region of atomic vapor cell. Now, let us refer to the DEM density plots in which two applied LG fields have equal OAMs, \({l}_{L}={l}_{R}=l\). A comparison between the related panels shows that the region of disentanglement has been grown for larger OAMs. It is worth to note that for LG modes with opposite sign OAMs, \({l}_{L}=\,{l}_{R}=l\), the maximal DEM doughnutlike region splits to \(4l\) symmetric segments. Other \(21\) planes in Fig. 8 show the DEM behavior in different points of the atomic vapor cell for different choices of OAM of two applied fields. An investigation on all panels of Fig. 8 confirms that the DEM patterns follow a common model which are extremely dependent on OAM and have \(\mathrm{2}{l}_{L}{l}_{R}\) maximal regions.
Conclusion
In conclusion, we studied quantum entanglement between an ensemble of threelevel atomic system and its spontaneous emissions. In the first scheme, we investigated the steadystate behavior of the DEM in a threelevel \(V\)type atomic system with the SGC effect under the multiphoton resonance condition. We found an intriguing result which illustrates an explicit dependency between the DEM density plot and the OAM of LG light beams. Moreover, we showed that the DEM density plot pattern is corresponding to the interference pattern of the two applied fields. In the second scheme, a microwave plane field was applied to the nondegenerate upper levels transition which leads to the maximal DEM. It was demonstrated that the number of maximal entanglement peaks is determined by the OAM of the applied LG fields. Both disentanglement and maximal DEM were simultaneously obtained in the second scheme. The presented results can be used in optical communications and information storage via preparing highdimensional Hilbert space.
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Acknowledgements
We would like to thanks A. Babazadeh for helpful discussions and comments on this work.
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All authors developed the research conceptions, analysed, and discussed the obtained results. Z.A.S. and R.A. performed the calculations and wrote the paper with major input from M.M.
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Amini Sabegh, Z., Amiri, R. & Mahmoudi, M. Spatially dependent atomphoton entanglement. Sci Rep 8, 13840 (2018). https://doi.org/10.1038/s41598018320518
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DOI: https://doi.org/10.1038/s41598018320518
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