Spatially dependent atom-photon entanglement

Abstract

The atom-photon entanglement using the Laguerre-Gaussian (LG) beams is studied in the closed-loop three-level V-type quantum systems. We consider two schemes with near-degenerate and non-degenerate 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 atom-photon 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 Laguerre-Gaussian 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 atom-photon entanglement in such a way that it may possess broad applications in high-dimensional quantum information processing and data storage.

Figure 1

Schematics of the three-level $V$-type atomic system with two near-degenerate excited states $\mathrm{|2}\rangle$ and $\mathrm{|3}\rangle$ and a ground state $\mathrm{|1}\rangle$. $2{\gamma }_1$ and $2{\gamma }_2$ are the spontaneous emission rates from upper levels to lower level which are in the order of MHz. Two applied LG fields, ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$, and the SGC make the proposed system a closed-loop system.

Figure 2

Steady-state behavior of the DEM versus $x$ for (a) $l_L=l_R=0$, (b) $l_L=l_R=1$ and (c) $l_L=l_R=2$. Used parameters are ${\gamma }_1={\gamma }_2=\gamma$, $\eta =0.99$, ${\mathrm{\Omega }}_{0_L}=7\gamma$, ${\Omega }_{0_R}=9\gamma$, $w_G=1mm$, $w_{LG}=270\mu m$ and ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=0$. Red lines indicate the maximum of DEM position in all panels.

Figure 3

Steady-state behavior of density matrix eigenstates versus $x$ for (a) $l_L=l_R=0$, (b) $l_L=l_R=1$ and (c) $l_L=l_R=2$. Solid, dashed and dash-dotted curves show the first, second and third eigenstates, respectively. The same parameters as in Fig. 2 are used in all panels. Red line, in each of panels, corresponds to the position of maximum of DEM.

Figure 4

DEM density plots as a function of $x$ 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 $2mm\times 2mm$ in which horizontal and vertical axes are $x$ and $y$ axes, respectively. Used parameters are ${\gamma }_1={\gamma }_2=\gamma$, $\eta =0.99$, ${\mathrm{\Omega }}_{0_L}=7\gamma$, ${\mathrm{\Omega }}_{0_R}=9\gamma$, $w_G=1mm$, $w_{LG}=270\mu m$ and ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=0$.

Figure 5

Density plot of the DEM versus $x$ and $y$, beyond multi-photon resonance condition, when two applied fields are considered as a plane-wave (right field) and a Gaussian (left field). Used parameters for the applied fields are ${\mathrm{\Omega }}_R=9\gamma$, ${\mathrm{\Omega }}_{0_L}=7\gamma$, $w_G=1mm$ and $\mathrm{\Delta }=\gamma$. The atomic parameters are ${\gamma }_1={\gamma }_2=\gamma$ and $\eta =0.99$.

Figure 6

DEM density plots as a function of $x$ and $y$ for a plane-wave laser as the right field and a LG field as the left field with different modes, $l_L=1$ and $-1$, (a,b), $l_L=2$ and $-2$, (c,d), $l_L=3$ and $-3$, (e,f), beyond multi-photon resonance condition. Size of each density plot is $1.4mm\times 1.4mm$ in which horizontal and vertical axes are $x$ and $y$ axes, respectively. The applied fields parameters are ${\mathrm{\Omega }}_R=9\gamma$, ${\mathrm{\Omega }}_{0_L}=7\gamma$, $w_{LG}=270\mu m$ and $\mathrm{\Delta }=\gamma$. The atomic properties are chosen to be same as in Fig. 5.

Figure 7

Schematics of the closed-loop three-level $V$-type atomic system with two non-degenerate excited states $\mathrm{|2}\rangle$ and $\mathrm{|3}\rangle$ and a ground state $\mathrm{|1}\rangle$ which can be prepared in the Rb Rydberg atoms. The spontaneous emission rate related to $\mathrm{|2}\rangle \leftrightarrow \mathrm{|3}\rangle$ transition is denoted by $2{\gamma }_3$; $2{\gamma }_1$ and $2{\gamma }_2$ are the spontaneous emission rates from upper levels to lower level which are in the order of MHz. Two applied LG fields, ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$ and one planar microwave field, ${\mathrm{\Omega }}_m$, establish a closed-loop atomic system.

Figure 8

DEM density plots as a function of $x$ 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 $2mm\times 2mm$ in which horizontal and vertical axes are $x$ and $y$ axes, respectively. Three applied fields characteristics are chosen to be ${\mathrm{\Omega }}_{0_L}=7\gamma$, ${\mathrm{\Omega }}_{0_R}=9\gamma$, ${\mathrm{\Omega }}_m=7\gamma$, $w_G=1mm$, $w_{LG}=270\mu m$ and ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R={\mathrm{\Delta }}_m=0$. The atomic system parameters are ${\gamma }_1={\gamma }_2=\gamma$ and ${\gamma }_3=2\gamma$.