Generation of polarization-entangled photon pairs from two interacting quantum emitters

Abstract

Entangled photon pairs are key elements in quantum communication and quantum cryptography. State-of-the-art sources of entangled photons are mainly based on parametric down-conversion from nonlinear crystals, which is probabilistic in nature, and on cascade emission from biexciton quantum dots, which finds difficulties in generating entangled photons in the visible regime. Here, we use the Wigner-Weisskopf theory to provide a demonstration that polarization-entangled photon pairs can be emitted from two interacting quantum emitters with two-level-system behavior and perpendicular transition dipole moments. These emitters can represent a large variety of systems (e.g., organic molecules, quantum dots, and diamond color centers) offering a large technological versatility, for example, in the spectral regime of the emission. We show that a highly entangled photon pair can be postselected from this system by including optical filters. Additionally, we verify that the photon entanglement is not significantly affected by small changes in the detection directions and in the orientation between the dipole moments.

Figures (11)

• Figure 1: Schematic representation of the two-photon emission from two initially inverted quantum emitters. The emitters (indexed by $j=1,2$) behave as two-level systems, with transition dipole moment $\boldsymbol{\mu}_j = \mu ( \cos\alpha_j \hat{\boldsymbol{x}} + \sin\alpha_j \hat{\boldsymbol{z}})$, and they are located at positions $\boldsymbol{r}_j$, with $\boldsymbol{r}_{12}=\boldsymbol{r}_{1}-\boldsymbol{r}_{2}$ oriented in the $z$-direction (axis indicated at left bottom). The relaxation of the emitters generates two photons in electromagnetic modes ($\boldsymbol{k}$, $s$) and ($\boldsymbol{k}'$, $s'$) with probability amplitude $c_{\boldsymbol{k} s , \boldsymbol{k}{'} {s'}}^{gg}$, where $s$ and $s'$ are the polarization modes and $\boldsymbol{k}$ and $\boldsymbol{k}'$ are the wave vectors. Additionally, $\theta$ and $\phi$ represent the polar and azimuthal angles, respectively, of the wave vector $\boldsymbol{k}$ in spherical coordinates. In Secs. \ref{['Section:Entanglement_generation']} and \ref{['Section:postselection']} we focus on the case of perpendicular transition dipole moments, with $\alpha_1 = -\alpha_2 = \pi/4$.
• Figure 2: Photon emission from the the symmetric and antisymmetric hybrid states. We fix $\alpha_1 = -\alpha_2 = \pi/4$ (see Fig. \ref{['Figure:1']}), which corresponds to perpendicular transition dipole moments. (a) Dependence on the distance $r_{12}$ between both emitters (normalized by the transition wavelength $\lambda_0= 618$ nm) of the coherent dipole-dipole coupling $V$ (black line) and of the dissipative coupling $\gamma_{12}$ (brown line). Both couplings $V$ and $\gamma_{12}$ are normalized by the spontaneous emission rate $\gamma_0$. (b) Schematic level structure and relaxation paths. The initial state $\ket{e e}$ can relax via the symmetric state $\ket{S}=(\ket{g e}+\ket{e g})/\sqrt{2}$ (transitions indicated with purple arrows) generating two photons polarized in the $\hat{\boldsymbol{x}}$ direction (corresponding to the direction of the transition dipole moment of the symmetric state, written in purple). $\ket{e e}$ can also relax via the antisymmetric state $\ket{A}=(\ket{g e}-\ket{e g})/\sqrt{2}$ (transitions indicated with green arrows), which leads to the emission of a photon polarized in the $\hat{\boldsymbol{z}}$ direction (opposite to the direction of the transition dipole moment of the antisymmetric state, written in green) and another photon polarized in the direction $-\hat{\boldsymbol{z}}$ (corresponding to the direction of the dipole moment of the antisymmetric state). (c) Radiation pattern of an electric point dipole oriented in the direction $\boldsymbol{\mu}_1 + \boldsymbol{\mu}_2 \propto \hat{\boldsymbol{x}}$ of the transition dipole moment of the symmetric state. $|\boldsymbol{E}|^2$ is the squared amplitude of the classical electric field generated by such electric-point dipole. (d) Radiation pattern of an electric point dipole oriented in the direction $\boldsymbol{\mu}_2 - \boldsymbol{\mu}_1 \propto -\hat{\boldsymbol{z}}$ (or equivalently, $\boldsymbol{\mu}_1 - \boldsymbol{\mu}_2 \propto \hat{\boldsymbol{z}}$) of the transition dipole moment of the antisymmetric state. The dashed gray arrows in (c,d) mark the direction of the $y$-axis. (e) Dependence of the probability density $D(\theta=\pi/2,\phi=\pi/2;\theta',\phi')$ on $\theta'$ and $\phi'$ for two DBATT molecules separated by a distance $r_{12}=0.075\lambda_0$ and embedded in naphthalene crystal with $n=1.5$. These molecules have spontaneous emission rate $\gamma_0 /(2\pi)= 21.5$ MHz and transition wavelength $\lambda_0 = 618$ nm.
• Figure 3: Characterization of the two-photon emission at directions $\hat{\boldsymbol{k}}=\hat{\boldsymbol{y}}$ and $\hat{\boldsymbol{k}}'=-\hat{\boldsymbol{y}}$ from two DBATT molecules with perpendicular transition dipole moments. These molecules have $\gamma_0 /(2\pi)= 21.5$ MHz and $\lambda_0 = 618$ nm and they are embedded in a naphthalene crystal with refractive index $n=1.5$. Additionally, we fix the intermolecular distance at $r_{12} = 0.075 \lambda_0$ (along the z axis), which yields $V\approx 3.5\gamma_0$. We plot the probability density of two-photon emission $P(\boldsymbol{k},s;\boldsymbol{k}',s')$ (in units of $\text{m}^2$) as a function of the photon frequencies $\omega_k =kc$ and $\omega_{k'}=k'c$ at (a) $s=s'=\hat{\boldsymbol{x}}$ and at (b) $s=s'=\hat{\boldsymbol{z}}$. (c) Relative phase $\delta$ between the two-photon probability amplitudes $c_{\boldsymbol{k} s , \boldsymbol{k}{'} {s'}}^{gg}(\infty)$ at $s=s'=\hat{\boldsymbol{x}}$ and at $s=s'=\hat{\boldsymbol{z}}$. On the right panel we show a zoom of this relative phase around $\omega_k =\omega_0 - V$ and $\omega_{k'}=\omega_0 + V$ (highlighted with a green box on the left panel).
• Figure 4: Schematic representation of the postselection procedure. The transition dipole moments of the emitters are assumed to be contained in the $xz$-plane and oriented perpendicularly to each other ($\alpha_1 = -\alpha_2 = \pi/4$ in Fig. \ref{['Figure:1']}). Blue circles represent photons emitted at frequency $\approx\omega_+ = \omega_0 + V$ from the interacting system, while red circles correspond to photons emitted at frequency $\approx\omega_- = \omega_0 - V$. Alice detects only photons emitted at $\hat{\boldsymbol{k}}=\hat{\boldsymbol{y}}$ and Bob does it at $\hat{\boldsymbol{k}}'=-\hat{\boldsymbol{y}}$. Additionally, Alice uses a filter with Lorentzian profile $F_A (\omega)$, with linewidth $\Gamma$ and central frequency $\omega_+$, whereas Bob uses a filter with Lorentzian profile $F_B (\omega)$, with linewidth $\Gamma$ and central frequency $\omega_-$.
• Figure 5: Characterization of the two-photon postselected state $\hat{\rho}$. We plot the dependence on the linewidth $\Gamma$ of the filters (normalized by the spontaneous emission rate $\gamma_0$) and on the distance $r_{12}$ between the two emitters (normalized by $\lambda_0$) of (a) $1-\mathcal{C}$ (where $\mathcal{C}$ is the concurrence), (b) $1-\mathcal{F}$ (where $\mathcal{F}$ is the the fidelity of $\hat{\rho}$ with respect to the Bell state $(\ket{\hat{\boldsymbol{x}}\hat{\boldsymbol{x}}}-\ket{\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}})/\sqrt{2}$), and (c) the normalizing factor $N$ of the density matrix divided by its maximum value $N_{\text{max}}$ (obtained within the range of filter linewidths and intermolecular distances analyzed). The two emitters are DBATT molecules, with $\gamma_0 /(2\pi)= 21.5$ MHz and $\lambda_0 = 618$ nm, which are embedded in a naphthalene crystal with refractive index $n=1.5$. The transition dipole moments of these molecules are contained in the $xz$-plane and oriented perpendicularly to each other ($\alpha_1 = - \alpha_2 = \pi/4$ in Fig. \ref{['Figure:1']}).