Coherent effects on two-photon correlation and directional emission of two two-level atoms

Sub- and superradiant dynamics of spontaneously decaying atoms are manifestations of collective many-body systems. We study the internal dynamics and the radiation properties of two atoms in free space. Interesting results are obtained when the atoms are separated by less than half a wavelength of the atomic transition, where the dipole-dipole interaction gives rise to new coherent effects, such as (a) coherence between two intermediate collective states, (b) oscillations in the two-photon correlation $G^{\left(2\right)}$, (c) emission of two photons by one atom, and (d) the loss of directional correlation. We compare the population dynamics during the two-photon emission process with the dynamics of single-photon emission in the cases of a $\Lambda$ and a V scheme. We compute the temporal correlation and angular correlation of two successively emitted photons using the $G^{\left(2\right)}$ for different values of atomic separation. We find antibunching when the atomic separation is a quarter wavelength $\lambda ∕4$. Oscillations in the temporal correlation provide a useful feature for measuring subwavelength atomic separation. Strong directional correlation between two emitted photons is found for atomic separation larger than a wavelength. We also compare the directionality of a photon spontaneously emitted by the two atoms prepared in phased-symmetric and phased-antisymmetric entangled states ${\mid \pm ⟩}_{{\mathbf{k}}_0}=e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_1}\mid a_1,b_2⟩\pm e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_2}\mid b_1,a_2⟩$ by a laser pulse with wave vector ${\mathbf{k}}_0$. Photon emission is directionally suppressed along ${\mathbf{k}}_0$ for the phased-antisymmetric state. The directionality ceases for interatomic distances less than $\lambda ∕2$.

Figure 1

(Color online) (a) Two particles (atoms), each with two levels separated by a distance $r$. In practice, the particles could be atoms or ions in a trap. (b) Timed excitation [12] using a laser with wave vector ${\mathbf{k}}_0$ and detection by detector $D$ $(=A,B)$ for the intensity distribution $I(\alpha ,r)$ or two-photon correlation $G^{\left(2\right)}({\alpha }_A,{\alpha }_B,r,\tau )$ using two detectors. (c) Double cascade (or ◇ scheme) composite four basis states for two-atom radiation system. If the ground state is neglected, we have the $\Lambda$ scheme with no Fano interference. (d) V scheme: The system can start from an entangled state composed of a superposition of the intermediate states with no photon and decay to the ground state, giving Fano interference.

Figure 2

(Color online) Transient dynamics for the full four-level scheme for $\Delta M=\pm 1$ transitions. (a) Populations $P^{(\pm )}$ in $\mid \pm ⟩=(1∕\sqrt{2})(\mid a_1,b_2⟩\pm \mid b_1,a_2⟩)\mid 1_{\mathbf{k}}⟩$ versus time for different interatomic distances. The populations increase rapidly at small times, reach a maximum at around $3∕\left(4\Gamma \right)$, and fall off gradually. A subtle feature is that at large $r$ the peaks of the two states oscillate closely around 0.25, the value at large interatomic distance. Note that the peak for $P^{(-)}$ (−−−−) is lower than the peak for $P^{(+)}$ (++++) when $r=n\lambda$ with $n$ being an integer. (b) Three-dimensional (3D) plots of $P^{(\pm )}$ show additional features. The two populations are different in the region $r\lesssim \lambda$. At $r\simeq \lambda ∕4$ the population in state ∣−⟩ is not depleted even at large times. (c) Transient populations for all states in the large-$r$ limit. The populations are the same around time $3∕\left(4\Gamma \right)$.

Figure 3

(Color online) Comparison of the populations in the bright state $P^{(+)}$ (solid line) and the dark state $P^{(-)}$ (dots) for the scalar photon with $g=\left(\sin kr\right)∕kr$ and the vector photon with $g^{\Vert }=[3∕{\left(kr\right)}^3](\sin kr-kr\cos kr)$ for linear polarization or $\Delta M=0$ and $g^{\perp }=\frac{3}{2}[\left(\sin kr\right)∕kr+\left(\cos kr\right)∕{\left(kr\right)}^2-\left(\sin kr\right)∕{\left(kr\right)}^3]$ for circular polarization or $\Delta M=\pm 1$. The result with the scalar photon gives physical insight but the results with the vector photon do not, since the nodes do not fall exactly at $n\lambda ∕2$.

Figure 4

(Color online) Probabilities $P^{(\pm )}$ [Eq. (19)] in the bright state ∣+⟩ (dots) and dark state ∣−⟩ (line) for a three-state system [inset of Fig. 1c], where the decay to state $\mid b_1,b_2,1_k,1_q⟩$ is not considered or neglected and $\mid \pm ⟩=(1∕\sqrt{2})(\mid a_1,b_2⟩\pm \mid b_1,a_2⟩)\mid 1_{\mathbf{k}}⟩$. We use the $\Delta M=0$ transition.

Figure 5

(Color online) Populations $P^{(\pm )}$ in the bright or symmetric entangled state ∣+⟩ and the dark or antisymmetric entangled state ∣−⟩ state for V scheme computed from Eq. (23) for $\Delta M=0$ transition. For $r⪡\lambda$, the ∣+⟩ state decays at a superradiant rate, and the ∣−⟩ state decays at a subradiant rate. The inset shows the V scheme with the collective states. (b) The populations computed using Eq. (25) for initial phased-entangled states ${\mid \pm ⟩}_{{\mathbf{k}}_0}=(1∕\sqrt{2})(e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_1}\mid a_1,b_2⟩\pm e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_2}\mid a_2,b_1⟩)$ prepared by timed excitation with a laser pulse incident at an angle ${\alpha }_0$. We have used ${\alpha }_0=0°$, which gives a maximum phase difference between the two atoms. The oscillations are due to the finite phase difference. Note that ${\alpha }_0=90°$, which corresponds to no phase difference gives the results in (a).

Figure 6

(Color online) Directional dependence of the (normalized) intensity $I(r,\alpha )∕I_m$ as a function of interatomic distance $r$ and observation angle $\alpha$ for different angles of ${\mathbf{k}}_0$ for states ${\mid \pm ⟩}_{{\mathbf{k}}_0}=(1∕\sqrt{2})(e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_1}\mid a_1,b_2⟩\pm e^{i{\mathbf{k}}_0\bullet {\mathbf{r}}_2}\mid b_1,a_2⟩)\mid 0⟩$. (a) ${\alpha }_0=0$ with initial phased-symmetric entangled state ${\mid +⟩}_{{\mathbf{k}}_0}$, (b) ${\alpha }_0=0$ with initial phased-antisymmetric entangled state ${\mid -⟩}_{{\mathbf{k}}_0}$, (c) ${\alpha }_0=90°$ with ${\mid +⟩}_{{\mathbf{k}}_0}$, and (d) ${\alpha }_0=90°$ with ${\mid -⟩}_{{\mathbf{k}}_0}$. The blue frame serves to highlight the directional correlation. The maximum value for all $\alpha$ and $r$ is $I_m$. We have used $\Delta M=0$ transition.

Figure 7

(Color online) Same as Fig. 6 but in cylindrical coordinate system. The red arrows indicate the direction of the excitation laser, ${\mathbf{k}}_0$.

Figure 8

(Color online) Schematic showing the four possible paths of photons from the two atoms to the detectors, corresponding to all the terms in the two-photon amplitude Eq. (32). The paths (c) and (d) arise when the interatomic distance $r$ is small compared to the transition wavelength.

Figure 9

(Color online) Two-photon temporal correlation $G^{\left(2\right)}\left(\tau \right)$ for different arrangements of detectors. (a) Detectors are along the interatomic axis $({\alpha }_A=0°$, ${\alpha }_B=180°$). (i) Destructive interference of the photons from two atoms spaced by odd integral of $\lambda ∕2$ leads to zero correlation. (ii) For $r<\lambda ∕2$, oscillations appear and become more rapid as $r$ decreases. The period of the oscillations decreases with increasing $r$, a useful feature for measuring subwavelength interatomic separation. (iii) Antibunching occurs around $r\simeq \lambda ∕4$ (highlighted by the red thick line) and can be understood as the destructive interference of two quarter waves. (b) Detectors are orthogonal to interatomic axis $({\alpha }_A={\alpha }_B=90°)$. For large atomic separation, the correlation decays exponentially to zero with $\tau =t_B-t_A$. Another interesting feature around $r\sim \lambda ∕2$ is where the finite or nonzero correlation occurs at large $\tau$. This is true even when the detectors are perpendicular to the interatomic axis. We have used the $\Delta M=0$ transition.

Figure 10

(Color online) Angular correlation is clearly seen for both cases where the first detector is along the interatomic axis or perpendicular to it, as first predicted by Dicke. In addition, the features are richer. (a) At half integrals of a wavelength, the correlation vanishes in all directions due to destructive interference. (b) For $\lambda ∕2<r<\lambda$ there is clear angular correlation of the second photon to the first photon going to detector $A$. For $r⪡\lambda ∕2$, the correlation is isotropic. The correlation is zero at half-integral wavelength only when both detectors are at right angles due to destructive interference. We have used the $\Delta M=0$ transition.