Pulse-area theorem in a single-mode waveguide and its application to photon echo and optical memory in ${\mathrm{Tm}}^{3+}:{\mathrm{Y}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$

We derive the area theorem for light pulses interacting with an inhomogeneously broadened ensemble of two-level atoms in a single-mode optical waveguide and present its analytical solution for Gaussian-type modes, which demonstrates the significant difference from the formation of $2\pi$ pulses by plane waves. We generalize this theorem to the description of photon echo and apply it to the two-pulse (primary) echo and the revival of silenced echo (ROSE) protocol of photon echo quantum memory. We implemented ROSE protocol in a single-mode laser-written waveguide made of an optically thin crystal ${\mathrm{Tm}}^{3+}$:${\mathrm{Y}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$ . The experimental data obtained are satisfactorily explained by the developed theory. Finally, we discuss the obtained experimental results and possible applications of the derived pulse-area approach.

Figure 1

Spatial scheme of interaction of light pulses with an ensemble of two-level atoms in a single-mode optical waveguide. $G(\mathrm{\Delta }/{\mathrm{\Delta }}_{\text{in}})$ is an inhomogeneous broadening with linewidth ${\mathrm{\Delta }}_{\text{in}}$, $c$ and $v_g$ are the speeds of light in free space and in the waveguide; $f\left({\mathbf{r}}_{\perp }\right)$ is a membrane function of light field in the the single-mode waveguide.

Figure 2

(a) Pulse area ${\mathrm{\Theta }}_p\left(z\right)$ of McCall-Hahn area theorem [5] versus optical depth $\varkappa z$ for different input pulse areas $[{\mathrm{\Theta }}_p\left(0\right):\pi \pm 0.1,3\pi \pm 0.1,5\pi \pm 0.1$]; (b) pulse area ${\mathrm{\Theta }}_p\left(z\right)$ of WPA theorem (39) versus optical depth $\varkappa z$ for different input pulse areas $[{\mathrm{\Theta }}_p\left(0\right):2\pi -0.2,4\pi -0.2,6\pi -0.2$]; the optical densities of atomic ensembles in the waveguide and in free space are assumed to be equal.

Figure 3

The atomic functions ${\mathrm{\Phi }}_{g,1,2}\left(\mathrm{\Theta }\right)$ for Gaussian, Bessel, and Bessel core membrane functions $f_{g,1,2}\left({\mathbf{r}}_{\perp }\right)$.

Figure 4

Spatial schemes and temporal sequences of light pulses in (a) the primary echo, (b) the revival of silenced echo (ROSE) protocol.

Figure 5

Pulse areas for single atomic group $M=1$ of signal pulse ${\mathrm{\Theta }}_s\left(z\right)$ (green dashed line), first control pulse ${\mathrm{\Theta }}_1\left(z\right)$ (black short dashed line), primary echo ${\mathrm{\Theta }}_e\left(z\right)$ (red solid line), and all echo pulses ${\mathrm{\Theta }}_{\mathrm{\Sigma },e}\left(z\right)$ (blue dotted line). (a) ${\mathrm{\Theta }}_s\left(0\right)+{\mathrm{\Theta }}_1\left(0\right)<2\pi$, (b) $2\pi <{\mathrm{\Theta }}_s\left(0\right)+{\mathrm{\Theta }}_1\left(0\right)<4\pi$. Inset conceptually illustrates formation of an $2\pi$-echo sequence in the waveguide at some moment of time with total pulse area ${\mathrm{\Theta }}_{\mathrm{\Sigma },e}=2\pi$, where $\tau$ is a time interval between the two exciting pulses, ${\tilde{v}}_g$ is a group velocity weakened by the light interaction with two-level atoms.

Figure 6

(a) Simplified experimental scheme, where IF is an interference filter, Obj is an objective, also there is one more objective installed inside the cryostat on the left side of the crystal, D is a diaphragm, AOM is an acousto-optic modulator, BS is nonpolarizing beam splitter, PBS is polarizing beam splitter, $\lambda /4$ is quarter-wave plate, $\lambda /2$ is half-wave plate, APD is an avalanche photodiode, SPAD is a single-photon avalanche photodiode, ${\alpha }_E=11.3^{\circ }$, ${\beta }_E=4.3^{\circ }$ are conventional Euler rotation angles. (b) Simulated full width at half-maximum $a$ of the laser-written waveguide eigenmode at different values of the refractive index contrast ${10}^{-4}<\mathrm{\Delta }n<8\times {10}^{-3}$ between the elliptical tracks and the core. The red dashed and blue solid curves correspond to widths along two orthogonal axes $x$ and $y$, respectively. Two inserts show the color map of normalized spatial distribution of the eigenmode's intensity for $\mathrm{\Delta }n={10}^{-3}$ and $\mathrm{\Delta }n=4\times {10}^{-3}$.

Figure 7

(a) The signal of the revived silenced echo magnified by 25 is shown at time 30 $\mu \mathrm{s}$ (orange curve) obtained in the waveguide in ${\mathrm{Tm}}^{3+}$:${\mathrm{Y}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$ crystal. The rephasing pulses are shown by the blue dotted curve. At $t=0$, $\mu \mathrm{s}$ input pulse and transmitted part are shown by black and red dashed curves, respectively. (b) The retrieval efficiency of input pulse (black squares) and effective coherence time $T_2$ of optical transition (red circles) of the waveguide optical memory versus intensity of two identical rephasing pulses $I=I_1=I_2$. The theoretical fits with Eq. (65) (black solid line) and plane-wave theory (blue dashed line) are presented.