Observation of coherent population transfer in a four-level tripod system with a rare-earth-metal-ion-doped crystal

Coherent population transfer in a laser-driven four-level system in a tripod configuration is experimentally investigated with a rare-earth-metal-ion-doped crystal $({\mathrm{Pr}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5)$. The population transfers observed here indicate that a main process inducing them is not optical pumping, which is an incoherent process inducing population transfer. Moreover, numerical simulation, which well reproduces the experimental results, also shows that the process inducing the observed population transfers is similar to stimulated Raman adiabatic passage (STIRAP) in the sense that this process possesses characteristic features of STIRAP.

Figure 1

Four-level system in a tripod configuration. $\Delta$ is the one-photon detuning from the $\mid 1⟩$-$\mid e⟩$ transition. ${\delta }_0$ and ${\delta }_2$ are the two-photon detunings from the $\mid 0⟩$-$\mid 1⟩$ and $\mid 1⟩$-$\mid 2⟩$ transitions, respectively. ${\Omega }_j\left(t\right)$ $(j=0,1,2)$ is the Rabi frequency corresponding to the $\mid j⟩$-$\mid e⟩$ transition.

Figure 2

Parameter dependence of the final population of $\mid 1⟩$ in the ideal tripod STIRAP processes. The Rabi frequencies are set as Eqs. (2, 3, 4). (a) Delay-time dependence. $T_d$ denotes the delay time of the pulse sequence $(T_d\equiv t_0-t_1)$. $T$ denotes the pulse width $(T\equiv T_0=T_1)$. The Rabi frequencies are set as ${\Omega }_0^{\left(0\right)}={\Omega }_1^{\left(0\right)}={\Omega }_2^{\left(0\right)}$. (b) Dependence on the peak Rabi frequency, ${\Omega }^{\left(0\right)}\equiv {\Omega }_0^{\left(0\right)}={\Omega }_1^{\left(0\right)}$, of the two pulses. $T_d$ is set to $T$ $(T\equiv T_0=T_1)$.

Figure 3

Schematic of the experimental setup. AOM, acousto-optic modulator. BS, beam splitter. PD, photodiode. Amp, amplifier. SA, spectrum analyzer. LO, local oscillator.

Figure 4

Wave forms of the three fields used in the experiment. The top, middle, and bottom diagrams correspond to the fields of frequencies ${\nu }_0$, ${\nu }_1$, and ${\nu }_2$, respectively. In step 2, the waveforms are set so that the Rabi frequencies given by Eqs. (2, 3, 4) are obtained (see Fig. 6 for details).

Figure 5

Energy-level diagram of ${\mathrm{Pr}}^{3+}$ ions doped in a YSO crystal. $\Delta$ denotes one-photon detuning. Two-photon detunings are ignored (not shown). $\mid {}^{1}D_2,\pm 1∕2⟩$, $\mid {}^{1}D_2,\pm 3∕2⟩$, and $\mid {}^{1}D_2,\pm 5∕2⟩$ correspond to $\mid e⟩$ in Fig. 1 for type-I, type-II, and type-III ions, respectively.

Figure 6

(Color online) Intensities of the fields for population transfer in step 2 in the present experiment. The origin of time corresponds to that in Fig. 4.

Figure 7

(Color online) Measured transmittance, $T_p\left(t\right)$, of the probe. The circles and the curves show the measured data and the theoretical fits, respectively. Figures (a)–(j) correspond to Figs. 6a, 6b, 6c, 6d, 6e, 6f, 6g, 6h, 6i, 6j, respectively. The origin of time is roughly set so that the intensity of the probe reaches the plateau at $t=0$.

Figure 8

Average population distribution, $\overline{P}\left(\Delta \right)$, of $\mid 1⟩$ of type-I and type-II ions, which was estimated from the fitting shown in Fig. 7. Figures (a)–(j) correspond to Figs. 6a, 6b, 6c, 6d, 6e, 6f, 6g, 6h, 6i, 6j, respectively.

Figure 9

Population of $\mid e⟩$ during the population transfer estimated by the simulation, which was averaged with respect to the two-photon detunings, ${\delta }_0$ and ${\delta }_2$. The one-photon detuning, $\Delta$, is set as $\Delta =0$. The origin of time is set to the time at which step 2 (population transfer) starts. Figures (a)–(j) correspond to Figs. 6a, 6b, 6c, 6d, 6e, 6f, 6g, 6h, 6i, 6j, respectively.

Figure 10

(Color online) Dependence of transferred population on the dephasing rate, ${\gamma }_p^{\prime }$ (log scale), for hyperfine transitions, which was calculated by the simulation. (a) Circles, triangles, and squares correspond to Figs. 6b, 6c, 6d, respectively. (b) Circles, triangles, and squares correspond to Figs. 6e, 6g, 6i, respectively. The one-photon detuning, $\Delta$, is set as $\Delta =0$.

Figure 11

(Color online) Calculated average population distribution, $\overline{P}$, of $\mid 1⟩$ of type-I and type-II ions when the population distributions for three types of ions are set as Eqs. (A9, A10, A11). The solid line shows $\overline{P}$ obtained from the definition (27) and Eqs. (A9, A10). The circles and the squares show $\overline{P}$ estimated by fitting a theoretical curve calculated with Eqs. (25, 26) to the calculated transmittance with Eqs. (17, 18). The circles and the squares correspond to the cases where $P_{\mathrm{III}}\left(\Delta \right)=0$ and $P_{\mathrm{III}}\left(\Delta \right)=1$, respectively. It has been assumed in the estimation that $\overline{P}$ is constant with respect to $\Delta$.

Figure 12

Calculated average population distribution, $\overline{P}\left(\Delta \right)$, of $\mid 1⟩$ of type-I and type-II ions by the numerical simulation. Figures (a)–(j) correspond to Figs. 6a, 6b, 6c, 6d, 6e, 6f, 6g, 6h, 6i, 6j, respectively.