Laser-induced coherent population trapping in C${}_{60}$

Coherent population trapping (CPT) and electromagnetically induced transparency in atoms have garnered enormous attention because of their conceptual novelty in quantum optics and potential applications in quantum information processing. Here, we show that CPT can occur in buckminsterfullerene ${\mathrm{C}}_{60}$. If we radiate ${\mathrm{C}}_{60}$ with a cw coupling field first and then with a weak probe field, ${\mathrm{C}}_{60}$ becomes much less absorptive. Turning off the coupling field restores the absorptive nature. Using two coupling pulses, we can prepare ${\mathrm{C}}_{60}$ in a dark state, which is transparent to the probe field. Such a dark state sensitively depends on the laser field amplitude, pulse durations, time delay between two coupling pulses, and laser chirps. Too strong fields refill the ground state, while too long durations overshoot the population and become less effective. There is an optimal delay between two pulses. By employing the first-principles method and solving the Liouville equation dynamically, we scan the vast laser parameter space and find a practical path to coherent population trapping in ${\mathrm{C}}_{60}$, which is of great importance for future experimental investigations.

Figure 1

Top: Schematic of CPT in ${\mathrm{C}}_{60}$. Bottom: (a) DFT energy spectrum of ${\mathrm{C}}_{60}$. The probe pulse is resonant between $T_{1g}$ ($|b\rangle$ state) and $H_u$ ($|a\rangle$ state). The coupling pulse is resonant between $T_{1g}$ ($|b\rangle$ state) and $T_{1u}$ ($|c\rangle$ state). (b) Part of the excited-state energy spectrum. The ground state is our $|a\rangle$ state. The lowest two-photon-allowed states are three states from 4 to 6 (or $|c\rangle$). The lowest one-photon-allowed state is the $|b\rangle$ state.

Figure 2

CW excitation. (a) Laser profiles. From $-200$ to 200 fs, only the coupling field is on. After 200 fs, both the coupling and probe fields are turned on. After 1100 fs, the coupling field is turned off. (b) Population in the ground state $|a\rangle$ (solid line), $|b\rangle$ (dotted lines), and $|c\rangle$ (long-dashed line).

Figure 3

Excitation with a single Gaussian pulse as the coupling pulse. (a) The coupling pulse is turned on at 100 fs. No coherent population trapping is formed. (b) The coupling pulse is turned on at 0 fs and overlaps with the cw probe. No CPT occurs. (c) The cw probe is turned on first at $-200$ fs, while the coupling pulse is turned on later and peaks at 0 fs. In contrast to the wisdom for atoms, coherent population trapping occurs.

Figure 4

(a) Three laser fields. The first two pulses are coupling pulses, with durations of ${\tau }_1=12$ fs and ${\tau }_2=55$ fs. The cw probe is sent in at 200 fs. (b) Population $\rho$ as a function of time for the $|a\rangle$, $|b\rangle$, and $|c\rangle$ states [19].

Figure 5

Effects of the laser amplitudes on the population. Solid lines, ground-state population ${\rho }_{aa}$; dotted lines, population ${\rho }_{bb}$ in $|b\rangle$; long-dashed lines, population ${\rho }_{cc}$ in $|c\rangle$. (a) $A_1=0.05$ V/Å and $A_2=0.05$ V/Å. (b) $A_1=0.10$ V/Å and $A_2=0.05$ V/Å. (c) $A_1=0.15$ V/Å and $A_2=0.05$ V/Å.

Figure 6

Effect of the laser-pulse amplitudes on the trapped population in the $|c\rangle$ state. (a) Population ${\rho }_{cc}$ as a function of the first pulse's amplitude, with that of the second pulse fixed at 0.05 $\mathrm{V}/\AA$. (b) Population ${\rho }_{cc}$ as a function of the second pulse's amplitude, with that of the first pulse fixed at 0.10 $\mathrm{V}/\AA$.

Figure 7

Effects of the laser-pulse duration on the population ${\rho }_{cc}$ in the $|c\rangle$ state. The duration is changed from (a) ${\tau }_2=15$ fs (bottom) to (e) 95 fs (top). The optimal value of the second pulse duration is 55 fs. The dotted line in the bottom of each figure is the laser pulse.

Figure 8

Laser duration effect on population trapping in the $|c\rangle$ state. (a) Population ${\rho }_{cc}$ as a function of the duration ${\tau }_1$ of the first pulse, with that of the second pulse fixed at 55 fs. (b) Population ${\rho }_{cc}$ as a function of the duration ${\tau }_2$ of the second pulse, with that of the first pulse fixed at 12 fs.

Figure 9

Population ${\rho }_{cc}$ as a function of the delay between the first and second pulses.

Figure 10

Impact of laser chirps on CPT. (a) Population in the $|c\rangle$ state as a function of the chirp ${\eta }_1$. (b) Time evolution of populations in the $|a\rangle$, $|b\rangle$, and $|c\rangle$ states. Here ${\eta }_1=0.00{\mathrm{fs}}^{-2}$ and ${\eta }_2=0.01{\mathrm{fs}}^{-2}$.

Figure 11

Proposed experimental setup. (Top) Continuous-wave excitation. The coupling wave is sent in first, and after a delay of 200 fs the probe wave is sent in. Switching off the coupling field, the medium absorbs the probe field. (Bottom) Pulse laser excitation. Here the first coupling pulse should have a longer pulse duration than the second coupling pulse. The time delay between them should be longer than 40 fs to induce the dark state.