Coherent population dynamics in rubidium atoms excited by resonant $0\pi$ pulses

We investigate the time dynamics of a four-level rubidium atomic system excited by a train of $0\pi$ pulses. Resonant $0\pi$ pulse shaping is achieved by propagation of weak femtosecond pulses through rubidium vapor and characterized using frequency-resolved optical gating and within the linear dispersion theory. The excitation of the Rb atoms by $0\pi$ pulse train is studied experimentally by modified direct frequency comb spectroscopy and theoretically within the density-matrix approach. We show that accumulation of populations and coherences, as typical pulse train excitation effects, strongly depend on the strength of the resonant $0\pi$ pulse shaping. Stronger pulse shaping reduces the coherent accumulation effects, eventually leading to the disappearance of the velocity selective optical pumping observed in the frequency domain.

Figure 1

Experimental scheme: ECDL, external cavity diode laser; FI, Faraday isolator; M, mirror; FM, folded mirror; S, beam stopper; A, analyzer; P, polarizer; PD, photodiode; FROG, experimental setup for SHG-FROG (including BBO crystal); cell 1, pulse reshaping; and cell 2, interaction cell.

Figure 2

Left panel: calculated pulse spectra of the weak fs pulse propagated through the resonantly $\left[5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=2\right)\right]$ excited 5-cm-long column of ${}^{87}\text{R}\text{b}$ low-density atomic vapor. The initial pulse shape is assumed to be hyperbolic-secant pulse with the electric field amplitude of $1\times {10}^6\text{V}/\text{m}$ and FWHM of 2817 GHz corresponds to the 100 fs Fourier-limited pulses. Right panel: calculated pulse spectra of the weak fs pulse for the narrow frequency range around pulse center.

Figure 3

Comparison of the experimental and theoretical absorption spectra of room-temperature rubidium vapor: (a) experimental optical thickness of the rubidium isotopic mixture in the natural abundance; (b) experimental optical thickness of the pure ${}^{87}\text{R}\text{b}$ isotope; and (c) calculated optical thickness of the pure ${}^{87}\text{R}\text{b}$ isotope.

Figure 4

Real part of the $0\pi$ pulse electric field retrieved from the pulse spectrum presented in Fig. 2 by applying the inverse Fourier transform. Inset: corresponding $0\pi$ pulse intensity profiles.

Figure 5

(Color online) SHG-FROG traces at (a) $24°\text{C}$ and (b) $115°\text{C}$ after propagation of the weak 100 fs pulse through the 5-cm-long column of rubidium vapor. The fs laser is centered at 795 nm (corresponding to $5{{}^{2}S}_{1/2}\to 5{{}^{2}P}_{1/2}$ excitation).

Figure 6

Calculated time evolutions of the ${}^{87}\text{R}\text{b}$ ${\rho }_{11}$, ${\rho }_{22}$ ground state hyperfine level populations in the case of resonant interaction with the $0\pi$ pulse train. $0\pi$ pulses are generated after the propagation of weak 100 fs Fourier-limited sech pulses through the 5-cm-long column of ${}^{87}\text{R}\text{b}$ atoms vapor at two different temperatures of cell 1 (cell 1 at $20°\text{C}$ and cell 1 at $50°\text{C}$). The central pulse frequency was set in resonance with ${}^{87}\text{R}\text{b}$ $5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=2\right)$ hyperfine transition.

Figure 7

Calculated time evolution of the ${}^{87}\text{R}\text{b}$ ${\rho }_{44}$ excited state hyperfine level populations in the case of resonant interaction with the $0\pi$ pulse train. $0\pi$ pulses are generated after the propagation of weak 100 fs Fourier-limited sech pulses through the 5-cm-long column of ${}^{87}\text{R}\text{b}$ atoms vapor at two different temperatures of cell 1 (cell 1 at $20°\text{C}$ and cell 1 at $50°\text{C}$). The central pulse frequency was set in resonance with ${}^{87}\text{R}\text{b}$ $5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=2\right)$ hyperfine transition.

Figure 8

Calculated ${}^{87}\text{R}\text{b}$ ${\rho }_{44}$ single $0\pi$ pulse excitation population dynamics. $0\pi$ pulses are generated after the propagation of weak 100 fs Fourier-limited sech pulses through the 5-cm-long column of ${}^{87}\text{R}\text{b}$ atoms vapor at different temperatures of cell 1 (cell 1 at $20°\text{C}$, cell 1 at $50°\text{C}$, and cell 1 at $100°\text{C}$). The central pulse frequency was set in resonance with ${}^{87}\text{R}\text{b}$ $5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=2\right)$ hyperfine transition.

Figure 9

Calculated ${}^{87}\text{R}\text{b}$ excited level population ${\rho }_{44}$ together with the corresponding $0\pi$ pulse electric field (real part) used for excitation. $0\pi$ pulses are generated after propagation through cell 1 at $100°\text{C}$. The central pulse frequency was set in resonance with ${}^{87}\text{R}\text{b}$ $5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=2\right)$ hyperfine transition.

Figure 10

Calculated Doppler-broadened ${}^{87}\text{R}\text{b}$ ground state hyperfine level populations, ${\rho }_{11}$ and ${\rho }_{22}$, for different atomic velocity groups in the case of resonant $\left(5{{}^{2}S}_{1/2}\to 5{{}^{2}P}_{1/2}\right)$ $0\pi$ pulse train excitation. Shown results are obtained for the $0\pi$ pulses generated after propagation through cell 1 at $20°\text{C}$ (upper panel) and $50°\text{C}$ (lower panel).

Figure 11

Calculated ground $\left({\rho }_{11},{\rho }_{22}\right)$ and excited $\left({\rho }_{33},{\rho }_{44}\right)$ state hyperfine level populations in one period of modulations, induced by $0\pi$ pulse train resonant excitation. The results are calculated for $0\pi$ pulses generated after propagation through cell 1 at $20°\text{C}$ (left panels) and $50°\text{C}$ (right panels).

Figure 12

Dependence of the changes in the probe laser transmission, $\Delta T$, induced by $0\pi$ pulse train excitation upon the strength of the $0\pi$ pulse shaping (i.e., temperature of cell 1 containing pure ${}^{87}\text{R}\text{b}$ isotope). The shaped $0\pi$ pulses interact with rubidium atoms in cell 2 containing rubidium isotopic mixture in the natural abundance. The probe laser was scanned across $5{{}^{2}S}_{1/2}\to 5{{}^{2}P}_{3/2}$ rubidium absorption line at 780 nm.

Figure 13

Comparison between measured and calculated $\Delta T$ for ${}^{87}\text{R}\text{b}$ $5{{}^{2}S}_{1/2}\left(F_g=1\right)\to 5{{}^{2}P}_{1/2}\left(F_e=0,1,2\right)$ transition line.