Experimental observation of double coherent stimulated Raman adiabatic passages in three-level $\Lambda$ systems in a cold atomic ensemble

Coherent stimulated Raman adiabatic passages (STIRAP) based on a dark state have many applications in diverse quantum systems. In the case of large single-photon detuning in three-level $\Lambda$ systems, one of the bright states almost does not involve the excited state and then acts as the dark state. In this paper, we report the experimental observation of STIRAP based on the dark state, as well as b-STIRAP based on the above bright state, with the same transfer efficiency in a three-level $\Lambda$ cold atomic ensemble. Moreover, both STIRAP and b-STIRAP are found to be immune to the time-varying ac Stark shift. As for a typical example to manipulate a superposition state, we realize an inverse operation on an initial superposition state, which cannot be achieved efficiently for STIRAP based on only the dark state. Our observed double STIRAP thus provides a feasible and robust manner to manipulate quantum states.

Figure 1

The simplified experimental setup for observing the double STIRAP. The laser-atom-coupling scheme is shown in the top panel. Hyperfine levels $|F=1,m_F=0\rangle$ and $|F=2,m_{F^{\prime }}=0\rangle$ of $5^2{S}_{1/2}$ of ${}^{87}\mathrm{Rb}$ are used as two ground states in the three-level $\Lambda$ system. The single-photon detuning $\Delta$ between Raman lasers and $|F=3\rangle$ of the excited state $5^2{P}_{3/2}$ of ${}^{87}\mathrm{Rb}$ is about 2.5 GHz.

Figure 2

Transfer efficiency vs the separation time $\Delta \tau$ between the pump pulse and the Stokes pulse. The red solid line is the theoretical simulation result, while the blue circles with error bars are the experimental data.

Figure 3

Adiabatic population dynamics of STIRAP (blue circles) and b-STIRAP (green stars) observed with time-resolved measurements. The red solid line shows theoretical results of both STIRAP and b-STIRAP.

Figure 4

(a) Measurement of the population dynamics of STIRAP with different ratios of Rabi frequencies between the pump and Stokes pulses. The lines are obtained by numerical calculations. (b) Measurement of the maximal transfer efficiency via the ratio ${\Omega }_P/{\Omega }_S$; the blue solid line is used to guide the eyes. The red dotted line is obtained from theoretical calculation based on the adiabatic approximation.

Figure 5

Experimental measurement of the two-photon bandwidth of STIRAP (red circles), b-STIRAP (green stars), and the $\pi$-pulse Raman transition (blue squares) while varying the frequency of the Stokes laser with the AOM. While the ratio of Rabi frequencies of the Raman transition is the same as for STIRAP, the rectangle pulses overlap in the time domain.

Figure 6

(a) Theoretical simulation of evolution during STIRAP with the initial superposition state $|{\Psi }_0\rangle =\frac{1}{2}|2\rangle -i\frac{\sqrt{3}}{2}|1\rangle$. The blue dotted, red solid, and black dot-dashed lines are derived by solving Schrödinger's equation with the Hamiltonian given by Eq. (1). The green dashed line is derived with $|\langle 2|{\Psi }_0\left(t\right)\rangle {|}^2$ with the analytical wave function $|{\Psi }_0\left(t\right)\rangle$. (b) Measurement result of the evolution of state $|2\rangle$. The solid curve is obtained using numerical simulation where a random fluctuation of $\pm 5\%$ in the Rabi frequencies is introduced. Such fluctuation will average the oscillations in the ideal STIRAP.