Microwave-stimulated Raman adiabatic passage in a Bose-Einstein condensate on an atom chip

We report the achievement of stimulated Raman adiabatic passage (STIRAP) in the microwave frequency range between internal states of a Bose-Einstein condensate magnetically trapped in the vicinity of an atom chip. The STIRAP protocol used in this experiment is robust to external perturbations as it is an adiabatic transfer and power efficient as it involves only resonant (or quasiresonant) processes. Taking into account the effect of losses and collisions in a nonlinear Bloch equations model, we show that the maximum transfer efficiency is obtained for nonzero values of the one- and two-photon detunings, which is confirmed quantitatively by our experimental measurements.

Figure 1

Hyperfine ground states of ${}^{87}\mathrm{Rb}$ involved in the STIRAP process, forming an upside-down $\mathrm{\Lambda }$ system. We transfer population from state $|F=2,m_F=2\rangle$ to state $|F=2,m_F=1\rangle$ via the unpopulated intermediate level $|F=1,m_F=1\rangle$. We write $|F=2,m_F=2\rangle =|e_1\rangle , |F=2,m_F=1\rangle =|e_2\rangle$, and $|F=1,m_F=1\rangle =|g\rangle$. The one- and two-photon detunings are respectively defined as $\mathrm{\Delta }\equiv ({\delta }_1+{\delta }_2)/2$ and $\delta \equiv {\delta }_1-{\delta }_2$.

Figure 2

Relative amplitude of the microwave fields $\alpha$ and $\beta$ used for the STIRAP protocol, as a function of time: the solid blue line and open blue circles correspond to $\alpha \propto {\mathrm{\Omega }}_1\left(t\right)$ tuned to the transition between $|e_1\rangle$ and $|g\rangle$; the red dashed line and filled red circles correspond to $\beta \propto {\mathrm{\Omega }}_2\left(t\right)$ tuned to the transition between $|e_2\rangle$ and $|g\rangle$. Circles are the measured values at the output of the amplifier and solid and dashed lines are plotted after the model of Eqs. (8).

Figure 3

(a) Vertical optical density (OD) profile of the BEC after the time of flight in $|F=2,m_F=2\rangle$ without the STIRAP and (b) optical density of the same BEC. (c) Optical density of the BEC after the time of flight in $|F=2,m_F=1\rangle$ with a 900-$\mu \mathrm{s}$ STIRAP sequence and (d) vertical optical density profile of the same BEC. Blue open circles stand for experimental data and the solid red line is a parabolic fit. In order to discriminate between the Zeeman sublevels of the $F=2$ manifold, during the time of flight we apply a magnetic-field gradient in the vertical direction of the plot. This gradient causes different accelerations for atoms with different $m_F$ numbers resulting in different spatial positions. Without the STIRAP sequence we produce a BEC of $(6.8\pm 0.6)\times {10}^3$ atoms and with the STIRAP sequence we produce a BEC of $(6.0\pm 0.5)\times {10}^3$ atoms. The transfer efficiency is around $87\%$.

Figure 4

Transfer efficiency $\eta$ as a function of the duration of the STIRAP sequence for $f_{\text{mw}}=6.838945$ GHz and $f_{\text{rf}}=860.5$ kHz (optimal parameters): the open blue circles correspond to experimental data; the solid red line corresponds to the model of Eqs. (4) and (5) with the fitted parameters given in Sec. 4; the cyan surface corresponds to the simulation with the fitted parameters and with noise as discussed in Sec. 4; and the dash-dotted magenta line corresponds to the model of Eqs. (4) and (5) with the fitted parameters and setting the one- and two-photon detunings to zero.

Figure 5

(a) Variation of the transfer efficiency $\eta$ as a function of $f_{\text{rf}}$ (which is related to the two-photon detuning $\delta$ by $\delta =4\pi f_{\text{rf}}+{\omega }_{02}-{\omega }_{01}$) for $\tau =2.5$ ms and $f_{\text{mw}}=6.838945$ GHz. (b) Variation of the transfer efficiency $\eta$ as a function of $f_{\text{mw}}$ (which is related to the one-photon detuning $\mathrm{\Delta }$ by $\mathrm{\Delta }=2\pi f_{\text{mw}}-({\omega }_{01}+{\omega }_{02})/2)$ for $\tau =2.5$ ms and $f_{\text{rf}}=860.5$ kHz. The open blue circles correspond to the experimental data; the solid red line corresponds to the model of Eqs. (4) and (5); and the cyan surface corresponds to the simulation with the fitted parameters and with noise as discussed in Sec. 4. In both sets of curves the frequency offset corresponds to the experimental parameters of Fig. 3 (optimal parameters). The vertical black dashed lines are the positions of the one- and two-photon resonances estimated theoretically from the fitted value of the magnetic field $B_0=2.446$ G.

Figure 6

(a) Variations of the eigenenergies as a function of time [Eqs. (13)] for $\delta =2\pi \times 10$ kHz, ${\mathrm{\Omega }}_2\left(0\right)=2\pi \times 14.4$ kHz, ${\mathrm{\Omega }}_1\left(\tau \right)=2\pi \times 43.4$ kHz and $\mathrm{\Delta }=$ 0; (b) same curves with $\mathrm{\Delta }=-6$ kHz; and (c) transfer efficiency (full simulation with fitted parameters and $\tau =2.5$ ms) as a function of the microwave frequency $f_{\text{mw}}$ and the ratio of the maximum Rabi frequencies ${\mathrm{\Omega }}_2\left(0\right)/{\mathrm{\Omega }}_1\left(\tau \right)$. The vertical white dashed line indicates $\mathrm{\Delta }=0$. The horizontal white dashed line indicates the ratio ${\mathrm{\Omega }}_2\left(0\right)/{\mathrm{\Omega }}_1\left(\tau \right)$ used in our experiment.