Fast-forward scaling of atom-molecule conversion in Bose-Einstein condensates

Robust stimulated Raman exact passages are requisite for controlling nonlinear quantum systems, with wide applications, ranging from ultracold molecules to nonlinear optics to superchemistry. Inspired by shortcuts to adiabaticity, we propose the fast-forward scaling of stimulated Raman adiabatic processes with nonlinearity involved, describing the transfer from an atomic Bose-Einstein condensate to a molecular one by controllable external fields. The fidelity and robustness of atom-molecule conversion are shown to surpass those of conventional adiabatic passages, assisted by the fast-forward driving field. Finally, our results are extended to fractional stimulated Raman adiabatic processes for the coherent superposition of atomic and molecular states.

Figure 1

Schematic of coherent two-color photoassociation of a Bose-Einstein condensate (BEC) in $\mathrm{\Lambda }$-type STIRAP, where energy levels $|1\rangle , |2\rangle$, and $|3\rangle$ present the electronic states for the atomic BEC, the excited molecular BEC, and the stable molecular BEC, respectively; ${\mathrm{\Omega }}_p$ and ${\mathrm{\Omega }}_s$ are Rabi frequencies for free-bound and bound-bound transitions.

Figure 2

Nonlinear STIRAP (a) for state transfer (b) from an atomic BEC to a molecular BEC, where ${\mathrm{\Omega }}_s$ (solid red lines) and ${\mathrm{\Omega }}_p$ (dashed blue lines) sequences are Gaussian type with amplitude ${\mathrm{\Omega }}_0=100$ (in units of $1/T_0$), $\tau =0.64T, T=1$ (in units of $T_0$), and $T_0=3.1\times {10}^{-5}$ s. The final population $|c_3{|}^2=0.9914$ is achieved in total time $10T$. For comparison, FF-assisted STIRAP (c) for state transfer (d) is also presented, where ${\mathrm{\Omega }}_s^{\mathrm{FF}}$ (solid red lines) and ${\mathrm{\Omega }}_p^{\mathrm{FF}}$ (dashed blue lines) with ${\mathrm{\Omega }}_0=5$ and other parameters are the same. Assisted by the FF driving field ${\mathrm{\Omega }}_c^{\mathrm{FF}}$ (dotted black lines), the final population $|c_3{|}^2=0.9999$ is achieved with total time $5T$.

Figure 3

The parameter $A_{\mathrm{nl}}$ is depicted to quantify the adiabatic condition, (6), where the parameters are used for nonlinear STIRAP (solid red line) and FF-assisted STIRAP (dashed blue line) in Fig. 2.

Figure 4

Using different values of $R\left(t\right)$, the modified pump and Stokes Gaussian pulses are presented for nonlinear STIRAP (a, b), together with the assisted FF driving field; ${\mathrm{\Omega }}_s^{\mathrm{FF}}$ (solid red lines) ${\mathrm{\Omega }}_p^{\mathrm{FF}}$ (dashed blue lines), and ${\mathrm{\Omega }}_c^{\mathrm{FF}}$ (dotted black lines). The parameters are the same as in Fig. 2. Here (a) and (b) correspond to the trigonometric and polynomial ansatzes of $R\left(t\right)$, respectively.

Figure 5

Nonlinear f-STIRAP (a) for generating the coherent state superposition (b), where ${\mathrm{\Omega }}_s$ (solid red lines) and ${\mathrm{\Omega }}_p$ (dashed blue lines) sequences are Gaussian type with amplitude ${\mathrm{\Omega }}_0=20$ (in units of $1/T_0$), $\tau =0.64T, T=1$ (in units of $T_0$), and $T_0=3.1\times {10}^{-5}$ s. The superposition of an atomic BEC and a molecular BEC with equal amplitudes is provided with total time $10T$. For comparison, FF-assisted f-STIRAP (c) for generating such a superposition (d) is also presented, where ${\mathrm{\Omega }}_s^{\mathrm{FF}}$ (solid red lines) and ${\mathrm{\Omega }}_p^{\mathrm{FF}}$ (dashed blue lines) with ${\mathrm{\Omega }}_0=5$ and the other parameters are the same. Assisted by the FF driving field ${\mathrm{\Omega }}_c^{\mathrm{FF}}$ (dotted black lines), the superposition of atomic and molecular BECs with equal amplitudes is achieved with total time $5T$.

Figure 6

Fidelity, $F=|\langle {\mathrm{\Psi }}_0\left(t_f\right)|\mathrm{\Psi }\left(t_f\right)\rangle {|}^2$, versus the amplitude ${\mathrm{\Omega }}_0$ (a) and the width $T$ (b) of Gaussian pulses, where ${\mathrm{\Psi }}_0\left(t_f\right)$ is the target state and $\left|\mathrm{\Psi }\right(t_f)\rangle$ is the final solution of the time-dependent Schrödinger equation. Here the fidelities of nonlinear STIRAP, nonlinear f-STIRAP, and FF-assisted STIRAP, denoted by solid black, dashed blue, and dotted red lines, and that of FF-assisted f-STIRAP (dash-dotted orange line) are undistinguishable. Other parameters are the same as in Figs. 2 and 5.