Nuclear-state engineering in tripod systems using x-ray laser pulses

Coherent superposition of nuclear states in tripod systems using three x-ray laser pulses is investigated theoretically. The laser pulses transfer the population from one ground state to an arbitrary superposition of other ground states using coincident pulses and stimulated Raman adiabatic passage techniques. The short wavelengths needed in the frame of the nuclei are achieved by envisaging an accelerated nucleus interacting with three x-ray laser pulses. This study exploits the Morris-shore transformation to reduce the tripod system into a coupled three-state $\mathrm{\Lambda }$-like system and a noncoupled state. We calculated the required laser intensities which satisfy the conditions of coincident pulses and adiabatic passage techniques. Considering the spontaneous emission from excited state $|4\rangle$ and unstable ground states $\left(|2\rangle ,|3\rangle \right)$ to other states, we have used a master equation for numerical study, and the final fidelity of desired states with respect to the tolerance of laser intensities is studied numerically.

Figure 1

(a) Linkage pattern of tripod system. All initial population is in state $|1\rangle$. (b) Linkage pattern of tripod system in Morris-Shore basis.

Figure 2

(a) Coincident pulses and (b) STIRAP schemes in the laboratory frame. In both schemes each pulse has a different frequency from one another.

Figure 3

Rabi frequencies in the nuclear rest frame (top frame) and diagonal elements of density matrix ${\rho }_{mm}(m=1,2,3,4)$ in accordance with the populations of states $|1\rangle ,|2\rangle ,|3\rangle$, and $|4\rangle$ (lower frame) vs time for ${}^{154}\mathrm{Gd}$ in the STIRAP frame using SXFEL parameters in master equation (8). The population of the state $|4\rangle$ is zero during the whole dynamics. The linewidths and branching ratios of states $|3\rangle$ and $|2\rangle$ are ${\mathrm{\Gamma }}_3=102\times {10}^{-3}$ meV, ${\mathrm{\Gamma }}_2=393\times {10}^{-6}$ meV, $B_{32}=1,B_{31}=0$, and $B_{21}=1$. The real intensities of XFEL pulses are $I_p=26.34\times {10}^{23}{\mathrm{W}/\mathrm{cm}}^2,I_Q=9.46\times {10}^{22}{\mathrm{W}/\mathrm{cm}}^2$, and $I_S=40.66\times {10}^{23}{\mathrm{W}/\mathrm{cm}}^2$; $\tau =1.5\times {10}^{-15}$ s. See the discussion in the text and Tables 2 and 3 for further parameters.

Figure 4

Rabi frequencies in the nuclear rest frame (first row) and diagonal elements of density matrix ${\rho }_{mm}(m=1,2,3,4)$ in accordance with the populations of states $|1\rangle ,|2\rangle ,|3\rangle$, and $|4\rangle$ (second row) vs time for ${}^{154}\mathrm{Gd}$ in the coincident pulse frame using SXFEL parameters in master equation (8). The linewidth and branching ratioes of states $|3\rangle$ and $|2\rangle$ are ${\mathrm{\Gamma }}_3=102\times {10}^{-3}$ meV, ${\mathrm{\Gamma }}_2=393\times {10}^{-6}$ meV, $B_{32}=1,B_{31}=0$, and $B_{21}=1$. The real intensities of XFEL pulses in the $k\text{th}$ step are $I_{P_k}=26.338\times {10}^{19}{sin}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2,I_{Q_k}=2.37\times {10}^{19}{cos}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2$, and $I_{S_k}=10.17\times {10}^{20}{cos}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2$. See the discussion in the text and Tables 2 and 3 for further parameters.

Figure 5

Final fidelity of desired state $\frac{1}{\sqrt{2}}\left(\right|2\rangle +|3\rangle )$ as a function of $\mathrm{\Delta }{\zeta }_P,\mathrm{\Delta }\left({\zeta }_{Q\left(S\right)}\right)$ so that ${\zeta }_P=(0.0002+\mathrm{\Delta }{\zeta }_P)\times {10}^{21}{\mathrm{W}/\mathrm{cm}}^2,{\zeta }_Q=(1.0293+\mathrm{\Delta }\left({\zeta }_Q\right))\times {10}^{21}{\mathrm{W}/\mathrm{cm}}^2$, and ${\zeta }_S=(6.757+\mathrm{\Delta }\left({\zeta }_S\right))\times {10}^{21}{\mathrm{W}/\mathrm{cm}}^2$ for ${}^{97}\mathrm{Tc}$ by using SXFEL parameters for $N=1$ (a) and $N=10$ (b) sequences of coincident pulses (in this case we consider the radioactive decay of state $|4\rangle )$. The highest values of the horizontal axis lead to calculated real intensities in the $k\text{th}$ step of $I_{P_k}=740.76\times {10}^{21}{sin}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2,I_{Q_k}=223.27\times {10}^{22}{cos}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2$, and $I_{S_k}=435.37\times {10}^{22}{cos}^2{\phi }_k{\mathrm{W}/\mathrm{cm}}^2$. See the discussion in the text and Tables 2 and 3 for further parameters.