Coherent control of the cooperative branching ratio for nuclear x-ray pumping

Coherent control of nuclear pumping in a three-level system driven by x-ray light is investigated. In single nuclei the pumping performance is determined by the branching ratio of the excited state populated by the x-ray pulse. Our results are based on the observation that in ensembles of nuclei, cooperative excitation and decay leads to a greatly modified nuclear dynamics which we characterize by a time-dependent cooperative branching ratio. We discuss prospects of steering the x-ray pumping by coherently controlling the cooperative decay. First, we study an ideal case with purely superradiant decay and perfect control of the cooperative emission. A numerical analysis of x-ray pumping in nuclear forward scattering with coherent control of the cooperative decay via externally applied magnetic fields is presented. Next, we provide an extended survey of nuclei suitable for our scheme, and propose proof-of-principle implementations already possible with typical Mössbauer nuclei such as ${}^{57}\mathrm{Fe}$. Finally, we discuss the application of such control techniques to the population or depletion of long-lived nuclear states.

Figure 1

The considered three-level system. (a) State $|E\rangle$ is populated by the SR pulse and can decay to the ground state $|G\rangle$ or to the isomeric state $|I\rangle$, presumed to be long lived. (b) The SR pulse couples states $|I\rangle$ and $|E\rangle$. The initial state $|I\rangle$ is assumed to be metastable. For both (a) and (b) the natural decay rates for the $|E\rangle \to |G\rangle$ and $|E\rangle \to |I\rangle$ transitions are ${\Gamma }_1$ and ${\Gamma }_2$, respectively. $|I\rangle$ and $|G\rangle$ could also be realized as different hyperfine sublevels of a single-nuclear state, as it is often the case in atomic quantum optics. This allows for proof-of-principle implementations in ${}^{57}\mathrm{Fe}$.

Figure 2

(a) Intensity of the coherent scattered light (solid red line), incoherent natural decay (dash-dotted magenta line), and superradiant decay in Eq. (19) (dashed blue line) for a nuclear sample of effective thickness $\xi =10$ and a single-nuclear transition driven by the SR pulse. (b) The hyperfine splitting of the nuclear levels produces quantum beats in the intensity of the scattered light. The quantum beat frequency is given by the hyperfine energy correction ${\Omega }_0=28.3\Gamma$, where $\Gamma$ is the natural width of the excited state.

Figure 3

The time-dependent coherent width ${\Gamma }_c(\tau )$ corresponding to the scattered intensity spectrum presented in Fig. 2b. See text for further explanations.

Figure 4

The population $N(\tau )/N_0$ of the three nuclear levels as a function of the dimensionless time parameter $\tau$ for unperturbed coherent decay. The single-nucleus branching ratio was taken for this case $b=0.5$ and the effective thickness of the sample $\xi =10$. The horizontal dotted lines indicate the steady state values.

Figure 5

The population $N(\tau )/N_0$ of the three nuclear levels as a function of time for suppressed coherent decay starting with $t_s=\pi /(2{\Omega }_0)$. The incoherent branching ratio is taken to be $b=0.5$ and the effective thickness of the sample is $\xi =10$. The horizontal dotted lines indicate the steady state values.

Figure 6

Excited state population for the unperturbed coherent decay (no switching) for several effective thickness parameters $\xi$.

Figure 7

Steady-state isomeric state population plotted against the effective thickness parameters $\xi$. The black dashed line shows the case of unperturbed coherent decay without switching. The red solid line shows corresponding results with coherent control of the branching ratio. The blue dash-dotted line indicates the single-particle result without cooperative decay.

Figure 8

The cooperative branching ratio $b_c(\tau )$ corresponding to an incoherent branching ratio of $b=0.5$ with and without switching at $\tau =0.055$ for an effective thickness parameter of $\xi =10$.