Coherence time extension in ${\text{Pr}}^{3+}$:${\text{Y}}_2$${\text{SiO}}_5$ by self-optimized magnetic fields and dynamical decoupling

Long coherence times are an essential prerequisite for implementations of quantum information technology. This requires techniques to control perturbing processes and hence prolong coherence times in quantum systems. In our work, we present systematic experimental investigations on prolongation of spin coherence times in a rare-earth ion-doped crystal. The approach is based on a combination of established coherence control techniques (i.e., zero first-order Zeeman shifts and simple dynamical decoupling), supported by automatic optimization of experimental control parameters, as well as precise characterization of the optimization loop and the strongly modified complex level structure by spin echoes and high-resolution Raman heterodyne spectroscopy. The spin-echo and Raman heterodyne data clearly prove successful optimization towards proper conditions of zero first-order Zeeman shifts, finally yielding a coherence time of 1 min, i.e., close to the theoretical limit set by the population lifetime in ${\text{Pr}}^{3+}$:${\text{Y}}_2$${\text{SiO}}_5$.

Figure 1

(a) Level scheme of a ${\mathrm{Pr}}^{3+}$ ion without external magnetic field ($B=0$) and at ${\vec{B}}_{\mathrm{ZEFOZ}}$ ($B\ne 0$). (b) Schematic pulse sequence for Raman heterodyne spectroscopy. (c) Schematic pulse sequence for spin-echo experiments.

Figure 2

Evolution of the magnetic field during the automatic optimization loop to determine the optimal ZEFOZ point. The loop applies Raman heterodyne spectroscopy and spin echoes to evaluate progress towards longer coherence times. Each data point indicates the result of one iteration step in the algorithm. The algorithm runs through 33 iterations before it converges to the optimal magnetic field, indicated by the red dot.

Figure 3

Variation of the signal pulse energy vs echo delay time for the case of the initial magnetic field ${\vec{B}}_{\mathrm{start}}=(732,173,219)\mathrm{G}$ (blue circles) and the optimized field ${\vec{B}}_{\mathrm{end}}=(741,177,215)\mathrm{G}$ (red squares). Solid lines represent fits based on Eq. (1), yielding $1/e$ coherence times of $T_2^{\mathrm{start}}=55\mathrm{ms}$ and $T_2^{\mathrm{opt}}=392\mathrm{ms}$. The $1/e$ level is indicated in the graph by a gray dashed line.

Figure 4

Zeeman spectra of the ground-state hyperfine structure around the optimized ZEFOZ point ${\vec{B}}_{\mathrm{end}}$, measured by Raman heterodyne spectroscopy. Variation of the transition frequency vs magnetic-field strength in the $x,y,z$ direction. Color coding in the contour plot indicates Raman signal powers. For each graph one component of the magnetic field was varied while the other two components were kept fixed. Thus, in the left graph the $(y,z)$ field components were set to $B_y=177\mathrm{G}$ and $B_z=215\mathrm{G}$ while the component $B_x$ was varied. In the middle graph, the $(x,z)$ field components were set to $B_x=741\mathrm{G}$ and $B_z=215\mathrm{G}$. In the right graph, the $(x,y)$ field components were set to $B_x=741\mathrm{G}$ and $B_y=177\mathrm{G}$. The white crosses indicate the ZEFOZ point.

Figure 5

Variation of the signal pulse energy vs total echo delay time at the optimized ZEFOZ configuration ${\vec{B}}_{\mathrm{end}}$ combined with dynamical decoupling for different cycling times. Solid lines represent fit functions according to a simple exponential decay.