Quantum Control and Sensing of Nuclear Spins by Electron Spins under Power Limitations

State of the art quantum sensing experiments targeting frequency measurements or frequency addressing of nuclear spins require one to drive the probe system at the targeted frequency. In addition, there is a substantial advantage to performing these experiments in the regime of high magnetic fields, in which the Larmor frequency of the measured spins is large. In this scenario we are confronted with a natural challenge of controlling a target system with a very high frequency when the probe system cannot be set to resonance with the target frequency. In this contribution we present a set of protocols that are capable of confronting this challenge, even at large frequency mismatches between the probe system and the target system, both for polarization and for quantum sensing.

Figure 1

The main problem. (a) Control and sensing of nuclear spins is achieved by satisfying the HH condition. The electron is driven with a RF ($\mathrm{\Omega }$) that is equal to the nuclear LF (${\omega }_l$). This results in dressed electron states that are on resonance with the LF, enabling the electron-nucleus spin interaction. (b) The electron is driven with a bounded RF, which is smaller than the LF ($\mathrm{\Omega }<{\omega }_l$) and thus no coupling can be achieved. This is a typical problem in the high magnetic fields regime. (c) We propose a set of protocols where even though the electron spin is driven with a bounded RF, $|\mathrm{\Omega }(t)|<{\omega }_l$, an effective dressed electronic energy gap that is equal to the LF is obtained. The effective electron-nucleus coupling strength decreases for a larger frequency mismatch ${\omega }_l-\mathrm{\Omega }$. Dotted lines (solid lines) indicate energy gaps (driving fields).

Figure 2

Polarization as a function of ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1$ in the strong and weak (inset) coupling regimes without BSS correction (blue) and with BSS correction (green). Strong coupling regime: without correction the polarization rate begins to sharply decrease at ${\mathrm{\Omega }}_2\approx {\mathrm{\Omega }}_1$. The correction enables us to maintain good polarization rates up to ${\mathrm{\Omega }}_2\approx 1.8{\mathrm{\Omega }}_1$. Weak coupling regime: The analysis takes noise into account. The polarization is effective up to ${\mathrm{\Omega }}_2\approx 1.4{\mathrm{\Omega }}_1$.

Figure 3

Coherence time ($T_2$) as a function of ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1$. Without the BSS correction (blue), at the regime of an efficient polarization, $T_2$ is decreased as ${\mathrm{\Omega }}_2$ is increased. The optimal $T_2$ is sharply peaked at ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1\approx 0.125$ with $T_2\approx 330\mu \mathrm{s}$ (not shown). With the BSS correction (green), a long $T_2$ time is maintained while increasing ${\mathrm{\Omega }}_2$. The optimal $T_2$ is peaked at ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1\approx 0.4$ with $T_2\approx 1000\mu \mathrm{s}$. The coherence time is a crucial parameter in the efficiency of control and estimation.

Figure 4

Polarization as a function of the nuclear LF in units of tenth of the coupling strength (blue line). The central resonance corresponds to ${\omega }_l={\mathrm{\Omega }}_0$. The two sidebands correspond to ${\omega }_l={\mathrm{\Omega }}_0\pm {\mathrm{\Omega }}_2$. The $y$ axis corresponds to the occupation of the nucleus when the initial state is the $|{\uparrow }_z⟩$ state, i.e., $P_N=1$. The main deep is broader than the side deeps because the coupling at the sideband frequencies is reduced. In contrast, the yellow line represent the analysis of the quantum sensing Hamiltonian, which is much narrower and is not limited by the coupling strength. The numerical simulations were performed with ${\mathrm{\Omega }}_0=1.5\mathrm{MHz}$, ${\mathrm{\Omega }}_1=0.1\mathrm{MHz}$, ${\mathrm{\Omega }}_2=1\mathrm{MHz}$, and $g=0.05{\mathrm{\Omega }}_2$.