Quantum Bath Control with Nuclear Spin State Selectivity via Pulse-Adjusted Dynamical Decoupling

Dynamical decoupling (DD) is a powerful method for controlling arbitrary open quantum systems. In quantum spin control, DD generally involves a sequence of timed spin flips ($\pi$ rotations) arranged to either average out or selectively enhance coupling to the environment. Experimentally, errors in the spin flips are inevitably introduced, motivating efforts to optimize error-robust DD. Here we invert this paradigm: by introducing particular control “errors” in standard DD, namely, a small constant deviation from perfect $\pi$ rotations (pulse adjustments), we show we obtain protocols that retain the advantages of DD while introducing the capabilities of quantum state readout and polarization transfer. We exploit this nuclear quantum state selectivity on an ensemble of nitrogen-vacancy centers in diamond to efficiently polarize the ${}^{13}\mathrm{C}$ quantum bath. The underlying physical mechanism is generic and paves the way to systematic engineering of pulse-adjusted protocols with nuclear state selectivity for quantum control applications.

Figure 1

Concept: adjusted pulses ($\delta \theta \ne 0$), nuclear state selective resonances and hyperpolarization. (a) Depiction of a central electron spin (e.g., the NV center in diamond) surrounded by a bath of nuclear spins (${}^{13}\mathrm{C}$). The contour lines represent the transverse hyperfine field ${\mathcal{A}}_{\perp }$, felt by the nuclear spins. (b) Schematic of the PolCPMG dynamical decoupling sequence, which comprises $N$ pulses separated by a period $\tau$. Each pulse rotates the electron spin around the $x$ axis by an angle $\theta =\pi +\delta \theta$, except for the initial and final pulses that rotate the spin by $\pi /2$ around the $y$ axis. (c),(d) Calculated Floquet phases $ϵ$ of the NV-${}^{13}\mathrm{C}$ coupled system periodically driven by the unit sequence shown in (b), as a function of $\tau$, with $\delta \theta =0$ (c) and $\delta \theta =\pi /10$ (d). Parameters are ${\omega }_L=1.9\mathrm{MHz}$ (corresponding to a magnetic field $B_z=1765\mathrm{G}$) and ${\mathcal{A}}_{\perp }/2\pi =180\mathrm{kHz}$, typical of the experiments described later. Dashed lines correspond to the uncoupled case (${\mathcal{A}}_{\perp }=0$). (e), (f) Coherence of the electron spin as a function of $\tau$ after a CPMG (f) and PolCPMG (g) sequence comprising $N=32$ pulses, with the electron spin initialized in $|X_{+}⟩$ and the nuclear spin initialized in $|\uparrow ⟩$ (orange), $|\downarrow ⟩$ (blue) or in a completely mixed state (purple). In (e) all the different cases are overlapped and shown in black. (g), (h) Time evolution of the nuclear spin state during the CPMG sequence at $\tau ={\tau }_0$ (g) and during the PolCPMG sequence at $\tau ={\tau }_{-}$ (h), with the system initialized as in (f). Here we used ${\mathcal{A}}_{\perp }/2\pi =380\mathrm{kHz}$ giving a full polarization transfer in a relatively small number of pulses ($N=16$), facilitating the visualization of the pulse-to-pulse evolution.

Figure 2

Observation of ${}^{13}\mathrm{C}$ hyperpolarization and nuclear state selective addressing. (a) The flip-angle adjustment $\delta \theta$ is controlled by the pulse duration $t_p$ relative to the nominal duration $t_p^0$ corresponding to a $\pi$ rotation. (b) Sequence used to probe the nuclear polarization. (c) NV spin coherence (defined as the probability of finding the NV in $|0⟩$ after the final $\pi /2$ pulse) measured with the sequence shown in (b) as a function of $\tau$ and $t_p$, with $R=0$ (top plot) and with $R=500$ at $\tau ={\tau }_{+}$ (middle) or $\tau ={\tau }_{-}$ (bottom). Parameters are ${\omega }_L\approx 1.9\mathrm{MHz}$, $t_p^0=40\mathrm{ns}$, and $N=32$. (d) Calculated NV spin coherence after a single application of PolCPMG, taking into account the ${}^{14}\mathrm{N}$ hyperfine structure and inhomogeneous broadening [24]. The NV is coupled to a single ${}^{13}\mathrm{C}$ spin (${\mathcal{A}}_{\perp }/2\pi =180\mathrm{kHz}$) initialized in a mixed state (top plot), in $|\uparrow ⟩$ (middle) and in $|\downarrow ⟩$ (bottom). In (c),(d), the dashed lines correspond to the resonance positions from Eq. (2).

Figure 3

Polarization dynamics. (a) Coherence spectra taken immediately after $R$ repetitions of the PolCPMG sequence at $\tau ={\tau }_{+}=288\mathrm{ns}$, for different values of $R$, with $N=32$. (b) Polarization of the ${}^{13}\mathrm{C}$ spin bath as a function of $R$ plotted in terms of the total sequence time, $T=RN{\tau }_{+}$, for different values of $N$. The polarization is normalized so that a value of $+1$ ($-1$) corresponds to all the ${}^{13}\mathrm{C}$ spins within the NV sensing volume being in the $|\uparrow ⟩$ ($|\downarrow ⟩$) state. (c) Polarization after $T=1\mathrm{ms}$ as a function of $N$. The dashed line is a numerical simulation for a single ${}^{13}\mathrm{C}$ (${\mathcal{A}}_{\perp }/2\pi =180\mathrm{kHz}$), including the ${}^{14}\mathrm{N}$ hyperfine structure and inhomogeneous broadening. (d) Polarization after $R=500$ cycles as a function of $t_p$, with $N=32$. For each value of $t_p$, $\tau$ was adjusted to the resulting ${\tau }_{+}$ (blue) or ${\tau }_{-}$ (orange) resonance. Dashed lines are numerical simulations.

Figure 4

Imaging nuclear polarization. (a) Experimental setup to image the nuclear polarization. The colored arrows depict the NV spins (green), polarized ${}^{13}\mathrm{C}$ (blue), and unpolarized ${}^{13}\mathrm{C}$ (red). (b) Polarization map of a $30\times 30\mu \mathrm{m}$ region under deliberately introduced gradients of $\mathrm{\Delta }\omega$ along $x$ and of $\mathrm{\Delta }\mathrm{\Omega }$ along $y$ for $t_p=46\mathrm{ns}$, $\tau ={\tau }_{\text{Pol}}=226\mathrm{ns}$, $N=32$, $R=400$. (c) PolCPMG spectra from two selected regions with low polarization (left panel) and high polarization (right panel). The dashed vertical line indicates the value of $\tau ={\tau }_{\text{Pol}}$. In each case, the two curves correspond to the spectrum without polarization ($R=0$, red curve) and with polarization ($R=400$, blue curve). The light-blue shading highlights the difference between the two curves. Solid lines are two-Lorentzian fits.