Suppression of Pulsed Dynamic Nuclear Polarization by Many-Body Spin Dynamics

In the burgeoning field of quantum technology, nuclear spins in solids and molecules have multiple facets. On the one hand, they function as long-lived qubits, constituting a building block for quantum computers [1, 2, 3, 4], quantum simulators [5, 6, 7], and quantum memories [8, 9, 10, 11]. On the other hand, a small ensemble of nuclear spins can be detected by utilizing an atom-sized quantum sensor such as a nitrogen-vacancy (NV) center in diamond [12]; nanoscale NMR spectroscopy realized this way will have a transformative impact on chemistry, biology, medicine, and other related disciplines [13, 14, 15]. It has also been demonstrated that nuclear spins themselves can serve as sensitive magnetometers [16].

In all applications, a prerequisite is to hyperpolarize nuclei at a level far beyond the thermal equilibrium, ideally to unity. Although the concept of dynamic nuclear polarization (DNP), in which the spin polarization is transferred from electrons to nuclei, can be traced back to the prediction and demonstration of the Overhauser effect in 1953 [17, 18] and various techniques have been developed since then [19, 20, 21, 22, 23], an additional challenge here is that qubits/sensors are often operated at low magnetic fields ($B_{0}<$ 100 mT) and at room temperatures. A promising approach is to engineer the electron–nuclear interaction via pulsed microwave control of the electron spin. For instance, the PulsePol sequence has been introduced as a broadband and robust method for the polarization transfer [24] and applied to recent experiments on quantum sensing and quantum simulation with NV centers [25, 26, 7]. Nonetheless, the near-unity hyperpolarization remains elusive, and it is pivotal to understand the suppression mechanisms.

In this work, we study higher-order many-body nuclear spin dynamics induced by a pulsed DNP sequence itself, with particular interest in a system with an optically polarizable electron spin embedded in a sparse nuclear spin bath. The NV center in diamond is a prime example, but other color centers in semiconductors [27, 28], molecular qubits [29, 30, 31], rare-earth ions in solids [10, 32], to name a few, are also relevant platforms. In a model system with a single $S=\frac{1}{2}$ electron spin interacting with multiple $I=\frac{1}{2}$ nuclear spins, we perform direct computations of the unitary dynamics during PulsePol and derive analytical expressions of the transition amplitudes of the nuclear spins, which are more involved to obtain via prevailing perturbative expansion methods based on average Hamiltonian theory [33, 34, 35, 36, 37, 38, 39]. After numerical simulations on the larger nuclear spins, we analyze the case of ¹³C nuclei coupled with a single NV center in diamond, including a concrete dataset from a spin-based quantum simulator reported in [7].

The system Hamiltonian under the secular approximation reads

where $m_{S}=\pm\frac{1}{2}$ is the electron spin state, $N_{\text{nuc}}$ is the number of nuclear spins, $f_{\text{n}}$ is the nuclear Larmor frequency, $\hat{I}_{z,x}^{(l)}$ are the operators of the $l$-th nuclear spin, and $A_{\parallel}$ ($A_{\perp}$) is the scalar hyperfine parameter parallel (perpendicular) to the external magnetic field $B_{0}$. Due to the hyperfine interaction, the nuclear spins precess at $f_{\text{p}}=\sqrt{(f_{\text{n}}\pm A_{\parallel}/2)^{2}+(A_{\perp}/2)^{2}}$ around the axis tilted by the angle $\theta=\arctan(A_{\perp}/2f_{\text{n}})$ from the direction of $B_{0}$. The inter-nuclear dipolar interactions, which induce a direct flip-flop between a pair of nuclei, are dropped from $\hat{H}$, partly because in a sparse nuclear bath this time scale is much longer than the pulse sequence time. More importantly, we show below that higher-order spin dynamics, driving indirect nuclear flip-flop and flip-flip mediated by the electron spin, play a crucial role in depolarization.

The PulsePol sequence [see also Fig. 1(a)] is given by

where $U_{x,y}=e^{-i\frac{\pi}{2}\hat{S}_{x,y}}$ and $\hat{U}_{t}=e^{-2\pi i\hat{H}\tau_{\text{pol}}/4}$ represent infinitesimally short $\pi/2$ pulses around the $x,y$ axes of the electron spin and the free evolution, respectively. The electron spin is initialized (optically) into $m_{S}=+\frac{1}{2}$ in advance and the unit sequence is repeated ${2N_{\text{pol}}}$ times. The pulse interval $\tau_{\text{pol}}$ is chosen so that the polarization transfer will be most efficient at a target frequency $f_{\text{t}}$, given by the resonance condition $f_{\text{t}}=3/2\tau_{\text{pol}}$.

To unveil higher-order spin dynamics in the simplest setting, we start our analysis with $N_{\text{nuc}}$ = 2 and uniform $A_{\perp}$. Furthermore, we assume $A_{\parallel}$ to be zero, as it primarily contributes to the shift of the precession frequency and does not play a major role in the polarization transfer. We set $f_{\text{n}}$ = 1 MHz (corresponding to 23.4866 mT for protons), $A_{\perp}$ = 0.3 MHz, $f_{\text{t}}=f_{\text{p}}$, and $N_{\text{pol}}$ = 1. At the end of PulsePol, the electron spin ends up in either flipped (polarization transferred) or unflipped. The transition probabilities $\hat{T}_{\alpha}=\bra{-\frac{1}{2}}\hat{U}\ket{+\frac{1}{2}}$ and $\hat{T}_{\beta}=\bra{+\frac{1}{2}}\hat{U}\ket{+\frac{1}{2}}$ of the electron spin can be broken down into

where $\hat{\sigma}_{z}=2\hat{I}_{z}$ is the Pauli operator and $\hat{\sigma}_{\pm}=(\hat{I}_{x}\pm i\hat{I}_{y})$ are the ladder operators, both on the nuclear spins, and $\bm{1}$ is the identity operator. The coefficients $\alpha_{j}$ and $\beta_{j}$ are the complex transition amplitudes, and the subscript $j$ indicates the corresponding nuclear spin dynamics, as sorted out in Fig. 1(b). The explicit forms of the transition amplitudes are given in Supplemental Material [40].

Six dynamics involving the ladder operators [except “phase flip” in Fig. 1(b)] are directly responsible for the (de)polarization, and their transition amplitudes as a function of $\tau_{\text{pol}}$, centered at the resonance, are shown in Fig. 1(c). $\alpha_{\pm}$ involve only one nuclear spin, with $\absolutevalue{\alpha_{-}}$ taking its maximum and $\absolutevalue{\alpha_{+}}$ falling to zero (node) near the resonance [41]. This is consistent with the average Hamiltonian treatment examined in previous studies [24, 25]. Were it not for other dynamics, the nuclear spin could be perfectly polarized in the direction anti-parallel to the magnetic field.

The situation changes when the dynamics involving two spins are in action. The most critical is the flip-flop that is accompanied by the phase flip of the second nuclear spin ($\alpha_{+,z}$). This behavior, counteracting the polarization process of $\alpha_{-}$, is similar to $\alpha_{+}$, but with non-zero transition amplitude near the resonance. It is observed from Fig. 1(c) that the different node positions of $\absolutevalue{\alpha_{+,z}}$ and $\absolutevalue{\alpha_{-}}$ make the coexistence of the flip-flop and flip-flip processes unavoidable. In addition, even when the electron spin is not flipped, there exists a process in which a flip-flip between the two nuclear spins occurs ($nn$ flip-flip, $\beta_{+,+}$). This process originates from a virtual electron–nuclear flip-flop followed by another virtual electron–nuclear flip-flip, thus bringing the electron spin back to the initial state. The $nn$ flip-flip suppresses the polarization as it changes the polarization of the nuclear spins regardless of the electron spin state. For $N_{\text{nuc}}\geq 2$, all the possible dynamics that arise from the combination of flip-flip and flip-flop can occur. Figure 2(a) shows $A_{\perp}$-dependence (equivalently, $\theta$-dependence) of the transition amplitudes.

All the coefficients grow gradually as increasing $A_{\perp}$. $\absolutevalue{\alpha_{-}}$ reaches its maximum at $A_{\perp}$ = 323 kHz, where we obtain $\absolutevalue{\alpha_{-}}:\absolutevalue{\alpha_{+}}:\absolutevalue{\alpha_{+,z}}:\absolutevalue{\beta_{+,+}}=0.69:0.02:0.13:0.19$. Given that $\beta_{+,\pm}$ ($\alpha_{+}$ and $\alpha_{\pm,z}$) are the second (third) order in $A_{\perp}$ [40], the ratio suggests that depolarization due to higher-order spin dynamics is no longer negligible.

Up to now, we have relied on the analytical expressions for $N_{\text{nuc}}=2$ to gain physical insight into the higher-order spin dynamics. We turn to numerical calculation of the nuclear polarization for $N_{\text{nuc}}\geq 2$. The computational procedure involves (i) the preparation of the initial density matrix $\rho$ with the electron spin in the pure $m_{S}=+\frac{1}{2}$ state and $N_{\text{nuc}}$ nuclear spins in a mixed state of $\bm{1}/2^{N_{\text{nuc}}}$, (ii) the time evolution $\rho^{\prime}=\hat{U}\rho\hat{U}^{\dagger}$, and (iii) the partial trace over the electron spin state $\rho_{\text{n}}^{\prime}=\mathrm{tr}_{\text{e}}(\rho^{\prime})$. After (iii), we evaluate the polarizations of the individual nuclear spins as $|2\langle\hat{I}_{z}^{(l)}\rangle|$, and the total polarization $P$ as $|\sum_{l}2\langle\hat{I}_{z}^{(l)}\rangle/N_{\text{nuc}}|$. We rerun (i)–(iii), using $\rho_{\text{n}}^{\prime}$ as the new initial state of the nuclear spins. The procedure is repeated $N_{\text{rep}}=1000$ times to reach a saturated value. We compute $P$ for $N_{\text{nuc}}=1,2,4,8$ with uniform $A_{\perp}$ up to 500 kHz. Matrix calculations were performed using MATLAB on a 56-core Intel CPU workstation.

It is known that in a coherent nuclear spin ensemble the collective dark state develops and saturates the polarization [42, 43, 44, 45, 46]. This effect is noticeable in the case of the homogeneous hyperfine coupling, because a flip of one nuclear spin, without the knowledge on which nuclear spin to have flipped, necessarily creates a quantum superposition across the ensemble. The higher-order spin dynamics and the formation of the dark state can spread cooperatively, and it is not straightforward to clearly separate the two effects. Nonetheless, as a way to formally distinguish them, we also evaluate $P$ of $\text{diag}(\rho_{\text{n}}^{\prime})$, where $\mathrm{diag}$ is defined as the operation that zeros all the off-diagonal elements of the density matrix. $\text{diag}(\rho_{\text{n}}^{\prime})$ is then fed into the next run. In this manner, the system is incoherent and no dark state is formed.

Figure 2(b) shows the simulated polarizations, with the upper (lower) panel for the coherent (incoherent) case. It is observed that the coherent case exhibits suppressed polarizations even for small $A_{\perp}$ and $N_{\text{nuc}}$, which is primarily due to the formation of the dark state. However, even in the incoherent case, the polarization drops quickly as increasing $A_{\perp}$. This is consistent with Fig. 2(a), in which $\alpha_{+}$ and $\alpha_{+,z}$ increase with $A_{\perp}$. The effect is more pronounced for larger $N_{\text{nuc}}$. On the other hand, in both cases, the polarization of $N_{\text{nuc}}=1$ is kept at (nearly) unity regardless of $A_{\perp}$. This corroborates our assertion that the origin of the observed imperfect polarizations is many-body in nature.

Having analyzed the roles of many-body effects in a model system, we are in a position to examine a more concrete system, the NV center in diamond. We first perform simulations with realistic conditions, prior to analyzing real experimental data. An important modification is that the electronic spin of the negatively charged NV center is $S=1$. Here, we use the $m_{S}=-1$ and 0 states, with the latter being free from the hyperfine interaction. The definition of $\theta$ is changed, accordingly, to $\theta=\arctan(A_{\perp}/f_{\text{n}})$.

We computationally prepare 500 different configurations of ¹³C clusters with $N_{\text{nuc}}$ = 6 (see [40] for detail) and simulate their polarization dynamics at $B_{0}$ = 40 mT, a typical value for experiments with NV centers. Figure 3(a) shows the scatter plot of simulated polarizations comparing the coherent and incoherent cases.

The horizontal axis $\langle A_{\perp}\rangle$ is defined as $\sum_{l=1}^{N_{\text{nuc}}}A_{\perp}^{(l)}/N_{\text{nuc}}$ in each cluster. There is a common tendency that the polarization diminishes with $\langle A_{\perp}\rangle$, but the achievable polarizations strongly depend on the details of the hyperfine parameters. We also observe that the coherent case has a broader distribution than the incoherent case. The histogram in the right-hand side is peaked at 0.84$\pm$0.07 (0.90$\pm$0.05) in the coherent (incoherent) case. The low polarization points in the coherent case (for similar $\langle A_{\perp}\rangle$) presumably arise from the formation of the dark state as in the simulation of Fig. 2(b), whereas we attribute the gradual decrease in the polarization for larger $\langle A_{\perp}\rangle$ to higher-order spin dynamics. We expect that the effect will become even stronger in larger systems ($N_{\text{nuc}}>$ 10) of recent experimental interest [6, 7, 8, 13, 14].

In spin systems with disordered hyperfine interactions, the influence of the dark state may be mitigated by introducing a disentangling operation [47]. A simple realization in the present setup is to reinitialize the NV center into the $m_{S}=-1$ state (optical initialization followed by a microwave $\pi$ pulse) after PulsePol and add a wait time of $2N_{\text{pol}}\tau_{\text{pol}}$, during which the respective nuclear spins will precess under different hyperfine fields and acquire different phases. This modified sequence is tested in Fig. 3(b) and shows clear improvement from the original sequence, with the peak of the histogram shifted from 0.84$\pm$0.07 to 0.88$\pm$0.07. The decreasing polarizations with $\langle A_{\perp}\rangle$, attributable to the higher-order dynamics, are still discernible.

It is of practical importance to examine how large the magnetic field should be to obtain near-unity hyperpolarization. The role of $A_{\perp}$ should become marginal as increasing $B_{0}$. Figure 4 shows $B_{0}$-dependence of the polarizations for 50 sets of ¹³C clusters with $N_{\text{nuc}}$ = 6.

In the incoherent case, the polarizations at $B_{0}=160$ mT fall on a narrow range of 0.99$\pm$0.01. On the contrary, the coherent cases show relatively large variations across different clusters, even when the disentangling operation is used. This reflects, as observed in Fig. 3, the strong dependence of the achievable polarizations on the details of the hyperfine parameters. Therefore, in a real system, applying large field of $B_{0}\gg 40$ mT may not be sufficient; a careful selection of ¹³C clusters with weak influence of the dark state will be crucial to achieve near-unity hyperpolarization with PulsePol.

The last analysis deals with the experimental data of [7], in which PulsePol was applied to 9 nuclei in a relaxation-free cryogenic environment. The averaged polarization was 0.64, but the calculation up to $N_{\text{nuc}}$ = 5 did not reproduce this value [7]. In Table 1, the first two rows list the hyperfine parameters of these 9 nuclei, numbered in descending order of the values of $A_{\perp}$.

We perform a full simulation with 9 spins, using the same parameter set as [7] ($B_{0}$ = 40.3553 mT, $N_{\text{pol}}$ =2, $\tau_{\text{pol}}$ = 8$\times$0.436 $\mu$s, $N_{\text{rep}}$ = 5000). The results are shown in the third and fourth rows. The average polarization in the coherent case is 0.77, close to the result of [7], with variations across 9 nuclear spins. High polarizations are obtained in the incoherent bath, consistent with small $\langle A_{\perp}\rangle$ of 13.5 kHz. More discussions are given in [40].

In summary, we analyzed the impact of higher-order nuclear spin dynamics induced by the pulsed DNP sequence, in conjunction with the formation of the collective dark state, both many-body in nature, on the nuclear polarization. The usefulness and limitation of disentangling operations were also discussed. We were able to understand the nuclear polarizations observed in the system of a single NV center in diamond reported in [7]. While we focused on a particular DNP sequence (PulsePol), we show in [40] that these dynamics are also responsible for another widely-used sequence (NOVEL [19]). In future studies, the effect of finite pulse lengths on higher-order spin dynamics will require a careful analysis. Systematic Hamiltonian engineering, applicable to multi-spin systems, may be used to achieve higher polarization even at low magnetic fields [48]. We also remark that, via an appropriate control over environmental nuclear spins which otherwise cause decoherence of an electron spin qubit, they can be converted into a resource valuable to quantum technology [42, 43, 44, 49, 50, 51]. Our work sheds light on collective engineering of nuclear spin ensemble as well as future designs of DNP protocols.

K.S. was supported by JSPS Grant-in-Aid for Scientific Research (KAKENHI) Grant No. JP22K03524.