Selective Decoupling and Hamiltonian Engineering in Dipolar Spin Networks

We present a protocol to selectively decouple, recouple, and engineer effective interactions in mesoscopic dipolar spin networks. In particular, we develop a versatile protocol that relies upon magic angle spinning to perform Hamiltonian engineering. By using global control fields in conjunction with a local actuator, such as a diamond nitrogen vacancy center located in the vicinity of a nuclear spin network, both global and local control over the effective couplings can be achieved. We show that the resulting effective Hamiltonian can be well understood within a simple, intuitive geometric picture, and corroborate its validity by performing exact numerical simulations in few-body systems. Applications of our method are in the emerging fields of two-dimensional room temperature quantum simulators in diamond platforms, as well as in molecular magnet systems.

Figure 1

Actuator based selective decoupling and recoupling in spin networks: (a) A NV center a few nanometers below the diamond surface, optically initialized and controlled by microwave irradiation, interacts with a spin network (e.g., fluorographene [28] or hBN [32]). (b) By toggling the NV center between the $|m=0⟩$ and $|m=-1⟩$ states, the effective (hyperfine) field for the spin network can be turned on and off, realizing two different, discrete Hamiltonians employed to engineer the effective interaction.

Figure 2

Actuator control of spin network: Tilt angle ${\theta }_{\mathrm{tilt}}^{(l+1)}$ after a single iteration of the protocol at a small initial tilt angle ${\theta }_{\mathrm{tilt}}=\pi /10$ is shown in (a) and (b). The regimes of high sensitivity [i.e., large slope in (b)] are reflected by the large white arc, with the strong dependence on ${\theta }_{\mathrm{tilt}}^{(l)}$ enabling spatial selectivity in the vicinity of an actuator. Iteration of the protocol, shown in (c), leads to exponential amplification of the effective tilt angle with the blue arrow depicting the effective rotation axis in the respective step. (d) Geometric picture of the decoupling to lowest Magnus order: the effective vectorial parts of the secular Hamiltonian (blue spheres) are successively rotated around the magic angle axis (blue diagonal arrow), leading to a cancellation of terms.

Figure 3

Decoupling and selective recoupling via local actuators. By locally disturbing the optimal decoupling field strength using 2 NV centers [as sketched in (a)], the electronic state of which can be toggled in synchrony with the ${\mathbf{B}}_1$ magnetic field, the effective interaction between two nuclear spins can be made spatially dependent. Furthermore, the disturbance of the perfect decoupling is essentially additive in contribution from both NVs. This is reflected by both the effective interaction strength ${\overline{\mathcal{S}}}_{1,2}$ shown in (b) and the decoupling ratio ${\mathcal{R}}_{1,2}$ in (c). The extent to which the decoupling and recoupling works is reflected by the temporal growth of the entanglement entropy, shown stroboscopically in (d) for two spins initially in the $|\uparrow ⟩\otimes |\downarrow ⟩$ state [corresponding parameter regimes indicated by crosses in (c)].

Figure 4

Selective decoupling in hexagonal spin networks. Qualitatively different spatial coupling topologies can be created within our protocol in conjunction with a local actuator (e.g., a NV center) located several nanometers (between 5.7 and 10 nm here) below a ${}^{13}\mathrm{C}$ hexagonal spin lattice. The decoupling ratio ${\mathcal{R}}_{j,j^{\prime }}$ (directly proportional to the effective interaction strength $||{\overline{\mathcal{H}}}_{j,j^{\prime }}||$) for all nearest neighbor spin pairs is shown in color-coded form with the spins located at the respective nodes. Depending on the chosen parameters, the decoupling protocol has a different length scale which can be exploited to give rise to different coupling structures. For instance, one can create a noninteracting patch (a) within an otherwise interacting network or vice versa, create a small interacting cluster in an otherwise essentially noninteracting lattice as shown in (b). Moreover, the parameters can be tuned to separate off an interacting patch shown in (c), or create an effective 1D ringlike lattice topology, shown in (d). For all subfigures $N_c=1$, the NV center is activated throughout the wait period $\tau$ (amplifying the effect on $\overline{\mathcal{H}}$) and located at $(x=0.355\mathrm{nm},y=0)$ (green circle) with variable depth $z$ and the ${}^{13}\mathrm{C}$ spins are located on a regular hexagonal sublattice with lattice constant $a_{\mathrm{lat}}=5a_{\mathrm{cc}}$, where $a_{\mathrm{cc}}=0.142\mathrm{nm}$ is the carbon-carbon bond length and the lattice constant of graphene. We choose this sublattice configuration to obtain a system in a weakly interacting regime, i.e., allowing the true decoupling ratio to be obtained from a pairwise treatment of spins and avoid many-body effects. A relative agreement of 99.5% or more is found with a six spins cluster calculation (including the nearest neighboring spins). For detailed parameters see Ref. [42].