Blueprint of optically addressable molecular network for quantum circuit architecture

First-principles calculations have been carried out using hybrid exchange density function theory (HDFT) with a 6–31G basis set in the Gaussian 09 code gaussian09 . The self-consistent field (SCF) procedure was converged to a tolerance of $10^{-6}$ a.u. ($\sim 0.3$ Kelvin). The broken-symmetry method wu2013 was used to allow spins to localize on the radicals and the convergence to a low-spin configuration. Electronic exchange and correlation are described using the B3LYP hybrid exchange density functional b3lyp , the advantages of which include a partial elimination of the electronic self-interaction error and a balance of the tendencies to delocalize one-electron wave-functions. The B3LYP density functional has previously been shown to provide an accurate description of the electronic structure and magnetic properties for both inorganic and organic compounds wu2015 ; zou2018 .

Supposing that an MRTS consists of two radicals and one spin coupler, and that the exchange interaction dominates over both dipolar interactions and spin-orbit coupling, then the Heisenberg spin Hamiltonian before optical excitations reads

where $J_{0}$ is the exchange interaction between radical spins $\hat{\vec{s}}_{1}$ and $\hat{\vec{s}}_{2}$, which can be computed as the energy difference between the triplet and broken-symmetry (BS) states of radicals, $J_{0}=2(E_{\mathrm{triplet}}-E_{\mathrm{BS}})$. When the spin coupler is in a triplet state after optical excitations, the spin Hamiltonian changes to

where $J_{1}$ ($J_{2}$) is defined as the exchange interaction between radical spins $\hat{\vec{s}}_{1}$ ($\hat{\vec{s}}_{2}$) and the triplet spin $\hat{\vec{S}}$. $J_{3}$ is the exchange interaction between two radical spins after optical excitations. To compute $J_{1}$, $J_{2}$, and $J_{3}$, we first find the total energies of the states $|a\rangle=|\uparrow\Uparrow\uparrow\rangle$, $|b\rangle=|\uparrow\Uparrow\downarrow\rangle$, $|c\rangle=|\downarrow\Uparrow\downarrow\rangle$, and $|d\rangle=|\downarrow\Uparrow\uparrow\rangle$, where $\uparrow$($\downarrow$) is defined as the spin state $|s=\frac{1}{2},s_{z}=\frac{1}{2}\rangle$ ($|s=\frac{1}{2},s_{z}=-\frac{1}{2}\rangle$) of a radical and $\Uparrow$ as $|S=1,S_{z}=1\rangle$ for the spin-coupler triplet. Then we can have

where $\Delta E_{ij}=E_{i}-E_{j}$ is defined as the energy difference between the states with the spin configurations $i$ and $j$ ($\in$ {$a,b,c,d$}).

Note that the broken-symmetry singlet and triplet states formed by two radical spins alone have the same $z$-component of spin as $|c\rangle$ and $|b\rangle,|d\rangle$, respectively, but much lower energies, owing to the triplet excitation energy ($\sim$ eV in general). The Kohn-Sham orbitals therefore need to be repopulated in order to drive the system into $|b\rangle$, $|c\rangle$, or $|d\rangle$. If two radicals are symmetric, then $E_{b}=E_{d}$ and therefore

The expectation value of the total spin operator $\hat{\vec{S}}_{\mathrm{total}}=\hat{\vec{s}}_{1}+\hat{\vec{S}}+\hat{\vec{s}}_{2}$ is $6$ for the state $|a\rangle$ (since it is a total spin eigenstate with $S_{\mathrm{total}}=2$), 3 for $|b\rangle$ and $|d\rangle$, and 2 for $|c\rangle$. We adopted that the exchange interactions are antiferromagnetic (AFM) when $J>0$ while ferromagnetic (FM) when $J<0$. The spin Hamiltonians in eq.1 and eq.2 and the computation of the exchange interactions therein can be easily generalized to the molecular structure with more radicals.

Here $J_{1}$ is equal to $J_{2}$ because the biTYY-DPA molecule has an inversion symmetry. $J_{0}$, $J_{1}$ and $J_{3}$ were computed for the dihedral angles between the phenyl ring and the anthracene, ranging from $0^{\circ}$ (coplanar) to $90^{\circ}$ (perpendicular) with $10^{\circ}$ increments. In all the calculations, we freezed the TYY radicals and rotate only the phenyl ring, which meant we assumed the dihedral angles at each end of the coupler are equal.

When the spin coupler is in its ground state, the two TYY radicals interact weakly through the spin polarisation (induced by the radical spin) on the coupler. After the optical excitation and ISC, the spin polarisation on the coupler is mostly dominated by the triplet spin, as shown in Fig.2(a-f). In our calculations, $J_{3}$ is three to four orders smaller than $J_{0}$ for all the dihedral angles studied (Fig.2g). We therefore neglect $J_{3}$ in what follows. When the dihedral angle is $0^{\circ}$, $J_{0}/k_{B}$ is predicted to be $16.8$ K and $J_{1}/k_{B}$ to be $-461.4$ K, which is above the room temperature. When the dihedral angle is $60^{\circ}$, which is a geometry that has been observed in the DPA crystal structure dpaangle , $J_{0}/k_{B}$ is predicted to be $15.2$ K and $J_{1}/k_{B}$ to be $-22.8$ K. When the dihedral angle is $90^{\circ}$, $J_{0}$ is computed to be negligible, however the dipolar interaction between the spins is estimated to be $\sim 10^{-4}$ K as the distance between radicals is $\sim 17\ \textup{\AA}$, and $J_{1}/k_{B}$ is predicted to be $-2.0$ K. The calculation results are consistent with the TREPR experiments for the optically driven transient magnetic properties of biTYY-DPA, which confirms the radical spins are aligned after the optical excitation as shown in the previous experiment teki2000 . At suitable temperatures the optical excitation and ISC can change the spin alignment of the radicals from AFM to FM with enhanced magnitude. This implies that biTYY-DPA not only has potential for quantum gate operations but also can work as an optically controlled single-molecule magnetic switch. This can potentially work at room temperature if the molecule can be prepared with a dihedral angle smaller than $20^{\circ}$ on a surface, as suggested in Fig.2g. The calculated spin densities reveal the nature of the exchange interactions between the radicals and the triplet on DPA. The spin densities of the three spin configurations $|a\rangle$, $|b\rangle$, and $|c\rangle$ for the dihedral angle of $0^{\circ}$ ($90^{\circ}$) are shown in Fig. 2a-c (d-f). $J_{1}$ decreases as the dihedral angle is increased because the rotation of the phenyl ring away from the coplanar geometry suppresses the delocalisation of the $\pi-$orbitals of DPA, which dominate the wave function of the triplet exciton. This is manifested by the much smaller spin densities on phenyl rings with the dihedral angle equal to $90^{\circ}$ than $0^{\circ}$, as shown in Fig. 2(d-f). The effect of phenyl rings on spin densities implies that we could engineer molecules to control spin densities thus interaction strength. In contrast to these large variations on the phenyl rings the spin densities on the radicals are insensitive to the dihedral angle: Mulliken population analysis yields a spin moment of $\sim 0.3\mu_{B}$ on the nitrogen and $\sim 0.5\mu_{B}$ on the oxygen for all the dihedral angles, which indicates the radical spins are well preserved although we have optical excitation on the spin coupler.

The mechanism for the strong radical-triplet exchange interaction $J_{1}$ can be understood using the Ovchinnikov’s topological spin alternation rule for $\pi$-conjugated systems ovchinnikov1978 ; lieb1989 : the spin-up and spin-down densities alternate on the neighbouring carbon atoms. The underlying physics of this rule is related to the formation of the spin polarisation induced by electron delocalisation in a $\pi$-conjugated system, similar to the indirect exchange mechanism in inorganic insulators anderson and the Lieb’s theorem for the bipartite graphene lattice lieb1989 . In the $|a\rangle$ state, the spin densities are consistent with the Ovchinnikov’s topological spin alternation rule, whereas in the $|b\rangle$ and $|c\rangle$ states the spin alternation pattern is violated when the spin-up densities meet at the junction between the phenyl ring and the TYY radicals (highlighted by the red arrows in Fig.2b,c). Therefore, the spin-aligned state $|a\rangle$ is favoured and the exchange interaction for the triplet excited state on DPA is FM. This mechanism based on the Ovchinnikov’s rule could be used to engineer the sign of the exchange interaction.

We have also computed the dynamics of the whole system based on the theory of open quantum systems described in §II.2. In Fig.3 (a), we show the time evolution of the off-diagonal term (coherence) for the reduced density matrix of the two radicals ($|\frac{1}{2},\frac{1}{2}\rangle\langle\frac{1}{2},-\frac{1}{2}|$), after tracing out the triplet manifold. We can see that the magnitude of the matrix element will increase when applying the optical driving field, and then will decay slowly as the triplet vanishes. This suggests the exchange interaction between radical and triplet induced will trigger the coherence or entanglement between the two radical spins. The diagonal term (population) for the $|\frac{1}{2},1,-\frac{1}{2}\rangle$ state in the density matrix (Fig.3b) shows a decaying Rabi-oscillation characteristics, indicating the coherence created during this process. In Fig.3 (c), the tomography for the magnitudes of the reduced density-matrix of the two radicals at the early stage ($T=6.2\ ps$) of the time evolution after applying the optical excitation has been plotted, in which we can clearly see the nonzero off-diagonal term for spin coherence. Here $|1\rangle=|\frac{1}{2},\frac{1}{2}\rangle_{L}|\frac{1}{2},\frac{1}{2}\rangle_{R}$, $|2\rangle=|\frac{1}{2},\frac{1}{2}\rangle_{L}|\frac{1}{2},-\frac{1}{2}\rangle_{R}$, $|3\rangle=|\frac{1}{2},-\frac{1}{2}\rangle_{L}|\frac{1}{2},\frac{1}{2}\rangle_{R}$, and $|4\rangle=|\frac{1}{2},-\frac{1}{2}\rangle_{L}|\frac{1}{2},-\frac{1}{2}\rangle_{R}$ ($L$ and $R$ label the radical on the left and right, respectively). We have also investigated the TREPR spectra at the early stage ($T=6.2\ ps$) in Fig.3(d-f). We have used different parameters for the exchange interaction between radical and triplet, which are comparable to the range of the exchange interactions computed by DFT as shown in §III.1. The exchange interaction between triplet and radical is $-10$ and $-10^{3}$ mT for Fig.3(d) and (e), respectively. We have also computed the powder-averaged spectra as shown in Fig.3(f). The main and side peaks have successfully been reproduced based on our open-quantum-system modelling, as compared with the experimental results teki2000 . Notice here that we haven’t distinguished emission and absorption. For the static magnetic field (X-band), the spin anisotropy, spin relaxation rates, the ISC rate, and spontaneous decay rate, we have used the same parameters as the previous work ma2022 When the exchange interaction is smaller than the static magnetic field ($\sim 350$ mT), there are more features in the spectra than those from the much larger exchange interaction. The side peaks stem from the triplet state of DPA while the central peaks (when the exchange interaction is -10 mT) originate from the radical spins, which are renormalized by the interactions with the triplet on DPA. Our EPR simulations are qualitatively also consistent with the experimental results in Ref.teki2000 , except that the authors therein distinguished the absorption and emission. The EPR spectra for the larger exchange interactions are saturated due to the limitation of the magnitude of the static magnetic field, which implies the EPR spectra will be dominated by the exchange interaction between the radicals and the triplet. Therefore, in this range of static magnetic field, we can only see the effect from the spin anisotropy ($D$ and $E$). For the large exchange interaction, we need other instruments to probe them, such as neutron scattering ns .

Quantum computer-aided design (QCAD) has been proposed recently for the quantum computing architectures based on silicon quantum dots gao2013 ; niquet2020 ; kyaw2021 , which has a huge potential to optimise the quantum circuit design, thus improving the design efficiency, although it is still in the early stage. In the mean time, the research on two-dimensional (2D) materials have been surging rapidly gibertini2019 ; ferrari2015 ; zhuang2015 . We could therefore generalize bi-radical to an example for molecular quantum-circuit architecture as shown in Fig.4. A two-dimensional molecular network consists of TYY radicals (qubits) and phthalocyanine (Pc, working as a spin coupler), which can be either chemically bonded by suitable bridging molecular structures or linked through Van der Waals forces. Here we have chose Pc as a spin coupler because (i) it has good optical properties porphyrinbooks and (ii) as a planar molecule, it is great building block to form two-dimensional network. On top of this molecular network, we could build nanophotonic devices controlling the spin and electronic states of the Pc molecules, i.e. inducing a transition from the singlet to the triplet, thus mediating the interactions among the radicals. The fabrication of the optical devices has been demonstrated recently by using an optical twizzer dsp2021 . Moreover, we can explore the triplet excited states, i.e., promoting the triplet from the ground state to the excited state, which is expected to allow us to further separate the radicals, ideally tens of nanometer distances, thus easing the design and fabrication of optical devices. We could not only take advantages from the recent development of the programable photonic quantum circuits bogaerts2020 , but also integrate spin qubits with photonics elshaari2020 . Regarding the feasibility of fabricating optical devices, one choice is to use nanometer-size single photon emitter recently developed such as the gold tips or nanoscale gaps on WSe₂ kern2016 ; ziegler2018 ; peng2020 . In addition, the quantum calligraphy, in which encoding strains into the 2D materials can be used to create and locate the single-photon emitters with a precision of nanometers rosenberger2019 . Also, the nanometer-sized single photon emitters can be fabricated by using quantum dots liu2021 . Another alternative for the optical devices is to use organic molecules such as Pc and dibenzoterrylene that have very good optical emission properties porphyrinbooks ; toninelli2021 , which is also compatible with spin couplers, thus further easing the integration with the molecular network. As shown in the above sections, the mediating couplings and quantum dynamics have been calculated, which suggests this proposal is theoretically feasible. The persistence of the exchange interaction between the radical and the triplet could be problematic for read-out. Further work will therefore be required either to identify mechanisms that can turn off the interaction after a defined time in order to read out quantum information, or to understand how to exploit the always-on interaction to perform quantum computation benjamin2003 .