Modeling Hyperfine Coupling in Molecular Magnets

Algorithms devised by Shor and Grover have demonstrated that a quantum computer factors numbers and searches databases much faster than a classical computer. Researchers have shown that molecular magnets are excellent candidates for quantum bits (qubits) [1], using either their nuclear or electron spins. Electron spins are advantageous due to their faster operational times; however, they decohere faster. Hyperfine coupling is the main cause of decoherence in magnetic molecules. This project tries to investigate the effect of hyperfine coupling on electron spin decoherence. Hyperfine coupling values were calculated using density functional theory and compared with experimental values. The Kubo-Anderson model was applied to link hyperfine coupling with the coherence time T2. The theoretical T2 was compared with experimental values. The simple theory of random frequency modulation is not adequate to explain decoherence time observed in a series of vanadyl complexes. Decoherence of electron spins in molecular complexes shows interesting physics that warrants further investigation. Introduction Quantum computing (QC) is of great interest to the scientific community due to its potential to solve quantum problems, which could lead to new breakthroughs in the fields of materials discovery, medicine, and machine learning [2]. A QC stores all the information of a wavefunction and uses quantum-mechanical principles, such as entanglement and superposition, to perform computations. Since it operates using the same quantum principles as atoms and molecules, QCs can calculate all of their quantum properties [2]. If a researcher can measure the bits before decoherence, they will obtain the direct quantum solution. Unlike traditional computers composed of bits with a value of either 0 or 1, quantum computers have qubits that are superpositions of two quantum states (see Fig. 1). One way to represent these quantum states is by using either an electron or nuclear spin. Fig. 1 A diagram illustrating the difference between classical and quantum computers [3]. Currently, nuclear spins are preferred to electron spins because nuclear spins are generally coherent for longer [4-5]. The method of controlling nuclear spins—Nuclear Magnetic Resonance (NMR)—requires a large magnetic field while electron spin-based QCs do not. To detect and control electron spins, Electron Paramagnetic Resonance (EPR) is used. Unfortunately, the theory of decoherence for EPR is not as developed as for NMR. This project took the first step in addressing this gap by applying established theory from NMR to EPR and finding out what can be applied. Fig. 2 The path that the magnetization vector ? ⃗⃗? points in. In NMR, there are two relaxation times commonly measured during experiments: T1 and T2. Fig. 2 shows the relaxation path of the magnetization vector. T1 is associated with the longitudinal relaxation of magnetization, in the direction of external magnetic field B0. T2 is associated with the transverse relaxation of magnetization, perpendicular to external magnetic field B0. T2 is used to determine how long the system stays coherent. This project focuses on finding a relationship between hyperfine coupling (HC) and T2. For a molecule to be used as a qubit, it needs a long relaxation time to hold and allow the quantum information to be measured. One mechanism that can induce decoherence is the hyperfine coupling—interactions between electron and nuclear spins. Random magnetic fields from nuclear spins can cause the electron spins to decohere. In the HC there are two contributing interactions: spin-dipole (SD) and Fermi contact (FC). The spin-dipole interaction is analogous to the interactions between dipoles in classical electrodynamics. In this case it is when the magnetic moments of the electron and nuclear spins interact. Fermi contact is when the electron and nuclear wavefunctions overlap. This happens through s orbital contributions of unpaired electrons. The HC has been shown to contribute to the decoherence in quantum dots [6]. To explain how HC contributes to decoherence in molecules, this project applied the Kubo-Anderson (random frequency-modulation) model—a general theory of relaxation [7-8]. This theory was applied by modeling nuclear spins as a perturbation. We used this to model the decoherence of electron spins in four vanadyl complexes, but the theoretical T2 values were off from the experimental T2 values. This is due to the diffusion barrier radius—when HC is so strong that the electron spin flip-flops do not occur on the timescale of decoherence—which needs to be accounted for. Once there is an accurate theory for decoherence, it may be possible to engineer molecules to maximize the coherence time. Computational Details One potential qubit candidate that has been identified is a type of vanadyl complex [9]. They all have the same charge and spin. Their electron spins were identified as potential qubits. This experiment focused on determining how the nuclear spin distances (from the hydrogen atoms to the vanadium atom) affected the electron spin decoherence. Charge: -2 Spin: 1⁄2 VC1