The spin coherence time T₂ describes the "quantum lifetime" of a qubit spin state in the presence of environmental noise. Interactions between the qubit spin and its environment can be described by the following many-body Hamiltonian:
\[
\hat{H} = \mathbf{SDS} + \mathbf{B} \gamma_e \mathbf{S} + \sum_i \mathbf{S} \mathbf{A}_i \mathbf{I}_i + \mathbf{B} \gamma_i \mathbf{I}_i + \sum_{i>j} \mathbf{I}_i \mathbf{J}_{ij} \mathbf{I}_j
\]
where \(\mathbf{SDS}\) is the zero-field splitting term, \(\mathbf{B} \gamma_e \mathbf{S}\) and \(\mathbf{B} \gamma_i \mathbf{I}_i\) are the Zeeman splitting terms, \(\mathbf{S} \mathbf{A}_i \mathbf{I}_i\) is the hyperfine term, and \(\mathbf{I}_i \mathbf{J}_{ij} \mathbf{I}_j\) is the nuclear dipolar interaction term.
From the Hamiltonian \(\hat{H}\) and resulting density matrix \(\rho\), the coherence function \(L(t)\) of the qubit can be computed
\[
L(t) = \frac{ \langle 1 | \rho(t) | 0 \rangle}{\langle 1 | \rho(0) | 0 \rangle }
\]
from which T₂ can be computed by fitting \(L(t)\) to a stretched exponential form:
\[
L(t) = \exp\left(\left( \frac{t}{T_2} \right)^{\eta}\right)
\]
In principle, \(L(t)\) can be solved using computational methods, such as the cluster correlation expansion (CCE) approach [1][2]. Here, T₂ is coputed using models developed in Refs. [3][4]. These models are fitted to CCE simulations and can therefore provide reliable estimates of T₂. In developing these methods, several assumptions are made:
-
The dominant decoherence mechanism is magnetic dipolar interactions between the qubit spin and nuclear isotopes present in the material. As a result, T₂ is largely a property of the host material, rather than the spin defect itself.
-
The secular approximation is invoked, which is permitted when \(\hat{S}_z\) is a good quantum number. This is generally a safe approximation in the high magnetic field limit.
For each naturally abundant isotope \(i\) of every element in the structure, the nuclear g-factor \(g_i\), nuclear spin quantum number \(I_i\), and natural abundance are read from the EasySpin database. The nuclear spin density \(n_i\) is computed from the lattice volume and natural abundance. The single-element coherence time scales as [3]:
\[
T_{2,\textrm{3D},i} = 1.46 \times 10^{18} \left|g_i\right|^{-1.64} I_i^{-1.1} n_i^{-1}
\]
Contributions from all isotopes are combined using a power-law formula [3]:
\[
T_{2,\textrm{3D}}^{-\eta} = \sum_i \left(T_{2,\textrm{3D},i}\right)^{-\eta}
\]
For 2D materials, a dimensional correction factor is applied [4]:
\[
T_{2,\textrm{2D},i} = C_{\alpha_\mathrm{2D}} n_i^{(2-\alpha_\mathrm{2D})/3} w^{-\alpha_\mathrm{2D}/3} T_{2,\textrm{3D},i}
\]
\[
T_{2,\textrm{2D}}^{-3/\alpha_\mathrm{2D}} = \sum_i T_{2,\textrm{2D},i}^{-3/\alpha_\mathrm{2D}}
\]
which, after fitting, simplifies to
\[
T_{2,\textrm{2D},i} = 0.94 n_i^{-0.28} w^{-0.95} T_{2,\textrm{3D},i}
\]
\[
T_{2,\textrm{2D}}^{-1.06} = \sum_i T_{2,\textrm{2D},i}^{-1.06}
\]
The thickness \(w\) is estimated using the van der Waals radii of the outermost atoms, and the 2D nuclear spin density is defined as:
\[
n^{\mathrm{2D}}_i = n^{\mathrm{3D}}_i \cdot d
\]
References
-
W. Yang, W. L. Ma and R. B. Liu,
"Quantum many-body theory for electron spin decoherence in nanoscale nuclear spin baths,"
Rep. Prog. Phys. 80, 016001 (2017).
DOI: 10.1088/0034-4885/80/1/016001
-
M. Onizhuk and G. Galli,
"Colloquium: Decoherence of solid-state spin qubits: A computational perspective,"
Rev. Mod. Phys. 97, 021001 (2025).
DOI: 10.1103/RevModPhys.97.021001
-
S. Kanai, F. J. Heremans, H. Seo, G. Wolfowicz, C. P. Anderson, S. E. Sullivan, M. Onizhuk, G. Galli, D. D. Awschalom, and H. Ohno,
"Generalized scaling of spin qubit coherence in over 12,000 host materials,"
Proc. Natl. Acad. Sci. 119, e2121808119 (2022).
DOI: 10.1073/pnas.2121808119
-
M. Y. Toriyama, J. Zhan, S. Kanai, and G. Galli,
"Strategies to search for two-dimensional materials with long spin qubit coherence time,"
npj 2D Mater. Appl. 9, 108 (2025).
DOI: 10.1038/s41699-025-00623-8