Digital Twin Simulations Toolbox of the Nitrogen-Vacancy Center in Diamond

TL;DR

The work introduces a Python-based digital twin for the NV center that numerically models spin dynamics under general optical, MW/RF, and environmental inputs using a non-perturbative time-dependent Hamiltonian in the laboratory frame. By solving the Lindblad master equation with a complete set of Hamiltonians ($\\hat{H}_0$, $\\hat{H}_1(t)$, $\\hat{H}_2(t)$) and flexible dissipation, the framework captures realistic pulse effects and phase evolution beyond rotating-frame approximations. Three demonstrations—two-qubit NV-$^{13}$C gates, dynamical decoupling sensing, and NV-based quantum teleportation—validate the tool against existing experiments and illustrate its ability to handle complex pulse sequences and phase accumulation. The open-source QuaCCAToo-based toolbox provides accessible, robust numerical modeling for quantum computing, sensing, and networks, with clear pathways for extending to other color centers and more sophisticated optical modeling.

Abstract

The nitrogen-vacancy (NV) center in diamond is a crucial platform for quantum technologies, where its precise numerical modeling is indispensable for the continued advancement of the field. We present here a Python library for simulating the NV spin dynamics under general experimental conditions, i.e. a digital twin. Our library accounts for electromagnetic pulses and other environmental inputs, which are used to solve the system's time evolution, resulting in a physical output in the form of a quantum observable given by fluorescence. The simulation framework is based on a non-perturbative time-dependent Hamiltonian model, where the states initialization and readout are postulated from the interaction with optical fields. By eliminating oversimplifications such as the adoption of rotating frames for the microwave and radio frequency fields, our simulations reveal subtle dynamics emerging from realistic pulse constraints. The software is illustrated with three examples and validated by comparing the simulations with experimental reports, relevant to the fields of quantum computing (conditional logic gates), sensing (dynamical decoupling sequences with coupled spins) and networks (state teleportation). Overall, this digital twin delivers a robust numerical modeling of the NV spin dynamics, with simple and accessible usability, which can be used for a wide range of applications.

Figures (7)

• Figure 1: Schematic representation of a color center digital twin. Color centers are point defects in solids that strongly interact with light and often possess a quantum-mechanical spin. The color center simulations software receives a series of physical inputs and generates an output, which is calculated numerically based on the time evolution of the system under the system's Hamiltonian. For the NV center in diamond, three inputs can be distinguished: optical (for initialization of spins), microwave and radio frequency pulses (for spin control) and environmental (such as temperature and external magnetic field). The output of the NV component is fluorescence, which can be related to a quantum-mechanical observable.
• Figure 2: (a) Optical pumping and readout mechanism of NV centers. The latter has triplet ($S=1$) and singlet ($S=0$) states within the band gap of diamond. The ground triplet states ($^3$A$_2$) can be driven from thermal equilibrium to the excited triplet states ($^3$E) by a non-resonant green laser. From there, the NV can decay back to $^3$A$_2$ through the emission of red fluorescence. Alternatively, it can undergo an intersystem crossing through the intermediate singlet states $^1$A$_1$ and $^1$E, leading to polarization of the $m_S=0$ spin sublevel. The same optical pumping mechanism can be used to measure the $m_S$ states populations. (b) Energy level diagram of NV$^-$. The dominant zero-field term splits the $m_S=0$ and $m_S=\pm1$ states, where the latter are further split by Zeeman interaction in the presence of an external magnetic field $\bold{B}_0$. Resonant MW pulses can drive coherent transitions between $m_S=0 \leftrightarrow \pm1$. Depending on the nitrogen isotope, $^{15}$N or $^{14}$N, the levels are further split, where transitions between $m_I$ dependent on $m_S$ can be achieved with resonant RF pulses.
• Figure 3: Simulated conditional rotations on (a) NV's electron spin $S$ and (b) nuclear spin $I^c$ from coupled $^{13}$C. (c) Energy level diagram of the system, neglecting the $m_S=+1$ state. By applying a resonant pulse with either one of the transitions, full Rabi oscillations are achieved to each spin, conditioned to the other. The decoherence of the nuclear spin is modeled by a collapse operator within the Lindblad master equation. The simulations reproduce the experimental results from F. Jelezko et al. (2004) polarization_population_2.
• Figure 4: (a) Hahn echo sequence simulation of NV-$^{13}$C system as performed experimentally by L. Childress et al. (2006) RWA_2. The fluorescence shows a characteristic spin echo envelope modulation, with a fast and a slow frequencies corresponding to the $^{13}$C nuclear spin Larmor frequencies at $\ket{m_S =+1}$ and $\ket{m_S =0}$ levels, respectively. The Hahn echo sequence refocuses static dephasings from the $^{14}$N nuclear spin and environment, enabling the detection of the $^{13}$C nuclear spin. (b) CPMG sequence simulation of a weakly coupled NV-$^{13}$C 13C_sensing_1 as experimentally performed by T. H. Taminiau et al. (2012) 13C_sensing_1. By repeating the $\pi$-pulse and free evolution of $\tau$ several times, the CPMG sequence is able to cancel out oscillating noises from the environment, enabling the sensing of the weakly coupled $^{13}$C nuclear spin. The intensities of the resonances show oscillations with the number of pulses $M$, which can be used to perform precise conditional gates to the $^{13}$C, exclusively by fast and precise MW pulses to the electron spin.
• Figure 5: (a) Schematic of the XY8-12 pulse sequence. It is composed of 8 intercalated $\pi$-pulses on $x$ and $y$ axes repeated $M$ times, thus canceling dephasings more efficiently than the CPMG sequence. (b) XY8-12 simulation of an NV sensing and external field $\bold{B}_2(t)$ with frequency $\omega_2=5.5$ MHz ambiguous_resonances. The NV's fluorescence observable shows a prominent resonant at $\tau_0=(2\omega_2)$ with multiple fringes. Apart from that, spurious harmonics are also present at fractions of $\tau_0$ due to the system's free evolution at the finite pulse lengths. (c) Simulation of RXY8-12 with a phase randomization in each XY8 block. With this random phase, the spurious harmonics are suppressed. This demonstrates the ability of the software to operate in arbitrary coupling regimes and time-dependent Hamiltonians. These simulations reproduce the experimental results presented by L. Tsunaki et al. (2025) ambiguous_resonances.