Quantum gyroscope based on three-dimensional rotation induced Berry phase

Abstract

Solid-spin defects in diamond provide long coherence times and room-temperature optical initialization and readout, making them an attractive platform for compact solid-state quantum gyroscopes. A central challenge for NV-based gyroscopes is that the rotation-induced signal is weak, while near-resonant operation, although enhancing the response, can induce nonadiabatic transitions that degrade the accumulated geometric phase and readout fidelity. Here we investigate a levitated diamond under three-dimensional rotation, in which intrinsic ${}^{14}\mathrm{N}$ nuclear spins associated with NV centers act as sensing qubits. We show that the rotation is encoded in a geometric (Berry) phase and identify a near-resonant regime with strongly enhanced phase response. To suppress the resulting nonadiabatic leakage, we introduce a counter-diabatic protocol derived from the Kato gauge potential. This enables robust geometric-phase accumulation and improves the sensitivity by four orders of magnitude relative to the conventional detuned protocol. We further evaluate the achievable sensitivity and the dominant experimental limitations, including decoherence and protocol overhead, thereby establishing a realistic route toward high-performance NV-based solid-state quantum gyroscopes.

Figures (5)

• Figure 1: (a) Schematic illustration of the rotation sensing setup using a levitated micro-diamond. The diamond spins with frequency $\omega_\gamma$ about its principal axis $z_D$ and precesses with frequency $\omega_\alpha$ about the trap symmetry axis $z_S$ at a tilt angle $\beta$. The entire system undergoes an external rotation with frequency $\Omega$ about the inertial axis $z_i$, which is parallel to $z_S$. The axis of the nitrogen-vacancy (NV) center, $z_{\text{NV}}$, forms an angle $\theta$ with $z_D$. (b) Time evolution of the NV nuclear spin energy levels over one rotation period $T=2\pi/\omega_\gamma$. The dashed blue and red curves represent the diagonal Hamiltonian terms $H_{++}$ and $H_{--}$, respectively. The solid black curves $\lambda$ denote the instantaneous eigenvalues. The parameters are chosen to satisfy the resonance condition, leading to avoid-crossings at the intersection points of the diagonal terms.
• Figure 2: (a) The Berry phase shift (relative to the phase at $\Omega=0$) as a function of the normalized system rotation frequency $\Omega/\omega_\gamma$. Three distinct regimes are presented: large detuning (blue line), near-resonance (orange line), and full resonance (green line). The phase shift in the large detuning regime remains negligible, while the near-resonance case exhibits the most significant sensitivity to $\Omega$. (b) Time evolution of the adiabatic parameter over one period. The adiabatic condition requires this parameter to be much less than $1$. The results indicate that the adiabatic condition breaks down for the full resonance and near-resonance regimes (where the parameter exceeds $1$), leading to a discrepancy between the theoretical Berry phase and the actual accumulated geometric phase.
• Figure 3: (a) Time dependence of the matrix elements of the required CD Hamiltonian (derived from the Kato gauge potential). The CD field induces a strong coupling between the $|+1\rangle$ and $|-1\rangle$ states (black solid line) with a peak amplitude on the order of $80$ kHz, while the couplings involving the $|0\rangle$ state (red and blue dashed lines) remain negligible. (b) The geometric phase as a function of the rotation rate $\Omega$ in the near-resonance regime. The green dashed line shows the theoretical prediction based on ideal adiabatic evolution. The orange solid line represents the numerical simulation without the CD term ("No CD"), which fails to follow the theoretical curve due to non-adiabatic transitions. The blue dotted line shows the simulation with the CD term ("With CD"), demonstrating that the inclusion of the Kato gauge potential effectively suppresses non-adiabatic errors and restores the expected geometric phase.
• Figure 4: (a) The slope of the geometric phase with respect to the rotation frequency, defined as $\partial_\Omega \varphi_g(s)$, plotted as a function of $\Omega$. The steep slope near $\Omega \approx 0$ indicates a high sensitivity to rotation. The inset shows the zoomed-in view within the practical measurement bandwidth (approximately $\pm 1$ Hz), where the response remains smooth. (b) The theoretical sensitivity $\eta$ of the rotation measurement versus $\Omega$. The calculation assumes an ensemble of $N=10^6$ NV centers. The system achieves a shot-noise-limited sensitivity of approximately $0.6 \, \mu\text{rad/s}/\sqrt{\text{Hz}}$ at the optimal working point. The inset highlights the sensitivity in units of $\mu \text{rad/s}/\sqrt{ \text{Hz}}$ in the weak transverse field regime.
• Figure 5: (a) The optimal measurement Time $T_m$ plotted as a function of the effective coherence time $\tau$. The effective coherence time is defined as $\tau = T_{1e}T_{2n}^*/(T_{1e}+T_{2n}^*)$, which accounts for both the electron spin relaxation ($T_{1e}$) and the nuclear spin dephasing ($T_{2n}^*$). The different colors represent varying overhead times $t_i$ required for initialization and readout processes. (b) The achievable sensitivity limits calculated at the optimal measurement times shown in (a). The results illustrate the dependence of the sensitivity on the system's coherence properties and the duty cycle constraints imposed by $t_i$.