Quantum Sensing using Geometrical Phase in Qubit-Oscillator Systems

Abstract

We present a quantum sensing protocol for coupled qubit-oscillator systems that surpasses the standard quantum limit (SQL) by exploiting a geometrical phase. The signal is encoded in the geometrical phase that is proportional to the area enclosed in oscillator phase space. This area is amplified through squeezing, enabling sensitivities beyond the SQL. Our method is independent of oscillator's initial state, amenable to sensing with high-temperature or logical error-corrected states. The protocol shows robustness to qubit Markovian noise and preserves its state-independence, underscoring its practicality for next-generation quantum metrology. We demonstrate application to force sensing beyond the SQL in longitudinally coupled systems, and to high-precision measurements of couplings and pulse calibration surpassing SQL in dispersively coupled circuit quantum electrodynamics (cQED) architectures.

Figures (5)

• Figure 1: We consider (a) longitudinally-coupled qubit-oscillator systems. (b) Free evolution of the coupled system in the oscillator phase space. (c) Sequence of operations on the oscillator and qubit for the geometric protocol and (d) the corresponding phase space motion.
• Figure 2: (a) Quantum Fisher information of the qubit to sense the force $\eta$ as a function of time. (b) The relative sensitivity of geometric protocol in comparison to standard free evolution scheme as a function of squeezing. Dashed line shows $\Delta\eta_r = 1$ as a reference.
• Figure 3: (a) Sequence of operations for the geometric method with phase $\theta = \int dt \omega(t)$, where the squeezing is achieved by suddenly changing the oscillator frequency $\omega(t)$ from $\omega$ to $\omega^\prime$. (b) Relative sensitivity of geometric protocol in comparison to standard free evolution with $\omega^\prime$ frequency as a function of squeezing.
• Figure 4: (a) We consider dispersively coupled qubit-oscillator systems. (b) Sequence of operations on the oscillator and qubit for the geometric protocol and (c) the corresponding phase space evolution. (c) Relative sensitivity in comparison to SQL for estimating either the displacement $\alpha$ or the dispersive coupling $\chi$ as a function of squeezing.
• Figure 5: Effect of noise for inertial force sensing. We consider typical parameters johnsson2016macroscopic where $\omega=1$, qubit-oscillator coupling $\gamma = 0.2$, with $10~\text{dB}$ of squeezing, the force $\eta=10^{-5}$ and the fock space cutoff $N=200$. (a) Wigner functions for the joint state at each step of the geometric protocol for all noise channels with strength $\lambda=0.05$. The last column is the final qubit state projected to the $x-y$ plane (marked by red arrow), with printed phase $\phi$ relative to the noiseless case (shown with blue arrow). For visual clarity the vectors are drawn with $100 \times$relative angle. (b) Numerical calculation of QFI for different initial states after 1 period of evolution for each noise channel as a function of increasing error rate $\lambda$.