Nonreciprocal Dispersive Coupling for Quantum Sensing

Abstract: Dispersive coupling is widely utilized for quantum information readout. Most prior studies have concentrated on reciprocal dispersive coupling. Here, we further construct nonreciprocal dispersive coupling and apply it to quantum sensing. For cavity photon number measurement, nonreciprocal dispersive coupling delivers higher precision than its reciprocal counterpart, and this advantage grows more pronounced with an increase in photon number. When directly measuring the single-photon driving strength, however, nonreciprocal dispersive coupling shows no superiority over reciprocal dispersive coupling. By converting the information of driving strength into cavity photon numbers via our proposed strategy, nonreciprocal dispersive coupling again outperforms reciprocal dispersive coupling in precision, with the advantage becoming increasingly significant at larger driving strength. This work presents a novel method to boost quantum sensing and enable the fabrication of ultra-precise quantum sensors.

• The paper shows that nonreciprocal dispersive coupling yields an exponential precision advantage over reciprocal coupling, validated via error propagation and quantum Fisher information analysis.
• It employs adiabatic elimination of a dissipative auxiliary bosonic mode to establish a unidirectional interaction between a cavity and a qubit, optimizing measurement performance under time constraints.
• The study introduces an indirect measurement strategy for single-photon driving strength that outperforms conventional methods, paving the way for ultra-precise quantum sensors.

Nonreciprocal Dispersive Coupling for Quantum Sensing

Introduction

This work systematically investigates nonreciprocal dispersive coupling in the context of quantum sensing, contrasting its metrological performance with that of conventional reciprocal dispersive coupling. The analysis is grounded in open quantum system dynamics achieved through adiabatic elimination of an intermediate bosonic mode, producing an effective unidirectional interaction between a cavity field mode and a qubit. The central focus is on evaluating measurement precision for two quantum sensing tasks: cavity photon number estimation and measurement of a resonant single-photon driving field. The study rigorously quantifies precision enhancements using both the error-propagation approach (via qubit Pauli operator measurements) and the quantum Fisher information formalism.

Model Construction and Dynamics

A nonreciprocal dispersive coupling is constructed by introducing an auxiliary bosonic mode with strong dissipation that mediates between a cavity mode and a qubit. Upon adiabatic elimination of this bosonic mode, the cavity-qubit system evolves according to a Lindblad master equation containing a nonreciprocal dissipator of the form $D[ae^{i\theta \sigma_z}]$. This structure ensures that the cavity coherently influences the qubit dispersively with no corresponding back-action. The resultant Markovian dynamics are analyzed using quantum Langevin equations, facilitating analytical solutions for observable dynamics and measurement statistics.

The effective parameters include the nonreciprocal dissipative coupling rate $\Gamma$, the reciprocal dispersive shift $\lambda$, and the cavity decay rate $\kappa$. The analysis is valid throughout the over-damped limit $\kappa_b \gg (\kappa, g)$, where $g$ is the coherent cavity-bosonic mode coupling.

Photon Number Estimation: Reciprocal vs. Nonreciprocal

Measurement of the cavity photon number is analyzed by initializing the cavity to a Fock state and interrogating the qubit using optimal Pauli measurements. Analytical expressions for the steady-state qubit coherence and the corresponding Pauli expectation values are derived. The measurement uncertainty $\delta^2 n$ is characterized as a function of system parameters and the photon number $n$.

For nonreciprocal coupling, the minimum uncertainty scales as $\delta^2 n \approx 2n$ in the $n \rightarrow 0$ limit, and $\Gamma$0 for large $\Gamma$1. The corresponding quantum Fisher information achieves $\Gamma$2 and $\Gamma$3 in these respective regimes.

Reciprocal dispersive coupling yields strictly larger uncertainties: $\Gamma$4 for small $\Gamma$5 and exponential degradation $\Gamma$6 for large $\Gamma$7, with quantum Fisher information $\Gamma$8 and $\Gamma$9.

The ratio of achievable uncertainties $\lambda$0 demonstrates a universal $\lambda$1 improvement for low photon number, while at large photon number nonreciprocal coupling delivers an exponential precision advantage. These results persist when including time-resource constraints—nonreciprocal coupling both improves steady-state measurement rate (reduced $\lambda$2) and measurement precision.

Measurement of Single-Photon Driving Strength

Direct measurement of the resonant single-photon driving strength $\lambda$3 yields no advantage for nonreciprocal dispersive coupling relative to reciprocal coupling. The derived uncertainties for weak and strong driving ($\lambda$4 and $\lambda$5) via Pauli measurements do not show enhancement with nonreciprocal interaction.

To address this, an indirect measurement strategy is introduced: encode $\lambda$6 into the stationary cavity photon number (via the relation $\lambda$7), then perform qubit-based readout. Employing this two-stage strategy, the nonreciprocal dispersive coupling achieves

$\lambda$8

with corresponding time-constrained uncertainties inversely scaling with the total measurement time and outperforming the reciprocal configuration for all $\lambda$9.

Key result: nonreciprocal dispersive coupling—when combined with indirect parameter encoding—surpasses reciprocal coupling in estimation precision, with a greater advantage at stronger drive.

Theoretical and Practical Implications

The analysis establishes the quantum Fisher information limits for both coupling regimes and explicitly demonstrates the optimality of qubit Pauli measurements in achieving these bounds. The exponential gain in sensing precision for nonreciprocal dispersive coupling at large photon numbers highlights a significant metrological advantage. Practically, the ability to achieve superior sensitivities with faster convergence to steady state enables the design of ultra-precise quantum sensors, particularly in platforms where time-resource constraints are critical, such as in superconducting circuits, circuit QED architectures, and hybrid quantum systems.

From a theoretical perspective, the construction via Lindblad-structured nonreciprocal interactions expands the quantum toolbox for engineered directionality and measurement back-action control. The findings clarify that the precision advantage is tied specifically to parameter regimes and measurement protocols matching the uni-directional readout topology.

Future investigations are well-positioned to generalize these principles to high-dimensional qudit systems, complex quantum networks, or multi-mode quantum field settings. Such developments are relevant not only for quantum metrology but also for quantum information processing tasks that demand stringent measurement back-action suppression or directionally selective coupling.

Conclusion

This work demonstrates that nonreciprocal dispersive coupling, constructed through dissipative engineering in cavity-QED architectures, yields substantial quantum sensing advantages when properly harnessed, especially for photon number measurements and related encoded parameter estimation tasks. While direct measurement of certain parameters does not benefit, a measurement strategy that leverages nonreciprocal directionality affords consistently higher precision. These insights support the development of metrologically optimal quantum devices and motivate further extension of nonreciprocal paradigms in quantum technology platforms.

Reference: "Nonreciprocal Dispersive Coupling for Quantum Sensing" (2606.04666)