The Multiparameter Frontier: Metrological Hierarchy and Robustness in Dispersive Quantum Interferometry

Abstract

We present a dispersive quantum thermometry protocol for simultaneous estimation of inverse temperature $β$ and interaction strength $x$ using a nonlinear Mach-Zehnder interferometer coupled to a thermal ancilla. We derive closed-form expressions for the quantum Fisher information matrix, establishing that metrological performance depends solely on the thermal visibility $\mathcal{V}(β)$ and its derivative. The output state remains diagonal in photon-number basis, making photon counting globally optimal and saturating the multiparameter quantum Cramér-Rao bound without adaptive feedback. Moving beyond ideal unitary evolution, we analyze protocol robustness under concurrent amplitude and phase damping. Using Fisher Information Susceptibility, we establish a clear hierarchy: NOON states offer maximal theoretical sensitivity but exhibit exponential fragility to loss, rendering them impractical. Squeezed vacuum states emerge as robust candidates for steady-state sensing, while cat states prove compelling for transient thermometry by retaining significant coherence after photon loss. We validate these predictions through digital quantum circuit implementation on IBM's \texttt{ibm_torino} processor. Experimental results confirm the predicted Fisher information landscape while revealing systematic noise-induced biases, demonstrating that current NISQ hardware can effectively benchmark fundamental trade-offs in multiparameter quantum sensing.

Figures (12)

• Figure 1: Schematic of the Mach--Zehnder interferometer (MZI) built with nonlinear beam splitters (BS) used in this work. A NOON state of $N$ photons enters the interferometer and, in one of the arms, the light field interacts dispersively with a two-level atom (sample) prepared in a thermal state at inverse temperature $\beta$. B and D denote detectors at the output ports.
• Figure 2: Quantum circuit used to simulate the Mach--Zehnder interferometer for $N=1$. Qubit $q_1$ encodes the thermal two-level atom, $q_2$ represents the interferometer output mode that is measured, and $q_3$ corresponds to the reference arm. Qubit $q_0$ is an ancilla used to prepare the atomic thermal state via the rotation $R$ and a CNOT. The dispersive interaction is implemented as a controlled-$U$ gate between $q_1$ and $q_2$, while Hadamard and CNOT gates model the beam splitters.
• Figure 3: Comparison between simulated and analytical interference probabilities. (a) Simulation results $P_0(\beta,x)$ obtained from $10^4$ circuit shots. (b) Analytical prediction from Eq. \ref{['eq:probabilities_Vstd']}. The color scale encodes the probability of detecting all photons at the reference output port.
• Figure 4: Dispersive information $Q_{xx}(\beta,x)$ as a function of inverse temperature and interaction strength for NOON probes with (a) $N=1$ and (b) $N=10$ photons. The information is suppressed for large positive $\beta$ (cold non-inverted regime, $\mathcal{V}\to0$), and it approaches its maximum $Q_{xx}^{\max}\simeq4N^2$ in the cold inverted limit $\beta\to-\infty$ where $\mathcal{V}\to1$. At $\beta=0$ one has $\mathcal{V}(0)=1/3$, so $Q_{xx}$ remains finite. Increasing $N$ enhances the maximal information approximately as $N^2$ and produces narrower interference fringes along $x$, leading to multiple equivalent operating points.
• Figure 5: Thermal information $Q_{\beta\beta}(\beta,x)$ for NOON probes with (a) $N=1$ and (b) $N=10$ photons. The information is concentrated in an intermediate $\beta$ band where the visibility varies most rapidly (i.e., where $|\mathcal{V}'(\beta)|$ is largest), and it is suppressed for $|\beta|\to\infty$ where the atom becomes nearly pure. The overall scale of $Q_{\beta\beta}$ is essentially independent of $N$, while larger $N$ produces a fine fringe structure along $x$ that mirrors the phase-sensitive lobes of $Q_{xx}$.