Thresholded Quantum Sensing with a Frustrated Kitaev Trimer

Abstract

We investigate the response of a Ramsey interferometric quantum sensor based on a frustrated, three-spin system (a Kitaev trimer) to a classical time-dependent field (signal). The system eigenspectrum is symmetric about a critical point, $|\vec{b}| = 0$, with four of the spectral components varying approximately linearly with the magnetic field and four exhibiting a nonlinear dependence. Under the adiabatic approximation and for appropriate initial states, we show that the sensor's response to a zero-mean signal is such that below a threshold, $|\vec{b}| < b_\mathrm{th}$, the sensor does not respond to the signal, whereas above the threshold, the sensor acts as a detector that the signal has occurred. This thresholded response is approximately omnidirectional. Moreover, when deployed in an entangled multisensor configuration, the sensor achieves sensitivity at the Heisenberg limit. Such detectors could be useful both as stand-alone units for signal detection above a noise threshold and in two- or three-dimensional arrays, analogous to a quantum bubble chamber, for applications such as particle track detection and long-baseline telescopy.

Figures (10)

• Figure 1: Kitaev trimer sensor structure. The thresholded quantum rectifiying sensor is based on a frustrated Kitaev trimer. Here, we illustrate a magnetic field impinging on the trimer from a particular direction (black waveform). Each vertex corresponds to a spin, and couplings along the edges encode interaction strengths.
• Figure 2: Kitaev trimer eigenspectrum. We plot the eigenenergies $\lambda_i$ as a function of the magnetic field amplitude $b$, with the field aligned along a cardinal axis. The eigenenergies associated with a standard sensor (green and gold lines) correspond to an interferometer with support on two linear branches. In contrast, the mousetrap sensor is associated with either the (asymmetric) gold-red pair or the green-purple pair of eigenenergies. Within the gray-shaded region, where the spectrum remains approximately linear, the rectifier behaves like a standard sensor. However, when $|b|>b_\mathrm{th}$, curvature of the eigenspectrum leads to an additional phase proportional to the second moment of the signal, $b^{2}(t)$. Thus, the sensor acts as a rectifier (i.e. becomes sensitive to the signal's second moment) only above the threshold $b_\mathrm{th}$.
• Figure 3: Trimer mousetrap sensor response. The response probability, $P[\phi_0]$, shown as a function of the signal amplitude, $\epsilon$, for a zero-mean, oscillatory signal with Gaussian envelope, $b(t) = \epsilon \exp(-t^{2}/s) \sin(2\pi k t)$, where $k = 0.05$, $s = 20$, $t_0 = -15$, and $t = 15$. Results are shown for $N=1$ and $N=3$ to illustrate how the response scales with the number of sensor copies. Both product state (PS) and entangled state (ES) are used as inputs for $N=3$. Note that a response is not incurred until $\epsilon$ is large enough ($\approx 0.3$) to be affected by the change in curvature of $\lambda_i$ near the asymmetric wells in the spectrum, with basins centered at $b = \pm 0.8$ (see Fig. \ref{['fig:Trimer']}).
• Figure 4: Mousetrap sensor response. The probability, $P[\phi_0]$, is shown as a function of time, $t$, for a range of signal amplitudes, $\epsilon$, for a zero-mean, oscillatory signal with Gaussian envelope, $b(t) = \epsilon \exp(-t^{2}/s) \sin(2\pi k t)$, where $k = 0.05$, $s = 20$, $t_0 = -15$, and $t = 15$. Insets show the signal $b(t)$ (top left in red) and the accumulated phase $\chi_\mathrm{mt}(t)$ (lower left in orange), both plotted on the same time scale as the main figure, with vertical ranges of $(-1,1)$ and $(-6,0)$, respectively. For small amplitudes ($\epsilon \lesssim 0.3$), the signal remains within the approximately linear region of the potential well, resulting in minimal phase accumulation and no integrated response. As $\epsilon$ exceeds the threshold near $0.3$, portions of the signal probe the nonlinear regime, where symmetry breaking in the spectrum leads to phase accumulation. The sensor thus acts as a signal rectifier responding only when the signal amplitude crosses $b_\mathrm{th}$, where it begins to integrate deviations from linearity.
• Figure 5: Omnidirectional mousetrap threshold response. This figure shows the probability $P[\phi_0]$ of a mousetrap remaining in its initial state as a function of the incident direction of the $b$ field, using the same Gaussian signal and parameters as in Figs. \ref{['fig2']} and \ref{['fig3']}. Due to the eight-fold symmetry of the response, only one octant of the sphere is shown. The response color mapping is indicated by the legend. At low amplitudes ($\epsilon \lesssim 0.3$), the sensor remains unresponsive across all directions (red), consistent with the plateau observed in Fig. \ref{['fig2']}. As $\epsilon$ increases, directional sensitivity begins to emerge around $\epsilon \approx 0.7$ (orange, yellow, green, blue), with significant anisotropic variations appearing in the range $0.7 < \epsilon < 1.5$, where the signal begins to probe the nonlinear regions of the spectrum.

Theorems & Definitions (1)

• Theorem 1