Probing Sensitivity near a Quantum Exceptional Point using Waveguide Quantum Electrodynamics

TL;DR

This work analyzes whether a quantum exceptional point (EP) can yield enhanced sensing in a passive PT dimer implemented with waveguide quantum electrodynamics. By engineering two superconducting qubits with a tunable coupler, the authors realize a non-Hermitian two-mode system with a passive EP at $g=γ/4$ and extract the complex eigenenergies as the coupling is varied, confirming the spectral coalescence and real-part emergence above the EP. Through pulsed and continuous-wave measurements of three observables, they show that, despite the sharp spectral response near the EP, there is no corresponding quantum-enhanced sensitivity under near-quantum-limited conditions; a three-mode master equation that includes the tunable coupler and Lamb-shift effects is needed to fit the data, and the maximum sensitivity occurs away from the EP. The results clarify the limitations of passive EP-based sensing and suggest that active PT-dimer implementations or alternative sensing schemes are required to achieve quantum-enhanced performance.

Abstract

Non-Hermitian Hamiltonians with complex eigenenergies are useful tools for describing the dynamics of open quantum systems. In particular, parity and time (\PT) symmetric Hamiltonians have generated interest due to the emergence of exceptional-point degeneracies, where both eigenenergies and eigenvectors coalesce as the energy spectrum transitions from real- to complex-valued. Because of the abrupt spectral response near exceptional points, such systems have been proposed as candidates for precision quantum sensing. In this work, we emulate a passive \PT~dimer using a two-mode, non-Hermitian system of superconducting qubits comprising one high-coherence qubit coupled to an intentionally lossy qubit via a tunable coupler. The loss is introduced by strongly coupling the qubit to a continuum of photonic modes in an open waveguide environment. Using both pulsed and continuous-wave measurements, we characterize the system dynamics near the exceptional point. We observe a behavior broadly consistent with an ideal passive \PT~dimer with some corrections due to the tunable coupler element. We extract the complex eigenenergies associated with the two modes and calculate the sensitivity as a function of the coupling strength. Confirming theoretical predictions, we observe no sensitivity enhancement near the quantum exceptional point. This study elucidates the limitations of exceptional-point systems as candidates for quantum-enhanced sensing.

Figures (9)

• Figure 1: Passive $\mathcal{PT}$ dimer experimental setup.a) False-colored optical micrograph of the device. The system includes qubits Q$_1$ (pink) and Q$_2$ (orange); Q$_2$ is coupled to a bidirectional coplanar waveguide (purple) with strength $\gamma/2\pi = 17MHz$, which terminates in a measurement amplification chain. We tune the coupling $g$ between the qubits using a tunable coupler (blue) to perform measurements near the exceptional point. b) Simplified model of the system. The qubits are modeled as two-level systems that are resonant at frequency $\omega/2\pi$ = 5 GHz coupled with tunable strength $g$. We observe the physics of the system near the exceptional point by measuring the population of Q$_1$ and by probing Q$_2$ through the waveguide.
• Figure 2: Time-domain qubit measurements across the exceptional point.a) Q$_1$ population time dynamics as a function of the relative coupling $\tilde{g}$. We initialize Q$_1$ in the excited state and then activate the coupling with varying strength and duration using square flux pulses applied to the tunable coupler. Tuning the coupling strength through the exceptional point ($g = \gamma/4$), the population time dynamics transition from exhibiting exponential decay to oscillations. b) Concurrent Q$_2$ coherence dynamics across the exceptional point. For this experiment, we initialize Q$_1$ in the $(\ket g + \ket e) / \sqrt{2}$ state, and we measure the coherence of Q$_2$ through heterodyne detection of the field-amplitude of the emission in the waveguide. c, d) Each time trace is fit to theory and plotted as a function of the relative coupling for comparison. The inset shows the measurement pulse sequence. e) Measured eigenenergies of the two-qubit system as a function of the relative coupling, traversing the exceptional point. The eigenenergies are extracted from fitting the time-domain measurements. At the EP ($\tilde{g} = 1$, red line), the qubits' eigenenergies are degenerate. Beyond the EP, they acquire a real component while maintaining a fixed imaginary offset equal to the average loss rate $\gamma/2$. The analytical eigenenergies derived from the two-qubit non-Hermitian Hamiltonian in Eq. \ref{['eq:simpleHamiltonian']} are shown with solid lines.
• Figure 3: Continuous-wave sensing across the exceptional point.a) Transmission spectroscopy of the passive $\mathcal{PT}$ dimer driven by a coherent probe through the waveguide. The hybridized qubit modes split in energy according to $2g$. b) Normalized maximum sensitivity $\eta$ of the continuous-wave transmission measurement to changes in relative coupling $\tilde{g}$ as defined by Eq. \ref{['eq:sensitivity']}. The measured observable is the emitted field amplitude $|\langle \hat{a}(\omega)\rangle|$. We calculate the sensitivity for all probe detunings to determine the maximum for each $\tilde{g}$. We measure approximately constant noise fluctuations of the transmission signal as a function of $\tilde{g}$. No sensitivity improvement is observed near the EP, as predicted by three-mode, continuous-wave master equation simulations (solid purple line).
• Figure 4: Sensitivity across the exceptional point in the time-domain.a) Maximum sensitivity of the Q$_1$ population measurement to variation in relative coupling $\tilde{g}$. The data for the measured observable $P_e^1(t)$ is shown in Fig. \ref{['fig:fig2']}a. For each $\tilde{g}$, we calculate the sensitivity for all interaction times to determine the maximum. The noise $\sigma(t, \tilde{g})$ is the standard error taken over 10,000 shots. We observe no sensitivity improvement near the EP---the decrease in sensitivity is the result of energy dissipation into the waveguide environment with increasing $\tilde{g}$. We model the sensitivity of the Q$_1$ population signal using the analytical expression in Eq. \ref{['eq:data_pop']} (solid line). b) Sensitivity $\eta_{\mathrm{Q_2}}$ of the Q$_2$ coherence measurement to changes in $\tilde{g}$ calculated using Eq. \ref{['eq:emission_sense']}. The data for the measured coherence observable $|\langle \hat{\sigma}_2^-(t) \rangle|$ is shown in Fig. \ref{['fig:fig2']}b. The noise over 4 million measurement shots is approximately constant as a function of $\tilde{g}$. We plot the sensitivity model obtained from master equation simulations of the three-mode system (solid orange line), in comparison with the analytical model of the simple $\mathcal{PT}$ dimer (dashed gray line).
• Figure S1: Experimental setup. Wiring schematic of the device and all electronics used to perform the experiment. Note that only one flux line configuration is shown (green), but each qubit and coupler is coupled to a flux line with separate, but identical, control electronics.