Experimental Determination of $\mathcal{P}\mathcal{T}$-Symmetric Exceptional Points in a Single Trapped Ion

Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal $(\mathcal{P}\mathcal{T})$ symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the $\mathcal{P}\mathcal{T}$-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.

Figure 1

(a) The energy level of ${{}^{171}\mathrm{Yb}}^{+}$ and the experimental setup. (b) Time evolution of the $|\uparrow ⟩$ state population without interspin coupling. The dissipation rate $\mathrm{\Gamma }$ is determined by an exponential fit. (c) Evolution of the $|\uparrow ⟩$ state population with both the coupling and dissipation beams. The data can be fit by a combination of a Rabi oscillation (gray dashed) and an exponential decay (black dashed). The evolution operator is defined as $\widehat{U}=e^{-i{\widehat{H}}_{\mathcal{P}\mathcal{T}}^{\prime }t}$. For panels (b) and (c), the black solid curve is the numerical simulation and the points are experimental data averaged from 1000 rounds of measurement. The error bars in $J$ and $\mathrm{\Gamma }$ are estimated by the standard deviation ($1\sigma$) of 20 rounds of experiment and fitting [41].

Figure 2

(a) Determination of EP from the measurement of $P_J(t)$ (black) and $P_{\mathrm{\Gamma }}(t)$ (gray). (b) In the $\mathcal{P}\mathcal{T}$-symmetry broken phase, we can calculate from $P_J(t)-P_{\mathrm{\Gamma }}(t)$ (black) the results of ${\sinh }^{-1}[P_{\mathrm{\Gamma }}(t)-P_J(t)]$ (gray), which show a clear linear time dependence with the slope being one of the eigenvalues. (c) The real (black) and imaginary (gray) parts of the eigenvalues. (d) The upper diagonal (black) and the imaginary part of the off-diagonal (gray) elements of the density matrix $\rho$ obtained in the $\mathcal{P}\mathcal{T}$-symmetric phase with $J/\mathrm{\Gamma }=10$. (e) The same results as in (d) for the $\mathcal{P}\mathcal{T}$-symmetry broken phase with $J/\mathrm{\Gamma }=0.826$. (f) The upper panel shows absolute values of the density matrix at $t=10\mu \mathrm{s}$ of (d) and the lower panel shows the same at $t=8\mu \mathrm{s}$ of (e). The left two plots are measured using quantum state tomography and the right are numerical simulations. For (a)–(e), the solid curves are numerical results and the dashed lines are interpolations.

Figure 3

(a) Evolution of the state population for $|\downarrow ⟩$ (black) and $|\uparrow ⟩$ (gray) for the periodically modulating Hamiltonian equation (6) with $J/\mathrm{\Gamma }=11.11$ and $\omega /J=0.84$. The effective Floquet Hamiltonian is in the $\mathcal{P}\mathcal{T}$-symmetry unbroken phase. (b) Population for the $\mathcal{P}\mathcal{T}$-symmetry broken phase with $J/\mathrm{\Gamma }=6.25$ and $\omega /J=1$. The solid curves are numerical simulations.

Figure 4

Phase diagram of the time-dependent Hamiltonian Eq. (8) obtained from measurement of $P_J(t)-P_{\mathrm{\Gamma }}(t)$. The red and yellow regions with positive $P_J(t)-P_{\mathrm{\Gamma }}(t)$ label the $\mathcal{P}\mathcal{T}$-symmetric phase, while the blue regions with negative value correspond to the $\mathcal{P}\mathcal{T}$-symmetry broken phase. To depict results with different $\mathrm{\Gamma }$ in the same plot, the unscaled data of $P_J(t)-P_{\mathrm{\Gamma }}(t)$ are shown without multiplying the scaling factor.