Quantum tomography of a third-order exceptional point in a dissipative trapped ion

TL;DR

This work demonstrates a PT- and anti-PT-symmetric third-order exceptional point (EP3) in a dissipative three-level trapped-ion system. By engineering two controlled loss channels and applying adiabatic elimination, the authors realize an effective non-Hermitian Hamiltonian $H_ ext{eff}$ with eigenenergies $E_\pm=\pm\sqrt{\Omega^2-\gamma^2}$ and $E_0=0$, yielding an EP3 at $\Omega=\gamma$. They observe the EP3 via non-Hermitian absorption spectroscopy and perform eigenstate tomography to directly show the coalescence of the three eigenstates; they also identify an intrinsic Liouvillian EP3 through quench dynamics, linking non-Hermitian and Liouvillian structures. The results illuminate the topological structure and potential sensing advantages of higher-order EPs in open quantum systems and pave the way for exploring non-Hermitian phenomena in multi-level quantum platforms. $

Abstract

The requirement for Hermiticity in quantum mechanics ensures the reality of energies, while the parity-time symmetry offers an alternative route to achieve this goal. Interestingly, in a three-level system, the parity-time symmetry-breaking can lead to a third-order exceptional point with distinctive topological properties and enhanced sensitivity. To experimentally implement this in open quantum systems, it is essential to introduce two well-controlled loss channels. However, the requirement for these two loss channels presents a challenge in experimental implementation due to the lack of methods to realize the dynamics governed by an effective non-Hermitian Hamiltonian. Here we address the challenge by employing two approaches to eliminate the effects of quantum jump terms so that the dynamics is governed by an effective non-Hermitian Hamiltonian in a dissipative trapped ion with two loss channels. Based on this, we experimentally observe the parity-time symmetry-breaking-induced third-order exceptional point through non-Hermitian absorption spectroscopy. In particular, we perform quantum state tomography to directly demonstrate the coalescence of three eigenstates into a single eigenstate at the exceptional point. Finally, we identify an intrinsic third order Liouvillian exceptional point associated with a parity-time symmetry breaking via quench dynamics. Our experiments can be extended to observe other non-Hermitian phenomena involving more than two levels and potentially find applications in quantum information technology.

Figures (7)

• Figure 1: Schematics of experimental configurations, and experimental results via non-Hermitian absorption spectroscopy.a, We use a $370$ nm laser beam A for cooling, optical pumping and state detection, $370$ nm lasers B and C to realize the dissipation in the $|1\rangle$ and $|2\rangle$ levels, respectively, and microwaves to generate the Hermitian part of the Hamiltonian. A $411$ nm laser is applied to drive the transition between the levels $|1\rangle$ and $|a\rangle$ for non-Hermitian absorption spectroscopy. Laser with wavelengths of $935$ nm, $3432$ nm, and $976$ nm are also utilized as auxiliary components in the experiments (see Supplementary Information S-1 for detailed descriptions of our experimental setup). The magnetic field has a small out-of-plane component in addition to the in-plane component so that $\bm{B}=\bm{B}_{\parallel}+\bm{B}_{\perp}$. b, The main energy levels and transitions used in our experiment. Transitions driven by the microwaves, the $370$ nm lasers B and C, and the $411$ nm laser are described by black, purple, and blue arrows, respectively. Spontaneous decay of the excited states is shown by dotted and dashed lines. Real c and imaginary d parts of complex eigenenergies obtained by theoretical calculations (solid lines) are shown together with experimental data points (circles). e, Real part of the eigenenergies near the EP3 (marked by the star) as a function of two additional detunings $\Delta_0$ and $\Delta_1$. f, Theoretical (solid lines) and experimental complex eigenenergies (circles) with respect to $\theta$ defined through $\Delta_0 = \Delta_r \cos \theta$ and $\Delta_1 = \Delta_r \sin \theta$ with $\Delta_r = 2\pi \times 0.020$ MHz. The energy bands are marked with the same colors as in e, and the starting points at $\theta = 0$ are marked by circles in e. In c-f, $\gamma = 2\pi \times 0.040\ \mathrm{MHz}$, $\Omega_a = 2\pi \times 0.004\ \mathrm{MHz}$, and the evolution time is $t_a = 200 \ \mu$s. The experimental results are averaged over 5 rounds of experiments (each contains 200 shots) with error bars being the standard deviation of the five experimental repetitions (error bars for some data points are smaller than the symbol size).
• Figure 2: Experimental results for eigenstate tomography.a-d, Variation of the normalized off-diagonal elements $\Delta |\rho_{j3}^\mathrm{n}|^2$ with respect to $\phi$ (or $\varphi$) and $\Omega/\gamma$. The initial state is $|u_z (\phi)\rangle$ in a, $|u_x (\phi)\rangle$ in b, and $|u_0 (\varphi)\rangle$ in c and d (see their definitions in the main text). In a,c,d, $j=2$, and in b, $j=1$. The circles indicate the zero points obtained by linearly interpolating the experimental data (see Methods Sec. D), and the black dashed lines represent the theoretical values: $\phi = \pm \cos^{-1} (\gamma / \Omega)$ for a, $\phi = \pm \cos^{-1} (\Omega/ \gamma)$ for b, and $\varphi = \tan^{-1} (\Omega/\gamma)$ for c and d. The crosses in b also denote zero points but are excluded in the calculation of inner products since they do not correspond to the eigenstates. e,f, Inner products of the eigenstates $|\psi_\pm\rangle$ and $|\psi_0\rangle$. The solid lines are obtained by Eq. (\ref{['TheoryEigenVectors']}), and the diamonds are experimental data.
• Figure 3: Results for observing a Liouvillian EP3 via quench dynamics. a, Experimental and b, theoretical values of $\rho_{12} - \rho_{01}$ with respect to the normalized evolution time $\gamma t$ and the ratio $\Omega/\gamma$. The dashed lines indicate the evolution results at the EP3. c, Curve fitting using the damped sinusoidal function for experimental data (circles) at $\Omega / \gamma = 5$. The inset shows the fitting results using the hyperbolic sine function at $\Omega / \gamma = 0.5$. The experimental data are averaged over 1000 repetitions. d, Fitted oscillation and decay factors $B_1$ and $B_2$ with respect to $\Omega / \gamma$. The theoretical results are $B_{1,2} = |\sqrt{(\Omega/\gamma)^2 - 1}|$ (see Method Sec. E). Error bars, some of which are smaller than the symbols, denote the $95\%$ confidence intervals of the fit.
• Figure 4: Spectral line and winding topology.a, The spectral line (the remaining population in $|a\rangle$ at the end of the dynamics at $t_a = 200 \ \mu$s with respect to the detuning $\delta_a$) for $\Omega/\gamma = 0.8$. Both the theoretical results (red line) and fitted ones (black line) are calculated according to Eq. (\ref{['Na_formu']}). The former ones are computed using the parameter $\Omega/\gamma = 0.8$, while the fitted results employ parameters obtained from a fitting procedure based on the experimental data (circles). b, Argument of $E_n(\theta) - E_B$ as a function of $\theta$ with $E_B =-0.016 - 0.032 \mathrm{i}$ plotted using the data in Fig. \ref{['fig1new']}f. The y-axis is shifted so that $\mathrm{arg} (E_0(0) - E_B) = 0$, and the $n$-th band is shifted along the x-axis by $2\pi n$ to show the spectral winding around $E_B$. The units of energy is $2\pi \times 1\ \mathrm{MHz}$. Here, $\gamma = 2\pi \times 0.040\ \mathrm{MHz}$ and $\Omega_a = 2\pi \times 0.004\ \mathrm{MHz}$. The experimental results are averaged over 5 rounds of experiments (each contains 200 shots) with error bars being the standard deviation of the 5 experimental repetitions.
• Figure 5: Results for quench dynamics.a, Experimental and b, theoretical results of an element $|\rho_{03}|$ of the density matrix with respect to the normalized evolution time $\gamma t$ and the ratio $\Omega/\gamma$. c, Curve fitting using the damped sinusoidal function for experimental data (circles) at $\Omega / \gamma = 5$. The inset shows the fitting results using the hyperbolic sine function at $\Omega / \gamma = 0.5$. The experimental data are averaged over 400 repetitions with error bars representing the standard deviation. d, Fitted oscillation and decay factors $B_1$ or $B_2$ with respect to $\Omega / \gamma$. The error bar has the same meaning as that in Fig. \ref{['fig3new']}d.