Experimental Witness of Quantum Jump Induced High-Order Liouvillian Exceptional Points

Abstract

The exceptional point has presented considerably interesting and counterintuitive phenomena associated with nonreciprocity, precision measurement, and topological dynamics. The Liouvillian exceptional point (LEP), involving the interplay of energy loss and decoherence inherently relevant to quantum jumps, has recently drawn much attention due to capability to fully capture quantum system dynamics and naturally facilitate non-Hermitian quantum investigations. It was also predicted that quantum jumps could give rise to third-order LEPs in two-level quantum systems for its high dimensional Liouvillian superoperator, which, however, has never been experimentally confirmed until now. Here we report the first observation of the third-order LEPs emerging from quantum jumps in an ultracold two-level trapped-ion system. Moreover, by combining decay with dephasing processes, we present the first experimental exploration of LEPs involving combinatorial effect of decay and dephasing. In particular, due to non-commutativity between the Lindblad superoperators governing LEPs for decay and dephasing, we witness the movement of LEPs driven by the competition between decay and dephasing occurring in an open quantum system. This unique feature of non-Hermitian quantum systems paves a new avenue for modifying nonreciprocity, enhancing precision measurement, and manipulating topological dynamics by tuning the LEPs.

Figures (5)

• Figure 1: Principle of the experiment for observing the movement of Liouvillian exceptional lines and points. (a) Transition from a second-order HEP to third-order LEPs with quantum jump. The second-order HEP (black) splits into two third-order LEPs (red) and three distinct second-order Liouvillian exceptional lines (LELs in purple). (b) Variation of the second-order and third-order LEPs with respect to $\alpha$. The colored surfaces are real parts of the complex eigenenergy Riemann surfaces with the orange, blue and green surfaces corresponding, respectively, to the eigenenergies $E_{1}$, $E_{2}$ and $E_{3}$. Three distinct second-order LELs, depicted in purple, are formed by the second-order LEPs, the third-order LEPs are localized at the crossing points where two of these second-order LELs intersect. (c) Schematic of the experimental setup with a single ultracold $^{\mathrm{40}}\mathrm{Ca}^{+}$ ion confined in a surface electrode trap (SET). AOM: acousto-optic modulator. AWG: arbitrary waveform generator. The AWG generates noise signals that are injected into the 729nm laser beam, which brings in dephasing. Another AOM is employed to control the power of the 854nm laser, which leads to decay. (d) Level scheme of the ion, where the lefthand side of the equator illustrates the experimental system with the mixture of a pure decay and a pure dephasing and the righthand side is for the equivalent two-level model. The solid lines with double arrows denote the transitions with Rabi frequencies driven by 729nm and 854nm lasers, and $\Delta$ is the detuning of the 729nm laser from the two-level resonance.
• Figure 2: Movement of second-order LEPs. (a) The real part and (b) the imaginary part of the complex eigenenergy Riemann surfaces when $\Delta=0$. The red and purple lines represent, respectively, the trajectories of the second-order LEP's movement dominated by dephasing and decay. Since the trajectories are largely hidden by Riemann surfaces, we draw short arrows for guiding the eye. (c) The red and purple lines are theoretical results of the trajectories of the LEP's movement. Dots with error bars stand for the experimentally observed second-order LEPs with different $\alpha$. (d-e) The orange, blue and green dots with error bars represent (d) the real and (e) the imaginary parts of the eigenvalues acquired from experimental measurement. The lines are calculated from master equation. Red (purple) area in each panel represents the movement range of the Lindblad LEP with $\alpha$ varying from $\alpha=0~(1)$ (i.e., the LEP due to dephasing or decay , labeled by a vertical dashed line) to the value labeled on the top of the panel. $E_{1}$ (orange line) and $E_{3}$ (green line) turn to be degenerate before the LEP is reached. For clarity, we replace the green solid line by the dashed line. The error bars are standard deviation representing the statistical errors of 14 000 measurements for each data point.
• Figure 3: Movement of third-order LEPs. (a) The real part and (b) the imaginary part of the complex eigenenergy Riemann surfaces when $\Delta/\Omega=1/\sqrt{8}$. The red and purple lines represent, respectively, the trajectories of the third-order LEP's movement dominated by dephasing and decay. Since the trajectories are largely hidden by Riemann surfaces, we draw short arrows for guiding the eye. (c) The red and purple lines are theoretical calculation of the trajectories of the LEP's movement. Dots with error bars stand for the experimentally observed third-order LEPs with different $\alpha$. In order to compare with the second-order LEPs, we added red and purple dashed lines when $\Delta=0$. (d-e) The orange, blue and green dots with error bars are (d) the real and (e) the imaginary parts of the eigenvalues acquired from experimental measurement. The lines are calculated from master equation. Red or purple area in each panel represents the movement range of the Lindblad LEP with $\alpha$ varying from $\alpha=0$ (i.e., the LEPs due to dephasing, labeled by a vertical dashed line) to the value labeled on the top of the panel. $E_{1}$ (orange) and $E_{3}$ (green) are degenerate before reaching LEP, while after crossing the LEP, $E_{1}$ (orange) and $E_{2}$ (blue) become degenerate when $\alpha<0.5$, and $E_{2}$ (blue) and $E_{3}$ (green) become degenerate when $\alpha>0.5$. For clarity, we replace some overlapped solid lines by dashed lines. The error bars represent standard deviation, i.e., the statistical errors of 14 000 measurements for each data point.
• Figure 4: Measured dependence of the effective decay rate $\gamma_0$ on the 854nm laser power in our experimental system. The data are well described by a cubic polynomial fit of the form $ax^2 + bx + c$, with a coefficient of determination $R^2 = 0.9996$. The fitted coefficients (95% confidence intervals) are: $a = -0.0643$, $b = 1.30$, and $c = 0.244$. The inset shows a representative decay curve measured at the 854-nm laser power of 3.86 $\mu$W. Each data point is obtained by averaging over 200 experimental repetitions.
• Figure 5: Measured dependence of the dephasing rate $\gamma_\phi$ on the amplitude of the applied white noise, specified in peak-to-peak voltage (Vpp). The data are well described by a cubic polynomial fit of the form $ax^2 + bx + c$, with the determination coefficient of $R^2 = 0.9800$. The fitted coefficients (95% confidence intervals) are: $a = 0.0882$, $b = 0.00940$, and $c = -0.0201$. The left and right insets display representative fits of the dephasing-induced population decay measured at white noise amplitudes of 3.5 V. Each data point is obtained by averaging over 200 experimental repetitions.