Observation of $\mathcal{PT}$-symmetric quantum coherence in a single-ion system

Parity-time ($\mathcal{PT}$)-symmetric systems, featuring real eigenvalues despite their non-Hermitian nature, undergo spontaneous $\mathcal{PT}$ symmetry breaking at the exceptional point (EP). The phase transition from $\mathcal{PT}$ symmetry to $\mathcal{PT}$ symmetry breaking has shown exotic functionalities in the classical realm, such as loss-induced transparency or lasing revival. This transition is also expected to produce a more unconventional effect in pure quantum $\mathcal{PT}$ devices. Here, we report experimental evidences of spontaneous $\mathcal{PT}$ symmetry breaking in a single cold ${}^{40}\mathrm{Ca}^{+}$ ion and, more importantly, a counterintuitive effect of perfect quantum coherence occurring at the EP. Excellent agreement between experimental results and theoretical predictions is identified. In view of the versatile role of cold ions in building a quantum memory or processor, our experiment provides an alternative platform to explore $\mathcal{PT}$-symmetric physics and utilize pure quantum EP effects.

Figure 1

The experimental system of a passive $\mathcal{P}T$-symmetric single ion. (a) Schematic diagram of the experimental setup. In the experiment, both 729- and 854-nm laser beams are switched on at the same time, and the quantum states of the ion at different times are read out by the electron shelving. Here, AOM denotes the acousto-optical modulator, PMT is the photomultiplier tube, and AWG is the arbitrary waveform generator. (b) The photograph of the ion trap. (c, d) The energy levels of the ${}^{40}\mathrm{Ca}^{+}$ ion, with the internal states $|0\rangle , |1\rangle$, and $|2\rangle$ corresponding to the energy levels ${}^2(m_J=-1/2), {}^2(m_J=+1/2)$, and ${}^2(m_J=+1/2)$, respectively. (e) The eigenvalues of $H_{\mathcal{P}T}$ vs $\gamma /\mathrm{\Omega }$. The projection on the back (dotted lines) and the left (thick lines) show the evolution of the imaginary parts and real parts, respectively. The projection on the bottom shows the evolution of the eigenfrequencies in the complex plane, and EP corresponds to $\gamma /\mathrm{\Omega }=1$.

Figure 2

(a, b) Dynamics of the $\mathcal{P}T$-symmetric system in the initial state $|0\rangle$ in the PTS phase (a) and PTB phase (b), respectively. The parameters are $\mathrm{\Omega }=2\pi \times 32\mathrm{kHz}$ and $\gamma =2\pi \times 1\mathrm{kHz}$ (a) and $\gamma =2\pi \times 47\mathrm{kHz}$ (b). (c, d) The dynamics of ${\rho }_{00}\left(t\right)$ of the experimental TLS for two phases show oscillatory (c) to steady-state behavior (d). (e) ${\rho }_{00}$ at a fixed time $t=2\pi /\mathrm{\Omega }$ vs the loss rate $\gamma$ with $\mathrm{\Omega }=2\pi \times 32\mathrm{kHz}$. The circles or marks are the experimental data, while the lines are from the theoretical fits. The error bars are the standard deviation of the measurements.

Figure 3

The order parameters ${\mathrm{\Sigma }}_Z$ (a) and ${\mathrm{\Sigma }}_Y$ (b) vs $\gamma$, with $\mathrm{\Omega }=2\pi \times 32\mathrm{kHz}$ and the initial state $|0\rangle$. The error bars indicate the estimated error from the fit to the dynamical equation for $\rho \left(t\right)$.