Dynamically Encircling an Exceptional Point in a Real Quantum System

The exceptional point, known as the non-Hermitian degeneracy, has special topological structure, leading to various counterintuitive phenomena and novel applications, which are refreshing our cognition of quantum physics. One particularly intriguing behavior is the mode switch phenomenon induced by dynamically encircling an exceptional point in the parameter space. While these mode switches have been explored in classical systems, the experimental investigation in the quantum regime remains elusive due to the difficulty of constructing time-dependent non-Hermitian Hamiltonians in a real quantum system. Here we experimentally demonstrate dynamically encircling the exceptional point with a single nitrogen-vacancy center in diamond. The time-dependent non-Hermitian Hamiltonians are realized utilizing a dilation method. Both the asymmetric and symmetric mode switches have been observed. Our Letter reveals the topological structure of the exceptional point and paves the way to comprehensively explore the exotic properties of non-Hermitian Hamiltonians in the quantum regime.

Figure 1

Asymmetric and symmetric mode switching by dynamically encircling the EP. (a)–(d) Encircling paths in the parameter space and prospective encircling trajectories in the eigenvalue Riemann sheets when the starting point locates at (a),(b) $\mathcal{P}\mathcal{T}$-symmetric phase and (c),(d) $\mathcal{P}\mathcal{T}$-symmetry broken phase. The left side of each diagram shows the encircling path. Green line denotes the Hamiltonian is in $\mathcal{P}\mathcal{T}$-symmetric phase and blue line means the Hamiltonian is in $\mathcal{P}\mathcal{T}$-symmetry broken phase. Red arc with arrow shows the encircling direction. The middle and right side of each diagram displays the prospective encircling trajectory. The red sheets represent $E_{+}$ and the blue sheets are $E_{-}$. The black lines indicate that the trajectory is on the Riemann plane, and the green lines denote that a transition between two sheets has occurred. The encircling initial states in the middle are (a),(b) $|{\alpha }_A⟩$ and (c),(d) $|{\alpha }_B⟩$, while the encircling initial states on the right side are (a),(b) $|{\beta }_A⟩$ and (c),(d) $|{\beta }_B⟩$. $|{\alpha }_A⟩$ and $|{\beta }_A⟩$ ($|{\alpha }_B⟩$ and $|{\beta }_B⟩$) are eigenstates of the initial Hamiltonian at starting point $A(B)$.

Figure 2

Realization of dynamically encircling the EP in a NV center. (a) Atomic structure of the NV center. (b) The hyperfine energy levels of the coupling system with NV electron spin and ${}^{14}\mathrm{N}$ nuclear spin. The four energy levels in the black box are utilized to form a two-qubit system. (c) Quantum circuit of the experiment. The coupling system is initialized to the state $|0{⟩}_e|1{⟩}_n$. Rotation $\pi /2$ along axis ${\widehat{n}}_1$ on the nuclear (nucl.) spin followed by rotation $\zeta$ along axis ${\widehat{n}}_2$ on the electron (el.) spin can prepare the coupling system to different initial states. Dynamically encircling the EP is realized by adding two selective microwave pulses [red arrows in (b)]. Then state tomography is implemented to measure the state $|\chi (t)⟩$ for further obtainment of the state $|\psi (t)⟩$.

Figure 3

Asymmetric mode switching by starting from $\mathcal{P}\mathcal{T}$-symmetric phase. (a),(b) Clockwise encircling the EP starting from eigenstates (a) $|{\alpha }_A⟩$ and (b) $|{\beta }_A⟩$. (c),(d) Counterclockwise encircling the EP starting from eigenstates (c) $|{\alpha }_A⟩$ and (d) $|{\beta }_A⟩$. The evolution of the encircling state is characterized by the overlaps $⟨{\alpha }_A|\psi (t)⟩$ (upper half of each diagram) and $⟨{\beta }_A|\psi (t)⟩$ (bottom half of each diagram). Black dots with error bars are experimental results, and gray lines are the simulation predications.

Figure 4

Symmetric mode switching by starting from $\mathcal{P}\mathcal{T}$-symmetry broken phase. (a),(b) Clockwise encircling the EP starting from eigenstates (a) $|{\alpha }_B⟩$ and (b) $|{\beta }_B⟩$. (c),(d) Counterclockwise encircling the EP starting from eigenstates (c) $|{\alpha }_B⟩$ and (d) $|{\beta }_B⟩$. The evolution of the encircling state is characterized by the overlaps $⟨{\alpha }_B|\psi (t)⟩$ (upper half of each diagram) and $⟨{\beta }_B|\psi (t)⟩$ (bottom half of each diagram). Black dots with error bars are experimental results, and gray lines are the simulation predications.