Dynamically encircling an exceptional point in a real quantum system

Quantum systems driven by non-Hermitian Hamiltonians exhibit exotic properties comparing to those governed by Hermitian HamiltoniansNP_Ganainy ; NPhoton_Feng . One of the distinct features is the existence of exceptional point (EP), at which both the eigenvalues and the corresponding eigenvectors coalesceCzech.J.Phys_Berry . Classical systems, such as optics and photonics, have provided fertile grounds to investigate EP-related novel phenomena and applicationsScience_Miri ; Nat.Mater_Ozdemir , such as single mode lasers with gain and lossScience_feng ; Science_Hodaei ; Science_Peng , unidirectional invisibilityPRL_Lin ; Nat.Mater_Feng , EP-enhanced mode splittingPRL_Wiersig ; Nature_Chen ; Nature_Hodael ; PRL_Liu and many othersNature_Regensburger ; PRL_Regensburger ; NatMater_Weimann ; NP_Peng . These fascinating processes were realized at or near an EP, but some other important properties of the EP can only be revealed when it is encircled in the parameter space. When the EP is encircled in a quasistatic manner, the two eigenvalues and the corresponding eigenvectors swap with each otherEur.phys_Heiss ; PRE_Heiss ; PRL_Dembowski ; Nature_Gao ; PRX_Ding . The effects due to dynamically encircling EPs have been investigated in classical systemsNature_Doppler ; Nature_Xu ; Nature_Yoon ; PRX_Zhang . Mode switching will emerge when the start points locate at different parity-time-symmetric ($\mathcal{PT}$-symmetric) phasesPRL_Lefebvre ; PRL_Atabek ; JPA_Uzdin ; JPA_Berry ; PRA_Gilary ; PRA_Thomas ; PRL_Hassan ; PRR_Pick . These mode switches are expected to play an important role in quantum controlPRR_Pick .

We take the following non-Hermitian Hamiltonian as an example to describe the mode switching phenomenon:

where $\gamma$ is a constant number chosen as $\gamma=1$, $\delta$ and $g$ are time-dependent real numbers. Eigenenergies of this Hamiltonian are $E_{\pm}=\pm\sqrt{g^{2}+\delta^{2}/4-1+i\delta}$, and a pair of EPs arises when $\delta=0$ and $g=\pm 1$. For different $\delta$ and $g$, the real and imaginary parts of $E_{\pm}$ are shown in Fig. 1, which displays a complex eigenvalue topology of two intersecting Riemann sheets wrapped around the EP. Dynamically encircling the EP in the parameter space is realized by the time-dependent parameters, $\delta(t)=0.5\sin[\theta(t)+\theta_{0}]$ and $g(t)=1+0.5\cos[\theta(t)+\theta_{0}]$, where $\theta(t)=\omega t$ is the encircling angle and $\theta_{0}$ defines the start point. The encircling direction is decided by the sign of $\omega$ as shown by the red arc with arrow. In Fig. 1a and b, $\theta_{0}=0$ and the start points locate at the $\mathcal{PT}$-symmetric phase. The encircling initial state is prepared to $|\alpha_{A}\rangle$ or $|\beta_{A}\rangle$ which is the eigenstate of the initial Hamiltonian at start point A. When the encircling direction is clockwise, both $|\alpha_{A}\rangle$ and $|\beta_{A}\rangle$ will evolve to $|\beta_{A}\rangle$ (Fig. 1a). When the encircling direction is counterclockwise, both $|\alpha_{A}\rangle$ and $|\beta_{A}\rangle$ will evolve to $|\alpha_{A}\rangle$ (Fig. 1b). The encircling final state depends on the encircling direction, which is taken as the asymmetric mode switching. In Fig. 1c and d, $\theta_{0}=\pi$ and the start points locate at the $\mathcal{PT}$-symmetry broken phase. The encircling initial state is prepared to $|\alpha_{B}\rangle$ or $|\beta_{B}\rangle$ which is eigenstate of the initial Hamiltonian at start point B. Whether the encircling direction is clockwise (Fig. 1c) or counterclockwise (Fig. 1d), both $|\alpha_{B}\rangle$ and $|\beta_{B}\rangle$ will evolve to $|\alpha_{B}\rangle$. This exhibits the symmetric mode switching.

Experimental investigations of phenomena via dynamically encircling EPs have been realized in classical systemsNature_Doppler ; Nature_Xu ; Nature_Yoon ; PRX_Zhang , and the research has remained elusive in the quantum domain. This is because the Hamiltonian of a closed quantum system is Hermitian while the evolution of dynamically encircling an EP is governed by the time-dependent non-Hermitian Hamiltonian. Recently, there have been experiments investigating dynamics of individual quantum systems driven by non-Hermitian HamiltoniansScience_Wu ; NP_M.Naghiloo ; PRB_Partanen . In these works, the Hamiltonians are time-independent, which prevent further investigating important physics driven by time-dependent non-Hermitian Hamiltonians. To overcome this obstacle, we have utilized a dilation method to realize time-dependent non-Hermitian Hamiltonians. Then dynamically encircling an EP has been realized in a single-spin system and mode switches have been observed.

The procedure of dynamically encircling an EP is described by the Schr$\ddot{\mathrm{o}}$dinger equation $i\partial_{t}|\psi(t)\rangle=H_{s}(t)|\psi(t)\rangle$. We introduce an ancilla qubit to construct the dilated state $|\Psi(t)\rangle=|\psi(t)\rangle\otimes|-\rangle+\eta(t)|\psi(t)\rangle\otimes|+\rangle$ (unnormalized for convenience), where $|-\rangle=(|0\rangle-i|1\rangle)/\sqrt{2}$ and $|+\rangle=-i(|0\rangle+i|1\rangle)/\sqrt{2}$ form an orthogonal basis of the ancilla qubit, and $\eta(t)$ is an appropriate operator given by the dilation method (see Supplementary Note II for details). The evolution of $|\psi(t)\rangle$ is embedded in the subspace of $|\Psi(t)\rangle$ where the state of the ancilla qubit is $|-\rangle$. The state $|\Psi(t)\rangle$ is governed by the Hermitian Hamiltonian, $H_{s,a}(t)$, which can be flexibly designed according to practical quantum systems. Here we choose

where $\Lambda(t)=\{H_{s}(t)+ib(t)I+[i\frac{d}{dt}\eta(t)+\eta(t)H_{s}(t)+ib(t)\eta(t)]\eta(t)\}M^{-1}(t)$, $\Gamma(t)=i[H_{s}(t)\eta(t)-\eta(t)H_{s}(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t)$, $M(t)=\eta^{\dagger}(t)\eta(t)+I$ and $b(t)$ is a real function (see Supplementary Note II for details). During the encircling process, the inner product of the encircling state, $\langle\psi(t)|\psi(t)\rangle$, covers several orders of magnitude since the evolution under the non-Hermitian Hamiltonian $H_{s}(t)$ is trace non-conservative. The probability of obtaining the ancilla qubit state $|-\rangle$, $P_{-}(t)$, will be small during some periods of evolution, which makes it difficult to continuously monitor the state evolution of dynamically encircling the EP though selecting the state $|-\rangle$ of the ancilla qubit (see Supplementary Note III for details). To solve this problem, we introduce an alternative measurement method. The dilated state can also be rewritten as $|\Psi(t)\rangle=[I-i\eta(t)]|\psi(t)\rangle\otimes|0\rangle-i[I+i\eta(t)]|\psi(t)\rangle\otimes|1\rangle$. The state of the system qubit is measured when the ancilla qubit state is $|0\rangle$, which is $|\chi(t)\rangle=[I-i\eta(t)]|\psi(t)\rangle$. Then the state $|\psi(t)\rangle$ is obtained by multiplying $[I-i\eta(t)]^{-1}$ to the state $|\chi(t)\rangle$ with proper renormalization.

Our scheme is demonstrated in a single nitrogen-vacancy ($\mathrm{NV}$) center in diamond (Fig. 2a). With a static magnetic field applied along the NV symmetry axis, the Hamiltonian of the $\mathrm{NV}$ center can be written as $H_{\mathrm{NV}}=2\pi(DS_{z}^{2}+\omega_{e}S_{z}+QI_{z}^{2}+\omega_{n}I_{z}+AS_{z}I_{z})$, where $D=2.87$ GHz is the zero-field splitting of the electron spin, $Q=-4.95$ MHz is the nuclear quadrupolar interaction, and $A=-2.16$ MHz is the hyperfine coupling between the electron spin and the nuclear spin. $\omega_{e}$ ($\omega_{n}$) denotes the Zeeman splitting of the electron (nuclear) spin. $S_{z}$ and $I_{z}$ are the spin-1 operators of the electron spin and the nuclear spin, respectively. The subspace spanned by $|0\rangle_{e}|1\rangle_{n},|0\rangle_{e}|0\rangle_{n},|-1\rangle_{e}|1\rangle_{n}$ and $|-1\rangle_{e}|0\rangle_{n}$ (black box in Fig. 2b) is encoded as a two-qubit system to construct Hamiltonian $H_{s,a}(t)$. The electron spin is chosen as the system qubit and the nuclear spin is selected as the ancilla qubit. By decomposing $\Lambda(t)$ and $\Gamma(t)$ in terms of Pauli operators, we can rewrite the Hamiltonian $H_{s,a}(t)$ in Eq.2 as $H_{s,a}(t)=\sum_{i=0}^{3}[A_{i}(t)\sigma_{i}\otimes I+B_{i}(t)\sigma_{i}\otimes\sigma_{z}]$, where $A_{i}(t)=\mathrm{Tr}[\Lambda(t)\cdot\sigma_{i}]/2$, $B_{i}(t)=\mathrm{Tr}[\Gamma(t)\cdot\sigma_{i}]/2$ are the decomposition coefficients, and $\sigma_{i}=I,\sigma_{x},\sigma_{y},\sigma_{z}$ for $i=0,1,2,3$ are Pauli operators. Two selective microwave pulses (red arrows in Fig. 2b) are applied to realize the Hamiltonian $H_{s,a}(t)$. The Hamiltonian of the pulses in the two-qubit system can be written as

The dilated Hamiltonian $H_{s,a}(t)$ can be constructed in an interaction picture when the amplitudes, angular frequencies and the phases of the two pulses satisfy the following relations, $\Omega_{1,2}(t)=2\sqrt{[A_{1}(t)\pm B_{1}(t)]^{2}+[A_{2}(t)\pm B_{2}(t)]^{2}}$, $\omega_{1,2}(t)=\omega_{\mathrm{MW}1,2}+2A_{3}(t)\pm 2B_{3}(t)$, and $\phi_{1,2}(t)=-\mathrm{arctan2}[A_{2}(t)\pm B_{2}(t),A_{1}(t)\pm B_{1}(t)]$ (see Supplementary Note IV for details).

The experiment was implemented on an optically detected magnetic resonance setup. The static magnetic field was set to 500 Gauss in order to polarize the NV center to state $|0\rangle_{e}|1\rangle_{n}$ by optical pumpingPRL_V.Jacques . When we choose $\eta(0)=\eta_{0}\cdot I$, the initial state of the two-qubit system has the form $|\Psi(0)\rangle=|\psi(0)\rangle\otimes(|-\rangle+\eta_{0}|+\rangle)$. The encircling initial state $|\psi(0)\rangle$ was set to the eigenstate, $|\alpha_{A}\rangle$ or $|\beta_{A}\rangle$ ($|\alpha_{B}\rangle$ or $|\beta_{B}\rangle$), of the initial Hamiltonian at start point A (B). $|\Psi(0)\rangle$ was prepared by rotation $R_{\widehat{n}_{1}}(\pi/2)$ on the nuclear spin followed by rotation $R_{\widehat{n}_{2}}(\zeta)$ on the electron spin (Fig. 2c), where $\widehat{n}_{1}=(1-\eta_{0}^{2},2\eta_{0},0)/(1+\eta_{0}^{2})$, $\widehat{n}_{2}$ and $\zeta$ depended on the choice of $|\psi(0)\rangle$. Then the system evolved under the dilated Hamiltonian $H_{s,a}(t)$ by applying two selective $\mathrm{MW}$ pulses given in Eq.3 to realize the dynamically encircling procedure. The total encircling time is set to be $T=15~{}\mu s$. Finally, a state tomography is implemented to measure the state $|\chi(t)\rangle$ when the ancilla qubit state is $|0\rangle$. The state $|\psi(t)\rangle$ is obtained by multiplying the operator $[I-i\eta(t)]^{-1}$ on the state $|\chi(t)\rangle$ with the maximum likelihood estimationPRA_Fiurasek and all the experimental fidelities of $|\chi(t)\rangle$ and $|\psi(t)\rangle$ in our experiment are close to 1 (see Supplementary Note III for details).

Fig. 3 displays the results of an asymmetric mode switching when the start point locates at the $\mathcal{PT}$-symmetric phase. The horizontal axes in Fig. 3 and 4 are rotation angle $\theta$ for clearly demonstration of the encircling process, where $\theta=\omega t$ with $\omega=\pm 2\pi/15~{}\mathrm{rad}\cdot\mu$s^-1. Fig. 3a and b show the cases of clockwise encircling. In Fig. 3a, we prepared the encircling initial state, $|\psi(0)\rangle$, to eigenstate $|\alpha_{A}\rangle$. The overlaps of $|\psi(0)\rangle$ with $|\alpha_{A}\rangle$ and $|\beta_{A}\rangle$ were $1.00(2)$ and $0.64(2)$, respectively. The non-zero overlap between $|\psi(0)\rangle$ and $|\beta_{A}\rangle$ is due to the non-orthogonality of the two eigenstates of $H_{s}(0)$. Then the encircling state followed the evolution of dynamically encircling the EP characterized by the overlaps with $|\alpha_{A}\rangle$ and $|\beta_{A}\rangle$. After the encircling process, the overlap between the encircling final state and $|\beta_{A}\rangle$ was $1.00(2)$. This shows that eigenstate $|\alpha_{A}\rangle$ will switch to $|\beta_{A}\rangle$ after encircling around the EP, as clarified by the sketch on the diagram. In Fig. 3b, the state was first prepared to $|\beta_{A}\rangle$ and the results show that it returned to $|\beta_{A}\rangle$ after the encircling process. The counterclockwise encircling cases are shown in Fig. 3c and d. The initial states were prepared to $|\alpha_{A}\rangle$ and $|\beta_{A}\rangle$ in Fig. 3c and d, respectively. After encircling the EP, the encircling states both evolved to $|\alpha_{A}\rangle$. All the measured evolutions in Fig. 3 show good agreement with the theoretical predictions. These results exhibit that the encircling initial state doesn’t influence the encircling final state. The encircling final state depends on the encircling direction. The evolutions displayed in Fig. 3 unambiguously certifies an asymmetric mode switching when the evolution starts from the $\mathcal{PT}$-symmetric phase.

Fig. 4 exhibits the results of a symmetric mode switching when the start point locates at the $\mathcal{PT}$-symmetry broken phase. The overlaps between the encircling state and the eigenstates, $|\alpha_{B}\rangle$ and $|\beta_{B}\rangle$, were used to describe the evolution. Fig. 4a and b show the cases of clockwise encircling and the initial states were prepared to $|\alpha_{B}\rangle$ and $|\beta_{B}\rangle$, respectively. The counterclockwise encircling cases are presented in Fig. 4c and d. For all these cases, the final states evolved to $|\alpha_{B}\rangle$ after the encircling process regardless of the initial state and the path direction. The experimental results agree well with the corresponding theoretical predictions. These results indisputably reveal a symmetric mode switching when initial Hamiltonian is $\mathcal{PT}$-symmetry broken.

In conclusion, dynamically encircling an EP in a single-spin system has been realized to explore the topological structure of the EP, and both the symmetric and asymmetric mode conversions have been observed. These mode switches provide a robust fashion to control quantum states and thus have great potential in quantum information precessingPRR_Pick . Furthermore, our successful demonstration of engineering time-dependent non-Hermitian Hamiltonians opens a door towards future investigation of complicated dynamical processes governed by time-dependent non-Hermitian Hamiltonians in the quantum realm, such as, encircling high-order EPsJ.Phys.A_Demange ; PRA_Schnabel , researching non-Hermitian geometric phaseJ.Math.Phys_Hossein ; PRA_Gong and studying non-Hermitian topological invariantsJ.Phys_Ananya .