Observation of the Knot Topology of Non-Hermitian Systems in a Single Spin

The non-Hermitian (NH) Hamiltonian has drawn intensive attention RMP_Bergholtz ; PRL_NHtopo1 ; PRX_NHtopo1 ; PRX_NHtopo2 ; Science_NHtopo ; PRL_NHtopo2 ; PRA_NHtopo due to its application in photonic Science_NHApli1 ; NP_NHApli ; Science_NHApli2 ; Science_NHApli3 ; Nature_PhotoNH ; Science_PhotoNH1 ; PRL_PhotoNH1 ; PRL_PhotoNH2 ; PRL_PhotoNH3 ; NaturePhotonics_PhotoNH ; NP_PhotoNH and acoustic NHAcous_Romain ; NHAcous_Tuo ; NHAcous_Zhi systems with gain and loss, open quantum systems JPA_OpenNH ; Nature_OpenNH ; PRL_OpenNHCar ; NP_OpenNHDiehl and condensed matter systems of quasi-particles with finite lifetime Arxiv_CondNH ; PRB_CondNH1 ; PRB_CondNH2 ; PRL_CondNH . In contrast to Hermitian systems, topological structures that are exclusive to NH systems give birth to many intriguing phenomena like NH bulk boundary correspondence PRL_Kai ; PRL_Dan and NH skin effects PRL_Nobuyuki . The emergence of NH nodal phases and gapped phases as well as their interplay with symmetry also adds new ingredients for NH topological band theory RMP_Bergholtz . Understanding the topology introduced by NH physics helps one to explore topologically robust quantities and non-trivial edge states. All these uncommon phenomena provide insights into NH systems.

More recently, researchers have found a framework based on homotopy theory PRB_Fan ; PRL_Haiping ; PRB_ZhiLi to classify the non-Hermitian topological phases. Compared with the K-theory PRX_NHtopo1 , this classification method does not need to assume specific symmetries and can reveal more topological invariants. To be precise, the Hamiltonian can be regarded as a mapping from the Brillouin zone to the space of energy bands and eigenstates. Thus the classification is done by finding all topologically non-equivalent mappings. Each class of mapping corresponds to a topological phase with a specific topological invariant. The non-Hermiticity of the Hamiltonian can generate complex eigenvalues, which can constitute the knot structures. The whole classification set of NH systems with knot topology is decomposed into several sectors, based on the braiding of eigenvalues, and each sector can be further classified with the eigenstate topology PRB_ZhiLi .

There are several experimental studies recently aiming at revealing the knot topology of the NH systems. For example, the braiding structure of energy bands has been observed in optical Nature_Fan , mechanical Nature_Harris and trapped ion PRL_Cao systems. A comprehensive demonstration of the knot topology, especially the essential information from the eigenstate topology requires coherent evolution under the NH Hamiltonian. It is difficult to realize NH Hamiltonians in quantum systems since closed systems are generally governed by Hermitian Hamiltonians. Apart from the difficulty in realizing a NH Hamiltonian, the inevitable decoherence of the quantum system tends to destroy the coherent evolution. In order to overcome these difficulties, we applied the universal dilation method Science_YW ; PRL_Wengang ; PRL_Uwe ; PRL_Kohei to realize the NH Hamiltonians and fabricated a 99.999% ${}^{\text{12}}$C isotope purified sample in which the decoherence had been significantly suppressed. Taking the one-dimensional (1D) NH lattice model as an example, we experimentally studied the knot topology in the NH Hamiltonian by measuring not only its energy eigenvalues, but also the corresponding eigenstates to reveal the global Berry phase. Compared with the NH topology shown in previous work PRL_Wengang on the NV center, which stems from encircling varying numbers of exceptional points, the knot topology investigated in our work arises from both the braiding of all energy bands and the eigenstate topology. Our platform is capable of dealing with other models and our results shed light on the complexity of topological features aroused by non-Hermiticity.

Here we consider a two-band NH 1D lattice model (Fig. 1(a)) whose Hamiltonian may be written as

where $a_{j}\ (b_{j})$ is the annihilation operator of sublattice $a\ (b)$ on the $j$th site, and the $\Gamma$’s are complex numbers representing the coupling strengths, i.e., $\Gamma^{0\pm}$ is the on-site interaction and $\Gamma_{1(2)}^{n\pm}$ is the hopping between sublattice $a\ (b)$ at site $j$ and lattice $b\ (a)$ at site $j+n$. Under the periodic boundary condition, the corresponding momentum space Hamiltonian is

This model holds plenty of topological phases under different parameter configurations that ensure bands being separable. For $m=1$, i.e., the nearest coupling case, the system may be in the unlink phase or the unknot phase. Because $k=0$ and $2\pi$ are equivalent due to the periodic condition, the energy bands form closed curves. When the parameters are set as $\Gamma^{0-}=-0.45,$ $\Gamma^{0+}=0.79,$ $\Gamma^{1-}_{1}=-0.30i,$ $\Gamma^{1-}_{2}=0.08i,$ and $\Gamma^{1+}_{1,2}=0$, the non-Hermitian system is in the unlink phase, where the two circles are unrelated, as shown by Fig. 1(b). If the parameters are changed to $\Gamma^{0-}=-0.21,$ and $\Gamma^{0+}=0.70$, the system will be in the unknot phase. There is only one circle because the two bands exchange (Fig. 1(c)). For $m=2$, the next nearest coupling enriches the topological phases in addition to the unlink and unknot. When the parameters are set as $\Gamma^{0-}=0.04,$ $\Gamma^{0+}=0.49,$ $\Gamma^{1-}_{1}=\Gamma^{1+}_{2}=-0.13i,$ $\Gamma^{1-}_{2}=\Gamma^{1+}_{1}=0.02i,$ $\Gamma^{2-}_{1}=-0.58i,$ $\Gamma^{2+}_{2}=-0.21i,$ $\Gamma^{2+}_{1}=0.03i,$ and $\Gamma^{2-}_{2}=0.09i$, the two eigenvalues form a non-trivial braiding structure. The energy bands braid around each other exactly once and this gives the Hopf link phase (Fig. 1(d)). The eigenstates also exhibit behaviors similar to those of the corresponding energy bands as shown in Fig. 1(e), 1(f) and 1(g). For larger $m$, the energy bands may braid more times, which leads to more phases. The classification based on homotopy theory has shown that all phases of this model are described by the braiding group $B(2)$ PRB_Fan ; PRB_ZhiLi .

We now study the topological phases of our model and the corresponding topological invariants in a quantum system based on the eigenvalues and eigenstates. Both the eigenvalues and the eigenstates can be obtained from the evolution under the NH Hamiltonian $H_{s}(k)=H^{(m)}(k)$. Let $\ket{\psi_{1,2}(k)}$ (the $k$ dependence is omitted for simplicity in the following) be the eigenstates with complex eigenvalues $E_{1,2}=E^{r}_{1,2}+iE^{i}_{1,2}$, where $E^{r}_{n}$ and $E^{i}_{n}$ are real and imaginary parts of $E_{n}$, respectively. For any initial state written as $\ket{\psi(0)}=c_{1}\ket{\psi_{1}}+c_{2}\ket{\psi_{2}}$, the evolution governed by $H_{s}(k)$ gives $\ket{\psi(t)}\propto c_{1}e^{-iE^{r}_{1}t+E^{i}_{1}t}\ket{\psi_{1}}+c_{2}e^{-iE^{r}_{2}t+E^{i}_{2}t}\ket{\psi_{2}}$. Without loss of generality, we assume $E^{i}_{1}>E^{i}_{2}$. The eigenvalues can be extracted from the time evolution of the populations of $\ket{\psi_{1,2}}$, and the steady state of NH evolution will be the eigenstate $\ket{\psi_{1}}$ since $E^{i}_{1}$ is larger than $E^{i}_{2}$. By implementing evolution governed by $-H_{s}$, the eigenstate $\ket{\psi_{2}}$ can be obtained, which is due to $-E^{i}_{2}>-E^{i}_{1}$ for $-H_{s}$.

The evolution under the NH Hamiltonian $H_{s}(k)$ can be realized based on the universal dilation method. The state of the system $\ket{\psi(t)}$ needs to evolve as $i\partial_{t}\ket{\psi(t)}=H_{s}(k)\ket{\psi(t)}$. By introducing an ancilla, the NH evolution can be realized in a subspace while the total Hamiltonian $H_{\rm tot}$ is Hermitian. The initial state that reads $\ket{0}_{s}\ket{-}_{a}+\eta(0)\ket{0}_{s}\ket{+}_{a}$ evolves to $\ket{\psi(t)}_{s}\ket{-}_{a}+\eta(t)\ket{\psi(t)}_{s}\ket{+}_{a}$ under the Hamiltonian $H_{\rm tot}$, where $\ket{\pm}$ are eigenstates of $\sigma_{y}$ and $\eta(t)$ is a properly chosen time-dependent operator. Thus in the $\ket{-}_{a}$ subspace, apart from a normalization constant, the evolution is strictly governed by $H_{s}(k)$.

We use a single nitrogen-vacancy (NV) center in diamond to experimentally realize the momentum space NH Hamiltonian for different phases with $k\in[0,2\pi]$ (see Appendix A for details of the experimental setup). The NV center is a type of point defect in diamond that is composed of a nitrogen atom and a neighbor vacancy as shown in Fig. 2(a). The ground-state of the NV center is a triplet state with an electronic spin $S=1$ that interacts with the nuclear spin $I=1$ of the substitutional ${}^{\text{14}}$N. We construct the dilated Hamiltonian $H_{\rm tot}$ in the subspace spanned by the states $\ket{m_{S},m_{I}}=\ket{0,1},\ket{0,0},\ket{-1,1}$, and $\ket{-1,0}$ as shown in Fig. 2(b). The energy levels are relabeled as $\ket{1}_{e}\ket{1}_{n},\ket{1}_{e}\ket{0}_{n},\ket{0}_{e}\ket{1}_{n}$, and $\ket{0}_{e}\ket{0}_{n}$, respectively. The electron spin is chosen to be the system and nuclear spin serves as the ancilla.

The pulse sequence used to realize the evolution under the NH Hamiltonian is shown in Fig. 2(d). The external static magnetic field was set to be 506 G. The state of the NV center was polarized to $\ket{0}_{e}\ket{1}_{n}$ by 532-nm laser pulses Polarization . After polarization, the initial state is prepared by the RF $\pi/2$ pulse $R_{\phi}(\pi/2)$, where the rotation axis lies in the XY plane and $\phi=$atan$[(\eta_{0}^{2}-1)/2\eta_{0}]+\pi/2$ is the angle between the rotation axis and the $x$ axis. Here we have chosen $\eta(0)=\eta_{0}I$. The evolution under $H_{\rm tot}$ is realized by applying two selective microwave (MW) pulses, and the coherent evolution should be long enough to drive the system to a steady state. To this end, $H_{s}$ is multiplied by an overall coefficient $\lambda$ to speed up the evolution and preserve the coherence. The strength of the MW pulses is proportional to the value of $\lambda$. However, too strong MW pulses may cause strong crosstalk between different subspaces. One possible solution is to use a strongly coupled ${}^{13}\text{C}$ nuclear spin. For the NV centers in diamond with ¹²C natural abundance, if one chooses $\lambda$ to maintain coherence, the system cannot reach the steady state because of crosstalk. Or if one overcomes the crosstalk by using weak MW pulses, the required evolution time is so long that the coherence will greatly decrease during evolution (see Appendix C). Therefore, it is challenging to study the knot topological features of NH systems with a ¹²C natural abundance diamond. To address this issue, we synthesized a diamond with 99.999% ${}^{\text{12}}$C isotope abundance by the chemical vapor deposition method. With this sample, the dephasing time of the NV center utilized in our experiment is $T_{2}^{\star}=78(7)$ $\upmu$s, as shown in Fig. 2(c). Such a long coherence time enables us to realize evolution under the NH Hamiltonian during which the coherence is preserved and the crosstalk is suppressed. To realize the evolution under $H_{\rm tot}$, we choose a proper rotating frame and use two selective microwave sequences with time-dependent amplitude and phase (see Appendix B). The eigenvalues are extracted from the population information by setting $R_{i}=I$ in the measurement sequence. The eigenstates of the NH Hamiltonian are reconstructed by applying $R_{i}=I,R_{-y}(\pi/2)$ and $R_{-x}(\pi/2)$ to the final state of evolution. Before readout of the photoluminescence ratio, we apply an RF $\pi/2$ pulse, which changes the basis of the ancilla from $\ket{\pm}_{n}$ to $\ket{0,1}_{n}$. Finally, selective $\pi$ pulses are applied to reverse the population of electrons or nuclear spin to get a set of equations that relate photoluminescence intensities to populations of each level. Solving this equation, the populations are obtained.

Both eigenvalues and eigenstates can be obtained from the measurement sequences. When $R_{i}=I$, we obtain a set of populations at different times by varying the evolution time under $H_{\rm tot}$. Renormalizing the population as $P_{1}=P_{\ket{1}_{e}\ket{1}_{n}}/(P_{\ket{1}_{e}\ket{1}_{n}}+P_{\ket{0}_{e}\ket{1}_{n}})$ gives the evolution of $P_{1}$ under the NH Hamiltonian. The corresponding quasi-momentum $k$ can be extracted by fitting $P_{1}$ under fixed model parameters. Figure 2(e) shows the result for the Hopf link phase with $k=0.6\pi$. The population evolution agrees with the theoretical prediction and gives $k_{\rm fit}=0.59(7)\pi$. The eigenvalues can thus be computed from $k_{\rm fit}$ and model parameters. By measuring the populations under different bases, we obtain the expectation $\braket{\sigma_{x,y,z}}$. Then we can reconstruct the steady state by $\rho=(I+\braket{\vec{\sigma}}\vec{\sigma})/2$, but the direct result may give a mixed state or even an unphysical state. Thus, we use the maximum likelihood estimation (MLE) to obtain a pure state close to the eigenstate (see Appendix D). Figure 2(f) shows the real and imaginary parts of the measured state and theoretical results for $k=0.6\pi$ in the Hopf link phase. Here $\rho_{1}=\ket{\psi_{1}}\bra{\psi_{1}}$ is the density matrix corresponding to $\ket{\psi_{1}}$. The fidelity is 99% for this state. All other states are obtained by following the same procedure and the fidelities are all higher than 97%.

The experimentally measured eigenvalues of different phases for various $k$ are plotted in Fig. 3. The $B(2)$ braiding behavior is characterized by the following definition of the winding number Nature_Fan :

This number $\nu$ reflects how many times the energy bands are braided. The term $\text{Tr}[H^{(m)}(k)]/2$ here can eliminate the dependence on the choice of reference energy Nature_Fan . So, the winding number defined in Eq. (3) only captures the mutual braiding of the eigenvalues under investigation. Every element of $B(2)$ can be expressed as $\tau^{n}$, where $n$ is an integer and $\tau$ is the generator. The one-to-one correspondence of $\nu=n$ exactly describes the $B(2)$ behavior. When $m=1,\Gamma^{0-}=-0.45,\Gamma^{0+}=0.79,\Gamma^{1-}_{1}=-0.30i,\Gamma^{1-}_{2}=0.08i,$ and $\Gamma^{1+}_{1,2}=0$, the system is in the unlink phase (Fig. 3(a)). The eigenvalues form two separated curves and $\nu=0$ in this phase. The two circles can be separated by a line, $\text{Im}(E)=0$ for example. This is just like a normal insulator in the Hermitian case. If we set the parameters as $\Gamma^{0-}=-0.21$ and $\Gamma^{0+}=0.70$, the system will be in the unknot phase, as shown by Fig. 3(b). The two bands interchange as $k$ goes from $0$ to $2\pi$ and in this phase $\nu=1$. Note that $k=0$ and $2\pi$ are equivalent points and in this phase $E_{1}(0)=E_{2}(2\pi)$. Thus two energy bands form a whole circle instead of two circles. For $m=2$, and $\Gamma^{0-}=0.04,\Gamma^{0+}=0.49,\Gamma^{1-}_{1}=\Gamma^{1+}_{2}=-0.13i,\Gamma^{1-}_{2}=\Gamma^{1+}_{1}=0.02i,\Gamma^{2-}_{1}=-0.58i,\Gamma^{2+}_{2}=-0.21i,\Gamma^{2+}_{1}=0.03i,$ and $\Gamma^{2-}_{2}=0.09i$, the eigenvalues form a non-trivial braiding pattern (Fig. 3(c)) and in this phase $\nu=2$. The two bands encircle each other and form a structure called the Hopf link. Although in this phase each band forms a circle, they cannot be separated by any line in the complex energy plane as done in the unlink phase.

Apart from the non-trivial topology of the energy bands, the global Berry phase determined by the eigenstates was also observed. The global Berry phase is $Q=\int_{0}^{2\pi}$Tr$[A(k)]dk$, where the non-abelian Berry connection is defined as $A_{mn}(k)=i\braket{\chi_{m}(k)}{\partial_{k}}{\psi_{n}(k)}$ Q_Liang . Here $\ket{\psi_{n}(k)}$ ($\ket{\chi_{m}(k)}$) are right (left) eigenstates of an NH Hamiltonian and satisfy the biorthogonal relation $\braket{\chi_{m}(k)}{\psi_{n}(k)}=\delta_{mn}$. The global Berry phase $Q$ can identify topological invariance in our model Q_Liang . For the unlink, unknot and Hopf link phases, $Q_{\rm ideal}=0,\pi$ and $2\pi$, respectively. Figure 4 shows the measured eigenstates projected on the XY plane and they show the same behavior as the eigenvalues. In the unlink phase, $\ket{\psi_{1,2}(k)}$ form two separated circles on the Bloch sphere, and so are the projections on the XY plane. In the unknot phase, the eigenstates form an end-to-end circle instead, just as the energy bands exchange each other. While in the Hopf link phase, each eigenstates form a closed loop, and the two loops intersect each other. After obtaining $\ket{\psi_{1,2}(k)}$ from MLE, $\ket{\chi_{1,2}(k)}$ can be solved from the biorthogonal relation. The experimental value of $Q$ can be obtained with the method in Ref. ComputeQ via $Q=\sum_{i,n}\text{Im}D_{i}^{n}$, where $D_{i}^{n}=\text{ln}\braket{\chi_{n}(k_{i+1})}{\psi_{n}(k_{i})}$ and $n=1$ and $2$ is the band index. The experimental results of $Q$ for the unlink, unknot and Hopf link phases are 0.00(2)$\pi$, 1.03(2)$\pi$ and 2.00(3)$\pi$, respectively, which agree well with theoretical predictions. Generally speaking, for $n$ bands models with eigenvalues $\{E_{i}\}$ and starting at $k=0$, the ordered set $(E_{1},E_{2},...,E_{n})$ goes to $(E_{\sigma(1)},E_{\sigma(2)},...,E_{\sigma(n)})$ as $k$ varies to $2\pi$. Then we have $e^{iQ}=(-1)^{P(\sigma)}$, where $P(\sigma)$ is the parity of the permutation. For our case, both the unlink phase and the Hopf link phase have even parity since each band returns to itself as $k$ goes from $0$ to $2\pi$. Thus in these two phases $Q=0~{}($mod $2\pi)$. Since the energy bands of the unknot phase exchange each other, the parity is odd with $Q=\pi~{}($mod $2\pi)$.

In conclusion, we have experimentally investigated the knot topology in an 1D NH model based on both eigenvalues and eigenstates. The knot structures of eigenvalues, including the unlink, unknot and Hopf link phases, were successfully observed, which manifest the B(2) braid group behavior. The global Berry phase was measured via high fidelity eigenstates, which serves as the knot invariant to identify the parity of band braiding. Our work makes NV center a desirable platform for investigating important non-Hermitian topology. The universality of our dilation method for arbitrary dimensional cases and the ground-state three-level structure of the NV center make it possible to explore the knot topology of three-band models. For 1D models with three bands, the knotted topological phases are described by the conjugate classes of $B(3)$ PRL_Haiping ; NatPhys_Bouhon , which host richer topological behaviors since $B(N)$ is not commutative when $N>2$. By introducing more momentum space parameters such as $\vec{k}=(k_{x},k_{y},k_{z})$, our platform can be utilized to investigate the knot topology of higher dimensional NH models PRB_Fan ; PRB_ZhiLi .

This work was supported by the National Key R&D Program of China (Grant Nos. 2018YFA0306600 and 2016YFB0501603), the National Natural Science Foundation of China (Grant No. 12174373), the Chinese Academy of Sciences (Grant Nos. XDC07000000 and GJJSTD20200001), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302200), Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000) and Hefei Comprehensive National Science Center. X. R. thanks the Youth Innovation Promotion Association of Chinese Academy of Sciences for their support. Ya Wang and Yang Wu thanks the Fundamental Research Funds for the Central Universities for their support.

Yang Wu, Yunhan Wang and Xiangyu Ye contributed equally to this work.