Observation of arbitrary topological windings of a non-Hermitian band

The discoveries of a wide variety of topological materials, with topological insulators [1, 2, 3, 4] and Weyl semimetals [5, 6] being two prominent examples, highlight the importance of topological band theory, which seeks to develop fundamental understandings of topological properties of energy band structures. Early efforts on topological band theory have focused on Hermitian Hamiltonians [7]. In recent years, motivated in part by the significant developments in topological photonics [8, 9, 10, 11, 12], where gain and loss are common features [13, 14, 15, 16], there have been significant emerging interests in developing topological band theory for non-Hermitian Hamiltonians [17, 18, 19, 20].

The energy bands of non-Hermitian Hamiltonians exhibit unique non-trivial topological features that are absent in Hermitian systems. In particular, since the energy bands of non-Hermitian Hamiltonians are in general complex, generically even a single energy band in one dimension can form a non-trivial loop in the complex plane, as characterized by integer non-zero winding numbers [18, 20]. This is in contrast with Hermitian systems, where non-trivial topology requires at least two bands, and moreover requires symmetry protection in one dimension [21]. Such non-trivial winding, which is unique to non-Hermitian systems, provides the topological underpinnings of remarkable phenomena such as the non-Hermitian skin effect [22, 23, 24, 25, 26, 27, 28] and necessitates the generalization of bulk-edge correspondence [29, 30, 31].

In spite of the central importance of energy band winding in the topological band theory of non-Hermitian systems, direct experimental observation and control of such winding have been lacking. Recent experiments have demonstrated the non-Hermitian skin effect [22, 23, 32], as well as the collapse of eigenspectrum as induced by the presence of an edge [33]. These experiments, however, only provide indirect evidence of energy-band winding in non-Hermitian systems. Moreover, strictly speaking, energy bands are defined only for infinite systems. For non-Hermitian systems, the eigenstates of a finite lattice can be qualitatively different from those in an infinite lattice. Thus, it is not a priori obvious how one can measure the band structure in experimentally feasible non-Hermitian lattices which are typically finite.

In this work, we experimentally implement a class of non-Hermitian Hamiltonians and measure its momentum-resolved complex band energy. Our implementation utilizes the concept of a synthetic dimension [34, 35, 36, 37, 38]. In particular, we use a synthetic frequency dimension [39, 40, 41, 42, 43, 44, 45] as formed by multiple frequency modes in a ring resonator under amplitude and phase modulations. Our measurement provides direct visualization of non-trivial topological band winding. Moreover, we demonstrate that the winding can be controlled by varying the parameters of the modulation, and highly complex winding as characterized by multiple different non-zero winding numbers can be achieved.

Specifically, in this paper we implement in the frequency synthetic dimension the 1D lattice Hamiltonian

where $\mathbf{a_{n}^{\dagger}}$ ($\mathbf{a_{n}}$) is the creation (annihilation) operator of the $n$-th lattice site with $m=1,2,\dots,M$ running over the coupling orders. This Hamiltonian becomes non-Hermitian if $\kappa_{+m}\neq\kappa_{-m}^{\ast}$ for some $m$ in Eq. (S26). A general non-Hermitian form of the coupling constant $\kappa_{\pm m}$ can be given by $\kappa_{\pm m}=C_{m}e^{\pm i\alpha_{m}}\pm\Delta_{m}e^{\pm i\beta_{m}}$ with the coupling strengths $C_{m},\Delta_{m}\geq 0$ and phases $\alpha_{m},\beta_{m}\in(-\pi,\pi]$, where $C_{m}e^{\pm i\alpha_{m}}$ is the Hermitian part and $\pm\Delta_{m}e^{\pm i\beta_{m}}$ is the anti-Hermitian part. The Hamiltonian of Eq. (S26) has been extensively explored theoretically [46, 18, 47]; however, it is challenging to implement such a Hamiltonian experimentally. The usual implementation of non-Hermitian Hamiltonians through the use of space-dependent gain/loss  [48, 13, 14, 15] does not straightforwardly create the non-Hermitian coupling in Eq. (S26). Non-Hermitian couplings were only realized very recently on platforms including pulse trains [22] and electronic circuits [33]; these demonstrations however implemented only a limited subset of non-Hermitian coupling, i.e. neareast-neighbor coupling (exclusively $m=1$) with restricted in-phase Hermitian and anti-Hermitian parts ($\alpha_{1}=\beta_{1}$). It remains an open question if one can realize general non-Hermitian couplings, which include a control over coupling strengths ($C_{m},\Delta_{m}$), phases ($\alpha_{m},\beta_{m}$) and long coupling ranges ($m>1$) [49, 50, 51, 52]. Our results in this paper indicate the significant potential of the platform of synthetic dimension to explore non-Hermitian topological physics and to implement Hamiltonians that are difficult to achieve in other means.

The Hamiltonian of Eq. (S26) exhibits a rich set of non-trivial topological behaviors that depend on the strength, phase and range of the coupling. We first consider the case with only nearest-neighbor coupling, where the only non-zero coupling constants are $\kappa_{-1}=C-\Delta\exp{(-i\phi)}$ and $\kappa_{+1}=C+\Delta\exp{(i\phi)}$ with $C,\Delta\in\mathbb{R}$, where the 1st term $C$ describes the Hermitian part and the 2nd term $\pm\Delta\exp{(\pm i\phi)}$ is the anti-Hermitian part. The special case with $\phi=0$ gives rise to the well-known Hatano-Nelson model [46], where the band winds along an ellipse in the complex plane, as shown in Fig. 1(a,bottom). For a reference energy $\epsilon\in\mathbb{C}$, a winding number of such a 1D band can be defined by [28, 18, 19]

For the Hatano-Nelson model [Fig. 1(a)], if the reference energy $\epsilon$ is in the interior of the winding loop, e.g. in the shaded area of Fig. 1(a,bottom), one obtains $v=1$. We note that such a non-trivial winding is a topological feature that is unique to non-Hermitian Hamiltonians. For a Hermitian Hamiltonian, $E(k)$ is restricted to the real axis, and $v=0$. We also note that the phase difference $\phi$ between the Hermitian and non-Hermitian parts of the coupling can change the shape of the loop. In the special example of $\phi=\pi/2$, $E(k)$ is restricted to a line and hence $v=0$, as shown by the green line in Fig. 1(a,bottom). More complex winding of the band structure with $|v|\neq 0,1$ can be achieved by introducing long-range coupling into the non-Hermitian Hamiltonian. As an example, in Fig. 1(b), we consider the case with $C=0$, $\Delta/C^{\prime}=\Delta^{\prime}/C^{\prime}=1$, $\phi=\pi$, $\theta=0$, and $\phi^{\prime}=\pi$, and $\phi^{\prime}=3\pi/4$, where $v=-2$ for a suitable choice of the reference energy. In general, the sign of $v$ is determined by the handedness of the winding, and $|v|$ depends on how many times the reference point $\epsilon$ is enclosed by the loop in each orientation.

To experimentally realize the Hamiltonian of Eq. (S26), we utilize the synthetic space as formed by the multiple frequency modes of a ring resonator. In the absence of modulation, the ring supports a set of resonant modes propagating along a specific direction [e.g. clockwise as shown in Fig. 1(c)]. These modes have frequencies $\omega_{n}=\omega_{0}+n\Omega$, where $n$ is an integer that indexes the mode. $\Omega$ is the free spectral range (FSR), as shown in Fig. 1(d,left). Hermitian coupling between nearest-neighbor frequency modes can be formed with the use of a phase modulation having an associated amplitude transmission factor $T_{Ph}=\exp{(-i\delta\Phi)}$ with $\delta\Phi(t)=A_{1}\cos\Omega t$. The strength of Hermitian coupling $C\propto A_{1}$. Similarly, anti-Hermitian coupling can be formed with the use of an amplitude modulation having a transmission factor $T_{Am}=1+B_{1}\sin(\Omega t+\phi)$. This gives rise to an anti-Hermitian coupling $\kappa_{\pm 1}=\pm\Delta\exp(\pm i\phi)$ with the strength $\Delta\propto B_{1}$. Here $\phi$ readily gives rise to the phase difference between the Hermitian and anti-Hermitian parts of the coupling. With simultaneous phase and amplitude modulations, the equations of motion are given by

where $a_{n}$ is the electric-field amplitude of the $n$-th frequency mode and $\tau$ is a slow-time variable counting the number of round trips that light has propagated in the resonator. This realizes the model of Fig. 1(a). Longer-range coupling can be achieved with modulators having modulation frequencies $m\Omega$ with $m>1$. Specifically, simultaneous implementation of the phase modulation $T_{Ph}=\exp{[-i\sum_{m}A_{m}\cos(m\Omega t+\alpha_{m})]}$ and the amplitude modulation $T_{Am}=1+\sum_{m}B_{m}\sin(m\Omega t+\beta_{m})$ will give rise to the Hamiltonian in Eq. (S26) where $\kappa_{\pm m}=C_{m}e^{\pm i\alpha_{m}}\pm\Delta_{m}e^{\pm i\beta_{m}}$ (for more details see Sec. S1A of the supplementary). Consequently, this method allows for the realization of an arbitrary complex band structure for such one-band 1D non-Hermitian lattices, as illustrated in Fig. 1(d) for a specific example. We note that the ability to simultaneously incorporate phase and amplitude modulations is essential for demonstrating topological winding. Hamiltonians with phase or amplitude modulation [53] alone do not exhibit non-trivial winding in their energy bands.

Now we illustrate how we directly measure the $k$-resolved non-Hermitian band structure in synthetic frequency space. As illustrated in Fig. 2(a), we use a fiber ring resonator undergoing both phase and amplitude modulations by electro-optic modulators (EOMs). We launch continuous-wave (CW) laser light into the ring via a beam splitter (BS1). A second beam splitter BS2 samples a small portion of the intracavity light for time-resolved detection (for more details on the experimental setup see Sec. S2 of the supplementary). The input laser frequency can be swept across a certain resonance of the unmodulated resonator to access various detunings $\delta\omega$, as illustrated in Fig. 2(b). Since the lattice space is formed by frequency modes, its reciprocal space is inherently time $t$ [50], with a round-trip time equivalent to one Brillouin zone; hence we can define $k=t\Omega$ as the quasimomentum. The detected signal $I(k,\delta\omega)$ can be related to the Green’s function, $G(k,\delta\omega)=i[\delta\omega-E(k)]^{-1}$, as

Here the band energy is

where $\gamma$ denotes all sources of loss in the absence of modulation. For a given $k$, $I$ is a Lorentzian function of $\delta\omega$. Then, at each $k$, it is possible to deduce the values of $\mathrm{Re}(E)$ and $|\mathrm{Im}(E)|$ by fitting $I(\delta\omega)$ to the form of Eq. (4). In our setup, we use a cavity with a sufficiently large loss, i.e. a sufficiently large positive $\gamma$, such that $\mathrm{Im}(E)<0$ is ensured for all $k$. For a $\gamma$ that is independent of $k$, $E(k)$ differs from the spectrum of $\mathbf{H}$ by a translation in the complex energy plane. The topological properties are not affected by such a translation.

To illustrate the specific process of the band-structure measurement, we show a representative experiment forming a lattice in synthetic frequency space as sketched in Fig. 2(c). There is a 1st-order Hermitian coupling $\kappa_{\pm 1}=C$ and a 2nd-order anti-Hermitian coupling $\kappa_{\pm 2}=\pm\Delta\exp{(\pm i\phi)}$. By scanning $k$ over one Brillouin zone and $\delta\omega$ over one FSR $\Omega$, we obtain the $I(k,\delta\omega)$ readouts as shown by a 2D color plot in Fig. 2(d). Here we show two representative vertical slices at $k=\pi/4$ and $k=3\pi/4$ and plot the corresponding $I(\delta\omega)$ data in Figs. 2(e,f), respectively. According to Eq. (4), $\mathrm{Re}(E)$ and $\mathrm{Im}(E)$ can be obtained by a Lorentzian fit for each $k$-slice of $I$ [purple curves in Figs. 2(e, f)]. We plot $\mathrm{Re}(E)$ and $\mathrm{Im}(E)$ thus obtained as the dots in Fig. 2(g). The red curves in Fig. 2(g) indicate the theoretically predicted form of the dispersion. The experimental results agree excellently with the theoretical predictions. Finally, we plot $E$ in the complex plane and visualize the winding [Fig. 2(h)], where the experimental data points and the theoretical fitting curve are both color-coded to indicate the values of $k$. For the specific dispersion in this example, the complex band winds into a bow-tie shape. For $k$ going from $0$ to $2\pi$, the left-side loop (light orange shading) is traversed clockwise once with the winding number $v=-1$, and the right-side counter-clockwise loop (light blue shading) has $v=1$.

Strictly speaking, band structures can only be defined for infinite systems that fulfill Bloch’s theorem, whereas all practical structures in experiments are finite. Thus, in seeking to measure experimentally a band structure, one is always attempting to infer the property of an infinite system through the measurement on a corresponding finite system. For the Hermitian system this does not present an issue. Within the energy range of the band in an infinite system, the eigenstates of the corresponding finite system are also extended and provide an increasingly better approximation of the eigenstates of the infinite system as the size of the finite system increases. For the non-Hermitian system, on the other hand, the situation is more subtle. The eigenstates of the finite system can be drastically different from those of the infinite system. As an extreme example, a finite non-Hermitian lattice truncated with an open boundary condition (OBC) can exhibit the non-Hermitian skin effect with all eigenstates localized on the edge, and the eigenspectra are thus drastically different from those of the corresponding infinite system  [28, 33]. Thus, it was noted that the winding number defined in Eq. (2) can not be simply identified from the eigenspectrum of a finite non-Hermitian system [18]. And hence it may appear surprising that through Eq. (4) we are able to measure the band structure of an infinite system through the experiments on a finite system. We emphasize that our experiments probe $E(k)$ as defined by Eq. (5). $E(k)$ is not an eigenenergy of the finite system, but rather the expectation value of the Hamiltonian of the finite system at state $\ket{k}$. The state $\ket{k}$ is always extended and moreover naturally approaches the Bloch state of an infinite system as the system size increases. Consequently, our experiments can provide a measurement of the band structure of an infinite system satisfying Bloch’s theorem. As an illustration, in the supplementary Sec. S1B and Fig. S1 we consider a numerical experiment where we apply our technique for the extreme case as mentioned above where a finite non-Hermitian lattice is truncated by the OBC. Even though such a finite lattice exhibits a non-Hermitian skin effect, our technique still provides a faithful measurement of the band structure of the infinite system.

We now present an example where the winding of the band can be straightforwardly controlled. In Fig. 3(a) we sketch a lattice with nearest-neighbor non-Hermitian couplings in the form $\kappa_{\pm 1}=C\pm\Delta\exp{(\pm i\phi)}$. Importantly, the topological winding is largely determined by the value of the phase $\phi$, which can readily be tuned in our experiments by varying the relative phase between the amplitude and phase modulations. In Figs. 3(b,c) we show the band structure measurement for $\phi=0$ accompanied by a theoretical prediction. This case serves as a faithful realization of the Hatano-Nelson model [46]. The complex band encircles an elliptical area in the counterclockwise direction [Fig. 3(d)] corresponding to a winding number $v=1$ (light blue shading). By contrast, if we have $\phi=-\pi/2$ [Figs. 3(e,f)], the topology is trivial - the band forms a line in the complex plane [Fig. 3(g)]. As we further choose $\phi=-3\pi/4$ [Figs. 3(h,i)], the band becomes a tilted ellipse with a clockwise handedness, as shown in Fig. 3(j). The winding number with respect to a reference energy inside the loop becomes $v=-1$.

A significant strength of synthetic dimensions lies in the ability to readily incorporate long-range couplings which can also be non-Hermitian. We harness this ability here to produce more complex windings. In Fig. 4(a) we sketch a lattice model with a Hermitian 1st-order coupling $\kappa_{\pm 1}=C$, and a skew-Hermitian 3rd-order coupling $\kappa_{\pm 3}=\pm\Delta\exp{(\pm i\phi)}$. This lattice has a dispersion $E(k)=2C\cos(k)+2i\Delta\sin(3k+\phi)-i\gamma$. Figs. 4(b) and (c) show the measurement results for $\phi=0$. The extracted complex band winding is shown in Fig. 4(c), which encloses three areas with the winding number $v=1,-1,1$ from left to right, respectively. As another example, in Fig. 4(d) the Hermitian coupling is varied to 2nd-order compared to the lattice in Fig. 4(a), and hence the dispersion becomes $E(k)=2C\cos(2k)+2i\Delta\sin(3k+\phi)-i\gamma$. We show a measurement and the extracted complex band winding with $\phi=-3\pi/4$ in Figs. 4(e,f), respectively. The winding loop encloses three areas with $v=1$ (light blue shaded), three with $v=-1$ (light orange shading) and two areas with $v=0$ (no shading). These examples act as longer-range generalizations of the instances shown Fig. 3. Additionally, since the winding number $v$ theoretically can take any integer value, to showcase the capability of implementing and measuring windings with $|v|>1$, we also present an instance as shown in Figs. 4(g–i). The lattice model illustrated in Fig. 4(g) incorporates non-Hermitian couplings of both 1st and 2nd orders, where we have $\kappa_{\pm 1}=\pm\Delta^{\prime}\exp{(\pm i\phi^{\prime})}$, $\kappa_{\pm 2}=C\pm\Delta\exp{(\pm i\phi)}$. The Bloch band is given by $E(k)=2C\cos(2k)+2i\Delta^{\prime}\sin(k+\phi^{\prime})+2i\Delta\sin(2k+\phi)-i\gamma$. In Fig. 4(h) we show the measurement for the case $\phi=\phi^{\prime}=\pi$, and the winding loop encloses not only areas with $v=-1$ (light orange shading) but also an area that winds twice clockwise, which corresponds to $v=-2$ (dark-orange shaded). More example experimental results can be found in the Sec. S3 of the supplementary.

In conclusion, we report the experimental observation of topological windings of the complex band energy in non-Hermitian lattices. We show that non-Hermitian lattice Hamiltonians in the frequency synthetic space can be implemented through simultaneous amplitude and phase modulations of a ring resonator supporting multiple resonant modes. Our realization of non-Hermitian lattice Hamiltonians in synthetic space brings unprecedented flexibility to arbitrarily tailor the complex band structure fully combining non-Hermitian, long-range and complex-valued couplings. Future work could consider generalizations of such complex band winding to higher dimensions, using both real and synthetic space, or to multi-band lattices, such as those involving exceptional points [17].

Supplementary Information