Interaction induced bi-skin effect in an exciton-polariton system

Introduction.—Exciton-polaritons are bosonic quasiparticles arising from the strong coupling between excitons and photons in microcavities Carusotto and Ciuti (2013); Deng et al. (2010); Keeling and Berloff (2011); Byrnes et al. (2014). The combination of photons and excitons allows polaritons to possess the characteristics of both its root parts, for example, low effective mass from the photonic part and strong nonlinearity from the excitonic part. Moreover, due to the photons escaping from the microcavity, exciton-polaritons are noticeably an open quantum (non-Hermitian) system, which requires additional pumping to maintain a steady state in the system Savvidis et al. (2001). This character makes the exciton-polariton system an ideal platform to study non-Hermitian transitions with gain and loss Nalitov et al. (2019); Xu et al. (2020). Although various research has been done in the Hermitian regime to realize polariton topological phases with the interplay between Zeeman shift resulting from the application of magnetic field and the transverse electric-transverse magnetic (TE-TM) splitting of the photonic modes Bardyn et al. (2015); Nalitov et al. (2015); Li et al. (2018); Gulevich et al. (2016); Klembt et al. (2018), accounting for nonlinearity Bardyn et al. (2016); Sigurdsson et al. (2017), and using the polarization splitting of elliptical micropillars Banerjee et al. (2018), several works have been reported in recent years to explore the non-Hermitian physics in the exciton-polariton system. Exceptional points (EPs) in the exciton-polariton system have been realized in Refs. Gao et al. (2015, 2018a, 2018b). Recently, the measurement of non-Hermitian topological invariants Su et al. (2020a), topological end mode lasing Comaron et al. (2020), and non-reciprocal polaritons Mandal et al. (2020) have been reported.

Non-Hermitian physics has attracted a tremendous attention in recent years due to the discovery of the non-Hermitian skin effect, where all the modes are localized to one end of a lattice  Bandres et al. (2018); Shen et al. (2018); Kawabata et al. (2019a); Longhi (2019); Ghatak and Das (2019); Hou et al. (2020); Weidemann et al. (2020); Lee and Thomale (2019); Hofmann et al. (2020); Zhou and Lee (2019); Liu et al. (2020a); Luo and Zhang (2019); Liu et al. (2020b); Okuma et al. (2020); Wang et al. (2020); Zhang et al. (2020); Li et al. (2020); Xiao et al. (2020); Song et al. (2019a, b, 2020); Lee et al. (2019a); Fu and Wan (2020); Okugawa et al. (2020). This is different to the case of Hermitian systems, where modes within a topological bandgap are localized at the edges of a finite sample due to bulk-boundary correspondence. Due to non-Hermiticity, the Bloch theory does not even hold approximately for finite sized non-Hermitian systems and one must turn to the generalized Brillouin zone (GBZ) theory based on a complex momentum to explain the non-Hermitian skin effect Okuma et al. (2020); Kunst et al. (2018); Kawabata et al. (2019b); Gong et al. (2018); Yao and Wang (2018); Lieu (2018). To realize the skin effect, the simplest model is the so-called Hatano-Nelson(HN) model Hatano and Nelson (1996, 1997) without disorder where the hopping in a lattice is different in different directions.

Along with the same idea of the HN model, the Su-Schrieffer-Heeger(SSH) model with asymmetric hopping has been proposed or realized in many systems like waveguides Zeuner et al. (2015); Yokomizo and Murakami (2020a), photonic lattices El-Ganainy et al. (2019); Xiao et al. (2020), circuits Helbig et al. (2020) etc. The non-Hermitian topological bulk-boundary correspondence can be obtained by using the GBZ theory and by solving the boundary equations with complex momentum Liu et al. (2019); Yao and Wang (2018); Yao et al. (2018); Okuma et al. (2020); Wang et al. (2020); Yang et al. (2020); Imura and Takane (2019); Zhang et al. (2020). Due to the chiral symmetry, the system will have an energy pair of ($E$,$-E$) and they will collapse at zero energy modes. The winding number can be well defined by the calculation with the periodic boundary energies with the GBZ. Meanwhile, more and more work is going to explain the topological origin of the skin modes and the GBZ is not necessary for line gap non-Hermitian Hamiltonians where the basis can be enlarged to construct a Hermitian Hamiltonian Zhang et al. (2020); Liu et al. (2020c); Okuma et al. (2020); Gong et al. (2018); Lee and Thomale (2019); Lee et al. (2019b). Although the winding in the whole Bloch region is still zero for zero energy pairs, the system can still have a topological non-trivial phase if energy pairs can be made to collapse at non-zero (complex) energy.

In this letter, we use the particle-hole symmetry of the fluctuation theory of exciton-polariton systems to realize “bi-skin” modes where all the wavefunctions are localized at both sides of a one-dimensional lattice. Being compatible with recent experiments, we consider a non-equilibrium polariton condensate corresponding to a plane-wave with non-zero momentum. By the Bogoliubov-de-Gennes(BdG) transformation Xu et al. (2017); Banerjee et al. (2020), the fluctuations of the condensate will have two modes with particle-hole symmetry. By considering a suitable BdG lattice by tight-binding theory, the bi-skin effect will appear with symmetric hopping between sites. To classify the symmetry and the topologies of the system, we calculate the winding number of the system without assuming the GBZ theory. Because of the particle-hole symmetry Okuma and Sato (2019), the energy spectrum will not collapse at zero modes but some purely imaginary energies. The different topological phases can be well defined by the winding of the EPs and the localization of the wavefunction. Finally, we analyze the stability of the system and calculate the energy dispersion, which can be observed via photoluminescence Utsunomiya et al. (2008); Pieczarka et al. (2015) or four-wave mixing spectroscopy Kohnle et al. (2011).

Model.—We consider a simple chain with symmetric hopping $J$ between nearest neighbours, formed by exciton-polariton micropillars, as shown in Fig. 1(a). Such chains have been realized experimentally Winkler et al. (2016). Typically the behavior of a polariton condensate is studied, which itself has no topological features in such a system. The system can be described by the driven-dissipative nonlinear Gross-Pitaevskii (GP) equation

Here, $\Delta=E_{0}-\hbar\omega_{p}$ is the onsite detuning between the bare polariton mode having energy $E_{0}$ and a coherent drive $F_{n}$ with frequency $\omega_{p}$. $g$ represents the nonlinear interaction strength and $\gamma_{C}$ is the effective dissipation. In order to find the optimum parameter range we move to the dimensionless units by making the following substitutions: $t\rightarrow t\hbar/\gamma_{C}$, $J\rightarrow J/\gamma_{C}$, $\Delta\rightarrow\Delta/\gamma_{C}$, $\psi_{n}\rightarrow\psi_{n}\sqrt{\gamma_{C}/g}$, and $F_{n}\rightarrow F_{n}\sqrt{g/\gamma_{C}^{3}}$. One of the main ingredients of our scheme is a plane-wave like stationary state $\psi_{0,n}=Ae^{ik_{p}n}$ (as shown in Fig. 2(a) ) realized by spatially modulating a coherent field Ohadi et al. (2018); Alyatkin et al. (2020); Pickup et al. (2020); Carusotto and Ciuti (2004) and the appropriate choice of $F_{n}$ is obtained by setting the time derivative in Eq. (1) to zero:

where $A$ is the square root of the density of the condensate and $k_{p}$ is the pumping momentum and $n\neq 1$ or $N$. Because of the boundary, we should set the first and the last site pumping by replacing $2J\cos(k_{p})$ in Eq. (2) with $Je^{ik_{p}}$ and $Je^{-ik_{p}}$, respectively.

To show the bi-skin effect we consider the fluctuations, by introducing the fluctuation amplitudes $u_{n}$ and $v_{n}$, and writing $\psi_{n}=\psi_{(0,n)}+u_{n}e^{iEt/\hbar}+v_{n}^{*}e^{-iE^{*}t/\hbar}$. The BdG modes as shown in Fig. 1(b) are derived by substituting $\psi_{n}$ into the mean-field GP equation, which yields the time-independent equations:

Here, $\Delta^{\prime}=\Delta+2\left|\psi_{(0,n)}\right|^{2}$ is an effective detuning. The BdG transformation ensures that $u_{n}$ and $v_{n}$ modes have the particle-hole symmetry, so the hopping energies for $u_{n}$ and $v_{n}$ are $\pm J$ and the inter hopping energies in one unit cell are $\pm\psi_{(0,n)}^{(*)2}$. We need to mention that the hopping strength between different pairs of neighboring sites have the same magnitude but differ in their phases because of the symmetry. The Bogoliubov matrix $H_{n}$ with $N$ sites obtained by the Eqs. (3)-(4) can be diagonalized by $V^{-1}\hat{H_{n}}V=\text{diag}\left(E_{N\times N},-E_{N\times N}^{*}\right)$ with $V$ the matrix formed by eigenvectors, meanwhile, $S_{x}V^{-1}\hat{H_{n}}VS_{x}=\text{diag}\left(E_{N\times N},-E_{N\times N}^{*}\right)$ where $S_{x}V$ can also be the eigenvectors of the system Xu et al. (2017); Banerjee et al. (2020), here, $S_{x,y,z}=\hat{I}_{N,N}\otimes\sigma_{x,y,z}$ and $\sigma_{x,y,z}$ are the Pauli matrices. On the other hand, the system has pseudo-Hermiticity Yokomizo and Murakami (2020b); Kawabata et al. (2019b) defined as $S_{z}\hat{H_{n}}S_{z}=\hat{H_{n}}^{\dagger}$. The particle-hole symmetry and pseudo-Hermiticity guarantee that the system can have a 1D topological transition Okuma et al. (2020); Yokomizo and Murakami (2020b) and the energy spectrum in real space is highly symmetrical.

The bi-skin modes.—The stationary states of the bulk polaritons can have non-zero momentum induced by the specific $F_{n}$ pumping laser and the nonlinearity makes the fluctuations modes $u_{n}$ and $v_{n}$ have a momentum shift. The momentum shift will just influence the phases of the energy spectrum under the periodic boundary condition (PBC), however, the energy spectrum under the open boundary condition (OBC) will completely change for the fluctuations at the edge can not go further and then become localized.

The model is equivalent to say that the hopping energies between the $u_{n}$ and $v_{n}$ sites are different, but we stress that this is an effect induced by nonlinear interactions, where the fluctuations are affected by the non-zero momentum of the considered polariton condensate (stationary state). The underlying physical system is still none other than a regular polariton lattice with symmetric Hermitian hopping between lattice sites. As is shown in Figs. 2(b) and (c), all eigenstates for $u_{n}$ and $v_{n}$ sites are localized at both sides. The energy spectrum of $u_{n}$ sites are $(E,E^{*})-i$ pairs, while the energy spectrum of $v_{n}$ sites are $(-E^{*},-E)-i$ pairs. All eigenvectors of $E-i$ and $-E^{*}-i$ will be localized at one side and wavefunctions of the $E^{*}-i$ and $-E-i$ will be localized at the opposite side. Remarkably, the localization here does not mean the polariton’s density localization but the fluctuations on the top of the polaritons.

Stability.— Our model is based on the fluctuations of exciton-polaritons with a non-zero momentum stationary state. Because of the symmetry of the system, the imaginary parts of the eigenenergies can be beyond zero. However, with sufficient dissipation, there will be a total energy shift $i\gamma_{C}$ that keeps all imaginary energy components below zero.

Figs 3(a) and (b) show the distribution of intensity of the real and imaginary parts of the fluctuation eigenstates in real space. All fluctuation eigenstates are localized at the edges of the system, while we find that those with least negative imaginary part are localized on the left side.

Because the imaginary parts of the energy spectrum are larger than the real parts, a fundamental problem is whether the spectrum can be observed experimentally. To prove this, we numerically calculate the BdG lattice’s energy dispersion in Fig. 3(c). We take $u_{n}$ and $v_{n}$ as perturbations with small random number compared with $\psi_{(0,n)}$ and solve the BdG equations. The dashed fitting curves have the same energy dispersion with diagonalization results but doubly shrinks the period for two inner sites here. The imaginary parts of the energy dispersion are different for the linewidth of the energy dispersion are different as a function of momentum. Remarkably, we solve the Eqs. (3)-(4) without coherent pumping here, which reveals our model is universal and can be extended to other systems.

Non-Hermitian topological invariants– The effective Hamiltonian can be written into the momentum space with use of the Fourier transformation $u_{n}\rightarrow\frac{1}{\sqrt{N}}\sum_{k}\hat{a}_{k}e^{ikn}$ and $v_{n}\rightarrow\frac{1}{\sqrt{N}}\sum_{k}\hat{b}_{k}e^{ikn}$ with $\hat{a}_{k}$ and $\hat{b}_{k}$ the annihilation operators for the momentum $k$, giving

where $\hat{I}$ is the $2\times 2$ identity matrix and the periodic bounary condition is considered and the basis vector is $(\hat{a}_{k},\hat{b}_{k-2k_{p}})$. The imaginary energy shift $i$ can be ignored for the theoretical study, however, it can determine the stability of two localization and is necessary to be observed by the energy dispersion. The non-Hermitian topology of $H_{k}$ with respect to the reference point $E_{R}$ is equivalent to the Hermitian topology of the following doubled Hamiltonian Lee et al. (2019a); Gong et al. (2018); Okugawa and Yokoyama (2019) with the basis vector $\tilde{v}=(\hat{a}_{k},\hat{b}_{k},\hat{a}_{k+2k_{p}},\hat{b}_{k-2k_{p}})^{\top}$ can be obtained in the supplementary materials where the open boundary condition is considered.

where $h(k,k_{p})=2J\left[\cos(k)\cos(2k_{p})\hat{\sigma}_{z}-\sin(k)\sin(2k_{p})\hat{I}\right]$ and $\hat{I}$ is the $2\times 2$ identity matrix. The absolute value of the fluctuations (on top of the condensate) hopping from $u_{n}$ to $v_{n}$ will move to the $-k_{p}$ direction ($\hat{a}_{k}\hat{b}_{k-2k_{p}}^{\dagger}$ term ), meanwhile the polariton fluctuations hopping from $v_{n}$ to $u_{n}$ will move to the $+k_{p}$ direction ($\hat{b}_{k}\hat{a}_{k+2k_{p}}^{\dagger}$ term). Therefore, the absolute squre of the eigenstates of the BdG lattice are localized at both sides of the lattice as shown in Figs. 2(b)-(c).

The construction of the Hermitian Hamiltonian in the enlarged Hilbert space not only allows interpretation with the Bloch theory Zhang et al. (2020); Lee et al. (2019a); Gong et al. (2018); Okugawa and Yokoyama (2019), but also highlights the differences of the four basis operators under the open boundary condition. The boundary will let the momentum be cut off at the different values and cause the fluctuations to move to the edge and then be localized. For example, the range of $k$ is $\left(-\pi,\pi\right)$, however, the range of $k\pm 2k_{p}$ is $\left(-\pi\pm 2k_{p},\pi\pm 2k_{p}\right)$. Some related work has been done in the SSH model with asymmetric hopping and interpreted with the GBZ theory Yokomizo and Murakami (2020b); Liu et al. (2020a); Yang (2020).

There are two EPs in the energy spectrum, so we need to calculate each winding separately. By unitary transition, the enlarged Hamiltonian can be written into a two blocks off-diagonal matrix and for each one we can define the winding around the reference energy $E_{R}$ Zhang et al. (2020)

The total winding number is zero for the vanishing of $E$ and $E^{*}$ because the winding direction is completely opposite. However, we can get the integer winding number $\pm 1$ or 0 in different topological regions. In our model, the reference energy is purely imaginary energies for the gapless $E$ and $-E^{*}$. The energies as a function of Hopping energies in the real space with OBC are shown in Figs. 4 (a) and (c). The critical points are labeled for the callapse of the imaginary energies.

The different topological phases can be distinguished by the eigenenergy spectrum of Eq.(5) and how many OBC energies in real space are encircled by the energy band in momentum space with PBC (see supplementary material):

• •

  Nontrivial phase $0<\left|J\right|<\frac{\left(A^{2}+\Delta^{\prime}\right)\sec(k_{p})}{2}$: All eigenstates are localized at both sides and all OBC energies are circled by the PBC energies.

• •

  Intermediate phase $\frac{\left(A^{2}+\Delta^{\prime}\right)\sec(k_{p})}{2}\leqslant\left|J\right|<\frac{\left(A^{2}-\Delta^{\prime}\right)\sec(k_{p})}{2}$: Parts of eigenstates are localized and parts of OBC energies including EPs are encircled by the PBC energies.

• •

  Trivial phase $J=0$ or $J\geqslant\frac{\left(A^{2}-\Delta^{\prime}\right)\sec(k_{p})}{2}$: Parts or no eigenstates are localized and no EPs are encircled by the periodic boundary spectrum.

Here, the critical points are calculated by the phases of the eigenvalues of Eq. (5) and this result is corresponding with the winding number obtained by Eq. (7) (see the supplementary materials). If the PBC energy dispersion has different linewidth (imaginary parts of the energies) as a function of momentum, the skin modes will appear. Although the skin modes can appear in the above three phases, the topological invariants are only $\pm 1$ for two EPs in the nontrivial phase and the intermediate phase. The energies profile under the PBC can not encircle all the energies under the OBC in the intermediate phase. Therefore, all eigenstates are localized only in non-trivial phase and the quantities of the localized eigenstates are corresponding with the quantities of OBC energies encircled by the PBC energies.

In the considered model all the calculations are done using the dimensionless variables. To get an idea about the physical parameters, we set $\gamma_{C}=0.1$ meV, which corresponds to a polariton lifetime around 3 ps. The coupling between the neighbouring micropillars becomes J=40$\mu$eV, which can be tuned by adjusting the overlap between the micropillars de Vasconcellos et al. (2011). The non-linearity induced blueshift becomes 0.1 meV which is a routine observation in GaAs based samples Estrecho et al. (2019). Given the room temperature polariton micropillar chain has been demonstrated experimentally Su et al. (2020b); Dusel et al. (2020), our scheme is also compatible with perovskite and organic samples. To make it more experimental friendly, the fluctuations can be made to have different polarization from that of the steady state by proper choice of the resonant driving and exciton-photon detuning Sigurdsson et al. (2017); Mandal et al. (2019).

Conclusion—We consider the behaviour of fluctuations on a polariton mean-field stationary state in a one-dimensional lattice of coupled micropillars. By taking the stationary state as a plane wave, which can be resonantly injected with a suitably patterned optical field, we find that interactions in the system allow the presence of a bi-skin effect where fluctuations are localized at both edges of the lattice. Even though the underlying hopping of the system is Hermitian, the interactions allow an effective non-reciprocal coupling between particle-hole fluctuations. This results in a non-trivial topology, confirmed by calculations of the winding number.

Acknowledgements— This work was supported by the Singaporean Ministry of Education, via the Tier 2 Academic Research Fund project MOE2018-T2-2-068.

Interaction induced bi-skin effect in an exciton-polariton system Xingran Xu Huawen Xu S. Mandal R. Banerjee Sanjib Ghosh T.C.H. Liew