## Abstract

We demonstrate the controlled coherent transfer of topological interface states in a one-dimensional non-Hermitian chain of interacting Bose-Einstein condensates. The topological protection stems from a spatially patterned pump in an open-dissipative system. As a test bed setup of the proposed phenomenon, we consider a chain of coupled micropillars with embedded quantum wells, possessing exciton-polariton resonances. The transfer of an interface state is driven by spatially localised, adiabatic pump modulation in the vicinity of the interface state. The stochastic calculations prove the coherent nature of the interface state transfer. For appropriate system parameters the coherence degree is preserved after multiple transitions, paving the way towards long-range transfer of a coherent quantum state.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Topological insulators (TI) are a class of materials that possess an energy bandgap and topologically protected low energy states [1,2]. Topological protection in these systems stems from the symmetry of the bulk, which is quantified by means of topological invariants. Bulk-boundary correspondence results in the protection of edge states, which hold promise for applications in dissipation-less communications and quantum computing.

While in standard TIs non-trivial topology results from the properties of a Hermitian Hamiltonian, recently a class of non-Hermitian topological systems attracted great interest [3]. These are of particular relevance to photonics, where open-dissipative effects are prevalent. The latter, on one side, makes photonic systems an optimal platform for probing phenomena emerging specifically in the non-Hermitian domain. In this context phenomena such as lasing of topological states have been demonstrated [4,5]. On the other side, recently it was proposed to use the non-Hermiticity of photonics as an efficient tool for controlling the topological properties of the system. This can be reached, particularly, through asymmetric coupling coefficients [6] or spatial modulation of gain-loss ratio in each site [7–9]. Arguably, the greatest fundamental interest lies in the investigation of topological states that result solely from the non-Hermiticity of the system, since these have no counterparts in the Hermitian case [10–17].

In this paper, we consider the question whether non-Hermiticity can be used for precise control of topological states. Similarly to electrons in crystalline media, electromagnetic waves in periodically patterned photonic structures form energy bands, which can lead to appearance photonic edge states [4,5,18–27]. The presence of topological protection suppresses the backscattering on disorder, thus generating an energy-efficient propagation channel. However, in such a setting there is no convenient way to externally control the direction or velocity of the wave packet. On the other hand, photonic implementations of non-Hermitian Hamiltonians relying on spatial modulation of external pumping allow to efficiently tune certain terms of the Hamiltonian. This opens the way to the control of topological states, which can be crucial for future applications, such as scattering-free optical interconnects, quantum computation, or Majorana state braiding [28].

The distinctive peculiarities of photonic TIs can be further extended in the regime of strong-light matter coupling in a microcavity with embedded quantum wells [29]. The emerging hybrid quasiparticles, called exciton-polaritons, are interacting via their excitonic component, allowing thus to achieve a strong nonlinear response in comparison to other photonic systems. Typically, etching of cavity is used to fabricate an array of coupled micropillars, mimicking the structure of a tight-binding Hamiltonian. In the majority of existing theoretical proposals [30–44] and experimental realizations [45–49] topological order emerges from Hermitian band engineering, whereas open-dissipative nature of the system serves only to create a non-equilibrium Bose-Einstein condensate of polaritons in each micropillar.

As we demonstrated recently, topological protection in a chain of coupled polariton micropillars can be achieved solely via the spatial modulation of external pump [9] in a system with equal hopping coefficients. Topological characterization of the system revealed the existence of multiple phases, with different number of end states. Here, we show that at the boundary of such phases a non-decaying topological interface state can be created. By means of adiabatic switching of the pump pattern, we induce a controllable transfer of the interface state. Moreover, by calculating the first-order correlation function, we demonstrate that such transfer is of coherent nature. We determine the optimal conditions for the coherent transfer, such as time dependence of the spatial pump pattern and the switching time. Our results are confirmed by stochastic simulations within the truncated Wigner approximation, which include the effect of quantum fluctuations.

## 2. Model

#### 2.1 Hamiltonian

The structure we consider consists of a chain of unit cells, each including four sites. The on-site potential within each cell is spatially modulated. The bulk of the chain is analogous as in our previous work [9]. Here, we consider interface states which emerge at the boundary between two phases supporting a different number of edge states. The Hamiltonian of the system reads

#### 2.2 Interface states and protocol of the transition

It was previously shown [9] that in the homogeneous case (i.e. $g_3=g_1$, $g_4=g_2$) the Hamiltonian (1) can exhibit one or two pairs of edge states, odd number of edge states, or be topologically trivial. In order to perform dynamical study of interface state behaviour in a dissipative system, one needs a single topological state to be the only non-decaying eigenstate. Hence, the imaginary part of interface state eigenenergy needs to be the highest among all the eigenmodes. Such a situation can be reached via judicious choice of parameters, with the interface state appearing on the boundary of two topological phases, exhibiting one and two pairs of edge states, respectively. Particularly, here we choose the values of parameters $g_1=-2\kappa$, $g_3=2\kappa$, $g_2=g_4=\kappa$, $\theta =\pi /3$, the length of the chain $N=40$, and the boundary unit cell $n_b=10$. We also uniformly reduce by $-i\gamma$ the on-site potentials of the system, so that only the interface state has positive imaginary part of eigenvalue. The shape of the interface state and the imaginary part of the energy spectrum are shown in Fig. 1(a), left and right panels, respectively, where $\gamma =1.35\kappa$. Notably, the interface state is “M”-shaped, almost completely localized in 5 sites, and centred exactly around the boundary site between the two phases.We proceed with the study of coherent transfer of an interface state. The protocol of transfer consists of several steps, which are illustrated as follows. For a better understanding, let us consider first the evolution in the mean field regime, as sketched in Fig. 1(b). We start with a random distribution, and eventually reach a nonequilibrium steady state configuration, if growth saturation is present. The latter naturally emerges in any system where pumping has a limited capacity. This state is labelled as “initial” in Fig. 1(b). Then, we gradually modify the pump pattern at the boundary of two phases, and thus shift the boundary by one unit cell. In the beginning of the “transient" stage the existing interface state ceases to be a non-decaying mode, and rapidly becomes depleted. The new interface state emerges at the new boundary between the phases, and becomes populated. Steady state of the configuration hosting the new shifted interface state is eventually reached in the “final" state. Yet, such a combination of two independent processes does not lead to a transfer of macroscopic population which retains the coherence of the condensate. To avoid this scenario, during the transition process we apply extra pump to the region of existing interface state, which replenishes the decaying population. In this case we find that the interface state is indeed shifted by one unit cell.

To verify this, we perform stochastic simulations, and calculate the spatially and temporally resolved degree of coherence between interface states at the initial and final stages of the process. The evolution of condensate can be described by a stochastic discrete Gross-Pitaevskii equation [9], in which sites are coupled to each other due to the presence of hopping. The corresponding set of equations reads

We subdivide the period of switching process $T_{\mathrm {tr}}$ into three parts of equal duration. At first, we gradually increase by $i\sigma$ the on-site potential of two last sites of the boundary unit cell $n_b$, which is the leftmost cell in Fig. 1(b). Second, we perform the switching of on-site potentials in the boundary unit cell $n_b+1$. Finally, in the third step we gradually turn off the supporting potential. Hence, the majority of on-site potentials in Eq. (2) are time-independent, except the vicinity of interface state. Particularly, for the unit cell $n_b+1$, the middle cell in Fig. 1(b), the switching is given by

In order to quantify the efficiency of the transition we calculate the coherence between the initially existing and the newly emerging interface states. As mentioned above, the interface state is mainly localised in 5 sites around the boundary between two phases. Accordingly, we introduce a vector describing this state

#### 2.3 Results

Numerical simulations were performed for the model of an exciton-polariton lattice, where the tunnelling rate is chosen as $\kappa =0.1$ meV. All the other quantities in the Hamiltonian are scaled relative to $\kappa$. Correspondingly, the temporal evolution is presented in ps. The parameter $d$ in the definition of white noise corresponds to the diameter of the micropillar, and is chosen as $d=3$ $\mu$m. The supporting on-site potential is $\sigma =3\gamma$, and the scaling factor of noise amplitude is chosen as $\beta =0.05$, corresponding to characteristic coherence time for polariton condensates [52]. The dependence of coherence degree on $\beta$ is discussed in Appendix B.

In Fig. 2 we present the evolution of the first order correlation function of (8) during a shift of a interface state by one unit cell, for different switching rates. We start at $t=0$ and perform the evolution for 500 ps with time independent Hamiltonian, during which we reach the nonequilibrium steady state. Note that from here onward, in all figures this part of the evolution is not shown. Then we gradually change the potentials in the boundary unit cells $n_b$ and $n_b+1$. The process is analogous to Landau-Zener transition [53,54]. One can expect that the lower is the transition speed, the higher will be the preserved degree of coherence due to a smaller perturbation of the steady state. This indeed is the case when increasing the switching time $T_{\mathrm {tr}}$ from 1 to 30 ps. However, due to non-Hermiticity of the Hamiltonian, the system is intrinsically of open-dissipative nature. This circumstance imposes a finite coherence time even in the steady state, stemming from the noise associated with input and output channels. In the context of exciton-polaritons these channels are represented by external incoherent pump and the finite lifetime of polaritons [29]. Correspondingly, the finite coherence time imposes the upper limit on the transition period. Thus, the interplay of these two factors determines the timescale optimal for coherent transfer of the interface state to the next unit cell.

We further analyse the impact of the supporting on-site potential applied during the transfer protocol, defined by Eqs. (5) and (6). In Fig. 3 we present the temporal evolution of the coherence degree in the presence and absence of the potential. First of all we study coherence in the absence of switching, i.e. $g_{n_b,0}(\tau )$. Blue and red curves correspond here to $g_{n_b,0} (\tau )$ with and without applying supporting potential, respectively. We note that the additional noise stemming from the supporting potential largely reduces coherence. On the contrary, the coherence in the case of shift by one unit cell [$g_{n_b,1}(\tau )$] is found to be very small in the absence of the supporting potential (see orange curve) whereas its presence allows to essentially preserve the coherence degree (purple line). It is remarkable that after the switching event, the degree and temporal evolution of the shifted interface state coherence (purple curve) is very close to that of coherence in the absence of switching (blue curve). This is another clear indication that the coherence is retained during the interface state transfer event.

In addition, we consider temporal dependence of the first order correlation function in the case of two consequent switching events, as shown in Fig. 4. The two switching events are separated by a large temporal window, during which the initial interface state becomes depleted, and the newly established interface state reaches to quasi steady state. Interestingly, the temporal profiles of the correlators $g_{n_b,1}$, $g_{n_b+1,1}$ [red and purple curves in Fig. 4(a)] are almost identical, indicating that the two switching events have same nature. Finally, after the second transition the correlator $g_{n_b,2}$ [orange curve in Fig. 4(a)] reaches the maximum value of 0.25. Evidently, this value shows that for the chosen parameters the multiple transitions will completely wash out the initial coherence.

In order to increase the conservation of coherence during the multiple transitions it is necessary the decrease the strength of the noise. However, in a realistic polariton model the latter has a limited range of allowed values. The dependence of coherence on scaling of the noise amplitude is discussed in Appendix B. A possible alternative could be the variation of the strength of supporting on-site potential. The dependence of coherence on the strength of supporting potential is shown in Fig. 5. The enhancement of supporting potential amplitude prevents the existing interface state from depletion, increasing thus the conservation of coherence. On the other hand, the stronger the pump, the more noise is introduced into the system, decreasing the coherence. The interplay of these factors determines the optimal rate of supporting potential $\sigma =3\gamma$, which is used in majority of the calculations.

## 3. Conclusion

In conclusion, we show the possibility of controllable coherent transport of topological interface states in a system of coupled Bose-Einstein condensates. We found that topologically protected interface state emerges solely due to spatial modulation of complex-valued on-site potential, while its temporal modulation causes the transfer of interface state. As a toy model we employ a system of interacting exciton-polariton condensates, where the on-site potential stems from the incoherent pump, and its spatial modulation provides the topological protection. We demonstrate that for the high purity systems with low noise amplitude a substantial coherence rate can be retained within a long-range transfer path, being an important prerequisite for practical applications.

## Appendix A: polariton model

A possible system for the realization of the proposed phenomena is a one-dimensional (1D) lattice of coupled micropillars [9]. Each micropillar contains a quantum well and is assumed to host a tightly bound exciton-polariton mode. In the mean-field approximation the evolution of the system can be described by discrete mean-field Gross-Pitaevskii equations

In the adiabatic approximation [55] we can write

In the first of Eqs. (9) pumping of the condensate is represented by the term $R n_{n,i}^{R}$, and losses by the decay rate $\gamma _c$. Considering quantum fluctuations only, the noise density is the sum of noise associated with these channels. Using Eq. (12), one has

Finally, substituting the Eqs. (12), (13) into Eq. (9), and adding the noise term (15), we reach at the Eq. (2) of the main text.

## Appendix B: scaling of the noise amplitude

Here we briefly discuss the dependence of first-order correlation function on the noise scaling factor $\beta$. For that, we study the temporal coherence of condensate in a single micropillar, defined as

## Funding

Ministry of Education and Science of the Russian Federation (14.Y26.31.0015); Narodowe Centrum Nauki (2016/22/E/ST3/00045, 2017/25/Z/ST3/03032).

## Acknowledgements

We acknowledge support from National Science Centre, Poland, grant No. 2016/22/E/ST3/00045 and grant No. 2017/25/Z/ST3/03032 under QuantERA program. VS acknowledges support from the mega-grant No. 14.Y26.31.0015 of the Ministry of Education and Science of the Russian Federation.

## Disclosures

The authors declare no conflicts of interest.

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