Chiral modes at exceptional points in exciton-polariton quantum fluids

TL;DR

This paper demonstrates how to generate chiral vortex modes with fixed handedness in exciton-polariton quantum fluids by exploiting exceptional points in a non-Hermitian resonator, enabling control over orbital angular momentum states.

Contribution

It introduces a method to produce chiral vortex modes at exceptional points in exciton-polariton systems, advancing control over topological states in quantum fluids.

Findings

Vortex states with fixed orbital angular momentum are generated near exceptional points.

Precise manipulation of the resonator's potential enables transition through the exceptional point.

The approach can be extended to create high-order orbital angular momentum states.

Abstract

We demonstrate generation of chiral modes -- vortex flows with fixed handedness in exciton-polariton quantum fluids. The chiral modes arise in the vicinity of exceptional points (non-Hermitian spectral degeneracies) in an optically-induced resonator for exciton polaritons. In particular, a vortex is generated by driving two dipole modes of the non-Hermitian ring resonator into degeneracy. Transition through the exceptional point in the space of the system's parameters is enabled by precise manipulation of real and imaginary parts of the closed-wall potential forming the resonator. As the system is driven to the vicinity of the exceptional point, we observe the formation of a vortex state with a fixed orbital angular momentum (topological charge). Our method can be extended to generate high-order orbital angular momentum states through coalescence of multiple non-Hermitian spectral degeneracies, which could find application in integrated optoelectronics.

Equations

$$i\hslash \frac{\partial \psi }{\partial t}=$$

$$\frac{\partial n_R}{\partial t}=$$

$$\begin{array}{rl}V(\text{r})=& V_R(\text{r})+i{V}_I(\text{r})\\ \approx & -\frac{g_{R}P(\text{r})}{{\gamma }_R}+i\frac{\hslash }{2}\left[\frac{RP(\text{r})}{{\gamma }_R}-{\gamma }_c\right].\end{array}$$

$$\left[-{\nabla }^2+(V^{\prime }+i{V}^{\prime \prime })\right]\psi =\tilde{E}\psi .$$

$$\hat{H}=\left[\begin{array}[]{cc}\tilde{E}_{n}&q\\ q^{*}&\tilde{E}_{n^{\prime}}\\ \end{array}\right],\quad\quad\tilde{E}_{n,n^{\prime}}=E_{n,n^{\prime}}-i\Gamma_{n,n^{\prime}},$$

Full text

Introduction. Exceptional points in wave resonators of different origin arise when both spectral positions and linewidths of two resonances coincide and the corresponding spatial modes coalesce into one Berry2004 ; Heiss2012 . Originally identified as an inherent property of non-Hermitian quantum systems Bender2007 ; Moiseyev2011 ; Bird15 , exceptional points have become a focus of intense research in classical systems with gain and loss Wierzig2015 , such as optical cavities An2009 , microwave resonators Dembowski2001 ; Dembowski2003 , and plasmonic nanostructures PlasmonicsEP2016 . The counterintuitive behaviour of a wave system in the vicinity of an exceptional point led to demonstrations of a range of peculiar phenomena, including enhanced loss-assisted lasing Peng2014 ; Rotter2014 , unidirectional transmission of signals Segev2010 , and loss-induced transparency Guo2009 .

Due to the nontrivial topology of the exceptional point, the two eigenstates coalesce with a phase difference of $\pm\,\pi/2$, which results in a well-defined handedness (chirality) of the surviving eigenstate Heiss2001 . This remarkable property of the eigenstate at the exceptional point was first experimentally demonstrated in a microwave cavity Dembowski2003 and, very recently, led to observation of directional lasing in optical micro-resonators Peng2016 ; vortex_laser . So far, the chirality of the unique eigenstate at an exceptional point has not been demonstrated in any quantum system.

In this work, we demonstrate formation of a chiral state at an exceptional point in a macroscopic quantum system of condensed exciton polaritons. Exciton polaritons are hybrid light-matter bosonic quasiparticles arising due to strong coupling between excitons and photons in semiconductor microcavities Deng_10 ; CiutiREV13 . Once sufficient density of exciton polaritons is injected by an optical or electrical pump, the transition to quantum degeneracy occurs, whereby typical signatures of a Bose-Einstein condensate emerge Deng_02 ; BEC06 ; BEC07 ; Deng_10 ; CiutiREV13 ; YamamotoREV14 . Radiative decay of polaritons results in the need for a continuous pump to maintain the population. This intrinsic open-dissipative nature of exciton-polariton condensates offers a new platform for study of non-Hermitian quantum physics. Several recent experiments have exploited the non-Hermitian nature of exciton-polariton systems Savvidis01 ; weak_lasing ; Lieb ; Gao2015 . Importantly, the existence of exceptional points and the associated topological Berry phase has been demonstrated in an optically-induced resonator (quantum billiard) for coherent exciton-polariton waves Gao2015 .

An optically-induced exciton-polariton resonator is a closed-wall potential arising due to injection of high-energy excitonic quasiparticles by an off-resonant optical pump and strong repulsive interaction between the excitonic reservoir and the condensate Bloch2010 . The size of the resonator is comparable to the de Broglie wavelength of the condensed exciton polaritons, and its geometry is defined by the spatial distribution of the optical pump Manni2011 ; Cristofolini13 ; Askitopoulos13 . Observation of the exceptional points in the spectra of exciton-polariton resonators is enabled by two characteristic features of this system. First, the optically confined exciton polaritons form a multi-mode condensate, i.e. they can occupy several single-particle energy states of the pump-induced effective potential Tosi12 . Secondly, the pump-induced potential is non-Hermitian, and both real (energy) and imaginary (linewidth) parts of its complex eigenenergies can be precisely controlled by adjusting parameters of the pump Gao2015 . As a result, two or more eigenstates of the system can be brought to degeneracy.

Here, we create a non-Hermitian trapping potential for exciton polaritons in the form of an asymmetric ring resonator, and observe condensation into several trapped modes. By changing the geometry of the pump, and therefore the overlap of the modes with the gain region, we tune the imaginary part of the optically-induced potential and observe the transition from crossing to anti-crossing of complex eigenvalues, which signals the existence of an exceptional point. Furthermore, the high-Q tapered microcavity used for our experiments SnokePRX2013 enables precise control over the ratio of the exciton and photon in the hybrid quasiparticle. We use this additional control parameter to drive the two lowest-lying dipole states of the system to a vicinity of an exceptional point and confirm the formation of a chiral mode – a charge one vortex – in analogy with microwave Dembowski2003 and optical Peng2016 ; vortex_laser resonators. In addition, we demonstrate formation of a mode with a higher-order topological charge (orbital angular momentum), when two coexisting exceptional points are tuned into close proximity of one another, thus opening the avenue for experimental tests of topological properties of higher-order exceptional points.

Experiment. We create exciton polaritons in a high-Q GaAs/AlGaAs microcavity similar to that used in Ref. SnokePRX2013 . The details about the experimental setup can be found in Supplemental Material (SM). By utilising a digital micromirror device (DMD), we morph the pump spot into the asymmetric ring shape shown schematically in Fig. 1. The pump simultaneously populates the system with exciton polaritons and forms a closed-wall potential due to the local blue shift in energy induced by the excitonic reservoir Bloch2010 . Similarly to Ref. Gao2015 , the inner area of the ring is kept constant and the varying widths of the potential walls, $W$ and $w$, affect the overlap between exciton-polariton condensate modes and the gain region. Analysis of the microcavity photoluminescence by means of energy resolved near-field (real-space) imaging allows us to obtain the spatial density distribution and energy levels corresponding to the condensate modes in the ring resonator. All experiments are performed in the strong coupling regime and the pump power is kept at around $1.5$ times that needed for condensation.

Due to the asymmetry of the potential walls imposed by the pumping geometry, as well as by the cavity gradient SnokePRX2013 , the eigenmodes of the optically-induced ring resonator resemble the Ince-Guassian modes Ince_Gaussian . Once the threshold for the condensation is reached, only a few modes of this resonator are occupied, and we focus on the two lowest-lying dipole modes $(1,1)$ with the orthogonal orientation of the nodal lines. Due to their orientation, the two dipole modes have different overlap with the exciton reservoir and are, in general, not energy degenerate and well separated from the other modes (see SM).

Because of the non-Hermitian nature of exciton polaritons, the eigenenergy of the modes in the ring potential is also complex, where the real part corresponds to the energy peak position, and the imaginary part corresponds to the linewidth. In our experiment, the peak positions corresponding to the two dipole modes can be tuned by changing the relative admixture of exciton and photon in the exciton polariton. This is achieved by keeping all parameters of the pump fixed and changing the relative position of the excitation beam and the sample, as a sizeable linear variation in the microcavity widths SnokePRX2013 results in variation of detuning between the cavity photon mode and bare exciton: $\Delta=E_{ph}-E_{ex}$. The change in the energies of the two dipole modes with changing detuning is shown in Fig. 2(a), where a clear crossing of the corresponding energy levels can be observed. (The corresponding dispersions below condensate threshold for varying detuning can be seen in SM). The transition to spectral degeneracy is accompanied by an avoided crossing of the imaginary parts of the eigenenergy (linewidths), as seen in Fig. 2(b). We stress that, due to the high quality factor of the microcavity and long lifetime of exciton polaritons in this microcavity ($\sim~{}200$ ps), the resonances have a very narrow linewidth, which helps to differentiate between closely positioned energy levels.

From the energy resolved spatial image of the cavity photoluminescence, we reconstruct the spatial probability density distribution of the polariton condensate wavefunction for each detuning value. Away from the degeneracy, the two dipole modes are clearly visible [insets in Fig. 2(a)]. As the two complex eigenvalues are tuned in and out of the degeneracy, we observe the characteristic exchange of the modes corresponding to the two energy branches. This kind of mode switch may indicate the existence of an exceptional point, so we vary a second control parameter in our system, which is the width of the right half-ring of the resonator, $w$ (see Fig. 1). As the value of this parameter is changed from $w=w_{1}$ to $w=w_{2}$, without changing the size of the resonator, $D$, we observe transition from crossing to anticrossing in energy and from anticrossing to crossing in the linewidths of the two resonances corresponding to the dipole modes [see Figs. 2(c,d)]. This transition confirms that an exceptional point exists in the parameter space $(\Delta,w)$ HeissPRE00 ; Wierzig2015 ; Gao2015 . Furthermore, by fixing the detuning at $-3.6$ meV, which corresponds to the energy crossing in Fig. 2(a), and tuning the right half-ring width from $w_{2}$ to $w_{1}$, we observe a vortex-like mode at $w_{1}<w<w_{2}$, as shown in Fig. 3(e).

In order to ascertain the nature of this state, we perform interferometry with a magnified ($\times 15$) reference beam derived from a small flat-phase area of the photoluminescence. First of all, the energy-resolved interferometric imaging confirms the phase structure of the two dipole modes Fig. 3 (a,b) and (c,d) away from the degeneracy point. Furthermore, the interference pattern shown in Fig. 3 (f) is stable for many minutes, and reveals a fork in the fringes which is a clear signature of a stable charge-one vortex. This measurement therefore confirms that a vortex with the topological charge one is formed in the vicinity of the spectral degeneracy, which is only possible if the two dipole modes coalesce with the $\pi$/2 phase difference. We note that due to the finite energy (frequency) resolution of spectroscopic measurements in our system, it is impossible to tune the system exactly to the exceptional point, whereby both real and imaginary parts of the complex egenenergy, as well as the eigenstates, would coalesce.

Theory. The full dynamics of an exciton-polariton condensate subject to off-resonant (incoherent) optical pumping can be described by the generalised complex Gross-Pitaevskii equation for the condensate wave function complemented by the rate equation for the density of the excitonic reservoir Wouters07 :

[TABLE]

where $P(\textbf{r})$ is the rate of injection of reservoir particles per unit area and time determined by the pump power and spatial profile of the pump beam, $g_{c}$ and $g_{R}$ characterise interaction between condensed polaritons, and between the polaritons and the reservoir, respectively. The decay rates $\gamma_{c}$ and $\gamma_{R}$ quantify the finite lifetime of condensed polaritons and the excitonic reservoir, respectively. The stimulated scattering rate, $R$, characterises growth of the condensate density.

Assuming that, under cw pumping, the reservoir reaches a steady state, $n_{R}(\textbf{r})=P(\textbf{r})/\left(\gamma_{R}+R|\psi(\textbf{r})|^{2}\right)$, and that the exciton-polariton density is small near the condensation threshold, Eq. (1) transforms into a linear Schrödinger equation for the condensate wavefunction, $\psi$, confined to an effective non-Hermitian potential Gao2015 :

[TABLE]

Both real and imaginary parts of this potential depend on the spatial profile of the pump $P(\textbf{r})$, as well as on the exciton-photon detuning, $\Delta$. As discussed in SM, the dependence on the detuning enters the model Eq. (1) through the dependence of its parameters on the Hopfield coefficient that characterises the excitonic fraction of the exciton polariton Deng_10 : $|X|^{2}=(1/2)\left[1+\Delta/\sqrt{\Delta^{2}+E_{R}^{2}}\right]$, where $E_{R}$ is the Rabi splitting at zero detuning.

The eigenstates and complex eigenvalues of the non-Hermitian potential (2) can be found by solving the non-dimensionalised stationary equation:

[TABLE]

where we have introduced the scaling units of length, $L=D$, energy $E_{0}=\hbar^{2}/(2m_{ph}L^{2})$, and time $T=\hbar/E_{0}$, $m_{ph}$ is the effective mass of the cavity photon, and $\tilde{E}$ is the complex eigenenergy normalised by $E_{0}/(1-|X|^{2})$. The normalised real and imaginary parts of the potential are: $V^{\prime}({\bf r})=V^{\prime}_{0}\Gamma_{X}P({\bf r})/P_{0}$, $V^{\prime\prime}({\bf r})=V^{\prime\prime}_{0}\left[\Gamma_{X}P({\bf r})/P_{0}-1\right]$, where $V^{\prime}_{0}=g_{ex}\gamma_{ph}/(E_{0}R_{0})$, $V^{\prime\prime}_{0}=\hbar\gamma_{ph}/(2E_{0})$, and $P_{0}=\gamma_{R}\gamma_{ph}/R_{0}$, where $\gamma_{ph}$ is the lifetime of the cavity photon, $g_{ex}$ is the stregth of the exiton-exiton interaction TassonePRB99 , and we have introduced a base value for the stimulated scattering rate $R_{0}$ (see SM). The dependence on detuning via the Hopfield coefficient is captured by the parameter $\Gamma_{X}=|X|^{2}/(1-|X|^{2})$. For simplicity, we approximate the shape of $P(\textbf{r})$ by Gaussian envelope functions, with the resulting shapes of $V^{\prime}({\bf r})$ and $V^{\prime\prime}({\bf r})$ shown in Fig. 4(a,d).

By solving the eigenequation (3) numerically and sorting values of $\tilde{E}$ in the ascending order of its real part, $E_{n}$, we obtain the corresponding hierarchy of eigenstates $\psi_{n}$. Figures 4(b) and (e) show the moduli of $\psi_{2}$ and $\psi_{3}$, respectively, which correspond to the slightly deformed dipole states $(1,1)$ with the orthogonal orientation of nodal lines. Importantly, the existence of these steady states of the exciton polariton condensate in the pump-induced potential is also confirmed by full dynamical simulations of the model equation (1) (see SM).

The dependence of the real and imaginary parts of the dipole modes eigenenergies on the experimental control parameters, $\Delta$ and $w$, is shown in Fig. 5. We can see that our simple linear model reflects the qualitative behaviour observed in the experiment (Fig. 2).

To understand why the experimental control parameters $\Delta$ and $w$ allow us to tune the system in and out of the vicinity of the exceptional point, we follow the standard approach BliokhPRL08 ; Heiss2012 ; Wierzig2015 ; Gao2015 , and construct a phenomenological couple-mode model for the two modes with the quantum numbers $n$ and $n^{\prime}$ near degeneracy (see SM for the details). The resulting effective two-mode interaction Hamiltonian can be written as follows:

[TABLE]

where $\tilde{E}_{n,n^{\prime}}$ are the complex eigenenergies of the uncoupled modes and $q$ characterises their coupling strength. The eigenvalues of the Hamiltonian (4) are $\lambda_{n,n^{\prime}}=\tilde{E}\pm\sqrt{\delta\tilde{E}^{2}+|q|^{2}}$, where $\tilde{E}=(\tilde{E}_{n}+\tilde{E}_{n^{\prime}})/2\equiv E+i\Gamma$, and $\delta\tilde{E}=(\tilde{E}_{n}-\tilde{E}_{n^{\prime}})/2\equiv\delta E-i\delta\Gamma$. The real and imaginary parts of $\lambda_{n,n^{\prime}}$ form Riemann surfaces with a branch-point singularity in the space of parameters $(\delta E,\delta\Gamma)$ Gao2015 , as shown in SM. At the exceptional points, $i\delta\tilde{E}_{EP}=\pm|q|$, the eigenvalues coalesce, $\lambda_{n}=\lambda_{n^{\prime}}$. The eigenstates also coalesce and form a single chiral state Heiss2001 ; Heiss2012 . In our system, the two eigenstates corresponding to $n=2$ and $n^{\prime}=3$ are dipole modes, and therefore the chiral state is a vortex with a well-defined topological charge one, as shown in Fig. 4(c,f) and Fig. 3(c,d).

The two parameters $(\delta E,\delta\Gamma)$ can be related to the experimental parameters $(\Delta,w)$. As discussed in SM, increasing $\Delta$ corresponds to increasing $\delta E$ and moves the system away from the spectral degeneracy, while growing $w$ corresponds to decreasing $\delta\Gamma$. Therefore the variable detuning and the width of the resonator’s wall allow us to control the approach to the exceptional point as demonstrated in the experiment (Fig. 2) and confirmed by theory (Fig. 5).

Conclusion. In summary, we have experimentally demonstrated the chirality of an eigenstate of a non-Hermitian macroscopic quantum coherent system of Bose-condensed exciton polaritons in the vicinity of an exceptional point. In our experiment, the chiral eigenstate is a vortex with a well-defined, deterministic topological charge (orbital angular momentum). We stress that, contrary to the previous demonstration of an exciton polariton vortex with a well-controlled charge chiral_lens , the shape of the optically induced ring resonator that creates a non-Hermitian potential for exciton polaritons in this work does not break the chiral symmetry. In such case, one would expect that the sign of the vortex charge would change randomly between realisations of the experiment Yulin ; Deveaud , which is not the case here. The deterministic nature of the vortex charge has been confirmed in our experiment by blocking the pump for a sufficient time interval (up to $1$ hour) to let the pump-injected reservoir disappear. After the pumping is resumed, the charge of the vortex remains the same. The controlled and reliable generation of a chiral state with a prescribed topological charge could find use in polariton-based integrated optoelectronic devices.

Last but not least, our technique of generating chiral states can be applied to higher-order modes in the non-Hermitian exciton-polariton system. In particular, if the inner size of the ring, $D$, is increased, higher-order modes can be populated by the condensate and brought to degeneracy by varying control parameters of the system. Moreover, two (or more) exceptional points can be simultaneously created and brought to a close vicinity of each other. As an example, the higher-order orbital momentum state formed in our experiment by hybridisation of two chiral modes, $\psi=\psi_{7}+i\,\psi_{8}+\psi_{9}+i\,\psi_{10}$, is shown in Fig. 6, together with the prediction of our linear theory. The interferometry image confirms that the vortices in the triple-vortex state have the same topological charge (see also a double-vortex state shown in SM). These results offer further opportunities to explore the physics of higher-order exceptional points EP3_1 ; EP3_2 ; EP3_3 ; EP3_4 and exceptional points clustering EP3_5 ; EP_clustering in an open quantum system.