Theory of exciton-polariton condensation in gap-confined eigenmodes

Exciton-polaritons arising in low-dimensional nanostructures, such as coupled quantum well (QW) excitons and confined photonic modes, have been shown to behave as a weakly interacting Bose gas [1]. Condensation phenomena in these systems have to be inevitably described in terms of a balance between driving and losses, due to photon leakage or exciton recombination. In this respect, a suitable generalization of the standard nonlinear Schrödinger equation, known as the non-equilibrium Gross-Pitaevskii equation (NEGPE), is able to largely account for the observed phenomenology including external driving and intrinsic losses, either under resonant or non-resonant pumping of the exciton-polariton field [2]. In particular, spectacular phenomena such as Bose-Einstein condensation [3, 4], superfluidity [5], formation of quantized vortices [6] have been observed in planar semiconductor microcavities, in which the photon field is confined in the QW plane between two high-reflectivity Bragg mirrors [7]. Besides showing interest for fundamental physics studies, exciton-polaritons have also been shown to potentially allow for useful applications due to their superior nonlinear properties, such as low-threshold lasers [8], all-optical switching [9], sensing [10], etc.

Recently, polariton condensation has been observed also in multi-layered QW heterostructures with periodic surface patterning, such as the one sketched in Fig. 1. In fact, it has been shown that top/bottom mirrors are not necessary to achieve the polariton condensation threshold, since out-of plane radiation losses associated to the photonic component of the polariton field can be fully engineered by the photonic lattice [11], even a one-dimensional one [12]. In particular, radiative losses of the photonic component can be fully suppressed by symmetry along certain directions, leading to the currently widespread concept of photonic bound states in the continuum (BIC) [13, 14]. When coupled to QW excitons, these modes lead to the formation of polariton eigenmodes that are dark for emission along specific spatial directions and at fixed energy [15], inheriting the topologically protected nature of the purely photonic BIC [16, 17, 18]. Polariton condensation has been observed in these systems by efficient relaxation into modes arising from the saddle-like dispersion of the lower polariton branch, i.e., not an absolute minimum of the dispersion relation [19, 20]. In these works, the condensation mechanism has been qualitatively interpreted as due to the creation of an effective potential well for negative mass polaritons, induced by the exciting laser spot over a finite width along the periodicity lattice, as well as to the fact that such confined states acquire a longer lifetime than other modes due to coupling to purely photonic BICs. However, no quantitative theory has been provided to account for the condensation in such BIC-like modes, to date. In particular, a rigorous discussion about the role of the negative mass branch in connection with the photonic-induced gap obtained as a consequence of the periodic refractive index modulation is still missing. This manuscript aims at filling this theoretical gap, by developing a model of out-of-equilibrium condensation in a multi-band exciton-photon coupled system with symmetry-dependent losses. In particular, we hereby unravel the critical role played by the negative mass and the energy gap between dark and bright modes in reaching the condensation onset.

Here is a short outline of the results presented in the paper. In Sec. II we describe the dynamical equations used to picture the onset of polariton condensation in the continuous-wave (CW) regime of a model Hamiltonian with a given number of photonic branches coupled to QW exciton modes. Then, in Sec. III we address in detail the behavior of our effective theory in a specific case, where the relevant physics can be traced back to the presence of two counter-propagating photonic modes. In order to understand the polariton condensation in this theoretical framework, we first characterize the polariton dispersion arising from the Hamiltonian diagonalization, by deriving polariton bands in Sec. III.1, and the effects of an external space-dependent potential $V(x)$ coupled to the particle density in Sec. III.2. In Sec. IV we show the results concerning the polariton condensation in discrete energy eigenvalues lying within the energy gap, which is ultimately due to the periodic photonic modulation. These eigenvalues correspond to spatially confined eigenmodes below the potential $V(x)$. In particular, after showing the results of our numerical simulations describing the behavior of the condensate population in the steady-state, we provide an analysis of the condensate loss rate (Sec. IV.1), as well as the spectral density describing the energy-momentum behavior of the light emitted by the condensate (Sec. IV.2). The behavior of such quantities in all the cases considered in the present work evidently supports the conjecture that polaritons are condensing into the discrete gap-confined eigenmodes previously characterized in Sec. III.2. Depending on the dark or bright nature of the negative mass branch out of which they originate, these modes display very different direct and reciprocal space profiles. In light of these results, in Sec. V we finally discuss the relevance of our findings and future perspectives.

In this section we introduce a general dynamical model that can be used to describe and characterize the onset of polariton condensation and its relation to the eigenstates properties in heterostructures like the one sketched in Fig. 1. We will assume CW pumping in this work, although the model is general. The system is assumed to be uniform in the direction transverse to $x$, consisting of several layers with different refractive indices, in the figure denoted as $n_{1}$ and $n_{2}$. For instance [19, 20], such a stack of different index material might correspond to a set of GaAs/Al_xGa_1-xAs layers, which give rise to the formation of quantum well (QW) excitons in the lower band-gap material [21, 22]. Due to the top surface patterning with spatial periodicity $a$, the free-propagating electromagnetic modes get folded, leading to the formation of gapped Bloch resonances [11, 12]. When coupled to QW excitons, these photonic modes are shown to give rise to the concept of photonic crystal polaritons [23], which have been evidenced in different systems and material platforms [24, 25, 26, 27], all of them falling within the same theoretical treatment [16]. In particular, it has been evidenced that a surface patterning is sufficient to induce gapped polariton branches with positive and negative effective masses around normal incidence and energies below the bare exciton resonance [19, 20]. It has been observed that polariton condensation occurs within a set of discretized quantum states appearing within such energy gaps. Their emission properties are shown to depend both on the spatial profile of the input light-source [19, 28], $P(x)$ in Fig. 1, and on the intrinsic emission features of the lower-lying polariton branch delimiting the band-gap. Similarly to the description of polariton condensation in the planar microcavity case, here it is reasonable to assume that particle scattering towards the condensed eigenstate can be described by means of an exciton reservoir non-linearly coupled to polariton states [29, 30, 31, 32]. However, in the present case it is also crucial to account for the mechanism leading to the formation of such discrete states within the polariton gap, as well as the peculiar emission properties shown by either the bare polariton bands or the polariton condensate modes. Moreover, the presence of different photonic modes may lead to some of them being in weak coupling with the exciton, but still participating in the relaxation dynamics and thus worth being taken into account. In this respect, as it is shown in the following, our approach seems to provide a quite complete picture allowing to account for the observed phenomenology, as well as predicting possible new outcomes, when including all these crucial aspects.

In our model, similarly to the standard approach adopted to describe condensation in microcavity systems [29, 33, 34], we assume $P(x)$ to be incoherently coupled to a reservoir population, $n(x,\,t)$, describing high-energy exciton states. By stimulated scattering, such a reservoir feeds lower-energy eigenmodes, thus replenishing the exciton-polariton population in these states. However, in order to capture the complex band-gap physics, emission properties as well as the formation of discrete levels, instead of using a single component wavefunction $\psi(x,\,t)$ (usually describing the lower branch polariton field), here we assume that the reservoir density couples to a set of relevant bare photonic and excitonic modes of the structure. In other words, the system dynamics is hereby described by means of a multi-component vector, $\overrightarrow{\psi}(x,\,t)$ formally reading

in which $A_{l}\equiv A_{l}(x,\,t)$ and $X_{l}\equiv X_{l}(x,t)$ ($l\in[1,2,\,\cdots,\,L]$) denote the wavefunction of the $l$-th photonic and $l$-th excitonic mode respectively. In particular, $2L$ is the total number of polariton branches participating in the system dynamical evolution.
By setting $n\equiv n(x,\,t)$, $\overrightarrow{\psi}\equiv\overrightarrow{\psi}(x,\,t)$, we assume the reservoir-polariton dynamics to be described by the following set of coupled differential equations

in which $\langle\vec{w},\,\vec{z}\rangle=\sum_{j}w_{j}^{*}z_{j}$ denotes the hermitian scalar product between the two vectors $\vec{w}$ and $\vec{z}$, with $w_{j}^{*}$ denoting the complex-conjugated of the component $w_{j}$.

The different terms in Eq. (2a) account for three main physical contributions: (i) the injection of population due to the CW driving through $P(x)$; (ii) the presence of a loss term controlled by the intrinsic finite-lifetime $\tau_{R}$ of the reservoir; (iii) a non-linear decay term proportional to the effective coupling $g>0$, that describes the stimulated population scattering from the reservoir to photon and exciton states. The $2L\times 2L$ matrix $G$ represents an operator satisfying the following constraints:

Equations (2ca) and (2cb) ensure that the reservoir density evolves as a real quantity, with last term acting as an effective loss (i.e., gain on the $\vec{\psi}$ field, as clear from Eq. (2b)). The normalization constraint in Eq. (2cc) guarantees for a particle-conserving dynamics. Indeed, if on the one hand $g$ controls the global scattering rate from the reservoir, on the other the model accounts for several decay channels, each controlled by an element of the matrix $G$. Since a particle scattered away from the reservoir must create a single particle into the light-matter part of the model, the $\{G_{jj}\}$ must add up to 1.

The coupled exciton-photon dynamics is encoded into Eq. (2b). The term proportional to the (non-Hermitian) $2L\times 2L$ matrix operator $\tilde{H}(x)$, given by

accounts for the peculiar complex energy-momentum dispersion laws of the relevant bare modes and their mutual linear interaction, i.e., $-iE(x)/\hbar-\Gamma(x)$, as well as the presence of an external space-dependent potential, $W(x)$, such that $W(x)=W^{\dagger}(x)$. In particular, for the same reason leading to Eq. (2c), in order to interpret $\Gamma(x)$ as a proper decay operator, one must require $\Gamma(x)$ to satisfy the following constraints:

Finally, the last term in Eq. (2b) accounts for the particle transfer from the reservoir to photonic and excitonic modes. In particular, the structure of the nonlinear terms proportional to $g$ in Eq. (2), and the constraints imposed on $G$, lead to the expected $g$-independent behavior of the time derivative of the total particle density, $N(x,\,t)\equiv n(x,\,t)+\langle\overrightarrow{\psi}(x,\,t),\,\overrightarrow{\psi}(x,\,t)\rangle$. By combining the two expressions in Eq. (2), the following rate equation for $N(x,t)$ is obtained

In the long-time limit, since the CW source does not depend on time (i.e., the only driven Fourier component of the reservoir density is the $\omega=0$ one), it is reasonable to assume that the dynamical system does not produce oscillating solutions (limit cycles). As a consequence, the steady-state (ss) configuration, i.e., the solutions satisfying $\frac{d}{dt}n(x,\,t)=\frac{d}{dt}\overrightarrow{\psi}(x,\,t)=0$, correspond to a fixed point of the dynamical system in Eq. (2). In other words, by recalling that due to Eq. (2cb) $\langle\vec{\psi},G\vec{\psi}\rangle\geq 0\,\forall\,\vec{\psi}$, we have the steady state relations

in which $n_{ss}$ and $\overrightarrow{\psi}_{ss}$ denote the steady-state reservoir density and exciton-photon configuration, respectively. As it is easy to verify by direct inspection, the system of coupled equations (2g) always admits the trivial configuration corresponding to an empty exciton-photon subsystem and a reservoir density proportional to the CW profile as a fixed point solution, i.e.,

Depending on the pump strength $P_{0}$ (see Fig. 1), we notice that such a solution might become linearly unstable, i.e., a small perturbation $\delta\overrightarrow{\psi}(x)$ placed on top of $\overrightarrow{\psi}_{ss}=0$ gives rise to a macroscopic increasing of the condensate density, i.e., $\langle\delta\overrightarrow{\psi}(x),\delta\overrightarrow{\psi}(x)\rangle$, pushing the system away from the trivial configuration in Eq. (2h). By linearizing the dynamical system in Eq. (2), we obtain the following expression for the exciton-photon density evolution:

which implies that the empty solution becomes locally unstable whenever

that is when the particle gain induced by the CW source overcomes particle losses.

We conclude this section with a final remark. Even though we have reported a general formalism to account for $L$ photonic modes coupled to $L$ exciton states, in what follows we will restrict our attention to the $L=2$ case. Indeed, if on the one hand all the expressions previously reported hold true in the general case, on the one other hand such case study is of particular interest in light of the recent experiments [19, 20, 28], where it has been observed that the relevant dynamics leading to the onset of condensation only involves a pair of polariton bands below the exciton energy.

As discussed in Sec. II, in order to describe the onset of condensation we first address the spectral properties of the system. In particular, since we can safely assume the QW exciton energy-momentum dispersion to be flat, the only non-trivial part concerns the representation of the counter-propagating gapped Bloch resonances, in which the gap is induced by the periodic dielectric modulation, e.g., as sketched of Fig. 1. Even though a complete characterization of such modes in a one-dimensional (1D) periodic dielectric structure can be performed through a Maxwell solver based, e.g., on a guided-mode expansion [12, 11], here we rather follow the simplified approach discussed in [35], where it was shown that close to normal incidence (i.e., around $k=0$), the dispersion of gapped photonic branches can be reproduced and approximated by means of two counter-propagating modes with linear dispersion that are diffractively coupled by an off-diagonal term $U$. The latter can be either positive or negative, which is actually related to the composition of the unit cell of the periodic 1D lattice (see, e.g., Ref. [19]). In fact, we assume it as an effective model parameter, but we keep in mind that it can be fully engineered through the characteristics of the periodic pattern. We then consider the following matrices to build Eq. 2d

and

with $\mathbb{1}_{4}$ being the $4\times 4$ identity operator. Notice that we hereby assume that the potential $V$ is the same for either photonic or excitonic components, for simplicity of computations and without loss of generality. In fact, while in practical cases one might differentiate between photonic and excitonic contributions, the qualitative behavior of results we are going to discuss do not depend from the choice made here. In this scenario, the wavefunction $\overrightarrow{\psi}(x,\,t)$ reduces to the following four-components vector:

This model describes two photonic modes separately coupled to two degenerate excitonic states, in the presence of an external potential $V(x)$ that is directly related to the presence of the CW source; $A_{+}$ and $A_{-}$ describe counter-propagating bands with linear dispersion whose slope is directly defined by their group-velocity, $v_{g}$. These modes cross at $k=0$ with energy $\hbar\omega_{A}$, and they have an intrinsic loss rate given by $\gamma_{A}$. The two modes $X_{+}$ and $X_{-}$ are then associated to a flat QW exciton resonance at energy $\hbar\omega_{X}$, whose bare loss rate is $\gamma_{X}$. The photonic modes are assumed to be linearly coupled via a term proportional to $\hbar U-i\hbar\tilde{\gamma}_{A}$, which converts right ($+$) propagating photons into left ($-$) propagating ones, and viceversa. In particular, in order to ensure that $\Gamma_{0}$ describes a proper decay term, one must require that $\gamma_{A}\geq|\tilde{\gamma}_{A}|$ (see Eq. (2e)). Each photonic component is assumed to be independently coupled to an exciton mode, with a coupling rate set by the Rabi frequency $\Omega_{R}/(2\pi)$.

As it is shown in the following, such a minimal model is already able to account for the onset of polariton condensation into discrete levels appearing inside the energy-gap between polariton bands with opposite effective-mass. In close agreement with experimental results reported in the literature, the properties of such levels is shown to depend on both the bare polariton dispersion and the pump characteristics. We first pay attention to the spectral properties of the non-Hermitian operator $\tilde{H}(x)$. For the sake of clarity, we proceed in two steps. In Sec. III.1, we first characterise the polariton dispersion in the absence of the external potential $V(x)$. The role of such space-dependent term in the appearence of a set of discrete levels is addressed in Sec. III.2.

As it is known from previous studies, polariton condensation is identified from accumulation of excitations in one specific state, usually the lowest energy one in planar microcavities, and this phenomenology is well captured by a one-dimensional single-mode GPE formulation. In this Section we show numerical results concerning the onset of polariton condensation in the steady-state regime, when the coupled multi-band exciton-photon system is connected to a particle reservoir driven by a spatially-dependent CW input source. The expression of such driving term is given as

in which $P_{0}$ is a time-independent pump-rate per $\mu$m, i.e., $[P_{0}]=$ ($\mu$m$\cdot$ps)^-1. In particular, following the discussion from the previous Section, it is hereby assumed that the pumping rate is linearly related to the barrier height, $V_{0}$ by the following relation:

in which $\eta$ is a parameter having the dimension $[\eta]=\mu$m^-1. Concerning the reservoir-polariton interactions, we assume the two excitonic modes to be similarly coupled to the reservoir density, and we made the same assumption for the photonic components for simplicity of computation and without loss of generality. In this scenario, the operator $G$ is diagonal and, due to the constraints reported in Eqs. (2cb) and (2cc), it has the following structure:

with $\alpha$ being an effective parameter such that $0\leq\alpha\leq 1$.

Since we are interested in the stationary solutions of Eq. (2) at large time $t$, and since the model always admits as a possible solution $\vec{\psi}(x)=0$, we consider an initial configuration characterized by a small population. By doing so, we are able to understand whether a tiny perturbation added to the “empty” solution actually leads to the appearance of a stable, macroscopically populated condensate. Details about the system initialization are given in App. C. Such initial configuration is then evolved in time by means of a standard numerical integration algorithm (i.e., explicit Runge-Kutta 4(5) method), and the convergence towards a possibly nonempty condensate configuration is monitored by looking at the time behavior of the total population, that is

where $A_{\sigma}\equiv A_{\sigma}(x,\,t)$ and $X_{\sigma}\equiv X_{\sigma}(x,\,t)$.

An example of time evolution is reported in Fig. 7, where the behavior of $N_{\psi}(t)$ as a function of time is again compared for the cases $U>0$ and $U<0$, assuming the same CW driving protocol and the same values for $V_{0}$. Even if the initial population injected in the light-matter subsystem is small (see App. C), independently of the sign of $U$, for sufficiently large values of $V_{0}$ the system dynamics tends towards a stable steady-state configuration displaying a macroscopic occupation, i.e.,

Interestingly, according to the results shown in Fig. 7, the convergence towards a nonempty condensate seems to depend on the sign of $U$. Indeed, if on the one hand for $V_{0}>1$ meV in both cases the system approaches a configuration with $N_{\psi,\,ss}>0$, on the other hand at exactly $V_{0}=1$ meV polariton condensation is only triggered for $U>0$. In particular, since $V_{0}$ is in one-to-one correspondence with $P_{0}$ (Eq. (2al)), such behavior suggests that in systems with $U>0$ condensation would occur for lower values of $P_{0}$, i.e., they would display a lower condensation threshold. An interesting prediction to be verified experimentally.

In order to understand whether this is indeed the case, we performed a characterization of the dependence of the steady-state occupation $N_{\psi,\,ss}$ on $V_{0}$. Results obtained for different values of the photonic losses are shown in Fig. 8. As in the previous cases, we compare the trend in data obtained for $U>0$ (triangles) and $U<0$ (squares), respectively. For what concerns the condensation threshold, data corresponding to $U>0$ do not display a significant dependence on $\hbar\gamma_{A}$. In particular, for all the cases considered the critical value of $V_{0}$ leading to condensation is slightly below 1 meV. On the contrary, at first glance, numerical results for $U<0$ display a considerably different behavior. In particular, the condensation threshold in this case shows more pronounced dependence on the photonic losses, going from $V_{0}\simeq 1$ meV to $\simeq 3$ meV while increasing $\hbar\gamma_{A}$ from $0.1$ to $0.3$ meV.

In addition, data obtained for values of $U$ having a different sign show a quantitatively different dependence on $V_{0}$. If on the one hand they clearly show that condensation is occurring in both cases, on the other hand the value of $U$ does not only affect the condensation threshold. In the next two sections we show that all these peculiar features of the model can be interpreted by looking at the properties of the states previously characterized in Sec. III.2. In particular, we first address the behavior of the losses of the condensed states. Then, we analyse the spectral properties of the radiation emitted from the condensate, and we show that it is peaked at energies corresponding to the discrete levels considered in the previous section.

We have reported an extensive theoretical analysis of the polariton condensation mechanism occurring in periodically patterned QW multilayers under incoherent driving. From a theoretical point of view, this work has required the generalization of the non-equilibrium Gross-Pitaevskii formulation under incoherent reservoir driving, originally proposed in Ref. [29], in order to account for multiple photonic bands interacting with the same QW exciton degenerate modes, which is an original contribution of the present work.

After analyzing the complex eigenmodes of the non-Hermitian Hamiltonian model including exciton-photon coupling and losses, we have numerically solved the driven-dissipative dynamics in the presence of a continuous wave Gaussian pump spot. We have modelled the effects of such a driving field as an effective potential barrier for the exciton-photon field, showing that it naturally induces the confinement of negative mass polaritons arising from the lower branch and becoming trapped due to the energy gap between the two polariton bands at energies below the exciton resonance.

In addition, we have studied the emission characteristics of these polaritonic branches, which are associated to the sign of the diffractive coupling term introduced to mimic the effect of the periodic pattern on photonic eigenmodes. In particular, the positive sign of such diffractive coupling term is associated with the lower, negative mass branch being dark at normal incidence (BIC condition) with minimal losses, while if it is negative this same branch becomes bright at normal incidence and its imaginary part is much larger. Analysing the behavior of the model as a function of the pump intensity (effectively described by the height of this potential barrier), we have found that condensation actually occurs in such gap-confined eigenmodes, which become spatially quantized as a function of the depth of the confining potential. We then make the relevant conclusion that condensation in these systems is indeed the result of such gap-confinement of negative mass polaritons, irrespective of their dark or bright emission at normal incidence. In particular, these results are in remarkable agreement with recent experimental findings corresponding to samples in which the diffractive coupling term is positive [19, 20]. Hence, we predict that a similar phenomenology is going to occur even in the opposite case of a negative diffractive coupling term, which motivates new experiments.

As a continuation of the present work, we envision applications of this theoretical framework to more complex pumping configurations, such as, e.g., periodic repetitions of a finite number of pumping spots, as recently reported in experiments [28]. In addition, a careful benchmarking of this effective model with experimental data might be a further test of the usefulness of the proposed approach. In particular, since experiments are often performed under pulsed excitations, an analysis of the solutions of the present model for pulsed driving, which is beyond the scope of the present manuscript, might be one of the possible next steps to undertake as a future project. We also notice that the model employed is a one-dimensional one, which essentially captures the relaxation mechanism in an effective way. On the other hand, the detailed dynamics along the transverse direction is completely neglected, which might still play a role in experiments. Hence, a possible further extension of the model might include the description of the modes dispersion along the transverse spatial direction as well.