## Abstract

We investigate theoretically the quantum oscillator-like states recently observed experimentally in polariton condensates. We consider a complex Gross–Pitaevskii-type model that includes the effects of self-interactions, and creation and decay of exciton-polaritons. We develop a perturbation theory for approximate solutions to this non-equilibrium condensate model and compare the results with numerically calculated solutions for both repulsive and attractive polariton–polariton interactions. While the nonlinearity has a weak effect on the mode selection, the condensate density profiles are modified at moderate gain strengths. We find the nonlinearity becomes more dominant when a very large gain of polaritons leads to an extended cloud with high condensate densities. Finally, we identify the relation of the observed patterns to the input pump configuration, and suggest this may serve as a generalized NOR gate in the tradition of optical computing.

## 1. Introduction

Bose–Einstein condensates (BECs) are prime examples of *nonlinear* quantum many-body systems because of interactions between coherent quantum particles. They are regarded as a macroscopic quantum phenomenon owing to their very high coherence and long-range order of all particles constituting this state of matter, a quantum state that typically is formed at very low temperatures [1–6]. Many set-ups of atomic BEC are for fixed particle numbers and can carry various topologically stable excitations ranging from solitons to giant vortices [3,7–10]. More recently, efforts have been made towards the realization of coherent *non-equilibrium* many-body systems that exchange particles and/or energy while at the same time obeying BEC properties, e.g. in solid-state systems [11,12]. One such system is the polariton quasi-particle BEC [2,4,6,11,12], which can be effectively described by a non-conservative nonlinear Schrödinger-type equation [6,13,14] formally generalizing the so-called Gross–Pitaevskii equation (GPE)—the model for the condensate wave function of dilute and weakly interacting atomic BEC [5,13,14].

Bose condensed polaritons have become feasible in semiconductor microcavities in the past decades [2,6,11,12]. Polaritons in semiconductor microcavities are superpositions between excitons (electron–hole pairs) and cavity photons, and as such the combination of the components determines the particle statistics [2,15]. These quasi-particles in a dilute regime show distinctive properties such as Bose–Einstein condensation, superfluidity, a finite non-zero speed of sound owing to the nonlinearity and the emergence of elementary topologically stable excitations such as solitons or quantum vortices [2,6,13,16,17]. In their very nature, polariton BECs are non-equilibrium systems—polaritons can be created via a local light field and decay after 1–100 ps mainly owing to leaking of the cavity photons [6,11,18]. The decay rate significantly depends on the quality of confinement of the light field [18] within the microcavity, and in this work, it is assumed to be constant over space. From a technological and experimental point of view, polariton BECs have several practical advantages over atomic BECs. For instance, BECs of dilute atoms must be realized at temperatures within the nano-Kelvin range [19–21], whereas polariton condensates can be observed even at room temperature in polymers [22] and in organic materials [23]. This is a consequence of the fact that polaritons have a very light mass owing to their inherent coupling to photons [6], and so have a much higher critical Bose condensation temperature. Restrictions on the polariton density, however, have to be taken into account to ensure their bosonic character is maintained [3].

In many settings, a polariton condensate is an effectively two-dimensional coherent many-body system that is tightly confined with regards to the optical axis of the semiconductor microcavity [2,11,12,24,25]. Depending on additional constraints the condensate may become essentially one dimensional [2,16]. The elementary excitations (nonlinear states or eigenmodes) within a one-dimensional system, with repulsive polariton–polariton interactions, are dark and grey solitons [6,16]. Grey solitons are stable excitations that move over long distances without changing their form and in principle with elastic scattering properties, but owing to the non-equilibrium character, they have a finite lifetime [26]. Dark solitons are the stationary special case of the grey soliton when the density depletes to zero at its minimum and with a *π* phase shift at the nodal point [5].

In this work, we are particularly interested in stationary and stable patterns within a one-dimensional polariton condensate, which become feasible when the gain and loss of particles is balanced, and once excited are similar to dark soliton [13] arrays [16] with respect to singularities at localized points—the oscillator modes of a harmonically trapped quantum state. These states form a family of eigenstates comparable to those in the linear case, but including the effects of growth and decay as well as nonlinear self-interactions. Physically, these stationary eigenstates are formed by local illumination of the semiconductor microcavity with a laser beam and the decay of the polaritons balances this gain. Experimental observations of oscillator modes in polariton condensates have been made in references [27,28], but a thorough theoretical description is lacking. The experimentally observed non-equilibrium quantum oscillator states were formed between two spatially separated pump spots of incoherently pumped polaritons, corresponding to two localized laser beams shining on the semiconductor microcavity and positioned within the two-dimensional plane. Between the pump spots, stationary spatial patterns were formed similar to those observed for a quantum harmonic oscillator, hence suggesting an effective one-dimensional problem.

The spatial patterns can be understood in a dynamical picture as follows. The polaritons generated at the spots move from the source peaks in all directions, because interactions of pump and reservoir with polaritons are repulsive [24]. So in between the pump spots, the bulk of polaritons are present and thus the condensate and its patterns form predominantly there. Depending on the pump strength, and the repulsiveness from the pump/reservoir, different modes can be observed; starting from the ground state of approximately Gaussian form (when nonlinearity plays a minor role). Experimentally, only the case for repulsive self-interactions has been considered so far. In this work, we predict—based on the recent experimental effort [29]—the states for attractive self-interactions as well and their dependence on the gain. To elucidate the role of nonlinear self-interactions for the pattern formation in quasi-harmonically trapped polariton condensates, we vary between attractive and repulsive interactions and the nonlinear character of polariton BEC will become dominant in a Thomas–Fermi (TF)-type regime, i.e. when condensate densities are very high and pumping is strong.

Our paper is organized as follows. First, we introduce the general model describing the polariton condensate in two dimensions and a simplified model for the effective one-dimensional investigations. Then, approximate analytical expressions for the condensate wave functions are presented and compared with numerical simulations. Subsequently, the dependence of the eigenvalues on pumping strength and the nonlinearity is presented. Finally, we introduce a generalized optical NOR device and summarize our findings.

## 2. Polariton condensate model

We describe the order parameter *ψ* of the polariton condensate by the state equation in the spin coherent case [14,30], which has proved to be an accurate description of recent experiments [13,24],
*m** is the effective mass of the condensed polaritons and *P* describes the geometry and magnitude of the pump spots, which are induced by linearly polarized laser beams [27,28]. The polariton reservoir is approximately given by [13,24]
*γ*_{C} denotes the globally constant decay rate of condensed polaritons, *γ*_{R} the relaxation of the reservoir, *β* is the scattering rate of polaritons into the condensate, *α*_{1} denotes the polariton–polariton interactions within the condensate, *g*_{P} is the repulsive interaction of the condensate with the pump and *g*_{R} the repulsive interactions of the condensate with the reservoir. Parameter values are chosen in accordance with recent experiments [13,16], i.e. effective mass *m**=5×10^{−5}*m*_{0}, where *m*_{0} is the mass of the free electron, *α*_{1}=0.001 ps^{−1} μm^{2}, *g*_{R}=0.022⋅ps^{−1} μm^{2}, *g*_{P}=0.07 ps^{−1} μm^{2}, *β*=0.05 ps^{−1} μm^{2}, *γ*_{R}=10 ps^{−1}, *R*_{R}=0.06 ps^{−1}⋅μm^{2}, *γ*_{C}=0.556 ps^{−1}. The pump geometry is assumed to be
*A* being the amplitude and the position of the two Gaussian pump spots is at *x*_{2} and *x*_{1}, respectively, which models the gain owing to local illumination of the semiconductor microcavity. If we neglect spatial extension in *y*, we simply write *P*(*x*). We note that the exact mathematical form does not significantly change the physics under consideration and is an approximation of the actual spot form used in recent experiments [13,24].

Taking this into account, and as outlined in §3 in more detail, we can effectively reduce (2.1) to a simplified model for the order parameter *ψ* of the polariton condensate given by [31]
*U* denotes the self-interaction strength, *γ*_{eff} the linear gain and *Γ* the nonlinear loss. The repulsive effect on the polaritons of the pump *P* and the reservoir *N*, defined in (2.1), is assumed to be approximated by *V* . For our analysis, we are interested in stationary states and so set *ψ*(*x*,*y*)=*ψ*(*x*), and rescale (2.4) as in reference [31], i.e. *ω* is the oscillator frequency and *l* and energy in units of *σ*_{1}=±1 (+ corresponds to attractive and − to repulsive self-interactions), *σ*_{2}=*Γ*/*U* and *V* (*x*)=*x*^{2} is the harmonic potential owing to the approximation of the pumping spots left and right of the condensate and the repulsive reservoir. We note that (2.5) is a nonlinear eigenvalue problem with real-valued eigenvalue *μ*_{1}, which we will determine numerically and analytically.

Let us identify the Carusotto–Wouters model as approximated in references [13,24,30] with the Keeling–Berloff model [31] by identifying the effective potential terms by
*V* =*kx*^{2} and the effective gain and loss terms
*N*=(*P*/*γ*_{R})(1−(*β*/*γ*_{R})|*ψ*|^{2}). To approximately match the potential terms, we neglect the density-dependent part of *N* (as it does not significantly affect the observed patterns), and we use a harmonic function that contains the maxima of the two Gaussian spots of (2.3). In figure 1, we show the correspondence between the pump spots and the harmonic approximation and remark that *k*∼*A*. To fit parameters in (2.6), we note that the density-dependent potential term can simply be matched *α*_{1}=*U*.

To identify the pump terms, we set *x*_{1}=−10, *x*_{2}=10 and 2*Γ*=*β*/*γ*_{R}. Consequently, the density-dependent pump parameter becomes *σ*_{2}=*Γ*/*U*=2.5, whereas the linear parameter is *Ω*=(0.165⋅*A*−0.278)/*k* with *k*=*m***ω*^{2}/2=0.092⋅*A* as in reference [10] fixing the potential strength *ω* via the choice of *A*. This identification yields the simplified equation (2.5) once the corresponding rescaling has been applied. Recall that sgn(*σ*_{1}) depends on the sign of the scattering length between condensed polaritons.

By tuning the properties of the pump (in particular, the geometry and pumping strength), and also having control over the nonlinear interaction strength *σ*_{1} through the use of Feshbach resonances [29], we see that the gain and loss coefficients (*σ*_{2} and *Ω*) can be varied widely. In the remainder of this work, we will set the density-dependent loss rate in rescaled units *σ*_{2}=0.3 and consider values for the pumping strength *μ*_{1} from *Ω*=0 to *Ω*=1, values that are in accordance with reference [31]. We remark that generally the loss rates for polariton condensates depend significantly on the actual experimental set-up, partly, because lifetimes vary from 1 to 100 ps and correspond to the quality of confinement within the microcavity [2,3,6,15,18], so even a quasi-equilibrium scheme is feasible in polariton condensates [15]. This allows us to examine the equilibrium limit in our nonlinear system seamlessly (*Ω*=0), and explore the nature of the system when gain and loss are of similar magnitudes but less than or equal to the magnitude of the nonlinear term. Setting *σ*_{1}=±1 means an increase of real-valued ‘nonlinear’ behaviour than if we used the initially introduced ‘natural’ parameters, which is justified in physical terms by using Feshbach resonances [3,10,14].

## 3. The polariton harmonic oscillator

First, we present results for simulations of the unreduced state equation (2.1). The pump geometry is assumed to be (2.3), which resembles two localized pump beams corresponding to laser beams shining on the semiconductor microcavity. These laser beams generate hot excitons which owing to energy relaxation drop into the lowest energy state and form the so-called polaritons, providing a local gain of these particles for the condensate at the pump spots. In figure 2*a,*^{1} we present the density profile |*ψ*|^{2} showing the harmonic oscillator *ground state* in-between two pump spots for *A*=8.5 roughly approximated by a Gaussian. Figure 2*b* shows the *first* excited harmonic oscillator-type state for *A*=11.5 obeying two distinct peaks of condensate density. Finally, figure 2*c* shows the *second* excited harmonic oscillator-type state for *A*=35 obeying three distinct peaks of condensate density. Increasing the amplitude allows us to access higher-order excited states. Flow occurs from regions with negative imaginary part to regions with positive imaginary part when the real part is positive (and the inverse when the real part is negative). Consequently, we see particles flow towards the maxima of the density profiles. This is as expected as linear gain dominates at low densities and nonlinear loss dominates at high densities.

Although we mainly aim to explain the patterns observed in reference [27], we note that the polariton harmonic oscillator can itself be implemented within a long one-dimensional nanowire similar to references [16,32] i.e. a microcavity with very strong transverse confinement. Additionally, an external in-plane trap can be imposed upon the effectively one-dimensional condensate as it has been experimentally shown for a sample consisting of three sets of four GaAs/AlAs quantum wells embedded in a GaAs/AlGaAs microcavity in references [33,34]. On the other hand, for the pump laser beams, there is, in principle, no restriction to the size of the area they are illuminating (Dr Hamid Ohadi, private discussion, Cavendish Laboratory, UK) and thus they can be assumed to be extended over a wide range, including the area where the condensate forms within the harmonic trap. So an almost homogeneous pump distribution can be provided on the one-dimensional geometry on top of a repulsive and approximately harmonic trap as described by the simplified model (2.5).

## 4. Perturbation theory

We now derive analytical expressions approximately solving (2.5). Considering (2.5) in the parameter limit *σ*_{1},*σ*_{2},*Ω*→0, we obtain the well-known quantum harmonic oscillator equation
*E*_{n}=1+2*n* with *n*∈{0,1,2,3,…}. These Hermite–Gauss polynomials are an orthonormal basis (ONB) of a Hilbert space. On the other hand, any wave function in a Hilbert space has an expansion in terms of Hermite polynomials of the form
*ϕ*_{n}(*x*) providing the ONB. By inserting this expansion in (2.5) and using (4.1), we obtain
*ϕ*_{m} and integrating over the whole space, we obtain the formula

We assume that for the *j*th nonlinear mode the largest coefficient in our expansion (4.4) is *B*_{j}. We take *B*_{n<j}=0 and assume that only basis wave functions of the same symmetry of the nonlinear mode contribute in the expansion, e.g. *ϕ*(*x*)=*ϕ*(−*x*), thus, we take *B*_{j+2k+1}=0 (where *μ*_{1},
*B*_{j}|^{2}:

We see that at leading order we have no information about the phase of the wave function. To proceed further and find simple expressions for the higher-order contributions, we assume that *B*_{j} is purely real and that *B*_{j}≫*B*_{j+2k}. This allows us to find the following expression for the higher-order complex coefficients:
*c* corresponds to the number of condensate particles similar to the case of fixed particle numbers [5].

*Thomas–Fermi-type regime:* starting from (2.5), we consider the regime where the kinetic energy becomes negligible compared with the external potential, nonlinearities and linear parameters, i.e.
*Ω* and *σ*_{2}, the TF regime significantly depends on the nonlinearity *σ*_{1} and the energy *μ*_{1} as formula (4.12) shows.

## 5. Comparison between numerical and analytical results

We now proceed to test our leading-order perturbation theory by comparing it with numerically calculated results. We use a modified squared-operator method [36] to find numerical solutions *ϕ*(*x*) to the equation (2.5).

In figure 3, we show the first three nonlinear modes for both repulsive (figure 3*a*–*c*) and attractive (figure 3*d,e*) interparticle interactions for the case *Ω*=0.2. The numerical solutions are given by the thick lines, with real and imaginary parts corresponding to the dashed and dash-dotted lines, respectively, while the density is given by the solid line. We see that, as in the case of the single-component atomic BEC [35], the *j*th mode has *j*−1 nodes of zero density where a *π* phase change occurs. However, unlike the atomic BEC case, the polariton condensate has a spatially dependent phase away from the phase singularity (evident through the spatially varying ratio of real to imaginary part in the wave function). This variable phase indicates that particle flow must occur in the condensate, with particles moving from regions where gain dominates, to regions where loss dominates.

We can compare the numerical results directly with the analytical expansion (4.4) assuming only the leading-order term is present, as given by equations (4.8) and (4.9). The analytical results are given by the thin solid lines in figure 3. More precisely, we can compare the analytically predicted density only with the numerical result; however, we include also the real part and imaginary part assuming that the imaginary part is zero. At this leading order, the nonlinearity plays no role in the form of the analytical wave function, so it is identical for the attractive and repulsive cases. Nevertheless, we see that it agrees remarkably well with the numerical solutions. We see that in the repulsive case the true solution lies outside the analytical result, whereas the inverse is true for the attractive case, as expected. Interestingly, we see that the agreement improves as we go to higher modes, suggesting the ground state is ‘more nonlinear’ in the sense that it deviates more strongly from the associated linear wave function.

In figure 4, we compare the analytical and numerical results at higher gain, given by *Ω*=0.5. We see that the pattern observed for *Ω*=0.2 becomes more evident, with the ground state showing the strongest deviation from the linear prediction. The higher-order modes continue to show remarkably good agreement.

The validity of the leading-order analytical prediction can be seen at a glance by comparing the predicted eigenvalues (4.8) with those found numerically. We see in figure 5 the dependence of the eigenvalues on gain parameter *Ω* for the first three modes (pairs of lines are arranged upper to lower and correspond to second, first and zeroth modes, respectively), for both repulsive (solid) and attractive (dashed) interactions. The analytical predictions are given by the associated thin lines. We see increasingly good agreement with mode order, with the second mode showing close agreement even up to *Ω*=1. The ground state, on the other hand, shows a discernible discrepancy by *Ω*=0.5. The vertical dash-dotted lines correspond to the values of *Ω* chosen to show the stationary states in figures 3 and 4. Overall, we see that the features of the stationary states are dictated predominantly by the gain and loss coefficients, rather than by the interparticle interactions.

## 6. NOR gate and an nNOR gate

The pattern formation within a one-dimensional nanowire can have technological application as a NOR gate. A NOR gate [37] is defined by the logic that two or more zero inputs yield an output 1 in a boolean logic system. Here, we can identify each pump spot (2.3) with 0 or 1 by defining that, if the laser beam is turned on, *A*≠0, the input corresponds to 0 and if not it corresponds to 1. The output will be the quantum harmonic oscillator patterns observed, if both pump spots are equal to 0. So we can identify all the harmonic oscillator patterns with an output 1, and thus we have established a simple feasible NOR gate by means of topologically stable excitations carried by a non-equilibrium polariton condensate within a semiconductor microcavity which stand in the tradition of optical computing [17,38].

This NOR gate can be generalized in the sense that we do not solely distinguish between ‘on’ corresponding to 0 and ‘off’ associated with 1 of both pump spots, but in addition include the experimental and numerical observation that different pump strengths lead to different numbers of density lobes. Therefore, we can enumerate the logical zeroes leading to different lobes states, i.e. ground state, first excited state, second excited state and so on. That is, if the pump spots have a certain amplitude *A* it leads to patterns of say

## 7. Conclusion

Motivated by a recent experiment, we considered a mean field model for polariton condensates to reproduce the observed patterns. Starting from this, we identified the general model with that of a simpler nonlinear non-equilibrium quantum harmonic oscillator. For this effective theory, we have presented a perturbative approach and so generalized the quantum harmonic oscillator theory to include the non-conservative character of the non-equilibrium polariton condensate. We introduced the family of excited states starting from the Gaussian ground state to the *n* node non-equilibrium quantum harmonic oscillator states. The analytical eigensolutions were compared with corresponding numerical simulations and showed excellent agreement. Owing to the recent experimental accessibility of Feshbach resonances in polariton systems, we further predicted the scenario of attractive self-interactions and outlined their role in nonlinear pattern formation. Particularly, the modification of the density formation was explicitly illustrated. In the course of our analysis, we showed that the energy eigenvalues depend on the pumping strength *Ω* linearly to the leading order, and conclude with an explanation of the experimentally observed patterns. Finally, we discussed the possibility for a generalized optical NOR device in polariton condensates using the underlying pattern formation logic.

## Ethics

This work only discusses ethical computer simulations and mathematical analysis.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

F.P. and T.A. conceived the mathematical models, interpreted the computational results, and wrote the paper. Both authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

F.P. was financially supported through his EPSRC doctoral prize fellowship at the University of Cambridge and a Schrödinger fellowship at the University of Oxford. T.J.A. thanks the Department of Applied Mathematics and Theoretical Physics for kindly supporting his visit to the University of Cambridge.

## Acknowledgements

F.P. would like to thank Natasha Berloff, Hamid Ohadi and Alexander Dreismann for stimulating discussions.

## Footnotes

↵1 The time evolution has been induced via a fourth-order Runge–Kutta method and a fourth-order finite differences scheme. We used a computational window of 226 space units in

*x*and along the transverse directions*y*. Typical space steps are about Δ*x*=0.1 and the timesteps Δ*t*≃1.5×10^{−4}.

- Received March 31, 2015.
- Accepted July 14, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.