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Effects of the linear polarization of polariton condensates in their propagation in codirectional couplers

We report on the linear polarization of polariton condensates in a codirectional coupler that allows evanescent coupling between adjacent waveguides. We observe polarization-dependent intensity oscillations in the output terminal of the coupler that we identify as the mode beating between the linear-polarized eigenmodes.


Introduction
Microcavity exciton polaritons have been, in the latest years, the subject of numerous investigations given their exceptional properties [1,2]. These properties, emerging from the strong coupling between their constituents, excitons and photons, allow polaritons to behave as bosonic particles. Their short lifetime, typically of the order of ps, is comparable or even smaller than their respective thermalization times, so in general these particles do not reach thermal equilibrium. However, a condensation similar to a Bose-Einstein one is observed when the particle density is increased [3][4][5][6][7]. Their very low effective mass (⁓10 −4 me, being me the free electron mass) can lead to condensation even at room temperature. This has been indeed the case in transition metal dichalcogenides, organic semiconductor materials and lead halide perovskites structures, where the enhancement of the exciton binding energy, characteristic of these compounds, has enabled the observation of strong light-matter coupling [8,9], polariton lasing and condensation [10][11][12][13][14][15] at room temperature. The ease of use and the flexibility offered by exciton polaritons unwrapped a variety of new proposed devices such as polariton interferometers [16], logic gates [17,18] or transistors [19][20][21]. Additionally, three dimensional polariton confinement has been achieved in microcavity pillars [22][23][24][25][26].
Using two dimensional lattices of these micropillars, it is possible to emulate graphene and its remarkable properties [27][28][29]. All these devices are built using refined lithographic techniques that ensure the polariton confinement along several directions. In the one-dimensional case, only a well-defined longitudinal path along which polariton condensates can travel remains [18,19,30,31].
In the present work, we will focus on semiconductor microcavity couplers. Such optical directional couplers are formed by parallel optical waveguides, closely spaced, so that energy exchange can occur between them [32]. The coupled power, limited by the mode's overlap in the coupler arms, is determined by the separation between the waveguides, the wavelength, the evanescence of the modes and the interaction length. These devices have been proved to be essential for splitting and combining light in photonic systems and have been used widely in the silicon-on insulator platform [33]. Quantum photonic waveguide circuits based on GaAs/Ga1-xAlxAs heterostructures have been demonstrated for the manipulation of quantum states of light [34]. These devices have been also exploited for guiding surface plasmon polaritons [35,36] and exciton polaritons [37][38][39]. More recently, we have reported on different on-chip routing devices: a counter-directional coupler [40] and a co-directional coupler for condensates of exciton-polaritons, studying the peculiarities of the polariton propagation [41] and how this is affected by the waveguides' energetic landscape [42]. A relevant factor is the spin state of the condensates after polariton's relaxation processes leading to their condensation [43,44].
Moreover, a spontaneous build-up of the linear polarization of the emitted light above the polariton condensation threshold has been reported both theoretically [45][46][47] and experimentally [4,[48][49][50][51]. The orientation of the polarization plane of the emission is pinned to a crystallographic axis of the microcavity [50,51]. This effect has been effusively observed for trapped polaritons using different trapping mechanism such as photonic disorder [52], stress [5] or annular optical confinement [53]. The study and control of the polarization state of polariton condensates has opened new possibilities of designing and improving spin-based devices [54][55][56][57]. Wire-shaped microcavities are particularly interesting in this respect because, due to their reduced symmetry, each polariton mode shows a polarization splitting into two modes polarized along and perpendicular to the wire axis [58].
Here, we theoretically and experimentally study the linear polarization of the emission of propagating polariton condensates in polaritonic co-directional couplers. Our results demonstrate a coupling between the adjacent waveguides that is not strongly dependent on polarization. However, we encounter striking polarization-dependent emission oscillations at the output terminal of the coupler. For a given set of perpendicular polarizations we find a phase shift between the oscillation's patterns. To better understand our experimental results, a dissipative Gross-Pitaevskii model is used to describe the polarization dynamics in the device.

Experimental details
The sample used in this work is a λ/2 cavity with top (bottom) distributed Bragg reflectors consisting of 23 (27) pairs of alternating layers of Al0.2Ga0.8As/AlAs. Three stacks of 4 GaAs quantum wells of 7 nm of nominal width are grown at the antinodes of the electromagnetic field inside the cavity. Low power measurements reveal a Q-factor of ~ 5000 and a Rabi splitting of 13.9 meV. The experiments reported here are performed in a region of the sample with a photon-exciton detuning δ ≈ −17 meV. The sample has been grown by molecular beam epitaxy and processed by reactive ion etching down to the QWs [41], creating a pattern of adjacent waveguides where, length (L), width (w) and separation (d), have been varied. Figure 1(a) shows a typical field of couplers, with different coupling lengths, formed by doubly bent waveguides with input and output terminals rotated ±45º from the longitudinal direction; the geometrical parameters are specified in Fig. 1(b). The part of the device where both waveguides remain parallel along the x-direction is dubbed coupling region: a few pairs of mirrors left in the region between the two arms enable the evanescent photonic coupling of polaritons between the guides [41]. For the experiments reported here, the dimensions of the selected device are L = 10 μm, w = 2 μm and d = 0.2 μm. The choice of these parameters allows the coupling of a large fraction of polaritons between the arms of the coupler. In our experiments, we nonresonantly pump the input terminal of the coupler with 2-ps pulses from a Ti:Al2O3 laser working at 1.664 eV, focusing the beam to a 4.5 μm diameter spot, with a microscope objective (NA = 0.40, f = 4 mm), impinging normally to the sample surface. The photoluminescence (PL) is collected through the same objective while the sample is kept at 12 K in a cold-finger, He flow cryostat, and detected with a CCD camera attached to a 0.5 m focal length imaging spectrometer. We ensure that the polariton condensation threshold (12 kW/cm 2 ) has been exceeded and that condensates propagate along the entire device pumping with a power density of 26 kW/cm 2 . , waveguide width (w= 6 µm) and waveguide separation (d = 0.6 µm). Input, output terminals and the coordinates axis are shown, corresponding to the nomenclature used in the text: x (y) parallel (perpendicular) to the main axis of the waveguides in the coupling region and x′ along the axes of the input and output terminals at !45° with respect to x and y, respectively

3.-Theoretical framework
To study the dynamics of exciton-polaritons theoretically we adopt a well-known model describing the coherent polaritons by two complex order parameter functions Ψ !,# for right (r) and left ( ) handed circularly polarized polaritons [55,59]. The coherent polaritons interact with baths of incoherent excitons having different spins. The excitons are characterized by their density !,# . The whole set of equations can be written as: In these equations, is the coupling parameter between the polaritons and the reservoirs of excitons, γ 0 is the coordinate dependent losses of the coherent polaritons. We assume that the polariton waveguides are formed by microstructuring creating a coordinatedependent effective potential, , for the polaritons. It is also considered that the Now we describe the condensates propagation in the coupling region and in the output terminal. Since the PL intensity in these regions is substantially lower than in the input terminal, we have spatially filtered the emission, so that the PL from the input terminal is removed. The excitation beam is vertically polarized ( % = 90º), i.e., parallel to the y axis.
The PL is analyzed using a linear polarizer at different angles, ranging from 6 = 0º (i.e., horizontal polarization) up to 180º in steps of 10º. For simplicity, only a summary of the PL for selected 6 is shown in Figure 3. Polariton condensates are generated in the bottom-left input terminal; when they arrive to the coupling region, -5 < x < 5 μm, a large fraction of the population is conveyed from the bottom to the top arm. After the coupling, polaritons continue propagating throughout the top arm until the edge of the waveguide at the output terminal, while only a minor fraction of the population remains in the bottom arm. By increasing either the length (L) or the spacing (d) between the arms, the fraction of coupled polaritons can be controlled [41]. Drastic intensity variations along the device are observed when the polarization of the emission is analyzed. We find a considerably large intensity when the polarization is analyzed at 0º. By contrast, a remarkable intensity reduction is observed around and above 30º. A further increase of 6 results in a slow PL recovery for 6 ≳ 90°.  130º): the emission shows just an exponential decay with propagation distance along the output terminal. This behavior is independent of the excitation laser polarization as borne out by our experiments, since the non-resonant excitation conditions in our case guarantee the erasing of polariton's spin memory during the relaxation processes (See also Figure   S1).
Let us now discuss how the theoretical model introduced above describes the effects   Note that y = 0 marks the center of the gap between the arms, therefore, the signal from y < 0 and y > 0 arises from the pumped and coupled arm, respectively. It is apparent that at the entrance, a larger polariton population is present in the pumped arm of the coupler  close to 90º, a weak signal is obtained at x1′ being the maximum now at x2′ ∼ 16 μm, i.e., out of phase from x1′ by half the beating distance (12 μm).
We can explain these polarization beatings by considering the TE-TM splitting in the waveguide and therefore taking into account the different group velocities for each polarization. In the framework of the mathematical model introduced above, the splitting lifts the degeneracy between the eigenmode polarized along (TE) and that polarized We would like to mention here that PL intensity fringes having a short spatial period are also clearly seen at the ends of the waveguides. These short-period oscillations are visible since polariton coherence is preserved during their propagation [61,62]. They originate from the reflection of the polaritons at the waveguide end resulting in the formation of counter-propagating polariton waves that interfere with the incoming ones [36].     Madrid, Spain.

I. Polarization degree of propagating polaritons.
Movie showing maps of propagating polaritons for their emission analyzed into its linearly polarized components.
Video S1. Maps of the polariton emission in a directional coupler characterized by L = 10 μm, w = 2 μm and d = 0.2 μm analyzed into its linear components for different analyzer angles ranging from θ d = 0° to 180° .

III. Simulations of the polariton dynamics.
Here we theoretically consider in detail the polaritons polarization dynamics. For the sake of mathematical convenience, we rewrite Eqs.  To check how accurately Eq. (S3) describes the dispersion characteristics of the guided polaritons we performed numerical simulations of Eqs. (S1)-(S2) and found out that Eq.
(S3) is very precise, especially for the fundamental mode.

III.1.2. Including TE-TM splitting.
In agreement with experimental findings [1], a simple perturbation analysis shows that TE-TM splitting lifts the degeneracy and each mode splits into two modes linearly polarized parallel and perpendicular to the waveguide axis [2]. For the case of the rectangular waveguide of infinite depth and width CD the dispersion of the modes can then be written as

S5
where the subindices , denote the directions of the polarization (the waveguide axis is oriented along x). To generalize this expression to a finite depth waveguide, one needs to consider the waveguide cut-off frequencies and to calculate the overlap integrals, using the waveguide eigenmodes. However, for the main conclusions discussed in this paper, it is sufficient to work with the modes of an infinite depth rectangular effective potential.

III.1.2.1. Horizontal and vertical excitation.
To confirm that the linearly polarized modes are the eigenmodes of the waveguides, we

III.1.2.2. Circular and diagonal excitation.
However, a very different dynamics can be observed when the waveguides are excited by light encompassing both x and y linear polarizations. We present the results for pumps circulary polarized and linearly polarized at 45º to the waveguide axis. Let us start the discussion with circulary polarized light. As seen in Figure S3,

S9
and S3 Stokes parameters. This effect is analogous to the decay of the beatings of the polaritons between neighboring waveguides discussed in in Ref. [4].

III.1.2.3. Mode structure across the waveguide.
An additional interesting fact is that the numerical simulations show that the polarization of the fundamental eigenmode is not linear at all points across the waveguide, see Fig. S2. As easily deduced from a second order perturbation analysis, if a mode is x/y-polarized in first order, the second order correction is y/x-polarized and has different parity. The structure of the fields, obtained from the results plotted in Fig. S2(a) and (b) at x=0, is shown in Fig.   S5(a/b) for x/y-polarized excitation and analyzed into its x/y-polarized components (black/ blue lines), where a small signal with orthogonal polarization to the main one is clearly seen.
As a consequence, a symmetric pump centered on the axis of the waveguide can excite a second order eigenmode, since the latter has a small even component in the polarization orthogonal to the polarization of the odd component. Additionally, this second order mode can also be excited if the pump is slightly displaced from the waveguide axis. The excitation of this second mode is weak but it is clearly seen in our simulations. The dependence on x of the maximum intensity, which occurs at y=0/0.2, for x/y polarized-emission, is shown in Fig.   S5(c) for x-polarized excitation. The equivalent dependence for y-polarized excitation is depicted in Fig. S5(d). At large x, both intensities decay with the same rate, attesting that they belong to the same mode. However, around x~ -5 the y/x polarized intensity in (c/d) shows conspicuous oscillations, arising from the interference of the fundamental and second order modes. These oscillations extend up to x ~ 0 in panel (d), attesting that some contribution from the second order mode is still present in the field structure depicted in panel (b).

III.2. Excitation by an incoherent pump.
Now, we turn to the study of the polariton dynamics pumped by an incoherent pump, creating a bath of incoherent excitons. This excitation disregards the term a /,2 in Eq. (S1) and considers only /,2 . in Eq. (S2). For our numerical simulations, we use the super-Gaussian , where @ is the intensity of the pump and 3I is its width. We use 3I = 0.9 and @ = 35, well above the polariton condensation threshold.
A weak random noise in the polariton field was also taken as initial conditions. To allow a direct comparison between the coherent and incoherent pump scenarios, the parameters were chosen so that the frequency of the excited polaritons is the same for both cases.
The two-dimensional reciprocal-space emission image, exciting with an incoherent pump, is shown in Fig. S4 (c): it is clearly seen that the wavevector, kx, of the propagating polaritons coincides with that found for coherent excitation.
The results of our simulations in the straight waveguide are presented in Fig. S6 (a), showing the distribution of x-and y-polarized polaritons for the stationary solution. It is clearly distinguished that the excited mode is x-polarized, however, as discussed above, the mode that has a dominating x-polarization of even parity also possesses a weak y-

S11
polarized component of odd parity. Thus, the polaritons are purely x-polarized only at the axis of the waveguide. Out-of-axis, their polarization is slightly elliptical, as borne out by the weak maxima of the y-polarized intensity close to the waveguide borders. In this case no beating in the y-polarization is seen (apart from the fringes caused by the reflection of the polaritons at the waveguide edges). This means that the incoherent pump excites only the fundamental mode with dominating x-polarization. The dispersion relation yields that this mode is the lowest in energy, therefore, for the chosen parameters, polariton condensation occurs at the thermodynamically favorable mode, in agreement with our experiments and with a previous theoretical analysis [5].
Having shown that in a straight waveguide an incoherent pump excites only one mode, and thus no polarization oscillations are possible, to explain the experimental results in our couplers, it is essential to simulate a waveguide which changes its direction. The profile used for the simulations is presented in Fig. S6

S12
wave ceases to be an eigenmode in section II: the polaritons start to redistribute over different horizontal (x↔) and vertical (y ↕) polarizations as demonstrated by the continuous changes of the Stokes parameters.
In section III, the waveguide is oriented at 45º to the x axis and the eigenmodes are linearly polarized, but with polarizations oriented at 45º and 135º degrees to this axis.
Because of the just mentioned polarization redistribution in section II, in general, in section III, the polaritons polarization is given by a linear combination these eigenstates.
These modes have different wavevectors and therefore the density of polaritons oscillates between x and y polarizations, as discussed in Section III.1.2.2. It is noticeable in Fig. S6 (b) that x-polarization gets depleted while polaritons appear in y-polarization. This explains satisfactorily the experimentally observed polarization oscillations in the output terminal.