Polariton Polarization Rectifier

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Sedov et al. Light: Science & Applications (2019) 8:79 Ofﬁcial journal of the CIOMP 2047-7538 https://doi.org/10.1038/s41377-019-0189-z www.nature.com/lsa

ARTICLE Open Access Polariton polarization rectiﬁer Evgeny S. Sedov1,2,3,YuriG.Rubo 4 and Alexey V. Kavokin1,2,5

Abstract We propose a novel photonic device, the polariton polarization rectiﬁer, intended to transform polariton pulses with arbitrary polarization into linearly polarized pulses with controllable orientation of the polarization plane. It is based on the interplay between the orbital motion of the polariton wave packet and the dynamics of the polariton pseudospin governed by the spatially dependent effective magnetic ﬁeld. The latter is controlled by the TE-TM splitting in a harmonic trap. We show that the unpolarized polariton pulse acquires linear polarization in the course of propagation in a harmonic trap. This gives the considered structure an extra function as a linear polarizer of polariton pulses.

Introduction transverse splitting on the pseudospin behaviour can be The concept of polariton devices is based on manip- described in terms of the effective quasimomentum- ulation by macroscopic coherent states of quasiparticles dependent in-plane magnetic field Ω(k)14. Methods for exciton-polaritons appearing under the strong coupling of controlling the spin degree of freedom are used in a set of excitons in semiconductor crystals with quantized light in devices referred to as spinoptronic devices3 for manip- optical microcavities1. Their properties combine the ulation by light polarization, including polariton polar- mobility and flexibility of photons2 and the controllability ization switches15, polarization filters of polariton flows16, of excitons3, which allows polariton devices to be com- polarization-controlled optical gates17, and the Berry- 18

1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; petitive with traditional photonic and electronic devices phase interferometer . for light control and signal processing. Polaritons inherit a Among established methods of controlling the polar- spin degree of freedom from both their constituents4, iton spin degree of freedom is the creation of an external which gives access to the polarization properties of the confining potential. Flat one-dimensional polariton emitted light. Controlling the polariton spin relaxation waveguides have been widely studied from the per- processes is an effective way to tailor light polarization. spective of the creation and propagation of spin-split The polariton pseudospin (Stokes vector) is in the heart of polariton condensates19 and solitonic polariton pulses20 various fundamental effects, including the optical spin as well as of accompanying effects, including backward – Hall effect5 7, polarization bistability8 and multi- Cherenkov radiation21, and the formation of polariza- stability9,10, spin currents of exciton polaritons6,11, etc. tion domains22. Recently, ring-shaped geometry was The most considerable mechanism of the polariton spin recognized in the study of polariton polarization beha- – relaxation is the precession of polariton pseudospins viour18,23 25. In this geometry, the in-plane effective induced by the splitting of the transverse electric and magnetic field acting upon a polariton pseudospin transverse magnetic photonic microcavity modes (TE-TM rotates as the polariton moves along the circle around splitting)12 along with the long-range electron and hole the symmetry axis of the ring-shaped confining potential 13 exchange interaction . The effect of the longitudinal- characterized by the frequency ωtr, which enriches the polarization dynamics with the complementary oscillation degree in addition to the Larmor precession Ω(k). Correspondence: Evgeny S. Sedov ([email protected]) Typically, the consideration of the polariton dynamics as 1Westlake University, 18 Shilongshan Road, Hangzhou 310024 Zhejiang well as the polarization dynamics in spin-split systems is Province, China performed within the adiabatic limit, assuming that the 2Institute of Natural Sciences, Westlake Institute for Advanced Study, 18 Shilongshan Road, Hangzhou 310024 Zhejiang Province, China spectral width of the polariton state Δω does not exceed Full list of author information is available at the end of the article.

© The Author(s) 2019 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a linktotheCreativeCommons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Sedov et al. Light: Science & Applications (2019) 8:79 Page 2 of 10

the characteristic frequency scales of the system, Δω < Results ωtr, Ω(k), as well as the separation of the energy levels, The concept of the device which is significant in the narrow confining potential. The geometry of the device setup is shown schemati- Furthermore, the limit wheretheLarmorfrequencyis cally in Fig. 1a. It consists of an optical microcavity with considered the largest frequency in the system, an embedded set of quantum wells operating in a strong- ΩðkÞΔω; ωtr, is used to separate different oscillation coupling regime. Arising exciton-polaritons are confined scales18. In this approach, the internal behaviour of the in an in-plane harmonic potential, which can be created polariton pulse as well as the interaction of polariton either for excitons via application of a local stress to the modes is adiabatically eliminated from consideration, sample26,27 or for photons by creating mesas28,29 with and its dynamics is described by the effective 1D thicknesses varying along their radius. The latter option is Hamiltonian for a classical spinning particle. The illustrated in Fig. 1a. In Fig. 1b, the potential landscape situation dramatically changes when the internal pulse across the structure is schematically shown in the reci- behaviour is not completely suppressed and polaritons procal space. Polaritonic systems also allow the use of are being filtered by their wave vectors such that their optical traps30 created by the non-resonant optical pump distribution changes in the radial direction. This sce- beam of a given shape. Polaritons are injected by a reso- nario may be realized in broad confining potentials, nant pulsed probe of energy hωp at a distance rp from the including the harmonic potential. centre of the trap tangential to its surface with the qua- In this paper, we consider the polarization dynamics of simomentum kp. To increase the polariton pulse lifetime, polariton pulses in a harmonic potential beyond the we introduce the subthreshold spatially homogeneous adiabatic limit. We demonstrate the splitting in real space non-resonant cw optical pumping of excitons. of an injected polariton pulse of any given polarization In the conservative limit, the system is described by the into two pulses of unequal intensity with orthogonal lin- effective Hamiltonian ear polarizations. The intensities of the resulting pulses 2k^2 can be manipulated, including complete suppression of ^ h ^ ^ ð1Þ H0 ¼ þ V ðÞþr hΩ Á S one of them. Based on the obtained peculiar polarization 2mÃ behaviour, we propose a novel spinoptronic device, the Ã ¼ =ðÞþ polariton polarization rectifier, which transforms the where m 2mlmt ml mt , with ml and mt being the arbitrary polarization of the optical pulse to a linear effective masses of longitudinal (or transverse-magnetic, polarization with controlled orientation of the polariza- TM) and transverse (or transverse-electric, TE) polariton ^ ^ ^ tion plane. We also show that the proposed device assigns modes, respectively; r ¼ ðÞx; y and k ¼ kx; ky are the linear polarization to a polariton pulse excited by an polariton position and quasimometum operators, unpolarized resonant probe. ðÞ¼r Ãω2 ð 2 þ 2Þ= respectively. V m tr x y 2 is the harmonic

Resonant probe abNon-resonant cw pump pulse

kp rp

rp kp ω p Cavity

QWs ω x ky y

kx Fig. 1 Schematic of the possible experimental conﬁguration of the polariton polarization rectiﬁer. a Sketch of the possible experimental

conﬁguration. The polariton condensate is excited by a resonant probe pulse (red split cone) at a distance rp from the centre of the harmonic trap with the wave vector kp and energy hωp, allowing the polariton pulse to propagate along a circular trajectory (dashed circle). The subthreshold non- resonant cw pump (pale green) is used to increase the pulse lifetime. b Potential landscape across the structure in the reciprocal space. The yellow arrows in both panels indicate the orientation of the linear polarization plane of the propagating polariton pulse at different positions along the propagation trajectory. The polarization of the probe pulse can be chosen arbitrarily. In panels (a) and (b), the circular polarization (blue arrow) is shown as an example. The blue concentric circles in both panels are given as a guide for the eye to indicate the harmonic trap Sedov et al. Light: Science & Applications (2019) 8:79 Page 3 of 10

potential characterized by the frequency ωtr. The polar- mode. The orientation of the corresponding polariton iton quantum state is described by the spinor pseudospins during evolution is schematically shown in T jΨðrÞi ¼ ½ΨþðrÞ; ΨÀðrÞ . We introduce the three- the inset of Fig. 2a. A resonantly excited polariton pulse of S^ ¼ 1 σ^ σ^ ¼ðσ^ ; σ^ ; σ^ Þ finite width and duration possesses the energy and qua- component spin operator 2 , where x y z is the vector of the Pauli operators. The TE-TM splitting simomentum spectrum, overlapping with the eigenstates gives rise to the directionally dependent effective mag- of the system. This implies that in the course of evolution, hi regardless of the initial polarization, the polariton pulse fi Ω^ ¼ Δ ^2 À ^2 ; Δ ^ ^ ; netic eld LT kx ky 2 LTkxky 0 in the tends to occupy the longitudinal and transverse eigen- microcavity plane, which causes precession of the polar- states and assigns linear polarization characteristics iton pseudospin withÀÁ the effective Larmor frequency to them. Ωð Þ Δ ¼ð= Þ À1 À À1 k . LT h 2 ml mt is the TE-TM splitting constant. Polarization dynamics of polariton pulses In the spin-degenerate case (ΔLT = 0), the Hamiltonian To reveal the polarization dynamics of polariton pulses (1) represents the 2D quantum harmonic oscillator and is in the proposed structure, we perform a set of numerical integrable. According to the correspondence principle31,at simulations based on the generalized Pauli equation for T high energies, the quantum treatment of a wave function the spinor jΨi ¼ ½Ψþðt; rÞ; ΨÀðt; rÞ , where Ψ ± ðt; rÞ are behaviour merges with the classical one of a single particle. the wave functions of the right- and left-circularly The dynamics of a polariton pulse is well described by the polarized polariton components; see details of the model classical equations for the trajectory of its centre of mass: in the Methods section. In our first numerical experiment, hir ðÞ¼hik ðÞ= Ã hik ðÞ¼À Ãω2 hir ðÞ dt t t m , dt t m tr t . The pulse we consider the behaviour of the polariton pulse excited propagates along the closed elliptical orbit, which degen- by the Gaussian probe pulse of duration wT = 5 ps reso- erates to a circular orbit when the initial conditions nant to the TE polariton branch, i.e., characterized by the hir ðÞ¼0 r0,andhik ðÞ¼0 k0, obey the condition of central frequency ωp = ω0 and the central wave vector kp equality of the kinetic and potential energies, = (0, kt). The excitation spectrum in comparison with the 2 2= Ã ¼ ð Þ h k0 2m V r0 . polariton dispersion is schematically shown in Fig. 2a. The In the presence of TE-TM splitting, the Hamiltonian (1) pulse is excited at position r0 = (r0, 0). We take the pump is not integrable, and the vast majority of semiclassical distance as r0 = 70 μm. For the kinetic energy of polar- trajectories are not closed. There remain, however, two itons to match the potential energy and the polariton important types of exact solutions, that is, those pulse to follow a closed circular trajectory, the corre- describing the polariton motion along circular orbits with sponding quasimomentum kt should be as large as À1 always either transverse (TE) or longitudinal (TM) spin. kt ’ 1:245 μm . A sufficiently large quasimomentum is The energies of these circular modes are defined by essential to reduce the effect of the zitterbewegung, i.e., the 2 2 = ¼ ð Þ h kl;t 2ml;t V r0 , resulting in different momenta for trembling of the trajectory of polaritons in real space due the same radius of the orbit r0. In Fig. 2a, we show the to the influence of the spin degree of freedom. As we have 32 energies of two polariton modes hωTM;TEðÞk; r0 ¼ shown in ref. , the zitterbewegung is less pronounced at 2 2 V ðr0Þþh k =2ml;t in the sample at position r0 = 70 μm. k>1 µm since its amplitude is inversely proportional to (Values of other parameters are given in the Methods the quasimomentum. section.) A circular trajectory of radius r0 is inherent to Figure 2c shows the evolution in time of the spatial polaritons belonging to the TM (upper red curve) and the intensity distribution ItðÞ¼; r Ψyðt; rÞΨðt; rÞ of the TE (lower blue curve) branches excited with the energy polariton pulse excited by the right-circularly polarized T hω0 ¼ hωTMðÞ¼kl; r0 hωTEðÞkt; r0 and the momentum kl probe pulse, jf i ¼ ½1; 0 . In Fig. 2g, the ring-shaped dis- and kt, respectively. tribution of intensity of the polaritonR pulse, integrated A Gaussian resonant pulse excites different eigenmodes over the time of observation, IðrÞ¼ Iðt; rÞdt, is pre- in the parabolic trap. For sufficiently strong TE-TM sented. The ring thickens near the polariton injection spot splitting, most of them propagate irregularly in the trap around r0. The evolution of the spatial distribution of the y ^ and destructively interfere with each other. In contrast, polarization components SjðÞ¼t; r Ψ ðÞt; r SjΨðÞt; r = the longitudinal and transverse wave packets are able to ItðÞð; r j ¼ x; y; zÞ of the polariton pulse is shown in Fig. propagate long azimuthal distances without substantial 2d–f, accompanied by the integratedR in time distribution y ^ changes in their shapes. Their pseudospins follow the of the polarization, SjðÞ¼r Ψ ðt; rÞSjΨðt; rÞdt=Iðt; rÞ,in direction of the effective magnetic field and are co- and Fig. 2h–j. Summarizing the main peculiarities of the fig- counterdirected to it for TM and TE modes, respectively. ures, we can claim that regardless of the initial circular This implies that the electric field oscillates in the azi- polarization, the latter is barely presented in the resulting muthal direction for the longitudinal (TM) mode and polarization map. The polariton pulse possesses linear oscillates in the radial direction for the transverse (TE) polarization, and the polarization plane twice rotates Sedov et al. Light: Science & Applications (2019) 8:79 Page 4 of 10

Initial polarizations: a kl kt b 2.5 y

2.0 x TM(kt ,r0)

) (meV) 1.5 0

k,r ES ( 0 1.0 TE(kl ,r0)

0.5 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 k (μm–1)

c de f I Sx Sy Sz 150

100 (ps) t 50

50 50 0 50 50 0 m) 0 m) 0 m) 0 m) –50 μ –50 μ –50 μ –50 μ ( ( ( ( y y y y x (μ 0 –50 x (μ 0 –50 x (μ 0 –50 x (μ 0 –50 m) 50 m) 50 m) 50 m) 50

ghij_ _ _ _

S S S ES I ESSx Sy Sz 50 1 1 m)

μ 0 ( y

–50 0 –1 –500 50 –500 50 –500 50 –500 50 x (μm) x (μm) x (μm) x (μm)

Fig. 2 Polarization behaviour of a polariton pulse excited resonantly to the TE dispersion branch. a Dispersion of two spin-split polariton modes hωTE;TMðk; r0Þ at position r0 = 70 μm. The origin zero energy corresponds to hωð0; 0Þ. The red blurry spot indicates the position and schematic spectral distribution of the probe pulse. The probe pulse of 5-ps duration is resonant to the TE polariton mode, i.e., kp = kt and ωp = ω0 at r0. The inset schematically shows the orientation in real space of the effective magnetic field Ω(k) (grey arrows) and the pseudospin of polaritons belonging to the TE (blue arrows) and TM (red arrows) branches. b Trajectories on the Poincaré sphere of the Stokes vector characterizing polarization of the polariton pulse for different polarizations of the probe pulse (indicated in the panel). The black arrows indicate the direction with respect to the evolution in time. c Evolution of the intensity and d–f polarization components of the polariton pulse. g Time-integrated spatial distribution of the normalized intensity and h–j polarization components of the polariton pulse. The polarization of the probe pulse is taken as right circular for (c)–(j) around the centre of the harmonic trap, adjusting the disappear at approximately a quarter of the first pass rotation of the effective magnetic field Ω^ ðkÞ. At the around the trap. beginning of the evolution, the polarization components The more important result is that under the considered Δ 2 experience fast oscillations with a frequency LTk0 , which excitation conditions, the polariton pulse acquires linear result in the appearance of the periodic ripples most polarization regardless of the initial polarization of the visible at the periphery of the polarization patterns in the probe. In Fig. 2b, we show trajectories of the normalized upper-right quadrant in the colour maps (Fig. 2h−j). The Stokes vector SðÞ¼t ΨjS^jΨ =hiΨjΨ on the Poincaré oscillations, however, possess a transient character and sphere for various initial conditions. In all numerical Sedov et al. Light: Science & Applications (2019) 8:79 Page 5 of 10

a kl kt b kl kt 2.5 ) 4 –1 3

m ps (l) 2.0 μ 2 vg ) ( (t) 0 v 1 g k,r

(

(k ,r )

g TM t 0 ) (meV) v 0 0 1.5 0.0 0.5 1.0 1.5 μ –1 k,r k ( m ) ( 0 1.0 TE(kl ,r0)

0.5 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 k (μm–1)

c de f

I Sx Sy Sz 150

100 (ps) t 50

0 50 50 50 50 0 m) 0 0 m) 0 –50 μ –50 m) –50 μ –50 m) ( μ μ x 0 –50 y 0 ( x 0 ( 0 ( (μ x (μ –50 y (μ –50 y x ( –50 y m) 50 m) 50 m) 50 μm) 50

ghij_ _ _ _ I Sx Sy Sz 50 1 1 m)

μ 0 ( y

–50

0 –1 –500 50 –500 50 –500 50 –500 50 x (μm) x (μm) x (μm) x (μm)

Fig. 3 Polarization behaviour of a polariton pulse excited resonantly to the TM dispersion branch. a Dispersion of two spin-split polariton modes hωTE;TMðk; r0Þ at position r0 = 70 μm in comparison with the spectrum of the probe pulse. The probe pulse of 5-ps duration is resonant to = ω = ω TE;TMð Þ the TM polariton mode, i.e., kp kl and p 0 at r0. The inset shows group velocities vg k of polaritons belonging to the TE and TM branches as functions of k. b Trajectories on the Poincaré sphere of the Stokes vector characterizing polarization of the two spatio-temporal components of the split polariton pulse. The black arrows indicate the direction with respect to the evolution in time. c Evolution of the intensity and d–f polarization of the two components of the polariton pulse. g Time-integrated spatial distribution of the normalized intensity and h–j polarization components of the polariton pulse. The polarization of the probe pulse is taken as right circular for (c)–(j). The dashed and solid closed black curves in panel (g) indicate the trajectories of the TE and TM components of the polariton pulse, respectively experiments presented, the degree of the circular polar- However, despite this overlap, the TM mode is not pre- ization in the polariton pulse after one period of rotation sented in the pulse dynamics in Fig. 2. around the harmonic trap does not exceed half a percent. In the next numerical experiment, we keep the probe In the considered excitation circumstances, the width of energy as hω0 but take the probe wave number as kp = kl the spatial spectrum of the probe pulse 2π=w such that the probe is now resonant to the TM branch; 0:77 μmÀ1 considerably exceeds the splitting of the TE see the comparison of the excitation spectrum with the À1 and TM branches, kt À kl 0:31 μm . Although the polariton dispersion in Fig. 3a. The initial polarization is pulse spectrum is centred with respect to the TE mode, it again taken as right circular. The polariton pulse nevertheless partially overlaps with the TM mode. dynamics and the dynamics of the polarization Sedov et al. Light: Science & Applications (2019) 8:79 Page 6 of 10

(k ,r ) (k ,r ) abckl kt kl kt TE l 0 0 TM t 0 1.0 100 1.0 50 0.8 0.8 10 0.6 5 0.6 TM I PL PL I I

TE 0.4 0.4 I 1 0.2 0.5 0.2

0.0 0.1 0.0 0.5 1.0 1.5 2.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.5 1.0 1.5 2.0 2.5 k (μm–1) k (μm–1) p (meV)

Fig. 4 The intensity of polariton pulses depending on the excitation conditions. a The dependence of the total time-integrated intensity of the polariton pulse on the probe wave number kp for two probe energies, hωp ¼ hω0 (black curve) and hωp ¼ hωTEðkl; r0Þ (blue curve). b The ratio of the time-integrated intensities of two spatial components, ITE and ITM, of the polariton wave packet as a function of the probe wave number at the probe energy hωp ¼ hω0. c The total time-integrated intensity of the polariton wave packet as a function of the probe energy hωp at two resonant quasimonenta, kl (orange curve) and kt (purple curve) components are presented in Fig. 3c–f, accompanied by TE component of the pulse that is distant from the the time-integrated intensity and polarization degree in excitation quasimomentum by kt − kp is excited with the Fig. 3g–j. In contrast to the above-considered case, the quasimomenta k on the periphery of the excitation pulse initial single wave packet, over the course of evolution, far from the central quasimomentum, kp < k. Hence, the splits into two components of orthogonal polarizations; central wave vector of the TE component of the pulse is see the two spiral trajectories in Fig. 3c–f. The more shifted to lower k relative to kt for which the kinetic and intensive wave packet component belongs to the reso- potential energy matching condition is not satisfied. This nantly excited TM branch, while the less intensive com- results in the elliptical shape of the trajectory. In the ponent belongs to the TE branch. considered excitation scheme, the orientation of the The two components of the polariton pulse demon- minor axis of the elliptical trajectory of the TE component strate different behaviours. The inset in Fig. 3a shows the of the polariton pulse coincides with the y axis. Hence, the group velocity of polaritons belonging to the TE and TM polarization of the TE pulse component near x = 0 (one ðÞTE;TM can briefly characterize it as S < 0) is concentrated inside branches, vg ðÞ¼k; r0 ∂kωTE;TMðk; r0Þ. It is clearly x the coloured region of the colour maps in Fig. 3h–j. seen that for the same probe energy hωp ¼ hω0, the group velocity of the TM component of the pulse, Figure 3b shows the trajectories of the normalized Stokes ðÞ ðÞ vector S(t) on the Poincaré sphere of the two components v l ¼ v TM ðÞk ; r , is larger than that of the TE compo- g g l 0 of the pulse. In both components, the circular polarization ðÞt ¼ ðÞTE ðÞ; ðÞl ðÞt nent, vg vg kt r0 : vg >vg . ispresentedbyasmallquantityoflessthan5%.Theresi- The considered case is also remarkable because the dual circular polarization is right circular for the TE-branch orthogonal linear polarizations of the polariton wave pulse and left circular for the TM-branch pulse. packet are separated in the radial direction: the Sx > 0 To further investigate the resonant character of the component is concentrated in the outer region, while the polarized polariton pulse excitation, in Fig. 4a, we plot the Sx < 0 component is concentrated in the inner region of dependence of the total intensity of two pulse compo- the highlighted area of the colour map in Fig. 3h. The nents integrated in time on the probe wave number kp for dashed and solid closed black curves in Fig. 3g show the two probe energies, hωp ¼ hω0 (black curve) and trajectories of the TE and TM components, respectively, hωTMðkt; r0Þ (blue curve). The latter energy corresponds of the polariton pulse describedR by theR parametric to the resonant energy on the TM branch at kt. The ratio dependence Y (X ) for X ðÞ¼t Iðt; rÞxdr= Iðt; rÞdr of the time-integrated intensities of two spatial compo- R j j Rj Aj Aj and Y ðÞ¼t Iðt; rÞydr= Iðt; rÞdr, where j = TE, TM; nents of the polariton wave packet as a function of the j Aj Aj ATE and ATM indicate the areas on the cavity plane probe wave number at the probe energy hωp ¼ hω0 is occupied by the corresponding components of the presented in Fig. 4b. Figure 4a shows that the total polariton pulse. The trajectory of the TM pulse compo- intensity of the polariton wave packet is nearly identical at nent is nearly circular, as the excitation is symmetric two resonant quasimomenta, kl and kt, corresponding to relative to the probe quasimomentum kp = kl, and the the cases considered in Figs. 2 and 3. However, the key kinetic energy matches the potential energy of the wave difference between the cases, that is, the very significant packet. In contrast, the TE pulse component propagates domination of one component of the polariton wave along the elliptical trajectory, with the major axis oriented packet over the other at kp = kt, is highlighted in Fig. 4b. along the x axis. This is connected with the fact that the The intensity of the dominant component at kt exceeds Sedov et al. Light: Science & Applications (2019) 8:79 Page 7 of 10

that of the other component by more than one and half is advantageous over the traditional polarizer due to the orders, while at kl, the superiority is less than three times. fact that full intensity of the probe is used to excite the Figure 4c shows the total time-integrated intensity of the polariton pulse, which implies reduced losses connected polariton wave packet as a function of the probe energy with reflection or absorption of the objectionable com- hωp at two resonant quasimonenta, kl (orange curve) and ponent of polarization. kt (purple curve). In both dependencies, the intensity reaches its maximum value near the resonance with the Discussion TE-polarized branch: close to ωt at kp = kl and close to ω0 In this paper, we report the concept of a spinoptronic at kp = kt; cf. with the dispersion in Fig. 3a. device able to transform polariton pulses with a given The above calculations have been performed under the polarization to linearly polarized pulses. The basis of the adiabatic limit, implying that the energy level separation, operating principle of the device is a separation of Δ 2 which is the TE-TM splitting LTkp , is large in compar- eigenmodes of the system with orthogonal linear polar- ison with the characteristic energy scales of the system, izations both in real and reciprocal space due to the including the spectral width of the polariton pulse. Let us mutual impact of the confining potential and the TE-TM now consider the polarization dynamics of the sub- splitting. We have shown that the spatial separation in picosecond polariton pulse possessing an energy spec- linear polarization is also valid for ultrashort (sub-pico- trum with a margin overlapping both the TE and TM second) pulses beyond the adiabatic limit. dispersion branches. For definiteness, we take the pulse One should note here that the effect of the separation of duration as wT = 200 fs. The other characteristics of the the linear polarizations of light can be observed even in a probe pulse are the same as those in Fig. 2, i.e., xp = x0, ωp pure photonic structure representing an empty optical = ω0 and kp = kt. The initial polarization is taken to be microcavity modifiedbyaconfining potential for photons right circular as well. The excitation spectrum is sche- in the cavity plane. However, the proposed system based on matically shown in Fig. 5a. In the considered excitation the strong-coupling regime has two undeniable advantages regime, the polariton wave packet predictably separates in terms of applications over the pure photonic system. into two spatial components with orthogonal polariza- In the pure photonic scheme with an empty micro- tions; see Fig. 5c–f. The time-integrated intensity and cavity, the lifetime of the optical pulse is fully determined polarization components in Fig. 5g–j show clear separa- by the quality factor of the cavity. In recent papers33,34,it tion of the trajectories of the wave packet components. was reported on ultra-high-Q-factor microcavities pro- The TE-branch component resonant to the probe exci- viding photon lifetimes of several hundreds of picose- tation follows the circular inner trajectory of radius r0 (the conds. The scheme proposed in our manuscript allows an black dashed ring in Fig. 5g), while the TM-branch extremely long lifetime of propagating polariton pulses, component propagates along the elliptic outer trajectory which in theory can be infinitely large even in micro- with the small ellipse axis close to 2r0 (the black solid cavities of relatively low quality. For this goal, we suggest ellipse in Fig. 5g). The trajectories of the Stokes vectors using the effect of the stimulated scattering from the characterizing polarization of the components of the reservoir of incoherent excitons to the coherent polariton polariton wave packet on the Poincaré sphere are shown state. The reservoir excitons are created optically by the in Fig. 5b. The conclusion made earlier is also valid subthreshold spatially homogeneous non-resonant cw beyond the adiabatic limit: the fraction of the circular pump, which itself does not create the polariton con- polarization becomes negligibly small in both components densate. When polaritons with the given quasimomentum of the polariton wave packet over the course of evolution. k are injected with the resonant probe pulse, the density of However, this fraction is noticeably smaller for the TE- polaritons with this k increases locally, which triggers the branch component. process of stimulated scattering to the k-state at the The discussion above is related to the operation with position of the pulse. The closer the non-resonant pump polariton pulses excited by a polarized input. The power is to the condensation threshold and the weaker remarkable peculiarity of the considered system is that a the polariton pulse, the more effective is the feeding of the polariton pulse excited by an initially unpolarized probe polariton pulse from the exciton reservoir. Remarkably, in acquires linear polarization in the course of propagation. the proposed scheme, the pulse lifetime is determined not The results of modelling confirming this statement are by the polariton or photon lifetime but by the inflow from shown in Fig. 6. The trajectory of the Stokes vector on the the exciton reservoir. In this regard, the pulse lifetime can Poincaré sphere degenerates to the closed one, analogous exceed that of polaritons and photons by many orders, to the characteristic of pulses excited by a polarized probe and achieving several nanoseconds for the lifetime of the (see Fig. 2b). Therefore, in addition to the allocated polariton pulses seems to be a routine procedure. function of the polarization rectifier, the considered sys- Another advantage of the polariton-based device over tem is also able to act as a polarizer for polariton pulses. It the pure photonic device is the controllability of the Sedov et al. Light: Science & Applications (2019) 8:79 Page 8 of 10

a kl kt b 2.5

2.0 TM(kt ,r0)

) (meV) 1.5 0

k,r ES ( 0 ω 1.0 TE(kl ,r0)

0.5 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 k (μm–1)

c def I Sx Sy Sz 150

100 (ps) t 50

100 100 100 100 0 50 50 50 50 –100 –100 –100 0 –100 0 m) 0 m) m) 0 m) –50 μ –50 μ –50 μ –50 μ –50 ( –50 ( –50 ( –50 ( 0 y 0 y 0 y 0 y x (μ 50 –100 x (μ 50 –100 x (μ 50 –100 x (μ 50 –100 m) 100 m) 100 m) 100 m) 100

ghij 100 _ _ _ _ I Sx Sy Sz 50 1 1 m)

μ 0 ( y –50

–100 0 –1 –100–50000–5050 50 100 –100–50 50 100 –100–50 50 100 –1000 100 x (μm) x (μm) x (μm) x (μm)

Fig. 5 Polarization behaviour of a sub-picosecond polariton pulse. a Dispersion of two spin-split polariton modes hωTE;TMðk; r0Þ at position r0 = 70 μm in comparison with the spectrum of the probe pulse. The ultrashort probe pulse of 200-fs duration is resonant to the TE polariton mode: kp = kt and ωp = ω0 at r0. b Trajectories on the Poincaré sphere of the Stokes vector characterizing polarization of the two spatio-temporal components of the split sub-picosecond polariton pulse. The black arrows indicate the direction with respect to the evolution in time. c Evolution of the intensity and d–f polarization of the two components of the sub-picosecond polariton pulse. g Time-integrated spatial distribution of the normalized intensity and h–j polarization components of the polariton pulse. The polarization of the probe pulse is taken as right circular for (c)–(j). The dashed and solid closed black curves in panel (g) indicate the trajectories of the TE and TM components of the polariton pulse, respectively

former. Due to the sensitivity of the exciton component to The proposed scheme has application potential far – an external impact (see, e.g. refs. 35 38), one can tune the beyond its use as a polarization rectiﬁer. In particular, we system to choose the required operating frequency range, have demonstrated that linear polarization is acquired by the TE-TM splitting energy, and the shape of the dis- a pulse excited by an unpolarized pulsed input during its persion surface. Remarkably, the tuning can be performed propagation. This allows the considered system to func- at any stage of working with the system, both in the stage tion as a polarizer for polariton pulses. The scheme can of structure growth and after the growth of the structure act as a laboratory tool for the study of polariton phe- is complete. nomena, including circular polarization supercurrents, Sedov et al. Light: Science & Applications (2019) 8:79 Page 9 of 10

We consider the reservoir to be pumped by a non- resonant homogeneous pump P slightly below the con- ¼ γ γ = γ densation threshold, P

Acknowledgements interference effects and spin-orbit interaction effects of This work is from the Innovative Team of the International Center for Polaritonics and is supported by Westlake University (Project No. long-living macroscopic polariton states. 041020100118). E.S.S. acknowledges support from the Grant of the President of the Russian Federation for state support of young Russian scientists (No. MK- Methods 2839.2019.2) and the RFBR (Grant No. 17-52-10006). Y.G.R. acknowledges support from CONACYT (Mexico) under Grant No. 251808. A.V.K. acknowledges We model the dynamics of the polariton pulses based the Saint Petersburg State University for research grant ID 40847559. on the generalized Pauli equation for the spinor T jΨi ¼ ½Ψþðt; rÞ; ΨÀðt; rÞ , where Ψ ± ðt; rÞ are the wave Author details 1 functions of the right- and left-circularly polarized Westlake University, 18 Shilongshan Road, Hangzhou 310024 Zhejiang Province, China. 2Institute of Natural Sciences, Westlake Institute for Advanced polariton components: ÂÃ Study, 18 Shilongshan Road, Hangzhou 310024 Zhejiang Province, China. ∂ jΨi ¼ ^ þ ð À γ Þ= jΨi þ j ð ; rÞi 3Vladimir State University, Gorky str. 87, Vladimir 600000, Russia. 4Instituto de ih t H0 ih RnR C 2 i F t Energías Renovables, Universidad Nacional Autónoma de México, 62580 ð2Þ Temixco, MOR, Mexico. 5Spin Optics Laboratory, Saint Petersburg State University, 1 Ulianovskaya, St. Petersburg 198504, Russia ^ where, in addition to the conservative Hamiltonian H0 Authors’ contributions given in (1), we introduce non-conservative processes of A.V.K. and Y.G.R. devised the concept. E.S.S. performed simulations, analysed the simulation results and wrote the manuscript, with input from all co- pumping and loss, with γC being the polariton decay rate. authors. A.V.K. initiated the study and guided the work. All authors discussed The polaritons are excited by the resonant probe and contributed to the paper. jFðt; rÞi ¼ FTðtÞFrðrÞjf i. To preserve the shape of the polariton pulse, we take the spatial component of Conflict of interest / ½Àðr À The authors declare that they have no conflict of interest. the probe in the Gaussianp formffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFr exp 2 2 Ã rpÞ =2w expðikprÞ of width w ¼ h=m ωtr. The pulse is shifted both in real and reciprocal space to rp and kp, Received: 25 February 2019 Revised: 16 July 2019 Accepted: 2 August 2019 respectively; see Fig. 1 for clarity. The temporal compo- / ½À 2= 2 ðÀ ω Þ nent is FT exp t 2wT exp i pt , where wT is the duration of the pulse and ωp is the probe frequency. The T References vector jf i ¼ ½fþ; fÀ defines polarization of the probe. 1. Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose–Einstein con- For better perception, in the model, we restrict ourselves densation. Rev. Mod. Phys. 82, 1489–1537 (2010). by choosing the characteristics of the probe pulse, r , k 2. Ballarini, D. et al. Macroscopic two-dimensional polariton condensates. Phys. p p Rev. Lett. 118, 215301 (2017). and ωp, such that the trajectory of the pulse is close to a 3. Liew, T. C. H., Shelykh, I. A. & Malpuech, G. Polaritonic devices. Phys.E:Low- – circular one taking rp = r0, ωp = ω0 and kp = kl,t. Dimens. 43,1543 1568 (2011). 4. Shelykh, I. A., Kavokin, A. V. & Malpuech, G. Spin dynamics of exciton polaritons in microcavities. Phys. Status Solidi B 242,2271–2289 (2005). To increase the pulse lifetime, we allow the reservoir of 5. Kavokin, A., Malpuech, G. & Glazov, M. Optical spin Hall effect. Phys. Rev. Lett. excitons to partially feed the polariton condensate; nR is 95, 136601 (2005). the density of the non-resonantly pumped reservoir of 6. Leyder, C. et al. Observation of the optical spin Hall effect. Nat. Phys. 3, 628–631 (2007). excitons, and R is the stimulated scattering rate describing 7. Kammann, E. et al. Nonlinear optical spin Hall effect and long-range spin the particle exchange between the reservoir and the pulse. transport in polariton lasers. Phys.Rev.Lett.109, 036404 (2012). Sedov et al. Light: Science & Applications (2019) 8:79 Page 10 of 10

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