Chiral condensates in a polariton hexagonal ring
Xuekai Ma, Yaroslav V. Kartashov, Alexey Kavokin, Stefan Schumacher
LLetter Optics Letters 1
Chiral condensates in a polariton hexagonal ring X UEKAI M A , Y AROSLAV
V. K
ARTASHOV , A
LEXEY K AVOKIN , AND S TEFAN S CHUMACHER Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn,Germany Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia Russian Quantum Center, Skolkovo 143025, Russia Westlake University, School of Science, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China Westlake Institute for Advanced Study, Institute of Natural Sciences, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China Spin Optics Laboratory, St-Petersburg State University, 1, Ulianovskaya, 198504 St-Petersburg, Russia College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA * Corresponding author: [email protected] September 18, 2020
We model generation of vortex modes in exciton-polariton condensates in semiconductor micropillars,arranged into a hexagonal ring molecule, in the pres-ence of TE-TM splitting. This splitting lifts the de-generacy of azimuthally modulated vortex modes withopposite topological charges supported by this struc-ture, so that a number of non-degenerate vortex statescharacterized by different combinations of topologi-cal charges in two polarization components appears.We present a full bifurcation picture for such vortexmodes and show that because they have different ener-gies, they can be selectively excited by coherent pumpbeams with specific frequencies and spatial configura-tions. At high pumping intensity, polariton-polaritoninteractions give rise to the coupling of different vortexresonances and a bistable regime is achieved. © 2020Optical Society of America http://dx.doi.org/10.1364/ao.XX.XXXXXX
Microcavity polaritons are bosonic quasiparticles with a finitelifetime on a picosecond scale. They may condense [1, 2], butstill experience spontaneous decay accompanied by the emissionof coherent light, the phenomenon called polariton lasing [3].Especially interesting is the situation where condensation andlasing occur in states carrying nonzero topological charges. Inthis case, spatial structuring of the microcavity potential en-ergy landscape, the presence of spin-orbit interaction (SOI), andstrong polariton-polariton interactions may dramatically affectthe emerging polariton states.SOIs of different physical origin have been widely stud-ied and play a crucial role in many areas of physics, includ-ing physics of semiconductors [4] and optics [5, 6]. It canalso strongly affect the behaviour of excitations in optoelec-tronic systems, such as semiconductor microcavities operatingin the regime of strong coupling between quantum-well excitonsand cavity photons, where exciton-polariton condensates areformed [7–9]. The SOI for polaritons arises from the splitting ofthe transverse-electric (TE) and transverse-magnetic (TM) modes of the cavity photons. Many interesting phenomena caused bythe TE-TM splitting were reported for microcavity polaritons,such as the formation of half-quantum vortices [10–13], opticalspin Hall effect [14, 15], skyrmions [16, 17], and topological in-sulators [18–24]. The resonant excitation of states with differenttopological charges for different spin components of a polari-ton condensate has been proposed [25]. In this context, it isimportant to develop the tools for topological engineering ofpolariton condensates that would enable on demand generationof vortices with specific topological charges [26].In this Letter, we study the formation of vortex polaritonmodes in micropillars arranged into a hexagonal structure(molecule) in the presence of TE-TM splitting. Vortex modeswith opposite topological charges in such a potential in the ab-sence of the TE-TM splitting are degenerate, i.e. they have iden-tical energies. The TE-TM splitting lifts the degeneracy affectingenergies of states in different ways: while several vortices withallowed topological charges acquire increased frequencies, mostof them are pushed to lower frequencies. This enables excitationof desired vortex states by coherent pumping with trivial phasedistribution. The excitation efficiency depends not only on thefrequency of the coherent pump, but also on its spatial config-uration and symmetry. We also investigate the influence of thenonlinearity caused by polariton-polariton interactions on themodes and demonstrate the bistability regime.The evolution of two polarization components of polaritoncondensates under the coherent pump can be described by thespinor Gross-Pitaevskii equations [18, 21]: i ¯ h ∂ Ψ ± ( r, t ) ∂ t = (cid:34) − ¯ h m (cid:79) ⊥ − i ¯ h γ c + g c | Ψ ± | + V ( r ) (cid:35) Ψ ± ( r, t )+ ∆ LT k (cid:0) ∂ x ∓ i ∂ y (cid:1) Ψ ∓ ( r, t ) + E ± ( r, t ) . (1) Here, the indices ± indicate the right-/left-circular polariza-tion components of polaritons, the effective polariton mass isgiven by m = − m e ( m e is the free electron mass), γ c = − is the polariton loss rate, g c = µ eV µ m denotes thepolariton-polariton interaction strength, ∆ LT represents the TE-TM splitting (leading to the SOI) intrinsically present in mi- a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p etter Optics Letters 2 Fig. 1.
Eigenstates of polaritons in a hexagonal molecule at different TE-TM splitting. (a) Dependence of eigenfrequencies ω (inTHz) of eigenmodes on the mode index n for different TE-TM splittings: ∆ LT = | Ψ ± | and phase arg ( Ψ ± ) distributions, marked bythe mode indices corresponding to the mode sequence in (a), for both polarization components Ψ + and Ψ − at ∆ LT = ∆ LT = k = µ m − , V ( r ) = ∑ V ( x − x n , y − y n ) is the potential energy landscape createdby micropillars with the diameter 2 d = µ m arranged into a ringwith a radius of R = µ m so that neighboring pillars slightlytouch each other [see the inset in Fig. 1(a)], V = V e − ( x + y ) / d describes the contribution from the individual pillar with depth V = − E ± ( r, t ) is the coherent pump.We first analyze the linear eigenstates of the hexagonalmolecule in the conservative regime, by setting γ c = E ± =
0. Assuming linear solutions of the form Ψ ± ( r, t ) = u ± ( r ) e − i ω t we obtain from Eq. (1) the eigenvalue problem¯ h ω u ± = − ¯ h m (cid:79) ⊥ u ± + Vu ± + ∆ LT k (cid:0) ∂ x ∓ i ∂ y (cid:1) u ∓ that we solvedto obtain eigenfrequencies ω and shapes u ± of the linear modes.If we consider only the lowest fundamental mode in each pillar(note that in a single micropillar the higher-order modes of thepolariton condensates with integer or fractional orbital angularmomenta can also be observed [27, 28]), the whole potentialwith six pillars give rise to 12 modes (Fig. 1). In the absence ofthe TE-TM splitting ∆ LT = Ψ + and Ψ − distributions can, forexample, have the same phase or π phase difference for thesame spatial distribution. Therefore there exist at least six setsof degenerated states. An additional degeneracy is connectedwith the fact that for a selected component Ψ + or Ψ − in thesix-pillar structure only vortices with topological charges (wind-ing numbers) of | m | ≤ + m and − m are degenerate too (their superpositiongives multipole states), with the exception for the state with m = m = m = ± ω = -5.665 THz) and m = ± ω = -5.286THz). These two latter sets of modes form quadruplets and ineach quadruplet only two pairs of modes have different densitydistributions. The modes with lowest and highest frequenciesform doublets with identical density distributions.The presence of the TE-TM splitting breaks the symmetry ofthe system and lifts the degeneracy of the vortex modes withopposite topological charges, see the green and blue dots inFig. 1(a) for different values of the TE-TM splitting. Remarkably,two lowest modes 1,2 and two highest modes 11,12 remain de-generate and experience a shift toward lower frequency valuesthat progressively increases with the increase of the TE-TM split-ting (compare green dots for ∆ LT = ∆ LT = m = ±
1, while modesemerging from the upper quadruplet carry vortices with charges m = ±
2. Note that phase singularities in modes 8 and 10 splitinto two singularities due to the presence of the TE-TM split-ting. If the splitting is significant, the shift of the modes in thefrequency domain may become so strong that the order of themodes changes completely: for example, the blue point for themode n = etter Optics Letters 3 Fig. 2.
Linear modes excited for different pump configurations.
Dependencies of the peak density ( ρ max in µ m − ) of Ψ + on thepump frequency ( ω in THz) for different pump configurations: (a) one-pillar excitation P , (b) two pump spots located symmet-rically P , (c) two pump spots asymmetrically located on even pillars P , (d) two pump spots in adjacent pillars P , (e) threesymmetric pump spots P , and (f) three pump spots in adjacent pillars P . Pump profiles are shown to the right of the ρ max ( ω ) dependencies. The potential landscape (dashed circles) is superimposed on top of the pump profile in (a). The green indices close tothe peaks correspond to the modes marked by the green numbers in Fig. 1(b). Here, E ± = ∆ LT = E ± ( r, t ) = E ± ( r ) e − i ω t , where ω is the frequency of the pump, and search for the stationarysolutions of Eq. (1) by solving the time-independent equation: (cid:34) − ¯ h m (cid:79) ⊥ − i ¯ h γ c + g c | u ± | + V ( r ) (cid:35) u ± + ∆ LT k (cid:0) ∂ x ∓ i ∂ y (cid:1) u ∓ + E ± ( r ) − ¯ h ω u ± = (2) For the homogeneous pump, only the fundamental mode [greennumber 1 in Fig. 1(b)] can be excited. To excite the vortex modes,we use a Gaussian pump beam given by E ± ( r ) = E ± e − r / d ,where E ± is the amplitude of the pump and pump beam width d = µ m is similar to that of the pillar [see Fig. 2(a)].For a single Gaussian pump beam located at an arbitrary (forexample, the upper) pillar, several non-degenerate modes canbe excited by changing pump frequency ω at ∆ LT = n =
1, 4, 5, 7, 10, 11from Fig. 1(b), leading to resonant spikes at corresponding eigen-frequencies, as shown in Fig. 2(a). The variety of excited modescan be efficiently controlled by using more than one pump spot.In all cases the excitation efficiency is given by the magnitude ofpump projection on the mode. For two symmetrically locatedpump spots [Fig. 2(b)], the phase of the condensate in thesetwo pillars should be the same to have maximal projection fora given polarization component, which leads to the excitationof the modes n =
1, 7, and 10. If we keep one pump spot in theupper pillar and move the second spot from the bottom pillarto its right neighbor [Fig. 2(c)], the two vortex-free modes 1 and11 are strongly enhanced. For adjacent pump spots in Fig. 2(d)the phase distribution of the mode 11 does not match the pumpdistribution anymore and disappears, while vortex-carryingmodes 5,6 with m = ± m = ± ρ max ( ω ) dependencies become tiltedand broaden with the increase of pump amplitude [Fig. 3(a)].In Fig. 2(d) one can see that the first two peaks at the left sideare very close to each other, so that they are tilted then the non-linearity versions start overlapping [see Fig. 3(b)] because thefirst stronger peak tilts more than the second weaker peak as thepump intensity increases. This leads to a bistability: the coexis-tence of two stable nonlinear states shown in Fig. 3(c,d) at thesame value of ω . Similar phenomena have been demonstratedin the similar potential landscapes for photon-like modes [30]or under non-resonant excitation [31]. Stable nonlinear statesoriginating from the non-degenerate modes 5 and 9 can be foundin Fig. 3(e,f). These solutions become strongly asymmetric. In-creasing pump strength leads to the splitting of the vortex witha higher topological charge m = ± m = ± etter Optics Letters 4 lasers where lasing from topologically protected modes wouldbe realised. Fig. 3.
Nonlinear modes.
Peak density ( ρ max in µ m − ) of Ψ + vs pump frequency ( ω in THz) for (a) one pump spot P with E ± = P with E ± = ∆ LT = Funding.
Deutsche Forschungsgemeinschaft (DFG) (No.231447078, 270619725); Russian Science Foundation (Project 17-12-01413- Π ). Westlake University, project 041020100118 andthe Program 2018R01002 funded by Leading Innovative andEntrepreneur Team Introduction Program of Zhejiang. Acknowledgements.
Paderborn Center for Parallel Comput-ing, PC ; A.K. acknowledges Saint-Petersburg State Universitysupport program, ID 40847559. Disclosures.
The authors declare no conflicts of interest.
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