Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling

Whittaker, Charles; Cancellieri, E.; Walker, P. M.; Gulevich, Dmitry R.; Schomerus, Henning; Vaitiekus, D.; Royall, B.; Whittaker, D. M.; Clarke, Edmund M.; Iorsh, I. V.; Shelykh, I. A.; Skolnick, M. S.; Krizhanovskii, D. N.

doi:10.1103/physrevlett.120.097401

Cited by 162 publications

(154 citation statements)

References 61 publications

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“…This indicates that the B-to-A and C-to-A tunnel coupling strongly depends on the polarization state of the polariton. A similar polarization behavior has been found by C. E. Whittaker et al under quasi-resonant excitation in a similar geometry [33].…”

supporting

confidence: 87%

Polariton condensation in S- and P-flatbands in a two-dimensional Lieb lattice

Klembt

Harder

Egorov

et al. 2017

Applied Physics Letters

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We study the condensation of exciton-polaritons in a two-dimensional Lieb lattice of micropillars. We show selective polariton condensation into the flatbands formed by S and Px,y orbital modes of the micropillars under non-resonant laser excitation. The real space mode patterns of these condensates are accurately reproduced by the calculation of related Bloch modes of S-and Pflatbands. Our work emphasizes the potential of exciton-polariton lattices to emulate Hamiltonians of advanced potential landscapes. Furthermore, the obtained results provide a deeper inside into the physics of flatbands known mostly within the tight-binding limit.Dispersionless energy bands or flatbands (FBs) appear in a large variety of condensed matter systems and are linked to a wide range of topological many-body phenomena such as graphene edge modes [1], the fractional quantum Hall effect [2][3][4][5] and flat band ferromagnetism [6][7][8].There is a variety of two-dimensional lattices that support flat energy bands [9][10][11], with the so-called Lieb lattice being on of the most straightforward examples [12]. Lieb lattices have been studied extensively in recent years and flatband states have been observed in photonic [13][14][15] as well as cold atom systems [16]. Creating artificial lattices in order to emulate and simulate complex many-body systems with additional degrees of freedom has attracted considerable scientific interest [17][18][19]. Exciton-polariton gases in periodic lattice potential landscapes have emerged as a very promising solid state system to emulate many-body physics [20,21]. Polaritons are eigenstates resulting of strong coupling between a quantum well exciton and a photonic cavity mode. The excitonic fraction provides a strong nonlinearity while the photonic part results in a low effective mass, allowing the formation of driven-dissipative BoseEinstein condensation [22,23]. These so-called quantum fluids of light [24] can be placed in an artificial lattice potential landscape using a variety of well developed semiconductor etching techniques [9,25,26], thin metal films [27], surface acoustic waves [28], or optically imprinted lattices [29,30]. In this work we investigate the polariton photoluminescence (PL) emission in a two-dimensional Lieb lattice ( Fig. 1(a)). Due to destructive interference of next neighbor tunneling J, flatbands form. Fig. 1(b) shows a tightbinding calculation of the first Brillouin zone (BZ) band structure, with the flatband dispersion highlighted in red. High symmetry points of the BZ are found in the inset.The two-dimensional polaritonic Lieb lattice was fabricated using an electron beam lithography process * sebastian.klembt@physik.uni-wuerzburg.de and a consecutive reactive ion etching step on an AlAs λ/2-cavity with three stacks of four 13 nm wide GaAs quantum wells (QWs) placed in the antinode of the electric field, with a 32.5 (36) fold AlAs/Al 0.20 Ga 0.80 As top (bottom) distributed Bragg reflector (DBR) (Fig. 1(c,d)). The Rabi splitting of the sample is 9.5 meV. Only the top D...

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confidence: 87%

Polariton condensation in S- and P-flatbands in a two-dimensional Lieb lattice

Klembt

Harder

Egorov

et al. 2017

Applied Physics Letters

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“…Thus, the increase of the strain-induced gap ε g shifts the existence domain of the DKh surface states to the region of large electron wave vectors, k s . It follows from equation(16) that the spectrum of the DKh surface states in strained HgTe is parabolic and described approximately by equation(19) for large wave vectors, k s , satisfying the condition k s g 2 g e  .…”

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confidence: 99%

Structure of surface electronic states in strained mercury telluride

Kyriienko

Shelykh

2019

New J. Phys.

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We present the theory describing the various surface electronic states arisen from the mixing of conduction and valence bands in a strained mercury telluride (HgTe) bulk material. We demonstrate that the strain-induced band gap in the Brillouin zone center of HgTe results in the surface states of two different kinds. Surface states of the first kind exist in the small region of electron wave vectors near the center of the Brillouin zone and have the Dirac linear electron dispersion characteristic for topological states. The surface states of the second kind exist only far from the center of the Brillouin zone and have the parabolic dispersion for large wave vectors. The structure of these surface electronic states is studied both analytically and numerically in the broad range of their parameters, aiming to develop its systematic understanding for the relevant model Hamiltonian. The results bring attention to the rich surface physics relevant for topological systems.

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“…Flat band networks have been proposed in one, two, and three dimensions and various flat band generators were identified [3][4][5][6]. Experimental observations of flat bands and CLS are reported in photonic waveguide networks [7][8][9][10][11][12][13][14][15], exciton-polariton condensates [16][17][18], and ultracold atomic condensates [19,20]. The tight binding network equations correspond to an eigenvalue problem EΨ l = − m t lm Ψ m .…”

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confidence: 99%

Topological flat Wannier-Stark bands

2018

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We analyze the spectrum and eigenstates of a quantum particle in a bipartite two-dimensional tight-binding dice network with short range hopping under the action of a dc bias. We find that the energy spectrum consists of a periodic repetition of one-dimensional energy band multiplets, with one member in the multiplet being strictly flat. The corresponding macroscopic degeneracy invokes eigenstates localized exponentially perpendicular to the dc field direction, and super-exponentially along the dc field direction. We also show that the band multiplet is characterized by a topological winding number (Zak phase), which changes abruptly if we vary the dc field strength. These changes are induced by gap closings between the flat and dispersive bands, and reflect the number of these closings.Recently much attention has been paid to flat bands in one-, two-and three-dimensional lattices with short range hoppings and non-trivial geometry [1]. Flat bands with finite range hoppings exist due to destructive interference leading to a macroscopic number of degenerate compact localized eigenstates (CLS) which have strictly zero amplitudes outside a finite region of the lattice [2]. Flat band networks have been proposed in one, two, and three dimensions and various flat band generators were identified [3][4][5][6]. Experimental observations of flat bands and CLS are reported in photonic waveguide networks [7][8][9][10][11][12][13][14][15], exciton-polariton condensates [16][17][18], and ultracold atomic condensates [19,20]. The tight binding network equations correspond to an eigenvalue problem EΨ l = − m t lm Ψ m . For bipartite lattices, the existence of flat bands and CLS is ensured through a proper usage of the protecting chiral symmetry [6]. For example, for the dice lattice shown in Fig. 1(a) and t lm = 1 the CLS consists of an empty C site which is surrounded by six excited A and B sites with alternating amplitudes ±1/ √ 6.When a dc field is added, quantum particles start to experience well studied Bloch oscillations (see e.g. [21,22]). Our results are therefore applicable for ultracold atoms in optical lattices where the electric field is substituted by a tilt of the lattice in the gravitational field [23] or accelerating the whole lattice [24]. The same type of perturbations can be arranged in optical waveguide arrays where the electric field is modeled by a curved geometry of the waveguides [25].In the present letter we report on a new family of flat bands which exist in dc biased bipartite lattices and which are not supported by CLS, despite of the short range hopping. We consider the dice lattice in the presence of a dc field oriented along the y-direction (Fig.1(a)). The absence of CLS indicates nontrivial topology. We compute winding numbers (Zak phase) which indeed show abrupt changes through conical intersection point degeneracies upon variations of the dc field strength. We also generalize to different orientations of the field F and other lattice geometries.

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Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling

Cited by 162 publications

References 61 publications

Polariton condensation in S- and P-flatbands in a two-dimensional Lieb lattice

Polariton condensation in S- and P-flatbands in a two-dimensional Lieb lattice

Structure of surface electronic states in strained mercury telluride

Topological flat Wannier-Stark bands

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