Bosonic Condensation and Disorder-Induced Localization in a Flat Band

Baboux, Florent; Li, Ge; Jacqmin, T.; Biondi, Matteo; Galopin, Élisabeth; Lemaı̂tre, A.; Gratiet, L. Le; Sagnes, I.; Schmidt, Sebastian; Türeci, Hakan E.; Amo, A.; Bloch, J.

doi:10.1103/physrevlett.116.066402

Cited by 314 publications

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“…Systems that exhibit flat bands have attracted considerable interest in the past few years, including optical [1,2] and photonic lattices [3][4][5][6], graphene [7,8], superconductors [9][10][11][12], fractional quantum Hall systems [13][14][15] and exciton-polariton condensates [16,17]. One interesting consequence of a flat band is the different scaling properties of its localization length when compared with a dispersive band [18][19][20], due to a multitude of degenerate states in the flat band.…”

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confidence: 99%

Parity-time symmetry in a flat-band system

2015

Phys. Rev. A

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In this paper we introduce Parity-Time (PT ) symmetric perturbation to a one-dimensional Lieb lattice, which is otherwise P-symmetric and has a flat band. In the flat band there are a multitude of degenerate dark states, and the degeneracy N increases with the system size. We show that the degeneracy in the flat band is completely lifted due to the non-Hermitian perturbation in general, but it is partially maintained with the half-gain-half-loss perturbation and its "V" variant that we consider. With these perturbations, we show that both randomly positioned states and pinned states at the symmetry plane in the flat band can undergo thresholdless PT breaking. They are distinguished by their different rates of acquiring non-Hermicity as the PT -symmetric perturbation grows, which are insensitive to the system size. Using a degenerate perturbation theory, we derive analytically the rate for the pinned states, whose spatial profiles are also insensitive to the system size. Finally, we find that the presence of weak disorder has a strong effect on modes in the dispersive bands but not on those in the flat band. The latter respond in completely different ways to the growing PT -symmetric perturbation, depending on whether they are randomly positioned or pinned.

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Section: Introductionmentioning

confidence: 99%

Parity-time symmetry in a flat-band system

2015

Phys. Rev. A

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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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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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“…Coupled cavity QED systems [20][21][22] have emerged as natural platforms to study many-body physics of open quantum systems. The current fabrication and control capabilities in solid-state quantum optics allows us to probe lattice systems [23][24][25][26][27][28][29][30][31] in the mesoscopic regime, providing a first glimpse into how macroscopic quantum behavior may arise far from equilibrium. It is therefore of interest to identify a physical system where a nonequilibrium phase transition (i) can be studied-at least in principle-in the thermodynamic limit, (ii) can be compared to an equilibrium analogue through a proper limiting procedure, and (iii) can be easily realized in an architecture that is currently available.…”

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Exotic Attractors of the Nonequilibrium Rabi-Hubbard Model

et al. 2016

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We explore the phase diagram of the dissipative Rabi-Hubbard model, as could be realized by a Ramanpumping scheme applied to a coupled cavity array. There exist various exotic attractors, including ferroelectric, antiferroelectric, and incommensurate fixed points, as well as regions of persistent oscillations. Many of these features can be understood analytically by truncating to the two lowest lying states of the Rabi model on each site. We also show that these features survive beyond mean field, using matrix product operator simulations. DOI: 10.1103/PhysRevLett.116.143603 Introduction.-A number of recent experimental breakthroughs [1][2][3][4] have spurred the investigation of nonequilibrium properties of hybrid quantum many-body systems of interacting matter and light. Characterized by excitations with a finite lifetime, when sustained by finiteamplitude optical drives they display steady-state phases that are generally far richer [5][6][7][8][9][10] than their equilibrium counterparts [11,12]. Critical phenomena in these open driven-dissipative systems often come with genuinely new properties and novel dynamic universality classes, even when an effective temperature can be identified [13][14][15][16][17], a statement that can be made robust in renormalization group calculations [18,19]. Coupled cavity QED systems [20][21][22] have emerged as natural platforms to study many-body physics of open quantum systems. The current fabrication and control capabilities in solid-state quantum optics allows us to probe lattice systems [23][24][25][26][27][28][29][30][31] in the mesoscopic regime, providing a first glimpse into how macroscopic quantum behavior may arise far from equilibrium. It is therefore of interest to identify a physical system where a nonequilibrium phase transition (i) can be studied-at least in principle-in the thermodynamic limit, (ii) can be compared to an equilibrium analogue through a proper limiting procedure, and (iii) can be easily realized in an architecture that is currently available.The Rabi-Hubbard (RH) model [32] represents the minimal description of coupled cavity QED systems, explicitly containing terms which do not conserve the particle number. These terms are relevant for the lowfrequency behavior of the coupled system and their inclusion lead, in equilibrium, to a Z 2 -symmetry breaking phase transition between a quantum disordered paraelectric phase and an Ising ferroelectric [32,33]. The equilibrium RH transition requires a sizable intercavity hopping or light-matter interaction, of the order of the transition

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Bosonic Condensation and Disorder-Induced Localization in a Flat Band

Cited by 314 publications

References 56 publications

Parity-time symmetry in a flat-band system

Parity-time symmetry in a flat-band system

Topological flat Wannier-Stark bands

Exotic Attractors of the Nonequilibrium Rabi-Hubbard Model

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