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Quantum coherence–driven self-organized criticality and nonequilibrium light localization

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Science Advances  16 Mar 2018:
Vol. 4, no. 3, eaaq0465
DOI: 10.1126/sciadv.aaq0465
  • Fig. 1 The conceived coherence-driven photonic nanosystem.

    The nonequilibrium system we consider is a quantum-coherently driven active nanophotonic heterostructure in its stopped-light regime. It consists of a 26-nm-thick quantum well (QW) bounded by aluminum (Al), with the upper cladding having a thickness of 50 nm. The QW is described by a three-level scheme (57), as shown in the lower inset. An incoherent incident pump (shown in red) sends electrons from level |1> to level |3> with a pump rate g, whereas a coherent drive (shown in white) couples the levels |2> and |3> with a Rabi frequency Ωa. Photons emitted in the |2>-|1> transition stimulate coherently the stopped-light (“heavy photon”) mode that the heterostructure supports (see Fig. 4). The pump beam deposits gain over the middle region highlighted translucently. Numerical Maxwell-Bloch-Langevin computations (61, 62) reveal the emergence of a sinc-shaped nonequilibrium-localized light field (shown in false color), in agreement with theory (see Fig. 2B, right).

  • Fig. 2 Synchronization in classical mechanics and in optics.

    (A) The dynamics and emergence of synchronization in many-body classical systems can be modeled using coupled mechanical oscillators (metronomes) (16). When a collection of metronomes is placed on a fixed basis (top), in which case they do not couple, they oscillate randomly (“disordered phase”), never synchronizing. In contrast, when the metronomes are placed on a basis that can freely move (bottom), in which case the metronomes couple, they eventually synchronize without any external intervention (that is, “spontaneously”), reaching an “ordered” state where they all move in phase. (B) In the active nanophotonic heterostructure that we study (Fig. 1), a large number of oscillating lasing modes couple nonlinearly in the gain region (translucent in Fig. 1) just as the metronomes of (A) couple nonlinearly via the moving base. Each lasing mode can be considered a “particle” in a many-body nonequilibrium system (27) and, on its own, is delocalized, oscillating in space with a spatial frequency βn (n = 1, 2,…) (left). These {βn} modes initially oscillate out of phase, giving rise to a noise pattern resulting from their superposition and interference (left). However, owing to their nonlinear coupling, they eventually synchronize in space after a transient time interval, giving rise to the emergence of a localized sinc function–shaped supermode (right). The onset of spatial synchronization of the {βn} modes is associated with a nonequilibrium first-order phase transition (see main text and Materials and Methods).

  • Fig. 3 Many-electron characteristics of the nanoplasmonic layers of the structure of Fig. 1.

    (A) Feynman diagrams illustrating schematically the polarization process (top) and renormalization of the electron-electron interaction strength, Ve, to Veff (bottom) in the considered plasmonic medium (Al). 1PI, one-particle irreducible. (B) Comparison between the real part of the macroscopic permittivity of Al (−Re{εAl}), as derived from the microscopic theory presented here (Eq. M1), the experimental data by Ehrenreich et al. (46), and the standard Drude formula (45).

  • Fig. 4 Complex band diagram of the nanophotonic heterostructure.

    Complex frequency dispersion band of the TM2 mode. Also shown schematically in the upper panel with a filled Lorentzian shape is the bandwidth of the QW gain (57), whereas highlighted are the excited {βn} states. The lower panel shows the group velocity vg of the TM2 mode, which becomes zero at two points in the region where β < 40 × 106 m−1. The inset in the lower panel presents an intuitive ray picture understanding of how positive feedback arises in the considered self-organized complex nanosystem.

  • Fig. 5 Spontaneous synchronization and emergence of attractors.

    (A) Onset of synchronization: temporal dynamics of two coupled {βn} states showing repeated cycles of asynchronous (AS) motion, followed by synchronization (S), and then temporal coherence buildup (CB). In all cases, the onset of spontaneous synchronization precedes that of activity (CB). (B) Emergence of attractors: dynamic evolution in the phase space (control parameter/inversion versus order parameter/photon number) of the localized light field of Fig. 1. After a transient period (0 < τ < 3), during which the light field remains robustly localized and the permittivity of the pumped core region dynamically changes (the control parameter ρ22 − ρ11 ∝ Im{εcopumped}), the system is attracted to a stable point, robust to perturbations. The inset shows the same result but without quantum coherence driving the system.

  • Fig. 6 Emergence of scaling and toward the quantum self-organized critical regime.

    (A) Order parameter (number of photons) versus the pump rate, g, with and without quantum coherence driving the nanosystem. The nonlinearly interacting {βn} lasing modes self-organize into a critical state, with a second-order (first-order) phase transition to a supercritical phase being observed without (with) the quantum-coherent drive. (B) Emergence of scale-invariant (fractal) power law in the order parameter versus the normalized coherent drive Embedded Image, where Embedded Image is the critical inversion. The computed critical isotherm exponent is δ ~ 3.07. The upper inset shows the variation of the critical inversion with the applied coherent field Ωa. For another suitably selected choice of parameters (see section S3), the control parameter (that is, the critical inversion Embedded Image, which plays the role of the effective temperature here) reduces to zero for a sufficiently strong drive Ωa (lower inset). That point is the quantum self-organized critical point (QSCP) of our nonequilibrium dynamical system, somewhat analogous to the quantum critical point (QCP) in equilibrium static systems in condensed matter (2125).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/3/eaaq0465/DC1

    table S1. Comparison between the two nonpotential (nontrivial) light localization schemes.

    fig. S1. Electron probability density for various potential configurations (63).

    fig. S2. Schematic illustration of the considered nanosystem along with its macroscopic characteristics.

    fig. S3. Negative Goos-Hänchen phase shift of a light ray upon reflection at a dielectric/plasmonic.

    section S1. Supplementary text on wave localization

    section S2. Effective thickness of and feedback formation in the plasmonic heterostructure

    section S3. Quantum coherence–driven three-level active medium

    section S4. Statistical independence of the nonequilibrium localization events

    References (6367)

  • Supplementary Materials

    This PDF file includes:

    • table S1. Comparison between the two nonpotential (nontrivial) light localization schemes.
    • fig. S1. Electron probability density for various potential configurations (63).
    • fig. S2. Schematic illustration of the considered nanosystem along with its macroscopic characteristics.
    • fig. S3. Negative Goos-Hänchen phase shift of a light ray upon reflection at a dielectric/plasmonic.
    • section S1. Supplementary text on wave localization
    • section S2. Effective thickness of and feedback formation in the plasmonic heterostructure
    • section S3. Quantum coherence–driven three-level active medium
    • section S4. Statistical independence of the nonequilibrium localization events
    • References (63–67)

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