Localization by virtual transitions in correlated disorder

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Localization by virtual transitions in correlated disorder

Alex Dikopoltsev, Hanan Herzig Sheinfux, and Mordechai Segev

Phys. Rev. B 100, 140202(R) – Published 11 October 2019

Abstract

Anderson localization is fundamentally relevant to all physical systems where coherent waves evolve in the presence of disorder. Physically, any disorder must have a finite spectral bandwidth, hence the disorder is always correlated. Here, we study the regime of localization mediated by virtual transitions in correlated disorder. We find that wave packets centered outside of the spectral extent of the disorder can localize, with localization length almost as short as for localization via first-order transitions. In this regime, virtual transitions lead to phenomena with profound significance, such as mobility edges, and strong localization in regions of momentum space where otherwise localization would be extremely weak. Remarkably, in two-dimensional systems, we show localization can occur in directions that cannot be reached by direct scattering from the disordered potential.

Received 5 June 2019
Revised 3 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.140202

Physics Subject Headings (PhySH)

Anderson localization Localization Photonics Weak localization

Atomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

Alex Dikopoltsev^1,*, Hanan Herzig Sheinfux^1,2, and Mordechai Segev ¹

¹Physics Department and Solid State Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel
²ICFO-Institut de Cie´ncies Foto´niques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

^*Corresponding author: alexdiko@technion.ac.il

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Issue

Vol. 100, Iss. 14 — 1 October 2019

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Images

Figure 1
Scattering transitions in momentum space. (a) Power spectrum $S_{V} (k_{x})$ of a 1D disordered potential concentrated in two finite bands. The arrows represent transitions in $k$ space, $k_{t}$ . The blue arrow represents a possible scattering process mediated by a spectral component with $k_{t} = k_{0} \hat{x}$ . The red arrows show transitions that are not supported by this potential. (b) Examples of scattering processes mediated by the $k_{0} \hat{x}$ spectral component of the disorder (blue arrow). The green curve is the phase mismatch $| Δ k_{z} (k_{x}) | = k_{x}^{2} / 2 n_{0} k$ . First-order phase-matched transition: a wave of wave number $- k_{0} / 2$ scatters into a wave at $k_{0} / 2$ , so that $Δ k_{z} (- k_{0} / 2) = Δ k_{z} (k_{0} / 2) .$ Second-order phase-matched transition: first, a wave scatters from $- k_{0}$ to a wave at $k_{x} = 0$ . This transition alone is not efficient due to high phase mismatch. However, a subsequent second transition from $k_{x} = 0$ to a wave at $k_{0}$ makes the overall transition efficient, which eventually leads to localization by second-order virtual transitions. The intermediate state at $k_{x} = 0$ is called “virtual” because it cannot be reached by a single phase-matched scattering event in this scheme.
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Figure 2
Simulations of beam propagation and localization in correlated disorder. (a) Power spectrum of disorder concentrated in two narrow regions $Δ k$ around wave numbers $k_{0}$ and $- k_{0}$ . (b)–(f) Simulated propagation of an initially Gaussian beam launched at various incidence angles corresponding to different $k_{in}$ values. The dashed-dotted yellow line marks the $x = 0$ line, while the dashed-dotted orange line marks the propagation direction in the absence of disorder (homogeneous medium). (b) At $k_{in} = 0$ the beam propagates on the z axis without expanding due to a second-order process. (c) At $0 < k_{in} < k_{0} / 2$ , the transitions are phase mismatched, so the beam propagates ballistically, unaffected by the disorder. (d) For $k_{in} = k_{0} / 2$ , multiple first-order transitions are phase matched, hence the beam is localized and exhibits on-axis propagation (e) $k_{0} / 2 < k_{new} < k_{0}$ , same as (c): the beam evolves ballistically on its initial trajectory, unaffected by the disorder. (f) Evolution in the regime of localization by virtual transitions. Initially, the wave packet expands but after some distance the expansion stops. (g) Effective width $w_{eff}$ as a function of propagation distance, representing an ensemble average over 100 realizations of disorder with a power spectrum as in (a). The beam incident at $k_{0} / 2 < k_{in} < k_{0}$ expands linearly as an undisturbed Gaussian beam. At $k_{in} = k_{0} / 2$ the wave packet initially expands slightly and then its $w_{eff}$ becomes constant as the beam localizes. At $k_{in} = k_{0}$ , the beam experiences intense expansion, but after some distance it eventually becomes localized, marking localization by virtual transitions.
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Figure 3
Left: Ensemble average of the output intensity distribution for various input wave numbers, $k_{in}$ . Right: beam cross sections for input wave numbers for which first-order and second-order localization occur, with the ballistic regimes in between. At $k_{0} / 2$ , the beam is localized by first-order phase-matched transitions, but localization occurs also at $k_{0}$ , induced by virtual transitions only. At other incidence angles the wave packet stays Gaussian (polynomial in the log-scale). The centers of output beams that would have propagated in the same system, but with a homogenous refractive index, are marked with a red dashed line.
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Figure 4
Effective beam width $w_{eff}$ for two types of correlated disorder: (a) when the disorder spectrum has a cutoff frequency at $2 k_{loc}$ , and (b) when the disorder spectrum has two narrow spectral bands around $2 k_{loc}$ , shifted from zero. The blue region marks the spectral extent of the disorder for each type ( $2 k_{loc}$ in the spectrum corresponds to $k_{loc}$ in the spectral extent). The plots in (a) and (b) are calculated for various strengths of disorder ( $Δ n$ ) and system sizes L. The upper bound on the inverse effective width is mainly defined by $l_{c}$ (determined by the strength of the disorder at specific wave numbers), while the lower bound is set by $L$ . (b) Localization of the first-order with short $l_{c}$ , in the vicinity of $k_{loc} \equiv k_{0} / 2$ and localization induced by virtual transitions at $2 k_{loc}$ and at 0 with an $l_{c}$ relatively small compared to $L$ .
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Figure 5
Localization on the $x$ axis strictly by virtual transitions in 2D correlated disorder. (a),(b) Example of a disordered potential $Δ n (x, y)$ whose FT is the spectrum in (b). The features consist of small scale and large scale structures (enlarged in the inset). The spectrum consists of finite regions of random coefficients located symmetrically on the vertices of a rectangle in $k$ space with size $[- k_{0}, k_{0}] \times [- \sqrt{3} k_{0}, \sqrt{3} k_{0}]$ . Yellow arrows mark possible transitions. (c) Effective beam width $w_{eff}$ , for 20 disorder realizations, for incidence angles: at (red), below (blue) and above (yellow) the angle associated with localization by virtual transitions. (d),(e),(f) Spectra of beams after 10 cm of propagation, (e) the angle where second-order virtual transitions are phase matched, (d),(f) lower and higher incidence angle, where both first and second-order transitions are phase mismatched. Red circles mark the central wave number at incidence and blue dashed circles mark the phase-matched $k$ . Insets: intensity cross section in the $x$ direction of the ensemble-averaged beam for 100 realizations of disorder in log scale for the three angles [inset (e) shows exponential decay, while (d) and (f) stay mainly Gaussian].
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