Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices
Yoav Lahini,1,* Assaf Avidan,1Francesca Pozzi,2Marc Sorel,2Roberto Morandotti,3 Demetrios N. Christodoulides,4 and Yaron Silberberg1
1Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel
2Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow, Scotland
3Institut national de la recherche´ scientifique, Universite´ du Que´bec, Varennes, Que´bec, Canada
4CREOL/College of Optics, University of Central Florida, Orlando, Florida, USA
(Received 19 April 2007; revised manuscript received 10 August 2007; published 10 January 2008) We experimentally investigate the evolution of linear and nonlinear waves in a realization of the Anderson model using disordered one-dimensional waveguide lattices. Two types of localized eigen- modes, flat-phased and staggered, are directly measured. Nonlinear perturbations enhance localization in one type and induce delocalization in the other. In a complementary approach, we study the evolution on short time scales of !-like wave packets in the presence of disorder. A transition from ballistic wave packet expansion to exponential (Anderson) localization is observed. We also find an intermediate regime in which the ballistic and localized components coexist while diffusive dynamics is absent. Evidence is found for a faster transition into localization under nonlinear conditions.
DOI: 10.1103/PhysRevLett.100.013906 PACS numbers: 42.25.Dd, 42.65.Tg, 72.15.Rn
The propagation of waves in periodic and disordered structures is at the foundation of modern condensed-matter physics. Anderson localization is a key concept, formu- lated to explain the spatial confinement due to disorder of quantum mechanical wave functions that would spread over the entire system in an ideal periodic lattice [1–4].
Although Anderson localization was studied experimen- tally, the underlying phenomena—the emergence of local- ized eigenmodes and the suppression of wave packet expansion—were rarely observed directly [5,6]. Instead, localization was usually studied indirectly by measure- ments of macroscopic quantities such as conductance [2], backscattering [7,8], and transmission [9,10].
An interesting issue concerns the effect of nonlinearity on Anderson localization. Nonlinear interactions between the propagating waves and nonlinearly accumulated phases can significantly change interference properties, thus fun- damentally affecting localization. The theoretical study of the nonlinear problem advanced using several approaches:
the study of the transmission problem [11], the study of the effect of nonlinear perturbations on localized eigenmodes [12], and the study of the effect of nonlinearity on wave packet expansion in the presence of disorder [13]. Only a few experiments were reported [5]. Recently, optical stud- ies enabled the study of wave evolution in nonlinear dis- ordered lattices [14–16], using a scheme discussed in [17,18]. In particular, Schwartz et al. [16] reported the observation of Anderson localization of expanding wave packets in 2D lattices.
In this work we investigate directly linear and nonlinear wave evolution in one-dimensional (1D) disordered pho- tonic lattices, using two different approaches. In the first part of this work, all the localized eigenmodes of a weakly disordered lattice are selectively excited. Nonlinearity is then introduced in a controlled manner, to examine its effect on localized eigenmodes. The second part of this
work presents a study of the effect of disorder on the evolution of !-like wave packets (single site excitations).
A transition from free ballistic wave packet expansion to exponential localization is observed, as well as an inter- mediate regime of coexistence. We then measure the effect of nonlinearity on this process.
Our experimental setup is a one-dimensional lattice of coupled optical waveguides patterned on an AlGaAs sub- strate [19,20], illustrated in Fig.1(a). Light is injected into one or a few waveguides at the input and can coherently tunnel between neighboring waveguides as it propagates along thezaxis. Light distribution is then measured at the output [see, for example, Fig. 1(b)–1(d)].
The equations describing light dynamics in these struc- tures are identical (in the linear limit) to the equations describing the time evolution of a single electron in a lattice under the tight binding approximation [19], i.e., a set of coupled discrete Schro¨dinger equations:
FIG. 1 (color online). (a) Schematic view of the sample used in the experiments. The red arrow indicates the input beam. (b)–
(d) Images of output light distribution, when the input beam covers a few lattice sites: (b) in a periodic lattice, (c) in a disordered lattice, when the input beam is coupled to a location which exhibits a high degree of expansion, and (d) in the same disordered lattice when the beam is coupled to a location in which localization is clearly observed.
PRL100, 013906 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending 11 JANUARY 2008
0031-9007=08=100(1)=013906(4) 013906-1 2008 The American Physical Society
Figure 1.13: On top, a drawing of the experimental photonic waveguide setup from a paper by Lahini et al. [87]. The waveguides are arranged along the x-axis, and the light propagates along the z-axis. The red arrow shows the direction of the input beam. The three images below show experimental output patterns. The first is for periodic waveguides. The bottom two show localization in disordered waveguides;
the two images correspond to cases where the input beam is aimed at weakly and strongly localized sites respectively.
the Aubry-Andr´e transition and explored the effects of nonlinearities upon Anderson and Aubry-Andr´e localized phases [87, 90]. Furthermore, they have been able to study differences in density-density correlation functions between light traveling in waveguides with diagonal and off-diagonal disorder13 [89]. Finally, very recently, they have pioneered the study of topological edge states in quasiperiodic photonic waveg- uide lattices [85, 137]. These developments have opened an exciting new frontier in localization physics.
2, we discuss the application of the strong-disorder renormalization group to the dirty boson problem by Altman, Kafri, Polkovnikov, and Refael [7, 9]. This work has raised the possibility that, in one dimension, the weak-disorder transition of Giamarchi and Schulz [58] could give way to a different critical behavior beyond a certain disorder strength. We analytically calculate the superfluid susceptibility (1.21) at the strong-disorder fixed point of Altman et al. Then, in Chapter 3, we numerically extend the approach of Altman et al. to higher dimensions, identify a new fixed point that governs the two-dimensional superfluid-insulator transition, and extract various universal properties of this transition. In Chapter 4, we turn our attention to the many-body localization problem and numerically investigate the real- time dynamics of a many-body quasiperiodic system. We provide evidence that the noninteracting Aubry-Andr´e transition becomes a many-body localization transition upon the introduction of interactions. We additionally develop toy models of the many-body localized and ergodic regimes and use our data to extract estimates for the phase boundary in the interactions versus hopping plane.
This thesis interpolates material from three papers by the author [74, 75, 73].
Chapter 2 uses material from References [74] and [75], both coauthored with David Pekker and Gil Refael. Meanwhile, Chapter 3 is based on Reference [75]. Finally, Chapter 4 is based on Reference [73], coauthored with Vadim Oganesyan, Gil Refael, and David Huse. Some material from each of these papers has also been incorporated into this introductory Chapter.
Chapter 2
Critical Susceptibility for
Strongly-Disordered Bosons in One Dimension
2.1 Introduction
2.1.1 Motivation
In their seminal work on one-dimensional dirty bosons, Giamarchi and Schulz showed that perturbative disorder is irrelevant at the clean transition between the Mott insulating and superfluid phases [58]. Their analysis remains hugely influen- tial in our understanding of the role of disorder in one-dimensional bosonic systems.
Nevertheless, it is possible that the picture developed by Giamarchi and Schulz is not complete. Since their perturbative method cannot handle strong disorder, it cannot tell us whether the nature of the superfluid-insulator transition changes in this regime.
This possibility motivated Altman, Kafri, Polkovnikov, and Refael to formulate a strong-disorder renormalization group (SDRG) for the one-dimensional dirty boson problem. Using this approach, Altman et al. were able to identify a novel fixed point,
which may describe the superfluid-insulator transition in the strong-disorder regime.
The transition proposed by Altman et al., while still being of Kosterlitz-Thouless type, occurs at a nonuniveral value of the Luttinger parameter. This difference from the transition of Giamarchi and Schulz is physically very significant: it suggests that, in the strong-disorder regime, superfluidity is destroyed by a process other than phase slips [7, 9].
The strong-disorder fixed point of Altman et al. remains controversial. It is not of the infinite-randomness class, and the SDRG procedure does not become asymp- totically exact near criticality. Therefore, it is important to check the conclusions of the SDRG using other methods. The initial Monte Carlo simulations that followed the work of Altman et al. did not yield evidence in favor of strong-disorder criticality [14], motivating some to conclude that the transition is always of the type found by Giamarchi and Schulz. More recently however, Hrahsheh and Vojta performed a new Monte Carlo analysis, focusing on a stronger-disorder regime than was explored in the earlier simulations [70]. Their measurements of two quantities, the Luttinger parameter and the superfluid susceptibility, appeared to show the first independent evidence of strong-disorder criticality.
As we mentioned above, the nonuniversal critical Luttinger parameter was calcu- lated by Altman et al. and was, in one sense, the key prediction of their work [9]. On the other hand, these authors did not calculate the critical superfluid susceptibility.
To make closer contact between the SDRG and the Monte Carlo results of Hrahsheh and Vojta, we undertake a calculation of the susceptibility in this chapter.
2.1.2 Preview of the Results
Our principal result is that, at the strong-disorder transition, the divergence of the superfluid susceptibility is characterized by an anomalous dimension:
Llim→∞
dlnχ
dlnL = 2−ηsd (2.1)
Here, the exponent:
ηsd ≈ 1 2π
q
yi+yi2 (2.2)
depends upon the bare disorder strength, parametrized by the quantity yi. We plot ηsd as a function of yi in Figure 2.1. The meaning of the parameter yi will become more clear when we introduce the attractor distributions of the SDRG flows in Sec- tion 2.3 below. For now, note that yi = 0 corresponds to a flat distribution of bare Josephson couplings. As yi increases, the bare Josephson coupling distribution be- comes progressively more strongly peaked near the RG scale, effectively reducing the disorder strength. Thus, the anomalous dimension monotonically increases as the disorder strength decreases.
The nonuniversal anomalous exponent (2.2) differs from the value at the transition of Giamarchi and Schulz, which is the Kosterlitz-Thouless value of η= 14 [58]. Asyi
is increased, eventually our estimate (2.2) will reach the Kosterlitz-Thouless value, presumably indicating the crossover to the weak-disorder regime. This crossover was indeed seen in the numerics of Hrahsheh and Vojta [70]. Regardless of the disorder strength, Hrahsheh and Vojta also observed that the anomalous dimension is related
0.2 0.4 0.6 0.8 1.0yi
0.05 0.10 0.15 0.20
hsd
Figure 2.1: The anomalous exponent ηsd(yi), as approximated in the small yi regime by (2.2).
to the Luttinger parameter as η ≈ 2K1 . Our estimate (2.2) approximately matches the Luttinger parameter of Altman et al. via this relation [9].
However, our work also points to the need for some caution in interpreting nu- merical results. When tuning the transition near yi = 0, the wideness of the initial distribution of Josephson couplings prohibits efficient self averaging of the anomalous exponent. Cleanly observing the “true” exponent ηsd requires exploring very large lattices, or equivalently, doing a very large amount of statistical sampling on smaller lattices.
2.1.3 Organization of the Chapter
We begin in Section 2.2 by introducing our model of dirty bosons, the disordered rotor model1, and its relationship to the more familiar Bose-Hubbard model (1.19).
Next, in Section 2.3, we describe the application of the strong-disorder renormal- ization group to this model. We present the renormalization group flows that were
1This is also the model that we will study in two dimensions in Chapter 3.
derived analytically by Altman et al. and also comment on the important relationship that these authors found between symmetry properties of the rotor Hamiltonian and the nature of the insulating phase.
We proceed to our calculation of the superfluid susceptibility in Section 2.4. We outline the general structure of our calculation, derive some intermediate results, and then piece together the calculation ofχand the anomalous dimensionη. At key stages in the calculation, we compare to a numerical implementation of the SDRG. Then, in Section 2.5, we discuss the relationship of our results with the work of Altman et al. and Hrahsheh and Vojta. Finally, in Section 2.6, we summarize and examine the implications of our work for the superfluid-insulator transition.