Observing two-particle Anderson localization in linear disordered photonic lattices.

We show how two-dimensional waveguide arrays can be used to probe the effect of on-site interaction on Anderson localization of two interacting bosons in one dimension. It is shown that classical light and linear elements are sufficient to experimentally probe the interplay between interaction and disorder in this setting. For experimental relevance, we evaluate the participation ratio and the intensity correlation function as measures of localization for two types of disorder (diagonal and off-diagonal), for two types of interaction (repulsive and attractive), and for a variety of initial input states. Employing a commonly used set of initial states, we show that the effect of interaction on Anderson localization is strongly dependent on the type of disorder and initial conditions, but is independent of whether the interaction is repulsive or attractive. We then analyze a certain type of entangled input state where the type of interaction is relevant and discuss how it can be naturally implemented in waveguide arrays. We conclude by laying out the details of the two-dimensional photonic lattice implementation including the required parameter regime.

The mobility edges and reentrant localization transitions are studied in one-dimensional dimerized lattice with non-Hermitian either uniform or staggered quasiperiodic potentials. We find that the non-Hermitian uniform quasiperiodic disorder can induce an intermediate phase where the extended states coexist with the localized ones, which implies that the system has mobility edges. The localization transition is accompanied by the symmetry breaking transition. While if the non-Hermitian quasiperiodic disorder is staggered, we demonstrate the existence of multiple intermediate phases and multiple reentrant localization transitions based on the finite size scaling analysis. Interestingly, some already localized states will become extended states and can also be localized again for certain non-Hermitian parameters. The reentrant localization transitions are associated with the intermediate phases hosting mobility edges. Besides, we also find that the non-Hermiticity can break the reentrant localization transition where only one intermediate phase survives. More detailed information about the mobility edges and reentrant localization transitions are presented by analyzing the eigenenergy spectrum, inverse participation ratio, and normalized participation ratio.

We study Anderson localization of light in one-dimensional (1D) and 2D photonic lattices. The influence of nonlinearity and disorder on Anderson localization in both systems is studied numerically. A sharp difference between localization in the linear and the nonlinear regimes is observed. Localization in the linear regime is more pronounced in the 2D lattice than in the 1D lattice. In the nonlinear regime, the coexistence of nonlinearity and disorder leads us to different conclusions. There are different nonlinear regimes in which 1D localization is more pronounced than 2D localization.

International audience

Article Figures and data Abstract eLife digest Introduction Results Discussion Materials and methods Appendix 1: Model Details Appendix 2 Thermodynamics of assembly around a fluid cargo References Decision letter Author response Article and author information Metrics Abstract We computationally study how an icosahedral shell assembles around hundreds of molecules. Such a process occurs during the formation of the carboxysome, a bacterial microcompartment that assembles around many copies of the enzymes ribulose 1,5-bisphosphate carboxylase/ oxygenase and carbonic anhydrase to facilitate carbon fixation in cyanobacteria. Our simulations identify two classes of assembly pathways leading to encapsulation of many-molecule cargoes. In one, shell assembly proceeds concomitantly with cargo condensation. In the other, the cargo first forms a dense globule; then, shell proteins assemble around and bud from the condensed cargo complex. Although the model is simplified, the simulations predict intermediates and closure mechanisms not accessible in experiments, and show how assembly can be tuned between these two pathways by modulating protein interactions. In addition to elucidating assembly pathways and critical control parameters for microcompartment assembly, our results may guide the reengineering of viruses as nanoreactors that self-assemble around their reactants. https://doi.org/10.7554/eLife.14078.001 eLife digest Bacterial microcompartments are protein shells that are found inside bacteria and enclose enzymes and other chemicals required for certain biological reactions. For example, the carboxysome is a type of microcompartment that enables the bacteria to convert the products of photosynthesis into sugars. During the formation of a microcompartment, the outer protein shell assembles around hundreds of enzymes and chemicals. This formation process is tightly controlled and involves multiple interactions between the shell proteins and the cargo – the enzymes and other reaction ingredients – they will enclose. Understanding how to control which enzymes are encapsulated within microcompartments could help researchers to re-engineer the microcompartments so that they contain drugs or other useful products. Recent studies have used microscopy to visualize how microcompartments are assembled. However, most of the intermediate structures that form during assembly are too small and short-lived to be seen. It has therefore not been possible to explore in detail how shell proteins collect the necessary cargo and then assemble into an ordered shell with the cargo on the inside. Experiments alone are probably not enough to understand the process, especially since microcompartment assembly can currently only be studied within live cells or cellular extract. Within these complex environments it is difficult to determine the effect of any individual factor on the overall assembly process. Perlmutter, Mohajerani and Hagan have now taken a different approach by developing computational and theoretical models to explore how microcompartments assemble. Computer simulations showed that microcompartments could assemble by two pathways. In one pathway, the protein shell and cargo coalesce at the same time. In the other pathway, the cargo molecules first assemble into a large disordered complex, with the shell proteins attached on the outside. The shell proteins then assemble, carving out a piece of the cargo complex. The simulations showed that many factors affect how the shell assembles, such as the strengths of the interactions between the shell proteins and the cargo. They also identified a factor that controls how much cargo ends up inside the assembled shell. Perlmutter, Mohajerani and Hagan found that, in addition to revealing how microcompartments may assemble within their natural setting, the simulations provided guidance on how to re-engineer microcompartments to assemble around other components. This would enable researchers to create customizable compartments that self-assemble within bacteria or other host organisms, for example to carry out carbon fixation or make biofuels. A future challenge will be to investigate other aspects of microcompartment assembly, such as the factors that control the size of these compartments. https://doi.org/10.7554/eLife.14078.002 Introduction Encapsulation is a hallmark of biology. A cell must co-localize high concentrations of enzymes and reactants to perform the reactions that sustain life, and it must safely store genetic material to ensure long-term viability. While lipid-based organelles primarily fulfill these functions in eukaryotes, self-assembling protein shells take the lead in simpler organisms. For example, viruses surround their genomes with a protein capsid, while bacteria use large icosahedral shells known as bacterial microcompartments (BMCs) to sequester the enzymes and reactions responsible for particular metabolic pathways (Kerfeld et al., 2010; Axen et al., 2014; Shively et al., 1998; Bobik et al., 1999; Erbilgin et al., 2014; Petit et al., 2013; Price and Badger, 1991; Shively et al., 1973; Shively et al., 1973; Kerfeld and Erbilgin, 2015). Within diverse bacteria, BMC functions have been linked to bacterial growth, carbon fixation, symbiosis, or pathogenesis (Kerfeld and Erbilgin, 2015). Other protein-based compartments are found in bacteria and archea (e.g. encapsulins (Sutter et al., 2008) and gas vesicles (Pfeifer, 2012; Sutter et al., 2008)) and even eukaryotes (e.g. vault particles (Kickhoefer et al., 1998)), while some viruses may assemble around lipidic globules (Lindenbach and Rice, 2013; Faustino et al., 2014). Thus, understanding the factors that control microcompartment assembly and encapsulation is a central question in modern cell biology. From the perspectives of synthetic biology and nanoscience, there is great interest in reengineering BMCs or viruses as nanoreactors that spontaneously encapsulate enzymes and reagents in vitro (e.g. Luque et al., 2014; Douglas and Young, 1998; Rurup et al., 2014; Patterson et al., 2014; Patterson et al., 2012; Zhu et al., 2014; Rhee et al., 2011; Rurup et al., 2014; Wörsdörfer et al., 2012; Comas-Garcia et al., 2014), or as customizable organelles that assemble around a programmable set of core enzymes in vivo, introducing capabilities such as carbon fixation or biofuel production into bacteria or other organisms (e.g. Kerfeld and Erbilgin, 2015; Bonacci et al., 2012; Parsons et al., 2010; Choudhary et al., 2012; Lassila et al., 2014). However, the principles controlling such co-assembly processes have yet to be established, and it is not clear how to design systems to maximize encapsulation. In this article we take a step toward this goal, by developing theoretical and computational models that describe the dynamical encapsulation of hundreds of cargo molecules by self-assembling icosahedral shells. Although our models are general, we are motivated by recent experiments on a type of BMC known as the carboxysome (Kerfeld et al., 2010; Schmid et al., 2006; Iancu et al., 2007; Tanaka et al., 2008). Carboxysomes are large (40–400 nm), roughly icosahedral shells that encapsulate a dense complex of the enzyme ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) and other proteins to facilitate the Calvin-Bensen-Bassham cycle in autotrophic bacteria (Price and Badger, 1991; Shively et al., 1973; Shively et al., 1973; Iancu et al., 2007; 2010; Kerfeld et al., 2010; Tanaka et al., 2008). Recently, striking microscopy experiments visualized β-carboxysome shells assembling on and budding from procarboxysomes (the condensed complex of RuBisCO and other proteins found in the interior of carboxysomes) (Cameron et al., 2013; Chen et al., 2013). Genomic analysis suggests that many BMCs with diverse functions assemble via similar pathways (Cameron et al., 2013; Kerfeld and Erbilgin, 2015). However, the mechanisms of budding and pinch-off to close the shell remain incompletely understood because of the small size and transient nature of assembly intermediates. Moreover, experiments suggest that α-carboxysomes (another form of carboxysome) assemble by a different mechanism, in which shell assembly encapsulates an initially diffuse pool of RuBisCO (Iancu et al., 2010; Cai et al., 2015). The factors determining which of these assembly pathways occurs are unknown. BMC assembly is driven by a complex interplay of interactions among the proteins forming the external shell and the interior cargo. It is difficult, with experiments alone, to parse these interactions for those mechanisms and factors that critically influence assembly pathways, especially due to the lack of an in vitro assembly system. Models which can correlate individual factors to their effect on assembly are therefore an important complement to experiments. Previous experimental and theoretical studies of encapsulation by icosahedral shells, e.g. the assembly of viral capsids around their nucleic acid genomes (e.g. Hu and Shklovskii, 2007; Kivenson and Hagan, 2010; Elrad and Hagan, 2010; Perlmutter et al., 2013; 2014; Mahalik and Muthukumar, 2012; Zhang et al., 2013; Zhang and Linse, 2013; Hagan, 2008; Devkota et al., 2009; Dixit et al., 2006; Borodavka et al., 2012; Dykeman et al., 2013; 2014; Zlotnick et al., 2013; Johnson et al., 2004; Patel et al., 2015; Cadena-Nava et al., 2012; Comas-Garcia et al., 2012; 2014; Garmann et al., 2014a; 2014b; Malyutin and Dragnea, 2013), have demonstrated that the structure of the cargo can strongly influence assembly pathways and products. However, BMCs assemble around a cargo which is topologically different from a nucleic acid — a fluid complex comprising many, noncovalently linked molecules. We demonstrate here that changing the cargo topology leads to new assembly pathways and different critical control parameters. We present phase diagrams and analysis of dynamical simulation trajectories showing how the thermodynamics, assembly pathways, and emergent structures depend on the interactions among shell proteins and cargo molecules. Within distinct parameter ranges, we observe two classes of assembly pathways, which resemble those suggested for respectively α- or β-carboxysomes. We find that tunability of cargo loading is a key functional difference between the two classes of pathways. Shells assembled around a diffuse cargo can be varied from empty (containing almost no cargo) to completely full, whereas assembly around a condensed, procarboxysome-like complex invariably produces full shells. While we find that the encapsulated cargo becomes ordered due to confinement, complete crystalline order in the globule before encapsulation inhibits budding. We discuss these results in the context of recent observations on carboxysome assembly, and their implications for engineering BMCs, viruses or drug delivery vehicles that assemble around a fluid cargo (e.g. Refs. [Kerfeld and Erbilgin, 2015; Parsons et al., 2010; Choudhary et al., 2012; Lassila et al., 2014; Luque et al., 2014; Douglas and Young, 1998; Rurup et al., 2014; Patterson et al., 2014; Patterson et al., 2012; Zhu et al., 2014; Rhee et al., 2011; Rurup et al., 2014; Wörsdörfer et al., 2012]). Results Our model system is motivated by icosahedral viral capsids and BMCs (Tanaka et al., 2008; Kerfeld et al., 2010). Since icosahedral symmetry can accommodate at most 60 identical subunits, formation of large icosahedral structures requires subunits to assemble into different local environments. The subunits can be grouped into pentamers and hexamers, with 12 pentamers at the icosahedron vertices and the remaining subunits in hexamers. Viruses typically assemble from small oligomers of the capsid protein, which we refer to as the basic assembly unit (Hagan, 2014). Recent AFM experiments demonstrated that hexamers are the basic assembly unit during the assembly of BMC shell facets (Sutter et al., 2016), and the carboxysome major shell proteins crystallize as pentamers and hexamers (Tanaka et al., 2008). Motivated by these observations, our model considers two basic assembly units, one a pentamer and the other a hexamer, with interactions designed so that the lowest energy structure corresponds to a truncated icosahedron with 12 pentamers and 20 hexamers (Figure 1). While BMCs generally have more hexamers, our model is intended to explore the general principles of assembly around a fluid cargo rather than model a specific system. Further details of the model and a thermodynamic analysis are given in section 3 and the appendices. Figure 1 Download asset Open asset Description of the model. (A) Each shell subunit contains 'Attractors' (green circles) on the perimeter, a 'Top' (tan circle, 'T') in the center above the plane, and a 'Bottom' (purple circle, 'B' below the plane). (B) Interactions between complementary Attractors drive subunit dimerization, with the Top-Top repulsions (tan arrow) tuned to favor the subunit-subunit angle in a complete shell. Complementary pairs of attractors are indicated by green arrows in (A) for the pentamer-hexamer interface and in (B) for the hexamer-hexamer interface. (C) Bottom psuedoatoms bind cargo molecules (terra cotta circles, 'C'), while excluder atoms (blue and brown pseudoatoms in (D)) placed in the plane of the pentagon experience excluded volume interactions with the cargo. (D) The positions of excluder atoms in the lowest energy shell geometry, a truncated icosahedron with 12 pentamers (blue) and 20 hexamers (brown). https://doi.org/10.7554/eLife.14078.003 To understand how assembly around multiple cargo molecules depends on the relative strengths of interactions between components, we performed dynamical simulations as a function of the parameters controlling shell subunit-subunit (εSS), shell subunit-cargo (εSC), and cargo-cargo (εCC) interaction strengths. All energy values are given in units of the thermal energy, kB⁢T. We focus on parameters for which shell subunit-subunit interactions are too weak to drive assembly in the absence of cargo (εSS≤4.5). Except where mentioned otherwise, the cargo diameter is set equal to the circumradius of a shell subunit. For the simulated density of cargo particles, the phase behavior (in the absence of shells) corresponds to a vapor at εCC=1.3, liquid-vapor phase coexistence for εCC∈[1.6,2.0] (the phase coexistence boundary is slightly below εCC=1.6), and a solid phase at εCC=3.0. We find that tuning εCC through phase coexistence dramatically alters the typical assembly process. Strong cargo interactions (εCC≥1.6) drive formation of a globule followed by assembly and budding of a shell, such as observed for β-carboxysomes (Figure 2A, Simulation Video 1), while under weak interactions (εCC<1.6) shell assembly usually proceeds in concert with cargo encapsulation (Figure 2B, Simulation Video 2), as suggested for assembly of α-carboxysomes. We now elaborate on these classes of assembly pathways, and how the resulting assembly products depend on parameter values. Figure 2 with 2 supplements see all Download asset Open asset Snapshots illustrating typical assembly trajectories. (A) Multi-step assembly involving an amorphous globule of cargo and shell subunits. (B) Single-step assembly, in which shell assembly drives local cargo condensation. and (C) when shell-cargo interactions are too weak to condense the cargo. The values of the cargo-cargo (εCC), shell subunit-cargo (εSC), and subunit-subunit (εSS) interaction strengths are listed above each panel (all energies are in units of the thermal energy kB⁢T), and the time (in units of 106 timesteps) is noted below each image. The color scheme here and throughout the manuscript is: Red=Cargo, Blue=Pentagon Excluder, Brown=Hexagon Excluder. Attractor and Bottom pseudoatoms are omitted to aid visibility. Videos of assembly trajectories are included below. https://doi.org/10.7554/eLife.14078.004 Video 1 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Animation of a typical simulation showing assembly around a cargo globule. Parameters are εCC=1.6, εSC=7, and εSS=2.5. https://doi.org/10.7554/eLife.14078.007 Video 2 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Animation of a typical simulation showing simultaneous assembly and cargo condensation. Parameters are εCC=1.3, εSC=9, and εSS=3.5. https://doi.org/10.7554/eLife.14078.008 Assembly and budding from a cargo globule We begin by discussing assembly behavior when the cargo-cargo interactions are strong enough to drive equilibrium phase coexistence (εCC≥1.6). Near the phase boundary (εCC=1.6) a system of pure cargo particles is metastable on the timescales we simulate. However, for εSC>4, adding shell subunits drives nucleation of a cargo globule with shell subunits adsorbed on the surface. The subsequent fate of the globule depends on parameter values; typical simulation end-states are shown as a function of parameter values in Figure 3. For moderate interaction strengths (2.5≤εSS≤3.5) the globule grows to a large size, typically containing at least twice the cargo molecules that can be packaged within a complete shell. Adsorbed shell subunits then reversibly associate to form ordered clusters. Once a cluster acquires enough inter-subunit interactions to be a stable nucleus, it grows by coagulation of additional subunits or other adsorbed clusters. For the parameter set corresponding to Figure 2A, nucleation is fast in comparison to cluster growth, and thus two nuclei grow simultaneously. The last three images show the system immediately preceding and following detachment of the lower shell. Missing only one of its 32 subunits, the shell is connected to the remainder of the droplet only by a narrow neck of cargo. Insertion of the final subunit breaks the neck and completes shell detachment. The complete shell contains 120–130 cargo particles, which is slighty above random close packing (≈120 particles) but below fcc density (≈150 particles, see appendix 1.2). Figure 3 with 3 supplements see all Download asset Open asset Results of assembly around a cargo globule. (A) The most frequently observed assembly outcome is overlaid on a color map of the theoretical free energy density difference Δ⁢fassem (Equation (3)) between assembled shells and the unassembled globule. Results are plotted against the shell-cargo adsorption strength εSC and the shell-shell interaction strength εSS for indicated values of the cargo-cargo interaction strength εCC. (B) Representative snapshots of the predominant assembly outcomes shown in (A). https://doi.org/10.7554/eLife.14078.009 Figure 3—source data 1 List of all simulation outcomes for Figures 3A,5A. https://doi.org/10.7554/eLife.14078.010 Download elife-14078-fig3-data1-v3.zip Figure 3—source data 2 Criteria used to categorize assembly outcomes. The sizes of each cargo globule and shell assemblage, and associations between shell assemblages and cargo globules, were determined by clustering. The outcome was then categorized according to the criteria listed in this table. https://doi.org/10.7554/eLife.14078.011 Download elife-14078-fig3-data2-v3.zip Increasing the shell-shell interaction strength drives faster shell assembly and closure, thus limiting the size of the globule before budding. For the largest interaction strength we simulated (εSS=4.5) the globule typically does not exceed the size of a single shell, and multiple globules nucleate within the simulation box (Figure 2—figure supplement 1). This observation could place an upper bound on shell-shell interaction strengths, since multiple nucleation events were rare in the carboxysome assembly experiments (Cameron et al., 2013) (however, we discuss potential complicating factors within the cellular environment below). To quantify the relationship between assembly mechanism and parameter values, we calculate an assembly order parameter, defined as the maximum number of unassembled subunits adsorbed onto a globule during an assembly trajectory. The order parameter is shown as a function of the interaction strengths in Figure 4. For εCC≥1.6 and εSS≤3 we observe large values of the order parameter (e.g. >32, the red and yellow regions in Figure 4), which formation of a large amorphous globule with the to carboxysome shell assembly (Cameron et al., 2013). Figure with 1 supplement see all Download asset Open asset of assembly on shell-cargo and shell-shell interaction The assembly order parameter, defined as the maximum number of unassembled shell subunits adsorbed on a globule at any during a is shown as a function of εSC and εSS for indicated values of the cargo-cargo interaction εCC. of adsorbed unassembled subunits the two step assembly mechanism (Figure whereas values to simultaneous assembly and cargo (Figure Other assembly products of the parameter ranges, we observe classes of outcomes. weak shell-shell interactions to drive For and the cargo vapor phase is and the system no cargo on simulated timescales discuss below). cargo-cargo or shell-cargo interactions in unassembled where a cargo globule forms but the shell subunits on its to εSS we observe assembly on the leading to complete shells or two classes of In the first one or more shells almost but to from the droplet within simulated for when the subunit-cargo interaction does not a strong enough for the last to the cargo and close the shell. strong interactions drive the other of in which an of assembled shells the system of free subunits before any shells are In this it is also to observe in which within shells. the cargo-cargo interaction multiple narrow the parameter that leads to complete assembly and detachment. cargo globules nucleate at multiple within the simulation the of the the of εSC required for cargo resulting in shells a parameter We also observe a we refer to as in which shell assembly to the globule thus does not even to the attached The is especially for when the cargo even in the absence of shell encapsulation. For and of the number of nucleation we typically observe into a large globule. shell assembly and cargo For the cargo forms an equilibrium vapor phase in the absence of shell subunits. However, above values of εSS and the diffuse cargo molecules drive nucleation of shell The subsequent assembly depends on the shell-cargo interaction For εSC (Figure assembly only a cargo leading to but empty shells. For εSC (Figure 2B, and Simulation Video 2), the shell-cargo interactions drive local of cargo molecules. assembly and cargo then in the mechanism for assembly of α-carboxysomes (Iancu et al., 2010). Thus, tuning the shell-cargo interaction dramatically cargo with a from empty to shells around This the equilibrium (Figure by a complete shell to cargo molecules. This effect is to the of vapor below its inside of In assembly around a globule only full shells. Figure with 2 supplements see all Download asset Open asset Results of assembly around a cargo with weak interactions (A) The most frequently observed assembly outcome as a function of εSS and The of outcomes for is shown in Figure supplement and a data containing the outcome for each at each parameter set is included (Figure 3—source data 1). (B) Representative snapshots for the outcomes shown in (A). The complete shell outcomes are shown with the and to enable the encapsulated cargo. (C) The number of cargo molecules encapsulated by shells assembled in simulations is to the results of equilibrium simulations The results are all complete shells any assembled at each of the simulations were performed for simulations with and assembled shells at and were up to Assembly of full shells pathway, Figure or Figure is typically two of faster than assembly of empty shells (Figure This the key that the cargo in shell during all of molecules initially shell nucleation by interactions among clusters. the of a condensed globule a large for adsorption of additional subunits, the of subunits to the capsid, thus its The condensed cargo of the last subunits, which are by as noted for simulations of empty capsids et al., Figure how the products of assembly around cargo with weak interactions depends on parameters. While moderate parameter values lead to complete assembly, weak εSC and εSS of Figure shell leading to the In the of large εSC but weak εSS the shell-cargo interaction small disordered globules cargo particles, lower of Figure while under strong subunit and weak cargo interactions shells nucleate but cannot condense the leading to the complete but assembly for assembly around a strong interactions lead to and shells. However, the predominant of is now shell the cargo is below its the condensed globule leads to a on the shell subunits, which can the shell and thus closure of a shell. model The free energy model the parameter values required for shell assembly with no parameters map in Figure Since it is an equilibrium model and only considers the free energy difference between complete and unassembled it cannot between parameter values that lead to complete assembly or at the but simulation However, the thermodynamic does suggest that the simulations resulting in shells would on a We not show Δ⁢fassem in Figure because the globule is than assembled shells for εCC=1.3, but the of shells in our simulations roughly the of the equilibrium (Figure supplement 1). of other parameters or To investigate the results above depend on within our we performed of additional we performed simulations in which the between cargo diameter in shell subunit size was shown in Figure supplement assembly is most for our cargo diameter which the model was but assembly occurs for cargo varied a factor of we performed assembly simulations with cargo molecules with a

Quantum localization within an energy-shell of a closed quantum system stands in contrast to the ergodic assumption of Boltzmann, and to the corresponding eigenstate thermalization hypothesis. The familiar case is the real-space "Anderson localization" and its many-body Fock-space version. We use the term "Hilbert-space localization" in order to emphasize the more general phase-space context. Specifically, we introduce a unifying picture that extends the semiclassical perspective of Heller, which relates the localization measure to the probability of return. We illustrate our approach by considering several systems of experimental interest, referring in particular to the Bosonic Josephson tunneling junction. We explore the dependence of the localization measure on the initial state, and on the strength of the many-body interactions using a novel recursive projection method.

It is well known that for ordinary one-dimensional (1D) disordered systems, the Anderson localization length $\xi$ diverges as $\lambda^m$ in the long wavelength limit ($\lambda\rightarrow \infty$ ) with a universal exponent $m=2$, independent of the type of disorder. Here, we show rigorously that pseudospin-1 systems exhibit non-universal critical behaviors when they are subjected to 1D random potentials. In such systems, we find that $\xi\propto \lambda^m$ with $m$ depending on the type of disorder. For binary disorder, $m=6$ and the fast divergence is due to a super-Klein-tunneling effect (SKTE). When we add additional potential fluctuations to the binary disorder, the critical exponent $m$ crosses over from 6 to 4 as the wavelength increases. Moreover, for disordered superlattices, in which the random potential layers are separated by layers of background medium, the exponent $m$ is further reduced to 2 due to the multiple reflections inside the background layer. To obtain the above results, we developed a new analytic method based on a stack recursion equation. Our analytical results are in excellent agreements with the numerical results obtained by the transfer-matrix method (TMM). For pseudospin-1/2 systems, we find both numerically and analytically that $\xi\propto\lambda^2$ for all types of disorder, same as ordinary 1D disordered systems. Our new analytical method provides a convenient way to obtain easily the critical exponent $m$ for general 1D Anderson localization problems.

The mobility edges which separate the localized energy eigenstates from the extended ones exist normally only in three dimensional systems. For one-dimensional systems with random on-site potentials, one never encounters mobility edges, where all the eigenstates are localized. However, there are two kinds of 1D systems such as correlated disordered models, and the systems of exponentially decaying hopping kinetics, features of mobility edges at some specific values become possible. We study in this paper the properties of the mobility edges in a one-dimensional p-wave superfluid on an incommensurate lattice with exponentially decaying hopping kinetics. Without the p-wave superluid, the system displays a single mobility edge, which separates the extended regime from the localized one at a certain energy. Without the exponentially decaying hopping term, the system displays a phase transition from a topological superconductor to an Anderson localization at a certain disorder strength, where no mobility edge exists. We are interested in the influence of the p-wave superfluid on the mobility edge. By solving the Bogoliubov-de Gennes equation, the eigenvalues and the eigenfunctions are obtained. In order to identify the extending or the localized properties of the eigenvectors, we define an inverse participation ratio IPR. For an extended state, IPRn～1/L which goes to zero at a large L, and for a localized one, IPRn being constant. Therefore, the IPR can be taken as a criterion to distinguish the extended state from the localized one, while the mobility edge is defined as the boundary between two different states. We find that, with a p-wave superfluid, the system changes from a single mobility edge to a multiple one, and the number of mobility edges increases with the increased superfluid pairing order parameter. To further obtain the energy or the location of the mobility edge, we investigate the scaling behavior of wave functions by using a multifractal analysis, which is calculated through the scaling index . The minimum value of the index, with the values min= 1, 0min1, and min= 0, mean the extended, critical, and localized states, respectively. For the two consecutive states, the minima of the scaling index min when extrapolating to the large size limit between 0 and 1 signal the mobility edge. By exploring the corresponding Bogoliubov quasi-particle wave functions for the system under open boundary conditions together with the multifractal analysis for the system under periodic boundary conditions, we identify two mobility edges for the system of the p-wave superfluid pairing. Furthermore, we will investigate how the existence of the mobility edges influences the p-wave superfluid, and identify the phase diagram at the given parameters. We will in the future try to understand the relationship between the topological superfluid and the mobility edges.

One investigates a light localization at laser pulse propagation in 1D (layered) or 2D nonlinear photonic structure. The light localization takes place due to appearance of spatial solitons which velocities depend on their intensities and location in the photonic crystal. This results in changing of velocities of sub-pulses which are not soliton and reflect or transmit through the nonlinear photonic crystal. We found out the formation of two types of solitons. One of them is appeared in separate layer. This soliton appear because of cubic nonlinear response. The second type of soliton spreads over some layers of the photonic crystal. This soliton is found out on the base of solving the eigenvalue function. We consider also the Anderson localization in photonic crystal with nonlinearity. The main result of this part of the investigation is a prevalence of nonlinear localization over the Anderson localization. The formation of soliton depends on input light intensity, duration of pulse and shape of elements from which the photonic crystal consists. In 2D case the soliton formation depends on focusing of laser beam and on presence of laser energy in neighbouring elements of the photonic crystal. These solitons move with slow velocity inside the elements of the photonic crystal. Under certain conditions the soliton can stop. This phenomenon can be used for data storage. The similar solitons can appear in many elements of the photonic crystal.

We analyze the existence and stability of nonlinear modes localized at phase-slip waveguide channels and their intersections in two-dimensional photonic lattices with rectangular symmetry. While in linear lattices only one localized mode of this type may exist under special conditions, nonlinearity supports a variety of localized modes including the modes which bifurcate from the symmetric states and describe nonlinear asymmetric localized states. We also study the surface modes localized at the waveguide channel edges in such lattices.

Based on the exact solution method, the ground state and quench dynamics properties of one-dimensional single-spin flipped Fermi gas with repulsion interaction are studied. With the Bethe wave function, the single-body correlation function and two-body correlation function of the ground state and those between different eigen-states can be reduced into a summation of simple functions, thereby greatly reducing the computational difficulty. For the system in the ground state, the single-body correlation functions and two-body correlation functions as well as momentum distributions for spin-up particles are investigated in real space with different interaction strengths. As the interaction strength increases, the number of nodes in the single-body correlation function remains unchanged, while the amplitude of oscillation decreases. Meanwhile, the number of peaks in the two-body correlation function increases by one due to interaction, indicating that the spin-down particle behaves as a spin-up particle. The momentum distribution becomes more smooth around Fermi surface with the interaction strength increasing. The interaction quench dynamics is investigated. The system is prepared in the ground state of ideal Fermi gas, and then the interaction strength is quenched to a finite positive value. The system evolves under time-dependent Schrödinger equation. The overlap between the initial state and eigen-state of post-quench interaction strength is expressed in the form of continued multiplication. The square of the modulus of this overlap, which represents the occupation probability, is calculated. We find that the occupation probabilities of the ground state and doubly degenerated excited state always have the first and the second largest value for an arbitrary interaction strength, respectively, which means that the difference in eigenenergy between these two states gives the primary period of oscillation. For relatively large particle number (&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ N\geqslant10$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), the primary period always does not change under different interaction strengths.It is found that in the case of interaction quenching, the momentum distribution and the correlation function show periodic oscillations. When the interaction strength is adjusted to a relatively small value, the oscillation periodicity is well-defined and the oscillation amplitude is small. The system can be approximated by a two-level model. When the interaction strength increases to a very large value, the oscillation periodicity worsens and the amplitude increases, but a primary period remains unchanged. Although the overall deviation is far from the initial state, it is very close to the initial state at time &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ t=mL^2/(2\pi\hbar)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. This is because the difference between most energy eigenvalues is almost an integral multiple of energy unit &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ 2\times\left(2\pi/L\right)^{2}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;.

Minimal attractors in digraph system models of neuronal networks

It is well known that Anderson transition is a disorder-induced metal-insulator transition.Contrary to this conventional wisdom, some investigations have shown that disorders could destroy the phase coherence of localized modes in flatbands, making the localized states melt into extended states. This phenomenon is called the inverse Anderson transition. While, to date, the experimental observation of inverse Anderson transitions is still lacking. In this work, we report the implementation of inverse Anderson transitions based on Aharonov-Bohm topolectrical circuits. Different types of disorders, including symmetric-correlated, antisymmetric-correlated and uncorrelated disorders, can be easily implemented in Aharonov-Bohm circuits by engineering the spatial distribution of ground settings. Through the direct measurements of frequency-dependent impedance responses and time-domain voltage dynamics, the inverse Anderson transitions induced by antisymmetric-correlated disorders are clearly observed. Moreover, the flat bands and associated spatial localizations are also fulfilled in clean Aharonov-Bohm circuits or Aharonov-Bohm circuits sustaining symmetric-correlated and uncorrelated disorders, respectively. Our proposal provides a flexible platform to investigate the interplay between the geometric localization and Anderson localization, and could have potential applications in electronic signal control.

Persistent revivals in a system of trapped bosonic atoms

We have explored the dipole-dipole mediated, resonant energy transfer reaction, $32{p}_{3/2}+32{p}_{3/2}\ensuremath{\rightarrow}32s+33s$, in an ensemble of cold $^{85}\mathrm{Rb}$ Rydberg atoms. Stark tuning is employed to measure the population transfer probability as a function of the total electronic energy difference between the initial and final atom-pair states over a range of Rydberg densities, $2\ifmmode\times\else\texttimes\fi{}{10}^{8}\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{10}^{9} {\mathrm{cm}}^{\ensuremath{-}3}$. The observed line shapes provide information on the role of beyond nearest-neighbor interactions, the range of Rydberg atom separations, and the electric field inhomogeneity in the sample. The widths of the resonance line shapes increase approximately linearly with the Rydberg density and are only a factor of 2 larger than expected for two-body, nearest-neighbor interactions alone. These results are in agreement with the prediction [B. Sun and F. Robicheaux, Phys. Rev. A 78, 040701(R) (2008)] that beyond nearest-neighbor exchange interactions should not influence the population transfer process to the degree once thought. At low densities, Gaussian rather than Lorentzian line shapes are observed due to electric field inhomogeneities, allowing us to set an upper limit for the field variation across the Rydberg sample. At higher densities, non-Lorentzian, cusplike line shapes characterized by sharp central peaks and broad wings reflect the random distribution of interatomic distances within the magneto-optical trap (MOT). These line shapes are well reproduced by an analytic expression derived from a nearest-neighbor interaction model and may serve as a useful fingerprint for characterizing the position correlation function for atoms within the MOT.