Nonlinear Dimer Research Articles

Surprising features of the energy-mismatched nonlinear dimer.

The quantum mechanical energy-mismatched two-state system with cubic nonlinearity in its governing equation is surprisingly rich in its dynamics and has relevance to a number of subdisciplines of physics ranging from polaron phenomena to Bose-Einstein condensation. We review some of them that have been discussed recently and describe some new results that have not, pointing out their relevance in possible experiments.

The Ever Elusive, Yet-to-Be-Discovered Twist-Bend Nematic Phase

The second, lower-temperature nematic phase observed in nonlinear dimer liquid crystals has properties originating from nanoscale, polar, intermolecular packing preferences. It fits the description of a new liquid crystal phase discovered by Vanakaras and Photinos, called the polar-twisted nematic. It is unrelated to Meyer’s twist-bend nematic, a meta-structure having a macroscale director topology consistent with Frank–Oseen elastic theory.

Discrete Breathers of Nonlinear Dimer Lattices: Bridging the Anti-continuous and Continuous Limits

In this work, we study the dynamics of an infinite array of nonlinear dimer oscillators which are linearly coupled as in the classical model of Su, Schrieffer and Heeger (SSH). The ratio of in-cell and out-of-cell couplings of the SSH model defines distinct $\textit{phases}$: topologically trivial and topologically non-trivial. We first consider the case of weak out-of-cell coupling, corresponding to the topologically trivial regime for linear SSH; for any prescribed isolated dimer frequency, $\omega_b$, which satisfies non-resonance and non-degeneracy assumptions, we prove that there are discrete breather solutions for sufficiently small values of the out-of-cell coupling parameter. These states are $2\pi/\omega_b$- periodic in time and exponentially localized in space. We then study the global continuation with respect to this coupling parameter. We first consider the case where $\omega_b$, the seeding discrete breather frequency, is in the (coupling dependent) phonon gap of the underlying linear infinite array. As the coupling is increased, the phonon gap decreases in width and tends to a point (at which the topological transition for linear SSH occurs). In this limit, the spatial scale of the discrete breather grows and its amplitude decreases, indicating the weakly nonlinear long wave regime. Asymptotic analysis shows that in this regime the discrete breather envelope is determined by a vector gap soliton of the limiting envelope equations. We use the envelope theory to describe discrete breathers for SSH- coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry, topologically nontrivial regimes, when the spectral gap is small. Our asymptotic theory shows excellent agreement with extensive numerical simulations over a wide range of parameters.

Classical Rabi oscillations induced by unbalanced dissipation on a nonlinear dimer

Classical Rabi oscillations (ROs) are a signature feature of an unbroken parity-time (PT)-symmetry in conservative systems. Notwithstanding, they do not persist in systems with intrinsic dissipation from the surroundings. Here, we report permanent ROs in a parametrically driven and dissipative nonlinear dimer in a region of parameters where the conservative system exhibits broken PT-symmetry. We show that the mechanism for robust ROs lies in a suitable combination of typical gain/loss (PT-symmetry) configuration with nonlinearity and dissipation. This study presents a route to generate and control robust ROs in out-of-equilibrium systems in future experiments.

Dynamical symmetry breaking through AI: The dimer self-trapping transition

The nonlinear dimer obtained through the nonlinear Schrödinger equation has been a workhorse for the discovery the role nonlinearity plays in strongly interacting systems. While the analysis of the stationary states demonstrates the onset of a symmetry broken state for some degree of nonlinearity, the full dynamics maps the system into an effective [Formula: see text] model. In this later context, the self-trapping transition is an initial condition-dependent transfer of a classical particle over a barrier set by the nonlinear term. This transition that has been investigated analytically and mathematically is expressed through the hyperbolic limit of Jacobian elliptic functions. The aim of this work is to recapture this transition through the use of methods of Artificial Intelligence (AI). Specifically, we used a physics motivated machine learning model that is shown to be able to capture the original dynamic self-trapping transition and its dependence on initial conditions. Exploitation of this result in the case of the nondegenerate nonlinear dimer gives additional information on the more general dynamics and helps delineate linear from nonlinear localization. This work shows how AI methods may be embedded in physics and provide useful tools for discovery.

Discovering nonlinear resonances through physics-informed machine learning

For an ensemble of nonlinear systems that model, for instance, molecules or photonic systems, we propose a method that finds efficiently the configuration that has prescribed transfer properties. Specifically, we use physics-informed machine learning (PIML) techniques to find the parameters for the efficient transfer of an electron (or photon) to a targeted state in a nonlinear dimer. We create a machine learning model containing two variables, χ D and χ A , representing the nonlinear terms in the donor and acceptor target system states. We then introduce a data-free physics-informed loss function as 1.0 − P j , where P j is the probability, the electron being in the targeted state, j . By minimizing the loss function, we maximize the occupation probability to the targeted state. The method recovers known results in the targeted energy transfer (TET) model, and it is then applied to a more complex system with an additional intermediate state. In this trimer configuration, the PIML approach discovers desired resonant paths from the donor to acceptor units. The proposed PIML method is general and may be used in the chemical design of molecular complexes or engineering design of quantum or photonic systems.

The fractional nonlinear {mathcal{PT}} dimer

We examine a fractional discrete nonlinear Schrodinger dimer, where the usual first-order derivative in the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and {mathcal{{PT}}}-symmetric, and for localized initial conditions we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear {{mathcal{{PT}}}} dimer is solved in closed form in terms of Mittag–Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, we see that the main effect of the fractional derivative is to produce a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of {mathcal{{PT}}} symmetry, the oscillations experience some amplification for gain/loss values below some threshold, while beyond threshold, the amplitudes of both sites grow unbounded. The presence of nonlinearity can arrest the unbounded growth and lead to a selftrapped state. The trapped fraction decreases as the nonlinearity is increased past a critical value, in marked contrast with the standard (non-fractional) case.

Nonlinear Nanophotonic Circuitry: Tristable and Astable Multivibrators and Chaos Generator

AbstractThe concept of lumped optical nanoelements (or metactronics), wherein nanometer‐scale structures act as nanoinductors, nanocapacitors, and nanoresistors, has attracted a great deal of attention as a simple toolbox for engineering different nanophotonic devices in analogy with microelectronics. While recent studies of the topic have been predominantly focused on linear functionalities, nonlinear dynamics in microelectronic devices plays a crucial role and provides a majority of functions, employed in modern applications. Here, the metactronics paradigm is extended and nonlinear dynamical modalities are added to those nanophotonic devices that have never been associated with optical nanoantennas. Specifically, it is shown that nonlinear dimer nanoantennae can operate in the regimes of tristable and astable multivibrators as well as chaos generators. The physical mechanism behind these modalities relies on the Kerr‐type nonlinearity of nanoparticles in the dimer enhanced by a dipolar localized surface plasmon resonance. This allows one to provide a positive nonlinear feedback at moderate optical intensities, leading to the desired dynamical behavior via tuning the driving field parameters. The findings shed light on a novel class of nonlinear nanophotonic devices with a tunable nonlinear dynamical response.

Enhanced nonreciprocal transmission through a saturable cubic-quintic nonlinear dimer defect

The transmission properties through a saturable cubic-quintic nonlinear defect attached to lateral linear chains is investigated. Particular attention is directed to the possible non-reciprocal diode-like transmission when the parity-symmetry of the defect is broken. Distinct cases of parity breaking are considered including asymmetric linear and nonlinear responses. The spectrum of the transmission coefficient is analytically computed and the influence of the degree of saturation analyzed in detail. The transmission of Gaussian wave-packets is also numerically investigated. Our results unveil that spectral regions with high transmission and enhanced diode-like operation can be achieved.

Polarization dynamics in twisted fiber amplifiers: a non-Hermitian nonlinear dimer model.

We study continuous-wave light propagation through a twisted birefringent single-mode fiber amplifier with saturable nonlinearity. The corresponding coupled-mode system is isomorphic to a non-Hermitian nonlinear dimer and gives rise to analytic polarization-mode dynamics. It provides an optical simulation of the semi-classical non-Hermitian Bose-Hubbard model and suggests its use for the design of polarization circulators and filters, as well as sources of polarized light.

Phase transitions in Wick-rotated [formula omitted]-symmetric optics

PT-symmetric models with a Wick rotation of time (t→±it) show spectral phase transitions that are similar to those of dissipative systems driven out of equilibrium. Optics can provide an accessible test bed to explore spectral phase transitions of Wick-rotated PT-symmetric models. This is shown by considering the transverse dynamics of laser light in optical cavities with variable reflectivity and tilted mirrors. Two specific examples are discussed: the optical analogue of the hydrodynamic Squire model of vorticity, and the Wick-rotated PT-symmetric nonlinear dimer model. In the latter case the spectral phase transition is associated with the universal phase locking–unlocking transition in Adler’s theory of coupled oscillators.

Frequency comb generation for wave transmission through the nonlinear dimer

We study dynamical response of the nonlinear dimer relative to monochromatic wave injected via the waveguide. We show existence of a domain in space of frequency and injected amplitude where the stationary solutions of the time evolution equations do not exist. We present time dependent solutions which show that scattering waves carry multiple harmonics with frequencies spaced equidistantly.

Nonreciprocal transmission through a saturable nonlinear asymmetric dimer.

We investigate the nonreciprocal diodelike behavior of a dimer with an asymmetric on-site potential and a saturable nonlinearity. The dimer is coupled to linear side chains. The spectra of transmission and the rectifying factor are analytically obtained using a backward iteration of the set of discrete nonlinear Schrödinger equations used to model the wave propagation through the nonlinear dimer. We show that the windows of bistable behavior leading to a pronounced nonreciprocal diodelike transmission become wider and displaced to higher input field intensities as the saturation coefficient increases. Further, saturation of the nonlinear response has opposite impacts on the rectifying action over short- and long-wavelength input signals within the second bistability window. In the first window, the rectifying action is not compromised by the saturation, thus showing that a weak contribution of high-order susceptibilities to the nonlinear response can improve the efficiency of the nonreciprocal transmission. The rectifying action of a dimer with an asymmetric nonlinearity is also discussed.

Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis

In the present work, we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT) symmetry, i.e. with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT-symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT-symmetric trimers, while both of them are absent in linear PT-symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT-symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT-symmetric quadrimers.

Nanoradar based on nonlinear dimer nanoantenna

We introduce the concept of a nanoradar based on the operation of a nonlinear plasmonic nanoantenna. The nanoradar action originates from modulational instability occurring in a dimer nanoantenna consisting of two subwavelength nonlinear nanoparticles. Modulation instability causes a dynamical energy exchange between the nanoantenna eigenmodes resulting in periodic scanning of the nanoantenna scattering pattern. Such nanoradar demonstrates a wide scanning sector, low operation threshold, and ultrafast time response being potentially useful for many applications in nanophotonics circuitry.

Resonance effects in the nonadiabatic nonlinear quantum dimer

The quantum nonlinear dimer consisting of an electron shuttling between two sites and in weak interaction with vibrations, is studied numerically under the application of a DC electric field. A field-induced resonance phenomenon between the vibrations and the electronic oscillations is found to influence the electronic transport greatly. For initial delocalization of the electron, resonance has the effect of a dramatic increase in the transport. Nonlinear frequency mixing is identified as the main mechanism that influences transport. A characterization of the frequency spectrum is also presented.

Dynamic transition and chaotic behavior in a nonlinear dimer driven by a laser field

We study the dynamic transition and chaotic behavior of a nonlinear dimer driven by a laser field. We observe the Rabi oscillation because of the periodic time-dependent external field, give out the critical curve which represents the self-trapping transition, show the electron's chaotic behavior at some certain parameters, and compare the critical dynamic features between the single photon and the multiphotons case.

Exact Solution of a Generalized Nonlinear SchrödingerEquation Dimer

We present exact solutions for a nonlinear dimer system definedthrought a discrete nonlinear Schrödinger equation that containsalso an integrable Ablowitz–Ladik term. The solutions areobtained throught a transformation that maps the dimer into a doubleSine–Gordon like ordinary nonlinear differentialequation.

Possibility of a quasiparticle localication at the dissipative dynamics of a nonlinear degenerate adiabatic dimer

Abstract The dynamics of a nonlinear (NL) degenerate dimer interacting with a quantum heat bath has been investigated. The exact form of the reduced nonlinear dissipative equation up to the second order of the subsystem-bath site-diagonal interaction has been solved and analytic expressions for the steady state have been derived. It has been shown that the quantal thermal bath enxures correct steady-state behavior of the subsystem density matrix including proper long-time probability redistribution. Genuine relaxation exist in case of non-equal NL coupling, too. This NL coupling results in an effective increase of the subsystem temperature (or equivalently in decreasing of the corresponding thermal gap). Moreover, the non-equal NL coupling results correctly in the excitation transfer to the site with stronger NL coupling. In the high-temperature region the effect of the NL coupling is destroyed.

Effects of a quantum heat bath on the dynamics of a nonlinear subsystem: reduced density matrix approach

Abstract The investigation of a reaction field induced nonlinear subsystem dynamics coupled to a quantum heat bath is presented. Projection technique along with time-convolutionless treatment of the total Liouville equation is applied and a new reduced nonlinear dissipative Liouville (RNDL) equation for the subsystem dynamics is derived. It is shown that the new RNDL equation, due to the reaction field nonlinearity, contains besides the coherent nonlinear and standard dissipative terms also an extra dissipative term which mediates the bath influence on the subsystem dynamics over the reaction field nonlinear coupling. The approach presented describes correctly quantum effects and the description is not restricted to limiting temperature regimes. The application of the method to the degenerate symmetric nonlinear dimer with Ohmic intersite coupling to the quantum heat bath has revealed that in the long-time limit the effect of the nonlinearity, with the tendency for localization in the intermediate time regime, is destroyed. The subsystem is driven to the thermodynamic equilibrium with a symmetric steady state.