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We study theoretically and experimentally the subharmonic entrainment (SHE) breather soliton in mode-locked lasers for the first time, in which the ratio of the breather period to the round-trip time is an integer. We build a non-Hermitian degeneracy map of breather soliton, and illustrate that SHE arises between the two exceptional points (EPs). We obtain SHE at the ratio of 20, observe the evolution of breather soliton when tuning the gain and/or cavity loss, and prove that this phenomenon can improve the stability of breather soliton. Our research brings insight into the EP physics of ultrafast lasers and makes the mode-locked laser a powerful test bed for non-Hermitian degeneracy, which may open a new course in ultrafast laser research.

A chiral-coupled atomic chain of two-level quantum emitters allows strong resonant dipole-dipole interactions, which enables significant collective couplings between every other emitters. We numerically obtain the steady-state phase diagram of such system under weak excitations, where interaction-driven states of crystalline orders, edge or hole excitations, and dichotomy of chiral flow are identified. We distinguish these phases by participation ratios and structure factors, and find two critical points which relate to decoherence-free subradiant sectors of the system. We further investigate the transport of excitations and emergence of crystalline orders under spatially-varying excitation detunings, and present non-ergodic butterfly-like system dynamics in the phase of extended hole excitations with a signature of persistent subharmonic oscillations. Our results demonstrate the interaction-induced quantum phases of matter with chiral couplings, and pave the way toward simulations of many-body states in nonreciprocal quantum optical systems., Comment: 4 figures in the main text

We predict subharmonic entrainment of breather-soliton oscillations to a periodic perturbation at the round-trip time ${T}_{R}$ in Kerr-nonlinear optical resonators; an integer ratio ${T}_{b}/{T}_{R}=N\ensuremath{\gg}1$ results for breathing period ${T}_{b}$. Rigid entrainment is observed with intermediate finesse ($\mathcal{F}\ensuremath{\sim}30--40$) for $N$ up to 20, and we propose a way to realize higher entrainment ratios at higher finesse. This nontrivial synchronization across the widely separated timescales of the photon lifetime and round-trip time points to a new direction for research in this field and may find application, for example, in the measurement of a pulse train repetition rate that is electronically inaccessible.

We report on the observation of subharmonic resonances in high-order wave mixing associated with the quantized vibrational levels of atoms trapped in a one-dimensional optical lattice created by two intense nearly counterpropagating coupling beams. These subharmonic resonances, occurring at $\ifmmode\pm\else\textpm\fi{}1/2$ and $\ifmmode\pm\else\textpm\fi{}1/3$ of the frequency separation between adjacent vibrational levels, are observed through phase-match angularly resolved six- and eight-wave mixing processes. We investigate how these resonances evolve with the intensity of the incident probe beam, which couples with one of the coupling beams to create anharmonic coherence gratings between adjacent vibrational levels. Our experimental results also show evidence of high-order processes associated with coherence involving nonadjacent vibrational levels. Moreover, we also demonstrate that these induced high-order coherences can be stored in the medium and the associated optical information retrieved after a controlled storage time.

Parametric subharmonic instability (PSI) of internal wave beams with nearly twice the inertial frequency due to background rotation in stratified fluid is studied theoretically. Such near-inertial PSI is possible for beams of general profile; generic PSI can arise only for quasimonochromatic beams.

A subharmonic structure in Josephson junctions appears due to Andreev reflections within the junction. Here we report on experimental observation of a subharmonic half-gap singularity in interlayer ...

We theoretically study coherent subharmonic (multi-photon) transitions of a harmonically driven spin. We consider two cases: magnetic resonance (MR) with a misaligned, i.e., non-transversal driving field, and electrically driven spin resonance (EDSR) of an electron confined in a one-dimensional, parabolic quantum dot, subject to Rashba spin-orbit interaction. In the EDSR case, we focus on the limit where the orbital level spacing of the quantum dot is the greatest energy scale. Then, we apply time-dependent Schrieffer-Wolff erturbation theory to derive a time-dependent effective two-level Hamiltonian, allowing to describe both MR and EDSR using the Floquet theory of periodically driven two-level systems. In particular, we characterise the fundamental (single-photon) and the half-harmonic (two-photon) spin transitions. We demonstrate the appearance of two-photon Rabi oscillations, and analytically calculate the fundamental and half-harmonic resonance frequencies and the corresponding Rabi frequencies. For EDSR, we find that both the fundamental and the half-harmonic resonance frequency changes upon increasing the strength of the driving electric field, which is an effect analogous to the Bloch-Siegert shift known from MR. Remarkably, the drive-strength dependent correction to the fundamental EDSR resonance frequency has an anomalous, negative sign, in contrast to the corresponding Bloch-Siegert shift in MR which is always positive. Our analytical results are supported by numerical simulations, as well as by qualitative interpretations for simple limiting cases., Comment: 16 pages, 7 figures

Viscous fluids are used as vibration dampers in applications ranging from isolation tables to cars; they are well understood and easy to engineer. An attractive alternative proposed for turbines, medical applications, and spacecraft is to use granular materials as dampers: Unlike fluids, they are nearly independent of temperature, can operate in harsh environments, and provide damping with no fixed anchor. Unfortunately, there are no design rules for these dampers, so the present paper characterizes effects of amplitude and frequency of oscillation on generic granular dampers. The authors determine broad operating ranges and find that, unlike their fluid counterparts, granular dampers strongly suppress higher-order dynamic resonances.

We show that a hybridization of the optical and material nonlinearities takes place near the threshold of the subharmonic generation in photorefractive crystals. It results in a critical (with a singularity) enhancement of the rate of spatial amplification of light waves and leads to a variety of new optical critical phenomena.

Low temperature transport measurements on superconducting film - normal metal wire - superconducting film (SNS) junctions fabricated on the basis of 6 nm thick superconducting polycrystalline PtSi films are reported. The structures with the normal metal wires of two different lengths L=1.5 $\mu$m and L=6$\mu$m and the same widths W=0.3$\mu$m are studied. Zero bias resistance dip related to pair current proximity effect is observed for all junctions whereas the subharmonic energy gap structure originating from phase coherent multiple Andreev reflections have occurs only in the SNS junctions with short wires., Comment: ReVTex, 4 pages, 4 eps figures included

Resonance regions similar to the Arnol'd tongues found in single oscillator frequency locking are observed in experiments using a spatially extended periodically forced Belousov-Zhabotinsky system. We identify six distinct 2:1 subharmonic resonant patterns and describe them in terms of the position-dependent phase and magnitude of the oscillations. Some experimentally observed features are also found in numerical studies of a forced Brusselator reaction-diffusion model.

A noninvasive optical recording technique is introduced to study the spontaneous contractile activity in self-assembled heart cell networks. Continuous monitoring throughout the lifetime of the network reveals spontaneous appearance of various rhythm disorders. Analysis of two typical patterns, namely, subharmonic structures and sudden alternations between two dominant rates, indicates the emergence of intrinsic pacemakers. A model of one or two slightly variable nonlinear oscillators, acting on an excitable element, is shown to reproduce the main experimental results.

The photochemistry of the mesopause region entails a driven chemical oscillator enforced by solar short-wave irradiation. Zero-dimensional calculations show that this oscillator is able to produce nonlinear reactions like cascades of subharmonics or chaos. We investigate what will happen if this system is subjected to atmospheric diffusion. We discuss the system response and introduce different kinds of bifurcations.

The stationary solution is obtained for the photorefractive subharmonic gratings excited in crystals of the sillenite family by a standing light interference pattern and an applied ac electric field. We show that the main subharmonic with doubled spatial period may become unstable against excitation of the subharmonic with quadrupled spatial period. The threshold condition for this bifurcation is found.

We calculate the ac and dc current-voltage $(I\ensuremath{-}V)$ relation when a tunnel barrier is placed in the normal region of a superconductor--normal-metal--superconductor $({\mathrm{S}}_{1}\mathrm{N}{\mathrm{S}}_{2})$ junction, where the two superconducting contacts have different gaps $({\ensuremath{\Delta}}_{2}g~{\ensuremath{\Delta}}_{1}g~0)$. The voltages at which subharmonic gap structure (SGS) appears in the zero-temperature $I\ensuremath{-}V$ are the same as previously found for the SGS in ballistic ${\mathrm{S}}_{1}\mathrm{N}{\mathrm{S}}_{2}$ junctions at a finite temperature. Some of the SGS's are strong enough to produce negative differential resistance. We determine the variation of the ac Josephson current with bias, junction transmissivity, gap ratio ${\ensuremath{\Delta}}_{1}/{\ensuremath{\Delta}}_{2}$, and junction length $L$. The ac Josephson current through a ballistic SNS junction can also be used to measure the superconducting phase difference $\ensuremath{\varphi}$.

Chaos synchronization of nonidentical unidirectionally coupled systems is investigated and characterized in terms of the entrainment ratios of unstable periodic orbits. The implications of subharmonic entrainment for generalized synchronization are discussed and illustrated for discrete and continuous systems.

In this paper we present a control technique for stabilizing chaotic motion to periodic orbits in a ${\mathrm{CO}}_{2}$ laser with electro-optic feedback. The control method is based on negative feedback of subharmonic components of the laser intensity signal. A detailed analysis, performed using the four-level model for the ${\mathrm{CO}}_{2}$ laser, allows us both to reproduce the experimental features and to compare our method with the time-delayed autosynchronization method introduced by Pyragas [Phys. Lett. A 170, 421 (1992)].

Simulating two-dimensional optical pattern formation in sodium vapor, we show that hexagonal patterns can be unstable against their own spatial harmonics or subharmonics. The instability results in ultrahexagons or subhexagons, respectively. The formation of ultrahexagons is due to interaction with a second solution branch with a wave number nearly resonant to the harmonics of the primary hexagon. The mechanism of the instabilities is discussed on the basis of order parameter equations obtained from symmetry considerations.

We theoretically predict and experimentally demonstrate the phenomenon of frequency entrainment in optically injected semiconductor lasers. The laser exhibits nonlinear subharmonic and ultraharmonic resonances for appropriate values of its frequency detuning from the externally injected signal.

We calculate the dc current-voltage ($I\ensuremath{-}V$) relation for a ballistic superconductor-normal metal-superconductor (${S}_{1}N{S}_{2}$) junction, where the two superconducting contacts have different energy gaps (${\ensuremath{\Delta}}_{2}g~{\ensuremath{\Delta}}_{1}g~0$). Several new features appear in the $I\ensuremath{-}V$ relation as it evolves from a $\mathrm{NS}$ junction (${\ensuremath{\Delta}}_{1}=0$) to the symmetric $\mathrm{SNS}$ junction (${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}$). The dc current ${I}_{0}$ flowing at small voltages is ${I}_{0}=4\frac{e{\ensuremath{\Delta}}_{1}}{h}$, and is different from the critical current ${I}_{c}$. The differential conductance changes abruptly at $eV={\ensuremath{\Delta}}_{2}\ensuremath{-}{\ensuremath{\Delta}}_{1}$ due to the loss of a single Andreev reflection from the larger-gap superconductor. Finally, the voltages ${\ensuremath{\Delta}}_{n}$ at which subharmonic gap structures appear in the $I\ensuremath{-}V$ relation for temperatures ${k}_{B}T\ensuremath{\simeq}{\ensuremath{\Delta}}_{2}$ depend on both energy gaps ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2}$. We find a systematic way to classify this subgap structure.

Direct numerical simulation is used to study the periodic generation and evolution of streamwise vortical structures in the near wake of a circular cylinder. It is observed that streamwise structures are formed due to stretching of core vorticity which escapes out of the core and due to stretching of small scale streamwise vorticity already present outside the core. Distinct hairpin structures are observed which play a central role in transferring vorticity out of the core. Furthermore, the hairpin structures are associated with a spanwise subharmonic mode and are thus manifestations of a period-doubling mechanism.

The stationary to drifting transition of a subharmonic wave pattern is studied in the presence of inhomogeneities and drift forces as the pattern wavelength is comparable with the system size. We consider a pinning-depinning transition of stationary subharmonic waves in a tilted quasi-one-dimensional fluidized shallow granular bed driven by a periodic air flow in a small cell. The transition is mediated by the competition of the inherent periodicity of the subharmonic pattern, the asymmetry of the system, and the finite size of the cell. Measurements of the mean phase velocity of the subharmonic pattern are in good agreement with those inferred from an amplitude equation, which takes into account asymmetry and finite-size effects of the system, emphasizing the main ingredients and mechanism of the transition.

Properties of the dynamical-mode-locking phenomena are studied in the ac driven overdamped Frenkel-Kontorova model with deformable substrate potential. Appearance of very large subharmonic steps due to deformation of the substrate potential significantly influences the stability and existence of harmonic steps. Strong correlation among harmonic and subharmonic steps has been observed in which the larger the width of half-integer steps, the smaller that of harmonic steps. Amplitude dependence of harmonic steps significantly changes with the deformation of the potential where deviation from the well-known Bessel-like oscillations appears. Strong influence of the frequency of the ac driving force on the appearance and size of subharmonic steps has been observed.

Current-voltage characteristics of Andreev contacts in ${\text{LaFeAsO}}_{0.9}{\text{F}}_{0.1}$ have been measured using the break-junction technique. The contacts were prepared at $T=4.2\text{ }\text{K}$ from polycrystalline samples and exhibit superconductor/normal conductor/superconductor-type behavior due to multiple Andreev reflections. Two sets of subharmonic gap structures were detected indicating the existence of two distinct superconducting gaps: ${\ensuremath{\Delta}}_{L}=(5.5\ifmmode\pm\else\textpm\fi{}1)\text{ }\text{meV}$ and ${\ensuremath{\Delta}}_{S}=(1\ifmmode\pm\else\textpm\fi{}0.2)\text{ }\text{meV}$.

As shown recently, it is possible to create, on a vibrating fluid interface, mobile emitters of Faraday waves [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. They are formed of droplets bouncing at a subharmonic frequency which couple to the surface waves they emit. The droplet and its wave form a spontaneously propagative structure called a "walker." In the present paper we investigate the large variety of orbital motions exhibited by two interacting walkers having different sizes and velocities. The various resulting orbits which can be circular, oscillating, epicycloidal, or "paired walkers" are defined and characterized. They are shown to result from the wave-mediated interaction of walkers. Their relation to the orbits of other localized dissipative structures is discussed.

Giant Shapiro steps have recently been observed in the I-V characteristics of Josephson-junction arrays, both with and without transverse magnetic fields. For magnetic fluxes per unit cell of the array \ensuremath{\Phi}=(p/q)${\mathrm{\ensuremath{\Phi}}}_{0}$, these steps occur at voltages per unit cell of \ensuremath{\Elzxh}\ensuremath{\omega}/2eq, where \ensuremath{\omega} is the frequency of the rf field. I present a calculation of this effect, restricted to certain values of f=p/q and certain current directions within the lattice. This calculation leads to results concerning the width of steps as a function of q and as a function of the amplitude of the driving field. It also leads to a prediction of subharmonic steps, at voltages of (\ensuremath{\Elzxh}\ensuremath{\omega}/2e)(n/qm).

Microconstrictions between spin-singlet superconductor, Pb or $\mathrm{Mg}{\mathrm{B}}_{2}$, tip and half-metallic ferromagnet, ${\mathrm{La}}_{0.65}{\mathrm{Ca}}_{0.35}\mathrm{Mn}{\mathrm{O}}_{3}$ (LCMO), plate have been realized by means of a point-contact inset. Measurements of the current-voltage characteristics and of the dynamic conductance $G(V)$ versus bias have been performed to probe mutual influence of superconductivity and half-metallic ferromagnetism. In the contacts, which we distinguish as proximity affected ones, a few principal effects have been observed. Namely, with decreasing temperature, a spectacular drop of the contact's resistance has been detected with an onset of the Pb or $\mathrm{Mg}{\mathrm{B}}_{2}$ superconductivity; for small voltages, an excess current and doubling of the normal-state conductance have also been found. We conclude that the underlying physical explanations for these results are the conversion from spin-singlet pairs to spin-triplet ones at the Pb $(\mathrm{Mg}{\mathrm{B}}_{2})∕\mathrm{LCMO}$ interface and a long-range proximity induced superconductivity of LCMO. Superconducting state of LCMO is also supported by the observation of the coherent multiple Andreev reflections (subharmonic gap resonances). We also found that the character of $G(V)$ vs voltage dependence corresponds to that for the induced superconducting energy gap of LCMO much larger than that of Pb or $\mathrm{Mg}{\mathrm{B}}_{2}$. All specific characteristics of proximity affected contacts suggest that the local triplet superconducting fluctuations are essentially sustained in LCMO and the singlet superconductors only fix the phase coherency of a superconducting state.

Localized period-doubling waves arise in circularly shaken granular beds contained in an annular channel. These solitary wave packets are accompanied by a locally increased particle density. The width and velocity of the granular wave pulse are measured as a function of the bed height. A continuum model for the material distribution, based on the measured granular transport velocity as a function of the bed thickness, captures the essence of the experimental findings.

The overdamped dynamics in the commensurate structures of the one-dimensional Frenkel-Kontorova model subjected to a parametrized deformable periodic substrate potential and driven by a periodic force is examined. It was found that when the shape of the substrate potential starts to deviate from the standard one, new subharmonic steps appear in the response function even in the structures with an integer value of average interparticle distance while the critical depinning force can even decrease for some values of system parameters. These novel phenomena could be particularly relevant for the charge-density wave systems, vortex lattices, and systems of Josephson-junction arrays.

Nematic electroconvection is studied under asymmetric periodic excitation with a driving electric field $E(t)=E(t+T)\ensuremath{\ne}\ensuremath{-}E(t+T∕2)$. A new dynamic regime, distinguished by subharmonic dynamics, is discovered in the pattern state diagram between conventional conductive and dielectric regimes. The spatial and temporal pattern characteristics are investigated experimentally. The dynamics, threshold fields, and selected pattern wavelengths at onset, calculated from a Floquet analysis of the linearized electrohydrodynamic equations with a test mode ansatz, are in good agreement with experimental results.

Using a new experimental geometry, we have proved for the first time that the generation of spatial subharmonic gratings in photorefractive crystals is not dependent on optical nonlinearity. We present results which confirm that the subharmonic gratings result from a parametric excitation of ultra low-frequency eigenmodes of a crystal by a time modulated fundamental grating.

A bifurcation of transition that destroys generalized chaos synchronization is considered. This transition frequently occurs in regimes of subharmonic chaos entrainment where synchronization can be abruptly terminated due only to an almost unnoticeable change in the shape of the driving attractor. We explore the main cause of this sensitivity and ascertain the mechanism behind this transition.

A comment on the Letter by R. Baumann {ital et} {ital al}., Phys. Rev. Lett. {bold 74}, 1597 (1995). The authors of the Letter offer a Reply.

Subharmonic steps in the dc $I\ensuremath{-}V$ characteristics are reported in the model compound Nb${\mathrm{Se}}_{3}$, when subjected to combined dc and ac drives. The steps occur at currents which correspond to frequencies ${\ensuremath{\omega}}_{\mathrm{int}}=(\frac{p}{q}){\ensuremath{\omega}}_{\mathrm{ext}}$, with $p$ and $q$ integers. These steps cannot be explained in terms of the familiar classical equation of motion in a single degree of freedom, because the inertial term is too small. The observed behavior is compared with the devil's staircase, and a fractal dimension of 0.91 \ifmmode\pm\else\textpm\fi{} 0.03 is extracted.

Experiments on both point-contact junctions (Nb-Nb) and on small area thin-film tunnel junctions (Sn-O-Sn) show that applied 4-mm radiation produces satellites associated with "subharmonic" gap structure as well as the familiar microwave-assisted tunneling structure associated with the superconducting energy gap itself. The location in voltage of all these structures is given by eV=(2Δ±nh ν) / m, where 2Δ is the superconducting energy gap, ν is the applied frequency, h is Planck's constant, e is the magnitude of the electronic charge, V is the dc voltage drop across the junction, and m and n are integers: m=1,2,3,… and n=0,1,2,…. The power dependence of the satellite structure and the microwave-assisted tunneling structure is consistent for all junctions tested with the expression Jn2(m e Vrf / h ν), where Jn(x) is the ordinary Bessel function of order n, Vrf is the amplitude of the induced microwave voltage drop accross the junction, and the other quantities are as defined above. A distinction between the explanations—multiparticle tunneling and nonlinear self-coupling—of the subharmonic gap structure cannot, however, be made on the basis of the observed power dependence. Hence other properties such as relative amplitude and line shape are considered and it is concluded that self-coupling mechanisms seem able to account for all the observations, whereas multiparticle tunneling may be discarded as the sole explanation of the structure.

Roughening transitions are explored as a function of the strength ${V}_{2}$ of a subharmonic potential, which can be varied with the chemical potential difference in a two-component crystal like galena (PbS) or one of its analogs (e.g., SnTe). The dependence of both the facet size and the step energy on ${V}_{2}$ are given, for small ${V}_{2}$, explicitly in terms of the surface stiffness at ${V}_{2}$=0. A close connection of the model to the F model is exploited to obtain hitherto inaccessible features of the F model phase diagram in direct and staggered fields.

We have studied the excess currents that flow for $Vl{\ensuremath{\Delta}}_{1}+{\ensuremath{\Delta}}_{2}$ in two superconductor (${S}_{1}\ensuremath{-}I\ensuremath{-}{S}_{2}$) tunnel junctions. The features of interest are "multiparticle" tunneling, which is observed as a current increase at ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2}$, and the subharmonic effects which occur at $\frac{2\ensuremath{\Delta}}{n}$ when ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}$. Strong evidence is presented that subharmonic junctions contain metallic shorts, and we show that the $\frac{2\ensuremath{\Delta}}{n}$ series can be observed even in the absence of single-particle tunneling. By extending the study to asymmetrical junctions (${\ensuremath{\Delta}}_{1}\ensuremath{\ne}{\ensuremath{\Delta}}_{2}$) we conclude that at present there is no satisfactory explanation of the subharmonic effects.

Azbel-Kaner cyclotron resonance was studied at 277 GHz and 1.5 \ifmmode^\circ\else\textdegree\fi{}K in a (100)-plane single-crystal specimen of Ni. Resonances from both the neck and belly of the Cu-like majority-spin $s\ensuremath{-}p$-band Fermi surface were seen. The neck yielded $\frac{{m}^{*}}{{m}_{0}}=0.35\ifmmode\pm\else\textpm\fi{}0.014$ when $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ was along $〈011〉$. Beat structure in the subharmonic resonances suggested that belly orbits on both the majority-and minority-spin $s\ensuremath{-}p$-band Fermi surfaces were observed for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ within 28\ifmmode^\circ\else\textdegree\fi{} of $〈001〉$. As $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ was rotated away from $〈001〉$ the heavier mass increased smoothly from $\frac{{m}^{*}}{{m}_{0}}=5.09 \mathrm{to} 5.9$ while the lighter mass increased from 4.33 to 4.86. Only one of the three pairs of symmetry-related minority-spin $d$-band hole pockets were seen. The pockets having their greatest dimension along $〈001〉$ yielded an effective mass which increased from $\frac{{m}^{*}}{{m}_{0}}=0.75\ifmmode\pm\else\textpm\fi{}0.03$ at $〈001〉$ to ${m}^{*}{m}_{0}=1.11$ when $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ was 34\ifmmode^\circ\else\textdegree\fi{} from $〈001〉$.

Measurements are presented which show that below the superfluid transition, the generation of first-sound subharmonics in the low-megahertz range quantatively follows the Feigenbaum universal convergence. In addition, by using ion-trapping techniques the physical nature of the onset of the first bifurcation sequence is identified as the threshold for the generation of quantum vortex line, not the threshold for the production of macroscopic classical turbulence, i.e., acoustic cavitation.