Matter Wave Optics

The Zitter Model of the Electron interprets Zitterbewegung as a real internal motion of the electron at the speed of light, giving rise to its spin and magnetic moment. Based on this model and relying solely on four basic assumptions, we have derived numerous properties of the electron directly from its internal structure, including the de Broglie frequency, magnetic and angular momentum, Compton wavelength, the quantum magnetic flux, the quantum Hall resistance, the Lorentz factor, and the Schwinger critical field limits. By introducing a fifth assumption (a secondary helical motion), we derived the amplitude of this motion and predicted an anomalous magnetic moment with a g-factor of 1.0011607, obtained purely from geometric considerations, without any numerical fitting or free parameters.

We propose to localize spin mixing dynamics in a spin-1 Bose-Einstein condensate by a temporal modulation of spin exchange interaction, which is tunable with optical Feshbach resonance. Adopting techniques from coherent control, we demonstrate the localization/freezing of spin mixing dynamics, and the suppression of the intrinsic dynamic instability and spontaneous spin domain formation in a ferromagnetically interacting condensate of 87 Rb atoms. This work points to a promising scheme for investigating the weak magnetic spin dipole interaction, which is usually masked by the more dominant spin exchange interaction.

We investigate quantum coherences in the presence of noise by entangling the spin and path degrees of freedom of the output neutron beam from a noisy three-blade perfect crystal neutron interferometer. We find that in the presence of dephasing noise on the path degree of freedom the entanglement of the output state reduces to 0, however the quantum discord remains nonzero for all noise values. Hence even in the presence of strong phase noise nonclassical correlations persist between the spin and the path of the neutron beam. This indicates that measurements performed on the spin of the neutron beam will induce a disturbance on the path state. We calculate the effect of the spin measurement by observing the changes in the observed contrast of the interferometer for an output beam postselected on a given spin state. In doing so we demonstrate that these measurements allow us to implement a quantum eraser and a which-way measurement of the path taken by the neutron through the interferometer. While strong phase noise removes the quantum eraser, the spin-filtered which-way measurement is robust to phase noise. We experimentally demonstrate this disturbance by comparing the contrasts of the output beam with and without spin measurements of three neutron interferometers with varying noise strengths. This demonstrates that even in the presence of noise that suppresses path coherence and spin-path entanglement, a neutron interferometer still exhibits uniquely quantum behavior.

We perform a numeric study (Worm algorithm Monte Carlo simulations) of ultracold two-component bosons in two-dimensional optical lattices. We study how the Mott insulator to superfluid transition is affected by the presence of a second superfluid bosonic species. We find that, at fixed interspecies interaction, the upper and lower boundaries of the Mott lobe are differently modified. The lower boundary is strongly renormalized even for relatively low filling factor of the second component and moderate (interspecies) interaction. The upper boundary, instead, is affected only for large enough filling of the second component. Whereas boundaries are renormalized we find evidence of polaron-like excitations. Our results are of interest for current experimental setups. Mo 163 (Postdeadline) Quantum information

We use gauge fixing to derive Proca equation from Maxwell’s classical electrodynamics in curved spacetime. Further restrictions on the gauge yield the Klein-Gordon equation for scalar bosons. The self-coupling of electromagnetic fields through spacetime curvature originates the inertia of wave packets for non-null field solutions, suggesting an electromagnetic origin of mass. We study the weak field limit of these solutions and prove that the electrovacuum can behave as a charged nonlinear optical medium.

A. Muñoz Mateo1,3,∗ Xiaoquan Yu2,3,† and Jun Nian4,5‡ 1 Departament de Fı́sica Quàntica i Astrofı́sica, Universitat de Barcelona, Martı́ i Franquès, 1, E–08028 Barcelona, Spain 2 Department of Physics, Centre for Quantum Science, and Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin, New Zealand 3New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry and Physics,Massey University, Auckland 0745, New Zealand 4Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 route de Chartres, 91440 Bures-sur-Yvette, France and 5C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, United States

Registration; Welcome 9:00 -9:40 Fatkhulla Abdullaev Dissipative periodic waves, solitons and breathers of the nonlinear Schrodinger equation with complex potentials 9:40 -10:10 Alexey Okulov Cold atoms trapping via helical interference patterns of the phase-conjugated Laguerre-Gaussian beam 10:10 -10:40 Usama Al Khawaja Soliton localization in a vibrating harmonic trap 10:40 -11:00 Coffee break 11:00 -11:40 Boris A. Malomed Suppression of the quantum-mechanical collapse by repulsive interactions 11:40 -12:10 Goran Gligoric The effect of dipole-dipole interaction on transition from the immiscible to miscible state in linearly coupled binary dipolar Bose-Einstein condensate 12:10 -12:40 Alexander Itin Many-particle Landau-Zener models and dynamics of quantum phase transitions 12:40 -13:10

The Berry curvature provides a powerful tool to unify several branches of science through their geometrical aspect: topology, energy bands, spin and vector fields. While quantum defects-phase vortices and skyrmions-have been in the spotlight, as rotational entities in condensates, superfluids and optics, their dynamics in multi-component fields remain little explored. Here we use two-component microcavity polaritons to imprint a dynamical pseudospin texture in the form of a double full Bloch beam, a conformal continuous vortex beyond unitary skyrmions. The Berry curvature plays a key role to link various quantum spaces available to describe such textures. It explains for instance the ultrafast spiraling in real space of two singular vortex cores, providing in particular a simple expression-also involving the complex Rabi frequency-for their intricate velocity. Such Berry connections open new perspectives for understanding and controlling highly-structured quantum objects, including strongly asymmetric cases or even higher multi-component fields.

Saktioto and Prof. Dr. Jalil Ali for all their guidance and support throughout the duration of this research and thesis writing. I am greatly indebted to them for the knowledge imparted and the precious time they allocated to guide me. Dr. Saktioto provided the overall framework of this studies. I would like to extend my sincere appreciation to my family especially mom and dad for their tender support, morally and financially. I would also like to thanks to members of the Institute of Advanced Photonics and Sciences (APSI) for their assistance. They had provided me with ample information, cooperation and help during the process of conducting my research.

Saktioto and Prof. Dr. Jalil Ali for all their guidance and support throughout the duration of this research and thesis writing. I am greatly indebted to them for the knowledge imparted and the precious time they allocated to guide me. Dr. Saktioto provided the overall framework of this studies. I would like to extend my sincere appreciation to my family especially mom and dad for their tender support, morally and financially. I would also like to thanks to members of the Institute of Advanced Photonics and Sciences (APSI) for their assistance. They had provided me with ample information, cooperation and help during the process of conducting my research.

This study investigates the stability and decay properties of solutions to nonlinear Schrödinger equations (NLSEs) with time-dependent coefficients. Employing a blend of analytical and numerical methods, we delve into how temporal variations in coefficients influence the dynamics of wave functions. Our analysis reveals that time-dependent coefficients significantly affect the stability and decay rates of solutions, uncovering conditions that lead to either enhanced stability or accelerated decay. The findings highlight the critical role of coefficient temporality in dictating the behavior of NLSE solutions. These insights not only advance our theoretical understanding of NLSEs but also bear implications for practical applications in fields modeled by these equations. Our research opens avenues for exploiting time-dependent behaviors in designing systems with desired dynamical properties.

Suppression of the quantum-mechanical collapse by repulsive interactions 11:40-12:10 Goran Gligoric The effect of dipole-dipole interaction on transition from the immiscible to miscible state in linearly coupled binary dipolar Bose-Einstein condensate 12:10-12:40 Alexander Itin Many-particle Landau-Zener models and dynamics of quantum phase transitions

We analyze coherence effects during the splitting of a quasi one-dimensional condensate into two spatially separated ones and their subsequent merging into a single condensate. Our analysis takes into account finite-temperature effects, where phase fluctuations play an important role. We show that, at zero-temperature, the two split condensates can be merged into a single one with a negligible phase difference. By increasing temperature to a finite value below the critical point for condensation (Tc), i.e., 0 ≤ T /Tc < 1, a considerable enhancement of phase and density fluctuations appears during the process of splitting and merging. Our results show that if the process of splitting and merging is sufficiently adiabatic, the whole process is quite insensitive to phase fluctuations and even at high temperatures, a single condensate can be produced.

Existing mathematical models very successfully determine the probabilistic evolution of particles and their wave characteristics. However, explanation of an underlying reality remains unsettled. This paper demonstrates the feasibility of a free-travelling electron with definite motion in its distinct space-time frame that is related to the detector's frame by the wave oscillations. These oscillations present fully-defined motion relative to the detector's frame. This would not invalidate existing mathematical treatments. However, it could provide a path to explaining an underlying reality.

condensates (BECs). This poster reports three recent experiments using this technique. First, we have realized the dc atom SQUID geometry of a BEC in a toroidal trap with two Josephson junctions. We observe Josephson effects, measure the critical current of the junctions, and find dynamic behavior that is in good agreement with the simple Josephson equations for a tunnel junction with the ideal sinusoidal current-phase relation expected for the parameters of the experiment. Second, we have used free expansion of a rotating toroidal BEC to create matter wave Bessel beams, which are of interest because perfect Bessel beams (plane waves with amplitude profiles described by Bessel functions) propagate without diffraction. Third, we have realized the basic circuit elements necessary to create complex matter wave circuits. We launch BECs at arbitrary velocity along straight waveguides, propagate them around curved waveguides and stadium-shaped waveguide traps, and split them coherently at y-junctions that can also act as switches.

condensates (BECs). This poster reports three recent experiments using this technique. First, we have realized the dc atom SQUID geometry of a BEC in a toroidal trap with two Josephson junctions. We observe Josephson effects, measure the critical current of the junctions, and find dynamic behavior that is in good agreement with the simple Josephson equations for a tunnel junction with the ideal sinusoidal current-phase relation expected for the parameters of the experiment. Second, we have used free expansion of a rotating toroidal BEC to create matter wave Bessel beams, which are of interest because perfect Bessel beams (plane waves with amplitude profiles described by Bessel functions) propagate without diffraction. Third, we have realized the basic circuit elements necessary to create complex matter wave circuits. We launch BECs at arbitrary velocity along straight waveguides, propagate them around curved waveguides and stadium-shaped waveguide traps, and split them coherently at y-junctions that can also act as switches.

What we believe is a new scheme for producing semidiscrete self-trapped vortices ("swirling photon droplets") in photonic crystals with competing quadratic (χ (2)) and selfdefocusing cubic (χ (3)) nonlinearities is proposed. The photonic crystal is designed with a striped structure, in the form of spatially periodic modulation of the χ (2) susceptibility, which is imposed by the quasi-phase-matching technique. Unlike previous realizations of semidiscrete optical modes in composite media, built as combinations of continuous and arrayed discrete waveguides, the semidiscrete vortex "droplets" are produced here in the fully continuous medium. This work reveals that the system supports two types of semidiscrete vortex droplets, viz., onsite-and intersite-centered ones, which feature, respectively, odd and even numbers of stripes, N. Stability areas for the states with different values of N are identified in the system's parameter space. Some stability areas overlap with each other, giving rise to the multistability of states with different N. The coexisting states are mutually degenerate, featuring equal values of the Hamiltonian and propagation constant. An experimental scheme to realize the droplets is outlined, suggesting new possibilities for the long-distance transmission of nontrivial vortex beams in nonlinear media.

The Dirac equation is a cornerstone of quantum mechanics that fully describes the behaviour of spin ½ particles. Recently, the energy momentum relationship has been reconsidered such that |E|^2 = |(m0c^ 2)| 2 + |(pc)| 2 has been modified to: |E| 2 = |(m0c^2)|^2-|(pc)|^2 where E is the kinetic energy, moc^2 is the rest mass energy and pc is the wave energy for the spin ½ particle. This has been termed the 'Hamiltonian approach' and with a new starting point, the original Dirac equation has been derived: and the modified covariant form found is where h/2π = c = 1. The behaviour of spin ½ particles is found to be the same as for the original Dirac equation. The Dirac equation will also be expanded by setting the rest energy as a complex number, |(m0c 2)| e^jωt

In this paper, we delve into the potential implications of the Aharonov-Bohm effect on the conventional interpretation of the wave function, suggesting that the effect may reveal deeper connections between quantum mechanics and electromagnetic potentials. We extend this exploration by investigating the relationship between zitterbewegung, the rapid oscillatory motion of electrons, and their observable motion. We propose that the electron's wave function could be interpreted as a manifestation of oscillatory changes in its electromagnetic potentials. These changes are influenced by  the electron’s motion through space. Furthermore, we consider how this interpretation can provide a physical basis for phenomena such as quantum interference and phase shifts, offering a new perspective on the role of electromagnetic potentials in quantum mechanics. Through this approach, we aim to bridge the gap between classical electromagnetic theory and quantum mechanical descriptions, potentially leading to novel insights into the nature of quantum systems.

In line with quantum wave-particle duality, the Hamiltonian energy will be the sum of the wave and particle energies. A revised assignment of the wave vector, k = j2π/λ, transposes the wave energy of a quantum particle to a potential imaginary energy, V = jpc, then the Hamiltonian operator, H, will be transposed from a sum of real and imaginary energies to be a complex vector. A ‘non-relativistic’ Hamiltonian energy vector, H, is advanced:
H = |mc2|.ejwt = (mv^2).cos(wt) +  jpc. sin(wt)
where ϴ = wt = tan^-1 ((pc)/(mv^2))
The modulus of the Hamiltonian, |H|, will be the vector sum of kinetic and potential energies, equivalent to the rest mass energy, that is the Lorentz invariant.
|H| = mc^2 = ((pc)^2 + (mv^2)^2)^-1/2
In addition, the Hamiltonian complex energy vector is advanced:
H = |mc^2|.ejwt= (mv^2)(∆v^2/c^2).cos(wt) + j.sin(wt).(∆u^2/c^2)p.c
where ϴ = tan-1 (V/T) = wt

Furthermore, it will be shown that the Hamiltonian vector can be derived from the Schwarzschild solution.

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We consider merger of two parallel toroidal atomic Bose-Einstein condensates with different vorticities in a three-dimensional (3D) trap. In the tunnel-coupling regime, Josephson vortices (rotational fluxons) emerge in the barrier between the superflows. When the barrier is gradually eliminated, we observe essentially three-dimensional evolution of quantum vortices, which may include the development of the Kelvin-Helmholtz instability at the interface between the rings, in the framework of a weakly dissipative Gross-Pitaevskii equation. An initially more populated ring, carrying a persistent current, can drag an initially non-rotating less populated one into the same vortex state. The final state of the condensate crucially depends on an initial population imbalance in the double-ring set, as well as on the shape of the 3D trapping potential, oblate or prolate. In the prolate (axially elongated) configuration, robust 3D hybrid structures may appear as a result of the merger of persistent currents corresponding to different vorticities.

Considerable progress in experimental studies of atomic gases in a toroidal geometry opens up novel prospects for the investigation of fundamental properties of superfluid states and creation of new configurations for atomtronic circuits. In particular, a setting with Bose-Einstein condensates loaded in a dual-ring trap suggests a possibility to consider the dynamics of tunneling between condensates with different angular momenta. Accordingly, we address the tunneling in a pair of coaxial three-dimensional (3D) ring-shaped condensates separated by a horizontal potential barrier. A truncated (finite-mode) Galerkin model and direct simulations of the underlying 3D Gross-Pitaevskii equation are used for the analysis of tunneling superflows driven by an initial imbalance in atomic populations of the rings. The superflows through the corresponding Bose-Josephson junction are strongly affected by persistent currents in the parallel rings. Josephson oscillations of the population imbalance and angular momenta in the rings are obtained for co-rotating states and non-rotating ones. On the other hand, the azimuthal structure of the tunneling flow demonstrates formation of Josephson vortices (fluxons) with zero net current through the junction for hybrid states, built of counter-rotating persistent currents in the coupled rings.

We consider the model of a dual-core spatial-domain coupler with χ (2) and χ (3) nonlinearities acting in two parallel cores. We construct families of symmetric and asymmetric solitons in the system with self-defocusing χ (3) terms, and test their stability. The transition from symmetric to asymmetric soliton branches, and back to the symmetric ones proceeds via a bifurcation loop. A pair of stable asymmetric branches emerge from the symmetric family via a supercritical bifurcation; eventually, the asymmetric branches merge back into the symmetric one through a reverse bifurcation. The existence of the loop is explained by means of an extended version of the cascading approximation for the χ (2) interaction, which takes into regard the XPM par of the χ (3) interaction. When the inter-core coupling is weak, the bifurcation loop features a concave shape, with the asymmetric branches losing their stability at the turning points. In addition to the two-color solitons, which are built of the fundamental-frequency (FF) and second-harmonic (SH) components, in the case of the self-focusing χ (3) nonlinearity we also consider single-color solitons, which contain only the SH component but may be subject to the instability against FF perturbations. Asymmetric single-color solitons are always unstable, whereas the symmetric ones are stable, provided that they do not coexist with two-color counterparts. Collisions between tilted solitons are studied too. OCIS numbers: 190.6135; 190.2620; 230.4320 I. INTRODUCTION The effect of the second-harmonic (SH) generation by monochromatic light was first demonstrated in 1961 [1], and has been a topic of great interest ever since [2, 3]. The work on this theme includes many studies dealing with solitons supported by the quadratic (χ (2)) nonlinearity [4, 5]. While the χ (2) interactions between the fundamental-frequency (FF) and SH fields are sufficient for the creation of solitons, the competition between the χ (2) nonlinearity and its cubic (χ (3)) counterpart, either self-focusing or defocusing, may be an essential factor affecting the efficiency of the FF SH conversion. In addition to the material Kerr effect, it was predicted [6, 7] and experimentally demonstrated [8, 9] that χ (3) nonlinearity can be engineered by means of the quasi-phase-matching (QPM) technique, which makes it possible to control the modulational instability [14] and pulse compression [10] in the medium. The χ (3) nonlinearity may also be induced by semiconductor dopants implanted into the χ (2) material [11]-[13]. Effects of competing χ (2) : χ (3) nonlinearities on spatial solitons were analyzed in detail [15]-[20]. It was predicted that stable two-color solitons, built of the FF and SH components, exist in media with the self-defocusing sign of the cubic nonlinearity, where the solitons do not exist in the absence of the χ (2) interactions [16]. It has also been shown that the self-focusing χ (3) term supports single-color solitary waves built solely of the SH component, but they are prone to destabilization by the χ (2) interactions. In fact, the single-color solitons are always unstable when they coexist with the two-color ones [19]. The soliton dynamics in symmetric dual-core systems, alias couplers, has also drawn a great deal of attention, starting from the analysis of solitons and their bifurcations in the model of twin-core optical fibers with the intra-core Kerr nonlinearity [21]-[28], which was recently followed by the consideration of the soliton stability in the coupler combining the Kerr terms with the parity-time (PT) symmetry, represented by equal gain and loss coefficients in the coupled cores [29, 30]. A similar analysis was reported for spatial solitons in planar dual-core symmetric waveguides with intrinsic χ (2) nonlinearity [31],[32]. In all the cases, the increase of the total energy or power of the soliton (in the temporal or spatial domain, respectively) leads to the symmetry-breaking bifurcation (SBB), the difference being that the bifurcation is subcritical or supercritical in the couplers with the χ (3) and χ (2) nonlinear terms, respectively. These bifurcations are important examples of the phenomenology of the spontaneous symmetry breaking of localized modes in nonlinear media [33]. These findings suggest one to consider dual-core systems with competing self-focusing and self-defocusing nonlinearities, where the increase of the total power may, at first, convert symmetric solitons into asymmetric ones through the SBB, which is followed, at larger powers, by an inverse symmetry-restoring bifurcation (SRB), if the self-defocusing dominates at high powers. A natural model of this type is based on the coupler with a combination of self-focusing cubic and self-defocusing quintic intra-core nonlinear terms [34]. The analysis has demonstrated the existence of the corresponding bifurcation loops. A similar effect was demonstrated in an allied model, based on the nonlinear Schrödinger equation including the combination of the cubic-quintic (CQ) terms and a double-well potential, which

A comparative review is given of some well-known and some recent results obtained in studies of two-and three-dimensional (2D and 3D) solitons, with emphasis on states carrying embedded vorticity. Physical realizations of multidimensional solitons in atomic Bose-Einstein condensates (BECs) and nonlinear optics are briefly discussed too. Unlike 1D solitons, which are typically stable, 2D and 3D ones are vulnerable to instabilities induced by the occurrence of the critical and supercritical collapse, respectively, in the 2D and 3D models with the cubic self-focusing nonlinearity. Vortex solitons are subject to a still stronger splitting instability. For this reason, a central problem is looking for physical settings in which 2D and 3D solitons may be stabilized. The review addresses in detail two well-established topics, viz., the stabilization of vortex solitons by means of competing nonlinearities, or by trapping potentials (harmonic-oscillator and spatially-periodic ones). The former topic includes a new addition, closely related to the recent breakthrough, viz., the prediction and creation of robust quantum droplets. Two other topics included in the review outline new schemes which were recently elaborated for the creation of stable vortical solitons in BEC. One scheme relies on the use of the spin-orbit coupling (SOC) in binary condensates with cubic intrinsic attraction, making it possible to predict stable 2D and 3D solitons, which couple or mix components with vorticities S = 0 and ±1 (semi-vortices (SVs) or mixed modes (MMs), respectively). In this system, the situation is drastically different in the 2D and 3D geometries. In 2D, the SOC helps to create a ground state (GS, which does not exist otherwise), represented by stable SV or MM solitons, whose norm falls below the threshold value at which the critical collapse sets in. In the 3D geometry, the supercritical collapse does not allow one to create a GS, but metastable solitons of the SV and MM types can be constructed. Another new scheme makes it possible to create stable 2D vortex-ring solitons with arbitrarily high S in a binary BEC with components coupled by microwave radiation. Some other topics are addressed briefly, such as vortex solitons in dissipative media, and attempts to create vortex solitons in experiments.

Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a challenging issue. Previously, no such states in Bose-Einstein condensates (BECs) or other physical settings were known. Dipolar BEC suggests a unique possibility to predict stable anisotropic vortex quantum droplets (AVQDs). We demonstrate that they can be created with the vortex' axis oriented perpendicular to the polarization of dipoles. The stability area and characteristics of the AVQDs in the parameter space are revealed by means of analytical and numerical methods. Further, the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to which spinning AVQDs can follow the rotation in clockwise and anticlockwise directions, are found. Collisions between moving AVQDs are studied too, demonstrating formation of bound states with a vortex-antivortex-vortex structure. A stability domain for such stationary bound states is identified. Unstable dipolar states, that can be readily implemented by means of phase imprinting, quickly transform into robust AVQDs, which suggests a straightforward possibility for the creation of these states in the experiment.

A method to realize controllable inversion of energy levels in a one-dimensional spin-orbit (SO)coupled two-component Bose-Einstein condensate under the action of a gradient magnetic field and harmonic-oscillator (HO) trapping potential is proposed. The linear version of the system is solved exactly. By adjusting the SO coupling strength and magnetic-field gradient, the energy-level inversion makes it possible to transform any excited state into the ground state. The full nonlinear system is solved numerically, and it is found that the results are consistent with the linear prediction in the case of the repulsive inter-component interaction. On the other hand, the inter-component attraction gives rise to states of superposition and edge types. Similar results are also reported for the system with the HO trap replaced by the box potential. These results suggest a possibility to realize any excited state and observe it in the experiment.

We present an analysis of stationary solutions for two-dimensional (2D) Bose-Einstein condensates (BECs) with the Rashba spin-orbit (SO) coupling and Zeeman splitting. By introducing the generalized momentum operator, the linear version of the system can be solved exactly. The solutions are semivortices of the Bessel-vortex (BV) and modified Bessel-vortex (MBV) types, in the presence of the weak and strong Zeeman splitting, respectively. The ground states (GSs) of the full nonlinear system are constructed with the help of a specially designed neural network (NN). The GS of the mixed-mode type appears as cross-attraction interaction increases. The spin texture of the GS is produced in detail. It exhibits the Néel skyrmion structure for the semivortex GS of the BV type, and the respective skyrmion number is found in an analytical form. On the other hand, GSs of the MBV and mixed-mode types do not form skyrmions.

This is an invited short update of the topic covered by the review article, which aims to briefly survey progress made in theoretical and experimental studies of multidimensional solitons since the publication of the review. The Commentary was invited as an addition to the original review article that will be reprinted in an e-book celebrating the 50th anniversary of Journal of Physics B (incorporating Journal of Optics B: Quantum and Semiclassical Optics).

We address dynamics of Bose-Einstein condensates (BECs) loaded into a one-dimensional four-color optical lattice (FOL) potential with commensurate wavelengths and tunable intensities. This configuration lends system-specific symmetry properties. The analysis identifies specific multi-parameter forms of the FOL potential which admits exact solitary-wave solutions. This newly found class of potentials includes more particular species, such as frustrated double-well superlattices, and bichromatic and three-color lattices, which are subject to respective symmetry constraints. Our exact solutions provide options for controllable positioning of density maxima of the localized patterns, and tunable Anderson-like localization in the frustrated potential. A numerical analysis is performed to establish dynamical stability and structural stability of the obtained solutions, which makes them relevant for experimental realization. The newly found solutions offer applications to the design of sch...

We observe and analyze formation, decay, and subsequent regeneration of ring-shaped clusters of (2+1)-dimensional spatial solitons (filaments) in a medium with the cubicquintic (focusing-defocusing) self-interaction and strong dissipative nonlinearity. The cluster of filaments, that remains stable over ≈17.5 Rayleigh lengths, is produced by the azimuthal modulational instability from a parent ring-shaped beam with embedded vorticity l = 1. In the course of still longer propagation, the stability of the soliton cluster is lost under the action of nonlinear losses. The annular cluster is then spontaneously regenerated due to power transfer from the reservoir provided by the unsplit part of the parent vortex ring. A (secondary) interval of the robust propagation of the regenerated cluster is identified. The experiments use a laser beam (at wavelength 800 nm), built of pulses with temporal duration 150 fs, at the repetition rate of 1 kHz, propagating in a cell filled by liquid carbon disulfide. Numerical calculations, based on a modified nonlinear Schrödinger equation which includes the cubic-quintic refractive terms and nonlinear losses, provide results in close agreement with the experimental findings.

A possibility of creation of stable optical solitons combining one continuous and one discrete coordinates, with embedded vorticity, in an array of planar waveguides with intrinsic cubic–quintic (CQ) nonlinearity is demonstrated. The same system may be realized in terms of the spatiotemporal light propagation in an array of tunnel‐coupled optical fibers with the CQ nonlinearity. In contrast with zero‐vorticity states, semidiscrete vortex solitons do not exist without the quintic term in the nonlinearity. Two types of the solitons, viz., intersite‐centered (IC) and onsite‐centered (OC) ones, with even and odd numbers N of actually excited sites in the discrete direction, are identified. The modes carrying the embedded vorticity and 2 are considered. In accordance with their symmetry, the vortex solitons of the OC type exhibit an intrinsic core, whereas the IC solitons with small N may have a coreless structure. Facilitating their creation in the experiment, the modes reported in the ...

We investigate dynamics of 2D chiral solitons of semi-vortex (SV) and mixed-mode (MM) types in spin-orbit-coupled Bose-Einstein condensates with the Manakov nonlinearity, loaded in a dual-core (double-layer) trap. The system supports two novel manifestations of Josephson phenomenology: one in the form of persistent oscillations between SVs or MMs with opposite chiralities in the two cores, and another one demonstrating robust periodic switching (identity oscillations) between SV in one core and MM in the other, provided that the strength of the inter-core coupling exceeds a threshold value. Below the threshold, the system creates composite states, which are asymmetric with respect to the two cores, or collapses. Robustness of the chirality and identity oscillations against deviations from the Manakov nonlinearity is investigated too. These dynamical regimes are possible only in the nonlinear system. In the linear one, exact stationary and dynamical solutions for SVs and MMs of the Bessel type are found. They sustain Josephson self-oscillations in different modes, with no interconversion between them.

We design a framework based on the spatial-domain copropagation of two light beams with mutually orthogonal polarizations and opposite transverse components of carrier wave vectors in a nonlinear waveguide with randomly varying birefringence, the averaging with respect to which introduces an effective Manakov nonlinearity in the system. The corresponding two-component system of nonlinear Schrödinger equations is derived, being similar to the system of coupled one-dimensional Gross-Pitaevskii equations for a binary spin-orbit-coupled Bose-Einstein condensate. The system may also include an effective Rabi coupling (direct linear mixing of the components) and a periodic potential, representing a photonic-crystal structure in the underlying waveguide. For self-focusing and self-defocusing signs of nonlinearity, soliton solutions of several symmetry types are obtained by means of numerical methods, and their stability is investigated, including gap solitons in the case when the periodic potential is present.

We construct families of fundamental, dipole, and tripole solitons in the fractional Schrödinger equation (FSE) incorporating self-focusing cubic and defocusing quintic terms modulated by factors cos 2 x and sin 2 x, respectively. While the fundamental solitons are similar to those in the model with the uniform nonlinearity, the multipole complexes exist only in the presence of the nonlinear lattice. The shapes and stability of all the solitons strongly depend on the Lévy index (LI) that determines the FSE fractionality. Stability areas are identified in the plane of LI and propagation constant by means of numerical methods, and some results are explained with the help of an analytical approximation. The stability areas are broadest for the fundamental solitons and narrowest for the tripoles.

It was recently found that, under the action of the spin-orbit coupling (SOC) and Zeeman splitting (ZS), binary BEC with intrinsic cubic nonlinearity supports families of gap solitons, provided that the kinetic energy is negligible in comparison with the SOC and ZS terms. We demonstrate that, also under the action of SOC and ZS, a similar setting may be introduced for BEC with two components representing different atomic states, resonantly coupled by microwave radiation, while the Poisson's equation accounts for the feedback of the two-component atomic wave function onto the radiation. The microwave-mediated interaction induces an effective nonlinear trapping potential, which strongly affects the purport of the linear spectrum in this system. As a result, families of both gap and embedded solitons (those overlapping with the continuous spectrum) are found, which are chiefly stable. The shape of the solitons features exact or broken skew symmetry. In addition to fundamental solitons (whose shape may or may not include a node), a family of dipole solitons is constructed too, which are even more stable than their fundamental counterparts. A nontrivial stability area is identified for moving solitons in the present system, which lacks Galilean invariance. Colliding solitons merge into a single one.

The general objective of the work is to study dynamics of dissipative solitons in the framework of a one-dimensional complex Ginzburg-Landau equation (CGLE) of a fractional order. To estimate the shape of solitons in fractional models, we first develop the variational approximation for solitons of the fractional nonlinear Schrödinger equation (NLSE), and an analytical approximation for exponentially decaying tails of the solitons. Proceeding to numerical consideration of solitons in *Manuscript Click here to view linked References fractional CGLE, we study, in necessary detail, effects of the respective Lévy index (LI) on the solitons' dynamics. In particular, dependence of stability domains in the model's parameter space on the LI is identified. Pairs of in-phase dissipative solitons merge into single pulses, with the respective merger distance also determined by LI.

A comparative review is given of some well-known and some recent results obtained in studies of two-and three-dimensional (2D and 3D) solitons, with emphasis on states carrying embedded vorticity. Physical realizations of multidimensional solitons in atomic Bose-Einstein condensates (BECs) and nonlinear optics are briefly discussed too. Unlike 1D solitons, which are typically stable, 2D and 3D ones are vulnerable to instabilities induced by the occurrence of the critical and supercritical collapse, respectively, in the 2D and 3D models with the cubic self-focusing nonlinearity. Vortex solitons are subject to a still stronger splitting instability. For this reason, a central problem is looking for physical settings in which 2D and 3D solitons may be stabilized. The review addresses in detail two well-established topics, viz., the stabilization of vortex solitons by means of competing nonlinearities, or by trapping potentials (harmonic-oscillator and spatially-periodic ones). The former topic includes a new addition, closely related to the recent breakthrough, viz., the prediction and creation of robust quantum droplets. Two other topics included in the review outline new schemes which were recently elaborated for the creation of stable vortical solitons in BEC. One scheme relies on the use of the spin-orbit coupling (SOC) in binary condensates with cubic intrinsic attraction, making it possible to predict stable 2D and 3D solitons, which couple or mix components with vorticities S = 0 and ±1 (semi-vortices (SVs) or mixed modes (MMs), respectively). In this system, the situation is drastically different in the 2D and 3D geometries. In 2D, the SOC helps to create a ground state (GS, which does not exist otherwise), represented by stable SV or MM solitons, whose norm falls below the threshold value at which the critical collapse sets in. In the 3D geometry, the supercritical collapse does not allow one to create a GS, but metastable solitons of the SV and MM types can be constructed. Another new scheme makes it possible to create stable 2D vortex-ring solitons with arbitrarily high S in a binary BEC with components coupled by microwave radiation. Some other topics are addressed briefly, such as vortex solitons in dissipative media, and attempts to create vortex solitons in experiments.

We elaborate one-and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including fundamental ones, 1D multipoles, and 2D vortices. Previously, solitons of this type were only produced by models with competing nonlinearities. The present setting may be implemented in optics and Bose-Einstein condensates. The 1D version gives rise to an exact analytical solution for stable flat-top solitons, and generic families may be predicted by means of the Thomas-Fermi approximation. Stability of the obtained flat-top solitons is analyzed by means of linear-stability analysis and direct simulations. Fundamental solitons and 1D multipoles with k = 1 and 2 nodes, as well as vortices with winding number m = 1, are completely stable. For multipoles with k ≥ 3 and vortices with m ≥ 2, alternating stripes of stability and instability are identified in their parameter spaces.

We consider the dynamical model of a binary bosonic gas trapped in a symmetric dual-core cigarshaped potential. The setting is modeled by a system of linearly-coupled one-dimensional Gross-Pitaevskii equations with cubic self-repulsive terms and quadratic attractive ones,which represent the Lee-Huang-Yang corrections to the mean-field theory in this geometry. The main subject is spontaneous symmetry breaking (SSB) of quantum droplets (QDs), followed by restoration of the symmetry, with respect to the identical parallel-coupled trapping cores, following the increase of the QD's total norm. The SSB transition and inverse symmetry-restoring one form a bifurcation loop, whose shape in concave at small values of the inter-core coupling constant, κ, and convex at larger κ. The loop does not exist above a critical value of κ. At very large values of the norm, QDs do not break their symmetry, featuring a flat-top shape. Some results are obtained in an analytical form, including an exact front solution connecting asymptotically constant zero and finite values of the wave function. Collisions between moving QDs are considered too, demonstrating a trend to merger into breathers.

In the framework of the Gross-Pitaevskii equation, we study the formation and stability of effectively two-dimensional solitons in dipolar Bose-Einstein condensates (BECs), with dipole moments polarized at an arbitrary angle θ relative to the direction normal to the system's plane. Using numerical methods and the variational approximation, we demonstrate that unstable Townes solitons, created by the contact attractive interaction, may be completely stabilized (with an anisotropic shape) by the dipole-dipole interaction (DDI), in interval θ cr < θ ≤ π/2. The stability boundary, θ cr , weakly depends on the relative strength of DDI, remaining close to the "magic angle", θm = arccos 1/ √ 3. The results suggest that DDIs provide a generic mechanism for the creation of stable BEC solitons in higher dimensions.

We review the state of the art and recently obtained theoretical and experimental results for two-and three-dimensional (2D and 3D) solitons and related states, such as quantum droplets, in optical systems, atomic Bose-Einstein condensates (BECs), and other fields-in particular, liquid crystals. The central challenge is avoiding the trend of 2D and 3D solitary states supported by the ubiquitous cubic nonlinearity to be strongly unstable-a property far less present in one-dimensional systems. Many possibilities for the stabilization of multi-dimensional states have been theoretically proposed over the years. Most strategies involve non-cubic nonlinearities or using different sorts of potentials, including periodic ones. New important avenues have arisen recently in systems based on two-component BEC with spin-orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. An important recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, which are stabilized by the presence of Lee-Hung-Yang quantum fluctuations around the underlying mean-field states. By and large, multi-dimensional geometries afford unique opportunities to explore the existence of complex self-sustained states, including topologically rich ones, which by their very nature are not possible in one-dimensional geometries. In addition to vortex solitons, these are hopfions, skyrmions, and hybrid vortex-antivortex complexes, which have been predicted in different models. Here we review recent landmark findings in this field and outline outstanding open challenges. List of acronyms: 1D one-dimensional 2D two dimensional 3D three dimensional BEC Bose-Einstein condensate FF fundamental frequency GPE Gross-Pitaevskii equation GVD group-velocity dispersion LHY Lee-Huang-Yang (correction to the mean-field dynamics of BEC) MF mean field MM mixed mode NLSE nonlinear Schrödinger equation SH second harmonic SOC spin-orbit coupling SV semi-vortex TS Townes' soliton VK Vakhitov-Kolokolov (stability criterion) 2.3. Multimode waveguides 2.4. Exciton-polaritons 2.5. Plasma filaments 3. Multidimensional states supported by nonlinear potentials 4. Multidimensional states in atomic BECs 4.1. Stabilization by optical lattices 4.2. Stabilization by spin-orbit coupling 4.3. Giant vortex solitons in binary BECs 4.4. Quantum droplets in binary BECs 5. Soliton states in liquid crystals and liquid ferromagnets 6. Conclusions and outlook 7. Bibliography 1.

We elaborate a method for the creation of two-and one-dimensional (2D and 1D) self-trapped modes in binary spin-orbit (SO)-coupled Bose-Einstein condensates (BECs) with the contact repulsive interaction, whose local strength grows fast enough from the center to periphery. In particular, an exact semi-vortex (SV) solution is found for the anti-Gaussian radial-modulation profile. The exact modes are included in the numerically produced family of SV solitons. Other families, in the form of mixed modes (MMs), as well as excited state of SVs and MMs, are produced too. While the excited states are unstable in all previously studied models, they are partially stable in the present one. In the 1D version of the system, exact solutions for the counterpart of the SVs, namely, semi-dipole solitons, are found too. Families of semi-dipoles, as well as the 1D version of MMs, are produced numerically.

We analyze a possibility of macroscopic quantum effects in the form of coupled structural oscillations and shuttle motion of bright two-component spin-orbit-coupled striped (one-dimensional, 1D) and semi-vortex (two-dimensional, 2D) matter-wave solitons, under the action of linear mixing (Rabi coupling) between the components. In 1D, the intrinsic oscillations manifest themselves as flippings between spatially even and odd components of striped solitons, while in 2D the system features periodic transitions between zero-vorticity and vortical components of semi-vortex solitons. The consideration is performed by means of a combination of analytical and numerical methods.