Energetics and Control of Ultracold Isotope-Exchange Reactions between Heteronuclear Dimers in External Fields

Topological States in Partially-PT -Symmetric Azimuthal Potentials Yaroslav V. Kartashov,1,2 Vladimir V. Konotop,3and Lluis Torner1,4 1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region 142190, Russia 3Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande 2, Edifício C8, Lisboa 1749-016, Portugal 4Universitat Politecnica de Catalunya, 08034 Barcelona, Spain (Received 25 May 2015; published 4 November 2015) We introduce partially-parity-time ðpPT Þ-symmetric azimuthal potentials composed from individual PT -symmetric cells located on a ring, where two azimuthal directions are nonequivalent in a sense that in such potential excitations carrying topological dislocations exhibit different dynamics for different directions of energy circulation in the initial field distribution. Such nonconservative ratchetlike structures support rich families of stable vortex solitons in cubic nonlinear media, whose properties depend on the sign of the topological charge due to the nonequivalence of azimuthal directions. In contrast, oppositely charged vortex solitons remain equivalent in similar fully-PT -symmetric potentials. The vortex solitons in the pPT - and PT -symmetric potentials are shown to feature qualitatively different internal current distributions, which are described by different discrete rotation symmetries of the intensity profiles. DOI: 10.1103/PhysRevLett.115.193902 PACS numbers: 42.65.Tg, 42.65.Jx, 42.65.Wi The evolution of nonlinear waves carrying topological phase dislocations is a physical problem of fundamental importance attracting attention in various areas of physics, including optics, matter waves, hydrodynamics, cavities, and electron beams. Such waves are of particular interest because of the salient role they play in numerous classical and quantum phenomena [1,2]. The propagation of vortexcarrying beams is especially intriguing in the presence of transverse modulation of the refractive index of the material [3].Conservative potentials arising due to such modulations play a strong stabilizing role for vortex states in nonlinear media. They arrest collapse for two-dimensional beams in cubic media and suppress azimuthal modulation instabilities of bright vortex solitons that are ubiquitous in uniform media with focusing nonlinearity [4,5]. The intensity distributions of vortex solitons in inhomogeneous media reflect the symmetry of the underlying potentials [6,7]. While in some potentials stable vortex solitons may maintain simple ringlike shapes [8], they become strongly modulated in conventional periodic systems, such as square [9,10], hexagonal [11], and honeycomb [12] optical lattices, and in photonic crystals. The discrete rotation symmetry of such potentials imposes restrictions on the available topological charges of most compact symmetric vortex states [13], which may not hold for extended excitations [14]. The common feature of such conservative potentials is the equivalence of two azimuthal directions, manifested in the identical parameters of vortex solitons with opposite topological charges. Aclassofnonconservative parity-time (PT )-symmetric potentials has a remarkable feature: the transition from a purely real to a complex eigenvalue spectrum, referred to as PT -symmetry breaking, occurs at a critical depth of the imaginary part of the structure [15]. Such a transition manifests itself in the qualitative modification of wave evolution. Optical guiding structures with a transversally symmetric refractive index and antisymmetric gain and loss landscapes provide unique platforms for the exploration of the effects associated with the PT -symmetry breaking, since they allow the realization of potentials where the PT -symmetry condition RðrÞ¼R∗ð−rÞholds [16].The breakup of PT symmetry in localized linear potentials was demonstrated in [17], and the concept is extended to periodic structures [18] and nonlinear states in such systems as isolated PT -symmetric waveguides [19], nondispersive and dispersive couplers [20], oligomers [21], and discrete arrays [22,23] as well as to continuous nonlinear [24], linear [19,25], and mixed [26] lattices. The spectrum of the complex potential may remain real even if the potential is not PT -symmetric [27] in one-dimensional or only partiallyPT -symmetric [28] in multidimensional problems. In contrast to conservative systems, where vortex-free modes do not feature internal currents, in PT -symmetric potentials such currents are necessary to achieve stable beam propagation. Thus, although PT -symmetric potentials may support stationary states with symmetric intensity distributions, there exists a certain selected direction in them, defined by the local currents. This phenomenon is responsible for the removal of degeneracy of vortices in discrete arrays with an embedded PT -symmetric defect [23]. In this Letter, we use this property to construct partially-PT -symmetric continuous azimuthal potentials from fully-PT -symmetric cells placed on a ring, where azimuthal directions become nonequivalent. We address PRL 115, 193902 (2015) PHYSICAL REVIEW LETTERS week ending 6 NOVEMBER 2015 0031-9007=15=115(19)=193902(7) 193902-1 © 2015 American Physical Society vortex solitons in such structures (for discrete arrays, see [23,29]) and show that, due to the nonequivalence of two azimuthal directions, the properties of vortex solitons depend not only on the absolute value of their topological charge, but also on its sign. We consider the evolution of paraxial beams in a focusing cubic medium with simultaneous transverse modulation of the refractive index and of the gain and losses, that is described by the dimensionless Schrödinger equation for the field amplitude q: i∂q ∂ξ¼−1 2∇2q−½preRreðη;ζÞ−ipimRimðη;ζÞq−qjqj2: ð1Þ Here ∇2¼∂2=∂η2þ∂2=∂ζ2is the Laplacian, η,ζare the transverse coordinates, ξis the normalized propagation distance, and the depths of the real pre and imaginary pim parts of the complex potential R¼preRre −ipimRim are determined by the complex refractive index profile δnre −iδnim, where δnre ∼preRre and δnim ∼pimRim. We build a complex azimuthal potential (see [30] for a conservative counterpart) by placing NGaussian waveguides that individually respect PT symmetry, equidistantly on a ring of the radius ρ: Rre ¼X N k¼1 e−½ðη−ρcos ϕkÞ2þðζ−ρsin ϕkÞ2=a2; Rim ¼X N k¼1 σk−1ðζcos ϕk−ηsin ϕkÞ ×e−½ðη−ρcos ϕkÞ2þðζ−ρsin ϕkÞ2=a2;ð2Þ where σ¼1,ϕk¼2πðk−1Þ=N, and ais the waveguide width. When σ¼−1, the potential is PT -symmetric, i.e., Rðη;ζÞ¼Rð−η;ζÞ¼R∗ðη;−ζÞ¼R∗ð−η;−ζÞ, meaning that the potential is even along the horizontal direction and PT -symmetric along the vertical direction, while for σ¼þ1it is still PT -symmetric vertically, but the invariance under the horizontal inversion is replaced by the invariance under simultaneous inversion of η and ζ:Rðη;ζÞ¼R∗ð−η;ζÞ¼R∗ðη;−ζÞ≠R∗ð−η;−ζÞ (Fig. 1). In the latter case, the potential is referred to as partially-PT (pPT ) -symmetric [28]. Such potentials have different orders of discrete rotation symmetries. We consider even values of Nfor which the PT -symmetric potential belongs to the CN=2;v point group of N=2rotations by the angles ϕσ¼−1¼4π=N and N=2mirror reflections, while pPT -symmetric potentials belong to the CNgroup of Nrotations by the angles ϕσ¼þ1¼2π=N. It is convenient to define the order of the rotational symmetry N0, so that N0¼Nfor σ¼þ1and N0¼N=2for σ¼−1. The essential difference between the potentials stems from their imaginary parts: in the pPT -symmetric potential, waveguides are oriented such that local currents from amplifying [bright spots in Fig. 1(b)] to absorbing [dark spots in Fig. 1(b)] domains inside each waveguide are pointed clockwise (such currents are most pronounced inside waveguides, where light intensity is larger, and are much weaker between waveguides), making this azimuthal direction nonequivalent to the counterclockwise direction. The profile of such potentials as a function of polar angle ϕat fixed radius rresembles a one-dimensional PT -symmetric lattice with a ratchetlike gain-loss landscape. In the PT -symmetric potential, gain and loss domains are exchanged in the neighboring waveguides [see Fig. 1(c)]; hence, current directions inside waveguides also alternate, making clockwise and counterclockwise directions equivalent. The difference in the discrete symmetries has a profound impact on the spectra of the linear eigenmodes qm¼wmðη;ζÞexpðibmξÞ, where bm¼bm re þibm im is the eigenvalue and m∈ℤdistinguishes the eigenmodes. The basic rotations by ϕσ¼þ1¼2π=N or by ϕσ¼−1¼4π=N, that leave unchanged the arrangements in each symmetry group, imply the existence of the nondegenerate fundamental state w0ðr; ϕÞ¼w0ðr; ϕþϕσÞcorresponding to a real eigenvalue b0. Higher-order eigenmodes can be represented as angular Bloch waves wmðr;ϕÞ¼eimϕUmðr;ϕÞ, where Umðr; ϕÞ¼Umðr; ϕþϕσÞ. Their eigenvalues, either real or appearing as complex conjugate pairs, are always degenerate. Let us introduce the operator ℋ¼ℋ0þipimRim in Eq. (1), where ℋ0¼−ð1=2Þ∇2−preRre is Hermitian. For σ¼−1, the operator ℋis PT -symmetric (i.e., PT H ¼HPT ). If its eigenvalue bmðℋwm¼bmwmÞ is real, then HðPT wmÞ¼bmðPT wmÞ, i.e., PT wm¼ ð−1Þwe−imϕU mðr; ϕþπÞis also an eigenmode corresponding to the same degenerate real bm. The same is true for the pPT -symmetric operator. For a complex bm,itis the state PT wN0−m¼ð−1ÞN0−me−iðN0−mÞϕU N0−mðr; ϕþπÞ that gives the second eigenmode for the same bm.ℋ0also possesses degenerate pairs of eigenvalues ~ bmðm≠0;N0=2Þ. The evolution of the two highest pairs of degenerate eigenvalues of ℋwith an increase of the imaginary part of the potential can be understood by using the simplified model accounting only for those two levels. Let us define FIG. 1 (color online). Profiles of real (a) and imaginary (b) parts of the pPT -symmetric potential, and (c) imaginary part of the PT -symmetric potential with N¼6. The real parts of potentials are identical. Here and in all contour plots, the profiles are shown within the η,ζ∈½−4.3;þ4.3window. PRL 115, 193902 (2015) PHYSICAL REVIEW LETTERS week ending 6 NOVEMBER 2015 193902-2 eigenmodes ~ wmand eigenvalues ~ bmof ℋ0ðℋ0~ wm¼ ~ bm~ wmÞand build matrix representation h~ wmjℋj~ wnifor complex operator ℋ. The elements of the resulting matrix are given by Hmn ¼~ bmδm;n þipimh~ wmjRimj~ wni. Using ϕσ periodicity of the Bloch modes and potential Rim, one obtains that h~ wmjRimj~ wni¼R∞ 0R2π 0~ w mRim ~ wnrdrdϕ¼ δm−n;N0Rm;n, where the coefficients Rm;n depend only on Rim and Um;nðr; ϕÞshapes within the angular interval ϕ∈½0;ϕσ. This implies that coupling of linear states with different vorticity m,nis possible with an increase of pim only for jm−nj¼N0. Thus, in the pPT -symmetric case with N¼N0¼6, the growing imaginary part ipimRim does not result in coalescence of the eigenvalues associated with m¼1and n¼2. In contrast, in a similar PT -symmetric potential with N0¼3, coalescence of the eigenvalues is possible. In this approximate model, the modified eigenvalues can be found from the matrix H¼ 2 6 6 6 6 6 4 ~ b100ipimR1;−2 0~ b1ipimR 1;−20 0ipimR1;−2~ b20 ipimR 1;−200 ~ b2 3 7 7 7 7 7 5 ð3Þ and are given by b1;2¼ð ~ b1þ~ b2Þ=2½ð~ b1−~ b2Þ2= 4−p2 imjR1;−2j21=2.Increasingpim leads to equality of two double degenerate eigenvalues at pcr im ¼ð ~ b1−~ b2Þ= 2jR1;−2j,thatremain double degenerate after that point but move into the complex plane at pim >p cr im.Atpim ¼pcr im the matrix Hcannot be represented in diagonal form but allows representation in the Jordan-block form H¼I⊗h, where Iis a 2×2identity matrix and his a 2×2matrix with elements h11 ¼h22 ¼ð ~ b1þ~ b2Þ=2,h12 ¼1,andh21 ¼0. This is an indication of the coexistence of two exceptional points at pim ¼pcr im [31],wheretwo pairs of vortex states with m¼1and m¼2simultaneously coalesce. The numerically calculated spectra, shown in Fig. 2for potentials with N¼6, support the above conclusions. In the PT -symmetric potential, one observes coalescence of two double degenerate real eigenvalues leading to the appearance of two double degenerate complex conjugate eigenvalues at pim ¼pcr im (we show only eigenvalues that can lead to symmetry breaking). In contrast, no such coalescence is observed in the pPT -symmetric potential, where noticeable imaginary parts bm im do not appear up to the point where the corresponding real parts bm re approach the edge of the continuous spectrum [Fig. 2(b)]. Such behavior suggests that stable vortex modes may form in the pPT -symmetric potential, where two azimuthal directions are nonequivalent, even in the regime where symmetry is already broken in the PT -symmetric case. Vortex solitons have the form q¼uðrÞexp½iϕðrÞþibξ, where r¼ðη;ζÞand u,ϕare the field modulus and phase, respectively. The latter defines the topological charge m¼ð2πÞ−1H∇ϕdl, where the integral is calculated over any closed contour surrounding the phase dislocation at r¼0. Equation (1) then yields bu ¼ð1=2Þ∇2u− j2=2u3þpreRreuþu3and ∇·j¼2pimRimu2, where we introduced the current j¼u2∇ϕ. Although the imaginary part of the potential enters only the equation for current, the latter does affect the soliton shape via the first equation. The rigorous simulations reveal that the charge rule jmj≤N=2−1(for even N) established in [13] and connecting the maximal topological charge of compact solitons with the order of discrete rotation symmetry of potential holds in the pPT -symmetric case. Furthermore, we consider a representative pPT -symmetric potential with N¼6supporting vortex solitons with charges up to jmj¼2. We set pre ¼5,a¼0.5, and ρ¼0.3Nand use pim as the main control parameter. Examples of solitons supported by such potentials are depicted in Figs. 3and 4. FIG. 2 (color online). Real (black curves) and imaginary (red curves) parts of the eigenvalues of linear eigenmodes of the PT - symmetric (a) and pPT -symmetric (b) structures versus pim at pre ¼5. Superscripts mindicate topological charges of beams that can be constructed by using linear combinations of corresponding eigenmodes. FIG. 3 (color online). Field modulus u(first row) and phase ϕ (second row) distributions for a soliton with in-phase spots (a), for the m¼þ1vortex soliton (b), and for a multipole soliton (c) in the pPT -symmetric structure. In all cases b¼1.8,pre ¼5, and pim ¼10. PRL 115, 193902 (2015) PHYSICAL REVIEW LETTERS week ending 6 NOVEMBER 2015 193902-3 All of them feature Npronounced bright spots. Solitons with in-phase and out-of-phase spots and m¼0are presented in Figs. 3(a) and 3(c), respectively. For pim ≠0, even these simplest m¼0solitons possess nontrivial phase distributions. Solitons with nonzero topological charges are shown in Figs. 3(b),4(a), and 4(b). The central physical result of this Letter is that the nonequivalence of two azimuthal directions in the pPT - symmetric potential causes substantial differences in the shapes and properties of oppositely charged vortex solitons with equal propagation constants. This is in sharp contrast to all previously reported findings on vortex solitons in conservative potentials, where two oppositely charged states are degenerate. The difference is illustrated in Figs. 4(a) and 4(b) for the m¼2states. These are the most stable vortex solitons in the pPT -symmetric structure with N¼6. The intensity and local phase modulations are substantially deeper for m¼þ2than for its m¼−2 counterpart. The origin of the difference is visible from phase distributions in the second row. Vortex solitons are characterized by the presence of a global current associated with vorticity. Such a current is counterclockwise for m>0and clockwise for m<0. Upon stationary propagation, the net gain and loss experienced by the solitons has to vanish; hence, local currents directed from amplifying into absorbing domains should appear in each waveguide. The direction of local currents is indicated in Fig. 4with short gray arrows. While for m>0the angular directions of all local and global currents are opposite, for m<0they coincide—see the vector current maps jðrÞin the last row. The difference in directions of local and global currents implies a difference in the current magnitudes jjðrÞj (third row), which, in turn, affect the field modulus distributions and lead to different soliton properties for a fixed b.In PT -symmetric potentials, the difference between states with opposite charges does not occur, because there the directions of local currents alternate in neighboring waveguides, so that, even if in one waveguide local and global currents have opposite directions, in the neighboring waveguide they coincide [second and fourth rows of Fig. 4(c)]. This leads to an additional azimuthal modulation of the current and field modulus distributions [first and third rows of Fig. 4(c)]. The current modulus distribution now contains two alternating types of spots. The field modulus distribution for the m¼−2vortex soliton can be obtained from that for the m¼þ2state simply by its rotation by an angle 2π=N; i.e., energy flows of such solitons remain identical—a consequence of the equivalence of two azimuthal directions in the PT -symmetric potential. The presence of additional azimuthal modulation reduces discrete rotation symmetry of the field modulus distributions in the PT -symmetric structure in comparison with its pPT -symmetric counterpart. Vortex solitons in the pPT -symmetric structure are characterized by their energy flows U¼∬jqj2dηdζ, whose dependencies on bare shown in Fig. 5(a). Solitons exist above a cutoff identical for opposite charges. The energy flow grows with band saturates at b→∞. Solitons with negative charges carry higher energy flows than solitons with positive charges. The difference δU¼Um¼−2− Um¼þ2acquires its maximal value for intermediate values of the propagation constants and then gradually vanishes when the soliton transforms into Nstrongly localized almost noninteracting bright spots concentrated within individual waveguides [Fig. 5(b)], consistent with physical expectations. A similar behavior was found for all studied pairs mof nonzero topological charges. The difference in energy flow δUbetween oppositely charged solitons always grows with the imaginary part of potential pim, and it can become comparable with U[Fig. 5(c)]. At large pim values, vortex solitons were found to exhibit considerable shape transformations. For example, Nspots can fuse into almost uniform ring at pim ≫pre. FIG. 4 (color online). Field modulus u(first row), phase ϕ (second row), current modulus jjj(third row), and vector current map (fourth row) for vortex solitons with m¼þ2(a) and m¼−2(b) in the pPT -symmetric structure and for a vortex soliton with m¼þ2(c) in the PT -symmetric structure. In all cases b¼1.8,pre ¼5, and pim ¼10. A white circle with an arrow in the phase distributions indicates the direction of the global current associated with vorticity, while short gray arrows indicate the direction of the local currents inside waveguides. PRL 115, 193902 (2015) PHYSICAL REVIEW LETTERS week ending 6 NOVEMBER 2015 193902-4 The nonequivalence of the azimuthal directions is manifested also in different stability properties of oppositely charged solitons. We found that m¼2vortex solitons are stable in a limited interval of propagation constants blow cr ≤b≤bupp cr , but such an interval differs substantially for solitons with positive and negative topological charges [compare domains of stability located between red dots in Figs. 6(a) and 6(b) and obtained by direct integration of Eq. (1) with perturbed inputs propagated up to ξ¼104]. The stability domain is nearly 3 times wider (in terms of pim) for m¼þ2solitons than for m¼−2ones. For small values of pim, the lower border of stability domain blow cr nearly coincides with the cutoff bco for existence shown in Fig. 6by black dots. For m¼þ2 solitons, the value blow cr starts departing from bco at pim >6, and at one point the lower border of stability domain fuses with upper border bupp cr . Beyond this point, solitons become unstable for any b. A similar scenario is found for m¼−2 solitons. The stability domain drastically expands when one approaches the conservative limit pim →0. We also found that m¼0solitons with in-phase spots and m¼1 solitons are unstable, while multipole modes, like those depicted in Fig. 3(c), can be stable. In summary, we introduced pPT -symmetric potentials where two azimuthal directions are nonequivalent and uncovered the important implications of such an effect in the properties of vortex solitons. The properties of the vortex soliton states supported by such potentials were found to depend on both the absolute value of their topological charge and on its sign, in complete contrast to all conservative potentials studied so far, where oppositely charged vortex solitons always exhibit equal properties. Note also that the potentials constructed here represent the first example of a PT -symmetric ratchet featuring a purely real spectrum in a certain parameter domain. V. V. K. was supported by FCT (Portugal) under Grants No. PTDC/FIS-OPT/1918/2012 and No. UID/FIS/ 00618/2013. [1] A. S. Desyatnikov, Y. S. Kivshar, and L. 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