The building principle of triatomic trilobite Rydberg molecules
TThe building principle of triatomic trilobite Rydberg molecules
Christian Fey, Frederic Hummel, and Peter Schmelcher
1, 2 Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: February 19, 2019)We investigate triatomic molecules that consist of two ground state atoms and a highly excitedRydberg atom, bound at large internuclear distances of thousands of ˚Angstroms. In the molecularstate the Rydberg electron is in a superposition of high angular momentum states whose probabilitydensities resemble the form of trilobite fossils. The associated potential energy landscape has anoscillatory shape and supports a rich variety of stable geometries with different bond angles andbond lengths. Based on an electronic structure investigation we analyze the molecular geometrysystematically and develop a simple building principle that predicts the triatomic equilibrium con-figurations. As a representative example we focus on Rb trimers correlated to the n = 30 Rydbergstate. Using an exact diagonalization scheme we determine and characterize localized vibrationalstates in these potential minima with energy spacings on the order of 100 MHz × h . I. INTRODUCTION
Ultralong-range Rydberg molecules (ULRM) are amanifestation of a novel type of chemical bond, wherea ground state atom is captured in the electronic cloudof a highly excited Rydberg atom [1]. In contrast to con-ventional diatomic molecules, an ULRM possess an os-cillatory potential energy surface and huge bond lengthsranging typically from a few hundreds to thousands ofBohr radii a . Based on the angular momentum l of theRydberg electron, two classes of ULRM can be distin-guished: Weakly bound non-polar ULRM that correlateto quantum-defect-split Rydberg states with low angu-lar momentum l <
3, as well as more deeply bound po-lar ULRM in which the Rydberg electron is in a super-position of hydrogen-like high- l states and may possesslarge electric dipole moments on the order of hundredsto thousands of Debye. In allusion to the shape of theirelectronic probability density, polar ULRM are furthersubdivided into ’trilobite’ molecules with dominant s -wave interaction [1] as well as ’butterfly’ molecules [2, 3]with dominant p -wave interactions. All of these specieshave been confirmed experimentally [4–8] via one- or two-photon association in ultracold samples of either Rb, Csor Sr. Experimental and theoretical research on ULRMdemonstrated novel possibilities to tailor molecular prop-erties via weak fields [7, 9–13] and to control atom-atominteractions [14, 15]. Furthermore, ULRM provide un-precedented access to the physics of electron-atom scat-tering [16–20] and ion-atom interactions [21–23].Having control over the density of the atomic sampleand the Rydberg excitation n , experiments are able tocreate and probe not only diatomic ULRM but also poly-atomic ULRM. These are bound states between one Ry-dberg atom and several ground state atoms. Although,originally predicted for polar high- l ULRM [24], experi-mental reasearch focussed so far exclusively on non-polartypes, that are more easily accessible via one- or two-photon transitions. Experiments with s -state ULRMconfirmed the existence of few-body states (trimers, tetramers, pentamers) as well as polaronic many-bodystates, both in excellent agreement with correspondingtheoretical models [11, 25–28].From a theoretical as well as from an experimentalpoint of view, the isotropy of the electronic wave func-tion in polyatomic s -state ULRM has certain advantages.It simplifies the theoretical models [26, 28] and grantshigh excitation efficiencies in the experiments, due tocomparatively large Franck-Condon factors. An obvi-ous drawback of this isotropy is, however, that thereis only weak control over the molecular geometry, espe-cially the angular geometry. This is different for poly-atomic p - and d -state ULRM [29, 30], which exist in linearand bent geometries. The Hilbert space of energeticallyavailable electronic states is here larger, and providesmore possibilities for the Rydberg electron to optimizeits wave function. Consequently, for polyatomic trilobiteULRM, with their large manifold of energetically degen-erate hydrogenic states, even more complex geometrieswith deeper potential wells are expected. Previous stud-ies predicted exotic properties of these molecules, suchas their capability to form Borromean like states[31] orthe appearance of quantum scars [32]. Furthermore thelarge electric dipole moments allow to tune their geom-etry via weak electric fields [33]. However, all of thesetheoretical works focused so far only on constrained ge-ometries, such as linear, planar or cubic configurations,or on polyatomic trilobite states in random configura-tions [32]. Consequently, a thorough understanding ofthe molecular geometry of even the simplest polyatomictrilobite ULRM, the trilobite trimer, is missing. In ourwork we aim at closing this gap. We analyze the fullthree-dimensional potential energy landscape of trilobitetrimers by means of an investigation of their electronicstructure and derive a simple building principle that ex-plains the resulting equilibrium positions. Subsequentlywe employ an exact diagonalization scheme to predictenergies and wave functions of bound vibrational states,which are relevant for spectroscopic measurements.This work is organized as follows. In Section II we a r X i v : . [ phy s i c s . a t o m - ph ] F e b present the electronic Hamiltonian of the molecular sys-tem and derive the corresponding potential energy sur-faces (PES). Furthermore we identify equilibrium posi-tions and explain their geometrical arrangement. In Sec-tion III we provide the theoretical framework for thedescription of vibrational states in these PES. Subse-quently, we present energies and probability densities ofvibrational states and discuss their properties. SectionIV contains our conclusions. II. ENERGY LANDSCAPE
A general polyatomic ULRM consists of an ionic core(here at the coordinate origin), a Rydberg electron at po-sition r and N neutral ground-state atoms at positions R i where i = 1 , . . . , N . A sketch of the setup for N = 2is presented in Fig. 1 (a). In the Born-Oppenheimer ap-proximation the adiabatic electronic Hamiltonian is givenby H = H + V where H describes the Rydberg electronin its ionic core potential while V is the interaction be-tween the Rydberg electron and the ground state atoms.In dependence of the electronic angular momentum l ,the eigenstates of H can be divided into low- l and high- l states. Due to their centrifugal barrier, high- l states (typ-ically l ≥
3) are shielded from the ionic core. To a goodapproximation they are given by hydrogen wave functions ϕ nlm ( r ) with energies − / (2 n ) (in atomic units) where n and m are the principal and the magnetic quantumnumber, respectively. However, due to the presence ofthe ground state atoms (perturbers) inside the Rydbergorbit, the hydrogenic states become coupled. We focus on Rb ULRM where this coupling is small compared to theenergy splitting between high- l ( l ≥
3) and low- l ( l < V = N (cid:88) j =1 πa [ k ( R j )] δ ( r − R j ) . (1)The energy dependence of the scattering length is ob-tained via modified effective range theory [36, 37] a ( k ) = a (0) + ( π/ αk with the electron wavenumber k , theRb(5 s ) polarizability α = 319 . a (0) = − . e -Rb(5s) triplet scatter-ing ( S ) [38]. In a semi-classical approximation the wavewave number is determined via k / − /R = − / (2 n ),where n is the principal quantum number of interest.Despite its simplicity, the Hamiltonian H captures theessential features of trilobite ULRM. Quantitative cor-rections originate from the Rb fine and hyperfine struc-ture, additional p -wave interactions as well as spin-spinand spin-orbit couplings [2, 3, 16, 39–42]. Furthermorethere exist non-perturbative methods relying on Greens’sfunction methods [3, 25, 43, 44].For dimers ( N = 1, R = R ) the contact interac-tion gives rise to an electronic eigenstate that strongly FIG. 1. (a) The Trilobite trimer consists of two ground stateatoms (red) at positions R / relative to the ionic core (blue).The Rydberg electron (here n = 30) is in a superpositionof the two trilobite states φ ( r ; R / ) (orange vs. gray den-sity). (b) Cuts of the two trimer PES (cid:15) ± ( R , R , θ ) for fixed R = R = 1563 a and θ = θ = 0 . π . The energy of thehydrogenic states with n = 30 is set to zero. These poten-tials are compared to the diatomic PES (cid:15) d ( R ) and (cid:15) d ( R ). Amagnification of the deviations is presented in the inset. (c)2D cut of the lower PES (cid:15) − ( R , R , θ ) where only θ = 0 . π is fixed. The colored lines mark cuts (cid:15) − ( R , R , θ ) (red-graydashed) and (cid:15) d ( R ) (solid blue) that are also visible in (b). localizes on the perturber and resembles the shape of atrilobite fossil, see Fig. 1 (a) for two examples. Perform-ing first order perturbation theory in the Hilbert spaceof quasi-degenerate hydrogenic states with n = n and l ≥
3, its wave function can be expressed as Ψ( r ; R ) = N φ ( r ; R ) with the trilobite orbital φ ( r ; R ) = n − (cid:88) l =3 l (cid:88) m = − l ϕ ∗ n lm ( R ) ϕ n lm ( r ) . (2)and the normalization constant N = φ ( R ; R ) − / [24].It is the superposition of all hydrogen states that min-imizes its energy by maximizing its density on the per-turber. This density is azimuthally symmetric aroundthe internuclear axis R . In particular, if R points alongthe z -axis, only terms with m = 0 contribute. The asso-ciated energy shift (first order) is given by (cid:15) d ( R ) = 2 πa [ k ( R )] φ ( R ; R ) . (3)This is the potential energy surface (PES) of themolecule. It is straightforward to show that the PESdepends only on R = | R | . An example for n = 30 (solidblue line) is presented in Fig. 1 (b). The deepest minimasupport a series of localized vibrational states [1].The electronic structure becomes altered when a sec-ond perturber is present ( N = 2). To obtain the elec-tronic trimer states efficiently, we separate the Hilbertspace with n = n and l ≥ φ ( r ; R j ) with j = 1 , R and R and do thus not probe the ground state atoms [31]. Aproof is provided in Appendix A. Consequently, withinfirst order perturbation theory, we can express the trimerstate as a linear combination of the two dimer solutions[24, 29, 31, 40] ψ ( r ; R , R ) = (cid:88) j =1 c j φ ( r ; R j ) (4)with coefficients c j that depend on R and R . Thissituation is visualized in Fig. 1 (a) schematically, wherewe take into account, that the shape of the trilobite state φ ( r ; R j ) depends explicitly on the nuclear coordinate R j .Eigenstates of this two-level system are determined bysolving the corresponding generalized eigenvalue problemfor H . The two resulting PES (cid:15) ± ( R , R ) = (cid:15) d ( R ) + (cid:15) d ( R )2 ± (cid:113) ( (cid:15) d ( R ) − (cid:15) d ( R )) + 4 c ( R , R ) , (5)can be expressed in terms of the diatomic potentials (3)and a term c ( R , R ) = 4 π a [ k ( R )] a [ k ( R )] | φ ( R ; R ) | . (6)The latter contains the trilobite orbital (2) as a functionof the two nuclear coordinates. It depends in addition to R = | R | and R = | R | , also on the relative angle θ =arccos [( R · R ) / ( R R )] and adds thus an anisotropy tothe PES. Furthermore it satisfies c ( R , R ) = c ( R , R ).Exemplary cuts of the PES (cid:15) ± ( R , R , θ ) are pre-sented in Fig. 1 (b) for fixed R = R = 1563 a and θ = θ = 0 . π but variable R (dashed gray-red anddashed-dotted yellow line). These potentials are com-pared to the corresponding diatomic PES (cid:15) d ( R ) and (cid:15) d ( R ) (solid blue and dashed black line), i.e. the PESwhen the presence of the second ground state atom isignored. While the trimer PES coincide with the dimer PES for very large and very small separations R , thereis an intermediate regime, here 900 a < R < a ,where one finds substantial deviations. These deviationsresult solely from the term c ( R , R ), which, based onthe structure of (5), can be interpreted as an effectivecoupling of the two dimer states φ ( r ; R ) and φ ( r ; R ).E.g. whenever one has c ( R , R ) = 0, there is no cou-pling and the two PES (cid:15) ± ( R , R , θ ) coincide with thediatomic PES. In this limit one has (cid:15) + ( R , R , θ ) = max( (cid:15) d ( R ) , (cid:15) d ( R )) (cid:15) − ( R , R , θ ) = min( (cid:15) d ( R ) , (cid:15) d ( R )) . (7)In contrast, a non-vanishing coupling c ( R , R ) intro-duces a level repulsion between the two diatomic PES.This is represented by the black arrows in the inset inFig. 1 (b). Importantly, in the lower PES (cid:15) − ( R , R , θ )this effect leads to an energy drop below the dimer PESand can therefore stabilize trimer states. This effectsis also visible in the higher-dimensional cut of the PES (cid:15) − ( R , R , θ ) in Fig. 1 (c). Level repulsion takes placein the region where R < a and R < a , whereit induces a rich oscillatory pattern with many radialminima, that are energetically well below the dimer PES(solid blue line). To provide some visual orientation, thedashed red line marks the curve (cid:15) − ( R , R , θ ) and linksFig. 1 (b) to Fig. 1 (c).The exemplary cuts of the PES in Fig. 1 (b) and (c)demonstrate that the coupling c ( R , R ) has a crucialimpact on the PES. In the following we study as to whichextent this mechanism affects the equilibrium configura-tion of the trimer (stable in R , R and θ ). To thisaim we evaluate (cid:15) − ( R , R , θ ) on a cubic grid and detectall local minima. Surprisingly, this yields a large set ofthe order of thousand equilibrium positions. We analyzethese positions in two steps. Firstly, we classify the equi-libria with respect to the coordinate R . Subsequently,in a second step, we focus on the structure with respectto the remaining coordinates R and θ .The histogram in Fig. 2 (a) presents the abun-dance of minima in dependence of the coordinate R . Due to the indistinguishability of the twoground state atoms, the histogram does not changeif one replaces R by R . As can be seen,the minima are not distributed homogeneously alongthe R axis but cluster around certain separations R c ∈ { , , , , , , , , } a marked by vertical dashed lines in Fig. 2 (b). Com-paring these values to the shape of the dimer PES in2 (b), we find that the positions of the strongest peaksin the histogram coincide with the equilibrium positionsof the dimer PES. Moreover, all peak positions can beidentified with the critical radii found in [45] at whichthe trilobite state φ ( r ; R ) satisfies semiclassical Einstein-Brillouin-Keller quantization conditions. The ellipticallyshaped densities of these states can be characterized bytwo integers ( n , n ) counting the nodes along differentelliptical directions. Exemplary probability densities for(0 ,
29) and (2 ,
27) are depicted in Fig. 2. We interpret
FIG. 2. Analysis of the positions of minima ( R , R , θ ) in the trilobite trimer PES (cid:15) − ( R , R , θ ) given in (5). (a) The histogramdepicts the number of minima as a function of their coordinate R . (b) Peak positions R c in the histogram (dashed lines) occurat radii R where the trilobite wave function φ ( r ; R ) is dominated by orbitals having integer number of nodes ( n , n ) alongtwo different elliptical directions [45]. The insets present exemplary probability densities for (0 ,
29) and (2 , (cid:15) d ( R ) (blue line) coincide with these radii R c . (c) Each dot represents the coordinates R and θ of a minimumin the subset with R ≈ a . The color encodes the potential well depths. The additional gray shading represents theelectronic density of the trilobite dimer | φ ( R ; R ) | associated to the subset with R = 1228 a and characterized by (2 , R c (dashed lines) intersect the peaks of the trilobite density, e.g. at theposition marked by the dashed red square. Bound vibrational states in this minimum are presented in Fig. 3 as an examplecase. this results in the following way: The radial structure oftrimer PES is governed dominantly by the dimer PES.For instance the cut (cid:15) ( R , R , θ ) in Fig. 1 (dashed gray-red line) is on a large scale well approximated by (7)(solid blue and dashed black line). However, the cou-pling c ( R , R ) induces oscillatory deviations and leadsto a substructure which is not captured by (7). Dueto these deviations the peaks in the histogram are notsharp but possess a certain width. Furthermore, whenEinstein-Brillouin-Keller quantization conditions are ful-filled, these deviations are sufficiently strong to induceminima at radii which are not stable in the diatomic sys-tem, e.g. at R = 800 a . The formation of ultralong-range Rydberg trimers with repulsive two-body interac-tion studied in [31] is a special example for this effect.Having analyzed the clustering with respect to the R coordinate we focus, in a second step, on the con-figuration of the remaining coordinates R and θ . Tothis aim we select subsets of minima sharing the same R ≈ R c , i.e. belonging to the same cluster. As an ex-ample Fig. 2 (c) depicts the minima of the cluster with R ≈ a . Every dot represents the coordinates of aminimum, i.e. R and θ , in the plane perpendicular to R . The dot color encodes the well depth ranging from -30 GHz to -15 GHz. In addition to the minima we presentthe trilobite density from (6) | φ ( R ; R ) | (shaded gray)as well as the regions where | R | coincides with clusterradii R c by dashed circles. Minima are expected to sup-port stable trimer states only if their depth is significantlylower than the depth of the dimer PES (cid:15) d (1228 a ) ≈ − l and m as described in (4). Bent equi-librium geometries are therefore absent in polyatomic s -state ULRM with almost isotropic PES but can occuralso in p - and d -state trimers [11, 25, 26, 29, 30] III. VIBRATIONAL STATES
In the following we focus on one of the trimer equi-librium positions in more detail and predict the sup-ported vibrational states. This is the minimum markedby the dashed red line in Fig. 2 (c) with coordinates
FIG. 3. Bound vibrational states localized in the minimum( R , R , θ ) = (1228 a , a , . π ), that is marked by thedashed red line in Fig. 2. (left) Angular cuts through thepotential minimum (dashed blue lines) and reduced angulardensities of vibrational states (solid lines). The offset is ad-justed to their binding energy. (right) Contour plots in the R - R plane of reduced radial densities for the two loweststates. Pictorial representation of the molecular geometryare used to interpret these densities. ( R , R , θ ) = (1228 a , a , . π ). This analysisserves as an example case to illustrate properties ofbound states that occur also in other minima of the PES.After separating the center-of-mass motion, the Hamil-tonian for the relative nuclear motion can be writtenas H rel = H vib + H rovib , where H vib describes pure vi-brational dynamics (depending only on R , R , θ ) and H rovib describes rotational as well as rovibrational dy-namics. The vibrational part reads [29, 46, 47] H vib = 1 m (cid:20) − ∂ ∂R − ∂ ∂R − cos θ ∂∂R ∂∂R (cid:21) − m (cid:18) R + 1 R − cos θR R (cid:19) (cid:18) ∂ ∂θ + cot θ ∂∂θ (cid:19) − m (cid:18) R R − R ∂∂R − R ∂∂R (cid:19) (cid:18) cos θ + sin θ ∂∂θ (cid:19) + (cid:15) − ( R , R , θ ) . (8)This Hamiltonian acts on wave functions χ ( R , R , θ ) be-ing normalized as ´ dR dR dθ sin θ | χ ( R , R , θ ) | = 1.The total angular momentum J of the nuclei is con-served and we focus on J = 0, for which case the rovi-brational part of the Hamiltonian vanishes. For Rb weuse m = 1 . · a.u. and consider only bosonic stateswhich satisfy, according to spin-statistics, χ ( R , R , θ ) = χ ( R , R , θ ).Eigenstates of (8) are obtained numerically in posi-tion space representation on a three-dimensional cubicgrid. For the R and R direction we use equidistant gridpoints and build the derivative operators via finite differ-ence expressions. For the θ degree of freedom we employa discrete variable representation (DVR) approach [48]. In a first step Legendre polynomials P l (cos θ ) are used asbasis functions to construct all kinetic energy operatorsrelated to θ as well as the operator cos( θ ). In a secondstep these operators are transformed into a new basis ofeigenstates of cos( θ ), which can be viewed as a discreteapproximation of position states | θ (cid:105) . This approach isnon-variational but has the advantage that the operatorof the PES (cid:15) − ( R , R , θ ) is diagonal, i.e. there is no needto evaluate overlap integrals. We achieve good conver-gence by using typically a set of 80 gridpoints in eachdimension.Fig. 3 presents reduced densities of the six en-ergetically lowest vibrational states χ ( R , R , θ ) thatlocalize in the potential well at ( R , R , θ ) =(1228 a , a , . π ). Out of all resulting eigenstates,the states shown in Fig. 3 were obtained by selectingthose having the largest integrated probability densityin the potential well considered. The angular densities( ´ dR dR | χ ( R , R , θ ) | ) of these states are depictedin Fig. 3 (left) together with an angular cut of thePES through the minimum (cid:15) − (1228 a , a , θ ). Alldepicted densities are very localized around the min-imum θ . The ground state has an energy of -16.2GHz and a Gaussian-shaped angular density. The corre-sponding radial density ( ´ dθ | χ ( R , R , θ ) | sin θ ) is pre-sented in Fig. 3 (right) and exhibits two peaks. Onenear ( R , R ) = ( R , R ) the other near ( R , R ) =( R , R ). This double-peak structure is a consequenceof the bosonic symmetry and implies that the moleculeis in a superposition of the two ’check-mark’ geometries,that are shown as cartoons in Fig. 3. The next higher vi-brational states appears approximately 100 MHz abovethe ground state. While its angular density resemblesthe density of the ground state, its radial density pos-sesses additional nodes that indicate an excited stretch-ing mode vibration, see pictorial representation in Fig.3. Even higher excited states exhibit combined bendingand stretching excitations. Altogether, the resulting vi-brational states demonstrate that the bent equilibriumgeometries of trilobite trimers are stable enough to sup-port a number of vibrational states and that their vibra-tional spacing is large enough to be resolved in currentexperimental setups [8]. IV. CONCLUSIONS
We presented the rich molecular structure of trilobitetrimers and derived a simple building principle that ex-plains their geometry. Starting from a stable diatomictrilobite , a robust trimer can be formed when the sec-ond ground state atom is placed in a density maximumof the diatomic trilobite wave function. For Rb trimerscorrelated to n = 30, we demonstrated that the result-ing potential minima are deep enough to support a se-ries of localized vibrational states with energy spacingson the order of 100 MHz. The plethora of equilibriumgeometries opens fascinating possibilities to control thearrangement of triatomic molecules but poses also a newchallenge to experiments: States in different moleculargeometries can have comparable vibrational energies andcan therefore hardly be distinguished spectroscopically.Future studies might therefore investigate as to whichextent external fields or optical lattices could serve asadditional selection mechanisms. Furthermore it mightbe interesting to generalize the derived building principle of trilobite trimers to tetramers or even larger clusters. Acknowledgments
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Rep. , 1 (2000). Appendix A: trilobite orbitals
To derive and to interpret the PES of the triatomicmolecule (5) we employed a two-dimensional basis thatconsists of the trilobite orbitals φ ( r ; R ) and φ ( r ; R ),see (4). One can show, that this approach is exact inthe sense, that it yields the same PES (5) and the sameelectronic states as the diagonalization of the interaction V in the larger basis set of hydrogenic states ϕ n lm ( r )with fixed n , 3 ≤ l ≤ n − | m | ≤ l . We proofthis by demonstrating that the interaction matrix V in(1) can be written as V = N (cid:88) j =1 πa [ k ( R j )] | φ j (cid:105) (cid:104) φ j | , (A1)where | φ j (cid:105) denotes the unormalized trilobite state withwave function (cid:104) r | φ j (cid:105) = φ ( r ; R j ) as defined in (2). Thenumber of ground state atoms N is in our case N = 2. From (A1) it becomes evident that all basis states whichare perpendicular to the trilobite orbitals φ ( r ; R ) and φ ( r ; R ) will not interact with the ground state atomsand will, therefore, not contribute to the trimer PES,nor to its molecular states.To proof (A1) we introduce the multiindex α =( n , l, m ) that labels all basis states compactly as ϕ α ( r ).The matrix elements of the delta potential of the j -thperturber read in this basis (cid:104) ϕ α | δ ( r − R j ) | ϕ α (cid:48) (cid:105) = ϕ ∗ α ( R j ) ϕ α (cid:48) ( R j ) . (A2)A particular property of this matrix is that all rows arelinear dependent, e.g. the first row ϕ ∗ ( R j ) [ ϕ ( R j ) , ϕ ( R j ) , ϕ ( R j ) , . . . ] (A3)is proportional to the second row ϕ ∗ ( R j ) [ ϕ ( R j ) , ϕ ( R j ) , ϕ ( R j ) , . . . ] (A4)etc. Consequently, the rank of the matrix representation(A2) is maximally one and there is, hence, maximallyone eigenstate of this matrix with a non-zero eigenvalue.This is the trilobite state (2) | φ j (cid:105) = (cid:88) α ϕ ∗ α ( R j ) | ϕ α (cid:105) (A5)with eigenvalue (cid:88) α | ϕ α ( R j ) | = (cid:104) φ j | φ j (cid:105) . For this reasonone can replace the delta potential (in the here consideredbasis set) by δ ( r − R j ) = | φ j (cid:105) (cid:104) φ j | ,,