A variational approach to Bogoliubov excitations and dynamics of dipolar Bose Einstein condensates

Transcription

1 IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. () 5 (pp) oi:./5-5///5 A variational approach to Bogoliubov excitations an ynamics of ipolar Bose Einstein conensates Manuel Kreibich, Jörg Main an Günter Wunner Institut für Theoretische Physik, Universität Stuttgart, D-55 Stuttgart, Germany Manuel.Kreibich@itp.uni-stuttgart.e Receive 5 October, in final form December Publishe January Online at stacks.iop.org/jphysb//5 Abstract We investigate the stability properties an the ynamics of Bose Einstein conensates with axial symmetry, especially with ipolar long-range interaction, using both simulations on gris an variational calculations. We present an extene variational ansatz which is applicable for axial symmetry an show that this ansatz can reprouce the lowest eigenfrequencies of the Bogoliubov spectrum, an also the corresponing eigenfunctions. Our variational ansatz is capable of escribing the roton instability of pancake-shape ipolar conensates for arbitrary angular momenta. After investigating the linear regime we apply the ansatz to etermine the ynamics an show how the angular collapse is correctly escribe within the variational framework. (Some figures may appear in colour only in the online journal). Introuction As an alternative to the irect numerical solution of the Gross Pitaevskii equation (GPE) an the Bogoliubov e Gennes equations (BDGE), which escribe the groun states an excitations of Bose Einstein conensates (BECs) [], Rau et al [, ] propose a variational approach with couple Gaussians to calculate stationary solutions an the stability of a conensate using the methos of nonlinear ynamics. Specifically, they use the time-epenent variational principle (TDVP) to map the GPE to a nonlinear ynamical system for the variational parameters whose fixe points correspon to groun states of the GPE. The stability can be analyse by means of the linearize equations of motion, i.e. in terms of the eigenvalues of the Jacobian. However, it remaine unclear whether there is a irect relationship between the eigenvalues of the BDGE an those of the Jacobian. In a recent work [] we analyse this problem for spherically symmetric systems an showe that a variational ansatz with couple Gaussians has limitations in that it can only escribe excitations with an angular momentum l. We presente an extene variational ansatz an showe that there is inee a goo agreement between the lowest eigenvalues of the BDGE an the Jacobian for arbitrary angular momenta, with or without aitional long-range interaction (LRI). A variational ansatz with a single Gaussian [5] cannot reprouce the biconcave structure of ipolar conensates reveale in certain parameter regions [, ]. Using the variational ansatz of couple Gaussians, however, it was possible to obtain non-gaussians shapes [,, ] an to reprouce the wave function an the energy of structure groun states. Aitionally, the stability was analyse which reveale that the collapse breaks the axial symmetry within the variational approach. However, the ansatz use in [] is only capable of escribing moes with angular momentum m = an m = (see iscussion in []). The angular collapse with m = symmetry, calculate numerically on a gri in [], is therefore not accessible to this variational ansatz of couple Gaussians. Several extensions of the Gaussian ansatz have been propose in the literature [ ], but they allow for no systematic inclusion of arbitrary angular momenta. It is the purpose of this work to exten the variational ansatz of couple Gaussians in such a way that an instability with an arbitrary projection of the angular momentum on the z axis can be escribe for systems with axial symmetry. We present an extene variational ansatz incluing angular exponentials 5-5//5+$. IOP Publishing Lt Printe in the UK & the USA

2 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al exp(imφ), which are eigenfunctions of the z component of the angular momentum operator. This enables us to escribe moes with arbitrary angular momentum within the variational framework, which will be emonstrate by applying the ansatz to conensates with contact interaction only, an with ipole ipole interaction (DDI). In section we iscuss the stability an excitations of BECs. We first present the metho of numerically solving the BDGE, which we nee for a comparison with the variational metho. The extene variational ansatz is escribe an applie to calculate the eigenfrequencies of elementary excitations. We compare the results with those obtaine from the full-numerical solution. In section we apply the variational ansatz to calculate the time evolution of a ipolar BEC an emonstrate that the ansatz can escribe the angular collapse. In section we raw conclusions an give an outlook on future work.. Stability an excitations The time epenence of a BEC is escribe in the mean-fiel approximation by the time-epenent GPE [] i ψ [ (r, t) = t M + M ω ( + λ z ) ] +N int (r, t) ψ(r, t), () where N is the particle number, int (r, t) = r V int (r r ) ψ(r, t) () the mean fiel prouce by the inter-particle interaction, ψ(r, t) the wave function of the conensate normalize to unity an M the mass of the atom species. The harmonic external trap is given by the trapping frequencies ω an ω z in the an z irection, an can be characterize by the trap aspect ratio λ = ω z /ω. We measure all lengths in units of the harmonic oscillator length a ho /Mω, all frequencies in units of ω, an the time in units of /ω. A stationary solution of equation () has the time epenence ψ(r, t) = ψ(r) exp( iμt/ ), where μ is the chemical potential of the conensate. The interaction potential is given by [] V int (r r ) = πa M δ(r r ) + μ μ (z z ) m (r r ). () π r r The quantity a is the s-wave scattering length an μ m the magnetic moment of the atom species an is, e.g., μ m = μ B (μ B the Bohr magneton) for 5 Cr []. The ipoles are assume to be aligne along the z axis by an external magnetic fiel. The interactions can be characterize by the imensionless parameters Na/a ho an D N μ μ π m a ho [5]. In this paper, we investigate BECs with an without LRI. For the conensate with DDI we use values of D = an λ =, since for these parameters structure groun states appear [, ]. For the BEC without LRI we use a trap aspect ratio of λ =, which was use in experiments []. The scattering length can be tune in a wie range via Feshbach resonances []. Thus, we use this quantity as a free parameter in our calculations. M.. Full-numerical treatment Besies the variational approach we also perform gri calculations to irectly solve the GPE an BDGE, which allows us to compare both methos. Before solving the BDGE, one first nees to fin the groun state of a BEC. In this work, we use the imaginary time evolution (ITE) for this task. We evolve an initial wave function on a gri in imaginary time τ efine by τ it, an as time evolves the wave function converges to the groun state. The time evolution on the gri is one via the split operator metho [], in which the time evolution operator in imaginary time is given by e Ĥ τ/ = e ˆT τ/ e ˆV τ/ e ˆT τ/ + O( τ ), () where ˆT an ˆV represent the kinetic an the potential terms of the mean-fiel Hamiltonian, respectively. The action of the kinetic part is trivial in Fourier space, whereas the action of the potential part is trivial in real space. Since we are ealing with a cylinrically symmetric system, we can use the Fourier Hankel algorithm of [5] to compute the necessary Fourier transforms for a time step an the ipolar integrals via the convolution theorem. To erive the BDGE, one starts from the usual ansatz for the perturbation of a stationary state ψ with chemical potential μ [] ψ(r, t) = [ψ (r) + λ(u(r) e iωt + v (r) e iω t )]e iμt/, (5) where ω is the frequency an λ the amplitue of the perturbation ( λ ). After inserting this ansatz into the time-epenent GPE (), while neglecting terms of secon orer in λ, an collecting terms evolving in time with exp( iωt) an exp(iωt), respectively, one obtains the BDGE [ ] ωu(r) = H μ + N r V int (r r ) ψ (r ) u(r) + N r V int (r r )ψ (r )u(r )ψ (r) + N r V int (r r )ψ (r )v(r )ψ (r), (a) [ ] ωv(r) = H μ + N r V int (r r ) ψ (r ) v(r) + N r V int (r r )ψ (r )u(r )ψ (r) + N r V int (r r )ψ (r )v(r )ψ (r), (b) where H = M + M ω ( + λ z ). Due to the cylinrical symmetry, we can make an ansatz for the solutions u(r) = e imφ u(, z), v(r) = e imφ v(,z), () where m is the usual quantum number of the projection of the angular momentum operator on the z axis. Since the Laplacian in cylinrical coorinates applie to the ansatz () yiels ) e imφ u(, z) = (,z m e imφ u(, z), () which only epens on m, there is a egeneracy of the eigenmoes for m >. Thus, we can restrict ourselves to

3 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al solutions with m. Another symmetry of the system is the reflection at the z = plane (z z). Therefore, the solutions can be escribe by the parity π z =± uner this reflection. We enote the set of quantum numbers with m π z. There are always solutions of the BDGE for ω = an the trapping frequencies, which represent the gauge moe an the centre-of-mass oscillations [, ]. We solve the linear, nonlocal BDGE () by calculating the matrix elements of the right-han sie in coorinate representation by means of the Fourier Hankel algorithm [5]. The lowest eigenvalues of the resulting high-imensional matrix are then calculate using the Arnoli iteration [], implemente in the Fortran ARPACK routines []. The solutions of the BDGE () yiel the frequencies an the shape of elementary oscillations of the conensates. But it may happen that the Bogoliubov spectrum of the groun state of a ipolar BEC contains imaginary frequencies [], which correspon to ynamical instabilities. A small perturbation of the groun state then leas to an exponential ecay. Note that the excitation spectrum may be influence by quantum fluctuations not inclue in our approach, especially for strong DDI [, ]. We calculate the groun states an Bogoliubov spectra for a BEC with an attractive short-range interaction (SRI) only (D = ) an a trap aspect ratio of λ =, an for a ipolar BEC with λ = an D =, for which it was shown that the structure groun state changes its stability in a pitchfork bifurcation [] at the scattering length Na/a ho.5. The structure groun states, where the maximum of particle ensity lies away from the centre, appear in isolate regions in parameter space with λ =,, 5,... []. For our choice of parameters, λ = an D =, the groun state assumes a biconcave structure for the scattering lengths.5 Na/a ho.. For both systems, groun states only exist for scattering lengths larger than the critical scattering length, a a crit.for the conensate with SRI, the critical scattering length is given by Na crit /a ho.5. BECs at small negative scattering lengths may exist, since the kinetic energy compensates the attractive interaction an stabilizes the conensate. The critical scattering length was probe experimentally in a conensate of 5 Rb, an significant eviations from the results of the GPE were measure [], which may be explaine by taking into account higher-orer nonlinear effects []. The critical scattering length of a ipolar BEC consierably epens on the trap geometry, in our case (λ =, D = ) its value is given by Na crit /a ho.5. Goo agreement between experiment an the results of the GPE was foun [5]. Figure shows the Bogoliubov spectra. Since both systems have alreay been consiere in the literature (see, e.g., [] for the BEC with SRI, an [5] for the ipolar BEC), we only give a short review. The spectrum of the BEC without LRI (figure (a)) can be unerstoo by taking the limit In [] a Gaussian ansatz with N G = was use. The scattering length, where the bifurcation takes place, however, converges slowly with an increasing number of Gaussians an may be ifferent in our calculations. (a) (b) m=, real m=, real m=, real m=, real 5 m=, real m=, real 5 Figure. Bogoliubov spectra of the groun states for (a) a BEC with short-range interaction (λ =, D = ) an (b) a ipolar BEC (λ =, D = ) as a function of the scattering length Na/a ho (shown are the real parts of the frequencies) with even (soli) an o (ashe) moes. Both spectra feature the lowest m = moe, whose frequency goes to zero when the scattering length reaches the critical value a crit, an the centre-of-mass oscillations at the trapping frequencies. Unique for the ipolar BEC are the lowest excitations for higher angular momenta m >, which turn unstable an mark a local collapse of the conensate. a, which yiels the exact eigenvalues of the cylinrically symmetric harmonic oscillator ω n,n z,m = n + m +λn z, () ω where n,z N are the quantum numbers for excitations along the an z irection, respectively. When ecreasing the scattering length towars the critical value, below which no stationary solutions exist anymore, the frequencies vary slightly, which means that in this regime the ynamics is ominate by the external trap, an the interaction acts as a quasi perturbation. Just slightly above the critical point, the lowest moe with m = rops to zero, which marks the global collapse of the conensate. The centre-of-mass oscillations, which constantly lie at the trapping frequencies, are represente by the lowest o m = moe (ω/ω = ) an the lowest even m = moe (ω/ω = ).

4 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al Re Im m=, real m=, real m=, real m=5, real m=, imag. m=, imag. m=, imag. m=, imag. m=5, imag Figure. The lowest moes of the Bogoliubov spectrum of the ipolar BEC from figure (b). The real an also imaginary parts of the frequencies are shown. In our resolution of the scattering length, the moes with m = anm = turn unstable simultaneously, but the latter has a higher imaginary frequency below the critical scattering length, which means that this moe ominates the collapse in the vicinity of the stationary solution. The Bogoliubov spectrum of the ipolar conensate (figure (b)) inicates a stable conensate when the scattering length is far from the critical point. However, for ecreasing scattering length, not only the moe with m = but also frequencies for higher angular momenta ecrease an rop to zero. At these points, those moes become unstable, as the eigenfrequencies are imaginary. Figure shows the real an imaginary parts of the lowest eigenfrequencies for the angular momenta m =,...,5. The moes with m = an m = are the first ones which turn unstable, an thus mark the local collapse of the conensate. This means, that collapse occurs via local ensity fluctuations in contrast to a global collapse, where the conensate collapse to the trap centre []. In section we iscuss the ynamics of such a local collapse, with the result that an extene variational ansatz is necessary to escribe the collapse. This is a unique feature of ipolar BECs an is relate to the biconcave structure of the groun state occurring for specific trapping geometries [, ]. These instabilities are referre to as rotons, or, more specifically, as iscrete angular rotons, since these excitations assume a shape with azimuthal noal surfaces []. Worth mentioning also is the avoie crossing in the lowest moes for the angular momenta m =,,, whereas for m = there is a regular crossing since ω = ω is an exact solution for every scattering length, which oes not allow for an avoie crossing... Variational approach The gri calculations are very accurate, if gri size an resolution are chosen carefully, but computationally expensive. The variational metho provies an alternative in that the computation time reuces significantly compare to gri calculations. With this motivation, Rau et al [,, ] use the variational ansatz N G ψ = e i(ak x x +A k y y +A k z z +γ k ) () k= to calculate stationary solutions an excitations of ipolar conensates. Here, N G is the number of Gaussians, A k x, Ak y an A k z escribe the withs an γ k the amplitue of each Gaussian. The quantities ( A k x, Ak y, Ak z,γk) are the variational parameters of this ansatz. In a recent article [] weshowe that the variational ansatz with couple Gaussians can only escribe moes with an angular momentum of l, m. For that reason we propose an extene variational ansatz with couple Gaussians an spherical harmonics, which can escribe moes with arbitrary angular momenta in spherically symmetric systems. We foun a goo agreement of the lowest eigenmoes with those obtaine by gri calculations. For the systems with axial symmetry consiere here, this ansatz is not applicable, since the angular momentum is no longer a goo quantum number an therefore the angular part of the excitations is not escribe by spherical harmonics. However, the z component of the angular momentum, represente by the quantum number m an the eigenfunctions exp(imφ), is still conserve. Using this knowlege we construct an extene variational ansatz, which is specific to cylinrically symmetric systems, an which is given by N G ( ψ = + m,p k m z p e )e imφ Ak A k z z p k z z γ k. k= m p=, () The complex variational parameters A k an Ak z escribe the withinthe an z irection, p k z is the isplacement along the z axis, an γ k escribes the amplitue of each Gaussian. With these parameters alone, the wave function oes not epen on the angular coorinate φ, thus the parameters ( A k, Ak z, pk z,γk) are responsible only for any m = excitation. To account for higher angular momenta an arbitrary parity π z, we inclue the sums over m an p in the brackets (π z =+belongs to p =, an π z = top = ). The amplitue of each of these excitations is given by the variational parameters m,p k. To erive the equations of motion for the variational parameters, we follow the proceure in [, ] an apply the Dirac Frankel McLachlan TDVP [, ]. It states that the norm of the ifference between the left- an right-han sies of the Schröinger or GPE I = iχ(t) Ĥψ(t) () must be minimize for a given variational ansatz ψ = ψ(z) with respect to the variational parameters z. The variation of χ, while ψ is fixe, an the ientification of χ with ψ leas to the necessary conition [] Kż = ih, () where the matrix K an the vector h are efine by ψ K ij = ψ ψ z, h i =. i z j z Ĥψ () i The remaining task is to evaluate all integrals appearing in equation (). Technical etails are presente in appenix A.

5 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al Since the groun state in the GPE correspons to a fixe point of the equations of motion (), we can fin the groun state of a system by a nonlinear root search, for which we require { iμ for zi γ ż i = k, (5) else, where μ is the chemical potential. Stability an excitations in the framework of the TDVP are given by the linearize equations of motion, for which one first nees to rewrite the complex equations of motion () into real ones by efining the new vector of variational parameters z as z = (Re z, Im z). The time epenence of a small perturbation δ z is then given by [] δ z i = z i δ z j J ij δ z j, () z j j z= z j with the Jacobian J evaluate at the fixe point z. These linearize equations of motion are as usual solve by a harmonic time epenence of the perturbation δ z(t) = e iωt δ z(), () which leas to the eigenvalue problem iωδ z() = Jδ z(). () A real ω belongs to an oscillatory motion, as for real eigenvalues ω of the BDGE. The symmetry of the system, which leas to the quantum numbers m an π z in the Bogoliubov spectrum, gives rise to a block structure of the Jacobian which is further explaine in appenix B. The eigenvalues of the Jacobian an the BDGE can be irectly compare. But is there also a relationship between the eigenvectors of the Jacobian δ z an the eigenfunctions of the BDGE u an v? For a given eigenvector δ z, a wave function can be constructe by taking the ifference of the excitation with variational parameters z + δ z an the wave function of the stationary solution with the parameters z, δψ( z,δ z) ψ( z + δ z) ψ( z ). () Since we are ealing with an excitation in the linear regime, one can Taylor expan the first summan in equation () to first orer in δ z. This yiels the function [] δψ = δ z ψ, () z z= z which is the scalar prouct of the eigenvector an the graient of the wave function with respect to the variational parameters evaluate at the stationary solution. Thus, every eigenvector δ z belongs uniquely to a function δψ. As the Jacobian J is real an a stable moe is represente by an imaginary eigenvalue iω, for every eigenvalue iω with ω > thereisalsoaneigenvalue iω, therefore we can construct corresponing functions δψ + an δψ which evolve in time as exp(iωt) an exp( iωt), respectively. They can be combine to the perturbation δψ = δψ e iωt + δψ + e iωt. () Comparing this with the Bogoliubov ansatz from equation (5) leas to the relationship u δψ, v δψ +, () which links the eigenvectors of the Jacobian with the eigenfunctions of the BDGE. (a) (b) m=, real m=, real m=, real m=, real m=, real m=, real Figure. Comparison of the Bogoliubov spectrum for a BEC with SRI from figure (a) with the eigenvalues of the Jacobian obtaine from the variational ansatz for even (a) an o (b) parity. We fin very goo agreement for the lowest two moes of each angular momentum an parity, only close to the critical scattering length eviations of the secon lowest moes for m > can be seen, which become larger for increasing angular momentum... Results for the BEC with scattering interaction We now apply the variational ansatz () to a conensate with an attractive SRI. Figure shows the results, where N G = 5 couple Gaussians have been use (soli lines), compare to the full-numerical calculations (otte lines). For the sake of clarity we istinguish between even moes in figure (a) an o moes in figure (b). One recognizes that the lowest moe of each angular momentum an parity can be perfectly reprouce by the variational ansatz. Interestingly, as with our previous ansatz for spherically symmetric systems [], the frequencies of the centre-of-mass oscillations (lowest o m = atω = ω,lowestevenm = atω = ω ) agree, within numerical accuracy, with an arbitrary number of couple Gaussians, even with N G =. For the secon lowest moes we also fin very goo agreement as long as the scattering length is not close to the critical value. For the case m =, π z = even more eigenvalues of the Jacobian are close to the numerical results. To emonstrate the power of our extene ansatz, we calculate eigenmoes for angular momenta up to m = 5

6 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al 5 numerical m π z Figure. Comparison of both spectra as in figure, but here for a fixe scattering length of Na/a ho =. an angular momenta up to m =. The variational ansatz also works for these moes an can escribe the lowest moes of the Bogoliubov spectrum, therefore we are able to establish a connection between both methos. (see figure ) for a fixe scattering length Na/a ho =.. Even for these angular momenta the lowest moes agree well. For the secon excitations an lower angular momenta we also fin goo agreement, however, the eviations of the variational an numerical results become larger for increasing angular momenta. Nevertheless, there is an obvious relationship between the eigenfrequencies of the Bogoliubov spectrum an the eigenvalues of the Jacobian, where our extene variational ansatz () specific to cylinrically symmetric systems has been use. Thus, the variational ansatz is a vali alternative to simulations on gris if one is intereste only in the lowest moes... Results for the ipolar BEC We now want to inclue the DDI in the calculations an especially verify if the extene variational ansatz can escribe the stability change for higher angular momenta m >. We first investigate the eigenmoes in the stable regime with a scattering length Na/a ho 5. Figure 5 shows the comparison of both spectra for the angular momenta m =,...,. With six couple Gaussians, the lowest moes of each angular momentum an parity (soli lines) are converge to their corresponing numerical frequencies (otte lines). As in the case of the BEC without LRI, the centre-of-mass oscillations agree within numerical accuracy. For the moes with even parity, there is goo agreement even for the secon lowest moes, where the ifferences become larger for rising scattering length an angular momentum. In the case of o parity, the eviations for the secon lowest moes become quite large for m, when the scattering length is large. We have also investigate eigenmoes with an angular momentum up to m = in the stable regime. Figure shows the results for a fixe scattering length Na/a ho =. The variational ansatz can reprouce the lowest moes even for these angular momenta an, since the scattering length is not (a) (b) 5 5 m=, real Gaussians m=, real Gaussians m=, real Gaussians m=, real Gaussians m=, real Gaussians Gaussians m=, real Gaussians Gaussians Figure 5. Comparison of the Bogoliubov spectrum for a ipolar BEC from figure (b) with the eigenvalues of the Jacobian obtaine from the variational ansatz. The moes with even (a) an o (b) parity in the stable regime with a scattering length Na/a ho 5are shown. Whereas the two lowest moes agree well for even parity, for o parity the secon lowest moes iffer quite clearly, especially for higher angular momenta m. too large, also the secon lowest moes. For the case m =, π z =, we fin the lowest three moes converge, an a goo agreement of the fourth lowest moe. These investigations show that the variational ansatz () can also reprouce the full-numerical Bogoliubov spectrum, if only the lowest moes are compare. We now investigate the interesting case of a small negative scattering length, where the Bogoliubov spectrum shows unstable roton moes for ifferent angular momenta (see figure ). We focus here on the case m =, which cannot be escribe by the ansatz of couple Gaussians equation (), hence the extene variational ansatz is necessary. Figure shows the lowest moes with m = obtaine using ifferent numbers of Gaussians (soli an ashe lines). For comparison the full-numerical results are also plotte (otte line). We fin that with an increasing number of couple Gaussians the accuracy of the variational calculations increases graually. Other angular momenta show a similar behaviour, the convergence is in general better for lower angular momenta.

7 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al 5 numerical Gaussians m π z Figure. Comparison of both spectra as in figure 5, but here for a fixe scattering length of Na/a ho =. For the lowest moe of each angular momentum an parity, there is a very goo agreement. For the secon lowest moes we fin ifferences, which become larger for increasing angular momenta. The corresponence is slightly better for even than for o moes. As in the case without LRI there is a link between the full-numerical Bogoliubov spectrum an the eigenvalues of the Jacobian resulting from the variational ansatz. Im 5 m= m= m= m= m= m=5 E mf / NE ho Figure. Imaginary frequencies of the excite state with N G = 5 Gaussians. At the critical point the excite state shares the same instabilities as the groun state with aitionally an m = instability, which is typical for the collectively excite state. The inset shows the tangent bifurcation in the mean fiel energy E mf /NE ho (N being the particle number) for the groun (soli) an excite (ashe) state, where E ho = ω is the energy quantum of the harmonic oscillator. Re Im Gaussians Gaussians Gaussians m=, imag. Gaussians Gaussians Gaussians Figure. The lowest moe with m = which turns unstable when the scattering length is ecrease. Compare are the full-numerical frequencies (otte line) with the results from the variational ansatz with a ifferent number of Gaussians (soli an ashe lines). The accuracy of the variational calculations increases graually with the number of couple Gaussians. Thus we can conclue that the variational ansatz () can escribe, though not quantitatively perfectly, the ynamical instabilities of the Bogoliubov spectrum for angular momenta m >. It is known that at the critical scattering length below which no stationary solution exists anymore, an unstable collectively excite state emerges in a tangent bifurcation [] in aition to the groun state (see the inset in figure ). For conensates without DDI, this excite state is unstable with respect to an m = excitation []. The variational approach offers the possibility to investigate this ipolar BEC state, which is not accessible to the full-numerical ITE metho, an its collapse mechanism. Figure shows the imaginary frequencies of the excite state obtaine with N G = 5 Gaussians. As both states merge at the critical scattering length, the same hols for the eigenfrequencies, an the excite state shares the instabilities of the groun state at the critical scattering length, except the aitional m = instability. In contrast to the case with SRI only or with monopolar LRI [], the excite state of the ipolar BEC for the trapping frequencies is unlikely to collapse with m = symmetry ue to the aitional instabilities for m >, which have a higher imaginary frequency an thus a larger collapse rate. The m = 5 excitation in figure changes its stability with increasing scattering length, in a similar way as the unstable moes of the groun state (see figure ). Note, however, that this oes not mean a global change of the stability ue to the other unstable moes. The excite state stays unstable for all scattering lengths consiere in our calculations..5. Shape of Bogoliubov excitations After linking the eigenvalues of the BDGE an the Jacobian of the time-epenent variational approach, we will check if there is also a irect corresponence between the eigenfunctions an eigenvectors of these two approaches. In section. we iscusse how to construct the functions δψ an δψ + from the eigenvectors an how they are relate to the functions u an v. For a better comparison of both methos, we use the usual normalization of the Bogoliubov functions r [ u v ] = [], an assume the same for the variational metho, r [ δψ δψ + ] =. We point out that in calculating δψ ± there is no ambiguity, except for a global phase, which we choose in such a way that the functions are real. Figure shows the comparison of both methos for

8 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al u, δ ψ u, δ ψ v, δ ψ + v, δ ψ m=, first excitation m=, first excitation 5 / a ho m=, first excitation m=, first excitation. 5 / a ho m=, secon excitation m=, secon excitation 5 / a ho m=, secon excitation m=, secon excitation 5 / a ho Figure. Raial epenence for the functions u, v an δψ, δψ + (re lines), respectively for ifferent values of z ( (blue crosses),.a ho (yellow squares),.a ho (green circles)), where N G = Gaussians have been use (see the text for the normalization). The first two excitations for the angular momenta m =, anafixe scattering length of Na/a ho = are shown. All functions are chosen to be real. There is very goo agreement, only for the secon m = excitation are there small eviations. some moes. The first two functions u with m = have one an two raial noes, respectively, because the lowest moe with m = represents the gauge moe, which is no physical excitation. For other angular momenta, the nth excitation in the irection has n raial noes. The asymptotic behaviour of all functions in figure for is m. The noal structure for the functions v is ifferent. All v for m = have one noe, an for the other angular momenta zero noes. For higher excitation numbers an angular momenta, the amplitue of v becomes smaller an smaller, inicating that in these limits the coupling between u an v in the BDGE can be neglecte, which is in accorance with previous calculations in the literature [,, 5]. For the secon excitation with m = we fin small eviations, which is to be expecte, since there are also eviations in the eigenfrequencies (cf figure ). By comparing more eigenmoes we can conclue: the better the agreement of the eigenfunctions, the better the accorance of the frequencies also. For higher moes, where there is no quantitative agreement between the eigenfrequencies, only the asymptotic behaviour an the noal structure can be escribe by the variational ansatz, even though there is at least qualitative agreement in those aspects. The comparison of the eigenfunctions of the BDGE with those obtaine from the eigenvectors of the Jacobian reveals very goo agreement. This is remarkable, since both functions originate from quite ifferent approaches of stability. The extene variational ansatz () cannot only quantitatively escribe the frequencies of the lowest moes, but also the shape of the oscillations, even for angular momenta, which were not accessible by the variational approach without the extension.. Dynamics The extene variational ansatz () was propose to fin a irect relationship between the eigenvalues of the Jacobian an the eigenfrequencies of the BDGE. We have erive the equations of motion using the TDVP to calculate the Jacobian. The equations of motion () are not only vali in the vicinity of a fixe point, but for all variational parameters for which the integrals in the equations of motion converge. Wilson et al [] simulate the ynamical evolution of a collapse for a ipolar conensate with biconcave structure by reucing the scattering length below the critical value, which yiels a symmetry breaking collapse (angular collapse) with m = symmetry. Rau et al [] simulate this scenario within the variational framework with an ansatz of couple Gaussians. However, as mentione in section., this ansatz cannot escribe wave functions that exhibit the m = symmetry. Motivate by these papers we will now integrate the equations of motion numerically with appropriate initial conitions to verify whether or not the extene variational ansatz is also applicable in calculating the ynamics of ipolar conensates an especially to escribe the angular collapse for arbitrary angular momenta, which is cause by the roton instability (cf sections. an.). To o so we only aress the collapse of groun states with biconcave structure. As a first example, we consier the system with N G = Gaussians an calculate the ynamics of an inuce collapse by lowering the scattering length below the critical value. As an initial state for t =, we take the stable groun state for a scattering length of Na/a ho = an a a small perturbation for a specific angular momentum by changing the variational parameter m,p= k= to a small value larger than zero, e.g. m,p= k= = 5, which is just a small perturbation of the groun state. We let the conensate evolve until tω =., after which the scattering length is linearly rampe own until tω =. to a value of Na/a ho =., which is below the critical scattering length. The conensate then evolves at a fixe scattering length, until it collapses to a point, from where on no further integration is possible. We calculate this scenario for an initial m = an m = excitation, which cannot be escribe by the variational ansatz without the extension an thus is an avance as compare to previous calculations of a collapsing conensate within the variational framework [].

9 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al (a) Figure. Moulus square ψ of the wave function of the collapsing ipolar conensate in a cut through the (z = )-plane. Bright (yellow) marks regions of high particle ensity, while ark (black) stans for low ensities. For the calculations N G = Gaussians were use. Figure shows the resulting ensity of the collapsing conensate for both cases. For an initial m = excitation, the collapsing clou exhibits the clear shape with an m = symmetry, the angular epenence is proportional to sin(φ). This result shows that the extene variational ansatz is not only able to escribe the instability in the vicinity of the groun state, but also the nonlinear ynamics given by the equations of motion (). If the initial state is perturbe by an m = excitation, the resulting ensity istribution shows a superposition of several angular momenta. There are altogether four ensity peaks, of which two are significantly larger than the other two. The appearance of four peaks correspons to an m = symmetry, but the istribution of the particle ensity clearly belongs to m =. Of the two large peaks, one is slightly higher than the other one which inicates an m = symmetry. The calculation shows that the ifferent angular momenta o not evolve freely, but instea are couple an influence each other. This is ue to the nonlinearity of the GPE () which carries over to the equations of motion for the variational parameters. There are many excitations one can a to the groun state as a perturbation. Above, we ae specific eigenmoes to investigate the behaviour of the extene variational ansatz when consiering the collapse ynamics. We now want to utilize a more physically motivate initial state. In [] the (b) authors ae moes with weights etermine by the Bose Einstein istribution, which leas to ψ (, z) + n,m nn,m N eπiα n,m [u n,m (, z) e imφ +vn,m (, z) e imφ ], () where ψ is the groun state. The weight of each excitation, where m is the angular momentum an n summarizes the other quantum numbers, is given by n n,m = (e ω n,m/k B T ). () Here, ω n,m is the energy of each excitation, k B the Boltzmann constant an T the temperature. The quantities α n,m in equation () etermine the initial phase of each excitation an are ranom numbers between an. The overall factor / N, with N the particle number, is necessary, since the wave function ψ is normalize to unity instea of N. We can now make use of the relationship between the eigenvectors of the Jacobian an the eigenfunctions of the BDGE (section.5) to prepare an initial state similar to that given by equation () within the variational framework, where instea of the functions, the eigenvectors of the variational parameters have to be ae to the groun state vector. For the following calculations we simulate a conensate consisting of N = 5 Cr atoms at a temperature of T = nk. We start with an initial wave function accoring to equation () an a scattering length of Na/a ho =, which is rampe own to a value of Na/a ho = over a time of tω =.. We performe several calculations with this scenario, for which the only ifferences are the initial phases α n,m.as the scattering length ecreases the particle ensity establishes its biconcave structure, which is a typical feature of ipolar BECs [ ]. If the scattering length further ecreases, the critical value is reache an the atomic clou starts to collapse to the region with the highest ensity, which is on a ring aroun the centre. Due to the initial excitations with angular momentum m >, which are now unstable, the symmetry is broken an several ensity peaks emerge. Figure shows the ensity istribution of four ifferent calculations shortly before collapse. Each conensate has three peaks locate almost equiistantly on a ring. But the istribution along the ring iffers: in figures (a) an (b) there are two large peaks (a) (b) (c) () Figure. Particle ensities of the collapsing clou at tω =. with high (low) ensity represente by bright (ark) regions. All calculations were performe with N G =. The ifferent ranom phases use to construct the initial state (see text) are the only ifferences in the four calculations. There are three ensity peaks each, but the overall rotation an the ensity istribution along the peaks are ifferent.

10 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al an a smaller one, while in figures (c) an () it is the other way roun. Aitionally the initial phases etermine the overall rotation of the collapse conensate in the (z = )- plane. We can conclue that both, the m = an m = moes, are responsible for the shape of the collapse clou, which correspons to the numerical solution of the BDGE (see figure ), in which both moes turn unstable almost simultaneously. A question, which naturally arises, is how the number of Gaussians influences the ynamics calculation. We performe similar calculations with N G = an N G = Gaussians. With four Gaussians we fin a ifferent behaviour in that a collapse with m = symmetry is favoure. However, with six Gaussians almost the same results as in the case with five Gaussians are obtaine. The only ifference is the time taken before the conensate collapses, which is shorter with six Gaussians. This can be explaine by the fact that the critical scattering length is shifte to a higher value an thus the critical point is reache earlier uring the simulation. The eigenfrequencies with our Gaussian ansatz o not fully agree with the full-numerical results (cf section.), however, we expect the variational results to not be completely converge with five or six Gaussians.. Conclusion an outlook We presente an extene variational ansatz, which is specific to cylinrically symmetric systems an which can escribe arbitrary angular momenta, an erive the equations of motion for the variational parameters using the TDVP. The eigenfrequencies of two ifferent systems (without an with DDI) were calculate using the variational ansatz an compare with the irect solutions of the BDGE. We foun a goo agreement of the lowest-lying moes an coul confirm the irect relationship of both methos. This extene our investigations of spherically symmetric systems [] an we coul show that this agreement is of generic type, inepenent of the symmetry an the interactions. The roton instability of a pancake-shape ipolar BEC was investigate using the extene ansatz. We coul fin instabilities for arbitrary values of the angular momenta, which is a great progress in comparison with the variational ansatz with couple Gaussians without the extension. With more an more Gaussians we obtaine graually more accurate results within the variational framework. Thus, our variational metho is a vali approximation to gri calculations. We further constructe functions of the eigenvectors of the Jacobian an foun that these functions agree very well with the solutions of the BDGE, as long as the corresponing eigenvalues agree. There is not only a irect link between the eigenfrequencies, but also between the eigenvectors an the eigenfunctions. After consiering the ynamics in the linear regime we irectly integrate the nonlinear equations of motion an showe that our variational ansatz, assuming that the initial conitions an the scenario of the time-epenent scattering length are appropriately chosen, is capable of escribing the angular collapse with arbitrary angular momenta. We use the relationship between the eigenvectors of the Jacobian an the eigenfunctions of the BDGE to construct an initial state which correspons to a conensate in thermal equilibrium within the mean fiel approximation, an simulate a realistic experiment of the angular collapse. We foun that the ranom initial phase istribution strongly influences the shape of the collapse atomic clou, which makes the resulting shape of an experiment irreproucible. Altogether it was shown that the extene variational ansatz is a significant progress compare to couple Gaussians, an as yet unaccessible results coul be obtaine. Our ansatz is a vali alternative to calculate eigenfrequencies an shapes of elementary excitations, if the lowest eigenmoes of stable conensates are consiere. The roton instabilities can be obtaine, but we coul not fin goo quantitative agreement, leaving our ansatz an approximation. In future applications of the extene variational ansatz it woul be possible to inclue the particle loss in the ynamics calculation, as the losses are necessary to accurately simulate a collapse scenario of a ipolar BEC [, ]. Furthermore, our ansatz is applicable to ipolar BECs in a one-imensional optical lattice [ ] to calculate the groun state, stability, excitations an the ynamics. The irect relationship between the eigenvectors of the Jacobian an the eigenfunctions of the BDGE allows several applications of the variational approach, e.g. to escribe a BEC at finite temperature, where a selfconsistent solution of the GPE an BDGE is necessary [5]. Acknowlegments This work was supporte by Deutsche Forschungsgemeinschaft. Appenix A. Calculation of the integrals for the variational ansatz This appenix is eicate to the integrals necessary for the matrix K an the vector h, which are efine by ( ) ψ K ij = ψ ψ z = ψ r (r) (r), (A.a) i z j z i z j ( ) ψ h i = ψ z Ĥψ = r (r) Ĥψ(r), (A.b) i z i to set up the equations of motion () of the variational ansatz () via the time-epenent variational principle. We write the variational ansatz in the form N G ψ = m,p k m z p e imφ e Ak A k z z p k z z γ k, (A.) k= m p=, where, k an k, are no variational parameters, but constants. We further introuce the abbreviations ψ z i z j ψ ψ z, z i Xψ i z j z Xψ (A.) i for the matrix elements, where X is an operator of the mean fiel Hamiltonian H = T + V ext + s + with the kinetic part, the external potential an the scattering an ipolar mean fiel potentials.

11 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al We o not give a etaile way on how to calculate all necessary integrals, nor o we present all results. We rather want to sketch the calculations an give results of some examples. For all etails we refer to the master thesis of one of the authors []. A.. Integrals of the K-matrix Before giving the results, we efine the abbreviations A kl Ak + (Al ), A kl z A k z + (Al z ), p kl z p k z + (pl z ), γ kl γ k + (γ l ) (A.) an the basic integrals I kl (m, n) m e Akl zz n e Akl z z p kl z z kl γ e kl γ Ɣ(m) = e ( ) A kl m Iz kl (n). (A.5) The integral can be ientifie with the gamma function by a simple substitution. The z integral is a Gaussian integral for n =, higher orers can be obtaine by the recursion formula I kl z (n + ) = p kl z I kl z (n). (A.) The integrals of the matrix K then yiel l m,p k m,p = πδm m I kl ( m +, p + p ), (A.a) l m,p A k = π p l A k m,p z = π I kl ( m +, p + p + ), (A.c) I kl ( m +, p + p ), (A.b) p l m,p p k z = π I kl ( m +, p + p + ), (A.) p l γ k m,p = π I kl ( m +, p + p ). p (A.e) The other integrals can similarly be expresse by equation (A.5). A.. Integrals of the kinetic term For the integrals of the kinetic term, the action of the Laplacian on the variational ansatz is necessary. This is most easily evaluate in cylinrical coorinates. The integrals can then be expresse by equation (A.5). For the parameters, e.g., one obtains l T m,p ψ / ω = π m k {[,p ( m +)A k k p +(p + )A k z ( p k ) ] z I kl ( m +, p + p ) + [ p p k z] I kl ( m +, p ) [ p k z z] Ak I kl ( m +, p + p + ) [ ( A k ] z) I kl ( m +, p + p + ) [ ( A k ] ) I kl ( m +, p + p ) }. (A.) A.. Integrals of the trapping potential The trapping potential is of the form V ext = M ω ( + λ z ). Thus, the corresponing integrals are easy to evaluate. As an example we obtain l Vext m,p ψ / ω = π k p k m,p (I kl ( m +, p + p ) + λ I kl ( m +, p + p + )). (A.) A.. Integrals of the scattering interaction The integrals for the interaction terms require some more attention. First an expression for the moulus square wave function ψ is neee. This enters the integrals for the interaction potentials. The scattering interaction is easier to calculate, since this kin of interaction is local, in contrast to the DDI. We further nee new abbreviations, which epen on four inices an which are efine by A ijkl A ij + Akl, γ ijkl γ ij + γ kl. Aijkl z A ij z + Akl z, pijkl z p ij z + pkl z, (A.) The new basic integral, I ijkl (m, n), can be obtaine from the former one, equation (A.5), by substituting all inices kl with ijkl. The scattering integral then reas l s m,p ψ / ω = π Na a ho ( j m,p ) i m,p k m,p δ m +m,m +m i, j,k m,m,m p,p,p ( I ijkl m + m + m + m +, p + p + p + p ). (A.) A.5. Integrals of the ipolar interaction In orer to calculate the integrals of the ipolar interaction, we first nee the expression for the mean fiel potential. This potential can be rewritten using Fourier transforms an the convolution theorem, which yiels [] (r) = F [Ṽ (k) ñ(k)], (A.) where the functions with tiles esignate the Fourier transforms of the ipole potential an the particle ensity. The transform of the ipole potential is well known to be [] Ṽ (k)/ ω = π ( ) k z k D/N. (A.) Our next task is to calculate ñ(k). Writing the square of the absolute value of the wave function with the given variational ansatz, one notices that the angular epenence is of the form exp(imφ) with some m Z. In[5] it was note that the three-imensional Fourier transform of such a function can be expresse as a one-imensional Hankel transform for the irection an a remaining one-imensional Fourier

12 J. Phys. B: At. Mol. Opt. Phys. () 5 M Kreibich et al transform in the z irection. The Hankel transform brings in the Bessel function, an the integral can be worke out by Taylor expaning the Bessel function. The Fourier transform is a Gaussian integral with linear terms an variable powers of z, for which an analytical expression can be foun in establishe tables of integrals [, equation..]. In the original integral, ψ, the integral over r is similar to the Fourier transform of ψ an can easily be expresse in the same way, which yiels the intermeiate result l m,p ψ = π e γ ijkl i, j,k m,m,m p,p,p ( m j ) i,p m,p m k,p σ m m σ m m im m m +m ( A kl ( A ij m m + m m + ) m m μ m,m μ m,m μ m,m α= β= ( μm,m α ) m m μ m,m )( μm,m β ) (A kl ) α ( ) A ij β ( ) μ m,m +μ m,m ( m m μ m,m ) μm,m α ( m m μ m,m ) μm,m β I k, (A.) where we efine σ m (sign m) m, { for sign m sign m μ m,m, (A.5) min( m, m ) otherwise, an use the Pochhammer symbol (a) n, which is efine by Ɣ(a + ) (a) n a(a )(a ) (a n + ) = Ɣ(a n + ). (A.) The remaining three-imensional integral I k reas I k = k e i(m +m m m )k φ e k /Aij e k /Akl k (α+β)+ m m + m m Ĩ z kl (p + p )Ĩz ij (p + p )Ṽ (k), (A.) with the z integrals Ĩ ij z (p) Ĩ kl z (p) zz p e Aij z z (p ij z +ik z )z, zz p e Akl z z (p kl z ik z)z, (A.a) (A.b) which are evaluate in [, equation..]. The Fourier transform of the ipole potential consists of two parts, V = (πkz /k π/) ω D/N V + V. The integral of I k containing V is equivalent to the integrals of the scattering interaction, an requires no further attention. The part of I k with V, which we enote by I k, can be rewritten to Ik / ω = π (D/N)δ m +m,m +m n! ( α ijkl ) n tt (n+) k z kz e (t )αijkl kz Ĩ z kl (p + p )Ĩz ij (p + p ). (A.) The Fourier transform of V s = πna/a ho δ(r) is πna/a ho, an thus proportional to V. This means that the whole integral of this part is equivalent to the integrals of the scattering interaction. The k z integral is of Gaussian type, an similar to Ĩ z kl an Ĩz ij. The remaining t integral can be expresse by the hypergeometric function F [], assuming all parameters p k z are zero. In the general case, the t integral can be efficiently evaluate using a Taylor series. Appenix B. Structure of the Jacobian The solutions of the BDGE () can be classifie by the quantum numbers m an π z, which express the symmetries of the system. Applying the TDVP an linearizing the equations of motion yiels an alternative escription of linear oscillations an stability, in which the Jacobian J plays a central role. The Jacobian, similar to the BDGE, also express the symmetries of the system an the quantum numbers, in that it assumes a block structure, which can in general be written as J = ( {J π z ±m} ) = J + J J + ± J ±, (B.)... where J π z ±m is the submatrix belonging to the angular momenta ±m an the parity π z. This block structure allows one to set up an iagonalize the Jacobian for each m an π z iniviually, analogously to solving the BDGE. The only ifference is that in the Jacobian the angular momenta m an m are couple, which is ue to the coupling of those angular momenta in the integrals necessary for the equations of motion (cf appenix A). A simple example, which shows that the coupling may not be neglecte, is the submatrix of the m =±, π z = Jacobian obtaine with the N G = Gaussian, for a ipolar BEC with λ = an D =.5. J + ± = ω,..5 (B.) where the variational parameters are orere as z = (Re,, Im,, Re,, Im, ) for a scattering length of Na/a ho =. The eigenvalues of the matrix (B.) are (+iω, +iω, iω, iω ). The eigenvectors of the eigenvalue +iω can then be combine to an eigenvector belonging to m = an another eigenvector for m =, the same hols for the eigenvalue iω. In general, for an arbitrary number of Gaussians an angular momenta, by taking appropriate linear combinations of the eigenvectors, eigenvectors belonging to a specific angular momentum +m or m can be obtaine. References [] Pitaevskii L P an Stringari S Bose Einstein Conensation (Oxfor: Oxfor University Press) [] Rau S, Main J an Wunner G Variational methos with couple Gaussian functions for Bose Einstein conensates with long-range interactions. I. General concept Phys. Rev. A