Bifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases
Patrick Köberle, Holger Cartarius, Tomaž Fabčič, Jörg Main, Günter Wunner
aa r X i v : . [ qu a n t - ph ] D ec Bifurcations, order, and chaos in the Bose-Einsteincondensation of dipolar gases
Patrick K¨oberle, Holger Cartarius, Tomaˇz Fabˇciˇc, J¨org Main,G¨unter Wunner
Institut f¨ur Theoretische Physik 1, Universit¨at Stuttgart, 70550 Stuttgart, Germany
Abstract.
We apply a variational technique to solve the time-dependent Gross-Pitaevskii equation for Bose-Einstein condensates in which an additional dipole-dipoleinteraction between the atoms is present with the goal of modelling the dynamics ofsuch condensates. We show that universal stability thresholds for the collapse of thecondensates correspond to bifurcation points where always two stationary solutionsof the Gross-Pitaevskii equation disappear in a tangent bifurcation, one dynamicallystable and the other unstable. We point out that the thresholds also correspond to“exceptional points”, i.e. branching singularities of the Hamiltonian. We analyse thedynamics of excited condensate wave functions via Poincar´e surfaces of section for thecondensate parameters and find both regular and chaotic motion, corresponding to(quasi-) periodically oscillating and irregularly fluctuating condensates, respectively.Stable islands are found to persist up to energies well above the saddle point of themean-field energy, alongside with collapsing modes. The results are applicable whenthe shape of the condensate is axisymmetric.PACS numbers: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky
At sufficiently low temperatures a condensate of weakly interacting bosons canbe represented by a single wave function whose dynamics obeys the dynamics of theGross-Pitaevskii equation [1, 2]. The equation is nonlinear in the wave function, andtherefore the solutions of the equation exhibit features not familiar from solutions ofordinary Schr¨odinger equations of quantum mechanics. As an example of the effectsof the nonlinearity, Huepe et al . [3, 4] demonstrated that for Bose-Einstein condensateswith attractive contact interaction, described by a negative s -wave scattering length a , bifurcations of the stationary solutions of the Gross-Pitaevskii equation appear. Theseauthors also determined both the stable (elliptic) and the unstable (hyperbolic) branchesof the solutions. In physical terms, the bifurcation points correspond to critical particlenumbers, above which, for given strength of the attractive interaction, collapse of thecondensate sets in. For Bose-Einstein condensates of Li [5, 6] and Rb atoms [7, 8]these collapses were experimentally observed.In those condensates the short-range contact interaction is the only interaction to beconsidered. By contrast, in Bose-Einstein condensates of dipolar gases [9–13] also a long-range dipole-dipole interaction is present. This offers the unique opportunity to study ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases and short-range interactions. Theachievement of Bose-Einstein condensation in a gas of chromium atoms [14], with alarge dipole moment, has in fact opened the way to promising experiments on dipolargases [15], which could show a wealth of novel phenomena [16–19]. In particular,the experimental observation of the collapse of dipolar quantum gases has also beenreported [20] which occurs when the contact interaction is reduced, for a given particlenumber, below some critical value using a Feshbach resonance.In this experimental situation it is most timely and appropriate to extend theinvestigations of the effects of the nonlinearity of the Gross-Pitaevskii equation todipolar quantum gases, and this is the goal of the present paper. To model theseeffects we will pursue, for the sake of simplicity, a variational ansatz. We do this inthe spirit of Refs. [3, 4, 21] where also variational techniques were applied to modelthe dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase bysolving the Gross-Pitaevskii equation without dipole-dipole interaction. In fact, quiterecently Parker et al. [22] have pointed out that in dipolar Bose-Einstein condensates theGaussian variational method gives excellent agreement with the full numerical solutionsof the Gross-Pitaevskii equation in wide ranges of the physical parameters. For oblatedipolar Bose-Einstein condensates the Gaussian approximation appears to agree onlyfor weak dipolar interactions. The approximation used may not be valid in the limitof interest and clearly further exact studies based on exact solutions of the dipolarGross-Pitaevskii equation need to be carried out to verify the results. Full numericalquantum calculations, for condensates with only the contact interaction [3,4,21] present,or with an additional attractive gravity-like 1 /r interaction [23–26], have confirmed thatproperties of the solutions of the Gross-Pitaevskii equation found in the variationalcalculations are recovered in the quantum calculations. We should therefore expectthat a simple variational approach will also capture essential features of the dynamicsof condensate wave functions of dipolar gases.We treat the problem in the “atomic” units provided by the magnetic dipole-dipoleinteraction, i.e., we measure lengths in units of the “dipole length” a d = µ µ m π ¯ h = α mm e µµ B ! a , (1)energies in units of E d = ¯ h / (2 ma ), frequencies in units of ω d = E d / ¯ h and time inunits of ¯ h/E d , respectively. In (1), α is the fine-structure constant, a the Bohr radius, m/m e the ratio of the atom and electron mass, and µ the magnetic moment. For Cr,with µ = 6 µ B ( µ B the Bohr magneton), one has a d = 91 a , E d = 1 . × − eV, and ω d / π = 4 . × Hz.In these “atomic” units, the Hartree equation of the ground state of a system of N identical bosons in an external trapping potential V trap = m ( ω r r + ω z z ) /
2, all in thesame single-particle orbital ψ , interacting via the contact interaction and the magneticdipole-dipole interaction, assumes the dimensionless form h − ∆ + γ r r + γ z z + N π aa d | ψ ( r ) | ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases N Z | ψ ( r ′ ) | − ϑ ′ | r − r ′ | d r ′ i ψ ( r ) = ε ψ ( r ) . (2)Here, ε is the chemical potential, a is the scattering length, and γ r,z = ω r,z / (2 ω d ) arethe dimensionless trap frequencies. Setting Ψ = √ N ψ , one recovers the familiar formof the time-independent Gross-Pitaevskii equation for dipolar quantum gases.In the above “atomic” units, the mean-field Hamiltonian in (2) obeys a scaling lawwith respect to the number N of atoms: Let ˜ ψ (˜ r ) be a solution of the (formal) one -bosonproblem for a given scattering length a/a d and trap frequencies ˜ γ r,z , H mf ( N = 1 , a/a d , ˜ γ r,z )(˜ r ) ˜ ψ (˜ r ) = ˜ ε ˜ ψ (˜ r ) , (3)then ψ ( r ) := N − / ˜ ψ (˜ r ), with r = N ˜ r , solves the N -boson problem for the samescattering length a/a d , H mf ( N, a/a d , γ r,z ) ( r ) ψ ( r ) = ε ψ ( r ) , (4)but with trap frequencies γ r,z = ˜ γ r,z /N and chemical potential ε = ˜ ε/N . Note that theparticle number scaling leaves the aspect ratio λ = γ z /γ r invariant. Thus, the physicalproperties of Bose condensates of dipolar quantum gases quite generally only depend onthe value of the scattering length a/a d and the particle number scaled trap frequencies N γ r,z or, alternatively, on the aspect ratio λ and the scaled geometric mean of the trapfrequencies N ¯ γ = N γ / r γ / z . We note that the dimensionless parameter D introducedby Dutta and Meystre [27] and Ronen et al . [28] to measure the effective strength ofthe dipole interaction in trap frequency units is related to our scaling parameters by D = ( N γ r / / .As an application of the particle number scaling law we emphasise that theexperimental results reported by Koch et al . [20] for the stabilisation of dipolar chromiumquantum gases for particle numbers ∼ ω/ π = 720 Hzcorrespond to a value of the scaled trap frequency N ¯ γ = 3 . × . Therefore theydirectly carry over to any pairs of particle numbers and mean trap frequencies with thisvalue of the scaled trap frequency and the same aspect ratios!To study the nonlinearity effects of the time-independent Gross-Pitaevskii equationfor dipolar gases we adopt the familiar variational ansatz of a (normalised) Gaussiantype orbital (e. g. [9, 18, 20]) ψ ( r ) = A exp h − ( A r r + A z z ) i (5)and exploit the time-dependent variational principle for the mean-field energy todetermine the width parameters A r and A z . It is well known that solutions can be foundonly in certain parameter ranges and that at critical values the condensate collapses.What – to the best of our knowledge – for dipolar quantum gases has gone unnoticedbefore is that at these stability thresholds actually two solutions of the extended Gross-Pitaevskii equation (2) disappear. The situation is depicted in figure 1 where thechemical potential is plotted as a function of the scattering length for different valuesof the trap aspect ratio λ and the fixed value of the scaled geometric mean of the trapfrequency of N ¯ γ = 3 . × . It can be seen that, as the scattering length is increased, ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases a/a d × × × × × × × N ε λ = 3.0 λ = 2.4 λ = 1.8 λ = 1.2 λ = 1.0 λ = 0.7 Figure 1.
Chemical potential for the mean trap frequency N ¯ γ = 3 . × , usedin the experiments of Koch et al . [20], and different values of the aspect ratio. Thetangential character of the bifurcations is particularly evident for the trap aspect ratios λ ≤ . −2 −1 −1 −2.0−1.5−1.0−0.50.0 a crit /a d λ N − γ a crit /a d Figure 2.
Stability thresholds, i.e. the critical scattering lengths a crit /a d , as a functionof the scaled trap parameter N ¯ γ and the aspect ratio λ . In the limit N ¯ γ → ∞ , λ → a crit /a d = 1 / for every value of λ two solutions are born in a tangent bifurcation, one correspondingto the ground state and the other to a collectively excited state. An inspection ofthe mean-field energies shows that the excited state corresponds to the branch of thechemical potential which diverges for a/a d → / N ¯ γ and λ is shown in figure 2. Large values of N ¯ γ correspond to the regimewhere the dipole-dipole interaction is dominant, as in the experiments of Koch et al . [20]. ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases N ¯ γ < N ε and the corresponding wave functionsare identical. The way to reveal the branch point singularity structure is to continuethe scattering length a/a d into the complex plane and to check a well-known propertyof exceptional points: If in traversing a full circle around the critical value a crit /a d atthe point of bifurcation, a/a d = ( a crit /a d ) + ̺ e iϕ , ϕ = 0 . . . π, a permutation of the twosolutions occurs, an exceptional point is located within the circular area [29].For the parameters of the Koch et al . [20] experiment we have performed suchan analysis, and for the case of an almost purely dipolar quantum gas the results areshown in figure 3. As one goes around a small circle in the complex plane with thecritical scattering length at the centre, the two solutions are permuted, i.e. we havean exceptional point. It must be said, however, that the usual way of experimentallyproving the occurrence of an exceptional point in open quantum systems, changing tworeal physical parameters to traverse a circle in the complex energy plane, cannot beapplied here. In the case of the Gross-Pitaevskii equation it is by continuing one realparameter, the scattering length, into the complex plane that the nature of the stabilitythresholds as exceptional points is revealed.We now analyse the dynamics of the condensate wave functions. To do so we startfrom the time-dependent Gross-Pitaevskii equation, which is (2) with the replacement ε → i ddt , and generalise the ansatz (5) to a time-dependent Gaussian type orbitalwith complex width parameters A r ( t ) and A z ( t ) and complex normalisation factor A ( t )(cf. [21]). We can then apply the time-dependent variational principle k i ˙ ψ ( t ) − Hψ ( t ) k = min. [32,33] to derive a system of ordinary nonlinear differential equations for the timeevolution of the real and imaginary parts of the variational parameters A r ( t ) and A z ( t ).These can be shown to be equivalent to canonical equations of motion belonging to atwo-dimensional nonintegrable autonomous Hamiltonian system. For the time evolution ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases -0.0012-0.0006 0.0000 0.0006 0.0012-0.0208 -0.0182 I m ( a / a d ) Re(a/a d )(a) -2000 -1000 0 1000 2000 464 466 468 I m ( N ε ) Re(N ε )/10 (b) Figure 3.
Chemical potentials N ε (b) of the two stationary solutions emerging in thetangent bifurcation for a circle (a) with radius ̺ = 10 − around a crit /a d = − . N ¯ γ = 3 . × , λ = 6 . one can therefore expect all the features familiar from nonlinear dynamics studies of suchsystems, including a transition to chaos. For brevity, the details of these calculationsare relegated to a subsequent paper. Here we shall concentrate on the results.We first investigate the stability of the two independent solutions of the stationaryGross-Pitaevskii equation which emerge from the bifurcations. To this end the fourequations of motion for the real and imaginary parts of A r and A z are linearised aroundthe values corresponding to the stationary states. In this way for each of the two stateswe obtain four eigensolutions ψ (lin) ∼ e κt of the linearised system of equations, witheigenvalues κ . For the ground state all eigenvalues turn out to be purely imaginary,proving that the state is indeed dynamically stable. For the collectively excited stateone also finds a positive real eigenvalue, and hence the state is dynamically unstable.Similar behaviour was found by Huepe et al . [3, 4] in their study of the stability ofbifurcating solutions with only an attractive contact interaction present.The system of nonlinear first-order differential equations also serves to investigatethe time evolution of any initial state of the condensate by following the correspondingtrajectories in the four-dimensional configuration space spanned by the coordinates ofthe real and imaginary parts of A r and A z . Since the total mean-field energy is a constantof motion the trajectories are restricted to three-dimensional hyperplanes, and theirbehaviour can most conveniently be visualised by two-dimensional Poincar´e surfaces ofsection defined by requiring one of the coordinates to assume a fixed value.We consider Poincar´e surfaces of section defined by the condition that the imaginarypart of A z ( t ) is zero. Each time the trajectory crosses the plane Im( A z ) = 0, the real andimaginary parts of A r ( t ) = A rr ( t ) + iA ir ( t ) are recorded. In figure 4 surfaces of section areplotted for seven different, increasing, values of the mean-field energy. Again the physicalparameters of the experiment of Koch et al . [20] are adopted, and the scattering lengthis fixed to a/a d = 0 .
1, away from its critical value. At these parameters, the variationalmean-field energy of the ground state is
N E gs = 4 . × and represents the localminimum on the two-dimensional mean-field energy landscape, plotted as a function ofthe (real) width parameters. The variational energy of the second, unstable, stationary ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases Figure 4.
Poincar´e surfaces of section of the condensate wave functions representedby their width parameters for seven different mean-field energies at the scaled trapfrequency N ¯ γ = 3 . × , aspect ratio λ = 6, and the scattering length a/a d = 0 . N E = 4 . × , . × , . × , . × , . × , . × , . × , respectively. state at these experimental parameters is N E es = 6 . × , it corresponds to the saddlepoint on the mean-field energy surface. Between these two energy values the motion onthe trajectories is bound, while for energies above the saddle point energy the motion onthe trajectories can become unbound: once the saddle point is traversed by a trajectory A r ( t ), A z ( t ), the parameters run to infinity, A rr ( t ) , A rz ( t ) → ∞ , meaning a shrinking ofthe quantum state to vanishing width, i.e. a collapse of the condensate takes place.The surface of section labelled 1 in figure 4 belongs to the energy of the ground state.At this energy the kinematically allowed region for the crossing points of the trajectoriesis confined to a single stable stationary point. At the next higher energy (surface ofsection labelled 2) the kinematically accessible region in configuration space has grown,and the initially stationary state has evolved into a periodic orbit (fixed point in thesurface of section), corresponding to a state of the condensate whose motion is periodic.The oscillations of the width parameters A r ( t ) and A z ( t ) represent oscillatory stretchings ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases r and z directions. The stable periodic orbit in the surfaceof section is surrounded by elliptical, quasi-periodic orbits, representing quasi-periodicoscillations of the condensate. The surface of section 3, at the next higher energy,reveals that bifurcations have occurred, creating new stable and unstable periodic states,manifested by the emergence of additional elliptical islands and separatrices in thesurface of section.The surface of section labelled 4 is the first in figure 4 with an energy value abovethe saddle point energy. Now chaotic orbits have appeared which surround the stableregions. In contrast to the (quasi-) periodic stretching oscillations of the condensatewithin the elliptical islands, the chaotic motion of the parameters describes a condensatewhich does not yet collapse but whose widths fluctuate irregularly.One might imagine that well above the saddle point energy stable condensate wavefunctions no longer exist. However, in the surfaces of section labelled 5, 6, and 7regular islands are still clearly visible. These stable islands are surrounded by chaotictrajectories. Since ergodic motion along these trajectories comes close to every pointin the configuration space, the chaotic motion sooner or later leads to a crossing of thesaddle point and then to the collapse of the condensate wave functions. It can be seenthat with growing energy above the saddle point the sizes of the stable regions graduallyshrink. The reason why the kinematically allowed regions surrounding the stable islandsare hardly recognisable any more in these surfaces of section is that high above the saddlepoint energy the chaotic motion becomes more and more unbound, which means thatthe trajectories cross the Poincar´e surfaces of section only a few times, if ever, beforethey escape to infinity and collapse takes place.It must be emphasised, however, that stable islands do persist even far above thesaddle point energy, implying the existence of quasi-periodically oscillating nondecayingmodes of the condensate wave functions.The prediction of such modes of energetically excited solutions of the time-dependent Gross-Pitaevskii equation for cold dipolar quantum gases is a result of ouranalysis. It would certainly be an intriguing and challenging task to examine whetherit is possible to prepare excited states of dipolar quantum gases of this type in boththe regular as well as in the chaotic regions, to distinguish between the two differentdynamics, and to access the stable regions high above the energy of the unstablestationary state. One way of creating the collectively excited states one might imagineis to prepare the condensate in the ground state, and then to non-adiabatically reducethe trap frequencies. Clearly experimental investigations along these lines are stronglyencouraged.One might ask, however, whether the Gross-Pitaevskii equation underlying ourcalculations is adequate at all to describe complex dynamics of this type in real dipolarquantum gases. In particular in the chaotic regime local density maxima might occur forwhich losses by two-body or three-body collisions would have to be taken into account.However, by virtue of the scaling law (4) parameter ranges can always be found where theparticle densities remain small even in these regimes and the Gross-Pitaevskii equation ifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases /r interaction [23],have shown that the nonlinear dynamical properties found in the variational calculationare recovered in the numerical calculations, and that the quantum behaviour may beeven richer. As a common feature of the propagation of quantum wave functions, thereal-time evolution of arbitrary condensate wave functions of dipolar gases will exhibitcomplicated fluctuations. An interpretation of their full dynamics, and in particular thesearch for periodic, quasiperiodic or chaotic structures, therefore will hardly be possiblewithout the guidance of the variational results obtained in this paper. Acknowledgments
This work was supported by Deutsche Forschungsgemeinschaft. H. C. is grateful forsupport from the Landesgraduiertenf¨orderung of the Land Baden-W¨urttemberg.
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