A simple analytical ansatz, which has been used to describe the intensity
profile of the similariton laser (a laser with self-similar propagation of
ultrashort pulses), is used as a variational wave function to solve the
Gross-Pitaevskii equation for a wide range of interaction parameters. The
variational form interpolates between the noninteracting density profile and
the strongly interacting Thomas-Fermi profile smoothly. The simple form of the
ansatz is modified for both cylindrically symmetric and completely anisotropic
harmonic traps. The resulting ground-state density profile and energy are in
very good agreement with both the analytical solutions in the limiting cases of
interaction and the numerical solutions in the intermediate regime.Comment: 4 pages, 3 figures, published versio

E. Lundh

F. Ö. Ilday

G. I. Barenblatt

K. N. Ilinski

L. P. Pitaevskii

M. Ö. Oktel

Murat Keçeli

English

arXiv

A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser (a laser with self-similar propagation of ultrashort pulses), is used as a variational wave function to solve the Gross-Pitaevskii equation for a wide range of interaction parameters. The variational form interpolates between the noninteracting density profile and the strongly interacting Thomas-Fermi profile smoothly. The simple form of the ansatz is modified for both cylindrically symmetric and completely anisotropic harmonic traps. The resulting ground-state density profile and energy are in very good agreement with both the analytical solutions in the limiting cases of interaction and the numerical solutions in the intermediate regime

Keçeli, M.

Ilday, F. Ö.

Oktel, M. Ö.

Bilkent University Institutional Repository

Ansatz from nonlinear optics applied to trapped Bose-Einstein condensatesMurat Keçeli, F. Ö. Ilday, and M. Ö. OktelDepartment of Physics, Bilkent University, 06800 Ankara, TurkeyReceived 30 November 2006; published 13 March 2007A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser alaser with self-similar propagation of ultrashort pulses, is used as a variational wave function to solve theGross-Pitaevskii equation for a wide range of interaction parameters. The variational form interpolates betweenthe noninteracting density profile and the strongly interacting Thomas-Fermi profile smoothly. The simple formof the ansatz is modified for both cylindrically symmetric and completely anisotropic harmonic traps. Theresulting ground-state density profile and energy are in very good agreement with both the analytical solutionsin the limiting cases of interaction and the numerical solutions in the intermediate regime.DOI: 10.1103/PhysRevA.75.035601 PACS numbers: 03.75.Hh, 42.65.TgIt is common for dynamical systems with weak couplingto show Gaussian behavior with respect to key parameters,whereas strongly coupled and highly nonlinear systems tendto exhibit power law dependences. A striking recent exampleof this is the ultrashort pulse formation in the similaritonlaser a laser with self-similar propagation of ultrashortpulses 1. Another nonlinear system, which exhibits a para-bolic density profile for strong interactions, is a trappedBose-Einstein condensate BEC. In Bose-Einstein conden-sation, the density of the condensate is analogous to the in-tensity of light in nonlinear optics NLO and the nonlineargoverning equation for this system, which is called theGross-Pitaevskii equation GPE, is very similar to the equa-tion for propagation of laser light in a nonlinear optical me-dium. Based on this similarity, it is natural to expect similarsolutions for these vastly different systems. Solitonlike andself-similar solutions of the nonlinear Schrödinger equationNLSE are important both in BEC and NLO.Solitonlike solutions arise when the nonlinearity is com-pensated by the dispersion and they are the only exact ana-lytical solutions of these NLSE’s, whereas self-similar solu-tions are asymptotic solutions that show up when the effectsof initial conditions die out but the system is still far from thefinal state 2. Although soliton-type solutions have been ex-tensively studied in both the NLO and BEC communities,self-similar solutions are not as comprehensively investi-gated. In optics, these types of solutions are used more ex-tensively from Raman scattering to pulse propagation in fi-bers, and it is shown that linearly chirped parabolic pulsesare exact asymptotic solutions of the NLSE with gain 3.Recently, we have demonstrated experimentally and numeri-cally that self-similar propagation of ultrashort parabolicpulses similaritons are stable in a laser resonator 1. Morerecently, we have developed a semianalytic theory of thesimilariton laser 4,5. Instrumental in this step was the in-troduction of an ansatz to describe the intensity profile of thispulse, which can be “tuned” to any condition, ranging fromweakly nonlinear Gaussian pulses to strongly nonlinearparabolic pulses. Here, motivated by the mathematicalsimilarity between the two systems, we apply the same an-satz to describe the density profile of a BEC in a quadratictrap. We show that this ansatz describes the system withexcellent accuracy, in the whole range from the noninteract-ing limit to the strongly interacting limit.Gross-Pitaevskii theory 6 gives a very successful de-scription of the ground state and excitations of BEC’s indilute atomic gases. The success of this theory implies thatthe condensate can be described accurately with a singlewave function and the interactions between the particles arethrough s-wave scatterings. The interaction of the particlesare then represented by the interaction strength g= 42am ,where a is the s-wave scattering length and m is the mass ofthe trapped particles. The theory reduces to a single equationthat describes the condensate wave function, known as theGPE, a type of nonlinear Schrödinger equation which arisesin many areas of physics like NLO and hydrodynamic theoryof fluids.The GPE can be obtained by minimizing the ground-stateenergy functional of the condensate,E = dr 22mr2 + Vrr2 +g2r4 , 1with respect to the wave function. The terms in the energyfunctional correspond to kinetic, trapping, and interaction en-ergies, respectively. The trapping potential can generally beapproximated with a harmonic potential for many of the ex-periments. The time-independent GPE follows as−22m2r + Vrr + gr2r = r , 2where  is chemical potential introduced as the Lagrangemultiplier for the normalization constraint 	dr nr=N,where nr= r2 is the density of the condensate.The nonlinearity of the GPE is due to an interaction be-tween particles and its effect becomes more pronounced asthe number of particles in the condensate increases which isthe case for current experiments where more than 107 par-ticles form the BEC. Since very few exact solutions of theNLSE, such as solitons, are known, many numerical algo-rithms 7–11 and variational methods 12–17 are developedto find ground-state solutions. Although variational methodsgive only an upper bound to the ground-state energy, theyrequire less calculation and can give accurate results if asuitable trial function is chosen. Another advantage of thePHYSICAL REVIEW A 75, 035601 20071050-2947/2007/753/0356014 ©2007 The American Physical Society035601-1variational principle is that it gives the functional form of thewave function which can be used to obtain other propertiesof the condensate.Therefore, many trial functions are proposed to obtain alower bound for the ground-state energy. Trial functions aregenerally chosen by adding parameters to a known approxi-mate analytical solution. The trivial approximate solutionsare obtained by looking at the limiting cases of the GPEwhere the nonlinearity is negligibly small or very high. As-suming an ideal Bose gas, Eq. 2 reduces to the Schrödingerequation where the chemical potential corresponds to the en-ergy eigenvalue, by neglecting the nonlinear term. With aharmonic trap the problem turns into a harmonic oscillatorproblem where the solution is a Gaussian. For the oppositecase where nonlinearity is dominant, the kinetic energy termcan be neglected in the GPE, and then the equation can eas-ily be solved for the density, nr=−Vr2g , when the right-hand side of the equation is positive. This approximation isknown as the Thomas-Fermi approximation TFA, and itshows that as the interaction increases, the density profilechanges from a Gaussian to a parabola for a harmonic trap.The TFA can be improved by adding the kinetic energy termobtained with the resulting wave function with a suitablecutoff 18 asEkinN= 15−2/512ln480 −54 . 3With this insight many trial functions are proposed forisotropic traps and anisotropic traps with cylindrical symme-try to describe the intermediate regime where neither theTFA and nor the ideal gas approximation is valid. Here, wemake use of a simple analytic function which has alreadybeen used in NLO where a similar behavior—Gaussian toparabolic—for the intensity profile of a similariton laser.The so-called similariton pulse in optics has a nearly para-bolic intensity profile to reduce the effect of Kerr nonlinear-ity. However, if the nonlinearity of the system is reduced, thepulse assumes the well-known Gaussian shape of dispersion-managed solitons. Therefore, the ansatz proposed in 4 todescribe these pulses has an adjustable profile between aGaussian and an inverted parabola:Snx = exp− k=1nx2kk . 4This function becomes a Gaussian when n=1,Sn=1x = exp− x2 , 5and turns into an inverted parabola when n goes to  sincethe summation in the exponent converges to ln1−x2 forx  1,Sn→ = exp− k=1x2kk = expln1 − x2 = 1 − x2. 6Moreover, the expansion converges so quickly that addingabout ten terms is enough to get a parabolic profile withsmooth ends. Besides, this function is easily integrablewhich makes it a good candidate for variational calculations.Motivated by these properties, we use the similariton an-satz, Eq. 4, as our trial wave function to minimize the en-ergy functional given in Eq. 1. To simplify the calculations,we non-dimensionalize the Gross-Pitaevskii functional byscaling length, energy, and wave function with oscillatorlength a= m , , and N /a3 , respectively see Table I.We first analyze the solution for a spherical harmonic trapVr= 12m2r2 and introduce the parameter Na /a whichis a measure of the strength of the interaction. With thisrescaling the energy functional becomesEN= 20d3rr2 + Vrr2 + 2r4 .7Ideally  can take any value between − to  since all theparameters are experimentally tunable. However, a negativescattering length, which means an attractive interaction,TABLE I. The values of the wave function at the center, theroot-mean-square sizes rrms, and chemical potentials are tabulatedin units of N /aw3 , aw, and , respectively. For comparison nu-merical results of Ref. 11 are given in parentheses. 0 rrms 0 0.4238 0.4238 1.2248 1.2248 1.5000 1.50000.2496 0.3969 0.3843 1.2794 1.2785 1.6805 1.67740.9986 0.3475 0.3180 1.3981 1.3921 2.0885 2.06502.4964 0.2515 0.2581 1.5355 1.5356 2.5803 2.58619.9857 0.1739 0.1738 1.8822 1.8821 4.0089 4.014149.926 0.1097 0.1066 2.5071 2.5057 7.2576 7.2484249.64 0.0665 0.0655 3.4152 3.4145 13.559 13.5532496.4 0.0330 0.0328 5.3855 5.3852 33.812 33.81010110210305101520βE/N0 5 10024FIG. 1. Color online Ground-state energy with respect to theinteraction parameter  obtained with the variational functionblack solid line. The resulting energy of a Gaussian variationalfunction is given by dotted red line and energy obtained withThomas-Fermi solution is given with dashed line green. ImprovedThomas-Fermi solution 18 is given by the dotted line blue. En-ergy per particle is given in units of . The inset is given for small values.BRIEF REPORTS PHYSICAL REVIEW A 75, 035601 2007035601-2causes collapse of the condensate when the particle numberis high. In this regime, our results agree with 12. In thepresent work, we concentrate on repulsive interaction. Withproper normalization the trial wave function has the formr = 14d3Iexpk=1nr/d2k2k , 8where d and n are our variational parameters withI=	0dr r2Snr which is an integral that can be calculatedanalytically for n=1,2 and numerically for n	2. Here theparameter d is responsible for the width of the condensate,which increases as the interaction increases, and n takes careof flattening of the central density. We minimize the energywith respect to d for different n values and chose the n thatgives the minimum energy. For d, we obtain a fifth-orderpolynomial equation where only one of the roots is physi-cally meaningful.We compare our results with the analytical approxima-tions as follows. For small  values, our trial function re-duces to a Gaussian and gives an exact result for =0, andfor large  our results agree well with the improved TFAresults as shown in Fig. 1. We also compare the resultingwave function with the numerical solutions obtained by thesteepest descent method for different  values in Fig. 2. Wealso tabulate our results and include the results of a recentnumerical analysis which minimizes the energy functionaldirectly by the finite-element method. Here it should benoted that tabulated kinetic, trap, and interaction energiessatisfy the virial theorem 2Ekin+2Etr−3Eint=0, to our nu-merical accuracy. It is also remarkable that even for large adding ten terms is enough to find a good approximation forthe wave function see Fig. 3 which shows the simplicity ofthe calculations.Using similar trial functions, we can also solve the GPEfor a cylindrical trap and a fully anisotropic trap. For thecylindrically symmetric trap, trial function takes the form,,z = C exp− k=1n /d2k2kexp− k=1nz z/dz2k2k , 9where C= N2d2dzIIz, I=	0dSn, and Iz=	− dzSnz. Wehave four variational parameters, but calculations are similarTABLE II. Results of our calculation for a cylindrically harmonic trap with =8. Energy and length units are N and a. The resultsof the numerical calculation in Ref. 9 are given in parentheses for comparison except for the last row. For =2165 the Thomas-Fermi resultfor the chemical potential is given in parentheses. xrms zrms Ekin Etr Eint 0.0000 0.7071 0.707 0.4204 0.42 1.2071 1.207 1.2071 1.207 0.0000 0.000 2.4142 2.4140.4330 0.7901 0.79 0.4374 0.44 1.0539 1.06 1.3894 1.39 0.2237 0.21 2.8907 2.880.8660 0.8500 0.85 0.4472 0.45 0.9976 0.98 1.5225 1.52 0.3500 0.36 3.2200 3.212.1650 0.9657 0.96 0.4707 0.47 0.8528 0.86 1.8188 1.81 0.6440 0.63 3.9596 3.944.3300 1.0892 1.08 0.4966 0.50 0.7337 0.76 2.1730 2.15 0.9595 0.96 4.8258 4.778.6600 1.2319 1.23 0.5332 0.53 0.6709 0.66 2.6549 2.64 1.3227 1.32 5.9712 5.9321.650 1.4798 1.47 0.5930 0.59 0.5314 0.54 3.5963 3.57 2.0432 2.02 8.2142 8.1443.300 1.7038 1.69 0.6536 0.65 0.4351 0.45 4.6121 4.57 2.7847 2.74 10.616 10.564.950 1.8447 1.84 0.6989 0.70 0.4128 0.41 5.3569 5.31 3.2960 3.26 12.361 12.286.600 1.9562 1.94 0.7319 0.73 0.3789 0.38 5.9693 5.91 3.7270 3.68 13.802 13.72165 3.7367 1.3297 0.1459 21.035 13.926 49.033 48.3290 1 2 3 4 500.020.040.060.080.1r(aω)ψ(√N/a ω3 )FIG. 2. Color online Wave function calculated with the steep-est descent method 9 is shown as a bold line blue whereas thesimilariton ansatz solution is given by the dashed line red for=100.0 500 100024681012βn0 500 1000123456βdFIG. 3. Change of variational parameters with interaction pa-rameter  is given. Left plot shows the number of terms in thesummation and the right one shows the change of width of thesimilariton ansatz in units of oscillator length a.BRIEF REPORTS PHYSICAL REVIEW A 75, 035601 2007035601-3to the isotropic case. We compare our results with the nu-merical results of Dalfovo and Stringari 9 in Table II. Cy-lindrically symmetric traps are the most common traps inBEC setups, and the aspect ratio obtained fromxrmszrmsis veryimportant to identify the BEC phase in these experiments. Itis shown in 9,12 that for the noninteracting case this ratio isequal to  and goes to  in the Thomas-Fermi limit. Thisresult is clearly seen from the values in Table II where=8, and it is also evident that convergence of the TFA isvery slow.There are also experiments with fully anisotropic traps19, and for this case the trial function can also be modifiedsimilarly with six variation parameters. The results are givenin Table III, where agreement with the results in 10 is ap-parent.In summary, an ansatz is introduced to investigate theground-state properties of a BEC at zero temperature forquadratic traps with arbitrary anisotropy. The ground-stateenergy and wave function are found to be very accurate any-where from the noninteracting case to the highly repulsiveone, as compared with numerical studies. The form of thetrial function changes from a Gaussian to a parabolasmoothly, and it successfully describes the intermediate re-gime of moderate interaction. Important quantities like as-pect ratio, chemical potential, and root-mean-square size ofthe clouds are calculated and compared to numerical studies9–11. With a slight modification the suggested form of thewave function can be applied to vortex states of the conden-sate. The time-dependent GPE can also be solved using asimilar form to study the growth dynamics.We have shown previously that a nonlinear system inoptics—namely, a high-energy femtosecond laseroscillator—exists stably between two extreme limits, corre-sponding to Gaussian pulse profiles for weak nonlinearityand parabolic profiles for strong nonlinearity. This indeedappears to be a common behavior observed in many systems,including the trapped BEC analyzed in this paper. We have asimple analytical function, which has the crucial property ofinterpolating any state between these extremes. In this paper,we have shown that the variational approach with the sameansatz yields excellent results for the BEC system in a qua-dratic trap. We believe that our approach can be generallyapplicable to other nonlinear systems in disparate fields.F.Ö.I. is supported by a TUBA-GEBIP grant andTUBITAK Grant No. 104T017. M.Ö.O. is supported by aTUBA-GEBIP grant and TUBITAK-KARIYER Grant No.104T165.1 F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys.Rev. Lett. 92, 213902 2004.2 G. I. Barenblatt, Scaling, Self-similarity, and Intermediate As-ymptotics Cambridge University Press, Cambridge, England,1996.3 M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley,and J. D. Harvey, Phys. Rev. Lett. 84, 6010 2000.4 C. Jirauschek, F. Ö. Ilday, and F. X. Kärtner unpublished.5 C. Jirauschek and F. Ö. Ilday unpublished.6 L. P. Pitaevskii, Sov. Phys. JETP 13, 451 1961; E. P. Gross,Nuovo Cimento 20, 454 1961; J. Math. Phys. 4, 195 1963.7 M. Edwards and K. Burnett, Phys. Rev. A 51, 1382 1995.8 P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards,Phys. Rev. A 51, 4704 1995.9 F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 1996.10 B. I. Schneider and D. L. Feder, Phys. Rev. A 59, 2232 1999.11 W. Bao and W. Tang, J. Comput. Phys. 187, 230 2003.12 G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 1996.13 K. N. Ilinski and A. Moroz, J. Res. Natl. Inst. Stand. Technol.101, 567 1996.14 V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein,and P. Zoller, Phys. Rev. Lett. 77, 5320 1996; Phys. Rev. A56, 1424 1997.15 A. L. Fetter, J. Low Temp. Phys. 106, 643 1997.16 J. L. Bohn, B. D. Esry, and C. H. Greene, Phys. Rev. A 58,584 1998.17 M. P. Singh and A. L. Satheesha, Eur. Phys. J. D 7, 3911999.18 E. Lundh, C. J. Pethick, and H. Smith, Phys. Rev. A 55, 21261996.19 M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K.Helmerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett.82, 871 1999.TABLE III. The chemical potentials per particle in units of are calculated using an ansatz modified for a completely anisotropictrap with =2 and g=2. The values of the interaction parameter are obtained from Ref. 19, and the values in parentheses corre-spond to the numerical solution given in 10. The variational pa-rameters nx,y,z and dx,y,z are given. nx ,dx ny ,dy nz ,dz 0 1, 1.000 1, 0.840 1, 0.707 2.207 2.2071.787 2, 1.753 2, 1.368 1, 0.817 3.604 3.5723.575 2, 1.956 2, 1.489 2, 1.160 4.385 4.3457.151 3, 2.433 2, 1.654 2, 1.258 5.492 5.42514.302 3, 2.780 3, 2.035 2, 1.384 7.010 6.90428.605 4, 3.310 3, 2.30 3, 1.687 9.049 8.90057.211 5, 3.880 4, 2.725 3, 1.896 11.78 11.57BRIEF REPORTS PHYSICAL REVIEW A 75, 035601 2007035601-4

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