THE LONGITUDINAL EXCITATION SPECTRUM OF AN ELONGATED BOSE-EINSTEIN CONDENSATE

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1 THE LONGITUDINAL EXCITATION SPECTRUM OF AN ELONGATED BOSE-EINSTEIN CONDENSATE M.C. RAPORTARU 1,, R. ZUS 2 1 Department of Computational Physics and Information Technologies, Horia Hulubei National Institute for Physics and Nuclear Engineering, Reactorului 30, Magurele, Ilfov, Romania 2 Faculty of Physics, University of Bucharest, Atomistilor 405, Magurele, Ilfov, Romania Corresponding author : mraportaru@nipne.ro Received September 15, 2017 Abstract. In this manuscript we show an efficient numerical approach for the longitudinal excitation spectrum of a quasi-one-dimensional collisionally-inhomogeneous Bose-Einstein condensate. We obtain by variational means the nonlinear ordinary differential equations of a generic longitudinal excitation and determine numerically the onset of instability, to identify which excitation is most likely to be observed experimentally. To quantify the efficiency of our approach, we study a cigar-shaped condensate with weak collisionally-inhomogeneous two-body interactions and show that the position of the resonance observed in the longitudinal excitation spectrum is higher than that observed for a condensate with homogeneous two-body interactions. Key words: Bose-Einstein condensates, longitudinal excitations, scientific computing. 1. INTRODUCTION The emergence of spatiotemporal patterns in spatially extended physical systems subjected to periodic forcing is a well-established research topic with applications in numerous fields such as hydrodynamics, reaction-diffusion processes, classical and quantum optics, biophysics, and, more recently, quantum mesoscopic systems [1]. Similarly, discrete systems with a large number of individual parts often exhibit a cooperative behavior which leads to structured states containing clear (possibly statistical) patterns, with examples such as sand dunes ripples, forest fires, models of bureaucracy, etc. [2]. In the case of Bose-Einstein condensates (BECs) [3, 4] pattern formation has received substantial interest due to the experimental robustness exhibited by these quantum gases and the computational amenability of the mean-field Gross-Pitaevskii (GP) equation which describes the 0 K dynamics of the condensates [5, 6]. Moreover, there is a plethora of results on the excitations supported by BECs, such as the recent ones on the nonclassical excitations reported in Ref. [7], the elementary excitations of quantum droplets [8] and the spectrum of excitations in a two-dimensional channel [9]. Finally, the numerical solution of the GP equation has enabled detailed study of the ground state and dynamics of Romanian Journal of Physics 63, 603 (2018) v.2.1* #338c7729

2 Article no. 603 M.C. Raportaru, R. Zus 2 Bose-Einstein condensates, and, equally importantly, verification and justification of effective lower-dimensional models, which are widely used to address various nonlinear phenomena. In particular, semi-implicit split-step Crank-Nicolson method is suitable for numerically solving the GP equation due to its absolute stability and favorable scaling of discretization errors. The early investigations into pattern-forming instabilities in ultra-cold quantum gases (see Ref. [10] for a review) received considerable interest after the prediction and subsequent observation of Faraday waves in cigar-shaped condensates. This research direction was afterwards consolidated by numerous results on density waves in fermionic ultra-cold gases, bosonic condensates with one or more atomic species that have either short-range or long-range (dipolar) interactions, etc. (see Ref. [11] and reference therein). The aforementioned results stem from two-fold investigations which combine full numerical computations with analytical ones which often reduce the initial partial differential equation to the level of a Mathieu equation whose spectrum is well known (see Ref. [12] for the main results). One result which has been cited often in the literature (see, for example, Ref. [13]) concerns the most unstable solution of the Mathieu equation which is commonly associated with the most unstable excitation that is observed experimentally. In this manuscript we extend the previous theoretical results by investigating the onset of instability beyond the solutions of the Mathieu equation, solving numerically the nonlinear ordinary differential equations for each excitation and determining the most unstable one at each moment in time. We consider a quasi-one-dimensional setup and a collisionallyinhomogeneous Bose-Einstein condensate, such that the variational equations which describe the dynamics of each excitation can be casted in a relatively simple analytical form, but our approach can be extended for other systems as well. The rest of the paper is structured as follows: in Section II we derive the variational equations which describe the radial dynamics of the condensate and that of the longitudinal wave, and in Section III we present our numerical results. Lastly, in Section IV we gather our final remarks and outline a few topics of future interest. 2. THE VARIATIONAL EQUATIONS The standard variational treatment of Bose-Einstein condensates draws from the similar calculations used in nonlinear optics for the propagation of light pulses in nonlinear media and has a long honored history. Here, we start from the timedependent Gross-Pitaevskii equation, namely i ψ t = 2 2m 2 ψ + V (r,t)ψ + g(r)n ψ 2 ψ (1)

3 3 The longitudinal excitation spectrum of an elongated Bose-Einstein condensate Article no. 603 where ψ = ψ(r,z,t), V (r,z,t) = mω 2 (t)r 2 /2, Ω(t) = Ω(1 + ɛsinωt), g(r,z) = g exp ( r 2 /2b 2) and consider a highly-elongated cylindrically-symmetric collisionallyinhomogeneous Bose-Einstein condensate whose wave function can be approximated by the usual Gaussian complex profile, for the radial component, and a periodic function of complex amplitude for the longitudinal excitation, i.e., ) ψ(r,z,t) = A(t)exp ( r2 2w 2 (t) + ir2 α(t) (1 + (u(t) + iv(t))coskz) (2) where the amplitude A is taken such that the wave function is normalized to unity over one period of the excitation, w(t) is the radial width of the condensate, α(t) is its canonically conjugate variable, u(t) and v(t) are real functions, k = 2π/p is the wave number of the longitudinal excitation and p is its period. The validity of the Gaussian part of the Ansatz for condensates with weak collisionally-inhomogeneous two-body interactions has been demonstrated by numerical means in Ref. [14], where it is noted, however, that for strong collisionally-inhomogeneous two-body interactions the radial profile resembles that of a coaxial cable with a low-density core and a high-density cover. This Ansatz can be extended to include an overall longitudinal envelope, as in Ref. [15], or non-gaussian radial Ansätze, as in Ref. [16], but the ensuing equations are quite cumbersome. For the Ansatz in equation (2), however, we obtain four ordinary differential equations which describe the dynamics of the radial width of the condensate and that of the complex amplitude of the longitudinal excitation of wave vector k, namely ( α = 1 2b 2 gρ ( 6u 2 ( v ) + 3u 4 + 3v 4 + 8v ) 2 πw 4 (4b 2 + w 2 )(u 2 + v α 2 m mω2 ( + 2 mw 4 4b4 gρ 6u 2 ( v ) + 3u 4 + 3v 4 + 8v ) ) πw 4 (4b 2 + w 2 )(u 2 + v 2, (3) ẇ = 2 αw m, (4) v ( 2b 2 gmρ ( v 2 7u 2)) u = 2πm w 2 (4b 2 + w 2 )(u 2 + v 2 + v ( 4πb 2 k 2 2 w 2 ( u 2 + v ) + πk 2 2 w 4 ( u 2 + v )) 2πm w 2 (4b 2 + w 2 )(u 2 + v 2 (5) u ( 2b 2 gmρ ( 3u 2 5v 2 8 )) v = 2πm w 2 (4b 2 + w 2 )(u 2 + v 2 u( 4πb 2 k 2 2 w 2 ( u 2 + v ) πk 2 2 w 4 ( u 2 + v )) 2πm w 2 (4b 2 + w 2 )(u 2 + v 2, (6)

4 Article no. 603 M.C. Raportaru, R. Zus 4 where we have discarded the explicit time dependent of Ω, α, w, u, and v on grounds of simplicity. The previous equations are intricate enough that they do not allow the standard analysis which yields the wave vector of the most unstable excitation through a Mathieu-type analysis. For this reason our approach is numerically oriented and relies on determining the {k,ω} pair for which the excitation is maximal, after a given evolution time. To this end, we solve the variational equations (3)- (6) numerically across a large region of the k ω plane (or equivalently the p ω plane, with p = 2π/k) where we expect the most unstable excitation to be located. As the equations are not stiff we can use a classical embedded Runge-Kutta method of order 4(5) which provides fast and accurate solutions [17]. We point out that for a typical experimental setup, i.e., one set of parameters {g,m,ρ,...}, we need around 10 6 points in the {p,ω} region of interest to determine confidently the most unstable excitation. We point out that the numerical solutions discussed above are independent from one another and that the computational load can be easily distributed over many CPU cores, thereby reducing the overall computing time. The parametric nature, with respect to k and ω, of equations (3)-(6) simplifies substantially the computations which effectively reduce to a series of parametric runs of the code which implements the Runge-Kutta method. Equations (3)-(6) extend the well-known ones in Ref. [18] into the collisionallyinhomogeneous regime, for which the existing variational equations [19] capture the u(t) and v(t) dynamics only to leading order. Let us also note that equations (3)- (6) stem from an energy minimization recipe which accounts for the impact of the radial dynamics on the longitudinal excitation, in striking contrast to similar the similar equations derived in Ref. [13] which appear after linearizing a perturbed ground state. 3. NUMERICAL RESULTS We start our numerical investigations considering a longitudinally-homogeneous quasi-one-dimensional collisionally-inhomogeneous 87 Rb Bose-Einstein condensate of m = kg, linear density ρ = atoms/m (which corresponds to a typical experimental setup of atoms over a longitudinal extent of 180 µm), g = 4π 2 /a s with a s = m, b = 20 µm, confined in a radial trap of Ω = 160(2π) Hz subjected to a periodic forcing of ɛ = 0.1 and ω = 210(2π) Hz, for which we consider times up to 150 ms. For this setting we show in Fig. 1 the spectrum of excitations as a function of time obtained through repeated numerical solutions of equations (3)-(6). The contour plot in Fig. 1 depicts the amplitude of the longitude excitations, i.e., the normalized maximum value of u 2 (t) + v 2 (t) over the interval [0,t], in a region of the p t plane that is experimentally relevant, where p is the period of the longitudinal wave and is in the µm regime, while t is time, taken in

5 5 The longitudinal excitation spectrum of an elongated Bose-Einstein condensate Article no. 603 the range ms, to quantify the instability onset times of the two waves. Fig. 1 Excitation spectrum of the condensate for a driving frequency ω = 170 (2π) Hz. Fig. 2 Excitation spectrum of the condensate for a driving frequency ω = 210 (2π) Hz. One immediately notices the two peaks of the contour plots, which get more prominent as t increases, that correspond to the Faraday and resonant wave, the first of which emerges faster. We point out that all other excitations grow very weakly and are of very small amplitude for all timescales which are experimentally relevant, so they can be effectively discarded. The spectrum of excitations shown in Fig. 1 is typical for non-resonant excitations in which the Faraday wave emerges faster than its resonant sibling, as can be seen from the color code. In Fig. 2 we depict a

6 Article no. 603 M.C. Raportaru, R. Zus 6 Fig. 3 Dispersion of excitations. typical resonant response for ω = 170 (2π) Hz in which one can easily notice that the resonant wave emerges faster than the Faraday one, in contrast to what has been observed in Fig. 1. Lastly, in Fig. 3 we show the spectrum of excitations in a region of the p ω plane that is experimental relevant plotting the normalized maximum value of u 2 (t) + v 2 (t) over the interval [0,150] ms. The results in Fig. 3 are in good agreement with what is known in the literature, but exhibit a very interesting shift in the position of the resonance which is not observed for ω = Ω = 160(2π) Hz but rather in the region [160(2π),180(2π)] Hz. 4. CONCLUSIONS AND OUTLOOK We have proposed a numerical recipe for determining the longitudinal excitation spectrum of a quasi-one-dimensional collisionally-inhomogeneous Bose-Einstein condensate that relies on solving the variational equations which capture the simplified dynamics of the condensate in a large region of the p ω plane. The proposed variational equations have provided an accurate picture of the instability onset of longitudinal excitations generated by radial modulations and allowed us to determine the one which is most likely to be observed experimentally. Moreover, the numerical results show that the position of the resonance observed in the longitudinal excitation spectrum is higher than that observed for a condensate with homogeneous two-body interactions.

7 7 The longitudinal excitation spectrum of an elongated Bose-Einstein condensate Article no. 603 Acknowledgements. For this work M.C.R. was supported by the Romania Ministry of Research and Innovation through PN /2017. The authors acknowledge fruitful discussions with A. Balaz and A.I. Nicolin. REFERENCES 1. M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993). 2. P. Bak, How nature works. The science of self-organized criticality (Copernicus Springer-Verlag, New York, 1999). 3. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2008). 4. V.S. Bagnato et al., Rom. Rep. Phys. 67, 5 (2015). 5. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez (eds.), Emergent Nonlinear Phenomena in Bose-Einstein Condensates (Springer-Verlag, Berlin, 2010). 6. A. Bogojevic, A. Balaz, and A. Belic, Phys. Rev. E 72, (2005); A. Bogojevic, I. Vidanovic, A. Balaz, and A. Belic, Phys. Lett. A 372, 3341 (2008). 7. A. Finke, New J. Phys. 18, (2016). 8. F. Wachtler and L. Santos, Phys. Rev. A 94, (2016). 9. I. V. Shunyaev, A. A. Elistratov, and Yu. E. Lozovik, Phys. Rev. A 94, (2016). 10. P.G. Kevrekidis and D.J. Frantzeskakis, Mod. Phys. Lett. B 18, 173 (2004). 11. J.B. Sudharsan, R. Radha, M.C. Raportaru, A.I. Nicolin, and A. Balaz, J. Phys. B: At. Mol. Opt. Phys. 49, (2016). 12. N.W. McLachlan, Theory and Application of Mathieu Functions (Oxford Univ. Press, New York, 1951). 13. A.I. Nicolin, R. Carretero-Gonzalez, and P.G. Kevrekidis, Phys. Rev. A 76, (2007); R. Nath and L. Santos, Phys. Rev. A 81, (2010). 14. A.I. Nicolin, A. Balaz, J.B. Sudharsan, and R. Radha, Rom. J. Phys. 59, 204 (2014); A. Balaz, R. Paun, A.I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, (2014) 15. A.I. Nicolin, Physica A 391, 1062 (2012). 16. A.I. Nicolin and R. Carretero-Gonzalez, Physica A 387, 6032 (2008); M.C. Raportaru, Rom. Rep. Phys. 64, 105 (2012). 17. J.C. Butcher, Numerical methods for ordinary differential equations (Wiley, New York, 2003). 18. A.I. Nicolin, Phys. Rev. E 84, (2011). 19. S. Balasubramanian, R. Ramaswamy, and A.I. Nicolin, Rom. Rep. Phys. 65, 820 (2013).