A Bose-Einstein condensate in an external potential consisting of a
superposition of a harmonic and a random potential is considered theoretically.
  From a semi-quantitative analysis we find the size, shape and excitation
energy as a function of the disorder strength. For positive scattering length
and sufficiently strong disorder the condensate decays into fragments each of
the size of the Larkin length ${\cal L}$. This state is stable over a large
range of particle numbers. The frequency of the breathing mode scales as
$1/{\cal L}^2$. For negative scattering length a condensate of size ${\cal L}$
may exist as a metastable state. These finding are generalized to anisotropic
traps

Sancisi, R

Schoenmakers, RHM

Swaters, RA

van Albada, TS

English

arXiv

University of Groningen

Kinematically lopsided spiral galaxies

T. Nattermann

V. L. Pokrovsky

A. I. Larkin

I. M. Lifshitz

Crossref

Bose-Einstein Condensates in Strongly Disordered Traps

University of Groningen Research Database

Asymmetries in the distribution of light and neutral hydrogen are often observed in spiral galaxies, Here, attention is drawn to the presence of large-scale asymmetries in their kinematics. Two examples of kinematically lopsided galaxies are presented and discussed. The shape of the rotation curve - rising more steeply on one side of the galaxy than on the other - is the signature of the kinematic lopsidedness. It is shown that kinematic lopsidedness may be related to lopsidedness in the potential, and that even a mild perturbation in the latter can produce significant kinematic effects. Probably at least half of all spiral galaxies are lopsided.</p

ARTS repository - University of Groningen

Nattermann, T.

Pokrovsky, V. L.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:0707.4444v1  [cond-mat.stat-mech]  30 Jul 2007Bose-Einstein Condensates in Strongly Disordered TrapsT. NattermannInstitut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, Zu¨lpicher Str. 77, D-50937 Ko¨ln, GermanyV.L. PokrovskyDepartment of Physics, Texas A&M University, College Station, Texas 77843-4242 andLandau Institute for Theoretical Physics, Chernogolovka, Moscow District, 142432, Russia(Dated: August 27, 2018)A Bose-Einstein condensate in an external potential consisting of a superposition of a harmonicand a random potential is considered theoretically. From a semi-quantitative analysis we find thesize, shape and excitation energy as a function of the disorder strength. For positive scatteringlength and sufficiently strong disorder the condensate decays into fragments each of the size of theLarkin length L. This state is stable over a large range of particle numbers. The frequency of thebreathing mode scales as 1/L2. For negative scattering length a condensate of size L may exist asa metastable state. These findings are generalized to anisotropic traps.PACS numbers: 03.75Hh, 03.75KkBose-Einstein condensation (BEC) is of great inter-est in a wide variety of systems including superfluidityand superconductivity. Its perhaps cleanest realizationis found in dilute ultracold gases of alkali atoms confinedin magnetic or optical traps which have been studied ingreat detail both experimentally [1, 2] and theoretically[3, 4] (and references therein).More recently these investigations have been extendedto traps formed by a superposition of a harmonic and arandom potential [5, 6, 7]. Whereas for harmonic trapsthe behavior of the condensate, e.g. its size, shape, ele-mentary excitations etc. is quite well understood, thisis not the case for traps including a finite amount ofdisorder. Most of the theoretical investigations of BECin random potentials use as a starting point a descrip-tion in momentum space (e.g. Bogoliubov transforma-tion or Beliaev-Popov perturbation theory) where thestudy of the typical disorder effects is notoriously diffi-cult [8, 9, 10, 11]. Disorder is there treated perturbativelywith ξ/L as a small parameter, where ξ and L denote thehealing and the Larkin length, respectively. As we willshow below in the case of sufficiently strong disorder thisparameter is of order one and perturbation theory breaksdown (but see [12]).In the present article we present a semi-quantitativeanalysis of BEC on the level of Larkin-Imry-Ma argu-ments [13, 14] to determine the size, shape and the fre-quencies of the breathing mode in weakly interactingBose-Einstein condensates in disordered traps. In addi-tion to the oscillator and the scattering length the Larkinlength appears as a new relevant length scale which dom-inates the properties of the condensate for weak interac-tion and strong disorder. Strong disorder reduces theanisotropy of the condensate. For attractive interactionthe condensate may exist in a metastable state.The model. Starting point is the Hamiltonian of inter-acting bosonsH =∫d3xΨ†(− ~22m∇2 + U(x) + 2π~2amΨ†Ψ)Ψ. (1)Here Ψ†(x) and Ψ(x) are the creation and annihilationoperators of the Bose field with∫d3x|Ψ|2 = N for theparticle number. m denotes the mass of the bosons and athe scattering length. The external potential representsa superposition of the harmonic trap and a Gaussian ran-dom potentialU(x) =m23∑i=1ω2i x2i + Udis(x). (2)The random potential is characterized by〈Udis〉 = 0, 〈Udis(x)Udis(x′)〉 = κ2δ(x − x′). (3)With this ansatz we assume that the correlation lengthof the disorder is smaller than all other length scales. Wewill briefly discuss other cases at the end of this article.In what follows we consider the state of the system atzero temperature. We will first examine the case of anisotropic 3-dimensional trap with ω1 ≈ ω2 ≈ ω3 = ωo.Assuming a compact condensate cloud of the radius Rthe condensate energy per particle ε can be written asε(R) ≈ ~22m(1R2+ 3aNR3+R2ℓ4− 43(R3L)1/2). (4)Here we introduced the oscillator length (the size of theharmonic oscillator ground state) ℓ = (~/(mωo))1/2. Ldenotes the Larkin length L = 16π~4/(27m2κ2) [13,14] in d = 3 dimensions. N is the total particle number.In the following we will set ~ = m = 1. In this notationthe healing length in the absence of disorder reads ξ =√R3/(6Na).The first and the second term in (4) describe the kineticenergy and the interaction of the particles, respectively.2Both terms favor the spreading of the condensate pro-vided the scattering length is positive. If the radius ofthe cloud is sufficiently large, namelyR≫ Ra = 3aN, (5)the interaction is negligible in comparison with the ki-netic energy. Neglecting the inhomogeneity of the con-densate, condition (5) can be rewritten as ξ ≫ R. Thethird and the fourth term describe the oscillator potentialand the typical potential well from the disorder, respec-tively, both cause its confinement. In the last term wehave taken into account that the typical value of the fluc-tuation of the potential in a region of linear size R scalesas κR−3/2. This term is larger than the kinetic energyonly on scales R & L. Similarly to the interaction, atsufficiently large radius of the cloud,R≫ RL = ℓ(ℓ/L)1/7, (6)the disorder can be neglected in comparison to the har-monic potential.In general the center of an attractive domain formed bystatic fluctuations of the random potential may not coin-cide with the center of the harmonic trap. Nevertheless,our estimates are correct since it is either the harmonictrap or the disorder which leads to the localization ofthe condensate. Experimentally localization by disorderis often observed by a sudden decrease of the oscillatorfrequency. In this case the condensate will be localizedin the coarse grained potential well closest to the origin.Condensate size. Below we determine the equilibriumsize R0 of the condensate as a function of the particlenumberN and the disorder strength from the equilibriumcondition ∂ε/∂R = 0. In the case of weak disorder andrelatively small particle number, such that ℓ ≪ L andN ≪ ℓ/(3a), we can ignore both the disorder and theinteraction and hence, from (4) R0 ≈ ℓ. At large particlenumber N > ℓ/(3a), the effect of the interaction sets inand R0 approaches the Thomas-Fermi radius [15]RTF(N) ∼(aN)1/5ℓ4/5 . (7)More interesting is the case of strong disorder, suchthat the oscillator length is much larger than the Larkinlength, ℓ≫ L. Then, for small particle number, the com-petition between the kinetic energy and the effective po-tential well resulting from the random potential on scaleR givesR0 ≈ L , Na≪ L. (8)Thus the size of the condensate is equal to the Larkinlength. We estimate the healing length as ξ ∼√L3/Na ≫ L from which it is plausible to concludethat the condensate is strongly depleted and the super-fluid stiffness vanishes [8].For larger particle numbers the interaction becomesimportant and the minimum of (4) is given byR0(N) ≈(92Na)2/3L1/3 , L ≪ aN ≪ ℓ( ℓL)5/7. (9)The (average) healing length is estimated from ξ ∼√R30/Na ∼ (L/Na)1/6R0 ≪ R0 from which we con-cludes that the condensate is weakly depleted only andthe superfluid stiffness remains finite [8]. At the up-per boundary for N the radius reaches RL. Finally, foreven larger particle number the radius crosses over tothe Thomas-Fermi radius. The crudeness of our energyexpression (4) does not allow accurate determination ofthe numerical pre-factors for all quantities derived from(4). If the disorder decreases, the Larkin-length L in-creases and the interval where (9) applies diminishes andeventually vanishes when L reaches the oscillator lengthℓ. In this case only the cross-over at Na ≈ ℓ discussedpreviously survives.Fragmentation. Next we consider the instability of acompact condensate against fragmentation. For N ≪L/a the ground state energy for increasing N decreasesas E ≈ −N/(6L2) until it reaches a plateau E ≈ E0 atN & N0 ≈ L/a where E0 ≈ −2/(27La) (compare (9)).Thus, at N > N0, the division of the condensate intoN/N0 fragments is energetically favorable. The energyof the fragmented condensate is E ≈ (N/N0)E0(N0) ≈−N/L2 < E0(N). Thus the fragmented condensate cor-responds to the true ground state. The total volumeof all fragments scales like ∼ NaL2 and is hence by afactor (L/(Na)) smaller than the volume of the com-pact state. The healing length ξ in the fragments isξ = (L3/(6Na))1/2 where N . N0, and hence ξ & L. Ac-cording to [8], in the region where ξ & L a finite fractionof the bosons are not in the condensate. The superfluidstiffness of the fragmented state most likely vanishes. ToLRFFIG. 1: State of dense fragments. Each fragment includesL/a particles. The minimal condensate radius is RF ∼(Na)1/3L2/3.describe the cross-over to the Thomas-Fermi behavior weconsider a state of dense fragments with the total radiusRF ∼ (Na)1/3L2/3. RF denotes the minimal size of ofthe fragmented state. If there is no confining parabolicpotential the fragments may be distributed over a muchlarger area to take advantage of the tails of the Gaus-3RNaLLR0(N)RTFRF(N)l((l/L)5/7 l((l/L)5FIG. 2: Condensate radius as a function of the particle num-ber. The bold line corresponds to the equilibrium fragmentedstate, the dashed line corresponds to the metastable compactstate.sian distribution of disorder. In a fragmented state withradius RF the oscillator energy per particle is of the or-der R2F ℓ−4 which has to be compared with the energyper particle −L−2 from the disorder. This gives for thecross-over particle numberNcoa ∼ ℓ (ℓ/L)5 . (10)This number is much larger then the particle cross-overnumber between (9) and (7) provided ℓ > L.The compact state described by (9) may occur as ametastable state which lives for a short period of timeafter a confining harmonic trap is switched off (or the os-cillator frequency ωo is strongly reduced). On larger timescales the condensate will lower its energy by decayinginto ∼ Na/L fragments each of the size L. This decayinto fragments may explain the striped phases found incigar-like traps [6].Breathing mode. Next we consider the breathing modefrequency ωB of the radial oscillations of the condensatearound the equilibrium configuration frommω2B = ε′′(R0). (11)Without disorder we find from (4) and (11) ωB = 2ωoin the non-interacting and ω =√5ωo in the interactingcase, respectively, in agreement with previous results [16].In the disordered case we obtain for Na≪ L≪ ℓωB ≈ 1√2(ℓL)2ωo ≫ ωo (12)and for L ≪ aN ≪ ℓ(ℓ/L)5/7 under the assumption of acompact non-fragmented condensateωB ∼ ℓ2(Na)7/6L5/6 ωo ≫ ωo. (13)Contrary to the pure case the interaction changes theradial oscillations of the condensate strongly.In the true equilibrium fragmented state the breathingmode will be controlled by the weak interaction betweenthe fragments. In this case the frequency gap will prob-ably decrease as ∼ R−2F .Negative scattering length. So far we assumed that thescattering length a is positive. In the case of negativescattering length as e.g. in the case of 7Li atoms [17],there is only a metastable state of finite radius Rc follow-ing from ∂ε/∂R|R=Rc = 0. This state becomes unstableat a critical particle number Nc. The condition for thedisappearance of the metastable minimum follows from∂ε(R)∂R=∂2ε(R)∂R2∣∣∣R=Rc= 0 (14)In the pure case L ≫ ℓ, condition (14) gives for thecritical particle number Nc ≈ 0.12ℓ/|a| in agreement with[17, 18, 19, 20].The strong disorder decreases the radius of themetastable droplet as it follows from the equilibrium con-dition(LR)1/2− 427NNc(LR)3/2= 1. (15)Here Nc denotes the critical particle number beyondwhich the metastable state disappearsNc =8243L|a| ≈ 0.033L|a| . (16)The radius of the metastable droplet in the region N <Nc decreases monotonically from Rc ≈ L for N ≪ Nc toRc ≈ 29L for N ≈ Nc.Anisotropic traps. So far we assumed that the trap isisotropic whereas experimentally often anisotropic trapsare considered. Our results can be easily extended tothese cases. For brevity we will consider here only thestrongly anisotropic situations (i) ω1 = ω2 = ωo ≪ ω3 =ω⊥ (d = 2) and (ii) ω1 = ωo ≪ ω2 = ω3 = ω⊥ (d = 1)corresponding to a pancake and a cigar-shape like trap,respectively. We assume that the the oscillator lengthobeys ℓ⊥ = (~/mω⊥)1/2 ≪ ℓ,L so that the extension ofthe condensate in the transverse direction is given by ℓ⊥.The extension R0 in the longitudinal direction followsthen from the minimum of the energy per particleε ≈ 12R2+3aN2Rdℓ3−d⊥+R22ℓ4− 2d(RdL4−dd )1/2, d = 1, 2.(17)Here we have introduced the d−dimensonal Larkin lengthLd = (Lℓ3−d⊥ )1/(4−d) and assumed that the correlationlength b of the random potential is small compared withthe transverse width of the trap, b≪ ℓ⊥. In the oppositecase b ≫ ℓ⊥, ℓ⊥ has to be replaced by ℓ2⊥/b in the defi-nition of Ld. From (17) we can read off the generalizedcross-over radiiRa ∼ ℓ⊥(ℓ⊥aN)1/(2−d), RL = ℓ(ℓLd) 4−d4+d. (18)For R≫ Ra the interaction and for R≫ RL the disordercan be neglected, respectively. For a d-dimensional trap4we find in analogy to (8) for small particle numbersR0 ≈ Ld , Na < ℓ⊥ (ℓ⊥/Ld)2−d . (19)In this region the healing length ξ & Ld such that afinite fraction of particles is not in the condensate andsuperfluidity is suppressed. For larger N (9) has to bereplaced byR0 ≈ ℓ⊥ (aN/ℓ⊥)2/d (Ld/ℓ⊥)(4−d)/d . (20)This formula is valid as long asaN < ℓ⊥ (ℓ⊥/ℓ)2−d (ℓ/Ld)(4−d)(d+2)4+d (21)For even largerN one find the generalized Thomas-FermibehaviorRTF (N) ≈ ℓ⊥(aNℓ4/ℓ5⊥) 12+d (22)However, we have to consider here again the instabilityof the compact domain state described by (20) againstfragmentation. The energy of the compact state scalesas −(1/aℓ⊥)(ℓ⊥/Ld)4−d whereas that of the fragmentedstate scales as −N/L2d. Thus the fragmented state hasthe lower energy until the radius of the set of dense frag-mentsRF (N) ∼ ℓ⊥(NaL2d/ℓ3⊥)1/d (23)reaches a value such that L−2d ∼ R2F /ℓ4 which happensatNco ∼(ℓ3−d⊥ /ℓ2−da)(ℓ/L)2+d (24)which generalizes (10), but is more realistic, especially atd = 1.In conclusion, we demonstrated that the disorderstrongly influences the size, shape and structure of thecondensate cloud. Sufficiently strong disorder leads tothe localization of the condensate fragments of linear sizeof the Larkin length. This result agrees with the conclu-sions about the suppression of the transport in elongatedBose-condensates derived in [5]. The increasing numberof particles enhances interaction and may delocalize thecondensate. Such a delocalization transition is especiallyrealistic in a strongly prolonged cigar-shape condensate.If the atoms in condensate attract each other, the dis-order decreases the critical number of particles at whichcollapse is developed.This work has been supported by DFG-projectNA222/5-2 (T.N. and V.P.) and by the DOE under thegrant DE-FG02-06ER 46278 (V.P.).Electronic mail: natter@thp.uni-koeln.de andvalery@physics.tamu.edu.[1] W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002). Nobellecture: When atoms behave as waves: Bose -Einsteincondensation and the atom laser.[2] E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875(2002), Nobel Lecture: Bose-Einstein condensation in adilute gas, the first 70 years and some recent experiments.[3] F. Dalfovo, S. Giorgini L. P. Pitaevskii and S. Stringari,Rev. Mod. Phys. 71, 463 (1999). Theory of Bose-Einsteincondensation in trapped gases.[4] A.J. Leggett, Rev. Mod. Phys. 73307 (2001). Bose-Einstein condensation in the alkali gases: Some funda-mental concepts.[5] D. Clement, A. F. Varn, M. Hugbart, J. A. Retter, P.Bouyer, L. Sanchez-Palencia, D. M. Gangardt, G. V.Shlyapnikov, and A. Aspect, Phys. Rev. Lett. 95, 170409(2005) Suppression of Transport of an Interacting Elon-gated Bose-Einstein Condensate in a Random Potential.[6] J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C.Fort, and M. Inguscio, Phys. Rev. Lett. 95, 070401 (2005)Bose-Einstein Condensate in a Random Potential.[7] C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno,D. S. Wiersma, and M. Inguscio, Phys. Rev. Lett. 95,170410 (2005) Effect of Optical Disorder and Single De-fects on the Expansion of a Bose-Einstein Condensate ina One-Dimensional Waveguide.[8] K. Huang, and H.F. Meng, Phys. Rev. Lett.69, 644(1992), Hard-sphere Bose gas in random external poten-tials.[9] S. Giorgini, L. Pitaevskii, and S. Stringari, Phys. Rev.B 49, 12938 (1994). Effects of disorder in a dilute Bosegas.[10] A. V. Lopatin and V. M. Vinokur Phys. Rev. Lett. 88,235503 (2002). Thermodynamics of the Superfluid DiluteBose Gas with Disorder.[11] G. M. Falco, A. Pelster, and R. Graham, Phys. Rev. A 75,063619 (2007), Phys. Rev. A 76, 013624 (2007). Collectiveoscillations in trapped Bose-Einstein-condensed gases inthe presence of weak disorder.[12] V.I.Yukalov, E.P.Yukalova, K.V.Krutitsky, R.Graham,arXiv:0705.3768 (2007). Bose-Einstein-condensed gasesin arbitrarily strong random potentials.[13] A.I. Larkin, Zh. Eksp. Theor. Fiz 58, 1466 (1970). Effectof inhomogeneities on the structure of the mixed state ofsuperconductors.[14] Y. Imry and S.K. Ma, Phys. Rev. Lett. 35, 1399 (1975).Random-Field Instability of the Ordered State of Contin-uous Symmetry.[15] E. Timmermanns, P. Tommasini and K. Huang, Phys.Rev. A 55, 3645 (1997) Variational Thomas-Fermi The-ory of a Nonuniform BEC at Zero Temperature.[16] S. Stringari, Phys. Rev. Lett. 77 2360 (1996). CollectiveExcitations of a Trapped Bose-Condensed Gas.[17] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.Hulet Phys. Rev. Lett. 75, 1687 (1995). Evidence ofBose-Einstein Condensation in an Atomic Gas with At-tractive Interactions.[18] E.V. Shuryak, Phys. Rev. A54, 3151 (1996). MetastableBose condensate made of atoms with attractive interac-tion.[19] Yu. Kagan, G. V. Shlyapnikov and J. T. M. Walraven,Phys. Rev. Lett. 76, 2670 (1996), Bose-Einstein Conden-sation in Trapped Atomic Gases.[20] Yu. Kagan, A. E. Muryshev, and G. V. Shlyapnikov,Phys. Rev. Lett. 81, 933 (1998). Collapse and Bose-Einstein Condensation in a Trapped Bose Gas with Neg-ative Scattering Length.

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Bose-Einstein Condensates in Strongly Disordered Traps

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