Table Of Content

Solitary waves and yrast states in Bose-Einstein condensed gases of atoms A. D. Jackson1, J. Smyrnakis2, M. Magiropoulos2, and G. M. Kavoulakis2 1The Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 2Technological Educational Institute of Crete, P.O. Box 1939, GR-71004, Heraklion, Greece (Dated: January 20, 2011) Considering a Bose-Einstein condensed gas confined in one dimension with periodic boundary 1 conditions, we demonstrate that, very generally, solitary-wave and rotational excitations coincide. 1 Thisexactequivalenceallowsustoestablishconnectionsbetweenanumberofeffectsthatarepresent 0 in these two problems, many of which havebeen studied using the mean-field approximation. 2 PACSnumbers: 05.30.Jp,67.85.-d,03.75.Lm n a J Introduction. Remarkable experimental developments to that of the yrast spectrum, but with a higher energy. 9 in the field of cold atomic gases now permit the realiza- However, in a recent study [15], Kanamoto, Carr, and 1 tion of systems not previously accessible. One example Ueda have provided numerical evidence that “quantum ] is the recentfabricationoftoroidaltraps that realizepe- solitonsintheLieb-LinigerHamiltonianarepreciselythe s riodic boundary conditions [1–5]. Tight, elongated traps yrast states”. The main result of the present study is to a g now permit the study of quasi-one-dimensional motion demonstrate analytically that solitonic and yrast states - sincethe energyassociatedwithtransverseexcitationsis are identical. t n muchhigherthantheinteractionenergy[6]. Averytight The present study contributes to the long-standing a toroidal trap is expected soon and will permit the study problemof the excitation spectrum of a one-dimensional u of periodic motion in a quasi-one-dimensional system. q Bose gas with periodic boundary conditions. In his sem- Such advances will make it possible to study many of . inal work Lieb [13] demonstrated that this spectrum has t the interestingphenomena predictedby the variousnon- a two branches. The usual Bogoliubov spectrum which is m linear models of such systems that have been studied in linear for long wavelengths and quadratic (i.e., single- recent decades. In the case of bosonic atoms, for exam- - particlelike)forhigherenergies. Itisshownherethatthe d ple, even the ground state is nontrivial. When the ef- otherbranch,whichhasbeenidentifiedascorresponding n fective interaction is attractive, a localized density max- tosolitary-waveexcitations[16,17],canalsoberegarded o imum forms for sufficiently strong coupling [7, 8]. When as the lowest-energy state of the gas for a fixed value of c the effective interaction is weakly repulsive, the density [ the angular momentum. is homogeneous. As the interaction strength increases, 2 however, the gas eventually enters the Tonks-Girardeau Inthefollowingwefirstdemonstratetheequivalenceof v state of hard-core bosons, in which the bosons resemble yrast states and soliton states for the case of an axially- 6 fermions in many respects [9, 10]. symmetric ring. This result is completely general and 1 Another interesting aspect of bosonic atoms confined makes no assumption about the explicit form of the 8 1 by a ring potential is their excitation spectrum [11–13]. many-body state. We then examine the limit of weak . Two fundamental forms of excitations, solitary-waveex- interactions, where the mean-field approximation is an 2 citations and rotational excitations, have been investi- excellentdescriptionofbothstates,anddiscussthelinks 1 gated. In the first case, one is interested in traveling- between various effects that appear in these two seem- 0 1 wavesolutionsforwhichthewavepropagatesaroundthe ingly different problems. : ringwithaconstantvelocityandwithoutchangeofform. v The equivalence of rotational and solitonic excitations. In the second case,one is interested in the lowest-energy i Let us start with the yrast states. For a given Hamilto- X stateofthesystemforagivenangularmomentum,which nian, Hˆ, the constraints of a fixed particle number and r is the so-called “yrast” state. a fixed angular momentum can be imposed with the in- a Clearly, both are excited states of the system, and it troduction of two Lagrange multipliers, µ and Ω, which is interesting to investigate if and how they are related. can be interpreted physically as the chemical potential Bloch’s theorem [14] implies that states of different val- and the angular velocity of the trapping potential. (We ues of the winding number are connected via excitation assume here that the gas has equilibrated in the rotat- of the center-of-mass. This implies that the dispersion ing trap). If Ψy(x1,x2,...,xN) is the yrast many-body relation, i.e., the energy of the system as function of the state, variations of the energy functional, angular momentum, is a periodic function on top of a parabolicenvelopefunctionduetocenter-of-massexcita- tions. Theyraststateis(bydefinition)thelowest-energy E(Ψy,Ψ∗y)= Ψ∗yHˆΨydx1...dxN stateforafixedangularmomentum. Therefore,thenaive Z expectation is that the dispersion relation appropriate for solitonic excitation should have a structure similar −µ Ψ∗yΨydx1...dxN −Ω Ψ∗yLˆΨydx1...dxN, (1) Z Z 2 ∗ with respect to Ψ yield atom-atom collisions (in one dimension), and where u= y ΩR as above. We now examine the consequences of this HˆΨy µΨy ΩLˆΨy =0. (2) equivalence. − − Consequences of the equivalence. The solitonic solu- Turning to the solitonic solutions, we assume that tions for a Bose-Einstein condensate confined to a ring these have the form potential with a repulsive effective interatomic interac- Φs(x1,x2,...,xN,t)=Ψs(z1,z2,...,zN)e−iµt/h¯, (3) tion, U0 > 0, are known to be Jacobi elliptic functions [26, 27] with density wherez =x ut,whereuisthevelocityofpropagation. i i Since i¯h∂Φs/∂−t=HˆΦs, n(z)=nmin+(nmax nmin)sn2 K(m)z m , (6) − πR | HˆΨ µΨ uPˆΨ =0, (4) (cid:18) (cid:19) s s s − − where Pˆ is the momentum operator. Comparison of wheresn(x|m)aretheJacobiellipticfunctions, nmin and nmax are the minimum and maximum values of the den- Eqs.(2) and (4) reveals that Ψ and Ψ satisfy the s y sity, and K(m) is the elliptic integral of the first kind. samedifferentialequationandthesameboundarycondi- The parameter m is determined by the equation tions. They are thus identical with the obvious equality u=ΩR, where R is the radius of the ring. 1/2 π R 1 λ Thisequivalenceisextremelygeneralandrequiresonly K(m)= − , (7) √2ξ0 m that the Hamiltonian is invariant under the transforma- (cid:18) (cid:19) tionxi zi. While the kinetic energyandthe two-body where ξ0 is the coherence lengthcorrespondingto a den- interact→ion evidently meet this requirement, this is not sity nmax, ¯h2/(2Mξ02) = nmaxU0, and λ = nmin/nmax. always the case in the presence of an external one-body The velocity of propagationis given by potential, V(x). The above results apply when V(x) is axiallysymmetricandtime-independent. Theimposition u √2qξ0 1 1 λ 1/2 πR √nminnmax = λ+ − dz. of strictaxialsymmetry has important effects onthe ex- act wave functions Ψ and Ψ . It forces them to have c R ± 2πR(cid:20) m (cid:21) Z−πR n(z) y s (8) anaxially-symmetricsingle-particledensity distribution, and it implies that the condensate is, in general, frag- Here c is the speed of sound in a homogeneous gas of mented. However, these conclusions do not necessarily densitynmax,Mc2 =nmaxU0,andq isthephasewinding reflectthebehaviourofrealphysicalsystems. Inpractice, number. According to the above argument, the yrast itisimpossibletoavoidweakanisotropiesinthetrapping states in mean-field approximation are given exactly by potential. Such symmetry-breaking anisotropies, even the same expressions. those which are vanishingly small in the limit of large Dispersion. We turn now to the dispersion relation, atom numbers, are sufficient to break the axial symme- i.e.,theenergyasfunctionofmomentum,appropriatefor try of the single-particle density and to restore an un- both solitons and yrast states. Given the solitary-wave fragmented condensate [20–25]. orderparameterin the form ψ =√neiφ, one can deter- s Inthefollowing,wefocusonthemean-fieldapproxima- mine the energy and momentum per particle according tion, which is valid for sufficiently weak interactions. At to the following formulas: mean-field level, we seek yrast states with a constrained astvaetreasgeofvaLˆl.ueItofshLô,urladthbeertnhoatnedintshisattinagltohnouhgahvitnhgeeyigreanst- E = 1 πR ¯h2 ∂√n 2+n ∂φ 2 + U0 n2 dz, state and its corresponding soliton state are rotationally N Z−πR"2M (cid:18) ∂z (cid:19) (cid:18)∂z(cid:19) ! 2 # symmetric, this rotational symmetry breaks within the (9) mean-field approximation for both of them. It has been and shown,however,inthelimitofweakinteractionsthatthe energies of the axially-symmetric and broken-symmetry πR yraststates are identical to leading orderin N, with dif- p=(h¯/N) n(∂φ/∂z)dz. (10) ferences of order 1/N [18, 19]. In the case of solitonic Z−πR solutions we seek travelling-wave solutions, which prop- The analytic form of φ(z), which is somewhat compli- agate with a constant velocity. As shown more gener- cated, is given in Ref.[27]. The desired E(p) follows im- allyabove,bothproblemsreducetothe samedifferential mediately given E(u) and p(u). Figure 1 shows the re- equation for the order parameters, with the result that sultsofthiscalculationfortwodifferentsituations. Only ψ =ψ =ψ. Specifically, y s the first two branches of this function are shown. Note −i¯hu∂∂ψx =−2¯hM2 ∂∂2xψ2 +(U0|ψ|2−µ)ψ. (5) tthhaetsEtr(u−ctpu)r=e eExp(pe)c.teCdlefarorlmy, tBhleocdhi’sspethrseioornemrelaatsiosntahteads above. In this equation we have assumed contact interactions, It is convenient to introduce the dimensionless quan- 2 where U0 is the matrix element for zero-energy elastic tity γ = NU0MR/(π¯h ), which is the ratio between 3 the interaction energy n0U0, with n0 being the homo- 11 geneous density n0 = N/(2πR), and the kinetic energy, q = -2 E0 = ¯h2/(2MR2). In Fig.1 the value of γ has been ) 10 chosen to be 120/π2 in the upper curve, and 3/(4π2) E0 ( 9 in the lower curve. In this plot the energy is mea- y q = -1 sured in units of E0 = h¯2/(2MR2), and the momen- erg 8 q = 0 tum is measured in units of p0 = ¯h/R. When p = 0, En 7 the energy E(p = 0) comes purely from the interaction energy, which is U0N/(4πR) or γ/2 in units of 6 E0. The maximum value of the momentum per atom 0 2 4 6 8 10 12 14 in each “quasi-periodic” interval is 2πp0. Finally, the Momentum (p ) 0 energy difference E(p = 2πp0) E(p = 0) = E0, and E(p = 4πp0) E(p = 2πp0) =−(22 1)E0. More gen- − − 5 erally, these energy differences are all the odd multiples of E0, due to the excitation of the center of mass, as 4 q = -2 ) expected from Bloch’s theorem. E0 ( 3 In the physics of solitary waves, it is well known that y g the velocity of propagation, u, of a solitary wave sat- r 2 q = -1 e isfies the equation u = ∂E/∂p. This equality can also n E 1 q = 0 be viewed from the point of view of the yrast states in mean-fieldapproximation. AssumethatE(l)istheyrast 0 energyperparticleasfunctionoftheangularmomentum 0 2 4 6 8 10 12 14 ′ perparticlelandthatE istheenergyinarotatingframe Momentum (p ) 0 ofreferenceforwhichthedensitydistributionoftheyrast ′ state is stationary. Given that E = E lΩ and that ∂E′/∂l = 0, it follows that Ω = ∂E/∂l o−r u = ∂E/∂p, FIG. 1: The dispersion relation E(p),for γ =120/π2 (upper which is the formula given above. plot), and γ =3/(4π2) (lower plot). The energy is measured InRef.[27]itwasarguedthat,whenthewindingnum- in units of E0 = h¯2/(2MR2), and the momentum in units ber q is zero, the velocity of propagation of a solitary of p0 = h¯/R. In the two curves both N/R and U0 have the samevalue,howeverinthelowercurvetheradiusis40times wave cannot vanish. This is obvious from the dispersion smaller than in thehigher one. relations of Fig.1. The region from p = 0 to p = πp0 (given by the left solid points in the two plots of Fig.1) corresponds to q = 0. The smallest possible value of u problem [27]. This effect is clearly seen in the lower plot (for q = 0) is attained at p = πp0 where its value is u = h¯/(2MR). At this value of p the density develops of Fig.1. a node, and a “dark” solitary wave forms. The density From the above remarks it is obvious that, as R de- is relatively insensitive to changes in the momentum in creases, there is a minimum value of the radius below the vicinity of this point, but there is a violent and dis- which u cannot vanish [26]. The condition for this is continuous change in the phase of the order parameter R/ξ =√6/2,whereξisthecoherencelengthcorrespond- that is associated with the change of the winding num- ing to a density n0 =N/(2πR). In the language of yrast ber. Moregenerally,the solitarywavesbecome darkand states,thissaysthatthereisacriticalcouplingofγ =3/2 the winding number changes whenever p is an odd mul- for having a metastable minimum at L/N = l = ¯h (or tiple of πp0. The first two such points, where q changes equivalently at p = 2πp0 [7, 8]. For smaller values of γ, − from0to 1andfrom 1to 2,aregivenassolidpoints the slope of the dispersion relation is positive as l ¯h − − − − → in the two plots of Fig.1. [or p (2πp0) ], and the possible metastable minimum → Itisalsointerestingtonotethattheinteractionenergy at this point disappears. (which scales like N/R) decreases relative to the zero- Energetic stability of the solitary waves. The creation point energy [which scales like ¯h2/(MR2)] as the radius of solitary waves in ring-like potentials will be of consid- of the ring decreases for a fixed value of N/R. The ratio erable interest. The intimate connection between soli- oftheseenergiesistheparameterγ,whichis (N/R)R2. ton and yrast states discussed here enables us to draw ∝ As R decreasesforfixedN/R,the interactionenergybe- a number of conclusions about the possible generation comes less important, and the gas approaches the limit of solitary waves and their energetic stability. The most ofanon-interactingsystem. InthelimitofsmallR,(i.e., obvious of these is that the lowest-energy state with a γ 1), the quasi-periodic intervals of the dispersion re- given L will also be a solitary-wave state. Ideally, one ≪ h i lationbecome linearwitha discontinuousfirstderivative would like to observe a family of solitary-wave solutions between them. The velocity of propagation of the soli- much like those observedin harmonic traps [28, 29], i.e., tary wave thus approachesconstant values, givenas odd densitydepressionstravellingwithavelocitywhichisless multiples of u = ¯h/(2MR), set by the non-interacting thanthatofsoundandwhichdecreasesasthedepression 4 becomes deeper. tation of a Bose gas moving in one dimension with pe- In the case of a repulsive effective interatomic interac- riodic boundary conditions, namely rotational and soli- tion, the curvature of the dispersion relation is negative tonic excitation. We have shown that yrast states and (as shown also in Fig.1) [30]. The “kink” states with a solitonic states are identical whenever the Hamiltonian momentumperparticlewhichisequaltoanintegermul- isaxially-symmetric. This resultisasimple consequence tiple of 2πp0 are the only possible local minima in the ofthemoregeneralobservationthattheproblemoffind- absence of rotation of the trap. The occupation of these ing wave functions that propagate with a fixed velocity states will give rise to persistent currents. Since these is equivalent to the problem of minimizing the energy states have a homogeneous density distribution and a subject to a constraintof fixedangular momentum. The non-zero circulation, they would be difficult to observe onlynecessaryconditionforthisequivalence,whichholds experimentally [31]. The same conclusion applies to a for all functional forms of the wave function, is the ro- rotatingtrap. Onewayto overcomethis difficulty would tational invariance of the Hamiltonian. This equivalence be to consider systems with attractive effective interac- permits the consideration of such systems from two dis- tions [15]. In this case the curvature of the dispersion tinct points of view, and we have noted certain aspects relation is positive [30], and persistent currents cannot of a unified picture that result from this observation in be stable. In contrast, solitary waves can be stabilized themean-fieldregime,whichisvalidforsufficientlyweak byrotatingthetrap. Theenergyinaframerotatingwith interactions. ′ ′ angularvelocityΩ is E =E lΩ,and E canhavelocal This connection between solitons and yrast states has − minima for a continuous range of l which is determined beensuggestedpreviously[15]andtheargumentsoffered by Ω. here are admittedly elementary. It would appear, how- Summary. In the present study we have presented a ever, that the rigorous equivalence demonstrated here is unified picture of two seemingly different forms of exci- neither generally known, nor adequately appreciated. [1] S.Gupta,K.W.Murch,K.L.Moore,T.P.Purdy,andD. M. Reimann,Phys. Rev.Lett. 86, 945 (2001). M. Stamper-Kurn,Phys.Rev.Lett. 95, 143201 (2005). [19] Elliott H. Lieb, Robert Seiringer, and Jakob Yngvason, [2] S.E.Olson,M.L.Terraciano,M.Bashkansky,andF.K. Phys. Rev.A 79, 063626 (2009). Fatemi, Phys. Rev.A 76, 061404(R) (2007). [20] D. Rokhsar, e-print arXiv:cond-mat/9812260. [3] I.LesanovskyandW.vonKlitzing,Phys.Rev.Lett.99, [21] M. Ueda and A. J. Leggett, Phys. Rev. Lett. 83, 1489 083001 (2007). (1999). [4] K. Henderson, C. Ryu, C. MacCormick, and M. G. [22] T.-L.HoandS.K.Yip,Phys.Rev.Lett.84,4031(2000). Boshier, New J. Phys. 11, 043030 (2009). [23] Xia-JiLiu,HuiHu,LeeChang,WeipingZhang,Shi-Qun [5] C. Ryu, M. F. Andersen, P. Cladé, V. Natarajan, K. Li, Yu-ZhuWang, Phys.Rev.Lett. 87, 030404 (2001). Helmerson, and W. D. Phillips, Phys. Rev. Lett. 99, [24] Ofir E. Alon, Alexej I. Streltsov, Kaspar Sakmann, and 260401 (2007). Lorenz S. Cederbaum, Europhys.Lett. 67, 8 (2004). [6] A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, [25] A.D.Jackson, G.M.Kavoulakis, andM.Magiropoulos, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Phys. Rev.A 78, 063623 (2008). Gupta,S.Inouye,T.Rosenband,andW.Ketterle,Phys. [26] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev.Lett. 87, 130402 (2001). Rev. A 62, 063610 (2000). [7] RinaKanamoto,HirokiSaito,andMasahitoUeda,Phys. [27] J. Smyrnakis, M. Magiropoulos, G. M. Kavoulakis, and Rev.A 68, 043619 (2003). A. D. Jackson, Phys. Rev.A 82, 023604 (2010). [8] G. M. Kavoulakis, Phys. Rev.A 69, 023613 (2004). [28] S.Burger,K.Bongs,S.Dettmer,W.Ertmer,andK.Sen- [9] L. Tonks,Phys. Rev.50, 955 (1936) gstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewen- [10] M. Girardeau, J. Math. Phys. 1, 516 (1960). stein, Phys.Rev.Lett. 83, 5198 (1999) [11] T. Tsuzuki, J. Low Temp. Phys.4, 441 (1971). [29] J. Denschlag,J.E.Simsarian, D.L.Feder,C.W.Clark, [12] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. 64, 1627 (1973) [Sov. Phys.JETP 37, 823 (1973)]. Helmerson,W.P.Reinhardt,S.L.Rolston,B.I.Schnei- [13] E. Lieb, Phys.Rev. 130, 1616 (1963). der, and W.D. Phillips, Science 287, 97 (2000). [14] F. Bloch, Phys.Rev.A 7, 2187 (1973). [30] A. J. Leggett, Rev.Mod. Phys.73, 307 (2001). [15] R. Kanamoto, L. D. Carr, and M. Ueda, Phys. Rev. A [31] Since a necessary (but not sufficient) condition for 81, 023625 (2010). metastability is that the density distribution is homoge- [16] P.P. Kulish, S.V.Manakov, and L. D.Faddeev,Theor. neous,whilerepulsiveinteractionsarenecessarytocreate Math. Phy.28, 615 (1976). persistent currents, themetastability of superflow works [17] M. Ishikawa and H. Takayama, J. Phys. Soc. Jpn. 49, against the observation of thesolitary waves with an in- 1242 (1980). homogeneous density distribution. [18] A. D. Jackson, G. M. Kavoulakis, B. Mottelson, and S.