Three-dimensional Gross-Pitaevskii solitary waves in optical lattices: stabilization using the artificial quartic kinetic energy induced by lattice shaking
M. Olshanii, S. Choi, V. Dunjko, A. E. Feiguin, H. Perrin, J. Ruhl, D. Aveline
TThree-dimensional Gross-Pitaevskii solitary waves in optical lattices: stabilizationusing the artificial quartic kinetic energy induced by lattice shaking
M. Olshanii a, ∗ , S. Choi a , V. Dunjko a , A. E. Feiguin b , H. Perrin c , J. Ruhl a , D. Aveline d a Department of Physics, University of Massachusetts Boston, Boston Massachusetts 02125, USA b Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA c Laboratoire de physique des lasers, CNRS, Universit´e Paris 13, Sorbonne Paris Cit´e, 99 avenue J.-B. Cl´ement, F-93430 Villetaneuse,France d Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Abstract
In this Letter, we show that a three-dimensional Bose-Einstein solitary wave can become stable if the dispersion law ischanged from quadratic to quartic. We suggest a way to realize the quartic dispersion, using shaken optical lattices.Estimates show that the resulting solitary waves can occupy as little as ∼ / N = 10 atoms, thus representing a fully mobile macroscopic three-dimensionalobject. Keywords:
Ultracold atoms, Matter waves, Solitary waves, Dispersion management, Shaken lattice
PACS:
1. Introduction
Creating mobile self-supporting three-dimensional (3D)matter waves—3D analogs of the 1D solitons first realizedin Refs. [2, 3]—is a long-standing goal of physics of ultra-cold gases [4, 5, 6, 7, 8]. 3D Bose-Einstein solitary wavesin continuum space are unstable because they violate [7]the Vakhitov-Kolokolov (VK) stability criterion [9]. Onestrategy for circumventing this problem is to create dis-crete solitary waves in optical lattices. These objects maybe stable for some sets of parameters [4, 7, 5], but are lo-calized on a limited number of sites. Thus they occupy a ∗ Corresponding author
Email address: [email protected] (M. Olshanii) A solitary wave is an isolated wave that maintains its shape dueto a balancing of dispersion and nonlinear attraction; in strict usage[1], a soliton is a solitary wave with a special property: when itcollides with another local disturbance—e.g. another soliton—thenasymptotically far from the collision, it regains its initial shape andvelocity. Regardless of how they may be realized, 3D solitary wavesare not expected to be solitons, because they are not expected toinstantiate any integrable systems. substantial portion of the Brillouin zone and so have lim-ited mobility. Another stabilizing strategy is to oscillatein time the interatomic coupling constant around a nega-tive value, but switching to positive for periods of time [6](the VK criterion does not apply to time-dependent non-linearities). However, here substantial atomic losses willlimit the lifetime of the 3D object. In a third approach,3D solitary waves are stabilized by varying the nonlinear-ity strength in space [8]; but then they are not free objectsmoving in a translationally invariant medium.Here we propose a way to satisfy the VK criterion bychanging the atomic dispersion law from quadratic to quar-tic. We suggest using shaking to couple the lowest and thefirst-excited energy bands in a 3D optical lattice so that thequadratic portions of the dispersions cancel out, and thequartic terms become dominant. This results in a highlymobile localized object. Preprint submitted to Elsevier September 17, 2018 a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p . Dispersion law and the stability of solitary mat-ter waves Let us discuss how the dispersion law enters the VKcriterion. Our main conclusion, given in Eq. (14) below,may also be reached using the well-known heuristic argu-ment based on how the kinetic and interaction energiesscale with the typical length (we will summarize this rea-soning following that equation); but because of the nov-elty of the situation, we prefer to use the more rigorousroute via the VK criterion. Consider a time-dependentnonlinear Schr¨odinger equation for the wavefunction of a d -dimensional Bose-Einstein condensate with a generalizedkinetic energy represented by a particular differential oper-ator of p -th ( p being even) degree (below we will proposea scheme, using ultracold gases, for realizing a physicalsystem described by Eq. (1) with d = 3 and p = 4): i (cid:126) ∂ Ψ ∂t = η d (cid:88) i =1 (ˆ p i ) p Ψ + g | Ψ | Ψ , (1)where η > g < p i ≡ − i (cid:126) ∂∂r i is the operator of the i thcomponent of momentum. Assume that Eq. (1) admitsnormalizable stationary solutionsΨ( r , t ) = exp[ − iµt/ (cid:126) ] ψ ( r ) : (2)their spatial part ψ ( r ) will then be governed by a time-independent nonlinear Schr¨odinger equation, η d (cid:88) i =1 (ˆ p i ) p ψ + g | ψ | ψ = µψη > , g < . (3)Any solution of (3) is a point of extremum, subject to thenormalization constraint that (cid:90) d d r | ψ | = N (4)be the number of particles, of the energy functional E = T + V . (5) Here T = η (cid:126) p d (cid:88) i =1 (cid:90) d d r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ p/ ψ∂r p/ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (6) V = 12 g (cid:90) d d r | ψ | (7)are the kinetic energy and the interatomic interaction en-ergy functionals, respectively. The corresponding varia-tional space is the space of continuous functions of thecoordinates; the chemical potential µ enters as the La-grange multiplier enforcing the constraint in Eq. (4). Wecan conjecture a stability criterion for localized station-ary solutions of Eq. (1) already at the level of the virialtheorem: regarding the interaction functional in Eq. (7)as a mean-field approximation of the pairwise interactionenergy with the d -dimensional δ -function as the potential,the virial theorem [10] would predict pT = − dV , and sub-sequently E = d − pd T . (8)Now, one would expect the energy of a stationary state tobe below the energy of a dilute, heavily delocalized cloud;the energy of the latter is close to zero, so the right-hand-side of Eq. (8) must be negative. At this point we con-jecture that for solitary waves to exist, the dispersion lawmust be sufficiently sharp, namely p > d . Next, we willconfirm this using the rigorous VK criterion.First, observe that if one multiplies the stationary non-linear wave equation in Eq. (3) from the left by Ψ ∗ , inte-grates it over all space, and combines the result with theexpression for the energy given in Eqs. (5, 6, 7), one ob-tains [7] E = µN − V . Combining this with Eq. (8), weobtain a relationship between the chemical potential andthe kinetic energy that, in particular, provides informationabout the sign of the chemical potential: µ = d − pd TN . (9)The VK criterion deals with continuous families of sta-tionary localized states parametrized by their norm. The2caling properties of the constituents of Eq. (3) imply thatmembers of such a family will be connected by scalingtransformations. Indeed, it is easy to show that if thereexists a stationary solution ψ ( r )—corresponding to thechemical potential µ and the number of atoms N —ofthe stationary wave equation in Eq. (3), then ψ ( r ) = λ ( N ) − p/ ψ ( r /λ ( N )) (10) µ = λ ( N ) − p µ , (11)with λ ( N ) = (cid:18) NN (cid:19) d − p , (12)is also a solution. According to the VK criterion, a memberof the family of localized stationary solutions in Eqs. (10, 11)is dynamically stable—in particular against collapse or dis-persion to infinity—if its chemical potential decreases withthe number of particles, dµdN < , (13)where the derivative is taken along the family. Using theresults in Eqs. (9 -12), it is easy to show that in our case, dµdN = − p (2 p − d ) d ( p − d ) T N (cid:18) NN (cid:19) dp − d . So Eq. (1) supports stable stationary solitary waves if p > d or 0 < p < d . (14)The second case does not currently seem physical. Recallthat p is required to be even; then, even for d = 3, thereis no p yielding the second inequality.The heuristic argument for the same conclusion goesas follows: let (cid:96) be the typical length scale of the atomiccloud. On dimensional grounds, the kinetic energy scalesas ∼ +1 /(cid:96) p , while the interaction energy, which is propor-tional to the density, scales as ∼ − /(cid:96) d . The matter wavewill be stable if the total energy has a minimum.Consider first the case p > d . Then the kinetic energydominates for small (cid:96) ( E → + ∞ ), and the interactionenergy for large (cid:96) ( E → (cid:96) , and assume, as is reasonable, that it is asmooth function of it. Then dE/d(cid:96) is negative for small (cid:96) ,and positive for large (cid:96) ; thus, for at least one value of (cid:96) , dE/d(cid:96) switches the sign from negative to positive, so thisvalue of (cid:96) is a local minimum. dE/d(cid:96) could cross zero atmultiple values of (cid:96) , but at the smallest and the largest ofthese, dE/d(cid:96) must cross from negative to positive values,in order to match the asymptotic behavior. In any case,there are local minima; moreover, because E ( (cid:96) ) is smooth,it follows that at least one of these local minima must bea global minimum.Now assume p < d . To show that E ( (cid:96) ) can have noglobal or local minimum, additional assumptions will beneeded: the reason is that while dE/d(cid:96) must in this caseswitch sign from positive to negative at least once (andif more than once, the first and the last switch must befrom positive to negative, to match the asymptotic be-havior), there could, conceivably, also exist intermediatezeros of dE/d(cid:96) where the switch is from negative to posi-tive. These would be local minima, which would mean apotentially stable matter wave. However, heuristically, itseems unlikely that E ( (cid:96) ) could have such an intricate struc-ture, with at least three local extrema. We can strengthenthis argument by assuming that the functional form ofthe matter field wavefunction ψ must belong to a one-parametric family, parametrized by (cid:96) . Then, on dimen-sional grounds, we must have ψ ( r ) = ϕ ( r /(cid:96) ) /(cid:96) d/ , where ϕ ( z ) does not depend on (cid:96) . From Eqs. (5-7) it then fol-lows that E ( (cid:96) ) = A/(cid:96) p − B/(cid:96) d , where A and B are positiveand do not depend on (cid:96) . Then E (cid:48) ( (cid:96) ) has a single zero, at (cid:96) = ( dB / pA ) / ( d − p ) , for which E (cid:48)(cid:48) ( (cid:96) ) = ( p − d ) pA (cid:18) dBpA (cid:19) p +2 p − d . So, if p < d , E (cid:48)(cid:48) ( (cid:96) ) <
0, so (cid:96) is a global maximum, andthere are no local or global minima. This seems the extentto which the heuristic argument can be pushed, and is onereason why it is useful to have the rigorous VK criterion.The standard 1D nonlinear Schr¨odinger equation ( p =3, d = 1) is known to support solitons [11, 12, 2, 3] andindeed it satisfies the first inequality in Eq. (14). To thecontrary, in 3D, the localized structures are known to col-lapse [13, 14]: and indeed, the combination p = 2 and d = 3 violates both inequalities in Eq. (14). We will nowsuggest a way to boost the dispersion sharpness to p = 4using dispersion management in optical lattices, and in sodoing stabilize the 3D solitary waves.
3. Realizing a quartic dispersion law in a shakenoptical lattice
Consider a single atom in a shaken 3D optical lattice:ˆ H = ˆ p m + W (cid:88) α = x, y, z cos[ κ ( r α − ξ cos ωt )] , (15)where m is the atomic mass, 2 W the lattice depth, κ =2 π/a lat the lattice wavevector, a lat the lattice spacing, ω the shaking frequency, ξ the shaking amplitude (shakingis applied in the grand-diagonal direction), and r x, y, z = x, y, z . We look for the solutions of the correspondingtime-dependent Schr¨odinger equation using a separation-of-variables ansatz: φ ( r , t ) = (cid:81) α = x, y, z φ ( α ) ( r α , t ). Thetime-dependent Schr¨odinger equations are identical for eachof the three factors. From now on, let us concentrate onthe time evolution of φ ( x ) . The results can be trivially re-cast for φ ( y ) and φ ( z ) , and we will assemble them into asingle 3D solution in the end.The time-evolution equation for φ ( x ) is i ∂∂t φ ( x ) ( x, t ) = ˆ H ( x ) φ ( x ) ( x, t ) (16)with ˆ H ( x ) = ˆ p x m + W cos[ κ ( x − ξ cos ωt )] , where ˆ H ( x ) enters the full Hamiltonian through ˆ H = ˆ H ( x ) +ˆ H ( y ) + ˆ H ( z ) . Assume that ξ (cid:28) a lat and Taylor expand ˆ H ( x ) to the first power in ξ :ˆ H ( x ) ≈ ˆ H ( x )0 + ˆ U ( x ) = ˆ p x m + W cos κx + κξW (sin κx )(cos ωt ) , where the operators ˆ H ( x )0 = ˆ p x m + W cos κx and ˆ U ( x ) = κξW (sin κx )(cos ωt ) are regarded as an “unperturbed Hami-tonian” and a “perturbation,” respectively. The corre-sponding time-dependent Schr¨odinger equation supportsFloquet-type solutions, φ ( x ) ( r , t ) = (cid:32) + ∞ (cid:88) n = −∞ φ ( x ) n ( r ) e inωt (cid:33) e − i E t/ (cid:126) , (17)where E is the Floquet quasi-energy. The states | χ ( x ) ( x ) (cid:105) = (cid:80) + ∞ n = −∞ φ ( x ) n ( x ) | n (cid:105) are the eigenstates of the Floquet Hamil-tonian: ˆ H ( x ) | χ ( x ) ( x ) (cid:105) = E ( x ) | χ ( x ) ( x ) (cid:105) (18)withˆ H ( x ) = ˆ p x m + W (cos κx ) + + ∞ (cid:88) n = −∞ n (cid:126) ω | n (cid:105)(cid:104) n | + 12 κξW (sin κx ) + ∞ (cid:88) n = −∞ ( | n + 1 (cid:105)(cid:104) n | + | n (cid:105)(cid:104) n + 1 | ) . (19)Here, the states | n (cid:105) lie in a Floquet Hilbert space.The Floquet Hamiltonian in Eq. (19) is also periodicin space. Following the Bloch theorem, we will be lookingfor the eigenstates of the Floquet Hamiltonian that havethe form | χ ( x ) , K x ( x ) (cid:105) = (cid:32) + ∞ (cid:88) (cid:96) = −∞ | χ ( x ) , K x , (cid:96) (cid:105) √ d exp[ i(cid:96)κx ] (cid:33) e iK x x , where K x is the x -component of the Bloch vector.Denote the eigenstates and the eigenvalues of the Flo-quet Hamiltonian in Eq. (19), labeled by indexes ¯ n =0 , ± , ± , . . . and s = 0 , , , . . . , as | χ ( x ) , K x , ¯ n, s ( x ) (cid:105) and E ( x ) , ¯ n, s ( K x ) respectively. For zero shaking, the eigen-states and eigenenergies are related to the eigenstates χ ( x ) , K x , s ( x ) and the eigenenergies E ( x ) , s ( K x ) of the sta-tionary lattice as: | χ ( x ) , K x , ¯ n, s ( x ) (cid:105) ξ → −→ χ ( x ) , K x , s ( x ) | n = ¯ n (cid:105) (20) E ( x ) , ¯ n, s ( K x ) ξ → −→ E ( x ) , s ( K x ) + (cid:126) ω ¯ n . (21)For a shaking frequency close to the transition frequencybetween the ground and the first excited energy bands in4he middle of the Brillouin zone, (cid:126) ω ≈ E ( x ) , s =10 ( K x = 0) − E ( x ) , s =00 ( K x = 0) , (22)these bands hybridize [15, 16, 17, 18, 19]. Observe that fora deep blue detuning, the bands will exchange roles and theparabolic dispersion law in the ground band will becomean inverted parabola. By continuity, for each shaking am-plitude, there will exist a shaking frequency for which thequadratic term in the dispersion law vanishes, the quar-tic term becoming dominant. Below, we analyze a sampleset of parameters where this phenomenon indeed happens.The resonance condition in Eq. (22) allows us to use de-generate perturbation theory, truncating the Hilbert spaceto a 2D space spanned by the states χ ( x ) , K x , s =00 ( x ) | n = 0 (cid:105) and χ ( x ) , K x , s =10 ( x ) | n = − (cid:105) , with the unperturbed energies E ( x ) , s =00 ( K x ) and E ( x ) , s =10 ( K x ) − (cid:126) ω , respectively. Our con-struction of a 3D quartic dispersion curve is completed bythe observation that the identical dispersion law will begenerated in the other two spatial directions, and that theresulting dispersion hypersurface is simply the sum of thethree.
4. An outline of a scheme for experimental real-ization
Let us now discuss a sample experimental realizationof the above scheme. Consider a set of parameters in-spired by the scheme used to induce a long-range ferro-magnetic order in an atomic gas [19], as follows. N = 10 Cs atoms are loaded in a 3D optical lattice with lat-tice spacing a lat = 532 nm and depth 2 W = 6 . E R , where E R = (cid:126) k / m is the recoil energy, and k = κ/ π/a lat is the wavevector of the lattice light. The interactions arerepulsive and some trapping will also be required at thisstage. We are going to postpone the choice of the scatter-ing length till later. Initially, there is no shaking. Thena low-frequency shaking starts, with the amplitude √ ξ ,in the grand diagonal direction, where ξ = 0 . a lat isthe shaking amplitude along each of the three 1D-lattice direction. Subsequently, the shaking frequency is slowlyramped to ω = 4 . E R / (cid:126) , a frequency slightly to the redfrom the interband spacing in the center of the Brillouinzone. For the same frequency, the edges of the zone un-dergo a complete adiabatic population inversion duringthe ramp. The final shaking frequency is tuned in sucha way that the quadratic terms in the dispersion law ofthe—heavily hybridized—ground band are canceled: theremaining curve is the 3D inverted quartic parabola de-picted in Fig. 1. Following the analogy with the 1D gapsolitons [20], we concentrate on the time-evolution of thecomplex conjugate of the wavefunction. Its time evolutionis governed by a non-inverted quartic dispersion law: i (cid:126) ∂ ˜Ψ ∂t = ˜ η (cid:126) (cid:32) ∂ ˜Ψ ∂x + ∂ ˜Ψ ∂y + ∂ ˜Ψ ∂z (cid:33) + ˜ g | ˜Ψ | ˜Ψ (23)˜ η > , ˜ g < , where ˜Ψ ≡ Ψ ∗ , ˜ η ≡ − η , and ˜ g ≡ − g . Accordingly, thestability of the solitary wave will require repulsive inter-actions; as an example, we set the scattering length to a S = 2 . a Bohr . At this point, trapping can be removed,following a numerically optimized schedule, and the re-sulting atomic cloud should be entirely self-supporting.The dispersion constant entering the time-dependent waveequation in Eq. (23) will have the value˜ η ≡ −
4! ( d E ( α ) , , ( K α ) /dK α ) K α =0 / (cid:126) , while the coupling constant in Eq. (23) will be given by˜ g ≡ − ζ × (4 π (cid:126) a S /m ), where ζ = a lat (cid:90) + a lat / − a lat / dr α |(cid:104) n = 0 | χ ( α ) , , , ( r α ) (cid:105)| ;here α is any of the three Cartesian directions, x , y , or z . Using our chosen set of parameters, this works out to˜ η = 0 . E R ( a lat / (cid:126) ) and ˜ g = − . E R a /N (see Eqs. (20)and (21) for notation). For the above set of parameters, aGaussian variational estimate predicts the wavevector dis-tribution diameter of ∆ K ≡ (cid:104) K (cid:105) ) / ≈ . π/a lat );for this degree of momentum localization, the solitary wave5 igure 1: (color online). Floquet energies E ( x ) , , ( K x ) (red dashedlines) and E ( x ) , − , ( K x ) (blue dotted line), the adiabatic descen-dants of the ground and the first excited bands, respectively, as func-tions of the Bloch vector K x . The parameters used correspond to asystem of Cs atoms in an optical lattice with depth 2 W = 6 . E R and spacing a lat = 532 nm, shaken in the grand-diagonal directionwith frequency ω = 4 . E R / (cid:126) and amplitude √ × . d . Thecurves are calculated using degenerate perturbation theory appliedto the Floquet Hamiltonian. For our set of parameters, quadraticterms in the ground band dispersion are canceled, and the formerlysub-leading inverted quartic dispersion (the red solid line in the inset)becomes dominant. Dispersion laws in the two remaining directions, y and z , are identical to the one along x . Inset: a magnified view ofthe peak of the Floquet energy E ( x ) , , ( K x ) (red dashed line), show-ing that it is very close to the pure—formerly sub-leading—invertedquartic dispersion (the red solid line). The axes labels of the mainplot are also the axes labels for the plot in the inset. would occupy only ∼ / (cid:15) dd = .
25 (see p. 70of Ref. [21]).
5. Conclusion and outlook
We have shown that a 3D Bose-Einstein condensatewith a quartic dispersion law satisfies the VK criterion forthe stability of localized structures in a dispersive non-relativistic self-focussing medium, and that such a disper-sion law can be realized in shaken optical lattices [15, 16,17, 18, 19]. Experimentally, as in the realization of band-gap solitons [20], our scheme will involve an inverse quarticdispersion and repulsive condensates.We see rotation sensing as the primary area of ap-plication of 3D solitary waves. Interferometry with self-attracting waves was a success from the onset [22] (seealso [23, 24, 25, 26]), showing a manyfold increase in thefringe visibility as compared to free waves. Further im-provements in sensitivity are expected for slow splittingof solitary waves on barriers [27] where the appearance ofmacroscopic quantum superpositions is predicted [28, 29].But while linear solitary wave interferometers are ideallysuited for acceleration sensing, Sagnac measurements re-quire a multi-dimensional geometry. There are waveguide-based proposals [30]; their principal drawback—in addi-tion to the fact that the loading of atoms into the guideis difficult, and that the requirements for the stability ofthe guiding fields are very strict—is the inflexibility ofthe interferometric scheme. In developing an interfero-metric scheme, one starts with a seed idea and then buildsupon it by successive modifications–usually strengtheningthe beamsplitters in some way—that result in increasesin the interferometric area (space-time for the gravitome-ters/accelerometers, and space-space for the rotation sen-sors). For waveguide-based schemes, this requires rebuild-ing the waveguide each time, with all the accompanyingoptimization and stabilization issues. In contrast, schemesbased on free-space propagation allow for easy implemen-tation of improvements. The present work paves the wayfor such schemes, by proposing a method to obtain solitarywaves that propagate in free space.6 cknowledgements
We thank David Campbell for his remarks. This workwas supported by grants from the US National ScienceFoundation (
PHY-1402249 ) and the US Office of NavalResearch (
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