Bose-Einstein Condensates in Superlattices
arXiv:nlin/0406063v1 [nlin.PS] 26 Jun 2004
Bose-Einstein Condensates in Superlattices Mason A. Porter [email protected] School of Mathematics and Center for Nonlinear Science Georgia Institute of Technology Atlanta, GA 30332, USA P. G. Kevrekidis Department of Mathematics and Statistics University of Massachusetts, Amherst MA 01003, USA February 26, 2008 Abstract We consider the Gross-Pitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigar-shaped Bose-Einstein condensates (BECs) in such lattice potentials. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability numerically. We find that the superlattice can operate as an efficient (symmetric and asymmetric) matter-wave splitter. We then focus on spatially extended wavefunctions in the form of modulated amplitude waves. With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice wavenumber as well as the two types of mixed resonances that result from interactions of both wavenumbers of the superlattice. In each case, we determine the resulting system’s equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions by direct simulations of the GP equation, identifying them as typically stable solutions of the model.
PACS: 05.45.-a, 03.75.Lm, 05.30.Jp, 05.45.Ac
1
Introduction
At low temperatures, particles in a dilute gas can reside in the same quantum (ground) state, forming a Bose-Einstein condensate.15, 20, 34, 50 This was first observed experimentally in 1995 with vapors of rubidium and sodium.3, 21 In 1
these experiments, atoms were confined in magnetic traps, evaporatively cooled to temperatures on the order of fractions of microkelvins, left to expand by switching off the confining trap, and subsequently imaged with optical methods.20 A sharp peak in the velocity distribution was observed below a critical temperature, indicating that Bose-Einstein condensation had occurred. Under the typical confining conditions of experimental settings, BECs are inhomogeneous, so condensation can be observed in both momentum and coordinate space. The number of condensed atoms N ranges from several thousand (or less) to several million (or more). The magnetic traps that confine them are usually approximated well by harmonic potentials. There p are two characteristic length scales: the harmonic oscillator length aho = ~/(mωho ) [which is on the order of a few microns], where ωho = (ωx ωy ωz )1/3 is the geometric mean of p the trapping frequencies, and the mean healing length χ = 1/ 8π|a|¯ n, where n ¯ is the mean particle density and a, the (two-body) s-wave scattering length, is determined by the atomic species of the condensate.5, 20, 38, 50 Interactions between atoms are repulsive when a > 0 and attractive when a < 0. For a dilute ideal gas, a ≈ 0. The length scales in BECs should be contrasted with those in systems like superfluid helium, in which the effects of inhomogeneity occur on a microscopic scale fixed by the interatomic distance.20 If considering only two-body, mean-field interactions, a dilute Bose-Einstein gas can be modeled using a cubic nonlinear Schr¨odinger equation (NLS) with an external potential, which is also known as the Gross-Pitaevskii (GP) equation. BECs are modeled in the quasi-one-dimensional (quasi-1D) regime when the transverse dimensions of the condensate are on the order of its healing length and its longitudinal dimension is much larger than its transverse ones.9–11, 20 In this regime, one employs the 1D limit of a 3D mean-field theory rather than a true 1D mean-field theory, which would be appropriate were the transverse dimension on the order of the atomic interaction length or the atomic size.6, 9–11, 58 When considering only two-body interactions, the condensate wavefunction (“order parameter”) ψ(x, t) satisfies a GP equation of the form i~ψt = −[~2 /(2m)]ψxx + g|ψ|2 ψ + V (x)ψ ,
(1)
where |ψ|2 is the number density,p V (x) is the external trapping potential, g = [4π~2 a/m][1 + O(ζ 2 )], and ζ = |ψ|2 |a|3 is the dilute gas parameter.5, 20, 38 Because the scattering length a can be adjusted using a magnetic field in the vicinity of a Feshbach resonance,25 the contribution of the nonlinearity in (1) is tunable.36 Potentials V (x) of interest include harmonic traps,20 periodic lattices2, 14, 18, 19, 27, 31, 45 and superlattices49, 57 (which can be either periodic or quasiperiodic), and superpositions of lattices or superlattices with harmonic traps. The existence of quasi-1D (“cigar-shaped”) BECs motivates the study of lower dimensional models such as Eq. (1). The case of periodic and quasiperiodic potentials without a confining trap along the dimension of the lattice is of particular theoretical and experimental interest. Such potentials have been used, for example, to study Josephson effects,2 squeezed states,47 Landau-Zener tunneling and Bloch 2
oscillations,45 and the transition between superfluidity and Mott insulation at both the classical17, 60 and quantum27 levels. Moreover, with each lattice site occupied by one alkali atom in its ground state, BECs in optical lattices show promise as a register in a quantum computer.54, 62 In experiments, a weak harmonic trap is typically used on top of the lattice or superlattice to prevent the particles from escaping. The lattice is also generally turned on after the trap. If one wishes to include the trap in theoretical analyses, V (x) is modeled by V (x) = V1 cos(κ1 x) + V2 cos(κ2 x) + Vh x2 ,
(2)
where κ1 is the primary lattice wavenumber, κ2 > κ1 is the secondary lattice wavenumber, V1 and V2 are the associated lattice amplitudes, and Vh represents the magnitude of the harmonic trap. (Note that V1 , V2 , Vh , κ1 , and κ2 can all be tuned experimentally.) The sinusoidal terms dominate for small x, but the harmonic trap otherwise becomes quickly dominant. When Vh ≪ V1 , V2 , the potential is dominated by its periodic contributions for many periods.14, 16, 22, 52 BECs in optical lattices with up to 200 wells have been created experimentally.48 In this work, we let Vh = 0 and focus on optical lattices and superlattices. Spatially periodic potentials have been employed in experimental studies of BECs2, 27, 29, 45, 47, 54 and have also been studied theoretically.1, 8–11, 16, 18, 22, 23, 37, 40, 42–44, 46, 51–53, 60, 61, 63 In recent experiments, BECs were loaded into superlattices with κ2 = 3κ1 .49 However, there has thus far been very little theoretical research on BECs in superlattices.24, 26, 41, 57 It is also worth noting that from the perspective of KAM theory, the value of the ratio κ2 /κ1 has important dynamical ramifications.28, 55 For example, if this ratio is irrational, the superlattice is quasiperiodic rather than periodic. We therefore consider both rational and irrational wavenumber ratios. We investigate in this study both localized and spatially extended waves of BECs in periodic and quasiperiodic superlattices. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability/instability numerically. We also show that the superlattice can operate as an efficient symmetric or asymmetric matter-wave splitter. In contrast, a regular lattice can only split matter waves symmetrically. We then focus on spatially extended wavefunctions in the form of modulated amplitude waves. We apply a coherent structure ansatz to Eq. (1), which yields a parametrically forced Duffing equation describing the spatial evolution of the field. We employ secondorder multiple scale perturbation theory7, 30, 55, 56 to study its periodic orbits (called “modulated amplitude waves” and denoted MAWs), and illustrate their dynamical stability with numerical simulations of the GP equation. We consider harmonic (1 : 1) resonances and two types of mixed resonances—resulting from, respectively, “additive” (2 : 1 + 1) and “subtractive” (2 : 1 − 1) mixing— all of which arise at the O(ε2 ) level of analysis. Previous perturbative studies of resonance phenomena in BECs have utilized an O(ε) analysis.51–53 Additionally, as mixed resonances arise from the interaction of multiple superlattice wavenumbers, they cannot occur in BECs loaded into regular lattices. 3
We structure our presentation as follows: In section 2, we discuss localized (solitary wave) solutions of the GP equation in superlattices. In section 3, we introduce modulated amplitude waves, and in section 4 we derive and solve “slow flow” dynamical equations that describe the resonance phenomena under consideration. We corroborate our results and test the stability of the MAWs with direct numerical simulations of the GP equation. Finally, in section 5, we summarize our findings and present our conclusions.
2
Localized Solutions
We first examine exponentially localized, solitary wave solutions and their stability and dynamical behavior in the presence of a superlattice potential. In particular, we let ~ = 2m = 1 and consider Eq. (1) with attractive interactions. With g = −1 (for simplicity), it is well-known that in the absence of a potential, the equation has an exact (bright) soliton solution of the form: √ √ (3) ψ(x, t) = eiωt 2ωsech ω(x − x0 ) ,
where ω = −µ is the frequency of this standing wave, µ denotes the BEC’s chemical potential, and x0 is the arbitrary position of its center. In our discussion of solitary waves, we focus on attractive interactions, as our theoretical considerations in the presence of the potential can only be applied in that case. For spatially extended states, we will consider both attractive and repulsive interactions. Extending the considerations presented herein for the dark solitary waves solitons known to exist for repulsive interactions35, 40, 41 is a challenging theoretical problem for future study. In the presence of the potential, Eq. (1) is a perturbed Hamiltonian system with perturbation energy given by Z ∞ E1 [ψ] = V (x)|ψ|2 dx . (4) −∞
Following the recent work of Kapitula, Kevrekidis, and Sandstede,33 we evaluate the reduced Hamiltonian by inserting Eq. (3) into Eq. (4) to obtain H(x0 ) =
X
j=1,2
2Vj
πκj cos(κj x0 ) κ π sinh 2√j ω
(5)
According to this recently-developed theory,32, 33 the perturbed wave chooses values of x0 so that H ′ (x0 ) = 0; that is, the selected center positions x0 are critical points of the reduced Hamiltonian. The small, nonzero linear stability eigenvalues (of the linearization around the solitary wave) in the presence of the perturbation are then given by λ2 = −ω 1/2 H ′′ (x0 ) ,
4
(6)
−20 4
1.4 −10 1.2
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Figure 1: The evolution of the unstable soliton for a superlattice potential with V2 = 2V1 = 2; κ2 = 3κ1 = 3. The initial condition consists of a solitary wave with ω = 1 centered around the maximum at x0 = 0. The unstable soliton splits into two symmetric fragments. as recently demonstrated.33 Hence, the soliton is unstable at the maxima of the superlattice potential. However, at the minima of this potential, the soliton is linearly stable. We tested this prediction through direct numerical simulations of Eq. (1) and obtained a variety of interesting conclusions indicating that the superlattice potential of Eq. (2) can be used as a soliton (or “condensate”) splitter. Moreover, it can be used for both symmetric and asymmetric splitting that should be experimentally observable. In our numerical simulations, it was necessary to utilize periodic boundary conditions, which imposed the use of rational ratios between κ1 and κ2 . In particular, the relation κ2 = 3κ1 , which has been achieved in experimental superlattices,49 was employed in the dynamical evolution described below. Placing the soliton in a potential maximum around which the potential is symmetric leads to a symmetric splitting of the solitary wave, as shown in Fig. 1 (in which V2 = 2V1 = 2). If, on the other hand, the potential is asymmetric around the maximum on which the soliton is placed, then the corresponding solitonic splitting is also asymmetric, as demonstrated in Fig. 2. Hence, the superlattice can be used as an efficient, controllable matter-wave splitter. If we initialize the soliton at the minimum of the potential, then in accord with the prediction of Eq. (6), the solitary matter wave persists as a robust structure at this local minimum (even if it is not also a global minimum); see Fig. 3 for an example of this behavior. It should be noted, however, that using very recent techniques,59 matter can be transferred to the deepest well.
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Figure 2: Same as Fig. 1, but now the soliton is initially centered around the maximum at x0 = −10.8. The unstable soliton splits into two asymmetric fragments.
−20
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Figure 3: Same as Fig. 1, but now the soliton is initially centered around the local minimum at x0 = −13.2. It remains there under time evolution.
6
3
Modulated Amplitude Waves
To decompose the solutions of interest, we use the ansatz ψ(x, t) = R(x) exp (i [θ(x) − µt]) .
(7)
When the above (temporally periodic) coherent structures (7) are also spatially periodic, they are called modulated amplitude waves (MAWs).12, 13 The orbital stability of MAWs for the cubic NLS with elliptic potentials has been studied by Bronski and co-authors.9–11 To obtain stability information about sinusoidal potentials, one takes the limit as the elliptic modulus k approaches zero.39, 55 When V (x) is periodic, the resulting MAWs generalize the Bloch modes that occur in the theory of linear systems with periodic potentials, as one is considering a nonlinear Floquet-Bloch theory rather than a linear one.4, 8, 18, 40, 56 In this work, we extend recent studies51, 52 of the dynamical behavior of MAWs of BECs in lattice potentials to superlattice potentials. Inserting (7) into Eq. (1), equating the real and imaginary components of the resulting equation, and defining S := R′ yields the following two-dimensional system of nonlinear ordinary differential equations: R′ = S , S′ =
c2 2mµR 2mg 3 2m − + 2 R + 2 V (x)R . R3 ~ ~ ~
The parameter c is defined via the relation θ′ (x) =
c , R2
(8)
which plays the role of conservation of “angular momentum,” as discussed by Bronski and coauthors.10 Constant phase solutions (i.e., standing waves), which constitute an important special case, satisfy c = 0. In the rest of the paper, we restrict ourselves to this class of solutions, so that R′ = S , 2mµR 2mg 3 2m + 2 R + 2 V (x)R . S′ = − ~ ~ ~
(9)
When V (x) = 0, Eq. (9) is integrable.51, 52 We consider the case with Vh = 0 (which implies, in practice, that the harmonic trap is negligible with respect to the superlattice potential for the domain of interest) and define 2mµ δ˜ := , ~ where
ε˜ α := −
2mg , ~2
2m V˜ (x) := − 2 V (x) ~
V˜ (x) = ε[V˜1 cos(κ1 x) + V˜2 cos(κ2 x)] ,
7
(10)
(11)
˜ α the parameters δ, ˜ , and V˜j are O(1) quantities, and the κj can either be commensurate (rational multiples of each other) or incommensurate, so that the superlattice can be, respectively, either periodic or quasiperiodic. We let κ2 > κ1 without loss of generality. ˜ α For notational convenience, we drop the tildes from δ, ˜ , and V˜j , so that Eq. (9) can be written R′′ + δR + εαR3 + εR[V1 cos(κ1 x) + V2 cos(κ2 x)] = 0 .
(12)
In this paper, we consider the case δ > 0 corresponding to a positive chemical potential.
4
Multiple Scale Perturbation Theory and Spatial Subharmonic Resonances
To employ multiple scale perturbation theory,7, 52, 56 we define “slow space” η := εx and “stretched space” ξ := bx = [1 + εb1 + ε2 b2 + O(ε3 )]x .
(13)
We expand the wavefunction amplitude as R = R0 + εR1 + ε2 R2 + O(ε3 )
(14)
and stretch the argument of the superlattice, so that V˜ (ξ) = V1 cos(κ1 ξ) + V2 cos(κ2 ξ) .
(15)
With this expansion, Eq. (12) becomes 2 2 ∂ 2 R0 ∂ 2 R1 2 ∂ R2 3 + ε + ε + O(ε ) 1 + b1 ε + b2 ε2 + O(ε3 ) ∂ξ 2 ∂ξ 2 ∂ξ 2 2 ∂ R0 ∂ 2 R1 ∂ 2 R2 + 2ε 1 + b1 ε + b2 ε2 + O(ε3 ) +ε + ε2 + O(ε3 ) ∂ξ∂η ∂ξ∂η ∂ξ∂η 2 2 2 ∂ R1 ∂ R2 ∂ R0 +ε + ε2 + O(ε3 ) + ε2 ∂η 2 ∂η 2 ∂η 2 3 + δ R0 + εR1 + ε2 R2 + O(ε3 ) + εα R0 + εR1 + ε2 R2 + O(ε3 ) + ε R0 + εR1 + ε2 R2 + O(ε3 ) [V1 cos(κ1 ξ) + V2 cos(κ2 ξ)] = 0 . (16)
To perform multiple scale analysis, we equate the coefficients of terms of different order (in ε) in turn. At O(1) = O(ε0 ), we obtain ∂ 2 R0 + δR0 = 0 , ∂ξ 2 which has the solution √ √ R0 (ξ, η) = A(η) cos( δξ) + B(η) sin( δξ) , 8
(17)
for some slowly-varying amplitudes A(η), B(η), equations of motion for which arise at O(ε). Equating coefficients at O(ε) yields √ ′ 3 √ ∂ 2 R1 2 2 δB − + δR = 2b δA − 2 αA(A + B ) cos( δξ) 1 1 2 ∂ξ 4 √ √ ′ 3 + 2b1 δB + 2 δA − αB(A2 + B 2 ) sin( δξ) 4 √ √ αA αB [−A2 + 3B 2 ] cos(3 δξ) + [−3A2 + B 2 ] sin(3 δξ) + 4 4 √ √ V1 A V1 A cos([κ1 − δ]ξ) + cos([κ1 + δ]ξ) + 2 2 √ √ V1 B V1 B sin([κ1 − δ]ξ) + sin([κ1 + δ]ξ) − 2 2 √ √ V2 A V2 A cos([κ2 − δ]ξ) + cos([κ2 + δ]ξ) + 2 2 √ √ V2 B V2 B (18) sin([κ2 − δ]ξ) + sin([κ2 + δ]ξ) . − 2 2 For R1 (ξ, η) to be bounded, the coefficients of the √ secular terms in Eq. (18) √ must vanish.7,√56 The harmonics cos( δξ) and sin( δξ) are always secular, √ whereas cos(3 δξ) and sin(3 δξ) are never secular. The other harmonics are secular only in the case of 2 : 1 subharmonic resonances.51, 52 As such spatial resonances have already been studied for quasi-1D BECs in periodic lattices, we will consider the situation in which (18) is non-resonant so that we can turn our attention to other resonant situations. The 2 : 1 resonance√can occur with √ respect to either the primary (κ1 = 2 δ) or secondary (κ2 = 2 δ) wavenumber of the lattice. [Our O(ε2 ) analysis below can be repeated in the presence of 2 : 1 resonances.] At O(ε), we can thus obtain either no resonance, a long-wavelength subharmonic resonance, or a short-wavelength subharmonic resonance. Equating the coefficients of the secular terms to zero in Eq. (18) yields the following equations of motion describing the slow dynamics: √ 3α A′ = −b1 δB + √ B(A2 + B 2 ) , 8 δ √ 3α B ′ = b1 δA − √ A(A2 + B 2 ) . 8 δ Equilibria of (19) correspond to periodic orbits of (1). One finds ! r 8b1 δ (A, B) = ± ,0 , 3α ! r 8b1 δ (A, B) = ± 0, , 3α
9
(19)
(20)
where we note that these two sets of equilibria are π/2 phase shifts of each other. More generally, equilibria satisfy A2 + B 2 =
8b1 δ , 3α
(21)
so that any other phase shift also yields an equilibrium of (19). However, we are more interested in the effects at order O(ε2 ), which we now analyze. At this second order of perturbation theory, BECs in superlattice potentials exhibit some dynamical behavior that cannot occur in BECs in simpler lattice potentials [where, for example, solutions of type of Eq. (21) straightforwardly arise53 ]. Equating coefficients at O(ε2 ) yields ∂ 2 R0 ∂ 2 R0 ∂ 2 R0 ∂ 2 R1 ∂ 2 R1 ∂ 2 R2 2 2 + δR = −(b + 2b ) − − 2b − 2 − 3αR R − 2b 2 2 1 1 1 1 0 ∂ξ 2 ∂ξ 2 ∂η 2 ∂ξ∂η ∂ξ 2 ∂ξ∂η − R1 V1 cos(κ1 ξ) − R2 V2 cos(κ2 ξ) , (22) where one inserts the expressions for R0 , R1 and their derivatives into the righthand-side of (22). To find the secular terms in Eq. (22), we compute √ √ R1 (ξ, η) = C(η) cos( δξ) + D(η) sin( δξ) + R1p (ξ, η) , √ √ R1p (ξ, η) = c1 cos(3 δξ) + c2 sin(3 δξ) 2 h i X √ √ √ √ cj3 cos([κj − δ]ξ) + cj4 cos([κj + δ]ξ) + cj5 sin([κj − δ]ξ) + cj6 sin([κj + δ]ξ) , + j=1
(23)
where j ∈ {1 , 2} and c1 = c2 = cj3 = cj4 = cj5 = cj6 =
α A(A2 − 3B 2 ) , 32δ α B(3A2 − B 2 ) , 32δ Vj A √ , 2κj (κj − 2 δ) Vj A √ , 2κj (κj + 2 δ) Vj B √ , 2κj (2 δ − κj ) Vj B √ . 2κj (κj + 2 δ)
(24)
Inserting (17) and (23) into (22) and expanding this equation trigonometrically yields 19 harmonics (that are also present for sines), which we list in Table 1. We indicate which of these harmonics are always secular, sometimes secular, and never secular. 10
At this order of perturbation theory, one finds 2 : 1 (primary subharmonic), 4 : 1 (secondary subharmonic), 1 : 1 (harmonic), 2 : 1+1 (additively mixed subharmonic), and 2 : 1−1 (subtractively mixed subharmonic) resonances. The first three types of resonances can occur with respect to either κ1 or κ2 , whereas the latter two require the interaction of both superlattice wavenumbers. Harmonic and mixed spatial resonances have not been analyzed previously for BECs, so we study these three situations in this paper. At O(ε), we considered the case √ without 2 : 1 resonances, so the associated resonance conditions (κj = ±2 δ) are necessarily not satisfied at the present [O(ε2 )] stage, as indicated in Table 1. Second-order subharmonic (4 : 1) resonances have been studied using elliptic function perturbation theory,51, 52 so we do √ not analyze them here. Their associated resonance conditions √ are κj = ±4 δ. The resonance relations for harmonic resonances are κj = ± δ. We will consider solutions that have harmonic resonance with respect to the primary lattice spacing κ1 . The resonance √ relation for additively mixed subharmonic resonances is κ2 + κ1 = ±2 √ δ, and that for subtractively mixed subharmonic resonances is κ2 − κ1 = ±2 δ. In the remainder of this section, we consider the non-resonant, harmonically resonant, and the two types of mixed resonant states in turn. It is also important to remark that with the slow spatial variable η = εx, the approximate solutions R(x) obtained perturbatively are valid for |x| . O(ε−1 ) despite the fact that we employ a second-order multiple scale expansion. By incorporating a third (“super slow”) scale ε2 x, which is even more technically demanding than our present perturbative procedure, one can obtain approximate solutions that are valid for |x| . O(ε−2 ).7 Before proceeding, we also note that in light of KAM theory, one expects different dynamical behavior (at least mathematically) depending on whether κ2 /κ1 is an integer, a rational number, or an irrational number. Only the situation κ2 = 3κ1 has been prepared experimentally, so we concentrate on that case in our numerical simulations. Our analytical work is valid for all real ratios κ2 /κ1 . We note additionally that we simulated these modulated amplitude waves using a numerical domain with periodic boundary conditions. This allows us to handle integer or rational values of κ2 /κ1 with appropriate selection of the domain parameters (so that the box size is an integer multiple of both spatial periods). However, quasiperiodic settings cannot be tackled numerically within this framework for the extended wave solutions considered in this section.
11
Label 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Harmonic √ cos( √δξ) cos(3√δξ) cos(5 δξ) √ cos([κ1 − √δ]ξ) cos([κ1 + √δ]ξ) cos([κ2 − √δ]ξ) cos([κ2 + √δ]ξ) cos([κ1 − 3√δ]ξ) cos([κ1 + 3√δ]ξ) cos([κ2 − 3√δ]ξ) cos([κ2 + 3√δ]ξ) cos([2κ1 − √δ]ξ) cos([2κ1 + √δ]ξ) cos([2κ2 − √δ]ξ) cos([2κ2 + δ]ξ) √ cos([κ1 + κ2 − √δ]ξ) cos([κ1 + κ2 + √δ]ξ) cos([κ1 − κ2 − √δ]ξ) cos([κ1 − κ2 + δ]ξ)
Assumed Assumed Assumed Assumed
Secular? Yes No No not in resonance not in resonance not in resonance not in resonance Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes
at at at at
O(ε) O(ε) O(ε) O(ε)
Resonance when secular N/A N/A N/A 2:1 2:1 2:1 2:1 4:1 4:1 4:1 4:1 1:1 1:1 1:1 1:1 2 : 1+1 2 : 1+1 2 : 1−1 2 : 1−1
Table 1: The harmonics in the right-hand-side of Eq. (22) after the formulas for R0 (17) and R1 (23) are inserted. We only list the cosines in this table, but the sines of these harmonics are present as well. We designate which harmonics are always secular, sometimes secular (under an appropriate resonance condition, which we indicate in the text), and never secular.
4.1
The Non-Resonant Case
In the non-resonant case, effective equations governing the O(ε2 ) slow evolution are 1 f1 (α, δ, κ1 , κ2 )B 2 + f2 (α, δ, κ1 , κ2 )A2 + f3 (α, δ, κ1 , κ2 , b1 ) D + f4 (α, δ, κ1 , κ2 )ABC ∆(δ, κ1 , κ2) +f5 (α, δ, κ1 , κ2 )B 5 + f6 (α, δ, κ1 , κ2 )A2 B 3 + f7 (α, δ, κ1 , κ2 )A4 B + f8 (α, δ, κ1 , κ2 , b2 )B , 1 D′ = − f1 (α, δ, κ1 , κ2 )A2 + f2 (α, δ, κ1 , κ2 )B 2 + f3 (α, δ, κ1 , κ2 , b1 ) C + f4 (α, δ, κ1 , κ2 )ABD ∆(δ, κ1 , κ2) +f5 (α, δ, κ1 , κ2 )A5 + f6 (α, δ, κ1 , κ2 )A3 B 2 + f7 (α, δ, κ1 , κ2 )AB 4 + f8 (α, δ, κ1 , κ2 )A , (25) C′ =
where ∆(δ, κ1 , κ2 ) = 256δ 3/2 16δ 2 − 4δκ21 − 4δκ22 + κ21 κ22
12
(26)
and f1 (α, δ, κ1 , κ2 ) = 3f2 (α, δ, κ1 , κ2 ) , f2 (α, δ, κ1 , κ2 ) = 96αδ[16δ 2 − 4δ(κ21 + κ22 ) + κ21 κ22 ] ,
f3 (α, δ, κ1 , κ2 , b1 ) = 256δ 2 b1 [−κ21 κ22 + 4δ(κ21 + κ22 ) − 16δ 2 ] , f4 (α, δ, κ1 , κ2 ) = 2f2 (α, δ, κ1 , κ2 ) ,
f5 (α, δ, κ1 , κ2 ) = 15α2 [−16δ 2 + 4δ(κ21 + κ22 ) − κ21 κ22 ] , f6 (α, δ, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, δ, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8 (α, δ, κ1 , κ2 , b2 ) = 64δ[V12 κ22 + V22 κ21 − 4δ(V12 + V22 + κ21 κ22 b2 ) + 16δ 2 b2 (κ21 + κ22 ) − 64δ 3 b2 ] . (27) In this case, the superlattice does not contribute to O(ε2 ) terms. Equilibrium solutions of (25) satisfy (f1 B 2 + f2 A2 + f3 )(f5 A5 + f6 A3 B 2 + f7 AB 4 + f8 A) − (f4 AB)(f5 B 5 + f6 A2 B 3 + f7 A4 B + f8 B) , f42 A2 B 2 − (f1 B 2 + f2 A2 + f3 )(f1 A2 + f2 B 2 + f3 ) (f1 A2 + f2 B 2 + f3 )(f5 B 5 + f6 A2 B 3 + f7 A4 B + f8 B) − (f4 AB)(f5 A5 + f6 A3 B 2 + f7 AB 4 + f8 A) D= , f42 A2 B 2 − (f1 B 2 + f2 A2 + f3 )(f1 A2 + f2 B 2 + f3 ) (28) C=
where one inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the spatial profile R = R0 + εR1 + O(ε2 ) used as the initial wavefunction in the numerical simulations of the full GP given by Eq. (1). A typical example of the √ non-resonant case is shown in Fig. 4, with V2 = 2V1 = 2 and κ2 = 3κ1 = 12 δ = 3π/(2b), where b is the stretching factor given by Eq. (13). In this simulation, we used b1 = b2 = 1 and ǫ = 0.1. It can be clearly seen that the relevant solution is dynamically stable, which we found to be generic in our numerical experiments. Simulations with rational κ2 /κ1 reveal similar phenomena.
4.2
Resonances
In this subsection, we consider harmonic resonances, additively mixed subharmonic resonances, and subtractively mixed subharmonic resonances. In the evolution equations for the slow dynamics, one inserts the appropriate resonance relation into ∆ and f1 –f7 . The function f8 has both the non-resonant contributions discussed above and additional resonant terms due to the superlattice. Note additionally that there is a symmetry-breaking in the resulting equations because the functional form of the lattice contains only cosine terms. 4.2.1
Harmonic Resonance with a Single Wavelength √ When κj = ± δ, there is a harmonic resonance. The effective equations governing the O(ε2 ) slow evolution in the presence of a harmonic resonance with 13
Figure 4: Same as (the left panel of) Fig. 1, but for the non-resonant spatially extended solution of Eq. (14) with C and D in Eq. (23) given by Eq. (25) [see text for parameter details].
14
respect to the primary lattice wave number κ1 are 1 f1 (α, κ1 , κ2 )B 2 + f2 (α, κ1 , κ2 )A2 + f3 (α, κ1 , κ2 , b1 ) D + f4 (α, κ1 , κ2 )ABC ∆(κ1 , κ2) +f5 (α, κ1 , κ2 )B 5 + f6 (α, κ1 , κ2 )A2 B 3 + f7 (α, κ1 , κ2 )A4 B + f8s (α, κ1 , κ2 , b2 )B , 1 D′ = f1 (α, κ1 , κ2 )A2 + f2 (α, κ1 , κ2 )B 2 + f3 (α, κ1 , κ2 , b1 ) C + f4 (α, κ1 , κ2 )ABD ∆(κ1 , κ2) +f5 (α, κ1 , κ2 )A5 + f6 (α, κ1 , κ2 )A3 B 2 + f7 (α, κ1 , κ2 )AB 4 + f8c (α, κ1 , κ2 )A , (29) C′ =
where ∆(κ1 , κ2 ) = 768κ31 (4κ21 − κ22 )
(30)
and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = 288ακ21 (κ22 − 4κ21 ) ,
f3 (α, κ1 , κ2 , b1 ) = 768κ41b1 (−κ22 + 4κ21 ) , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) , f5 (α, κ1 , κ2 ) = 45α2 (−κ22 + 4κ21 ) , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + 32V12 (κ22 − 4κ21 ) ,
f8c (α, κ1 , κ2 ) = fnon (α, κ1 , κ2 ) − 160V12 (κ22 − 4κ21 ) ,
fnon (α, κ1 , κ2 ) = 192κ21(V22 − 4κ21 κ22 b2 + 16κ41 b2 ) .
(31)
If considering a harmonic resonance with respect to the secondary lattice wavenumber κ2 , one obtains the appropriate equations for O(ε2 ) slow evolution by switching the roles of κ1 and κ2 in f8s and f8c . We recall as well that one needs an O(ε2 ) analysis to study such resonances. Note that the form of equations (31) corresponds to (27) except for the extra terms in f8c and f8s that arise from the superlattice. The equilibria of (29) are given by Eq. (28) except that one inserts the functions from (31). Additionally, the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One also inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the spatial profile R = R0 + εR1 + O(ε2 ) to use as an initial condition in direct numerical simulations of Eq. (1). A typical example of the single-wavelength resonant case is shown in Fig. 5, √ with V2 = 2V1 = 2 and κ2 = 4κ1 = 4 δ = π/b, where b is the stretching factor of Eq. (13); we used b1 = b2 = 1 and ǫ = 0.1. The resulting quasiperiodic (in space) patterns were generically found to persist in the dynamics of the system as stable (temporally oscillating) solutions. 15
Figure 5: Same as (the left panel of) Fig. 1, but for the harmonic resonant case with respect to the primary lattice wavelength. The solution given by Eq. (14) is used as an initial condition with C and D in Eq. (23) given by Eq. (29) with the functions (30)-(31) [see text for parameter details].
16
4.2.2
Resonances Due to Interactions Between the Two Wavelengths
Studying BECs in a superlattice rather than in a periodic lattice allows one to examine the spatial resonances caused by interactions between the two lattice wavelengths. As with harmonic resonances, an O(ε2 ) calculation is required to perform the analysis in√this case. When κ2 + κ1 = ±2 δ, one has an additively mixed subharmonic resonance. The effective equations governing the O(ε2 ) slow evolution in this case are (29) with ∆(κ1 , κ2 ) = 32κ1 κ2 (κ1 + 2κ2 )(2κ1 + κ2 )(κ1 + κ2 )3 (32) and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = −24ακ1 κ2 [2(κ41 + κ42 ) + 9(κ31 + κ32 ) + 14κ21 κ22 ] ,
f3 (α, κ1 , κ2 , b1 ) = 16κ1 κ2 b1 [2(κ61 + κ62 ) + 13κ1 κ2 (κ41 + κ42 ) + 34κ21 κ22 (κ21 + κ22 ) + 46κ31 κ32 ] , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) , f5 (α, κ1 , κ2 ) = 15α2 κ1 κ2 [5κ1 κ2 + 2(κ21 + κ22 )] , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) − fres (α, κ1 , κ2 ) , f8c (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + fres (α, κ1 , κ2 ) , fnon (α, κ1 , κ2 ) = 16[13κ21 κ22 b2 (κ41 + κ42 ) + 46κ41 κ42 b2 + 5κ21 κ22 (V12 + V22 ) + 2κ1 κ2 (V22 κ21 + V12 κ22 + κ61 b2 + κ62 b2 ) + 34κ31 κ32 b2 (κ21 + κ22 ) + 4κ1 κ2 (V12 κ21 + V22 κ22 ) + V12 κ41 + V22 κ42 ] , fres (α, κ1 , κ2 ) = 32V1 V2 [7κ21 κ22 + (κ41 + κ42 ) + 4κ1 κ2 (κ21 + κ22 )] .
(33)
Note that all the terms in fres are proportional to V1 V2 , as they arise from the effects of interacting resonances. Equilibria in this situation again satisfy (28) except that one now inserts functions from (32)-(33). Again, the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One again inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the initial spatial profile R = R0 + εR1 + O(ε2 ). A typical example of the case where the resonance occurs due to interactions between√two wavelengths is simulated in Fig. 6, with V2 = 2V1 = 2 and κ2 = 3κ1 = 3 δ/2 = 3π/(8b), where b is again given by Eq. (13) with b1 = b2 = 1 and ǫ = 0.1. The resulting complex patterns were found to persist as stable dynamical structures (with √ periodic time dynamics). When κ2 − κ1 = ±2 δ, one has a subtractively mixed subharmonic resonance. The effective equations governing the O(ε2 ) slow evolution in this case are again (29), with ∆(κ1 , κ2 ) = 32κ1 κ2 (κ1 − 2κ2 )(2κ1 − κ2 )(κ1 − κ2 )3 , 17
(34)
Figure 6: Same as (the left panel of) Fig. 1, but for the resonant case due to (additively mixed) interactions with two wavelengths. The solution of Eq. (14) is used as initial condition with C and D in Eq. (23) given by Eq. (29) with the functions (32)-(33) [see text for parameter details].
18
and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = 24ακ1 κ2 [−2(κ41 + κ42 ) + 9(κ31 + κ32 ) − 14κ21 κ22 ] ,
f3 (α, κ1 , κ2 , b1 ) = 16κ1 κ2 b1 [2(κ61 + κ62 ) − 13κ1 κ2 (κ41 + κ42 ) + 34κ21 κ22 (κ21 + κ22 ) − 46κ31 κ32 ] , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) ,
f5 (α, κ1 , κ2 ) = 15α2 κ1 κ2 [−5κ1 κ2 + 2(κ21 + κ22 )] , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) − fres (α, κ1 , κ2 ) ,
f8c (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + fres (α, κ1 , κ2 ) , fnon (α, κ1 , κ2 ) = 16[−13κ21κ22 b2 (κ41 + κ42 ) − 46κ41 κ42 b2 − 5κ21 κ22 (V12 + V22 )
+ 2κ1 κ2 (V22 κ21 + V12 κ22 + κ61 b2 + κ62 b2 ) + 34κ31 κ32 b2 (κ21 + κ22 ) + 4κ1 κ2 (V12 κ21 + V22 κ22 ) − V12 κ41 − V22 κ42 ] ,
fres (α, κ1 , κ2 ) = 32V1 V2 [−7κ21 κ22 − (κ41 + κ42 ) + 4κ1 κ2 (κ21 + κ22 )] .
(35)
As in the resonance due to additive mixing, all the terms in fres are proportional to V1 V2 . Equilibria in this case again satisfy (28) except that one inserts the functions from (34)-(35). Recall once more that the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One also inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain a spatial profile R = R0 + εR1 + O(ε2 ) to utilize as an initial wavefunction in numerical simulations of (1). In this case, the numerical results were found √ to yield similar (stable) temporal dynamics as in the case with κ2 + κ1 = ±2 δ.
5
Conclusions
In this work, we analyzed spatial structures in the Gross-Pitaevskii (GP) equation in superlattice potentials. This model describes cigar-shaped Bose-Einstein condensates (BECs) in optical lattices and superlattices. We investigated both localized states and spatially extended states of the mean field model. In the former case, we established the stability of bright solitary waves and indicated how the superlattice potential can be controllably used as both a symmetric and asymmetric matter-wave splitter. The achievement of asymmetric splitting is particular to BECs loaded into superlattice potentials, in contrast to regular optical lattices, which can only be used for symmetric splitting. In our study of spatially extended wavefunctions, we derived amplitude equations describing the evolution of spatially modulated states of the BEC. We used second-order multiple scale perturbation theory to study spatial harmonic resonances with respect to a single lattice wavenumber, as well as additively and subtractively 19
mixed subharmonic resonances. Harmonic resonances are a second-order effect that can occur in regular periodic lattices, but mixed resonances require the presence of a superlattice. In each situation, we determined the resulting dynamical equilibria, which represent spatially periodic solutions, and examined the stability of the corresponding solutions via direct simulations of the GP equation. In every case considered, the solutions (non-resonant, resonant with a single wavelength, and resonant due to interactions between two wavelengths) were found numerically to be dynamically stable under time-evolution of the Gross-Pitaevskii equation. From a theoretical standpoint, extending the considerations presented herein to the case of localized solutions in the presence of a positive scattering length (repulsive interaction) would appear to be a directly relevant next step. From an experimental perspective, realization of the states presented in this work in currently set-up experiments appears to be within reach.
Acknowledgements We wish to acknowledge Todd Kapitula for numerous useful interactions and discussions during the early stages of this work. P. G. K. gratefully acknowledges support from NSF-DMS-0204585, from the Eppley Foundation for Research, and from an NSF-CAREER award.
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Bose-Einstein Condensates in Superlattices Mason A. Porter [email protected] School of Mathematics and Center for Nonlinear Science Georgia Institute of Technology Atlanta, GA 30332, USA P. G. Kevrekidis Department of Mathematics and Statistics University of Massachusetts, Amherst MA 01003, USA February 26, 2008 Abstract We consider the Gross-Pitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigar-shaped Bose-Einstein condensates (BECs) in such lattice potentials. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability numerically. We find that the superlattice can operate as an efficient (symmetric and asymmetric) matter-wave splitter. We then focus on spatially extended wavefunctions in the form of modulated amplitude waves. With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice wavenumber as well as the two types of mixed resonances that result from interactions of both wavenumbers of the superlattice. In each case, we determine the resulting system’s equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions by direct simulations of the GP equation, identifying them as typically stable solutions of the model.
PACS: 05.45.-a, 03.75.Lm, 05.30.Jp, 05.45.Ac
1
Introduction
At low temperatures, particles in a dilute gas can reside in the same quantum (ground) state, forming a Bose-Einstein condensate.15, 20, 34, 50 This was first observed experimentally in 1995 with vapors of rubidium and sodium.3, 21 In 1
these experiments, atoms were confined in magnetic traps, evaporatively cooled to temperatures on the order of fractions of microkelvins, left to expand by switching off the confining trap, and subsequently imaged with optical methods.20 A sharp peak in the velocity distribution was observed below a critical temperature, indicating that Bose-Einstein condensation had occurred. Under the typical confining conditions of experimental settings, BECs are inhomogeneous, so condensation can be observed in both momentum and coordinate space. The number of condensed atoms N ranges from several thousand (or less) to several million (or more). The magnetic traps that confine them are usually approximated well by harmonic potentials. There p are two characteristic length scales: the harmonic oscillator length aho = ~/(mωho ) [which is on the order of a few microns], where ωho = (ωx ωy ωz )1/3 is the geometric mean of p the trapping frequencies, and the mean healing length χ = 1/ 8π|a|¯ n, where n ¯ is the mean particle density and a, the (two-body) s-wave scattering length, is determined by the atomic species of the condensate.5, 20, 38, 50 Interactions between atoms are repulsive when a > 0 and attractive when a < 0. For a dilute ideal gas, a ≈ 0. The length scales in BECs should be contrasted with those in systems like superfluid helium, in which the effects of inhomogeneity occur on a microscopic scale fixed by the interatomic distance.20 If considering only two-body, mean-field interactions, a dilute Bose-Einstein gas can be modeled using a cubic nonlinear Schr¨odinger equation (NLS) with an external potential, which is also known as the Gross-Pitaevskii (GP) equation. BECs are modeled in the quasi-one-dimensional (quasi-1D) regime when the transverse dimensions of the condensate are on the order of its healing length and its longitudinal dimension is much larger than its transverse ones.9–11, 20 In this regime, one employs the 1D limit of a 3D mean-field theory rather than a true 1D mean-field theory, which would be appropriate were the transverse dimension on the order of the atomic interaction length or the atomic size.6, 9–11, 58 When considering only two-body interactions, the condensate wavefunction (“order parameter”) ψ(x, t) satisfies a GP equation of the form i~ψt = −[~2 /(2m)]ψxx + g|ψ|2 ψ + V (x)ψ ,
(1)
where |ψ|2 is the number density,p V (x) is the external trapping potential, g = [4π~2 a/m][1 + O(ζ 2 )], and ζ = |ψ|2 |a|3 is the dilute gas parameter.5, 20, 38 Because the scattering length a can be adjusted using a magnetic field in the vicinity of a Feshbach resonance,25 the contribution of the nonlinearity in (1) is tunable.36 Potentials V (x) of interest include harmonic traps,20 periodic lattices2, 14, 18, 19, 27, 31, 45 and superlattices49, 57 (which can be either periodic or quasiperiodic), and superpositions of lattices or superlattices with harmonic traps. The existence of quasi-1D (“cigar-shaped”) BECs motivates the study of lower dimensional models such as Eq. (1). The case of periodic and quasiperiodic potentials without a confining trap along the dimension of the lattice is of particular theoretical and experimental interest. Such potentials have been used, for example, to study Josephson effects,2 squeezed states,47 Landau-Zener tunneling and Bloch 2
oscillations,45 and the transition between superfluidity and Mott insulation at both the classical17, 60 and quantum27 levels. Moreover, with each lattice site occupied by one alkali atom in its ground state, BECs in optical lattices show promise as a register in a quantum computer.54, 62 In experiments, a weak harmonic trap is typically used on top of the lattice or superlattice to prevent the particles from escaping. The lattice is also generally turned on after the trap. If one wishes to include the trap in theoretical analyses, V (x) is modeled by V (x) = V1 cos(κ1 x) + V2 cos(κ2 x) + Vh x2 ,
(2)
where κ1 is the primary lattice wavenumber, κ2 > κ1 is the secondary lattice wavenumber, V1 and V2 are the associated lattice amplitudes, and Vh represents the magnitude of the harmonic trap. (Note that V1 , V2 , Vh , κ1 , and κ2 can all be tuned experimentally.) The sinusoidal terms dominate for small x, but the harmonic trap otherwise becomes quickly dominant. When Vh ≪ V1 , V2 , the potential is dominated by its periodic contributions for many periods.14, 16, 22, 52 BECs in optical lattices with up to 200 wells have been created experimentally.48 In this work, we let Vh = 0 and focus on optical lattices and superlattices. Spatially periodic potentials have been employed in experimental studies of BECs2, 27, 29, 45, 47, 54 and have also been studied theoretically.1, 8–11, 16, 18, 22, 23, 37, 40, 42–44, 46, 51–53, 60, 61, 63 In recent experiments, BECs were loaded into superlattices with κ2 = 3κ1 .49 However, there has thus far been very little theoretical research on BECs in superlattices.24, 26, 41, 57 It is also worth noting that from the perspective of KAM theory, the value of the ratio κ2 /κ1 has important dynamical ramifications.28, 55 For example, if this ratio is irrational, the superlattice is quasiperiodic rather than periodic. We therefore consider both rational and irrational wavenumber ratios. We investigate in this study both localized and spatially extended waves of BECs in periodic and quasiperiodic superlattices. We examine both stable and unstable solitary wave solutions and illustrate their dynamical stability/instability numerically. We also show that the superlattice can operate as an efficient symmetric or asymmetric matter-wave splitter. In contrast, a regular lattice can only split matter waves symmetrically. We then focus on spatially extended wavefunctions in the form of modulated amplitude waves. We apply a coherent structure ansatz to Eq. (1), which yields a parametrically forced Duffing equation describing the spatial evolution of the field. We employ secondorder multiple scale perturbation theory7, 30, 55, 56 to study its periodic orbits (called “modulated amplitude waves” and denoted MAWs), and illustrate their dynamical stability with numerical simulations of the GP equation. We consider harmonic (1 : 1) resonances and two types of mixed resonances—resulting from, respectively, “additive” (2 : 1 + 1) and “subtractive” (2 : 1 − 1) mixing— all of which arise at the O(ε2 ) level of analysis. Previous perturbative studies of resonance phenomena in BECs have utilized an O(ε) analysis.51–53 Additionally, as mixed resonances arise from the interaction of multiple superlattice wavenumbers, they cannot occur in BECs loaded into regular lattices. 3
We structure our presentation as follows: In section 2, we discuss localized (solitary wave) solutions of the GP equation in superlattices. In section 3, we introduce modulated amplitude waves, and in section 4 we derive and solve “slow flow” dynamical equations that describe the resonance phenomena under consideration. We corroborate our results and test the stability of the MAWs with direct numerical simulations of the GP equation. Finally, in section 5, we summarize our findings and present our conclusions.
2
Localized Solutions
We first examine exponentially localized, solitary wave solutions and their stability and dynamical behavior in the presence of a superlattice potential. In particular, we let ~ = 2m = 1 and consider Eq. (1) with attractive interactions. With g = −1 (for simplicity), it is well-known that in the absence of a potential, the equation has an exact (bright) soliton solution of the form: √ √ (3) ψ(x, t) = eiωt 2ωsech ω(x − x0 ) ,
where ω = −µ is the frequency of this standing wave, µ denotes the BEC’s chemical potential, and x0 is the arbitrary position of its center. In our discussion of solitary waves, we focus on attractive interactions, as our theoretical considerations in the presence of the potential can only be applied in that case. For spatially extended states, we will consider both attractive and repulsive interactions. Extending the considerations presented herein for the dark solitary waves solitons known to exist for repulsive interactions35, 40, 41 is a challenging theoretical problem for future study. In the presence of the potential, Eq. (1) is a perturbed Hamiltonian system with perturbation energy given by Z ∞ E1 [ψ] = V (x)|ψ|2 dx . (4) −∞
Following the recent work of Kapitula, Kevrekidis, and Sandstede,33 we evaluate the reduced Hamiltonian by inserting Eq. (3) into Eq. (4) to obtain H(x0 ) =
X
j=1,2
2Vj
πκj cos(κj x0 ) κ π sinh 2√j ω
(5)
According to this recently-developed theory,32, 33 the perturbed wave chooses values of x0 so that H ′ (x0 ) = 0; that is, the selected center positions x0 are critical points of the reduced Hamiltonian. The small, nonzero linear stability eigenvalues (of the linearization around the solitary wave) in the presence of the perturbation are then given by λ2 = −ω 1/2 H ′′ (x0 ) ,
4
(6)
−20 4
1.4 −10 1.2
|u(x,0)|2
1.6
−15
2
0
−2
−5 −4 −20
x
1
−15
−10
−5
0
5
10
15
20
5
10
15
20
x
0 0.8 3
10 0.4 15
0.2
|u(x,100)|
0.6
2
5 2 1 0 −1 −2
20
0
10
20
30
40
50
60
70
80
90
100
t
−3 −20
−15
−10
−5
0
x
Figure 1: The evolution of the unstable soliton for a superlattice potential with V2 = 2V1 = 2; κ2 = 3κ1 = 3. The initial condition consists of a solitary wave with ω = 1 centered around the maximum at x0 = 0. The unstable soliton splits into two symmetric fragments. as recently demonstrated.33 Hence, the soliton is unstable at the maxima of the superlattice potential. However, at the minima of this potential, the soliton is linearly stable. We tested this prediction through direct numerical simulations of Eq. (1) and obtained a variety of interesting conclusions indicating that the superlattice potential of Eq. (2) can be used as a soliton (or “condensate”) splitter. Moreover, it can be used for both symmetric and asymmetric splitting that should be experimentally observable. In our numerical simulations, it was necessary to utilize periodic boundary conditions, which imposed the use of rational ratios between κ1 and κ2 . In particular, the relation κ2 = 3κ1 , which has been achieved in experimental superlattices,49 was employed in the dynamical evolution described below. Placing the soliton in a potential maximum around which the potential is symmetric leads to a symmetric splitting of the solitary wave, as shown in Fig. 1 (in which V2 = 2V1 = 2). If, on the other hand, the potential is asymmetric around the maximum on which the soliton is placed, then the corresponding solitonic splitting is also asymmetric, as demonstrated in Fig. 2. Hence, the superlattice can be used as an efficient, controllable matter-wave splitter. If we initialize the soliton at the minimum of the potential, then in accord with the prediction of Eq. (6), the solitary matter wave persists as a robust structure at this local minimum (even if it is not also a global minimum); see Fig. 3 for an example of this behavior. It should be noted, however, that using very recent techniques,59 matter can be transferred to the deepest well.
5
x
2
4
1.8
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−16
1.6
−14
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−12
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|u(x,0)|2
−18
0
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−4 −30
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x 4
|u(x,100)|2
−20
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0
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−4 −30
100
−25
−20
−15
t
−10
x
Figure 2: Same as Fig. 1, but now the soliton is initially centered around the maximum at x0 = −10.8. The unstable soliton splits into two asymmetric fragments.
−20
4
3
−15
|u(x,0)|2
3.5
2.5
x −5
2 0
10
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t
0
−20
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|u(x,100)|
0.5
−25
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1.5
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−4 −30
2
−10
2
2
0
−2
−4 −30
−25
−20
−15
−10
x
Figure 3: Same as Fig. 1, but now the soliton is initially centered around the local minimum at x0 = −13.2. It remains there under time evolution.
6
3
Modulated Amplitude Waves
To decompose the solutions of interest, we use the ansatz ψ(x, t) = R(x) exp (i [θ(x) − µt]) .
(7)
When the above (temporally periodic) coherent structures (7) are also spatially periodic, they are called modulated amplitude waves (MAWs).12, 13 The orbital stability of MAWs for the cubic NLS with elliptic potentials has been studied by Bronski and co-authors.9–11 To obtain stability information about sinusoidal potentials, one takes the limit as the elliptic modulus k approaches zero.39, 55 When V (x) is periodic, the resulting MAWs generalize the Bloch modes that occur in the theory of linear systems with periodic potentials, as one is considering a nonlinear Floquet-Bloch theory rather than a linear one.4, 8, 18, 40, 56 In this work, we extend recent studies51, 52 of the dynamical behavior of MAWs of BECs in lattice potentials to superlattice potentials. Inserting (7) into Eq. (1), equating the real and imaginary components of the resulting equation, and defining S := R′ yields the following two-dimensional system of nonlinear ordinary differential equations: R′ = S , S′ =
c2 2mµR 2mg 3 2m − + 2 R + 2 V (x)R . R3 ~ ~ ~
The parameter c is defined via the relation θ′ (x) =
c , R2
(8)
which plays the role of conservation of “angular momentum,” as discussed by Bronski and coauthors.10 Constant phase solutions (i.e., standing waves), which constitute an important special case, satisfy c = 0. In the rest of the paper, we restrict ourselves to this class of solutions, so that R′ = S , 2mµR 2mg 3 2m + 2 R + 2 V (x)R . S′ = − ~ ~ ~
(9)
When V (x) = 0, Eq. (9) is integrable.51, 52 We consider the case with Vh = 0 (which implies, in practice, that the harmonic trap is negligible with respect to the superlattice potential for the domain of interest) and define 2mµ δ˜ := , ~ where
ε˜ α := −
2mg , ~2
2m V˜ (x) := − 2 V (x) ~
V˜ (x) = ε[V˜1 cos(κ1 x) + V˜2 cos(κ2 x)] ,
7
(10)
(11)
˜ α the parameters δ, ˜ , and V˜j are O(1) quantities, and the κj can either be commensurate (rational multiples of each other) or incommensurate, so that the superlattice can be, respectively, either periodic or quasiperiodic. We let κ2 > κ1 without loss of generality. ˜ α For notational convenience, we drop the tildes from δ, ˜ , and V˜j , so that Eq. (9) can be written R′′ + δR + εαR3 + εR[V1 cos(κ1 x) + V2 cos(κ2 x)] = 0 .
(12)
In this paper, we consider the case δ > 0 corresponding to a positive chemical potential.
4
Multiple Scale Perturbation Theory and Spatial Subharmonic Resonances
To employ multiple scale perturbation theory,7, 52, 56 we define “slow space” η := εx and “stretched space” ξ := bx = [1 + εb1 + ε2 b2 + O(ε3 )]x .
(13)
We expand the wavefunction amplitude as R = R0 + εR1 + ε2 R2 + O(ε3 )
(14)
and stretch the argument of the superlattice, so that V˜ (ξ) = V1 cos(κ1 ξ) + V2 cos(κ2 ξ) .
(15)
With this expansion, Eq. (12) becomes 2 2 ∂ 2 R0 ∂ 2 R1 2 ∂ R2 3 + ε + ε + O(ε ) 1 + b1 ε + b2 ε2 + O(ε3 ) ∂ξ 2 ∂ξ 2 ∂ξ 2 2 ∂ R0 ∂ 2 R1 ∂ 2 R2 + 2ε 1 + b1 ε + b2 ε2 + O(ε3 ) +ε + ε2 + O(ε3 ) ∂ξ∂η ∂ξ∂η ∂ξ∂η 2 2 2 ∂ R1 ∂ R2 ∂ R0 +ε + ε2 + O(ε3 ) + ε2 ∂η 2 ∂η 2 ∂η 2 3 + δ R0 + εR1 + ε2 R2 + O(ε3 ) + εα R0 + εR1 + ε2 R2 + O(ε3 ) + ε R0 + εR1 + ε2 R2 + O(ε3 ) [V1 cos(κ1 ξ) + V2 cos(κ2 ξ)] = 0 . (16)
To perform multiple scale analysis, we equate the coefficients of terms of different order (in ε) in turn. At O(1) = O(ε0 ), we obtain ∂ 2 R0 + δR0 = 0 , ∂ξ 2 which has the solution √ √ R0 (ξ, η) = A(η) cos( δξ) + B(η) sin( δξ) , 8
(17)
for some slowly-varying amplitudes A(η), B(η), equations of motion for which arise at O(ε). Equating coefficients at O(ε) yields √ ′ 3 √ ∂ 2 R1 2 2 δB − + δR = 2b δA − 2 αA(A + B ) cos( δξ) 1 1 2 ∂ξ 4 √ √ ′ 3 + 2b1 δB + 2 δA − αB(A2 + B 2 ) sin( δξ) 4 √ √ αA αB [−A2 + 3B 2 ] cos(3 δξ) + [−3A2 + B 2 ] sin(3 δξ) + 4 4 √ √ V1 A V1 A cos([κ1 − δ]ξ) + cos([κ1 + δ]ξ) + 2 2 √ √ V1 B V1 B sin([κ1 − δ]ξ) + sin([κ1 + δ]ξ) − 2 2 √ √ V2 A V2 A cos([κ2 − δ]ξ) + cos([κ2 + δ]ξ) + 2 2 √ √ V2 B V2 B (18) sin([κ2 − δ]ξ) + sin([κ2 + δ]ξ) . − 2 2 For R1 (ξ, η) to be bounded, the coefficients of the √ secular terms in Eq. (18) √ must vanish.7,√56 The harmonics cos( δξ) and sin( δξ) are always secular, √ whereas cos(3 δξ) and sin(3 δξ) are never secular. The other harmonics are secular only in the case of 2 : 1 subharmonic resonances.51, 52 As such spatial resonances have already been studied for quasi-1D BECs in periodic lattices, we will consider the situation in which (18) is non-resonant so that we can turn our attention to other resonant situations. The 2 : 1 resonance√can occur with √ respect to either the primary (κ1 = 2 δ) or secondary (κ2 = 2 δ) wavenumber of the lattice. [Our O(ε2 ) analysis below can be repeated in the presence of 2 : 1 resonances.] At O(ε), we can thus obtain either no resonance, a long-wavelength subharmonic resonance, or a short-wavelength subharmonic resonance. Equating the coefficients of the secular terms to zero in Eq. (18) yields the following equations of motion describing the slow dynamics: √ 3α A′ = −b1 δB + √ B(A2 + B 2 ) , 8 δ √ 3α B ′ = b1 δA − √ A(A2 + B 2 ) . 8 δ Equilibria of (19) correspond to periodic orbits of (1). One finds ! r 8b1 δ (A, B) = ± ,0 , 3α ! r 8b1 δ (A, B) = ± 0, , 3α
9
(19)
(20)
where we note that these two sets of equilibria are π/2 phase shifts of each other. More generally, equilibria satisfy A2 + B 2 =
8b1 δ , 3α
(21)
so that any other phase shift also yields an equilibrium of (19). However, we are more interested in the effects at order O(ε2 ), which we now analyze. At this second order of perturbation theory, BECs in superlattice potentials exhibit some dynamical behavior that cannot occur in BECs in simpler lattice potentials [where, for example, solutions of type of Eq. (21) straightforwardly arise53 ]. Equating coefficients at O(ε2 ) yields ∂ 2 R0 ∂ 2 R0 ∂ 2 R0 ∂ 2 R1 ∂ 2 R1 ∂ 2 R2 2 2 + δR = −(b + 2b ) − − 2b − 2 − 3αR R − 2b 2 2 1 1 1 1 0 ∂ξ 2 ∂ξ 2 ∂η 2 ∂ξ∂η ∂ξ 2 ∂ξ∂η − R1 V1 cos(κ1 ξ) − R2 V2 cos(κ2 ξ) , (22) where one inserts the expressions for R0 , R1 and their derivatives into the righthand-side of (22). To find the secular terms in Eq. (22), we compute √ √ R1 (ξ, η) = C(η) cos( δξ) + D(η) sin( δξ) + R1p (ξ, η) , √ √ R1p (ξ, η) = c1 cos(3 δξ) + c2 sin(3 δξ) 2 h i X √ √ √ √ cj3 cos([κj − δ]ξ) + cj4 cos([κj + δ]ξ) + cj5 sin([κj − δ]ξ) + cj6 sin([κj + δ]ξ) , + j=1
(23)
where j ∈ {1 , 2} and c1 = c2 = cj3 = cj4 = cj5 = cj6 =
α A(A2 − 3B 2 ) , 32δ α B(3A2 − B 2 ) , 32δ Vj A √ , 2κj (κj − 2 δ) Vj A √ , 2κj (κj + 2 δ) Vj B √ , 2κj (2 δ − κj ) Vj B √ . 2κj (κj + 2 δ)
(24)
Inserting (17) and (23) into (22) and expanding this equation trigonometrically yields 19 harmonics (that are also present for sines), which we list in Table 1. We indicate which of these harmonics are always secular, sometimes secular, and never secular. 10
At this order of perturbation theory, one finds 2 : 1 (primary subharmonic), 4 : 1 (secondary subharmonic), 1 : 1 (harmonic), 2 : 1+1 (additively mixed subharmonic), and 2 : 1−1 (subtractively mixed subharmonic) resonances. The first three types of resonances can occur with respect to either κ1 or κ2 , whereas the latter two require the interaction of both superlattice wavenumbers. Harmonic and mixed spatial resonances have not been analyzed previously for BECs, so we study these three situations in this paper. At O(ε), we considered the case √ without 2 : 1 resonances, so the associated resonance conditions (κj = ±2 δ) are necessarily not satisfied at the present [O(ε2 )] stage, as indicated in Table 1. Second-order subharmonic (4 : 1) resonances have been studied using elliptic function perturbation theory,51, 52 so we do √ not analyze them here. Their associated resonance conditions √ are κj = ±4 δ. The resonance relations for harmonic resonances are κj = ± δ. We will consider solutions that have harmonic resonance with respect to the primary lattice spacing κ1 . The resonance √ relation for additively mixed subharmonic resonances is κ2 + κ1 = ±2 √ δ, and that for subtractively mixed subharmonic resonances is κ2 − κ1 = ±2 δ. In the remainder of this section, we consider the non-resonant, harmonically resonant, and the two types of mixed resonant states in turn. It is also important to remark that with the slow spatial variable η = εx, the approximate solutions R(x) obtained perturbatively are valid for |x| . O(ε−1 ) despite the fact that we employ a second-order multiple scale expansion. By incorporating a third (“super slow”) scale ε2 x, which is even more technically demanding than our present perturbative procedure, one can obtain approximate solutions that are valid for |x| . O(ε−2 ).7 Before proceeding, we also note that in light of KAM theory, one expects different dynamical behavior (at least mathematically) depending on whether κ2 /κ1 is an integer, a rational number, or an irrational number. Only the situation κ2 = 3κ1 has been prepared experimentally, so we concentrate on that case in our numerical simulations. Our analytical work is valid for all real ratios κ2 /κ1 . We note additionally that we simulated these modulated amplitude waves using a numerical domain with periodic boundary conditions. This allows us to handle integer or rational values of κ2 /κ1 with appropriate selection of the domain parameters (so that the box size is an integer multiple of both spatial periods). However, quasiperiodic settings cannot be tackled numerically within this framework for the extended wave solutions considered in this section.
11
Label 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Harmonic √ cos( √δξ) cos(3√δξ) cos(5 δξ) √ cos([κ1 − √δ]ξ) cos([κ1 + √δ]ξ) cos([κ2 − √δ]ξ) cos([κ2 + √δ]ξ) cos([κ1 − 3√δ]ξ) cos([κ1 + 3√δ]ξ) cos([κ2 − 3√δ]ξ) cos([κ2 + 3√δ]ξ) cos([2κ1 − √δ]ξ) cos([2κ1 + √δ]ξ) cos([2κ2 − √δ]ξ) cos([2κ2 + δ]ξ) √ cos([κ1 + κ2 − √δ]ξ) cos([κ1 + κ2 + √δ]ξ) cos([κ1 − κ2 − √δ]ξ) cos([κ1 − κ2 + δ]ξ)
Assumed Assumed Assumed Assumed
Secular? Yes No No not in resonance not in resonance not in resonance not in resonance Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes Sometimes
at at at at
O(ε) O(ε) O(ε) O(ε)
Resonance when secular N/A N/A N/A 2:1 2:1 2:1 2:1 4:1 4:1 4:1 4:1 1:1 1:1 1:1 1:1 2 : 1+1 2 : 1+1 2 : 1−1 2 : 1−1
Table 1: The harmonics in the right-hand-side of Eq. (22) after the formulas for R0 (17) and R1 (23) are inserted. We only list the cosines in this table, but the sines of these harmonics are present as well. We designate which harmonics are always secular, sometimes secular (under an appropriate resonance condition, which we indicate in the text), and never secular.
4.1
The Non-Resonant Case
In the non-resonant case, effective equations governing the O(ε2 ) slow evolution are 1 f1 (α, δ, κ1 , κ2 )B 2 + f2 (α, δ, κ1 , κ2 )A2 + f3 (α, δ, κ1 , κ2 , b1 ) D + f4 (α, δ, κ1 , κ2 )ABC ∆(δ, κ1 , κ2) +f5 (α, δ, κ1 , κ2 )B 5 + f6 (α, δ, κ1 , κ2 )A2 B 3 + f7 (α, δ, κ1 , κ2 )A4 B + f8 (α, δ, κ1 , κ2 , b2 )B , 1 D′ = − f1 (α, δ, κ1 , κ2 )A2 + f2 (α, δ, κ1 , κ2 )B 2 + f3 (α, δ, κ1 , κ2 , b1 ) C + f4 (α, δ, κ1 , κ2 )ABD ∆(δ, κ1 , κ2) +f5 (α, δ, κ1 , κ2 )A5 + f6 (α, δ, κ1 , κ2 )A3 B 2 + f7 (α, δ, κ1 , κ2 )AB 4 + f8 (α, δ, κ1 , κ2 )A , (25) C′ =
where ∆(δ, κ1 , κ2 ) = 256δ 3/2 16δ 2 − 4δκ21 − 4δκ22 + κ21 κ22
12
(26)
and f1 (α, δ, κ1 , κ2 ) = 3f2 (α, δ, κ1 , κ2 ) , f2 (α, δ, κ1 , κ2 ) = 96αδ[16δ 2 − 4δ(κ21 + κ22 ) + κ21 κ22 ] ,
f3 (α, δ, κ1 , κ2 , b1 ) = 256δ 2 b1 [−κ21 κ22 + 4δ(κ21 + κ22 ) − 16δ 2 ] , f4 (α, δ, κ1 , κ2 ) = 2f2 (α, δ, κ1 , κ2 ) ,
f5 (α, δ, κ1 , κ2 ) = 15α2 [−16δ 2 + 4δ(κ21 + κ22 ) − κ21 κ22 ] , f6 (α, δ, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, δ, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8 (α, δ, κ1 , κ2 , b2 ) = 64δ[V12 κ22 + V22 κ21 − 4δ(V12 + V22 + κ21 κ22 b2 ) + 16δ 2 b2 (κ21 + κ22 ) − 64δ 3 b2 ] . (27) In this case, the superlattice does not contribute to O(ε2 ) terms. Equilibrium solutions of (25) satisfy (f1 B 2 + f2 A2 + f3 )(f5 A5 + f6 A3 B 2 + f7 AB 4 + f8 A) − (f4 AB)(f5 B 5 + f6 A2 B 3 + f7 A4 B + f8 B) , f42 A2 B 2 − (f1 B 2 + f2 A2 + f3 )(f1 A2 + f2 B 2 + f3 ) (f1 A2 + f2 B 2 + f3 )(f5 B 5 + f6 A2 B 3 + f7 A4 B + f8 B) − (f4 AB)(f5 A5 + f6 A3 B 2 + f7 AB 4 + f8 A) D= , f42 A2 B 2 − (f1 B 2 + f2 A2 + f3 )(f1 A2 + f2 B 2 + f3 ) (28) C=
where one inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the spatial profile R = R0 + εR1 + O(ε2 ) used as the initial wavefunction in the numerical simulations of the full GP given by Eq. (1). A typical example of the √ non-resonant case is shown in Fig. 4, with V2 = 2V1 = 2 and κ2 = 3κ1 = 12 δ = 3π/(2b), where b is the stretching factor given by Eq. (13). In this simulation, we used b1 = b2 = 1 and ǫ = 0.1. It can be clearly seen that the relevant solution is dynamically stable, which we found to be generic in our numerical experiments. Simulations with rational κ2 /κ1 reveal similar phenomena.
4.2
Resonances
In this subsection, we consider harmonic resonances, additively mixed subharmonic resonances, and subtractively mixed subharmonic resonances. In the evolution equations for the slow dynamics, one inserts the appropriate resonance relation into ∆ and f1 –f7 . The function f8 has both the non-resonant contributions discussed above and additional resonant terms due to the superlattice. Note additionally that there is a symmetry-breaking in the resulting equations because the functional form of the lattice contains only cosine terms. 4.2.1
Harmonic Resonance with a Single Wavelength √ When κj = ± δ, there is a harmonic resonance. The effective equations governing the O(ε2 ) slow evolution in the presence of a harmonic resonance with 13
Figure 4: Same as (the left panel of) Fig. 1, but for the non-resonant spatially extended solution of Eq. (14) with C and D in Eq. (23) given by Eq. (25) [see text for parameter details].
14
respect to the primary lattice wave number κ1 are 1 f1 (α, κ1 , κ2 )B 2 + f2 (α, κ1 , κ2 )A2 + f3 (α, κ1 , κ2 , b1 ) D + f4 (α, κ1 , κ2 )ABC ∆(κ1 , κ2) +f5 (α, κ1 , κ2 )B 5 + f6 (α, κ1 , κ2 )A2 B 3 + f7 (α, κ1 , κ2 )A4 B + f8s (α, κ1 , κ2 , b2 )B , 1 D′ = f1 (α, κ1 , κ2 )A2 + f2 (α, κ1 , κ2 )B 2 + f3 (α, κ1 , κ2 , b1 ) C + f4 (α, κ1 , κ2 )ABD ∆(κ1 , κ2) +f5 (α, κ1 , κ2 )A5 + f6 (α, κ1 , κ2 )A3 B 2 + f7 (α, κ1 , κ2 )AB 4 + f8c (α, κ1 , κ2 )A , (29) C′ =
where ∆(κ1 , κ2 ) = 768κ31 (4κ21 − κ22 )
(30)
and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = 288ακ21 (κ22 − 4κ21 ) ,
f3 (α, κ1 , κ2 , b1 ) = 768κ41b1 (−κ22 + 4κ21 ) , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) , f5 (α, κ1 , κ2 ) = 45α2 (−κ22 + 4κ21 ) , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + 32V12 (κ22 − 4κ21 ) ,
f8c (α, κ1 , κ2 ) = fnon (α, κ1 , κ2 ) − 160V12 (κ22 − 4κ21 ) ,
fnon (α, κ1 , κ2 ) = 192κ21(V22 − 4κ21 κ22 b2 + 16κ41 b2 ) .
(31)
If considering a harmonic resonance with respect to the secondary lattice wavenumber κ2 , one obtains the appropriate equations for O(ε2 ) slow evolution by switching the roles of κ1 and κ2 in f8s and f8c . We recall as well that one needs an O(ε2 ) analysis to study such resonances. Note that the form of equations (31) corresponds to (27) except for the extra terms in f8c and f8s that arise from the superlattice. The equilibria of (29) are given by Eq. (28) except that one inserts the functions from (31). Additionally, the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One also inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the spatial profile R = R0 + εR1 + O(ε2 ) to use as an initial condition in direct numerical simulations of Eq. (1). A typical example of the single-wavelength resonant case is shown in Fig. 5, √ with V2 = 2V1 = 2 and κ2 = 4κ1 = 4 δ = π/b, where b is the stretching factor of Eq. (13); we used b1 = b2 = 1 and ǫ = 0.1. The resulting quasiperiodic (in space) patterns were generically found to persist in the dynamics of the system as stable (temporally oscillating) solutions. 15
Figure 5: Same as (the left panel of) Fig. 1, but for the harmonic resonant case with respect to the primary lattice wavelength. The solution given by Eq. (14) is used as an initial condition with C and D in Eq. (23) given by Eq. (29) with the functions (30)-(31) [see text for parameter details].
16
4.2.2
Resonances Due to Interactions Between the Two Wavelengths
Studying BECs in a superlattice rather than in a periodic lattice allows one to examine the spatial resonances caused by interactions between the two lattice wavelengths. As with harmonic resonances, an O(ε2 ) calculation is required to perform the analysis in√this case. When κ2 + κ1 = ±2 δ, one has an additively mixed subharmonic resonance. The effective equations governing the O(ε2 ) slow evolution in this case are (29) with ∆(κ1 , κ2 ) = 32κ1 κ2 (κ1 + 2κ2 )(2κ1 + κ2 )(κ1 + κ2 )3 (32) and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = −24ακ1 κ2 [2(κ41 + κ42 ) + 9(κ31 + κ32 ) + 14κ21 κ22 ] ,
f3 (α, κ1 , κ2 , b1 ) = 16κ1 κ2 b1 [2(κ61 + κ62 ) + 13κ1 κ2 (κ41 + κ42 ) + 34κ21 κ22 (κ21 + κ22 ) + 46κ31 κ32 ] , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) , f5 (α, κ1 , κ2 ) = 15α2 κ1 κ2 [5κ1 κ2 + 2(κ21 + κ22 )] , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) − fres (α, κ1 , κ2 ) , f8c (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + fres (α, κ1 , κ2 ) , fnon (α, κ1 , κ2 ) = 16[13κ21 κ22 b2 (κ41 + κ42 ) + 46κ41 κ42 b2 + 5κ21 κ22 (V12 + V22 ) + 2κ1 κ2 (V22 κ21 + V12 κ22 + κ61 b2 + κ62 b2 ) + 34κ31 κ32 b2 (κ21 + κ22 ) + 4κ1 κ2 (V12 κ21 + V22 κ22 ) + V12 κ41 + V22 κ42 ] , fres (α, κ1 , κ2 ) = 32V1 V2 [7κ21 κ22 + (κ41 + κ42 ) + 4κ1 κ2 (κ21 + κ22 )] .
(33)
Note that all the terms in fres are proportional to V1 V2 , as they arise from the effects of interacting resonances. Equilibria in this situation again satisfy (28) except that one now inserts functions from (32)-(33). Again, the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One again inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain the initial spatial profile R = R0 + εR1 + O(ε2 ). A typical example of the case where the resonance occurs due to interactions between√two wavelengths is simulated in Fig. 6, with V2 = 2V1 = 2 and κ2 = 3κ1 = 3 δ/2 = 3π/(8b), where b is again given by Eq. (13) with b1 = b2 = 1 and ǫ = 0.1. The resulting complex patterns were found to persist as stable dynamical structures (with √ periodic time dynamics). When κ2 − κ1 = ±2 δ, one has a subtractively mixed subharmonic resonance. The effective equations governing the O(ε2 ) slow evolution in this case are again (29), with ∆(κ1 , κ2 ) = 32κ1 κ2 (κ1 − 2κ2 )(2κ1 − κ2 )(κ1 − κ2 )3 , 17
(34)
Figure 6: Same as (the left panel of) Fig. 1, but for the resonant case due to (additively mixed) interactions with two wavelengths. The solution of Eq. (14) is used as initial condition with C and D in Eq. (23) given by Eq. (29) with the functions (32)-(33) [see text for parameter details].
18
and f1 (α, κ1 , κ2 ) = 3f2 (α, κ1 , κ2 ) , f2 (α, κ1 , κ2 ) = 24ακ1 κ2 [−2(κ41 + κ42 ) + 9(κ31 + κ32 ) − 14κ21 κ22 ] ,
f3 (α, κ1 , κ2 , b1 ) = 16κ1 κ2 b1 [2(κ61 + κ62 ) − 13κ1 κ2 (κ41 + κ42 ) + 34κ21 κ22 (κ21 + κ22 ) − 46κ31 κ32 ] , f4 (α, κ1 , κ2 ) = 2f2 (α, κ1 , κ2 ) ,
f5 (α, κ1 , κ2 ) = 15α2 κ1 κ2 [−5κ1 κ2 + 2(κ21 + κ22 )] , f6 (α, κ1 , κ2 ) = 2f5 (α, δ, κ1 , κ2 ) , f7 (α, κ1 , κ2 ) = f5 (α, δ, κ1 , κ2 ) , f8s (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) − fres (α, κ1 , κ2 ) ,
f8c (α, κ1 , κ2 , b2 ) = fnon (α, κ1 , κ2 ) + fres (α, κ1 , κ2 ) , fnon (α, κ1 , κ2 ) = 16[−13κ21κ22 b2 (κ41 + κ42 ) − 46κ41 κ42 b2 − 5κ21 κ22 (V12 + V22 )
+ 2κ1 κ2 (V22 κ21 + V12 κ22 + κ61 b2 + κ62 b2 ) + 34κ31 κ32 b2 (κ21 + κ22 ) + 4κ1 κ2 (V12 κ21 + V22 κ22 ) − V12 κ41 − V22 κ42 ] ,
fres (α, κ1 , κ2 ) = 32V1 V2 [−7κ21 κ22 − (κ41 + κ42 ) + 4κ1 κ2 (κ21 + κ22 )] .
(35)
As in the resonance due to additive mixing, all the terms in fres are proportional to V1 V2 . Equilibria in this case again satisfy (28) except that one inserts the functions from (34)-(35). Recall once more that the expressions for C and D have f8s rather than f8 as a prefactor for B and f8c rather than f8 as a prefactor for A. One also inserts an equilibrium value of A and B from Eq. (21). One then inserts the equilibrium values of A, B, C, and D into (17) and (23) to obtain a spatial profile R = R0 + εR1 + O(ε2 ) to utilize as an initial wavefunction in numerical simulations of (1). In this case, the numerical results were found √ to yield similar (stable) temporal dynamics as in the case with κ2 + κ1 = ±2 δ.
5
Conclusions
In this work, we analyzed spatial structures in the Gross-Pitaevskii (GP) equation in superlattice potentials. This model describes cigar-shaped Bose-Einstein condensates (BECs) in optical lattices and superlattices. We investigated both localized states and spatially extended states of the mean field model. In the former case, we established the stability of bright solitary waves and indicated how the superlattice potential can be controllably used as both a symmetric and asymmetric matter-wave splitter. The achievement of asymmetric splitting is particular to BECs loaded into superlattice potentials, in contrast to regular optical lattices, which can only be used for symmetric splitting. In our study of spatially extended wavefunctions, we derived amplitude equations describing the evolution of spatially modulated states of the BEC. We used second-order multiple scale perturbation theory to study spatial harmonic resonances with respect to a single lattice wavenumber, as well as additively and subtractively 19
mixed subharmonic resonances. Harmonic resonances are a second-order effect that can occur in regular periodic lattices, but mixed resonances require the presence of a superlattice. In each situation, we determined the resulting dynamical equilibria, which represent spatially periodic solutions, and examined the stability of the corresponding solutions via direct simulations of the GP equation. In every case considered, the solutions (non-resonant, resonant with a single wavelength, and resonant due to interactions between two wavelengths) were found numerically to be dynamically stable under time-evolution of the Gross-Pitaevskii equation. From a theoretical standpoint, extending the considerations presented herein to the case of localized solutions in the presence of a positive scattering length (repulsive interaction) would appear to be a directly relevant next step. From an experimental perspective, realization of the states presented in this work in currently set-up experiments appears to be within reach.
Acknowledgements We wish to acknowledge Todd Kapitula for numerous useful interactions and discussions during the early stages of this work. P. G. K. gratefully acknowledges support from NSF-DMS-0204585, from the Eppley Foundation for Research, and from an NSF-CAREER award.
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