Kinetic Thomas–Fermi solutions of the Gross–Pitaevskii equation

Optics Communications 282 (2009) 1472–1477

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Kinetic Thomas–Fermi solutions of the Gross–Pitaevskii equation M. Ölschläger a, G. Wirth a, C. Morais Smith b, A. Hemmerich a,* a b

Institut für Laser-Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Institute for Theoretical Physics, Utrecht University, 3508 TD Utrecht, The Netherlands

a r t i c l e

i n f o

Article history: Received 1 October 2008 Received in revised form 1 December 2008 Accepted 19 December 2008

PACS: 32.80.Hd 03.75.Hh 03.75.Nt 11.15.Ha 75.10.Jm

a b s t r a c t Approximate solutions of the Gross–Pitaevskii (GP) equation, obtained upon neglection of the kinetic energy, are well known as Thomas–Fermi solutions. They are characterized by the compensation of the local potential by the collisional energy. In this article we consider exact solutions of the GP-equation with this property and definite values of the kinetic energy, which suggests the term ‘‘kinetic Thomas– Fermi” (KTF) solutions. Despite their formal simplicity, KTF-solutions can possess complex current density fields with unconventional topology. We point out that a large class of light-shift potentials gives rise to KTF-solutions. As elementary examples, we consider one-dimensional and two-dimensional optical lattice scenarios, obtained by means of the superposition of two, three and four laser beams, and discuss the stability properties of the corresponding KTF-solutions. A general method is proposed to excite twodimensional KTF-solutions in experiments by means of time-modulated light-shift potentials. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction A significant portion of the physics encountered in Bose– Einstein condensates (BECs) of ultra-cold atomic or molecular gases can be well described by a mean field approximation, which replaces the complicated many-body wave function by a single mean field obeying the famous Gross–Pitaevskii (GP) equation [1]. This equation complements the Schrödinger equation by a non-linear term, which accounts for two-body interactions. In analogy to single particle physics, stationary states are determined by the time-independent GP-equation

h

i T^ þ VðrÞ þ gjwðrÞj2 wðrÞ ¼ lwðrÞ; 2

ð1Þ

h where T^   D denotes the kinetic energy operator, l is the chem2m ical potential, VðrÞ is the potential energy, and g is a constant, which is determined by the binary collision cross-section in the s-wave approximation [2]. Not only the ground state of the system is often described with remarkable precision by means of the GP-equation but also the low energy excitation spectrum, including vortices and solitons [1,3]. Solutions w of Eq. (1) may be characterized in terms of the following useful quantities: the local phase S defined via w ¼ jwjeiS , the particle density q  jwj2 , the current density j  h=mÞrS. ði h=2mÞðwrw  w rwÞ, and the velocity field v  j=q ¼ ð Recall that r  j ¼ 0 and r  v ¼ 0 if q–0, whereas the latter relation indicates that vortex filaments can only occur at density nodes.

* Corresponding author. Tel.: +49 40 8998 5162; fax: +49 40 8898 5190. E-mail address: [email protected] (A. Hemmerich). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.12.054

In this paper we consider a class of excited stationary solutions of the GP-equation, which are characterized by a compensation of the local collisional and potential energies up to a spatially constant term, i.e.

VðrÞ ¼ V 0  gjwðrÞj2 :

ð2Þ

This property is well known for approximate solutions derived in the so called Thomas–Fermi regime [1], where the kinetic energy in the GP-equation is neglected. Here, however, we are interested in exact solutions wðrÞ of Eq. (1) subject to Eq. (2), thus requiring that wðrÞ is an eigenfunction of the kinetic energy operator, which suggests the term ‘‘kinetic Thomas–Fermi (KTF) solutions”. KTFsolutions require some degree of non-linearity. They arise at the boundary between the weak interaction regime, where the dynamics retains Schrödinger character, and the strong interaction regime, where the inherent non-linearity of the GP-equation dominates. A natural environment for the emergence of KTF-solutions are optical lattices [4–6] (i.e. ultracold gases subjected to periodic light-shift potentials), where they correspond to excited states at the edge of the first Brillouin zone. The interest in KTF-solutions in two- and three-dimensional (2D, 3D) optical lattices results from the combination of their remarkable formal simplicity with the possibility of non-zero current density fields, which can acquire unconventional spatial topologies, for example, that of a vortex–antivortex sheet [7] in a 2D square lattice. This article comprises a study of KTF-solutions in optical lattices regarding their stability and their accessibility in experiments. In Section 2 we point out that any light-shift potential arising in an arbitrary monochromatic light-field with spatially constant polarization permits KTF-solutions of Eq. (1). In

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Section 3 we present a general discussion of the stability of KTFsolutions, which is applied to examples of particular interest for experiments in Section 4. Here, we concentrate on 1D and 2D optical lattice scenarios readily obtained in experiments by means of the superposition of two, three and four laser beams. For the case of the 1D lattice we point out that the corresponding KTF-solution arises at the boundary between the regimes of linear Bloch bands and non-linear Bloch bands, characterized by additional loop structures [8–10]. In Section 5 we generalize the considerations of Ref. [7] showing that for any 2D KTF-solution a suitable time-varying light-shift potential can be found, in order to drive the required current density. Finally, in Section 6 this general scheme is applied to the examples of Section 4. 2. KTF-solutions in arbitrary light-shift potentials KTF-solutions arise in a large class of light-shift potentials [11] according to the following scheme: pffiffifficonsider an arbitrary monochromatic light-field Eðr; tÞ  ð1= 2ÞðEðrÞeixt þ E ðrÞeixt Þ with EðrÞ  EðrÞ ^, where EðrÞ is a complex scalar and ^ is a spatially constant complex polarization vector satisfying ^ ^  ¼ 1. Maxwells 2   rEðrÞ ¼ 0. equations imply ðD þ k ÞEðrÞ ¼ 0 with k ¼ x=c and ^ The light-field Eðr; tÞ gives rise to a light-shift potential VðrÞ  RðaÞ hjEðr; tÞj2 i ¼ RðaÞ jEðrÞj2 , where a denotes the complex polarizability of the particles, which is assumed to be scalar, and the triangular brackets denote the time-average over one oscillation cycle. We now introduce the wave function wðrÞ  p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ Erec w with the single-photon RðaÞ=g EðrÞ, which satisfies Tw 2 hkÞ =2m. Moreover, Eq. (2) holds with V 0 ¼ 0 recoil energy Erec  ð and thus Eq. (1) is satisfied with l ¼ Erec . In brief, monochromatic light-fields with spatially constant polarization give rise to KTFsolutions of the GP-equation with a particle density proportional to the time-averaged intensity. This assertion may be generalized to include a wider class of light-fields admitting certain types of polarization gradients. Note that due to the vectorial character of the electric field, there is in general more than one light-field yielding the same KTF-solution. For repulsive collisional interaction, g > 0 and thus RðaÞ > 0 is required, which corresponds to normal dispersion, and thus a negative light-shift potential, which is obtained for negative detuning of the light-field with respect to the relevant atomic transition. In this case, the density maxima arise in the potential minima, i.e. the confining potential stabilizes the gas against collisional pressure. For attractive collisions and positive detuning, the density maxima coincide with the potential maxima, such that the repelling potential force counteracts collisional implosion of the gas. 3. Stability analysis Because KTF-solutions generally describe excitations, a central question in regard to their physical significance concerns their stability. The stability of solutions w of Eq. (1) may be considered via R 3 ^ þ ðV  lÞjw j2 þ the grand canonical potential K½we   d r½we Tw e e 4 gjwe j =2 for we  w þ ev with e 2 R, and an arbitrary normalized wavefunction v. Use of Eq. (2) and l ¼ Erec yields @@e K½we e¼0 ¼ 0 and

@2 K½we e¼0 ¼ @ e2

Z

3 d r½2v ðT^  Erec Þv þ g ðwv þ w vÞ2 ;

ð3Þ

Stability requires that

@2 K½we e¼0 > 0: @ e2

ð4Þ

Henceforth, we restrict ourselves to superpositions of N optical travelling waves sharing the same polarization vector ^ with arbitrary amplitudes and wave-vectors km ; m 2 f1; . . . ; Ng with k ¼ jkm j for all m and k ¼ 2p=k denoting the wavelength. The corresponding

PN

KTF-solution has the form w  m¼1 wm eikm r with spatially constant complex amplitudes wm . Different choices of km correspond to a rich variety of periodic and quasi-periodic light-shift potentials [5], including triangular, hexagonal or square lattice geometries. If the lattice potential is periodic, it suffices that each unit cell separately satisfies Eq.(4). In each unit cell we may then expand the arbitrary P Q perturbation v ¼ v3=2 n1 ;...;nN 2Z C n1 ;...;nN Nm¼1 einm km r in a Fourier-series with respect to the Bravais lattice, with v denoting the unit cell volP ume. Normalization requires n1 ;...;nN 2Z jC n1 ;...;nN j2 ¼ 1. For the kinetic R 3 P term in Eq.(3) we find d r2v ðT^  Erec Þv ¼ 2Erec n1 ;...;nN 2Z jC n1 ;...;nN j2 PN 2 ^ ^ ½ð m¼1 nm km Þ  1 with km  km =k. This term takes values larger than zero, if the expansion of v comprises terms C n1 ;...;nN of sufficiently PN ^ high order m¼1 nm km , reflecting the fact that high frequency perturbations possess large kinetic energies. Let us assume repulsive collisions, i.e. g > 0. The collisional term gðwv þ w vÞ2 is positive 2 then and @@e2 K½we e¼0 becomes positive for perturbations with kinetic energies exceeding Erec , which thus do not contribute to possible instabilities of wðrÞ. If wðrÞ comprises Fourier terms up to maximally first order, we may limit the Fourier expansion of relevant perturbations v to second order. For attractive collisions (g < 0), the negative collisional term gðwv þ w vÞ2 does not permit stability. As discussed for specific examples below, stable KTF-solutions typically  denotes the mean particle density.  > Erec , where q require g q 2 h a=m in terms of the s-wave scattering length Expressing g  4p a and the atomic mass m, this leads to a > amin with 2 . Inserting typical values (k ¼ 2p 106 m1 ; amin  k =8pq 20 3 q ¼ 10 m ) yields amin  300 a0 (a0  Bohr-radius), which is well in reach of experiments, if necessary by exploiting a Feshbach resonance [1]. 4. Examples of KTF-solutions in optical lattices In the following, we apply the previous general considerations to three elementary examples: a 1D optical lattice and 2D optical lattices with triangular and square geometries. The collisional interaction is assumed to be repulsive and correspondingly the required light-shift potentials are negative. 4.1. 1D lattice We begin with a 1D optical lattice composed of two counterpropagating travelling waves sharing the same polarization, thus  sin2 ðkxÞ with the mean yielding a light-shift potential VðxÞ ¼ 2V  potential well depth p V ffiffiffiffiffiffiffiffi > 0. The corresponding KTF-solution can  c sinðkxÞ, where in accordance with Eq. be written as /ðxÞ  i 2q   c  V=g. (2) q The simplicity of this example lets us directly see that it marks the boundary between the regime of weak interaction, where the description in terms of Bloch states yields a conventional dispersion relation, and the strong interaction case, where the band structure acquires additional loops at the edges and the centre of the first Brillouin zone, with the consequence of inherently non-linear hysteretic dynamics [8–10]. We suspect that this observation generalizes to be a generic property of KTF-solutions in optical lattices also in 2D and 3D. To clarify this point we remind ourselves (following Ref. [8]) that the Bloch states near the edge of the first Brillouin zonepmay within a two-level ffiffiffiffi be approximated  ðcos a eikð1þjÞx þ sin a eikð1þjÞx Þ with j picture as /a;j ðxÞ ¼ q denoting the (small) deviation of the quasi-momentum from its value k at the zone edge (jjj 1). The variational parameter a is determined by subjecting /a;j ðxÞ to Eq. (1), which leads to the system of equations

    q 2Erec þ 3  2 V  2l sinð2aÞ þ V ¼ 0 q c   q  4Erec j  V cosð2aÞ sinð2aÞ  V cosð2aÞ ¼ 0: q c

ð5Þ

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a

c

μκ

μκ μ+

μ+

μ0

μ−

μ− κ

κ

b

d

μκ μ+ μ0 μ−

κ

 1 Fig. 1. The first and second Bloch bands derived from Eq. (5) are shown in (a), (b) and (c). (a) illustrates thep linear ffiffiffiffiffiffiffiffi regime (q ¼ 2 qc ), (b) shows the boundary between3 the ¼q  c ). The cusp at j ¼ 0 corresponds to the KTF-solution /ðxÞ  i 2q  c sinðkxÞ. (c) illustrates the non-linear regime (q c)  ¼ 2q linear and the non-linear regime (q  rec .  ¼ Erec . In (d) the eigenvalues of MðzÞ for N ¼ 2 are plotted versus the mean well depth z ¼ V=E characterized by a loop structure in the lower band. For all cases V

In Fig. 1 we plot the chemical potential lðjÞ, solution to Eq. (5), in  c (a), q ¼q c q  c (c). The case q ¼q  c (b), characterized by a cusp aris(b) and q ing at the zone edge (j ¼ 0), separates the regimes with (c) and without (a) a loop structure in the dispersion function lðjÞ. By setting j ¼ 0 in Eq. (5) one may directly determine the solutions at the   arise for zone boundary. Two solutions l ¼ Erec  12 ð2  3 qqc  1ÞV Pq  c an additional solution (the top of  . If q arbitrary values of q   becomes possible, for which the loop) l0 ¼ Erec  ð1  qqc ÞV c holds. The corresponding wave function sinð2aÞ ¼pqqffiffiffiffiffiffi  sinðkxÞ is identical with the KTF-solution /ðxÞ at the /0 ðxÞ  i 2q  c and the chemical potential becomes ¼q critical density q l0 ¼ Erec in this case. Let us next inspect the stability of the KTF-solution /ðxÞ. Upon P making a Fourier expansion of v ¼ k1=2 n2fN;...;Ng C n einkx up to 2 Nth order on the unit cell ½0; k we may express @@e2 K½we e¼0 ¼ c MðzÞ c as a bilinear form with respect to the vector c  ða; bÞ, where a  R½ðC N ; . . . ; C N Þ; b  I½ðC N ; . . . ; C N Þ and

MðzÞ 

MðþÞ ðzÞ

0

0

MðÞ ðzÞ

z-axis. We admit an arbitrary phase lag h between the oscillations of the two standing waves. The specific case h ¼ p=2 has been discussed in Ref. [7]. Apart from the interesting additional physics accessible, the inclusion of other values of h in our analysis is essential for experiments, because h can only be controlled with finite precision. Light-field configurations of this type have been extensively applied in numerous previous experiments [12]. pffiffiffiffi ih=2  ðe sinðkxÞþ The corresponding KTF-solution is wh ðx; yÞ  q  denoting the mean particle density. According eih=2 sinðkyÞÞ with q

a

c

b

d

!

ð6Þ

 rec and the symmetric matrices Mð Þ ðzÞ   c =Erec ¼ V=E with z ¼ g q n;m 1 2 ðn þ z  1Þ dn;m þ 4 z ðdnþ2;m þ dn2;m þ dn;mþ2 þ dn;m2 Þ 14 z ðdnþ2;m þ dn2;m þ dn;mþ2 þ dn;m2 Þ. Stability requires that all eigenvalues of MðzÞ exceed zero. Limiting the Fourier expansion to second order N ¼ 2 yields the eigenvalues shown in Fig. 1d. The lowest eigenvalue crosses zero at z ¼ 1:22, where zErec denotes the mean potential well depth. We also have extended the Fourier expansion up to third order obtaining the same result for the lowest lying branch of the eigenvalues in Fig. 1, thus confirming that in fact higher than second order terms in the expansion of v are irrelevant for the stability. 4.2. 2D square lattice Next, we discuss a 2D optical square lattice scenario composed of two optical standing waves ^ z sinðkxÞ and ^ z sinðkyÞ, oriented along the x- and y-axes with linear polarizations parallel to the

Fig. 2. Local particle density qh ðx; yÞ for h ¼ 0 (a), h ¼ p=4 (b), and h ¼ p=2 (c). Black (white) indicates low (high) particle density. In (d) the particle current density jh ðx; yÞ is shown for h – np. The white dashed (c) and grey (d) rectangles shows a k=2  k=2 sized plaquette. In all graphs an area corresponding to 3  3 plaquettes are shown.

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r  v h ðx; yÞ ¼ ^z 2p

 X h np mp nþm ; ð1Þ d x  ; y  n;m2Z m k k

b

π

Fig. 4. Stability diagram for the family of KTF-solutions wh ðx; yÞ  pffiffiffiffi ih=2  and light-shift q ðe sinðkxÞ þ eih=2 sinðkyÞÞ with mean particle density q  ðsin2 ðkxÞ þ sin2 ðkyÞ þ 2 cosðhÞ sinðkxÞ sinðkyÞÞ. potential V h ðx; yÞ ¼ V

ð7Þ

showing that v h ðx; yÞ represents a pure vortex–anti-vortex lattice with vortex-filaments at positions kx; ky 2 p Z. For g > 0 the vortices are pinned at the potential maxima, which correspond to the density minima of the lattice (black regions in Fig. 2b and c), in accordance with results obtained for plain vortex lattices prepared in large scale traps and subsequently exposed to an optical lattice potential [13,14]. For a superfluid, we expect the particle density to vanish at the poles of the velocity field on a spatial scale determined by the healing length. We briefly discuss the implications for the vortex cores of the h ¼ p=2-case, which have a radius on the order of Rcore  1=k.  Þ1=2 , upon use of Eq. (2) Defining the healing length as n  ð8paq 2   ¼ Erec =V with mean potential well one may write ðknÞ ¼ Erec =g q  Thus, the condition that the core size exceeds the healing depth V.  length n < Rcore is equivalent to Erec < V. R 3 R 3 The mean kinetic energy per particle T h  d r qh m2 v 2h = d rqh connected with the velocity field v h increases from zero to its maximal value p2 Erec as h is tuned from zero to p=2. The total kinetic energy per particle is Erec independent of h. The difference Erec  T h reflects the quantum pressure, corresponding to the h-dependent degree of localization. The angular momentum per particle is R 3 R 3 h sinðhÞ ^ z. The small difference Lh  d r r  jh = d r qh ¼ ð8=p2 Þ  between the factor 8=p2 and unity accounts for the fact that the velocity fields from different vortices with opposite sense of rotation yield some cancellation. We may consider the stability of wh ðx; yÞ following the same procedure as in the 1D case illustrated in the context of Eq. (6). Here, a Fourier expansion up to second order leads to a 26  26 matrix, which is diagonalized in order to obtain the (g > 0) stability diagram shown in Fig. 4. As is seen in the figure, the vortical particle flux jh tends to stabilize wh ðx; yÞ. The largest stability range  rec J 3:1 ) arises for h ¼ ðn þ 1Þp corresponding to Fig. 2c and (V=E 2 Fig. 3c. For h ¼ np the flux vanishes and instability occurs for any

a

π θ

to Eq. (2) the required light-shift potential is V h ðx; yÞ ¼ 2 2  ¼ gq  . ðkxÞ þ sin ðkyÞ þ 2 cosðhÞ sinðkxÞ sinðkyÞÞ with V Vðsin Despite its formal simplicity, wh ðx; yÞ possesses remarkable properties. For h ¼ np with integer n, wh ðx; yÞ represents a two-dimensional array of stationary solitons separated by nodal lines, where the particle density qh ðx; yÞ becomes zero (black regions in Fig. 2a). For values of h – np the particle density nodes are points (cf. Fig. 2b and c) and a periodic pattern of vortical fluxes arises with alternating rotational sense for adjacent plaquettes. This is illustrated in Fig. 2d, which shows the particle flux density for  sinðhÞ r  ^ h=mÞ q z sinðkxÞ sinðkyÞ. h – np, given by jh ðx; yÞ ¼ ð While jh scales with sinðhÞ, its spatial structure is the same for all h. In contrast, the spatial structure of the corresponding velocity field v h depends on h. This is shown in Fig. 3, where v 2h is plotted for h ¼ p=10 (a), h ¼ p=4 (b) and h ¼ p=2 (c). If h tends to zero, each contour for some fixed value of jv h j approaches the diagonal nodal line structure of the h ¼ 0 density distribution (cf. Fig. 2a). Nevertheless, for all h – np, one obtains

c

Fig. 3. The square of the velocity field v 2h is plotted for h ¼ p=10 (a), h ¼ p=4 (b) and h ¼ p=2 (c). The same areas as in Fig. 2 are shown.

a

b

c

Fig. 5. The particle density q/ ðx; yÞ (a), the flux density j/ ðx; yÞ (b), and the square of the velocity field v 2/ ðx; yÞ (c) are plotted. Black (white) indicates low (high) values. The equilateral triangles (with k=2 side length) indicate a single plaquette of the lattice.

potential well depth. For negative g-values, wh ðx; yÞ becomes unstable for all h. 4.3. 2D triangular lattice As a third example, we briefly discuss a triangular lattice composed of three travelling waves propagating within the xyplane with linear polarizations parallel to the z-axis and wavevectors km ¼ k fcosð2pm=3Þ; sinð2pm=3Þg; m 2 f1; 2; 3g, mutually Thepcorresponding KTF-solution is enclosing paffiffiffiffiffiffiffiffi 120 ffiffi  ffi ikx angle.  =3ðe þ eikx=2 cos 23 ky Þ. In contrast to the superw/ ðx; yÞ  q position of four travelling waves, there is no free phase parameter here. Inserting complex amplitudes for the three superimposed travelling waves would merely yield a spatial shift of the resulting field [5]. In Fig. 5 we show the particle density q/ ðx; yÞ (a), the flux density j/ ðx; yÞ (b) and the square of the velocity field v 2/ ðx; yÞ (c). We encounter a situation very similar to the case of the square lattice at h ¼ p=2 discussed in Fig. 2c and d and Fig. 3c, obtaining a vortex–anti-vortex lattice with vortex-filaments at the nodes of the particle density. 5. Excitation of KTF-solutions In order to study KTF-solutions in experiments, a technique is required to excite them. A well-known method to produce excitations with a definite momentum in BECs is stimulated Raman scattering (SRS). Two intersecting laser beams with common polarizations, frequencies x and x þ X, and k-vectors k1 ; k2 are employed to produce a travelling intensity grating moving at a speed cg  X=Dk with Dk  jk1  k2 j. This light grating yields a

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corresponding moving light-shift potential, which can excite a the Bogoliubov ffidispersion velocity field, if X and Dk match withq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð h D k=2mÞ2 þ cs with the relation [15,16], which requires c g pffiffiffiffiffiffiffiffiffiffiffiffiffi  sound velocity cs  qg=m. In the regime of phonon-like excita g), the resonance condition is approximated tions (ð hDkÞ2 =2m q by cg ¼ cs . SRS can be extended to yield excitations with more complex spatial geometries. For example, the KTF-solution wh ðx; yÞ ^;  ^ in momentum space, comprises the four components  hk x hk y ^; y ^ denote the unit vectors in x- and y-directions. The exciwhere x tation of each momentum component requires a pair of counterpropagating laser beams with frequencies x and x þ X, i.e. in total eight beams are necessary forming a pair of crossed bichromatic standing waves. Similar multi-beam variants of SRS have been proposed as a means to excite vortex or skyrmion states in a BEC [17,18]. In the following, we discuss the use of SRS to excite arbitrary 2D KTF-solutions wðx; yÞ ¼ jwðx; yÞj eiSðx;yÞ in the xy-plane, which may possess velocity fields with a complex geometry. Since wðx; yÞ solves the Helmholtz equation p we ffiffiffi may consider the monochroixt ixt  with matic light-field pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE1 ðx; y; tÞ  ð1= 2Þ½Eðx; yÞe þ E ðx; yÞe z and the corresponding light-shift potenEðx; yÞ  g=RðaÞ wðx; yÞ ^ 2 tial Vðx; yÞ ¼ RðaÞjEðx; yÞj2 ¼ gjwðx; yÞj pffiffiffi. Consider the following bichromatic extension E2 ðx; y; tÞ  ð1= 2Þ½EX ðx; y; tÞeixt þ EX ðx; y; tÞeixt  with EX ðx; y; tÞ  cosðgÞ Eðx; yÞ þ sinðgÞ eiXt E ðx; yÞ (with some g 2 ½0; p=2). Since E1 ðx; y; tÞ by definition solves Maxwells equation, this holds for E2 ðx; y; tÞ in very good approximation, if X x is assumed. Note that in the following examples X=x is on the order of 1011 . The bichromatic field E2 ðx; y; tÞ yields the light-shift potential V X ðx; y; tÞ ¼ RðaÞjEX ðx; y; tÞj2 ¼ Vðx; yÞð1þ sinð2gÞ cosð2Sðx; yÞ  XtÞÞ. Hence, V X ðx; y; tÞ is a sum of the stationary light-shift potential Vðx; yÞ satisfying Eq. (2) and a modulation term V mod ðx; y; tÞ  sinð2gÞVðx; yÞ cosð2Sðx; yÞ  XtÞ with an experimentally adjustable modulation strength sinð2gÞ. V mod ðx; y; tÞ provides a light-shift grating moving according to the wave-vector field Kðx; yÞ  2rSðx; yÞ directly proportional to the velocity field v ðx; yÞ of wðx; yÞ. Thus, temporary application of V mod ðx; y; tÞ should be a means to excite the KTF-solution wðx; yÞ, if  hX is adjusted to match the energy difference between wðx; yÞ and the ground state in the lattice potential Vðx; yÞ [21]. Since we are interested in solu is on the same ortions wðx; yÞ, for which the collisional energy g q der as the kinetic energy, we cannot directly apply the theory of Ref. [21] to calculate the excitation efficiency. An extension of Ref. [21] to the regime of significant collisional interaction is a difficult venture, which requires further research. In case of a stable KTF-solution, the fact that we can provide the appropriate timedependent potential, which drives the required current density, is nevertheless a strong indication, that this KTF-solution can be efficiently excited. 6. Examples of bichromatic light-shift potentials In the following, we illustrate the SRS-method in case of the example wh ðx; yÞ ¼ jwh ðx; yÞj eiSh ðx;yÞ introduced in the second paragraph of Section 4. The required bichromatic light-field is tÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  cosðgÞ Eh ðx; yÞ þ sinðgÞ eiXt Eh ðx; yÞ with Eh ðx; yÞ  EX h ðx; y;p wh ðx; yÞ g=RðaÞ ^ z. The corresponding bichromatic light-shift pomod ðx;y;tÞ with V mod ðx;y;tÞ ¼ tential becomes V X h ðx;y;tÞ ¼ V h ðx;yÞþV h h V h ðx;yÞsinð2gÞcosð2Sh ðx;yÞ  XtÞ. The time-evolution of the moduðx; y; tÞ is illustrated in Fig. 6 for h ¼ p=2, indicating that lation V mod h it acts as a collection of microscopic rotors, which apply angular momentum with alternating sign within the plaquettes of the square lattice V p=2 ðx; yÞ. In the vicinity of each maximum of V p=2 ðx; yÞ (replotted in (a) from Fig. 2c) the micro-rotor term V mod p=2 ðx; y; tÞ provides a quadrupole potential rotating with alternating directions for adjacent plaquettes. In (b)-(j) V mod p=2 ðx; y; tÞ is

a

b

c

d

e

f

g

h

i

j

Fig. 6. In (a) V p=2 ðx; yÞ is replotted from Fig. 2c. White (black) regions indicate minima (maxima). In (b)–(j) the micro-rotor potential V mod p=2 ðx; y; tÞ is shown for Xt ¼ np=8 with n 2 f0; 1; . . . ; 8g. The same area as in (a) is shown. The dashed white squares indicate a k=2  k=2-sized plaquette.

shown for Xt ¼ np=8, with n 2 f0; 1; . . . ; 8g, thus illustrating a XR t ¼ p=2 clockwise rotation with angular frequency XR ¼ X=2. Let us briefly estimate the resonance condition. The angular momentum applied to each plaquette is approximately given by m XR r2 , where r  k=4 is the distance from the centre to the edge of the plaquette. Excitation of vortices requires an angular momenh and thus  h X  ð8=p2 Þ 2Erec . tum of  h per plaquette, i.e. m XR r2   For rubidium atoms and a convenient lattice wavelength (k = 1030 nm) X=2p ¼ 3:5 kHz. It has been recently pointed out that within a description of the optical lattice with the periodic potential V p=2 ðx; yÞ in terms of a Bose–Hubbard model [19], the rotor potential V mod p=2 ðx; y; tÞ simulates a staggered magnetic field alternating for adjacent plaquettes [20]. Finally, we note that the modulation term ðx; y; tÞ obtained for the triangular KTF-solution w/ ðx; yÞ of V mod / Fig. 5 (cf. Section 4) displays a similar behavior as that observed in Fig. 6 for the square lattice: Centered at each density node, microscopic rotors with quadrupolar shape apply angular momentum to the triangular plaquettes of Fig. 5. Experimentally, the generation of the bichromatic light-field EX h ðx; y; tÞ, appropriate for the excitation of wh ðx; yÞ, is straightforward using the optical set-up illustrated in Fig. 7a, thus extending a method proven practicable in previous experiments [12]. The monochromatic components Eh ðx; yÞ and eiXt Eh ðx; yÞ are produced in two nested Michelson-interferometers, each with its two branches folded under 90° angle. Two laser beams with adjustable amplitudes, frequency separation and linear polarizations perpendicular to the drawing plane in Fig. 7 are used to couple both interferometers, which comprise piezo-electrically driven mirrors for servo control of the optical path length differences and thus the temporal phase differences between the standing waves produced in each interferometer branch. The requirement, that the spatial

a

b ω Ω

cold atoms

ω

ω Ω

ω

Fig. 7. (a) Optical set-up consisting of two nested Michelson-interferometers. PZT = piezo-electric transducer, M = mirror, BS = beam splitter, AOM = acoustooptic frequency modulator, PD = photo diode, S = servo electronics. (b) Beam configuration for triangular lattice.

M. Ölschläger et al. / Optics Communications 282 (2009) 1472–1477

fields in the two interferometers are complex conjugates, is realized by setting their temporal phase differences to h and h, respectively. This corresponds to the adjustment of optical path length differences Dl ¼ hk=2p. The frequency difference X and the modulation strength sinð2gÞ are controlled by the acousto-optic frequency modulator shown in Fig. 7a. Providing the bichromatic light-field for excitation of the triangular KTF-solution w/ ðx; yÞ requires the superposition of six light beams with frequencies x and x þ X according to the sketch in Fig. 7b. All beams share the same linear polarization perpendicular to the drawing plane in Fig. 7b. Three beams with the same frequency p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x are used to produce the light-field E/ ðx; yÞ ¼ z. The phase differences between these beams g=RðaÞ w/ ðx; yÞ ^ determine the position of the nodes of E/ ðx; yÞ. To ensure that the three beams at frequency x þ X yield the corresponding field eiXt E ðx; yÞ, their phase differences need to be the same as those between the x-beams. Unfortunately, because all beams with the same frequency travel along different paths, an interferometric realization of this condition is not easily implemented. 7. Conclusions In summary, we have introduced a family of stationary solutions of the Gross–Pitaevskii equation (denoted as kinetic Thomas–Fermi-solutions (KTF)) with definite values of the kinetic energy, for which the local collisional energy is compensated by the potential energy. Such solutions are particularly relevant in the context of optical lattice scenarios, where they represent excited states at the edge of the first Brillouin zone. KTF-solutions in two- and three-dimensional optical lattices, despite their striking formal simplicity, typically possess non-zero current density fields, which can acquire unconventional spatial topologies, for example, that of a vortex–antivortex sheet. Conditions for the stability of KTF-solutions have been discussed and a general method has been proposed to excite KTF-solutions in experiments by means of time-modulated light-shift potentials. We have applied our general considerations to a few elementary examples: a 1D optical lattice, a 2D square lattice and a 2D triangular lattice, however, they should apply to more complex lattice geometries including quasi-periodic lattices.

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