Diffractive focusing of a uniform Bose-Einstein condensate
Patrick Boegel, Matthias Meister, Jan-Niclas Siem?, Naceur Gaaloul, Maxim A. Efremov, Wolfgang P. Schleich
DDiffractive focusing of a uniform Bose-Einsteincondensate
Patrick Boegel , Matthias Meister , , Jan-Niclas Siemß , ,Naceur Gaaloul , Maxim A. Efremov , , and Wolfgang P.Schleich , , Institut für Quantenphysik and Center for Integrated Quantum Science andTechnology (IQ ST ), Universität Ulm, D-89069 Ulm, Germany Institute of Quantum Technologies, German Aerospace Center (DLR), D-89077Ulm, Germany Institut für Theoretische Physik, Leibniz Universität Hannover, D-30167 Hannover,Germany Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167Hannover, Germany Hagler Institute for Advanced Study at Texas A & M University, Texas A & MAgriLife Research, Institute for Quantum Science and Engineering (IQSE), andDepartment of Physics and Astronomy, Texas A & M University, College Station,Texas 77843-4242, USAE-mail: [email protected], [email protected]
Abstract.
We propose a straightforward implementation of the phenomenon ofdiffractive focusing with uniform atomic Bose-Einstein condensates. Both, analytical aswell as numerical methods not only illustrate the influence of the atom-atom interactionon the focusing factor and the focus time, but also allow us to derive the optimalconditions for observing focusing of this type in the case of interacting matter waves.
Keywords : Bose-Einstein condensate, diffractive focusing, non-linear matter waves.
Submitted to:
New J. Phys.
Version: 18 February 2021 a r X i v : . [ qu a n t - ph ] F e b iffractive focusing of a uniform Bose-Einstein condensate
1. Introduction
When focusing an electromagnetic wave, the position and the strength of the focus aretypically controlled by a lens which imprints a position-dependent phase on the incomingwave. However, focusing is possible even without a lens, namely by employing theconcept of diffractive focusing, where the focus is a consequence of the non-trivial shapeof the initial wave function. In this article we extend this concept to non-linear matterwaves and show how it can be experimentally realized with Bose-Einstein condensates(BECs).Already in 1816 A. J. Fresnel realized [1, 2] that light passing through a circularaperture creates a bright spot on the symmetry axis before it starts to spread. Thistype of focusing is nowadays known as diffractive focusing and was originally describedfor light by the two-dimensional (2D) paraxial Helmholtz equation [3].The same effect occurs for matter waves in one or multiple dimensions. Indeed,since the equations for electromagnetic fields within the paraxial approximation havea form similar to the Schrödinger equation of a free particle, diffractive focusing wasstudied [4, 5] and successfully observed for water waves and plasmons [6], as well as foratoms [7], electrons [8], neutrons [9], and molecules [10].In all these cases, the effect of diffractive focusing manifests itself [11] providedthe initial wave function is ( i ) a real-valued one and ( ii ) has a non-Gaussian shape.Moreover, this kind of focusing can be very useful for the collimation of waves, such aswater waves and x -rays, for which no ordinary lenses exist.In contrast to the studies mentioned above, in the present article we explore thisphenomenon for an atomic BEC [12, 13] in a regime where the atom-atom interactionplays a key role. Our paper has a twofold objective: ( i ) We generalize the diffractivefocusing effect to the case of interacting waves , and ( ii ) we demonstrate that, in thisregime, a rather straightforward implementation is possible.For this purpose, we consider an atomic BEC with a rectangular initial wavefunction emulating the case of previous studies that analyzed matter wave diffraction outof a rectangular slit [4, 5, 6]. In the laboratory, such shapes can be realized with BECsconfined to so-called box potentials leading to uniform ground-state-density distributionsdue to the repulsive atom-atom interactions. The required potentials can for instancebe generated by blue-detuned light sheets, in some cases combined with higher-orderLaguerre-Gaussian (LG) laser beams [14, 15, 16].When the trap is switched-off, the freely evolving rectangular matter waveundergoes diffractive focusing in complete analogy to a matter wave being diffractedby a rectangular aperture in the near-field or paraxial regime. The particularity ofthe realization we are discussing in this study is that there is no need for a dedicatedaperture, but the box potential itself acts as the aperture and forms the required non-trivial initial state. Hence, the size of the aperture is given by the characteristic lengthof the box potential or by the size of the BEC itself.This straightforward implementation of diffractive focusing occurs in all spatial iffractive focusing of a uniform Bose-Einstein condensate
2. Theoretical foundations
In order to demonstrate the effect of diffractive focusing with a BEC we consider a quasi-1D setup that contains all relevant aspects and allows for an elementary presentation ofthe core features. To this end, we effectively freeze the dynamics in two dimensions, andanalyze the focusing of an appropriately shaped wave function in the third dimension.In this section we first introduce the effective 1D GPE that governs the dynamicsof the BEC. We then present a special form of the box-shaped trapping potential basedon a higher order LG mode.
To arrive at a quasi-1D BEC consisting of N atoms of mass m , we start from the 3DGPE [20] i (cid:126) ∂∂t ψ ( r , t ) = (cid:20) − (cid:126) m ∂ ∂ r + V ( r , t ) + gN | ψ ( r , t ) | (cid:21) ψ ( r , t ) (1)for the macroscopic wave function ψ = ψ ( r , t ) which is normalized according to the iffractive focusing of a uniform Bose-Einstein condensate (cid:90) d r | ψ ( r , t ) | = 1 , (2)where r ≡ ( x, y, z ) is the position vector with the Cartesian coordinates x , y and z .Here we assume that the atoms are interacting via a contact potential whosestrength is determined by the s -wave scattering length a s , resulting in the interactionconstant g ≡ π (cid:126) a s m . (3)In addition, the external potential V ( r , t ) ≡ V ⊥ ( x, y ) + V Box ( z, t ) (4)consists of the harmonic trap V ⊥ ( x, y ) ≡ m (cid:0) ω x x + ω y y (cid:1) (5)in the transverse directions determined by the trap frequencies ω x and ω y , as well as thebox potential V Box = V Box ( z, t ) along the z -axis enforcing a rectangular ground state.Throughout this paper we consider the case when the longitudinal characteristiclength L z of the external potential is much larger than the transverse one L ⊥ ≡ (cid:112) (cid:126) /mω ⊥ with ω ⊥ ≡ √ ω x ω y . In Appendix A we show with the help of Appendix B that in thislimit there is no dynamics in the transverse direction, that is in the x - y plane, as longas N a s (cid:28) L z .Based on these assumption we derive in Appendix A the effective 1D GPE [21] i (cid:126) ∂∂t ϕ ( z, t ) = (cid:18) − (cid:126) m ∂ ∂z + V Box ( z ) + ˜ g | ϕ | (cid:19) ϕ ( z, t ) (6)for the wave function ϕ = ϕ ( z, t ) along the z -direction. Here the effective interactionstrength ˜ g ≡ gN c ⊥ (7)is determined by the interaction constant g , Eq. (3), the number of particles N , andthe coupling parameter c ⊥ , which in general is not a constant and originates from thenon-linear coupling between the transverse ( x - y plane) and the longitudinal ( z -axis)dynamics.In Appendix A we derive also analytical expressions for c ⊥ in two limiting cases,namely for ( i ) almost non-interacting and ( ii ) weakly interacting atoms. When ≤ N a s (cid:28) L ⊥ (cid:28) L z , the atom-atom interaction is so small that there is nocoupling between the dynamics in the transverse and the longitudinal degrees of freedom,resulting in the expression c ⊥ = 12 πL ⊥ . (8) iffractive focusing of a uniform Bose-Einstein condensate L ⊥ (cid:28) N a s (cid:28) L z , we can apply the Thomas-Fermiapproximation [20] for the ground state of the 3D GPE, Eq. (1), and arrive at theformula c ⊥ = 12 πL ⊥ (cid:18) L z N a s (cid:19) (9)for the parameter c ⊥ .We conclude by noting that in section 3.3 we perform full 3D numerical simulationsbased on Eq. (1), to test the validity of the effective 1D description. We find that for theparameters considered in this article the 1D GPE, Eq. (6), with the coupling parameter c ⊥ given by Eq. (9) describes correctly the dynamics of the BEC. We realize the box potential V Box by a LG laser beam, more precisely by the radiallysymmetric LG l mode with the intensity profile [15] I l ( ρ ) = 2 πl ! Pw (cid:18) ρ w (cid:19) l exp (cid:18) − ρ w (cid:19) , (10)where l = 0 , , , ... is the order of the mode, w and P are the waist and the power ofthe beam, respectively. Here ρ ≡ (cid:112) y + z measures the distance from the beam axis,as depicted in Fig. 1.By choosing the waist w of the LG beam to be much larger than the characteristiclengths of the harmonic trap along the transverse directions, that is when w √ l (cid:29) L ⊥ ,as shown in Fig. 1, we can neglect the cylindrical symmetry of the beam profile and usethe effective 1D intensity profile I l ( ρ = z ) along the z -axis instead.In addition, if the laser frequency is far blue-detuned from the atomic resonance,the associated optical dipole potential reads [15, 22] V l ( z ) = (cid:126) Γ ∆ I l ( z ) I s . (11)Here Γ , ∆ and I s denote the decay rate, the detuning, and the saturation intensityrespectively.With the help of the explicit expression, Eq. (10), of the intensity, we find with I l ( ρ = z ) the formula V l ( z ) = 2 l πl ! (cid:126) Γ ∆ PI s w (cid:18) zw (cid:19) l exp (cid:18) − z w (cid:19) (12)for the trapping potential caused by the LG mode.If the chemical potential of the ground state is much lower than the maximum V l ( z l ) of the trapping potential located at z l ≡ w (cid:112) l/ , we can approximate Eq. (12) aroundthe potential minimum at z = 0 and obtain the power-law V l ( z ) ∼ = 2 l πl ! (cid:126) Γ ∆ PI s w (cid:18) zw (cid:19) l (13) iffractive focusing of a uniform Bose-Einstein condensate Figure 1.
Trap arrangement for diffractive focusing of a uniform BEC. The harmonictrap V ⊥ ≡ V ⊥ ( x, y ) , Eq. (5), (orange) and the LG potential V l = V l ( (cid:112) y + z ) ,Eq. (12), (blue) cause a confinement for the atoms in the transverse ( x - y plane) andthe longitudinal ( z -axis) directions, respectively, yielding a cigar-shaped ground-state-density distribution (green) of atoms. for the trapping potential V l .In the following we choose our parameters such that this power-law approximationis valid and Eq. (13) can be used to describe the box potential. Moreover, we notethat similar box-like trapping potentials can also be realized by combining appropriateHermite-Gauss beams with Gauss beam endcaps [14], or by employing blue-detunedpainted potentials [16].
3. Diffractive focusing
In this section we first identify the parameters of the LG potential V l , Eq. (13),which allow us to create a quasi-1D BEC ground state wave function with the desiredrectangular shape. By taking this state as the initial one, we then study in detail theeffect of the atom-atom interaction on the diffractive focusing. Finally, we compare theresults of our quasi-1D model to a full 3D treatment. We start with the discussion of the ground state of a quasi-1D BEC in the LG potentialgiven by Eq. (13). Throughout our article we consider Rb atoms [23, 24] and emphasize iffractive focusing of a uniform Bose-Einstein condensate Table 1.
Parameters and their values used in our numerical simulations. Here a denotes to the Bohr radius. number of atoms N scattering length a s a trap frequencies ω ⊥ = ω x = ω y π · . · Hz transverse length L ⊥ . · − m longitudinal length, beam waist L z ∼ = w · − m effective frequency ω z π · .
52 Hz detuning ∆ π · . · Hz decay rate Γ π · . · Hz saturation intensity I s
16 W · m − laser power P . that the values of all relevant parameters listed in Table 1 are accessible in a state-of-theart experiment.According to Table 1 we find for the ratio N a s L ⊥ ∼ = 245 , (14)which implies that the atoms are indeed weakly interacting.Since in this case the parameter c ⊥ is given by Eq. (9), we obtain from Table 1 thevalue c ⊥ ∼ = 1 . · m − . (15)With the help of the imaginary-time propagation method [25] we have solved Eq. (6)numerically, and have obtained the wave function ϕ l = ϕ l ( z, t ) for the ground state ofa BEC in the LG box potential V l given by Eq. (13) for different values of l , namely l = 2 , , .As displayed in Fig. 2 (a), for increasing values of l the potential V l becomes morerectangular, that is flatter at z = 0 and steeper at the positions z = ± w . Since wefulfill the condition of the Thomas-Fermi approximation [20], the wave function of theground state has the form of the inverse potential, as shown in Appendix C. Thus, whenthe potential gets more rectangular, the ground state is more rectangular, too.However, Fig. 2 (b) clearly shows that the overlap between the ground-state wavefunction ϕ l and a perfectly rectangular wave function ϕ R ( z ) ≡ √ w Θ ( w − | z | ) , (16)of width w depicted by the dashed line, is optimal for the state with l = 10 . Here Θ denotes the Heaviside step function.In Appendix C we derive the implicit equation (cid:18) l (cid:19) l l l ! = 8 √ π (cid:126) ∆mL ⊥ Γ I s w P (cid:18) N a s w (cid:19) , (17) iffractive focusing of a uniform Bose-Einstein condensate Figure 2.
Creation of a rectangular wave function of a BEC with the help of anoptical box potential. We show the LG potential V l = V l ( z ) (a) of mode order l givenby Eq. (13), together with the normalized probability density P l ≡ | ϕ l ( z, | (b) of thecorresponding ground state for the values of l = 2 , and . The maximal overlapbetween the ground-state wave function ϕ l and a wave function ϕ R , Eq. (16), of perfectrectangular shape shown by the dashed line, occurs for l = 10 . determining the optimal l for the set of parameters of our envisioned experiment.With the parameters listed in Table 1, Eq. (17) predicts l ∼ = 9 . , which agrees verywell with our numerical simulations. As a result, we consider l = 10 throughout ourfurther analysis. Now we are in the position to analyze the effect of the atom-atom interaction ondiffractive focusing. We start with the ground state of the BEC in the trapping potential V ⊥ + V and then switch off only the LG potential V while simultaneously changingthe scattering length a s from its initial value, a s = 100 a , to its final one a (f) s via aFeshbach resonance [26]. Here a is the Bohr radius.The resulting time evolution in the z -direction, calculated from Eq. (6) with a split-step algorithm [27], is shown in Fig. 3 for two different values of the final scatteringlength, namely for a (f) s = 0 (a, c) and a (f) s = 0 . a (b, d). The maximum of thedistribution appears at z = 0 .Since the initial wave function ϕ ( z, has a nearly rectangular shape, wecharacterize the focusing effect by the focusing factor f ≡ max t | ϕ (0 , t ) | max z | ϕ ( z, | , (18) iffractive focusing of a uniform Bose-Einstein condensate Figure 3.
Influence of the atom-atom interaction on the phenomenon of diffractivefocusing of a uniform BEC apparent in the time evolution of the normalized 1D densitydistribution | ϕ ( z, t ) | / | ϕ (0 , | . The initial state ϕ ( z, is the ground state of thetrapping potential V ⊥ + V and the dynamics (a, b) takes place after switching offonly the longitudinal potential V and changing the scattering length a s from its initialvalue a s = 100 a to its final one a (f) s = 0 (a), or a (f) s = 0 . a (b). The bottom row(c, d) presents the corresponding density distributions at t = 0 (blue line) and at thefocal time t = t f (orange dashed line). For both choices of the scattering length thefocusing appears at z = 0 for the focal times t f = 0 . /ω z and t f = 0 . /ω z with thefocusing factors f = 1 . and f = 1 . . which describes the increase of the amplitude of the probability density during thedynamics in comparison with its initial value.From Fig. 3, we note the factors f = 1 . and f = 1 . for a (f) s = 0 and a (f) s = 0 . a , respectively. Moreover, the repulsive atom-atom interaction ( a (f) s > )results in ( i ) a decrease of the focal time t f , and ( ii ) a faster spreading of the wavefunction directly after the focus.Figure 4 displays the dependence f = f (cid:16) a (f) s (cid:17) and t f = t f (cid:16) a (f) s (cid:17) for three differentvalues of l , with the initial state being the ground state of the complete potential iffractive focusing of a uniform Bose-Einstein condensate Figure 4.
Dependence of the focusing factor f (a) and the focal time t f (b) of auniform BEC on the final scattering length a (f) s for different orders l of the LG mode,with the initial state being the ground state of the potential V ⊥ + V l . For comparisonthe dashed line represents the case where the initial wave function is given by ϕ R ,Eq. (16), and of perfect rectangular shape. For a growing final scattering length a (f) s the focusing factor f and the focal time t f both decrease rapidly. Therefore, diffractivefocusing is only visible for weak atom-atom interactions. V = V ⊥ ( x, y )+ V l ( z ) . For a growing interaction strength the focusing factor and the focaltime decrease rapidly. Hence, the diffractive focusing is only visible if the atom-atominteraction is very weak during the dynamics.The focusing effect occurs for any of the LG potentials displayed in Fig. 2. However,only the initial wave function ϕ l ( z, with l = 10 provides the focusing parameters,which are closest to the ones resulting from ϕ R , Eq. (16).Furthermore, Fig. 5 presents a contour plot of the focusing factor f for differentvalues of the initial and final scattering lengths a s and a (f) s with l = 10 . In case I,that is a s = 0 , there is no atom-atom interaction and the ground state is given by asine function, yielding no focusing at all. For larger values of a s , the desired initialrectangular shape starts to form, and the focusing factor increases as illustrated by thecases II and III for a s = 10 a and a s = 100 a , respectively. However, for any values a s > , the focusing factor f decreases for growing values of a (f) s , as displayed in Figs. 4and 5.To summarize, in order to observe the phenomenon of diffractive focusing for aquasi-1D BEC, we require ( i ) a large initial scattering length a s for preparing a nearlyrectangular state, and ( ii ) a small final scattering length a (f) s to have a measurable effectduring the dynamics. Hence, experimentally the use of Feshbach resonances [26] ismandatory to tune the atom-atom interaction in the desired way. iffractive focusing of a uniform Bose-Einstein condensate Figure 5.
Contour plot of the focusing factor f as a function of the initial and finalscattering lengths a s and a (f) s . For the final value of the scattering length a (f) s = 0 . a ,the three images I, II, and III display the initial density distribution | ϕ ( z, | (solidblue line) corresponding to the ground state of the potential V = V ⊥ ( x, y ) + V ( z ) ,and the density distribution | ϕ ( z, t f ) | at the focal time t f (dashed orange line), forthe initial scattering lengths a s = 0 , a s = 10 a , and a s = 100 a indicated by thestars in the contour plot. In these three cases, the focusing factors are f I = 1 (with ω z t I f = 0 ), f II = 1 . (with ω z t II f = 0 . ), and f III = 1 . (with ω z t III f = 0 . ),which demonstrate that f depends only weakly on the initial scattering length a s > .However, a vertical cut through the contour plot at a fixed value of a s = 100 (redline) reveals that f depends strongly on the final scattering length a (f) s , in completeagreement with the dependence shown in Fig. 4 (a). We conclude this discussion of the phenomenon of diffractive focusing in a BEC bybriefly examining to what degree the effective 1D GPE given by Eq. (6) describes thefree propagation of the quasi-1D BEC. For this purpose, we again start from the groundstate of the BEC in the trapping potential V = V ⊥ ( x, y ) + V ( z ) and then switch offthe LG potential V , while simultaneously changing the scattering length to its finalvalue a (f) s .First, we solve Eq. (6) for the wave function ϕ = ϕ ( z, t ) numerically and obtain iffractive focusing of a uniform Bose-Einstein condensate Figure 6.
Time evolution of the normalized 1D density at z = 0 , with the transversewave function Φ given by Eq. (A.14), (orange curve), or Eq. (A.21), (blue curve). Forcomparison, the green curve displays the time evolution of the normalized integrateddensity P ( z, t ) given by Eq. (19), where ψ = ψ ( x, y, z, t ) is based on the numericalsolution of the 3D GPE defined by Eq. (1). For all cases, the initial state is givenby the ground state of the trapping potential V ⊥ + V and the dynamics occurs afterswitching off only the longitudinal potential V and instantaneously changing thescattering length a s from its initial value a s = 100 a to its final one a (f) s = 0 (a), or a (f) s = 0 . a (b). the time dependence of the normalized 1D density | ϕ (0 , t ) | / max z | ϕ ( z, | at z = 0 .Here we consider the two cases of almost non-interacting and weakly interacting atomsdepicted in Fig. 6 by the orange and blue curve, respectively. For these limits theparameter c ⊥ , Eq. (A.12), determined by the transverse wave function Φ = Φ ( x, y ) isof the form of Eqs. (8) and (9).Then, we perform the full 3D numerical simulation of the GPE given by Eq. (1) forthe wave function ψ = ψ ( r , t ) . The time evolution of the normalized integrated density P (0 , t ) / max z ( P ( z, at z = 0 , with P ( z, t ) ≡ (cid:90) d x d y | ψ ( r , t ) | , (19)is displayed by the green curve in Fig. 6.As a result, for our trap configuration, with the relevant parameters listed in Table1, the quasi-1D approximation is very reliable and in excellent agreement with the resultsof the full 3D simulation.A comparison between the curves corresponding to ( i ) almost non-interacting atoms(orange line) and ( ii ) weakly interacting atoms (blue line) with the full 3D curve revealsthat the ground state obtained within the Thomas-Fermi approximation and leading iffractive focusing of a uniform Bose-Einstein condensate c ⊥ given by Eq. (9), is more accurate in describing thedynamics of the system. This statement holds true even for small values of a (f) s , as longas the ground state was created for large values of the initial scattering length a s .We note that due to the change of the scattering length at t = 0 the transversewave function does not describe the ground state anymore and the system undergoescollective excitations. In our 3D simulations we observed such breathing oscillations inthe transverse direction. However, as a consequence of the large misanthropy of thetrapping potential, the time scale of the transverse dynamics is much shorter comparedto the longitudinal motion and, thus, the influence of these fast oscillations on the slowerlongitudinal dynamics mostly averages out for the parameters considered in this article.
4. Diffractive focusing viewed from Wigner phase space
This section illuminates the phenomenon of diffractive focusing of a BEC from quantumphase space. For this purpose we first recall the essential ingredients of the Wignerformulation [19] of quantum mechanics and then study classical trajectories in theabsence and the presence of an atom-atom interaction. This elementary approachprovides us with a deeper insight into the dynamics of the Wigner function for aninteracting matter wave.
The Wigner function [19] corresponding to the wave function ϕ = ϕ ( z, t ) is defined as W ( z, p ; t ) ≡ π (cid:126) (cid:90) ∞−∞ dy exp (cid:18) i (cid:126) py (cid:19) ϕ ∗ (cid:16) z + y , t (cid:17) ϕ (cid:16) z − y , t (cid:17) , (20)where p is the momentum.Integration of W over p , or over z yields the relations (cid:90) ∞−∞ d pW ( z, p ; t ) = | ϕ ( z, t ) | , (21)or (cid:90) ∞−∞ d zW ( z, p ; t ) = | ˜ ϕ ( p, t ) | , (22)connecting the marginals of W to the probability density distributions | ϕ ( z, t ) | and | ˜ ϕ ( p, t ) | in position and momentum space, respectively [19]. Here ˜ ϕ ( p, t ) ≡ √ π (cid:126) (cid:90) ∞−∞ d z exp (cid:18) − i (cid:126) pz (cid:19) ϕ ( z, t ) (23)is the momentum representation of the wave function ϕ = ϕ ( z, t ) .Although the Wigner function W is normalized, that is (cid:90) ∞−∞ dz (cid:90) ∞−∞ dp W ( z, p ; t ) = 1 , (24)these properties do not imply that W is always positive. Indeed, the Wigner function isa quasi-probability distribution [19] and its negative parts reflect the quantum featuresof the system under consideration. iffractive focusing of a uniform Bose-Einstein condensate Instead of deriving and solving the dynamical equation for the Wigner functioncorresponding to the 1D GPE, Eq. (6), we obtain the time-dependent Wigner functiondirectly from the definition, Eq. (20), of W in terms of the time-dependent wave function ϕ = ϕ ( z, t ) determined by solving Eq. (6) numerically.In order to visualize the dynamics in phase space, we take a point { z, p } inphase space and find the "classical trajectories" { Z ( t ) , P ( t ) } governed by the Hamiltonequations ddt Z ( t ) = ∂∂P H ( Z, P ; t ) , (25) ddt P ( t ) = − ∂∂Z H ( Z, P ; t ) (26)subjected to the initial conditions Z (0) ≡ z and P (0) ≡ p . Here the classicalHamiltonian H ( Z, P ; t ) ≡ P m + ˜ g | ϕ ( Z, t ) | (27)corresponds to the 1D GPE, Eq. (6), without the trapping potential.We emphasize that the use of Eqs. (25), (26) and (27) implies the knowledge of thewave function ϕ = ϕ ( z, t ) at all times obtained by numerically solving Eq. (6). Before we consider the case of interacting particles, we first recall [5, 28] the interaction-free dynamics ( a (f) s = 0 ) of the Wigner function W , where the initial state is the groundstate of the complete trapping potential V = V ⊥ + V , as discussed in Section 3.1. InFigs. 7 (a)-(d) we display the Wigner functions for four different times, where the red andblue colors correspond to positive and negative values of W = W ( z, p ; t ) , respectively.According to Eqs. (21) and (22), the integration over the momentum or the positionvariable provides us with the position distribution | ϕ ( z, t ) | (lower sub-figure), or themomentum distribution | ˜ ϕ ( p, t ) | (left sub-figure).Figure 7 brings out most clearly the origin of the phenomenon of diffractive focusing.Indeed, at t = 0 , the Wigner function W = W ( z, p ; 0) exhibits both positive and negativevalues. During the free expansion, t > , the parts of the Wigner function correspondingto p > ( p < ) move to the right (left) along the straight lines { z + pt, p } , displayedin Fig. 7 (d) by different colors for two different initial points in phase space. Theselines are parallel to the z -axis and describe the free classical motion, resulting from Eqs.(25), (26) and (27) with ˜ g = 0 .At the time of focusing, t = t f , the position distribution | ϕ ( z, t f ) | features anarrow maximum at z = 0 . Indeed, integration over p in Eq. (21) for fixed position z ,yields a maximum only at the values of z , which correspond to the maximal values of W = W ( z, p, t f ) , displayed by dark red color. According to Fig. 7 (a)-(d), this is theline z = 0 in phase space. In other words, focusing takes place at z = 0 , because at iffractive focusing of a uniform Bose-Einstein condensate Figure 7.
Diffractive focusing of a uniform BEC viewed from Wigner phase space. Weillustrate the time evolution of the Wigner function corresponding to the ground stateof the trapping potential V ⊥ + V after switching off only the longitudinal potential V , and changing the scattering length a s from its initial value a s = 100 a to itsfinal one a (f) s = 0 (left column), or a (f) s = 0 . a (right column). Here the red colorsindicate large positive values of W = W ( z, p ; t ) and the blue ones mark domains ofquantum phase space where W assumes negative values as suggested by the color-codeto the right of (h). The corresponding position and momentum distributions | ϕ ( z, t ) | and | ˜ ϕ ( p, t ) | are shown in the lower and left sub-figures, respectively. The classicaltrajectories { Z ( t ) , P ( t ) } governed by Eqs. (25), (26) and (27) are displayed by differentcolors for different initial points in phase space. The focal time t f is a function of a (f) s as shown in Fig. 4. iffractive focusing of a uniform Bose-Einstein condensate t = t f all negative parts of the initial Wigner function W ( z, p ; 0) have moved away fromthe p -axis [4, 5]. However, they now subtract from the positive parts of the wings andmake the distribution in space even narrower.For t > t f , the negative parts of W ( z, p, have moved further away from the line z = 0 . Since the positive parts of the original Wigner function are at lower momentathan the negative ones, they move slower and are therefore left at the center of the phasespace. They are the origin of the spreading of the position distribution | ϕ ( z, t ) | . Next, we discuss the nonlinear time evolution of the Wigner function in the case of anon-vanishing atom-atom interaction, namely for the final value of the scattering length a (f) s = 0 . a . Figure 7 (e) presents the same initial Wigner function W ( z, p ; 0) as Fig.7 (a).In contrast to the case of no atom-atom interaction, Figs. 7 (f)-(h) indicate thatthe parts of the Wigner function corresponding to p > ( p < ) do not only move tothe right (left) but also move up (down). This effect can be explained as follows.For short times, t (cid:28) t f , we can neglect in Eq. (6) the kinetic energy term ( − (cid:126) / m ) ∂ /∂z compared to the interaction term ˜ g | ϕ ( z, t ) | and arrive at the nonlinearequation i (cid:126) ∂∂t ϕ ( z, t ) ∼ = ˜ g | ϕ ( z, t ) | ϕ ( z, t ) , (28)which unfortunately is still not easy to solve.However, we note that Eq. (28) conserves the quantity | ϕ ( z, t ) | , that is ∂ | ϕ ( z, t ) | /∂t = 0 , resulting in the simplified equation i (cid:126) ∂∂t ϕ ( z, t ) = ˜ g | ϕ ( z, | ϕ ( z, t ) (29)with the solution ϕ ( z, t ) = ϕ ( z,
0) exp (cid:18) − i (cid:126) ˜ g t | ϕ ( z, | (cid:19) . (30)Thus, for t (cid:28) t f , the wave function ϕ = ϕ ( z, t ) picks up only a position-dependent phase determined by the initial distribution | ϕ ( z, | and the effectiveinteraction strength ˜ g . The gradient − ˜ gt∂ | ϕ ( z, | /∂z of this phase defines the increasein momentum, which is a function of z .This result also follows from Eqs. (25) and (27) and is displayed in Fig. 7 (h) by theclassical trajectories corresponding to different initial points in phase space. Hence, dueto this increase in momentum, the negative parts of the Wigner function get deformedand move faster away from the center compared to the case ˜ g = 0 , resulting in the focusappearing at earlier times. This behavior is also confirmed by Fig. 3 (b).For t > t f , the position distribution | ϕ ( z, t ) | spreads further, as shown in Fig. 7(h), and reduces its amplitude. Thus, the interaction term ˜ g | ϕ ( z, t ) | in Eq. (6) getssmaller compared to the kinetic energy term, and the evolution of the wave function iffractive focusing of a uniform Bose-Einstein condensate ϕ = ϕ ( z, t ) can be described solely by the Schrödinger equation. This effect is illustratedin Fig. 7 (h) by the classical trajectories, which in the long-time limit are again parallelto the z -axis.
5. Conclusions and outlook
In this article we have studied the phenomenon of diffractive focusing of interacting matter waves employing analytical as well as numerical methods. We have proposed astraightforward implementation of this effect with an atomic BEC confined by a box-liketrap as realized for instance in Ref. [16]. The interaction of the atoms forming a BECleads to the non-linearity in the GPE and is an essential ingredient in the preparationof a rectangular wave function, previously studied [4, 5, 6] in the context of Schrödingerwaves and obtained by a rectangular slit.As benchmarks, we have identified the focusing factor and the focus time which bothare functions of the strength of the atom-atom interaction. These measures allow us toderive the optimal conditions for observing this type of self-focusing of a BEC. Havingidentified the origin of diffractive focusing for interacting matter waves, illuminated bythe time evolution of the Wigner function in phase space, we conclude that the cleanestrealization occurs when the atom-atom interaction is switched off during the dynamicsby a magnetic Feshbach resonance.For the sake of simplicity we have restricted our treatment to a quasi-1D case.However, the effect of diffractive focusing takes also place for higher dimensions andcould be realized with a 3D box potential [16] generated by blue-detuned laser light. Inanalogy to the non-interacting case we expect the focusing factor to be enhanced in thissituation.We conclude by emphasizing that diffractive focusing can be used to generatebright sources of matter waves for dedicated applications in lithography and precisionmeasurements [29]. A more detailed discussion of these points goes beyond the scope ofthis article and has to be postponed to a future publication.
Acknowledgments
This project is supported by the German Space Agency (DLR) with funds providedby the Federal Ministry for Economic Affairs and Energy (BMWi) under Grant Nos.50WP1705 (BECCAL), 50WM1862 (CAL) and 50WM2060 (CARIOQA) as well as bythe Deutsche Forschungsgemeinschaft (German Research Foundation) through CRC1227 (Dq-mat) within Project No. A05. M.A.E. is thankful to the Center for IntegratedQuantum Science and Technology ( IQ ST ) for its generous financial support. The researchof the IQ ST is financially supported by the Ministry of Science, Research and Arts,Baden-Württemberg. W.P.S. is grateful to the Hagler Institute for Advanced Study atTexas A & M University for a Faculty Fellowship, and to Texas A & M AgriLife Researchfor the support of this work. iffractive focusing of a uniform Bose-Einstein condensate Appendix A. Dynamics of a Bose-Einstein condensate in a cigar-shapedtrap
We devote this appendix to the derivation of the effective 1D GPE describing thenon-linear dynamics of a BEC along the longitudinal direction of a highly anisotropiccigar-shaped trapping potential. Here we approximate the complete wave function bythe product of the transverse time-independent wave function of only the transversecoordinates x and y , the time-dependent longitudinal wave function of the longitudinalcoordinate z , as well as a time-dependent phase factor. This approach allows us toderive analytical formulas for the effective 1D interaction strength for (i) almost non-interacting and (ii) weakly interacting atoms. Appendix A.1. Decoupling of transverse and longitudinal dynamics
To derive an equation governing the dynamics of a quasi-1D BEC which consists of N atoms of mass m , we start from the 3D GPE [20] i (cid:126) ∂∂t ψ ( r , t ) = (cid:20) − (cid:126) m ∂ ∂ r + V ( r , t ) + gN | ψ ( r , t ) | (cid:21) ψ ( r , t ) (A.1)for the BEC wave function ψ = ψ ( r , t ) with the external potential V ( r , t ) ≡ V ⊥ ( x, y ) + V Box ( z, t ) (A.2)being the sum of a harmonic trap V ⊥ ( x, y ) ≡ m (cid:0) ω x x + ω y y (cid:1) (A.3)in the transverse directions determined by the trap frequencies ω x and ω y , and a boxpotential V Box = V Box ( z, t ) yielding trapping in the z -direction.In this article we consider a highly anisotropic cigar-shaped trapping geometrydefined by the relation L ⊥ (cid:28) L z , (A.4)where L z is the longitudinal characteristic length of the external potential and L ⊥ ≡ (cid:112) L x L y is the transverse one with L x ≡ (cid:112) (cid:126) /mω x and L y ≡ (cid:112) (cid:126) /mω y .In this case, as shown in Appendix B, the total energy per particle of the BEC isapproximately given by the relation EN ∼ = (cid:126) ω ⊥ (cid:18) N a s L z (cid:19) / (A.5)with ω ⊥ ≡ √ ω x ω y .Hence, for ≤ N a s (cid:28) L z , (A.6)the total energy per particle E/N is much smaller than the characteristic energy scale (cid:126) ω ⊥ of the transverse direction, making it impossible to drive collective excitations in iffractive focusing of a uniform Bose-Einstein condensate ψ ( r , t ) ≡ Φ ( x, y ) ϕ ( z, t ) exp (cid:18) − i (cid:126) ε t (cid:19) (A.7)can be approximated by the product of the real-valued wave function Φ = Φ ( x, y ) describing the ground state in the transverse direction, the wave function ϕ = ϕ ( z, t ) along the z -direction, and a time-dependent phase factor, where the constant ε shallbe determined later as to simplify the equations.Moreover, the function Φ is chosen to be normalized, that is (cid:90) d x d y Φ = 1 . (A.8)When we insert our ansatz, Eq. (A.7), into the 3D GPE, Eq. (A.1), we obtain theidentity Φ (cid:18) i (cid:126) ∂ϕ∂t (cid:19) + ε Φ ϕ = (cid:20) − (cid:126) m (cid:18) ∂ Φ ∂x + ∂ Φ ∂y (cid:19) + V ⊥ Φ (cid:21) ϕ + Φ (cid:18) − (cid:126) m ∂ ϕ∂z + V Box ϕ (cid:19) + gN | ϕ | Φ Φ ϕ. (A.9)Finally, we multiply both sides of Eq. (A.9) from the left by Φ , integrate over x and y , and arrive at the non-linear equation i (cid:126) ∂∂t ϕ = (cid:18) − (cid:126) m ∂ ∂z + V Box ( z ) + ˜ g | ϕ | (cid:19) ϕ (A.10)for the longitudinal wave function ϕ = ϕ ( z, t ) , with the effective interaction strength ˜ g ≡ gN c ⊥ (A.11)determined by the interaction constant g , Eq. (3), the number of particles N , and theintegral c ⊥ ≡ (cid:90) d x d y Φ . (A.12)Here we have made use of Eq. (A.8) and have chosen the constant ε ≡ (cid:90) d x d y Φ (cid:20) − (cid:126) m (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + V ⊥ ( x, y ) (cid:21) Φ (A.13)to simplify Eq. (A.10).As a result, we have derived the 1D GPE (A.10) which describes the longitudinaldynamics of a quasi-1D BEC characterized by the two inequalities Eqs. (A.4) and (A.6).In order to employ Eq. (A.10), we first have to find the ground-state wave function Φ of the transverse direction and then evaluate the effective interaction strength ˜ g , Eq.(A.11).According to Eqs. (A.4) and (A.6), there exist two distinct cases where both Φ and ˜ g can be calculated analytically, namely the limit of almost non-interacting atoms, ≤ N a s (cid:28) L ⊥ (cid:28) L z , and the case of weakly-interacting atoms, L ⊥ (cid:28) N a s (cid:28) L z . Inthe next sections we consider these two situations. iffractive focusing of a uniform Bose-Einstein condensate Appendix A.2. Almost non-interacting atoms
For almost non-interacting atoms with ≤ N a s (cid:28) L ⊥ (cid:28) L z , we neglect the interactionterm gN | ψ | in the 3D GPE (A.1), and the equation becomes approximately separablein the coordinates x , y , and z . As a result, the transverse wave function Φ in theansatz, Eq. (A.7), for ψ coincides with the wave function Φ (ho)0 ( x, y ) ≡ (cid:112) πL x L y exp (cid:18) − x L x − y L y (cid:19) (A.14)of the ground state of a 2D harmonic oscillator.By inserting Eq. (A.14) into the definitions for c ⊥ , Eq. (A.12), and ε , Eq. (A.13),and performing the integration over x and y , we obtain the explicit expressions c ⊥ = 12 πL x L y ≡ πL ⊥ (A.15)for the parameter c ⊥ and ε = 12 (cid:126) ( ω x + ω y ) (A.16)for the constant ε .This approach to quasi-1D BECs has already been discussed in similar ways byother groups [30, 31, 32]. Our results exactly coincide with their findings for the sameorder of approximation. Appendix A.3. Weakly interacting atoms
In the case of weakly interacting atoms, that is for L ⊥ (cid:28) N a s (cid:28) L z , the interaction term gN | ψ ( r , t ) | in the 3D GPE (A.1) is the leading one and we can apply the Thomas-Fermiapproximation [20] by neglecting the kinetic term when determining the ground-statewave function. Starting from the stationary solution ψ ( r , t ) = φ ( x, y, z ) exp (cid:18) − i (cid:126) µt (cid:19) , (A.17)we thus obtain the ground-state wave function φ ( x, y, z ) = (cid:115) µ − V ( x, y, z ) gN Θ [ µ − V ( x, y, z )] . (A.18)Here Θ denotes the Heaviside function and µ = (cid:18) mgN ω x ω y πL z (cid:19) (A.19)is the chemical potential derived in Appendix B when the box potential is approximatedby infinitely high potential walls separated by L z .As a result, within the Thomas-Fermi approximation, the total wave function φ = φ ( x, y, z ) given by Eq. (A.18) is again the product φ ( x, y, z ) = Φ ( x, y ) ϕ ( z ) (A.20) iffractive focusing of a uniform Bose-Einstein condensate Φ ( x, y ) ≡ (cid:20) L z µ − V ⊥ ( x, y ) gN (cid:21) Θ [ µ − V ⊥ ( x, y )] (A.21)and the longitudinal wave function ϕ ( z ) ≡ √ L z Θ ( L z − | z | ) , (A.22)with V ⊥ = V ⊥ ( x, y ) and µ given by Eqs. (A.3) and (A.19), respectively.By inserting Eq. (A.21) into Eq. (A.12), we obtain the explicit expression c ⊥ ∼ = 12 πL ⊥ (cid:18) L z N a s (cid:19) (A.23)for the parameter c ⊥ .Moreover, inserting Eq. (A.21) into Eq. (A.13), and neglecting the second-orderderivatives over x and y , we arrive at the formula ε ∼ = (cid:90) d x d y V ⊥ ( x, y ) Φ = 13 µ (A.24)for the constant ε , where µ is given by Eq. (A.19).We emphasize that the expression, Eq. (A.23), for c ⊥ is still obtained in the regimewhere the motion along the transverse direction is effectively frozen out ( N a s (cid:28) L z ),but in contrast to the previous case the interaction between the particles is taken intoaccount when determining the shape of the transverse ground-state wave function Φ .The comparison between the solutions of the effective 1D GPE (A.10) with c ⊥ given byEq. (A.23), and the 3D GPE (A.1), presented in Fig. 6, shows that, for the parametersconsidered in this article, our expression, Eq. (A.23), for c ⊥ describes the dynamics moreaccurately than the standard formula, Eq. (A.15), corresponding to weakly interactingatoms.We conclude this discussion by noting that the case of even stronger atom-atominteraction, when L z (cid:28) N a s , can be treated in the similar way. Indeed, the dynamicsalong the transverse direction is then much faster compared to the longitudinal directiondue to the relation L ⊥ (cid:28) L z . Here one can perform the adiabatic approximation [31, 32]to factorize the total wave function and to describe the longitudinal dynamics of thequasi-1D BEC. Appendix B. Thomas-Fermi approximation: chemical potential and energyof a Bose-Einstein condensate
The decoupling of the longitudinal and transverse degrees of freedom analyzed inAppendix A rests on the estimate, Eq. (A.5), of the total energy of the BEC perparticle in terms of the characteristic energy of the transverse motion. In this appendixwe use the Thomas-Fermi approximation and derive this estimate by first obtaining theanalytical expression for the chemical potential of a BEC governed by the 3D GPE (A.1).By elementary integration of the relation between the chemical potential and the energy,we then arrive at the desired estimate. iffractive focusing of a uniform Bose-Einstein condensate Appendix B.1. Chemical potential
The chemical potential µ of a BEC follows from the normalization condition I = (cid:90) d x d y d z | φ ( x, y, z ) | = 1 (B.1)of the Thomas-Fermi wave function [20] φ ( x, y, z ) = (cid:115) µ − V ( x, y, z ) gN Θ [ µ − V ( x, y, z )] . (B.2)Here Θ is the Heaviside function and µ is the chemical potential of the BEC which isdetermined byIn order to derive an analytical expression for µ , we approximate V Box by thepotential of infinitely high walls V Box ( z ) ∼ = (cid:40) , | z | ≤ L z ∞ , | z | > L z (B.3)separated by L z .According to Eq. (B.2), only the points { x, y, z } obeying the inequality V ( x, y, z ) ≤ µ contribute to the integral in Eq. (B.1). Hence, the regions of integration in Eq. (B.1)are given by x b x + y b y ≤ − L z ≤ z ≤ L z , (B.4)where b x ≡ µ/ ( mω x ) and b y ≡ µ/ ( mω y ) .By introducing the polar coordinates x ≡ b x r cos θ and y ≡ b y r sin θ with ≤ r ≤ and ≤ θ ≤ π , we arrive at I = b x b y gN (cid:90) L z − L z d z (cid:90) r d r (cid:90) π d θ (cid:0) µ − µr (cid:1) , (B.5)where we have used the identity V ⊥ ( b x r cos θ, b y r sin θ ) = µr , which then leads us to I = π µb x b y L z gN . (B.6)With the definitions of b x and b y together with the normalization condition,Eq. (B.1), we find the explicit expression µ = (cid:18) mgN ω x ω y πL z (cid:19) (B.7)for the chemical potential of a BEC being confined by an infinitely high box potentialalong the z -axis and two harmonic potentials along the x - and y -direction. We emphasizethat Eq. (B.7) is valid for arbitrary length scales L ⊥ and L z of the external potentials. iffractive focusing of a uniform Bose-Einstein condensate Appendix B.2. Energy
A similar calculation can be performed to find the total energy [20] E = N (cid:90) d x d y d z (cid:104) V ( x, y, z ) | φ ( x, y, z ) | + g | φ ( x, y, z ) | (cid:105) (B.8)of a BEC within the Thomas-Fermi approximation.However, the more convenient approach consists of inserting the result, Eq. (B.7),for the chemical potential into the definition [20] µ = d E d N , (B.9)which we can directly integrate to obtain E = 23 N µ. (B.10)This relation [33] coincides with the one for a purely harmonically trapped BEC intwo dimensions.Inserting Eqs. (B.7) and (3) for the interaction constant g into Eq. (B.10), we arriveat the expression EN = 2 √ (cid:126) ω ⊥ (cid:18) N a s L z (cid:19) / . (B.11)With √ / ≈ . , this yields the estimate EN ∼ = (cid:126) ω ⊥ (cid:18) N a s L z (cid:19) / (B.12)for the total energy per particle within the Thomas-Fermi approximation. Appendix C. Box potential: Optimal order parameter of LG mode
In the approach outlined in the main body of our article we approximate the potentialwell in the z direction by a LG mode of large order l . We have found numerically forthe parameters of Table 1 the optimal value l = 10 . Indeed, in this case the overlapbetween the wave function corresponding to the potential provided by the LG mode, anda rectangular one supported by a box with infinitely high and step walls, is maximal.In this appendix we derive an analytical formula for the optimal value l in terms of theparameters of the setup.Our argument relies on the Thomas-Fermi approximation and we progress in twosteps: ( i ) We first connect the chemical potential to the parameters of the potential, and( ii ) then maximize the overlap of the two wave functions by matching the maximum ofthe Thomas-Fermi wave function with the height of the rectangular one.Within the Thomas-Fermi approximation, the stationary solution ϕ ( z, t ) = ϕ T F ( z ) exp (cid:18) − i (cid:126) µ T F t (cid:19) (C.1) iffractive focusing of a uniform Bose-Einstein condensate i (cid:126) ∂∂t ϕ ( z, t ) = (cid:18) − (cid:126) m ∂ ∂z + V Box ( z ) + ˜ g | ϕ | (cid:19) ϕ ( z, t ) (C.2)reads [20] ϕ T F ( z ) = (cid:20) µ T F − V l ( z )˜ g (cid:21) Θ [ µ T F − V l ( z )] , (C.3)where we have used the potential V l = V l ( z ) , Eq. (13), for V Box in Eq. (C.2).The chemical potential µ T F is determined by the normalization condition (cid:90) ∞−∞ dz | ϕ T F ( z ) | = 1 . (C.4)According to Eq. (13) the LG potential V l can be approximated by V l ( z ) ≈ v l (cid:18) zw (cid:19) l , (C.5)where we have introduced the abbreviation v l ≡ l πl ! (cid:126) Γ ∆ PI s w . (C.6)The Thomas-Fermi distance z T F defined by the condition µ TF = V l ( z TF ) followsfrom the potential V l given by Eq.(C.5) as z T F = w (cid:18) µ T F v l (cid:19) l . (C.7)Hence, the normalization condition, Eq. (C.4), of the Thomas-Fermi wave function,Eq. (C.3), takes the form µ T F ˜ g (cid:90) z TF − z TF dz (cid:18) − z l z lT F (cid:19) = 4 l l + 1 µ T F ˜ g z T F . (C.8)When we combine Eqs. (C.7) and (C.8), we obtain the expression µ T F = (cid:18) l (cid:19) l l +1 (cid:18) v l ˜ g l l w l (cid:19) l +1 (C.9)for the chemical potential in terms of the order l and the parameters w and v l of theLG mode.Since ϕ T F is normalized, the best overlap with the perfectly rectangular wavefunction ϕ R ( z ) ≡ √ w Θ ( w − | z | ) (C.10)of width w is achieved, if | ϕ T F (0) | = | ϕ R (0) | . (C.11)Since according to the definition, Eq. (C.5), we find V l (0) = 0 , the matchingcondition, Eq. (C.11), of the two wave functions given by Eqs. (C.3) and (C.10) reads µ T F ˜ g = 12 w , (C.12) iffractive focusing of a uniform Bose-Einstein condensate (cid:18) l (cid:19) l v l = ˜ g w . (C.13)Finally, we insert Eqs. (3), (7), (9), and (C.6) into Eq. (C.13) and arrive at theimplicit equation (cid:18) l (cid:19) l l l ! = 8 √ π (cid:126) ∆mL ⊥ Γ I s w P (cid:18) N a s w (cid:19) , (C.14)for the optimal l as a function of the parameters of our setup. Here we have used therelation L z = w indicated in Table 1. References [1] Fresnel, A. J.
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