Laser Physics Letters       LETTER Statics and dynamics of a self-bound dipolar matter-wave droplet To cite this article: S K Adhikari 2017 Laser Phys. Lett. 14 025501   View the article online for updates and enhancements. Related content Stable and mobile excited two-dimensional dipolar Bose–Einstein condensate solitons S K Adhikari - Stable and mobile two-dimensional dipolar ring-dark-in-bright Bose–Einstein condensate soliton S K Adhikari - Two-dimensional bright and dark-in-bright dipolar Bose–Einstein condensate solitons on a one-dimensional optical lattice S K Adhikari - Recent citations Frontiers in multidimensional self-trapping of nonlinear fields and matter Yaroslav V. Kartashov et al - Hydrodynamic model of a Bose–Einstein condensate with anisotropic short-range interaction and bright solitons in a repulsive Bose–Einstein condensate Pavel A Andreev - Self-trapped quantum balls in binary Bose–Einstein condensates Sandeep Gautam and S K Adhikari - This content was downloaded from IP address 186.217.236.55 on 20/05/2019 at 18:37 https://doi.org/10.1088/1612-202X/aa532e http://iopscience.iop.org/article/10.1088/0953-4075/47/22/225304 http://iopscience.iop.org/article/10.1088/0953-4075/47/22/225304 http://iopscience.iop.org/article/10.1088/1612-2011/13/3/035502 http://iopscience.iop.org/article/10.1088/1612-2011/13/3/035502 http://iopscience.iop.org/article/10.1088/1612-2011/13/3/035502 http://iopscience.iop.org/article/10.1088/1612-2011/13/8/085501 http://iopscience.iop.org/article/10.1088/1612-2011/13/8/085501 http://iopscience.iop.org/article/10.1088/1612-2011/13/8/085501 http://dx.doi.org/10.1038/s42254-019-0025-7 http://dx.doi.org/10.1038/s42254-019-0025-7 http://iopscience.iop.org/1555-6611/29/3/035502 http://iopscience.iop.org/1555-6611/29/3/035502 http://iopscience.iop.org/1555-6611/29/3/035502 http://iopscience.iop.org/1555-6611/29/3/035502 http://iopscience.iop.org/0953-4075/52/5/055302 http://iopscience.iop.org/0953-4075/52/5/055302 https://oasc-eu1.247realmedia.com/5c/iopscience.iop.org/156371774/Middle/IOPP/IOPs-Mid-LPL-pdf/IOPs-Mid-LPL-pdf.jpg/1? 1 © 2016 Astro Ltd Printed in the UK 1. Introduction After the observation of Bose–Einstein condensate (BEC) [1, 2] of alkali atoms, there have been many experimental studies to explore different quantum phenomena involving matter wave previously not accessible for investigation in a controlled environment, such as, quantum phase transition [3], vortex-lattice formation [4], collapse [5], four-wave mixing [6], interference [7], Josephson tunneling [8], Anderson local- ization [9] etc. The generation and the dynamics of self-bound quantum wave have drawn much attention lately [10]. There have been studies of self-bound matter waves or solitons in one (1D) [10] or two (2D) [11, 12] space dimensions. A soli- ton travels at a constant velocity in 1D, due to a cancellation of nonlinear attraction and defocusing forces [13]. The 1D soliton has been observed in a BEC [10]. However, a two- or three-dimensional (3D) soliton cannot be realized for two- body contact attraction alone due to collapse [13]. There have been a few proposals for creating a self-bound 2D and 3D matter-wave state which we term a droplet exploit- ing extra interactions usually neglected in a dilute BEC of alkali atoms [1]. In the presence of an axisymmetric nonlocal dipolar interaction [14] a 2D BEC soliton can be generated in a 1D harmonic [11] or a 1D optical-lattice [12] trap. Maucher et  al [15] suggested that for Rydberg atoms, off-resonant dressing to Rydberg nD states can provide a nonlocal long- range attraction which can form a 3D matter-wave droplet. In this Letter we demonstrate that a tiny repulsive three-body interaction can avoid collapse and form a stable self-bound dipolar droplet in 3D [16]. There have been experimental [17] Laser Physics Letters S K Adhikari Statics and dynamics of a self-bound dipolar matter-wave droplet Printed in the UK 025501 LPLABC © 2016 Astro Ltd 14 Laser Phys. Lett. LPL 10.1088/1612-202X/aa532e 2 Laser Physics Letters Statics and dynamics of a self-bound dipolar matter-wave droplet S K Adhikari1 Instituto de Física Teórica, UNESP—Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil E-mail: Adhikari@ift.unesp.br Received 20 November 2016, revised 5 December 2016 Accepted for publication 6 December 2016 Published 23 December 2016 Abstract We study the statics and dynamics of a stable, mobile, self-bound three-dimensional dipolar matter-wave droplet created in the presence of a tiny repulsive three-body interaction. In frontal collision with an impact parameter and in angular collision at large velocities along all directions two droplets behave like quantum solitons. Such a collision is found to be quasi elastic and the droplets emerge undeformed after collision without any change of velocity. However, in a collision at small velocities the axisymmeric dipolar interaction plays a significant role and the collision dynamics is sensitive to the direction of motion. For an encounter along the z direction at small velocities, two droplets, polarized along the z direction, coalesce to form a larger droplet—a droplet molecule. For an encounter along the x direction at small velocities, the same droplets stay apart and never meet each other due to the dipolar repulsion. The present study is based on an analytic variational approximation and a numerical solution of the mean-field Gross–Pitaevskii equation using the parameters of 52Cr atoms. Keywords: Bose–Einstein condensate, Gross–Pitaevskii equation, soliton, dipolar atoms (Some figures may appear in colour only in the online journal) Astro Ltd IOP Letter 1 www.ift.unesp.br/users/Adhikari 2017 1612-202X 1612-202X/17/025501+8$33.00 doi:10.1088/1612-202X/aa532eLaser Phys. Lett. 14 (2017) 025501 (8pp) publisher-id doi mailto:Adhikari@ift.unesp.br http://crossmark.crossref.org/dialog/?doi=10.1088/1612-202X/aa532e&domain=pdf&date_stamp=2016-12-23 http://www.ift.unesp.br/users/Adhikari http://dx.doi.org/10.1088/1612-202X/aa532e 2 S K Adhikari and theoretical [18] studies of the formation of a trapped dipo- lar BEC droplet. In fact, for dipolar interaction stronger than two-body contact repulsion, a dipolar droplet has a net attrac- tion [19, 20]; but the two-body contact repulsion is too weak to stop the collapse, whereas a three-body contact repulsion can eliminate the collapse and form a stable stationary drop- let. Such a droplet can also be formed in a nondipolar BEC (details to be reported elsewhere) [21]. We study the frontal collision with an impact parameter and angular collision between two dipolar droplets. Only the collision between two integrable 1D solitons is truly elastic [10, 13]. As the dimensionality of the soliton is increased such collision is expected to become inelastic with loss of energy in 2D and 3D. In the present numerical simulation at large velocities all collisions are found to be quasi elastic while the droplets emerge after collision with practically no deforma- tion and without any change of velocity. Due to axisymmetric dipolar interaction, two droplets polar- ized along the z direction, attract each other when placed along the z axis and repel each other when placed along the x axis and the collision dynamics along x and z directions has different behaviors at very small velocities. For a collision between two droplets along the z direction, the two droplets form a single bound entity in an excited state, termed a 3D droplet molecule [22]. However, at very small velocities for an encounter along the x direction, the two droplets repel and stay away from each other due to dipolar repulsion and never meet. The dipolar interaction potential, being not absolutely integrable, does not enjoy well defined Fourier transform that would appear for an infinite system [23]. Therefore, to get meaningful results, it is necessary either to regularize this potential, or, which is equivalent, to deal only with finite systems, where the system size plays the role of an effective regularization. That is, as soon as atomic interactions include dipolar forces, only finite systems are admissible. In other words, the occurrence of dipole forces prescribes the sys- tem to be finite, either being limited by an external trapping potential or forming a kind of a self-bound droplet. The con- ditions of stability of such droplets are studied in the present manuscript. 2. Mean-field model The trapless mean-field Gross–Pitaevskii (GP) equation for a self-bound dipolar droplet of N atoms of mass m in the pres- ence of a three-body repulsion is [2, 24] ∫ φ π φ φ φ φ ∂ ∂ = − ∇ + | | + | | + | |′ ′ ⎡ ⎣⎢ ⎤ ⎦⎥ t t m aN m N K a N U t t r R r r r i , 2 4 2 3 , d , , 2 2 2 2 2 3 4 dd dd 2 ħ ħ ħ ħ( ) ( ) ( ) ( ) (1) ħ ( ) µ µ π θ ≡ = − a m U R R 12 , 1 3 cos ,dd 0 d 2 2 dd 2 3 (2) where a is the scattering length, ( )−= ′R r r , θ is the angle between the vector R and the polarization direction z, µ0 is the permeability of free space, µd is the magnetic dipole moment of each atom, and K3 is the three-body interaction term. This mean-field equation has recently been used by Blakie [24]2 to study a trapped dipolar BEC. We can obtain a dimensionless equation, by expressing length in units of a scale l and time in units of ħ/τ≡ml2 . Consequently, (1) can be rewritten as ( ) ( ) ( ) ( )∫ φ π φ φ φ φ ∂ ∂ = − ∇ + | | + | | + | |′ ′ ⎡ ⎣⎢ ⎤ ⎦⎥ t t aN K N a N U t t r R r r r i , 2 4 2 3 , d , , 2 2 3 2 4 dd dd 2 (3) where K3 is expressed in units of ħ /l m4 and φ| |2 in units of l−3 and energy in units of ħ /( )ml2 2 . The wave function is nor- malized as ( )∫ φ| | =tr r, d 12 . For an analytic understanding of the formation of a drop- let a variational approximation of (3) is obtained with the axisymmetric Gaussian ansatz: [25–27] ( ) / / ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ φ π ρ = − − ρ ρ − w w w z w r exp 2 2 , z z 3 4 1 2 2 2 2 2 (4) where ρ = +x y2 2 2, ρw and wz are the radial and axial widths, respectively. This leads to the energy density per atom: ∫ φ π φ φ φ φ = |∇ | + | | + | | + | | | |′ ′ E Na K N a N U r r r r r R r r 2 2 6 3 2 d , 2 4 3 2 6 dd 2 dd 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) (5) and the total energy per atom ( )∫≡ EE r rd [26]: [ ( )] /π κ π κ= + + + − = ρ ρ ρ ρ − E w w K N w w N a a f w w w w 1 2 1 4 18 3 2 , , z z z z2 2 3 2 3 4 2 dd 2 (6) ( ) ( ) ( )κ κ κ κ κ κ κ κ = + − − = − − f 1 2 3 d 1 , d atanh 1 1 . 2 2 2 2 2 (7) In (6), the first two terms on the right are contributions of the kinetic energy of the atoms, the third term on the right cor- responds to the three-body repulsion, and the last term to the net attractive atomic interactions responsible for the forma- tion of the droplet for | | >a add. The higher order nonlinearity (quintic) of the three-body interaction compared to the cubic nonlinearity of the two-body interaction, has led to a more singular repulsive term at the origin in (6). This makes the system highly repulsive at the center ( →ρw w, 0z ), even for a small three-body repulsion, and stops the collapse stabilizing the droplet. The stationary widths ρw and wz of a droplet correspond to the global minimum of energy (6) [26, 27] [ ( )] π κ π + − + = ρ ρ ρw N a a g w w K N w w 1 2 2 4 18 3 0, z z 3 dd 3 3 2 3 5 2 (8) 2 The term droplet formation in [24] refer to a sudden increase of density of a dipolar BEC in a trap, whereas the present droplet is self-bound without a trap. Laser Phys. Lett. 14 (2017) 025501 3 S K Adhikari [ ( )] π κ π + − + = ρ ρw N a a c w w K N w w 1 2 2 4 18 3 0, z z z 3 dd 2 2 3 2 3 4 3 (9) ( ) ( ) ( ) ( ) ( ) ( ) κ κ κ κ κ κ κ κ κ κ κ κ = − − + − = + − − − g c 2 7 4 9 d 1 , 1 10 2 9 d 1 . 2 4 4 2 2 2 4 2 2 2 3. Numerical results Unlike the 1D case, the 3D GP equation  (3) does not have an analytic solution and different numerical methods, such as split-step Crank–Nicolson [28] and Fourier spectral [29] meth- ods, are used for its solution. We solve the 3D GP equation (3) numerically by the split-step Crank–Nicolson method [28] for a dipolar BEC [27, 30] using both real- and imaginary-time propagation in Cartesian coordinates employing a space step of 0.025 and a time step upto as small as 0.000 01. In numer- ical calculation, we use the parameters of 52Cr atoms [26], e.g. =a a15.3dd 0 and m = 52 amu with a0 the Bohr radius. We take the unit of length l = 1 μm, unit of time ħ/τ≡ =ml2 0.82 ms and the unit of energy ħ /( ) = × −ml 1.29 102 2 31 J. The scattering length a can be controlled experimentally, independent of the three-body term K3, by magnetic [31] and optical [32] Feshbach resonances and we mostly fix a = −20a0 below. In figures 1 we show the 2D contour plot of energy (6) as a function of widths ρw and wz for different N and K3. This figure highlights the negative energy region. The white region in this plot corresponds to positive energy. The minimum of energy is clearly marked in figures 1. For a fixed scattering length a, (8) and (9) for variational widths allow solution for the number of atoms N greater than a critical value Ncrit. For =a a a15.3dd 0 (dipolar) and for a > 0 (nondipolar) no droplet can be formed. Laser Phys. Lett. 14 (2017) 025501 4 S K Adhikari droplets of larger sizes. In contrast to a local energy minimum in 1D [10] and 2D [11] solitons, the 3D droplets correspond to a global energy minimum with E < 0, viz figures 1, and are expected to be stable. The stability of the droplets is confirmed (details to be reported elsewhere) by real-time simulation over a long time interval upon a small perturbation. and extreme inelastic collision with the formation of droplet molecule is possible for v < 1. To test the solitonic nature of the droplets, we study the frontal head-on collision and collision with an impact param- eter d of two droplets at large velocity along x and z axes. To set the droplets in motion the respective imaginary-time wave functions are multiplied by ( )± vxexp i and real-time simula- tion is then performed with these wave functions. Due to the axisymmetric dipolar interaction the dynamics along x and z axes could be different at small velocities. At large velocities the kinetic energy Ek of the droplets is much larger than the internal energies of the droplets, and the latter plays an insig- nificant role in the collision dynamics. Consequently, there is no qualititative difference between the collision dynamics along x and z axes and that between the collision dynamics for different impact parameters at large velocities. As veloc- ity is reduced, the collision becomes inelastic resulting in a deformation and eventual destruction of the individual drop- lets after collision. At very small velocities, the dipolar energy plays a decisive role in collision along x and z axes, and the dynamics along these two axes have completely different characteristics, viz figure 9. Figure 3. Variational (line) and numerical (chain of symbols) (a) rms sizes ρ z,rms rms and (b) energy | |E versus the number of 52Cr atoms N in a droplet for two different K3: 10−38 m6 s−1 and 10−37 m6 s−1. The physical unit of energy for 52Cr atoms can be restored by using the energy scale × −1.29 10 31 J. Figure 4. Variational (v, line) and numerical (n, chain of symbols) reduced 1D densities ( )ρ x1D and ( )ρ z1D along x and z directions, respectively, and corresponding energies of a 52Cr droplet with a = −20a0 for different N and K3: (a) = = −N K10 000, 103 37 m6 s−1, (b) N = 3000, = −K 103 37 m6 s−1, (c) =N 10 000, = −K 103 38 m6 s−1, and (d) N = 3000, = −K 103 38 m6 s−1. Laser Phys. Lett. 14 (2017) 025501 5 S K Adhikari The collision dynamics of two droplets of figure  4(b) ( = = −N K3000, 103 37 m6 s−1) moving along the x axis in opposite directions with a velocity ≈v 38 each and with an impact parameter d = 2 is shown in figures 6(a)–(f ) by suc- cessive snapshots of 3D isodensity contour of the moving droplets. A similar collision dynamics of the same droplets moving along the z axis with a velocity ≈v 37 each with impact parameter d = 2 is illustrated in figures  7(a)–(f ). The droplets come close to each other in figures 6(b) and 7(b), coalesce to form a single entity in figures 6(c)–(d) and 7(c)–(d), form two separate droplets in figures 6(e) and 7(e). The drop- lets are well separated in figures 6(f ) and 7(f ) without visible deformation/distortion in shape and moving along x and z axes with the same initial velocity showing the quasi elastic nature of collision. The frontal head-on collision is also quasi elastic. To study the angular collision of two droplets of figure 4(b), at t = 0 two droplets of are placed at =± =x z3, 1, respec- tively, and set into motion towards the origin with a veloc- ity ≈v 40 each by multiplying the respective imaginary-time wave functions by ( )± +x zexp i50 9.5i and performing real- time simulation. Again the isodensity profiles of the droplets before, during, and after collision are shown in figures 8(a)– (b), (c)–(d) and (e)–(f ), respectively. The droplets again come out after collision undeformed conserving their velocities. Figure 5. The 3D isodensity ( ( )φ| |r 2) of the droplets of (a) figures 4(a)–(d) (b). The dimensionless density on the contour in figures 5 and 6–8 is 0.001 which transformed to physical units is 109 atoms cc−1. Figure 6. Collision dynamics of two droplets of figure 4(b) placed at =± =∓x z4, 1 at t = 0 moving in opposite directions along the x axis with velocity ≈v 38, illustrated by 3D isodensity contours at times (a) t = 0, (b) = 0.042, (c) = 0.084, (d) = 0.126, (e) = 0.168, (f ) = 0.210. The velocities of the droplets are shown by arrows. Figure 7. Collision dynamics of two droplets of figure 4(b) placed at =± =∓x z1, 4.8 at t = 0 moving in opposite directions along the z axis with velocity ≈v 37 by 3D isodensity contours at times (a) t = 0, (b) = 0.052, (c) = 0.104, (d) = 0.156, (e) = 0.208, (f ) = 0.260. Laser Phys. Lett. 14 (2017) 025501 6 S K Adhikari Two dipolar droplets placed along the x axis with the dipole moment along the z directions repel by the long range dipolar interaction, whereas the two placed along the z axis attract each other by the dipolar interaction. This creates a dipolar barrier between the two colliding droplets along the x direc- tion. At large incident kinetic energies, the droplets can pen- etrate this barrier and collide along the x direction. However, at very small kinetic energies (v < 1), for an encounter along the x direction the droplets cannot overcome the dipolar bar- rier and the collision does not take place. There is no such barrier for an encounter along the z direction at very small velocities and the encounter takes place with the formation of a oscillating droplet molecule. To illustrate the different nature of the dynamics of collision along x and z directions at very small velocities we consider two droplets of figure 4(d) ( = = −N K3000; 103 38 m6 s−1). For an encounter along the z direction at t = 0 two droplets are placed at =±z 3.2 and set in motion in opposite directions along the z axis with a small velocity ≈v 0.5. The dynamics is illustrated by a 2D contour plot of the time evolution of the 1D density ( )ρ z t,1D in figure 9(a). The two droplets come close to each other at z = 0 and coalesce to form a droplet molecule and never separate again. The droplet molecule is formed in an excited state due to the liberation of binding energy and hence oscillates. For an encounter along the x direction at t = 0 two droplets are placed at =±x 1.6 and set in motion in opposite directions along the x axis with the same velocity ≈v 0.5. The dynamics is illustrated by a 2D contour plot of the time evolution of the 1D density ( )ρ x t,1D in figure 9(b). The droplets come a lit- tle closer to each other due to the initial momentum. But due to long-range dipolar repulsion they move away from each other eventually and the actual encounter never takes place. In col lision dynamics of nondipolar BECs and in collision of dipolar BEC along z direction the BECs never exhibit this peculiar behavior. A semi-quantitative estimate of the dipolar repulsion of the collision of two droplets along the x axis at small velocities can be given by the variational expression for energy per atom (6) for a fixed wz, e.g. ( ) [ ( )]π κ π = + + − ρ ρ ρ ρ − E w w K N w w N a a f w w 1 2 18 3 2 , z z 2 3 2 3 4 2 dd 2 (10) where we have removed the wz-dependent constant term. Equation  (10) gives the energy well felt by an individual atom approaching the droplet along the x axis. The single approaching atom will interact with all atoms of the droplet distributed along the extention of the droplet along the z direc- tion (∼0.8, viz figure 4(d)). The most probable z value of an atom in the droplet to interact with the approaching atom is /∼ ≈z w 2 0.5.zrms In figure 10 we plot ( )ρE w versus ρw with the parameters of the droplet of figure 4(d) employed in the dynamics shown in figure  9. We find in this figure  that for small wz the energy well is entirely repulsive. For medium values of wz an attractive well with a repulsive dipolar barrier appears and for large wz a fully attractive well appears without the dipolar barrier, which is also the case of an approaching atom along the z axis. For the probable wz values there is a dipolar energy barrier of height ∼0.2 near ∼ρw 2 to 3. For the dynamics in figure 9, the approacing atom has an energy of / /= =v 2 0.5 2 0.1252 2 , which is smaller than the height of the dipolar barrier at ∼ρw 2 to 3. Hence the approaching dipolar droplet in figure 9(b) turns back when the distance between the two droplets is ∼2. In the collision along z direction there is no dipolar barrier and the encounter takes place at all velocities. Figure 8. Collision dynamics of two droplets of figure 4(b) placed at =± =x z4, 1 at t = 0 moving towards origin with velocity ≈v 40 by 3D isodensity plots at times (a) t = 0, (b) = 0.042, (c) = 0.084, (d) = 0.126, (e) = 0.168, (f ) = 0.210. Figure 9. (a) 2D contour plot of the evolution of 1D density ( )ρ z t,1D versus z and t during the collision of two droplets of figure 4(d) initially placed at =±z 3.2 at t = 0 and moving towards each other with velocity ≈v 0.5. (b) 2D contour plot of the evolution of 1D density ( )ρ x t,1D versus x and t during the encounter of the same droplets initially placed at =±x 1.6 at t = 0 and moving towards each other with velocity ≈v 0.5. Laser Phys. Lett. 14 (2017) 025501 7 S K Adhikari 4. Summary We demonstrated the creation of a stable, stationary self-bound dipolar BEC droplet for a tiny repulsive three-body contact interaction for < | |a add and study its statics and dynamics employing a variational approximation and numerical solu- tion of the 3D GP equation (1). The droplet can move with a constant velocity. At large velocities, the frontal collision with an impact parameter and the angular collision of two drop- lets are found to be quasi elastic. At medium velocities, the collision is inelastic and leads to a deformation or a destruc- tion of the droplets after collision. At very small velocities, the collision dynamics is sensitive to the anisotropic dipolar interaction and hence to the direction of motion of the drop- lets. The collision between two droplets along the z direction leads to the formation of a droplet molecule after collision. In an encounter along the x direction at very small velocities, the two droplets repel and stay away from each other avoiding a collision. It seems appropriate to present a classification of the droplet formation in different parameter domains, e.g. scat- tering length a, dipolar length add, the strength of three-body interactions K3, and the number of atoms N. In the absence of dipolar interaction ( =a 0dd ), a droplet can be formed for attractive atomic interaction (a < 0). In all cases there is a minimum number of atoms Ncrit for the droplet formation, which increases as the three-body interaction K3 increases or the scattering length a increases corresponds to less attraction, viz figure 2. There is no upper limit for the number of atoms to form a droplet. A similar panorama exists for the formation of a dipolar droplet with the exception that the dipolar droplet can be formed for