Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves
M. Olshanii, D. Deshommes, J. Torrents, M. Gonchenko, V. Dunjko, G. E. Astrakharchik
SSciPost Physics Submission
Triangular Gross-Pitaevskii breathers and Damski shockwaves
M. Olshanii , D. Deshommes , J. Torrents , M. Gonchenko , V. Dunjko , G. E.Astrakharchik Department of Physics, University of Massachusetts Boston, Boston Massachusetts 02125,USA Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, E08028 Barcelona,Spain Departament de F´ısica, Universitat Polit`ecnica de Catalunya, E08034 Barcelona, Spain* [email protected] 1, 2021
Abstract
The recently proposed hydrodynamic solution [arXiv:2011.01415] for the two-dimensional triangular Gross-Pitaevskii breathers observed in recent experiments[Phys. Rev. X 9, 021035 (2019)] is reinterpreted in terms of the Damski shockwave. We demonstrate exact agreement between the two, for a time interval0 < t < T /
8, with T the period of the harmonic trapping potential. When thetriangular shape is first imposed on the cold atoms at t = 0, it results in infinitegradients of the density and velocity fields in the hydrodynamic description at theboundaries of the atomic cloud. This singularity is related to an underlying wave-breaking catastrophe outlined by Damski [Phys. Rev. A 69, 043610 (2004)]. Atthe level of hydrodynamics, such a divergence leads to an infinite force acting onthe boundary atoms. At the time instant t = T /
8, a different catastrophe occurs:the outer edges of three separate shock waves, originating from the three sidesof the initial triangle, collide at the origin, where the density gradient becomesdiscontinuous. Despite the divergence of the boundary force at t = 0, the inter-pretation in terms of the Damski shock wave shows that this force vanishes at alltimes between the two catastrophes. Therefore, for the time interval 0 < t < T / Contents t > a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b ciPost Physics Submission2 Thomas-Fermi hydrodynamics 53 A side remark: the bulk density 64 Damski shock waves 65 A general map between hydrodynamic solutions, induced by scale invari-ance 86 A particular scale-invariance-induced map to be used 107 The SGZ solution vs. Damski shock waves 118 Discussion and summary 12References 13
The present work originates from the recent serendipitous experimental discovery of triangular-shaped two-dimensional (2D) breathers—periodically pulsating objects—in experiments with2D harmonically trapped Bose condensates [1]. In this generation of experiments, it is possibleto impose essentially any initial shape on the cloud. Reference [1] used uniformly filled triangles,squares, pentagons, hexagons, disks, and some other shapes as initial conditions. It was foundthat out of all the shapes considered, two of them—the circle and the equilateral triangle—showperiodic revivals, further interpreted as
2D Gross-Pitaevskii breathers .In the present work, we will concentrate on the triangular-shape breather. A Bosecondensate is initially prepared in a flat-bottom corral in the shape of an equilateral triangle.When the condensate is subsequently released in a 2D harmonic trap, the outer edge of thecondensate first starts expanding. At the same time, the flat-density patch in the center ofthe atomic cloud starts shrinking in area and increasing in density, while the transition region between the flat patch and the zero-density edge expands in size. See Fig. 1 for an illustrationof the cloud geometry. It has been seen both experimentally and in Gross-Pitaevskii simulationthat at a time t = T , the central flat-density patch disappears and the condensate acquires hexagonal symmetry.Here and below, T ≡ π/ω is the period of the applied harmonic trap, which has frequency ω .The flat patch then reappears, and at t = T , ciPost Physics Submission xy z n ( z , t ) SW edge, bothinner and outer, t = 0 density, t < T /8 outer SW edgeinner SW edge2D SW edge, bothinner and outer, t = T /4 Figure 1: The geometry of the problem. We show both the initial corral (solid line) and itsupside-down revival at t = T (dashed line), as well as the density along the down-verticalray ( x = 0 , y ≡ − z < t = T /
8, the bulk portion of the density distribution disappearsand the one-dimensional theory stops being valid.one again finds a flat-density triangular shape, but this time oriented upside down. At t = 3 T , the central flat-density patch disappears again and a hexagon reappears. At t = T , the condensate returns to its initial shape . See Fig. 1 for the general layout.To suppress oscillations of the moment of inertia in experiments [2], the size of the trianglewas chosen in such a way that the initial trapping energy was exactly equal to the sum of thekinetic and interaction energies. In particular, this condition guarantees that the size of theupside-down triangle at t = T / t = T / t = T / t = 0 and t = T / i (cid:126) ∂∂t ψ = − (cid:126) m ∆ ψ + g | ψ | ψ + mω r ψ (cid:90) | ψ | d r = Nψ = ψ ( r , t ) . (1)3 ciPost Physics Submission Here and below, m is the atomic mass, g is the Gross-Pitaevskii coupling constant, N is thenumber of atoms, and ω is the trapping frequency. The initial shape of the cloud is a uniformlyfilled equilateral triangle [1, 4, 5], ψ ( r , t = 0) = ground state of an equilateral-triangle-shaped corral of side L . On the one hand, the breather has signatures of being a purely classical Thomas-Fermihydrodynamic phenomenon [6], with no quantum quantum effects being relevant. On the otherhand, the abrupt density drop at the edges of the initial configuration seemingly generates aninfinite force. Two questions arise here.(i) Does the force at the edges retain the divergence at later moments of the evolution, t > t >
0, does the hydrodynamic initial condition at t = 0 fullydetermine the further hydrodynamic evolution, or does the latter depend on the parameter ξ/L , which lies beyond hydrodynamics?Here and below, ξ ≡ (cid:126) / √ mgn is the Gross-Pitaevskii healing length, which sets the quantumlower bound on the size of spatial features, and n ≈ N/ ( √ / L is the initial density insidethe breather.An exact hydrodynamic solution [5], introduced below, shows that (i) the force is finite at t >
0. In the present article, we demonstrate that (ii) the solution [5] is self-consistent, withno dependence on the healing length. t > Thanks to an ingenious insight, the authors of Ref. [5] found a formal solution of the 2DThomas-Fermi hydrodynamic equations that reproduces the “triangular breather” phenomenoneverywhere in space and time except at the boundary at integer multiples of
T / only for triangular ones, there is an exact map between anideal 2D gas with a flat phase space (coordinate-momentum) density distribution and a 2DThomas-Fermi hydrodynamics. The map allows one to construct the hydrodynamic solutionsinside the zero-density edge.
In 2004, Bogdan Damski suggested a simple singular solution of the 1D Thomas-Fermihydrodynamic equations [7], mappable to a wave catastrophe (a shock wave) in a nonlineartransport equation. Curiously, at the outer edge of the shock wave front , the density gradientvanishes at all times, except at the very time of the catastrophe.The solution that interests us starts from a catastrophe, with nonzero density only to theleft of the outer edge. The density distribution is a left half of a parabola with an initialinfinite curvature. However, this curvature decreases with time.4 ciPost Physics Submission
The above solution can be supplemented by a constant density at minus infinity thatconnects to the original parabola with a cusp. This cusp is the inner edge of the shock wavefront .Our hypothesis is that in the experiment [1] , the area between the flat middle and the edgeis a version of the Damski shock wave . For a slow spatial variation of the wavefunction, one may neglect the second spatial derivativeof the magnitude of the wavefunction | ψ | in the Gross-Pitaevskii equation (1) and arrive atthe time-dependent Thomas-Fermi hydrodynamics [6]: ∂∂t n + ∇ · ( n v ) = continuity equation ∂∂t v + ( v · ∇ ) v = − m ∇ ( gn ) − ω r Euler’s equation , (2)where n ( r , t ) = | ψ ( r , t ) | is the time-dependent density profile and v ( r , t ) = [ ψ ∗ ( r , t ) ∇ ψ ( r , t ) −∇ ψ ∗ ( r , t ) ψ ( r , t )] / [2 imn ( r , t )] is the velocity field. The initial conditions are n ( r , t = 0) = (cid:26) n for r ∈ equilateral triangle with side length L = 2 √ R µ v ( r , t = 0) = 0 (3)Here the characteristic velocity is related to the interaction strength of the atoms and to thedensity V µ ≡ (cid:114) gn m , and sets the typical length scale in the harmonic trap, R µ ≡ V µ ω . The above equations can be explicitly solved resulting in [5] n ( r , t ) = 13 √ mg Area (cid:20)
Triangle (cid:104) V r ( t ) r + v ,r ( r , t ) , V r ( t ) r + v ,r ( r t ) , V r ( t ) r + v ,r ( r , t ) (cid:105) ∩ Triangle (cid:104) V v ( t ) r + v ,v ( r , t ) , V v ( t ) r + v ,v ( r , t ) , V v ( t ) r + v ,v ( r , t ) (cid:105)(cid:21) . , (4)5 ciPost Physics Submission Here Triangle[ a , b , c ] is a triangle with vertices a , b , c and Area[ F ] is the spatial area of ageometric shape F ; further, r = (cid:16) , − √ (cid:17) , r = (cid:16) + 12 , + 12 √ (cid:17) , r = (cid:16) − , + 12 √ (cid:17) , V r ( t ) = 2 √ V µ cot[ ωt ] , V v ( t ) = 2 √ V µ tan[ ωt ] , v ,r ( r , t ) = + r ω sin[ ωt ] , and v ,v ( r , t ) = − r ω cos[ ωt ] . The solution (4) of Ref. [5] perfectly reproduces the time-evolution observed in the experiment[1]. The only remaining question is what to do with the apparent failure of the hydrodynamicequations (2) at the instances of time when the density develops a sharp edge: at t = 0, t = T /
4, and, in general, at t = integer × T /
4. A related question is how hydrodynamics canlead to a solution with a discontinuity. The answer, as we will see below, is shock waves.
An almost trivial observation that we will nonetheless use below is as follows. All the way to t = T /
8, the solution (4) features a central region of a flat density n bulk ( t ). In this region, theinteraction force vanishes, leaving only the force of the external trap. The atoms there arefreely falling towards the center. It is easy to show that the resulting density behaves as n bulk ( t ) = n cos ( ωt ) . (5)Accordingly, the velocity field becomes v bulk ( r , t ) = − r tan( ωt ) . (6) The article [7] poses the following question: what are the initial conditions for a general one-dimensional set of hydrodynamic equations such that the resulting solutions can be mapped tothe solutions of the nonlinear transport equation ∂∂t ν + ( ν ∂∂z ) ν = 0, and as such show a wavecatastrophe? Indeed, the local chemical potential does not necessarily have the linear form µ ( n ) = g n that is familiar from the mean-field description of Bose-Einstein condensates.Instead, µ ( n ) can rather substantially deviate from the linear dependence, especially in theone-dimensional case where a quadratic dependence is reached in the Tonks-Girardeau limit ofstrong interactions. 6 ciPost Physics Submission The one-dimensional Thomas-Fermi hydrodynamics in the absence of a trap reads ∂∂t n + ∂∂z ( n v ) = 0 continuity equation ∂∂t v + (cid:18) v ∂∂z (cid:19) v = − m ∂∂z ( g n ) . Euler’s equation (7)The Damski shock wave, modified for our purposes, reads n ( z, t ) = (cid:40) mg (cid:18) z − ( Z G + V G t ) t (cid:19) (cid:41) × (cid:26) z ≤ Z G + V G t (cid:27) v ( z, t ) = (cid:26) (cid:18) z − ( Z G + V G t ) t (cid:19) + V G (cid:27) × (cid:26) z ≤ Z G + V G t undeterm. otherwise (cid:27) , (8)where Z G and V G are arbitrary Galilean boosts.We observe the following:1. The force at the zero-density point Z n =0 ( t ) = Z G + V G t , where n ( Z n =0 ( t ) , t ) = 0 , is zero (since there we have ∂∂z n = 0), and it moves at a constant velocity V n =0 ( t ) = V G as a result.2. Let us select a density value n , . The point at which the density reaches this value, Z n = n , ( t ) = Z G + (cid:18) − (cid:114) g n , m + V G (cid:19) t such that n ( Z n = n , ( t ) , t ) = n , , also moves at constant velocity: V n = n , ( t ) = − (cid:114) g n , m + V G . Now, select a velocity value v , and require that the solution (8) reaches this velocity v , at the point Z n = n , ( t ). Interestingly, this can be fulfilled at all times, simply by setting V G = 2 (cid:114) g n , m + v , . ciPost Physics Submission Now, the solution (8) can be amended as follows (see also Fig. 1): n ( z, t ) = n , for z ≤ Z inner ( t ) (cid:26) mg (cid:16) z − Z outer ( t ) t (cid:17) (cid:27) for Z inner ( t ) ≤ z ≤ Z outer ( t )0 for z ≥ Z outer ( t ) v ( z, t ) = v , for z ≤ Z inner ( t ) (cid:16) z − Z outer ( t ) t (cid:17) + V outer for Z inner ( t ) ≤ z ≤ Z outer ( t )undetermined for z ≥ Z outer ( t ) , (9)with the positions and the velocities of the outer and inner edges given by V outer = 2 (cid:114) g n , m + v , , (10) Z outer ( t ) = Z edge,0 + V outer t , (11) V inner = − (cid:114) g n , m + v , , and (12) Z inner ( t ) = Z edge,0 + V inner t , (13)where Z edge,0 is the arbitrary initial position of the shock wave front, infinitely narrow at thisinstance.Four more observations:3. Velocity v , D with which the atoms move at the edge is different from the velocity V ofthe edge itself, similarly to the difference between the real and phase velocities.4. The solution (9) remains exact at all times, not only in the beginning of the evolution.5. If a mapping to an ideal gas exists (similar to the one proposed in Ref. [5]), then even inthe presence of a trap, the outer edge of the shock-wave front will still experience zeropressure from the other atoms, because it will be mapped to a single free atom at theedge of the ideal gas cloud. In this case the trajectory of the outer edge is that of asingle particle in a trap.6. The formula (12) for the velocity of the inner edge of the shock wave front can be provenfor any discontinuity in the derivative of the density, using matter conservation alone.However, this conclusion is only valid in one dimension: it can be shown that additionalterms in the continuity equation destroy this relationship in the case of non-straight 2Dedges. Pitaevski and Rosch discovered a particular symmetry of 2D Bose-condensates [2]. Thissymmetry stems from the fact that in two dimensions, the coupling constant g in Ref. (1) hasthe same dimensionality as the diffusion constant (cid:126) / (2 m ) appearing in the kinetic energy and8 ciPost Physics Submission as such does not induce a length scale. As a result, the following three observables form aclosed algebra: the Hamiltonian, the moment of inertia (proportional to the hyperradius), andthe generator of scaling transformations. Empirical consequences are that (a) the dynamics ofthe moment of inertia separates form that of the rest of the system, and (b) there emergesan additional integral of motion, namely the Casimir invariant for the above algebra. Theseproperties allow us to relate the dynamics of any two systems that have the same hyperangulardynamics—the dynamics complementary to the dynamics of the hyperradius—but differenthyperradial one and, more generally, different dependence of their Hamiltonians on thehyperradius.Let us first introduce the hyperradius R : R ( t ) ≡ (cid:18)(cid:90) n ( r , t ) r d r (cid:19) . (14)The new integral of motion that is preserved by the hydrodynamic equations (2) in the 2Dcase is represented by the square of a generalized hyperangular momentum: L ≡ m R ( t ) { E kinetic-hyperangular ( t ) + E interaction ( t ) } , (15)where E kinetic-hyperangular ( t ) = (cid:90) n ( r , t ) m v ( r , t )2 d r − m ˙ R E interaction ( t ) = (cid:90) gn ( r , t ) d r E interaction ( t ) = (cid:90) gn ( r , t ) d r are respectively the kinetic hyperangular and interaction energies.The dynamics of the hyperradius is governed by the equation of motion¨ R = L m R − ω R . (16)The motion generated by equation (16) is an isochronous (meaning that the period doesnot depend on the energy) but polychromatic oscillation of universal base frequency 2 ω . Astationary fixed point of (16) is R = (cid:114) L mω . (17)Note that at this point, the sum of the kinetic hyperangular and interaction energies equalsthe trapping energy: E kinetic-hyperangular + E interaction = E trapping , where E trapping ( t ) = (cid:90) n ( r , t ) mω r d r . At the level of the hydrodynamic equations (2), the map between two motions sharingthe same hyperangular dynamics looks as follows. Consider two sets of the 2D hydrodynamic9 ciPost Physics Submission equations (2), generally corresponding to two different trapping frequencies, ω and ω butwith the same coupling constant g : ∂∂t , n , + ∇ , · ( n , v , ) = ∂∂t , v , + ( v , · ∇ , ) v , = − m ∇ , ( g n , ) − ω , r , , (18)where ∇ , ≡ ∂/∂ r , . It can be straightforwardly verified that there is a one-to-one corre-spondence between the solutions of the first and the second sets, given by R ( t ) n ( r , t ) = R ( t ) n ( r , t ) R ( t ) (cid:18) v ( r , t ) − r ddt ln[ R ( t )] (cid:19) = R ( t ) (cid:18) v ( r , t ) − r ddt ln[ R ( t )] (cid:19) , (19)where r R ( t ) = r R ( t ) dt R ( t ) = dt R ( t ) , (20)with t = 0 ⇔ t = 0 . (21) A particular case of the general map (19)–(21) is given by ω = ω ω = 0 n ( r ,
0) = n in ( r ) n ( r ,
0) = n in ( r ) (22) v ( r ,
0) = v ( r ,
0) = , where n in ( r ) is the initial density, the same for both systems. Notice that the two systemsshare the same initial value of the hyperradius, R (0) = R (0) ≡ R (0) , and the same value of the Casimir invariant, L = L ≡ L (0) . We will identify the System 1 with the system described by Eqs. (2), subject to theinitial conditions (3). Recall that these initial conditions were chosen in such a way that thehyperradius R resides at the stationary point: R ( t ) = R (0) ≡ L (0) mω . ciPost Physics Submission The identification of System 1 is completed by setting( n, v ) = ( n , v )( r , t ) = ( r , t ) . As for System 2, we will take it to be the same as System 1 except that there is no trappingpotential. We get R ( t ) R (0) = (cid:112) ωt ) = 1cos( ωt ) r = (cid:112) ωt ) r = r cos( ωt ) t = 1 ω tan( ωt ) , (23)and, accordingly, n ( r , t ) = 1cos ( ωt ) n ( r , t ) v ( r , t ) = 1cos( ωt ) ( v ( r , t ) − r ω sin( ωt ) cos( ωt )) . (24) vs. Damski shock waves
Let us emphasize that System 2, subject to the map (22)–(24), describes free propagation fromthe initial condition (3), depicted in Fig. 1 as a solid line. We now focus our attention to thecenter of the base of the initial triangle, at ( x = 0 , y = − L / (2 √
3) = − R µ ). In free propagationfrom a triangle, the left and right vertices bounding the base cannot have an immediate effecton the dynamics in the center. As a result, for a period of time, the propagation in the basecenter, under the “free” System 2, will effectively be a free one-dimensional propagation. Thisis the point where the Damski shock wave emerges as a description of the dynamics.Let us make the following association: gn (( x = 0 , y = − z ) , t ) m = g n ( z , t ) m ( v (( x = 0 , y = − z ) , t )) y = − v ( z , t ) , where n ( z, t ) and v ( z, t ) describe the Damski shock wave (9), with Z edge,0 = L / (2 √
3) = R µ ≡ V µ ωg n , m = gn m ≡ V µ v , = 0 . (25)The bulk density n , remains constant both in space and in time. The front of the shockwave is a half-parabola, with a center at Z outer ( t ) = R µ (1 + 2 ωt )11 ciPost Physics Submission and the bulk interface at Z inner ( t ) = R µ (1 − ωt ) . At t = 1 /ω , the inner edge of the shock wave front reaches the origin and the one-dimensionaltheory collapses. Now observe that according to the map (23), t = 1 /ω corresponds to theactual time of t = T /
8, which is exactly the instance when the bulk disappears in the exact2D solution (4). In general, the shock wave (9)–(13), with the association (25), under themap (22)–(24), can be shown to reproduce the solution (4) at ( x = 0 , y = − z ) exactly, for aperiod of time 0 ≤ t ≤ T /
8. Figure 2 corroborates this correspondence. - - - y / L n ( x = y , t = T ) / n SGZ vs Damski shockwave for the mid - base of a Δ Figure 2: The shock wave theory (blue, dashed) vs exact hydrodynamics (red). The densityis plotted at t = T /
16, along the vertical symmetry axis of the triangle.We expect that the other points on the base of the original triangle, along with theircounterparts on the other two sides, will also behave as one-dimensional shock waves, but witha solution that stops being valid before
T / t = T /
8, one may nevertheless expect that the collapse of the solution at the origin attime
T / t = T / t = ∞ ) and taking advantage of Observation4 above, one can show that the outer edge reaches y = − z = − L / √
3, which is exactly theposition of the lower vertex of the upside-down triangle in Fig. 1.
We have interpreted the exact solution [5] for the triangular breather observed in the experi-ment [1] in terms of Damski shock waves [7]. This interpretation remains valid for the timesin 0 < t < T / t = 0 can be further reinterpreted as a wave breakingcatastrophe [7]. The hydrodynamic equations can not be directly continued to the t < ciPost Physics Submission domain. Indeed, the experiment [1] and its Gross-Pitaevskii model [1, 5] suggest that the exacttime evolution is time reversal invariant with respect to the instance t = 0. This propertyimplies that at this point, the Galilean boost V G (see (8)) that determines the velocity of theouter edge of the shock wave (10-11) reverses sign and, as such, undergoes a sudden jump.Such a discontinuity can not be supported by the hydrodynamic equations, signifying a failurethereof.A related phenomenon occurs at the time t = T /
8. This is the moment when the inneredge (12-13) of the shock wave reaches the origin, where it meets the two other shock waveedges, originating from the two other sides of the initial triangle. At this instant, the regionoccupied by the bulk (5-6) shrinks to a point. Again, the time reversal invariance suggestedby both the experiment and the Gross-Pitaevskii numerics implies that at t = T /
8, the bulkvelocity gradient in (6) reverses sign , along with the velocity of the inner edge (12), thussignifying another breakdown of the hydrodynamic description.
Acknowledgements
We are immeasurably grateful to Jean Dalibard for numerous discussions and as well forproviding access to unpublished numerical data, and to Z.-Y. Shi for useful discussions.
Funding information
This work was supported by the NSF (Grants No. PHY-1912542, andNo. PHY-1607221) and the Binational (U.S.-Israel) Science Foundation (Grant No. 2015616).M.G. is partially supported by the Spanish grant PGC2018-098676-B-I00 (AEI/FEDER/UE)and the Juan de la Cierva-Incorporaci´on fellowship IJCI-2016-29071. G.E.A. acknowledgesfinancial support from the Spanish MINECO (FIS2017-84114-C2-1-P), and from the Secretariad’Universitats i Recerca del Departament d’Empresa i Coneixement de la Generalitat deCatalunya within the ERDF Operational Program of Catalunya (project QuantumCat, Ref. 001-P-001644).
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