arXiv:1601.00071v1 [cond-mat.quant-gas] 1 Jan 2016 rsoyi eotd[0.Ee huhtesuperfluids the though Even [20]. reported spec- tunneling is scanning troscopy using super- lattice two-dimensional vortex a conducting in a melting paper of recent observation interesting direct II an type In for [17]. details [17–19]. in state superconductors investigated glass vortex been vortex has the a transition disorder to This transition of a presence melts undergoes in lattice lattice and liquid vortex vortex high- the into for presence particular superconductors, In in temperature fluctuation, general. thermal in significant melting in systems of lattice sys- two-dimensional lattice ideal of the vortex an mechanism of provide the of optical it understanding melting as the for problem of tem by important study an pinned is The be BEC also that [16]. shown can have lattice vortices we Recently, hidden pinned [15]. the a gets lattice in lattice optical placed vortex the is to the BEC rotating lattice the optical When co-rotating rotat- [14]. exper- in see the lattices BEC, superconductors vortex on ing of review studies II a the theoretical For and type mimics [13]. imental fields a which magnetic in in [12] placed lattice lattice vortex vortex Abrikosov triangular tized micro- their [11]. and quantum etc and glass superconductivity magnetism 10], Bose [9, 8], properties scopic [7, superfluid-Mott- transition Anderson ideal [6], insulator as behavior an such superfluid as problems [5], can emerged several localization BEC studying has con- for potential in Bose-Einstein ground random potential [2–4]. in Random speckles random- densates optical atoms by a [1]. ultracold created Such be lattice in optical realized system. in be the a can long-range breaks field-induced-order of disorder that when symmetry shown enhanced been continuous be has can it the correlations over Recently averaging survive disorder. effects strong the real-life contrary expectation, way and In to unavoidable controlled are a impurities disorder. this random in and systems in display interaction to because between possible attention interplay is much it got system has potential dom ail oaigBCcetshgl ree quan- ordered highly creates BEC rotating rapidly A ran- in (BEC) Condensates Bose-Einstein of study The iodrIdcdVre atc etn nBs-isenC Bose-Einstein in Melting Lattice Vortex Induced Disorder ASnmes 37.m 37.k 78.j 67.40.Vs 67.85.Hj; 03.75.Kk; 03.75.Lm; numbers: PACS an vortices. of profile the absence factor of The structure pair the each lattices. presence from optical demonstrated in po two is melts impurity lattice of also random lattice superposition vortex of the the presence by that inside In show vortices We hidden impurities. with melts. of hole cluster giant a of st type by the new also a and wi that lattice the impurity vortex show impurity Abrikosov single free single the impurity a of the of position in the presence on in depends that distortion show We disorder. or esuytevre atc yaisi rsneo igeim single of presence in dynamics lattice vortex the study We 1 eateto hsc,PnihryUiest,Puducherr University, Pondicherry Physics, of Department 2 eateto hsc,S ueUiest,Pn 107 Ind 411007, Pune University, Pune SP Physics, of Department .Mithun T. 1 .Porsezian K. , 1 ie otxi uefli n yeI superconductor form quan- type-II same a and on the superfluid the acting a has that force in fact Magnus vortex the the tized to or due force is and transverse superconductors superfluid the type-II simi- between The the dynamics vortex in behaviour. the similar vortices) in show of larity also (pinning systems impurity the of of both dynamics presence the in is Similarly, BEC vortices field rotating lattice. magnetic triangular applied the Abrikosov in both equilibrium superconductors the type-II in and example, configuration For lattice systems vortex these behaviour. in very similar dynamics are show vortex superconductors the and systems, helium) different liquid and (BEC density, frno muiiso iodri h ytm o this For system. the presence in the disorder by or induced impurities is random which of BEC in melting lattice investigated. be can mechanism processes where such systems of vortex ideal and as considered [20] are The fields lattices research several [31]. across systems two-dimensional impact in have the studied of melting is of rapidly mechanism understanding cooper- a exchange the [30]. in ring to BECs lattice ative due pancake vortex BEC the the quasi-two-dimensional of BEC of rotating of edge melting center outer quantum the the The in at form of can vortex instead liquid potential of vortex radial investigation that quartic in shows The trapped [28]. BEC in [29]. condensate melting BEC melt lattice the to in used of lattice been rotation vortex also have fast compression can anisotropic to BEC The in due lattice vortex place of has take melting It the that reported. shown [24–27] are been in fluctuations quantum while to theoretically, vor- due explored melting well superfluid is two-dimensional lattice the tex dislocation-mediated of the melting [23] thermal reference In way vor- controlled the dynamics. on a tex disorder in and display interaction between to a interplay possible the provides is BEC it Therefore where system. system nat- the of in presence impurities the in ural of dynamics because vortex difficult Studying is superconductor 22]. [21, velocity vortex the n ihayt Dey Bishwajyoti and nti ril epeetanwmcaimo vortex of mechanism new a present we article this In egho h muiyptnil ethen We potential. impurity the of rength h itga ftedsacsbetween distances the of histogram the d hrsett h oiin ftevortices the of positions the to respect th K ogrneodri h etdvortex melted the in order long-range fpedrno oeta generated potential pseudorandom of tcnb rae ntevre lattice vortex the in created be can it eta rdsre h otxlattice vortex the disorder or tential otxltiegt itre n the and distorted gets lattice vortex uiya ela admimpurities random as well as purity steqatzdcruainvco and vector circulation quantized the is 004 India. 605014, y f = 2 ρ K ia. × V ondensate where ρ stemass the is V is 2 we study numerically the effects of random potential on additional lattice to the main optical lattice Vimp = 2 2 2 2 the vortex lattice of a rotating BEC. Disorder is intro- V0[cos (kx)+cos (ky)] + V1[cos (k1.r)+cos (k2.r)] [35]. duced in the system by the external potential associated The randomness of the potential is determined by the ra- with the random impurities. In order to understand the tio of the wavelengths of the main and additional optical role played by the impurities we also study the effect of lattice. The origin of the dissipation term (γ) in Eq. (1) a single impurity on the vortex lattice in BEC. Our nu- is due to the presence of thermal component present in merical calculations based on the Gross-Pitaevskii equa- the trap or in other words it is related to the collision tion (GPE) show that a single vortex can be trapped by of the noncondensed atoms with the condensate atoms. a single impurity. The vortex lattice gets distorted in To include dissipation, the dynamical equation given by presence of a single impurity. The extent of the vortex the Gross-Pitaevskii equation (GPE) is modified by in- lattice distortion depends on the position of the impu- cluding the γ-term in the equation. For the derivation of rity (commensurate or incommensurate with respect to the modified GPE see [36]. The value of the parameter the undistorted Abrikosov vortex lattice) as well as the γ has been obtained by fitting theoretical results with strength of the impurity potential. We also show that a experiments to be γ =0.03 [36]. Thereafter this particu- cluster of impurities can create a giant hole containing lar value of the parameter (γ =0.03) have been used by hidden vortices in the vortex lattice. Finally, we show several authors [16, 28, 37–39] etc. for studying vortex the disorder induced vortex lattice melting in presence of dynamics in BEC. The damping term helps faster con- random impurity potential and also the pseudorandom vergence to the equilibrium state. The small variation of potential generated by the superposition of two optical γ value only effects the convergence time. In our system lattices. this term is introduced for faster convergence of the sys- The dynamics of the system is described by the two- tem to the equilibrium vortex lattice state. In Eq. (1), dimensional (2D) dimensionless time-dependent Gross- the spatial coordinates, time, condensate wave function, −1 Pitaevskii Equation (2DGPE) rotational frequency and energies are in units of a0, ω⊥ , −3/2 1 2 2 2 a0 , ω⊥ and ¯hω⊥ respectively, where a0 = p¯h/mω⊥. (i − γ)ψt = [− (∂ + ∂ )+ V (x,y) − µ + p|ψ| − ΩLz ]ψ (1) 2 x y The 2DGPE (Eq. (1)) is solved numerically using the split-step Crank-Nicolson method. In our simulation, we where V (x, y) = Vtrap + Vimp, Ω is the rotational fre- set small spatial step ∆x = ∆y = 0.08 and time step quency and Lz is the angular momentum in z direction. ∆t = 0.001 for Fig. 1, while ∆x = ∆y = 0.04 and ∆t = We consider the condensate is trapped in a harmonic 0.0005 for Figs. 2-8. The parameters are chosen from the 1 2 4 87 plus quartic trap potential Vtrap = 2 (r + λr ). The ad- experiments on Rb [37, 40]. The dissipation parameter vantage of harmonic plus quartic trap potential over the is set to γ =0.03 and λ =0.01. Eq. (1) is propagated in harmonic potential is that in the former case the con- imaginary time by replacing t as t →−it until the ground densate can be rotated with rotational frequency larger state solution is achieved and thereafter it is propagated than the trap frequency (Ω > 1) [32, 33]. However we in real-time. In order to conserve the norm of the wave show that the impurity-mediated vortex lattice melting function for nonzero value of γ, the chemical potential is is possible even for Ω < 1. Further we have shown that treated as time-dependent and is adjusted at each time the vortex lattice melting is also possible for harmonic step. trap potential. Vimp is the impurity potential. For a We characterize the distortion of the vortex lattice in single nonmagnetic impurity at position (x0,y0) we take presence of impurity in terms of (i) the structure factor 2 2 ik.r the impurity potential as Vimp = V0 exp[− (x − x0) + S(k)= R dr|ψ(r,t)| e (ii) the impurity potential en- 2 2 (y − y0) /(σ/2) ] where V0 is the strength of the im- ergy hVimpi [15, 20] and (iii) the histograms of distances purity potential and σ is width of the potential. Such a between each pair of vortices [29]. We calculate the varia- single defect can be created experimentally [2]. For ran- tion of these quantities with the strength of the impurity dom nonmagnetic impurities the impurity potential or potential. disorder is defined by an independent random variable To show the pinning of the single vortex by an im- uniformly distributed over [−V0, V0] at each spatial posi- purity we study the dynamical evolution of the rotating tion r, where V0 denotes the strength of the disorder [34]. condensate in presence of an impurity or pinning center. Experimentally it is possible to create random impurities We track the positions of the single vortex at different using laser speckle method [11]. Another way to create times of the dynamical evolution. Fig. 1 shows the po- such disorder or pseudorandom potential is by superpos- sitions of the single vortex at different times in presence ing two optical lattices of incommensurate periods. The of a single impurity at (0, −0.8). From the figures in the disorder in this case is represented by the bichromatic two panels we can see that the single vortex enters the potential Vimp = V0[cos(2πβx + φ) + cos(2πβy + φ)], condensate at the top and finally gets pinned to the im- where β = k2/k1 is the irrational ratio of the wave purity by following a spiral path. The pinning potential vectors of the two optical lattices and φ is their phase prevents the vortex movement to other places by provid- differences. Although the potential is deterministic, it ing the increased effective potential as discussed in [41]. mimics disorder in finite-size system [11]. Pseudoan- Similar pinning of a single vortex (flux line) by a pinning dom potential can also be created by adding a second center also takes place in type-II superconducting films 3
4.0 t=10000 (a) t=12000 (b) t=12400 (c) t=12490 (d) t=12500 (e) 0.016
0.012 0.0 y 0.008
−4.0 0.004 x 0 −4.0 0.0 4.0 t=12510 (f) t=12520 (g) t=12530 (i) t=12540 (h) t=12550 (i) 0.016
0.012
0.008
0.004
0
FIG. 1: (Color online) (a)-(i) Condensate density |ψ|2 showing the time evolution of the single vortex position. The vortex gets pinned to the single impurity placed at the position (0, −0.8) following a spiral path. Here V0 = 20 and Ω = 0.479.
0.01 (a) (b) (c) (d) (e) 4.0 0.008
0.006 0.0 y 0.004 −4.0 0.002 x 0 −4.0 0.0 4.0 (f) (g) (h) (i) (j)
2.5 0.003 α 1 0.0 α 0.002 ky 2 −2.5 0.001
kx 0 −2.5 0.0 2.5
FIG. 2: (Color online) Condensate density |ψ|2 depicted in upper panel shows the vortex lattice, while the lower panel shows the structure factor profile of each vortex lattice situating above its top. Here (a) and (f) are |ψ|2 and its structure factor without impurity, while (b)-(e) 2 show |ψ| for single impurity which is kept at the positions (x0 = 1.8,y0 = 0.12),(x0 = 0.4,y0 = 0.023), (x0 = 0.9,y0 = 0.06), and (x0 = 1.5,y0 = 0.1) respectively, and (g)-(j) are the corresponding structure factor profile. Angles α1 and α2 are measured as marked in (f). Here V0 = 5 and Ω = 1.1.
[42]. We now show how the vortex lattice gets distorted shows periodic peaks of regular hexagonal lattice [15]. as the position of the single impurity is varied. In ab- The structure factor profile for Fig. 2(a) is shown in sence of impurity the vortex lattice in rotating BEC is Fig. 2(f) where we can see periodic peaks of regular a triangular Abrikosov lattice where the vortex lattice hexagonal structure. Figs. 2(b-e) show the vortex lattice positions have translational periodicity or long-range or- pattern when a single impurity position is varied from der as shown in Fig. 2(a). In addition to condensate a commensurate position (Fig. 2(b)) to various incom- density profile we have also plotted the density structure mensurate positions (Figs. 2(c-e)). Here commensurate factor profile. The structure factor provide us informa- position of the single impurity means it coincides with a tion about the periodicity of the condensate density. For vortex position of the undistorted Abrikosov lattice Fig. Abrikosov triangular lattice, the structure factor profile 2(a). In presence of impurity the vortex lattice gets dis- 4
90 0.5 purity and the distortion is minimum when the impurity
80 position commensurate with the undistorted vortex lat- 0.4 tice positions. The vortex lattice distortion is maximum 70 when the impurity is placed in the center of the two vor- 60 tices of the Abrikosov vortex lattice (Fig. 2(d)). From 0.3 the corresponding structure factor profile (Fig. 2(i)) we max 50 | 2
α can see that there are no periodic peaks and no regular -
1 40 α | 0.2 hexagonal lattice structure. Fig. 3 shows the variation of 30 the angular difference |α1 −α2|max and the impurity po- Eimpurity/Eimpurity0 20 0.1 tential energy Eimpurity = hVimpi of the distorted vortex 10 lattice with the strength of the single impurity potential. The angles are defined in Fig. 2(f). Here |α1 − α2|max 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 represents the maximum of the difference in angles. The V0 increase in the angular difference |α1 −α2|max show that the vortex lattice gets increasingly distorted with the strength of the impurity potential V0 (red line). Sim- ilarly the impurity potential energy E initially FIG. 3: (Color online) Variation of the angular distortion |α1 − impurity 0 α2|max (red line) and the lattice energy (green dots) with V0, where decreases quite rapidly with increasing V (green dots) V0 is the strength of the single defect for Ω = 1.1. Single defect is and finally it attains nearly a constant value showing at the position (0.9, 0). Here Eimpurity0 is the impurity potential that the vortex lattice has got pinned to the impurity energy when Ω = 0. [16]. It is known that a ‘giant hole’ created in BEC can be used to study controlled circular superflow and phase ordered and the long-range order or lattice translational slippage in superfluids and also the transverse Tkachenko et al invariance is lost. This is reflected in the corresponding modes in the vortex lattice [43]. Kasamatsu created structure factor profiles shown in Figs. 2(g-j) respec- a giant hole in the condensate density of a BEC trapped tively. From these figures we can see that the distortion in a harmonic plus quadratic potential [33] by rotating et al of the vortex lattice depends on the position of the im- the condensate with a frequency Ω > 1. Engels created a giant hole in a rapidly rotating dilute BEC by removing atoms from the rotating condensate using a tightly focused resonant laser [12]. We, on the other (a) hand, show that a giant hole can be created in the con-
0.01 3 25 (a) (b)
25 20 4.0 2 20 15 0.008 15 10 1 0.006 0.0 V(x,y) 10 5 y 0 5 0 0.004 0 −1 4 −4.0 2 0.002 −2 −4 0 −2 y 0 −2 x −3 x 2 0 4−4 −4.0 0.0 4.0 0.01 (c) (d) 0.012 0.008 0.01 0.018 (b) 0.008 0.016 0.006 4.0 0.014 0.006 0.012 0.004 0.0 y 0.01 0.004 0.008 0.002 0.002
−4.0 0.006 0.004 0 0 x 0.002 0 −4.0 0.0 4.0 FIG. 5: (Color online) (a) Giant hole in the condensate density |ψ|2 for a cluster of impurities kept at the center for Ω = 1.1, (b) FIG. 4: (Color online) (a) The potential of a cluster of 6 impurities corresponding phase profile, (c) giant hole for a cluster of impurities placed in the harmonic plus quartic trap potential, and (b) the placed away from the center for Ω = 1.1, and (d) giant hole for 2 condensate density |ψ| in absence of rotation. Here V0 = 20. cluster of impurities kept at the center for Ω = 0.95. Here V0 = 20. 5
0.014 8.0 (a) 0.03 (b) (a) 0.01 (b) 0.02
0.012 4.0 4.0 0.025 0.008 0.01 0.015 0.02 0.0
0.0 0.008 y y 0.006 0.015 0.01 −4.0 0.006 0.004 0.01 0.005 −8.0 −4.0 0.004 0.005 0.002 x 0.002 0 0 x −8.0 −4.0 0.0 4.0 8.0 −4.0 0.0 4.0 0 0 2 (c) 0.018 (d) 0.025 FIG. 8: (Color online) Condensate density |ψ| showing melted 0.016 vortex lattice (a) for the anharmonic trap potential, and (b) for 0.014 0.02 the harmonic trap potential. Here Ω = 0.9 and V0 = 80 0.012 0.015 0.01
0.008 8.0 0.01 (a) 0.04 (b) 0.05 0.006 0.035 4.0 0.004 0.005 0.03 0.04 0.002 0.025 0.0 0.03 0 0 y 0.02
−4.0 0.015 0.02 (Color online) Condensate density |ψ|2 for various strength 0.01 FIG. 6: −8.0 0.01 V0 V0 V0 0.005 of the random impurity potential. (a) = 10, (b) = 30 , x (c) V0 = 50, and (d) V0 = 80. Here Ω = 1.1. −8.0 −4.0 0.0 4.0 8.0 0 0 30.0 (c)
20.0 5.0 (a) 0.005 counts 10.0 0.004 0.0
y 0.003
k 0.0 (d) 0.002 20.0 0.001
−5.0 0 counts 10.0 −5.0 0.0 5.0
kx 0.0 0.0 4.0 8.0 12.0 40.0 (b) distance 30.0
20.0 FIG. 9: (Color online) Condensate density |ψ|2 showing melted counts vortex lattice (a) for biochromatic potential, and (b) for combined 10.0 potential (main lattice+secondary lattice), (c) histogram plot for 0.0 Fig. (a), and (d) histogram plot for Fig. (b). (c)
20.0 densate by a cluster of impurities. For this we place a
counts 10.0 cluster of impurities, a central impurity surrounded by five other impurities, at the center of the system. The 0.0 effective potential (trap + impurity) is shown in Fig. 0.0 4.0 8.0 12.0 4(a). It may be noted that in contrast to the case in distance [33] where the trap potential is a ‘Mexican hat’, in our case the effective potential is a ‘Mexican hat’ together with five peaks along the circular ring at the bottom of FIG. 7: (Color online) (a) The structure factor profile showing the the hat. The condensate density in this effective poten- degree of distortion of the melted vortex lattice of Fig. 6(d), (b) tial in absence of rotation is shown in Fig. 4(b). Fig. the histogram plot for Fig. 2(a), and (c) histogram plot for Fig. 6 (d). 5(a) shows a giant hole at the center of the rotating con- densate. On comparison of Fig. 5(a) with Fig. 2(a) (the undistorted vortex lattice) we see that in Fig. 5(a) 6 three visible vortices are missing. However the average disorder or the pseudorandom potential created by the angular momentum of these two vortex lattices remains superposition of two optical lattices as mentioned above. almost same, namely, < Lz >= 21.4 for lattice in Fig. Fig. 9(a) shows the melted vortex lattice in presence of 5(a) and < Lz >= 20.8 for lattice in Fig. 2(a). It the pseudorandom bichromatic optical lattice potential may be mentioned here that according to Feynman rule Vimp = V0[cos(2πβx + φ) + cos(2πβy + φ)] and Fig. 9(c)
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