arXiv:2004.09973v1 [cond-mat.quant-gas] 19 Apr 2020 u bevto fteetodffrn odtn modes, Goldstone different two these care- through of identified observation is for- phase ful The supersolid phase. the supersolid of the mation in are array droplets that quantum confirming of rigidity, phase the revealing modes, dnie h uesldpoete ntedplrquan- dipolar the of al. in gases et tum properties Chomaz supersolid [5]. the the term with quartic Gross-Pitaevskii identified additional dynamics extended an mean-field of the presence the by describing found described are equation, well supersolid dipolar be the to of re- waves properties superfluid is The matter a in trap BEC. of immersed the interference structure, the from crystalline producing with BEC expand, experiments, to the leased Ancilotto both and In been Roccuzzo has [9]. by This supported well at- interactions. theoretically and [6] dipolar short-range long-range repulsive al. BECs, tractive competitive et dipolar to Tanzi strongly due and in arising phase [5] supersolid have the al. phase droplets achieved B¨ottcher et quantum state revealed. of ground arrays been incoherent of and regimes herent distinct Three diagram experiments different in [5–8]. condensates dipolar in identified atomic cold of 5]. its area [4, to the systems avenues in alternate emerged observation inconclusive, potential liq- being in Helium Observation uid resistance. zero without pres- displaying flow latter of the and the state viscosity with component, a superfluid order a is crystalline of supersolid ence a A manifests that [3]. with investigated matter realization 2], extensively possible [1, one its the back for being time Helium some liquid quite the conjectured been has ∗ § ‡ † [email protected] [email protected] [email protected] [email protected] u ta.[]osre w o nryGoldstone energy low two observed [8] al. et Guo eyrcnl,tetasetsproi hs a been has phase supersolid transient the recently, Very matter of phase supersolid the of existence Possible uesldbhvo noedmninlsl-rpe Bose self-trapped one-dimensional in behavior Supersolid − eua oeEnti odnae(E) co- (BEC), condensate Bose-Einstein regular a ihls .Parit, K. Mithilesh 2 ob xcl n-hr ftecntn background. constant the of one-third exactly domains parameter be different to in droplet self-trapped the and oteuiutu nia ae.Tesltosyedtecon the yield solutions The waves. cnoidal ubiquitous connec M¨obius the transform, a to through governing obtained Schr¨odinger equation, are free-floa nonlinear dynamics, self-trapped amended the in the form of Bloch desired the (BE of condensate tions Bose-Einstein dipolar in recently observed 166 nttt fApidSine,AiyUiest,Sector-1 University, Amity Sciences, Applied of Institute uesldi neoi tt fmte,soigcrystallin showing matter, of state exotic an is Supersolid .INTRODUCTION I. Er and 1 164 eateto hsclSine,Ida nttt fScien of Institute Indian Sciences, Physical of Department Dy n eerhKlaa oapr714,Ws egl India Bengal, West Mohanpur-741246, Kolkata, Research and [7]. 1, ∗ ag Tyagi, Gargi Dtd pi 2 2020) 22, April (Dated: 2, † heedaSingh, Dheerendra h mne olna croigreuto,necessar- Schr¨odinger equation, nonlinear amended the solutions droplet droplet [22]. multiple approach large have variational flat-top from may exact indicated and an has nonlinearity [11] It quadratic solutions a QFs. and from non- BEC arising mean-field Schr¨odinger regular repulsive like nonlinear a linearity has amended which the (NLSE), equation by modelled been of dimensions. that unlike higher is pressure, in repulsion two quantum attraction the mean-field the from as balance to distinct 21], required is [10, dimension dimensions one three demonstrated of and been case also The have [20]. transitions soli- droplet Bright to of verification. tons some experimental investigated, found been have spin have which and [19], bosons fermions degenerate polarized attractive Bose- of [18], droplets regime Bose- Fermi crossover dimensional explored: in been have droplets Bose droplets quantum as- the Various of of its [17]. pects role dynamics particularly expansion the and properties, mode examined system collective also the and determining al., gases QFs feature dipolar et general strongly a Chomaz is of stabilization LHY droplets. that demonstrated liquid quantum these beyond to leads self-trapped (LHY) (QFs) mean- fluctuations Lee-Huang-Yang the quantum of [16] and mean-field Balance energy trap. a (MF) of field absence the quan- in self-trapped liquid the gas of ideal tum in possibility an Bose the observed them explore making dipolar to been ground [15], in liquid have quantum [12–14] magnetic They and condition free-floating [11]. the form quan- self-trapped droplets where BEC, tum component two a in main ultracold in possibility trap the a investigating without worth scenario. phase supersolid is realizing it supersolid of Hence, transient the form for phase. and released interfere was two Goldstone trap to other the of the matter-waves from of distortion case BEC the the to in experiments, led earlier, mentioned trap As the modes. of However, presence vibrations. the supersolid residual the the and of phase modes condensed sound distinct the to corresponding ee ehv dnie xc lc aesltosof solutions wave Bloch exact identified have we Here, have droplets quantum self-trapped shaped cigar The do- parametric a identified has [10], Petrov Recently, igqatmfli.Orgnrlsolutions general Our fluid. quantum ting )i rp ee epeeteatsolu- exact present we Here, trap. a in C) 5 od-033 ta rds,India Pradesh, Uttar Noida-201303, 25, h oetrsda odnaei found is condensate residual lowest The . igawd ls fsproi solutions supersolid of class wide a ting h enfil n eodmean-field beyond and mean-field the re ihasprudbackground, superfluid a with order e tn odnae uesldbehavior supersolid condensate, stant 1, ‡ n rsnaK Panigrahi K. Prasanta and eEducation ce Enti condensate -Einstein 1, § 2

3/2 ily accompanied by a non-vanishing condensate density, √2M g It has constant solution: ψconst = π~ δg , arising from showing clearly the supersolid phase of quantum droplet the balance of MF and BMF contributions. The plane µ system. We employ M¨obius transform [23, 24] to con- wave solution, ψ = √P exp(ikx i ~ t), which is prone nect a wide class of desired solutions with the cnoidal to modulational instability (MI),− with functions, representing nonlinear periodic density waves, which manifest in diverse physical systems. Our proce- dure yields a unified picture of the uniform condensate, √2Mg3/2 Mg3 µ¯ √P = + , (4) self-trapped and periodic supersolid phases in different ± 2π~ δg ± s2π2~2 δg2 δg parameter domains. 2 2 The paper is organized as follows. In Sec. II, theory whereµ ¯ = µ ~ k . The steady-state solution is per- of quantum droplets is briefly described, leading to the − 2M amended NLSE, governing its dynamics in one dimen- turbed by small amplitude and the resulting MI has been sion. In Sec. III, we present supersolid solutions; a Bloch investigated in [25, 26]. MI manifests at wave number kP , wave, necessarily possesing a constant background. Their where Ω becomes complex properties in various configurations are highlighted. Fi- 1 √2M √P nally, we conclude with the summary of results and future Ω= k2 ~2k2 4M g3/2 δgP , v P P ~ directions for investigation. ± 2M u − π 2 − !! u t (5) II. DYNAMICS OF QUANTUM DROPLETS IN with kP is the wavenumber of the small perturbation. MI 2 2 1/3 ONE DIMENSION 2π ~ P 2 occurs for g> M δg and   Quantum droplets have been shown to occur in the bi- nary BECs, with repulsive intra- (g , g ) and attractive √2M √P ↑↑ ↓↓ k < 2 g3/2 δgP k . (6) inter-component interaction (g ), in the vicinity of the | P | v π~ 2 − ≡ 0 ↑↓ u ! MF collapse instability point. Specifically, they appear u t in the regime, 0 < δg = g + √g g √g g , with ↑↓ ↑↑ ↓↓ ↑↑ ↓↓ MI is maximum at kP = k0/√2 and k0 is maximum at ≪ 3 attractive inter-component interaction g < 0 and repul- 3M g ↑↓ µ = 2 2 . The regime in the parameter domain, sive intra-component average interaction g = √g g > − 8 π ~ δg 0. The energy density of such homogeneous mixture↑↑ ↓↓ has where MI occurs is conducive for inhomogeneous solu- been obtained [11], tions. In the following section, we investigate the pos- sible inhomogeneous solutions for the amended NLSE, explicitly showing the presence of a supersolid phase. (g1/2n g1/2n )2 = ↑↑ ↑ − ↓↓ ↓ E 2 1/2 1/2 III. SUPERSOLID PHASE √g g (g + √g g ) (g n + g n )2 + ↑↑ ↓↓ ↑↓ ↑↑ ↓↓ ↑↑ ↑ ↓↓ ↓ (g + g )2 ↑↑ ↓↓ We consider propagating Bloch function type solutions [27]: 2√M 3/2 ~ (g n + g n ) , (1) − 3π ↑↑ ↑ ↓↓ ↓ x vt µ where n and n are densities of the two components, ψ = ψ0 − ,m exp ikx i ~ t . (7) ↑ ↓ ξ − related by n = n = n g /g . The first two terms     ↑ ↓ ↑↑ ↓↓ ~k in Eq. (1) are the MF contribution and the last term Here v = M and ψ0 is a periodic function with m being represents the LHY-BMFp correction. With the assump- the modulus parameter: tion, g = g g , we have n = n = n and ↑↑ ∼ ↓↓ ↑ ↓ ~2 √2M 0 < δg g g . The energy density reduces to, ∂2ψ + δgψ3 g3/2ψ2 µψ¯ =0. (8) ≪ ↑↑ ∼ ↓↓ − 2M x 0 0 − π~ 0 − 0 2 Henceforth, we take ψ to be a real periodic function δgn 2√M 3/2 0 = (gn) , (2) without loss of generality, taking advantage of the global E 2 − 3π~ U(1) symmetry of the system. For finding general solu- 3 2 2 having the equilibrium density n0 = 8g /(9π δg ) and tions, appropriate M¨obius transformations are employed, chemical potential µ = δgn /2. The validity of has 0 − 0 E to connect the solution space to the ubiquitous cnoidal been justified by diffusion Monte-Carlo simulation [11]. waves [28], satisfying: f” af λf 3 = 0 [23]. The general The modified Gross-Pitaevskii equation, with cubic solution has the form, ± ± MF and quadratic BMF nonlinearty, has the form,

δ x vt A + Bf ( − ) ~2 √2M ξ ~ 2 3/2 ψ0(x, t)= x vt (9) i ψt = ψxx + δg ψ ψ g ψ ψ. (3) 1+ Df δ( ) −2M | | − π~ | | −ξ 3

Balancing of nonlinearity with dispersion leads to δ = 1 a. and δ = 2 as possible choices. Here, we consider δ =1as 2 it has richer structure, which allows for diverse boundary 1.4 10 conditions to be satisfied. We now start with the periodic solution, 1.0

0.6 x vt n(x) ψ (x, t)= A + B sn − ,m , (10) 0 ξ   0.2 and find that it is necessarily accompanied by positive 3 2 -4 -3 -2 -1 0 1 2 3 4 √2M g / background A = 3π~ δg , having one-third the value of x/ the uniform background. It is worth emphasizing that b. 2 the supersolid solutions does not smoothly trace it to the 2.5 10 constant condensate in the limit m 0 when B = 0. 2.0 → Its amplitude, B = 2m A, never exceeds that of the ± m+1 1.5 background in the allowedq energy range of the modulus parameter 0 0, showing clearly that for the quan- ± m+1 tum supersolidq  immersed in a residual BEC [5, 6, 9]. The FIG. 1. (Color online) The density of supersolid phase as 2 a function of scaled position is depicted for different values density of the residual BEC n = A2 1 2m > res − m+1 of intra-component interaction (g) and modulus parameter, 0. This diffused matter-wave density rules outq the sce- with δg = 0.1 (black: m = 0.2 and red: m = 0.5), for both nario of one atom per site thereby overtaking the Penrose the panels. The value of intra-component interaction, g = 10 and Onsagar criterion [29–31]. for panel (a) and g = 12 for panel (b). Panel (c) shows the The density of the supersolid phase is shown in Fig. pictorial representation of several quantum droplets immersed 1, as a function of scaled position variable x/ξ, in the in a residual BEC, with boundaries from the both sides (solid vertical lines), confined in a box of finite length. comoving frame. It is evident that inter-droplet spac- ing (peak-to-peak distance) and number of quantum droplets are inter-related. With increasing repulsive intra-component interaction g, number of droplets in- creases and their size decreases for a fixed value of x/ξ. We now proceed to study the general periodic solution Further, keeping all the physical parameters constant, of the amended NLSE, increase in the modulus parameter increases the inter- droplet spacing. The second exact solution is of the sinusoidal form: x vt A + B sn −ξ ,m ψ0 = , (13) A x vt  ψ0 = , (12) 1+ D sn −ξ ,m x vt 1+ D sin −ξ    ~ with the amplitudes, A = 3π µ¯ and D = √2Mg3/2 A straightforward but tedious calculation yields, 9π2~2 δg ~2 1 3 µ¯. The healing length ξ = and D = 2m/(m + 1), B = DX , with X = 4M g 2Mµ¯ ± ± − 3 − ± 4M g 2 µq = 2 2 . Hence, the minimum chemicalq poten- g2 g2p 4g1µ /2g1. The value of D, 0 < D < 1, min − 9π ~ δg ± − | | tial is again identical to that of the self-trapped droplet shows the non-singular nature of the general solutions. p 1 ML [11]. The possible solutions of A and ξ ( ξ2 = ~2 ) are listed 4 below: a. 3 1.5 10 g=5 g=10 1.0 2 2 g=12 1 g2X+ 6 (g2/4g1 µ¯) (g2X+ g2/2g1 µ¯) A = ± − − − , 0.5 ± 2 g2 +3g1 g2 4g1µ¯ p − E/N (14a) 0 p 2 -0.5 2 g2X 6 (g2/4g1 µ¯) (g2X µ¯) A = − ± − − − , (14b) -40 -20 0 20 40 ± 2 g2p 3g1 g2 4g1µ¯ k − − b. p 2 2.8 10 3 g2X+/2 µ¯ 3 (g2/4g1 µ¯) (g2X+ µ¯) m=0.2 m=0.5 L = − ± − − , 2.0 ± − p(m + 1) m=0.7 (14c) 1.0 2 E/N g2X+/2 µ¯ 3 (g2/4g1 µ¯) (g2X µ¯) 0 L = − ± − − − . ± p(m + 1) (14d) -1.0 -40 -20 0 20 40 k 2 2 The chemical potential: (i) µ> g2 and (ii) 0.414 g2 < − 4g1 − g1 µ< 0. The minimum chemical potential is less than that FIG. 2. (Color online) The energy per atom of the super- of obtained for the self-trapped supersolids [Eq. (10) and solid as a function of k is depicted for different values of re- (12)] or quantum droplets [11]. pulsive intra-component interaction and modulus parameter, It is worth mentioning that, for the case of cigar- with m = 0.5. It shows that increasing repulsive interaction shaped BEC, the Gross-Pitaevskii (GP) equation does while keeping the competitive intra- and inter-component in- not yield a constant background with periodic modu- teraction constant, δg = 0.1, decreases the total energy per atom. The value of intra-component interaction is g = 10 for lation. The presence of both mean-field and BMF en- panel (b). Increasing the value of modulus parameter leads ergies are crucial for the existence of these solutions to broader dispersion, due to periodicity of the solution. here. The soliton trains observed in the attractive BEC are composed of BEC droplets in the absence of a su- perfluid background [32]. For the case of NLSE with The dispersion relation for supersolid phase is demon- a phase locked source, periodic solutions with constant strated in Fig. (2), for different repulsive inter- background are permissible [23]. In optical fibers anal- component interaction in panel a. It exhibits the de- ogous periodic solutions in the temporal domain have crease in energy per atom with increasing g. In panel been experimentally observed as frequency combs [33]. b, the dispersion relation is shown for different moduli Recently, NLSE with cubic-quadratic nonlinearity and parameter, showing comparative flattening of the energy phase locked with a source has been investigated, wherein per atom for greater moduli. This happens due to for- the parity even and kink type solutions have been iden- mation of front-like droplet, with infinite period when tified [34]. The cubic-quadratic equation with a phase- modulus parameter approaches unity. locked source have also been investigated through the fractional linear transformations, for attractive Kerr non- linearity and repulsive BMF correction [35]. The behav- V. CONCLUSION ior of solitons under nonlinearity management has been explored [36]. In the ensuing sections, the dispersion re- lation for supersolid is investigated, distinctly showing In conclusion, we have obtained the exact supersolid that increase of repulsive inter-component interaction de- solutions in one-dimensional self-trapped BEC. The su- creases the energy. persolid phase occurs in a free-floating condition, unlike the case of dipolar BEC, wherein the condensate with appropriately tuned interactions needs to be released, and the interference of matter-wave results in supersolid IV. ENERGY AND MOMENTUM OF THE formation with superfluid background in the exact pe- SUPERSOLID PHASE riodic solution. In the present case, exact Bloch waves, quantum droplets appear in a periodic array with inter- We now investigate the dispersion relation of the super- spacing between two droplets constant, immersed in a solid phase. The energy and momentum density [37, 38] residual BEC background, that establishes inter-droplet can be computed for a supersolid in a box of finite length. coherence [5, 39]. The chemical potential for two solu- The expressions for the momentum and energy are pro- tions is identical and for the general solution it is less, vided in the supplementary information. which is more favourable. 5

VI. ACKNOWLEDGEMENTS from the REDX, Center for Artificial Intelligence, IISER Kolkata, funded by Silicon Valley Community Founda- We thank Prof. R. B. MacKenzie for fruitful discus- tion through grant number 2018-191175 (5618). G. Tyagi sion. M. K. Parit acknowledges the financial support is thankful to IISER Kolkata for hospitality.

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SUPPLEMENTARY INFORMATION: SUPERSOLID BEHAVIOR IN ONE-DIMENSIONAL SELF-TRAPPED BOSE-EINSTEIN CONDENSATE

Energy and momentum of the supersolid phase

In this section, we have provided the analytical expression for the energy and momentum. The energy and momen- tum density [37, 38] can be computed, E = dx, where is the Hamiltonian density H H ~2 R δg 2√2M = ∂ ψ 2 + ψ 4 g3/2 ψ 3 , (15) H 2M | x | 2 | | − 3π~ | |

i~ and momentum P = 2 (ψ∂xψ∗ ψ∗∂xψ) dx. The analytical expressions for the momentum and energy are presented − 3 2 2 2 √2M g / 2m 3~ π (m+1) δg ~ below for the solution-I,R with A = ~ , B = A, ξ = 3 , and D1 = . 3π δg ± m+1 2M g 2M q q 1. Number of atoms

The number of atoms N = ψ 2 dx. | | R tv x N(x)= A2m + B2 (x tv)+ B2ξE am − m m − ξ    
x tv x tv +2 AB√ mξ log dn − m √mcn − m . (16) ξ − ξ     

2. Energy density

E + E + E + E + E λ E(x)= 1 2 3 4 5 × E (17a) 6m2ξ2 E =2 3A2m2ξ2(x tv) A2g Ag + D k2 + B4g (m + 2)ξ2(x tv) (17b) 1 − 1 − 2 1 1 − E =2 B2m 9Aξ2(2Ag g )(x tv)+ D tv 3k2ξ2 +2m +1 + D x 3k2ξ2 + m 1 (17c) 2 1 − 2 − 1
− 1
− x tv E =2B 2ξE am − m m ξ2 9Am(2Ag g ) 2B
2g (m + 1) + D m 3


k2ξ2 + m +1 (17d) 3 ξ − 1 − 2 − 1 1 −     x tv x tv
x tv

E = B2mξcn − m dn − m 3Bξ2(4Ag g )+2 B2g ξ2 + D m sn − m (17e) 4 ξ ξ 1 − 2 1 1 ξ        3 3 2 2 2 2
E5 =3B√mξ 8A g1m 6A g2m +4A B g1(m +1)+ D1k m B g2(m + 1) (17f) − − x tv x tv λ = log dn − m √mcn − m ,

(17g) E ξ − ξ     

3. Momentum density

~k tv x P(x)= A2m + B2 (x tv)+ B2ξE am − m m m − ξ     
x tv x tv ~k +2AB√mξ log dn − m √mcn − m = N(x). (18) ξ − ξ m     

For supersolids confined in a box of finite length L (from L/2 to L/2): number, N = N (L/2) N( L/2), energy, E = E(L/2) E( L/2), and momentum, P = P (L/2) P−( L/2). − − − − − − This figure "QD_SCM2.png" is available in "png" format from:

http://arxiv.org/ps/2004.09973v1