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Supersolid Behavior in One-Dimensional Self-Trapped Bose-Einstein Condensate

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Supersolid Behavior in One-Dimensional Self-Trapped Bose-Einstein Condensate Supersolid behavior in one-dimensional self-trapped Bose-Einstein condensate 1, 2, 1, 1, Mithilesh K. Parit, ∗ Gargi Tyagi, † Dheerendra Singh, ‡ and Prasanta K. Panigrahi § 1Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur-741246, West Bengal, India 2Institute of Applied Sciences, Amity University, Sector-125, Noida-201303, Uttar Pradesh, India (Dated: April 22, 2020) Supersolid is an exotic state of matter, showing crystalline order with a superfluid background, observed recently in dipolar Bose-Einstein condensate (BEC) in a trap. Here, we present exact solu- tions of the desired Bloch form in the self-trapped free-floating quantum fluid. Our general solutions of the amended nonlinear Schr¨odinger equation, governing the mean-field and beyond mean-field dynamics, are obtained through a M¨obius transform, connecting a wide class of supersolid solutions to the ubiquitous cnoidal waves. The solutions yield the constant condensate, supersolid behavior and the self-trapped droplet in different parameter domains. The lowest residual condensate is found to be exactly one-third of the constant background. I. INTRODUCTION corresponding to the distinct sound modes of the residual condensed phase and the supersolid vibrations. However, Possible existence of the supersolid phase of matter the presence of the trap led to distortion of Goldstone has been conjectured quite some time back [1, 2], with modes. As mentioned earlier, in the case of the other two the liquid Helium being the one extensively investigated experiments, the BEC from the trap was released for the for its possible realization [3]. A supersolid is a state of matter-waves to interfere and form transient supersolid matter that manifests a crystalline order with the pres- phase. Hence, it is worth investigating the possibility ence of a superfluid component, the latter displaying zero of realizing supersolid phase without a trap in ultracold viscosity and flow without resistance. Observation in liq- scenario. uid Helium being inconclusive, alternate avenues to its Recently, Petrov [10], has identified a parametric do- potential observation emerged in the area of cold atomic main in a two component BEC, where self-trapped quan- systems [4, 5]. tum droplets form [11]. They have been observed in Very recently, the transient supersolid phase has been the free-floating condition [12–14] in dipolar Bose gas identified in dipolar condensates in different experiments and magnetic quantum liquid [15], making them an ideal [5–8]. Three distinct regimes of ground state phase ground to explore the possibility of the self-trapped quan- diagram a regular Bose-Einstein condensate (BEC), co- tum liquid in the absence of a trap. Balance of the mean- herent and− incoherent arrays of quantum droplets have field (MF) energy and Lee-Huang-Yang (LHY) beyond been revealed. B¨ottcher et al. [5] and Tanzi et al. [6] mean-field [16] quantum fluctuations (QFs) leads to these achieved the supersolid phase in strongly dipolar BECs, self-trapped quantum liquid droplets. Chomaz et al., arising due to competitive repulsive short-range and at- demonstrated that LHY stabilization is a general feature tractive long-range dipolar interactions. This has been of strongly dipolar gases and also examined the role of theoretically well supported by Roccuzzo and Ancilotto QFs determining the system properties, particularly its [9]. In both the experiments, BEC from the trap is re- collective mode and expansion dynamics [17]. Various as- leased to expand, with the interference of matter waves pects of the quantum droplets have been explored: Bose- producing crystalline structure, immersed in a superfluid Bose droplets in dimensional crossover regime [18], Bose- BEC. The properties of the dipolar supersolid are found Fermi droplets of attractive degenerate bosons and spin to be well described by the extended Gross-Pitaevskii polarized fermions [19], have been investigated, some of equation, describing the mean-field dynamics with the which have found experimental verification. Bright soli- presence of an additional quartic term [5]. Chomaz et al. tons to droplet transitions have also been demonstrated identified the supersolid properties in the dipolar quan- [20]. The case of one dimension is distinct from the two 166 164 arXiv:2004.09973v1 [cond-mat.quant-gas] 19 Apr 2020 tum gases of Er and Dy [7]. and three dimensions [10, 21], as mean-field repulsion is Guo et al. [8] observed two low energy Goldstone required to balance the quantum pressure, unlike that of modes, revealing the phase rigidity, confirming that array attraction in higher dimensions. of quantum droplets are in the supersolid phase. The for- The cigar shaped self-trapped quantum droplets have mation of the supersolid phase is identified through care- been modelled by the amended nonlinear Schr¨odinger ful observation of these two different Goldstone modes, equation (NLSE), which has a repulsive mean-field non- linearity like regular BEC and a quadratic nonlinearity arising from QFs. It has an exact flat-top large droplet ∗ [email protected] solutions [11] and may have multiple droplet solutions † [email protected] indicated from variational approach [22]. ‡ [email protected] Here, we have identified exact Bloch wave solutions of § [email protected] the amended nonlinear Schr¨odinger equation, necessar- 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.
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