Dynamics of 1D and 3D quantum droplets in parity-time-symmetric harmonic-Gaussian potentials with two competing nonlinearities
Introduction
Recently, quantum droplets (QDs) [1], [2], as the soliton-like states and a new kind of liquid matter, originating from the finely tuned balance between the mutual attractive and repulsive forces, have been paid more and more attention in the studies of ultracold atoms and superfluids due to their peculiar features including flexibility and observability and so on [1], [2], [3], [4], [5], [6], [7], [8], [9]. However, in the multi-dimensional cases, the critical and supercritical collapse occur owing to the instability of self-trapped localized modes built by usual cubic nonlinearity which represents attractive interactions between atoms [10], [11]. As a correction to the mean-field (MF) dynamics with attractive interaction of Bose–Einstein condensates (BECs), the Lee–Huang–Yang (LHY) repulsive effect [12] originating from the leading-order correction due to quantum fluctuations was proposed to stabilize the self-trapped states, which allows stable droplets to be predicted in binary BECs [1], [2]. Furthermore, QDs have been experimentally observed in dipolar bosonic gases through the competition of long-range dipole–dipole attraction and LHY repulsion, such as the dipolar condensate of 164Dy, homonuclear mixtures of 39K, and heteronuclear mixture of 41K and 87Rb [13], [14], [15], [16], [17], [18], [19], [20].
What calls for special attention is that in a lower-dimensional case QDs not only survive, but also become more ubiquitous and remarkable [1], [2]. In one-dimensional (1D) geometry, the density profiles of QDs can be described by the effective 1D Gross–Pitaevskii (GP) equations with the repulsive cubic and attractive quadratic nonlinearities [2], [3], the latter term representing the LHY correction is different from the repulsive quartic nonlinearity in three-dimensional (3D) geometry [1], [5], [6], [9], [13]. Particularly, two physically different regimes forming in 1D binary Bose gases are determined analytically, corresponding to small droplets of an approximately Gaussian shape and large “puddles” with a broad flat-top plateau [3].
Since the concept of parity-time () symmetry was first proposed by Bender and Boettcher in 1998 [21], where the parity operator and time reversal operator are defined as , and , respectively, the non-Hermitian quantum mechanics [22], [23] has caught more and more attention, which makes the non-Hermitian Hamiltonians with various of complex potentials probably admit the fully-real spectra and/or exhibit some new types of stably nonlinear modes in Kerr and even pow-law nonlinear media [24], [25], [26], [27]. Especially, an optical experiment of quantum mechanics can be realized through a judicious inclusion of gain or loss regions in waveguide geometries [27] such that a large number of phenomena related to -symmetry including the celebrated spontaneous symmetry breaking can be observed experimentally [28], [29], [30]. Therefore, a variety of optical potentials were verified to support stable soliton solutions of some nonlinear wave models, such as the Scarf-II potential [26], [27], [31], [32], Gaussian potential [33], [34], [35], [36], harmonic potential [37], [38], [39], [40], optical lattice potential [27], [41], [42], [43], etc.
More recently, Li et al. [44] observed symmetry breaking transitions in a dissipative Floquet system of ultracold Li atoms. In fact, some experimental design ideas have been proposed about the experimental realization of symmetry in BECs. For example, Klaiman et al. [45] presented a quantum scheme (similarly to the known waveguide experiments [28], [29], [30]) on a BEC in a double-well potential, where the gain and loss distributions were realized by injecting atoms into one well and coherently removing atoms from another one, respectively. Kreibich et al. [46] proposed an experiment for a quantum mechanical realization of a -symmetric system by embedding this system in a larger multi-well system, where the -symmetric currents were implemented by an accelerating BEC with a titled optical lattice. Chen and Malomed [47] presented that the local amplification of the BEC wave function may be provided by a locally placed matter-wave laser based on a dual-core trap. And Labouvie et al. [48] described an experiment with a weakly interacting BECs trapped in a 1D optical lattice with localized loss created by a focused electron beam. -symmetry broken states in BECs in static optical lattice potentials were also realized by setting suitable initial currents between neighboring sites [49].
Owing to the modulation of external potential and the availability of optical lattices for experiments with binary Bose–Einstein condensate, the dynamics of 1D droplets trapped in optical lattice potential and the spontaneous symmetry breaking of 1D droplets in a dual-core trap were studied [50], [51], [52], [53], [54]. Moreover, the QDs were also theoretically verified to appear in the dipolar condensates trapped in the 3D harmonic [55] and anharmonic [56] oscillators. However, to the best of our knowledge, the QDs trapped in complex -symmetric non-periodic optical potentials were not discussed in the multi-dimensional cases before.
Motivated by the aforementioned discussions about the QDs and effects of -symmetric potentials, in this paper we would like to introduce the -symmetric optical harmonic-Gaussian potentials (see Eqs. (11), (21)) into the amended GP equations including the LHY correction (1D and 3D). The forms of chosen potentials with the purposes to produce exact solutions of the 1D and 3D amended GP equations, respectively (see Eqs. (12), (23)). The profiles of QDs and their stability will be discussed in detail. These results will enlighten us on better understanding of the related physical mechanism of QDs in complex -symmetric potentials. The main results of this paper can be summarized as follows:
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The complex -symmetric harmonic-Gaussian potential is introduced into the amended GP equations including the LHY correction in both 1D and 3D geometries. As a result, exact solitons are found for certain parameters for amended GP equations with the harmonic-Gaussian potential in both 1D and 3D geometries, and they are shown to be stable for some parameter regions.
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Collisions between two quantum droplets are investigated in the -symmetric harmonic-Gaussian potential in detail, and there exist elastic collisions for certain parameters. And the strong LHY correction term will ensure the stable dynamical behaviors of the droplets.
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The stable excitations of stationary quantum droplets in the harmonic-Gaussian potential are studied by the density-modulation with a finite wave number in a single droplet and changing the potential parameters and strength of the cubic nonlinearity as functions of time.
The remainder of this paper is organized as follows. In Section 2, we introduce the amended GP equations with - symmetric potentials for the QDs and the general theory. In Section 3, we introduce the -symmetric harmonic-Gaussian potential, and analyze the entirely-real spectra of the corresponding non-Hermitian Hamiltonians (i.e., linear spectral problems) for different parameters. We also discuss the 1D static QDs for the -symmetric harmonic-Gaussian potential and analyze the stabilities of exact solutions and numerical solutions. Then dynamical behaviors of the QDs are investigated for the harmonic-Gaussian potential including collisions and excitations, respectively. Moreover, we also study the influence of the LHY correction term on the stability of QDs. In Section 4, we discuss the 3D static QDs for the 3D -symmetric harmonic-Gaussian potential and their stabilities, as well as the effect of the LHY correction term on the QD stability. Finally, some conclusions and discussions are presented in Section 5.
Section snippets
Amended GP equations with potentials and general theory
The 3D QD model with -symmetric potentials.—In the binary BECs with mutually symmetric spinor components trapped in the 3D -symmetric optical potentials, the underlying time-dependent GP equation with the LHY correction (a combination of the usual MF cubic nonlinearity and a quartic defocusing/self-focusing term) can describe the dynamics of the QDs as follows [1], [5], [6], [9], [13]: where the complex wave function
-symmetric phase transitions
In this section, we mainly introduce the meaningful 1D -symmetric non-periodic optical potentials, and concentrate on the celebrated -symmetry unbreaking and breaking phenomena in the linear regime where and are the eigenvalue and localized eigenfunction, respectively, that is, we consider the linear case of Eq. (6) without the nonlinear terms and (In the other words, we close the nonlinear terms and of Eq. (6)). One can
3D stationary solitons and stability
In this section, we will firstly give the corresponding 3D harmonic-Gaussian potential. And then, the 3D exact nonlinear localized modes for amended GP equation (2) with the harmonic-Gaussian potential are found. Besides, a family of new 3D fundamental droplets can be found around them by numerical iteration technique used previously. Moreover, we use direct numerical evolution simulations to investigate the stability of these 3D droplets.
3D harmonic-Gaussian potential.—Firstly, we
Conclusions and discussions
In conclusion, we have introduced the -symmetric harmonic-Gaussian potentials into the amended GP equations (1D and 3D), and obtained stable droplets with different shapes. In the 1D case, we have demonstrated that the harmonic-Gaussian potential can admit fully-real linear spectra in certain potential parameter space, with apparent symmetry breaking behaviors. And, we also study the stability of exact and numerical solutions by the spectral stability analysis and the direct wave
CRediT authorship contribution statement
Jin Song: Conceptualization, Methodology, Software, Investigation, Analysis, Visualization, Writing – original draft. Zhenya Yan: Conceptualization, Methodology, Analysis, Supervision, Funding acquisition, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the two anonymous referees for their insightful comments and suggestions. The work was supported by the National Natural Science Foundation of China (No. 11925108).
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