In recent years, a new type of self-bound quantum droplets (QDs) has attracted much attention in the field of ultracold atoms. Based on the theoretical suggestions described in the [1], QDs were experimentally created in dipolar bosonic gases of erbium [2], in dysprosium [3], and in mixtures of two atomic states of ³⁹K with appropriate signs of the inter-component and intra-component interactions [4]. Owing to the balance between the attraction generated by the dipole-dipole interaction and the Lee−Huang−Yang (LHY) repulsion caused by the quantum fluctuation [5], QDs has been observed in the experiment of the dipole boson gas[6-10]. Another possibility for the creation of QDs was realized by means of the balance of the inter-component attraction and LHY repulsion in the binary boson mixture [11-13].

Theoretically, the LHY correction takes different forms in different spatial dimensions in the modified Gross-Pitaevskii equations. In the one-dimensional (1D) geometry, the competition of the quadratic LHY attraction and the effective cubic mean-field repulsion can form 1D QDs. QDs in 1D free space are studied and two different physical regimes are found, corresponding to small droplets of an approximately Gaussian shape and large puddles with a broad flat-top plateau, respectively [14]. Small droplets collide quasielastically, while large colliding droplets may merge or suffer fragmentation, depending on their relative velocity. Optical lattice is an effective tool to deal with matter-wave solitons in Bose-Einstein condensates (BECs) and optical solitons in nonlinear media. Therefore, the introduction of optical lattice potential will influence the dynamics of QDs. It has been demonstrated that there are two kinds of physically regimes in optical lattices, namely, on-site QDs and off-site QDs [15]. Both on-site QDs regardless of the value of norm and off-site QDs for the norm exceeding a critical value are completely stable. Unlike free space, the slowly moving small QDs in optical lattices merge after the collision. Multi-stable QDs are also studied in optical lattices [16]. A continuous family of odd-symmetric and even-symmetric QDs exists in both first finite gap and semi-infinite gap. In addition, spontaneous symmetry breaking of QDs in a dual-core couplers was studied [17]. QDs feature spatial density profiles of two different types: bell-shaped and flat-top ones, for relatively small and large values of N, respectively. The spontaneous symmetry breaking of QDs occurs and the symmetry is restored with the increase of the total norm. Collisions between moving QDs trend to merge into breathers. Very recently, the stability and collision of ${ \mathcal P }{ \mathcal T }$-symmetric QDs have been studied in dual-core couplers with balanced gain and loss [18]. It is found that the stability of symmetric QDs depends critically on the competition of gain and loss, inter-core coupling, and optical lattice potential. Slowly moving ${ \mathcal P }{ \mathcal T }$-symmetric QDs tend to merge into breathers, while the fast-moving ones occur quasielastic collision and suffer fragmentation for small and large values of N, respectively.

Recently, QDs in 1D systems have been generalized to 2D and 3D by modifying the form of the LHY term [19-22]. There are many potential values for stable QDs, such as manipulations of quantum information [23] and matter-wave interferometry [24]. Therefore, it is of great significance to study the stability of QDs. On the other hand, the dynamics of BECs in optical lattices have been widely studied [25], and the study of ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices originates from quantum mechanics. For real eigenenergies and probability conservation, it is usually assumed that the Hamiltonian operator be Hermitian in quantum mechanics [26]. However, it is proved that non-Hermitian Hamiltonian can also have entirely real spectra so long as they guarantee ${ \mathcal P }{ \mathcal T }$ symmetry in the pioneering work [27]. The ${ \mathcal P }{ \mathcal T }$ symmetry and non-Hermitian systems are in extensively investigated in the theory and experiments in recent decades. The asymmetric light transport was systematically investigated [28]. The unidirectional absorber and unidirectional laser were proposed [29]. In general, the parity operator $\hat{P}$ is defined as $\vec{p}\to -\vec{p}$, $\vec{x}\to -\vec{x}$ ($\vec{p}$, $\vec{x}$ stand for momentum and position operators, respectively) and the time operator $\hat{T}$ is defined as $\vec{p}\to -\vec{p}$, $\vec{x}\to \vec{x}$, $\vec{{\rm{i}}}\to -\vec{{\rm{i}}}$. A ${ \mathcal P }{ \mathcal T }$-symmetric optical lattice potential V(x) is realized by the condition V(x) = V^*(−x). The most interesting feature of this pseudo-Hermitian Hamiltonian is the existence of a critical threshold beyond which a phase transition occurs due to spontaneous ${ \mathcal P }{ \mathcal T }$ symmetry breaking [30-34]. In this regime, the spectrum is no longer real. Because the equivalency between Schrödinger equation and the equation describing the propagation of light, optics [35-40] and BECs [41-44] in ${ \mathcal P }{ \mathcal T }$-symmetric systems have been reported in the past years.

In the present work, we study the stability and collisions of 1D QDs formed by Bose gas in ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. This paper is structured as follows. In section 2, we introduce the model, and ${ \mathcal P }{ \mathcal T }$-symmetric phase transition is revealed. In section 3, the stability of the ${ \mathcal P }{ \mathcal T }$-symmetric QDs is studied. The ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibits oscillatory stability with the increase of N and g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase. The ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase. The collisions of QDs in ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices are investigated in section 4. The collisions of ${ \mathcal P }{ \mathcal T }$-symmetric QDs with equal norms are consistent with that of QDs in optical lattices. However, the collisions of droplets with unequal norms are completely different from that in free space. In addition, the collision between a stable and an unstable ${ \mathcal P }{ \mathcal T }$-symmetric QDs tends to merge into breathers after the collision. Finally, the paper is concluded by section 5.

We investigate the system that QDs are formed in the binary condensate with mutually symmetric spinor components trapped in the 1D ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. The dynamics of the QDs in this system can be described by the modified Gross-Pitaevskii equations, including the LHY correction in the scaled form.

where g > 0 represents the strength of the cubic self-repulsion, $V(x)={V}_{0}{\sin }^{2}(x)$ and $W(x)={W}_{0}\sin (2x)$ are the real part and imaginary part of complex lattice potential, respectively. V₀ and W₀ satisfy the ${ \mathcal P }{ \mathcal T }$-symmetry, i.e. V(−x) = V(x), W(−x) = −W(x). V₀ and W₀ are the amplitudes of the real lattice and imaginary lattice, respectively. Stationary solutions of equation (1) are sought in the form ψ(x, t) = φ(x)e^−i$\mu$t, where φ(x) is the complex function and $\mu$ is the chemical potential. In this case, φ(x) satisfies

to determine the linear stability properties of the QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices, we consider small perturbations on the solutions of equation (1) in the form,

where u and ν are the infinitesimal perturbations, the imaginary part of λ represents an unstable growth rate. The superscript * represents the complex conjugation. After substitution into equation (1) and linearizing, we obtain the following linear eigenvalue problem

where

Evidently, the QDs are linearly unstable if λ has an imaginary parts, but they are stable if λ is real.

The linear property of periodic potential can be understood by studying the linear problem of equation (2), i.e. $\tfrac{1}{2}{\partial }_{{xx}}\phi -[V(x)+{\rm{i}}W(x)]\phi =\mu \phi $. Since the potentials V(x) and W(x) of equation (1) are π-periodic, according to Floquet Bloch theorem, the form of eigenfunction is φ = Φ_k(x)e^ikx, where Φ_k(x + π) = Φ_k(x) and k stands for the real Bloch momentum. In figure 1(a), we show the typical profiles of the real and imaginary parts of ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. It is worth noting that the eigenvalues of the complex potential band structure are usually complex. However, for ${ \mathcal P }{ \mathcal T }$-symmetric periodic potential, as long as the parameter of system is lower than the phase transition point, the eigenvalues are completely real. Figure 1(b) shows the linear band gap structure with real part strength V₀ = 0.3, and imaginary part strength W₀ = 0.01, W₀ = 0.15, and W₀ = 0.3, respectively. Above the threshold value W₀ = 0.15 (phase transition point), the eigenvalue spectra become complex, and all bloch bands merge together concurrently. Figure 1(c) shows the imaginary parts of complex eigenvalues occurs at W₀ = 0.3. There exists complex eigenvalues and QDs are unstable. In the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ > 0.15), $\mu$ exists non-zero imaginary part. So, any QDs are unstable to perturbations and there is an overall instability effect on propagation. In the following, we discuss the dynamics of QDs in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ < 0.15).

In this section, we discuss the spatial profiles and stability of QDs in ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. A set of numerical solutions are obtained by using the the imaginary-time-integration method. Fixing the strength of ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices as V₀ = 0.3 and W₀ = 0.05, we depict the profiles of static droplets for different values of condensate norm $N={\int }_{-\infty }^{\infty }| \psi {| }^{2}{\rm{d}}x$, which is proportional to the number of atoms in the condensate, in figures 2(a) and (b). Borrowing the concept of ${ \mathcal P }{ \mathcal T }$-symmetric solitons [39], we term the wave-function profiles of these self-bound states satisfying the property of φ^*(x) = φ(−x) as ${ \mathcal P }{ \mathcal T }$-symmetric QDs. QDs in the system have two different types of spatial profiles, viz., approximately Gaussian-shaped QDs [see figure 2(a)] and multi-humped QDs [see figure 2(b)] for relatively small and large values of N, respectively.

Below, the stability of ${ \mathcal P }{ \mathcal T }$-symmetric QDs is considered. In reality, the system can not be completely isolated, and the dynamics of the system is expected to be robust to small fluctuations. Only if perturbed QDs can survive for a sufficiently long time, QDs can be observed in the experiment. Therefore, it is very important to study the stability of QDs. We can study stabilization of the ${ \mathcal P }{ \mathcal T }$-symmetric QDs through scanning a broad range of N and g. Figure 3(a) shows the linear stability analysis of ${ \mathcal P }{ \mathcal T }$-symmetric QDs with different values of condensate norm N and the strength of the cubic self-repulsion g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ = 0.05). The QDs exhibit oscillating stability behavior for different strength of the cubic self-repulsion g and norm N. Fixing the parameter as g = 1, we study stabilization of the ${ \mathcal P }{ \mathcal T }$-symmetric QDs through scanning a broad range of N in figure 3(d), corresponding to the dashed line of the longitudinal section in figure 3(a). The stability of ${ \mathcal P }{ \mathcal T }$-symmetric QDs with different norm N have oscillating behavior. The linear stability analysis results may be also verified by direct numerical evolution simulations, which can be realized by using the split-step Fourier method with 3% random noise added into the initial conditions. The real-time dynamic evolution of ${ \mathcal P }{ \mathcal T }$-symmetric QDs for N = 1.2 and N = 2.5 are depicted in figures 3(e) and (f), respectively, corresponding to the black pentagram marks of figure 3(d). Fixing the parameter as N = 4 corresponding to the dashed line of the cross section of figure 3(a), we study the oscillating stability behavior of the ${ \mathcal P }{ \mathcal T }$-symmetric QDs through scanning a broad range of g in figure 3(g). The results can be also verified by the real-time dynamic evolution for g = 0.57 and g = 0.8, as shown in figures 3(h) and (i), corresponding to the black pentagram marks of figure 3(g). Moreover, it is worth noting that the ${ \mathcal P }{ \mathcal T }$-symmetric QDs become robustly stable at sufficiently large values of g. Figure 3(b) shows the linear stability analysis of ${ \mathcal P }{ \mathcal T }$-symmetric QDs with different values of condensate norm N and the strength of the cubic self-repulsion g in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ = 0.2). For the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase, the stable QDs can not be found regardless of the values of N and g. As predicted, QDs are unstable because the parameter of system exceeds the phase transition point and the band gap is closed and complex eigenvalues occur. In order to further clarify the effect of gain and loss on the stability of ${ \mathcal P }{ \mathcal T }$-symmetric QDs, the phase diagram of the stability of ${ \mathcal P }{ \mathcal T }$-symmetric QDs with respect to the intensity of imaginary part W₀ and the cubic self-repulsion g in figure 3(c) for N = 4. As expected, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ > 0.15), and the ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibit oscillating stability behavior with the increase of g for fixing N and W₀ in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ < 0.15). The physical origin of oscillatory stability with the increase of N and g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase maybe is the competitive interplay between the QDs in optical lattice and its gain and loss, which must ensure ${ \mathcal P }{ \mathcal T }$ symmetry of the system satisfied. These results are novel with respect to that of QDs in ${ \mathcal P }{ \mathcal T }$-symmetric dual-core couplers [18].

One of the characteristics of a soliton is that its shape does not change when it collides with another soliton. From this point of view, verifying a persistence of the shape of droplets involved in the pairwise collisions is very important problem. The collision dynamics of QDs in optical lattices have been proved that slow-moving ground state QDs tend to merger into breathers after collision, while fast-moving QDs may lead to quasi-elastic collision or splitting, which depends on the value of k. We simulated the collision and solved the initial condition of equation (1)

where ψ₁(x) and ψ₂(x) are the stationary shapes of QDs with normalization N₁ and N₂, respectively. k is a kick that determines the velocity of droplet. This ansatz approximates a solution comprising two initial QDs located at x₀ and −x₀. When W₀ is relatively small, the collision dynamics of QDs with equal norms is consistent with that in real optical lattices [15]. For a relatively small value of k, the Gaussian-shaped QDs tend to merger into breathers after the collision. With the increase of k, the quasi-elastic collision is restored. For multi-humped QDs, a newly merged quiescent breather occurs after the collision. For the fast-moving QDs, they undergo fragmentation and a majority of the particles keep in the moving ones. We also analyze collision dynamics of QDs with unequal norms in figures 4(a) and (b). In figure 4(a), a Gaussian-shaped droplet (N = 2) collides with a multi-humped droplet with N = 8. It is found that the large droplet forms quiescent breather and the trajectory of the small droplet is deflected, and the two droplets become highly excited, exhibiting internal periodic vibrations. The situation is quite different for QDs in free space [14]. With the increase of k, the two droplets undergo fragmentation after the collision. Unstable solitons can transform into breather-like objects [45]. In figure 4(c), a stable droplet (N = 5.5) collides with an unstable droplet with N = 4. An interesting phenomenon occurs, two droplets tend to merger into breathers after the collision at a relatively small value of k. Besides, the collision between two droplets with N = 4 corresponding to g = 0.57 and g = 0.8 is shown in figure 4(d), respectively. QDs form two breathers of unequal norms after the collision at a relatively small value of k.

We study the stability and collisions of ${ \mathcal P }{ \mathcal T }$-symmetric QDs forming in a binary bosonic condensate trapped in ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. It is found that the stability of ${ \mathcal P }{ \mathcal T }$-symmetric QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric system depends strongly on the values of the imaginary part W₀ of the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. Due to band gap closure and complex eigenvalues occur, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase (W₀ > 0.15). The ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibit oscillatory stability with the increase of N and g. Finally, collisions between stable ${ \mathcal P }{ \mathcal T }$-symmetric QDs with unequal norms are systematically studied in ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices. The slowly moving QDs tend to merge into breather after the collision. The quasi-elastic collision for fast-moving small QDs, while large droplets undergo fragmentation and the particles in colliding QDs are kept in the moving ones. However, for collisions of droplets with unequal norms, compared with a large puddle droplet collides with a small droplet, the large droplet becomes highly excited, while the small droplet remains essentially in an unperturbed shape in the free space, the moving slowly large and small stable ${ \mathcal P }{ \mathcal T }$-symmetric QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices are highly excited after the collision [14]. With the increase of k, both of the droplets undergo fragmentation after the collision. Besides, the collision between a stable and an unstable ${ \mathcal P }{ \mathcal T }$-symmetric QDs, tend to merge into breathers after the collision.

Significantly, compared with conservative systems, the appearance of ${ \mathcal P }{ \mathcal T }$ symmetry concept has brought a variety of influences on theoretical and experimental researches in the past decade [46-49]. The effect of ${ \mathcal P }{ \mathcal T }$-symmetry on the stability of QDs is still lacking. Therefore, extending the concept of ${ \mathcal P }{ \mathcal T }$ symmetry to QDs can broaden more unknown secrets of QDs.