Self-bound liquid droplets in one-dimensional optical speckle potentials

We present a comprehensive description of the equilibrium properties of self-bound liquid droplets in one-dimensional optical speckle potentials at both zero and finite temperatures. Using the Bogoliubov theory we calculate analytically the equation of state, fluctuations induced by disorder, and the equilibrium density. In particular, we show that the peculiar competition between the speckle disordered, the interactions and the Lee-Huang-Yang quantum fluctuations may strongly affect the stability and the formation of the self-bound droplet. We address also the static and dynamical properties of such a disordered droplet using the generalized disorder-dependent Gross-Pitaevskii equation. Notably, impacts of a weak speckle potential are treated numerically for both small droplets of an approximately Gaussian shape and large droplets with a flat-top plateau.


Introduction
Since their discovery by Petrov [1], self-bound droplets in binary Bose-Einstein condensates (BECs) have attracted considerable attention as they have opened new routes to study the many-body problem with a high degree of control.Such an exotic state stabilizes due to the delicate balance between mean-field attraction and beyond mean-field repulsion provided by the Lee-Huang-Yang (LHY) corrections.This exquisite stabilization relies critically on the dimensionality of the system; the LHY fluctuations are repulsive in higher dimensions while being attractive in one-dimension (1D).Experimentally, ultradilute quantum liquids were observed in both homonuclear Bose mixtures of 39 K [2][3][4], and heteronuclear bosonic mixture of 41 K-87 Rb and 23 Na-87 Rb [5].
On the other hand, disorder is a key ingredient of the microscopic (and macroscopic) world, all realistic physical systems unavoidably imply some level of disorder.Even a small amount of disorder can lead to exciting new phenomena that do not have any clean equivalent, such as the Anderson localization [21].In the last decade there were extensive theoretical and experimental efforts for understanding the role of the interplay of disorder and interactions in ultacold atoms (see e.g.[22][23][24][25][26][27][28]).Nonetheless, the physics of Bose-Bose mixtures in disorder is still not completely understood due to the coupling between the various components which introduces extra degrees of freedom leading to unpredictable behavior and disclosing novel phase transitions [29][30][31]).Most recently the ground-state properties of quantum droplets in 3D and 2D disordered potentials have been investigated in [32,33].However, to the best of our knowledge, the physics of 1D disordered ultradilute quantum liquids has never been explored before.
The aim of this paper is then to investigate the impacts of 1D weakly random potential created by optical speckles on a self-bound droplet of Bose mixtures.Major features of such optical speckles are: (i) long-ranged, (ii) non-Gaussian potentials, (iii) their amplitude, geometry and correlation length can be readily adjusted.These features allow us to explore quantum droplets with a high degree of control.
We shed light in particular on how quantum and disorder fluctuations affect the stability and equilibrium properties of the self-bound droplets.Using the Bogoliubov theory [33], we derive useful analytical expressions for the equation of state, the equilibrium density, and the glassy fraction in terms of the disorder parameters (strength and correlation length) at both zero and finite temperatures.In particular, we find a strong dependence of the ground-state energy, the equilibrium density, the droplet depletion due to disorder, and the stability of droplet on the disorder parameters.It is revealed that stability of the droplet requires small disorder strength and large disorder correlation length (i.e.exceeds the healing length of the condensate).
We investigate in addition the equilibrium and the dynamics of such a disordered droplet in both Gaussian-like and flat-top regimes.To this end, we use the generalized disorder-dependent Gross-Pitaevskii equation (GGPE) which we derive self-consistently in the frame of the Hartree-Fock-Bogoliubov formalism [18].The numerical simulation of the GGPE points out that the disorder tends to significantly affect the droplet profiles and its width.
The rest of the paper is organized as follows.In section 2 we review some of the main steps of the Bogoliubov-Huang-Meng theory for Bose-Bose mixtures in disordered potentials.Section 3 addresses the effects of a 1D disorder speckle potential on the bulk properties of 1D self-bound droplets by applying the developed Bogoliubov-Huang-Meng theory.Section 4 is devoted to the static and dynamical properties of such disordered droplets by numerically solving the underlying GGPE for small and large dropets.We finish by drawing conclusions in section 5.

Bogoliubov-Huang-Meng theory
We consider a weakly interacting Bose mixture with equal masses m 1 = m 2 = m, subjected to a weak disorder potential U. The disorder potential is assumed to have vanishing ensemble averages ⟨U(r)⟩ = 0 and a finite correlation of the form ⟨U(r)U(r ′ )⟩ = R(r − r ′ ).The Hamiltonian including all point-like interactions can be written in terms of the creation and annihilation operators â † j,k and âj,k , where the subscript j = {1, 2} refers to the jth component of the mixture, as [33]: â † j,p âj,p−q âj,k+q + where E k = h2 k 2 /2m, V is a quantization volume, and U k is the Fourier transform of the external random potential U(r).
Under the Bogoliubov prescription, the ground-state of ultradilute Bose mixtures (or quantum droplets) is assumed to contain most of the atoms, we replace the operators âj,0 and â † j,0 by a c-number, i.e. âj,0 = â † j,0 = N j , where N j is the number of particles.Using the normalization relation: where n j = N j /V is the density of each component.In equation ( 2) we kept only quadratic terms in â † j,k̸ =0 , âj,k̸ =0 up to the second-order in the coupling constants.We also assumed that for weak enough disorder, disorder fluctuations decouple in the lowest order [22,23].As a result we ignored the terms U k−p â † j,k âj,p with both k = 0 and p = 0.
Hamiltonian (2) can be diagonalized making the extended Bogoliubov-Huang-Meng transformation where the Bogoliubov quasiparticle amplitudes are given by: which are chosen to be real without loss of generality, and the disorder translations β j,k are defined by the equations Note that the transformation ( 3) is canonical and the quasi-particle annihilation, bj,k , and creation, b † j,k , operators must obey the usual Bose commutation relations: It allows us to decouple the quantum and random variables and get the bilinear form of the Hamiltonian.
The resulting diagonalized Hamiltonian varies with each realization of the random potential.Thus, the final bilinear Hamiltonian of disordered binary BECs can be obtained by performing the disorder ensemble average [22].This yields: , where ε k± are the Bogoliubov spectra and E is the ground-state energy [33].
From now on we consider a symmetric mixture with, equal densities n 1 = n 2 = n/2, equal intraspecies g 1 = g 2 = g and interspecies g 12 = g 21 coupling constants.Therefore, the Bogoliubov excitation energies take the form ε k± = E 2 k + 2E k nδg ± , where δg ± = g(1 ± g 12 /g) [34].The ground-state energy of this system including the disorder corrections can be written in d-dimensions as: The leading term is the mean-field energy.The subleading term gives the correction to the ground-state energy due to the external random potential.The last term accounts for the LHY quantum corrections to the ground-state energy.The disorder fluctuations (i.e.droplet depletion induced by the disorder potential) arises from the accumulation of density near the potential minima and density depletion around the maxima.It is defined as [30,31,33] which is obtained by performing the disorder ensemble average.Noteworthy, the validity of the present approach requires the condition n R± ≪ n i.e the disorder fluctuations are small.

Self-bound droplets
Assuming the weak-coupling regime where the correlation length is much larger than the mean interparticle separation [35].The Bogoliubov theory can be then safely used in low-dimensional systems at both zero and finite temperatures.
Let us then consider a self-bound droplet subjected to a 1D weak speckle disorder which is often used with ultracold atoms experiments [36][37][38][39][40]. Experimentally, effective 1D speckle disorder can be created when coherent light is diffracted by a ground glass diffuser with a quasi-1D slit and focused by a convergent lens  9) as a function of the density n for various values of the disorder strength for g12/g = −0.7 and σ/ξ = 0.6.(Bottom) Ground-state energy of a 1D liquid from equation ( 9) as a function of the density n for various values of the correlation length for g12/g = −0.7 and R = 0.6.[36][37][38][39][40].The speckle disorder is characterized by the the following real-space correlation function : R(x) = U 2 R sinc 2 (x/σ), where U R is the disorder strength and σ is the disorder correlation length.In Fourier space, it can be written as [24,41]: where Θ denotes the Heaviside function.Indeed the function R(k) has a sharp high-momentum cutoff at k = 2/σ.Inserting equation ( 8) into equation ( 6), one obtains for the ground-state energy: where n (0) = 16gm/ 9π 2 h2 (δg + /g) 2 is the 1D equilibrium density [6], R = U R /|ε B |, and |ε B | = h2 /ma 2  12 is the binding energy of dimers composed from atoms from different component.For R = 0, the energy (9) reduces to the result of a clean droplet [6,17].
Figure 1 shows that by increasing R and decreasing σ/ξ, the minimum and maximum start to disappear indicating that the liquid phase becomes unstable.Note that the liquid droplet evaporates even for very small R.This can be understood from the intrinsic strong correlation between particles arising from the geometric confinement in 1D.Remarkably, the 1D droplet remains robust even for relatively small correlation length, σ/ξ, on contrary to the 3D droplet which evaporates for small σ/ξ (disappearance of the local minimum in the ground-state energy) [33].We see also from the bottom panel that the disorder correlation length has a minor effect at higher densities (n/n (0) ≳ 1) which is not the case for 3D disordered droplets.The droplet depletion due to disorder effects n R /n = ± n R± /n can be computed via equation ( 7) its behavior in terms of the disorder strength R is depicted in figure 2(a).We see that n R increases monotonically with R signaling that the atoms occupy the localized state giving rise to the transition to a new quantum phase.Consequently, the presence of a small amount of disorder may significantly affect the stability and the formation of the droplet.
The validity criterion of the present approach requires the condition n R /n ≪ 1.For instance, in the case of g 12 /g = −0.7 and σ/ξ = 0.6, one has n eq /n (0) = 0.7, and hence the critical disorder strength below which the droplet is stable must be R ≪ 3.183.
In figure 2(b) we present the equilibrium density n eq /n (0) of a 1D dirty quantum liquid, found from the minimum of the ground-state energy (9).As expected, the equilibrium density decreases in the limit of small disorder correlation length.In such a regime the disorder effects dominate both the LHY and the mean-field forces.
It is well known that ignoring entirely thermal fluctuation effects is a difficult task in current experiments.Therefore, it is instructive to analyze the role of low but finite temperatures on the equation of state and on the stability of the quantum droplet.At finite temperature the free energy is given by [34,42]: By integrating this expression over momentum we arrive which takes the following dimensionless form: where E is given in equation ( 9) and the subleading term accounts for thermal corrections.Figure 3(a) shows that the free energy exhibits first a maximum in the limit of small densities.After this maximum is exceeded, the droplet spreads spontaneously until it reaches the minimum free energy value.The state corresponding to this minimum is a stable quantum droplet phase, while the maximum corresponds to an unstable state.The two solutions start to desapear at a critical temperature T ≃ 0.35|ε B |.For T > 0.35|ε B |, the quantum droplet becomes unstable and eventually evaporates.
In figure 3(b) we plot the critical temperature T c associated with the free energy minimum as a function of the disorder strength for different values of the disorder correlation length.We see that T c decreases with R while it increases with σ notably for large R due to the interply of the disorder and LHY effects.It is worth noticing that the equilibrium density n eq /n 0 , disorder fraction, n R , and the critical temperature T c of 1D droplet are less sensitive to correlation length, σ/ξ, except in the regime of large R, in contrast to the 3D case.This can be attributed to the particular form/feature of the 1D speckle disorder potential which is different from the other correlated disorders.

Static and dynamical properties
Assuming that the condensate ϕ varies smoothly at the scale of the extended healing length.As a consequence, we can include corrections due to quantum, thermal and disorder fluctuations locally as nonlinear terms in the GPE and treat them classically.The functional energy reads: It is convenient to introduce characteristic units of length x = x/ξ, time t = ht/(mξ 2 ), and energy E 0 = h2 /(mξ 2 ), where ξ = (πh 2 / √ 2mg) δg + /g is the extended healing length.This gives for the wavefunction The regime of interest consists of setting δg + /g ∼ 0 and δg − /g ≈ 2 [6], thus the dimensionless functional energy turns out to be given where Ũ = U/E 0 .The droplet wavefunction is then given by the solution of the 1D disorder-dependent GGPE which can be derived using i In the absence of the disorder potential, equation (17) reduces to that derived in the literature [6].
For the disorder potential we choose a Gaussian spikes defined as [43]: where M is the number of impurities, Ũ0 is the strength of the spike, and σ is a dimensionless width.This speckle potential is generated using a set of random numbers which are then mapped into the interval [−L, L] by a linear transformation.

Flat-top droplet
In order to obtain the stationary localized states, we perform the numerical integration of the static GGPE by adopting the split-step Fourier spectral method.Stationary states with chemical potential, µ, can be computed using ϕ(x, t) = ϕ(x) exp(−iµ t), where ϕ(x) is an equilibrium solution.The results are shown in figure 4. We see from figures 4(a) and (b) that in the absence of disorder, the droplet reaches its flat top shape without any special structure.However, modulations in the density of the self-bound droplet are observed at finite disorder even as small as Ũ0 = 0.02 in the plateau region.These modulations manifest themselves as a deformation of the quantum liquid droplet.As the strength, Ũ0 , and the correlation length, σ, of the disorder are increased, the density fluctuations become larger leading to the emergence of a localized fraction.The same behavior holds true in the 3D case [33].When Ũ0 becomes sufficiently strong one can expect that the droplet fragments into multiple mini droplets.An important remark is that the disorder effects are practically suppressed near the edge of the droplet whatever the values of the disorder strength and the temperature, most likely due to the surface effects which are striking enough to eliminate the influence of the disorder potential in this region.
Having obtained the stationary solution, we evolve it in time by solving the full time-dependent GGPE (17).In figure 5 we plot the spatiotemporal density of the evolution of a disordered droplet for several values of the disorder strength.It is clearly visible that the droplet is significantly sensitive to disorder and fragments into several localized modes (mini droplets) even for small disorder strength (see figure 5(a)).Note that the number of the droplets depends on the disorder parameters (strength and correlation length).Increasing further the disorder strength, the long-range order is destroyed and thus, the droplet is completely evaporated (see figure 5(b)).
To better understand the time evolution of the droplet, we investigate its mean width, q.It can be calculated from the disorder-dependent GGPE (17) using q = ´+∞ −∞ dxx 2 |ϕ(x, t)| 2 /2N.We present the numerical results in figure 4(c).For weak disorder strength, the droplet width increases and then saturates.It exhibits slight oscillations in the long time limit.We see also that the presence of a weak disorder leads to shift the width from its clean counterpart at larger times, t ≳ 100 (see red and orange curves).

Small droplet
In the case of a small droplet, our simulations depict smooth density profiles (see figures 6(a) and (b)) indicating that the disorder slightly deforms the small droplet compared to the flat top case.Notably, we observe that the droplet size shrinks as the disorder strength is increased and the amplitude reduces as the disorder correlation length gets larger.This can be attributed to the fact that the disorder is not strong enough to balance the LHY forces.
Here we discuss the time evolution of a small droplet.Figure 7(a) shows that for a very weak disorder, the small droplet remains almost stable during its time evolution.The disordered droplet becomes narrower and develops small density modulations near the center as the disorder strength increases signaling that the action of the disorder is effective as shown in figure 6(b).In such a situation many atoms, spewing more energy due to the disorder, leave the droplet and spread out thus, disperse over time.Figure 7(c) shows that in the presence of a weak disorder, the width increases with time revealing that the droplet expands rapidly.

Conclusions
In summary, we investigated effects of a 1D optical speckle disorder on the bulk properties of a droplet state in symmetric Bose mixtures.Useful expressions have been obtained for the ground-state energy, the droplet depletion due to disorder, and the free energy.We showed that a stable droplet can survive only for small values of disorder strength and large correlation length.At finite temperatures, we demonstrated that the critical temperature beyond which the liquid droplet dissolves strongly depends on the disorder parameters.
We analyzed in addition the statics and dynamics of a disordered quantum droplet using the generalized disorder-dependent GPE which we derived self-consistently.Our numerical simulation points out that, even small amount of disorder may substantially deform the atomic density and modify the ground-state and dynamical properties of the droplet especially in the plateau region.We found that during its spatiotemporal evolution, the droplet exhibits small oscillations and splits into many mini structures for increasing disorder.
The findings of the present work not only provide unique opportunities for quantitatively testing and understanding of dirty droplets but constitute also an ideal platform for future experiments.

Figure 1 .
Figure 1.(Top) Ground-state energy of a 1D liquid from equation (9) as a function of the density n for various values of the disorder strength for g12/g = −0.7 and σ/ξ = 0.6.(Bottom) Ground-state energy of a 1D liquid from equation (9) as a function of the density n for various values of the correlation length for g12/g = −0.7 and R = 0.6.

Figure 2 .
Figure 2. (a) Disorder fraction nR = ∑ ± n R± inside the droplet as a function of the disorder strength R for σ/ξ = 0.6 and g12/g = −0.7.(b) Equilibrium density neq/n 0 of a of a 1D dirty quantum liquid as a function of the disorder strength for various values of σ/ξ.