Stochastic Growth Dynamics and Composite Defects in Quenched Immiscible Binary Condensates

We study the sensitivity of coupled condensate formation dynamics on the history of initial stochastic domain formation in the context of instantaneously quenched elongated harmonically-trapped immiscible two-component atomic Bose gases. The spontaneous generation of defects in the fastest condensing component, and subsequent coarse-graining dynamics, can lead to a deep oscillating microtrap into which the other component condenses, thereby establishing a long-lived composite defect in the form of a dark-bright solitary wave. We numerically map out diverse key aspects of these competing growth dynamics, focussing on the role of shot-to-shot fluctuations and global parameter changes (initial state choices, quench parameters and condensate growth rates). We conclude that phase-separated structures observable on experimental timescales are likely to be metastable states whose form is influenced by the stability and dynamics of the spontaneously-emerging dark-bright solitary wave.

In experiments with single-component atomic Bose-Einstein condensates (BECs), the spontaneous formation of vortices [26] and dark solitons [27] during the condensation phase transition has been observed, with the latter providing quantitative confirmation of the KZ scaling law [28][29][30][31].Studies of the KZ mechanism in multi-component BECs have focused on the formation of two same-species condensates in a tunnel-coupled configuration [32], and on quantum phase transitions in binary condensates [33,34] and spinor Bose gases [35][36][37][38][39][40].Work on homogeneous spinor Bose gases has revealed the presence of two distinct timescales in the process of domain formation [37,38]: a fast timescale on which initial domains form via the KZ mechanism, and a much longer nonlinear evolution which gradually erases the initial domain memory.Importantly, studies of non-equilibrium dynamics in spinor gases suggest that this erasure may potentially occur over timescales longer than typical experimental system lifetimes [39,40].An important, related, question arising from studies of two-component BEC experiments in the immiscible regime -for example using 87 Rb-41 K [41], 87 Rb-85 Rb [42], or 87 Rb-133 Cs [43,44] -is the extent to which initial conditions and intrinsic system dynamics affect the observ-able density profiles.
In this Letter we provide strong evidence that the density profiles observed in immiscible two-component BEC experiments are determined by the stochastic history of spontaneous defect formation, and subsequent defect dynamics, during condensate growth (shown schematically in Fig. 1).In particular, we show how the dynamics of spontaneously generated defects in the fastest-growing condensate often leads to a spontaneously-generated dark-bright solitary wave structure [46][47][48][49][50], and that the form of the immiscible density profiles observable in experiments depends critically on the dynamical stability of this dark-bright solitary wave.We have confirmed that this process is relatively robust to perturbations such as experimental trap asymmetries between components, while  , ω (Rb) ⊥ , ω (Cs) z , ω (Cs) ⊥ ) = 2π × (3.89, 32.2, 4.55, 40.2) Hz, with longitudinal and transverse shifts of ∼1 µm in their centers [43,45].Scattering lengths are taken to be (a Rb,Rb , a Cs,Cs , a Rb,Cs ) = (100, 280, 650) × a 0 , where a 0 is the Bohr radius.In the example shown here, µ Rb = µ Rb , µ Cs = 7.34µ Cs .The initial condition is a thermalized state with no definite condensate mode (r ∼ 1).
remaining sensitive to thermal fluctuations from shot to shot.Our results are in broad qualitative agreement with the experiment of Ref. [43], which manifested qualitatively distinct phase-separated structures from shot to shot while following a fixed experimental procedure.Although theoretical work in the mean field description [45] has investigated the sensitivity of equilibrium profiles to trap asymmetries between components and atom number variations in this context, the appearance of qualitatively distinct phase-separated structures from shot to shot is not explained by a mean-field description.
Our analysis is based on a sudden temperature and chemical potential quench of an equilibrated finite-temperature twocomponent atomic cloud, the components of which do not initially possess a single macroscopically-occupied mode.In a significant number of numerical runs (see Fig. 2 for a typical example), the longest-surviving spontaneously-generated defect in the fastest-condensing component acts as a mobile microtrap [51] for the second component.In the case that the surviving defect is a dark soliton, growth of the bright (infilling) component in the microtrap (stabilizing the dark soliton against decay [52]) leads to a spontaneously-generated mean-field-stabilized dark-bright soliton.As our system is only weakly one-dimensional (µ i ω (i) ⊥ ) we also observe phase structures consistent with vortex and solitonic vortex [53] defects at early times, which may initially infill to create a vortex-bright soliton [54] rather than a dark-bright soliton.However, microtraps created by all types of defect generically lead to a dark-bright solitary wave structure.Over longer timescales, the resulting dark-bright solitary wave forms a fixed domain wall [55,56].The interplay between conden-sate growth in the first component, defect formation and dissipative evolution, and condensate growth of the second component -alongside increasing mean-field repulsion between the co-forming immiscible condensates -leads to interesting features which should be observable in current experiments.This Letter initially focuses on the crucial role of stochastic fluctuations during condensate growth, before examining the dependence of the dynamics on changes in the global quench parameters.
Many dynamical two-component BEC simulations have been based on coupled ordinary [46,47,57,58] or dissipative [52] Gross-Pitaevskii equations (GPEs), and recent works have also used classical field [59], or truncated Wigner [33,34] methods.To best capture the crucial effects of thermal fluctuations during condensate growth, we use coupled stochastic projected Gross-Pitaevskii equations [40,60,61], in the form where i labels the species (here 87 Rb or 133 Cs), and tering lengths a ii (intraspecies) and a i j (interspecies), atomic masses m i , and reduced mass M i j .Here, ψ i is intended to describe the evolution of highly-occupied modes in the "c-field" region defined by projector P, consisting of single-particle modes φ l with energies below a carefully-selected energy cutoff [62].The c-field is coupled to the above-cutoff reservoir at temperature T via growth (described by the species-dependent rates γ i ) and noise (described by dW i [63]) processes.To sim-   and (e-h) the condensate density profiles after 1 s (insets 2 s).After this time the Rb defect has either: been stabilized by Cs infilling (e); fully decayed within Rb (f); or decayed to the left (g) or right (h) edge of the condensate.Quench parameters are as in Fig. 2. ulate two-component condensate formation, we first obtain a thermalized initial state by numerically propagating Eq. ( 1) to equilibrium for initial chemical potentials µ i and common temperature T 0 , and then instantaneously quench these parameters to final values T < T 0 and µ i ≥ µ i .We extract the timedependent condensate mode using the Penrose-Onsager (PO) criterion [64], by seeking a macroscopically-occupied eigenmode in short-time averages over single trajectories [60].We choose trap configurations, temperatures, and condensation timescales consistent with a recent 87 Rb-133 Cs (Rb-Cs hereafter) BEC experiment [43], performed with the species in overlapping elongated harmonic traps with slightly displaced (by ∼1 µm) centers [45].
Figure 2 shows snapshots of the c-field density (a) and condensate mode density (b) in a single simulation of Eq. ( 1).The essential dynamical features are well-captured by the c-field density.Figure 2 also shows the dynamics of the condensate atom numbers, N 0 , (c), and the occupation ratio, r, of the (PO) condensate to the next-most-highly occupied single-particle mode (d).The initial (quasi-condensate) state has no single macroscopically-occupied mode (r ∼ 1), but r increases rapidly following the quench, signalling the suppression of phase fluctuations during condensate growth [15].The curves in (c) and (d) reveal three characteristic evolutionary stages: a short onset time with strongly non-equilibrium dynamics (for 0.1 s), followed by rapid relaxation to a metastable equilibrium state (here, a rapid Rb growth up to t ∼ 0.5 s), before a subsequent slow evolution towards a global equilibrium.
In the Rb-Cs two-component BEC experiment of Ref. [43], Rb displayed a tendency to condense first , with Cs sympathet-ically cooled through collisions.We have chosen quench parameters consistent with such behaviour (see Fig. 2 caption), and find that when Cs condenses the Rb condensate number decreases slightly (our simulations do not include evaporative cooling).Preferential conditions for rapid Cs condensate growth arise when a spontaneous dark-bright solitary wave forms close to the trap center, leading eventually to a fixed domain wall (as in Fig. 2).At longer times, a Cs condensate also appears at the edges of the system, slightly compressing the Rb condensate towards the trap center.
The full evolution is shown in Fig. 3(a)-(d) for a range of different numerical runs (loosely corresponding to different experimental runs) based on the same quench sequence.Figure 3(a) shows the case of Fig. 2, clearly revealing the evolutionary stages of slow-moving defect (in this case, a dark soliton), near-stationary dark-bright solitary wave, and eventual static domain wall structure.Figures 3(b)-(d) reveal alternative (and roughly equally-likely) outcomes realized for different post-quench dynamical noise sequences (of the same mean amplitude), describing thermal fluctuations.Figure 3(c) shows slower Cs growth in an initially shallower, and hence more rapidly moving, dark solitary-wave microtrap.In contrast to Fig. 3(a), where dark solitary wave decay and stabilisation occur at the trap center, Figs.3(c,d) show dark solitary wave decay and stabilisation occurring either to the left (c), or to the right (d), illustrating the crucial role of the motion of the decaying dark solitary wave [65].Figure 3(b) reveals the appearance of multiple defects at early times [see also Fig. 2 (t=0.06s)],which in this case decay rapidly without providing a microtrap facilitating Cs growth; in this case Cs condenses in the regions of low Rb density.Figure 3(e-h) shows the condensate profile after 1s (insets 2s), consisting of either a large Cs structure in the middle separating the Rb [inset to (e)], or a large Rb structure enclosed by Cs [insets to (f), (g), (h)].The latter case appears to be the most probable long-term outcome for the chosen quench parameters; long-time evolution (potentially beyond accessible experimental timescales) of case (a) eventually also yields such a structure.
For simplicity, results presented so far are based on an identical initial state, with different dynamical noise sequences used in the post-quench evolution.We have confirmed that our qualitative findings remain unchanged in simulations with identical quench parameters but a different (random) thermalized initial state, as would be expected in different experimental runs.However, we do find that initial states can differ in their propensity to generate, at short post-quench evolution times, a deep dark solitary wave in Rb, facilitating efficient Cs condensation [case (a) of Fig. 3].Irrespective of the stabilization or decay of the dark-bright solitary wave, condensate atom number evolution is the same (within statistical variations); moreover, for a given set of quench parameters, the long-term evolution tends to favour the dominance of either Rb or Cs.This is in qualitative agreement with experimental observations [43] and dissipative GPE simulations [66].We have also performed detailed numerical simulations of the dissipative GPE to verify the importance of dark-bright solitary waves in the early stages of formation, finding that dynamical mean-field stabilisation occurs rapidly even for perfectly imprinted dark-bright solitons, and that multiple imprinted darkbright solitons very rapidly coalesce into a single long-lived dark-bright soliton.
In Figure 4 we show the regimes of dynamical evolution arising for different quench parameters, focusing on the dependence of N 0 on variations of the chemical potentials µ i and growth rates γ i .With γ Rb = γ Cs held fixed, we find Rb dominates the evolution for a broad range of µ Rb > µ Rb [Fig.4(a)].Here, the spontaneously-generated defects tend to decay within Rb, with Cs failing to condense into a dark-bright solitary wave and instead condensing gradually at the trap edges [Fig.4(c)].For quenches with large µ Cs [Fig.4(a), µ Cs = 11.9µCs ] this role is reversed, with Cs growing very rapidly and the Rb condensate gradually disappearing [Fig.4(d)].In these quenches, we find that Cs may grow within Rb to form a dark-bright solitary wave, around the Rb condensate, or both.However, irrespective of this initial evolution, we find that prior to its complete disappearance, the Rb condensate always lies within the Cs condensate, in qualitative agreement with experimental findings [43].We have also investigated the effect of the growth rate on the condensate evolution (standard estimates [60] suggest that γ Cs ∼ 10γ Rb ), and find that increasing the ratio of γ Cs /γ Rb , for fixed chemical potentials, favours the rapid growth of Cs [Fig.4(b)].
Small independent transverse shifts (∼1 µm) in the trap minima for each species were previously found, in the context of mean-field theory, to change the form of the (metastable) equilibrium condensate densities [45].Our present results suggest that fluctuations strongly suppress the effects of these asymmetries.In particular, while different dynamical noise sequences (of fixed mean amplitude) can alter the qualitative structure of the metastable density profiles seen after 1 s (see Fig. 3), the presence or absence of trap asymmetry (for identical noise sequences) appears to only alter the details, not the qualitative structure, of these density profiles.
Our discussion has focused on the importance of fluctuations and global quench features for fixed cutoffs.However, our theoretical model does not account for dynamics at energies above the cutoff, which could play a role during the coupled evolution.The highly nonlinear nature of the evolution (the coupled dissipative model, without noise, already has 8 independent parameters) is such that minor changes in the relative quench parameters (e.g., µ Cs /µ Rb , γ Cs /γ Rb ) lead to an entirely different phase space trajectory, and probing such a multi-dimensional parameter space is numerically prohibitive.Nonetheless, although we do not expect quantitative agreement with the experiment of Ref. [43] from our model, the generic forms of phase-separated structure shown in Fig. 3 for well-chosen quench parameters are qualitatively similar to those observed in the experiment, and occur on timescales of the same order.
In conclusion, we have studied the role of fluctuations, and spontaneous defect formation, in the growth of trapped, immiscible two-component condensates.Our central finding is that the stochastic dynamics of spontaneously-generated defects may, via formation of dark-bright solitary waves, directly determine the (metastable) pattern of immiscible phases that can be observed on experimental timescales.These effects are robust to variations in parameters associated with experimental reproducibility, such as small changes in atom numbers, or the presence of minor trap asymmetries, and are in qualitative agreement with previous experimental observations.The dynamics explored here should be observable in detail in current cold atom experiments, by systematically monitoring many single-trajectory evolutions for identically-formed pre-quench thermalized states.

FIG. 2 .
FIG. 2. (color online) Typical numerical evolution of a quenched two-component system.(a) and (b) show post-quench evolution, up to 2 s, of 2D column densities (n col , n 0,col ) and 1D integrated density profiles (n int , n 0,int ) for c-field and Penrose-Onsager condensate respectively.(c) and (d) show the evolution of (respectively) the condensate number, N 0 , and the occupation ratio, r, between the condensate and the nextmost-highly occupied single-particle mode.All simulations in this Letter (unless otherwise specified) use initial temperature T 0 = 80 nK, initial chemical potentials µ Rb /k B = 2.13 nK, µ Cs = 0.956µ Rb , an instantaneous temperature quench to T = 20 nK, and rates γ Rb = γ Cs = 0.263 s −1 (equivalent to γ i /k B T = 10 −4 ).Both components are confined in harmonic traps with frequencies (ω (Rb) z

FIG. 3 .
FIG. 3. (color online) Evolution for four different post-quench random noise sequences, with identical pre-quench state.(a-d) show the c-field density evolution [shown here via n(z) i = n i (x = 0, y = 0, z)],and (e-h) the condensate density profiles after 1 s (insets 2 s).After this time the Rb defect has either: been stabilized by Cs infilling (e); fully decayed within Rb (f); or decayed to the left (g) or right (h) edge of the condensate.Quench parameters are as in Fig.2.