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Faraday patterns in coupled one-dimensional dipolar condensates

We study Faraday patterns in quasi-one-dimensional dipolar Bose-Einstein condensates with parametrically driven dipolar interactions. We show that in the presence of a roton minimum in the excitation spectrum, the emergent Faraday waves differ substantially in two- and one-dimensional geometries, providing a clear example of the key role of confinement dimensionality in dipolar gases. Moreover, Faraday patterns constitute an excellent tool to study non-local effects in polar gases, as we illustrate for two parallel quasi-one-dimensional dipolar condensates. Non-local interactions between the condensates give rise to an excitation spectrum characterized by symmetric and anti-symmetric modes, even in the absence of hopping. We show that this feature, absent in non-dipolar gases, results in a critical driving frequency at which a marked transition occurs between correlated and anti-correlated Faraday patterns in the two condensates. Interestingly, at this critical frequency, the emergent Faraday pattern stems from a spontaneous symmetry breaking mechanism.


I. INTRODUCTION
Inter-particle interactions play an essential role in the physics of ultra-cold gases. Although in many experiments these interactions may be approximated by a contact potential, there is a rapidly growing interest in a novel type of cold gases in which electric or magnetic dipole-dipole interactions (DDI) are crucial for the occuring phenomena. These so-called dipolar gases include atoms with large magnetic moments [1][2][3], polar molecules [4][5][6] and Rydberg gases [7]. The distinct nature of the dipolar interactions leads to a wealth of novel physics, including a geometry-dependent stability and a roton-like minimum in the excitation spectrum (for reviews see e.g. [8,9]).
Interestingly, the long-range dipolar interactions result in an inherent non-local nature of dipolar gases, particularly striking in deep optical lattices. For non-dipolar systems, gases trapped in different sites of a deep lattice do not interact with each other. Hence, for a vanishing inter-site hopping, different sites may be considered as independent, uncorrelated experiments. In contrast, inter-site DDI play a substantial role even in the absence of hopping. Recent lattice experiments have shown that the inter-site dipolar interactions are a key element in the dynamics [10], as well as in the stability and collapse of dipolar condensates [11,12].
Faraday patterns constitute a paradigmatic example of pattern formation in periodically driven systems [13,14] ranging from classical fluids [15], through multimode lasers [16] and superfluid Helium [17]. Interestingly, Faraday patterns may be observed in BECs by modulating the nonlinearity resulting from the interatomic interactions [18][19][20][21][22][23], as shown in recent experiments [24]. Faraday patterns in Bose-Einstein condensates (BECs) may be directly linked to the spectrum of elementary excitations, and in this sense provide an excellent insight into fundamental properties of the condensates. In nondipolar gases the Faraday pattern selection is determined uniquely for each modulation frequency due to the monotonically growing character of the excitation energy [25]. Interestingly, this is not anymore the case for dipolar BECs with a roton-like minimum in the excitation spectrum [26]. As a result, it has been shown that Faraday patterns in two-dimensional (2D) dipolar condensates present remarkable qualitative novel features [27] In this paper, we analyze quasi-one-dimensional (quasi-1D) dipolar condensates with periodically driven dipolar interactions. We demonstrate that Faraday patterns provide a clear example of the non-trivial role of confinement dimensionality in dipolar gases, showing that in the presence of a roton-like minimum in the excitation spectrum, Faraday patterns in a quasi-1D trap differ significantly with respect to the 2D case [27]. Moreover, Faraday patterns provide as well an excellent tool for a study of non-local effects in dipolar condensates, as we illustrate with two parallel quasi-1D BECs, in the absence of tunneling. The non-local dipolar interactions between both BECs lead to an unfolding of the excitation spectrum into symmetric and anti-symmetric modes with respect to the transposition of the two condensates. We show that, as a consequence, at a critical driving frequency a transition between correlated (symmetric) and anti-correlated (anti-symmetric) Faraday patterns in the two BECs occurs. For the critical driving the emergent Faraday pattern differs from one realization to another, resulting from a spontaneous symmetry breaking mechanism.
The paper is structured as follows. In Sec. II we introduce the model for periodically driven quasi-1D dipolar condensates. Section III is devoted to Faraday patterns in a single quasi-1D BEC, with a focus on the differences as compared to 2D condensates. Section IV is dedicated to the effects of the inter-condensate dipolar interactions arXiv:1207.1999v3 [cond-mat.quant-gas] 12 Jul 2012 on the Faraday pattern selection in two parallel disjoint quasi-1D dipolar condensates. We conclude in Sec. V.

II. MODEL
We consider in this paper quasi-1D dipolar BECs, either in a single trap (Sec. III) or in two parallel traps (Sec. IV). Since the former case may be considered as a particular realization of the latter, we present in this Section the general formalism for parallel quasi-1D BECs aligned along the z axis, and separated along the y axis by a distance ∆. We assume the potential barrier separating both quasi-1D BECs sufficiently large to suppress any hopping between them. Each condensate experiences a strong harmonic confinement of frequency ω ⊥ in the x-y plane and no confinement along the z direction. The atoms possess a magnetic dipole moment µ (the results are equally valid for electric dipoles) oriented by an external field along the y axis. We employ dimensionless expressions, using units of frequency ω ⊥ and length l ⊥ = /M ω ⊥ , with M the particle mass. Due to the strong x-y confinement we assume that the system remains in the ground state of the x-y harmonic oscillator (this condition is self-consistently verified), and employ the non-local non-linear Schrödinger formalism developed in Refs. [28,29] for a stack of quasi-1D dipolar BECs, to obtain the coupled equations for the wave functions ψ j (z) in traps j = 1, 2: Short-range interactions are characterized by the coupling constant g = g 3D n 0 /2π ω ⊥ l 3 ⊥ , where n 0 is the linear density, and g 3D = 4πa sc 2 /M , with a sc the s-wave scattering length. The DDI are determined by the coupling where J n (x) are the Bessel functions of the first kind. In the following we consider a parametric modulation of the dipole-dipole interactions where α characterizes the modulation strength. Such modulation may be implemented with intensity oscillations of the polarizing electric field for the case of polar molecules, or with additional transverse magnetic fields, which lead to a precession of the dipole moment orientation, for the case of magnetic dipoles. The modulation of g d induces Faraday waves. With the aim of examining the growth of such patterns, we introduce the following ansatz for the wave functions: which describes well the physics of the pattern in the linear regime, where the modulation is weak and we may consider each momentum component q of the pattern separately. In Eq. (4), we introduce the complex amplitude A j (t) = u j (t) + iv j (t), which determines the perturbation from the initial homogeneous solution is the chemical potential. Inserting Eqs. (3) and (4), into Eq. (1), and linearizing in A j we arrive at the system of equations describing the modulation dynamics

III. FARADAY PATTERNS IN A SINGLE QUASI-1D DIPOLAR BEC
We consider in this Section the case of a single condensate, being particularly interested in the differences between the emergent Faraday patterns in a quasi-1D trap and those predicted in Ref. [27] for a 2D condensate.
Employing a similar Bogoliubov analysis as the one presented in Ref. [28], we obtain the spectrum of elementary excitations in the considered case (see Fig. 1): where with Ei(x) the exponential integral function. Using Eqs. (5) and (6), we arrive at the corresponding Mathieu equation [30]: Following Floquet theorem [31], the solutions of Eq.
is the Floquet characteristic exponent, which can be found numerically. If the real part σ ≡ Re(σ) > 0, the homogeneous quasi-1D BEC becomes dynamically unstable and Faraday patterns emerge. The typical wave length of the pattern will be determined by the most unstable mode, i.e., that with the largest σ. In the limit of small driving amplitude, α → 0, the properties of the pattern are governed by momenta q obeying parametric resonances n (q) = nω.
Contrary to non-dipolar BECs with a monotonic spectrum (q), dipolar gases may offer a more complex rotonmaxon spectrum [26] (Fig. 1). As a consequence of this non-monotonic character, for a specific range of ω, between the roton and maxon frequencies (ω r and ω m , respectively) there are three values q 1 < q 2 < q 3 satisfying the resonance condition (q) = ω. Figure 2 shows the stability diagram for a driving frequency in this particular window. As expected, for small amplitudes α, the three instability tongues (white regions) correspond exactly to q 1,2,3 (Fig. 1). This opens an interesting question about which of the three modes dominates the pattern formation. For a 2D geometry, Ref. [27] showed that when modulating dipole-dipole interactions, the most unstable mode corresponds to the intermediate momentum q 2 < q r , with q r the roton momentum. Crucially, as we show below, this is not the case in a quasi-1D dipolar condensate. This striking contrast between quasi-1D and 2D predictions illustrates once more the key role played by the trapping geometry in dipolar gases.
The problem of the most unstable mode is best understood employing a series expansion of the Floquet exponent with small parameter b(q, ω, α) [20,32,33], which, for the first parametric resonance (q) = ω, yields σ b(q, ω, α)/2 ∝ q 2 F 0 (q). Remarkably, in contrast to the 2D case, we find that the most unstable mode corresponds to the largest momentum q 3 > q r (solid line in Fig. 2). Fig. 3 depicts a momentum of the most unstable mode as a function of the driving frequency ω. The plot confirms that for all frequencies within the window ω m < ω < ω r the momentum characterizing the most unstable mode is larger than the roton momentum, contradicting the prediction for a 2D pancake geometry [27].
For ω < ω r , alike in the 2D case, the observed modulations are dominated by higher resonances with q in the vicinity of q r . However, unlike the 2D scenario, even in this regime the most unstable mode in a quasi-1D BEC is characterized by q > q r . We emphasize that the different nature of the Faraday pattern reported here stems solely from the quasi-1D character of the condensate, which leads to a specific momentum dependence of b(q, ω, α) that differs from that in 2D. We have simulated numerically the time evolution of the non-local non-linear Schrödinger equation (1) with the parametrically driven nonlinearity, according to Eq. (3). The emergent pattern has been examined by means of Fourier transform of the condensate density, which confirmed the results for the most unstable mode that we obtained within the Mathieu analysis.

IV. FARADAY PATTERNS IN TWO 1D DIPOLAR BECs
We now turn to the study of Faraday patterns in two parallel quasi-1D dipolar BECs. For non-dipolar BECs, in the absence of hopping, each BEC behaves independently, and hence an experiment with two BECs reduces to two uncorrelated experiments with a single condensate. The situation is radically different in dipolar BECs, since, despite the absence of hopping, the non-local character of the dipolar potential gives rise to a coupling between the two BECs, with the strength of the intercondensate interactions governed by F 1 (k z ). These nonlocal interactions lead to a collective character of the elementary excitations that are shared among the two quasi-1D condensates [28,34]. Consequently, the excitation spectrum unfolds into two branches which correspond, respectively, to symmetric and antisymmetric states with respect to the transposition of traps j = 1 ↔ j = 2. Interestingly, this implies that a periodic modulation of the dipolar interactions yields two different parametric resonances for each driving frequency ω = ± (q ± ), even in the absence of the roton minimum (see Fig. 4). Note, that the patterns are characterized not only by their momentum q ± but also by their symmetric (+) or antisymmetric (−) character. In analogy to Sec. III, the double solution raises a fundamental question about which of these two modes is the most unstable, and hence provides the dominant Faraday pattern. We stress that this nontrivial physics stems directly from the inter-condensate interactions, which lead to the splitting between the two branches in the spectrum, being a qualitatively new feature of dipolar condensates.
Similarly to the previous section, we employ Eqs. (5) for j = 1, 2, and the spectra (10). In turn, we obtain two decoupled Mathieu equations for the symmetric and anti-symmetric combinations u ± = u 1 ± u 2 : to which we apply the Floquet analysis employed in the study of Eq. (8). As in the case of a single BEC, the first parametric resonances ± (q ± ) = ω are characterized by the Floquet exponent σ ± b ± (q, ω, α)/2 ∝ q 2 (F 0 (q) ± F 1 (q)), and the emerging Faraday pattern is determined, for each driving frequency separately, by the mode with the largest σ. Remarkably, the involved mo- For ω well below (above) ωc we observe a single peak at q+ (q−) indicating that a symmetric (anti-symmetric) Faraday pattern emerges (see insets). In the vicinity of ωc (shaded region), both modes are equally unstable and we observe the two corresponding peaks occuring simultaneously in the Fourier transform (see text). mentum dependence of F 0 (q)±F 1 (q) leads to an intricate relation between the Floquet exponents and the driving frequency ω, as presented in Fig. 5.
Crucially, the curves σ ± (ω) cross at a critical frequency ω c . In consequence, we expect a distinct transition, as a function of the driving frequency ω, between the symmetric Faraday pattern for ω < ω c and the anti-symmetric pattern for ω > ω c . Such transition is marked by an abrupt change of the patterns from a maximum-maximum alignement (correlated patterns) to a maximum-minimum alignement (anticorrelated patterns), as depicted in the corresponding insets of Figs. 6 and 7. Moreover, for ω = ω c the patterns in both condensates exhibit a pronounced change of the wavelength of the modulation, from 1/q + (ω c ) to 1/q − (ω c ).
This transition has been confirmed by means of direct numerical simulations of Eqs. (1), with the parametric driving governed by Eq. (3). As for a single condensate, we Fourier transform the density of each condensate to obtain the dominant momenta of the emergent Faraday patterns. The results, in the vicinity of the critical frequency ω c , are depicted in Fig. 6, where, on top of the spectra ± , for each driving frequency ω we indicate with a circle the momentum value where the numerically evaluatedn j (k z ) shows a marked maximum. We find that, in agreement with the results for σ ± (ω) presented in Fig. 5, for ω well below ω c the pattern presents a single momentum component at q + , being characterized by a corre-lation between the patterns in both quasi-1D BECs. In contrast, for ω well above ω c a single momentum component q − is observed, and the patterns in the two quasi-1D BECs are anti-correlated.
In order to quantify the transition between correlated and anti-correlated patterns we introduce the correlation coefficient where S nj (z) = n j (z) − n j , with n j the average density in a trap j. Pattern correlation is then characterized by r > 0, whereas anti-correlation leads to r < 0. Fig. 7 illustrates the radically different time evolution of the correlation coefficient below and above the critical driving ω c . Clearly, for frequencies sufficiently smaller (larger) than ω c the system arrives at perfectly correlated (anticorrelated) pattern with |r| = 1. An interesting scenario occurs for driving frequencies in the vicinity of the critical ω c (shaded region in Figs. 5 and 6), where both the symmetric pattern with wavelength 1/q + (ω c ) and the anti-symmetric pattern with wavelength 1/q − (ω c ) are equally unstable. As a result, the Fourier transform of the density in each quasi-1D BEC shows a simultaneous appearance of both momentum peaks, q + and q − (see Fig. 6).
Note that at ω = ω c , not only + (q + ) = − (q − ) but also b + (q + , ω, α) = b − (q − , ω, α) and hence the two Mathieu equations (11) for u + and u − become identical. This  Fig. 6 (ωc = 0.055). The upper curve, which corresponds to ω = 0.025 < ωc, approaches r = 1 indicating a perfectly correlated pattern in both quasi-1D traps. The lower curve, which corresponds to ω = 0.08 > ωc, reaches r = −1 proving a perfect anti-correlation between the Faraday patterns in the two traps. The insets show the corresponding numerical results for the density distribution nj(z) for t = 11000, with the bright (dark) colors indicating density maxima (minima). Naturally, for sufficiently long times, well beyond the linear regime, the Faraday patterns and their correlations are eventually destroyed.
symmetry is however spontaneously broken in experiments due to quantum and thermal fluctuations, which lead to different initial conditions (populations) for both modes, that change randomly from one realization to another. This spontaneous symmetry breaking mechanism is best studied quantitatively by considering the relative weight of the momentum peaks at q + and q − in the Fourier-transform of the densityn(k z ). To this end, we define the imbalance parameter χ(t) =n (q + , t) −n(q − , t) n(q + , t) +n(q − , t) .
For ω well below or above ω c , once the pattern emerges, χ(t) = ±1. In the vicinity of ω c , however, the imbalance parameter χ(t) shows a clear periodicity with frequency 2ω (see Fig. 8). Note that these oscillations do not result from non-linear competition, as they occur well within the linear regime. In fact, the 2ω oscilla-tions of χ(t) originate in different, spontanously chosen, initial conditions for u + and u − , which lead to their different time evolution that can be well approximated by u ± (t) = (u c ± cos(ωt) + u s ± sin(ωt)) exp(σωt), where u c/s ± are the constants determined by the initial conditions. Furthermore, spontaneous symmetry-breaking leads to a different result for the imbalance χ(t) from one realization to another, what we have confirmed by considering small random differences in the initial conditions for our numerical simulations of Eqs. (1).

V. CONCLUSIONS
Faraday patterns in dipolar BECs are crucially dependent on the unique properties of the dipole-dipole interactions. In particular, due to the long-range anisotropic nature of the dipolar interactions, the character of the Faraday patterns depends strongly on the dimensionality of the condensates. We have shown that for periodically modulated dipolar interactions, Faraday patterns in 2D and 1D geometries differ substantially in the presence of a roton minimum in the excitation spectrum. Moreover, for parallel quasi-1D dipolar BECs, the intercondensate interactions lead, even in the absence of hopping, to an excitation spectrum characterized by symmetric and anti-symmetric modes. This, in turn, gives rise at a critical driving frequency to a marked transition between correlated and anti-correlated Faraday patterns in the two condensates. Interestingly, at this transition point the Faraday pattern selection stems from a spontaneous symmetry breaking mechanism.

VI. ACKNOWLEDGEMENT
We acknowledge funding by the German-Israeli Foundation, the Cluster of Excellence QUEST and the DFG (SA1031/6).