{'fraction_non_alphanumeric': 0.040221291866028706, 'fraction_numerical': 0.01450358851674641, 'mean_word_length': 4.168727460072128, 'pattern_counts': {'":': 0, '<': 19, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stability of the density-wave state of a dipolar condensate in a pancake trap

We study a dipolar boson-fermion mixture in a pancake geometry at absolute zero temperature, generalizing our previous work on the stability of polar condensates and the formation of a density-wave state in cylindrical traps. After examining the dependence of the polar condensate stability on the strength of the fermion-induced interaction, we determine the transition point from a ground-state Gaussian to a hexagonal density-wave state. We use a variational principle to analyze the stability properties of those density-wave state.


I. INTRODUCTION
The recent realization of a Bose-Einstein condensate of chromium atoms [1,2] opens up the study of quantum-degenerate gases that interact via the long range, anisotropic magnetic dipole interaction. This is an anisotropic and long-range interaction that leads to the appearance of a wealth of new properties past those characteristic of systems with isotropic interactions [3]. In the case of 52 Cr s-wave interactions are normally much stronger than the dipolar part, but it has been shown experimentally that by applying a magnetic field the s-wave scattering length can be lowered to values comparable to the Bohr radius [4], so that the dipole interaction dominates the system. In the regime where the dipole-dipole interaction is dominant the condensate is characterized by the existence of a metastable state whose properties depends on the trapping geometry [5,6,7,9,10,11], as was experimentally demonstrated in Ref. [8]. A roton feature has also been predicted to exist in these systems for appropriate parameters. There would considerable interest indeed in accessing the associated roton instability [12,13,14], as it is characterized by the spontaneous generation of a periodic density modulation in the condensate [15]. Unfortunately that state is unstable against collapse, but we found recently that this difficulty can be circumvented by the addition of a small fraction of non-interacting fermions, resulting in a significant stabilization of the bosonic system in cylindrical traps [16]. The simultaneous trapping of bosonic and fermionic isotopes of chromium [17] also points to an interesting directions, with the possibility to observe novel ground and metastable phases in quantum degenerate polar bosonfermion mixtures.
This paper extends our previous study of dipolar boson-fermion mixtures from cylindrical traps to pancake geometries, showing that dipolar bosons can be stabilized considerably by increasing the boson-fermion s-wave scattering length, and more importantly the densitywave states of dipolar bosons can be stable for typical, zero-field boson-fermion interaction strengths. Section II describes the induced potential created by fermions on bosons, and Sec. III discusses the general properties of the energy functional as a function of boson-fermion interaction using a variational Gaussian ansatz. Section IV presents an analysis of the transition point to the densitywave state for various combinations of trap aspect ratios and boson-boson contact interaction strengths. Section V introduces a density-wave ansatz in the form of a series of shifted Gaussians. The stability of these density-wave states is determined by varying the width of each Gaussian to seek minima in the interaction energy for various boson-boson s-wave interaction. We find a condition for the stability of these types of density modulated states. Finally, Section VI is a summary and outlook.

II. FERMION-INDUCED INTERACTION IN POLAR CONDENSATES
We consider a mixture of N b dipolar bosons of mass m b and N f single-component fermions of mass m f confined in a pancake-shaped trap characterized by a tight harmonic potential of frequency ω z along the z-axis and a softer harmonic potential of frequency ω ⊥ in the transverse direction. A polarizing external electric or magnetic field, is taken to be along the z-axis . The dipole-dipole interaction between two bosonic particles separated by a distance r is then where g dd is the dipole-dipole interaction strength. In the mean-field approximation, the energy functional for the order parameter φ(r) of the dipolar condensate can be expressed as is the sum of the kinetic energy and the trapping potential and λ = ω ⊥ /ω z . The second term in the energy functional (2) denotes the contact interaction between bosons, characterized by the strength g = 4π 2 a bb /m b with a bb being the s-wave scattering length, and the third term describes the nonlocal dipole-dipole interaction between bosons. Finally, the last term E ind accounts for the fermion-induced interaction V ind (k) between bosons, given in linear response theory [18] as follows.
We assume a contact boson-fermion interaction of strength g bf = 2π 2 a bf /m r , where a bf is the bosonfermion s-wave scattering length and m r = m b m f /(m b + m f ) is the reduced mass. The boson-fermion interaction energy has the form g bf n f (k)n(−k)d 3 k, with n(k) and n f (k) being the bosonic and fermion densities in the momentum space. The linear response of the fermions to a bosonic density n(k) can be expressed as n f (k) = V ind (k)n(k), so that for the effect of the fermions on the bosonic energy functional is The explicit form of the induced potential is where χ f is the density response function [18]. It is related to the dynamical structure factor S(k, ω), which is the probability of exciting particle-hole pairs with momentum k out of the Fermi sea, by For a non-interacting singlecomponent Fermi system we have where p = |p|, k f is the Fermi momentum, the excitation energy is ω 0 pk = pk cos θ/m f + k 2 /(2m f ), and θ is the relative angle between p and k. This expression assumes that the fermions are locally free, so that there is a local Fermi sphere in momentum space. In this local density approximation, k f is given by [19] k f ℓ z = 1.9N where ℓ z = /(m b ω z ) is the oscillator length in the z direction.
Following Ref. [16] the induced potential V ind (k) is given by where ν = k f m f /(π ) 2 is the three-dimensional fermionic density of states, and k 2 = k 2 x + k 2 y + k 2 z is the square of the fermionic momentum. The induced potential is attractive for very low momenta and goes to zero with increasing momenta.
Throughout this paper, we consider the parameters corresponding to a 52 Cr-53 Cr mixture, and fix the number of fermions to be N f = 10 3 unless otherwise stated. The zero-field bosonic s-wave interaction for 52 Cr is taken to be a bb = 103a 0 , and a bf = 70a 0 for the boson-fermion scattering length [20].

III. STABILITY OF THE GAUSSIAN DIPOLAR CONDENSATE
We now proceed to determine the stability of the dipolar condensate, using a variational ansatz in the parameter space of the fermion-induced interaction and dipolar strength. The variational wave function is taken as the Gaussian with d, d z being the variational parameters and d 3 r|φ(r)| 2 = 1. Substituting this Gaussian ansatz and its Fourier transform into Eqs. (2) and (4) yields the energy E g of the condensate as The derivation of algebraic forms of the interaction-energy terms and of the function F are straight-forward but lengthy, and their derivations are relegated to an Appendix. In Eq. (10) we have also introduced the effective three-dimensional dipole-dipole interaction The coefficients g < and g > find their origin in the momentum-independent contact interaction, given by the summation of the s-wave boson-boson scattering and the constant terms in the induced interaction (8). They are given explicitly by Finally is due to the nonlocal induced interaction. We remark that the effective contact interaction consists of the boson-boson contact interaction as well as the momentum-independent part of the dipole-dipole interaction and of the induced interaction. From Eqs. (10) and (11) we have that for low momenta It is known that in boson-fermion mixtures without dipolar interaction, phase separation occurs when g s becomes negative [22,23,24]. In this paper, in contrast, we restrict our considerations to the case g s > 0 by changing the boson-boson contact interaction g, so that phase separation does not take place.
The energy functional (10) was minimized with respect to η and d z /ℓ z for various parameter values, with our results summarized in Fig. 1. Without the fermion induced interaction, g ind = 0, the ground-state energy of the system is not bounded from below, and the strong trapping in z direction creates a local minimum in the energy landscape as a function of d z and η [ Fig.1 (a)]. For finite induced interactions g ind > 0, we find in contrast that the energy landscape is characterized by two minima, as shown in Fig. 1 (b), (c), and (d). In Fig. 1 is a Gaussian state with narrow width in the transverse x-y plane. The additional local minimum close to With increasing boson-fermion interaction g ind , though, the ground-state energy corresponding to the global minimum at η > 1 approaches that of the local minimum [ Fig. 1 (c)], and these minima eventually reach equal energies at a critical value of g ind . The new ground state past that point is characterized by the parameters η < 1 and d z ∼ ℓ z [ Fig. 1 (d)], that is, it is a wide Gaussian in x-y plane.
Experimentally one may either vary the boson-boson s-wave scattering length while keeping the boson-fermion scattering length constant, or vary g bf with constant s-wave interaction g. In a boson-fermion mixture of chromium isotopes, typically g ind ∼ 10 −3 is small, which corresponds to an energy landscape that resembles that of Fig. 1(b) and the broad Gaussian corresponds to a metastable state. Such a metastable state has been achieved experimentally in experiments by the Stuttgart group [8]. A system that offers the potential to reach the regime of Fig. 1(d) is provided by a mixture of bosonic 87 Rb and fermionic 40 K with the scattering length of rubidium atoms tuned close to zero. In that mixture, a zero-field scattering length a bf ≈ 250a 0 [21] gives g ind ∼ 0.2.

IV. TRANSITION TO A DENSITY WAVE
The excitation spectrum of condensates dominated by a dipolar interaction is predicted to exhibit a roton minimum [12,13]. As a consequence, a bosonic densitymodulated state in the x-y plane may arise as a local minimum, and it may actually have a lower energy than the metastable Gaussian state and for high enough bosonic densities [15]. This section discusses the transition to the appearance of such a state as a function of number of bosonic particles for various combinations of trap ratio.
We proceed by introducing the new variational wave function that describes a density-wave structure with triangular symmetry, where φ(x, y, z) is defined in Eq. (9), n is an integer, andk 0 = k 0 d z ≪ η = d z /d. Substituting this trial wave function (13) into Eq. (1), we find that the scaled excess energy of the density-modulated state relative to the Gaussian state is approximately given by where n d ≡ η 2 ℓ z /d z [25]. The normalization condition reads a 2 0 + 3(a 2 1 + a 2 2 + a 2 3 )/2 = 1, and the effective transverse potential is In evaluating Eq. (14) we kept only the first four terms n = 0, . . . , 3 of Eq. (13) as the energy converges at the transition point. The magnitude of the error in that approximate expression is estimated to be of the order of e = exp(−k 2 0 η 2 /4). In the subsequent calculations we consider values ofk 0 such that this error is less than or on the order of 10 −6 .
We numerically determine the variational parameters that minimize the excess energy ǫ(k 0 , a 1 , a 2 , a 3 ) as a function of g < for fixed n d and g 3d . These results are summarized in Fig. 2, which shows the critical number of bosonic particles N b such that ǫ has a minimum fork 0 = 0. By tuning the trap aspect ratio λ = ω ⊥ /ω z to higher values, the critical number of bosons N b required to achieve ǫ < 0, that is, a transition from a Gaussian to a densitywave metastable state with lower energy, is lowered as a result of an increased n d . In addition, we note that the value ofk 0 determined by the present variational calculation matches closely the position of roton minima at the excitation spectrum of the condensate as in Ref. [15]. We also checked the energy with density wave having square symmetry and find that the triangular one always has lower energy.

V. DENSITY WAVE STABILITY
In the previous section we determined the transition to the density-wave instability, assuming the ansatz (13). As already mentioned, this transition is related to the existence of a roton minimum in the excitation spectrum. However, assuming such a density wave does not provide an answer to the question of its stability against collapse. This is the issue that we address now, restrict- ing our considerations to the regime where the bosonfermion interaction strength is close to its zero-field value a bf = 70a 0 . This corresponds in the case of 52 Cr-53 Cr isotopes to g ind ∼ 10 −3 . This is the parameter regime characterized by a Gaussian metastable state, see Sec. II and Fig. 1(b).
Once the transition to the density-wave state has occurred the condensate can also be described as a superposition of shifted Gaussian wave functions within the two-dimensional Z ≡ x + iy plane, where j is the index of the lattice site, Z j the lattice vectors generating periodic density modulations, ξ the width of each density peak, M the total number of density peaks, and l the distance between neighboring density peaks. It is given by l = 2π/k 0 where k 0 is the position of roton minimum in the excitation spectrum of the condensate.
Using the results of Sec. III to investigate the stability of each Gaussian in Eq. (16), we now show that the presence of fermions substantially stabilizes the densitywave state, noting that a stable density wave should be characterized by non-zero values of ξ/d z and d z /ℓ z .
From Eq. (16), the total interaction energy is found to be [26] where we have assumed that ξ ≪ l, i.e., that neighboring Gaussian peaks in Eq. (16) have little overlap.
Here E < 1 , E > 1 , E < 2 , E > 2 , E < 3 and F have the same form as in the Appendix, but σ is now a function of (ξ, d z , θ), σ = d 2 z cos 2 θ + ξ 2 sin 2 θ. The function f (l, ξ, d z ) is responsible for the particular geometry of the densitymodulated state. Other terms, on the other hand, just arise from the energy of each individual Gaussian and we can thus apply the results of Sec. III to the present analysis.
Assuming a triangular crystal and including only the interaction between nearest neighbors, we have withk = kd z , the effective two-dimensional potential V eff (k) is defined in Eq. (15), and J 0 is the zeroth-order Bessel function of the first kind. We now discuss the minimum of the energy functional E int (ξ, d z ) as a function of the effective strength of the contact interaction g s , which is varied by changing g < , see Eq. (12). Without boson-fermion interaction, the interaction energy is a monotonically increasing function of ξ with E int (ξ → 0, d z ) → −∞. As a result each gaussian of the density wave Eq. (16) collapses. A typical example of the dependence of the interaction energy on ξ is shown as the dotted curve in Fig. 3(a). With non-zero boson-fermion interaction energy, in contrast, the interaction energy (17) exhibits a minimum at a finite ξ as illustrated by the solid curve in Fig. 3(a) f or d z = 1.8ℓ z and g < = 0.17. The existence of that minimum implies the stability of each Gaussian, that is, the stability of the wave function (16).
We can determine the dependence of the gaussian widths ξ on the s-wave boson-boson scattering length by changing g < . To do this we first minimize Eq. (17) as a function of g < , treating d z and ξ as variational parameters. Figure 3 (b) shows the resulting value ξ/ℓ z as a function of g < for λ = 0.25, N f = 1000 and ℓ z = 0.15µm. We observe that ξ decreases for smaller values of the contact interaction g < , and for g < 0.14, the minimum-energy state corresponds to the collapsed state, ξ = 0. Below that point the density wave is unstable. This behavior can be understood from the effective potential by first integrating out the z direction to obtain the effective potential (ℓ z /d z ) 3 V eff (k ⊥ d z ), where V eff is given by Eq. (15) and k 2 ⊥ = k 2 x + k 2 y . The width d z of the Gaussians is determined by the position of the minimum in Eq. (17). For g < 0.14, V eff (k ⊥ d z ) is at-tractive for all high momenta, but at the critical value g < ≈ 0.14, it approaches zero as k ⊥ → ∞, as shown by the dashed curve in Fig. 4. Hence the energy is always minimized when the wave function has zero width in the x-y plane, i.e., ξ → 0. For g < > 0.14 the effective potential is repulsive both at low and high momenta, and attractive in-between as illustrated by the solid curve in Fig. 4. Subsequently the energy is minimized for a finite value of ξ.
To find the range of parameter space characterized by the appearance of density-wave states, we need to consider, in addition to stability arguments, the transition point discussed in Section III. Table 1 summarizes the range of s-wave scattering lengths and critical numbers of 52 Cr atoms necessary to be inside stable density-wave regime.

VI. CONCLUSION
In summary, we have analyzed the stability of a dipolar bosonic condensate mixed with non-interacting fermions in a pancake trap at T = 0. We found that the fermions help stabilize the condensate for a significant range of boson-boson and boson-fermion interaction strengths. We then investigated the transition of the system from a Gaussian-like to the density-wave ground state as a function of number of bosons, strength of the contact interaction, and trap aspect ratio. Our central result is the use of a variational ansatz to show that while in a purely bosonic system the density-wave state is always unstable it can be stabilized by the admixture of even a small boson-fermion interaction.
In particular, this study leads us to the conclusion that a pancake-shaped 87 Rb-40 K mixture, which has a large boson-fermion s-wave scattering length, should be absolutely stable in the dipole dominated regime. By tuning the s-wave scattering length it is possible to reach a situation characterized by the appearance of a roton instability in the excitation spectrum, leading to the existence of a stable density-wave state. However, due to the small dipole moment of rubidium atoms the transition to this stable density-wave regime needs a substantial number of atoms, of the order of 10 6 . Future work will discuss the effect of the dipolar nature of the fermionic isotopes on the condensate -with chromium atoms in mind -, the existence and stability of density waves, as well as possible extensions to rotating systems.
For k < 2k f and in spherical coordinate, the interaction energy including the s-wave and induced interaction is given by where g < is defined by Eq. (11). This integral contain contribution from both the local and nonlocal part of the induced interaction. After integrating over k and rearranging the terms we obtain the expression Here σ = d 2 z cos 2 θ + d 2 sin 2 θ is a function of (θ, d z , d), erf(y), erfc(y), Γ, and γ denote the error function, complementary error function, gamma function, and incomplete gamma function, respectively.
Similarly, the contribution for k > 2k f to the interaction energy that contains both the s-wave boson-boson interaction and the nonlocal induced interaction is where g > defined in Eq. (11). In this equation the first term in the bracket stems from the s-wave boson-boson interaction and the last term from attractive nonlocal part of the induced interaction. Again after evaluating the integral over k we get The total interaction energy is the sum of the contributions in Eq. (A2), (A4), (A6) and is given in Eq. (10). In this present paper we took n = 1, . . . , 3 in the series in Eqs. (A5), (A7) as the energy E g in Eq. (10) converges with this inclusion.