Rayleigh-Taylor instability and mushroom-pattern formation in a two-component Bose-Einstein condensate

The Rayleigh-Taylor instability at the interface in an immiscible two-component Bose-Einstein condensate is investigated using the mean-field and Bogoliubov theories. Rayleigh-Taylor fingers are found to grow from the interface and mushroom patterns are formed. Quantized vortex rings and vortex lines are then generated around the mushrooms. The Rayleigh-Taylor instability and mushroom-pattern formation can be observed in a trapped system.


I. INTRODUCTION
When a layer of a lighter fluid lies under that of a heavier fluid, the translation symmetry on the interface is spontaneously broken and the interface is modulated due to the Rayleigh-Taylor instability (RTI) [1,2,3,4]. Waves on the interface then grow into complicated patterns with mushroom shapes [3,5]. The RTI plays crucial roles in a variety of nonequilibrium phenomena, ranging from convection of water in a kettle to supernova explosions [6].
In the present paper, we investigate the RTI and ensuing dynamics in a phase-separated two-component Bose-Einstein condensate (BEC). Recently, there has been a growing interest in the interface properties of such BECs. For instance, the Kelvin-Helmholtz instability [4,7,8], which occurs at the interface between two fluids with a relative velocity, has been observed in a 3 He superfluid system [9,10]. The Kelvin-Helmholtz instability is also predicted in a two-component BEC of atomic gases [11]. When a magnetic field is applied to a magnetic fluid (a colloidal suspension of fine magnetic particles), the surface is deformed by the Rosensweig instability [12] and grows into a pattern of crests. Such a surface phenomenon can be theoretically shown to occur also at the interface in a two-component BEC with a dipole-dipole interaction [13]. Analytical expressions of the interface tension in a two-component BEC have been derived in Refs. [14,15,16].
The present paper reveals that the RTI emerges at the interface between two immiscible BECs that are pushed toward each other by, e.g., a magnetic-field gradient. We first consider an ideal flat interface, and numerically show that the interface becomes deformed by the RTI to grow into the well-known mushroom pattern. The significant difference between this phenomenon and that in classical fluids is that the vortices under the caps of the mushrooms are quantized. Thus, in three dimensions (3D), quantized vortex rings are generated around the mushrooms. Bogoliubov analysis shows that the excitation spectrum of the interface modes closely resembles that for classical fluids. We also propose a realistic BEC system in a harmonic trap and show that the RTI can be observed experimentally.
This paper is organized as follows. Section II A numerically formulates the problem and demonstrates the RTI and ensuing dynamics for an ideal system. Section II B gives the Bogoliubov spectrum of the interface modes. Section III analyzes a trapped system. Section IV provides conclusions to the study.

II. RAYLEIGH-TAYLOR INSTABILITY AT AN IDEAL INTERFACE
A. Mean-field dynamics The system considered here is a zero-temperature twocomponent BEC described by the Gross-Pitaevskii (GP) equations, where ψ j , m j , and V j are the macroscopic wave function, atomic mass, and external potential, respectively, for the jth (j = 1, 2) component. The interaction parameters g jj ′ are given by where a jj ′ and m jj ′ are the s-wave scattering length and reduced mass, respectively, between components j and j ′ . We assume that the interaction parameters satisfy the phase-separation condition, For concreteness, we employ the hyperfine states |F, m F = |1, 1 and |1, −1 of a 87 Rb atom for components 1 and 2, respectively. According to the scattering lengths reported in Ref. [17], we have a 11 = a 22 = 100.4a B and a 12 = 101.3a B with a B being the Bohr radius, and then the condition for the phase separation in Eq. (3) is satisfied. The spin-exchange dynamics |1, 1 , |1, −1 → |1, 0 , |1, 0 can be suppressed by, e.g., the microwave-induced quadratic Zeeman effect [18], which can lift the energy of the |1, 0 state. When the hyperfine spins are parallel to the magnetic field B, the magnetic-field gradient exerts forces ±µ B ∇|B|/2 on the two components in opposite directions, where µ B is the Bohr magneton.
In order to clearly demonstrate the RTI, we first consider a 3D system without a trapping potential. We prepare a quasi-stationary state with a field gradient B ′ ≡ dB/dz > 0, which is uniform in the x-y direction. Components 1 and 2 are located in the z < 0 and z > 0 regions, respectively, and their flat interface is located at the z = 0 plane. The Stern-Gerlach force pushes the two components toward each other. We add small random seeds to the initial state, which numerically breaks the translation symmetry in the x-y direction and triggers the RTI. We assume periodic boundary conditions in the x and y directions. Figure 1 shows the time evolution of the density and phase profiles on a plane perpendicular to the initial flat interface. The two components are pushed toward each other (arrows in Fig. 1), and the interface starts to modulate due to the RTI (second row of Fig. 1). Subsequently, the amplitude of the wave on the interface grows to form the mushroom shapes (third row of Fig. 1). We can see that there are quantized vortices under the caps of the mushrooms (red circle in Fig. 1). After that, the vortices enter into the mushroom patterns, giving rise to complicated dynamics (fourth row of Fig. 1). Figure 2 (a) shows the isodensity surface of component 1 at t = 310 ms. We can see that Rayleigh-Taylor fingers and mushroom patterns flourish at the interface. One of the mushroom shapes is magnified in Fig. 2 (b), in which the topological defect under the cap of the mushroom is indicated by the red ring. The vortex rings are generated by the upward flow of the atoms around the center and downward flow at the periphery of the cap of the mushroom shape.

B. Bogoliubov analysis
Before studying the Bogoliubov spectrum, we recall the dispersion relation for the RTI in classical fluids. Let us consider a situation in which inviscid incompressible fluids produce a flat interface perpendicular to the direction of gravity, where the densities of the lower and upper fluids are ρ 1 and ρ 2 with ρ 1 < ρ 2 . From the linear analysis, the interface mode has a dispersion relation of [4] where g is the gravitational constant and σ is an interface-tension coefficient. If gravity is absent, ω is real for all k and proportional to k 3/2 . In the presence of the gravitational force, there is always a range of k in which ω is imaginary, and hence the interface is dynamically unstable. The range of instability is given by and the most unstable wave number k 0 is We perform the Bogoliubov analysis by decomposing the wave function as where f j (z) is a quasi-stationary state with a flat interface around the z = 0 plane and µ j is the chemical potential. For the case of Fig. 1, f 1 (z) and f 2 (z) are symmetric with respect to the interface on the x-y plane and µ 1 = µ 2 ≡ µ because g 11 = g 22 . The small deviation φ j (r) from the ground state is written as where the wave vector is assumed to be in the x direction without loss of generality. The mode functions u j,k (z) and v j,k (z) satisfy the Bogoliubov-de Gennes equations, where (j, j ′ ) = (1, 2) and (2, 1), m is the mass of 87 Rb, , and f j is assumed to be real. Figure 3 shows the Bogoliubov spectrum obtained by numerically diagonalizing Eq. (9). In Fig. 3, we plot only the lowest mode, which corresponds to the mode localized near the interface. The second lowest mode has much larger energy. When the field gradient B ′ is zero, the excitation energy is real for all wavelengths λ (red line in Fig. 3). The logarithmic plot in the inset of Fig. 3 indicates that ω is proportional to k 3/2 , which is in agreement with Eq. (4) with g = 0. In the presence of the field gradient B ′ , the Bogoliubov spectrum becomes imaginary for λ −1 smaller than a critical value λ −1 c , which is λ −1 c ≃ 2.1 × 10 4 m −1 for B ′ = 5 mG/cm and λ −1 c ≃ 3.8 × 10 4 m −1 for B ′ = 15 mG/cm. The longwavelength modes are always unstable for B ′ = 0, as in the RTI in classical fluids.
The analytic expression of the interface tension in a phase-separated two-component BEC has been derived in Refs. [14,15,16]. For a 12 /a − 1 ≪ 1, where a ≡ a 11 = a 22 , the interface tension σ has the form, where n is the atom density. Using the characteristic density 4 × 10 14 cm −3 for n and substituting Eq.  Fig. 3. Using Eqs. (6) and (10), the most unstable wavelength is estimated to be ≃ 40 µm for B ′ = 20 mG/cm, which is in qualitative agreement with the wavelength of the interface modulation in Fig. 1.

III. DYNAMICS IN A HARMONIC TRAP
We next consider a system confined in an axisymmetric harmonic potential V trap = m[ω 2 xz (x 2 + z 2 ) + ω 2 y y 2 ]/2. The radial and axial trap frequencies are ω xz = 2π × 100 Hz and ω z = 2π × 5 kHz, and the potential has a tight pancake shape. The initial state is the ground state of the GP equation for B ′ = 0, in which the interface is parallel to the x axis and components 1 and 2 (the hyperfine states |1, 1 and |1, −1 of 87 Rb) are localized in the z < 0 and z > 0 regions, respectively. A small random noise is added to the initial state to trigger the dynamical instability. At t = 0, the field gradient B ′ = 1.3 G/cm is applied in the z direction. Figure 4 shows the time evolution of the column density |ψ j | 2 dy of each component, obtained by 3D simulation of the GP equation. At t ≃ 80 ms, the interface starts to modulate with a wavelength ≃ 8-10 µm due to the RTI. This wavelength is in qualitative agreement with ≃ 8.7 µm estimated using Eqs. (6) and (10) with the peak density n ≃ 3.2 × 10 15 cm −3 . The modulation on the interface then develops into the mushroom pat-terns at t ≃ 100 ms. After that, the system evolves in a complicated manner and eventually components 1 and 2 are interchanged, localizing in the z > 0 and z < 0 regions, respectively.
In Fig. 4, one can see that both components invade around to the back of each other and the periphery of the condensate is disturbed. This is because the repulsive interaction between the two components is weak in the low-density periphery region, and the two components pass through each other. For a spherical trap, the passthrough phenomenon is more severe and a much larger number of atoms (> 10 9 ) is needed to clearly realize the RTI. If we use a square-well potential produced by, e.g., a flat-top beam [19], the atomic density becomes more uniform on the interface, which suppresses the pass-through, realizing an ideal RTI as discussed in Sec. II A. Increasing the inter-component repulsion [20] using the Feshbach resonance can also suppress the pass-through.

IV. CONCLUSIONS
We have shown that the RTI and mushroom-pattern formation occur in a two-component phase-separated BEC, as in classical fluids. The significant difference between the quantum RTI and the classical RTI is that quantized vortex lines and vortex rings are formed under the caps of the mushrooms. The Bogoliubov analysis showed that the excitation spectra of the interface modes are very similar to those of classical fluids. We proposed a possible experiment to observe the phenomena in a realistic trapped system.
Various phenomena related to fluid instabilities may be reproduced in BECs with renewed interest. For example, splashing of drops and crown formation [21] is considered to be related to instabilities which include the RTI. Droplet formation by the Plateau-Rayleigh instability [22] in BECs is also an interesting future problem.