Rashbon condensation in a Bose gas with Rashba spin-orbit coupling

We show that in a two-component Bose gas with Rashba spin-orbit coupling (SOC) two atoms can form bound states (Rashbons) with any intra-species scattering length. At zero center-of-mass momentum there are two degenerate Rashbons due to time-reversal symmetry, but the degeneracy is lifted at finite in-plane momentum with two different effective masses. A stable Rashbon condensation can be created in a dilute system with weakly attractive intra-species and repulsive inter-species interactions. The critical temperature of Rashbon condensation is about six times smaller than the BEC transition temperature of an ideal Bose gas. Due Rashba SOC, excitations in the Rashbon condensation phase are anisotropic in momentum space.

Introduction. In recent several years one major progress in ultracold atom physics was the realization of spin-orbit coupling (SOC) in Bose-Einstein condensation [1] and ultracold Fermi gases [2,3]. In contrast to the intrinsic SOC of electrons in atoms, SOC in neutral atoms refers to the coupling between spin and center-of-mass momentum of atoms. Bose gases with SOC have displayed very rich phase diagrams. In experiments where the SOC is an equal-weight combination of Rashba and Dresselhaus SOCs, several phase transitions, including magnetic to spin-mixed states and normal to magnetics states, were observed in Bose gases [1,4,6], consistent with theoretical studies [5]. In a uniform Bose gas with Rashba SOC, competition between plane-wave and spin-stripe phases were predicted [7][8][9].
New phases, such as half-vortex and skyrmion-lattice phases, were predicted in trapped systems [10][11][12][13]. In this paper, we are going to show that a stable pairing state can appear in a two-component Bose gas with Rashba SOC.
In contrast to the well-observed BCS-BEC crossover in Fermi gases [14], pairing state of Bose gas is an exotic but never observed phenomena. Although Feshbach molecules of Bose atoms have been created in experiments [15][16][17], rapid particle-loss rate due to strong inelastic collision near the resonance severely limits the molecule lifetime, making it impossible to reach a condensed state. Despite experimental difficulties, the pairing state of a Bose gase has been explored theoretically [18][19][20], but it was found unstable even with a weak attractive interaction away from the resonance [21][22][23].
The successful creation of SOC offers a new opportunity to realize the pairing state in a Bose gas. As in the fermion case, Rashba SOC changes the atom density of states (DOS) which has strong effects on pairing and produces Rashbon [24], the two-body bound state with negative scattering length. In the following, we first study the two-body bound states of bosons with Rashba SOC. We find that at zero center-of-mass momentum bound states (Rashbons) can exist with arbitrary intra-species scattering length, while SOC has virtually no effect on the bound state created by the inter-species interaction. Next, we study the possibility of Rashbon condensation in a Bose gas with Rashba SOC. We find that Rashbon condensation can be stabilized by a repulsive inter-species interaction. The Rashbon condensation may be realized in a dilute Bose gas with weak intra-species attraction and inter-species repulsion. One signature of this phase is the anisotropic excitation spectrum.
The Rashbon transition temperature is about six times smaller than the ideal BEC temperature.
Model. We consider a two-component Bose gas with Rashba SOC, described by the c kσ represents the annihilation operator of a boson with wave-vector k and spin σ, S(k ⊥ ) = 2 κ(k x + ik y )/m, κ is the strength of Rashba SOC, k ⊥ is the projection of k in the x-y plane, and ǫ k = 2 k 2 /2m. The single-atom HamiltonianĤ 0 can be easily diagonalized, yielding helical excitations with energies given by ξ k± = ǫ k ± 2 κk ⊥ /m. The s-wave interaction between atoms is given bŷ where V is the volume. The inter-species coupling constants satisfy g ↑↓ = g ↓↑ = 4π 2 a ′ /m, where a ′ is the inter-species scattering length. In the following, we consider only the symmetric case with two identical intra-species coupling constants, g ↑↑ = g ↓↓ = 4π 2 a/m, where a is the intra-species scattering length. In this symmetric case, the system is invariant under Two-body bound states. We first study two-body bound states described by the wave where ψ σσ ′ (k, k ′ ) = ψ σ ′ σ (k ′ , k) is a coefficient. By solving the eigenvalue problem, H|Ψ q = E q |Ψ q , we can obtain wavefunction and eigenenergy of bound states. At q = 0, the eigenequation can be further written as where ψ ′ k is a four component-vector given by M k is the matrix of eigenenergy minus kinetic energy and SOC, E k = E 0 − 2ǫ k , and G is the matrix of coupling constants, Define the vector Q = G k ψ ′ k /V , from Eq. (4) we can obtain an equation for Q, Using the symmetry S(k ⊥ ) = −S(−k ⊥ ), we find that where . Eq. (7) has three different solutions, two intra-species bound states with Q 3 = Q 4 = 0 and one inter-species bound state with Q 1 = Q 2 = 0 and Q 3 = Q 4 . Due to the symmetry ψ ↓↑ (k, −k) = ψ ↑↓ (−k, k), the solutions always satisfy Q 3 = Q 4 which is also guaranteed by the G-matrix elements G 34 = G 43 = G 33 = G 44 in Eq. (7). The G-matrix can also be chosen as a diagonal matrix with G 33 = G 44 = g ↑↓ , but then the unphysical solution with Q 3 = Q 4 has to be taken out by hand.
For the two degenerate bound states at q = 0 created by the intra-species interaction, their eigenenergy E 0 is determined from the equation where the first r.h.s. term is due to T -matrix correction. Eq. (9) shows that these bound states are Rashbons which can exist with any intra-species interaction, whereas in a simple Bose gas without SOC two-body bound states only exists in the repulsive regime. The binding energy defined by E b = −E 0 − 2ǫ κ is presented in Fig. 1(a) where ǫ κ = 2 κ 2 /2m.
When the intra-species interaction is tuned from attraction to repulsion, the binding energy monotonically increases with 1/(κa). We find that in the limit of κa → 0 − , the binding energy has the asymptotic form 0.132ǫ κ , much smaller than that in the fermion case [25]; when κa → 0 + , E b → 2 /(ma 2 ), recovering the result of a dilute Bose gas without SOC.
The degeneracy of Rashbons at q = 0 is protected by time-reversal symmetry. One Rashbon wavefunctions is given by where N is a normalization constant. The other Rashbon wavefunction can be obtained by . Rashbon appearance in the attractive regime is due to the increase in atom DOS at low energies by SOC. The density of state of the lower helicity excitation ξ k− is a constant at energy minimum ξ k− = −ǫ κ for k ⊥ = κ and k z = 0, which leads to an infrared divergence at zero binding energy on r.s.h. of Eq. (9) and consequently Rashbon appearance in the attractive regime. Rashbons in the weakly attractive regime may be helpful for experimental observation. Since the system is far away from resonance, the particle loss rate due to inelastic collision may be suppressed.
At q ⊥ = 0, the bound-state eigenergy problem cannot be reduced to a simple equation.
We numerically solve for bound state energies, and find that the two Rashbons have two different effective masses, as shown in Fig. 1 (b). The lift of Rashbon degeneracy is not surprising, because two Rashbons are no longer connected by time-reversal symmetry at finite q ⊥ and the Rashbon degeneracy is no longer protected by time-reversal symmetry.
The two Rashbon effective masses behave differently with the intra-species scattering length a. The bigger effective mass m * + reaches maximum at resonance, while the smaller effective mass m * − decreases monotonically with 1/(κa). In the limit a → 0 − , we obtain m * + = 8m and m * − = 8m/3; at resonance, m * + = 9.29m and m * − = 2.36m; in the limit a → 0 + , both effective masses recover the results without SOC, m * ± → 2m. When the in-plane momentum q ⊥ exceeds a critical value q c , the Rashbon dissociates into excited atoms. We find that the critical wavevector q c is different for different Rashbons, approximately satisfying the condition for Rashbon dissociation in the effective-mass approximation, E 0 + 2 q 2 c± /(2m * ± ) ≈ −2ǫ κ . The critical momenta vanish in the limit of weakly attractive interaction a → 0 − , and diverge in the opposite limit a → 0 + .
For the bound state created by the inter-species interaction, its eigenenergy at q = 0 is which is the same as that without SOC and gives the same result E b = 2 /(ma ′ 2 ), whereas in the fermion case the inter-species bound state is strongly affected by SOC [25]. The wave function of this bound state is also the same as that without SOC, where N ′ is a normalization factor.
The qualitative difference between Rashbon and the inter-species bound state can be explained in terms of symmetries of their wavefunctions as given in Eq. (10) and (12). introduced. Rashbon condensation is not directly coupled to the condensation of interspecies bound states. In the dilute limit with weakly attractive intra-species interaction and repulsive inter-species interaction, the Rashbon binding energy is much smaller than the binding energy of the inter-species bound state. In the following, we consider the system with only Rashbon condensation and focus on the spin-balanced case, g ↑↑ = g ↓↓ and |∆ ↑↑ | = |∆ ↓↓ | = ∆. In general there may be a phase difference between ∆ ↑↑ and ∆ ↓↓ . Without losing generality we define ∆ ↑↑ = e iθ ∆, ∆ ↓↓ = e −iθ ∆ and ∆ > 0. The mean-field Hamiltonian of the Rashbon condensation phase is given by where B + k is the field operator with four components for κ/n 1/3 = 60, the order parameter ∆ = 0.0075ǫ κ is much smaller than the binding energy density of each spin component, the matrix H k is given by ξ k = ǫ k − µ + 2g ↑↑ n + g ↑↓ n, and µ is chemical potential.
The mean-field Hamiltonian Eq. (14) can be diagonalized by generalized Bogoliubov transformation. The single-particle excitations form two branches with excitation energies given by where ϕ k = φ k + θ and φ k = arg(k x + ik y ). The pairing order parameters and density can be obtained self-consistently, yielding the following equations at zero temperature We numerically solve Eq. (16) and find that the mean-field solution always exist in the dilute limit n → 0, as shown in Fig. 2.
In Rashbon condensation, quasi-particle excitation energies given in Eq. (15) are anisotropic, dependent on the angle ϕ k = φ k + θ. This anisotropy is stronger at low energies when k ⊥ is near κ, as shown in Fig. 3(a). At higher energies, the anisotropy becomes weaker and eventually disappears. This anisotropic effect is caused by the coupling between pairing order parameters ∆ ↑↑ and ∆ ↓↓ due to SOC. For a spin-up atom with wavevector k, SOC can flip its spin down with a phase φ k . This phase becomes φ k + π for the spin-up atom with opposite wave-vector −k. These two spin-flips can turn an atom pair from total spin-up to total spin-down states with phase 2φ k + π. If 2φ k + π + 2θ = 2lπ where l is an integer, spin-flips are encouraged and the quasi-particle energy ε k− is at minimum. If 2φ k + 2θ = 2lπ, spin-flips are discouraged and the quasi-particle energy is at maximum. As shown in Fig. 3(b), the quasi-particle energy ε k− shows a periodic behavior as a function of φ k with period π.
In the following, we focus on Rashbon condensation in the dilute limit with attractive intra-species interaction, κ ≫ n 1/3 and (−a) −1 ≫ n 1/3 . Since in the dilute limit the distance between Rashbons is the largest length scale, the structure of Rashbons is not affected by the weak interaction between Rashbons, which is very similar to the BEC limit of BEC-BCS crossover in Fermi gases. In this limit, Eq. (16) can be solved analytically, and we find that the order parameter ∆ is much smaller than Rashbon binding energy, The attractive intra-species interaction tends to make the system unstable. If the Rashbon condensation is stable, the positive compressibility condition ∂µ/∂n > 0 must be satisfied.
We find that in the dilute limit this stability condition is given by κ(a ′ +2a) > 3/2. Therefore a repulsive inter-species interaction with κ > 3/(2a ′ + 4a) ≫ n 1/3 is required to stabilize the Rashbon condensation in a dilute Bose gas with Rashba SOC.
In the Rashbon condensation phase, in addition to single-particle excitations, there are also pair excitations. At the transition temperature T c of Rashbon condensation, pair excitations are quadratically dispersed. In the dilute limit with attractive intra-species interaction, they have effective masses approximately as same as those of Rashbons in vacuum. Since in this limit the Rashbon binding energy is much bigger than k B T c , single-particle excitations can be neglected at T c , and only excited Rashbons contribute to the density at T c , where s = ±, E q± ≈ E 0 + 2 q 2 z /(4m) + 2 q 2 ⊥ /(2m * ± ) are Rashbon energies in the effective mass approximation, and ′ denotes the summation over q for |q ⊥ | ≤ q c± . From Eq. (17), we obtain the transition temperature where T a = 2πζ − 2 3 ( 3 2 ) 2 n 2/3 /(k B m) is the critical temperature of an ideal Bose gas and ζ(x) is the Riemann zeta function. Eq. (18) shows that the transition temperature of Rashbon condensation T c in the dilute limit is about six times smaller than the BEC transition temperature of an ideal Bose gas.
In current experiments in 87 Rb, the strength of Rashba SOC is limited by the wavelength of the Raman laser λ = 804.1nm, κ ≤ 7.8 × 10 6 m −1 [1]. With background intra-species scattering length a bg = 100a 0 and density of the order of 10 13 cm −3 [26], the dilute region of Rashbon condensation is hardly reachable. With the new proposal to generate Rashba SOC [27,28], if κ can be enhanced to 2 × 10 8 m −1 and scattering lengths can be tuned to a = −95a 0 and a ′ > 330a 0 , Rashbon condensation may be observed around 29nK with n = 10 13 cm −3 in 87 Rb.
Discussion and conclusion. We have shown that Rashbon condensation can be mechanically stable in a dilute Bose gas with Rashba SOC and weakly attractive intra-species interaction. In this dilute region, we expect that the particle loss rate is suppressed because of its density dependence. As shown in experiments on 85 Rb in the dilute region [15,17], the loss rate of Feshbach molecules is much smaller than the molecule binding energy. Now with the help of Rashba SOC, the Rashbon binding energy is exponentially small, and the lifetime of dilute Rashbon condensation is expected to be long enough for experimental observations.
There are a lot of interesting questions to be answered about Rashbon condensation.
Collective excitations in this phase are worth to explore. Another important question is whether or not at a higher density there is a quantum phase transition between Rashbon condensation and mixture of atom and Rashbon condensates. We plan to address these issues in future studies.
In summary, we find that two Bose atoms with Rashba SOC can form a Rashbon with any intra-species interaction. In contrast, the bound state created by the inter-species interaction is not affected by SOC. At zero center-of-mass momentum there are two degenerate Rashbons with the degeneracy protected by time-reversal symmetry. The degeneracy is lifted at finite in-plane momentum with two different effective masses. We explore the possibility of Rashbon condensation in a dilute Bose gas with Rashba SOC and attractive intra-species interaction. We find that Rashbon condensation can be stabilized by a repulsive inter-species interaction. In Rashbon condensation, the single-particle excitation energy is anisotropic, due to coupling between pairing order parameters by SOC. The transition temperature of Rashbon condensation is about six times smaller than that of BEC in an ideal Bose gas.
Acknowledgement. We would like to thank Z. Q. Yu, W. Zhang, and T.-L. Ho for