Validity of single-channel model for a spin-orbit coupled atomic Fermi gas near Feshbach resonances
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Validity of single-channel model for a spin-orbit coupled atomic Fermi gas nearFeshbach resonances
Jing-Xin Cui , , Xia-Ji Liu , Gui Lu Long , , and Hui Hu ∗ Department of Physics, Tsinghua University, Beijing 100084, China ARC Centre of Excellence for Quantum-Atom Optics,Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology, Melbourne 3122, Australia Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China (Dated: July 23, 2018)We theoretically investigate a Rashba spin-orbit coupled Fermi gas near Feshbach resonances,by using mean-field theory and a two-channel model that takes into account explicitly Feshbachmolecules in the close channel. In the absence of spin-orbit coupling, when the channel coupling g between the closed and open channels is strong, it is widely accepted that the two-channel modelis equivalent to a single-channel model that excludes Feshbach molecules. This is the so-calledbroad resonance limit, which is well-satisfied by ultracold atomic Fermi gases of Li atoms and Katoms in current experiments. Here, with Rashba spin-orbit coupling we find that the condition forequivalence becomes much more stringent. As a result, the single-channel model may already beinsufficient to describe properly an atomic Fermi gas of K atoms at a moderate spin-orbit coupling.We determine a characteristic channel coupling strength g c as a function of the spin-orbit couplingstrength, above which the single-channel and two-channel models are approximately equivalent. Wealso find that for narrow resonance with small channel coupling, the pairing gap and molecularfraction is strongly suppressed by SO coupling. Our results can be readily tested in K atoms byusing optical molecular spectroscopy.
I. INTRODUCTION
As a realization of non-abelian gauge fields in neutralcold atoms [1–5], spin-orbit (SO) coupled atomic gaseshave attracted a lot of attentions in recent years. TheSO coupled bosonic gas of Rb atoms was first achievedby Spielman’s group at National Institute of Standardsand Technology (NIST) in early 2011 [1]. The SO cou-pled atomic Fermi gas has also been realized most re-cently at Shanxi University [4] and at Massachusetts In-stitute of Technology (MIT) [5] with K and Li atoms,respectively. These novel atomic gases have many in-teresting properties inherent to spin-orbit coupling, andhave potential applications in future quantum technol-ogy. A well-known example is the emulation of the long-sought topological superfluids and Majorana fermions [6–8], which lie at the heart of topological quantum infor-mation and computation [9, 10].Most of previous theoretical studies on SO coupledFermi gases are based on a single-channel model [6, 11–31]. In this model, the interaction between atoms is de-scribed by a single parameter, i.e., the s -wave scatteringlength a s . The scattering length can experimentally betuned by using Feshbach resonances. As a result, in theabsence of SO coupling, the Fermi gas can cross froma Bose-Einstein condensate (BEC) over to a Bardeen-Cooper-Schrieffer (BCS) superfluid [32], when the scat-tering length changes from positive to negative values.With SO coupling, the picture of BEC-BCS crossover ∗ Electronic address: [email protected] may be qualitatively altered. For example, for a partic-ular Rashba-type SO coupling, a new two-body boundstate - referred to as rashbon - is formed [11–14]. Byincreasing the SO coupling strength, the system maychange from a BCS superfluid to a BEC of rashbons,even on the BCS side with a negative scattering a s < [12–14]. The pairing gap of this system will be signifi-cantly enhanced due to the increased density of state atthe Fermi surface [14]. An anisotropic superfluid due tothe Rashba SO coupling has also been predicted [13].A more realistic and complete description of ultracoldatomic Fermi gases near Feshbach resonances, however,should be the two-channel model, which includes bothatoms in the open channel and Feshbach molecules inthe closed channel [33]. In this model, in addition to thebackground s -wave scattering length a bg between atoms,two other parameters are used in order to fully describethe interaction. These are the detuning energy of Fes-hbach molecules ν and the channel coupling strength g between molecules and Fermi atoms. Therefore, the in-teraction of the system consists of two parts. The non-resonant part is the contact interaction between atomswith the strength determined by the background scatter-ing length, while the resonant interaction is induced bythe coupling between molecules and atoms. Near Fesh-bach resonances without SO coupling, it is known thatthe single-channel and two-channel models are essentiallyequivalent when the channel coupling strength g is largeenough [34, 35]. This is the so-called broad resonancecondition, satisfied by the Fermi gases of K and Liatoms, which are so far the two main systems used in thecold-atom laboratory.In this paper, we aim to examine the equivalence ofthe single-channel and two-channel models for a RashbaSO coupled Fermi gas near Feshbach resonances. Thisis by no means obvious, as fermionic pairing is notablyaffected by SO coupling at the BEC-BCS crossover. Weuse mean-field theory and focus on the most interestingresonant limit. Our results show that in the presenceof SO coupling, the broad resonance condition is muchmore difficult to achieve. As a result, for an ultracoldatomic Fermi gas of K atoms, which is known to bewell described by the single-channel model without SOcoupling, we may have to use a two-channel model al-ready at a moderate SO coupling strength.Our paper is organized as follows. In the next section(Sec. II), we introduce the model Hamiltonian. In Sec.III, we diagonalize the Hamiltonian by using mean-fieldtheory to obtain the grand thermodynamic potential andsolve the resulting coupled mean-field equations. In Sec.IV, we discuss the equivalence between the single-channeland two-channel models. In Sec. V, we show how to testexperimentally the difference between the two models,by using optical molecular spectroscopy [36]. Finally, wesummarize in Sec. VI.
II. MODEL HAMILTONIAN
We consider a three-dimensional (3D) resonantly-interacting atomic Fermi gas with Rashba-type SO cou-pling, described by the two-channel model Hamiltonian, H = H SO + H m + H I , (1)where H SO and H m stand for the non-interacting Hamil-tonian of SO coupled atoms in the open channel andof Feshbach molecules in the closed channel, respec-tively. The interaction Hamiltonian H I = H am + H aa includes both the atom-molecule coupling between thetwo channels ( H am ) and the background interaction be-tween open-channel atoms ( H aa ).For atoms, we take the following single-particle RashbaSO Hamiltonian, H SO = ~ k m + ~ m λ k ⊥ · σ ⊥ , (2)where k ⊥ ≡ ( k x , k y ) and σ ⊥ ≡ ( σ x , σ y ) are respectivelythe in-plane momentum and in-plane Pauli matrix, and λ is the Rashba SO coupling strength. Note that, thestandard representation of the Rashba SO coupling isgiven by λ ( k y σ x − k x σ y ) [13]. Here, for convenience wehave performed a spin-rotation to rewrite the Rashbaterm into a slightly different but fully equivalent form λ ( k x σ x + k y σ y ) [14]. In the second quantized form, H SO = X k σ ǫ k a † k σ a k σ + ~ λk ⊥ m (cid:16) e − iϕ k a † k ↑ a k ↓ + H.c. (cid:17) , (3) where a † k σ is the creation operator for atoms with mo-mentum k in the spin state σ , ǫ k ≡ ~ k / (2 m ) and ϕ k ≡ arg( k x + ik y ) . To diagonalize this single-particleHamiltonian, we introduce the field operators in the he-licity basis labeled by “ ± ”, (cid:18) h k + h k − (cid:19) = 1 √ (cid:18) e − iϕ k e iϕ k − (cid:19) (cid:18) a k ↑ a k ↓ (cid:19) , (4)with which the single-particle Rashba Hamiltonian be-comes diagonal, H SO = X k (cid:16) ǫ k + h † k + h k + + ǫ k − h † k − h k − (cid:17) . (5)Note that, in the helicity basis, the single-particle dis-persion relation now breaks into two branches: ǫ k + = ǫ k + ~ λk ⊥ / (2 m ) for the upper branch and ǫ k − = ǫ k − ~ λk ⊥ / (2 m ) for the lower branch. To describe Feshbachmolecules, we use annihilation operators b q . The energyof molecules is denoted as ν , which after renormaliza-tion [33] is related to the detuning energy from thresholdof Feshbach resonance B , i.e., ν = ∆ µ ( B − B ) , where ∆ µ ≡ µ a − µ m is the magnetic moment difference be-tween the atomic ( µ a ) and bound molecular state ( µ m )[33]. The Hamiltonian of Feshbach molecules may bewritten as, H m = 2 ν X q b † q b q . (6)Finally, the interaction Hamiltonian is given by H I = H aa + H am , where H aa = U bg X kk ′ q a † q / k ↑ a † q / − k ↓ a q / − k ′ ↓ a q / k ′ ↑ (7)is the non-resonant interaction between atoms, withstrength given by the background s -wave scatteringlength after renormalization [33], U bg = 4 π ~ a bg /m , and H am = g X kq (cid:2) b † q a q / k ↑ a q / − k ↓ + H.c. (cid:3) (8)is the resonant interaction between atoms and molecules,with strength parameterized by g . After renormaliza-tion, the magnitude of the channel coupling strength g isrelated to the width of the Feshbach resonance W , i.e., g ≡ p ∆ µW U bg . We note that, in the two-channel modelone may define an effective s -wave length [35], a s = a bg (cid:18) − WB − B (cid:19) = a bg − g ν m π ~ . (9) III. MEAN FIELD THEORY
We use the standard mean-field theory to solve thetwo-channel model Eq. (1), by assuming that all themolecules and Cooper pairs condense into the zero-momentum state. Thus, we set q = 0 in the interactionHamiltonian H I . Here, we have excluded the possibil-ity of an inhomogeneous superfluid phase (i.e., q = 0 ),which may exist in the presence of an in-plane Zeeman-field [37]. This is consistent with the two-body calcula-tion [13, 27, 38] that the ground state of two particles inour Hamiltonian always has zero center-of-mass momen-tum. Following the procedure in Ref. [33], we introducethe following field parameters: φ m = h b i , (10) p = X k h a k ↑ a − k ↓ i , (11) f = X k D a † k ↑ a k ↑ E = X k D a † k ↓ a k ↓ E , (12)where φ m is the molecular field in the closed channel, p isthe pairing field, and f is half of the number of fermionicatoms in the open channel. The interaction Hamiltonian H aa can therefore be written as H aa U bg ≃ X k σ f a † k σ a k σ − X k (cid:16) pa † k ↑ a †− k ↓ + H.c. (cid:17) − | p | − f . (13)Similarly, we approximate H am as H am ≃ − g X k (cid:16) φ m a † k ↑ a †− k ↓ + H.c. (cid:17) . (14)Thus, within mean-field the total Hamiltonian is givenby, H = − U bg (cid:0) | p | + f (cid:1) + X k τ ǫ k τ h † k τ h k τ + X k σ U bg f a † k σ a k σ +2 ν | φ m | − X k h ( U bg p + gφ m ) a † k ↑ a †− k ↓ + H.c. i , (15)where τ ≡ ± is the index of helicity branch. By definingan order parameter ∆ = − ( U bg p + gφ m ) and rewriting allthe field operators in the helicity basis, the total mean-field Hamiltonian becomes H = − U bg (cid:0) | p | + f (cid:1) + 2 ν | φ m | + X k τ ( ǫ k τ + U bg f ) h † k τ h k τ − ∆2 X k h e − iϕ k h † k + h †− k + + e iϕ k h † k − h †− k − + H.c. i . (16)To determine the variational field parameters ( φ m , p , and f ), we diagonalize K = H − µ N by using Bogoliubovtransformation and calculate the grand thermodynamic potential Ω . Here, N ≡ P k σ a † k σ a k σ + 2 P k b † k b k is theoperator of total number of atoms and µ is the chemi-cal potential. Using the field operators for Bogoliubovquasiparticles [33], α k + and α k − , K takes the diagonalform, K = X k τ E k τ α † k τ α k τ − U bg (cid:0) | p | + f (cid:1) + 2 ( ν − µ ) | φ m | + X k (cid:20) ( ξ k + U bg f ) − E k + + E k − (cid:21) , (17)where ξ k ≡ ǫ k − µ and the energies of Bogoliubov quasi-particles E k ± are given by, E k ± = s(cid:18) ξ k ± ~ λk ⊥ m + U bg f (cid:19) + | ∆ | . (18)At temperature T , it is straightforward to write downthe grand thermodynamic potential, Ω = X k (cid:20) ξ k + U bg f − E k + + E k − (cid:21) − U bg (cid:0) | p | + f (cid:1) +2 ( ν − µ ) | φ m | − k B T X k τ ln (cid:20) e − E k τkBT (cid:21) . (19)The field parameters ( φ m , p , and f ) must satisfy thecoupled self-consistent equations, ∂ Ω /∂f = 0 , ∂ Ω /∂p =0 , and ∂ Ω /∂φ m = 0 . Furthermore, the chemical potentialis determined by the total number of atoms N , i.e., N = − ∂ Ω ∂µ = 2 f + 2 φ m . (20)These four coupled equations can be solved to obtainthe pairing order parameter ∆ = − ( U bg p + gφ m ) andchemical potential µ . IV. RESULTS AND DISCUSSIONS
To clearly contrast the two-channel model with single-channel model, we focus on the resonant limit and neglectthe back-ground interaction. By setting U bg = 0 (andtherefore ∆ = − gφ m ) and renormalizing the energy ofmolecules ν by using [33], ν = 2 ν + X k g ǫ k , (21)we obtain the coupled gap equation and number equationin the two-channel model ( τ ≡ ± ), ν − µ ) g = X k "X τ / − n k τ E k τ − ǫ k , (22) N − g = X k " − X τ (cid:18) − n k τ (cid:19) ǫ k τ − µE k τ , (23)where n k ± ≡ / ( e E k ± /k B T + 1) is the Fermi-Dirac distri-bution function. In contrast, the gap and number equa-tions in the single-channel model are given by [13], − m π ~ a s = X k "X τ / − n k τ E k τ − ǫ k , (24) N = X k " − X τ (cid:18) − n k τ (cid:19) ǫ k τ − µE k τ , (25)respectively. By recalling from Eq. (9) that the effec-tive s -wave scattering length in the two-channel modelis − π ~ a s /m = g / (2 ν ) , it is clear that the gap andnumber equations in both models have the same struc-ture. However, additional terms, − µ/g and − /g ,appear in the two-channel gap and number equations, re-spectively. For a finite SO coupling constant λ , if g → ∞ , µ/g and /g go to zero. Then, the equations ofthe two models become exactly the same. This is thesame as the situation without SO coupling. In otherwords, the single-channel model and two-channel modelcoincide with each other in the broad resonant limit, asthey should be. However, for a finite channel couplingstrength g , if λ is sufficiently large, deep two-body boundstate (i.e., rashbon) appears, with a divergent chemicalpotential ( µ → −∞ ; see Fig. 1(b) below). Thus, we cannot neglect µ/g in Eq. (22) anymore and the two mod-els are no longer equivalent. In this strong SO couplinglimit, we anticipate a qualitative difference between thesingle-channel and two-channel models.Let us turn to detailed numerical calculations. Forsimplicity, we consider the case in which the temperatureis zero and the system is exactly at Feshbach resonance( ν = 0 and a − s = 0 ). To characterize the width of Fes-hbach resonances, we introduce a dimensionless channelcoupling constant, g = 2 m ~ k / F g, (26)where k F = (3 π N/V ) / is the Fermi wavelength. Wetake the energy and length in the units of the Fermi en-ergy E F = ~ k F / m and the inverse Fermi wavelength k − F , respectively. The SO coupling strength is measuredby the dimensionless parameter λ/k F . In experiments,the typical magnitude of SO coupling strength is at theorder of the Fermi wavelength, i.e., λ = O ( k F ) .Fig. 1 reports the evolution of the pairing gap (a)and chemical potential (b) with decreasing the resonancewidth. For comparison, the prediction of single-channelmodel is also shown by solid lines. As seen from Fig. 1(a),for systems with small SO coupling strength, the single-channel model and two-channel model give the same re-sults when g is large enough (i.e., g > ). In the caseof Li atoms with g ≃ (the purple dot line in Fig.1), we can not see the difference from the single-channelprediction. This means that the broad resonance condi-tion is always valid for Li atoms. However, for a smaller λ /k F ∆ / E F Single Channel Modelg =600 Lig =50 Kg =20g =5 (a)0 1 2 3 4 5−10−8−6−4−202 λ /k F µ / E F (b) Figure 1: The pairing gap ∆ (a) and chemical potential µ (b)as a function of the Rashba SO coupling strength λ for dif-ferent atom-molecule coupling g or resonance width at zerotemperature and at Feshbach resonance. g , e.g., K atoms with g ≃ , the pairing gap deviatesclearly from the single-channel prediction at the typicalexperimental SO coupling strength λ/k F = 3 , althoughthe two models give essentially the same pairing gap inthe absence of SO coupling. With increasing the SO cou-pling strength, the difference between the two models be-comes more significant. For even smaller g (i.e., g = 5 ),it is interesting that the dependence of the pairing gapon SO coupling strength changes qualitatively. The pair-ing gap starts to decrease with increasing SO couplingstrength and vanishes at sufficiently large SO coupling.This dramatic change is somehow not anticipated, asthe pairing gap is always enhanced by SO coupling in thesingle-channel model. It is closely related to anisotropicsuperfluidity caused by the Rashba SO coupling. As dis-cussed in Ref. [13], due to SO coupling the fermionicsuperfluid has mixed singlet and triplet components.The fraction of triplet pairing grows with increasing theSO coupling strength. Thus, within the single-channelmodel, the amplitude of pairing gap reflects both sin-glet and triplet pairing strengths, and increases as theSO coupling increases. In the two-channel model, how- ∆ / E F λ /k F =0 λ /k F =3 λ /k F =5(a) λ /k F g c (b) Figure 2: (a) The pairing gap ∆ as a function of the resonancewidth g for different Rashba SO coupling strength λ/k F . (b)The critical resonance width g (below which the pairing gapdiffers more than 5% from the prediction of the single-channelmodel) for different λ/k F . ever, the most important resonant-interaction Hamilto-nian H am is of s -wave character and hence favors the sin-glet pairing. As the triplet pairing is favored by RashbaSO coupling, the resonance width and SO coupling haveopposite effects on the pairing gap and destruct with eachother. The destruction becomes very pronounced withdecreasing the resonance width, leading to a completelysuppressed pairing gap at large SO coupling and narrowresonance width.The suppression of pairing gap can also be mathemati-cally understood from the two-channel gap equation, Eq.(22), where we may treat ν − µ ) as the effective en-ergy detuning of Feshbach molecules. By increasing theSO coupling, the chemical potential will diverge to −∞ ,and hence the effective energy detuning is pushed up tothe BCS limit. As a result, the pairing gap becomes sig-nificantly suppressed.Now, it is clear that the broad resonance limit becomesmuch more difficult to reach in the presence of RashbaSO coupling. To quantitatively characterize the broadresonance condition, we show in Fig. 2(a) the behavior λ /k F | φ m | / n Figure 3: The molecular fraction | φ m | /N as a function ofthe Rashba SO coupling λ/k F for K atoms where g = 50 . of the pairing gap ∆ as a function of the resonance width g , for some selected values of λ/k F . As g increases, thepairing gap ∆ grows rapidly at first, and then saturatesto the prediction of single-channel model. Quantitatively,we may define a critical g c , in such a way that above g c the relative difference in the pairing gaps predicted bythe two models is less than . Fig. 2(b) presents g c asa function of SO coupling strength. It gives a qualitativephase diagram. Above g c we may safely use the single-channel model to describe the Rashba SO coupled atomicFermi gas near Feshbach resonances. While below g c , thetwo-channel model must be adopted. For K atoms with g ≃ , we find that the single-channel model becomesinsufficient at a moderate Rashba SO coupling, λ ∼ k F . V. EXPERIMENTAL RELEVANCE
To experimentally test our predictions, we consideroptical molecular spectroscopy, which projects Feshbachmolecules onto a vibrational level of an excited molecule.The rate of excitations enables a precise measurementof the fraction of the closed-channel Feshbach moleculesin the paired state, although the fraction could be ex-tremely small [36]. Near resonance, the paired state maybe treated as dressed molecules [35, 36], | dressed i = e iφ p − Z m | open i + p Z m | closed i , (27)where Z m can be identified as the component fractionof Feshbach molecules, i.e., Z m = 2 | φ m | /N . As anconcrete example, in Fig. 3, we show the fraction for K atoms (with g ≃ ) as a function of SO couplingstrength. As the SO coupling increases, the population ofFeshbach molecule is almost flat at first. However, afterthe coupling reaches a critical value λ ≃ k F , it growsvery fast.The impact of SO coupling on the population of Fesh-bach molecules is best seen in Fig. 4, where we present | φ m | / n λ /k F =0 λ /k F =3 λ /k F =5 0 5 1001020 λ /k F g m a x Figure 4: The molecular fraction as a function of the reso-nance width g for different Rashba SO coupling. The insetshows g max (see text for definition) as a function of the SOcoupling strength. the fraction as a function of the resonance width at sev-eral SO coupling strengths. For a given non-zero SOcoupling, the fraction is a non-monotonic function of theresonance width. By decreasing g from the broad reso-nance limit, the fraction first grows then drops to zero,as a result of the competition between SO coupling andresonance width, as mentioned earlier. In the limit of nar-row resonance, the vanishing molecular fraction is con-sistent with the suppression in the pairing gap at largeSO coupling, as shown in Fig. 1(a), due to the relation ∆ = − gφ m . In contrast, in the absence of SO couplingthe molecular fraction increases steadily with decreasingthe resonance width. We may define a characteristic g max at which the fraction reaches its peak value. As shown inthe inset of Fig. 4, when λ/k F is zero, namely there is noSO coupling, g max = 0 , and the population reaches unityin the limit of g = 0 . As the SO coupling increases, g max increases. We emphasize that for small resonancewidth g , even a small SO coupling could lead to a strongsuppression of the population of Feshbach molecules. VI. SUMMARY
In conclusion, we have investigated a Rashba spin-orbit coupled Fermi gas near Feshbach resonances, by using a two-channel model. When the spin-orbit cou-pling strength is small and Feshbach resonance is broad,the two-channel model is equivalent to the single-channelmodel, as we may anticipate [34, 35]. However, for agiven resonance width, if the SO coupling strength issufficiently large, these two models are no longer equiv-alent. Moreover, for a narrow resonance the pairing gapand the fraction of Feshbach molecules are strongly sup-pressed by SO coupling. We could test these predictionsby measuring experimentally the molecular fraction us-ing optical molecular spectroscopy [36]. We have charac-terized quantitatively the equivalence of the two modelsby introducing a critical resonance width, above whichthe two models are approximately the same. By cal-culating the dependence of the critical resonance widthon the spin-orbit coupling strength, we have found thatthe single channel model may break down for Rashbaspin-orbit coupled K atoms at a moderate spin-orbitcoupling strength.Our results are obtained within mean-field theory,which is known to provide a qualitative picture ofresonantly-interacting atomic Fermi gases. For quanti-tative purpose, the crucial pairing fluctuation must beincluded. This may be addressed by using many-body T -matrix theories in the future [35, 39, 40]. Acknowledgments
Jing-Xin Cui and Gui Lu Long were supported by theNational Natural Science Foundation of China (GrantNo. 11175094), National Basic Research Program ofChina (NFRP-China Grant No. 2009CB929402 and No.2011CB9216002), and Tsinghua University Initiative Sci-entific Research Program. Xia-Ji Liu and Hui Hu weresupported by the ARC Discovery Projects (Grant No.DP0984637 and No. DP0984522) and the NFRP-China(Grant No. 2011CB921502). [1] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature(London) 471, (2011).[2] R. A. Williams, L. J. LeBlanc, K. Jiménez-García, M. C.Beeler, A. R. Perry, W. D. Phillips, and I. B. Spielman,Science , 314 (2012).[3] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B.Yan, G.-S. Pan, B. Zhao, Y. Deng, H. Zhai, S. Chen, and J.-W. Pan, Phys. Rev. Lett. , 115301 (2012).[4] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.Zhai, and J. Zhang, Phys. Rev. Lett. , 095301 (2012).[5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. ,095302 (2012).[6] C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. , 160401 (2008).[7] X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A ,021603(R) (2012).[8] X.-J. Liu and H. Hu, Phys. Rev. A , 033622 (2012).[9] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[10] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[11] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B ,094515 (2011).[12] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys.Rev. B , 014512 (2011).[13] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett. , 195304 (2011).[14] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. , 195305(2011).[15] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. ,195303 (2011).[16] W. Yi and G.-C. Guo, Phys. Rev. A , 031608(R)(2011).[17] J. Zhou, W. Zhang, and W. Yi, Phys. Rev. A , 063603(2011).[18] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, andS. Das Sarma, Phys. Rev. B , 140510(R) (2011).[19] M. Iskin and A. L. Subaşi, Phys. Rev. Lett. , 050402(2011).[20] T. Ozawa and G. Baym, Phys. Rev. A , 043622 (2011).[21] K. Zhou and Z. Zhang, Phys. Rev. Lett. , 025301(2012).[22] L. Han and C. A. R. Sá de Melo, Phys. Rev. A ,011606(R) (2012).[23] K. Seo, L. Han, and C. A. R. Sá de Melo, Phys. Rev. A , 033601 (2012).[24] L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys.Rev. A , 033633 (2011).[25] L. He and X.-G. Huang, Phys. Rev. Lett. , 145302(2012).[26] L. He and X.-G. Huang, Phys. Rev. B , 014511 (2012).[27] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A ,063618 (2011).[28] X.-J. Liu, Phys. Rev. A , 033613 (2012).[29] X. Yang and S. Wan, Phys. Rev. A , 023633 (2012).[30] P. Zhang, L. Zhang, and W. Zhang, Phys. Rev. A ,042707 (2012).[31] P. Zhang, L. Zhang, and Y. Dong, Phys. Rev. A ,053608 (2012).[32] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod.Phys. , 1215 (2008).[33] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo,and R. Walser, Phys. Rev. Lett. , 120406 (2001).[34] R. B. Diener and Tin-Luo Ho,arXiv:cond-mat/0405174v2.[35] X.-J. Liu and H. Hu, Phys. Rev. A , 063613 (2005).[36] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack,and R. G. Hulet, Phys. Rev. Lett. , 020404 (2005).[37] Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo,arXiv:1208.2029 (2012).[38] L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700(2012).[39] H. Hu, X.-J. Liu, and P. D. Drummond, Europhys. Lett. , 574 (2006).[40] X.-J. Liu and H. Hu, Europhys. Lett.75