Polarons in a ferromagnetic spinor Bose-Einstein condensates
PPolarons in a ferromagnetic spinor Bose-Einstein condensates
Xiao-Lu Yu ∗ and Boyang Liu Institute of Theoretical Physics, Beijing University of Technology Beijing 100124, China (Dated: January 22, 2021)We investigate the polarons formed by immersing a spinor impurity in a ferromagnetic state of F = 1 spinor Bose-Einstein condensate. The ground state energies and effective masses of thepolarons are calculated in both weak-coupling regime and strong-coupling regime. In the weaklyinteracting regime the second order perturbation theory is performed. In the strong coupling regimewe use a simple variational treatment. The analytical approximations to the energy and effectivemass of the polarons are constructed. Especially, a transition from the mobile state to the self-trapping state of the polaron in the strong coupling regime is discussed. We also estimate thesignatures of polaron effects in spinor BEC for the future experiments. I. INTRODUCTION
The polaron is a state formed in a system of asingle distinguishable particle interacting with mediumatoms, which can be used to describe various systemsin traditional solid states [1, 2], quantum liquids [3, 4],advanced materials [5], chemistry[6, 7] and biophysics[8, 9]. Recently, an impurity atom immersed in a Bose-Einstein condensate (BEC) has been experimentally re-alized [10, 11], and there are intensive interests focusingon the physics of Bose polarons[12–25]. The main moti-vations of these simulations are to explore the hithertoinaccessible strong coupling regime in polaron physics byutilizing the remarkable tunabilities of experimental tech-niques in the ultracold gas systems [26].The purpose of the present paper is to generalize thepolaron effects in a spinor impurity-spinor BEC system.The spin of the alkali atoms is essentially free in anoptical trap, producing a rich variety of spin textures.The spin-polaron effects arise from spin dependent in-teractions between an impurity and BEC atoms, whichhas several features distinct from the scalar BEC. First,the scalar BEC systems only provide a gapless phononmode. There are many collective modes in a spinorBEC, which could be gapless or gapped, coupling withthe impurity to induce polaron effects. This properlyprovides the scenario to simultaneously simulate opti-cal and acoustic polaron of solid-states [1, 27, 28] withspinor quantum gases. Second, the energies and effectivemasses of polarons among different spin components willsplit, due to spin-exchange collisions between an impu-rity and BEC atoms. These are characteristic phenomenain spin-polaron physics, which plays an important role instrongly correlated electron materials [5, 29]. Third, withthe techniques of optical Feshbach resonance [30], it ispossible to manipulate polaron with different spin com-ponents, such as the formation of self-trapping states inthe strong-coupling regime.The method of the present paper is to map the spinorimpurity-BEC system onto Fr¨ohlich Hamiltonian of orig-inal polaron problem, then analogous effects are derived ∗ [email protected] in quantum gas systems. Fr¨ohlich Hamiltonian is oneof the simplest examples of a quantum field theoreticalproblem, since it basically consists of a single particle in-teracting with a Bose field. There are many variationsof Fr¨ohlich Hamiltonian, which are studied thoroughlyin [31, 32]. However, a spinor generalization of Fr¨ohlichHamiltonian has not been explicitly spelled out in anyreal physical system to our knowledge, which is what weattempt to do here.The paper is organized as follows. The spinor gen-eralization of Fr¨ohlich polaron Hamiltonian is describedin Sec. II. In the weak-coupling regime, we illustratesthe polaron arising from spin-dependent interactions bya perturbation theory in Sec. III. With a simple varia-tional treatment in Sec. IV, we construct analytical ap-proximations to the energy and effective mass of polaronin the strong-coupling regime. The splitting of energyand effective mass among different spin components areemphasized. Sec. V contains experimental discussionsand conclusions. II. POLARON HAMILTONIAN
Let us consider a system of homogenous spin-1 Bosegas with s-wave interaction. This system can be de-scribed in the second quantized Hamiltonian [33] as H BEC = (cid:88) k ,f (cid:15) k a † k ,f a k ,f + 12 (cid:88) k ,f δ k + k , k + k × ( c a † k ,f a † k ,f a k ,f a k ,f + c a † k ,f a † k ,f F f f (cid:48) · F f f (cid:48) a k ,f (cid:48) a k ,f (cid:48) ) , (1)where f is a hyperfine spin index with value ± , F is astandard spin-1 matrix vector, and (cid:15) k = k m is the kineticenergy of BEC atoms with mass m .When an impurity atom also with hyperfine spin 1 isimmersed in a spin-1 BEC of a different type atom, thecontact potential between the impurity and BEC can bewritten [30] V IB = λ + β F · S + γ ˆ P . (2)Here, S represents spin-1 matrix vector of an impu-rity, λ = ( g + g ) / a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n tion, β = ( g − g ) / γ = (2 g − g + g ) / g F = 2 πa IBF /µ in the first order ofperturbation theory, where a IBF is the s-wave scatteringlength in the total spin F channel with the relative mass µ = mM/ ( m + M ). ˆ P is a projection operator of spinsinglet channels [34]. Correspondingly, the singlet stateof two spin-1 particles is ( | , − (cid:105) − | , (cid:105) + |− , (cid:105) ) / √ , , T , which emerges withthe condition c < a k, a k, a k, − = √ n + ˜ a k, ˜ a k, ˜ a k, − , (3)with Bogoliubov transformation [35, 36] b k , = (cid:115) (cid:15) k + c n + ω k , ω k , ˜ a k , + (cid:115) (cid:15) k + c n − ω k , ω k , ˜ a †− k , ,b k , = ˜ a k , ,b k , − = ˜ a k , − . (4)The operators b k ,f are the annihilation operators ofthree Bogoliubov excitations in spin-1 BEC with dis-persions as the following [33]: (i) gapless density mode ω k , = (cid:112) (cid:15) k ( (cid:15) k + 2( c + c ) n ), (ii) ferromagnetic spinwave mode ω k , = (cid:15) k , (iii) gapped spin quadrupole mode ω k , − = (cid:15) k + 2 | c | n , where c = 4 π (2 a BB + a BB ) / (3 m )and c = 4 π ( a BB − a BB ) / (3 m ). a BBF is the s-wave scat-tering length in the total spin F channel.Then we can express the total Hamiltonian as H tot = E g + E IB + H pol . (5)Here, the first term is the ground state energy of BEC.The second term is the mean-field interaction energy be-tween an impurity and the BEC, which is diagonal in thehyperfine spin states of the impurity as E IB = n λ + β λ
00 0 λ − β + γ (6)The last term has the similar form as Fr¨ohlich polaronHamiltonian[1] H pol = p M + (cid:88) k ,f ω k ,f b † k ,f b k ,f + (cid:88) k ,f ( V k ,f b k ,f e i k · r + h.c. ) . (7)Here and below, we take unit volume and (cid:126) = 1. M and r are the mass and coordinate of the impurity atom. Theground state of the spinor BEC is treated as a static meanfield, above which excitations are modeled as a bath ofthree different types of boson quasiparticles. A directcalculation gives coupling matrixes between an impurity atom and collective modes of spinor BEC as V k , = (cid:112) n W k λ + β λ
00 0 λ − β + γ ,V k , = √ n β β − γ ,V k , − = √ n γ , (8)where W k = ξk √ ξk ) and the healing length of thespinor BEC is defined as ξ = √ mn ( c + c ) . These ma-trixes are generalizations of the coupling function in thecontext of an impurity immersed in a scalar BEC [13, 14].Correspondingly, relevant collision processes are cate-gorized by the interspecies spin-exchange interactions.There are clear physical meanings for these coupling ma-trixes. Since the majority of atoms are condensed in f = 1 component, the matrix V k , has to be diagonal dueto the coupling between density wave and an impurityatom. For spin-exchange processes, we should notice thatan impurity and BEC atoms are distinguishable. With-out the statistical constrain of identical bosons, there aretwo kinds spin-exchange processes corresponding to V k , and V k , − , in contrast to the identical spin-1 case whereonly one spin-changing process is allowed [33]. Morespecifically, V k , term exchanges (cid:126) spin angular momen-tum accompanied by spin fluctuations, and V k , − termexchanges 2 (cid:126) spin angular momentum accompanied byspin quadrupolar fluctuations.We define the dimensionless coupling strengths be-tween an impurity and BEC as α f = a f a BB ξ , (9)with a = a IB , a = ( a IB + a IB ) / a − = (2 a IB +3 a IB + a IB ) /
6. Correspondingly, the mean-field energycan be expressed in terms of scattering lengths a f as E IB,f = 2 πn a f /µ . It is well known that the Fr¨ohlich-like Hamiltonian (7) resists exact analytical solutions hasqualitatively different characteristics in weak-coupling( α f (cid:28)
1) and strong-coupling ( α f (cid:29)
1) regimes, whichwill becomes clear in the following investigation.
III. PERTURBATIVE TREATMENTS IN THEWEAK-COUPLING REGIME
Let us treat the last term of (7) H (cid:48) = (cid:88) k ,f ( V k ,f b k ,f e i k · r + h.c. ) (10)as a perturbation [37], which changes the number ofquasiparticles. Therefore, the first order perturbed en- (a) (b)FIG. 1. (a) The impurity with momentum p and virtual col-lective modes with momentum k corresponds to second-orderperturbation theory, where there are three different vertexescoupling to collective modes given by V k ,f . (b) The impuritydissipative energy by creating a spin wave mode, which is theonly emission process at small momentum. ergy vanishes in the vacuum state of quasiparticles. Con-sidering the second order perturbation processes of differ-ent spin components, all initial states have an impurityof momentum p and no collective excitations [38]. In theintermediate state | n (cid:105) , momentum of the impurity andcollective excitations are ( p − k ) and k , correspondingto the initial energy E = p M and the intermediate en-ergy E n,f = ( p − k ) M + ω k ,f , as shown in Fig. 1(a). So weobtain the second order perturbed energy (cid:52) E f = (cid:88) n | (cid:104) n | H (cid:48) | (cid:105) | E − E n,f . (11)The effective mass of the impurity can be obtainedby performing the small momentum expansion of aboveequation E f ( p ) = (cid:52) E f (0) + p M ∗ f + O (cid:0) p (cid:1) . (12)To extract the polaron energy (cid:52) E f (0) experimentally[10, 11], we need to take into account the initial mean-field shift E IB , because what we actually measure in ex-periments is the total energy shift of an impurity due tointeractions with BEC as (cid:52) f = E IB,f + (cid:52) E f (0) , (13)For weak interactions, the energies are well described bythe mean-field shift. The perturbed corrections can beviewed as the first step beyond the mean-field results.For an impurity atom at f = 1 state, we have (cid:52) E ( p ) = ( λ + β ) n (cid:88) k W kp M − ω k , − ( p − k ) M , (14)which is analogous to the acoustic polaron in solid-statephysics. In the second-order perturbation theory, we must take account of the fact that g = πa IB µ is not ex-act, but only valid up to first-order. In the second-order,we have[20] 2 πa IB µ = g − g (cid:88) k µk . (15)With this cautionary remark, we are able to subtractultraviolet divergence in (14) from the mean-field shiftat the same order, and obtain the finite result of polaronenergy at f = 1 state (Here and below, we take the energyunit as Mξ ) (cid:52) E (0) = α I ( ˜ m ) , (16)where I ( ˜ m ) = (1 + ˜ m ) ˜ m (cid:90) ∞ dx π × ( 11 + ˜ m − x ˜ mx √ x + 2 + x ) (17)with mass ratio ˜ m = m/M . This is essentially the sameresults as the polaron in scalar BEC [14].For an impurity atom at f = 0 state, there are twopossible intermediates states in the second order pertur-bation theory as (cid:52) E ( p ) = λ n (cid:88) k W kp M − ω k , − ( p − k ) M + β n (cid:88) k p M − ω k , − ( p − k ) M . (18)With the same procedure of subtractions, the energy ofa static polaron at f = 0 state becomes (cid:52) E (0) = α I ( ˜ m ) . (19)In contrast to f = 1 state, no matter how small the initialvelocity of the impurity at f = 0 state is, it dissipatesenergy by creating a a spin wave mode (magnon) withgapless quadratic dispersion as shown in Fig. 1(b). Inother words, the critical velocity of the impurity at f = 0state is zero, which is distinct from the sound velocity at f = 1 state. In the ferromagnetic phase of spinor BEC,the superfluid current will decay through development ofspin textures, due to the spin-gauge symmetry [33].There is an additional contribution to polaron energyfrom the coupling between an impurity and gapped spinquadrupole excitations for f = − (cid:52) E − ( p ) = ( λ − β + γ/ n (cid:88) k W kp M − ω k , − ( p − k ) M + ( β − γ n (cid:88) k p M − ω k , − ( p − k ) M + ( γ n (cid:88) k p M − ω k , − − ( p − k ) M . (20)The polaron energy at f = − (cid:52) E − (0) = α − I ( ˜ m ) + (cid:115) a BB − a BB )3 a BB × ( 2 a IB − a IB + a IB a IB ) α √ m ˜ m (21)Combining the polaron energies with correspondingmean-filed shifts for three spin components, we obtainthe total energies of the impurity, which are illustratedin Fig. 2(a). We should notice that total energy shifts aredominated by the mean-field contributions in the weak-coupling regime.Let us now to calculate the effective mass of the impu-rity for three internal states, which are given by M ∗ f M = 11 − M | E (cid:48)(cid:48) f (0) | . (22)The singularity at | E (cid:48)(cid:48) f (0) | = 1 /M may indicate a disruptchange of ground state in the strong coupling regime, al-though the perturbation theory has been broken downbefore the singularity point [32]. The mean-field shiftdoes not contribute to the effective masses, and there arenot any ultraviolet divergences in the following calcula-tions.For an impurity atom at f = 1 state, a straightforwardcalculation up to the second order perturbation shows M ∗ M = 1 + α I ( ˜ m ) , (23)where I ( ˜ m ) = 4(1 + ˜ m − ) π × (cid:90) ∞ x dx ( ˜ m − √ x + x ) √ x . (24)The enhancement of effective mass relative to the baremass indicates the lower mobility of the impurity, due tothe formation of screening quasiparticles around. Cor-respondingly, the effective mass of an impurity atom at f = 0 and f = − M ∗ M = 1 + α I ( ˜ m ) ,M ∗− M = 1 + α − I ( ˜ m ) + (cid:115) a BB a BB − a BB ) × ( 2 a IB − a IB + a IB a IB ) α ˜ m √ m . (25)In the weak-coupling regime, the splittings of effectivemass among different spin components are within a fewpercent as shown in Fig. 2(b). We should notice thatthese results are coincident with the intermediate cou-pling theory of Lee-Low-Pines up to the order of α f [39].Mobility of the impurity depends on the hyperfine spin f = = =- - - - - - a / ξ Δ f f = = =- - - a / ξ M f * / M ( a )( b ) FIG. 2. (Color online) Tuning Na- Rb ( ˜ m = 3 .
78) scatter-ing length a IB while keeping a IB = 84 . a B and a IB = 81 . a B fixed [30], where a B is the Bohr radius. The scattering lengthsof Rb are a BB = 101 . a B and a BB = 100 . a B [40]. Withthe typical density of BEC n = 10 cm − , the healing length ξ = 270 nm. Three coupling constants α f < . | a IB /ξ | < .
1, where we expect perturbation the-ory to be reliable. (a). The total energy shifts of an impurityamong different spin components. (b). Effective mass amongdifferent spin components. states, since the screening effects by the surrounding col-lective modes depends on the different types of disper-sion. The splitting effective mass of different spin statesis a characteristic phenomena in spin-polaron physics,which plays an important role in strongly correlated elec-tron materials [5, 29].
IV. POLARON ENERGY AND EFFECTIVEMASS IN THE STRONG-COUPLING REGIME
Beginning with Landau’s speculation that an electronin an ionic crystal can be trapped by lattice distortionproduced by the electron itself [41], subsequent worksgave rise to the strong-coupling polaron theory. Follow-ing the approach of Pekar and Landau [42], we suggest aproduct ansatz for the trial wave function of Hamiltonian(7) | Ψ (cid:105) = | φ (cid:105) | χ (cid:105) | ψ (cid:105) | ψ (cid:105) | ψ − (cid:105) , (26)where | φ (cid:105) and | χ (cid:105) are normalized spatial and spinor func-tions of the impurity variable only, and | ψ f (cid:105) that of col-lective modes variables only. The variational principlerequires the minimization of energy (cid:104) Ψ | H pol | Ψ (cid:105) = − (cid:104) φ | ∇ M | φ (cid:105) + (cid:88) k ,f ω f ( k ) (cid:104) ψ f | b † k f b k f | ψ f (cid:105) + (cid:88) k ,f ( G ∗ f ( k ) (cid:104) ψ f | b k f | ψ f (cid:105) + h.c. ) , (27)with the definition G ∗ f ( k ) ≡ (cid:104) φ | e i k · r | φ (cid:105) (cid:104) χ | V k f | χ (cid:105) . (28)It is convenient to introduce displaced collective modesoperators B f ( k ) ≡ b k f + G f ( k ) ω f ( k ) , (29)which satisfies the canonical commutation rules. Thenthe polaron energy is calculated as (cid:104) Ψ | H pol | Ψ (cid:105) = − (cid:104) φ | ∇ M | φ (cid:105) − (cid:88) k ,f | G f ( k ) | ω f ( k )+ (cid:88) k ,f ω f ( k ) (cid:104) ψ f | B † k f B k f | ψ f (cid:105) . (30)The second term is always nonnegative, because B † k f B k f is a quadratic Hermitian operator and ω f ( k ) (cid:62) (cid:12)(cid:12)(cid:12) ψ f (cid:69) satisfying B f ( k ) (cid:12)(cid:12)(cid:12) ψ f (cid:69) = 0, the variationalenergy becomes (cid:104) Ψ | H pol | Ψ (cid:105) = − (cid:104) φ | ∇ M | φ (cid:105) − (cid:88) k ,f | G f ( k ) | ω f ( k ) . (31)Generally, a spinor wave function | χ (cid:105) of the impurity isa superposition of three internal states. In the following,we only concern about the impurity at a hyperfine spineigenstate, where the last term of (31) is determined bythe diagonal elements of V k . In order to evaluate theground state energy, we have to specify the concrete formof trial wave function φ ( r ) of the impurity. The simplestchoice is a hydrogen-like ground state wave function [1] φ ( r ) = (cid:115) t πξ e − tr/ξ , (32)where t is a dimensionless variational parameter relatedto the radius of polaron √ < r >ξ = √ t . (33)Substituting (32) into (31), the variational energy for animpurity at a spin f eigenstate can be evaluated straight-forwardly as E f = t − ˜ α f I ( t ) , (34) α f ˜ = α f ˜ = α f ˜ = α f ˜ = - - t ℰ f FIG. 3. (Color online) The variational energy is shown asa function of the variational parameter at different couplingstrength. The variational energy start to develop a local min-imum when ˜ α f > .
8. The polaron energy becomes negativeat a critical coupling strength ˜ α f = 15 .
8, which indicates atransition from an unbound state to a localized state. Corre-sponding to the minimum of energy, there is a discontinuityon the variational parameter t , jumping from zero to 1.39. where ˜ α f ≡ α f (1 + ˜ m )(1 + ˜ m − ) and I ( t ) = 2 π t (cid:90) ∞ x dx (1 + x ) (1 + 2 t x ) . (35)The strong-coupling polaron corresponds to the solu-tions of negative variational energy, which is determinedby the value of coupling constant ˜ α f . Only when ˜ α f exceeds some critical value, the minimum of variationalenergy will be lower than zero. The impurity atom cantransit from a mobile state to a localized state, so callself-trapping state. This essential feature is shown inFig.3, where the variational treatment predicts that theimpurity start to localize at the critical coupling strength α ∗ f = 15 . m )(1 + ˜ m − ) . (36)Let us consider a much lower density gas of spinor im-purity atoms immersed in a spinor BEC. The impurity-BEC interactions are characterized by three scatteringlengths, while the interactions among impurity atomsare safely neglected. Inferring from (36) and (9), wecan manipulate the formation of polarons for differentspin components by tuning the heteronuclear scatteringlengths. As shown in Fig. 4(a) for a Na- Rb sys-tem, we plot the total energy of the impurity defined as (cid:52) f = E IB,f + E f . Increasing a IB while keeping a IB and a IB fixed, the polaron of f = 1 will first develop a lo-calized state at a IB /ξ = 0 .
23. Polarons later becomeslocalized at a IB /ξ = 0 .
44 and a IB /ξ = 1 .
28 for f = 0and f = −
1, respectively. From (33), polarons can self-localize in a region smaller than the healing length, whichcould also be identified as the system crossing over fromthe weak to strong coupling regime.The above solution of the polaron energy is static,where the impurity is centered around origin and trappedby BEC density distortion. In order to study the effec-tive mass in strong coupling regime, it is assumed thatthe impurity state function (cid:12)(cid:12)(cid:12) ˜ φ (cid:69) moves with a small ve-locity u . Then we minimize the following quantity [43] (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) H pol − u · P (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) = − (cid:68) ˜ φ (cid:12)(cid:12)(cid:12) ∇ M (cid:12)(cid:12)(cid:12) ˜ φ (cid:69) + i u · (cid:68) ˜ φ (cid:12)(cid:12)(cid:12) ∇ (cid:12)(cid:12)(cid:12) ˜ φ (cid:69) + (cid:88) k ,f ( G ∗ f ( k ) (cid:104) ψ f | b k f | ψ f (cid:105) + h.c. )+ (cid:88) k ,f ( ω f ( k ) − u · k ) (cid:104) ψ f | b † k f b k f | ψ f (cid:105) , (37)where the Lagrange multiplier u is the velocity ofpolaron and the total momentum operator is P = − i (cid:126) ∇ + (cid:80) k ,f k b † k ,f b k ,f . Redefining the displaced opera-tor ˜ B f ( k ) ≡ b k f + G f ( k ) ω f ( k ) − u · k , the phonon ground state issolved by ˜ B f ( k ) (cid:12)(cid:12)(cid:12) ˜ ψ f (cid:69) = 0. Then, the variational energyup to the second order of velocity is found to be E f ( u ) = − (cid:104) φ | ∇ M | φ (cid:105) + 12 M u − (cid:88) k ,f | G f ( k ) | ω f ( k ) + (cid:88) k ,f ( u · k ) | G f ( k ) | ω f ( k ) . (38)The effective mass of spin f eigenstate can be expressedin terms of the variational parameter as M ∗ f M = 1 + ˜ m ˜ α f I ( t ) , (39)where I ( t ) = 83 π t (cid:90) ∞ x dx (1 + x ) (1 + 2 t x ) . (40)By the minimization of variation energy (34), we couldobtain the best choice of variational parameter t for everycoupling constant ˜ α f above the critical value. As shownin Fig. 4(b) , the polarons effective mass can increasewith several order of magnitudes, in contrast with theweak-coupling regime where the enhancements of effec-tive mass are within few percents. The large splittingsof effective mass among different spin components aremanifested on the strong-coupling regime, which couldbe detected by measuring the mobility of impurity. Weshould also notice that there is a stronger enhancementof the effective mass for a lighter impurity, due to thefactor ˜ m in (39). It has been pointed out that the largeeffective mass regime is difficult to realize in most solidmaterials [44]. Therefore, the large scale enhancement ofthe effective mass in the ultracold gases will open up theprospect to investigate this regime hitherto inaccessiblein the solids. f = = =- - - - - - a / ξ Δ f f = = =- a / ξ M f * / M f ( a )( b ) FIG. 4. (Color online) Relevant parameters are chosen asFig.2. (a). The energy shifts of an impurity among differentspin components are plotted as functions of the scatteringlength in the strong-coupling regime. Dashed lines indicatethe mean-field shift before critical coupling strengths. (b).Effective mass among different spin components. The hori-zontal dashed line at M ∗ f /M = 10 . V. DISCUSSIONS AND SUMMARIES
In the current experimental conditions, we estimatetypical parameters for our system as following. The mostimportant one is the dimensionless coupling constant α f defined by (9). For the s-wave scattering betweenspinor gases of Na and Rb, it has been measuredthat ( λ, β, γ ) = 2 π (cid:126) a B /µ × (78 . , − . , .
06) [30], where a B is the Bohr radius. With the typical density of BEC n = 10 cm − and scattering lengths a BB = 100 . a B [40], we notice that the healing length ξ ≈
270 nm andthe energy unit of polaron 1 / ( M ξ ) ≈
290 nK. For thesevalues, we obtain the coupling constants α f ≈ .
01, sothe weak-coupling perturbation theory is suitable. Sinceboth β and γ are much smaller than λ in this experimen-tal setup, the splitting of polaron energies and effectivemass among different hyperfine spin states are within fewpercents.We have assumed spatial homogeneity in our investiga-tion so far. Let us consider the effect of a harmonic trapon the dimensionless coupling constant α f . The densityof BEC is well approximated by the Thomas-Fermi ap-proximation for a large cloud [45]. It is straightforwardto show that the radius of cloud R ∝ ( a BB / ¯ a ) / ¯ a , where¯ a = 1 / √ M ¯ ω is a trap length and ¯ ω is a trap frequency.Therefore, we find the coupling constant α f in the pres-ence of a harmonic trap as α f ∝ ( a BB / ¯ a ) / ( a f /a BB ¯ a ).In order to explore the properties of polarons in strong-coupling regime, there are three ways to enhance the cou-pling strength, which are increasing impurity-BEC scat-tering length a f , decreasing BEC scattering length a BB ,or tighten up the harmonic trap [14]. Experimentally, allthe three ways can be well explored. In our proposals,we mainly focus on tuning a IB with current techniques ofthe optical Feshbach resonance for spinor gases. For ex-ample, when we are able to increase a IB up to a factor of10, we enhance the effective coupling constant α f up to afactor of about 100, which could provide an experimentalaccess to the strong-coupling regime α f ≈
1. We shouldnotice that it is much more challenging to control thespinor system near the resonance. An alternative tech-nique to tune the spin-dependent interaction is to usethe microwave-induced Feshbach resonance [46], in orderto reduce atomic losses near the resonance. Although there is still no such kind of measurement performed ina spinor impurity-BEC system, we expect that the largesplittings of polaron energies and effective masses amongdifferent spin components in the strong-coupling regimewill be observed in the near future experiment.In summary, we reveal polaron effects where a spin-1 impurity atom immersed in a ferromagnetic phase ofspinor BEC. In the weak-coupling regime, the contribu-tions from different types of collective modes are stud-ied with the perturbation theory. In the strong-couplingregime, the gapless phonon mode plays a dominant rolein the formation of self-trapping states. We predict crit-ical values of coupling constants by a variational treat-ment, where the impurity start to transit from a mobilestate to a localized state. The properties of polarons arebest manifested on the splittings of energies and effectivemasses among different spin components. Our results areof particular significance for exploring the new featuresof polaron effects in spinor BEC, which are very distinctfrom the scalar case.
VI. ACKNOWLEDGMENTS
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