Polaronic and dressed molecular states in orbital Feshbach resonances
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Polaronic and dressed molecular states in orbital Feshbachresonances
Junjun Xu and Ran Qi Department of Physics, University of Science and Technology Beijing, Beijing 100083, China Department of Physics, Renmin University of China, Beijing 100872, ChinaReceived: date / Revised version: date
Abstract.
We consider the impurity problem in an orbital Feshbach resonance (OFR), with a single excitedclock state | e ↑(cid:105) atom immersed in a Fermi sea of electronic ground state | g ↓(cid:105) . We calculate the polaroneffective mass and quasi-particle residue, as well as the polaron to molecule transition. By including oneparticle-hole excitation in the molecular state, we find significant correction to the transition point. Thistransition point moves toward the BCS side for increasing particle densities, which suggests that thecorresponding many-body physics is similar to a narrow resonance. The behaviours of a system with impurities are of par-ticular interest in condensed matter physics. One of thefamous examples is the Kondo effect, where the electricresistivity changes dramatically due to the magnetic im-purities at low temperature [1]. Theoretically, we can con-sider an impurity model as minority spin- ↑ particles im-mersed in a spin- ↓ Fermi sea. This system may be in aquasi-particle state, where each single impurity with thesurrounding spin- ↓ particles forms a polaron; or it couldbe in a dressed molecular state, where the impurity couldform cooper pairs with the spin- ↓ fermions.Such a simplified model is especially realizable in coldatomic systems, due to the newly development of lasercooling and trapping[2,3]. Furthermore, the realization ofmagnetic Feshbach resonances allows one to study thecrossover from Bose-Einstein condensation (BEC) side toBardeen-Cooper-Schrieffer (BCS) side continuously [4,5].On the deep BCS side, the ground state is the polaronicstate, while the dressed molecule with pairing around Fermienergy is energetically not favarable. On the deep BECside, the molecule is deeply bounded and becomes theground state. This impurity related problems have beenstudied in various experimental and theoretical researches[6,7,8,9,10,11,12,13,14].Recent studies of many-body physics across narrowFeshbach rsonances have revealed rich new physics due tothe strong energy dependence in scattering amplitude[15,16].For example, the polaron to molecule transition will moveto the BCS side for sufficient narrow resonance width[7,17,18,19]. However, most narrow resonances discoveredin alkli atoms is difficult to control due to the smallness a e-mail: [email protected] b e-mail: [email protected] of ∆B (i.e. the resonance width in magnetic field). Re-cent discovery of orbital Feshbach rsonance (OFR) in al-kali earth Yb atoms has opened a door for a new classof resonance system [20,21,22]. A lot of exciting physics,for example, the massive Leggett mode, is hopefully to beobserved in such systems [23]. More recent studies haveshown that the superfulid transition temperature of Fermigas across such resonances behaves similar to that acrossa narrow resonance while the very large value of ∆B al-lows great controllability in many-body system [24]. Soone natural question is how the low temperature proper-ties behave in such systems. We address this question byinvestigate polaronic and molecular states in this paper.Similar problems have been studied by Chen et al with avariational method [25]. We will show that such a varia-tional method is equivalent to the T -matrix approxima-tion. Furthermore, we will include one particle-hole exci-tation in the molecular state and get significant correctionto the polaron to molecule transition point.The OFR consists of an orbital electronic ground state | g (cid:105) and an excited clock state | e (cid:105) with the nuclear spin- ↑ or ↓ [20,21,22]. This system has an open channel | g ↓ + e ↑(cid:105) and a close channel | g ↑ + e ↓(cid:105) with the energy shift be-tween this two channels δ tuned by the magnetic field. Weconsider here a single | e ↑(cid:105) impurity immersed in a Fermisea | F S g ↓ (cid:105) of | g ↓(cid:105) atoms. The coupling between these twochannels cause the atoms to convert between the open andclose channels. Different from the magnetic Feshbach res-onances, the OFR can have two kinds of particle-hole andpairing contribution in the polaronic and molecular state,which will be investigated in the following.This paper is organized as follows. First, we look atthe polaron effective mass and quasi-particle residue, theresults of which show narrow resonance behaviour in theOFR. The particle-hole and pairing contribution from theopen and close channels in the polaronic and molecular a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Junjun Xu, Ran Qi: Polaronic and dressed molecular states in orbital Feshbach resonances state are also considered. Then we investigate the polaronto molecule transition in this system. The transition pointis found to move toward the BCS side when increasing theparticle density. At last, we give our conclusions.
We consider an alkali earth system with a Fermi sea of | g ↓(cid:105) atoms and a single | e ↑(cid:105) impurity. The Hamiltoniancan be written as H = H + V + V , (1)where H = (cid:88) k ε k a † k ,e ↑ a k ,e ↑ + (cid:88) k ( ε k − µ F ) a † k ,g ↓ a k ,g ↓ + (cid:88) k ( ε k + δ/ (cid:16) a † k ,e ↓ a k ,e ↓ + a † k ,g ↑ a k ,g ↑ (cid:17) is the non-interaction Hamiltonian for open and close chan-nels with ε k = ¯ h k / (2 m ) and µ F = E F is the chemicalpotential of | g ↓(cid:105) atoms. The interaction part is V = g (cid:88) kpq (cid:16) a † k / q ,e ↑ a † k / − q ,g ↓ a k / − p ,g ↓ a k / p ,e ↑ + a † k / q ,e ↓ a † k / − q ,g ↑ a k / − p ,g ↑ a k / p ,e ↓ (cid:17) ,V = g (cid:88) kpq (cid:16) a † k / q ,e ↑ a † k / − q ,g ↓ a k / − p ,g ↑ a k / p ,e ↓ + H . c . (cid:17) with g and g gover the inter and intra coupling be-tween open and close channels. They are connected to thespin singlet and triplet interaction strength g + and g − as g = ( g + + g − ) / g = ( g − − g + ) / /g ± = m/ (4 π ¯ h a ± ) − (cid:80) k / (2 (cid:15) k ).Here δ is the energy shift between the two channels whichcan be tunned by the magnetic field. The effective two-body s -wave scattering length a s = [ − a + (cid:113) mδ/ ¯ h ( a − a )] / ( a (cid:113) mδ/ ¯ h −
1) with the scattering length of interand intra interaction a = ( a + + a − ) / a = ( a − − a + ) / δ = ¯ h /ma [20].For the following comparison, we plot a s as a functionof detuning δ for single-component particle densities n = k F / (6 π ) = 0 . × cm − and 5 × cm − in Fig.1. Hereafter we choose the experimental parameter a + =1900 a and a − = 200 a for Yb. [21,22].The single impurity can form a polaronic state with thesurrounding particles, exciting particle-hole excitations inthis system. In a many-body calculation, the polaron en-ergy corresponds to the pole of G ( p , ω ) at p = 0, i.e., E = (cid:60) [ Σ (0 , E )], where G ( p , ω ) = 1 / [ ω − ε p − Σ ( p , ω )] isthe single particle propagator of the impurity and Σ ( p , ω )is the self-energy. Basically, one can expand Σ ( p , ω ) inthe number of particle-hole excitations. As noticed in [9], δ / E F - - - / k F a s ( a ) δ / E F - - - / k F a s ( b ) Fig. 1.
The s -wave scattering length a s as a functionof detuning δ . The red and blue lines in (a) and (b) arefor single-component densities n = k F / (6 π ) = 0 . × cm − and 5 × cm − respectively.the one particle-hole approximation in the polaronic statecorresponds to the ladder self-energy Σ ( p , ω ) = (cid:90) d q (2 π ) Θ ( q − k F ) T oo ( p + q , ω + ε q ) , (2)where T oo ( q , E + ε q ) ≡ (cid:88) q 2, where r is the effective range. Since r is negative here, as k increases, one needs a larger − /a s to produce the same energy-dependent scattering length a s ( k ) [24]. This will lead to the shift of inverse effectivemass and quasi-particle residue toward the BCS side asone increases the particle densities, which are similar asprevious studies on narrow Feshbach resonances [19]. Inthe deep BEC regime at around detuning δ = 0, we see adiscrepancy of this shift. We attribute this to the failure ofeffective range expansion and a high-order correction be-yond effective range may be needed in the phase shift. Asillustrated in Fig. 2(c) and (d) the quasi-particle residuedominates in the deep BCS regime. This is because theparticle-hole contribution from close channel is not ener-getically favourable due to a large detuning at deep BCSregime, and the open channel contribution with pairingaround Fermi energy has higher energy than the quasi-particle contribution. When go to the deep BEC side, as δ decreases to zero, the polaron reduces its energy by pair-ing at zero energy in the close channel, resulting the closechannel particle-hole contribution as the dominant term. The single impurity can also form a dressed molecularstate by pairing in the open or close channel. Chen et - - - / k F a s P o / P c ( a ) - - - - / k F a s P o / P c ( b ) Fig. 3. The open and close channel contribution P o = (cid:80) k>k F | B k | (solid lines), P c = (cid:80) p | C p | (dashed lines)in a molecular state. The red and blue colors are definedas in Fig. 1. al have considered such dressed molecular state withoutincluding particle-hole excitations [25], and they find thedimer energy corresponds to the Thouless pole of our T -matrix T oo defined in Eq. (3) for q = 0, i.e.,1 − g χ + ∆χ o χ c = 0 , (10)where χ o/c = χ o/c (0 , ω ) | ω = E , χ = χ o + χ c , and we havedefined E = µ − µ F as the relative dimer energy. Similarly,the pairing from open and closed channel contributionscan be written as (cid:88) k>k F | B k | = g ∂ ω χ o g ∂ ω χ o + ( g − ∆χ o ) ∂ ω χ c , (11) (cid:88) p | C p | = ( g − ∆χ o ) ∂ ω χ c g ∂ ω χ o + ( g − ∆χ o ) ∂ ω χ c . (12)We show these two contributions for different particle den-sities in Fig. 3. At the deep BCS regime where the detun-ing δ is large, the open channel molecule is energeticallyfavourable and dominates. Similar like the polaron case,when go to the BEC side, the detuning δ becomes smalland this energy detuning is overcome by the energy gainfrom pairing at Fermi sea in the open channel to zeroenergy pairing in the close channel. So there will be acrossover from open channel to close channel contribu-tions. We see the cross point moves toward the BCS sidefor increasing particle densities in Fig. 3.Based on past experience in the single impurity prob-lem [17], the particle-hole excitation has significant in-fluence on the location of polaron to molecule transitionpoint, thus to get the transition point it is necessary toinclude one particle-hole excitation in the molecular state.In this case, the molecular state wavefunction can be writ-ten as | Ψ M (cid:105) = (cid:18) (cid:88) k (cid:48) B k a † k ,e ↑ a †− k ,g ↓ + (cid:88) p C p a † p ,e ↓ a †− p ,g ↑ + (cid:88) kk (cid:48) q (cid:48) B k (cid:48) kq a † q − k − k (cid:48) ,e ↑ a † k (cid:48) ,g ↓ a † k ,g ↓ a q ,g ↓ + (cid:88) pkq (cid:48) C pkq a † q − k − p ,e ↓ a † p ,g ↑ a † k ,g ↓ a q ,g ↓ (cid:19) | F S g ↓ (cid:105) , (13)where the prime means the summation is restricted to k ( k (cid:48) ) > k F , and q < k F . The last two terms on the right Junjun Xu, Ran Qi: Polaronic and dressed molecular states in orbital Feshbach resonances include the particle-hole contributions to open and closechannel molecules. Similar like previous sections, the vari-ation of (cid:104) Ψ M | H − µ | Ψ M (cid:105) gives the following coupled equa-tions E k B k = g (cid:88) k (cid:48) B k + g (cid:88) p C p − g (cid:88) k (cid:48) q (cid:48) B k (cid:48) kq − g (cid:88) pq (cid:48) C pkq , (14) E p C p = g (cid:88) p C p + g (cid:88) k (cid:48) B k , (15) E k (cid:48) kq B k (cid:48) kq = g (cid:88) k (cid:48)(cid:48) (cid:48) ( B k (cid:48)(cid:48) kq − B k (cid:48)(cid:48) k (cid:48) q ) − g B k − B k (cid:48) )+ g (cid:88) p ( C pkq − C pk (cid:48) q ) , (16) E pkq C pkq = g (cid:88) p (cid:48) C p (cid:48) kq − g B k + 2 g (cid:88) k (cid:48) (cid:48) B k (cid:48) kq , (17)where E k = E − (cid:15) k , E p = E − δ − (cid:15) p , E k (cid:48) kq = E − (cid:15) q − k − k (cid:48) − (cid:15) k (cid:48) − (cid:15) k + (cid:15) q , and E pkq = E − δ − (cid:15) q − k − p − (cid:15) p − (cid:15) k + (cid:15) q . Here E = µ − µ F is the relative dimer energy. Bydefining η = g − / (cid:80) k (cid:48) B k + (cid:80) p C p ), ξ = g + / (cid:80) k (cid:48) B k − (cid:80) p C p ), η kq = 2 g (cid:80) k (cid:48) (cid:48) B k (cid:48) kq + g (cid:80) p C pkq , and ξ kq =2 g (cid:80) k (cid:48) (cid:48) B k (cid:48) kq + g (cid:80) p C pkq , the coupled equations canbe re-written as E k B k = η + ξ − (cid:88) q (cid:48) η kq , (18) E p C p = η − ξ, (19) E k (cid:48) kq B k (cid:48) kq = 12 ( η kq − η k (cid:48) q ) − g B k − B k (cid:48) ) , (20) E pkq C pkq = ξ kq − g B k . (21)From Eq. (18) and (19) we have η = − g − − g + g − χ c − g χ + ∆χ o χ c (cid:88) kq (cid:48) η kq E k , (22) ξ = − g + − g + g − χ c − g χ + ∆χ o χ c (cid:88) kq (cid:48) η kq E k , (23)where χ o = χ o (0 , E ) = (cid:80) k (cid:48) /E k , χ c = χ c (0 , E ) = (cid:80) p /E p , χ = χ o + χ c , and ∆ = g − g . Plug η and ξ into Eq. (18) we have B k = − (cid:88) k (cid:48) q (cid:48) (cid:48) η k (cid:48) q (cid:48) γE k E k (cid:48) − (cid:88) q (cid:48) (cid:48) η kq (cid:48) E k , (24) where γ = 1 /T oo (0 , E ) = 1 /a − χ o with a = ( g − ∆χ c ) / (1 − g χ c ). From Eq. (20) and (21) we have η kq = 11 − g χ o kq (cid:2) g χ c kq ξ kq − ( g χ o kq + g χ c kq ) B k − g (cid:88) k (cid:48) (cid:48) η k (cid:48) q E k (cid:48) kq + g (cid:88) k (cid:48) (cid:48) B k (cid:48) E k (cid:48) kq (cid:35) , (25) ξ kq = 11 − g χ c kq (cid:2) g χ o kq η kq − g g χ kq B k − g (cid:88) k (cid:48) (cid:48) η k (cid:48) q E k (cid:48) kq + g g (cid:88) k (cid:48) (cid:48) B k (cid:48) E k (cid:48) kq (cid:35) , (26)where χ o kq = χ o ( q − k , E + (cid:15) q − (cid:15) k ) = (cid:80) k (cid:48) (cid:48) /E k (cid:48) kq , χ c kq = χ c ( q − k , E + (cid:15) q − (cid:15) k ) = (cid:80) p /E pkq , and χ kq = χ o kq + χ c kq .Plug ξ kq in Eq. (26) into Eq. (25) we have(1 − g χ o kq ) η kq = g χ o kq χ c kq − g χ c kq η kq − (cid:34) g − ∆χ c kq − g χ c kq (1 + g χ o kq ) − g (cid:35) B k − g − ∆χ c kq − g χ c kq (cid:88) k (cid:48) (cid:48) η k (cid:48) q E k (cid:48) kq + g ( g − ∆χ c kq )1 − g χ c kq (cid:88) k (cid:48) (cid:48) B k (cid:48) E k (cid:48) kq . So by plunging B k into above equation and after somesimplification we get the final integral equation γ E η kq = (cid:88) k (cid:48) q (cid:48) (cid:48) η k (cid:48) q (cid:48) γE k (cid:48) E k + (cid:88) q (cid:48) (cid:48) η kq (cid:48) E k − (cid:88) k (cid:48) (cid:48) η k (cid:48) q E k (cid:48) kq , (27)where γ E = 1 /T oo ( q − k , E + (cid:15) q − (cid:15) k ) = 1 /a E − χ o kq ,with a E = ( g − ∆χ c kq ) / (1 − g χ c kq ). We see the scatteringlength corresponds to a s = a ( E → 0) = a E ( k → , q → , E → q = 0 as an approximation. We show the polaronand molecule energy for two different densities in Fig. 4.For comparison, we also show the molecule energy with-out particle-hole excitations as dashed lines. The polaronto molecule transition is shown in Fig. 5 for increasingparticle densities. The inclusion of one particle-hole exci-tation in the molecular state makes significant correctionto the transition point, as shown in Fig. 5(b), where thedashed line indicates the result without particle-hole ex-citation in the molecular state. In the zero-density limit,the energy dependence in scattering amplitude can be ne-glected and our results recovers that of a single-channelmode as it should be. As the density increases, the po-laron to molecule transition point goes toward the BCSside for increasing densities, which is similar to previousstudies in narrow Feshbach resonances [17,18,19]. To conclude, we have studied the polaronic and dressedmolecular states in a OFR, considering a single | e ↑(cid:105) atom unjun Xu, Ran Qi: Polaronic and dressed molecular states in orbital Feshbach resonances 5 - - - - - - - - - - - - / E F - /( k F a s ) E / E F - /( k F a s )( a ) ( b ) Fig. 4. The polaron and molecule energy as a function ofinteraction strength for different densities. The solid linesare polaron energies including one particle-hole excitation,while the dashed/dash-dotted lines are molecule energieswithout/with one particle-hole excitation. Here the redand blue colors are defined as in Fig. 1. - - - - - - - - - - - - ( a ) ◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ - - - - - ( b ) E / E F - /( k F a s ) - / ( k F a c ) n [ cm - ] Fig. 5. The polaron to molecule transition for differ-ent densities. (a) The solid/dash-dotted lines are po-laron/molecule energies including one particle-hole exci-tation, with the red and blue colors are defined as in Fig.1 and the black lines are results of single-channel limits.(b) The critical interaction strength as a function of par-ticle density. For comparing, we also plot the results with-out particle-hole excitation in the molecular state as thedashed line in (b).immersed in a | g ↓(cid:105) Fermi sea. Our T -matrix approxima-tion is found to be equivalent to the variational wave func-tion method. We find significant correction to the polaronto molecule transition point by including one particle-holeexcitation in the molecular state. The transition pointmoves toward the BCS side for increasing particle den-sities. The effective mass and quasi-particle residue is alsoconsistent with the finite effective range behaviour. Differ-ent from single-channel Feshbach resonances, in the OFRwe have two pairing contributions, corresponding to theopen and close channels. The crossover from open to closechannel pairing moves toward the BCS side for increasingparticle densities at both polaronic and molecular states.Note that our results only include one particle-holeexcitation. Studies in narrow Feshbach resonances haveshown that the inclusion of a second particle-hole excita-tion only makes slight changes to the system [17]. Acknowledgments We are grateful to Matthias Punk for his help on the nu-merical calculation. JX is supported by NSFC (No. 11504021),and FRFCU (No. FRF-TP-17-023A2). RQ is supported byNSFC (No. 11774426), the Fundamental Research Fundsfor the Central Universities, and the Research Funds of Renmin University of China under Grants No. 15XNLF18and No. 16XNLQ03. Author contribution JX performed the numerical calculation and RQ providedthe method. 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