Confinement-induced Resonance of Alkaline-earth-metal-like Atoms in Anisotropic Quasi-one-dimensional Traps
CConfinement-induced Resonance of Alkaline-earth-metal-like Atoms in AnisotropicQuasi-one-dimensional Traps
Qing Ji, Ren Zhang, ∗ Xiang Zhang, † and Wei Zhang
1, 3, ‡ Department of Physics, Renmin University of China, Beijing 100872, China Department of Applied Physics, School of Science,Xi’an Jiaotong University, Shanxi 710049, China Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,Renmin University of China, Beijing 100872, China (Dated: September 5, 2018)We study the confinement-induced resonance (CIR) of
Yb atoms near an orbital Feshbachresonance in a quasi-one-dimensional tube with transversal anisotropy. By solving the two-bodyscattering problem, we obtain the location of CIR for various anisotropy ratio and magnetic field.Our results show that the anisotropy of the trapping potential can serve as an additional knob to tunethe location of CIR. In particular, one can shift the location of CIR to the region attainable in currentexperiment. We also study the energy spectrum of the system and analyze the properties of CIRfrom the perspective of bound states. We find that as the orbital Feshbach resonance acquires twonearly degenerate scattering channels, which in general have different threshold energies, CIR takesplace when the closed channel bound state energy becomes degenerate with one of the thresholds.
I. INTRODUCTION
Confinement-induced resonance (CIR) plays a key rolein the quantum simulation of 1D quantum models withcold atoms [1–7], by means of which the effective inter-action strength for a quantum atomic gas confined inan elongated quasi-1D geometry can be resonantly en-hanced. As was firstly proposed by Olshanii [1], thelow-energy two-body scattering processes within a quasi-1D trapping potential can be well described by an ef-fective 1D model, where the 1D interaction strength isintimately related to both the three-dimensional (3D) s -wave scattering length a s and the external confinement.Later, it is realized that this phenomenon can be under-stood in the framework of Feshbach resonance, where theground state in the strongly confined transversal plane isregarded as the open channel, and the excited states as awhole assume the closed channel. The effective 1D inter-action becomes resonant as the bound state in the closedchannel degenerates with the open channel threshold [2].Since in the dilute limit, the properties of the system aredominated by two-body processes, this observation pavesthe avenue towards the simulation of 1D quantum many-body Hamiltonian with tunable interaction in a quasi-1Dconfiguration. With the aid of CIR, the Tonks-Girardeaugas and super Tonks-Girardeau gas have been exploredexperimentally in cold alkali-metal atoms [8, 9].The study of CIR has also been then extended to alkali-metal atoms in quasi-1D confinement with transversalanisotropy [3–7]. The presence of anisotropy can serve asanother knob to tune the effective 1D interaction strengthand the position of CIR [4–6]. Besides, the anharmonicity ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] of the transversal trapping potential is suggested to becrucial to explain the splitting of resonances observedin experiments [3, 7]. Besides, the inclusion of syntheticspin-orbit coupling can provide another tuning parameterthat can alter the position of CIR [10, 11].Recently, the ultracold quantum gas of alkaline-earth-metal-like (AE) atoms has attracted great attention ow-ing to its unique advantages in quantum simulation. Inaddition to the electronic ground state S , the AE atomhas an extremely long-lived clock state P whose single-atom lifetime can be as long as many seconds. Theground and clock state manifolds hence can serve as anadditional degree of freedom, which is referred as orbit .For fermionic AE atoms, the nuclear spin is nonzero anddecouples from the electronic degree of freedom in theground and clock states. As the interatomic interaction isdominated by electronic wave functions, the nuclear spindegree of freedom supports an SU( N ) symmetry, where N denotes the number of possible nuclear spin states.This property paves the route towards the investigationof fermionic systems with a plethora of SU( N ) symmetryin AE atoms [12–20].Another distinctive feature of AE atoms is the ex-istence of spin-exchange scattering between the groundstate S and clock state P [17–19], which is not onlycrucial for the technique of orbital Feshbach resonance(OFR) to tune the interatomic interaction [21–23], butalso a essential ingredient for the simulation of Kondophysics [12, 24–27]. It is then of great interest to find away to enhance the spin-exchange interaction strength,and in turn the Kondo temperature to the extent achiev-able in current experiments. One possible method toachieve this goal is to tune the system through a CIRin a quasi-1D confinement. [25, 26] To one’s delight, theresonant enhancement of spin-exchange scattering nearCIR has been demonstrated in Yb atoms [28]. In suchexperiment, the ground state and clock state atoms aretrapped in isotropic quasi-1D tubes simultaneously by a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p magic wave length lasers along the x - and y -directions,while the clock state atoms are further trapped by an ex-ternal potential along the z -direction. Thus, the intensityof lasers along the transversal ( x – y ) and axial( z ) direc-tions can be viewed as two knobs to control the locationof CIR, which enriches the tool box for the simulationwith AE atoms.In the current manuscript, we investigate the two-bodyscattering problem of AE atoms in a quasi-1D confine-ment with transversal anisotropy, i.e., the confinementstrengths along the x - and y -directions are in general dif-ferent. By investigating the low-energy two-body scatter-ing processes, we show that the transversal anisotropy ofthe trapping potential can serve as another tuning knobto manipulate the location of CIR. For alkali-metal atomsin quasi-1D confinement, CIR can be regarded as a Fes-hbach resonance where the transversal ground state andexcited states as a whole assume the open and closedchannels, respectively [2, 5, 10] However, for AE atomsnear an OFR, there are four atomic states involved andhence two scattering channels which are slightly shiftedby Zeeman effect. As a result, by confining the AE atomsin a quasi-1D trap, the two channels would acquire differ-ent open channel threshold energies. A natural questionwould be which one of the thresholds should be inter-sected by the closed channel bound state energy whenCIR occurs. By comparing with the solution of scatter-ing states, we show that CIR takes place when the closedchannel bound state energy is degenerate to only one ofthe threshold energies.The remainder of the manuscript is organized as fol-lows. In Sec. II, we present the formalism of low-energytwo-body scattering between two AE atoms near anOFR, first in a quasi-1D trap with transversal anisotropy,and then using an effective 1D model. By matching so-lutions of the two models, we obtain the effective inter-action strength in the 1D model in Sec. III. Then, wediscuss in Sec. IV the energy spectrum of the two-bodybound state and analyze CIR from the perspective ofbound state. Finally, we summarize in Sec. V. II. SCATTERING STATES
The leading goal of dimension reduction from quasi-one to one dimension is to find a concise 1D effec-tive model Hamiltonian which captures the low-energyphysics of the original quasi-1D Hamiltonian by integrat-ing out the degrees of freedom with high energy. For aFermi gas at low temperature, the energy regime of in-terest is usually close to the Fermi surface, which lieswithin the low-energy regime for a dilute system. Thus,a natural scheme is to match the low-energy scatteringamplitude close to the scattering threshold of quasi-1Dsystem to that of a 1D model. In this section, we solvethe scattering problem for AE atoms close to an OFRin both quasi-1D and 1D geometries, and determine theposition of CIR from the resulting scattering amplitude. g e z x y a x a y FIG. 1: Energy levels of alkaline-earth-metal-like atomstrapped in an anisotropic quasi-one-dimensional tube. Theblue and red balls denote atoms in the ground state S andclock state P , respectively. The up and down arrows de-note different nuclear spin states. The trapping potential isproduced by a laser with magic wave length which can trapthe ground state and clock state simultaneously. a x and a y are the typical lengths of the trapping potential along the x -and y -direction, respectively, which can be tuned by changingthe laser intensity and profile. The Zeeman energy differencebetween the | g ↑ ; e ↓(cid:105) and | g ↓ ; e ↑(cid:105) channels is δ ≡ δ e − δ g ,where δ g and δ e are the Zeeman energy of the ground stateand clock state manifolds, respectively. A. Quasi-one-dimension
First we would like to introduce some basic propertiesof AE atoms. Considering two AE atoms in the ground S and clock P states (denoted respectively by the or-bital | g (cid:105) and | e (cid:105) states) with different nuclear spin states(denoted by pseudo-spin | ↑(cid:105) and | ↓(cid:105) ) confined in a quasi-one-dimensional tube as shown in Fig. 1, one can definetwo sets of basis {| + (cid:105) , |−(cid:105)} and {| g ↑ ; e ↓(cid:105) , | g ↓ ; e ↑(cid:105)} ,which are |±(cid:105) ≡ (1 / | ge (cid:105) ± | eg (cid:105) )( | ↑↓(cid:105) ∓ | ↓↑(cid:105) ) and | gs ; es (cid:48) (cid:105) ≡ (1 / √ | gs (cid:105)| es (cid:48) (cid:105) − | es (cid:48) (cid:105)| gs (cid:105) ) with { s, s (cid:48) } = {↑ , ↓} . The two sets of basis are related to each other viathe unitary transformation |±(cid:105) = (1 / √ | g ↑ ; e ↓(cid:105) ∓ | g ↓ ; e ↑(cid:105) ). One of the salient features of the AE atoms isthat the interaction Hamiltonian is diagonal in the ba-sis of {| + (cid:105) , |−(cid:105)} , i.e., ˆ V = V + ( r ) | + (cid:105)(cid:104) + | + V − ( r ) |−(cid:105)(cid:104)−| .The associated s -wave scattering length of V ± ( r ) are de-noted by a s ± . For Yb atoms, a s + ≈ a and a s − ≈ . a with a the Bohr’s radius. On the otherhand, the free Hamiltonian is diagonal in the basis of {| g ↑ ; e ↓(cid:105) , | g ↓ ; e ↑(cid:105)} . Thus the total Hamiltonian can bewritten asˆ H Q1D = (cid:20) − (cid:126) µ ∇ r + µ ω x x + µ ω y y + V ( r ) (cid:21) ( P ↑↓ + P ↓↑ )+ δ P ↑↓ + V ( r )( S ex + S † ex ) , (1)where µ is the reduced mass of the two atoms, and ω x and ω y are the trapping frequencies along the x - and y -directions, respectively. To quantify the anisotropystrength of the tapping potential, we define the ratioof ζ = ω x /ω y . The specific case of ζ = 1 means thetrapping potential is isotropic in the transverse plane.The projection operators P ss (cid:48) and S ex are defined as P ss (cid:48) = | gs ; es (cid:48) (cid:105)(cid:104) gs ; es (cid:48) | and S ex = | g ↑ ; e ↓(cid:105)(cid:104) g ↓ ; e ↑ | ,respectively. The Zeeman energy difference betweenthe | g ↑ ; e ↓(cid:105) and | g ↓ ; e ↑(cid:105) channels is denoted by δ = ( m ↑ − m ↓ )( δµ ) B , where ( δµ ) = 2 π (cid:126) × Yb atoms. The interaction potential V ( r ) =( V + ( r ) + V − ( r )) / V ( r ) = ( V − ( r ) − V + ( r )) / V i =0 , ( r ) = 2 π (cid:126) a si µ δ ( r ) ∂∂r r (2)with a s = ( a s + + a s − ) / a s = ( a s − − a s + ) / (cid:15) and theZeeman energy δ is much smaller than the energy gapof the harmonic trap, i.e. { (cid:15), δ } (cid:28) { (cid:126) ω x , (cid:126) ω y } . Hence,when the two atoms are faraway, the interaction potentialdecays to zero and the system is dominated by the freeHamiltonian. The two atoms thus reside in the groundstate | n x = 0 , n y = 0 (cid:105) of the transverse trapping poten-tial with n x and n y the quantum numbers of the cor-responding harmonic potentials. When the two atomscollide in a short distance, the interaction potential be-comes important and couples the ground state to excitedstates of the trapping potential.The general scattering wave function associated withthe Hamiltonian Eq. (1) takes the following form | Ψ( r ) (cid:105) = Ψ ( ↑↓ ) ( r ) | g ↑ ; e ↓(cid:105) + Ψ ( ↓↑ ) ( r ) | g ↓ ; e ↑(cid:105) , (3)where the functions Ψ ( ↑↓ ) ( r ) and Ψ ( ↓↑ ) ( r ) are given byΨ ( ↑↓ ) ( r ) = (cid:104) αe ik ( ↑↓ ) z + f ( ↑↓ ) e ik ( ↑↓ ) | z | (cid:105) ζ / φ ( (cid:112) ζx ) φ ( y )+ (cid:48) (cid:88) ( n x ,n y ) B ( ↑↓ ) n x ,n y e − κ ( ↑↓ ) nx,ny | z | φ n x ( (cid:112) ζx ) φ n y ( y ); (4)Ψ ( ↓↑ ) ( r ) = (cid:104) βe ik ( ↓↑ ) z + f ( ↓↑ ) e ik ( ↓↑ ) | z | (cid:105) ζ / φ ( (cid:112) ζx ) φ ( y )+ (cid:48) (cid:88) ( n x ,n y ) B ( ↓↑ ) n x ,n y e − κ ( ↓↑ ) nx,ny | z | φ n x ( (cid:112) ζx ) φ n x ( y ) . (5)Here, α and β denote the possible amplitudes of the in-cident atoms in the | g ↑ ; e ↓(cid:105) and | g ↓ ; e ↑(cid:105) channels, re-spectively. Considering the zero-range pseudo-potentialin which the scattering state with odd parity along the z -direction fails to feel the potential, we only focus on theeven parity part such that the summation (cid:80) (cid:48) runs overall even combinations of ( n x , n y ) except the ground state n x = n y = 0. Besides, φ ( x ) is the ground state wave function of 1D harmonic oscillator, and the parameters k ( ↑↓ ) , k ( ↓↑ ) , κ ( ↑↓ ) n x ,n y and κ ( ↓↑ ) n x ,n y are defined as k ( ↑↓ ) = (cid:112) µ ( (cid:15) − δ ) / (cid:126) ,κ ( ↑↓ ) n x ,n y = (cid:113) µ ( n x (cid:126) ω x + n y (cid:126) ω y − (cid:15) + δ ) / (cid:126) ,k ( ↓↑ ) = (cid:112) µ(cid:15)/ (cid:126) ,κ ( ↓↑ ) n x ,n y = (cid:113) µ ( n x (cid:126) ω x + n y (cid:126) ω y − (cid:15) ) / (cid:126) . (6)The scattering amplitudes f ( ↑↓ ) and f ( ↓↑ ) , as wellas the coefficients B ( ↑↓ ) n x ,n y and B ( ↓↑ ) n x ,n y are to be deter-mined by solving the Shr¨odinger equation H | Ψ( r ) (cid:105) = E | Ψ( r ) (cid:105) . To this end, one can perform the oper-ation lim ε → (cid:82) + ε − ε dz (cid:82)(cid:82) ∞−∞ dxdyζ / φ ∗ n x ( √ ζx ) φ ∗ n y ( y ) onthe both hand sides of the Shr¨odinger equation, then ob-tains the relations, f ( ↑↓ ) = 2 π | φ (0) | ζ / ik ( ↑↓ ) (cid:16) a s η ( ↑↓ ) + a s η ( ↓↑ ) (cid:17) , (7) B ( ↑↓ ) n x ,n y = − πζ / κ ( ↑↓ ) n x n y φ ∗ n x (0) φ ∗ n y (0) (cid:16) a s η ( ↑↓ ) + a s η ( ↓↑ ) (cid:17) , (8) f ( ↓↑ ) = 2 π | φ (0) | ζ / ik ( ↓↑ ) (cid:16) a s η ( ↓↑ ) + a s η ( ↑↓ ) (cid:17) , (9) B ( ↓↑ ) n x ,n y = − πζ / κ ( ↓↑ ) n x n y φ ∗ n x (0) φ ∗ n y (0) (cid:16) a s η ( ↓↑ ) + a s η ( ↑↓ ) (cid:17) , (10)where η ( ss (cid:48) ) = ∂∂z (cid:104) z Ψ ( ss (cid:48) ) ( x = 0 , y = 0 , z ) (cid:105)(cid:12)(cid:12)(cid:12) z → + . Sub-stituting Eqs. (7-10) into Eqs. (4) and (5) and furtherinto the expression of η ( ss (cid:48) ) , one finds the equations for η ( ss (cid:48) ) , (cid:112) ζ (cid:20) ia y k ( ↑↓ ) + Λ ( ↑↓ ) (cid:21) (cid:16) a s η ( ↑↓ ) + a s η ( ↓↑ ) (cid:17) = αζ / √ π − a y η ( ↑↓ ) , (11) (cid:112) ζ (cid:20) ia y k ( ↓↑ ) + Λ ( ↓↑ ) (cid:21) (cid:16) a s η ( ↓↑ ) + a s η ( ↑↓ ) (cid:17) = βζ / √ π − a y η ( ↓↑ ) , (12)where Λ ( ss (cid:48) ) = ∂∂z (cid:16) zλ ( ss (cid:48) ) ( z ) (cid:17)(cid:12)(cid:12)(cid:12) z → + with λ ( ss (cid:48) ) ( z ) = (cid:48) (cid:88) ( n x ,n y ) a y πκ ( ss (cid:48) ) n x ,n y | φ n x (0) | | φ n y (0) | e − κ ( ss (cid:48) ) nx,ny | z | . (13)We would like to stress that the derivative ∂/∂z andthe summation over the excited states (cid:80) (cid:48) can not beinterchanged as the summation is not uniformly conver-gent. Furthermore, it can be proved that in the limit of -20-10010 g = ( h = a s ! ) g + g - g c -1 a y =a s ! -20-10010 10 -1 a y =a s ! a y =a s ! E E b E th a y =a s ! E E b E th a y =a s ! E E b E th a y =a s ! E E b E th (a) = 1 (d) = 10(b) = 5 (c) = 6 : FIG. 2: Effective one-dimensional interaction g ’s as functions of trapping length a y . In our calculation, we take Yb atomas an example and use the following parameters a s + = 1900 a and a s − = 219 . a , ω x = ζω y and δ = 0. Note that g c isalways zero since the | + (cid:105) and |−(cid:105) channels are decoupled in the absence of effective magnetic filed. The location of CIR can bemanipulated by changing the anisotropy ratio. In each panel, the closed channel bound state energy E b and the open channelthreshold E th are shown in the insets, with the energy unit (cid:126) /µa s − . The position of the intersection between E b and E th coincides with the location of CIR determined by a scattering state analysis. z → + , λ ( ss (cid:48) ) ( z ) can be expanded as λ ( ss (cid:48) ) ( z ) = C − z + C + C z + · · · , (14)which implies the existence of Λ ( ss (cid:48) ) [29]. The singularitypart of λ ( ss (cid:48) ) ( z ) can be understood in the following way:when the distance of the two colliding atoms approacheszero, the interaction energy will dominate the behaviorof the atoms. In this regime, the wave function of thetwo atoms is not sensitive to the trapping potential, andhence should acquire the same singular behavior as in thethree-dimensional geometry. In our calculation, we per-form the summation over excited states numerically andfit λ ( ss (cid:48) ) ( z ) according to the series expansion in Eq. (14)to extract the zeroth-order term C .By solving Eqs. (11) and (12) and substituting the ex-pressions for η ( ↑↓ ) and η ( ↓↑ ) into Eqs. (7) and (9), onefinally obtains the explicit expression of the scatteringamplitudes (cid:18) f ( ↑↓ ) f ( ↓↑ ) (cid:19) = − ( I + iAP ) − (cid:18) αβ (cid:19) , (15)where the parameters I , A and P take the form I = (cid:18) (cid:19) , P = (cid:18) k ( ↑↓ ) k ( ↓↑ ) (cid:19) , (16) A = − a y √ ζ (cid:34)(cid:18) a s a s a s a s (cid:19) − + √ ζa y (cid:18) Λ ( ↑↓ )
00 Λ ( ↓↑ ) (cid:19)(cid:35) . (17) It can be seen that the interaction effect is manifested bythe presence of scattering lengths a s and a s in A . Weemphasize that Eq. (15) is valid only in the low-energylimit of { (cid:15), δ } (cid:28) { (cid:126) ω x , (cid:126) ω y } . What one needs to do fol-lowing is to construct an effective 1D Hamiltonian whichcan reproduce the scattering amplitude Eq. (15) in thesame energy regime. B. One-dimension
Now we construct an effective 1D model for the quasi-1D Hamiltonian Eq. (1) and analyze the correspondingscattering problem. The interaction part of this effectivemodel should be written asˆ V = [ g + | + (cid:105)(cid:104) + | + g − |−(cid:105)(cid:104)−| + g c | + (cid:105)(cid:104)−| + h . c . ] δ ( z ) , (18)where g + and g − are the effective 1D interactionstrengths corresponding to V + ( r ) and V − ( r ), respectively,and g c denotes the coupling between the | + (cid:105) and |−(cid:105) channels. We would like to stress that unlike the quasi-1D model, the effective interaction is in general not di-agonal in the 1D case even in the basis of {| + (cid:105) , |−(cid:105)} .The underlying reason is that in the presence of magneticfield, the free Hamiltonian is non-diagonal in the basis of {| + (cid:105) , |−(cid:105)} . When the excited states are integrated out,the coupling effect between | + (cid:105) and |−(cid:105) induced by thefree Hamiltonian can also be manifested in the effectiveinteraction, leading to a nonzero g c . However, in the ab- -15-55151 10 100-15-5515 g + g - g c FIG. 3: Effective one-dimensional interaction g ’s as functionsof anisotropy ratio ζ . Here, we take ω y = 100 kHz, and theZeeman shift as (a) δ = 0 . (cid:126) ω y and (b) δ = − . (cid:126) ω y . Otherparameters are same as in Fig. 2. In this case, g c is driven tobe divergent near the confinement-induced resonance. sence of magnetic field, the | + (cid:105) and |−(cid:105) channels decou-ple, and the quasi-1D Hamiltonian Eq. (1) can be reducedto two separate single-channel models with their corre-sponding interactions g ± . CIR within each channel thentakes place at the position given by the single-channelcalculation [4–7]. A similar effect has been discussed inthe exploration of CIR in spinor atoms. [30]In the basis of {| g ↑ ; e ↓(cid:105) , | g ↓ ; e ↑(cid:105)} , the effective 1D Hamiltonian can be written asˆ H = (cid:18) − (cid:126) µ d dz (cid:19) ( P ↑↓ + P ↓↑ ) + δ P ↑↓ + (cid:34) g + + g − + 2 g c P ↑↓ + g + + g − − g c P ↓↑ + g − − g + S ex + S † ex ) (cid:35) δ ( z ) , (19)where P ss (cid:48) and S ex have the same definition as that inEq.(1). Similar to the case of quasi-1D system, the gen-eral scattering wave function of Eq. (19) can be writtenas | Ψ ( z ) (cid:105) = (cid:16) αe ik ( ↑↓ ) z + f ( ↑↓ )1D e ik ( ↑↓ ) | z | (cid:17) | g ↑ ; e ↓(cid:105) + (cid:16) βe ik ( ↓↑ ) z + f ( ↓↑ )1D e ik ( ↓↑ ) | z | (cid:17) | g ↓ ; e ↑(cid:105) . (20)After some straightforward calculation, one can obtainthe scattering amplitudes f ( ↑↓ )1D and f ( ↓↑ )1D of the 1Dmodel, (cid:32) f ( ↑↓ )1D f ( ↓↑ )1D (cid:33) = − ( I + iA P ) − (cid:18) αβ (cid:19) , (21)where I and P are given in Eq. (16), and A is definedas A = − (cid:126) µ (cid:18) g + + g − + 2 g c g − − g + g − − g + g + + g − − g c (cid:19) − . (22)By comparing Eq. (22) with Eq. (15), and requiring f ( ↑↓ )1D = f ( ↑↓ ) and f ( ↓↑ )1D = f ( ↓↑ ) , one finally obtains theequation of the effective 1D interaction strengths (cid:18) a s a s a s a s (cid:19) − + √ ζa y (cid:18) Λ ( ↑↓ )
00 Λ ( ↓↑ ) (cid:19) = 4 (cid:126) √ ζµa y (cid:18) g + + g − + 2 g c g − − g + g − − g + g + + g − − g c (cid:19) − . (23)For the particular case of ζ = 1, Λ ( ↑↓ ) and Λ ( ↓↑ ) can besimplified as ζ H (1 / , δ/ (cid:126) ω y ) and ζ H (1 / , ζ H ( a, d ) denotes the Hurwitz-Zeta function.Thus, Eq. (23) reproduces the result in Ref. [25]. Foranother particular case that the magnetic field is zero,the | + (cid:105) and |−(cid:105) channels decouple and the solution ofEq. (23) can be significantly simplified as g ± = 2 (cid:126) √ ζµa y (cid:0) a y /a s ± + √ ζ Λ (cid:1) , g c = 0 , (24)where Λ ≡ Λ ( ↑↓ ) = Λ ( ↓↑ ) . Furthermore, it can be proved that in the isotropic case of ζ = 1, Λ = C = − ζ H (1 / , III. EFFECTIVE INTERACTION
For the general case with transversal anisotropy anda finite magnetic field, one can obtain the effective 1Dinteraction strengths by solving Eq. (23) numerically. InFig. 2, we present the variation of the effective interactionstrengths g ± and g c by varying the transverse trappingpotential for a set of different anisotropy ratios with Zee- -1 -0.5 0 0.5 1-0.500.51 -0.5 0 0.50.40.50.60.70.8-2 -1 0 1 20.511.522.53 -2 -1 0 1 26.577.588.59(a) =0.1(c) =1 (d) =15(b) =0.259 FIG. 4: Closed channel bound state energy E b (black solid) as a function of δ for various anisotropy ratios. CIR takes placewhen the closed channel bound state energy E b is equal to either E L th (blue) or E H th (red). Other parameters are same as inFig. 3. man shift δ = 0. In this figure, the transverse trap ischaracterized by the length scale a y ≡ (cid:112) (cid:126) / ( µω y ) associ-ated with the y -axis harmonic potential. In each panel,the resonance at the lower end of a y /a s − is associatedwith the potential V − ( r ), while the one at the higherend of a y /a s − is associated with V + ( r ). Specifically, inFig. 2(a), the system is isotropic in the transverse planewith ζ = 1 and the locations of the two resonances re-cover the result of Olshanii with a y /a s − = − ζ H (1 / , a y /a s + = − ζ H (1 / , ζ = 5, where the locations of the two res-onances are shifted to the right and reach a y /a s − = 1 . a y /a s + = 1 .
6, respectively. If the anisotropy isfurther enhanced, the locations of CIRs move back tothose of the isotropic case at about ζ = 6 . a y /a s − = 1 . a y /a s + = 1 . ζ = 10. This observa-tion shows that in the aid of transversal anisotropy, thelocation of CIR can be tuned to a large extent, which isfacilitative for the control of CIR in AE atoms in exper-iments.In Fig. 3, we show the effective 1D interactions as func-tions of anisotropy ratio ζ for various magnetic fields. Inthis calculation, we take ω y = 100 kHz and increase ω x tovary the anisotropy ratio ζ . As the Zeeman field couplesboth the | + (cid:105) and |−(cid:105) channels, the interactions g ± and g c all become divergent at CIR. In Fig. 3(a) and 3(b), weshow the CIR associated with the |−(cid:105) channel, while theZeeman shifts are taken as δ = 0 . (cid:126) ω y and δ = − . (cid:126) ω y ,respectively. First of all, it is clear that the location of CIR can be changed by tuning the magnetic field, whichworks together with the anisotropy ratio to provide aversatile toolbox to tune CIR to experimentally favor-able regimes. Secondly, g c is driven to be divergent nearCIR. This phenomenon implies that the effective 1D in-teraction no longer respects to the nuclear spin rotationalsymmetry and couples the | + (cid:105) and |−(cid:105) channels by con-taining non-zero g c . IV. BOUND STATE ANALYSIS
CIR can also be understood as a Feshbach resonance.The ground state of the transversal trapping potentialwith n x = n y = 0 is considered as the open channel,while all the excited states as a whole compose the closedchannel. The resonance takes place when the bound stateenergy of the closed channel becomes degenerate with theopen channel threshold.The bound state energy E b of the closed channelcan be obtained by solving the Shr¨odinger equation P † c ˆ H P c | Ψ b (cid:105) = E b | Ψ b (cid:105) with ˆ H = ˆ H Q1D + µω z z / P c the projection operator associated with the closed chan-nel [5, 29], and then take the limit ω z →
0. The resultingequation reads (cid:18) a s a s a s a s (cid:19) − + √ ζa y (cid:18) Λ ( ↑↓ )
00 Λ ( ↓↑ ) (cid:19) = 0 , (25)where Λ ( ↑↓ ) = F ( E b + δ ) / √ πζ and Λ ( ↓↑ ) = F ( E b ) / √ πζ -2-1012 / = h ! y c = 0 : c = 12 : FIG. 5: Locations of CIR in the parameter plane spanned by ζ and δ . In the regions of ζ (cid:46) .
259 and ζ (cid:38) .
158 (blue),CIR takes place when the closed channel bound state energyintersects with the lower threshold E L th , while in the regionof 0 . (cid:46) ζ (cid:46) .
158 (red), CIR happens when the closedchannel bound state intersects with the higher threshold E H th .Other parameters are same as in Fig. 3. with F ( E ) = (cid:90) ∞ dt (cid:32) √ ζe −E t (cid:112) (1 − e − ζt )(1 − e − t ) t − √ ζte −E t + 1 t / (cid:33) . (26)Notice that this expression is consistent with Eq. (23) bysetting the right-hand-side to zero. The reason for thisconsistency is that when CIR takes place, i.e., g (cid:48) s → ∞ ,there always exit a bound state near the threshold energy.By solving Eq. (25), one obtains the closed channelbound state energy as shown in the insets of Fig. 2 for δ = 0 and in Fig. 4 for the case of finite δ . Notice thatin all cases the intersection between the closed channelbound state energy and the open channel threshold takesplace at the same position of diverging effective 1D in-teraction, i.e., the location of CIR. This concludes thatthe results obtained by solving a scattering problem areconsist with those of the bound state analysis.As in the vicinity of an OFR, the Zeeman energies ofthe | g ↑ ; e ↓(cid:105) and | g ↓ ; e ↑(cid:105) channels are detuned by δ ,one can respectively define a threshold energy for eachof them, i.e., E ( ↑↓ )th = (cid:126) ( ω x + ω y ) / δ and E ( ↓↑ )th = (cid:126) ( ω x + ω y ) /
2. For convenience, we can further define thelower threshold and the higher threshold energy as E L th =min( E ( ↑↓ )th , E ( ↓↑ )th ) and E H th = max( E ( ↑↓ )th , E ( ↓↑ )th ), denotedby the blue and red lines in Fig. 4, respectively. Noticethat the difference between E L th and E H th , which is theZeeman energy shift δ , is in general much smaller thanthe trapping frequency of the transversal confinement. Thus, both of the two scattering channels are relevant tothe quasi-1D low-energy scattering process.In Fig. 4, we show the closed channel bound stateenergy as a function of Zeeman energy for variousanisotropy ratios. Our results show that in the regionsof ζ (cid:46) .
259 and ζ (cid:38) . E b = E L th , while in the region of0 . (cid:46) ζ (cid:46) . E b = E H th . At the critical points ζ c = 0 .
259 and12 . δ = 0. We also notice that the positionsof CIR are symmetric with respect to δ = 0, which re-flects the symmetry between the | g ↑ ; e ↓(cid:105) and | g ↓ ; e ↑(cid:105) channels with opposite Zeeman energy shift. The twobranches of CIR with varying anisotropy ratio and theZeeman energy detuning are summarized in Fig. 5. V. CONCLUSION
In summary, we study the two-body problem ofalkaline-earth-metal-like atoms confined in quasi-one-dimensional tube with transversal anisotropy. By solvingthe scattering problem, we find that the anisotropy ra-tio of the trapping potential can serve as a knob to tunethe location of confinement-induced resonance. This ex-tra controllability can work together with the Zeemanenergy to facilitate the tuning of CIR in experiments.Then we explore the CIR from the viewpoint of boundstate. Since in the vicinity of an orbital Feshbach reso-nance, the detuning of the two channels are in generalmuch smaller than the transversal trapping frequency,both scattering channels are of relevance in low-energyphysics, such that two open channel threshold energies E L th and E H th can be defined. We find that CIR takesplace when the closed channel bound state degeneratesto either E L th or E H th . Our results can be readily checkedin current experiments, where the position of CIR canbe determined by measuring the atom-loss rate and thebound state binding energy can be extracted from theradio frequency spectrum. Acknowledgement.
This work is supported bythe National Key R&D Program of China (Grant No.2018YFA0306501(W.Z.), 2018YFA0307601 (R.Z.)), theNational Natural Science Foundation of China (GrantNos. 11434011, 11522436, 11704408, 11774425, 11804268(RZ)), and the Research Funds of Renmin University ofChina (Grant Nos. 16XNLQ03, 18XNLQ15). X.Z. ac-knowledges support from the National Postdoctoral Pro-gram for Innovative Talents (Grant No. BX201601908)and the China Postdoctoral Science Foundation (GrantNo. 2017M620991). [1] M. Olshanii, Phys. Rev. Lett. , 938 (1998). [2] T. Bergeman, M. G. Moore, and M. Olshanii, Phys. Rev. Lett. , 163201 (2003).[3] E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Reich-sollner, V. Melezhik, P. Schmelcher, and H.-C. N¨agerl,Phys. Rev. Lett. , 153203 (2010).[4] S.-G. Peng, S. S. Bohloul, X.-J. Liu, H. Hu, and P. D.Drummond, Phys. Rev. A , 063633 (2010).[5] W. Zhang and P. Zhang, Phys. Rev. A , 053615 (2011).[6] V. S. Melezhik and P. Schmelcher, Phys. Rev. A ,042712 (2011).[7] S. Sala, P.-I. Schneider, and A. Saenz, Phys. Rev. Lett. , 073201 (2012).[8] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, andM. Rigol, Rev. Mod. Phys. , 1405 (2011)[9] Xi-Wen Guan, Murray T. Batchelor, and Chaohong Lee,Rev. Mod. Phys. , 1633 (2013)[10] R. Zhang and W. Zhang, Phys. Rev. A , 053605 (2013).[11] Y.-C. Zhang, S.-W. Song, and W.-M. Liu, Sci. Rep. ,4992 (2015).[12] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S.Julenne, J. Ye, P. Zoller, E. Demler, M. D. Lukin and A.M. Rey, Nature Phys. , 289 (2010).[13] S. Taie, R. Yamazaki, S. Sugawa, Y. Takahashi, Nat.Phys. , 825 (2012).[14] M. A. Cazalilla, A. M. Rey, Rep. Prog. Phys. , 124401(2014).[15] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F.Sch¨afer, H. Hu, X. J. Liu, J. Catani, C. Sias, M. Inguscioand L. Fallani, Nature Phys. , 198 (2014).[16] C. N. Yang and Y-Z. You, Chin. Phys. Lett. , 020503(2011).[17] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S.Safronova, P. Zoller, A. M. Rey, J. Ye, Science, , 1467(2014).[18] F. Scazza, C. Hofrichter, M. H¨ofer, P. C. De Groot, I. Bloch, and S. F¨olling, Nature Phys. , 779 (2014) andNature Phys. , 514 (2015).[19] G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L.Livi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D.Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L.Fallani, Phys. Rev. Lett. , 120402 (2014) and Phys.Rev. Lett. , 239903 (2015).[20] C. Hofrichter, L. Riegger, F. Scazza, M. H¨ofer, D. R. Fer-nandes, I. Bloch, and S. F¨olling, Phys. Rev. X , 021030(2016)[21] R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, Phys. Rev.Lett. , 135301 (2015).[22] G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias,J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. , 265301 (2015).[23] M. H¨ofer, L. Riegger, F. Scazza, C. Hofrichter, D. R.Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S.F¨olling, Phys. Rev. Lett. , 265302 (2015).[24] M. Nakagawa and N. Kawakami, Phys. Rev. Lett. ,165303 (2015).[25] R. Zhang, D. Zhang, Y. Cheng, W. Chen, P. Zhang, andH. Zhai, Phys. Rev. A , 043601 (2016)[26] Y. Cheng, R. Zhang, P. Zhang, and H. Zhai, Phys. Rev.A , 063605 (2017)[27] M. K.-Nagy, Y. Ashida, T. Shi, C. P. Moca, T. N.Ikeda, S. F¨olling, J. I. Cirac, G. Zar´and, E. A. Demler,arXiv:1801.01132[28] L. Riegger, N. Darkwah Oppong, M. H¨ofer, D. R. Fer-nandes, I. Bloch, and S. F¨olling, arXiv:1708.03810[29] Z. Idziaszek and T. Calarco, Phys. Rev. A , 022712(2006).[30] X. Cui, Phys. Rev. A90