Bound states of an ultracold atom interacting with a set of stationary impurities
BBound states of an ultracold atom interacting with a set of stationary impurities
Marta Sroczyńska and Zbigniew Idziaszek Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland (Dated: July 21, 2020)In this manuscript we analyse properties of bound states of an atom interacting with a set ofstatic impurities. We begin with the simplest system of a single atom interacting with two staticimpurities. We consider two types of atom-impurity interaction: (i) zero-range potential representedby regularized delta, (ii) more realistic polarization potential, representing long-range part of theatom-ion interaction. For the former we obtain analytical results for energies of bound states. Forthe latter we perform numerical calculations based on the application of finite element method.Then, we move to the case of a single atom interacting with one-dimensional (1D) infinite chain ofstatic ions. Such a setup resembles Kronig-Penney model of a 1D crystalline solid, where energyspectrum exhibits band structure behaviour. For this system, we derive analytical results for theband structure of bound states assuming regularized delta interaction, and perform numerical cal-culations, considering polarization potential to model atom-impurity interaction. Both approachesagree quite well when separation between impurities is much larger than characteristic range of theinteraction potential.
I. INTRODUCTION
Hybrid systems of ultracold atoms and trapped impu-rities like ions [1–9] or Rydberg atoms [10, 11] have beenthe subject of intense experimental and theoretical stud-ies over the past years [12]. They have been proposedfor quantum simulations [13–15], quantum computations[16–18], realization of new mesoscopic quantum states[19, 20], probing quantum gases [21–24] or fundamen-tal studies of low-energy collisions and molecular states[25–36]. By tuning the geometric arrangement of theimpurities, it is possible to simulate various solid-stateand molecular systems [37–40]. Several experiments havebeen focused on studying controlled chemical reactions atultra-low temperatures in such systems [5, 41–45].In this work we are considering two systems. The firstsystem contains two static impurities, while the second isa 1D linear crystal of static impurities. We consider twodifferent potentials for atom-impurity interactions, repre-senting two distinct physical systems: atomic impuritiesin the ultracold gas and hybrid atom-ion system. Forthe former we assume regularized delta potential, whilefor the latter we take polarization potential represent-ing long-range part of the atom-ion interaction, whichwe regularize at small distances imposing a short-rangecut-off. The regularized delta potential models only s -wave scattering at ultralow energies and depends onlyon a single parameter: the s -wave scattering length. Itszero-range character allows for analytical solution of thecorresponding Schrödinger equation for arbitrary set ofdelta-like scatterers [40].The atom-ion interaction, which has a long-range be-haviour, can be modeled by including only the long-rangepart given by the polariziation potential − C /r and ashort-range boundary condition. The latter can be rep-resented either by a short-range phase introduced in theframework of the quantum-defect theory [25], or by reg-ularizing the short-range divergence with some regular-izing function [46]. In this work we choose the latter option, assuming parametrization of the atom-ion po-tential by the long-range dispersion coefficient C anda cut-off radius b . For such a potential one can solve 1Dradial Schrödinger analytically and express the scatteringlength in terms of C and b parameters [47]This work is structured as follows. The potentialswhich we are considering are introduced in sec. II. Insec. III we solve the Schrödinger equation for an atominteracting with two impurities and analyse the resultsfor different values of the short-range scattering length.We perform our analysis for atomic impurities, when theatom-impurity interaction is modeled with delta pseu-dopotential, and for ionic impurities, when we assumeatom-impurity interaction in the form of the polarizationpotential. In sec. IV we consider an infinite chain of ionicimpurities. First, we solve the Schrödinger equation nu-merically using finite element method and we discuss nu-merical solutions of the Schrödinger equation for differentvalues of atom quasi-momentum in 1D periodic system.Then, we derive analytic formula determining energies ofbound states for regularized delta potential, and studybehaviour of energy bands versus scattering length anddistance between impurities. We finish in sec. V present-ing some final conclusions. II. ATOM–IMPURITY INTERACTIONA. Pseudopotential
Within the ultracold regime, where mainly s -wavescattering takes place for bosonic or distinguishable par-ticles, we can model the atom–impurity interaction bythe Fermi pseudopotential [48, 49] given by V ( r ) = gδ ( r ) ∂∂r r, (1) a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l where g depends on the 3D s -wave scattering length a and g = 2 π (cid:126) m a. (2)Note that only m atom mass enters the coupling con-stant as we assume that impurities are stationary, andthe reduced mass µ = m . Such a potential can serve asa good approximation of a physical potential providedthat the distance between impurities L is large compar-ing to the characteristic range of the interaction R n ofthe power-law potential V ( r ) = − C n /r n : L (cid:29) R n . Inthe case of atom-ion potential, R = √ µC / (cid:126) , whilefor van der Waals potential between neutral atoms R =(2 µC / (cid:126) ) / [50]. For modelling of bound-states we haveto impose another constrain: a (cid:29) R n , which is equivalentto the following condition E b (cid:28) E n , where the character-istic energy is E n = (cid:126) / (2 µR n ) [50]. This, expresses thefact that the pseudopotential can be used to reproducebound states in the universal limit, with binding energies E b that are close to the threshold [51, 52]. Going beyondthe above mentioned conditions, requires inclusion of theenergy-dependent scattering length in (2) [53–55]. B. Regularized atom–ion interaction potential
We will also consider more realistic potential, suchas polarization potential between atoms and ions. Thelong–range part of the atom–ion potential is given by V ( r ) r →∞ −−−→ − C /r . With this potential we can asso-ciate the characteristic length and energy scales, thatare used further in this work: R ∗ = √ mC / (cid:126) and E ∗ = (cid:126) / m ( R ∗ ) [56]. Here, we will use regularizedversion of this long–range potential in the form of Lenzpotential [47], which is finite for r → : V ( r ) = − C ( r + b ) , (3)where b is the parameter that can be related to the scat-tering length a [47] a ( b ) = R ∗ (cid:115) (cid:18) bR ∗ (cid:19) cot π (cid:115) (cid:18) R ∗ b (cid:19) . (4)This dependence is shown in Fig. 1. We observe that, ac-cording to formula (4), one value of the scattering lengthcan be reproduced by many values of b . The scatteringlength dependence on b exhibits several resonances thatare related to crossing the dissociation threshold by thebound states supported by (3). The number of boundstates n is related to the cut-off parameter b , by the fol-lowing rule: b ∈ ( b n − , b n ) , where b n = 1 / √ n − . - -
505 b / R * a / R * FIG. 1. Scattering length as a function of the regularizationparameter given by Eq. (4) for the regularized atom–ion in-teraction potential (3).
III. SYSTEM WITH TWO IMPURITIES
We investigate the bound states of the system con-taining of a single atom that interacts with two impu-rities placed symmetrically along z -axis, such that theirpositions are ± d = (0 , , ± d ) and the distance betweenthem is d . We assume that each impurity interacts onlywith the atom, and we do not take into account theirmutual interactions. We will study the dependence ofbound state energies on the scattering length and on thedistance between impurities.The Hamiltonian of such a system is H = − (cid:126) m ∆ + V ( r − d ) + V ( r + d ) , (5)where V denotes the atom–impurity interaction, whichis given by two different atom–impurity potentials intro-duced in sec. II. A. Atom–impurity interaction modeled by thepseudopotential
We solve the Schrödinger equation, using the Green’sfunction technique. The Green’s function for the three-dimensional scattering in free space reads (see e.g. [57]) G ( r , r (cid:48) ) = A e ik | r − r (cid:48) | | r − r (cid:48) | , (6)where A = − m/ π (cid:126) . Let us denote r = | r + d | and r = | r − d | , so that we have G ( − d , r ) = A e ikr r ≡ G ( r ) (7) G ( d , r ) = A e ikr r ≡ G ( r ) , (8)where for convenience we have also introduced a short-ened notation G ( r ) for the Green’s function. In thecase of Fermi pseudopotential, the Hamiltonian can besolved analytically [40], in principle for arbitrary arrange-ment of the impurities. In order to find the energies of thesystem, we have to solve the following set of equations: (cid:40) k = g (cid:8) ∂∂r r ( k G ( − d , r ) + k G ( d , r )) (cid:9) r →− d k = g (cid:8) ∂∂r r ( k G ( − d , r ) + k G ( d , r )) (cid:9) r → d , (9)which can be expressed using the notation with r and r : k = g (cid:110) ∂∂r r ( k G ( r ) + k G ( r ) (cid:111) r → k = g (cid:110) ∂∂r r ( k G ( r ) + k G ( r ) (cid:111) r → . (10)Let us now calculate the derivatives of the Green’s func-tion that appear in the first equation and their values inthe limit of r → : (cid:18) ∂∂r r G ( r ) (cid:19) r → = A (cid:18) ∂∂r r e ikr r (cid:19) r → == A (cid:18) ∂∂r e ikr (cid:19) r → = A ik (cid:0) e ikr (cid:1) r → = A ik. (11)Then, we have (cid:18) ∂∂r r G ( r ) (cid:19) r → = A (cid:18) ∂∂r r e ikr r (cid:19) r → == A (cid:18) e ikr r + r ∂∂r e ikr r (cid:19) r → = A e ik d d . (12)Derivatives of the Green’s function and their limits for r → , appearing in the second equation can be calcu-lated in an analogous way. Now we insert the obtainedresults into the system of equations (10): k = g A (cid:16) k ik + k e ik d d (cid:17) k = g A (cid:16) k e ik d d + k ik (cid:17) . (13)Above expression (13) can be rewritten in a matrix formas (cid:32) g A ik − g A e ik d d g A e ik d d g A ik − (cid:33) (cid:18) k k (cid:19) = 0 . (14)This system of equations has solutions provided that thedeterminant of the matrix is equal to zero. From thiscondition we get two independent solutions: g A (cid:18) ik ± e ik d d (cid:19) − . (15)Since we are looking for bound states, the wavenumber k = iκ where κ is real, the energy E = − (cid:126) κ m and κ = (cid:112) − mE/ (cid:126) . Taking into account that g A = − a ,we can rewrite the above expression as − κ ± e − κ d d = 1 a . (16) The energy levels can now be found numerically for givenvalue of the scattering length and d . At the threshold E = κ = 0 Eq. (16) yields: ± d = 1 a , ( E = 0) . (17)From this we see that, at the distance d = | a | / the newbound state either appears or disappears at the thresh-old, depending on the sign of the scattering length.Let us now consider two limiting cases. In the limit d → , from Eqn. (16) we obtain κ d → −−−→ ± d , (18)which diverges as d is going to zero. This singularbehaviour results from the Green’s function in the off-diagonal terms, which are not regularized by ∂∂r r opera-tor and as a consequence yields divergence at d → . It ispossible to reformulate a regularization operator in theway that it correctly reproduces the limit of two deltafunctions [32]. We note, however, that the limit d → corresponds physically to combining two impurities in asingle molecular complex, which in principle would havea different scattering length than a sum of two scatteringlengths of the separate objects.In the case where the separation of the impurities isvery large ( d → ∞ ), the term e − dκ / d goes to zero andwe get aκ = 1 , which implies the existance of the boundstate for positive values of the scattering length E d →∞ −−−→ − (cid:126) ma (19)and no bound states in the case of a < .Fig. 2 compares bound state energies evaluated fromEq. (16) for different values of the scattering length. Forpositive scattering lengths, at large distances the boundstate energies are degenerate, and tend to the energy ofa single bound state (19). In contrast for negative scat-tering lengths, there are no bound states at large dis-tances, as the separate delta potential does not supportany bound states for a < . Nevertheless, at distance d < | a | / , two impurities posses a single bound state,crossing the threshold at d = | a | / . Exactly, at the samedistance, for positive scattering lengths, one of the boundstates disappears at the threshold, and for d < | a | / , twoimpurities support again only a single bound state. Wenote, that for d > | a | / , Eq. (16) is not valid for negativescattering lengths. B. Atom–impurity interaction modeled by theregularized atom–ion potential
In this case, we cannot solve the Hamiltonian analyt-ically and we have to rely on numerics. We begin bylooking for eigenstates for a single ion, using two dif-ferent numerical methods: Numerov algorithm and finite - - - / R * E / E * a / R * =- / R * =- / R * = / R * = FIG. 2. Energy spectrum resulting from (16) - the energylevels of a system consisting of an atom interacting with twoimpurities by the delta pseudopotential with different scatter-ing length: a/R = − (blue), a/R ∗ = − (black), a/R ∗ = 1 (red), a/R ∗ = 5 (orange). element method. This comparison helps to adjust the pa-rameters of the grid in the finite element method, whichwe later use to solve the two–ions case. Single ion case
The interaction potential for a single ion is sphericallysymmetric. Therefore, the wave function can be decom-posed as ψ ( r ) = R ( r ) Y lm ( θ, φ ) , where R ( r ) is the ra-dial part and Y lm ( θ, φ ) is the spherical harmonic, withquantum numbers l and m , representing the angular mo-mentum and its projection on the z -axis, respectively. Inorder to find the bound states, we only need to solve theradial part of the Schrödinger equation. It is convenientto look for R ( r ) /r , which simplifies the Laplacian oper-ator, but does not affect the energies. The Hamiltonianto solve reads: H = − (cid:126) m d dr + (cid:126) m l ( l + 1) r − C ( r + b ) . (20) Numerov method.
With Numerov algorithm we solve theSchrödinger on the grid of equally spaced points between r = r min and r = r max , assuming that the wave functionvanishes at the boundaries. In principle, r max shouldbe much larger than R ∗ and a . For our computationswe take r min = 0 and r max = 20 R ∗ . The solutions for l = 0 , , are shown in Fig. 4. Finite element method.
In this case, we are solving thefollowing Schrödinger equation − (cid:126) m ∆ ψ − C ( ρ + z + b ) ψ = Eψ. (21)It is convenient to rewrite the above equation in the cylin- - z max z max ρ max FIG. 3. An example grid used for the finite element method.The grid size is determined by the local de Broglie wavelengthand it becomes very dense in the vicinity of the ion at z = 0 and ρ = 0 . drical coordinates − (cid:126) m (cid:18) ∂ ∂z + ∂ ∂ρ + 1 ρ ∂∂ρ (cid:19) ψ + C ( ρ + z + b ) ψ = Eψ. (22)where additionally we assumed m = 0 symmetry of thesolutions.In order to find the energy levels of the system, wesolve Eq. (22) numerically using finite element methodimplemented in the Mathematica software [58]. We per-form calculations for a single ion placed at the originof the coordinate system, in a rectangular box, with − z max ≤ z ≤ z max and ≤ ρ ≤ ρ max . The value of ρ max and z max should be relatively large comparing tothe scattering length in order to not affect the boundstate wave functions by the boundary conditions. Forour computations we take z max = 8 R ∗ and ρ max = 8 R ∗ .We assume Dirichlet boundary conditions ψ = 0 alongall the boundaries except ρ = 0 , where we set von Neu-mann boundary condition: ∂∂ρ ψ ( ρ = 0 , z ) = 0 . The reg-ularization parameter b is set such that one bound stateis supported for a given scattering length. It is worthnoting that close to the ion, the potential is getting rela-tively deep and the corresponding wave function becomesquickly oscillating in that region. To address this issue wehave used variable grid size related to the local de Brogliewavelength λ ( r , E ) = 2 π/ (cid:112) m | E − V ai ( ρ, z ) | / (cid:126) ), byassuming that area of a single cell in the grid fulfils ∆ ≤ λ ( r , E ) /N . We have tested several values of N parameter, observing that numerical calculations startconverging for N (cid:38) in the case of the atom–ion po-tential supporting one bound state and N (cid:38) for deeperpotentials supporting two bound states. An example grid - - - - - -
10 b / R * E / E * FEMNumerov, l = = = FIG. 4. Energies of bound states in a regularized atom–ionpotential for different values of b computed using Numerovalgorithm (blue, red and orange correspond to the angularmomentum l = 0 , , , respectively) and finite element algo-rithm (black). is shown in Fig. 3.Fig. 4 shows the energies of bound state obtained usingboth methods. We note, that both numerical approachesgive almost identical results, which convinces regardingthe numerical convergence of both methods. Two ions case
We now turn to the system of two ions. We solve theSchrödinger equation with the Hamiltonian (5), using thefinite element method with the same boundary conditionsas in the single ion case. The value of the cut-off param-eter b is chosen such, that the potential is relatively shal-low, and only one or two bound states are supported. Incontrary to the pseudopotential model, now we obtain fi-nite results for both small and large separations betweenthe impurities.In Fig. 5 we plot the energies of bound states for differ-ent values of the scattering length a , and the cut-off pa-rameter b as a function of distance d between impurities.In addition we include predictions of the pseudopotentialmodel (16). For a > and d → ∞ , impurities do not seeeach other and the bound states energies tends asymp-totically to the values for a single impurity (dashed line),calculated from (20) using Numerov method. Boundstates for polarization potential behaves basically in asimilar way as for pseudpotential. At some finite dis-tance, which is now different for positive and negativescattering length, bound states for the polarization po-tential cross the threshold, and below that characteris-tic distance, the system supports only a single shallowbound state. We note that for large scattering lengths a = ± R ∗ , the crossing point is similar for potentialssupporting one and two bound states. In contrast, for a = ± R ∗ , the crossing point is quite different between these potentials, and it also deviates from the pseudopo-tential prediction d = | a | / . This is probably due to thefinite size effects when a ∼ R ∗ . We suppose that replace-ment of the scattering length by the energy–dependentone [53–55], would possibly improve the agreement, atleast for the pseudopotential model.Similar discrepancies can be observed at large dis-tances for a = R ∗ , where all three calculations predictvarious asymptotic values for the bound state of a sepa-rated impurity. The agreement, is much better for highervalue of the scattering length a = 5 R ∗ . When the dis-tance between impurities is getting close to zero, thebound states calculated for various models show differentbehaviour. In such case our models break down and wedo not show this limit in the plot. For the pseudopo-tential model this happens, because d is not any morelarge comparing to R ∗ , and the conditions for the ap-plicability of the pseudopotential approximation are nolonger fulfilled. For the regularized atom–ion potential,at distances d comparable to the cut-off parameter b , thepotentials starts strongly too overlap, and in this caseresults depend on b , determining the number of boundstates in the regularized potential. In all the panels weobserve the deeply lying bound states supported by theregularized atom-ion potential. Their energies, however,substantially depend on the number of bound states sup-ported by the potential, and even at the same value of thescattering length, they differ. Those deeper lying boundstates are not the target of our analysis. IV. PERIODIC SYSTEM
Here, we consider an atom interacting with an infinitechain of equally spaced static ions. The interaction V ai is given by the regularized atom-ion potential (sec. II B).Similarly to the two–ion case, we neglect the interactionbetween the ions. The Hamiltonian reads H = − (cid:126) m ∆ − ∞ (cid:88) n = −∞ V ( r − d n ) , (23)where d n = (0 , , nL ) is the position of n -th ion and L is the distance between the neighbouring ions (period).The ions are placed along z -axis.Exploiting the fact that the system is axially symmet-ric and periodic along z -axis, and taking into accountthe Bloch theorem, we can write the wave function incylindrical coordinates ρ and z in the following form ψ ( r ) = e iqz u q ( ρ, z ) e imφ , (24)where q is the quasi-momentum. In the following weconsider only the eigenstates with the symmetry m = 0 .Substituting (24) into the Schrödinger equation with the a / R * =- - - - - E / E * a ) - - - - - a / R * = - - - - E / E * b ) - - - - a / R * =- - - - - - -
50 d / R * E / E * c ) - - - - - a / R * = - - - - - / R * E / E * d ) - - - - FIG. 5. Energy spectrum as a function of d (half of the distance between the impurities) for different values of the scatteringlength a and corresponding regularization parameter b supporting one bound state (blue color) or two bound states (red color):(a) a/R ∗ = − , b/R ∗ = 0 . (blue), b/R ∗ = 0 . (red), (b) a/R ∗ = 1 , b/R ∗ = 0 . (blue), b/R ∗ = 0 . (red), (c) a/R ∗ = − , b/R ∗ = 0 . (blue), b/R ∗ = 0 . (red), (d) a/R ∗ = 5 , b/R ∗ = 0 . (blue), b/R ∗ = 0 . (red). Thegreen line shows the energy spectrum calculated with pseudopotential. The dashed gray line corresponds to the bound statein the large d limit, calculated for a single impurity. Hamiltonian (23), leads to the following equation for u q − (cid:126) m (cid:18) ∂ ∂z + ∂ ∂ρ − q + 2 iq ∂∂z + 1 ρ ∂∂ρ (cid:19) u q ( ρ, z )+ − ∞ (cid:88) n = −∞ V ( r − d n ) u q ( ρ, z ) = Eu q ( ρ, z ) . (25) A. Atom–impurity interaction modeled by theregularized atom–ion potential
In order to find the energy levels of the system, wesolve Eq. (25) numerically using finite element method,in a similar manner as described for the two–ion system.We perform calculations for an ion placed in the position d = (0 , , L/ in the rectangular box with z ∈ [0 , L ] and ρ ∈ [0 , ρ max ] . The value of ρ max should be large com-paring to the scattering length in order to not affect thebound state wave functions, and we take ρ max = 6 R ∗ for a/R ∗ = ± and ρ max = 10 R ∗ for a/R ∗ = ± . For ρ = ρ max we assume Dirichlet boundary conditions: u q ( ρ max , z ) = 0 , while for ρ = 0 we assume von Neu-mann boundary condition ∂∂ρ u ( ρ = 0 , z ) = 0 . Function u q should be periodic in z direction, so for z = 0 and z = L we set periodic boundary conditions. The regular-ization parameter b is set such that one bound state issupported for a given scattering length.In Fig. 6 we show how the energy levels change with thedistance between the neighboring ions for different valuesof the scattering length and some selected values of thequasi-momentum q . We start with discussing the case of a > , i.e. Fig. 6b and Fig. 6d. At large distances be-tween the neighbouring impurities, the energy levels fordifferent q converge to the same limit, and the band be-comes very narrow. This asymptotic value is given by theenergy of the bound state associated with a single impu-rity. As the distance L between the impurities decreases,the energy band becomes wider and some bound statescrosses the threshold, starting with the quasi-momentum q = π/L .For a < (Fig. 6a and Fig. 6c), the energy bands haveeven more complex structure. At large separations be- a / R * =- - - - - - - - E / E * a ) - - - - q = = π L q = π L q = π L q = π L a / R * =
11 2 3 4 5 6 7 - - - - E / E * b ) q = = π L q = π L q = π L q = π L single ion a / R * =- - - - -
100 L / R * E / E * c ) - - - - - q = = π L q = π L q = π L q = π L a / R * =
52 4 6 8 10 12 - - - - - - / R * E / E * d ) - - - - - q = = π L q = π L q = π L q = π L single ion FIG. 6. Energy levels of an atom interacting with periodic system of impurities as a function of the period for differentvalues of scattering length and corresponding regularization parameter: (a) a/R ∗ = − , b/R ∗ = 0 . , (b) a/R ∗ = 1 , b/R ∗ = 0 . ,(c) a/R ∗ = − , b/R ∗ = 0 . , (d) a/R ∗ = 5 , b/R ∗ = 0 . . The atom–impurity interaction is modeled bythe regularized atom–ion potential. The insets show zoom on the spectrum close to E = 0 . Red lines denote the solutions of(25) with q = 0 and blue lines are the results of (25) with q = π/L . Gray dotted, dot–dashed and dashed lines correspond to q = π/ (4 L ) , q = π/ (2 L ) , q = 3 π/ (4 L ) , respectively. tween the impurities, for each quasi-momentum there is asingle bound state, which at L → ∞ tends to the energyof bound state localized on a single ion. This representsdeeply lying bound state of the atom–ion potential, andclose to the threshold there are no bound states in thisregime, similarly to the two–ion system. As the ion sepa-ration decreases, some bound states crosses the thresholdentering from the continuum, and later different energybands start to overlap. This process actually begins forbound state with q = 0 and continues to q = π/L . ascan be seen in the panel a) ( a = − R ∗ ). For a = − R ∗ ,probably due to the finite range effects, this behaviouris quite different. We can observe that between boundstates with q = 0 and q = π/L , there are no other statescrossing the threshold.Fig. 7 shows some exemplary wave functions of thebound states. Presented wave functions are, to large ex-tend, spherically symmetric. B. Atom–impurity interaction modeled by thepseudopotential
We now turn to the analytical calculation of the energyspectrum for an atom interacting with a chain of impuri-ties, where the interaction is modeled by the pseudopo-tential (1). We solve the problem using Green’s functiontechnique, starting from the Lippmann-Schwinger equa-tion (see e.g.[57]). This yields ψ ( r ) = (cid:90) d r (cid:48) G ( r , r (cid:48) ) ∞ (cid:88) n = −∞ V ( r (cid:48) − d n ) ψ ( r (cid:48) ) , (26)where we drop inhomogeneous term, which is not impor-tant for the bound states. In order to calculate the in-tegral, we insert the atom–impurity interaction potential(1) into (26), which gives ψ ( r ) = g ∞ (cid:88) n = −∞ G ( r , d n ) γ n , (27) FIG. 7. Wave functions for (a) a/R ∗ = 1 , L/R ∗ = 2 . , q = π/L ( E/E ∗ = − . ), (b) a/R ∗ = 1 , L/R ∗ = 2 . , q = 0 ( E/E ∗ = − . ). An impurity is placed at ( z, ρ ) = ( L/ , . where γ n = (cid:18) ∂∂r n r n ψ ( r ) (cid:19) r → d n (28)and r n = r − d n . Since the potential is periodic alongthe z -axis, using Bloch theorem we can rewrite the wave-function ψ as ψ ( r ) = e iqz φ ( r ) , (29)where φ is periodic and satisfies φ ( r ) = φ ( r − d n ) . Sub-stituting (29) into the expression (28) for γ n , we get γ n = (cid:18) ∂∂r n r n e iqz φ ( r ) (cid:19) r → d n = C e iqnL , (30)where C = (cid:18) ∂∂r n r n φ ( r n ) (cid:19) r n → . (31)The specific value of C is not important, as it drops outin the further calculations. Since regularization operatorremoves /r singularity from the short-range behaviourof the wavefunction, we can assume that C is finite. Now, we inserting the wave function ψ defined in (27) into thedefinition of γ n (28), which leads to γ n = g (cid:32) ∂∂r n r n ∞ (cid:88) n (cid:48) = −∞ G ( r , d n (cid:48) ) γ n (cid:48) (cid:33) r n → == g γ n β ( E ) + (cid:88) n (cid:48) (cid:54) = n G ( d n , d n (cid:48) ) γ n (cid:48) , (32)where we have introduced β ( E ) = (cid:18) ∂∂r rG ( r + d n , d n ) (cid:19) r → . (33)We have obtained two expressions for γ n : (30) and (32),which yields the following equation C e iqnL = g C β ( E ) e iqnL + (cid:88) n (cid:48) (cid:54) = n e iqn (cid:48) L G ( d n , d n (cid:48) ) . (34)We can now simplify (34), dividing both sides by C andmultiplying by e − iqnL , which gives g β ( E ) + (cid:88) n (cid:48) (cid:54) = n e iq ( n (cid:48) − n ) L G ( d n , d n (cid:48) ) . (35)The value of Green’s function in (35) is G ( d n , d n (cid:48) ) = A e ik | n − n (cid:48) | L L | n − n (cid:48) | (36)while β ( E ) is β ( E ) = (cid:18) ∂∂r r A e ikr r (cid:19) r → = A ik = −A κ, (37)where κ = ik is real for eigenstates with negative ener-gies. After inserting (37) and (36) into the right–handside of (35) we obtain g β ( E ) + (cid:88) n (cid:48) (cid:54) = n e iq ( n (cid:48) − n ) L G ( d n , d n (cid:48) ) == aL (cid:0) κL + ln { (1 − e − κL + iqL )(1 − e − κL − iqL ) } (cid:1) , (38)where we have used the series expansion of the logarithmfunction in order to make the summation ∞ (cid:88) n =1 z n n = − ln(1 − z ) . (39)This holds, provided that | z | < (in our case | z | = | exp( − κL ) | , so the condition κ > has to be satisfied).Finally, we need to solve La = ln (cosh( κL ) − cos( qL )) + ln 2 (40)for κ , which brings the following solution κ = 1 L arcosh (cid:18) cos( qL ) + 12 e L/a (cid:19) . (41)The solutions of this equation are shown in Fig. 8, pre-senting energy bands of bound states for different valuesof the scattering length and the quasi-momentum, as afunction of the impurity spacings. Basically, we observevery similar behaviour as in the case of atom–ion poten-tial, except the fact that delta pseudopotential does notsupport bound states for a < . Due to the same argu-ment, there are no deep bound states in the spectrum asobserved for atom–ion potential. For negative scatteringlengths, the plots present only the curves for relativelysmall q , because for larger q , Eq. (40) predicts imaginary κ , when cos( qL ) + e L/a < .In Fig. 9, we plot bound state energies for positivevalues of the scattering length and some selected quasi-momenta, comparing two types of atom-impurity inter-actions considered in the paper. We observe that in thecase of ionic chain, the pseudopotential method worksdefinitely worse than for the two–ion system. Similarlyto the case of two impurities, the asymptotic value at L → ∞ obtained from numerics for atom–ion potentialis slightly lower than for the pseudopotential, which isdue to the finite range effects. V. SUMMARY
In this work we have considered bound states of anatom interacting with different setups of static impuri-ties. First, we calculated energies of bound states fortwo delta pseudopotentials and show that they can evenexist for negative values of the scattering length, whichis not possible for a single atomic impurity. Such boundstates, however, exist only when the distance betweenimpurities is smaller than some characteristic value ofthe order of the scattering length. Similar behaviour is observed when we consider long-range polarization po-tential. On the other hand, for positive values of thescattering length and at large distance between impuri-ties, there are two solutions for bound–state energies. Inthe asymptotic limit they tend to the energy of a singleatom-impurity molecular state. At smaller distances, thedegeneracy is lifted and at some characteristic distancebetween impurities, one of the bound disappears at thethreshold. Calculations performed for the atom-ion po-larization potential exhibits a similar behaviour.For an infinite chain of ionic impurities, we roughly ob-serve an analogous behaviour as for two ions. In this casebound states aggregate into bands. For positive values ofthe scattering length, the energy bands at large separa-tions between ions correlate with energies of a separateatom–ion bound states. For negative values of the scat-tering length, the shallowest energy band disappears atlarge ion separations. Finally, we extended our analyticalcalculations performed for two impurities to the case of1D infinite chain of delta-like impurities. We derived rel-atively simple analytical equation determining the energylevels of bound states for this system.In the future investigations we intend to include theenergy-dependent scattering length in the delta pseu-dopotential [53, 54], which would allow to account forthe finite-range effect of the potential. Assuming theenergy-dependence appropriate for the polarization po-tential, in principle we should be able to better reproducethe numerical calculations performed with finite-elementmethod for ionic chain, and explain the behaviour of theenergy bands for smaller values of a . This would require,however, generalization of the energy-dependent scatter-ing length for the polarization potential to the negativeenergies, which so far has been only realized for van derWaals interactions [55]. VI. ACKNOWLEDGEMENTS
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