Particle-hole bound states of dipolar molecules in optical lattice
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Particle-hole bound states of dipolar molecules in optical lattice
Yi-Cai Zhang , Han-Ting Wang , Shun-Qing Shen , and Wu-Ming Liu Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics and Centre of Theoretical and Computational Physics,The University of Hong Kong, Pokfulam Road, Hong Kong, China (Dated: June 20, 2018)We investigate the particle-hole pair excitations of dipolar molecules in optical lattice, which canbe described with an extended Bose-Hubbard model. For strong enough dipole-dipole interaction,the particle-hole pair excitations can form bound states in one and two dimensions. With decreasingdipole-dipole interaction, the energies of the bound states increase and merge into the particle-holecontinuous spectrum gradually. The existence regions, the energy spectra and the wave functionsof the bound states are carefully studied and the symmetries of the bound states are analyzed withgroup theory. For a given dipole-dipole interaction, the number of bound states varies in momentumspace and a number distribution of the bound states is illustrated. We also discuss how to observethese bound states in future experiments.
PACS numbers: 05.30.Jp, 03.75.Hh, 03.65.Ge
I. INTRODUCTION
Cold and ultracold molecules are attracting more andmore attention due to their broad applications in thefields of high-precision measurement, quantum chemistry,quantum information and many-body physics [1]. Recentyears the experimental techniques for cold and ultracoldmolecules have been developed greatly. The homonuclearmolecular Bose-Einstein condensation (BEC) was real-ized experimentally [2, 3] and the quantum state withexactly one molecule at each site of an optical lattice wascreated with a Feshbach resonance and STIRAP (stimu-lated Raman adiabatic passage) techniques [4, 5]. Mean-while the ultracold heteronuclear molecules were also pro-duced [6–10].The heteronuclear molecules, such as SrO, RbCs, orNaCs, prepared in their electronic and vibrational groundstates, have considerable permanent electric dipole mo-ment. The dipole-dipole interaction between moleculescan be generated by the external applied electric field[1]. Moreover, the magnitude of the dipole-dipole inter-action can be controlled by the strength of the electricfield [11, 12]. The tunable long range dipole-dipole inter-action may significantly modify the ground state and col-lective excitations of trapped condensates. For example,in trapped dipolar gases, it was shown theoretically [13]that the mean-field inter-particle interaction and, hence,the stability diagram are governed by the trapping ge-ometry. With increasing dipolar interaction, the groundstate of rotating atomic Bose gases undergoes a series oftransitions between vortex lattices of different symme-tries: triangular, square, “stripe”, and “bubble” phases[14]. In rapidly rotating Fermion gas, the dipole-dipoleinteraction may even result in the fractional quantumHall-like states [15].The particles with long-range interactions in opticallattice can be described with extended Hubbard model[16–18]. Compared with the regular Bose-Hubbard model with on-site interaction, the extended Bose-Hubbard model has richer ground state phases, such asMott insulator, particle density wave, superfluidity orsupersolid phase [19–26]. On the other hand, the gap-ful particle and hole excitations in the insulating phase[27–29] may bind together and form bound states dueto the long-range interactions. Although the excitons(holon-doublon pairs) in one-dimensional fermionic Hub-bard model have been extensively studied [30], the ex-citons in higher dimensions, and especially, in bosonicHubbard models are less studied.In this paper, we study the particle-hole pair excita-tions of dipolar bosonic molecules in optical lattice, espe-cially, the possible bound states due to the dipole-dipoleinteraction. The paper is organized as follows. In Sec. II we introduce the extended Bose-Hubbard model todescribe the polarized bosonic molecules in optical lat-tice, and derive the eigen equations to describe the sin-gle particle-hole pair excitation. In Sec. III , the ex-istence regions, the energies and the wave functions ofthe particle-hole bound states are calculated in one andtwo dimensions. The symmetries of the bound states areanalyzed and possible experimental observation of thesebound states is also discussed. A summary is presentedin Sec. IV . II. THE PARTICLE-HOLE BOUND STATES INTHE EXTENDED BOSE-HUBBARD MODEL
Considering only the nearest-neighbor interaction V tosimulate the effect of dipole-dipole interaction, we writethe extended Bose-Hubbard model as, (a) x V o ( x ) V Utt (b)
FIG. 1. Sketches of optical lattice and interactions of ultra-cold heteronuclear molecules. (a) The landscape of opticallattice potential in the x-y plane. The square optical latticepotential V o ( x, y ) = V opt [ sin ( k x x ) + sin ( k y y )] can be cre-ated by counter propagating far detuned laser beams, where V opt is the depth of lattice and k x ( y ) is the laser wave vectoralong x ( y ) direction. (b) The interactions between dipolarmolecules in the optical lattice (see equation(1)). The ar-rows on the molecules indicate the polarization of the electricdipole moments. H = H + H ,H = U X ~r n ~r ( n ~r −
1) + V X ~r,~σ n ~r n ~r + ~σ ,H = − t X ~r,~σ ( b † ~r b ~r + ~σ ) , (1)where n ~r = b † ~r b ~r is the number operator at site ~r with b ~r ( b † ~r ) the annihilation (creation) operator of particle. ~σ denotes the nearest neighbor vectors of site ~r . The on-site interaction U , the nearest neighbor interaction V andthe hopping t are expressed as U = g Z d~x | W ( ~x ) | ,V = Z d~xd~x ′ | W ( ~x − ~r ) | V dd ( ~x − ~x ′ ) | W ( ~x ′ − ( ~r + ~σ )) | ,t = − Z d~xW ∗ ( ~x − ~r )( − ~ m ∇ + V ( ~x )) W ( ~x − ( ~r + ~σ )) , (2)where W ( ~r ) is Wannier function corresponding to thelowest energy band and V dd ( ~x ) = C dd π − cos θ | ~x | is thedipole-dipole interaction of two particles separated witha distance | ~x | . For the electric dipole-dipole interaction, C dd = d ǫ with d the electric dipole moment, ǫ is vacuumpermittivity. We assume that the polarization of dipolemoment is along the z axial direction. In the followingstudies, we consider particles confined in one-dimensional chain or two-dimensional square lattice and treat hoppingterm H as a perturbation.In the atomic limit ( t = 0), n ~r is a good quantumnumber and the eigenstates of H can be written as directproduct of number states. When the filling factor is one, V ≥ zV < U , the Mott insulating ground state canbe written as | ... i with the ground state energy E = N zV , where z is the coordination number and Nthe number of lattice sites. The excited states containone or more particle-hole pairs. A single particle-holepair state with a hole at r and a particle at r ′ can berepresented as | r, r ′ i = | r r ′ ... i . Taking intoaccount the hopping term t , the particle and hole willmove in the lattice. Similar to the two-magnon statesin ferromagnetic system [31], the single particle-hole pairstate can be written as a linear combination of | r, r ′ i : | ψ i = X ~r,~r ′ φ ~r,~r ′ | ~r, ~r ′i , (3)where φ ~r,~r ′ will be determined by solving the approxi-mate eigen equation in single particle-hole pair subspace h ~r, ~r ′| H | ψ i = E h ~r, ~r ′| ψ i . (4)Calculating H | ~r, ~r ′ i and considering the boundary con-dition φ ~r,~r = 0 (a particle and a hole do not share thesame lattice site), we get − X ~σ ( tφ ~r + ~σ,~r ′ + 2 tφ ~r,~r ′ + ~σ ) = εφ ~r,~r ′ + X ~σ δ ~r ′ − ~r,~σ [ V φ ~r,~r ′ + ( − t ) φ ~r + ~σ,~r ′ + ( − t ) φ ~r,~r ′ − ~σ ] − δ ~r ′ − ~r, X ~σ ( tφ ~r + ~σ,~r ′ + 2 tφ ~r,~r ′ + ~σ ) , (5)where δ ~r ′ − ~r,σ is the Kronecker delta function and ε = E − E − U . The particle-hole excitation energy is ω = E − E = ε + U .Owing to the translational invariance, it is convenientto apply a transformation φ ~r,~r ′ = √ N e i ~K · ~R φ ( ~ρ ), where ~R = ~r + ~r ′ and ~ρ = ~r ′ − ~r denote the center-of-mass andthe relative coordinates respectively and ~K the total mo-mentum of the particle and hole. At fixed ~K , the eigenequation (5) becomes − X ~σ [ te i ~K · ~σ φ ( ~ρ − ~σ ) + 2 te i ~K · ~σ φ ( ~ρ + ~σ )] = εφ ( ~ρ ) + X ~σ δ ~ρ,~σ [ V φ ( ~ρ ) + ( − te i ~K · ~σ − te i − ~K · ~σ ) φ ( ~ρ − ~σ )] − δ ~ρ, X ~σ [ te − i ~K · ~σ φ ( ~σ ) + 2 te i ~K · ~σ φ ( ~σ )] . (6)Considering the finite range character of the nearestneighbor interaction V , we utilize the Green’s function −1 −0.5 0 0.5 1−10−8−6−4−20246 K/ π ε /t V=8tV=5tV=3tV=2tContinuous Spectrum
FIG. 2. Energy spectra of the bound states of particle-hole pair in one dimension for different nearest-neighbor in-teractions V = 2 t (black circles), V = 3 t (red triangles), V = 5 t (black diamonds) and V = 8 t (red squares) respec-tively ( K = K x ). As comparison, the continuous spectrumof a particle-hole pair excitation is also shown (green shadedregion). approach to solve the eigen equation. To this end, weintroduce an effective Hamiltonian H eff = H ph, + V ph , (7)where H ph, = − t X ~ρ,~σ e i ~K · ~σ/ | ~ρ ih ~ρ − ~σ | − te i ~K · ~σ/ | ~ρ ih ~ρ + ~σ | ,V ph = X ~σ (2 te − i ~K · ~σ/ + te i ~K · ~σ/ ) | ~σ ih | + h.c. + X ~σ ( − V ) | ~σ ih ~σ | , to describe the motion of a particle around a hole, where | ~ρ i is basis set in relative coordinate spaces. The ~K -dependent H ph, describes the kinetic energy of the par-ticle and V ph denotes the interaction between particleand hole. With | φ i = P ~ρ φ ( ~ρ ) | ~ρ i , it is easy to reproduceEq. (6) from the eigen equation H eff | φ i = ε | φ i . After aFourier transformation, the kinetic energy is obtained as ε ( ~K, ~q ) = − t X ~σ e i ( ~K/ ~q ) · ~σ − t X ~σ e i ( ~K/ − ~q ) · ~σ , (8)where ~q is the relative momentum of particle and hole.When a particle is adjacent to a hole, the energy offset − V in V ph may result in the formation of particle-holebound states. −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 700.10.20.30.40.50.60.70.80.9 ρ | φ R ( ρ ) | FIG. 3. Probability density of a bound state in real spacein the case of V = 8 t and K = 0. The probability densitydecreases exponentially with increase of distance. Introducing retarded Green functions G =lim η → + / ( ε − H ph, + iη ) and G = lim η → + / ( ε − H eff + iη ), we could calculate G through the Lippmann-Schwinger equation G = G + G V ph G . In the realspace, the Green’s function G has a general form h ~ρ ′ | G | ~ρ i = lim η → + X m φ m ( ~ρ ′ ) φ ∗ m ( ~ρ ) ε − E m + iη = G ( ~ρ ′ , ~ρ, ε ) , where the contributions of the continuous spectra are ne-glected and E m ’s are the eigenvalues of the bound stateswith φ m ’s the corresponding eigenfunctions [32]. We candetermine the eigenvalues and eigenfunctions by analyz-ing the poles and residues of the obtained Green’s func-tion.From Eq. (7), we have h ~ρ ′ | G | ~ρ i = lim η → + π ) d Z π − π d d qe i~q · ( ~ρ ′ − ~ρ ) ε − ε ( ~K, ~q ) + iη , and h ~ρ ′ | G | ~ρ i = h ~ρ ′ | G | ~ρ i + X ~ρ , ~ρ h ~ρ ′ | G | ~ρ ih ~ρ | V ph | ~ρ ih ~ρ | G | ~ρ i . (9)For specific ~ρ ′ = 0 or σ , Eq. (9) can be reduced toa set of simultaneous linear equations. Green’s func-tions h ~ρ ′ | G | ~ρ i with ~ρ ′ = 0 or σ can thus be obtainedexactly. The residues of h | G | ~ρ i is always vanishing for V/t d FIG. 4. The mean size d of bound states as a function of V atK=0 in one dimension. When the nearest-neighbor interac-tion V approaches the critical value V = 3 t , the mean size ofthe bound state becomes divergent and the bound state ener-gies merge into the continuous spectrum (see the red trianglesin Fig.2), indicating the disintegration of the bound states. non-vanishing bound states energies ε in our calcula-tions, which is consistent with the boundary condition φ (0) = 0. We present our results in the next section. III. BOUND STATES IN ONE AND TWODIMENSIONSA. Bound states in one dimension
In the Mott insulating phase, the excitation energy ofa single particle/hole was calculated as ω p,h = ± [ − ε U ( n −
12 ) + 2 dV n − µ ]+[( ε + ε U ( n + 12 ) + U (10)with the dynamical Gutzwiller approach [29]. Up to thefirst order of tU , the single particle-hole pair excitationenergy is ω = ω p ( ~k p ) + ω h ( ~k h ) = U + ε ( ~K, ~q ) (11)with the total momentum ~K = ~k p + ~k h and relative mo-mentum ~q = ( ~k h − ~k p ) /
2. For the extended Bose-Hubbardmodel, however, a particle-hole pair may form a boundstate due to the interaction (See Eq. (7)). In the follow-ing calculations, we take the lattice constant as 1 and use t > t/U ≪ zV < U are assumed to make sure that the ground stateof the system is the deep Mott insulating phase.In one dimensional case, ε ( ~K, ~q ) = − t ˆ x cos( q + θ x ) with t ˆ x = t q cos ( K x ) + sin ( K x ) and θ ˆ x = −3 −2 −1 0 1 2 343210−1−2−3−4 xs wave y 00.511.522.533.5−3 −2 −1 0 1 2 343210−1−2−3−4 xp x wave y −2.5−2−1.5−1−0.500.511.522.5 −3 −2 −1 0 1 2 343210−1−2−3−4 xd x −y wave y −5−4−3−2−1012345 FIG. 5. The wave functions of bound states in two dimen-sional case at ~K = (0 ,
0) and V = 12 t (non-normalized). s -, p x - and d x − y -wave symmetries can be observed respectively. arctan ( tan ( K x )). The top and bottom boundaries ofparticle-hole continuous spectrum areΩ t ( b )1 ( K ) = ± t ˆ x , (12)(see Fig.2). Compared with particle-hole pairs in fermionHubbard model with half filling [34], the minimum ofband width of the continuous spectrum is non-zero at K = ± π . This is because that there are no particle-holesymmetries in Bose-Hubbard model.For a specific nearest-neighbor interaction V and totalmomentum K , we search for bound state solutions out-side the continuum. In the case of ε < − t ˆ x , the free V/t ε /t s wavep x(y) waved x −y waveThe boundry of continuous spectrum ε =−V (a) FIG. 6. Variations of the energies of different bound states intwo dimension at ~K = (0 , V and cross the particle-holecontinuum ( − t at ~K = (0 , V . Green function is calculated as [33] h ~ρ ′ | G | ~ρ i = lim η → + π ) d Z π − π d d qe i~q · ( ~ρ ′ − ~ρ ) ε − ε ( ~K, ~q ) + iη = αβ | ρ − ρ ′ | γ ρ − ρ ′ , (13)where α = − √ ε − t x , γ = e iθ ˆ x and β = e − κ with κ = arccosh ( | ε | t ˆ x ).With ρ ′ = 0, ±
1, Eq. (9) is reduced to 3 linear equa-tions. The Green functions of h− | G | ρ i , h | G | ρ i and h | G | ρ i can be exactly obtained as h− | G | ρ i = θ ( − ρ )( β/γ ) − ( ρ +1) − β ( t ˆ x − V β ) ρ = 0 h | G | ρ i = 0 , h | G | ρ i = θ ( ρ )( βγ ) ρ − − β ( t ˆ x − V β ) ρ = 0 , (14)where θ ( ρ ) is Heaviside step function. From these Greenfunctions, we obtain two degenerate bound states corre-sponding to the particle on the left (right) of the holerespectively. The bound state energy is ε = − V − t x V . (15)Accordingly, β is simplified as β = t ˆ x V < V >t r cos ( K x sin ( K x . (16) −1 −0.5 0 0.5 1−20−15−10−5051015 K/ π ε /t s wave (V=12t)p x(y) wave (V=12t)d x −y wave (V=12t)s wave (V=6t)p x(y) wave (V=6t)d x −y wave (V=6t)continuous spectrum FIG. 7. Dispersion relations of bound states along the line of K x = K y = K for V = 6 t and V = 12 t . As comparison, theparticle-hole continuum is also shown (green shaded region). When 0 ≤ V ≤ t , the interaction between particle andhole is too weak to bind them together and no boundstate is found. When t < V < t , two degenerate boundstates exist in the regions of 2 arccos q V − t t < | K x | ≤ π .The region get larger with the increase of V . When V ≥ t , bound states may be found in the whole first Brillouinzone. In Fig.2, we show the spectra of the bound statesfor the interaction V = 2 t , 3 t , 5 t and 8 t, respectively.The energies of the bound states decrease with increasing V . At V = 2 t , the calculated existence intervals are0 . π < | K x | ≤ π .From the residues of the Green function, we can alsoextract the bound state wave functions. For example,the bound state with a particle on the right of a hole iswritten as φ R ( ρ ) = C ( t ˆ x V ) ( ρ − e − iθ ˆ x ( ρ − ρ ≥ φ R ( ρ ) = 0 ρ < , (17)where C = q − t x V is the normalized constant. InFig.3, we show the probability distribution of the wavefunction for K = 0 and V = 8 t .The mean size of the bound state is calculated as d = h φ R ( ρ ) | ρ | φ R ( ρ ) i = V V − t x . In Fig.4, we show themean size of the bound state as a function of the nearest-neighbor interaction V at K x = 0. The stronger is theinteraction, the smaller is the size and the closer do theparticle and hole bind. B. Bound states in two dimension
Before presenting the numerical results, we discuss thesymmetries of the effective Hamiltonian H eff in Eq. (7)in details. After a gauge transformation | ~ρ i ′ = | x, y i ′ = e − i ( xθ ˆ x + yθ ˆ y ) | x, y i = e − i ( xθ ˆ x + yθ ˆ y ) | ~ρ i (18)with θ ˆ x = arctan ( tan ( K x )) and θ ˆ y = arctan ( tan ( K y )),we could remove the phase factors in the Hamiltonianand get H ′ eff = H ′ ph, + V ′ ph , (19) H ′ ph, = − t ˆ x X ~ρ,~σ x | ~ρ i ′ h ~ρ − ~σ x | ′ − t ˆ y X ~ρ,~σ y | ~ρ i ′ h ~ρ − ~σ y | ′ ,V ′ ph = − V X ~σ | ~σ i ′ h ~σ | ′ + t ˆ x X ~σ x | ~σ x i ′ h | ′ + h.c. + t ˆ y X ~σ y | ~σ y i ′ h | ′ + h.c., where ~σ x and ~σ y denote the nearest neigh-bors along the x and y directions, respec-tively, and t ˆ x = t q cos ( K x ) + sin ( K x ) and t ˆ y = t q cos ( K y ) + sin ( K y ) are effective hoppingsalong the x and y directions.When K x = ± K y , t ˆ x = t ˆ y , H ′ eff has symmetries of D group. According to the irreducible representations ofthis group, the bound states can be classified and labeledwith s , p x ( y ) and d x − y wave, respectively. Among them,the s wave belongs to an identical representation A of D group, the degenerate p x and p y waves form a two di-mensional irreducible representation E of D group, andthe d x − y wave belongs to an irreducible representation B of D group.When K x = ± K y , t ˆ x = t ˆ y , the symmetry reduces to D , a subgroup of D . For simplicity, we still label thefour bound states with s , p x ( y ) and d x − y . Differently,here all the s and d wave belong to identical represen-tations A of D group, while p x ( p y ) wave belongs toan irreducible representation B ( B ) of D group. Thedegeneracy of p x and p y waves is lifted.With ρ ′ = (0 , ± , ± G ( ~ρ, ~ρ ′ ) can be expressed with elliptic integrals[32, 35]. Although no brief solutions of < ρ | G | ρ ′ > couldbe found, we may factorize the particle-hole bound stateequations and analyze the existence conditions for thebound states along the symmetric lines of K x = ± K y = K [35]. Considering the asymptotic behaviours of ellipticintegrals, we get the thresholds of the interaction V asfollows: V cr,s ( K ) = t ˆ x for s wave, V cr,p ( K ) = π π − t ˆ x for p x ( y ) wave and V cr,d ( K ) = π − π t ˆ x for d x − y wave. Fromthese thresholds, the existence region of every boundstate is obtained respectively. For s wave, we have2 arccos r V − t t ≤ | K | ≤ π, V ∈ ( t, t ] . When
V > t , the s wave bound state may be found inthe whole Brillouin zone. The existence region of p wavebound states is2 arccos s ( π − π V ) − t t ≤ | K | ≤ π, V ∈ ( ππ − t, ππ − t ] . When
V > ππ − t , the existence region extends to thewhole Brillouin zone. For d wave, existence region isexpressed as2 arccos s ( − ππ V ) − t t ≤ | K | ≤ π, V ∈ ( π − π t, π − π t ] . When
V > π − π t , the d wave existence interval is thewhole Brillouin zone.Now we present the numerical results in two dimen-sions. At ~K = (0 , V ≥ V cr,d (0) = 10 . t . The wave functions with s − , p x − and d x − y − wave symmetry for V = 12 t areshown in Fig. 5. With the decrease of V , the high-est d x − y -wave, the degenerate p x ( y ) -waves and the low-est s -wave merge into the continuous spectrum one byone. Finally, all the bound states disappear when V ≤ V cr,s (0) = 3 t . In Fig. 6, we illustrate the variations ofthe bound state energies with the changes of V .We then search for bound state solutions along theline of K x = K y = K . As shown in Fig. 7, there arefour bound states for all K ∈ [ − π, π ] at V = 12 t . At V = 6 t , s -wave exists in the whole region, while p x ( y ) -and d x − y -wave states appear at 0 . π < | K | ≤ π and0 . π < | K | ≤ π respectively. Decreasing V further,we find that the regions of the bound states shrink, andall the bound states disappear when V < V cr,s ( ± π ) =1 t . Similar results are obtained along the line of K x = − K y = K .Away from the lines of K x = ± K y , the degeneracy of p x and p y bound states is lifted, as mentioned before. Ata given V , different bound states have different existenceregions in ~K space. Consequently, the number of boundstates vary in the ( K x , K y ) space. As an example, weshow the number distribution of the bound states for V = 6 t in Fig. 8. C. Discussion of observations of the bound states
Inelastic light scattering directly measures the dynami-cal structure factor S ( ~q, ω ), the Fourier transformation ofdensity correlations. Bragg spectroscopy has been pro-posed to detect quantum phases in optical lattice andsuccessfully applied to measure the excitation spectra(phonons of BEC), the composition of the excitations andthe Higgs-type amplitude mode in the superfluid conden-sate, as well as the particle-hole excitation energies in theMott-insulator state [36–42]. When this technique is uti-lized in the study of ultra-cold polar molecules in the −1 −0.5 0 0.5 1−1−0.500.51 K x / π K y / π n=4 n=1 n=2 n=3 FIG. 8. The number distribution of bound states in the firstBrillouin zone at V = 6 t . As shown, there are more boundstats at the corners than that at the center in the first Bril-louin zone. optical lattice, we may expect extra resonance peaks cor-responding to the particle-hole bound states lying outsidethe particle-hole continuum.Compared with the traditional solid state counter-parts, the deep Mott insulating state with t/U ≪ zV < U can be realized by tuning the inter-action parameters t , U and V in ultra-cold dipolarmolecules in the optical lattices. In deep optical lat-tice the hopping is approximately evaluated as t =(4 / √ π ) E r ( V opt /E r ) / exp [ − V opt /E r ) / ], with V opt =18 ∼ E r the optical lattice depth and E r the lattice re-coil energy [43]. Under the condition of asa ⊥ ≪
1, where a s is the s -wave scattering length and a ⊥ = q ~ mω ⊥ with ω ⊥ transverse trapping frequency in one dimension or trap-ping frequency of the z direction in two dimensions, theon-site interaction is estimated as U D = q π ~ ω ⊥ a s /l and U D = √ π ~ a s ma z (2 π ) l with l = ( E r V opt ) aπ and a the latticeconstant [43, 44].Taking Bose molecule Na Li for example, we mayestimate the typical values of t , U and V for observingthe particle-hole bound state. The molecule preparedin the ground state has a permanent electric moment d = 0 . D [45]. With a lattice constant a ∼ . µm , a s /a ∼ .
01 and the transversal tapping frequency ω ⊥ ∼ π × Hz, we have t ∼ nK , U D ∼ nK and U D ∼ nK . The nearest-neighbor interaction V can be tunedas V ∼ nK with an applied electric field. IV. SUMMARY
In summary, we have investigated bound states ofparticle-hole pair resulting from the dipole-dipole inter-action between polar molecules in the optical lattice. Fora large enough dipole-dipole interaction, two degeneratebound states, which correspond to a particle on the leftand the right of a hole, are shown to exist in one dimen-sion. While in two-dimensional case, four bound states,with s -, p x ( y ) - and d x − y - symmetry respectively, arefound along the lines of K x = ± K y . Away from the linesof K x = ± K y , the degeneracy between p x and p y waves islifted. With decreasing the nearest-neighbor interaction V , the energies of the bound states increase and mergeinto the particle-hole continuum gradually. The wavefunctions, the dispersion relations and the existence re-gions of the bound states are studied in details. For agiven nearest-neighbor V , the number of bound statesis different in different regions of ~K space and a num-ber distribution of bound states is given for the nearest-neighbor V = 6 t in two dimensions. The possible exper-imental observation of the particle-hole bound states isalso discussed.Electron-hole bound state excitations (excitons) havebeen extensively studied for many years and very re-cently, the excitonic Bose-Einstein condensates have beenrealized experimentally in semiconductors [46, 47]. Wehope our study on the particle-hole bound states in bosonic systems would enrich our understanding of ele-mentary excitations in quantum many-body systems andstimulate more efforts on the bound state phenomena invarious systems such as magnets, superconductors andatomic systems. Acknowledgements:
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