Analytical Mean-Field Approach to the Phase Diagram of Ultracold Bosons in Optical Superlattices

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1 Laser Physics, Vo. 5, No. 2, 25, pp Origina Text Copyright 25 by Astro, Ltd. Copyright 25 by MAIK Nauka /Interperiodica (Russia). PHYSICS OF COLD TRAPPED ATOMS Anaytica Mean-Fied Approach to the Phase Diagram of tracod Bosons in Optica Superattices P. Buonsante, V. Penna, *, and A. Vezzani 2 Dipartimento di Fisica and nità INFM, Poitecnico di Torino, Corso Duca degi Abruzzi 24, I-29 Torino, Itay *e-mai: penna@poito.it 2 Dipartimento di Fisica & nità INFM, niversità degi Studi di Parma, Parco Area dee Scienze 7/a, I-43 Parma, Itay Received September 29, 24 Abstract We report a mutipe-site mean-fied anaysis of the zero-temperature phase diagram for utracod bosons in reaistic optica superattices. The system of interacting bosons is described by a Bose Hubbard mode whose site-dependent parameters refect the nontrivia periodicity of the optica superattice. An anaytic approach is formuated based on an anaysis of the stabiity of a fixed point of the map defined by the sef-consistency condition inherent in the mean-fied approximation. The experimentay reevant case of the two-period one-dimensiona superattice is briefy discussed. In particuar, it is shown that, for a specia choice of the superattice parameters, the haf-fiing insuator domain features an unusua oophoe shape that the singe-site mean-fied approach fais to capture.. INTRODCTION Athough originay introduced for iquid heium in confined geometries [], the Bose Hubbard (BH) mode proves successfu in describing utracod atoms trapped in optica attices [2]. In this framework, the sites of the ambient attice correspond to the oca minima of the effective optica potentia created by counterpropagating aser beams, and the height of the potentia barriers between adjacent minima, which is proportiona to the aser intensity, determines the hopping ampitude in the BH mode. Such a direct reation aows for unprecedented experimenta contro of the mode parameters and pays a key roe in experiments aimed at reveaing the superfuid insuator transition characterizing the BH mode [3]. In genera, the superposition of simpe optica attices with commensurate attice constants gives rise to confining potentias characterized by a richer periodicity the so-caed superattices [4]. Reference [5] reports on a recent experiment in which a simpe D optica superattice is created by superimposing two D optica attices [6] and a cigar-shaped magnetic potentia providing for confinement in the transverse direction. Each of these optica attices is created by the interference pattern of two aser beams crossing at a given ange. The attice constants, determined by the crossing ange, are chosen to be d and d 2 3d, so that the superce of the resuting optica potentia contains three oca minima. Foowing this scheme, a -periodic D superattice i.e. a attice characterized by a -site superce can be created by a suitabe adjustment of the crossing anges determining d and d 2. Foowing the tight-binding-ike approach of [2], a system of utracod akai atoms confined in a D optica superattice comprising M sites is described by the foowing BH Hamitonian: H M k ---n 2 k ( n k ) ( µ v k )n k -- t k ( a k a k a k a k ), () where, a k, and n k a k a k are, respectivey, the boson creation, annihiation, and number operators reevant to the site abeed k. As for the Hamitonian parameters, > accounts for on-site repusion (proportiona to the atomic scattering ength), µ is the grand canonica chemica potentia, v k is the oca potentia at site k, and t k is the hopping ampitude between adjacent sites k and k. The -site periodicity of the superattice yieds t k s t k tτ k, v k s v k v ν k, (2) where s,, M/ abes the superces and t, v are scaing coefficients directy reated to the intensity of the aser beams that give rise to the optica potentia. As is we known, in the homogeneous case Hamitonian () is characterized by the superfuid insuator quantum phase transition []. In more detai, the competition between the on-site repusion and the kinetic energy proportiona to and t, respectivey gives rise to a zero-temperature phase diagram in the µ/ t/ pane consisting of an extended superfuid phase and a series of adjacent Mott-insuator obes. In the atter, the system is remarkaby characterized by a commensurate popuation, i.e., by an integer fiing. Severa numerica and anaytica approaches have been adopted for the study of such a zero-temperature phase diagram. We refer the reader to [7] for a brief review of such techniques. a k 36

2 362 BONSANTE et a. Recenty, some attention has been devoted to the phase diagram of superattice BH modes [8 ]. In genera, incompressibe Mott domains are expected to occur in correspondence with critica fractiona fiing. In the case of D -periodic superattices, such critica fiings are integer mutipes of. Furthermore, it has been shown that, when the oca potentias v j are not a different from each other, some of the Mott domains exhibit an unusua oophoe shape [2]. In this paper, we study the zero-temperature phase diagram of Hamitonian () by adopting a mutipe-site mean-fied approach, thus generaizing the technique introduced in [3]. The atter provides satisfactory quaitative resuts for the quantum phase transition occurring in the homogeneous case, but fais to predict the oophoe insuator domains that may appear in the case of superattices []. We mention that a two-site mean-fied approach is adopted in [7] for the study of homogeneous attices. We furthermore show that, in genera, the zero-temperature phase diagram can be worked out by anayzing the stabiity of a particuar fixed point of the map defined by the mean-fied sefconsistency condition []. Many such anayses can be carried out anayticay based on a perturbative expansion of the spectrum of the mean-fied Hamitonian. This aows one to determine the phase diagram by soving a numerica probem that is much ess demanding than the standard iterative procedure used to dea with the origina sef-consistency equations. Furthermore, in some specia cases, entirey anaytica resuts can be obtained. In particuar, our method provides for an anaytica description of the Mott-obe boundaries of the homogeneous case [, 4]. Expoiting our method, we anayze the reaistic case of a 2 D superattice. In particuar, we study the insuator domain corresponding to the critica fiing f /2, showing that its usua obe shape shrinks at the bottom and turns into a oophoe as the potentia offset v 2 v between the sites of the superce vanishes. Furthermore, we provide an exact anaytic description of the boundaries of such a oophoe domain. 2. MLTIPLE-SITE MEAN FIELD In the simpe case of the usua attice,, quaitative information about the zero-temperature phase diagram of Hamitonian () can be obtained by making use of the singe-site mean-fied approach introduced in [3]. Denoting by the expectation vaue in the ground state, it is assumed that, for every k, a k a k a k a k a k a k a k a k. (3) This aows one to recast Hamitonian () as the sum M of M singe-site Hamitonians, H, where (4) and the so-caed superfuid parameters are to be determined sef-consistenty as α k a k a k. (5) After the transationa invariance characterizing the system is taken into account, α k α k α, the Hamitonians in Eq. (4) are decouped and become formay identica. The origina probem thus reduces to the study of one singe-site Hamitonian. In this framework, the Mott-insuator domains in the µ/ t/ phase diagram are characterized by the vanishing of the superfuid order parameter. Indeed, in this case it is easy to check that the oca density of bosons n k is pinned at an integer vaue and the system is incompressibe. In inhomogeneous structures transationa invariance is ost, and the singe-site Hamitonians in Eq. (4) are couped by sef-consistency conditions (5). This singe-site mean-fied approach gives fairy satisfactory quaitative resuts for superattices whose superce features oca potentias v k that are a different from each other [], but it fais to capture the oophoe-shaped insuator domains that appear when some of these potentias are equa [2]. A more structured approach that takes into account the nontrivia periodicity of a -periodic superattice invoves adopting approximation (3) every th site. By doing so, Hamitonian () becomes the sum of identica -site Hamitonians, one for each superce, and, as in the singe-site approach, the origina probem reduces to the study of one such superce Hamitonian. Dropping the superce index, the atter reads where k ---n 2 k ( n k ) µn k t( α k α k )( a k a k α k ) k k k ---n 2 k ( n k ) ( µ v k )n k t k ( a k a k a k a k ) k t [ α ( a a ) α ( a a ) 2α α ], (6) α j a j a j, j,. (7) This approach is expected to give satisfactory resuts if approximation (3) is adopted for the hopping terms characterized by the owest hopping ampitude, t < t h. The superfuid parameters defined in Eq. (5) are rea, since the boson operators in Eq. (4) have a rea representation on the usua Fock basis. LASER PHYSICS Vo. 5 No. 2 25

3 ANALYTICAL MEAN-FIELD APPROACH TO THE PHASE DIAGRAM 363 As in the singe-site approximation, the Mott-insuator phase is characterized by vanishing superfuid parameters: α α. In this situation, the mean-fied Hamitonian commutes with the tota number of bosons (in the superce), n. Hence, the expectation vaue k k of the atter on the ground-state is fixed to an integer vaue determined by the Hamitonian parameters and, quite interestingy, the fiing of the system f M M n is a mutipe of. k k The most standard approach to the mean-fied probem defined by Eqs. (6) and (7) consists of an iterative numerica procedure. In more detai, the superfuid parameters appearing in mutipe-site Hamitonian (6) at a given iteration are determined evauating Eq. (7) on the ground state of the previous iteration. The procedure is arrested when the vaue of the superfuid parameters does not change significanty between two subsequent iterations. 3. ANALYTICAL APPROACH The standard iterative procedure iustrated above shows that soving the sef-consistency probem in Eqs. (6) and (7) amounts to finding a stabe fixed point of the map α ' F ( α, α ) (8) α ' F ( α, α ), where F j (α, α ) a j, j,. Note that the choice α α, corresponding to the Mott-insuator phase, is a fixed point of the map in Eq. (8) for any vaue of the Hamitonian parameters µ,, {t k }, {v k }. Indeed, as we mentioned in the previous section, in this situation the ground state of the system beongs to a fixed-number subspace and the expectation vaues in Eq. (7) necessariy vanish. This means that the insuator domains are characterized by choices of the Hamitonian parameters that make the fixed point α α stabe. According to the standard criterion, this happens when the absoute vaues of the eigenvaues of the Hessian matrix α F ( α, α ) α F ( α, α ) α F ( α, α ) α F ( α, α ) α α (9) are smaer than. Note that the Hessian matrix is competey determined by a first-order expansion of F j in the parameters α, α, which can in turn be obtained from a first-order expansion of the ground state of in the same parameters. Since the term t α α in Eq. (6) does not contribute first-order corrections to the ground state of, it can be discarded without oss of generaity. After this, the desired first-order approximation can be obtained using t as the perturbative parameter, since it mutipies a of the first-order terms in α and α appearing in Eq. (6): () () Since the unperturbed Hamitonian commutes with the tota number of bosons (in the superce), its eigenstates beong to fixed-number subspaces. Denoting by φ h such eigenstates and by h the reevant eigenvaues, the first-order approximation to the ground state of is ψ φ t ψ, where φ is the unperturbed ground state and This means that where t V, V α ( a a ) α ( a a ). ψ h φ h V φ h φ h. F j ( α, α ) ψ a j ψ φ a j ψ ψ a j φ φ a j a j ψ α c j α c j, φ c jk t h a j a j φ φ h a k a k φ h h (2) (3) (4) Therefore, as we have mentioned, the Hessian matrix in Eq. (9) is determined in terms of the coefficients appearing in first-order approximation (3) and defined in Eq. (4), and the condition for the stabiity of fixed point α α is c ± c c. (5) Since the coefficients in Eq. (4) depend on the Hamitonian parameters in Eqs. () and (2) through the eigenvaues and eigenstates of, inequaity (5) aows one to determine the regions of the µ/ t/ pane pertaining to the Mott-insuator phase. According to the above discussion, within such a phase the (integer) number of bosons in each superce is N φ n k φ and k corresponds to the fractiona fiing f N/. Note that the study of the phase space by inequaity (5) is much ess demanding than the standard iterative procedure briefy iustrated in the previous section. Indeed, for a given choice of the Hamitonian parameters, the atter invoves the iterative diagonaization of a matrix whose size is d, where d k C k k ( k )! is the dimension of the subspace reevant to k! ( )! k bosons in sites and C provides a cutoff for the (in principe) infinite Hibert space of the probem. LASER PHYSICS Vo. 5 No. 2 25

4 364 BONSANTE et a. t/.2. /2.2 /2..2. /2.2 /2. Conversey, no iterative procedure is required for the study of inequaity (5). Indeed, it is sufficient to diagonaize ony the three (independent) bocks of reevant to the tota numbers of bosons N, N, and N, where N is the ce popuation characterizing the Mott domain under investigation. Furthermore, as we iustrate in the foowing, in some simpe cases inequaity (5) can be studied in a competey anaytica way. 4. RESLTS: 2 SPERLATTICE In this section we consider the reaistic case of a 2 D superattice, which can be created as in [5] by superimposing two homogeneous D optica attices with attice constants d and d 2 2d. The insuator domains reevant to the owest fractiona fiings (dark gray), as evauated by means of a numerica study of inequaity (5), are dispayed in the figure for the parameter choice., τ., τ 2.3, v., and, from the top to the bottom pane, v 2.2,.6,.3,.. Note that the width at t/ of the haf-fiing insuator domain equas the energy offset between the sites of the same superce, v 2 v []. As the atter µ/ Superfuid camping around a fractiona insuator domain for a 2 superattice BH Hamitonian. Each pane shows the region of the phase diagram containing the insuator domains (dark gray areas) reevant to the ower critica fiings, which are aso shown. The insets contain a pictoria representation of the reevant effective optica potentia. As the energy offset between the attice sites in the same superce decreases (from top to bottom), the superfuid phase (white) camps around the haf-fiing insuator domain, which assumes an unusua oophoe shape. vanishes, the insuator domain assumes an unusua oophoe shape [2]. In this specia case, (v v 2, bottom pane of the figure), the study of inequaity (5) invoves the diagonaization of 2 2 matrices and can be carried out in a competey anaytica way. After some cacuations, it is possibe to show that the oophoe-domain border is determined by the foowing equation: 3τ 2 ( τ τ 2 )t 3 [ ( 2τ 2 τ ) µ ( 5τ 2τ 2 )]τ t 2 µ 2 ( τ τ 2 )t µ 2 µ 3. (6) A simpe anaysis shows that the oophoe domain disappears when no positive t satisfies the preceding equation for µ, i.e., when τ < 2τ 2. However, this is an artifact introduced by the mean-fied approximation, which is known to provide at best quaitative information. As is shown in [2], the oophoe domain can be proven to exist for any τ 2 τ by resorting to the exact mapping between the hard-core imit (t/ ) of Hamitonian () and the mode for spiness noninteracting fermions on the same superattice. 5. CONCLSIONS In this paper, we introduce a mutipe-site meanfied approach to the study of the zero-temperature phase diagram for utracod bosons in reaistic onedimensiona superattices. A perturbative expansion in the hopping ampitudes between neighboring superces aows one to recast the sef-consistency constraints invoved in this approach into a probem that requires a numerica effort much ess demanding than the usua iterative procedure. Reying on such a mutipe-site mean-fied approach, we suppy some expicit resuts for the experimentay reevant case of a two-periodic superattice. In particuar we show that, as the energy offset between the sites of the superce decreases, the superfuid phase camps around the haf-fiing insuator domain, which assumes an unusua oophoe shape for a vanishing energy offset. Our mutipe-site mean-fied approach shows that this camping effect of the superfuid phase ikewise occurs around a of the fractiona-fiing insuator domains of the 2 superattice. Simiar and even more compex camping effects can be shown to occur on more structured superattices [2]. ACKNOWLEDGMENTS The work of P.B. was entirey supported by the MRST project Quantum Information and Quantum Computation on Discrete Inhomogeneous Bosonic Systems. A.V. aso acknowedges partia financia support from the same project. LASER PHYSICS Vo. 5 No. 2 25

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