State engineering of impurities in a lattice by coupling to a Bose gas
SState engineering of impurities in a lattice bycoupling to a Bose Gas
Kevin Keiler and Peter Schmelcher , Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg, LuruperChaussee 149, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg, LuruperChaussee 149, 22761 Hamburg, GermanyE-mail: [email protected], [email protected]
Abstract.
We investigate the localization pattern of interacting impurities,which are trapped in a lattice potential and couple to a Bose gas. For smallinterspecies interaction strengths, the impurities populate the energetically lowestBloch state or localize separately in different wells with one extra particle beingdelocalized over all the wells, depending on the lattice depth. In contrast, forlarge interspecies interaction strengths we find that due to the fractional filling ofthe lattice and the competition of the repulsive contact interaction between theimpurities and the attractive interaction mediated by the Bose gas, the impuritieslocalize either pairwise or completely in a single well. Tuning the lattice depth,the interspecies and intraspecies interaction strength correspondingly allows for asystematic control and engineering of the two localization patterns. The sharpnessof the crossover between the two states as well as the broad region of theirexistence supports the robustness of the engineering. Moreover, we are able tomanipulate the ground state’s degeneracy in form of triplets, doublets and singletsby implementing different boundary conditions, such as periodic and hard wallboundary conditions.
Keywords : many-body physics, correlations, Bose gas, optical lattice, impurities a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t tate engineering of impurities in a lattice by coupling to a Bose Gas
1. Introduction
The interest in the properties and dynamics of ultracold atomic mixtures has been sub-stantially increasing in the last few decades. This is not only due to their high degreeof controllability, especially of the underlying trapping potentials and inter-atomic in-teractions [1,2], but also because they show a plethora of intriguing phenomena. Theserange from pair-tunnelling effects in lattices [3, 4] to phase separation processes [5–8]and composite fermionization [9–11], and go as far as the spontaneous generation ofsolitons [12, 13]. Compared to systems with a single species the multi-componentcase of different bosons [14, 15] allows for correlations to appear not only within onetype of species, but especially between different bosonic species. One-dimensionalsystems are of particular interest, since they allow for strong correlations in the di-lute regime [16, 17] which is related to the inverse scaling of the effective interactionstrength to the density [18, 19].A specific case of bosonic mixtures is given by the immersion of a minority species,consisting of a few particles and typically called impurities, into a majority speciesof many particles. Such setups have been studied theoretically [20–29] and experi-mentally [30–34] for a single impurity, as simulator for polaron physics, as well asfor many impurities [35–38]. Especially the latter case is of immediate interest sincethe bath, into which the impurities are immersed, mediates an effective attractive in-teraction between the impurities, leading to a clustering of these very particles [39–41].One could think of exploiting this mediated interaction in order to configure the impu-rities, e.g. in a lattice, in a controlled and systematic manner. Such state preparationsare indeed relevant for applications e.g. in quantum information processing and atom-tronics [42–44]. One pathway is to use a small number of minority atoms in order toinfluence a larger number of majority atoms. In a triple-well structure this is achievedby increasing the number of minority atoms in the central well, exploiting the increaseof atom-atom interactions, and thereby initiating and enhancing tunnelling from theleft to the right well of the majority atoms [45]. Pushing this to the extreme, itis in principle possible to implement a single-atom-transistor, which shall serve as aswitch [46–50]. In other atomtronic switching devices, often a triple-well is consid-ered, where one identifies the wells as source, gate and drain. The middle well whichis called the gate serves as a mediator of particle transfer between the outer wells, i.e.the source and the drain [51,52]. Such electronic analogues have the potential to serveas building blocks for cold atom-based quantum computation [53, 54] as well as atomchip technologies [55]. A problem most of these systems suffer from, is the fact thatthey are considered to be isolated and would lose their coherence when coupled to an tate engineering of impurities in a lattice by coupling to a Bose Gas tate engineering of impurities in a lattice by coupling to a Bose Gas
2. Setup and methodology
Our system consists of a mixture of two bosonic species. The bosonic A species istrapped in a one-dimensional lattice with periodic or hard wall boundary conditions.It is immersed in a Bose gas of a second B species of bosons obeying the same boundaryconditions but without the lattice potential. This setup lies within reach of currentexperimental techniques, since beyond controlling the dimensionality, various trappingpotentials for the atoms can be achieved, including in particular one-dimensionalring geometries [60] and box potentials [61]. The optical lattice potential for theA atoms (impurities) does not affect the Bose gas, which is achievable by choosingthe corresponding laser wavelengths and atomic species [62]. Thereby, we create atwo-component system with each species being trapped individually. Furthermore,we introduce a coupling Hamiltonian ˆ H AB between the two species. Both subsystemsare confined to a longitudinal direction, accounting for the one-dimensional character,and excitations in the corresponding transversal direction are energetically suppressedand can therefore be neglected. This finally results in a Hamiltonian of the formˆ H = ˆ H A + ˆ H B + ˆ H AB . The Hamiltonian of the A species reads Figure 1.
Sketch of the two-component mixture. The impurities interactrepulsively via an intraspecies contact interaction of strength g AA and via aninterspecies interaction of strength g AB with the atoms of the Bose gas. Due tothe latter coupling a Bogoliubov mode (red) in form of a phonon in the Bose gasof species B (yellow) mediates an attractive (long-range) interaction between theimpurities (blue). tate engineering of impurities in a lattice by coupling to a Bose Gas H A = (cid:90) L dx ˆ χ † (x) (cid:104) − (cid:126) m A d dx + V sin (cid:16) πk x L (cid:17) + g AA ˆ χ † (x) ˆ χ (x) (cid:105) ˆ χ (x) , (1)where ˆ χ † is the field operator of the lattice A bosons, m A their mass, V the latticedepth, g AA the intraspecies interaction strength, k the number of wells in the latticeand L is the length of the system. The B species is described by the Hamiltonian ofthe Lieb-Liniger model [63–65] for periodic boundary conditionsˆ H B = (cid:90) L dx ˆ φ † (x) (cid:104) − (cid:126) m B d dx + g BB ˆ φ † (x) ˆ φ (x) (cid:105) ˆ φ (x) , (2)where ˆ φ † is the field operator of the B species, g BB > m B is the correspondingmass. Moreover, we assume equal masses for the species m A = m B . The interactionbetween the species A and B is given byˆ H AB = g AB (cid:90) L dx ˆ χ † (x) ˆ χ (x) ˆ φ † (x) ˆ φ (x) , (3)where g AB is the interspecies interaction strength. The interaction strengths g α ( α ∈ { A, B, AB } ) can be expressed in terms of three dimensional s-wave scatteringlengths a Dα , when assuming the above-mentioned strong transversal confinement withthe same trapping frequencies ω σ ⊥ = ω ⊥ for both species σ ∈ { A, B } . In this case it ispossible to integrate out frozen degrees of freedom, leading to a quasi one-dimensionalmodel with g α = 2 (cid:126) ω ⊥ a Dα .Throughout this work we consider a triple-well and focus on the scenario of smallparticle numbers with four impurities N A = 4, thereby having fractional filling in thelattice, and N B = 10 atoms in the Bose gas. The interaction among the latter atomsis set to a value where the depletion is negligible in case of no interspecies coupling,i.e. g BB /E R λ = 6 . × − , with E R = (2 π (cid:126) ) / m A λ being the recoil energy and λ = 2 L/k the optical lattice wavelength.Our numerical simulations are performed using the ab-initio
Multi-Layer Multi-Configuration Time-Dependent Hartree method for bosonic (fermionic) Mixtures (ML-MCTDHX) [66–68], which is able to take all correlations into account. Within ML-MCTDHX one has access to the complete many-body wave function which allowsus consequently to derive all relevant characteristics of the underlying system. Inparticular, this means that we are able to characterize the system in terms of numberstates by projecting onto an appropriate basis [69, 70]. Besides investigating thequantum dynamics it allows us to calculate the ground (or excited) states by usingeither imaginary time propagation or improved relaxation [71], thereby being able to tate engineering of impurities in a lattice by coupling to a Bose Gas
3. State control and engineering
Let us analyze the ground state of our mixture in dependence of the lattice depth V ,the interspecies coupling strength g AB and the intraspecies interaction strength g AA .As a first step, we calculate the ground state using ML-MCTDHX, thereby obtainingthe full wave function. In order to be able to interpret the wave function, we project ina second step the numerically obtained ground state wave function onto number states | (cid:126)n A (cid:105) ⊗ | (cid:126)n B (cid:105) . The number states | (cid:126)n A (cid:105) for the A species are spanned by generalizedWannier states [72, 73], whereas the number states for the Bose gas of species B areeither plane waves or infinite square well eigenstates, i.e. the eigenstates of the kineticenergy operator using either periodic or hard wall boundary conditions. As a result,we gain a clear insight into the ground state in the different regimes, which will bedefined by the distribution of the A species atoms among the Wannier states or Blochstates. In the following, tensor products | (cid:126)n A (cid:105) ⊗ | (cid:126)n B (cid:105) with different number states | (cid:126)n A (cid:105) will be called configurations. Additionally, we explore the effect of hard wall boundaryconditions compared to periodic ones, thereby revealing how they affect the groundstate properties. In the following, we explore the ground state of the system with periodic boundaryconditions for varying V and g AB , fixing the intraspecies interaction strength to g AA /E R λ = 0 . g AA is performed in section3.2. In order to extract information out of the complete many-body wave function,we project onto the above-mentioned number states | (cid:126)n A (cid:105) ⊗ | (cid:126)n B (cid:105) and determine theprobability of being in the number state | (cid:126)n A (cid:105) for the impurity A species, irrespectiveof the number state configurations of the B species, namely P ( | (cid:126)n A (cid:105) ) = (cid:88) i |(cid:104) (cid:126)n Bi | ⊗ (cid:104) (cid:126)n A | Ψ (cid:105)| , (4) tate engineering of impurities in a lattice by coupling to a Bose Gas {| (cid:126)n Bi (cid:105)} could be any number state basis set of the Bose gas with fixed particlenumber and | Ψ (cid:105) is the total many-body ground state wave function. In order toassociate the impurity state | n A , n A , n A (cid:105) with a spatial distribution we construct thenumber states either with a generalized Wannier basis of the lowest band or thecorresponding Bloch basis set, indicated in the following by the subscript W or Bl ,respectively. In principle it is necessary to consider also Wannier states of higher bands,but it turns out that the many-body ground state wave function is approximately welldescribed by a superposition of tensor products of number states | (cid:126)n A (cid:105) ⊗ | (cid:126)n B (cid:105) with theFock space of the A species restricted to the lowest band. Number states spanned byWannier or Bloch states of higher bands do not contribute. The order of the entriesin the number state, built of Wannier states, is connected to the localization of theWannier states in the wells from left to right, e.g. n describes the number of A atomsin the left localized Wannier state of the lowest band. In contrast to that, the orderingfor the number states, built of Bloch states, follows the energy of the Bloch states ofthe lowest band, e.g. n corresponds to the energetically lowest Bloch state. Thetransformation between Bloch and Wannier states is given by w bR (x) = 1 √ k (cid:88) p exp( − ipR ) φ bp (x) , (5)where w R (x) is the Wannier state associated with the position R of the correspondingwell, k the number of lattice sites, p the momentum of the Bloch states φ p (x) and b the index of the band.Figure 2 shows that the many-body ground state can be divided into fourdifferent regions. For small lattice depths and interspecies interaction strengths allfour particles of the A species populate the energetically lowest Bloch state [figure2(c)]. Increasing the lattice depth for small g AB three lattice atoms will localizeseparately in different wells with the one extra particle being delocalized over all thewells [figure 2(a)]. This can be seen by the fact that the corresponding probability isgiven by P ( | , , (cid:105) W ) = , meaning that the remainder of the probability is equallydistributed over the states | , , (cid:105) and | , , (cid:105) , resulting in the state | Ψ (cid:105) M = 1 √ (cid:104) | , , (cid:105) W ⊗ | ψ B (cid:105) + | , , (cid:105) W ⊗ | ψ B (cid:105) + | , , (cid:105) W ⊗ | ψ B (cid:105) (cid:105) . (6)Due to translational symmetry each of the states contributes equally with a probabilityof . The reader should note that the structure of the wave function, given inequation 6, can only be uncovered by the procedure of projection onto number states | (cid:126)n A (cid:105) ⊗ | (cid:126)n B (cid:105) and is not explicitly given by the numerical simulation. Therefore, | Ψ (cid:105) M is not the exact result of ML-MCTDHX but an approximation to it. Our approach isto perform the correlation-including ML-MCTDHX calculations and to subsequently tate engineering of impurities in a lattice by coupling to a Bose Gas Figure 2.
Probability of finding the impurity A species atoms (a) being localizedseparately in different wells with the one extra particle being delocalized over allthe wells, (b) all localized in a single well, (c) residing in the energetically lowestBloch state and (d) localized pairwise in two different wells in dependence of thelattice depth V and the interspecies interaction strength g AB for g AA /E R λ =0 . analyze them.Interestingly, the increase of g AB leads to two different regions in the configurationspace, in contrast to the case of g AA = 0 [59]. For g AA = 0, increasing the lattice depthand the interspecies interaction strength, the ground state wave function undergoesa transition from an uncorrelated to a highly correlated state, which manifests itselfin the localization of the lattice atoms in the latter regime of large lattice depths andinterspecies interaction strengths. This means that all A atoms cluster in a singlewell, while the B atoms are expelled from it. This clustering can be understood interms of an attractive induced impurity-impurity interaction, which is mediated bythe B species (cf. figure 1). In contrast to that, allowing for a repulsive intraspeciesinteraction of strength g AA between the A atoms counteracts the induced interaction.However, these two types of interactions do not simply add up, since the inducedinteraction is of long-range type, whereas the intraspecies interaction is of contacttype. As a result, depending on the choice of the lattice depth V and the interspeciesinteraction strength g AB the impurities of species A either accumulate all in one well[figure 2(b)], which is already happening in the case of g AA = 0, or pairwise in twodifferent wells [figure 2(d)]. Apparently, in the case of pairwise localization of the tate engineering of impurities in a lattice by coupling to a Bose Gas g AB is in both regions threefold degenerate (triplet). This essentially means that inthe corresponding region the ground states are either given by | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) , | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) and | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) or by (7) | , , (cid:105) W ⊗ | Ψ B (cid:105) , | , , (cid:105) W ⊗ | Ψ B (cid:105) and | , , (cid:105) W ⊗ | Ψ B (cid:105) . (8)The states {| Ψ iB (cid:105)} and {| ¯Ψ iB (cid:105)} [74] are each normalized to unity and incorporatethe localization effect of the A species by e.g. spatially avoiding the impuritiescorrespondingly, which can be seen in the one-body density (cf. section 3.3, figure5). Due to the degeneracy of the ground state one can choose such superpositions ofthe states in equation 7 and 8, which preserve the translational invariance of the totalHamiltonian. For this reason the probabilities in figures 2 (a),(b),(d) are bounded bya maximum value of 1 / | , , (cid:105) W ⊗ | Ψ B (cid:105) in one regime. In this sense, with the lattice depth of individual wellswe have introduced an additional control parameter for the manipulation of impurityconfigurations.So far, we gained insight into the state configurations that are populated by theimpurities. While this finding itself allows for a systematic control of the impuritiesin the lattice, it is nevertheless of interest in which way the correlation with the Bosegas impacts this very species. Therefore, we also analyze the probability distribution P ( | (cid:126)n B (cid:105) ) of the number states | (cid:126)n B (cid:105) that build the corresponding B species states {| Ψ iB (cid:105)} and {| ¯Ψ iB (cid:105)} . We find that for each of the strongly coupled degenerate groundstates the B species states {| Ψ iB (cid:105)} and {| ¯Ψ iB (cid:105)} each populate the same number stateswith equal probability, i.e. |(cid:104) (cid:126)n B | Ψ iB (cid:105)| = |(cid:104) (cid:126)n B | Ψ jB (cid:105)| with i, j ∈ { , , } , whilediffering only in a relative phase for each coefficient of the number state. This is whyit is sufficient to show only the probability distribution of one B species state for thestates in equations 7 and 8. As a basis for the number states | n B , n B , n B , n B , n B (cid:105) wechoose plane waves (1 / √ L ) exp( iκ x), with wave vectors κ = 2 πz/L ( z = 0 , ± , ± , ... ),where n corresponds to the κ = 0 mode, n , n correspond to z = ± n , n to z = ±
2. It turns out that the population of all higher momentum states is negligible, tate engineering of impurities in a lattice by coupling to a Bose Gas Figure 3.
Probability distribution for the states of the B species for g AA /E R λ =0 . g AB /E R λ = 0 .
135 and (a) V /E R = 18 or (b) V /E R = 22 .
5. (a)describes ground states with a complete localization of A atoms in a single well(cf. equation 8), whereas (b) describes ground states with a pairwise localizationin two wells (cf. equation 7). which we checked explicitly by projecting onto the corresponding number states. Infigure 3, we see that due to the correlation with the impurity species the Bose gascan no longer be described by a single number state with all particles occupying the κ = 0 mode [76]. Interestingly, it is also not sufficient to consider only single particleexcitations. It is rather necessary to consider up to four particle excitations for theB species states in both regimes. Comparing the B species states in the two regimes,one finds that they strongly populate the same number states, but differ w.r.t. thequantitative distribution among those number states. For example, in figure 3(a)the number states | , , , , (cid:105) and | , , , , (cid:105) are less populated than the numberstate | , , , , (cid:105) , whereas their probability exceeds that of | , , , , (cid:105) in figure3(b). These findings support the choice of our treatment of the many-body problemusing a method that is in particular capable of taking all necessary correlations intoaccount. An approximation of the many-body Hamiltonian which relies on few-particleexcitations, will not capture the localization pattern presented in figure 2. In the previous subsection, we have identified four different number stateconfigurations for the impurity species, while assuming a fixed intraspecies interactionstrength of g AA /E R λ = 0 . g AA . In order to explore the range of validity of this crossover,we fix the lattice depth as well as the interspecies interaction strength such that for g AA = 0 we arrive at a degenerate subspace of ground states given by equation 8,instead of the one given by equation 7 for g AA /E R λ = 0 . g AA = 0 the correlated region, which is split into two sub-regions for g AA /E R λ = 0 . tate engineering of impurities in a lattice by coupling to a Bose Gas P ( | (cid:126)n A (cid:105) ), while varying the couplingstrength g AA for fixed V and g AB . In figure 4, we see that for small g AA the correlatedregion is well described by a single triplet of ground states, as in the case of g AA = 0,where all impurities accumulate in a single well. A further increase of the intraspeciesinteraction strength leads to a break-up of the cluster into two pairs (equation 7)and finally results in a ground state with all A atoms localized separately in differentwells with the one extra particle being delocalized over all the wells (cf. equation 6).Essentially, this means that one needs a certain intraspecies interaction strength g AA between the impurities in order to arrive at a crossover diagram as in figure 2. Belowthat critical value the crossover is well captured by the g AA = 0 case, including, if g AA (cid:54) = 0, a regime for small g AB and large V where the ground state is given by | Ψ (cid:105) M (equation 6). In other words, for small g AA the crossover diagram in figure 2 willconsist of three different regimes, where the two regimes in figure 2(b) and (d) willmerge into a single one, describing complete localization of the impurities in a singlewell (equation 8). Thus, the ground state comprising pairwise localization of the Aatoms will not exist in this case.Qualitatively, the crossover in figure 4 can be understood again in terms of acompetition between the attractive induced interaction and the repulsive intraspeciescontact interaction. For small g AA the induced interaction is dominating the behaviourof the A species atoms, leading to their complete localization in a single well.At a certain interaction strength g AA this is no longer the case, resulting in apairwise accumulation of A atoms. Apparently, the different nature of the competinginteractions (long-range and contact) does not lead to a trivial reduction of the four-impurity cluster to a three-impurity cluster, but rather leaves this out as a possibilityand directly favours a two-impurity cluster. Astonishingly, the crossover between thedifferent configurations is very sharp, such that the system occupies only one of thetriplets without superposing them. Furthermore, it is possible to control the width ofthe plateau of the pairwise localization by adjusting the interspecies coupling strength g AB correspondingly. The plateau (red) corresponding to a degenerate manifold of Figure 4.
Probability distribution of the number state configurations of theimpurities in dependence of the intraspecies coupling strength g AA for V /E R =22 . g AB /E R λ = 0 .
084 or (b) g AB /E R λ = 0 . tate engineering of impurities in a lattice by coupling to a Bose Gas g AB [cf. figure 4(b)]. This also means that a smaller value of the interspeciesinteraction strength g AB leads to a smaller critical value of g AA [cf. figure 4(a)] atwhich the transition to the ground state | Ψ (cid:105) M takes place (black plateau, equation 6).In essence, we find that by tuning the intraspecies interaction strength, we are ableto control and engineer the localization of the A atoms in the lattice. The sharpnessof the crossovers allows for a clear and systematic way of choosing the ground states,while the broad plateaus make the triplets robust with respect to fluctuations of g AA .In this sense, the coupled system serves as a transistor-like switching device for numberstate preparation of impurities in a lattice. Our findings in the previous sections so far relied on the fact that we assumed periodicboundary conditions. It is therefore of immediate interest in which way the localizationpattern in figure 2(b) and (d) depends on the choice of the boundary conditions.For this reason, we consider solely values of the lattice depth V and interspeciesinteraction strength g AB of the crossover diagram such that we arrive at the groundstate configurations in equations 7 and 8. The corresponding values are given intable 1. Subsequently, for those two regimes we change the boundary conditions tohard wall boundary conditions. Obviously, this change will break the translationalsymmetry of the Hamiltonian. Instead, the Hamiltonian now obeys parity symmetry,suggesting that the former ground state degeneracy of a triplet might now be givenby a doublet. Indeed, for the states comprising complete localization of the A atomsin a single well (equation 8) we arrive at the subspace of degenerate ground states,where the atoms of species A localize in the outer wells, i.e. | , , (cid:105) W ⊗ | Ψ B (cid:105) and | , , (cid:105) W ⊗| Ψ B (cid:105) [77]. However, the degenerate ground state with pairwise localizationfor periodic boundary conditions does not exhibit any degeneracy for hard wallboundary conditions anymore. In this sense, the ground state in that regime is given bya non-degenerate parity symmetric ground state of the form | , , (cid:105) W ⊗| ¯Ψ B (cid:105) (singlet).Investigating the one-body density of the Bose gas of species B, one gets anintuition for the reason why in one region the ground state triplet becomes a doubletand in the other it becomes a singlet for hard wall boundary conditions. The one-bodydensity shows that the Bose gas accumulates in the centre of the box potential [78](not shown here), because it is energetically favourable for the majority of the bosonsof the Bose gas to occupy the energetically lowest eigenmode of the box potential andnot accumulate to either side of the box. As a consequence, the A atoms localizein the outer wells due to the repulsive coupling to the B species, thereby avoidingoccupation of the middle well. For the ground states comprising complete localizationof A atoms in a single well, the restriction of the A atoms to the outer wells allows tate engineering of impurities in a lattice by coupling to a Bose Gas Figure 5.
One-body density ρ (1) ,iB for the states of the B species in table 1 with g AA /E R λ = 0 . g AB /E R λ = 0 .
135 and (left column) V /E R = 18 or (rightcolumn) V /E R = 22 .
5. The first row is based on periodic boundary conditions,the second row is based on hard wall boundary conditions and the third oneincorporates a repulsive Gaussian potential for the B species in the centre of thebox in addition to the hard wall boundary conditions. The dotted line is a sketchof the lattice potential, indicating the position of the wells [77]. for two possible many-body states out of three in equation 8, whereas for the groundstates with pairwise localization of A atoms only a single state in equation 7 obeysthis restriction.Following the above line of argumentation, one might now ask whether it ispossible to change the singlet ground state into a doublet and vice versa by forcing theBose gas out of the centre and thereby to either side of the box. In order to achievethis we implement a repulsive Gaussian potential for the B species in the middle ofthe box which is localized in the middle well of the lattice and has an amplitude A that is approximately twice as large as V , i.e. A /E R = 50. It turns out that thisprocedure has the effect we aim for, leaving an imprint on the one-body density of thecorresponding B species states [cf. figure 5 (e) and (f)], which is defined as ρ (1) ,iB (x) = (cid:90) dx ... dx N B | Ψ iB (x , x , ..., x N B ) | , (9)integrating over all B atoms except for one.In figure 5, we show the one-body density ρ (1) ,iB ( i ∈ { , , } ) of the states of theB species, following the nomenclature in table 1. Figure 5(e) shows that the one- tate engineering of impurities in a lattice by coupling to a Bose Gas Table 1.
Degenerate subspaces of the ground state for different boundaryconditions for g AB /E R λ = 0 .
135 and g AA /E R λ = 0 . V lead either to a complete localization of A atoms in a singlewell or to a pairwise localization [77]. boundary conditions V /E R = 18 V /E R = 22 . | , , (cid:105) W ⊗ | Ψ B (cid:105) | , , (cid:105) W ⊗ | ¯Ψ B (cid:105)| , , (cid:105) W ⊗ | Ψ B (cid:105) | , , (cid:105) W ⊗ | ¯Ψ B (cid:105)| , , (cid:105) W ⊗ | Ψ B (cid:105) | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) hard walls | , , (cid:105) W ⊗ | Ψ B (cid:105) | , , (cid:105) W ⊗ | ¯Ψ B (cid:105)| , , (cid:105) W ⊗ | Ψ B (cid:105) hard walls and repulsive Gaussian potential | , , (cid:105) W ⊗ | Ψ B (cid:105) | , , (cid:105) W ⊗ | ¯Ψ B (cid:105)| , , (cid:105) W ⊗ | ¯Ψ B (cid:105) body density of the B species indeed exhibits a minimum in the centre of the box,leading to an accumulation to both sides. Consequently, the A atoms accumulatein the middle well, such that only one many-body ground state out of equation 8fulfills this restriction, namely | , , (cid:105) W ⊗ | Ψ B (cid:105) . In contrast to that, for pairwiseimpurity localization forcing the B species out of the centre of the box allows for twopossible ground states from equation 7 [cf. figure 5(f)], namely | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) and | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) . Table 1 summarizes the engineering of the degenerate subspace ofthe ground state in dependence of the boundary conditions for pairwise and completelocalization of the A atoms in a single well. Figure 5(a) resembles the case of g AA = 0,where the particles of species B are expelled from the well where all the A atomsfully localize, leaving an imprint on the one-body density. In figure 5(b) the Bspecies atoms need to be expelled from two wells due to the pairwise localizationof A atoms. Because of the fact that there are less impurities per well the herebyreduced interspecies interaction allows for a larger one-body density of species B inthe region of the pairwise occupied wells. Figure 5 (c) and (d) are similar to (a) and(b) except for a shifting of the density closer to the centre of the box potential. Thisis simply an effect of the change to hard wall boundary conditions. Apart from twospecific ground state configurations (figure 5(b) red and 5(d) green) it is possible toidentify any of the many-body ground states in table 1 just by analyzing the positionof the one-body density of the B species states with respect to the lattice potential -irrespective of the boundary conditions.Thus, we are able to engineer the character of degeneracy of the ground state bychoosing the boundary conditions correspondingly (in combination with a Gaussianpotential). Combining this with the fact that it is possible to switch between differently tate engineering of impurities in a lattice by coupling to a Bose Gas V , g AB and g AA , one might thinkof an (adiabatic) particle transfer of the following type. Initially, we prepare theground state in one of the doublet states (using hard wall boundary conditions), e.g. | , , (cid:105) W ⊗ | Ψ B (cid:105) . Increasing the intraspecies interaction strength g AA adiabaticallythe ground state will reconfigure to the singlet | , , (cid:105) W ⊗ | ¯Ψ B (cid:105) . Essentially, this canbe interpreted as a transfer of two impurities from the left to the right well.
4. Conclusions
We have shown that it is possible to manipulate the configuration space of latticetrapped impurities with fractional filling immersed in a Bose gas. For small interspeciesinteraction strengths, the impurities populate the energetically lowest Bloch stateor localize separately in different wells with the one extra particle being delocalizedover all the wells, depending on the lattice depth. In contrast, for large interspeciesinteraction strengths and depending on the lattice depth and intraspecies couplingwe find that the impurities either localize pairwise or completely in a single well ofthe lattice. Astonishingly, in dependence of the intraspecies and interspecies couplingas well as the lattice depth the system switches between those two internal stateconfigurations, allowing for an engineering of the impurity distribution in a systematicand controlled manner. Furthermore, the change from periodic to hard wall boundaryconditions will reconfigure the ground state from a triplet to either a doublet for groundstates where the A atoms fully localize in one well, or to a singlet for ground stateswhere they localize pairwise in one well. We can exploit this degeneracy even furtherin order to select individual states out of the manifold by applying a small asymmetryto the lattice potential. Eventually, we are not only able to let the impurities clusterin a certain way, but also manipulate in which wells they accumulate. Additionally,we are able to influence the ground state’s character of degeneracy. In the spirit ofatomtronics, we have developed a switching device for many-body state preparation,thereby controlling the accumulation of impurities in a lattice. This analysis is alsoapplicable for a larger number of particles in the environment, while still remaining inthe few particle regime (we have tested this for N B ∈ [10 , g AB . However, such a particleincrease will also increase the attractive induced interaction for a given choice of V and g AB , thereby shifting the transition region. Increasing the number of impurities,the impurities might form multi-atom clusters of different types depending on theparameter regime due to the long-range character of the induced interaction. We haveadditionally performed calculations for three impurities in the same setup. In thiscase the splitting into the two regions of pairwise and complete localization for large g AB does not occur. Instead, for large interspecies interaction strengths we find only tate engineering of impurities in a lattice by coupling to a Bose Gas Acknowledgments
The authors appreciate fruitful and insightful discussions with K. Sengstock. K.K. acknowledges helpful discussions with J. Schurer and M. Pyzh. P. S. gratefullyacknowledges funding by the Deutsche Forschungsgemeinschaft in the framework ofthe SFB 925 ”Light induced dynamics and control of correlated quantum systems”and support by the excellence cluster ”The Hamburg Centre for Ultrafast Imaging-Structure, Dynamics and Control of Matter at the Atomic Scale” of the DeutscheForschungsgemeinschaft. K. K. acknowledges a scholarship of the Studienstiftung desdeutschen Volkes.
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