Entangling lattice-trapped bosons with a free impurity: impact on stationary and dynamical properties
Maxim Pyzh, Kevin Keiler, Simeon I. Mistakidis, Peter Schmelcher
EEntangling lattice-trapped bosons with a free impurity: impact on stationary anddynamical properties
Maxim Pyzh, ∗ Kevin Keiler, Simeon I. Mistakidis, and Peter Schmelcher
1, 2, † Center for Optical Quantum Technologies, Department of Physics,University of Hamburg, Luruper Chaussee 149, 22761 Hamburg Germany The Hamburg Centre for Ultrafast Imaging, Universität Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany
We address the interplay of few lattice trapped bosons interacting with an impurity atom in abox potential. For the ground state, a classification is performed based on the fidelity allowing toquantify the susceptibility of the composite system to structural changes due to the intercomponentcoupling. We analyze the overall response at the many-body level and contrast it to the single-particle level. By inspecting different entropy measures we capture the degree of entanglementand intraspecies correlations for a wide range of intra- and intercomponent interactions and latticedepths. We also spatially resolve the imprint of the entanglement on the one- and two-body densitydistributions showcasing that it accelerates the phase separation process or acts against spatiallocalization for repulsive and attractive intercomponent interactions respectively. The many-bodyeffects on the tunneling dynamics of the individual components, resulting from their counterflow,are also discussed. The tunneling period of the impurity is very sensitive to the value of theimpurity-medium coupling due to its effective dressing by the few-body medium. Our work providesimplications for engineering localized structures in correlated impurity settings using species selectiveoptical potentials.
I. INTRODUCTION
Multicomponent quantum gases can be experimentallystudied with a high degree of controllability in the ultra-cold regime [1, 2]. Specifically, two-component mixturesof bosons or fermions can be trapped in various speciesselective external geometries [3, 4]. Few-body ensemblescan be realized in particular in one-dimension (1D) [5, 6]while the scattering lengths are tunable through Fes-hbach and confinement induced resonances [7, 8]. In1D bosonic mixtures the adjustability of the intercom-ponent interactions gives rise to intriguing phenomenasuch as phase-separation processes [9, 10] in the repul-sive regime, formation of bound states, e.g., droplet con-figurations [11, 12] for attractive interactions as well asquasiparticle-like states in highly particle imbalanced sys-tems [13, 14].In this latter context, an impurity species is embed-ded in an environment of the majority species called themedium. The presence of a finite impurity-medium cou-pling leads to an effective picture where the impurityproperties deviate from the bare particle case exhibiting,for instance, an effective mass [15–18] and induced inter-actions [19–22] mediated by the medium. The resultantstates are often called polarons [23, 24] and have beenexperimentally realized mainly in higher-dimensions [25–29] and to a lesser extent in 1D [13, 30] using spectro-scopic schemes. Since these settings consist of a few-body subsystem they naturally show enhanced correla-tion properties, especially in 1D, rendering their many-body treatment inevitable. In particular, the emergent ∗ [email protected] † [email protected] impurity-medium entanglement can lead to spatial un-dulations of the medium. This mechanism is manifested,for instance, as sound-wave emission [17, 31] and collec-tive excitations [32, 33] of the host or the formation of abound state [34–36] between the impurity and atoms ofthe medium for attractive interspecies interactions.Another relevant ingredient is the external trapping ge-ometry that the two components experience. Indeed, forharmonically trapped and homogeneous systems remark-able dynamical features of impurity physics include thespontaneous generation of localized patterns [17, 37, 38],inelastic collisional aspects of driven impurities [39–41] with the surrounding and their relaxation at longtimescales [42–44]. On the other hand, when a latticepotential is introduced the situation becomes more com-plicated giving rise, among others, to doped insulatorphysics [45, 46] and impurity transport [47–49]. Appar-ently, configuring one component by manipulating its ex-ternal trap while leaving the other intact, e.g. by us-ing a species selective external potential, it is possible tocontrol the response of the unperturbed component viathe impurity-medium interaction [50, 51]. For instance,operating in the lowest-band approximation it has beendemonstrated that a lattice trapped impurity interact-ing with a homogeneous host exhibits besides tunnelingdynamics [52] also self-trapping events [53, 54] and caneven undergo Bloch-oscillations [55]. The opposite case,where the medium resides in the lattice, provides an ex-perimental probe of the impurity-medium collision pa-rameters [56] and interaction strength [57].In this work by considering an impurity in a box po-tential and a lattice trapped few-body medium we ex-amine how the latter affects the impurity’s spatial distri-bution by means of (de-)localization for different latticedepths and intercomponent interactions. Indeed, a lat- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b tice trapped medium can reside either in a superfluidor an insulating-like phase [45], a fact that is expectedto crucially impact the impurity’s configuration and viceversa [58]. To address the ground state properties andquantum quench dynamics of the above-discussed impu-rity setting we utilize the multi-layer multi-configurationtime-dependent Hartree method for atomic mixtures(ML-MCTDHX) [59–61]. This variational method en-ables us to account for the relevant correlations of themixture and operate beyond the lowest-band approxima-tion for the medium.Focusing on the ground state of the system and in or-der to testify its overall response for varying intercom-ponent interactions we determine the fidelity betweenthe coupled and decoupled composite system both atthe many-body and the single-particle level. Note thatin impurity settings this observable is commonly termedresidue [23, 24] enabling us to identify e.g. the polaronformation, while the influence of the impurity-mediumentanglement in this observable is still an open issue. Itis demonstrated that despite the fact that the total entan-gled state may strongly deviate from its decoupled con-figuration, this effect is arguably less pronounced or evendiminished at the single-particle level. Furthermore, weshowcase that the build-up of impurity-medium entangle-ment is sensitive to the interplay between the intercompo-nent interactions and the lattice depth [45]. Interestingly,stronger interactions do not necessarily lead to a largeramount of entanglement, whereas the state of the major-ity species may undergo substantial structural changes,which remain invisible at the single-particle level. More-over, we identify the imprint of the background on theimpurities and vice versa by relying on one- and two-bodydensity distributions evincing a rich spatial structure ofthe components with respect to the lattice depth as wellas the inter- and intracomponent interactions. In particu-lar, it is argued that for repulsive (attractive) interactionsthe impurity delocalizes (localizes) around the central lat-tice site. The delocalization of the impurity is accompa-nied by its phase-separation with the majority compo-nent [62], where the impurity tends to the edges of thebox for a superfluid background or exhibits a multi-humpstructure for an insulating medium. We further analyzehow much the intercomponent correlations are actuallyinvolved in the structural changes observed in the spatialprobability distributions. To this end we compare den-sity distributions of the numerically exact ground state tothe corresponding ones of an approximate non-entangledground state. We identify that the entanglement-inducedcorrections accelerate phase-separation at repulsive cou-plings and generally slow down spatial localization at at-tractive interactions.Finally, we monitor the non-equilibrium dynamicsof the mixture. We prepare the system in a phase-separated, i.e. disentangled configuration, and quenchthe intercomponent interactions to smaller values re-sulting in the counterflow of the components and thustriggering their tunneling dynamics and the consequent build-up of entanglement. The majority component playsthe role of a material barrier for the impurity [49, 63]which performs tunneling oscillations whose period de-pends strongly on the impurity-medium interaction. Themany-body nature of the tunneling process of the com-ponents is testified by invoking the individual naturalorbitals constituting the time-evolved many-body state.Our presentation is structured as follows. In Section IIwe introduce the impurity setting and in Section III wediscuss our many-body treatment to tackle its groundstate and dynamics. The ground state properties of thedelocalized impurity and the lattice trapped medium areaddressed in Section IV. We analyze the fidelity betweenperturbed and unperturbed (reduced) density operators,quantify the degree of entanglement and visualize its im-pact on single- and two-body density distributions of eachspecies for different intra- and intercomponent interac-tions and lattice depths. The non-equilibrium dynam-ics of the mixture following a quench of the impurity-medium coupling to smaller values is discussed in Sec-tion V. We provide a summary of our results and elabo-rate on future perspectives in Section VI. II. SETUP AND HAMILTONIAN
We consider a single impurity particle immersed ina few-body system of ultracold bosons. Both compo-nents reside in a quasi-1D geometry ensured by a strongtransversal confinement [13]. Along the longitudinal di-rection the N A majority species atoms of mass m A aretrapped inside a lattice of depth V with l sites and length L with hard-wall boundary conditions. The impurityatom of mass m B is subject to a box potential of the samelength. The species-dependent trapping has been suc-cessfully demonstrated experimentally [3, 4]. The inter-particle interactions are of s-wave contact type with g AA denoting the majority-majority interaction strength and g AB the majority-impurity coupling. Both may be tunedindependently by a combination of Feshbach and confine-ment induced resonances [7, 8]. Furthermore, we assumeequal masses m A = m B , which corresponds to a mixtureof the same isotope with the particles being distinguish-able due to two different hyperfine states [64–69]. Byintroducing R ∗ = L and E ∗ = (cid:126) / ( mL ) as length andenergy scales we arrive at the following rescaled many-body Hamiltonian: H = − ∂ ∂y − N A (cid:88) i (cid:18) ∂ ∂x i + V sin ( πlx i ) (cid:19) + g AA N A (cid:88) i
In order to account for effects stemming from inter-particle correlations we rely on the Multi-Layer Multi-Configurational Time-Dependent Hartree Method foratomic mixtures (ML-MCTDHX), for short ML-X [59–61]. This ab-initio method has been successfully appliedto solve the time-dependent Schrödinger equation of var-ious experimentally accessible and extensively studiedsystems. The core idea of this method lies in expandingthe many-body wave-function in terms of product statesof time-dependent single-particle functions [70, 71]. Thisbecomes beneficial, when the number of basis configu-rations with considerable contribution to the state fluc-tuates weakly during the time propagation, whereas theconfigurations themselves do change. Taking a variation-ally optimal basis at each time-step allows us to cover thehigh-dimensional Hilbert space at a lower computationalcost compared to a time-independent basis.The wave function ansatz for a given system is de-composed in multiple layers. On the first layer, calledtop layer, we separate the degrees of freedom of the bi-nary mixture into product states of majority and im-purity species functions | Ψ σi ( t ) (cid:105) with σ ∈ { A, B } and i ∈ { , . . . , S } : | Ψ( t ) (cid:105) = S (cid:88) i =1 (cid:112) λ i ( t ) | Ψ Ai ( t ) (cid:105) ⊗ | Ψ Bi ( t ) (cid:105) . (2)Here, the time-dependent coefficients λ i ( t ) , normalizedas (cid:80) Si =1 λ i ( t ) = 1 , determine the degree of entanglementbetween the components [72]. The choice of S = 1 resultsin the so-called species mean-field (SMF) approximation,meaning that no entanglement is assumed between thecomponents [15]. In that case the intercomponent corre-lations, if present, are neglected and every component iseffectively subject to an additional one-body potential in-duced by the fellow species [49, 62]. In this work, we puta special emphasis on the impact of the entanglementon several one- and two-body quantities by comparing the numerically exact ground state to the correspondingSMF approximation.On the second layer, called species layer, eachspecies function | Ψ σi ( t ) (cid:105) is expanded in terms of species-dependent symmetrized product states of single-particlefunctions (SPFs) | ϕ σj ( t ) (cid:105) with j ∈ { , . . . , s σ } , accountingfor the bosonic nature of our particles and abbreviatedas | (cid:126)n σ (cid:105) = | n σ , . . . , n σs σ (cid:105) : | Ψ σi ( t ) (cid:105) = (cid:88) (cid:126)n σ | N σ C i,(cid:126)n σ ( t ) | (cid:126)n σ ( t ) (cid:105) . (3)In this expression the sum is performed over all config-urations (cid:126)n σ | N σ obeying the particle-number constraint (cid:80) s σ i =1 n σi = N σ . On the third and final layer, called prim-itive layer, each SPF is represented on a one-dimensionaltime-independent grid [73].The Dirac-Frenkel variational principle [74] is subse-quently applied to the above ansatz in order to derivethe coupled equations of motion for the expansion co-efficients λ i ( t ) , C i,(cid:126)n σ ( t ) and the SPFs | ϕ σj ( t ) (cid:105) . Finally,performing imaginary time-evolution one arrives at theground state wave-function (Section IV), whereas the realtime-propagation allows to study the non-equilibrium dy-namics of an arbitrary initial state (Section V). The re-sults to be presented below have been obtained by using ( S, s A , s B ) = (4 , , functions/SPFs on the top/specieslayers as well as grid points on the primitive layer.We have carefully checked the convergence behavior ofour results by comparing to simulations with a largernumber of orbitals ( S, s A , s B ) = (6 , , and found nosignificant changes for the quantities of interest.In the following we will often refer to the reduced j -body density operators ˆ ρ σj of species σ and the intercom-ponent reduced ( j + k ) -body density operator ˆ ρ σ ¯ σj + k ob-tained from the many-body density operator ˆ ρ = | Ψ (cid:105) (cid:104) Ψ | : ˆ ρ σj = tr Nσ \ j { tr N ¯ σ { ˆ ρ }} , (4) ˆ ρ σ ¯ σj + k = tr Nσ \ j { tr N ¯ σ \ k { ˆ ρ }} , (5)where N σ \ j stands for integrating out N σ − j coordi-nates of component σ and ¯ σ (cid:54) = σ . Of particular interestare the reduced one-body density operators ˆ ρ A and ˆ ρ B aswell as the reduced two-body intra- and intercomponentdensity operators ˆ ρ A and ˆ ρ AB respectively, since they de-termine the expectation values of various experimentallyaccessible local one- and two-body observables, such asthe average particle position, the inter-atomic distanceor the wave-packet width. IV. IMPACT OF INTERCOMPONENTCOUPLING ON GROUND STATE PROPERTIES
In Section IV A we analyze to which extent the many-body wave-function as well as the reduced one-body den-sity operators are modified by the intercomponent inter-action. To this end we analyze the fidelity between theinteracting and non-interacting (reduced) density opera-tors, which is a measure of their closeness. We find thatwith increasing absolute value of the interaction strengththe system is more robust w.r.t. changes on the one-bodyas compared to the many-body level. Moreover, eachcomponent is affected differently depending on the lat-tice depth and majority interaction strength.Subsequently, in Section IV B we quantify the degree ofentanglement by means of the von-Neumann entropy andidentify parameter regions with substantial inter-particlecorrelations. Interestingly, increasing the absolute valueof the intercomponent coupling does not always result instronger entanglement. In fact, there are parameter re-gions where a strongly interacting ground state becomesalmost orthogonal to the non-interacting one and thecomponents remain to a good approximation disentan-gled.Finally, we combine insights from Sections IV Aand IV B to identify interesting parameter regimes andperform an in-depth analysis of the underlying physicalphenomena in Section IV C. In particular, we inspect howthe spatial representation of density operators is alteredand compare those to the corresponding SMF results.The latter allows us to spatially resolve the correctionsto the SMF densities induced by the entanglement andinterpret its impact as acceleration or deceleration of theundergoing processes, e.g., the phase separation or local-ization.
A. Fidelity for quantifying the impact of theintercomponent interaction
First, we aim to analyze how the intercomponent cou-pling g AB impacts the ground state of non-interactingspecies (NIS) at g AB = 0 . For this purpose, we evaluatethe fidelity [75] of two density operators ˆ ρ and ˆ σ definedas: F (ˆ ρ, ˆ σ ) = (cid:18) tr (cid:113)(cid:112) ˆ ρ ˆ σ (cid:112) ˆ ρ (cid:19) = F (ˆ σ, ˆ ρ ) . (6)We start with the fidelity between a NIS many-bodydensity ˆ ρ = | Ψ (cid:105) (cid:104) Ψ | and a many-body density ˆ ρ g = | Ψ g (cid:105) (cid:104) Ψ g | for some finite coupling g AB (Fig. 1). Sinceboth density operators describe pure states, Eq. (6) re-duces to F mb = | (cid:104) Ψ | Ψ g (cid:105) | . This measure, F mb , is alsoknown as the polaron residue studied in the context ofphonon dressing of an impurity particle immersed in abath of majority atoms [23, 24].For a weakly interacting ( g AA = 0 . ) majority com-ponent [Fig. 1a] we observe that the many-body fidelityat a fixed lattice depth decreases monotonously with themodulus of the coupling strength g AB . At deep latticesthe rate of its reduction is larger, a behavior which iseven more pronounced at strong negative g AB , wherethe interacting state becomes almost orthogonal to thenon-interacting one ( g AB = − and V = 1000 ). The black dashed line encircles a parameter region of insta-bility where the SMF ansatz collapses to a configurationwith broken parity symmetry. For a moderately inter-acting ( g AA = 3 . ) majority component [Fig. 1b] themany-body fidelity becomes much more stable. Contrar-ily to Fig. 1a the rate of reduction with g AB is larger atshallow lattices instead. Finally, for a moderately deep( V = 500 ) lattice [Fig. 1c] we observe a peculiar fastdecay around g AA ≈ − starting at g AB < − . Addi-tionally, at g AA ≈ − and positive g AB there is a smallpronounced decay region (black dashed circle), which isabsent in the SMF approximation.Next, we analyze the fidelity between a free impuritydescribed by a pure state | Φ (cid:105) (cid:104) Φ | and an entangled one ˆ ρ B , in general being a mixed state [Fig. 2]. Eq. (6) thensimplifies to F B = | (cid:104) Φ | ˆ ρ B | Φ (cid:105) | . This measure allowsto judge to which extent the impurity atom is still a "free"particle of mass m B . We emphasize that it should not beconfused with a polaron quasi-particle having a renormal-ized effective mass. We observe that F B follows overall asimilar pattern as the many-body fidelity F mb , but witha significantly slower decay rate. Though there are somestrong qualitative differences, see in particular Fig. 2c.Namely, the abrupt decay of F mb around g AA ≈ − atnegative g AB [Fig. 1c] is absent in F B along with thesmall decay region at positive g AB (black dashed circle).From this we anticipate that the majority component isresponsible for these features in F mb .For the above reason, we now investigate the comple-mentary fidelity F A = F (ˆ ρ A ( g AB = 0) , ˆ ρ A ) , i.e., betweenmixed states characterizing a majority particle in the NISstate ˆ ρ A ( g AB = 0) and in the interacting state ˆ ρ A [Fig. 3].This quantity captures to which extent a majority par-ticle is still in a mixed state induced solely by the in-traspecies interaction strength g AA . In case of a weak g AA [Fig. 3a] F A is notably affected only at deep lattices V > and strong negative coupling g AB < − . Forlarge g AA [Fig. 3b] we observe that the intercomponentcorrelations are not strong enough to overcome the in-traspecies ones, thus barely affecting the mixedness ofthe NIS majority state, since F A ≈ in the whole range − < g AB < and < V < . In Fig. 3c we find ev-idence that the majority component is indeed responsiblefor the particular decay patterns observed in the many-body fidelity F mb , which were absent in F B . Overall,the majority component demonstrates a higher level ofrobustness at the single-particle level as compared to theimpurity. − − g AB V b) Figure 1. Fidelity | (cid:104) Ψ | Ψ g (cid:105) | between a many-body state | Ψ (cid:105) at g AB = 0 and a many-body one | Ψ g (cid:105) at finite g AB , for a) g AA = 0 . , b) g AA = 3 . and c) V = 500 as a function of the majority-impurity coupling g AB and the lattice depth V (a,b) orthe interaction strength of the majority atoms g AA (c). All quantities are given in box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) with L denoting the extension of the box trap. Regions encircled by black dashed lines indicateparameter regions with substantial qualitative differences to the SMF ansatz. − − g AB V a) − − g AB V b) − − − g AB − − − g AA c) F B Figure 2. Fidelity F B = | (cid:104) Φ | ˆ ρ B | Φ (cid:105) | between a free impurity particle | Φ (cid:105) at g AB = 0 and an entangled one ˆ ρ B at finite g AB , for a) g AA = 0 . , b) g AA = 3 . and c) V = 500 and varying majority-impurity coupling g AB and the lattice depth V orthe interaction strength of the majority atoms g AA . All quantities are expressed in box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) while L is the extension of the box trap. B. Entropy measures for quantifying the degree ofcorrelations
As we have seen in the previous section, an initiallydisentangled composite system may be drastically influ-enced by the intercomponent coupling. However, it is farfrom obvious to which extent the correlations are actu-ally involved when the ground state undergoes structuralchanges [76]. For instance, a strongly interacting groundstate may in fact just represent a different disentangledstate or a state seemingly unaffected by the coupling mayfeature substantial correlations which guarantee its ro-bustness. To investigate these intriguing possibilities weperform a further classification based on the degree ofinter-particle correlations.To quantify the degree of correlations in our impuritysystem we use the von-Neumann entropy of the reduceddensity operators [77]. Here, we distinguish between the entanglement entropy S vN of the reduced density oper- ator ˆ ρ σ of species σ [9, 45] and the fragmentation en-tropy S σvN of the reduced one-body density operator ˆ ρ σ of species σ [51, 78, 79]. The former, ˆ ρ σ , is obtained bytracing the density operator ˆ ρ of the composite many-body system over one of the species, while the latter, ˆ ρ σ , by additionally tracing ˆ ρ σ over all of the particlesof the remaining component except one. In the pres-ence of correlations the resulting reduced density oper-ator will describe a mixed state. The entanglement en-tropy is caused by intercomponent correlations whereasthe fragmentation entropy is primarily a signature of in-tracomponent ones, though it can be greatly impactedonce the intercomponent correlations become dominant.Explicitly, the entanglement and fragmentation entropiesare given as: − − g AB V a) − − g AB V b) Figure 3. Fidelity F A = F (ˆ ρ A , ˆ ρ A ( g AB = 0)) between mixed states characterizing a majority particle when the medium isdisentangled ˆ ρ A ( g AB = 0) and entangled ˆ ρ A with the impurity atom, for a) g AA = 0 . , b) g AA = 3 . and c) V = 500 as afunction of the majority-impurity coupling g AB and the lattice depth V (a,b) or the interaction strength of the majority atoms g AA (c). All quantities are provided in box units of characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) with L being theextension of the box trap. S vN = − tr (ˆ ρ σ ln ˆ ρ σ ) = − S (cid:88) i =1 λ i ln λ i with ˆ ρ σ = tr ¯ σ (ˆ ρ ) = S (cid:88) i =1 λ i | Ψ σi (cid:105) (cid:104) Ψ σi | , (7) S σvN = − tr (ˆ ρ σ ln ˆ ρ σ ) = − s σ (cid:88) i =1 n σi ln n σi with ˆ ρ σ = tr N σ − (ˆ ρ σ ) = s σ (cid:88) i =1 n σi | Φ σi (cid:105) (cid:104) Φ σi | . (8)In these expressions, λ i and | Ψ σi (cid:105) denote the natural pop-ulations and natural orbitals of the spectrally decom-posed ˆ ρ σ , while n σi and | Φ σi (cid:105) are the natural popula-tions and natural orbitals of the spectrally decomposed ˆ ρ σ [59, 71]. Also, S and s σ are the number of speciesorbitals and single-particle functions respectively, N σ isthe number of σ component particles and σ (cid:54) = ¯ σ .In the following, we display the species entanglement S vN from Eq. (7) [Fig. 4] and the majority fragmen-tation S AvN from Eq. (8) [Fig. 5] as a function of themajority-impurity coupling g AB and the lattice depth V or the interaction strength of the majority atoms g AA .In case the entanglement entropy S vN is close to zero,the corresponding subsystems are to a very good ap-proximation disentangled. Thus, making a SMF ansatzin Eq. (2) would greatly facilitate numerical calculationswhile providing quantitatively good results for physicalobservables. On the other hand, already moderate val-ues of entanglement may have an impact on some phys-ical quantities with measurable differences to the SMFapproximation. Regarding the fragmentation entropy ofinteracting majority atoms S AvN it is highly non-trivialto predict how their intrinsic mixedness, caused by the intra-particle interactions g AA , can be changed by theintercomponent coupling g AB .
1. Weakly repulsive interacting majority component
For a weakly interacting majority component with g AA = 0 . , the entanglement entropy S vN [Fig. 4a] dis-plays two different behaviors depending on the sign of thecoupling strength. For positive g AB it increases graduallywith increasing coupling strength g AB , with the build-upbeing faster for a deeper lattice [51]. This is related tothe onset of phase separation taking place sooner for adeeper lattice with increasing g AB (see also the discus-sion in Section IV C). Turning to negative g AB the en-tanglement entropy first grows gradually with decreasingcoupling strength g AB , but then, for larger V below somethreshold value, the entanglement reduces to almost zero( g AB < − and V > ). Apart from the above men-tioned pattern the overall behavior of S vN in Fig. 4a isvery similar to the one observed in the correspondingmany-body fidelity [Fig. 1a].The fragmentation entropy of the majority component S AvN [Fig. 5a] at g AB = 0 is larger for a deeper lattice.The reason is that the ratio of the intraspecies interac-tion energy and the single-particle energy of the majoritycomponent increases with a larger V or g AA . In the limitof an infinitely deep lattice or an infinitely strong in-traspecies repulsion we expect full fermionization, mean-ing that the one-body density operator becomes a mixedstate with a uniform distribution of natural orbitals andthe fragmentation entropy of the majority componentreaches the value ln( N A ) ≈ . . However, we observethat we are operating far away from that limit, since max S AvN < . .At positive g AB , as the entanglement entropy S vN builds up [Fig. 4a], the fragmentation entropy S AvN of themajority component at g AB = 0 is more robust to varia- − − g AB V a) − − g AB V b) − − − g AB − − − g AA c) S v N Figure 4. Entanglement entropy S vN , see Eq. (7), for a) g AA = 0 . , b) g AA = 3 . and c) V = 500 with varying majority-impurity coupling g AB and the lattice depth V (a,b) or the interaction strength of the majority atoms g AA (c). All quantitiesare given in box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) with L denoting the extension of the boxtrap. − − g AB V a) − − g AB V b) − − − g AB − − − g AA c) S A v N Figure 5. Fragmentation entropy S AvN , see Eq. (8), for a) g AA = 0 . , b) g AA = 3 . and c) V = 500 with respect to themajority-impurity coupling g AB and the lattice depth V (a,b) or the interaction strength of the majority atoms g AA (c). Allquantities are provided in terms of box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) while L denotesthe extension of the box trap. tions of g AB at deeper lattice depths compared to shal-low lattices [Fig. 5a]. Once the entanglement becomesstrong enough to overcome intracomponent correlations,the fragmentation entropy of the majority atoms startsto increase with a fast rate (e.g. V = 1000 , g AB > ). Atnegative g AB , if the medium features a small fragmen-tation entropy at g AB = 0 ( V < ), then S AvN risesfirst with decreasing g AB , reaches a local maximum andfinally drops to very small values at a sufficiently strongcoupling strength. In contrast, if the fragmentation en-tropy of the decoupled majority component has alreadyreached a moderate magnitude ( V > ), then the ini-tial fragmentation is gradually reduced with decreasing g AB , until finally both entropies become negligibly small( g AB < − ). Once that happens, the resulting many-body state becomes to a good approximation a disentan-gled composite state with a condensed majority compo-nent.
2. Moderately repulsive interacting majority component
The entanglement entropy S vN of a moderately inter-acting majority medium at g AA = 3 . in Fig. 4b dis-plays the same qualitative behavior as the many-bodyfidelity F mb shown in Fig. 1b. Contrary to g AA = 0 . theentanglement is overall less pronounced and builds upfaster at shallow lattice depths instead. Such a compara-tively weak entanglement leaves only a minor imprint onthe fragmentation of the majority component S AvN , seeFig. 5b, manifested as a weak dependence on the cou-pling g AB . The fragmentation of the majority species issubstantial compared to g AA = 0 . Fig. 5a at the samelattice depth. Nevertheless, the fermionization limit isnot yet reached, since max S AvN ≈ . . The intercompo-nent correlations are not strong enough to overcome theintraspecies ones in accordance with the robustness ofthe majority component observed on the one-body levelin Fig. 3b. From this we expect a rather small impact ofentanglement on observables, which depend solely on themajority particle distribution.
3. Attractively interacting majority component
Finally, we analyze the dependence of the above-described entropy measures on the intraspecies interac-tion strength g AA for a moderately deep lattice depth V = 500 [Fig. 4c and Fig. 5c]. Since repulsive interactionshave been already amply covered, we here concentrate onnegative g AA and g AB .As it can be readily seen, there is a parameter sector at g AB < and g AA > − containing high values for the en-tanglement entropy S vN [Fig. 4c]. This sector displays asimilar behavior to S vN in Fig. 4a at negative couplings,namely starting from the decoupled regime, the entan-glement grows with decreasing g AB , only to drasticallydecrease below some negative threshold value of g AB .This threshold for g AB lies at lower values the higherthe intracomponent interaction strength g AA is. We findthat this abrupt decay of S vN coincides with the oneobserved in the many-body fidelity F mb [Fig. 1c]. Thissuggests that the disappearance of intercomponent corre-lations leads to an increased susceptibility of the systemto g AB variation. The other decay region, present in F mb at g AA ≈ − and negative g AB , is missing in the entan-glement entropy S vN . Form this we infer that it can beunderstood within the SMF picture. Additionally, thereis also another much smaller sector characterized by ahigh entanglement entropy at g AB > and g AA ≈ − . Itis directly related to structural changes observed in F mb and F A at the same values [Fig. 1c and Fig. 3c], whichwould have been absent in the SMF picture. Apart fromthat, below g AA < − the entanglement entropy amongthe components is either absent or of minor relevance.Previously, we have mentioned that an isolated major-ity species, which interacts repulsively ( g AA > ), fea-tures a higher degree of fragmentation the larger g AA is.In the case of attractive interactions ( g AA < ), how-ever, the situation is different. Namely, starting from g AA = 0 the fragmentation entropy tends first to increasewith decreasing g AA , but then decreases up to the pointof describing approximately a condensed state again [seeFig. 5c at g AB = 0 ]. Regarding the impact of the in-tercomponent coupling g AB on S AvN we observe overallvery similar patterns as for the entanglement entropy S vN [Fig. 4c]. Regions where both entropic measures S vN and S AvN are of small magnitude remind of the cor-responding sectors in Fig. 4a and Fig. 5a at
V > and g AB < − . C. Single- and two-particle density distributions
The measures of fidelity and entropy discussed in theprevious sections are very useful in identifying parame-ter regions being substantially impacted and/or highlycorrelated indicating regimes of high interest for furtherinvestigation. However, they do not provide insights intothe actually undergoing processes. To get a better under-standing we ask for the impact on measurable quantitiessuch as the one-body and two-body density distributionfunctions, which can be accessed by fluorescence imagingwith a quantum gas microscope [80–84].In the following, ρ σ ( z ) describes the probability den-sity to find a single particle of species σ at position z ,while ρ σ ¯ σ ( z , z ) denotes the probability density to si-multaneously measure one particle of species σ at posi-tion z and another one of the same or different species ¯ σ at position z . The expectation value of any local ob-servable depending on up to two degrees of freedom canbe evaluated as an overlap integral with the appropriateprobability density. Since many local observables oftendepend only on the distance between the particles, i.e. O ( z , z ) = O ( z − z ) , we replace ρ σ ¯ σ ( z , z ) by the prob-ability density ρ σ ¯ σr ( r ) to measure two particles belongingto the same or different species at a relative distance r independent of their individual positions. To this end weperform a coordinate transformation R = ( z + z ) / and r = z − z giving the following identity: (cid:90) ρ σ ¯ σ ( z , z ) dz dz = (cid:90) ρ σ ¯ σ ( r, R ) dr dR. (9)Then we define: ρ σ ¯ σr ( r ) = (cid:90) ρ σ ¯ σ ( r, R ) dR. (10)Our first goal here is to investigate how the above men-tioned quantities are affected in parameter sectors dis-playing strong susceptibility to structural changes iden-tified in Section IV A and, in particular, whether the den-sity distributions are capable to capture the undergoingchanges in the many-body state.Our second goal is to extract the impact of the en-tanglement. To this end we compare the above densitydistributions obtained from the variational ML-X calcu-lations to the ones where the SMF ansatz is assumed.The latter will be distinguished by a tilde sign placed ontop of the corresponding quantities. In the following, weshall evince that a large entanglement entropy identifiedin Section IV B has indeed a notable impact, but not al-ways on all of the above mentioned density distributions.Thus, it may enhance or impede the effects coming fromthe induced SMF potential, such as phase separation andlocalization, or affect the bunching properties of the ma-jority component. a1a2 b1b2 c1 d1d2c2 Figure 6. Upper panels: one-body probability densities ρ A ( x ) , ρ B ( y ) [Eq. (4)] and distance probability distributions ρ AAr ( x − x ) , ρ ABr ( x − y ) [Eq. (10)] at g AA = 0 . , V = 100 and for various values of g AB (see legend). Lower panels: difference betweenprobability densities obtained from the variational ML-X simulations and the SMF ansatz, the latter distinguished by a tildesign. All quantities are given in box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) with L being theextension of the box trap.
1. Weakly repulsive interacting majority component
For a shallow lattice ( V = 100 ) we observe in Fig. 6that the majority component (panel a1) at g AB = 0 occu-pies mainly the central site (at z = 0 ) and the two inter-mediate ones (at z = ± . ), while ρ AAr (panel c1) featuresan almost Gaussian shape due to weak intraspecies corre-lations. At moderate positive couplings ( g AB > ) bothquantities are only slightly affected in accordance withthe robustness of F A in this interaction regime [Fig. 3a].At moderate negative couplings ( g AB < − ) both ρ A and ρ AAr shrink with decreasing g AB indicating an increasedbunching tendency of the majority atoms towards thecentral lattice site. The impact of entanglement hereis moderate. It leads to an increased probability for themajority component to occupy the two intermediate sites,while disfavoring the central site (panel a2). Thus, it actsas an inhibitor of localization at negative g AB and coun-teracts changes induced by the SMF potential at positive g AB . Furthermore, entanglement favors bunching of themajority particles independent of the sign of the coupling(panel c2).The decoupled impurity particle (panel b1) occupiesthe ground state of the box potential. At moderate pos-itive couplings it develops two humps and forms a shellaround the majority component density, a signature ofphase separation [45, 62] further confirmed by the ap-pearance of two humps in ρ ABr (panel d1). At negativecouplings ρ B and ρ ABr shrink with decreasing g AB ac-cumulating around the trap center. The entanglementfavors the process of phase separation at positive cou-plings and bunching between the two species at negative couplings (panel d2), while slowing down the shrinkingof ρ B at negative coupling (panel b2). We also remarkthat upon reaching a certain threshold value of g AB > ,the SMF solution experiences breaking of parity sym-metry, causing substantial differences to the many-bodysymmetry-preserving solution (not shown).For a deep lattice ( V = 1000 ) in Fig. 7 the major-ity component (panel a1) at g AB = 0 displays an al-most uniform distribution over all the lattice sites, while ρ AAr (panel c1) features a multi-hump structure due tostronger intraspecies correlations [cf. Fig. 5a)]. At mod-erate positive couplings ( g AB > ) the width of ρ A and ρ AAr is only slightly increased, again in accordance withthe robustness of F A [Fig. 3a]. Thus, the majority com-ponent, experiencing the presence of a repelling impurityatom, shows a slight enhancement of the already presentdelocalization over the lattice. At moderate negative cou-plings ( g AB < − ) both ρ A and ρ AAr shrink with decreas-ing g AB to the extent where all atoms occupy predom-inantly only the central site ( g AB < − ). Such a largedifference to the non-interacting ground state is in accor-dance with the observations made in F A [Fig. 3a].The impact of entanglement is structurally differentcompared to a shallow lattice (panels a2 and c2). Atpositive couplings, the entanglement greatly increasesthe probability for the majority atoms to be found atthe central site, while decreasing the probability at outer sites ( z = ± . ) and being indifferent to the intermedi-ate sites (panel a2). Additionally, it favors the bunch-ing of the majority particles at the same or neighboringsites and disfavors them being more than two sites apart(panel c2). At negative couplings, it acts in a similar0 a1a2 b1b2 c1 d1d2c2 Figure 7. Upper panels: one-body probability densities ρ A ( x ) , ρ B ( y ) [Eq. (4)] and distance probability distributions ρ AAr ( x − x ) , ρ ABr ( x − y ) [Eq. (10)] at g AA = 0 . , V = 1000 and for various values of g AB (see legend). Lower panels: difference betweenprobability densities obtained from many-body ML-X calculations and SMF ansatz, the latter distinguished by a tilde sign. Allquantities are given in box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) with L denoting the extensionof the box trap. way as in the case of shallow lattices, except that for asufficiently strong coupling strength ( g AB < − ), whereboth entropy measures are of small magnitude [see Figs. 4and 5a], the SMF ansatz is in good accordance with themany-body solution.The impurity particle (panel b1) at positive couplings( g AB > ) first develops two humps, but then as the cou-pling increases, the relative distance between those peaksgrows, while the humps themselves become flatter. Thereis a strong signature of an onset of a four-peak structureat g AB = 5 . This is in accordance with the increasingrelative distance between the species (panel d1) and thefact that the majority atoms are distributed uniformlyover all the lattice sites in contrast to g AA = 0 . , wherethe majority component was occupying mainly the cen-tral and the intermediate sites. At negative couplings( g AB < − ) ρ B and ρ ABr shrink with decreasing g AB .The entanglement favors the process where the impu-rity atom moves from the box center to its boundariesindependently of the sign of the coupling (panel b2). At g AB < − . it plays only a minor role, the same as forthe majority component. Regarding ρ ABr , at positivecouplings the entanglement favors the process of phaseseparation by pushing the impurity particle more thantwo sites apart from a majority atom (panel d2). Atnegative couplings it enhances the bunching between thetwo species, even when the entanglement entropy is verysmall (e.g. at g AB = − . ).
2. Moderately repulsive interacting majority component
Considering our findings regarding fidelity and entropymeasures we investigate here only shallow lattices at pos-itive couplings (Fig. 8), where the structural changescaused by the coupling and the entanglement entropy S vN may have a sizable impact on density distributions.The decoupled density of the majority component (panela1) has three pronounced humps at the central ( z = 0 )and intermediate sites ( z = ± . ). The profile is overallmore spread compared to a weakly interacting majority[cf. Fig. 6 panel a1]. Indeed, it is most beneficial for twoparticles to occupy neighboring sites (see the two humpsin panel c1). The majority component gets only a weakfeedback from the presence of a repulsive impurity atom,even at coupling strengths comparable to g AA in accor-dance with the robustness of F A in Fig. 3b. The roleof the entanglement is also rather weak, though qualita-tively different to g AA = 0 . in Fig. 6. Thus, it increasesthe probability for the majority particle to be found atthe region enclosed between the two intermediate sites,while decreasing the probability to be detected outsideof that region (panel a2). Furthermore, it favors particledistances of a half lattice constant ( a l = 0 . R ∗ ) (panelc2).The impurity particle (panel b1) experiences phaseseparation similar to Fig. 6 (panel b1), i.e., upon increas-ing g AB it develops two humps with a minimum at thetrap center. Then, those humps separate and flatten,until finally they would form a four-hump structure withthree local minima located at the position of the threepeaks in the majority component density (compare to1 a1a2 b1b2 c1 d1d2c2 Figure 8. Upper panels: one-body probability densities ρ A ( x ) , ρ B ( y ) [Eq. (4)] and distance probability distributions ρ AAr ( x − x ) , ρ ABr ( x − y ) [Eq. (10)] at g AA = 3 . , V = 100 and for different values of g AB (see legend). Lower panels: difference betweenprobability densities obtained from the many-body ML-X calculations and SMF ansatz, the latter distinguished by a tilde sign.All quantities are provide in terms of box units with characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) while L is theextension of the box trap. panel a1). The separation between the species is againclearly manifested as two humps in ρ ABr with favored dis-tance of a lattice constant ( a l = 0 . R ∗ ) (panel d1). Theentanglement affects the impurity atom in quite an op-posite way when compared to the majority component(panel b2), i.e., it decreases the probability for the impu-rity atom to be found at the region enclosed between thetwo intermediate sites, while increasing the probabilityto lie outside of that region. Additionally, similar to thebehavior at weaker g AA [cf. Fig. 6 panel d2], the entan-glement accelerates the phase separation process (paneld2).
3. Attractively interacting majority component
Finally, we concentrate on negative intraspecies inter-actions g AA , namely a weak negative g AA = − . at neg-ative g AB [Fig. 9], contained in the parameter sector withsubstantial entanglement entropy [Fig. 4c].In Fig. 9 a decoupled majority atom where g AB = 0 is localized at the central ( z = 0 ) and intermediate ( z = ± . ) wells (panel a1). Even though the major-ity atoms are attracted to each other, the probabilityto be one or even two wells apart is still sizable (panelc1). With decreasing g AB both ρ A and ρ AAr shrink to aGaussian. The impact of entanglement is quite differentcompared to the previously considered cases. Thus, at g AB > − . the entanglement slows down the processof ρ A localization at the central well (panel a2). Thestrongest impact is reached around g AB ≈ − . , wherethe entanglement entropy is largest for the given valueof intracomponent interaction g AA = − . [Fig. 4c]. Be-low g AB < − . , as the entanglement entropy suddenly drops, so does the difference to the SMF ansatz. The in-tercomponent correlations favor clustering of the major-ity atoms at − . < g AB < and g AB < − . , whereasat − . < g AB < − . , where the entanglement entropyis largest, they inhibit the clustering (panel c2).The impurity density ρ B shows a similar behavior asthe majority component density (panel b1), also in termsof the role of the entanglement (panel b2). The widthof ρ ABr shrinks with decreasing g AB (panel d1), whilethe entanglement enhances the bunching between the twospecies (panel d2). V. QUENCH INDUCED TUNNELINGDYNAMICS
Having analyzed in detail the ground state proper-ties of our system, we subsequently study the dynam-ical response of a single impurity coupled to a latticetrapped species upon quenching the interspecies interac-tion strength g AB . To this end we prepare the system inits ground state for V = 500 , g AB = 6 . and g AA = 0 . ,leading to the formation of a two-fold degeneracy in theground state and the two species phase separate [45]. Inthis sense, the ground state one-body density is givenby a superposition state of two parity-symmetry brokenconfigurations, where the density of the first one is de-picted in Fig. 10a and the second one corresponds to itsparity-symmetric (with respect to x = 0 ) counterpart. Itis possible to remove this degeneracy in order to selectany of the states in the respective degenerate manifold.Technically, this is done by applying a small asymmetry,e.g. a tilt, to the lattice potential, thereby breaking theparity symmetry and energetically favoring one of the2 a1a2 b1b2 c1 d1d2c2 Figure 9. Upper panels: one-body probability densities ρ A ( x ) , ρ B ( y ) [Eq. (4)] and distance probability distributions ρ AAr ( x − x ) , ρ ABr ( x − y ) [Eq. (10)] at g AA = − . , V = 500 and for various values of g AB (see legend). Lower panels: differencebetween probability densities obtained from the variational ML-X simulations and SMF ansatz, the latter distinguished by atilde sign. All quantities are expressed in box units of characteristic length R ∗ = L and energy E ∗ = (cid:126) / ( mL ) while L beingthe extension of the box trap.Figure 10. (a) One-body density ρ σ ( x ) of the initial state configuration for V = 500 , g AA = 0 . and g AB = 6 . at t = 0 .Temporal evolution of (b) the averaged position of the impurity (cid:104) ˆ X B (cid:105) [see eq. (11)], (c) the one-body density of the majorityspecies and (d) the one-body density of the impurity upon quenching the interspecies interaction strength to g AB = 4 . . above-mentioned states [49].To trigger the dynamics starting from the initial stateconfiguration illustrated in Fig. 10a we quench the inter-species interaction strength to a smaller value. As a rep-resentative example of the emergent tunneling dynamicsof each species we present the temporal evolution of thecorresponding one-body densities in Fig. 10c,d followinga quench to g AB = 4 . , while keeping fixed V = 500 and g AA = 0 . . In this case the impurity performs an oscil-latory motion which is reminiscent of the tunneling of aparticle in a double-well. This can be attributed to thelifting of the degeneracy for smaller interspecies inter-action strengths. For a post-quench value of g AB = 4 . the initially prepared state has a substantial overlap withthe post-quench ground state and the first excited statesuch that in the course of the dynamics the system willoscillate between those two. This is similar to a single particle in a double-well which is prepared as a super-position of the first doublet and undergoes a tunnelingbetween the sites. Correspondingly, the majority specieswill undergo a collective tunneling in the lattice geome-try [49, 52]. Thus, the probability distribution of a singlemajority species particle will oscillate between the initialdistribution [Fig. 10a] and its parity-symmetric coun-terpart. Due to the repulsive nature of the interspeciescoupling the two species move in opposite directions suchthat they end up in phase-separated configurations afterhalf a period. Note that the oscillation period, being theenergy gap between the two energetically lowest eigen-states of the post-quench Hamiltonian (not shown here),depends on the post-quench g AB . This can be easily veri-fied by monitoring the temporal evolution of the averaged3position of the impurity [13] which is defined as (cid:104) ˆ X B (cid:105) = (cid:90) L/ − L/ dxρ B ( x ) x. (11)For various post-quench g AB we find that the impuritywill occupy its parity-symmetric counterpart, reflectedin the decrease of (cid:104) ˆ X B (cid:105) towards negative values, whilethe oscillation decreases with smaller g AB [Fig. 10b].In order to gain insight into beyond mean-field effectswe investigate the natural populations n σj [see eq. (8)]which indicate the degree of fragmentation of the sub-system [9, 70]. For simplicity here we present the popu-lations of the first two dominantly populated natural or-bitals while using six orbitals in the actual calculations.The initial depletion of both subsystems is rather small,i.e. n A ≈ . and n B ≈ . , such that any decrease ofthese populations upon quenching g AB is due to dynami-cal many-body effects. We find that for both subsystemsdominantly two natural orbitals contribute during thedynamics [Fig. 11c,d], while the ones of the medium areless impacted by the quench. For the natural popula-tions of the impurity signatures of an oscillation can beobserved, where n B initially decreases and revives backtowards n B ≈ . , while n B initially increases and af-terwards drops back to nearly zero. In order to attributethe occupation of the additional natural orbital to phys-ical processes, we analyze the spatial distribution of thenatural orbitals Φ Bj ( x, t ) [see eq. (8)] themselves focusingon the impurity [Fig. 11a,b]. In Fig. 11a we observe thatthe first natural orbital corresponds to the oscillatory be-havior of the one-body density of the impurity, but lack-ing the smooth transition between the phase-separatedconfigurations [see Fig. 10d]. The first natural orbitaldominates during the dynamics and we can interpret itsbehavior as corresponding to the presence of the phase-separated density configurations. Consequently, the sec-ond natural orbital [Fig. 11b], resembling the mirror im-age of the first one, contributes to deviations from thissolution. Due to its structure we can deduce that it isresponsible for initiating the transport of the impurity,thereby allowing for the counterflow of the two species.Note that the presence of more than one natural orbitalduring the dynamics is a clear signature that mean-fieldtheory would not provide an accurate description of thesystem dynamics. Hence, the fact that | Φ B (cid:105) is occupiedis a manifestation of many-body effects, influencing themotion of both species. VI. SUMMARY AND OUTLOOK
In this work we analyze the static and dynamical prop-erties of a few-body particle-imbalanced bosonic mix-ture at zero temperature. Importantly, the componentsare exposed to different one-dimensional external trapswhere the majority species is subject to a finite latticepotential while the single impurity is trapped in a box of the same extension as the lattice. We study the responseof the composite system upon the variation of majority-impurity coupling g AB and majority component inter-nal parameters being either the lattice depth V or themajority-majority interaction strength g AA .To quantify the response of static properties we em-ploy the fidelity between two density operators describingground states at zero and a finite intercomponent interac-tion g AB . We contrast the response at the many-body tothe single-particle level. We observe that the compositesystem is quite robust to the variation of the intercom-ponent interaction at strongly repulsive g AA , while beingfragile at strongly attractive g AB and deep lattices V aswell as when g AA is weakly attractive and g AB is stronglyattractive. Upon comparison to the fidelities betweenthe corresponding reduced one-body density operators ofeach component, we not only observe that each species isaffected to a much smaller degree, but they also responddifferently. Thus, for the impurity atom the deviationfrom the box ground state increases smoothly with in-creasing absolute value of g AB , while the reduced densityof the majority component remains very robust to g AB variations except for the above mentioned parameter re-gions where the many-body fidelity exhibits significantstructural changes in the ground state.Next, we have been performing a further classificationof our system based on entropy measures. Namely, wequantify the amount of entanglement and intraspeciescorrelations deposited in the binary mixture by evaluat-ing the von-Neumann entropy of the respective subsys-tems. Interestingly, we find that our composite system isonly weakly entangled for parameter regions which un-dergo substantial structural changes. Additionally, weobserve that while the entanglement entropy continu-ously grows with increasing repulsive g AB , it does notbehave the same for attractive g AB , where it reaches alocal maximum at a finite value of g AB < . Anotherpeculiar observation is that the fragmentation entropy ofthe majority component undergoes a strong variation forparameter regions, where the fidelity measure does notshow any evidence of majority particles being affected bythe intercomponent interaction. Even though the mixedcharacter of the reduced density of the medium suffersfrom substantial changes, it remains un-observable on thesingle-particle level.To visualize our observations stemming from the fi-delity measure we show the one-body density distribu-tions of each component along with the probability distri-butions for two particles of the same or different speciesto be measured at a relative distance from each other.These quantities are usually accessible in state-of-the-artultracold atom experiments and determine the expecta-tion values of local one- and two-body observables. In-deed, strong deviations appearing in the fidelity at thesingle-particle level are also clearly visible in the corre-sponding one-body density. At positive couplings we ob-serve an interspecies phase separation where the impurityis pushed to the box edges, while leaving the majority4 Figure 11. Temporal evolution of the density of (a) the first and (b) the second natural orbital Φ Bj ( x, t ) [see eq. (8)] of theimpurity and (c), (d) the natural populations n σj of both subsystems upon quenching the interspecies interaction strength g AB of the ground state in Fig. 10a to g AB = 4 . . component intact. At negative couplings both compo-nents tend to increase their localization at the centralwell.To further quantify our conclusions stemming from theentanglement measure we rely on the difference betweenthe above probability distributions and the correspond-ing ones when assuming a disentangled state in our calcu-lations. Again, we find strong deviations for parametersdisplaying high entanglement entropy values. Thus, atpositive couplings the entanglement favors the process ofphase separation, while at negative couplings it gener-ally, but not always, counteracts the localization of bothspecies.Quenching the interspecies interaction strength we areable to induce a dynamical process which for the impu-rity is reminiscent of the tunneling of a single particlein a double well potential. This can be attributed tothe lifting of the degeneracy for the corresponding post-quench Hamiltonian as well as the substantial overlapof the initial state configuration with the post-quenchground state and the first excited state. Due to the re-pulsive interspecies interaction also the majority specieswill undergo a tunneling in the lattice geometry such thatthe two species move in opposite directions, ending up inphase-separated configurations after half a period. Weidentify the presence of two dominant natural orbitalsfor the impurity species during the dynamics, where thefirst one corresponds to phase-separated configurations inthe respective one-body density, while the second one re-sembles the mirror image of the first one. The presenceof an additional natural orbital emphasizes the many- body character of the dynamics, thereby influencing themotion of the impurity.There are various promising research directions thatare worth pursuing in the future. Indeed, the general-ization of our findings for an increasing particle numberin the medium or larger lattice potentials as well as therole of the lattice filling factor is desirable. Also, a moreelaborated analysis on the possibly emerging impurity-medium bound states or the engineering of droplet-likeconfigurations in such settings at strong intercomponentattractions would be important. Furthermore, it wouldbe intriguing to study the persistence and possible al-terations of the identified spatial configurations in thepresence of finite temperature which will impact the co-herence of the lattice bosons [85, 86]. Another perspec-tive is to investigate the relevant radiofrequency spec-trum [31, 42] in order to capture the emergent polaronproperties including their lifetime, residue and effectivemass especially in the attractive interaction regimes ofbound state formation. ACKNOWLEDGMENTS
M. P. and K. K. gratefully acknowledge a scholarship ofthe Studienstiftung des deutschen Volkes. S. I. M. grate-fully acknowledges financial support in the framework ofthe Lenz-Ising Award of the Department of Physics ofthe University of Hamburg. [1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[2] I. Bloch, (Cambridge University Press, 2017) p. 253.[3] T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi,Phys. Rev. A , 021601 (2009).[4] K. Henderson, C. Ryu, C. MacCormick, and M. G.Boshier, New J. Phys. , 043030 (2009).[5] F. Serwane, G. Zürn, T. Lompe, T. Ottenstein, A. Wenz,and S. Jochim, Science , 336 (2011). [6] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel,V. Scarani, P. Treutlein, and N. Sangouard, Science ,441 (2016).[7] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[8] T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. , 1311 (2006).[9] S. I. Mistakidis, G. C. Katsimiga, P. G. Kevrekidis, andP. Schmelcher, New J. Phys. , 043052 (2018). [10] M. Pyzh and P. Schmelcher, Phys. Rev. A , 023305(2020).[11] D. S. Petrov and G. E. Astrakharchik, Phys. Rev. Lett. , 100401 (2016).[12] L. Parisi and S. Giorgini, arXiv:2003.05231 (2020).[13] J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio,F. Minardi, A. Kantian, and T. Giamarchi, Phys. Rev.A , 023623 (2012).[14] F. Meinert, M. Knap, E. Kirilov, K. Jag-Lauber, M. B.Zvonarev, E. Demler, and H.-C. Nägerl, Science , 945(2017).[15] S. I. Mistakidis, A. G. Volosniev, N. T. Zinner, andP. Schmelcher, Phys. Rev. A , 013619 (2019).[16] L. P. Ardila and S. Giorgini, Phys. Rev. A , 033612(2015).[17] F. Grusdt, G. E. Astrakharchik, and E. Demler, New J.Phys. , 103035 (2017).[18] H. Tajima and S. Uchino, New J. Phys. , 073048(2018).[19] A. S. Dehkharghani, A. G. Volosniev, and N. T. Zinner,Phys. Rev. Lett. , 080405 (2018).[20] S. I. Mistakidis, A. G. Volosniev, and P. Schmelcher,Phys. Rev. Research , 023154 (2020).[21] J. Takahashi, H. Tajima, E. Nakano, and K. Iida,arXiv:2011.07911 (2020).[22] F. Brauneis, H.-W. Hammer, M. Lemeshko, and A. G.Volosniev, arXiv:2101.10958 (2021).[23] P. Massignan, M. Zaccanti, and G. M. Bruun, Rep. Progr.Phys. , 034401 (2014).[24] R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina,and E. Demler, Rep. Progr. Phys. , 024401 (2018).[25] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau,P. Schauss, S. Hild, and D. Bellem, Nat. Phys. , 235(2013).[26] Z. Z. Yan, Y. Ni, C. Robens, and M. W. Zwierlein, Sci-ence , 190 (2020).[27] F. Scazza, G. Valtolina, P. Massignan, A. Recati, A. Am-ico, A. Burchianti, C. Fort, M. Inguscio, M. Zaccanti, andG. Roati, Phys. Rev. Lett. , 083602 (2017).[28] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M.Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, andJ. J. Arlt, Phys. Rev. Lett. , 055302 (2016).[29] M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. Wal-raven, R. Grimm, J. Levinsen, M. M. Parish, R. Schmidt,M. Knap, and E. Demler, Science , 96 (2016).[30] A. Wenz, G. Zürn, S. Murmann, I. Brouzos, T. Lompe,and S. Jochim, Science , 457 (2013).[31] S. I. Mistakidis, G. M. Koutentakis, F. Grusdt,H. R. Sadeghpour, and P. Schmelcher, arXiv:2011.13756(2020).[32] S. I. Mistakidis, G. M. Koutentakis, G. C. Katsimiga,T. Busch, and P. Schmelcher, New J. Phys. , 043007(2020).[33] A. Boudjemâa, N. Guebli, M. Sekmane, and S. Khlifa-Karfa, J. Phys.: Cond. Matt. , 415401 (2020).[34] L. A. P. Ardila, G. E. Astrakharchik, and S. Giorgini,Phys. Rev. Research , 023405 (2020).[35] A. Camacho-Guardian, L. A. P. Ardila, T. Pohl, andG. M. Bruun, Phys. Rev. Lett. , 013401 (2018).[36] K. Mukherjee, S. I. Mistakidis, S. Majumder, andP. Schmelcher, Phys. Rev. A , 053317 (2020).[37] G. Bougas, S. I. Mistakidis, and P. Schmelcher,arXiv:2009.08901 (2020).[38] H. Tajima, J. Takahashi, E. Nakano, and K. Iida, Phys. Rev. A , 051302 (2020).[39] S. I. Mistakidis, F. Grusdt, G. M. Koutentakis, andP. Schmelcher, New J. Phys. , 103026 (2019).[40] K. Mukherjee, S. I. Mistakidis, S. Majumder, andP. Schmelcher, Phys. Rev. A , 023615 (2020).[41] F. Theel, K. Keiler, S. I. Mistakidis, and P. Schmelcher,arXiv:2009.12147 (2020).[42] S. I. Mistakidis, G. C. Katsimiga, G. M. Koutentakis,T. Busch, and P. Schmelcher, Phys. Rev. Research ,033380 (2020).[43] T. Lausch, A. Widera, and M. Fleischhauer, Phys. Rev.A , 023621 (2018).[44] S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev.Lett. , 150601 (2009).[45] K. Keiler, S. I. Mistakidis, and P. Schmelcher,arXiv:2004.12719 (2020).[46] A. Bohrdt, F. Grusdt, and M. Knap, New J. Phys. ,123023 (2020).[47] Z. Cai, L. Wang, X. Xie, and Y. Wang, Phys. Rev. A ,043602 (2010).[48] T. H. Johnson, S. R. Clark, M. Bruderer, and D. Jaksch,Phys. Rev. A , 023617 (2011).[49] F. Theel, K. Keiler, S. I. Mistakidis, and P. Schmelcher,New J. Phys. , 023027 (2020).[50] K. Keiler and P. Schmelcher, New J. Phys. , 103042(2018).[51] K. Keiler, S. Krönke, and P. Schmelcher, New J. Phys. , 033030 (2018).[52] K. Keiler and P. Schmelcher, Phys. Rev. A , 043616(2019).[53] M. Bruderer, W. Bao, and D. Jaksch, Europhys. Lett. , 30004 (2008).[54] T. Yin, D. Cocks, and W. Hofstetter, Phys. Rev. A ,063635 (2015).[55] F. Grusdt, A. Shashi, D. Abanin, and E. Demler, Phys.Rev. A , 063610 (2014).[56] C. Weber, S. John, N. Spethmann, D. Meschede, andA. Widera, Phys. Rev. A , 042722 (2010).[57] S. Will, T. Best, S. Braun, U. Schneider, and I. Bloch,Phys. Rev. Lett. (2011).[58] H. Tajima, J. Takahashi, S. I. Mistakidis, E. Nakano, andK. Iida, arXiv:2101.07643 (2021).[59] L. Cao, V. Bolsinger, S. Mistakidis, G. Koutentakis,S. Krönke, J. Schurer, and P. Schmelcher, The J. Chem.Phys. , 044106 (2017).[60] L. Cao, S. Krönke, O. Vendrell, and P. Schmelcher, J.Chem. Phys. , 134103 (2013).[61] S. Krönke, L. Cao, O. Vendrell, and P. Schmelcher, NewJ. Phys. , 063018 (2013).[62] S. I. Mistakidis, G. C. Katsimiga, G. M. Koutentakis,T. Busch, and P. Schmelcher, Phys. Rev. Lett. ,183001 (2019).[63] A. C. Pflanzer, S. Zöllner, and P. Schmelcher, J. Phys.B: At. Mol. and Opt. Phys. , 231002 (2009).[64] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell,and C. E. Wieman, Phys. Rev. Lett. , 586 (1997).[65] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman,and E. A. Cornell, Phys. Rev. Lett. , 1539 (1998).[66] H.-J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. In-ouye, A. P. Chikkatur, and W. Ketterle, Phys. Rev. Lett. , 2228 (1999).[67] P. Maddaloni, M. Modugno, C. Fort, F. Minardi, andM. Inguscio, Phys. Rev. Lett. , 2413 (2000).[68] K. M. Mertes, J. W. Merrill, R. Carretero-González, D. J. Frantzeskakis, P. G. Kevrekidis, and D. S. Hall, Phys.Rev. Lett. , 190402 (2007).[69] C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher,M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, andK. Sengstock, Nature Physics , 496 (2008).[70] A. U. J. Lode, C. Lévêque, L. B. Madsen, A. I. Streltsov,and O. E. Alon, Rev. Mod. Phys. , 011001 (2020).[71] O. E. Alon, R. Beinke, and L. S. Cederbaum,arXiv:2101.11615 (2021).[72] R. Horodecki, P. Horodecki, M. Horodecki, andK. Horodecki, Rev. Mod. Phys. , 865 (2009).[73] J. Light, I. Hamilton, and J. Lill, J. Chem. Phys. ,1400 (1985).[74] A. Raab, Chem. Phys. Lett. , 674 (2000).[75] R. Jozsa, J. Mod. Opt. , 2315 (1994).[76] W. Wang, V. Penna, and B. Capogrosso-Sansone, NewJ. Phys. , 063002 (2016).[77] I. Bengtsson and K. Życzkowski, (Cambridge UniversityPress, 2017).[78] R. Roy, A. Gammal, M. C. Tsatsos, B. Chatterjee,B. Chakrabarti, and A. U. J. Lode, Phys. Rev. A ,043625 (2018). [79] S. Bera, S. K. Haldar, B. Chakrabarti, A. Trombettoni,and V. K. B. Kota, European Phys. J. D , 1 (2020).[80] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I.Gillen, S. Fölling, L. Pollet, and M. Greiner, Science ,547 (2010).[81] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau,I. Bloch, and S. Kuhr, Nature , 68 (2010).[82] A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Sa-lomon, I. Bloch, and C. Gross, Phys. Rev. Lett. ,263001 (2015).[83] M. Hohmann, F. Kindermann, T. Lausch, D. Mayer,F. Schmidt, E. Lutz, and A. Widera, Phys. Rev. Lett. , 263401 (2017).[84] M. Pyzh, S. Krönke, C. Weitenberg, and P. Schmelcher,New J. Phys. , 053013 (2019).[85] F. Lingua, B. Capogrosso-Sansone, F. Minardi, andV. Penna, Sc. Rep. , 1 (2017).[86] K. W. Mahmud, E. N. Duchon, Y. Kato, N. Kawashima,R. T. Scalettar, and N. Trivedi, Phys. Rev. B84