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New J. Phys. 20 (2018) 063014 https://doi.org/10.1088/1367-2630/aac718

PAPER

Emergence of junction dynamics in a strongly interacting Bosemixture

REBarfknecht1,2, A Foerster2 andNTZinner1,3

1 Department of Physics andAstronomy, AarhusUniversity, NyMunkegade 120,Denmark2 Instituto de Física, Universidade Federal do RioGrande do Sul, Av. BentoGonçalves 9500, Porto Alegre, RS, Brazil3 Aarhus Institute of Advanced Studies, AarhusUniversity, DK-8000AarhusC,Denmark

E-mail: rafael@phys.au.dk and rafael.barfknecht@ufrgs.br

Keywords: strongly interacting systems, Bosemixtures, spin dynamics

AbstractWe study the dynamics of a one-dimensional system composed of a bosonic background and oneimpurity in single- and double-well trapping geometries. In the limit of strong interactions, thissystem can bemodeled by a spin chainwhere the exchange coefficients are determined by thegeometry of the trap.We observe non-trivial dynamics when the repulsion between the impurity andthe background is dominant. In this regime, the system exhibits oscillations that resemble thedynamics of a Josephson junction. Furthermore, the double-well geometry allows for an enhancementin the tunneling as compared to the single-well case.

1. Introduction

The experimental investigation of ultracold atomic systems hasmade possible the realization of severalcelebratedmodels in quantummechanics and condensedmatter. These experiments are characterized by therigorous control over the parameters of the system and by precisemeasurement techniques. Themanipulationof optical traps, for instance, allows for the construction of different confining geometries [1], fromone-dimensional tubes [2, 3] to lattice systems [4, 5]. Traps consisting of twowells, in particular, have beenextensively employed in recent experiments [6–10]. They are of special interest in the study the Josephson effect[11] in cold atoms, where Bose–Einstein condensates placed in such potentials [12, 13] are considered in analogyto superconductors [14, 15]. The high degree of precision demonstrated in these experiments also extends to thenumber of particles under consideration and to the strength of the interactions between them. Two-componentfermionic systemswith only a few atoms [16–18] and paradigmaticmodels such as the infinitely repulsiveTonks–Girardeau gas [19, 20] can also be realized and studied in the lab. Combining these features, otherexperiments have recently exploredmulticomponent strongly correlated gases [21], which have shown severalexotic properties [22].

The limit of strong interactions has been a particularly favored starting point in the theoretical study of one-dimensional systemswith internal degrees of freedom, due to the possibility ofmapping theHamiltonian to aneffective spin chain [23–27]. One of the key features of thismapping is that the exchange coefficients of the spinchain are solely determined by the trapping potential, and powerful numericalmethods to calculate thesecoefficients are now available [28, 29]. Approaching the problemof few atoms in a trap in the stronglyinteracting regime has also provided knowledge of the fundamental properties of quantummagnetism [30–35].On the other hand, themany-body case in the limit of total population imbalance—the ‘impurity’ problem—

also presentsmany interesting features, such as quantumflutter [36] andBloch oscillations in the absence of alattice [37–41] (the latter having been recently observed in experiments [42]).

In this workwe study a strongly interacting system composed of an impurity and a bosonic background insingle- and double-well potentials. Differentmethods have been employed to tackle themany-body problemofbosons in a double-well, even outside themean-field regime [43–47]. Addressing the subject from a few-bodyperspective [48–52], however,might lead to new insight on the properties of these systems.Herewe show that, in

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the regimewhere the repulsion between the impurity and the background is dominant, the system can exhibitnon-trivial dynamical effects: the impurity undergoes Josephson-like oscillations when initialized at the edge ofthe system, and can have its tunneling enhancedwhen a barrier is present. These effects provide new perspectivesin the study of spin state transfer and quantum transport in one-dimensional systems, and should be observableusing current experimental techniques.

2. Systemdescription andHamiltonian

Weconsider the problemof an impurity confined in the presence of a background of strongly interactingbosons.We assume the impurity is a boson in an internal state defined by ñ∣ , while the remaining identicalbosons are described by ñ∣ . Consequently, all atoms have the samemassm. Two-component Bose gases can berealized experimentally using, for instance, 87Rb atoms in different hyperfine states such as = = - ñ∣F m2, 1F

and = = ñ∣F m1, 1F [53, 54]. The number of identical bosons is given by N , while the total system size is= +N N 1. TheHamiltonian for this problem iswritten as

å å åd k d= + - + -= =

<

( ) ( ) ( ) ( )H H x g x x g x x , 1i

N

ii

N

ii j

N

i j1

01

where thefirst sum involves the single-particleHamiltonianH0 (see below)which is the same for bothcomponents, while the remaining terms account for the contact interactions. The coordinates are denoted by xfor the impurity and x i for the remaining bosons. The interaction strength is defined as g for the impurity-background interactions, and asκg for the background-background interactions. Those parameters can beexperimentallymanipulated using Feshbach [55] or confinement induced resonances [56]. The single-particleHamiltonian in equation (1) is given by = - +¶

¶( ) ( )H x V x

m x0 2

2 2

2 . Here,V(x) is a double-well potential (see

figure 1) expressed as w= -( ) (∣ ∣ ˜)V x m x b1

22 2, whereω is the trapping frequency. The parameter b̃ denotes the

displacement of the twominima of thewells with respect to the origin, and also defines the size of the barrier atthis point as w=( ) ˜V m b0 1

22 2

.We then refer to b̃ as the ‘barrier parameter’. Bymaking =b̃ 0 wenaturallyrecover the harmonic single-well potential. Although this formof potential has analytical solutions in terms ofparabolic cylinder functions [57], we obtain the single-particle wave functions and energies through numericaldiagonalization (see appendix A.1).Wewill focus on the behavior of the spatial distributions and the impuritydynamics in the repulsive case ( k >g , 0), for different choices of the intraspecies interaction parameterκ andthe barrier parameter b̃ .While cases of attractive interactions ( k <g , 0) can in principle be explored, theproperties of the system in this regime reproduce only highly excited states related to the so-called Super Tonks–Girardeau gas [58, 59]. Simulating the dynamics of systemswith attractive interactions would likely requiretaking into account the formation of bound pairs, an effect which is beyond the scope of the formalism employedhere. Throughout this work, wewill consider all quantities in harmonic oscillator units; therefore, length, energyand time are given in units of w=l m , w andω−1, respectively.While the intraspecies interactionparameterκ is dimensionless, the parameter g is considered in units of ml2 . For simplicity, we alsomake thebarrier parameter dimensionless by rescaling it as = ˜b b l. In our calculations, we set g=20 and assumethat w= = =m 1.

In the limit of strong interactions ( g 1), Hamiltonian (1) can bewritten, up to linear order in g1 , as thefollowingXXZ spin chain (see appendix A.2 for details)

Figure 1. Sketch of the 1Dbosonic systemwith an impurity in the double-well potential. The parameter b̃ sets the position of theminimumof eachwell and also the size of the barrier between them at x=0. In the limit of strong interactions, the system can bemapped to a spin chainwhere the coefficientsα are determined by the shape of the trap.

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New J. Phys. 20 (2018) 063014 REBarfknecht et al

å s sak

s s= - - + +=

-+ +⎡

⎣⎢⎤⎦⎥( · ) ( ) ( )H E

g

1

2

1, 2s

i

Ni i i

zi

zi

01

11 1

where E0 is the energy of the system in the limit of infinite repulsion and si denotes a Paulimatrix acting on site i.The coefficientsαi, often called geometric coefficients, are calculated using thewave function for a systemofNspinless fermions, which is constructed as the Slater determinant of the lowest occupied orbitals in the trap (seeappendix A.3). The spatial part of thewave function for a bosonic system is then obtained bymeans of theFermi–Bosemapping [60]. A comparative study of the spatial distributions for a strongly interacting few-bodybosonic system in the double-well has been presented in [61].

3. Spin densities

To obtain the probability densities for each component in the system, wemust combine the spatial distributionsof the atoms in the trapwith the probability ofmagnetization of the corresponding site for an eigenstate of thespin chain described by equation (2). The spin densities are therefore given by

år r= =

( ) ( ) ( )( ) ( )x x , 3i

Ni

1

with r r= ( )( ) ( )m xi i i , where r ( )xi describes the individual atomic densities (see appendix A.4), while ( )m i

denotes the probability of each site in a spinwave function cñ∣ having spin up or down. The quantities r( )x andr( )x thus describe the spatial distributions of the impurity and the background bosons, respectively. Infigure 2(a)we show the results for the spin densities of a 3+1 system for b=0 (single-well) and b=2 (double-well) at different values of the intraspecies parameterκ.

The cases where b=0 correspond to the results expected for the harmonic trap, which have been broadlycovered for bosonic and fermionic systems in previousworks [25, 32, 62–64]. Forκ<1 the repulsion betweenthe impurity and the background is larger than the background repulsion. This causes the impurity to be pushedto the edges of the system, which is an effect also found in the case of aweakly interacting background [65]. Inthis regime, the system exhibits Ising-type ferromagnetic correlations, and is characterized by a nearlydegenerate ground state [35]. Atκ=1 all interactions are equal and the spin densities show theHeisenberg-typeferromagnetic profiles expected for isospin bosons [31]. In this case both distributions display the samecharacteristic Tonks–Girardeau spatial densities, but scaled to the number of particles in each species.Whenκ>1, the repulsion between the background bosons dominates, andwe observe predominantlyantiferromagnetic correlations, where the impurity is placed near the center of the trap. In the double-wellpotential, all densities are depleted in the center of the system, but this is not the only relevant effect. For the caseofκ<1 the impurity has now a larger probability of being near the center of the trap (as compared to the single-well case), since the background density is strongly reduced in this region. It is even possible to expect aconfiguration (specially for a larger number of background bosons)where the impurity is completely localizednear the barrier between thewells (see figure 2(b)). This effect is directly related to the imbalance in the

Figure 2. (a) Spin densities for the ground state of a 3+1 system at different values of the barrier parameter b and of the backgroundinteraction parameterκ (the values for each panel are determined by the labels on the rows/columns). The red curve corresponds tothe background density r( )x while the blue curve corresponds to the impurity density r( )x . (b) Spin densities for a 9+1 system atκ=0.2 and two different choices of b. For a larger value of b, the impurity has a greater probability of being placed near the barrier, atthe center of the system. The density for each component is normalized to its corresponding number of particles.

3

New J. Phys. 20 (2018) 063014 REBarfknecht et al

numerical values of the geometrical coefficients at the edges and near the center of the system as b is increased.For the cases ofκ=1 , again the impurity and the background densities have the same shape, aside fromnormalization. Atκ>1, we observe a similar configuration, with a small bias of the impurity toward the centerof the trap.

4.Dynamics

Wenow turn to the dynamics of the impurity after being initialized at the left edge of the system. Thecorresponding initial spin state is therefore given by ñ∣ . Since this is not an eigenstate of the spin chain,we can expect the spin state to evolve in time governed by equation (2), andwe denote it by c ñ∣ ( )t . A thoroughstudy of spin state transfer in traps of different shapes has been done byVolosniev et al in [24]. It has been shown[66–68] that transfer is optimized by consideringκ=2 (which turns equation (2) into anXXHamiltonian)witha set of exchange coefficients where a µ -( )j N jj . Herewe focus on the tunneling times for the impuritybetween thewells (or between the left and right sides of the system in the case of a single-well)when thebackground repulsion is smaller than the repulsion between the impurity and the background (κ<1).Wepoint out that, sincewe do not consider any other external perturbations, like trap or interaction quenches, wecan assume that the spatial distributions remain in the ground state. This also allows us to consider only themanifold of equation (2)with lowest energy. To quantify the dynamics of the impurity, we calculate its averageposition as

ò rá ñ = ( ) ( ) ( )x t x t x x, d , 4

where r( )x t, is the time dependent spin density calculatedwith the spin state c ñ∣ ( )t .When considering theregime ofκ<1, we observe that the projections of the initial wave function on the two lowest eigenstates aredominant when compared to the case of higher excited states. This allows us to attempt a two-level descriptionfor the time evolution of the spinwave function; we thuswrite y = +w-∣ ( )⟩ ∣ ⟩ ∣ ⟩t c g c eeg

te

i J , where ñ∣g and ñ∣edenote the two eigenstates of the spin chainwith lower energy, and cg and ce are the projections of the initial wavefunction over these states. The frequencyωJ=Ee−Eg is given by the gap between the energies of the firstexcited stateEe and the ground stateEg.

Infigure 3, we present the results for á ñ( )x t in the single-well (b= 0) and double-well potentials (b= 2),with two choices ofκ<1, also showing a comparisonwith the two-level description in each of these cases. Atb=0, we notice that themotion of the impurity between the edges of the system is verywell captured by the

Figure 3. Impurity average position as a function of time for different values ofκ and b. The solid red curves show the exact results,while the blue dashed curves are obtainedwith the two-level description (see text).

4

New J. Phys. 20 (2018) 063014 REBarfknecht et al

two-level approximation. This behavior clearly resembles the oscillations in population expected for a bosonicJosephson junction described as amany-body system in a double-well. In the present context, however, thebarrier is composed by the repulsive background gas.We expect these results to hold even in the case ofmorethan one impurity, provided that the system is imbalanced (that is, the background gasmust have a largernumber of particles). In this situation, an initial state described by theminority species completely localized ateither side of the trap should have its time evolution governedmainly by the two lowest energy eigenstates. In thesingle-well case withweaker intraspecies interaction (κ=0.2) the tunneling of the impurity is suppressed. Here,the behavior of the background approaches that of an ideal Bose gas, where the atoms tend to ‘bunch up’ in thecenter of the trap.Now, comparing the single- and double-well cases, we see that, forκ=0.5, the presence ofthe barrier slows down the tunneling of the impurity. Furthermore, we observe oscillations on a smaller scale,due to a larger overlap between the initial state and the excited states of the spin chainHamiltonian. Atκ=0.2,however, we get an enhanced tunneling of the impurity when considering a double-well as opposed to the single-well case. This effect has been also foundwith a different choice of double-well potential [24]. Onemightinterpret it as a splitting of the background gas by the barrier in such away that the impurity is able to tunnelthrough faster than it would in the absence of the barrier. However, if we consider a single-particle problemwhere an atom is initialized in the left well, it is clear that increasing the barrier sizewould only lead toexponential suppression in the tunneling frequencies.We therefore conclude that the accelerated tunnelingobserved in the regimeswe consider is only possible due to the presence of the bosonic background, and thusconstitutes amany-body effect.

To get an understanding of this behavior over a larger parameter space, we plot infigure 4 the energy gapbetween the ground state and thefirst excited state for several values ofκ and b. The non-monotonic behavior ofthe gap as a function of b indicates that, for smallκ, there is some choice of barrier size that increases that energygap, and therefore enables a higher tunneling frequency between thewells. Asκ increases, however, we see thatthis behavior disappears and the presence of the barrier only reduces the gap, thusmaking the tunneling slower.

5. Increasing N

As afinal example, we consider a case wherewe increase the number of background bosons to =N 5 (toobserve similar effects as in the case of =N 3, we choose tomaintain an even total number of atoms). Theinitial state is once again defined by the impurity placed at the left edge of the system, that is, ñ∣ .Wekeep the intraspecies repulsion parameter fixed atκ=0.25. Infigure 5(a)we once again show the results for theaverage position of the impurity as a function of time. For the single-well, the tunneling times are so long that theimpurity is effectively frozen at the left edge of the system. In this case, an analogy can be drawn to the self-trapping regime in a few-body system as presented in [48]. For b=3, however, we again notice that a fastermotion of the impurity from the left to the right well is induced. The difference in the results with andwithoutthe barrier is even clearer than in the case of =N 3. This can also be seen in the energy gap between the twolowest states, as presented infigure 5(b): a very pronounced curve shows the increase in this quantity for smallκand >b 1.We point out that the final time (t=104 in units ofω−1) considered in the present case isfive timeslarger than in the case of =N 3. Time scales for harmonically trapped systems are set by the inverse frequency,which in present experiments with few-body cold atoms is of around 100 μs [18]. The total times obtained inexperimental setups can therefore be decreased by considering tighter traps.

Figure 4.Energy gap between the two lowest states of the spin chain for different values of the barrier parameter b and backgroundinteractionκ for the 3+1 case.

5

New J. Phys. 20 (2018) 063014 REBarfknecht et al

6. Conclusions

Wehave studied the static and dynamic properties of an impurity in the presence of a background of bosons insingle-well and double-well geometries. The ground state spin densities are described by a combination of thespatial distributions in the limit of infinite repulsion and the eigenstates of a spin chain.We have shown that theposition of an impurity initialized at the left edge of the systemdisplays oscillations similar to the ones observedin Josephson junctions. Additionally, for weaker background interactions, the tunneling of the impurity can infact be enhanced by the introduction of a barrier. Thismany-body effect is only possible in the presence of abackground.We interpret it as the increase of the gap between the two lowest energy states, which governs thelow-frequency dynamics of the system.Our results open new perspectives on the study of quantum transport inone-dimensional systems, hinting at the possibility of realizing a bosonic Josephson junction in the completeabsence of an artificial barrier.Moreover, the inclusion andmanipulation of double-well potentials and evenlatticesmay allow for the optimized transfer of spin states.

Acknowledgments

The authors thankXiaoling Cui andArtemVolosniev for reading and commenting on themanuscript, andTomasz Sowiński for fruitful discussions. The following agencies—ConselhoNacional deDesenvolvimentoCientífico e Tecnológico (CNPq), theDanishCouncil for Independent ResearchDFFNatural Sciences and theDFF Sapere Aude program—are gratefully acknowledged forfinancial support.

Appendix. Exact solutions, geometrical coefficients and spatial distributions in the limitof infinite repulsion

In this appendixwe present the solutions obtainedwith numerical diagonalization for the single-particle in adouble-well, mapping between a strongly interacting one-dimensional system and the spin chainHamiltoniandescribed by equation (2), expressions for the geometrical coefficients and densities in the impenetrable limit,and the analytical formof the eigenstates and eigenvalues of the spin chain in the 3+1 case. The single-particleenergies and numerical values for the exchange coefficients are calculated using the open-source codeCONAN[29]. Some of these quantities depend on the spinless fermionwave function F ¼( )x x, , N1 , which is constructedas the Slater determinant of theN lowest-lying orbitals of the trapping potential. The energyE0 from equation (2)is then simply the sumof the energies of these individual states.

Figure 5. (a)Average position of the impurity as a function of time for the 5+1 case, in the single- (b = 0, red solid curve) anddouble-well (b = 3, blue dashed curve). The background interaction parameter is set asκ=0.25. (b)Energy gap between the twolowest states of the spin chain for different values of the barrier parameter b and background interactionκ.

6

New J. Phys. 20 (2018) 063014 REBarfknecht et al

A.1. Single-particle solutions in the double-wellThe eigenvalues and eigenstates of a particle in a double-well were obtained through numerical diagonalizationof theHamiltonianH0 using the 50 lowest states of the harmonic oscillator (b= 0) as basis. Infigure A1wepresent these solutions for different values of the barrier parameter b. In panel (a), we showhow each pair ofstates becomes degenerate as the barrier size is increased. This is reflected in the eigenstates shown in panel (b): atlarger values of b, the ground state and the first excited state have the same probability distribution, differing onlyin parity.

A.2.Mapping the strongly interacting system to a spin chainHamiltonianIn this sectionwe showdetails of themapping betweenHamiltonian (1) and the spin chain described byequation (2). Although different approaches have been used to describe thismapping [23, 26, 32, 33], we focuson the one presented in [24].We start by considering amore general bosonicHamiltonian of the form

å åå å

å

d k d

k d

= + - + -

+ -

= = =

< ¢ ¢

< ¢ ¢

( ) ( ) ( )

( ) ( )

H H x g x x g x x

g x x , A.1

i

N

ii

N

j

N

i ji i

N

i i

j j

N

j j

10

1 1

where the total number of particles is given by = + N N N . In the limit of infinite interactions ( ¥g ), theeigenstates ofHamiltonian (1) can be described by

åY = F=

({ }) ( )( )

a P x x, , A.2k

L N N

k k i j1

,

0

where the sum runs over the = +

( )( )L N N,N N

Npermutations of the coordinates, and Pk is the permutation

operator. In this expression, F0 is simply thewave function in the impenetrable limit, with coordinates fixed as

{ }x i and { }x j , with = ¼ i N1, , and = ¼ j N1, , . To investigate the behavior of the energy at very strong (butfinite) interactions, we use theHellmann–Feynman theorem, which gives

åå å

å

d k d

k d

¶¶

= áY - Yñ + áY - Yñ

+ áY - Yñ

= =

< ¢ ¢

< ¢ ¢

∣ ( )∣ ∣ ( )∣

∣ ( )∣ ( )

E

gx x x x

x x , A.3

i

N

j

N

i ji i

N

i i

j j

N

j j

1 1

where thefirst termon the right-hand side accounts for interactions between different bosons, while theremaining terms arise from interactions between identical bosons. The conditions for the derivatives at thecontact point between two particles are given by

Figure A1. (a) Single-particle energies of the 4 lowest eigenstates of the double-well potential as a function of the barrier parameter b.The dotted curves show the results calculatedwithCONAN (see text), while the symbols are the values obtained by numericaldiagonalization. The inset shows the energy E0 of the spinless fermionwave function as a function of b forN=4. (b)The four lowestsingle-particle eigenstates for different values of the barrier parameter b. Solid blue, dashed red, dotted green and dot-dashed blackcurves correspond to ground state, 1st, 2nd and 3rd excited states, respectively.

7

New J. Phys. 20 (2018) 063014 REBarfknecht et al

k

k

¶Y¶

-¶Y¶

= Y =

¶Y¶

-¶Y¶

= Y =

¢-

+

¢

¢-

+

¢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

( ) ( )

x xg x x

x xg x x

2 ,

2 A.4

i ii i

j jj j

for identical bosons and

¶Y¶

-¶Y¶

= Y =

-

+

⎛⎝⎜

⎞⎠⎟ ( ) ( )

x xg x x2 , A.5

i ji j

for a distinguishable pair. In the expressions abovewe have+ - = +x x 0m n , while- - = -x x 0m n .Combining equations (A.3)–(A.5), we get

k k¶¶

= + + ( )E

g

K

g

K

g

K

g, A.6

2 2 2

where

ò ò

ò ò

ò ò

ò ò

ò ò

ò ò

å

å

å

d

d

d

=

- -

Y

=

- -

Y

=

- -

Y

= = ¶Y¶

¶Y¶

-

+

< ¢ ¶Y¶

¶Y¶

-

+

¢

< ¢ ¶Y¶

¶Y¶

-

+

¢

¢

¢

( )

( )

( )

( )

∣ ∣

( )

∣ ∣

( )

∣ ∣

K

x x x x x x

x x x x

K

x x x x x x

x x x x

K

x x x x x x

x x x x

d , , d d , , d

4 d , , d d , , d,

d , , d d , , d

4 d , , d d , , d,

d , , d d , , d

4 d , , d d , , d,

i j

N NN N x x i j

N N

i i

NN N x x i i

N N

j j

NN N x x j j

N N

1, 1

,1 1

2

1 12

1 1

2

1 12

1 1

2

1 12

i j

i i

j j

where the denominator introduces a normalization factor. Integratingwith respect to gwe obtain the followingenergy functional

k k= - + + ⎛

⎝⎜⎞⎠⎟ ( )E E

K

g

K

g

K

g, A.70

where E0 is the energy in the limit of infinite repulsion, andwe neglect terms of higher order in ( g1 ). Byintroducing thewave function described by equation (A.2) in the expression above, we obtain

å å å åå

= -+ +a

k k=-

=- -

=-

=-

=

( )( )

( ) ( ) ( )

( )E EA A A

aA.8

i

N

g k

L N Nik k

L N Nik k

L N Nik

k

L N Nk

01

1

1

1, 1 2

1

2, 2

1

, 2

1

, 2

i

with

= - = = ( ) ( ) ( ) ( )A a a A a A a, , , A.9ik ik ik ik ik ik ik2 2 2

where aik represents the coefficients in equation (A.2)multiplying termswith neighboring and particles atposition i and +i 1, while the remaining terms have the same role, for , and pairs. The purpose of suchterms is to account for the energy contribution of exchanging two neighboring particles with particular spinprojections. The coefficientsαi are now independent of spin, and can bewritten as

ò

òa =

F

< < - -¶F

¶ =

< < -

∣ ( )∣

( )

( )x x

x x x x x

d ... d

d d , , , ,, A.10i

x x xN

x x x

x x x

x x xN i N

11 1

, , , ,2

11 0 1

2

N

i N

NN i

N

1 2

0 1

1 2

where F ( )x x x, , , ,i N0 1 is again thewave function present in equation (A.2) (wherewe have omitted the spinindices). Since this wave function is defined in the region determined by a particular order of the coordinates, itis enough to calculate the integrals in one particular sector, such as < < -x x x 1N1 2 .

8

New J. Phys. 20 (2018) 063014 REBarfknecht et al

Now, let us consider a spin chainHamiltonian defined as

å k k= - P + P + P

=

-

+

+

+⎜ ⎟⎛

⎝⎞⎠ ( )H E J

1 1, A.11s

i

N

ii i i i i i

01

1, 1 , 1 , 1

where s sP = -+ +( · )i i i i, 1 1

21 is the operator that exchanges neighboring spinswith different projections

and s sP = P = ++

+ +( )i i i i

zi

zi, 1 , 1 1

21 have the same action, but for identical spins. A generic spin state can now

bewritten as

åcñ = ñ=

∣ ∣ ( )( )

a P , A.12k

L N N

k k N N1

,

1 1

where once again the sum runs over the permutations of the N and N spins. Calculating the expected value ofHamiltonian (A.11) as c cá ñ∣ ∣H , we obtain

å å å åå

c cá ñ = -+ +

k k=-

=- -

=-

=-

=

( )∣ ∣ ( )

( ) ( ) ( )

( )H EJ A A A

a, A.13

i

Ni k

L N Nik k

L N Nik k

L N Nik

k

L N Nk

01

1

1

1, 1 2

1

2, 2

1

, 2

1

, 2

where the coefficients Aik ,Aik and Aik have the samemeaning as in equation (A.9), andå =

( )akL N N

k1, 2 introduces

a normalization factor. It becomes clear that the energy functionals given by equations (A.8) and (A.13) areidentical if a=J gi i . Furthermore, by rewriting equation (A.11) in terms of the Paulimatrices, we obtain

å s sak

s s= - - + +=

-+ +⎡

⎣⎢⎤⎦⎥( · ) ( ) ( )H E

g

1

2

1, A.14s

i

Ni i i

zi

zi

01

11 1

which is the spin chainHamiltonian described in equation (2) of themain text.We conclude then that theeigenvalue problems definedwith equations (A.8) and (A.13) are identical, which validates, for a stronglyinteracting system, themapping betweenHamiltonians (1) and (2).

A.3. Geometrical coefficientsDifferentmethods have been developed for calculating the coefficients in equation (A.10), in particularexploring the determinant formof F ¼( )x x, , N0 1 . Here, we use the open-source codeCONAN [29] to calculatethem, considering the double-well potential with different barrier parameters. An equivalent approach, basedonChebyshev polynomials, has been published in [28]. Infigure (A2), we show results for the geometricalcoefficients in the cases ofN=4, 5 and 6.Due to the parity invariance of the trapping potentials consideredhere, we haveαN=α1,αN− 1=α2,K, whichmeanswemust calculate, atmost,N/2 coefficients in each case.

In the single-well cases (b= 0), the coefficients are simply the ones expected for the given number of particlesin a harmonic trap. Particularly, forN=4we haveα1≈1.78,α2≈2.34. As b is increased, themajor change isobserved for evenN in the central coefficient, which vanishes for b 3. In this limit, we have the two sides of thesystemdescribed by almost completely separate harmonic trapswithN/2 in eachwell. The values of thecoefficients for harmonic traps in this situation have analytical expressions, given byα=π/2 forN=2 anda p= ( )3 2 23 3 forN=3. The situation is not the same for oddN. In this case, there is no central coefficientto vanish as the barrier is increased.Wewould expect such a system to exhibit different spatial densities anddynamics.

A.4. Spatial correlations in the impenetrable limitIn this sectionwe describe the spatial densities for a given number of atomsN in the impenetrable limit. Thesedensities reproduce the results expected for a Tonks–Girardeau gas or a gas of spinless fermions, and are

Figure A2.Geometrical coefficients for three different particle numbers in a double-well. The circlesmark the numerical values ofeach coefficient. The dotted lines connect the coefficients corresponding to the same values of b, which is increased from b=0 (top—lighter colors) to b=3 (bottom—darker colors).

9

New J. Phys. 20 (2018) 063014 REBarfknecht et al

characterized by a chain of localized atoms. The individual distributions are calculatedwith the followingexpression [32]

òr d= - F ¼ ¼G

( ) ( )∣ ( )∣ ( )x x x x x x x xd ... d , , , , , A.15iN i i N1 0 1

2

where the integral is performed in the regionΓ=x1<x2<K<xN. The quantity r ( )xi then gives the spatialdistribution of the atomwith index i. A formula for calculating these densities has been obtained byDeuretzbacher et al [23], and is written as

årl

l=¶¶

¶¶

- l=

-

=

⎛⎝⎜⎜

⎞⎠⎟⎟( ) [ ( ) ]∣ ( )x

xc B xdet 1 , A.16i

j

N

ji

j

j0

1

0

where = - - -- - -

-( ) ( )!( )!( )! !

cji N j

i N j i j

1 1

1

N 1

and thematrixB(x) is composed by the single-particle states superpositions

ò j j=-¥

( ) ( ) ( )b x y y ydmnx

m n . Infigure A3we show these spatial distribution for the cases ofN=4 andN=6as the barrier parameter b is increased.

A.5. Eigenvalues and eigenstates of the spin chainHamiltonianWepresent here the analytical expression for thematrix formofHamiltonianHs for the case of

= + = N N3 1. The choice of basis is given by ñ{∣ , ñ∣ , ñ ñ∣ ∣ }, :

=

-

- -

- -

-

ka a ak

a

a a k a ak

a

a a k a ak

a

a ka a ak

+ +

+ +

+ +

+ +

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

( )

( )

( )

( )

( )

H

0 0

0

0

0 0

. A.17s

g g

g g g

g g g

g g

2

2

2

2

1 1 2 1

1 1 1 2 2

2 1 1 2 1

1 1 1 2

In the expression abovewemade the simplification that, due to the parity invariance of the trappingpotential,α3=α1. Diagonalizing thismatrix we obtain the following eigenvalues

a k a a k ak

a k a a k ak

a k a k a k a k ak

a k a k a k a k ak

=- + + + +

=+ + + +

=- + - + + + +

=+ - + + + +

( )

( )

( ) ( )

( ) ( )( )

g

g

g

g

2,

2,

1 2,

1 2, A.18

112 2

22

1 2

212 2

22

1 2

312 2

22 2

1 2 2

412 2

22 2

1 2 2

Figure A3. Spatial distributions in the impenetrable limit for the case of (a)N=4 and (b)N=6.

10

New J. Phys. 20 (2018) 063014 REBarfknecht et al

and the respective non-normalized eigenstates

ca k a a

a ka k a a

a k

ca a k a

a ka a k a

a k

ca k a k a k

a ka k a k a k a

a k

ca k a k a k

a ka k a k a k a

a k

ñ =+ + + +

ñ=- + - +

ñ= -- - + - + - - +

ñ= -+ - + - - + - - +

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∣

∣

∣( ) ( ) ( )

∣( ) ( ) ( )

( )

1, , , 1 ,

1, , , 1 ,

1,1 1

,1

, 1 ,

1,1 1

,1

, 1 . A.19

112 2

22

2

1

12 2

22

2

1

22 1

2 222

1

2 12 2

22

1

32 1

2 222 2

1

12 2

22 2

2 2

1

412 2

22 2

2

1

12 2

22 2

2 2

1

The eigenvalues and eigenstates presented above can be compared to the ones in [24] by takingαi/g=Ji.Wehave also omitted the constant energy termE0 in these expressions. In the regime ofκ<1, the two lowest energystates are given by ò3 (ground state) and ò1 (first excited state).We can therefore write an analytical expression forthe frequencyωJ that defines the tunneling of the impurity between thewells. It is given by

wa k a k a a k a k

k=

- + + + -( )( )

g

1, A.20J

2 12 2

22

12 2

22 2

where againwe consideredα3=α1. Infigure A4we show the behavior of this frequencywith increasingκ in thecase of b=0. The result shown here can be directly related to the line defined by b=0 infigure 4 of themain

text. For smallκ, we get w = a kaJ g2

12 2

2, while forκ?1 (approaching the fermionic limit for the background gas) it

becomes constant: w =a a a a- + +

J g

2 1 12

22

.

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