Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap
F. Nur ¨Unal, ∗ B. Het´enyi, and M. ¨O. Oktel
Department of Physics, Bilkent University, Ankara, Turkey (Dated: September 17, 2018)The dynamics of a single impurity interacting with a many particle background is one of thecentral problems of condensed matter physics. Recent progress in ultracold atom experiments makesit possible to control this dynamics by coupling an artificial gauge field specifically to the impurity.In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a singleatom. We show that an external magnetic field coupled to the impurity is a versatile tool to probethe impurity dynamics. Using Bethe Ansatz (BA) we calculate the eigenstates and correspondingenergies exactly as a function of the flux through the trap. Adiabatic change of flux connects theground state to excited states due to flux quantization. For repulsive interactions, the impuritydisturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This dragtransfers momentum from the impurity to the background and increases the effective mass. Theeffective mass saturates to the total mass of the system for infinitely repulsive interactions. Forattractive interactions, the drag again increases the effective mass which quickly saturates to twicethe mass of a single particle as a dimer of the impurity and one fermion is formed. For excited stateswith momentum comparable to number of particles, effective mass shows a resonant behavior. Weargue that standard tools in cold atom experiments can be used to test these predictions.
PACS numbers:I. INTRODUCTION
Ultracold atom systems are effectively used as a testbed for condensed matter models. They are preferredbecause of the high degree of control in experiments suchas tunable interactions, impurities and direct measure-ments by optical techniques. Certain theoretical modelsof condensed matter such as resonant interactions [1] orbosonic Mott transition [2] have been realized for the firsttime using cold atoms. Many models of one dimensionalsystems have been realized using two dimensional opticallattices to form narrow tubes [3–5].One of the powerful theoretical tools to describe onedimensional systems is the Bethe Ansatz (BA). BA so-lution has been generalized to many integrable models,e.g. systems with multiple components, different statis-tics or spin [6–10]. This exact solution method has beenemployed to explain experimental data on a number ofinstances [4, 11]. However, as BA methods are restrictedto one dimension, they have not been used to describesystems where an external artificial gauge field is present.In one dimension, such an external magnetic field canbe disregarded by using a gauge transformation, unlessthe one dimensional system closes onto itself. Thus, ifthe particles are confined to a ring as opposed to a tube,the artificial magnetic field will significantly effect thephysics. Such rings, in the form of toroidal traps, havebeen realized experimentally [12–19]. Although none ofthese experiments have included an artificial gauge fieldso far.In this work, we consider such a toroidal trap contain-ing non-interacting fermions and describe the behavior ∗ [email protected] of a single charged impurity interacting with backgroundatoms. We argue that an artificial magnetic field couplingto the impurity is an efficient way to probe the polaronstate forming due to the interactions. Artificial magneticfields are created by coupling light to the internal statesof the atoms [20–22]. Hence, they are highly specific tothe internal state making it possible to create effectivemagnetic fields coupling only to one type of atom.The charged particle is expected to drag the unchargedfermions along with itself around the ring. Because of theinteractions between the impurity and the backgroundatoms a collective excitation usually called a polaron isformed [23]. This excitation will couple to the externalmagnetic field with the charge of the impurity particle,however, its mass will critically depend on the interactionstrength. The amount of angular momentum carried bythe impurity and the uncharged fermions also depend onthe total external flux through the ring. By changing theartificial magnetic field strength, it is possible to accessexcited states of the system adiabatically. We show thatan artificial magnetic field coupling specifically to theimpurity would be a very effective tool to probe polaronphysics.We describe this system exactly using a Bethe Ansatz(BA) solution for contact interactions which are justi-fied for cold atoms as the dominant scattering is s-wave.For strongly attractive interactions, the impurity forms abound state with one of the background fermions and thephysics reduces to the motion of a dimer with twice themass of the particle. In the other limit of infinitely re-pulsive interaction, effective mass saturates to total par-ticle number. We calculate the energy and momentumdistributions, total transferred momentum and the effec-tive mass for all interaction strengths. We believe theseresults can be experimentally checked with state of theart toroidal traps and techniques for artificial gauge field B FIG. 1. (Color online) A simple illustration of the system. N − β = qRA ~ . generation.The paper is organized as follows. In the next section,we define the model, introduce the notation and reviewearlier studies. In Section III, we solve the system fortwo particles and then generalize to any particle numberusing the BA. Sec. IV contains the analytical solution ofthe BA equations in certain limits and comparison withnumerical solutions. We present our results for severalquantities such as energy, angular momentum and effec-tive mass of the charged particle. We give our conclusionsalong with a brief discussion of possible experiments inSec. V. II. THE MODEL
The first quantized Hamiltonian for one charged par-ticle among N − H = 12 m (cid:16) ~ i ∂∂x − qA (cid:17) − ~ m N X j =2 ∂ ∂x j +2 c N X j =2 δ ( x − x j ) . (1)All particles are assumed to be on a ring of radius R,0 ≤ x i ≤ πR . The position of the charged particle is x and A is the vector potential in the symmetric gauge.The Hamiltonian can be made dimensionless by using,˜ x j = x j R , ˜ E = E mR ~ , ˜ c = c mR ~ and β = qRA ~ . β isthe total magnetic flux through the ring in units of fluxquantum q/h . Dropping the tildes H = (cid:16) − i ∂∂x − β (cid:17) − N X j =2 ∂ ∂x j + 2 c N X j =2 δ ( x − x j ) . (2)The effect of the magnetic field can be shifted to theboundary conditions by a gauge transformation [24].Namely, when the first particle makes a full circle aroundthe ring the wave function gains a phase factor of e iβ π where the periodic boundary conditions (PBCs) for theuncharged particles remain unaffected by the gaugingprocess, H → e − iβx H e iβx . (3)Apart from the twisted BCs, the δ -function interactioncan be handled as a two-sided boundary condition (BC)between two different regions of N-particle space corre-sponding to different permutations of particles. The dis-continuity relation at the boundary x = x j (which isobtained by passing to the center of mass and relativecoordinates and then integrating the Hamiltonian) is( ∂ j − ∂ ) ψ (cid:12)(cid:12)(cid:12) x As the interactions are reduced to BCs, the wave func-tion for a given permutation of the particles is a super-position of plane waves. As the collisions of equal massparticles in one dimension conserve magnitudes of the in-coming momenta, the interacting problem is integrable.Hence, in a given region only a finite number of planewaves are needed to construct the wave function. Tomake our notation clear, we first start with the case ofone charged particle with one neutral particle. A. N=2 Particles For two particles we have 2! = 2 regions and the wavefunction in these regions is expressed as follows:Ψ ( x , x ) = (12) e i ( k x + k x ) + (21) e i ( k x + k x ) , Ψ ( x , x ) = (12) e i ( k x + k x ) + (21) e i ( k x + k x ) , (5)where we use parenthesis with a subscript to indicate thecoefficients of plane waves. In this notation numbers inthe parenthesis indicate the order the wave vectors k , k are distributed to the coordinates in the exponent and thesubscript indices indicate the ordering of the particles onthe ring, i.e. Ψ means x < x . At x = x , the wavefunctions in the two regions should be equal whereas theirderivative should obey Eq.4. Equating the coefficients ofeach plane wave on both sides, we obtain: BCs: at x = x ,(12) + (21) = (12) + (21) , (6)(12) − (21) = (12) (cid:16) s (cid:17) + (21) (cid:16) − s (cid:17) , (7)where s = i ( k − k ) /c . Combined BCs give (12)(21) = s s − s − s (12)(21) . (8)Allowed values for k , k are found by applying the PBCs.PBC for one of the particles gives the BA equation. BCs: at 2 π as x : 0 → π, Ψ ( x = 0) = Ψ ( x = 2 π ),(12) = (12) e ik π , (21) = (21) e ik π . (9)as x : 0 → π, Ψ ( x = 0) = e iβ π Ψ ( x = 2 π ),(12) = (12) e i ( k + β )2 π , (21) = (21) e i ( k + β )2 π . (10)Combining the two BCs at 2 π , we obtain another con-straint k + k + β = n , for n ∈ Z . This is a reflection ofthe total angular momentum conservation in the system.Eqs.8 and Eq.10 have non-trivial solutions only when thedeterminant below vanishes, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s − e − i ( k + β )2 π s − s − s − e − i ( k + β )2 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (11)Solution of this determinant gives the BA equation, α = c (cid:16) π α + n − β ) (cid:17) + c (cid:16) π α − n + β ) (cid:17) , (12)where energy is E = ( n − β ) + α for α = k − k . For thetwo-particle case, this problem can also be solved exactlywithout using the BA [27], c = α (cid:16) cos( π ( n − β ))sin( πα ) − cot( πα ) (cid:17) . (13)These two equations analytically reproduce each otherand the numerical results match perfectly (Fig.2).Extension of this method to N particles is straightfor-ward if cumbersome. −10 −5 0 5 10012345 Ground State1st Excited State2nd Excited StateBA from N particle result N=2 β =0.2n=0 (total momentum) (dimensionless) E n e r g y L e v e l s ( i n u n i t s o f h / m R ) Interaction Strength c FIG. 2. (Color online) Energy of the lowest three statesvs. interaction strength for N = 2 particles, for zero to-tal angular momentum. Only scattering states are displayed.Energy is calculated by three different methods. Lines arefrom Eq.13 direct analytical solution without employing BA,which is algebraically same with the two-particle BA calcula-tion (Eq.12). Diamonds are from the general N -particle BAcalculation (Eq.21). −1.5 −1 −0.5 0 0.5 1 1.501234 β = 0= 1= 2=−1=−2= 3=−3nnnnnnnc=-0.05N=2 E n e r g y L e v e l s ( i n u n i t s o f h / m R ) FIG. 3. (Color online) Ground state energy vs. flux β for N = 2 particles with total angular momentum n . Eigenstatesfor flux β with total angular momentum n are also eigenstatesfor flux β +1 with total angular momentum n +1. The systemcan be analyzed by considering flux values between − / <β < / n . As the flux is increased by one adiabatically,the system evolves to a higher excited state which has onemore unit of total angular momentum. As can be observed,the crossings between different eigenstates is not a problemfor adiabatic evolution since only states with different totalangular momentum n are degenerate. B. N-1 Fermions, One Charged Particle The distinguishable charged particle is denoted againby x and the wave function is defined in N ! regions corresponding to different permutations [9]. In each oneof these regions the wave function consists of N ! planewaves in its most general form without imposing the an-tisymmetry between the fermions. As a total we have N ! × N ! coefficients:Ψ ... = (123 . . . ) ... e i ( k x + k x + k x + ... ) + (213 . . . ) ... e i ( k x + k x + k x + ... ) + . . . Ψ ... = (123 . . . ) ... e i ( k x + k x + k x + ... ) + (213 . . . ) ... e i ( k x + k x + k x + ... ) + . . . Ψ ... = (123 . . . ) ... e i ( k x + k x + k x + ... ) + (213 . . . ) ... e i ( k x + k x + k x + ... ) + . . . ... ... (14)where k , k , . . . k N are distinct wavenumbers. BCs at x = x are not effected by the addition of other fermionsat the end of the sequence: (123 . . . )(213 . . . ) ... = s s − s − s (123 . . . )(213 . . . ) ... . (15)BCs at 2 π follow the same logic;as x : 0 → π, Ψ ...N ( x = 0) = Ψ ...N ( x = 2 π ),yielding (123 . . . ) ...N = (123 . . . ) ...N e ik π , (213 . . . ) ...N = (213 . . . ) ...N e ik π . (16)Number of independent coefficients decreases consid-erably by requiring antisymmetry upon exchange offermions. Every coefficient of a plane wave in region x < x < x < . . . < x N is identical with the co-efficient of the same plane wave in region x < x The cot πk term in Eq.21 diverges at every integer k,thus, regardless of the value of β (or λ ) there is a rootbetween every consecutive integer (Fig.5). By changingthe value of λ , all roots can be adjusted so that the to-tal angular momentum constraint is satisfied. All theeigenstates in this problem can be labeled identically bychoosing N distinct integers corresponding to the differ-ent branches of cot πk and the total angular momentum n ∈ Z . The energy of an eigenstate is simply the sum ofsquares of all wavenumbers E = N X i =1 k i . (22)For the simplest case of β = 0, the ground state corre-sponds to λ = 0 and the total angular momentum n = 0.The roots k are distributed symmetrically around zerofor even N, hence, automatically satisfy the total angularmomentum condition. The wavevectors for the groundstate are in the N branches of cot from − N/ N/ λ , N roots which areon the same branches of cot can be generated so as tocreate an eigenfunction with non-zero total angular mo-mentum ( n = 0). Second, at least one of the roots canbe chosen to reside on a branch that is not occupied forthe ground state. For such a particle-hole excitation λ must be adjusted to ensure the total angular momentumconstraint.Inclusion of the magnetic field affects only the totalangular momentum constraint. As that constraint is de- −5 −4 −3 −2 −1 0 1 2 3 4 5−40−2002040 k c c o t ( π k ) k − λ FIG. 5. (Color online) A representation of graphical solutionto BA equation. The cot πk term in Eq.21 diverges at everyinteger k and there is a root between every consecutive integerindependently from λ ( β ). By changing the value of λ , allroots can be adjusted so that the total angular momentumconstraint is satisfied. fined only up to an integer ( n ), the problems with valuesof β differing by an integer are identical. Eigenstates forflux β which have total angular momentum n are alsoeigenstates for flux β + 1 which have total angular mo-mentum n +1. This is a restatement of flux quantization.We can analyze the system by considering flux values be-tween − / < β < / n can be degenerate in energy.The BA equation Eq.21 can be very efficiently solvedonce the regions for the roots are determined. We usedthe Newton-Raphson algorithm to find a solution withina particular region. As all the roots depend monotoni-cally on λ , another Newton-Raphson search is employedto satisfy the total angular momentum condition. Wehave found numerical solutions for systems of up to 10000particles with high accuracy.Although numerically solving the BA equation is effi-cient and accurate, an analytic solution can provide moreinsight about the physics of the system. Analytic formu-lae for energy, angular momentum and effective mass alsowould be desirable to make correspondence with experi-mental observations.In the limit of strong interactions 1 /c ≪ N/c ≫ 1, such an analytic form can beobtained by approximating the roots of the BA equa-tion. In this limit, because the cot diverges quickly nearintegers, most of the roots are close to integers. Apartfrom the few roots near k ∼ λ , the deviation of the root∆ from an integer s is small [9]. Solving for this smalldeviation we find that the roots occur at k + s = s + 1 π acot 2 c ( s − λ ) ,k − s = − s − π acot 2 c ( s + λ ) , s = 0 , , . . . , N − , (23)where acot is defined in the continuous region (0 , π ) forEqs.23 to be accurate guesses. Here we have restricted s to analyze the ground state and excited states with rootson the same cot branches. Applying the total angularmomentum condition we get, n − β = N/ − X s =0 k s = 1 π N/ − X s (cid:16) acot 2 c ( s − λ ) − acot 2 c ( s + λ ) (cid:17) = c π Z x F d x (cid:16) acot( x − b ) − acot( x + b ) (cid:17) , (24)with b = 2 λ/c and x F = ( N − / /c . Here the initialassumption of strong interactions and large particle num-bers allow us to approximate the sum by an integral. Forthe ground state and the first few excited states n − β is small compared to N and the integral can be approxi-mated as n − β = c π Z x F + bx F − b d x atan x ≈ cbπ atan x F . (25)Through this relation b , hence λ , is obtained for any fluxvalue, allowing us to find expressions for all the roots ina self consistent way. A. Energy Using these expressions for the roots, the total energyis E = N/ − X s =0 k s = N/ − X s (cid:26) s + 2 sπ (cid:16) acot 2 c ( s − λ ) + acot 2 c ( s + λ ) (cid:17) + 1 π (cid:16) (acot 2 c ( s − λ )) + (acot 2 c ( s + λ )) (cid:17)(cid:27) . (26)The first term above is the total ground state energy of N − E = cb ( n − β ) + c x F − c π (cid:26)(cid:16) ( x F + b ) + 1 (cid:17) atan( x F + b ) + (cid:16) ( x F − b ) + 1 (cid:17) atan( x F − b ) − x F (cid:27) , (27)with b = π ( n − β ) c atan( x F ) . This approximate form for energy successfully repro-duces numeric results for particle numbers as small as 4throughout all the interaction range. Ground state en-ergy as a function of interaction strength is plotted fora typical case in Fig.4 for 1000 particles at β = 0 . δ -function interactionsupports one bound state in one dimension. Correspond- ing imaginary wavevectors appear as solutions of the BAequation. For k = α + iσ with ( α, σ ) ∈ R , the BA equa-tion has only two roots with σ = 0. The charged parti-cle is bound with only one of the background fermions.When 1 / | c | ≪ 1, the complex roots are at k = λ ± ic/ E = − c ( b + 1)2 + cb ( n − β )+ c x F c π (cid:26)(cid:16) ( x F + b ) +1 (cid:17) atan( x F + b )+ (cid:16) ( x F − b ) +1 (cid:17) atan( x F − b ) − x F (cid:27) , (28)with b = ( n − β ) c (cid:16) − π atan( x F ) (cid:17) . −10000 −8000 −6000 −4000 −2000 0−0.2−0.15−0.1 c L N=100n=0 β =0.2 −50 0−0.2−0.19−0.18 c L (a) c L N=100n=0 β =0.20 50−0. −0.19−0.18 c L (b) FIG. 6. Angular momentum of the impurity vs. interac-tion strength for N = 100 particles for (a) attractive, (b)repulsive interactions from the analytic calculation, numeri-cal solutions produce the same results. In the non-interactinglimit, the impurity carries all the angular momentum ( n − β ). L saturates to almost zero for infinitely strong repulsive in-teractions as the total angular momentum is shared equallybetween all particles. The same behavior holds for excitedstates. For strongly attractive interactions, L saturates tohalf the total value signifying dimer formation with one back-ground particle. The insets in both figures focus on the weakinteraction limit. B. Angular Momentum To understand the physics of the system and make cor-respondence to possible experiments, it is important tocalculate other measurable quantities. In particular, for this system we are interested in how the dynamics of theimpurity particle is affected by the fermion background.To this end, it is instructive to calculate angular momen-tum carried by the impurity L and the related effectivemass. As the impurity is interacting with the fermions,this effective mass is not only the mass of the impuritybut also gets a contribution from the fermions draggedalong with it. Such a compound object is generally calledthe polaron state or dimer state especially for attractiveinteractions.As stated above, one of the most interesting physicalquantities in this system is the angular momentum car-ried by the charged particle, represented by the operatorˆ P = − i ∂∂x . As this particle is coupled to the exter-nal magnetic field, ˆ P is the canonical momentum notthe kinetic momentum. However, canonical momentumis the quantity that is generally measured by expansionimaging in artificial magnetic field experiments. The ex-pectation value of h ˆ P i = L is easily obtained by takingthe derivative of the total energy with respect to flux, L = − ∂ ∆ E∂β . (29)Using the approximate form for the energy Eq.27, weobtain L = π ( n − β )atan x F − c x F (cid:26) ( x F + b )atan( x F + b ) − ( x F − b )atan( x F − b ) (cid:27) . (30)This form is valid for positive c and easy to interpret.In the non-interacting limit, the canonical momentum ofthe charged particle is fixed by the external flux. Hence,all the angular momentum is carried by the charged par-ticle. As interaction is turned on, the charged particledrags the background fermions and transfers some of itsangular momentum to them. Stronger interactions in-crease the fraction of the transferred angular momentumand in the limit of infinitely repulsive interactions angu-lar momentum is equally shared by N particles. On theother hand, for strong attractive interactions, L satu-rates to half of the angular momentum in the systemproving the formation of a dimer with one backgroundfermion.The behavior of L is displayed in Fig.6 and Fig.7as a function of c and β . Even for N = 100 parti- cles, the difference between numerical calculation of thederivative and the expression given above is negligible.As a function of interaction strength, the rapid decreaseand eventual saturation of L validates the scenario dis-cussed above. The linear dependence on flux is expected,however, the slope of L decreases as interaction getsstronger. This slope carries valuable information as it isrelated to the effective mass of the composite excitationformed by the impurity and background fermions. C. Effective Mass We define the effective mass as m ∗ = 2 ∂ ∆ E∂β . (31) −10−8−6−4−202 β L c=10c=100c=1000c=10000 N=100n=0 FIG. 7. (Color online) The angular momentum of the chargedparticle vs. flux at varying interaction strength for N = 100particles. As expected L depends linearly on flux. The slopeof the line decreases with increasing interaction strength in-dicating higher values of effective mass of the impurity. m * c −2000 0 β =0.2−400011.52 c −400 −200 011.52 β =0.2 β =0.6 β =5.2 m * (a) x10 c m * β =0.2 β =5.2 β =10.2N=50n=0 (b) FIG. 8. (Color online) Effective mass of the impurity vs. in-teraction strength for N = 50 particles and zero total angularmomentum n = 0. (a) For attractive interactions, m ∗ satu-rates to twice the mass of the impurity due to the formationof a tightly bound pair. The inset shows the behavior aroundzero interaction in more detail. For attractive interaction, theeffective mass (given by Eq.32) is almost insensitive to fluxchange. (b) For repulsive interactions, m ∗ converges to N . Asflux β increases, this saturation gets faster. The dependenceon the flux is more prominent for small particle numbers. In the non-interacting limit, the effective mass is equalto m , however, its behavior is very different for attrac-tive and repulsive interactions. As repulsive interactionsare increased, it gets harder for the impurity to tunnelthrough the fermions and the dragged particles increasethe effective mass. The increase in the effective masssaturates only when all the particles are moving togetherwith the impurity. Thus, at large repulsive interaction,the effective mass reaches N m . For weak attractive inter-actions, the first effect is once again the drag increasingthe effective mass. However, the attractive δ -functionhas a single bound state in one dimension. Thus, the im-purity captures one of the background fermions and asthe size of the bound state gets smaller, Pauli exclusioneffectively repels the other fermions. The effective massfor attractive interactions increases and reaches 2 m forinfinitely attractive interaction where a dimer is formedfrom the impurity and one fermion. For attractive inter-actions, the analytical expression in the strongly inter-acting limit is useful to calculate the dimer mass, ∂ ∆ E∂β = 21 − atan( x F ) π − − atan( x F ) π ) + 12 π (1 − atan( x F ) π ) (cid:26) atan( x F + b ) + atan( x F − b ) + x F + b x F + b ) + x F − b x F − b ) (cid:27) . (32)We calculated the effective mass numerically and analyt-ically. For the ground state, the above scenario is vali-dated by these calculations (see Fig.8). The dependenceof the effective mass on the external magnetic field isstrongest for small particle number as this limit is thestrongly interacting limit in one dimension. As the num-ber of fermions increases, effective mass in the ground state has weak dependence on β . In this case, effectivemass is essentially determined locally as the ability forthe impurity particle to complete a full rotation is ham-pered.The utility of an external magnetic field is the accessit provides to excited states through adiabatic pumping.Excited states in this system are expected to be stable −800−400 c m * N=50n=0 β =15.2 FIG. 9. Resonant behavior in m ∗ for β comparable to N . When the drag effect applied by the background parti-cles overcomes the driving force of the magnetic field, m ∗ canbecome negative. When the second derivative of the energywith respect to flux becomes zero, m ∗ diverges. This diver-gence does not change the infinitely strong interaction limit. due to angular momentum conservation. It is thus rea-sonable to expect effective mass measurements to be car-ried out on such states in a cold atom setting. For theexcited states, with angular momentum | n | < N/ c in-creases. However, for higher excited states, there is reso-nant behavior (Fig.9). Due to the nature of BA solution,a state for which all the roots are on the cot branchesfrom − N/ N/ N/ D. Correlations Apart from the single particle properties related to theimpurity, it is instructive to look at global properties tounderstand how the external particle disturbs the one di-mensional Fermi liquid. A common way to visualize thedisturbance in the Fermi sea is to plot the deviation of thedistance between the BA roots (wavevectors) from one.For an undisturbed Fermi sea, this deviation is alwaysone. For a weakly interacting impurity, the deviationis confined to a narrow region in k-space around n − β (see Fig.10). This is expected as the impurity carrying N=100n=0 β =10.2 (k j +k j+1 )/2 −50 −40 −30 −20 −10 0 10 20 30 40 500.981.021.041.061.08 c=10c=50c=10000 / ( k j + − k j ) FIG. 10. Effective momentum density in k-space for differentinteraction strengths. In the strongly interacting limit, thedistance between adjacent BA roots (wavevectors) is one. Forsmall c, the roots are closer to each other around ( n − β ).The impurity carrying n − β units of angular momentum inthe non-interacting limit first disturbs the fermions which aremomentum matched to that value. n − β units of angular momentum in the non-interactinglimit first starts dragging fermions which are matched inmomentum. As the strength of repulsion increases, sodoes the effected region in k-space, however, the devi-ation gets smaller. For infinitely repulsive interactions,the impurity becomes indistinguishable from the back-ground fermions (Fig.10). For highly excited states where n is comparable to N, particle-hole excitations complicatethis picture similar to the effect we discussed for the ef-fective mass.Another important physical property is the two-particle correlation function. Although for δ -function in-teractions only the value of this function at zero deter-mines the interaction energy, its general form is exper-imentally accessible through Hanbury-Brown-Twiss[29]type measurements. This correlation also can be re-garded as the real-space form of the bound state cre-ated by the impurity. To calculate the two-particle cor-relation function, we need to determine the coefficientsof the plane waves in each region. Following McGuire[9] we choose the first coefficient in the first region x < x < . . . < x N ,(123 ..N ) ..N = (1 − e i πk ) . (33)Other coefficients in this region determined by BCs yieldvery similar expressions. The wavenumber associatedwith the distinguishable particle appears in the exponentand the sign of the permutation multiplies the coefficient:(213 ..N ) ..N = − (1 − e i πk ) , (34)(312 ..N ) ..N = (1 − e i πk ) (35)...0 x −x g ( x , x ) c=1 β =0 c=1 β =0.2 c=1 β =0.8c=100 β =0c=100 β =0.8 N=2n=0 FIG. 11. (Color online) Two-particle correlation function for N = 2 particles. As expected, g at zero separation decreaseswith increasing interaction strength. For weak interactions,the correlation function at zero decreases with increasing flux.However, for strong interactions, the correlation is almostinsensitive to flux change due to the fermionization of thecharged particle. The coefficients in other regions are related to the samecoefficient in the first region with a phase factor deter-mined by the PBCs. This phase factor is a full circlerotation around the ring of the particles that x has topass to be in the given region. Thus, the momenta be-longing to the particles that x has passed multiply thecoefficient, e.g.(21354 ..N ) ..N = (1 − e i πk ) e i π ( k + k ) . . . (36)In the simple form, the wave functions are not normal-ized, but we normalize the correlation function at theend. The two-particle correlation function in any state isgiven as g ( x , x ) = Z π d x · · · Z π d x N Ψ ∗ Ψ . (37)For the N = 2 particle case, the correlation function issimply the absolute square of the wave function. As couldbe expected, the correlation function is highly affected bythe flux for the two-particle case. Using the numericallyand analytically found wavenumbers in the expression, g ( x , x ) = 4 − πk ) − πk ) − πk ) sin( πk ) cos( k − k )( x − x − π ) , (38)we observe that inclusion of the flux generally decreasesthe two-particle correlation function (Fig.11). However,if the interactions are strong enough so that the particlesare almost fermionized, this decrease is very small. It isalso notable that although the flux breaks time-reversalsymmetry and the wave functions choose a direction on x −x g ( x , x ) c=1 β =0 c=1 β =0.8 c=1 β =12.2N=50n=0 (a)(b) x −x g ( x , x ) g ( x , x ) c=100 β =0c=100 β =12.2N=50n=0 x −x x −x g ( x , x ) FIG. 12. (Color online) Two-particle correlation function for N = 50 particles. (a) For weak interactions, Friedel oscil-lations occur as interference of two waves with wavelengthsrelated to k F − β and k F + β . (b) At strong interactions, thecorrelation becomes zero at zero separation since the impurityis effectively indistinguishable and g and the frequency ofFriedel oscillations are almost insensitive to flux change. the ring, correlation function is even with respect to x − x . This property holds for any particle number.For the general N particle case, we arrange the wavefunction in a better form to evaluate the integrals. Weassign two wavenumbers to x and x , and the rest of theparticles are represented by a Slater determinant sincethey are indistinguishable fermions. For example, if wehave k , k associated with x , x respectively, the Slaterdeterminant is represented by D indicating the use ofall wavenumbers except k and k in the exponents, D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ik x e ik x e ik x . . .e ik x e ik x e ik x . . . ... e ik x N e ik x N e ik x N . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (39)1Hence, the wave function in the first region can be writtenas,Ψ = (cid:16) (12 . . . N ) ...N + (21 . . . N ) ...N (cid:17) D + (cid:16) (13 . . . N ) ...N + (31 . . . N ) ...N (cid:17) D + . . . (40)Integrating Ψ ∗ Ψ over x , . . . , x N , the Slater determinantsare orthogonal in large particle number limit, as the outerroots of cot ’s are very close to integer values. The corre-lation function is then expressed as a sum over pairs ofmomenta associated with x and x , g ( x ) = N X t The problem of a single impurity interacting with afermion background has attracted attention of the con-densed matter society for years. In this paper, we arguethat an artificial gauge field coupling exclusively to theimpurity is an effective tool to probe the physics of thissystem at any interaction strength. We consider a Fermigas in a narrow ring trap and an artificial magnetic fieldcoupling only to a single impurity. We solve this systemexactly by using the BA for contact interactions and cal-culate the dependence of measurable quantities on theexternal magnetic flux. We observe this dependence fortotal energy, angular momentum of the charged particle,effective mass and the two-particle correlation function.Using an artificial magnetic field in this system has twoadvantages. The usual measurement tools such as expan-sion imaging become probes of thermodynamic quantitiesby comparing measurements at different flux values. Forexample, the change of the momentum carried by the im-purity caused by the magnetic field is a direct probe of theeffective mass of the impurity. The second advantage isobtained by adiabatically increasing the flux value. Al-though the Hamiltonian of the system is periodic withflux, adiabatic evolution connects the ground state atzero flux to excited states at integer flux. In a cold atomexperiment, such excited states can be expected to havelong lifetimes due to total angular momentum conserva-2tion. Thus, we have calculated the physical propertiesfor not only the ground state but also for excited statesadiabatically connected to it.Our results show that the system can be described bya simple physical picture. The charged particle interact-ing with the background particles drags them along withitself around the ring. In the non-interacting limit, all ofthe angular momentum in the system is carried by the im-purity. As interactions are turned on, fermions which areclose to the impurity in momentum are disturbed moreand start to gain momentum. At the limit of infinitelyrepulsive interactions, the charged particle is effectivelyindistinguishable from the background and the total mo-mentum is shared equally between all particles. The paircorrelation function also confirms this picture. The valueof the correlation function near zero is mostly insensitiveto the external flux while away from the correlation holefrequency of the Friedel oscillations sensitively dependson it. For strongly repulsive interactions, the effectivemass saturates the total mass of the particles since it isdragging all of the background fermions along with itselfaround the ring.For attractive interactions, the impurity forms a boundpair with one of the fermions. The effect of dimer for- mation can be clearly seen in the angular momentumand the effective mass. For infinitely strong attractiveinteractions, angular momentum carried by the impuritysaturates half the value of total angular momentum andthe effective mass saturates twice the mass of the particlewhich confirm the presence of the dimer as a compositeparticle.The physical properties calculated in this paper areexperimentally accessible through the standard tools ofultracold atom experiments. While artificial magneticfields have been demonstrated in a variety of settings,they have not been used in combination with a toroidaltrap to our knowledge. We believe our exact resultswould be relevant for such an experiment. 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As expected, the corre-lation function at zero separation g (0) decreases withincreasing interaction strength until it saturates to zero.The other important feature of the correlation functionis the Friedel oscillations [30] reflecting the sharpness ofthe Fermi surface in one dimension.The two-particle correlation function is a local quan-tity while the external flux changes the system propertiesglobally. For any pair to feel the effect of the flux, one ofthe particles must go a full circle through the ring. Hence,as could be expected, the effect of the artificial magneticfield on the correlation function decreases as the numberof particles increases or if they interact strongly. Evenfor the lowest lying excited states, we find the primaryeffect of the flux is on the Friedel oscillations while theshape of the correlation hole is unchanged.Finally, we calculate a thermodynamic quantity whichis also related to g (0). Derivative of the energy withrespect to interaction strength c gives us interaction po-tential, so, the kinetic and interaction contributions tothe total energy can be separated. In Fig.13, one cannotice that the interaction energy makes a peak andthen decreases for increasing c . The initial increase isexpected, however, as interactions become stronger, thetendency of the fermions to avoid the impurity dominatesand the impurity is effectively fermionized. This is appar-ent in the δ -function BC Eq.4. Additionally, the interac-tion potential is equal to the correlation function at zero g (( x − x ) = 0) times the interaction strength whichreproduces the results obtained by taking the derivativeof the total energy. c K i n e t i c E n e r g y β =0.2 β =10.2 N=50n=0 I n t e r a c t i o n P o t e n t i a l β =0.2 β =10.22c g (0) , β =0.22c g (0) , β =10.2 c FIG. 13. (Color online) Kinetic energy of the particles vs.interaction strength for β = 0 . β = 10 . N = 50 par-ticles. The inset shows the interaction potential contributionto the total energy. The initial increase in interaction energyfollows the increase in the interaction strength. However, be-yond a certain strength, the tendency of the fermions to avoidthe impurity is more dominant. These plots are obtained bytaking the derivative of the total energy with respect to c . Al-ternatively, the interaction potential energy is also obtainedby using the two-particle correlation function at zero separa-tion. Both results are plotted in the inset showing remarkableagreement. V. CONCLUSION