Multicomponent Strongly Interacting Few-Fermion Systems in One Dimension
Artem G. Volosniev, Dmitri V. Fedorov, Aksel S. Jensen, Nikolaj T. Zinner, Manuel Valiente
aa r X i v : . [ qu a n t - ph ] A p r Few-Body Systems (EFB22) manuscript No. (will be inserted by the editor)
Artem G. Volosniev · Dmitri V. Fedorov · Aksel S. Jensen · Nikolaj T. Zinner · Manuel Valiente
Multicomponent Strongly Interacting Few-FermionSystems in One Dimension
Received: date / Accepted: date
Abstract
The paper examines a trapped one-dimensional system of multicomponent spinless fermionsthat interact with a zero-range two-body potential. We show that when the repulsion between particlesis very large the system can be approached analytically. To illustrate this analytical approach weconsider a simple system of three distinguishable particles, which can be addressed experimentally. Forthis system we show that for infinite repulsion the energy spectrum is six fold degenerate. We alsoshow that this degeneracy is partially lifted for finitely large repulsion for which we find and describecorresponding wave functions.
Keywords strongly interacting few-body systems · one dimensional harmonic traps · multicomponentfermions · Tonks-Girardeau gas
Experimental study of few-body physics in cold atomic gases is a complicated task since such systemsusually have a relatively large particle density and, hence, many-body correlations should be taken intoaccount. Only very recently setups with small particle numbers were realized in Heidelberg [1; 2; 3]where ground state systems of a few fermionic atoms, Li , were prepared in a quasi-one-dimensionaltrap. Such setups pave the way for the experimental study of few-body correlations where accuratetheoretical description can be obtained through advanced numerical investigation [4; 5]. This paperoverviews the newly developed method [6] to study the mentioned experimental setups in the limit ofstrong repulsion between particles without applying complicated numerical routines. This analyticalapproach gives a description of the Tonks-Girardeau gas [7; 8; 9] of a few particles in a trap. Moreover,as a by-product results obtained with this method can be used as a reference point for numericalcalculations.The structure of the paper is the following: in section 2 we introduce the Hamiltonian that is widelyused to describe the relevant experimental setups [1; 3; 4; 5; 6; 10; 11], in section 3 we illustrate theapproach using the simple system of three distinguishable particles in a harmonic trap, that to thebest of our knowledge was not addressed before. There we find and describe eigenstates of such systemin the limit of strong interparticle interaction. Artem G. Volosniev · Dmitri V. Fedorov · Aksel S. Jensen · Nikolaj T. ZinnerDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, DenmarkManuel ValienteSUPA, Institute for Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UKE-mail: [email protected]
This paper considers N particles of equal mass m in one spatial dimension. Additionally, the followingassumptions are applied: i ) the particles can be divided into classes of identical spinless fermions; ii )the system is trapped by some external potential, V ext ; iii ) the interparticle interaction is assumed tobe of zero range, V = gδ ( x i − x j ), where g is a strength parameter and x i , x j are the coordinates ofparticles i and j . These assumptions lead to the following Hamiltonian H = − ¯ h m N X i =1 ∂ ∂x i + N X i =1 V ext ( x i ) + g X i>j δ ( x i − x j ) . (1)This model Hamiltonian is often used to study the relevant experimental setup of a few fermionic atoms Li in different hyperfine states [1; 3; 4; 5; 6; 10; 11]. This model Hamiltonian allows one to exploreanalytical approach only in limiting cases, e.g. two particles in a harmonic trap [12] or N particleswithout an external confinement [13], such that the theoretical study for the relevant experiments isusually provided using different numerical techniques, e.g. [4; 5]. It is shown in [6] that one can findeigenstates for such Hamiltonian in the case of very strong repulsion between particles, i.e. if 1 /g → − ¯ h m X i =1 ∂ ∂x i + mω X i =1 x i + gδ ( x − x ) + gδ ( x − x ) + gδ ( x − x ) ! Ψ = EΨ, (2)where Ψ is the wave function, E is the energy of the system, ω is the frequency of the harmonicoscillator, g is assumed to be large and positive, such that 1 /g →
0. Eq. (2) can be recast into the freeSchr¨odinger equation plus the boundary conditions (cid:18) ∂Ψ∂x i − ∂Ψ∂x j (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x i − x j =+0 − (cid:18) ∂Ψ∂x i − ∂Ψ∂x j (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x i − x j = − = 2 gm ¯ h Ψ (cid:12)(cid:12)(cid:12)(cid:12) x i = x j , (3)where i, j = 1 , , i = j . Eqs. (2) and (3) contain all ingredients that are needed to describe ageneral method [6]; moreover, the system of three distinguishable particles is interesting on its ownrights as experimentally relevant. /g = 0.To solve eq. (2) for large values of g we first notice that if 1 /g = 0, then eq. (3) yields Ψ | x i = x j = 0,which means that eq. (2) can be seen as an equation for free particles that cannot penetrate throughone another. This problem formally resembles the case of three identical spinless fermions with the onlydifference that now the total wave function not need to have a continuous derivative when two particlesmeet. This possibility to have a discontinuous derivative follows from eq. (3). However, since theparticles cannot penetrate through one another it is enough to solve the Schr¨odinger equation only fora given ordering of the particles, i.e. x < x < x , with the boundary conditions Ψ x = x = Ψ x = x = 0.These boundary conditions can be satisfied only for the energies from the eigenspectrum of spinlessfermions { E , E , ... } , which implies that the total wave function for 1 /g = 0 can be built using thewave function of spinless fermions Ψ F for a given energy E i Ψ = a Ψ F for x < x < x a Ψ F for x < x < x a Ψ F for x < x < x a Ψ F for x < x < x a Ψ F for x < x < x a Ψ F for x < x < x (4) where a i are real coefficients. To obtain the wave function (4) we assumed that the energy E i corre-sponds to only one wave function Ψ F . The extension of the method for the more general situation whenthe energy E i corresponds to more than one wave function is discussed in refs. [6; 14]. The possibilityto build the wave function in the form of eq. (4) implies that for 1 /g = 0 the energy spectrum of eq. (2)is six fold degenerate, since all six coefficients a i in eq. (4) are linearly independent.3.2 Case of finitely large repulsion between particles, 1 /g → Method . To find the behavior of the energy in the vicinity of 1 /g = 0 we use the Hellmann-Feynmantheorem to obtain the derivative of the energy ∂E∂g = R d x d x ( Ψ ) x = x + R d x d x ( Ψ ) x = x + R d x d x ( Ψ ) x = x h Ψ | Ψ i . (5)Next we notice that from eq. (3) it follows that Ψ | x i = x j ∼ /g + o (1 /g ), which allows us to concludethat E ∼ E i − K/g + o (1 /g ), where the parameter K is given by K . = lim g →∞ g ∂E∂g = lim g →∞ g R d x d x ( Ψ ) x = x + R d x d x ( Ψ ) x = x + R d x d x ( Ψ ) x = x h Ψ | Ψ i . (6)The limit in this equation can be easily taken, since ( Ψ ) x i = x j ∼ /g + o (1 /g ), which yields K = ¯ h m R (cid:0) ( ∂∂x − ∂∂x ) | x − x =+0 Ψ − ( ∂∂x − ∂∂x ) | x − x = − Ψ (cid:1) d x d x h Ψ | Ψ i + (7)¯ h m R (cid:0) ( ∂∂x − ∂∂x ) | x − x =+0 Ψ − ( ∂∂x − ∂∂x ) | x − x = − Ψ (cid:1) d x d x h Ψ | Ψ i +¯ h m R (cid:0) ( ∂∂x − ∂∂x ) | x − x =+0 Ψ − ( ∂∂x − ∂∂x ) | x − x = − Ψ (cid:1) d x d x h Ψ | Ψ i . Using the wave function (4) the value of K can be written as K = γ ( a − a ) + ( a − a ) + ( a − a ) + ( a − a ) + ( a − a ) + ( a − a ) a + a + a + a + a + a , (8)where γ is defined as γ = ¯ h m R x References 1. Friedhelm Serwane (2011), Deterministic preparation of a tunable few-fermion system, PhD thesis,Ruprecht-Karls-University, Heidelberg, Germany.2. Friedhelm Serwane et al. (2011), Deterministic preparation of a tunable few-fermion system, Science Phys.Rev.Lett. Phys.Rev.Lett. Phys. Rev. J. Math. Phys. Phys. Rev. Phys. Rev. A Foundations of Physics J. Math. Phys.5