Polaron residue and spatial structure in a Fermi gas Christian Trefzger, Yvan Castin To cite this version: Christian Trefzger, Yvan Castin. Polaron residue and spatial structure in a Fermi gas. EPL - Euro- physics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2013, 101, pp.30006. hal-00747260v2 HAL Id: hal-00747260 https://hal.archives-ouvertes.fr/hal-00747260v2 Submitted on 9 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. epl draft Polaron residue and spatial structure in a Fermi gas Christian Trefzger and Yvan Castin Laboratoire Kastler Brossel, Ecole´ Normale Sup´erieure and CNRS, UPMC, 24 rue Lhomond, 75231 Paris, France PACS 03.75.Ss { Degenerate Fermi gases Abstract {We study the problem of a mobile impurity of mass M interacting via a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass m. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue Z, and the impurity-to-fermion pair correlation function G(x). We show that G(x) deviates from 4 unity at large distances as −(A4 + B4 cos 2kFx)=(kFx) , where kF is the Fermi momentum; since A4 > 0 and B4 > 0, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality 4 catastrophe; at distances between ∼ 1=kF and the asymptotic distance where the 1=x law applies, they reveal that G(x) exhibits an intriguing multiscale structure. Introduction. { The physics of atomic Fermi gases Model. { We consider in three dimensions an ideal has recently experienced a fast development, thanks to the gas of N same-spin-state fermions of mass m, enclosed in a Feshbach resonance technique that allows to tune the s- cubic quantization volume V with periodic boundary con- wave scattering length a of the interaction, and to obtain ditions. The gas is perturbed by an impurity, that is a dis- highly degenerate strongly interacting Fermi gases [1, 2]. tinguishable particle, of mass M. The impurity interacts The first realizations of spin polarized configurations [3,4] with each fermion resonantly on a s-wave broad or narrow asked for theoretical interpretation of the experimental Feshbach resonance, as described by a two-channel model results, and shortly afterwards it was proposed that, in [17{19], with a Hamiltonian H^ written in [11]: The parti- the strongly polarized case, the minority atoms dressed cles exist either in the form of fermions or impurity in the by the Fermi sea of the majority atoms form a normal open channel, or in the form of a tightly bound fermion- gas of quasiparticles called polarons [5, 6], which agrees to-impurity molecule in the closed channel. These two with the experimental phase diagram [3,7]. Strictly speak- forms are coherently interconverted by the interchannel ing, such polarons should be called \Fermi polarons", so coupling of amplitude Λ 1. We restrict to the zero-range as to distinguish from the traditional condensed-matter- (or infinite momentum cut-off) limit, so that the inter- physics polaron, i.e. an electron coupled to a bosonic bath action is characterized by the s-wave scattering length a of phonons [8]. and the non-negative Feshbach length R∗ [20]. In terms of The basic single-polaron properties, such as its binding the effective coupling constant g and of the interchannel energy to the Fermi sea and its effective mass, are now coupling Λ, one has well understood, both for broad [5, 6, 9, 10] and narrow 2π 2a π 4 [11{13] Feshbach resonances. Here we study the polaron ~ ~ g = and R∗ = 2 2 ; (1) ground state properties going beyond the energy search, µ Λ µ for both types of resonances. We study the quasiparticle where µ = mM=(m + M) is the reduced mass. A broad residue Z, already investigated experimentally in [14, 15] Feshbach resonance corresponds to R∗ = 0. and theoretically in [12, 16], and the intriguing issue of the density distribution that surrounds the impurity, as Polaronic ansatz. { Whereas the ground state of characterized by the pair correlation function: This gives the system presents two branches, a polaronic branch and access to the spatial extension of the polaron, of potential a dimeronic branch [9,11{13,16,21,22], we restrict to the important consequences on the properties of the polaronic 1We can neglect the open-channel fermionic interaction if its gas. (background) scattering length abg is ≈ the van der Waals length. p-1 Christian Trefzger and Yvan Castin M/m = 0.1505 M/m = 1 M/m = 6.6439 polaronic branch. The ground-state of the N fermions is 1 1 1 the usual Fermi sea (\FS"), of energy EFS(N). We de- 0.8 (a1) 0.8 (b1) 0.8 (c1) 0.6 0.6 0.6 = 0 termine the ground state of a single impurity interacting * Z 0.4 0.4 0.4 R F with the N fermions using the unexpectedly high-accuracy k 0.2 0.2 0.2 approximation proposed in [6] (and generalized to encom- 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 pass the narrow resonance case in [11,13]), that truncates 1 1 1 the Hilbert space to at most one pair of particle-hole ex- 0.8 (a2) 0.8 (b2) 0.8 (c2) 0.6 0.6 0.6 2 = 1 * citations of the Fermi sea . For a zero total momentum, Z 0.4 0.4 0.4 R F this corresponds to the ansatz 0.2 0.2 0.2 k 0 0 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -6 -4 -2 0 2 4 ^y X ^y X ^y y 1 1 1 j poli = φ d0 + φqbqu^q + φkqdq−ku^ku^q jFSi; 0.8 (a3) 0.8 (b3) 0.8 (c3) q 0.6 0.6 0.6 k;q = 10 * Z 0.4 0.4 0.4 R (2) F ^y y ^y 0.2 0.2 0.2 k where dk,u ^k and bk are the creation operators of an im- 0 0 0 -10 -5 0 5 10 -10 -5 0 5 10 -14 -12 -10 -8 -6 -4 purity, a fermion and a closed-channel molecule of wave 1/(k a) 1/(k a) 1/(k a) vector k. The prime above the summation symbol means F F F that the sum is restricted to q belonging to the Fermi sea Fig. 1: (Color online) Quasiparticle residue (6) for various of N fermions, and to k not belonging to that Fermi sea. mass ratios M=m (columns a,b,c) and various Feshbach lengths The successive terms in (2) correspond in that order to (rows 1,2,3). Black line (solid for Z > 1=2, dashed for the ones generated by repeated action of the Hamiltonian Z < 1=2): Numerical solution. Red line: Second order weakly H^ on the Λ = 0 polaronic ground state. One then has to attractive expansion (22), that diverges for R∗ > 0 when s of minimize the expectation value of H^ within the ansatz (2), Eq. (19) tends to unity, and for R∗ = 0 when 1=(kFa) ! 0. −1 Circles in the inset [(kFa) 2 [−3; 0];Z 2 [0:3; 1]] of (c2): with respect to the variational parameters φ, φq and φkq, with the constraint h j i = 1. Expressing φ in Experimental data of [15] for kFR∗ ' 0:9474. Vertical dot- pol pol kq ted lines [11]: Polaron-to-dimeron crossing point; on the left terms of φ and φ in terms of φ, as in [11], one is left with q q (resp. right) of this line, the ground state is polaronic (resp. a scalar implicit equation for the polaron energy counted dimeronic) [for (a1), there is a very thin zone to the right of with respect to EFS(N); in the thermodynamic limit: this line where the ground state is trimeronic [25]]. Z 0 d3q 1 ∆Epol ≡ Epol − EFS(N) = 3 ; (3) (2π) Dq function [9]. Within the polaronic ansatz (2), it was shown where the prime on the integral over q means that it is in [16] that simply Z = jφj2. We first write the amplitude restricted to the Fermi sea q < k , with the Fermi mo- F φq in terms of the denominator (4) as [11]: mentum kF related as usual to the mean density of the Fermi sea ρ = N=V by k = (6π2ρ)1=3. The function of A F φq = (5) the energy in the denominator of the integrand is Dq where A is a normalization factor. Then using Eq. (5) and 1 µk µ2R µ F ∗ the coupled equations for φ, φ and φ [11], we get the Dq = − 2 2 + 4 ∆Epol + "q q kq g π ~ π~ m following expression for the residue: Z 0 d3k0 1 2µ + 3 − 2 02 ; (4) Z 0 3 (2π) Eq−k0 + "k0 − "q − ∆Epol ~ k 2 1 d q 1 Z ≡ jφj = 1 + 2 3 2 2 2 2 2 Λ (2π) Dq where " = ~ k for the fermions, E = ~ k for the k 2m k 2M 0 3 3 2 −1 0 Z d kd q 1=D impurity, and the prime on the integral over k means + q : (6) 0 (2π)6 E + " − " − ∆E that it is restricted to k > kF.