PHYSICAL REVIEW A 98, 013626 (2018) Attractive Fermi polarons at nonzero temperatures with a finite impurity concentration Hui Hu, Brendan C. Mulkerin, Jia Wang, and Xia-Ji Liu Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Victoria 3122, Australia (Received 29 June 2018; published 25 July 2018) We theoretically investigate how quasiparticle properties of an attractive Fermi polaron are affected by nonzero temperature and finite impurity concentration in three dimensions and in free space. By applying both non- self-consistent and self-consistent many-body T -matrix theories, we calculate the polaron energy (including decay rate), effective mass, and residue, as functions of temperature and impurity concentration. The temperature and concentration dependencies are weak on the BCS side with a negative impurity-medium scattering length. Toward the strong attraction regime across the unitary limit, we find sizable dependencies. In particular, with increasing temperature the effective mass quickly approaches the bare mass and the residue is significantly enhanced. At temperature T ∼ 0.1TF ,whereTF is the Fermi temperature of the background Fermi sea, the residual polaron-polaron interaction seems to become attractive. This leads to a notable down-shift in the polaron energy. We show that, by taking into account the temperature and impurity concentration effects, the measured polaron energy in the first Fermi polaron experiment [Schirotzek et al., Phys.Rev.Lett.102, 230402 (2009)] could be better theoretically explained. DOI: 10.1103/PhysRevA.98.013626 I. INTRODUCTION Experimentally, the first experiment on attractive Fermi polarons was carried out by the Zwierlein group at Mas- Over the past two decades, ultracold atomic gases have pro- sachusetts Institute of Technology (MIT) in 2009 using 6Li vided an ideal platform to understand the intriguing quantum many-body systems [1]. The simplest example is probably a atoms [20]. The polaron energy and residue were determined moving impurity immersed in a background medium that is by using radio-frequency (rf) spectroscopy, in the vicinity of often well understood [2,3]. In this so-called polaron problem, the unitary limit and in the strong attraction BEC regime. The the interaction between impurity and medium can be tuned attractive polaron picture was used later to understand the arbitrarily by using Feshbach resonances [4]. The motion of radio-frequency spectrum of a quasi-two-dimensional Fermi the impurity is then addressed by low-energy excitations of gas [21]. The existence of repulsive Fermi polarons was 40 the background medium and its fundamental properties are experimentally confirmed in 2012 by immersing K impurity profoundly affected [2]. In the case of fermionic impurity with inaFermiseaof6Li atoms near a narrow Feshbach resonance finite density and concentration, the emergence of a Fermi [22]orbyusing40K atoms in two dimensions [23]. Most liquid behavior is anticipated [5,6]. recently, a careful analysis of the quasiparticle properties of Theoretically, it turns out that the polaron problem can repulsive Fermi polarons in 6Li systems was performed at the be well approximately solved by using a variational ansatz European Laboratory for Non-linear Spectroscopy (LENS), that includes only one-particle-hole excitation, as proposed by Florence [24]. The experimental observation of attractive and Chevy in 2006 in his seminal work [7]. For a noninteracting repulsive Bose polarons has also been reported [25,26]. single-component Fermi sea as the background medium, when In all these experiments, the experimental data were com- the impurity-medium scattering length a is tuned to the unitary pared with the theoretical predictions of a single impurity limit (a →∞), Chevy’s ansatz predicts a polaron energy at zero temperature [2]. The unavoidable nonzero temper- EP −0.607εF [7,8], where εF is the Fermi energy of the ature and finite impurity concentration in real experiments Fermi sea, which is very close to the exact diagrammatic are anticipated to give negligible corrections. However, those Monte Carlo (Diag-MC) result of EP =−0.615(1)εF [9–12]. corrections have never been carefully examined, except the This perfect agreement may result from a nice cancellation of idealized case of 1D Fermi polarons [27], where the exact the higher-order contributions, as checked by the next order solution is available. A possible reason is that the current calculation with the inclusion of two-particle-hole excitations polaron theory relies heavily on Chevy’s variational approach [13]. The simple variational ansatz was later used to discover [7], which is unfortunately difficult to handle the nonzero repulsive Fermi polarons in the metastable upper branch [14] temperature and finite impurity concentration [28]. and to describe Bose polarons [15]. At the level of including The purpose of the present work is to address the effects two-particle-hole excitations, Chevy’s ansatz has also been of nonzero temperature and finite impurity concentration for applied to the medium systems with bosonic degrees of attractive Fermi polarons in three dimensions and in free freedom, such as a Bose-Einstein condensate (BEC) [16]ora space, by using both non-self-consistent [8,29–33] and self- Bardeen-Cooper-Schrieffer (BCS) superfluid [17,18]. In those consistent T -matrix theories [34–37]. In the limit of zero cases, the interplay between polarons and the resulting Efimov temperature and a single impurity and in three dimensions, the trimer was explored [16–19]. non-self-consistent T -matrix theory is shown to be equivalent 2469-9926/2018/98(1)/013626(9) 013626-1 ©2018 American Physical Society HUI HU, BRENDAN C. MULKERIN, JIA WANG, AND XIA-JI LIU PHYSICAL REVIEW A 98, 013626 (2018) to Chevy’s variational approach [8,32]. At weak attractions by using the well-established many-body T -matrix theories on the BCS side of the impurity and medium scattering [38–40], which amount to summing up all the ladder-type resonance, we find that the effects of nonzero temperature and diagrams. In this approximation, the self-energy of the finite impurity concentration are indeed negligible, confirming impurity atom is given by [8], the previous anticipation. However, across the resonance and (0) toward the strong attraction regime, a nonzero temperature = kB T G↑ (q − k,iνn − iωm)(q,iνn ), (5) may significantly reduce the effective mass and enhance the q,iνn residue of attractive Fermi polarons. The polaron energy also shows a considerable temperature dependence. In particular, where the vertex function can be written through the Bethe- Salpeter equation as at the typical experimental temperature T ∼ 0.1TF , where TF is the Fermi temperature of the Fermi sea, the polaron- 1 polaron interaction may become attractive, leading to a sizable (q,iνn ) = , (6) U −1 + χ(q,iν ) downshift in the polaron energy, which may be experimentally n resolved. Indeed, by taking into account the temperature and and the pair propagator χ(q,iνn)is impurity concentration effects, we find that the measured (0) polaron energy in the first Fermi polaron experiment at MIT χ = kB T G↑ (q − k,iνn − iωm)G↓(k,iωm ). (7) [20] could be better theoretically understood. k,iωm Here, ν ≡ 2nπk T with integer n are the bosonic Matsubara II. MANY-BODY T-MATRIX THEORIES OF ATTRACTIVE n B frequencies. Eqs. (4)–(7) form a closed set of equations, FERMI POLARONS which have to be solved self-consistently. We refer to it as We consider a two-component three-dimensional Fermi gas the self-consistent T -matrix theory, or the “G0↑G↓” theory of mass m with a large spin polarization (i.e., n↑ = n n↓), according to the structure in the pair propagator. We note which is described by the model Hamiltonian [7,8], that in a strong-coupling theory, the self-consistency does not necessarily guarantee a better or more accurate theory [39,40]. = − † + − † H [(k μ)ck↑ck↑ (k μ↓)ck↓ck↓] For attractive Fermi polarons, it is actually more useful to take k a non-self-consistent treatment of the many-body T -matrix U † † theory, as suggested by the comparison with numerically + ck↑cq−k↓cq−k ↓ck ↑, (1) V exact Diag-MC simulations [9–11,13] or exact Bethe ansatz q,k,k solutions [27]. That is, we simply use a noninteracting impurity 2 2 where k ≡ h¯ k /(2m), μ and μ↓ are the chemical potentials Green function, of spin-up and spin-down atoms, respectively, and U<0 1 is the bare attractive interatomic interaction strength, to be G(0)(k,iω ) = , ↓ m − − (8) renormalized in terms of the s-wave scattering length a, iωm (k μ↓) according to in the pair propagator Eq. (7). At zero temperature and in the 1 m m single-impurity limit, this non-self-consistent T -matrix theory = − . (2) U 4πh¯ 2a h¯ 2k2 or the “G0↑G0↓” theory exactly recovers Chevy’s variational k approach [8]. In the large spin polarization limit, we treat the spin-down In this work, we explore both non-self-consistent and self- atoms as impurities and assume that, at the first order of the consistent T -matrix theories, as both of them are not justified impurity concentration x = n↓/n, the background medium in the strong-coupling limit [39,40]. There is no reason to of spin-up atoms is not affected by interaction. As a result, expect that the self-consistent theory works better than the (0) μ can be taken as the chemical potential μ (T ) εF = non-self-consistent approach, or vice versa. The applicability h¯ 2(6π 2n)2/3/(2m) of an ideal Fermi gas at low temperatures of the two theories should be benchmarked by numerically and the thermal Green’s function of spin-up atoms is exact calculations such as Diag-MC simulations or accurate experimental measurements.