Bose polarons at finite temperature and strong coupling Pietro Massignan Outline ✦ Quantum mixtures ✦ Impurities in an ideal Fermi sea (“Fermi polarons”) ✦ Impurities in a weakly-interacting Bose gas (“Bose polarons”) ✦ Zero-temperature physics ✦ New features appearing at non-zero temperature Collaborators Nils Guenther Maciej Lewenstein Georg M. Bruun Quantum Mixtures in CondMat 3He-4He ultracold gaseous mixtures: FF (BEC-BCS crossover) [Zwerger, Lecture Notes in Phys.] B+FF superfluids (coherent/damped dynamics) ENS-Paris BB mixtures (ultradilute quantum liquid droplets) Stuttgart, Innsbruck, Barcelona spinor gases, SU(N) invariant systems Kyoto, Florence, Munich, … quantum magnets, quantum Hall systems, spin-liquids quark-gluon plasma neutron stars Very different microscopically, but ∃ common emergent and universal many-body descriptions. 4 Universality in Quantum Mixtures 20 orders of magnitude difference in temperature Coulomb plasma but similar transport properties! e.g., shear viscosity/entropy density: [Adams et al., NJP 2012] 5 Many-body systems (from Richard Mattuck’s book) 6 Quasi-Particles No chance of studying real particles Landau: of importance are the collective excitations, which generally behave as quasi-particles! a QP is a “free” particle with: @ q. numbers (charge, spin, ...) @ renormalized mass @ chemical potential @ shielded interactions @ lifetime 7 Ultracold atoms chemical composition interaction strength temperature periodic potentials physical dimension atom-light coupling exotic interactions (x-wave, spin-orbit) dynamics disorder periodic driving (shaken optical lattices) 8 Imbalanced Fermi gases Two-component Fermi gas with N↑≫N↓: a strongly-interacting system, or an ensemble of weakly-interacting quasi-particles (a Fermi liquid) 3 m N 5/3 E = ✏F N 1+ # + N Ep + ..., 5 " m⇤ N # " ✓ " ◆ # kinetic energy chemical potential (energy) kinetic energy of one polaron of the Fermi sea of the polarons (m* is their effective mass) 9 Spectrum of Fermi polarons low power RF: high power RF: repulsive pol. high power is needed to couple to the MH continuum, attractive pol. due to a small FC overlap molecule+hole continuum experiments: MIT (2009), ENS-Paris (2009), Innsbruck (2012), Cambridge (2012), Innsbruck (2016), LENS (2017) theory: Chevy, Recati, Combescot, Zwerger, Enss, Schmidt, Bruun, Pethick, Zhai, Levinsen, Parish, Castin, … 10 review: PM, Zaccanti and Bruun, Rep. Prog. Phys. (2014) Repulsive polarons • most general case (equal masses, broad resonance) • ∃ meta-stable quasiparticle at E>0 • long-lived, even close to unitarity • measures of E, meff and Z in very good agreement with simple theory • polaron-polaron interactions negligible [Scazza, Valtolina, PM, Recati et al., PRL 2017] 11 Impurities in a Bose gas weak RF pulse + quick decoherence: a few |2> impurities in a bath of |1> atoms JILA: Hu, …, Cornelll and Jin, PRL (2016) Aarhus: Jørgensen, …, Bruun and Arlt, PRL (2016) T=0 theory: Rath, Schmidt, Das Sarma, Bruun, Levinsen, Parish, Giorgini, … 12 Fermi vs. Bose - weakly-interacting Bose gas non-interacting Fermi sea (kNaB≪1) smooth crossover from Temperature BEC phase transition at Tc degenerate to classical Impurity ground state polaron/molecule transition smooth crossover Three-body physics negligible important role Stability rather stable mixture rapid three-body losses 13 Definition of the problem • Bath treated with Bogoliubov theory 2 1/3 kn =(6⇡ n) 2/3 units: 2 2⇡ n En = kn/2mB » critical temperature: Tc = 0.436En m ⇣( 3 ) ⇡ B ✓ 2 ◆ 3/2 » condensate density: n0 = n[1 (T/Tc) ] − » bath chemical potential: µ = n B TB 0 » bath vacuum scattering matrix: =4⇡a /m TB B B B B » dispersion of the excitations: Ek = ✏k (✏k +2µB) B 2 q » free bosons: ✏k = k /2mB • Impurity-bath coupling: finite temperature Green’s functions (non-perturbative!) • Polaron energy: !p = ✏p +Re[⌃(p, !p)] 1 • Polaron residue: Z = p 1 @ Re[⌃(p, !)] − ! |!p 14 Diagrammatic scheme 1 Impurity Green’s function: (p,i! )= G j (p,i! ) 1 ⌃(p,i! ) G0 j − − j = + BEC boson T T T>0: important diagram excited boson 1 1 impurity Ladder T-matrix: (p,i! )− = − ⇧(p,i! ) missing in ladder approx: T j Tv − j ˜ = + ˜ + ˜ T T T 1 1 Extended T-matrix: ˜(p,i! )− = − ⇧(p,i! ) n (p,i! ) T j Tv − j − 0G0 j ⌃ = + ˜ = + + T T T T TT ⌃0 ⌃1 ⌃L Extended| self-energy:{z } | ⌃{z=}⌃0 +| ⌃1 {z } = + (impurity dressed by BEC only) T T=0 ladder: Rath and Schmidt, PRA 2013 Perturbation theory at T>0: Levinsen, Parish, Christensen, Arlt, and Bruun, arXiv:1708.09172 Extended T>0 diagrammatic scheme: Guenther, PM, Lewenstein and Bruun, arXiv:1708.08861 15 Varying coupling strength Aarhus: knaB =0.01 0.0 ● ● ● ● T=0 ● ● -0.2 ● ● T=0.5Tc ● -0.4 ● n E ω↑ T=Tc / -0.6 ● ω ωL -0.8 ● ω↓ dots: T=0 ladder theory by -1.0 Rath and Schmidt, PRA 2013 -1.2 1 1 0 1 2 3 4 Z 0.5 0 -1 0 1 2 3 4 -1/kna Guenther, PM, Lewenstein, and Bruun, arXiv:1708.08861 16 Varying temperature Weak attraction 0.0 -0.2 (k a = 1) ω↑ n -0.1 -0.4 − ω↓ -0.6 0 30 -0.2 ωL n -0.2 Energy: E / -0.4 ω -0.3 -0.6 0 30 -0.4 -0.5 1 Residue: Z 0.5 0.0 0.5 1.0 1.5 T/Tc Guenther, PM, Lewenstein, and Bruun, arXiv:1708.08861 17 Varying temperature Strong attraction 0.0 -0.4 (unitarity) ω↑ -0.2 -0.8 ω↓ -1.2 -0.4 0 30 ω- -0.4 Energy: n E / -0.6 -0.8 ω -1.2 -0.8 0 30 -1.0 1 Residue: Z 0.5 0.0 0.5 1.0 1.5 T/Tc Guenther, PM, Lewenstein, and Bruun, arXiv:1708.08861 18 Understanding fragmentation !0 =Re[⌃(p =0, !0)] Weak coupling, -0.02 knaB 0.10 -0.03 low temperature behavior of Re(Σ): 0.08 n E / -0.04 r 0.06 Σ (kna = 0.1) 0.04 − -0.05 dashed: T =0.05Tc 0.02 solid: T =0.1Tc -0.06 0 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 ω/En 3 d k fk ⌃1(!) ⇡ (2⇡)3 1 n0 v− ⇧(k, ! + Ek) !+E ✏ on-shell for: ! + Ek = ✏k + ⌃0(k, ! + Ek) Z T − − k− k ! + n 0TB n ⇡! n ( ) exTv − 0 Tv − TB non-condensed fraction a a : equal splitting a a : single polaron | | & B | | . B 1/2 ! , !0[1 (Z0nex/n0) ] ! n v " # ' ± " ' T Z , ZL/2 " # ' in accord with perturbation theory [Levinsen et al., arXiv:1708.09172] 19 Repulsive polarons Weak repulsion 20 (kna = 1) T=0 T=0.1TC 15 T=0.4TC n E ) T=1TC Spectral function: 10 0, = T=1.1TC k ( A T=2TC 5 T=10TC 0 0.0 0.2 0.4 0.6 0.8 1.0 /En Guenther, PM, Lewenstein, and Bruun, arXiv:1708.08861 20 Conclusions • Bose polarons greatly differ from Fermi ones • No polaron/molecule transition • Fundamental role played by the BEC, and the associated large low- energy density of states • Non-perturbative treatment is crucial • The T=0 attractive polaron fragments into two quasiparticles at T>0 • The upper of the two negative energy excitations disappears at Tc • The ground state quasiparticle remains well-defined across Tc 0.0 -0.2 ω↑ -0.1 -0.4 ω↓ -0.6 0 30 -0.2 ωL n -0.2 E / -0.4 ω -0.3 -0.6 0 30 -0.4 -0.5 1 Z 0.5 Thank you! 21 0.0 0.5 1.0 1.5 T/Tc.