Bose Polarons at Finite Temperature and Strong Coupling

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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

Z 0.5 Thank you! 21

0.0 0.5 1.0 1.5

T/Tc