Supplemental Material: “Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms”

Andr´eSchirotzek, Cheng-Hsun Wu, Ariel Sommer, and Martin W. Zwierlein Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: May 1, 2009) In this supplemental material we state, starting with the variational Ansatz by Chevy, key prop- erties of the polaron, such as the energy E↓ and the quasiparticle residue Z, and we calculate its RF spectrum using Fermi’s Golden Rule. We connect this approach and its implication for finite impu- rity concentration to the results of Fermi liquid theory and the T-matrix formalism used in [1–4]. Furthermore, details are provided about the extraction of the quasiparticle residue Z.

X Polaron wavefunction, energy and quasiparticle 1 E↓ = f(E↓, q) (5) residue V q

We start with the hamiltonian for a dilute two compo- These all depend on the function f(E, q) with nent mixture of fermionic atoms interacting via the van- 1 1 X 1 der-Waals potential V (r) [5]. Thanks to the diluteness of f −1(E, q) = + (6) g0 V ²k − ²q + ²q−k − E the system, the potential is of short range R compared k>kF to the interparticle distance 1/kF , so kF R ¿ 1. Its It is a measure of the interaction strength between spin Fourier transform V (k) is thus essentially constant, g0, up and spin down, modified by the presence of the spin up below kF and rolls off to zero at a momentum on the or- Fermi sea. As usual, g can be replaced by the physically der of 1/R À kF . The many-body Hamiltonian for the 0 system is then observable scattering length a for collisions between spin 1 m 1 P 1 up and down via [5] g = 4π~2a − V k 2² . X X 0 k † g0 † † ˆ q q µ ¶ H = ²kckσckσ + c q c q ck0+ ↓c−k0+ ↑ k+ 2 ↑ −k+ 2 ↓ 2 2 mkF π V 0 −1 k,σ k,k ,q f (E, q) = 2 2 − 1 + (7) (1) 2π ~ 2kF a X µ ¶ Here, the label σ denotes the spin state ↑,↓, ²k = 1 1 1 2 2 † − ~ k /2m, V is the volume of the system and the c ,ckσ V ²k − ²q + ²q−k − E 2²k kσ k>kF are the usual creation and annihilation operators for fermions with momentum k and spin σ. The trial wave- The integral in above expression is convergent and gives function for the Fermi polaron with zero momentum pro- ½ µ ¶¾ posed by F. Chevy in [6] is −1 mkF π E q f (E, q) = 2 2 − 1 + I , (8) 2π ~ 2kF a EF kF Z ∞ µ µ 2 ¶ ¶ k>kXF x 2x + 2xy − ² † I(², y) = dx ln − 1 |Ψi = ϕ0 |0i |FSi + ϕkqc cq↑ |q − ki |FSi 2 ↓ ↑ k↑ ↓ ↑ 1 2y 2x − 2xy − ² qkF 2 1 2 norm hΨ|Ψi = |ϕ0| + |ϕkq| = 1. That is, E↓ y qkXF 2 in this approximation. The expression for the quasipar- |ϕkq| δ(~ω + E↓ − ²k + ²q − ²q−k) ticle residue Z of a single spin down impurity in Eq. 4 qgases This structure of the RF spectrum is a generic fea- in [1–4], among others. Chevy’s wavefunction offers a ture of quasiparticle spectra. In Fermi liquid theory, the simple way of calculating the RF spectrum of a single propagator of a quasiparticle is approximated as a pole impurity. P at energy E(k) > 0 (relative to the ground state energy), ˆ † The RF operator V = ~ΩR k ck,f ck,↓ +h.c. promotes lifetime 1/γ(k) and residue Z plus an incoherent spec- the impurity into the free final state |fi (energy Ef ) with- trum [3, 8] out momentum transfer [5]. In the experiment, the final internal state is the second lowest hyperfine state of 6Li. Z GR(k, ω) = + GR,inc(k, ω) (13) Fermi’s Golden Rule for the impurity starting in state − ~ω + E(k) + i~γ(k) − |Ψi is The spectral function is given by A−(k, ω) = ¯D E¯ − 1 ImGR(k, ω) = Z 1 ~γ(k) + Ainc(k, ω) 2π X ¯ ¯2 π − π (~ω+E(k))2+~2γ(k)2 − Γ(ω) = ¯ f|Vˆ |Ψ ¯ δ(~ω − (E − E )) (11) ~ f ↓ which tends to f inc A−(k, ω) = Zδ(~ω + E(k)) + A− (k, ω) (14) Where ω is the RF offset from the bare atomic transition frequency between the internal states labeled by ↓ and in the limit of small damping of the quasiparticle. A−(k, ω) measures the probability that removing a par- f. One possible final state is |0i ≡ |0if |FSi↑, i.e. a zero momentum particle in the final state plus a perfect ticle with momentum k will cost an energy −~ω. The RF spectrum in linear response is given by [2] Fermi sea of up spins, with energy E|0i = 0 relative to the X Fermi energy EF of the environment. Other possible final 2 † Γ(ω) = 2π~ΩR A−(k, ²k − µ − ~ω)nF (²k − µ − ~ω) states are |q, ki ≡ |q − ki c cq↑ |FSi with q < kF f k↑ ↑ k and k > kF , i.e. a particle with momentum q − k in (15) the final state and a Fermi sea with a hole at q and where µ is the chemical potential of the quasiparticle and an excited environment particle above the Fermi sea at βx nF (x) = 1/(e + 1) is the Fermi function that tends to k. The energy of these states is E|q,ki = ²k − ²q + ²q−k θ(−x) at zero temperature. This is intuitively under- relative to the environment Fermi energy EF . The matrix stood: For a given momentum k, the RF photon with elements are energy ~ω has to provide the energy ²k − µ (relative to D ¯ ¯ E the initial chemical potential µ) to create a free parti- ¯ ˆ ¯ 0 ¯V ¯ Ψ = ~ΩR ϕ0 cle in the final state. The rest, ~ω − ²k + µ, is used D ¯ ¯ E to remove a particle from the initial state (probability ¯ ˆ ¯ q, k ¯V ¯ Ψ = ~ΩR ϕkq A−(k, ²k − µ − ~ω)) if there exists such a particle (factor 3 nF (²k − µ − ~ω)). Eq. 15 is equivalent to Fermi’s Golden Calculation of the incoherent background Rule Eq. 11 [9]. In the case where the spectral func- tion is dominated by a quasiparticle peak, the spectrum Using Eq. 3, we can write the incoherent part of the becomes spectrum as: X Γ(ω) = 2π~Ω2 Z δ(² + E(k) − µ − ~ω) k>kXF R k inc 2 2 k Γ (ω) ≡ 2π~ΩR |ϕkq| δ(~ω + E↓ − ²k + ²q − ²q−k) inc qkXF = R f(E , q)2δ(~ω + E − ² + ² − ² ) Connecting to our case of a single quasiparticle with µ = ~ ω2 V2 ↓ ↓ k q q−k qkF mkF ~ω+E↓ q 2 2 K( , ) 8π ~ 2EF kF y2 −y2  + − for y > 1 , Polaron spectral function  y − y2 −1 with K(², y) = +  y for y− < 1 < y+ ,  To connect the single particle and the Fermi liquid de- 0 for 1 > y+ . scription, we calculate the propagator GR(k, ω) for the q − y y2 removal of a single spin down impurity from the wave- and y± = ± 2 + 4 + ² (20) function |Ψi. By definition, The incoherent spectrum is then R † iHt/ˆ ~ ³ ´ iG (k, t) = hΨ| c e c |Ψi θ(t) (18) ~ω+E↓ − k↓ k↓ Z 1 y2K , y inc 2 ZEF 2EF Γ (ω) = πΩR dy ³ ³ ´´2(21) ~ω2 E Inserting a complete set of eigenstates, this gives 0 1 − π − I ↓ , y 2kF a EF X R 2 iEf t/~ iG−(k, t) = |hf| ck↓ |Ψi| e θ(t) (19) One can check that the total spectrum obeys the sum f rule Z ∞ The state ck↓ |Ψi is void of any spin down impurity and 2 dωΓ(ω) = 2π~ΩR (22) has non-vanishing matrix elements only with either the −∞ unperturbed spin up Fermi sea, |FSi (if k = 0), or with ↑ and in particular that the total weight of the incoher- particle-hole excitations |q, k0i = c† c |FSi (in the k0↑ q↑ ↑ √ ent background is proportional to 1 − Z, which is not 0 case k = q − k ). These matrix elements are ϕ0 = Z obvious from the form in Eq. 21. For RF frequencies and ϕk0q resp., the corresponding energies Ef = 0 and close to threshold ~ω + E↓ ¿ 2EF , the hole momen- Ef = ²k0 − ²q relative to EF . So one has: tum q and the particle momentum k must be close to each other to fulfill energy conservation, i.e. they have 0 kX>kF to be close to the Fermi momentum. The double sum R 2 i(²k0 −²q)t/~ iG−(k, t) = (Zδk,0+ δk,q−k0 |ϕk0q| e )θ(t) over q and k thus gives a phase space suppression on 2 q 0. This is just the Fermi liquid form density of states√ above threshold gives a spectrum pro- R 2 of G− but for a single quasiparticle with zero momentum portional to ~ω + EB/ω . For large RF energies, large (E(0) = 0 in (13)), as described by |Ψi. The spectral particle momenta k are involved, the suppression due to function is the Fermi seap becomes negligible and the spectrum be- 2 haves like ~ω + E↓/ω , as for a molecule of binding A−(k, ω) = Zδ(~ω)δk,0 + energy E↓. This is natural as for large momenta, we are 0 probing short-range physics which involves at most two kX>kF 2 δk,q−k0 |ϕk0q| δ(~ω + ²k0 − ²q) particles, a spin up environment atom and the impurity.

q

RF Spectrum of a finite concentration of impurities the environment around zero RF offset (the resonance for

Since polarons are found to be weakly interacting, they will form a Fermi sea filled up to the impurity Fermi mo- mentum kF ↓. The fact that the dispersion of polarons m E(k) = m∗ ²k differs from that of a free particle due to the effective mass m∗ 6= m leads to broadening of the RF spectra. The RF photon has to supply the difference in m kinetic energies (1 − m∗ )²k between the initial and the m 2 2 final state, with a maximal shift (1− m∗ )~ kF ↓/2m. The

spectral shape is easily obtained: The spectral function atom transfer / a.u. at momentum k will be dominated by polarons that oc- cupy that momentum state. The coherent part of the coh spectral function is thus A− (k, ω) = Zδ(~ω + E(k)) with E(k) = −~2k2/2m∗ = − m ² relative to the im- m∗ k 0 1 2 3 4 5 purity Fermi energy. The coherent part of the spectrum rf offset / ε then becomes F X Γcoh(ω) = 2π~Ω2 Acoh(k, ² − E − ~ω) FIG. 1: Determination of the quasiparticle residue Z. Impu- R − k ↓ rity spectrum (red), environment spectrum (blue) and spec- k tral response of non-interacting atoms (black dashed), folded where the sum extends up to the impurity Fermi momen- over from negative RF offsets. tum kF ↓. With the free, 3D density of states ρ(²), this is Z non-interacting atoms) adds some weight to the environ- EF ↓ coh 2 m ment background at the position of the polaron peak. To Γ (ω) = 2π~ΩR d² ρ(²)Zδ(² − E↓ − ~ω − ∗ ²) 0 m remove this effect in the determination of Z, the part of µ ¶ the environment’s response at negative frequency offset 2 Z ~ω + E↓ = 2π~Ω ρ × is folded towards the positive side (dashed line in Fig. 1) R 1 − m 1 − m m∗ m∗ and subtracted from the environment spectrum. As it ³ m ´ θ (1 − )E − ~ω − E (23) turns out, this procedure changes the value for Z by less m∗ F ↓ ↓ than 5% for all spectra in Fig. 2 of the main paper. This coherent part of the spectrum starts at the polaron ground state energy ~ω = |E↓|, then grows like a square m root and jumps to zero when ~ω − |E↓| = (1 − m∗ )EF ↓. ∗ On resonance, where m ≈ 1.2, this occurs at ~ω−|E↓| = [1] M. Punk and W. Zwerger, Phys. Rev. Lett. 99, 170404 0.2x2/3E ≈ 0.04E for x = 0.1. This is still smaller F ↑ F ↑ (2007). than the Fourier width of the RF pulse used in the exper- [2] P. Massignan, G. M. Bruun, and H. T. C. Stoof, Phys. iment of about 0.1EF . The size of the jump is given by Rev. A 78, 031602 (2008). 2 Z 2π~ΩR m ρ(EF ↓) and reflects the impurity Fermi sur- [3] M. Veillette, E. G. Moon, A. Lamacraft, L. Radzihovsky, 1− m∗ face in the RF spectrum. This behavior of the coherent S. Sachdev, and D. E. Sheehy, Phys. Rev. A 78, 033614 part of the spectrum was found in [1] and was discussed (2008). [4] W. Schneider, V. B. Shenoy, and M. Randeria, preprint recently in [4]. It is intriguing that the sharpness of the arXiv:0903.3006 (2009). Fermi surface and its discontinuity should, at least in [5] W. Ketterle and M. Zwierlein, in Ultracold Fermi Gases, principle, be observable in the RF spectrum. Proceedings of the International School of Physics ”En- rico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006, edited by M. Inguscio, W. Ketterle, and C. Salomon (IOS Determination of Z from experimental spectra Press, Amsterdam, 2008). [6] F. Chevy, Phys. Rev. A 74, 063628 (2006). In order to extract the quasiparticle residue Z, we de- [7] R. Combescot, A. Recati, C. Lobo, and F. Chevy, Phys. Rev. Lett. 98, 180402 (2007). termine the area under the peak that is not matched by [8] P. Nozi`eres, Theory of Interacting Fermi Systems, Ad- the environment’s response and divide by the total area vanced Book Classics (Addison-Wesley, Reading, MA, under the impurity spectrum (see spectrum in the inset 1997). of Fig. 4 in the main body of the paper). Due to the [9] A. Fetter and J. Walecka, Quantum Theory of Many- Fourier width of the probe pulse, the strong response of Particle Systems (McGraw-Hill, New York, 1971).