Observation of a Smooth Polaron–Molecule Transition in a Degenerate Fermi Gas

Observation of a smooth polaron–molecule transition in a degenerate Fermi gas

Gal Ness,1 Constantine Shkedrov,1 Yanay Florshaim,1 2, 3 2, 3 2, 3 1, Oriana K. Diessel, Jonas von Milczewski, Richard Schmidt, and Yoav Sagi ∗ 1Physics Department, Technion – Israel Institute of Technology, Haifa 32000, Israel 2Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Strasse. 1, 85748 Garching, Germany 3Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München,Germany (Dated: September 29, 2020) Understanding the behavior of an impurity strongly interacting with a Fermi sea is a long-standing challenge in many-body physics. When the interactions are short-ranged, two vastly different ground states exist: a polaron quasiparticle and a molecule dressed by the majority atoms. In the single-impurity limit, it is predicted that at a critical interaction strength, a first-order transition occurs between these two states. Experiments, however, are always conducted in the finite temperature and impurity density regime. The fate of the polaron-to-molecule transition under these conditions, where the statistics of quantum impurities and thermal effects become relevant, is still unknown. Here, we address this question experimentally and theoretically. Our experiments are performed with a spin-imbalanced ultracold Fermi gas with tunable interactions. Utilizing a novel Raman spectroscopy combined with a high-sensitivity fluorescence detection technique, we isolate the quasiparticle contribution and extract the polaron energy, spectral weight, and the contact parameter. As the interaction strength is increased, we observe a continuous variation of all observables, in particular a smooth reduction of the quasiparticle weight as it goes to zero beyond the transition point. Our observation is in good agreement with a theoretical model where polaron and molecule quasiparticle states are thermally occupied according to their quantum statistics. At the experimental conditions, polaron states are hence populated even at interactions where the molecule is the ground state and vice versa. The emerging physical picture is thus that of a smooth transition between polarons and molecules and a coexistence of both in the region around the expected transition. Our findings establish Raman spectroscopy as a powerful experimental tool for probing the physics of mobile quantum impurities and shed new light on the competition between emerging fermionic and bosonic quasiparticles in non-Fermi-liquid phases.

I. INTRODUCTION complete loss of quasiparticle behavior [18]. In contrast, for mobile impurities, the energy cost related to the im- In order to understand the motion of an electron purity recoil stabilizes the formation of Fermi polarons through an ionic lattice, Landau suggested treating the with well-defined quasiparticle properties [19]. electron and the phonons that accompany its movement One of the simplest scenarios in which Fermi polarons as a new quasiparticle named ‘polaron’ [1]. The con- naturally emerge is in ultracold gases, where a small num- cept of the polaron was later found to be applicable in ber of (mobile) spin impurities can be immersed in a sys- many other systems, including semiconductors [2], high- tem of free fermions of the opposite spin. Such ultracold, temperature superconductors [3], alkali halide insulators spin-imbalanced systems are ideally suited to explore po- [4], and transition metal oxides [5]. In such systems, polaron physics [14, 20, 21] owing to their extremely long larons appear as weakly- or strongly-coupled quasiparti- spin-relaxation times and tunability of the s-wave scat- cles that are classified as large or small, depending on tering length, a, between the impurity and the majority the size of the distortion they generate in the underlying atoms via Feshbach resonances [22]. crystalline structure of the material [6]. Understanding the properties of polarons coupled to a bosonic bath is Initial experiments with spin-imbalanced Fermi gases still an ongoing effort in areas ranging from solid-state in harmonic confinement revealed phase separation into three regions at unitary interactions (a ). It was physics [7, 8] and ultracold atoms [9–11], to quantum → ∞ chemistry [12]. observed that phases arrange according to the varying The concept of polarons becomes also a powerful tool local density, with an inner core of a spin-balanced su- for our understanding of the properties of quantum im- perfluid being separated from a second shell of a partially purities interacting with a fermionic environment. In this polarized normal gas, and a third shell of a fully polar- 3 ized gas [23–25]. It was Chevy who first pointed out arXiv:2001.10450v3 [cond-mat.quant-gas] 27 Sep 2020 context, applications range from ions in liquid He [13], and mixtures of cold atomic gases [14] to excitons in- [26] that the radius between the outer and intermediate teracting with electrons in atomically thin semiconduc- shells is related to the solution of the Fermi polaron prob- tors [15–17]. Strikingly, in fermionic systems an infinite lem [27]. For weak attractive interactions (a < 0), the number of low-energy excitations leads to the Anderson ground state is a long-lived quasiparticle dressed by the orthogonality catastrophe for immobile impurities and a majority particles, forming the attractive polaron. Be- yond the Feshbach resonance, at a > 0, it was found that a metastable polaronic state also exists energetically far up in the excitation spectrum [28–35]. This so-called re- ∗ Electronic address: [email protected] pulsive polaron becomes, however, progressively unstable 2 towards unitary interactions. is still open. Fermi polarons have well-defined momenta with a nar- In this work, we address this question both theoret- row dispersion relation that is described by a renormal- ically and experimentally. Our experiments are per- ized effective mass [29, 36]. The attractive polaron performed with a spin-imbalanced, ultracold Fermi gas in sists as the ground state even as the interactions increase the BEC-BCS crossover regime [22, 46]. To gain detailed towards unitarity. However, for still stronger interac- insight into the behavior of the quasiparticles, we employ tions, the system favors a molecular ground state dressed a novel spectroscopic method based on a two-photon Ra- by the majority fermions [27, 37, 38] (see Fig. 3(a) be- man transition. Raman spectroscopy allows us to clearly low). It was predicted that the energies of the polaron identify the coherent response of the polarons, and deter- 1 and molecular states cross around (k a)− 0.9 —with mine some of its key properties, including its energy and F c ≈ kF being the Fermi wave vector of the majority— leading spectral weight. We compare our results with a theoret- to a sharp, first-order transition between the two ground ical model that takes into account the thermal occupa- states [27, 29, 37–40]. Contrasting claims for a smooth tion of polarons and molecules at finite momenta. Both crossover were also put forward [41–44]. our theoretical model and the measurements consistently The Fermi polaron problem represents the limiting show that a finite impurity concentration and tempera- case of a spin-imbalanced Fermi gas. Therefore, the nature have a striking effect on the transition: they smooth ture of the polaron-to-molecule transition has profound it and lead to a regime where polarons and molecules theoretical implications for the phase diagram of the coexist. spin-imbalanced BEC-BCS crossover [38, 45–48]. While After describing the experimental setup in Section II, at zero temperature, the polaron-to-molecule transition we briefly introduce our theoretical model in Section III was predicted to be pre-empted by phase separation be- and present the calculated Raman spectra. Based on tween the superfluid and the normal phases [49], at finite this, in Section IV we develop a fitting routine for the temperature, increased thermal fluctuations are expected experimental spectra which allows us to extract physical to suppress the superfluid and restore the polaron-to- quantities, such as the polaron energy, the quasiparticle molecule transition. Experimentally, the Fermi polaron spectral weight, and the contact parameter. The experi- was initially studied by radio-frequency (rf) spectroscopy, mental results are presented and discussed in Section V, where the attractive polaron was identified by a narrow while in Section VI we give a detailed theoretical deriva- peak appearing exclusively in the minority spectrum [50]. tion of our model. In particular, we show how the quanti- The spectral weight of this peak was interpreted as the ties accessible in the experiment are computed. Finally, quasiparticle residue (or weight) Z, which quantifies how in Section VII, we summarize our results, discuss their similar the polaron remains to the non-interacting impu- implications for the many-body physics of cold Fermi rity particle. Accordingly, it is determined by the overlap gases and outline directions for future work. between the polaron wavefunction and its non-interacting impurity state. When Z is zero, the quasiparticle description is no longer valid. In the experiment, Z was observed II. THE EXPERIMENTS to continuously decrease and vanish above a certain interaction strength in contrast to theoretical predictions The experiments are performed with a harmonically- based on the Chevy Ansatz wavefunction [38]. trapped ultracold gas of 40K atoms. The system is ini- A different approach to measure Z was employed in tially prepared in an incoherent mixture of the two low- Ref. [31]. Here, Z was determined from coherent oscilla- est Zeeman states denoted by 1 and 2 (see Fig. 1), tions between the polaron and a non-interacting impurity with the majority of atoms being| i in state| i1 . The cool- state. In this approach, the coherent oscillations address ing sequence is similar to the one described| i in Ref. [56], the polaronic state even when the attractive polaron is here modified to produce a spin-polarized ensemble of an excited state above the molecular ground state. This 100, 000 atoms in the state 1 at a temperature of ∼ | i made it possible to measure the weight Z across the full T 0.2 TF , where TF is the Fermi temperature. This polaron-to-molecule transition, and it was found that Z is achieved≈ by terminating the evaporation cooling at a 1 indeed does not vanish beyond (kF a)c− . Further proper- magnetic field of 201.75G, below the Feshbach resonance ties of attractive and repulsive Fermi polarons were also (B0 202.14G [56]), where three-body processes remove determined, including their effective mass [51, 52], en- all the≈ atoms in state 2 . The magnetic field is then ergy [31, 32, 50, 52], thermodynamics [53], equation of ramped adiabatically to| i the BCS side of the Feshbach state [54], and formation dynamics [55]. However, de- resonance (204.5G), where the interaction between the spite these tremendous efforts, the fate of the polaron- states 1 and 2 , parametrized by the s-wave scattering to-molecule transition at realistic conditions, namely fi- length|ai, is weak.| i To introduce the impurities, we use nite temperature and impurity concentration, remains a short (few microseconds) rf pulse that transfers a very unknown. Importantly, the key question whether the small fraction of the atoms from state 1 to 2 . This first-order polaron-to-molecule transition prevails at fi- is followed by a hold time of 100ms during| i which| i the nite impurity density, and separates sharply a phase of two states fully decohere. Finally, the magnetic field is polarons from a gas of dressed molecular quasiparticles, ramped adiabatically to its final value where we wait an- 3

2 (a) 4 P3/2 not observe any signiﬁcant diﬀerence between the two ∆ 40K datasets, in what follows we shall present their results together.

D2 line Raman 1 Raman 2 The main innovation in our experiments is the use of + 766.7nm ω1, σ ω2, π Raman spectroscopy. In conventional rf spectroscopy, the photon momentum is negligible, and the atomic momentum is essentially unchanged in the transition. As 2 4 S1/2 a result, the transition probability depends only weakly 3 ω 9 2 | i on the atom velocity and the maximal signal is attained F = 2 5 1 | i mF = for atoms that are not at rest. In particular, at finite | i m = 7 − 2 9 F − 2 impurity density, the measured peak depends on tem- mF = 2 − perature [11] (see Appendix B). In contrast, in a Raman zˆ process, the momentum change is significant compared to (b) the atomic momentum, and consequently, the transition ˆ rate depends on the atomic velocity. As we show fur- ~ yˆ ω2, k2 ther below, the Raman spectrum reflects the projection B~ xˆ of the polaron momentum distribution along the two-

7◦

1 (k a)− = 0.66, 0.06, 0.75 F − − 7◦ 0.6 ) 2 2 ˆ / ω 3 ( 0.5 ω ) ω1, ~k1 P 1 45◦ ω (

0.4 P 0 0 10 20 30 FIG. 1. Experimental setup. (a) Level diagram of the relevant states in 40K. The majority of atoms occupy state 0.3 2 1 = F = 9/2, mF = 9/2 in the 4 S1/2 manifold, and their| i interaction| with the− minorityi atoms in state 2 = 0.2 | i F = 9/2, mF = 7/2 can be tuned in the vicinity of the |Feshbach resonance− ati B 202.14G [56]. The counter- 0.1 propagating Raman beams≈ (wiggly blue lines) are pulsed for 500µs and couple atoms in state 2 to state 3 = Transition probability, 0 | i | i F = 9/2, mF = 5/2 , which is initially unoccupied. After- | − i 0 5 10 15 20 wards, we detect the number of atoms in state 3 [56]. The Raman two-photon detuning, ω (εF /¯h) single-photon Raman detuning is ∆ = 2π |54i.78(8)GHz − × relative to the D2 transition. (b) 3D sketch of the beam configuration in the experiment. Two Raman beams with FIG. 2. Raman spectra of an imbalanced Fermi gas in orthogonal polarization (blue lines with arrowheads) overlap the BEC-BCS crossover. The Raman transition probes the atomic cloud (yellow), which is being held in an elon- the minority atoms, which represent around 4% of the whole gated crossed-beams optical dipole trap (red lines). The op- cloud. The data is fit with the function given in Eq. (6) (solid lines). The dark shaded area under the graph is tical trap oscillation frequencies are ωradial = 2π 238(3)Hz × the combined contribution from molecules and the incoher- and ωaxial = 2π 27(2)Hz, in the radial and axial directions, respectively. The× gravitational acceleration direction is zˆ. ent part of the polaron wavefunction, Pbg (ω; Tbg,Eb), while − the light shaded area is the coherent polaron contribution, 0 0 Pcoh ω; Tp, pol, m∗ . We extract the polaron energy, pol, from the peak position (dotted vertical lines), shifted by the other 3.3ms before pulsing the Raman beams for 500µs. recoil energy due to the finite Raman momentum transfer of q¯ 1.9 kF . The second and third graphs from the bottom The minority concentration x, can be defined globally are≈ vertically shifted for clarity by 0.1 and 0.3, respectively. by x = N / N, with N (N) being the total number 1 I I In the inset, we depict the data at (kF a)− = 0.75 multiplied of impurity (bath) atoms, or alternatively by averaging by ω3/2 to demonstrate the power-law scaling of the high- over its local value x in the harmonic trap (see Ap- h i frequency tail. Each point is an average of three experiments, pendix A). Since the local density nI (r) depends on kF a, and the area below each curve is normalized to unity. Since x changes even when x is kept constant. To ensure the measurement is done up to some maximal frequency, to thereh i are no systematic deviations in the experiments properly normalize the data we must account for the miss- due to this effect, we have repeated the measurements ing spectral weight in the unmeasured tail. This is done by 3/2 twice: once keeping x at approximately 0.04, which gives adding to the normalization the integral of ω− up to an 2 2 1 energy cutoff of ¯h /mr , where r0 181 a0 is the effective x 0.23 at (kF a)− = 0.85, and a second time main- 0 ≈ tainingh i ≈ the same value of x by varying x. Since we did range of the interparticle potential [22]. h i 4 photon Raman wavevector which, due to the symmetry species have the same mass m and their free dispersion 2 of the resulting spectrum, allows us to uniquely identify relation is given by εp = p /2m (unless indicated oth- the coherent contribution of the polarons. erwise, we work in units ofh ¯ = kB = 1). The inter- As illustrated in Fig. 1, in our setup two Raman beams action between impurity and bath particles is modeled couple atoms in the minority state 2 to a third state by the contact interaction in the last term of Eq. (1), 3 , which is initially unoccupied. The| i beam parameters which is an excellent approximation for open-channel |arei the same as described in Ref. [57]. We denote their dominated Feshbach resonances as employed in our ex- frequencies by ω1 and ω2 and their wavevectors by k1 periment [22]. Its strength U is related to the s-wave and k2. The measurement is performed by recording the scattering length a by the Lippmann–Schwinger equation 1 1 P number of atoms transferred to the state 3 versus the U − = m/4πa V − 1/2εk. | i − k two-photon detuning, ω = ω1 ω2 E0/¯h, where E0 Polarons and molecules.— The physics of polarons is the bare transition energy between− − states 2 and 3 . and molecules can be qualitatively understood in terms To achieve the utmost sensitivity, we measure| thei atoms| i of two sets of variational wavefunctions that approximate using a high-sensitivity fluorescence detection scheme we the exact eigenstates of the Hamiltonian in Eq. (1). On have recently developed [56, 57]. the one hand, the formation of polarons is well described In Fig. 2 we depict three representative experimental by an Ansatz that systematically expands the many-body 1 datasets taken on the BCS side ((kF a)− = 0.66, blue wavefunction in terms of particle-hole excitations of the 1 − circles), unitarity ((kF a)− = 0.06, red squares) and Fermi sea. It was found [27, 39, 60, 61] that accounting 1 − only for a single such excitation in form of the Chevy on the BEC side ((kF a)− = 0.75, black triangles). The spectrum is symmetric on the BCS side, but becomes Ansatz [26, 62], asymmetric towards unitarity with a tail at high frequen- p p X 0 p cies that grows to be the dominant spectral feature on ψP = α0 dp† FSN + αk,qdp† +q kck† cq FSN (2) | i | i − | i the BEC side. As we will see below, the symmetric part k,q of the spectrum is associated with the coherent response already yields a remarkably good approximation for the of polarons, while the asymmetric contribution is due to p p attractive polaron at momentum p. Here, α and α the polaron incoherent part as well as molecules. The 0 k,q are variational parameters, and primed sums indicate peaked response of the quasiparticles arises since for the that the summation is taken over momenta fulfilling coherent contribution of the polarons the Raman tran- k > k and q < k . The state FS denotes the sition rate is proportional to the one-dimensional mo- F F N zero| | temperature| | Fermi sea of N majority| i particles (the mentum distribution, which is symmetric at equilibrium Fermi wave vector k , and energy ε refer to the major- (k k invariant, neglecting effective mass variation). F F ity ensemble). Crucially, the first term describes the so- Molecules,→ − on the other hand, are dissociated by the Ra- called coherent part of the polaron wavefunction. At low man process. Similar to the incoherent contribution from polaron momenta it determines the quasiparticle weight polarons, this opens another degree of freedom, namely p Z = α 2. It results in a coherent quasiparticle peak in the relative motions of the two atoms, which gives rise p 0 the spectroscopic| | measurements, while the second term, to the asymmetric energy tail in the Raman spectra. As describing the entanglement of the impurity with bath can be seen in the inset of Fig. 2, this tail features a degrees of freedom, leads to an incoherent background power-law scaling of ω 3/2 at large ω, as expected from − of asymmetric shape. Importantly, while the polaron the Tan contact relations [46, 58, 59]. quasiparticle weight Zp is finite for all interactions, the polaron becomes an excited state beyond a critical in- III. FERMI POLARON MODEL teraction strength. It is thus not occupied and hence —at the polaron-to-molecule transition— a jump in the spectral response is expected. In order to analyze the experimentally measured Ra- On the other hand, to lowest order, molecular states man spectra in terms of the physics of polarons and at momentum p can be described by an Ansatz with molecules, we consider the Hamiltonian describing a variational parameters βp of the form [37, 38, 62] system of fermionic impurities immersedH in a homoge- k neous fermionic bath, p X 0 p ψM = βk c† kdk† +p FSN 1 . (3) | i − | − i k X X = ε c c + ε d d p p† p p p† p Here, a fermion is removed from the Fermi surface and H p p is paired with the impurity particle. Both expressions in U X Eqs. (2) and (3) can be systematically improved in their + cp† +qcpdp† 0 qdp0 . (1) V − p,p0,q accuracy by entangling a larger number of particle-hole excitations of the Fermi sea with the quantum impurity. Here V is the system volume, and the operators cp† and The minimization of the energy functional p p d denote fermionic creation operators of bath ( 1 ) and ψ E ψ with respect to the variational p† h P/M |H − | P/M i impurity ( 2 ) particles, respectively (see Fig. 1).| i Both parameters allows one to determine the renormalized | i 5

(a) att (0) (0) rep(0) (b) ¯ ¯ ¯ pol mol pol Acoh Ainc Amol 1 ) (1) (kF a)− = 0.6 ω 0.2 ( ¯

0 A ) F ε (1) 4 0.1 0.1 1 − ) (2) (2) (kF a)− = 1.2 F Energy ( ε 1 − c ) ∆E ( 0 a 1 .

F (3) (3) (kF a)− = 1.6 k

8 Transition probability, − 0 1 2 ( 0 0 1 1.27 2 0 10 20 30 40 1 Interaction, (kF a)− Raman two-photon detuning, ω (εF /¯h)

FIG. 3. Theoretical Raman spectra and quasiparticle energies. (a) Quasiparticle energies of different polaron and molecule states. The solid lines display the energy of the molecule (red), the attractive (blue) and the repulsive (black) polaron at zero momentum as a function of the interaction strength, calculated from the variational states in Eqs. (2) and (3). The 1 crossing of the polaron and molecule energies at (kF a)c− 1.27 results in a first-order transition from a polaronic to a molecular ground state. The blue circles show the peak position≈ of the coherent part of the Raman spectrum. The comparison with the solid blue line makes it evident that the zero-momentum polaron energy can reliably be extracted using this approach. For the excited polaron and molecular states, the energy minimum does not always appear at zero momentum. We show the corresponding lowest energies of the attractive polaron and the molecule at finite momentum as dashed lines. Since the energies are almost degenerate, we indicate the interaction strengths at which the energy minimum switches to finite momentum by a diamond symbol. The inset shows the difference between the zero-momentum energy and the minimal energy for both polaron (blue) and molecule (red) branches. (b) Many-body Raman spectra of the imbalanced Fermi gas for three interaction strengths, calculated from an occupation average for fixed impurity concentration NI = 0.15 N and temperature T = 0.2 TF . The blue-, red- and grey-shaded regions represent the coherent and incoherent polaronic contributions from (2) as well as the molecular contributions from (3), respectively. The coherent part yields an almost symmetric lineshape that is peaked at the polaron energy shifted by the constant Raman two-photon recoil energy (vertical dotted line), while the incoherent and molecular parts lead to an asymmetric continuum. For clarity, the second and third graphs from the bottom are shifted by 0.04 and 0.1, respectively.

dispersion relations pol(p) and mol(p) of the polaron are Ei = pol(p), mol(p) , respectively. and the molecule, respectively. The wavefunctions { P } The operator VˆR = (f † d + h.c.) describes the in Eqs. (2) and (3) predict that the energies of the p p+q¯ p transition with relative two-photon momentum q¯ = k2 polaron and molecule cross at an interaction strength − 1 k1 from an interacting impurity state ( 2 in the experi- − | i (kF a)c 1.27 (see Fig. 3(a)). The deviation from ment) at momentum p, to a hyperfine state (created by ≈ 1 the state-of-the-art prediction of (kF a)− 0.90(4), c ≈ fk†; state 3 in the experiment) that is decoupled from obtained from diagrammatic Monte Carlo (diagMC) the fermionic| i environment and governed by the Hamilto- calculations [27, 39, 63], is mostly due to the simple P nian f = εp(cp† cp + fp†fp). The final states f of approximation taken for the molecular Ansatz that H p | i energy Ef are thus given by non-interacting continuum neglects particle-hole dressing of the molecular state states such as f † FS or f †c† c FS . [38]. p | N i p k q | N i Raman spectroscopy.— For a single impurity, the In experiments, the number of impurities NI = nI V 2 is finite. Treating this case theoretically simplifies at Raman transition rate Γ(ω, i) = 2πΩe (ω, i) is given by Fermi’s golden rule as A sufficiently low impurity concentration, where one may assume that impurities occupy polaronic and molecular X 2 eigenstates of the form of Eqs. (2) and (3), respectively. (ω, i) = f VˆR i δ [ω (Ef Ei)] , (4) A h | | i − − f Importantly, this allows polaron states to be occupied at finite momentum even in the regime where the molecule is where (ω, i) denotes a Raman spectrum and Ωe is the the ground state, and vice versa, for the molecule before effectiveA Rabi frequency. Here, the impurity resides in the polaron-to-molecule transition (for a detailed discus- an initial state i that is characterized by a conserved sion see Section VI). Each of these occupied states con- momentum p and| i may be either polaronic or molecu- tributes to the total normalized Raman signal which is p p lar; i.e. i = ψP , ψM (such as, e.g., approximately thus obtained by averaging over occupation numbers of given by| Eqs.i {| (2) andi | (3)).i} The corresponding energies polarons and molecules with associated Fermi and Bose 6 distribution functions 1. Coherent polaron peak Pcoh(ω): a roughly symmetric peak due to the coherent part of the polaron. 1 X ¯(ω) = pol(ω, p) nF [pol(p)] It contains information about quasiparticle proper- A NI A · p ties such as the polaron energy pol, the spectral 1 X weight Z, and the effective mass m∗. In partic- + mol(ω, p) nB [mol(p)] . (5) ular, we find that the coherent part spectrum is NI A · p proportional to the polaronic momentum distribu- Here, (ω, p) (ω, i = ψp ) and analogously for tion, and its peak position gives the polaron en- Apol ≡ A | P i ergy at zero momentum 0 (plus the recoil energy mol(ω, p). Similar to the single impurity case, the full pol many-impurityA Raman signal is connected to the Raman from the two-photon Raman transition, see Eq. (17) ¯ 2 ¯ below). This correspondence emerges from the fi- rate by Γ(ω, i) = 2πΩeNI (ω, i). Model prediction.—A Theoretical Raman spectra nite transferred photon momentum and thus is not based on wavefunctions Eqs. (2) and (3) are shown in affected by thermal shifts as observed in rf mea- Fig. 3(b) for three interaction strengths before and after surements [11] (see Appendix B). This is evident the polaron-to-molecule transition. Overall, the calcu- from the theoretical analysis in Fig. 3(a), where a lated spectra are qualitatively very similar to the mea- comparison between the calculated polaron energy sured ones shown in Fig. 2. Each spectrum is composed (solid blue) and the extracted peak position of the of a polaronic and a molecular contribution. As can be coherent Raman response (blue circles) is shown. seen, the polaronic contribution is separated into a coher- 2. Background signal P (ω): an asymmetric line- ent (blue shading) and an incoherent part (black shad- bg shape extending to high frequencies that contains ing). The former arises from the coherent part of the po- the combined response arising from the incoherent laron wavefunction proportional to αp 2, and thus gives | 0 | part of the polaron, as well as from molecules that access to the quasiparticle weight Zp of polarons. The are dissociated by the Raman lasers. In a wave- incoherent polaron contribution, in turn, is due to the function picture, the former corresponds to con- second term in Eq. (2), leading to a highly asymmetric tributions as given by the second term in Eq. (2). lineshape. The molecular contribution features a simi- We find that up to a rescaling by a factor, the lar asymmetric lineshape as Raman transitions dissoci- shapes of the background spectra from polarons ate molecules into relative momentum states described p and molecules are similar. by the variational parameters βk in Eq. (3). Both the incoherent polaron and molecular contributions involve Based on this identification, we are able to develop a fit superposition states at large momenta (and thus probing model for the transition probability that is largely model- short-distance physics); hence, their spectra feature the independent, and reflects the lineshape of the coherent 3/2 characteristic high-frequency ω− tails which were and background signals experimentally observed. ∼ ¯ 0 ¯ Our calculations show that Raman spectroscopy pro- P (ω) = ZPcoh(ω; Tp, pol, m∗) + (1 Z)Pbg(ω; Tbg,Eb) . vides a tool for a clear dissection of the polaron − (6) state. This is due to the fact that —in contrast to Here, both contributions Pcoh and Pbg are normalized rf spectroscopy— there is a qualitative difference be- to unity. The ‘many-body polaron weight’ Z¯ quantifies tween an almost symmetric response arising from the co- the weight of the coherent polaron peak for a system herent polaron contribution (blue shading in Fig. 3(b)) with a finite density of impurities. In the limit T, x and a combined asymmetric response stemming from the 0, it reduces to the polaron quasiparticle residue Z for→ incoherent part of the polaron wavefunction and from interactions where the polaron is the ground state. Next, molecules (black and red shading, respectively). It is we determine suitable shape functions for Pcoh and Pbg. this clear distinction between the coherent and incoher- Raman coherent polaron peak.— The Raman ent/molecular response that enables us to experimentally spectrum of a single polaron can be expressed as extract polaron quasiparticle properties with high detail using Raman spectroscopy. pol(ω, k) = Z coh(ω, k) + (1 Z) inc(ω, k) , (7) A A − A where the coherent part can be approximated at low mo- IV. QUANTITATIVE ANALYSIS OF menta as EXPERIMENTAL RAMAN SPECTRA (ω, k) = δ [ω k q + (k)] , (8) Acoh − +¯ pol In order to quantitatively analyze the experimental Z is approximated as a momentum independent quasi- data, we devise a fit model for the lineshape of the Ra- 0 man transition amplitude. Here we make use of the fact particle weight, and the dispersion pol(k) = pol + 2 that, although the theoretical spectra are approxima- k 2m∗ is parametrized by an effective mass, m∗, and 0 tions, they reveal general characteristics of the response, the polaron energy, pol. The function Pcoh accounts only namely that it is composed of two main contributions: for the Raman response arising from (ω) with m∗, Acoh 7

0 pol, and a polaron temperature Tp being fit parameters. As a final step, in order to obtain the probability Pcoh, As described below, the incoherent response arising from the response Γ¯coh has to be normalized to unity. We will be attributed to P . arrive at Ainc bg Within this model it is useful to recognize that the 0 m P (ω; T , , m∗) = n [k (ω)] . (13) total number of impurities can be interpreted as a sum coh p pol q¯ P q¯ of impurities contributing to the coherent response Ncoh, Background Raman signal.— We find that the as well as impurities in the incoherent polaron part Ninc experimental spectral lineshape arising from the inco- and molecules Nmol, herent polaron part and the molecules are fit well by N = N + (N + N ) . (9) the response of a thermal gas of molecules (see Ap- I coh inc mol pendix E). Indeed, this model covers well the overall spectral weight of the background and allows us to in- The left-hand side, NI , is responsible for the full signal, corporate the ω 3/2 tail. P (ω), while Ncoh = ZN¯ I yields the contribution ZP¯ coh − The fit function∼ P (ω) is derived by considering a pair in Eq. (6). The sum (Ninc + Nmol) = (1 Z¯)NI , in turn, bg − of atoms bound as a molecule with a binding energy gives (1 Z¯)Pbg. Moreover, the number of polarons is − E and a center-of-mass momentum k . The Raman given by Npol = Ncoh + Ninc with Ncoh = ZNpol. Thus, b cm within a model with momentum independent quasiparti- process dissociates the pair and changes the center-of- mass momentum to k + q¯. In addition, the unbound cle weight Z Zk=0, one has Z¯ = ZNpol/NI . cm The coherent≡ part of the polaron Raman spectrum fermions acquire a relative momentum krel krel . En- ergy conservation yields ≡ | | coh is related to the coherent contribution of the full, A ¯ ¯ ¯ many-impurity Raman response Γ(ω) = Γcoh(ω)+Γbg(ω) q¯2 qk¯ k2 = mω mE cm,q¯ , (14) by [64, 65] rel − b − 4 − 2 X ˆ Γ¯ (ω) = 2πΩ2 (ω, k) n [ (k),T ] , (10) where kcm,q¯ kcm q¯. coh e Acoh · F pol p The probability≡ F· (k ) that a pair will be dissociated k rel with a relative momentum krel is determined by its rela- R ¯ 2 tive envelope wavefunction, resulting in [67] such that dωΓcoh(ω) = 2πΩeNcoh. Since the concentration of impurities is finite, polarons 2 π− √mEb can be found at non-zero momenta [29, 66]. As Eq. (8) F (krel) = 2 . (15) 2 q¯2 shows, a polaron with a momentum k gives a coherent mEb + krel + krel,q¯q¯+ 4 contribution to the Raman signal if (k) = εk q ω, pol +¯ − which can be solved for kq¯ (ω) k q¯ˆ. In particular, For kcm we assume a thermal Boltzmann dis- if m = m, this yields a linear relation≡ · between ω and tribution at temperature Tbg, G(kcm,Tbg) = ∗ 2 3/2 k /4mTbg kq¯. Otherwise, the solution has a weak dependence on (4πmTbg)− e− cm , which allows us to de- 2 2 2 (1 m/m∗) and k k kq¯. However, this depen- rive an analytical fit model. The combined probability − ⊥ ≡ − dence is only noticeable for kq¯ close to kF , and when the to find a pair with an initial kcm,q¯, and final krel, is effective mass is substantially larger than the bare mass then given by the product of F and G. To obtain the (see Appendix E). Neglecting this small effect, we obtain Raman transition probability as a function of frequency, we change variables from (kcm,q¯, krel) to (kcm,q¯, ω) using m 0 q¯ ˆ ˆ kq¯ (ω) = ω + pol , (11) (14). Finally, integration over the angle krel q¯ and kcm,q¯ q¯ − 2 yields the normalized Raman transition probability· for the background signal (for details see Appendix D), whereq ¯ q¯ . ≡ | | Evaluation of Eq. (10) shows that the coherent polaron Pbg (ω; Tbg,Eb) = (16) Raman rate is proportional to the one-dimensional mo- k2 s 2mω˜/q cm,q¯ Z p 4mTbg mentum distribution of polarons in the direction of q¯, 2Eb 2m 2mω˜ kcm,q¯qe¯ − 2 dkcm,q¯ − , Γ¯coh(ω) = 2πm Ω NcohnP [kq¯ (ω)] q¯. In the local den- 3 2 2 2 e π Tbg 4mEbq¯ + (¯q + kcm,q¯q¯ 2mω) sity approximation (LDA) this distribution is given by −∞ − (for details see Appendix C) 2 whereω ˜ ω E q¯ . This integral does not have an ≡ − b − 4m 3 analytic solution, but it can be readily calculated numer- 5 2 2 0 − 2 ! 6Tp εF pol kq¯ ically. − 2m∗T nP [kq¯ (ω)] = Li 5 ζPe− p . Note that in this fit model, T and E are effective ¯ 2 bg b − √πxZkF εF m/m∗ − temperatures and binding energies. Since Pbg also de- (12) scribes the incoherent polaron contribution, Eb can be Here x is the global impurity concentration, Li5/2 is the interpreted as the molecular binding energy only 0 mol ( µ)/Tp 1 − polylogarithm function, and ζP = e− pol− is the in the limit of large (kF a)− . The effective temperature fugacity of polarons. The chemical potential µ is tuned Tbg compensates for the absence of Pauli blocking in the so that Eq. (10) is normalized to the number of polarons molecular model, and therefore should not be interpreted that contribute to the coherent part of the response. as the physical temperature of molecules. 8

0 1 0 Pol ¯ Eb Z − 0.75 ) F

ε 2 − 0.5

Polaron temperature, Tp (TF ) Energy ( 0.4 0.25 4 − Quasiparticle weight, 0 0 0.5 0 0.5 1 1.5 0.5 0 0.5 1 1.5 − 1 − 1 Interaction, (kF a)− Interaction, (kF a)−

FIG. 4. Measured polaron and pairs binding energies. FIG. 5. Measured quasiparticle weight. The quasipar- 0 The polaron energies, pol, (blue circles) are obtained using ticle weight, Z¯, is shown for different interaction strengths. Eq. (17) from the position of the spectral peak, ω0. The Blue circles mark the weight extracted from the Raman spec- theoretical prediction obtained from the variational Ansatz tra by identifying the nearly symmetric spectral lineshape Eq. (2) is shown as a dashed blue line. The Eb parameter arising from the coherent polaron contribution. Z¯ is smoothly (red squares) is determined by fitting the Raman spectra decreasing towards the polaron–molecule transition, in agree- 1 with Eq. (6). For (kF a)− > 0.5, it is in good agreement ment with our theoretical prediction, averaged over the har- with the energy obtained from the simple molecular Ansatz monic trap using the LDA (solid line). For comparison, we Eq. (3) (solid red line). The dotted red line shows the result also present the trap-averaged prediction for a single impurity of an improved molecular Ansatz [38]. Note that the theo- at T = 0 (dotted line) obtained from Ref. [38]. retical polaron and molecule energies are averaged over the harmonic trap (see Appendix A), and as a result, they cross 1 at a (kF a)− slightly lower than the predicted polaron-to- spectrum onset. We fix Tbg = 2 TF , which yields a min- molecule transition in a homogeneous gas. Inset: extracted ¯ polaron temperatures. The approximate majority tempera- imal systematic error in extracting Z (see Appendix E). ture is denoted by the dashed line. The polaron effective mass, m∗, is strongly coupled to the polaron temperature. To make the fit robust, we set m∗ to the trap-averaged theoretical value at k 0, calcu- → V. EXPERIMENTAL RESULTS lated from Eq. (2). We find that the effective mass mod- ifies the extracted polaron weight and molecular binding energy only marginally. In fact, setting m to the bare The peak of the coherent polaron spectrum, as given ∗ mass leads to a maximal deviation of less than 0.4 σ. We in Eq. (13), is at k = 0. According to Eq. (11), this q¯ are left with three free fitting parameters: Z¯, E , and T . maximum is attained for b p Examples of fits are shown in Fig. 2 (solid lines). Over- q¯2 all, we find excellent agreement between the fits and ω = 0 . (17) 0 2m − pol the measured spectra throughout the whole interaction range. The light and dark shaded areas beneath the For interactions below the transition point, the most sig- curves are the spectral contributions of the coherent part nificant contribution to the spectral peak stems from the of the polaron and the background, respectively. The coherent part of polarons. Thus, from ω0 we can deter- BCS-side data (blue circles) are dominated by a nearly 0 ¯ mine the polaron energy, pol. We find the peak position symmetric quasiparticle peak with Z = 0.91(3), while (dotted vertical lines in Fig. 2) by fitting the points above the BEC-side data (black triangles) are dominated by the the median with a skewed Gaussian [68]. The resulting asymmetric pair dissociation spectra leading to a small polaron energies are plotted in Fig. 4 (blue circles). We coherent weight Z¯ = 0.18(2). The unitary data (red compare the data and find excellent agreement with the squares) shows both the symmetric peak and an asym- predictions of our theoretical model (dashed line), which metric tail. The quasiparticle weight is Z¯ = 0.58(3), close in turn are close to diagMC and T-matrix calculations to the value of 0.47(5) which was measured for 6Li atoms 1 [20, 27, 36, 38, 39]. Beyond (kF a)− = 0.9 the weight of with rf spectroscopy [50]. the coherent peak is small. Thus we restrain the fit in The coherent polaron spectral weight, Z¯, extracted 0 this regime by using the value for pol obtained from the from the fits is shown in Fig. 5. It approaches unity on 1 Chevy Ansatz. the far BCS side, as expected. For increasing (kF a)− , we Next, we extract the polaron weight Z¯ by fitting the observe a smooth decrease of Z¯. We compare the result measured spectra with Eq. (6). The effective temperature to the calculation of our theoretical model in the LDA 1 parameter, Tbg, controls the sharpness of the background (solid line) and find good agreement for (kF a)− > 0.4. 9

Indeed, as shown in Section VI, it is essential to account 20 for the coexistence of polarons and molecules. To demonstrate the crucial role of the ﬁnite impurity density and ) F k

temperature, we also plot in Fig. 5 the prediction for a I 15 single impurity at zero temperature in the LDA (dotted N line). Our data clearly disagrees with this result, and (2 in particular, does not exhibit a sudden change at the C 10 polaron-to-molecule transition as predicted in the single polaron limit. 5

The polaron temperature parameter, Tp, is shown in Contact, the inset of Fig. 4. We find Tp to be around 0.25 TF , slightly higher than the measured majority temperature 0 of approximately 0.2 TF (marked by a dashed line). A 0.5 0 0.5 1 1.5 − 1 moderate systematic decrease of the extracted temper- Interaction, (kF a)− ature is visible as the interaction strength is increased. We attribute this behavior to the reduction in the quasi- FIG. 6. Measured Tan contact. The contact coefficient particle population due to a lower quasiparticle lifetime C is shown for different interaction strengths, obtained using at high momenta. Eq. (18) with Eb and Z¯ extracted from the Raman spectra. We now turn to examine the Eb parameter, which is The theory for a single impurity at zero temperature predicts extracted from the background signal of the Raman spec- a discontinuous change in C at the polaron-to-molecule transi- tra. The results are presented as red squares in Fig. 4. tion [38]. While this discontinuity is smoothed when averaged We also show the theoretical predictions obtained from for a harmonically trapped gas using the LDA (dotted line), the simple variational Ansatz (Eq. (3)) in the LDA (solid an abrupt change is expected to remain. The data signifi- red). Including particle-hole dressing of the molecular cantly deviates from this prediction. Instead, it agrees well state [38] leads to a further lowering of the molecule en- with our calculation taking into account the finite tempera- ergy (dotted red line). Since in the far BEC limit, the ture and coexistence of polarons and molecules (solid line), see Section VI. The two red squares indicate data measured by molecules dominate the Raman response, the parameter the MIT group [11] using rf spectroscopy of a unitary, homo- Eb regains its physical interpretation as the molecular 6 geneous Li gas at T = 0.17 TF (lower point) and T = 0.29 TF binding energy (up to a contribution on the order of F (upper point). arising from the neglect of the presence of a Fermi surface). In this region, we find good agreement between the data and the theoretical predictions. in particular, it does not show any sudden change as pre- Finally, we extract the Tan contact, C, from our ex- dicted in the single impurity limit. We also indicate the perimental data. The contact coefficient is related to contact measured by the MIT group with a homogeneous the tail of the momentum distribution of the quasiparti- 6Li gas using rf spectroscopy [11] (red squares), which cles and measures the short-distance correlations between agrees with our measurements to within the experimen- bath and impurity particles [58, 69–75]. Moreover, the tal uncertainty. contact relates the high-momentum tail to various many- body quantities, such as the thermodynamic pressure and 3/2 quantifies the spectral weight in the universal ω− tail VI. THEORETICAL RAMAN SPECTRA of the Raman spectra [46, 59]. Being related to the derivative of the ground state en- As shown in the previous section, we find no experi- ergy, in the single-impurity limit at T = 0, C is expected mental evidence for a discontinuity in the extracted ob- to jump at the polaron-to-molecule transition [38]. In our servables. We now demonstrate that this observation is fitting model, the tail appears in the spectral contribution consistent with a finite impurity density theory, which of incoherent polarons and molecules. Thus the contact inherently features a first-order transition in the single- is related to the parameters of our model by [72, 76] impurity limit. In fact, discontinuities predicted in this p limit are smoothed out by a finite impurity density. This C = 4π(1 Z¯) Eb/ 2εF . (18) − effect becomes further amplified at finite temperature. Here C is given in units of 2NI kF such that the high- In order to incorporate the finite impurity density and frequency tail of the spectrum approaches P (ω) temperature in the calculation of Raman spectra, we p 2 3/2 → C εF √2π ω [77]. adopt an effective quasiparticle approach. In this model, The results for C are shown as blue circles in Fig. 6. quasiparticle states —obtained in the single-impurity For comparison, we plot the trap-averaged theoretical limit— are occupied thermally according to their quan- prediction for the contact in both the single-impurity tum statistics. More precisely, we consider the polaron limit (dotted line) and in the many-impurity case (solid and molecule states, given by Eqs. (2) and (3), to be pop- line), as discussed in the following section. The data is ulated according to the Fermi-Dirac and Bose-Einstein 1 in excellent agreement with the many-body model, and, distributions n (, T ) = (exp [( µ)/ T ] 1)− , re- F/B − ± 10

spectively. Here, µ denotes the chemical potential which nI =0 nI =0.15 n nI =0.15 n T =0.2 T determines the impurity density at temperature T via T =0 T =0 F 1 1 X nI (µ, T ) = (nF [pol(p),T ] + nB [mol(p),T ]) . V p (19) Note that the impurity temperature and chemical potential µ are set independently of the bath. Speciﬁcally, 0.5 ) 0.5 1 F − (kF a)− = 0.6 for all calculations in this section, µ is tuned to yield ε 1 an impurity density of nI = 0.15 n at a ﬁnite impurity − temperature T = 0.2 TF [78].

Energy ( 1.5 In Fig. 7, the polaron contribution to the total impu- Polaron contribution − 0 0.5 1 rity density in the initial state is shown as a function Momentum, p (kF ) 1 0 of the interaction strength (kF a)− . Evidently, in the single-impurity limit at zero temperature (blue circles) 0.5 0 0.5 1 1.5 2 − 1 the system undergoes a sharp transition from a purely Interaction, (kF a)− 1 polaronic to a purely molecular state at (k a)− 1.27. F ≈ Still at zero-temperature but at finite impurity den- FIG. 7. Calculated polaron contribution. Fraction of sity (red squares), the system is purely polaronic up to impurity particles propagating as polarons as a function of 1 1 interaction strength (k a)− . The impurity particles which (kF a)− 1.1. At this interaction strength the chemi- F cal potential≈ reaches the minimum of the molecular dis- are not polaronic are bound to a bath particle, thus forming persion and, henceforth, it remains pinned to that value a molecule. In the single-impurity limit and T = 0 (blue), there is a sharp transition between a polaron and a molecule (for an illustration of this effect, we show in the inset 1 at (kF a)−c 1.27. Finite impurity density (0.15 n, red) leads of Fig. 7 the dispersion relations of polarons (blue) and ≈ 1 1 to smoothing of the transition for (k a)− < (k a)− . When molecules (red) as well as the impurity chemical potential F F c the temperature is increased (0.2 TF , black), the polaronic µ (black); for a more detailed discussion see Appendix F). 1 1 branch is populated also for (kF a)− > (kF a)− and the sharp 1 c Accordingly, for 1.1 < (kF a)− < 1.27 molecules begin to transition disappears. The inset shows the polaron (blue) and 1 condense in the lowest-lying∼ molecular∼ state while the po- molecule (red) dispersions for nI = 0.15 n and (kF a)− = 0.6, laron Fermi surface shrinks and eventually vanishes at the along with the chemical potential at T = 0 (dashed black) 1 and T = 0.2 T (solid black). At T = 0.2 T , polarons and polaron-to-molecule transition. In this range of (kF a)− , F F polarons and molecules coexist, even at T = 0. Beyond molecules are populated only by thermal excitations, while at the transition, the system forms a molecular condensate T = 0 the chemical potential is above the minimum of the polaron dispersion such that a well-defined polaronic Fermi within the bath of the remaining majority atoms. surface forms. Finally, at finite temperature (black triangles) polarons and molecules coexist as a thermal mixture. This blurs the transition and leads to a smooth interpolation be- We note that in order to accurately incorporate states tween polaron and molecule dominated regimes. Note with a finite lifetime or continuum states, a solution of that the temperatures considered in this work exceed the full imbalanced problem would be necessary. the critical temperature for Bose-Einstein condensation As discussed in Section III, we compute the Ra- of molecules, which thus form a purely thermal gas. man spectra given by Eq. (5) for finite impurity den- In our calculations, we only occupy quasiparticle states sity and temperature by summing the single-impurity with an infinite lifetime and finite quasiparticle weight. Raman spectra over all impurity momenta, weighted by This ensures that the initial state has an infinite lifetime their occupation probability. The single-impurity Raman as expected for an equilibrium state, and also that the spectra are obtained by computing the matrix elements quasiparticle picture remains valid. As a consequence, f VˆR i in Eq. (4) for the AnsätzeEqs. (2) and (3). This h | | i we cut off the quasiparticle populations of the polaronic yields and molecular states at momenta where they no longer p p 2 (ω, ψ ) = α δ [ω εp+q¯ + pol(p)] (20) feature poles on the real frequency axis. A P 0 − For the polaron, this momentum cutoff occurs when X 0 p 2 + αk q δ [ω εp+q¯+q k εk + εq + pol(p)] p 2 , − − − the energy becomes positive or α0 vanishes. For the k,q molecule, however, this cutoff occurs| | when the dispersion intersects the continuum of states delimited by a for the polaron, and 2 p parabola of the form ( p kF ) . In fact, this condi- (ω, ψ ) = (21) | | − A M tion causes the slight dent in the polaron contribution X 0 p 2 1 βk δ[ω εk εp k+q¯ + mol(p) + εF ] at (kF a)− 0.4 visible in Fig. 7, as beyond that value − − − molecules have≈ a well-defined dispersion for all momenta k and thus do not have a cutoff. Similarly, for the polaron for the molecule. These expressions make explicit the 1 its cutoff condition changes at around (kF a)− > 0.3. three contributions that make up the many-body Raman ∼ 11

spectrum as we have discussed in the previous sections, nI =0 nI =0.15 n nI =0.15 n (a) T =0 T =0 T =0.2 TF namely a coherent and incoherent polaron part as well 1 ¯ as a molecular part. Note that our Raman spectra are Z normalized such that they sum to unity once integrated over frequency ω. As discussed in Section III, in Fig. 3, we exemplarily show many-body Raman spectra for three 1 (kF a)− across the transition. In the following, we de- 0.5 scribe how such Raman spectra give access to quasiparticle properties in the regime of ﬁnite impurity concentration. The polaron Z-factor can be obtained from the self- 1 Quasiparticle weight, 0 − energy of the impurity via Zp = 1 ∂ωΣ(ω, p) ω=ωp , (b) where ω is a pole in the retarded− Green’s function| of

p ) the quasiparticle at momentum p [79]. The momentum- F k

I 20 dependent weight Zp can, alternatively, be obtained from N the overlap of the non-interacting wavefunction with the (2 1.2 1.3 p 2 interacting one, Zp = α0 [80]. In the molecular state, C the impurity is bound| to| a bath particle, leading to a 10 vanishing Z-factor in the thermodynamic limit [38]. Similar to the single-impurity quasiparticle residue, Z¯ Contact, is given by the spectral weight of the coherent part of the 0 Raman spectra (blue-shaded area in Fig. 3). It can be 0.5 0 0.5 1 1.5 2 − 1 calculated from the single impurity residues via Interaction, (kF a)− ¯ 1 X Z = Zp nF [pol(p),T ] . (22) Calculated quasiparticle weight and contact. NI · FIG. 8. p The quasiparticle weight (a) and contact coefficient (b) are As evident in Fig. 8(a), in the single-impurity limit Z¯ shown for different interaction strengths. In the single- impurity limit (blue), the transition between polaron and features a sharp jump at the polaron-to-molecule tran- 1 sition where it drops to zero as the polaron is not pop- molecule at (kF a)− 1.27 leads to a sharp jump between the polaronic and molecular≈ residues and contacts. As in ulated anymore. Importantly, in this limit Z¯ reduces ¯ Fig. 7, at T = 0 and finite impurity density (0.15 n, red) the to the zero-momentum polaron residue, Z = Z0, before transition is smoothed, and eventually blurred at finite im- the transition. At finite impurity density and T = 0, purity density and temperature (0.2 TF , black). The inset in this jump is smoothed with the many-body weight again the lower figure shows a magnification around the transition dropping to zero at the transition. point. At finite temperature and density, the transition is completely blurred with Z¯ being lowered on the polaronic side compared to T = 0 and the single impurity Based on the single-impurity coefficients (Appendix G), limit. This is due to the circumstance that, first, some the many-impurity contact coefficient of the full many- impurity particles are propagating as molecules with a body spectrum is determined by vanishing residue and, second, also finite-momentum po- X C = C [p, (p)] n [ (p),T ] larons with a lower residue Zp < Z0 contribute to the pol pol · F pol many-body weight Z¯. p X As shown in Fig. 5, the predicted smooth behavior of + C [p, (p)] n [ (p),T ] . (23) ¯ mol mol B mol the many-body weight Z is consistent with the experi- p · mental observation. The overestimation of the theoretical values for Z¯ in the polaron-dominated interaction The contact C is shown in Fig. 8(b) as a function of 1 1 regime for (kF a)− < 0.4 can be attributed to several (kF a)− at finite impurity density for T > 0 and T = 0, reasons. Firstly, the∼ single-impurity polaron weight Z along with the prediction in the single-impurity limit at will be reduced when higher-order terms are included in T = 0. As can be seen, these scenarios differ signif- the wavefunction Ansatz Eq. (2). Secondly, due to the ne- icantly only around the polaron-to-molecule transition. glect of finite-lifetime molecular states, the polaron con- While the single-impurity limit features a discontinuity, tribution in the initial state is overestimated. Thirdly, already the finite density graph at T = 0 shows a smooth the disregard of finite-lifetime polarons leads to an effec- transition between the polaronic and molecular contacts. tive population transfer to low-momenta polaron states At finite temperature this transition is further blurred. which, again, results in a higher quasiparticle weight. This is in line with the experimental observation shown The large-frequency behavior of the single-impurity in Fig. 5, where the measured data are compared to the and many-body Raman spectra is governed by a power- trap-averaged, theoretical prediction for C (solid black law proportional to the Tan contact C (see Section V). line). 12

VII. DISCUSSION fects, which could, for instance, describe an instability of a Fermi polaron gas towards p-wave superfluidity [81, 82], In this work, we have investigated the attractive Fermi presents a formidable theoretical challenge. While the polaron problem at finite impurity density and tempera- finite density of impurities can be included in quantum ture, employing a novel Raman spectroscopy technique. field theory approaches, the systematic study of polaron– The main advantage of this approach compared to rf polaron interactions requires the inclusion of extended spectroscopy is that the momentum transfer imparted sets of vertex functions [48, 83, 84] that are beyond the by the two-photon transition is significant relative to the reach of mean-field approximations. Similarly, the devel- atomic momentum. As a result, Raman spectroscopy al- opment of wavefunction-based approaches that system- lows us to directly probe the previously inaccessible mo- atically include a finite number of impurities is challeng- mentum distribution of polarons. In order to maintain ing. Here, a major task is the inclusion of higher-order a good signal to noise ratio when working at a low im- particle-hole excitations that become crucial not only in purity density, we additionally employ a high-sensitivity order to account for induced correlations, but also to en- fluorescence detection scheme with which we can reliably sure that the polaron dressing of each individual impurity measure signals of only a few atoms [56]. This allows is fully accounted for [85]. us to probe the polaron-to-molecule transition at finite While the development of such approaches remains impurity density in previously unattainable regimes. an outstanding challenge, it holds promise to shed fur- To extract physical quantities from the data, we have ther light on the nature of the phase diagram of highly- developed a simple fitting model that leverages the sep- imbalanced quantum gases [47–49]. Our findings sug- aration of the Raman spectra into two contributions: gest that close to the interaction where the polaron-to- the nearly symmetric coherent polaron response, and an molecule transition takes place, polarons and molecules asymmetric background arising from the incoherent re- coexist when the temperature is above the critical tem- sponse of polarons and from molecules. From the mea- perature of molecular Bose-Einstein condensation. At sured Raman spectra we obtain the polaron energy, the lower temperatures, the phase diagram is not yet un- quasiparticle spectral weight, and the contact parameter. derstood and contrasting predictions have been made. In order to gain a better understanding of our mea- On the one hand, at zero-temperature, the polaron-to- surements, we have devised a theoretical model based molecule transition marks the endpoint of a fermionic on a variational description of polarons and molecules polaron phase, where its finite Fermi surface volume that takes into account finite impurity density and tem- vanishes and a polarized superfluid phase consisting of perature. The physical picture that arises from our molecules is expected to take over [38, 86]. On the experimental and theoretical observations is intriguing: other hand, considering the strong atom-dimer interac- All measured quantities show a smooth transition with tions close to the transition point [87], it has been pre- no sudden changes around the predicted polaron-to- dicted that the system might become unstable towards molecule transition. As we show theoretically, this is phase separation between superfluid and normal phases explained by the population of polarons and molecules [49]. The application of Raman spectroscopy in an im- at finite momenta, resulting from the finite impurity balanced Fermi gas at lower temperature [88] and ho- density and temperature. The excellent agreement be- mogeneous traps [89–92] might help to distinguish these tween the theoretical model and the experimental data scenarios and allow one to experimentally determine, e.g., strongly suggests a coexistence phase of polarons and the transition temperature towards phase separation. molecules around the interaction strength where the first- Furthermore, away from the transition point a plethora order transition in the single-impurity limit takes place. of phases has been discussed in the literature, ranging We stress that this coexistence region and the smooth from p-wave pairing of polarons to the FFLO phase transition from polarons to molecules is a general char- [46, 47, 93, 94]. Interestingly, our results in the im- acteristic of any realistic scenario where many impurities purity limit already hint at some of these possibilities. are present. As shown in Fig. 3, we find that excited molecules in To interpret our experimental data we have employed the polaronic regime feature a dispersion relation with a quasiparticle theory, in which the many-body Hilbert a minimum at finite momentum (see also Refs. [33, 95] space of impurity particles is spanned by single-particle and a recent discussion in Ref. [44]). This effect may be states obtained from variational wavefunctions. The ap- regarded as a precursor of the long-sought-after FFLO proximate nature of this approach is reflected by the phase in the imbalanced BEC-BCS crossover [93, 94, 96– fact that polarons and molecules are effectively cre- 98], which emerges due to the macroscopic occupation ated by composite impurity-bath operators that main- of such molecular states at finite momentum. Intrigu- tain their respective commutation relations only approx- ingly, our calculation of the momentum-resolved spec- imately. Correspondingly, both impurity-induced corre- tral function of polarons shows that they as well feature lations between majority fermions, as well as quasiparti- a roton-like minimum at finite momentum in their ex- cle interactions (polaron–polaron, polaron–molecule, and cited state. This raises the question of whether these molecule–molecule), induced by the Fermi sea, are ne- finite-momentum states can be prepared in a controlled glected. The accurate inclusion of such correlation ef- way, which could subsequently lead to the formation of 13 a metastable, non-equilibrium polaron gas with a non- exciting perspective of studying the fate of the polaron- trivial Fermi surface topology [99, 100]. to-molecule transition in highly imbalanced Bose-Fermi mixtures. This question has recently become the fo- One possible way to study these questions is the ex- cus of experimental and theoretical studies of interact- tension of Raman spectroscopy to Raman injection spec- ing exciton-electron gases in two-dimensional transition- troscopy. Similar to rf injection spectroscopy [31, 32, metal dichalcogenides [104], where the coexistence be- 101, 102], the system is initially prepared in a weakly- tween molecular exciton-electron bound trion states and interacting state and driven to a state where impurities Fermi-polarons may lead to novel electronic and optical strongly interact with their environment. Thus, with Ra- properties [15–17, 105–107]. man injection one can prepare polarons at a specific momentum. This enables a direct measurement of key polaron properties, such as the momentum-dependent ef- ACKNOWLEDGMENTS fective mass, residue and lifetime. In addition, Raman injection could potentially facilitate the population and We thank Felix Werner and Wilhelm Zwerger for in- observation of the elusive finite-momentum polaron and spiring discussions and valuable input. This research was molecular states as precursors of exotic phases in the supported by the Israel Science Foundation (ISF), grant BEC-BCS crossover. Moreover, Raman injection may No. 1779/19, and by the United States - Israel Bina- also provide a promising means to probe unoccupied ex- tional Science Foundation (BSF), grant No. 2018264. citation branches of the spectral function [65]. Such ex- R. S. is supported by the Deutsche Forschungsgemein- periments could enable the controlled study of the mo- schaft (DFG, German Research Foundation) under Ger- mentum relaxation rate of polarons, which is currently many’s Excellence Strategy – EXC-2111 – 390814868. investigated as a pathway for the realization of polaron- G. N. is supported by the Helen Diller Quantum Center polariton-induced optical gain in two-dimensional semi- at the Technion. O. K. D. and J. v. M. are supported by conductor heterostructures [103]. a fellowship of the International Max Planck Research Finally, in cold atom experiments, both fermionic and School for Quantum Science and Technology (IMPRS- bosonic impurities can be implemented. This opens the QST).

[1] L. D. Landau, “On the motion of electrons in a crystal vation of Attractive and Repulsive Polarons in a Bose- lattice,” Phys. Z. Sowjetunion 3, 664 (1933). Einstein Condensate,” Phys. Rev. Lett. 117, 055302 [2] G. Lindemann, R. Lassnig, W. Seidenbusch, and (2016). E. Gornik, “Cyclotron resonance study of polarons in [11] Zoe Z. Yan, Yiqi Ni, Carsten Robens, and Martin W. GaAs,” Phys. Rev. B 28, 4693–4703 (1983). Zwierlein, “Bose polarons near quantum criticality,” [3] N. F. Mott, “Polaron models of high-temperature super- Science 368, 190–194 (2020). conductors,” J. Phys.: Condens. Matter 5, 3487–3506 [12] Mikhail Lemeshko and Richard Schmidt, “Molecular (1993). impurities interacting with a many-particle environ- [4] R. D. Popp and R. B. Murray, “Diffusion of the Vk- ment: From ultracold gases to helium nanodroplets,” in polaron in alkali halides: Experiments in NaI and RbI,” Theoretical and Computational Chemistry Series (Royal J. Phys. Chem. Solids 33, 601–610 (1972). Society of Chemistry, 2017) pp. 444–495. [5] S. Moser, L. Moreschini, J. Jaćimović,O. S. Bariˇsić, [13] Jun Kondo and Toshio Soda, “Theory of ion mobility H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bost- in liquid 3He,” J. Low Temp. Phys. 50, 21–32 (1983). wick, E. Rotenberg, L. Forró, and M. Grioni, “Tun- [14] Pietro Massignan, Matteo Zaccanti, and Georg M. able polaronic conduction in Anatase TiO2,” Phys. Rev. Bruun, “Polarons, dressed molecules and itinerant ferro- Lett. 110, 601–610 (2013). magnetism in ultracold Fermi gases,” Rep. Prog. Phys. [6] David Emin, Polarons (Cambridge University Press, 77, 034401 (2014). 2012). [15] Meinrad Sidler, Patrick Back, Ovidiu Cotlet, Ajit Sri- [7] Alexandre S. Alexandrov and Jozef T. Devreese, Ad- vastava, Thomas Fink, Martin Kroner, Eugene Dem- vances in Polaron Physics (Springer Berlin Heidelberg, ler, and Atac Imamoglu, “Fermi polaron-polaritons in 2010). charge-tunable atomically thin semiconductors,” Nat. [8] Weng Hong Sio, Carla Verdi, Samuel Poncé, and Feli- Phys. 13, 255–261 (2016). ciano Giustino, “Polarons from first principles, without [16] Dmitry K. Efimkin and Allan H. MacDonald, “Many- supercells,” Phys. Rev. Lett. 122, 246403 (2019). body theory of trion absorption features in two- [9] Ming-Guang Hu, Michael J. Van de Graaff, Dhruv dimensional semiconductors,” Phys. Rev. B 95, 035417 Kedar, John P. Corson, Eric A. Cornell, and Debo- (2017). rah S. Jin, “Bose polarons in the strongly interacting [17] Christian Fey, Peter Schmelcher, Atac Imamoglu, and regime,” Phys. Rev. Lett. 117, 055301 (2016). Richard Schmidt, “Theory of exciton-electron scattering [10] Nils B. Jørgensen, Lars Wacker, Kristoffer T. Skalm- in atomically thin semiconductors,” Phys. Rev. B 101, stang, Meera M. Parish, Jesper Levinsen, Rasmus S. 195417 (2020). Christensen, Georg M. Bruun, and Jan J. Arlt, “Obser- [18] P. W. Anderson, “Infrared catastrophe in Fermi gases 14

with local scattering potentials,” Phys. Rev. Lett. 18, larons in a two-dimensional Fermi gas,” Phys. Rev. Lett. 1049–1051 (1967). 122, 193604 (2019). [19] Achim Rosch, “Quantum-coherent transport of a heavy [36] R. Combescot, A. Recati, C. Lobo, and F. Chevy, “Nor- particle in a fermionic bath,” Adv. Phys. 48, 295–394 mal state of highly polarized Fermi gases: Simple many- (1999). body approaches,” Phys. Rev. Lett. 98, 180402 (2007). [20] FrédéricChevy and Christophe Mora, “Ultra-cold po- [37] Christophe Mora and FrédéricChevy, “Ground state of larized Fermi gases,” Rep. Prog. Phys. 73, 112401 a tightly bound composite dimer immersed in a Fermi (2010). sea,” Phys. Rev. A 80, 033607 (2009). [21] Richard Schmidt, Michael Knap, Dmitri A. Ivanov, [38] M. Punk, P. T. Dumitrescu, and W. Zwerger, “Polaron- Jhih-Shih You, Marko Cetina, and Eugene Demler, to-molecule transition in a strongly imbalanced Fermi “Universal many-body response of heavy impurities gas,” Phys. Rev. A 80, 053605 (2009). coupled to a Fermi sea: a review of recent progress,” [39] N. V. Prokof’ev and B. V. Svistunov, “Bold diagram- Rep. Prog. Phys. 81, 024401 (2018). matic Monte Carlo: A generic sign-problem tolerant [22] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite technique for polaron models and possibly interacting Tiesinga, “Feshbach resonances in ultracold gases,” Rev. many-body problems,” Phys. Rev. B 77, 125101 (2008). Mod. Phys. 82, 1225–1286 (2010). [40] R. Combescot, S. Giraud, and X. Leyronas, “Analytical [23] Martin W. Zwierlein, Andre Schirotzek, Christian H. theory of the dressed bound state in highly polarized Schunck, and Wolfgang Ketterle, “Fermionic superflu- Fermi gases,” Europhys. Lett. 88, 60007 (2009). idity with imbalanced spin populations,” Science 311, [41] D. M. Edwards, “A smooth polaron–molecule crossover 492–496 (2006). in a Fermi system,” J. Phys.: Condens. Matter 25, [24] Guthrie B. Partridge, Wenhui Li, Ramsey I. Kamar, 425602 (2013). Yean-an Liao, and Randall G. Hulet, “Pairing and [42] Jin-Ge Chen, Tian-Shu Deng, Wei Yi, and Wei Zhang, phase separation in a polarized Fermi gas,” Science 311, “Polarons and molecules in a Fermi gas with orbital 503–505 (2006). Feshbach resonance,” Phys. Rev. A 94, 053627 (2016). [25] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, [43] Hiroyuki Tajima and Shun Uchino, “Many Fermi po- and W. Ketterle, “Observation of phase separation in a larons at nonzero temperature,” New J. Phys. 20, strongly interacting imbalanced Fermi gas,” Phys. Rev. 073048 (2018). Lett. 97, 030401 (2006). [44] Xiaoling Cui, “Fermi polaron revisited: polaron- [26] F. Chevy, “Universal phase diagram of a strongly in- molecule transition and coexistence,” arXiv preprint teracting Fermi gas with unbalanced spin populations,” (2020), 2003.11710. Phys. Rev. A 74, 063628 (2006). [45] Predrag Nikolićand Subir Sachdev, “Renormalization- [27] Nikolay Prokof’ev and Boris Svistunov, “Fermi-polaron group fixed points, universal phase diagram, and 1/N problem: Diagrammatic Monte Carlo method for diver- expansion for quantum liquids with interactions near gent sign-alternating series,” Phys. Rev. B 77, 020408 the unitarity limit,” Phys. Rev. A 75, 033608 (2007). (2008). [46] Wilhelm Zwerger, ed., The BCS-BEC Crossover and the [28] Xiaoling Cui and Hui Zhai, “Stability of a fully magne- Unitary Fermi Gas, Lecture Notes in Physics, Vol. XVI tized ferromagnetic state in repulsively interacting ul- (Springer, 2012). tracold Fermi gases,” Phys. Rev. A 81, 041602 (2010). [47] W. Zwerger, “Strongly interacting Fermi gases,” Pro- [29] Richard Schmidt and Tilman Enss, “Excitation spectra ceedings of the International School of Physics “Enrico and rf response near the polaron-to-molecule transition Fermi” 191, 63–141 (2016). from the functional renormalization group,” Phys. Rev. [48] Bernhard Frank, Johannes Lang, and Wilhelm Zwerger, A 83, 063620 (2011). “Universal phase diagram and scaling functions of im- [30] P. Massignan and G. M. Bruun, “Repulsive polarons balanced Fermi gases,” J. Exp. Theor. Phys. 127, 812– and itinerant ferromagnetism in strongly polarized 825 (2018). Fermi gases,” Eur. Phys. J. D 65, 83–89 (2011). [49] S. Pilati and S. Giorgini, “Phase separation in a polar- [31] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, ized Fermi gas at zero temperature,” Phys. Rev. Lett. P. Massignan, G. M. Bruun, F. Schreck, and R. Grimm, 100, 030401 (2008). “Metastability and coherence of repulsive polarons in a [50] AndréSchirotzek, Cheng-Hsun Wu, Ariel Sommer, and strongly interacting Fermi mixture,” Nature 485, 615– Martin W. Zwierlein, “Observation of Fermi polarons in 618 (2012). a tunable Fermi liquid of ultracold atoms,” Phys. Rev. [32] Marco Koschorreck, Daniel Pertot, Enrico Vogt, Bernd Lett. 102, 230402 (2009). Fröhlich, Michael Feld, and Michael Köhl,“Attractive [51] S. Nascimbène, N. Navon, K. J. Jiang, L. Tarruell, and repulsive Fermi polarons in two dimensions,” Na- M. Teichmann, J. McKeever, F. Chevy, and C. Sa- ture 485, 619–622 (2012). lomon, “Collective oscillations of an imbalanced Fermi [33] Richard Schmidt, Tilman Enss, Ville Pietilä, and Eu- gas: Axial compression modes and polaron effective gene Demler, “Fermi polarons in two dimensions,” Phys. mass,” Phys. Rev. Lett. 103, 170402 (2009). Rev. A 85, 021602 (2012). [52] F. Scazza, G. Valtolina, P. Massignan, A. Recati, [34] Vudtiwat Ngampruetikorn, Jesper Levinsen, and A. Amico, A. Burchianti, C. Fort, M. Inguscio, M. Za- Meera M. Parish, “Repulsive polarons in two- ccanti, and G. Roati, “Repulsive Fermi polarons in a dimensional Fermi gases,” Europhys. Lett. 98, 30005 resonant mixture of ultracold 6Li atoms,” Phys. Rev. (2012). Lett. 118, 083602 (2017). [35] N. Darkwah Oppong, L. Riegger, O. Bettermann, [53] Zhenjie Yan, Parth B. Patel, Biswaroop Mukherjee, M. Höfer,J. Levinsen, Meera M. Parish, I. Bloch, and Richard J. Fletcher, Julian Struck, and Martin W. S. Fölling,“Observation of coherent multiorbital po- Zwierlein, “Boiling a unitary Fermi liquid,” Phys. Rev. 15

Lett. 122, 093401 (2019). [71] Eric Braaten and Lucas Platter, “Exact relations for a [54] N. Navon, S. Nascimbène,F. Chevy, and C. Salomon, strongly interacting Fermi gas from the operator prod- “The equation of state of a low-temperature Fermi uct expansion,” Phys. Rev. Lett. 100, 205301 (2008). gas with tunable interactions,” Science 328, 729–732 [72] Eric Braaten, Daekyoung Kang, and Lucas Platter, (2010). “Universal relations for a strongly interacting Fermi gas [55] M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. M. near a Feshbach resonance,” Phys. Rev. A 78, 053606 Walraven, R. Grimm, J. Levinsen, M. M. Parish, (2008). R. Schmidt, M. Knap, and E. Demler, “Ultrafast many- [73] FélixWerner and Yvan Castin, “General relations for body interferometry of impurities coupled to a Fermi quantum gases in two and three dimensions: Two- sea,” Science 354, 96–99 (2016). component fermions,” Phys. Rev. A 86, 013626 (2012). [56] Constantine Shkedrov, Yanay Florshaim, Gal Ness, An- [74] FélixWerner and Yvan Castin, “General relations for drey Gandman, and Yoav Sagi, “High-sensitivity rf quantum gases in two and three dimensions. ii. bosons spectroscopy of a strongly interacting Fermi gas,” Phys. and mixtures,” Phys. Rev. A 86, 053633 (2012). Rev. Lett. 121, 093402 (2018). [75] R. Rossi, T. Ohgoe, E. Kozik, N. Prokof’ev, B. Svis- [57] Constantine Shkedrov, Gal Ness, Yanay Florshaim, and tunov, K. Van Houcke, and F. Werner, “Contact and Yoav Sagi, “In situ momentum-distribution measure- momentum distribution of the unitary Fermi gas,” Phys. ment of a quantum degenerate Fermi gas using Raman Rev. Lett. 121, 130406 (2018). spectroscopy,” Phys. Rev. A 101, 013609 (2020). [76] Yoav Sagi, Tara E. Drake, Rabin Paudel, Roman Cha- [58] Shina Tan, “Large momentum part of a strongly corre- purin, and Deborah S. Jin, “Breakdown of the Fermi lated Fermi gas,” Ann. Phys. 323, 2971–2986 (2008). liquid description for strongly interacting fermions,” [59] Yusuke Nishida, “Probing strongly interacting atomic Phys. Rev. Lett. 114, 075301 (2015). gases with energetic atoms,” Phys. Rev. A 85, 053643 [77] William Schneider and Mohit Randeria, “Universal (2012). short-distance structure of the single-particle spectral [60] R. Combescot and S. Giraud, “Normal state of highly function of dilute Fermi gases,” Phys. Rev. A 81, 021601 polarized Fermi gases: Full many-body treatment,” (2010). Phys. Rev. Lett. 101, 050404 (2008). [78] The value of nI = 0.15 n is taken as the typical value of [61] Weizhe Edward Liu, Jesper Levinsen, and Meera M. the experimentally realized x . Parish, “Variational approach for impurity dynamics [79] A. A. Abrikosov, Methods ofh Quantumi Field Theory in at finite temperature,” Phys. Rev. Lett. 122, 205301 Statistical Physics (Dover Publications Inc., 1975). (2019). [80] If the polaron is a stable quasiparticle (Im(ωp) = 0), [62] Christian Trefzger and Yvan Castin, “Impurity in a its Z-factor within the polaron Ansatz (2) is given by Fermi sea on a narrow Feshbach resonance: A varia- αp 2. These definitions are, however, not equivalent if | 0 | tional study of the polaronic and dimeronic branches,” the polaron acquires a finite lifetime (Im(ωp) = 0). This Phys. Rev. A 85, 053612 (2012). discrepancy can be seen from comparing the6 definition [63] Kris Van Houcke, FélixWerner, and Riccardo Rossi, of Zp using the self-energy to the normalization con- p 2 “High-precision numerical solution of the Fermi polaron dition for α0 as they differ in the placement of the problem and large-order behavior of its diagrammatic absolute value| | bars. series,” Phys. Rev. B 101, 045134 (2020). [81] Aurel Bulgac, Michael McNeil Forbes, and Achim [64] Tung-Lam Dao, Antoine Georges, Jean Dalibard, Schwenk, “Induced P -wave superfluidity in asymmetric Christophe Salomon, and Iacopo Carusotto, “Measur- Fermi gases,” Phys. Rev. Lett. 97, 020402 (2006). ing the one-particle excitations of ultracold fermionic [82] Yusuke Nishida, “Casimir interaction among heavy atoms by stimulated Raman spectroscopy,” Phys. Rev. fermions in the BCS-BEC crossover,” Phys. Rev. A 79, Lett. 98, 240402 (2007). 013629 (2009). [65] Tung-Lam Dao, Iacopo Carusotto, and Antoine [83] A. Camacho-Guardian, L. A. PeñaArdila, T. Pohl, and Georges, “Probing quasiparticle states in strongly in- G. M. Bruun, “Bipolarons in a Bose-Einstein conden- teracting atomic gases by momentum-resolved Raman sate,” Phys. Rev. Lett. 121, 013401 (2018). photoemission spectroscopy,” Phys. Rev. A 80, 023627 [84] R. Rossi, T. Ohgoe, K. Van Houcke, and F. Werner, (2009). “Resummation of diagrammatic series with zero con- [66] Martin Veillette, Eun Gook Moon, Austen Lamacraft, vergence radius for strongly correlated fermions,” Phys. Leo Radzihovsky, Subir Sachdev, and D. E. Sheehy, Rev. Lett. 121, 130405 (2018). “Radio-frequency spectroscopy of a strongly imbalanced [85] Xiang Li, Enderalp Yakaboylu, Giacomo Bighin, Feshbach-resonant Fermi gas,” Phys. Rev. A 78, 033614 Richard Schmidt, Mikhail Lemeshko, and Andreas (2008). Deuchert, “Intermolecular forces and correlations me- [67] Cheng Chin and Paul S. Julienne, “Radio-frequency diated by a phonon bath,” The Journal of Chemical transitions on weakly bound ultracold molecules,” Phys. Physics 152, 164302 (2020). Rev. A 71, 012713 (2005). [86] Subir Sachdev and Kun Yang, “Fermi surfaces and Lut- [68] Aaron L. Stancik and Eric B. Brauns, “A simple asym- tinger’s theorem in paired fermion systems,” Phys. Rev. metric lineshape for fitting infrared absorption spectra,” B 73, 174504 (2006). Vib. Spectrosc. 47, 66–69 (2008). [87] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, [69] Shina Tan, “Generalized virial theorem and pressure re- “Scattering properties of weakly bound dimers of lation for a strongly correlated Fermi gas,” Ann. Phys. fermionic atoms,” Phys. Rev. A 71, 012708 (2005). 323, 2987–2990 (2008). [88] Rianne S. Lous, Isabella Fritsche, Michael Jag, [70] Shina Tan, “Energetics of a strongly correlated Fermi Bo Huang, and Rudolf Grimm, “Thermometry of a gas,” Ann. Phys. 323, 2952–2970 (2008). deeply degenerate Fermi gas with a Bose-Einstein con- 16

densate,” Phys. Rev. A 95, 053627 (2017). [106] M. M. Glazov, “Optical properties of charged excitons [89] Biswaroop Mukherjee, Zhenjie Yan, Parth B. Patel, Zo- in two-dimensional semiconductors,” The Journal of ran Hadzibabic, Tarik Yefsah, Julian Struck, and Mar- Chemical Physics 153, 034703 (2020). tin W. Zwierlein, “Homogeneous atomic Fermi gases,” [107] Atac Imamoglu, Ovidiu Cotlet, and Richard Schmidt, Phys. Rev. Lett. 118, 123401 (2017). “Exciton-polarons in two-dimensional semiconductors [90] Klaus Hueck, Niclas Luick, Lennart Sobirey, Jonas and the Tavis-Cummings model,” arXiv preprint Siegl, Thomas Lompe, and Henning Moritz, “Two- (2020), 2006.15963. dimensional homogeneous Fermi gases,” Phys. Rev. [108] C. Lobo, A. Recati, S. Giorgini, and S. Stringari, “Nor- Lett. 120, 060402 (2018). mal state of a polarized Fermi gas at unitarity,” Phys. [91] Biswaroop Mukherjee, Parth B. Patel, Zhenjie Yan, Rev. Lett. 97, 200403 (2006). Richard J. Fletcher, Julian Struck, and Martin W. [109] Christophe Mora and FrédéricChevy, “Normal phase of Zwierlein, “Spectral response and contact of the unitary an imbalanced Fermi gas,” Phys. Rev. Lett. 104, 230402 Fermi gas,” Phys. Rev. Lett. 122, 203402 (2019). (2010). [92] Markus Bohlen, Lennart Sobirey, Niclas Luick, Hauke [110] Brendan C. Mulkerin, Xia-Ji Liu, and Hui Hu, “Break- Biss, Tilman Enss, Thomas Lompe, and Hen- down of the Fermi polaron description near Fermi de- ning Moritz, “Sound propagation and quantum-limited generacy at unitarity,” Ann. Phys. 407, 29–45 (2019). damping in a two-dimensional Fermi gas,” Phys. Rev. [111] Hiroyuki Tajima and Shun Uchino, “Thermal crossover, Lett. 124, 240403 (2020). transition, and coexistence in Fermi polaronic spectro- [93] Peter Fulde and Richard A. Ferrell, “Superconductivity scopies,” Phys. Rev. A 99, 063606 (2019). in a strong spin-exchange field,” Phys. Rev. 135, A550– [112] Weizhe Edward Liu, Zhe-Yu Shi, Jesper Levinsen, and A563 (1964). Meera M. Parish, “Radio-frequency response and con- [94] I. A. Larkin and Y. N. Ovchinnikov, “Nonuniform state tact of impurities in a quantum gas,” Phys. Rev. Lett. of superconductors,” h. Eksp. Teor. Fiz , 1136 (1964). 125, 065301 (2020). [95] Richard Schmidt, From few-to many-body physics with [113] Matthias Punk, Many-particle physics with ultracold ultracold atoms, Ph.D. thesis, Technische Universität gases, Dissertation, Technische UniversitätMünchen, München (2013). München (2010). [96] D. T. Son and M. A. Stephanov, “Phase diagram of a cold polarized Fermi gas,” Phys. Rev. A 74, 013614 (2006). Appendix A: Local density approximation for [97] Yean an Liao, Ann Sophie C. Rittner, Tobias Paprotta, spin-imbalanced gas Wenhui Li, Guthrie B. Partridge, Randall G. Hulet, Ste- fan K. Baur, and Erich J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 To account for the non-uniform atomic density, we (2010). compare measurements to average quantities calculated [98] Francesco Piazza, Wilhelm Zwerger, and Philipp in the LDA. To this end, we assume that the distribution Strack, “FFLO strange metal and quantum criticality of the majority atoms is not affected by the presence of in two dimensions: Theory and application to organic the minority atoms; hence it can be calculated as for non- superconductors,” Phys. Rev. B 93, 085112 (2016). [99] Kun Yang and Subir Sachdev, “Quantum criticality of a interacting fermions. The minority density distribution, Fermi gas with a spherical dispersion minimum,” Phys. nI (r), is calculated by taking into account the interac- Rev. Lett. 96, 187001 (2006). tions with the majority atoms through a renormalization 0 [100] Subir Sachdev, “Topological order, emergent gauge of the confining potential: V2 (r) = V1 (r) (1 /εF ), − pol fields, and Fermi surface reconstruction,” Rep. Prog. where V1 (r) is the potential felt by the majority atoms Phys. 82, 014001 (2018). [108]. We neglect the weak interactions between po- [101] C. A. Regal and D. S. Jin, “Measurement of positive and larons [109]. The expected value of any observable, A, is negative scattering lengths in a Fermi gas of atoms,” then given by the minority-weighted local density aver- 90 R 3 Phys. Rev. Lett. , 230404 (2003). d rA(r)nI (r) [102] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadz- age: A = R 3r r . Notice that when we compare h i d nI ( ) ibabic, Tarik Yefsah, Waseem S. Bakr, and Mar- 1 experimental results to theory as a function of (kF a)− , tin W. Zwierlein, “Spin-injection spectroscopy of a spin- the Fermi wave vector k is that of the trap, namely the 109 F orbit coupled Fermi gas,” Phys. Rev. Lett. , 095302 local k at the center of the trap. (2012). F [103] Li Bing Tan, Ovidiu Cotlet, Andrea Bergschneider, Richard Schmidt, Patrick Back, Yuya Shimazaki, Mar- tin Kroner, and Ata ¸c Imamo˘glu,“Interacting˙ polaron- Appendix B: Dependence of the coherent spectrum polaritons,” Phys. Rev. X 10, 021011 (2020). peak on the transferred photon momentum [104] Kin Fai Mak, Keliang He, Changgu Lee, Gwan Hy- oung Lee, James Hone, Tony F. Heinz, and Jie Shan, In this appendix, we examine the correlation between “Tightly bound trions in monolayer MoS2,” Nat. Mater. 12, 207–211 (2012). the peak position of the coherent polaronic spectrum and [105] Ovidiu Cotlet¸, Falko Pientka, Richard Schmidt, Gergely the energy of the zero-momentum attractive polaron. As Zarand, Eugene Demler, and Atac Imamoglu, “Trans- experimentally observed in Ref. [11], at finite impurity port of neutral optical excitations using electric fields,” density there is a temperature-dependent shift of the rf Phys. Rev. X 9, 041019 (2019). spectrum peak position relative to the zero-momentum 17

Tp = 0.1 TF , 0.15 TF , 0.2 TF Appendix C: Raman transition rate of the coherent polaron contribution in the LDA | |

0.61 In this appendix we provide more detail on the deriva- − 0.61 − tion of the Raman rate for the coherent polaron contri- ) bution under the LDA. F 0.62 ε − Eq. (10) gives the rate for a homogeneous system. 0.63 Here, we treat the case of a harmonically trapped gas. 0.62 − − 0.64 The occupation averaged coherent response is given by

Energy ( − 2 Z 0 0.1 0.2 0.3 2πΩe 3 X Γ¯coh (ω) = d r coh (ω, k) (C1) Polaron temperature, Tp (TF ) V A k 0.63 " 0 ! # − m 2 pol 2 | | nF pol(k) µ + ωho 1 r ,Tp , 0 0.1 0.2 0.3 1.9 × − 2 − εF Photon momentum transfer,q ¯ (kF ) where coh(ω, k) is given in Eq. (8), and ωho = A1/3 FIG. 9. Coherent spectral peak position at unitar- F /(6N) denotes the geometrically-averaged harmonic ity. Using the full many-body model for the Raman spec- trapping frequency. In cylindrical coordinates the in- tra (first term of Eq. (20)), we plot the coherent peak po- tegral over k within Eq. (C1) decomposes into a two- sition vs. transferred photon momentum for polaron tem- dimensional integral of k over directions perpendicular peratures Tp = 0.1 TF (blue circles), 0.15 TF (red squares), ⊥ 1 to q¯, and an integral over kq¯ along the direction of q¯. and 0.2 TF (black triangles) at (kF a)− = 0 with nI = The condition imposed by the δ-function within coh is 0.15 n. We observe convergence to the zero-temperature value then given by A pol = 0.6066 εF (dashed line) for photon momentum trans- − fer larger than 0.1kF . Specifically, with Raman spectroscopy 2 2 2 kq¯ + k 1 1 kq¯q¯ 0 q¯ in our experiment (¯q = 1.9 kF ), no shift is expected. Inset: ⊥ + ω pol + = 0 , (C2) temperature dependence of the spectral peak position with 2 m − m∗ m − − 2m conventional rf spectroscopy (¯q = 0). and can be solved for kq¯. At low temperatures and for most interaction strengths, we find that k (1 m/m∗) q¯ for momenta − polaron energy. Our theoretical model captures correctly which are not suppressed by the Fermi distribution. We thus neglect the first term proportional to (1 m/m∗) this phenomenon and shows that it is absent in Raman − spectroscopy. in Eq. (C2) when evaluating the δ-function in Eq. (C1). Carrying out the integrations in Eq. (C1) we then obtain In order to calculate this shift, we compute the peak that position of the coherent polaron contribution (first term ¯ 2 of Eq. (20) within Eq. (5)) as a function of the photon Γcoh(ω) = 2πmΩeNcoh nP [kq¯ (ω)]/q ¯ , (C3) transfer momentum. The results at unitarity, shifted by the recoil energy, are shown in Fig. 9 for three rel- where nP [k] is given in Eq. (12) and kq¯(ω) is given evant temperatures. In rf spectroscopy (¯q 0), we find in Eq. (11). Correspondingly, the fugacity ζP = → 0 µ T that the peak is shifted to energies lower than the zero- e− ( pol− )/ p within Eq. (12) is set by the normalization R ¯ 2 momentum polaron energy (pol, dashed line). As the dωΓcoh(ω) = 2πΩeNcoh, which gives photon transfer momentum increases, however, this shift 3/2 rapidly vanishes. Already forq ¯ > 0.1 kF , a value easily  0  ¯ εF εF reached in Raman spectroscopy experiments∼ even with a xZ − pol Li ( ζ ) = ∗ . (C4) 3 2  m 2  small angle between the Raman beams, the shift is neg- − − 6 m Tp ligible. This establishes a major advantage of Raman spectroscopy over conventional rf spectroscopy. In the inset of Fig. 9, we calculate the shift in the Appendix D: Raman transition rate of the rf spectroscopy peak at unitarity as a function of tem- incoherent polaronic and molecular contributions perature, at finite impurity density. The calculated rf shift initially decreases and then increases with temper- Here we derive the response of a thermal ensemble of ature. This non-monotonic dependence follows the trend molecules, each of which is made of a single impurity and of the initial-to-final state energy gap at the maximally- a single bath particle and considered to be in vacuum. As populated momentum value. Interestingly, even without a wavefunction Ansatz for the molecule, we use considering a finite-temperature reservoir, we find that k X k > cm cm † † for Tp 0.1 TF , the shift increases with temperature, as ψ = γl c ldl+kcm 0 . (D1) | i − | i observed∼ in Ref. [11] and discussed in Refs. [43, 110–112]. l 18

0.5 0.1 ) ) ω ω ( ( bg coh 0.4 P P

0.3 1 1 (kF a)− = 0.6 (kF a)− = 0 0.2 1 (kF a)− = 1.2 1 (kF a)− = 0.3 0.1 1 (kF a)− = 1.6 1 (kF a)− = 0.6 Transition probability, Transition probability, 0 0 0 5 10 15 0 10 20 30 40 Raman two-photon detuning, ω (εF /¯h) Raman two-photon detuning, ω (εF /¯h)

FIG. 10. Comparison of coherent part Raman spectra. FIG. 11. Comparison of background Raman spectra. Using the full many-body model for the Raman spectra (first Using the full many-body model for the Raman spectra (inco- term of Eq. (20), solid lines), and the approximation due to herent and molecular terms in Eq. (5), solid lines), and the ap- Eq. (13) (dashed lines), the coherent Raman rate is shown for proximation obtained by fitting the simplified model Eq. (13) three interaction strengths. In these calculations we use the to the theoretical spectra (dashed lines), Raman spectra are effective mass and polaron energy obtained from the polaron shown for three interaction strengths. For clarity, the second Ansatz. For clarity, the second and third graphs from the and third graphs from the bottom are shifted by 0.02 and bottom are shifted by 0.1 and 0.2, respectively. 0.04, respectively.

Note that the only difference to Eq. (3) is that, here, we of the impurity problem. To this end, we compare the do not consider the Fermi sea of background particles. two parts of the fitting function to theoretical calcula- This simplification is made in order to obtain a closed- tions. form expression for the fitting function which is feasible Coherent polaron response.— In Fig. 10 we to calculate numerically. Using Eq. (D1) as the initial present a comparison between the first part of the fitting state i for Eq. (4), one obtains | i model, namely Pcoh (dashed lines), and the full solution of our theoretical model introduced in Section III (solid ω, ψkcm = (D2) A lines). As can be seen, the approximation is excellent at X 2 k2 1 kcm cm unitarity and at (kF a)− = 0.3. Closer to the predicted γl δ ω εl εkcm l+q + Eb + . − − − 4m transition, minor differences develop at k k . There l q ≈ F are two causes to this behavior. First, the increase of m∗ kcm leads to a small asymmetry. Second, the polarons do not The variational parameter γl has to be obtained by the minimization of ψkcm E ψkcm . Within this calcula- populate high momentum states since the width of the tion the relative momentumh |H− after| thei Raman dissociation excitation branch increases dramatically as the momentum increases, leading to a narrowing of the theoretical is given by krel = l + (kcm + q¯)/2. After changing vari- Raman spectrum. Importantly, the center peak position ables l to krel, evaluating the δ-function implements the energy conservation of Eq. (14). The probability F (k ) coincides for both spectra, which allows us to use Eq. (17) rel 0 of Eq. (15) is then given directly by the matrix element for the extraction of pol. γkcm 2. Averaging the Raman spectral func- Background signal.— Here, we analyze the applica- krel (kcm+q¯)/2 | − | bility of the second part of our fitting function, P , to fit tion over all momenta k weighted by a thermal Boltz- bg cm the background spectrum that combines the incoherent mann distribution yields and molecular contributions. In Fig. 11, we compare the Z best fit of Pbg to the background signal, as calculated ¯(ω) = dk ω, ψkcm G (k ,T ) . (D3) A cm A · cm bg by the full many-body model. Overall they match well, especially at high frequencies. The difference at low fre- The final expression in Eq. (16) is obtained by subsequent quencies stems from the neglect of the majority specie’s integration of kcm over directions perpendicular to q¯. Fermi surface in the fitting model. We should consider systematic errors in extracted observables that may arise due to this approximation. Tbg Appendix E: Validation of the fitting model affects almost solely the low-frequency part of the spectrum. Therefore, it should be chosen to compensate for In this appendix we discuss the applicability of our fit- the absence of Pauli blocking in our fitting model and ting model, i.e., Eqs. (13) and (16), to the Raman spectra minimize errors in the extracted Z¯. The effective bind- 19

T = 0.2T , T , 2T , 4T mol pol nI =0 nI =0.15 n nI =0.15 n bg F F F F T =0 T =0 T =0.2 T 1 F 2 1/2 Z¯simulated Z¯ﬁtted −

¯ 1 Z

D E µ

0.75 5% − min

E 0.5 0.5 0 0 2 4 Tbg (TF ) 0.25 0 Energy gap, Quasiparticle weight,

0 0.5 0 0.5 1 − 0.5 0 0.5 1 1.5 2 1 − 1 Interaction, (kF a)− Interaction, (kF a)−

FIG. 12. Fitting simulated background Raman spectra FIG. 13. Polaronic and molecular energy gap. Energy with fixed effective temperature Tbg. The black line de- gap between the lowest-lying polaron/molecule modes and the notes the homogeneous many-body quasiparticle weight com- chemical potential vs. interaction strength. The blue lines puted for T = 0.2 TF , x = 0.15. Errorbars mark the fitted show the gap for the case of a single impurity at T = 0, while residue using an effective temperature of 0.2 TF (yellow di- the red and black lines show the gaps for finite density at amonds), TF (blue circles), 2 TF (black triangles), and 4 TF T = 0 and T > 0, respectively. The solid lines with circle (red squares). Inset: Root-mean-square deviation of the ex- markers denote the gap between the lowest-lying molecular tracted residue from the computed one, exhibiting a minimal state and the chemical potential, whereas the dashed lines deviation at approximately 2 TF , the value we use for fitting with triangle markers denote the polaronic gaps. the experimental data.

the molecule are populated according to fermionic and ing energy, Eb, on the other hand, affects mainly the bosonic statistics, respectively. high-frequency part of the spectrum, where the fit and Tuning the chemical potential µ yields a van- numerical data are in excellent agreement. ishing density. As the chemical potential→ −∞ is increased, We find the optimal value for Tbg by fitting theoreti- the two bands begin to be populated accordingly such cal Raman spectra of the background signal (incoherent that a small impurity density begins to form. Tuning the polaron and molecule) due to Eq. (5) at eight interaction chemical potential further, eventually it will reach the strengths. In Fig. 12, we plot the fit results for the quasi- minimum of the lower-lying dispersion. If the polaron particle residue, obtained with four exemplifying values dispersion is lower-lying, it may surpass the minimum, of Tbg. The effect of varying Tbg is a systematic shift of effectively forming a Fermi surface. Tuning even further, the residue. The inset of Fig. 12 presents the root-mean- the chemical potential will reach the minimum of the square difference between the simulated and the fitted molecule dispersion. As we populate according to bosonic quasiparticle residue as a function of the fixed value for statistics, at sufficiently low temperature, the molecules the effective temperature. We observe a minimal discrep- will thus begin to condense in the corresponding state ancy at Tbg 2 TF . Fig. 12 clearly shows that even at and a molecular BEC will form within the fermionic bath. ≈ ¯ sub-optimal values of Tbg, the qualitative behavior of Z The chemical potential may therefore not surpass this ¯ does not change. The reason for this is that Z measures minimum and will remain pinned to it even for larger the spectral weight of the roughly symmetric peak, and impurity densities. therefore it is rather insensitive to variations in the fitting In Figure 13 the energy gap between the chemical po- procedure. tential and the lowest-lying polaronic (triangles, dashed) and molecular (circles, solid) states is shown for different interaction strengths. We show this gap for the case of Appendix F: Setting the chemical potential a single impurity (blue), at finite density (nI = 0.15 n) and T = 0 (red) as well as at finite density and finite In this appendix we elaborate on the behavior of temperature (T = 0.2 TF , nI = 0.15 n, black). In the the chemical potential with respect to the polaron and single impurity limit the, the chemical potential is tuned molecule dispersions, intending to highlight the mecha- to the minimum of pol(p) below the transition and to nism by which the bands are populated. Throughout the minimum of mol(p) above. Thus, below the transi- this manuscript, we tune the chemical potentials such tion the gap of the molecular dispersion marks the energy that under application of Eq. (19) the respective densi- gap between the polaronic ground state and the excited ties are reproduced, which means that the polaron and molecular state, and vice versa above the transition. 20

At finite density and zero temperature, below the tran- Appendix G: Single-impurity contact coefficients sition a Fermi surface of polarons forms. When the chemical potential approaches the lowest-lying molecular state 1 Here we provide generalized expressions for the single- (solid, red) at around (k a)− 1.1 molecules begin to F ≈ impurity contact at finite momentum, as derived for condense. Beyond the transition, the polaronic Fermi p = 0 in Refs. [38, 113]. They read surface vanishes (positive energy gap) and all impurities condense in the molecular ground-state. At finite Cpol [p, pol(p)] = (G1) temperature, a polaronic Fermi surface forms initially 2 p 2 1 1 X 0 m α ((kF a)− < 0.2), but eventually the polaron and the 0 | | 2 V 0 molecule are∼ both populated thermally, as visible from q 1 1 P 1 U V k pol(p) εk εq−k+p+εq their positive energy gaps. Note that the lowest point of − − − the dispersion relations does not necessarily lie at p = 0. for the polaron, and

Cmol [p, mol(p)] = (G2) 1 " 2#− 1 X 0 1 m2 V (p) + ε εk εp k k mol F − − + for the molecule. From this, the full many-body contact coeﬃcient C can be calculated via Eq. (23).