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

Understanding the behavior of a spin impurity strongly-interacting with a Fermi sea is a long-standing challenge in many-body physics. A central question is under what conditions the impurity can be described as a quasiparticle, called polaron. For short-range interactions and zero temperature, most theories predict that beyond a critical interaction strength, a first-order phase transition will occur from a polaronic ground state to a molecular one. We study this question with a spin-imbalanced ultracold Fermi gas with tunable interactions, utilizing a novel Raman spectroscopy probing technique that allows us to isolate the quasiparticle contribution. We find that for increasing interactions, there is a smooth transition from a polaronic to a molecular response, with no evidence of a first-order phase transition. Nonetheless, the polaron contribution almost vanishes above the expected transition point. From the Raman spectra, we determine the polaron energy, molecule binding energy, and the contact parameter. The energies agree well with theoretical predictions and connect smoothly around the vanishing point of the polaron. The contact parameter follows the molecular branch, in contrast to the prediction that it will have a clear change of behavior as the ground state changes its nature. The overall emerging physical picture based on our measurements is of a smooth transition between polarons and molecules and coexistence of both in the region around the expected phase-transition.


I. INTRODUCTION
To understand the motion of an electron through an ionic lattice, Landau suggested treating the electron and the phonons that accompany its movement as a new quasiparticle named "polaron" [1]. The useful concept of the polaron was later found to be applicable in many other systems, including semiconductors [2], hightemperature superconductors [3], alkali halides insulators [4], transition metal oxides [5], and 2D materials embedded in a cavity [6]. Polarons can be weakly or stronglycoupled, depending on their binding energy, and they are classified as large or small, depending on the size of the distortion they generate in the material periodic potential [7]. Understanding their properties is still an ongoing effort [8][9][10]. One of the simplest scenarios in which polarons naturally emerge is when a single spin impurity is immersed in a system of free fermions with the opposite spin. Ultracold gases are ideally suitable to explore polaron physics [11], thanks to extremely long spin-relaxation times and tunability of the s-wave scattering length, a, between the impurity and the majority atoms via a Feshbach resonance [12].
Experiments with spin-imbalanced Fermi gases at unitarity (a → ∞) revealed phase separation into three regions: an inner core of spin-balanced superfluid, a second shell of a partly polarized normal gas, and a third shell of fully polarized gas [13][14][15]. Chevy pointed out that the radius between the outer and intermediate shells is related to the solution of the impurity problem [16]. For weak attractive interactions, the ground state is a long-lived quasiparticle dressed by the majority particles * Electronic address: yoavsagi@technion.ac.il (attractive polaron). For repulsive interactions, a metastable polaronic state also exists (repulsive polaron), but it becomes progressively unstable towards unitarity. Polarons have well-defined momenta with a narrow dispersion relation and a renormalized effective mass [17,18]. The attractive polaron ground state persists even as the interactions increase towards unitarity, but for still larger interactions the system favors a molecular ground state dressed by the majority spins. Most theories predict that the transition between these two different ground states is of first-order nature and will occur at a critical interaction strength of (k F a) −1 c ≈ 0.9 (k F is the Fermi wave-vector) [18][19][20][21][22][23], but a contrasting claim for a smooth crossover was also put forward [24]. At a finite impurity concentration, the phase diagram becomes more complex. At T = 0, the polaron-molecule phase transition may be pre-empted by phase separation between the superfluid and the normal phases, both of which can be balanced or polarized [25]. At finite temperature, however, increased thermal fluctuations are expected to suppress the superfluid and restore, at least qualitatively, the polaron-molecule transition.
Experimentally, the attractive polaron was initially studied by rf spectroscopy and identified by a narrow peak appearing only in the minority spectrum [26]. The spectral weight of this peak was interpreted as the quasiparticle residue, Z, which measures the overlap of the state with a non-interacting impurity state. Z was observed to continuously decrease and vanish above a certain interaction strength. When Z is zero the quasiparticle description is no longer valid. The measured Z did not agree with theoretical predictions based on the Chevy ansatz wave-function [21], and it did not exhibit a jump in Z, which would be expected in a first-order transition. A different approach to measure Z was employed in Ref. [27], using coherent oscillations between the po-laron and a non-interacting state. In that measurement, Z did not vanish beyond (k F a) −1 c , most likely because the coherent oscillations may involve the polaronic state even when it is not the ground state. Other parameters of both the attractive and repulsive polarons were also determined, including the effective mass [28,29], equation of state [30], energy [26,27,29,31], and formation dynamics [32]. Whether the ground state undergoes a phase transition at a finite temperature is still an open question.
In this work, we address this question with a spin-imbalanced ultracold Fermi gas in the BEC-BCS crossover regime [12,33]. To this end, we employ a novel spectroscopic method based on a two-photon Raman transition. In conventional rf spectroscopy, the photon momentum is negligible and the atomic momentum is essentially unchanged in the transition. As a result, the transition probability depends only weakly on the atom velocity and the maximal signal is attained for atoms that are not at rest. In contrast, in a Raman process the momentum change is significant compared to the atomic momentum, hence the transition rate depend on the atomic velocity. For quasi-particles, the Raman spectrum directly reflects the momentum distribution. Hence, we can separate the spectrum into two parts, symmetric and asymmetric, which correspond to polarons and dressed molecules, respectively. Each of them gives access to important observables, and their relative weight gives the quasiparticle residue. In addition, the momentum sensitivity allows us to identify the zero-momentum polaron, and determine its energy. We also extract from the Raman spectra the contact parameter, and determine its dependence on the interaction strength. Based on the measurements, our main finding is that polarons smoothly vanish as 1/k F a increases, accompanied by a growing contribution from a molecular-like state. This behavior supports a smooth transition and not a first-order phase transition.

II. THE EXPERIMENTS
Our experiments are performed with a harmonicallytrapped ultracold gas of 40 K atoms prepared in an incoherent mixture of the two lowest energy states |1 and |2 , with the majority of atoms being in state |1 (Fig. 1). The cooling sequence is similar to the one described in Ref. [34], only here we tune the parameters such that at the end of the optical evaporation stage 65% of the atoms are in state |1 . Then, we ramp the magnetic field to a value of 201.75G, where three-body processes remove all the atoms in state |2 . We are left with ∼ 100, 000 atoms in state |1 held in an optical dipole trap at a temperature of T /T F ≈ 0.2, where T F is the Fermi temperature. The magnetic field is ramped adiabatically to the BCS side of the Feshbach resonance, where the interactions are weak (204.5G-206G). A short (few microseconds) rf pulse transfers a very small fraction of the atoms to statex 14G [34]. The counter-propagating Raman beams (wiggly blue lines) are pulsed for 500µs and couple atoms in state |2 to state |3 = |F = 9/2, mF = −5/2 , which is initially unoccupied. Afterward, we detect the number of atoms in state |3 [34]. The single-photon Raman detuning is ∆ = −2π × 54.78(8)GHz relative to the D2 transition. (b) 3D sketch of the beam configuration in the experiment. Two Raman beams (blue lines with arrowheads) with orthogonal polarization overlap the atomic cloud (yellow), which is being held in an elongated crossed-beams optical dipole trap (red lines). The optical trap oscillation frequencies are ω radial = 2π × 238(3)Hz and ω axial = 2π × 27(2)Hz, in the radial and axial directions, respectively. The gravitational acceleration direction is −ẑ.
|2 , and we wait another 100ms for the two states to fully decohere. Finally, the field is ramped adiabatically to its final value, where we wait another 3.3ms before giving the 500µs Raman pulse.
To account for the non-uniform atomic density, we compared all measurements to the average quantities calculated in the local density approximation. To this end, we assume the distribution of the majority atoms is not affected by the presence of the minority atoms; hence it can be calculated as for non-interacting fermions. The minority density distribution, n 2 (r), is calculated by taking into account the interactions with the majority atoms through a renormalization of the confining potential: V 2 (r) = V 1 (r) 1 − Ep E F , where V 1 (r) is the potential felt by the majority atoms and E p is the polaron energy [35]. We neglect the weak interactions between polarons [36]. The expected value of any observable, A, is then given by the minority-weighted local density . Specifically, the minority concentration, x = n 2 / n 1 , can be either calculated globally by x = N 2 / N 1 , with N i being the total number of atoms at state |i , or by averaging over its local value, x . Since n 2 (r) depends on k F a through E p , x changes even when x is kept constant. To ensure there are no systematic deviations in the experiments due to this effect, we have repeated the measurements twice: one time keeping x at approximately 0.04, which gives x ≈ 0.22 at (k F a) −1 = 0.9, and a second time maintaining x ≈ 0.23. We did not observe any significant difference between the two datasets, therefore in what follows we shall present their results together.
The two Raman beams couple atoms in the minority state |2 to a third state |3 , which is initially unoccupied (Fig. 1a). The beam parameters are the same as in Ref. [37]. We denote their frequencies by ω 1 and ω 2 and wave-vectors byk 1 andk 2 . The measurement is performed by recording the number of atoms transferred to state |3 versus the two-photon detuning, ω = ω 1 − ω 2 − E 0 /h, where E 0 is the bare transition energy between states |2 and |3 . To achieve the utmost sensitivity, we measure the atoms using a high-sensitivity fluorescence detection scheme we have recently developed [34,37].
In Fig. 2 we depict three representative datasets taken on the BCS side ((k F a) −1 = −0.66, blue circles), unitarity ((k F a) −1 = −0.06, red squares) and BEC side ((k F a) −1 = 0.75, black triangles). The spectrum is symmetric on the BCS side, but becomes asymmetric towards unitarity with a tail at high frequencies that grows to be the dominant spectral feature on the BEC side. The symmetric and asymmetric parts of the spectrum are associated with the response of polarons and pairs, respectively. The reason for the symmetric response of the quasiparticles is that the Raman transition rate in this case is proportional to the one-dimensional momentum distribution, which is symmetric at equilibrium (k → −k). Pairs, on the other hand, are dissociated by the Raman process, which opens another degree of freedom, namely the relative motions of the two atoms. This gives rise to the asymmetric energy tail in the Raman spectra. As can be seen in the inset of Fig. 2, this tail has a powerlaw scaling of ω −3/2 at large ω. This is expected, since the photon energy is negligible compared to the relative kinetic energy, hence the behavior is very similar to rf spectroscopy [38][39][40].

III. QUANTITATIVE ANALYSIS OF RAMAN SPECTRA
Each Raman spectrum is normalized to unity such that it has the meaning of transition probability. The data is (1) (solid lines). The dark shaded area under the graph is the pairs' contribution P mol (ω, T mol , E b ), while the light shaded area is the quasiparticles' contribution Pp (ω, Tp, Ep, m * ). The ratio of the latter to the total area under the graph is the quasiparticle residue Z. We extract the polaron energy, Ep, from the peak position (dotted vertical lines). The second and third graphs from the bottom are vertically shifted for clarity by 0.1 and 0.3, respectively. Each point is an average of three experiments, and the area below each curve is normalized to unity. Since the measurement is done up to some maximal frequency, we must account for the missing spectral weight in the unmeasured tail. This is done by adding to the normalization the integral of ω −3/2 up to an energy cut-off ofh 2 /mr0, where r0 is the effective range of the inter-particle potential [12]. In the inset we depict the data at (kF a) −1 = 0.75 multiplied by ω 3/2 to demonstrate the power-law scaling of the high-frequency tail. fitted with a model: where P p (ω) and P mol (ω) are the transition probabilities of polarons and pairs, respectively. Our model has six parameters. For the polarons, T p is the temperature, E p is their energy (or chemical potential), and m * is the effective mass. For pairs, T mol is the temperature and E b is the binding energy. E p is not a fitting parameter since it is determined directly from the polaron peak position, as we explain below. We have found that in the expected range of values for the polaron effective mass, it has a small effect on Z and E b (see Appendix B). The effective mass, however, is strongly coupled to the polaron temperature, and therefore we do not leave it as a free parameter but rather set its value to the Chevy ansatz calculation [17]. This leaves us with four fitting parameters: Z, E b , T p and T mol . In what follows, we develop the expressions for P p (ω) and P mol (ω), and in the next section we present the results of the fits.
When state |3 is very weakly interacting with the other atoms, as is the case in 40 K, the Raman transition rate is given by [41,42]: withk being the wave-vector of the interacting minority atom before the Raman transfer. Here and onward we use the natural Fermi units, namely E F , k F and T F of the majority atoms, and effective mass will be in units of the bare mass. µ is the chemical potential,q =k 1 −k 2 is the relative two-photon wave-vector, Ω e is the effective Rabi frequency of the Raman transition, A k , ω is the spectral function, and n F (E) is the Fermi-Dirac distribution function. The total Raman rate is obtained by summing over all available momentum states: To proceed with the analysis, we have to provide the spectral function. We employ here a simple model that was found to be in excellent agreement with the measured spectral function of a balanced Fermi gas in the BEC-BCS crossover [43]. The model includes two contributions: a coherent narrow dispersion of polarons with well-defined momentum, and a broad response due to incoherent thermal mixture of pairs. As we show below, we find that this model fits the Raman data very well and enables the extraction of important observables.
Raman rate for polarons. It is instructive to start with non-interacting particles.
In this case, we take A k , ω = δ ω −k 2 + µ , where δ is the Dirac delta function. Γ k , ω is non-zero only when Thus, there is a linear relation between ω and k q ≡k ·q. The total Raman transfer rate, Γ (ω), is then proportional to n 2 (k q ), with n 2 (k) being the one-dimensional momentum distribution of the minority atoms along the direction ofq (see Appendix A and Ref. [37]). The spectral function of polarons can be written as: where Z is the quasiparticle residue, and (k) is the polaron dispersion relation given by k = E p +k 2 /m * . The first term describes coherent polarons with well defined momenta, while the second term, A inc k , ω , describes an incoherent mixture of collision processes between the polarons and majority particles in the vicinity of the Fermi surface [17,26]. When the concentration of impurities is finite, the polarons form their own Fermi surface and therefore can be found at non-zero momentum even without considering these collisions [44]. Similar to the free fermions case, a polaron with a wave-vectork contributes to the Raman signal ifk k q (ω). Unlike the free fermions case, the solution has a weak dependence on k 2 ⊥ and 1 − 1 m * , where k 2 ⊥ = k 2 − k 2 q . However, this dependence is only noticeable when k q is close to k F and when the effective mass is substantially larger than the bare mass (see Appendix A). Therefore, it is an excellent approximation to use: The coherent part of the spectral function then yields in Eq. (3) a rate which is proportional to the onedimensional momentum distribution of the polarons in the direction ofq, which in the local density approximation is given by where k q (ω) is the solution of Eq. (5), x = N 2 /N 1 is the impurities concentration, Li 5/2 is the polylogarithm function, and ζ 2 is the fugacity of the minority atoms. To ensure normalization ∞ −∞ n 2 (k q )dk q = 4π 3 , the fugacity must satisfy the equation: The incoherent part of the spectral function is expected to produce a tail of high momentum polarons [44], which should be symmetric to k q → −k q . In the Raman spectra data, however, we do not observe such a symmetric tail which is not accounted by the polaron momentum distribution discussed above. In particular, in all datasets the signal falls rapidly for k q < −k F (which corresponds to the region near and below zero Raman detuning). This is true even when the polaron peak is small and Z is close to zero. We therefore conclude that the incoherent excitations in the polaron state are either too small to be observed or relax and occupy the molecular branch. Accordingly, we omit the contribution of A inc to the Raman transfer rate. The final expression for the Raman transition probability due to polarons is thus where the 2q factor corresponds to the integration conversion dk → dω by Eq. (5).
Raman rate for pairs. The second part of the model is based on a thermal distribution of bound |1 -|2 pairs. We consider a pair with a binding energy E b and a centerof-mass wave-vectork cm . The Raman process dissociates the pair and changes the center-of-mass wave-vector tō k cm +q, and in addition, the unbound fermions acquire a relative wave-vectork rel . Energy conservation yields: where k cm,q is the projection ofk cm along the direction ofq. The probability that a pair will be dissociated with a relative wave-vector k rel is given by [38]: For k cm,q we take a thermal Gaussian distribution, G, with a temperature T mol . For the temperatures at which we typically work, it is an excellent approximation of the molecular Bose distribution. The combined probability to find a pair with an initial k cm,q and then k rel after the dissociation is given by the multiplication of the F and G. To relate this to the Raman transition, we change variables from k cm,q , and k rel to k cm,q , and ω using Eq. (9). Finally, we integrate over k cm,q to obtain the Raman transition probability for pairs, 2 denotes a shifted detuning. This integral does not have an analytic solution, but it can be readily calculated numerically.

IV. RESULTS
The maximum transition rate as given in Eq. (8) is obtained for k q = 0, which allows us to directly probe the zero-momentum polaron and gives its energy. According to Eq. (5), this maximum is attained for Hence, by determining ω 0 we can determine the polaron energy. We find the peak position by fitting the points above the median to a skewed Gaussian [45] (dotted vertical lines in Fig. 2). The resulting polaron energies are plotted in Fig. 3 (blue circles). We compare our data and find very good agreement with theoretical calculations based on the Chevy ansatz (dashed line), which in turn are very close to diagrammatic Monte-Carlo and T-matrix calculations [11,17,[19][20][21].
We find that our model provides an excellent description of the data throughout the BEC-BCS crossover (solid lines in Fig. 2). The light and dark shaded areas beneath the line are the spectral contributions of polarons and molecules, respectively. The BCS-side data (blue circles) with Z = 0.89(9) is dominated by the symmetric quasiparticle peak, while the BEC-side data (black triangles) with Z = 0.09(5) is dominated by the asymmetric pair dissociation spectra. The unitary data (red squares) shows both the symmetric peak and an asymmetric tail, and the quasiparticle residue is Z = 0.43(3), very close to the value of 0.47(5) that was measured for 6 Li atoms with rf spectroscopy [26].
Polaron energies (blue circles) and pairs binding energies (black triangles) for various interaction strengths (kF a) −1 . The polaron energies, Ep, are obtained using Eq. (12) and the position of the spectral peak, ω0, which is determined by fitting the highest 50% points in each spectrum to a skewed Gaussian [45]. We plot the polaron energies only in the range where the polaron actually exists -at interactions weaker than the vanishing point of the linear fit in Fig. 5, (kF a) −1 c = 0.91. We find good agreement with local density approximation calculations based on the Chevy ansatz (dotted blue line) [16,21]. The pairs binding energy is determined by fitting the Raman spectra with Eq. (1). For (kF a) −1 > 1, E b is in good agreement with the mean-field prediction (solid black line): [20]. As a reference, we also plot the two-body binding energy −2 (kF a) −2 (dashed black line).
The quasiparticle residue extracted from the fits is shown as blue circles in Fig. 4. The residue on the BCS side approaches unity, as expected. For increasing (k F a) −1 , we observe a gradual decrease of Z. Above a certain interaction strength, (k F a) −1 c , the values of Z are close to zero. We do not observe any sharp change of Z that would indicate a first-order phase transition. By fitting a linear function to the decreasing points (solid line in Fig. 4), we obtain (k F a) −1 c = 0.91 (7). This is in good agreement with the value of 0.90(2) calculated by diagrammatic MC [19,20] and 0.84 calculated by variational approach [21]. For comparison, we plot in Fig. 4 the data of the MIT group [26] (red squares), which were extracted from rf spectroscopy measurements. Their data show similar behavior, albeit with a somewhat smaller value of (k F a) −1 c = 0.76 (2). In order to test whether the behavior of Z could be explained by the density inhomogeneity of the trapped gas, we calculate the trap average Z using the Chevy ansatz [16,21] in a local density approximation. Our data clearly disagree with the result of this calculation (dashed line). Note that since the quasiparticle residue essentially measures the spectral weight of the symmetric peak, it is insensitive to variations in the fitting procedure and the behavior displayed in Fig. 4 stays qualitatively the same.  (7) it is very close to zero. For comparison, we plot the theoretical prediction based on the Chevy ansatz (dashed black line) [16,21] averaged over the harmonic trap using the local density approximation. We also plot the experimental data of the MIT group [26] (red squares).
We now turn to examine the molecules binding energy, E b , which is extracted from pair response part in the Raman spectra and is plotted in Fig. 3 as black triangles.
c , our data agree with a theoretical mean-field calculation of the pair binding energy (solid black line) [20]. For comparison, we also plot the two-body binding energy in vacuum (dashed black line), which, as expected, is smaller than the measured energies since it does not include the interactions between the molecules and the surrounding Fermi sea. The polaron energies connect smoothly to the pairs binding energy around (k F a) −1 c . The molecules binding energy is related to another important quantity that can shed light on the polaronmolecule phase transition -the contact parameter, C. The contact quantifies short-range correlations of unlike fermions [33]; hence it is expected to jump at a first-order phase transition between an unpaired (polaronic) and paired ground states [21]. Since our measurements are done in a trap, the discontinuous change in C is smeared, but even so a clear change in its behavior is expected in the transition (solid black line in Fig. 5). A measurement of the contact can therefore point to a change in the nature of the many-body state. The contact associated with the molecules in our model can be written as [39,43] where C is given in units of 2N 2 k F . We plot the result of this calculation as blue circles in Fig. 5. The extracted contact is in reasonable agreement with the local den-  13), with E b and Z extracted from the Raman spectra, versus the interaction strength. The theory of a single impurity at zero temperature (dashed line) predicts a discontinuous change in C due to the polaron-to-molecule phase transition [21]. When the same theory is calculated for a harmonically trapped gas using the local density approximation (solid line), this feature is smoothed, but the change of behavior should be clearly observable. Our data is significantly different from the prediction in the polaron regime. The two red squares indicate data points measured by the MIT group [46] using rf spectroscopy of homogeneous unitary 6 Li gas at T = 0.17TF (lower point) and T = 0.29TF (upper point).
sity calculation based on a molecular variational wavefunction [21] for (k F a) our data seems to follow the molecular branch and is markedly larger than the contact expected for polarons, as calculated by the Chevy variational ansatz. For comparison, we plot in Fig. 5 the contact measured by the MIT group with a homogeneous 6 Li gas using rf spectroscopy [46] (red squares).

V. DISCUSSION
In this work we have investigated the attractive Fermi impurity problem using a new Raman spectroscopy method. The main difference of this technique relative to rf spectroscopy is that the momentum transfer imparted by the two-photon transition is significant compared to the atomic momentum, i.e. about twicehk F in our experiments. Through the conservation of energy and momentum, the Raman transition rate depends on the occupied single-particle spectral function [41,42]. Since we probe an inherently small signal coming only from the impurity species, we employed a high-sensitivity fluorescence detection scheme, with which we can reliably measure signals of only few atoms [34]. Using a simple model for the spectral functions of polarons and thermal ensemble of pairs, we extracted from the Raman spectra several important observables. In particular, the velocity selective nature of the Raman transition creates a one-to-one correspondence between the Raman detuning and the polaron momentum. This allows us to identify the zero-momentum polaron and determine its energy, E p . It also naturally divides the spectrum into a symmetric part, which corresponds to the one-dimensional momentum distribution, and asymmetric part, which corresponds to pair dissociation. The relative weights allow determination of the quasiparticle residue, Z. From the tail of the asymmetric spectrum we extract the pair binding energy, E b . Finally, we determine the contact parameter, C, of the imbalance gas using two methods: from the measured Z and E b , and by performing an independent rf spectroscopy measurement.
The physical picture that arises from our measurements is quite intriguing. The measured energies of both polarons and pairs are in excellent agreement with the prediction of the theory. However, we do not observe a signature of a first-order phase transition that would manifest itself as a clear change of slope in Z and C. The residue shows a linear decreasing trend, and above (k F a) −1 = 0.91 it is consistent with zero. The quasiparticle gradually vanishes for increasing interaction. At the same time, the contact follows the molecule branch, even in the region where the residue is non-zero. This points to a co-existence of both polarons and pairs in this region.
There are three main differences between our experi-ments and theoretical models: the gas is at a finite temperature, the impurity concentration is finite, and the gas is not homogeneous due to the harmonic potential. We have calculated the effect of the latter using local density approximation, and while it definitely widens any sharp feature in the data, the polaron-molecule phase transition should still be clearly visible in both Z and C. We can therefore rule-out the trapping potential from being the explanation of our observations. The finite impurity concentration means polarons populate higher momentum states and have a Fermi surface. In the region where the polaron and molecule branches are very close in energy, this means polarons have available energy to be excited to the molecule branch. This effect is made even stronger by the finite temperature of polarons. Further theoretical investigation is required to understand how the fermionic quasiparticles are excited into the molecule branch which is bosonic in nature. In addition, near the phase transition, the excited branches (polaron or molecule), are very long lived [18]. All these factors contribute to the population of both states near the polaron-to-molecule phase transition which can explain our observations.
For simplicity, we carry the calculation here for a homogeneous system. Eq. (3) can be written in an integral form: Γ(ω) = 3N1 4π Γ(k, ω)d 3k . We use this expression together with Eq. (4) to proceed with the calculation of the latter's first (coherent) term. We then divide the integration overk into the direction ofq and the perpen- In these calculations we use the effective mass and polaron energy obtained from the Chevy ansatz [17]. dicular directionk ⊥ . The integral over k q yields: where k q k ⊥ , ω is a solution to the equation: (A2) As expected, the Raman signal is proportional to the volume (or the number of atoms).
For non-interacting particles we take Z = m * = 1. This simplifies considerably Eq. (A1) and we obtain (A3) Note that in this case k q is only a function of ω. We arrive at the result: where we identify the integral of the Fermi function over the perpendicular direction as the momentum distribution in the direction of q: n q (k q ), and the normalization with the majority k F contributes the N 2 / N 1 factor. Therefore, as k q is only determined by ω, we get that the Raman rate is proportional to the one-dimensional momentum distribution in the direction ofq. It is easy to verify that Γ(ω) given by Eq.(A4) obeys the sum rule ∞ −∞ Γ (ω) dω = πΩ 2 e N 2 [47,48]. For interacting particles, m * = 1 and the denominator in Eq. (A1) depends on k q k ⊥ , ω and therefore cannot The polaron residue and molecular binding energy extracted with different fit parameters. Blue circles denote the results with T mol as a free fit parameter and effective mass given by the Chevy ansatz [17], as in the case of the red squares the effective mass was fixed to the bare mass. In the resulted residues we find linear fits that cross zero at (kF a) −1 = 0.91 (7) and (kF a) −1 = 0.89(8) respectively. Keeping the effective mass according to [17] and also fixing T mol to 0.2TF (black triangles) or 0.5TF (green pentagons) we find the Z vanishing points at (kF a) −1 = 0.97 (9) and (kF a) −1 = 0.89(8) respectively. Inset: Binding energy for the different constants choices, presents weak dependence.
be taken out of the integral. However, at low temperatures and for most interaction strengths k q 1 − 1 m * q where the Fermi function is not negligible. Therefore, we can replace Eq. (A1) with the approximated expression Eq. (A4), and solve k q (ω) assuming k ⊥ = 0. To demonstrate this point, we plot in Fig. 6 the comparison between a full numerical integration of the exact expression in Eq. (A1) and the approximation of Eq. (A4). As can be seen in the figure, the approximation is excellent at unitarity and at (k F a) −1 = 0.5, and only at (k F a) −1 = 0.9, near the predicted transition, there are some minor but noticeable differences that exist close to k q ≈ k F . Importantly, the center position is the same for both spectra, which allows us to use Eq. (12) for the extraction of E p . The Raman rate for the coherent polarons is therefore given by Γ p (ω) = Z 3N2Ω 2 e 4q n 2 [k q (ω)] and is proportional to the polaronic momentum distribution.

Appendix B: Effective mass and temperatures
Our detection technique enhances the separation between the symmetric polaronic contribution and the tailed molecular one. The precise extraction of physical observables yet depends on the choice of expected spectra. We implemented a simple model as described in Section III, and here we present its robustness to various choices of problem parameters.
As inferred in Appendix A, the contribution of polaron effective mass that differs from the bare mass leads to slight variations in the expected spectra. As can be seen from Eq. (6), the polaron effective mass consistently appears as a multiplicative factor alongside the temperature, thus these two parameters are strongly coupled in the fitting. In order to extract the polaron temperature we therefore fix the effective mass according to the Chevy ansatz calculation [17]. In Fig. 7 we present the effect of this choice on the extracted observables Z and E b . For this comparison, we repeat the analysis with m * = 1. We find that the results using the effective mass (blue circles) differ only slightly from those extracted by fitting with the bare mass (red squares). Once the polaron mass is fixed, we can extract its temperature, as denoted by blue circles in Fig. 8. In the region where the polaronic spectral weight is significant, we find T p to be around 0.3T F , moderately higher than the initial sample temperature of ≈ 0.2T F .
The molecular temperature mostly changes the steepness of the rise at the low-frequency edge of the molecular spectrum. The fitted T mol is shown as black triangles in Fig. 8. When the polaronic spectral weight is appreciable, the molecular contribution is overwhelmed by the polaronic peak, and our method is less sensitive to the former's temperature. This results in a larger scatter in the extracted T mol . For larger (k F a) −1 , T mol approaches 0.4T F . Since our model for molecules neglects manybody effects and interactions, T mol should not directly match the ensemble temperature. Finally, we note that the scatter in both temperatures affects only weakly the robust observables E p , E b and Z. The black triangles (green pentagons) in Fig. 7 present the resulted observables when we fix the molecular temperature to 0.2T F (0.5T F ). For both cases, slight reductions of the residue and binding energy are observed, but the general result kept consistent.