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

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.

Popular Summary

One of the fundamental problems in many-body physics is understanding the behavior of an impurity immersed in a sea of fermions. The impurity can interact with nearby particles to create an effective quasiparticle known as a polaron, or it can form a bound pair with one of those particles to create a molecule. Theories predict that when the strength of interactions is increased, an abrupt transition will occur between these two states, however, the question of whether this happens under realistic experimental conditions has remained open for many years. We address this question, experimentally and theoretically, and find that an abrupt transition occurs only in the theoretical scenario of a single impurity at zero temperature. Otherwise, the system undergoes a smooth transition.

We develop a novel probing technique—based on two-photon Raman spectroscopy—that is sensitive to atomic velocity, which gives us access to the momentum distribution of polarons. With this technique, we investigate a dilute ultracold gas of neutral fermionic atoms whose interactions can be tuned. We then compare our measurements to a theoretical model of polaron and molecule quasiparticles that are thermally occupied according to their quantum statistics. We find that a finite temperature and impurity density leads to a smooth transition between polarons and molecules and a coexistence of both in the region around the expected transition.

Our results establish Raman spectroscopy as a powerful tool for probing the physics of mobile quantum impurities and shed light on the competition between emerging fermionic and bosonic quasiparticles.

Figure 1

Experimental setup. (a) Level diagram of the relevant states in ${}^{40}\mathrm{K}$. The majority of atoms occupy state $|1⟩=|F=9/2,m_F=-9/2⟩$ in the $4^2{\mathrm{S}}_{1/2}$ manifold, and their interaction with the minority atoms in state $|2⟩=|F=9/2,m_F=-7/2⟩$ can be tuned in the vicinity of the Feshbach resonance at $B\approx 202.14\mathrm{G}$ [56]. The counterpropagating Raman beams (wiggly blue lines) are pulsed for $500\mu \mathrm{s}$ and couple atoms in state $|2⟩$ to state $|3⟩=|F=9/2,m_F=-5/2⟩$, which is initially unoccupied. Afterward, we detect the number of atoms in state $|3⟩$ [56]. The single-photon Raman detuning is $\mathrm{\Delta }=-2\pi \times 54.78(8)\mathrm{GHz}$ relative to the $D_2$ transition. (b) 3D sketch of the beam configuration in the experiment. Two Raman beams with orthogonal polarization (blue lines with arrowheads) 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 ${\omega }_{\text{radial}}=2\pi \times 238(3)\mathrm{Hz}$ and ${\omega }_{\text{axial}}=2\pi \times 27(2)\mathrm{Hz}$, in the radial and axial directions, respectively. The gravitational acceleration direction is $-\widehat{z}$.

Figure 2

Raman spectra of an imbalanced Fermi gas in the BEC-BCS crossover. The Raman transition probes the minority atoms, which represent around 4% of the whole cloud. The data are fit with the function given in Eq. (6) (solid lines). The dark shaded area under the graph is the combined contribution from molecules and the incoherent part of the polaron wave function, $P_{\mathrm{bg}}(\omega ;T_{\mathrm{bg}},E_b)$, while the light shaded area is the coherent polaron contribution, $P_{\mathrm{coh}}(\omega ;T_p,{\epsilon }_{\mathrm{pol}}^0,m^{*})$. We extract the polaron energy ${\epsilon }_{\mathrm{pol}}^0$ from the peak position (dotted vertical lines), shifted by the recoil energy due to the finite Raman momentum transfer of $\overline{q}\approx 1.9k_F$. The second and third graphs from the bottom are vertically shifted for clarity by 0.1 and 0.3, respectively. In the inset, we depict the data at ${(k_{F}a)}^{-1}=0.75$ multiplied by ${\omega }^{3/2}$ to demonstrate the power-law scaling of the high-frequency tail. 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, to properly normalize the data we must account for the missing spectral weight in the unmeasured tail. This is done by adding to the normalization the integral of ${\omega }^{-3/2}$ up to an energy cutoff of ${\hslash }^2/m{r}_0^2$, where $r_0\approx 181a_0$ is the effective range of the interparticle potential [22].

Figure 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) and 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 crossing of the polaron and molecule energies at ${(k_{F}a)}_c^{-1}\approx 1.27$ 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 $N_I=0.15N$ and temperature $T=0.2T_F$. The blue-, red-, and gray-shaded regions represent the coherent and incoherent polaronic contributions from Eq. (2) as well as the molecular contributions from Eq. (3), respectively. The coherent part yields an almost symmetric line shape 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.

Figure 4

Measured polaron and pairs binding energies. The polaron energies ${\epsilon }_{\mathrm{pol}}^0$ (blue circles) are obtained using Eq. (17) from the position of the spectral peak ${\omega }_0$. The theoretical prediction obtained from the variational Ansatz Eq. (2) is shown as a dashed blue line. The $E_b$ parameter (red squares) is determined by fitting the Raman spectra with Eq. (6). For ${(k_{F}a)}^{-1}>0.5$, it is in good agreement with the energy obtained from the simple molecular Ansatz Eq. (3) (solid red line). The dotted red line shows the result of an improved molecular Ansatz [38]. Note that the theoretical polaron and molecule energies are averaged over the harmonic trap (see Appendix pp1), and as a result, they cross at a ${(k_{F}a)}^{-1}$ slightly lower than the predicted polaron-to-molecule transition in a homogeneous gas. Inset: extracted polaron temperatures. The approximate majority temperature is denoted by the dashed line.

Figure 5

Measured quasiparticle weight. The quasiparticle weight $\overline{Z}$ is shown for different interaction strengths. Blue circles mark the weight extracted from the Raman spectra by identifying the nearly symmetric spectral line shape arising from the coherent polaron contribution. $\overline{Z}$ is smoothly decreasing toward the polaron-molecule transition, in agreement with our theoretical prediction, averaged over the harmonic trap using the LDA (solid line). For comparison, we also present the trap-averaged prediction for a single impurity at $T=0$ (dotted line) obtained from Ref. [38].

Figure 6

Measured Tan contact. The contact coefficient $C$ is shown for different interaction strengths, obtained using Eq. (18) with $E_b$ and $\overline{Z}$ extracted from the Raman spectra. The theory for a single impurity at zero temperature predicts a discontinuous change in $C$ at the polaron-to-molecule transition [38]. While this discontinuity is smoothed when averaged for a harmonically trapped gas using the LDA (dotted line), an abrupt change is expected to remain. The data significantly deviate from this prediction. Instead, they agree well with our calculation taking into account the finite temperature and coexistence of polarons and molecules (solid line); see Sec. 6. The two red squares indicate data measured by the MIT group [11] using rf spectroscopy of a unitary, homogeneous ${}^6\mathrm{Li}$ gas at $T=0.17T_F$ (lower point) and $T=0.29T_F$ (upper point).

Figure 7

Calculated polaron contribution. Fraction of impurity particles propagating as polarons as a function of interaction strength ${(k_{F}a)}^{-1}$. The impurity particles which are not polaronic are bound to a bath particle, thus forming a molecule. In the single-impurity limit and $T=0$ (blue), there is a sharp transition between a polaron and a molecule at ${(k_{F}a)}_c^{-1}\approx 1.27$. Finite impurity density ($0.15n$, red) leads to smoothing of the transition for ${(k_{F}a)}^{-1}<{(k_{F}a)}_c^{-1}$. When the temperature is increased ($0.2T_F$, black), the polaronic branch is populated also for ${(k_{F}a)}^{-1}>{(k_{F}a)}_c^{-1}$ and the sharp transition disappears. The inset shows the polaron (blue line) and molecule (red line) dispersions for $n_I=0.15n$ and ${(k_{F}a)}^{-1}=0.6$, along with the chemical potential at $T=0$ (dashed black line) and $T=0.2T_F$ (solid black line). At $T=0.2T_F$, polarons and molecules are populated only by thermal excitations, while at $T=0$ the chemical potential is above the minimum of the polaron dispersion such that a well-defined polaronic Fermi surface forms.

Figure 8

Calculated quasiparticle weight and contact. The quasiparticle weight (a) and contact coefficient (b) are shown for different interaction strengths. In the single-impurity limit (blue), the transition between polaron and molecule at ${(k_{F}a)}^{-1}\approx 1.27$ leads to a sharp jump between the polaronic and molecular residues and contacts. As in Fig. 7, at $T=0$ and finite impurity density ($0.15n$, red) the transition is smoothed, and eventually blurred at finite impurity density and temperature ($0.2T_F$, black). The inset in (b) shows a magnification around the transition point.

Figure 9

Coherent spectral peak position at unitarity. Using the full many-body model for the Raman spectra [first term of Eq. (20)], we plot the coherent peak position versus transferred photon momentum for polaron temperatures $T_p=0.1T_F$ (blue circles), $0.15T_F$ (red squares), and $0.2T_F$ (black triangles) at ${(k_{F}a)}^{-1}=0$ with $n_I=0.15n$. We observe convergence to the zero-temperature value ${\epsilon }_{\mathrm{pol}}=-0.6066{ϵ}_F$ (dashed line) for photon momentum transfer larger than $0.1k_F$. Specifically, with Raman spectroscopy in our experiment ($\overline{q}=1.9k_F$), no shift is expected. Inset: temperature dependence of the spectral peak position with conventional rf spectroscopy ($\overline{q}=0$).

Figure 10

Comparison of coherent part Raman spectra. Using the full many-body model for the Raman spectra [first term of Eq. (20), solid lines] and the approximation due to Eq. (13) (dashed lines), the coherent Raman rate is shown for three interaction strengths. In these calculations we use the effective mass and polaron energy obtained from the polaron Ansatz. For clarity, the second and third graphs from the bottom are shifted by 0.1 and 0.2, respectively.

Figure 11

Comparison of background Raman spectra. Using the full many-body model for the Raman spectra [incoherent and molecular terms in Eq. (5), solid lines] and the approximation obtained by fitting the simplified model Eq. (13) to the theoretical spectra (dashed lines), Raman spectra are shown for three interaction strengths. For clarity, the second and third graphs from the bottom are shifted by 0.02 and 0.04, respectively.

Figure 12

Fitting simulated background Raman spectra with fixed effective temperature $T_{\mathrm{bg}}$. The black line denotes the homogeneous many-body quasiparticle weight computed for $T=0.2T_F$, $x=0.15$. Error bars mark the fitted residue using an effective temperature of $0.2T_F$ (yellow diamonds), $T_F$ (blue circles), $2T_F$ (black triangles), and $4T_F$ (red squares). Inset: root-mean-square deviation of the extracted residue from the computed one, exhibiting a minimal deviation at approximately $2T_F$, the value we use for fitting the experimental data.

Figure 13

Polaronic and molecular energy gap. Energy gap between the lowest-lying polaron and molecule modes and the chemical potential versus interaction strength. The blue lines show the gap for the case of a single impurity at $T=0$, while the red and black lines show the gaps for finite density at $T=0$ and $T>0$, respectively. The solid lines with circle markers denote the gap between the lowest-lying molecular state and the chemical potential, whereas the dashed lines with triangle markers denote the polaronic gaps.