## Tuning the atomic pairing

Cold atomic gases are extremely flexible systems; the ability to tune interactions between fermionic atoms can, for example, cause the gas to undergo a crossover from weakly interacting fermions to weakly interacting bosons via a strongly interacting unitary regime. Murthy *et al.* studied this crossover in a gas of fermions confined to two dimensions. The formation of atomic pairs occurred at much higher temperatures in the unitary regime than previously thought.

*Science*, this issue p. 452

## Abstract

The nature of the normal phase of strongly correlated fermionic systems is an outstanding question in quantum many-body physics. We used spatially resolved radio-frequency spectroscopy to measure pairing energy of fermions across a wide range of temperatures and interaction strengths in a two-dimensional gas of ultracold fermionic atoms. We observed many-body pairing at temperatures far above the critical temperature for superfluidity. In the strongly interacting regime, the pairing energy in the normal phase considerably exceeds the intrinsic two-body binding energy of the system and shows a clear dependence on local density. This implies that pairing in this regime is driven by many-body correlations, rather than two-body physics. Our findings show that pairing correlations in strongly interacting two-dimensional fermionic systems are remarkably robust against thermal fluctuations.

Fermion pairing is the key ingredient for superconductivity and superfluidity in fermionic systems (*1*). In a system with *s*-wave interactions, two scenarios can occur: In the first one, as realized for weakly attractive fermions that are described by the theory of Bardeen-Cooper-Schrieffer (BCS), formation and condensation of pairs both take place at the same critical temperature (*T*_{c}) (*2*). In the second scenario, fermion pairing accompanied by a suppression of the density of states at the Fermi surface occurs at temperatures exceeding the critical temperature. Finding a description of this so-called pseudogap phase, especially for two-dimensional (2D) systems, is thought to be a promising route to understanding the complex physics of high-temperature superconductivity (*3*–*6*).

The Bose-Einstein condensation (BEC)–BCS crossover of ultracold atoms constitutes a versatile framework with which to explore the normal phase of strongly correlated fermions (Fig. 1A). The crossover smoothly connects two distinct regimes of pairing: the BEC regime of tightly bound molecules and the BCS regime of weakly bound Cooper pairs. In 2D (unlike 3D) systems with contact interactions, a two-body bound state with binding energy *E*_{B} exists for arbitrarily small attractions between the atoms. The interactions in the many-body system are captured by the dimensionless parameter ln(*k*_{F}*a*_{2D}), where *k*_{F} is the Fermi momentum and *a*_{2D} is the 2D scattering length. As we tune the interaction strength from the BEC [large negative ln(*k*_{F}*a*_{2D})] to the BCS side [large positive ln(*k*_{F}*a*_{2D})], the character of the system smoothly changes from bosonic to fermionic (*7*). A strongly interacting region lies in between these two weakly interacting limits, where *a*_{2D} is of the same order as the interparticle spacing (~*k*_{F}^{–1}). Previously, a matter-wave focusing method was used to measure the pair momentum distribution of a 2D Fermi gas across the crossover, leading to the observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition at low temperatures (*8*, *9*). An outstanding question concerns the nature of the normal phase above the critical temperature—specifically, how does the normal phase cross over from a gapless Fermi liquid on the weakly interacting BCS side to a Bose liquid of two-body dimers on the BEC side? Is there an interaction regime in which pairing is driven by many-body effects rather than the two-body bound state? Although previous cold atom experiments have explored this problem both in 3D (*10*–*14*) and 2D (*15*, *16*) systems, a consensus is yet to emerge (*3*, *7*, *17*–*20*).

We addressed these questions by studying the normal phase of such a 2D ultracold Fermi gas trapped in a harmonic potential. The underlying potential leads to an inhomogeneous density distribution, and therefore we can use the local density approximation to directly measure the density dependence of many-body properties. We performed our experiments with a two-component mixture of ^{6}Li atoms with ~3 × 10^{4} particles per spin state that were loaded into a single layer of an anisotropic harmonic optical trap. The trap frequencies were ω* _{z}* ≈ 2π × 6.95 kHz and ω

*≈ 2π × 22 Hz in the axial and radial directions, leading to an aspect ratio of about 300:1. We reached the kinematic 2D regime by ensuring that the thermodynamic energy scales, temperature (*

_{r}*T*), and chemical potential (μ) are smaller than the axial confinement energy (

*21*,

*22*). We tuned the scattering length

*a*

_{2D}by means of a broad magnetic Feshbach resonance (

*23*).

To investigate fermion pairing in our system, we used radio-frequency (RF) spectroscopy. We performed experiments with the three lowest-lying hyperfine states of ^{6}Li, which at low magnetic fields are given by , , and . We started with a two-component mixture of atoms in hyperfine states or (fig. S1) (*21*). A RF pulse transferred atoms from state to the third unoccupied hyperfine state , and we subsequently imaged the remaining density distribution in . The idea underlying this technique is that the atomic transition frequencies between hyperfine states are shifted by interactions or pairing effects in an ensemble. For example, a state of coexisting pairs and free atoms (Fig. 1B) will lead to two energetically separated branches in the RF spectrum, from which we can gain quantitative information on pairing and correlations in the many-body system. Creating initial samples in either or allows us to access a wide range of interaction strengths and minimize final state interaction effects (*21*).

In our inhomogeneous 2D system, the local Fermi energy depends on the local density *n*(*r*) in each spin state according to *E*_{F}(*r*) = (2πℏ^{2}/*m*)*n*(*r*), where *m* is the mass of a ^{6}Li atom (*24*). As a consequence, the thermodynamic quantities *T*/*T*_{F} and ln(*k*_{F}*a*_{2D}) also vary spatially across the cloud. We applied the thermometry developed in (*21*, *25*) to extract these local observables. We measured the local spectral response (*26*) by choosing a RF pulse duration (τ_{RF} = 4 ms) that is sufficiently short to prevent diffusion of transferred atoms, but also sufficiently long that we obtained an adequate Fourier limited frequency resolution δω_{RF} ≈ 2π × 220 Hz (fig. S2) (*21*). In Fig. 1C, we show a typical absorption image of the 2D cloud that is used as a reference and another with a RF pulse applied at a particular frequency. The difference between the two images features a spatial ring structure, which qualitatively shows that for a given frequency, the depletion of atoms in initial state occurs at a well-defined density/radius. By performing this measurement for a range of RF frequencies, we can tomographically reconstruct the spatially resolved spectral response function*I*(*r*, ω_{RF}) = [*n*_{0}(*r*) – *n*′(*r*, ω_{RF})]/*n*_{0}(*r*)where *n*_{0}(*r*) and *n*′(*r*, ω_{RF}) are the density distribution of atoms in state without and with the RF pulse, respectively. An example of the tomographically reconstructed spectra, taken at ln(*k*_{F}*a*_{2D}) ~ 1.5, is shown in Fig. 1D. The frequency of maximum response depends smoothly on the radius and therefore the local density. Such density-dependent shifts may arise from pairing effects, in which the effective binding energy between fermions is dependent on the density of the medium, or Hartree shifts, which are offsets in the spectrum caused by the mean-field interaction energy with no influence on the binding energy between fermions. The position of the RF absorption peak alone (Fig. 1D) does not serve as a reliable observable by which to distinguish between these two effects because it lacks a suitable reference energy that already incorporates Hartree shifts (*21*). One way to obtain this reference scale is to measure the RF transitions from both bound and free branches to the third unoccupied state (*21*). However, we found that in the temperature regime (*T*/*T*_{F} < 1.5) explored in our experiments, the thermal occupation of the free (unpaired) branch is too low to be observed.

In order to achieve a sufficient population of the unpaired branch, we applied the quasiparticle spectroscopy method pioneered in (*27*) for the measurement of the superfluid gap of a 3D Fermi gas. Although our system is in the normal phase, the same technique can be used to determine the pairing gap. The key idea of this method lies in creating a slightly spin-imbalanced mixture so that the excess majority atoms necessarily remain unpaired owing to the density mismatch. These unpaired atoms (or dressed quasi-particles) contribute a second absorption maximum in the RF response function in addition to the one from pairs. We refer to the energy difference between the two branches in the spectrum as the pairing energy Δ*E*. In our experiments, we created a slight spin-imbalance using a sequence of Landau-Zener sweeps (*21*), where and are densities in hyperfine states and , respectively. We show typical density profiles of majority and minority components in Fig. 2A.

The pairing energy Δ*E* allows us to distinguish between two different pairing scenarios. If Δ*E* coincides with the energy *E*_{B} of the dimer state, we are in the two-body regime. By contrast, we associate the situation of a density (*E*_{F})–dependent Δ*E*, exceeding *E*_{B}, with many-body pairing, in which the relative pair wave function is strongly altered by the presence of the surrounding medium of interacting fermions. In Fig. 2, B and C, we illustrate these two scenarios using ideal single-particle dispersion relations in the BEC and BCS limits at zero temperature; both limits have free and bound branches. The RF photons drive transitions from these branches to the continuum. The transition of bound pairs occurs with a sharp onset at a threshold RF frequency at which the dissociated fragments have no relative momenta. Higher-frequency RF photons provide relative momenta to the transferred particles, which leads to a slowly decaying tail in the spectrum (*21*). This leads to the highly asymmetric feature seen in the spectrum in Fig. 2, D and E. On the other hand, the transition of unpaired particles leads to a symmetric peak because it does not involve a dissociation process.

The crucial difference between the BEC and BCS regimes arises from the fact that the energy minimum of the free branch occurs at *k* ~ 0 on the BEC side and *k* ~ *k*_{F} on the BCS side. Although the RF spectra in the two limits seem qualitatively similar, the fundamental difference in their dispersion appears as an energy difference between the two branches. Whereas on the BEC side, Δ*E* ~ *E*_{B} independent of local density, Δ*E* ~ Δ + *E*_{B} on the BCS side, where is the many-body gap parameter from BCS theory. In the latter case, Δ*E* is necessarily larger than *E*_{B} and density dependent. Although this idealized picture provides some intuition for RF spectroscopy in a 2D Fermi gas, the actual dispersion relations in the strongly interacting region and at high temperatures may not follow this mean-field description. However, the behavior of Δ*E*—particularly its deviation from *E*_{B}—is still a reliable indicator for pairing beyond two-body physics.

In Fig. 3, A and B, we show the measured spectra *I*(*r*, ω) for magnetic fields 670 and 690 G using a mixture, which corresponds to central values of ln(*k*_{F}*a*_{2D}) ~ –0.5 and ln(*k*_{F}*a*_{2D}) ~ 1, respectively. The response from unpaired quasiparticles appears at frequency ω_{RF} ~ 0, whereas the pairing branch with an asymmetric line shape appears at larger frequencies. Examples of spectra at fixed radii are shown in Fig. 3, C and D. We fit these local spectra with a combined fit function that includes a symmetric Gaussian (for the quasiparticle peak) and an asymmetric threshold function (for the paired peak) that is convolved with a Gaussian to account for spectral broadening arising from finite RF frequency resolution and final state effects (*28*). We present a detailed account of our data analysis in (*21*). The choice of fit function has a systematic effect on the quantitative results presented here, which cannot be eliminated at this point because a reliable theoretical prediction of the shape of the spectral function exists only in the weakly coupled BEC (*18*) and BCS (*29*) limits.

At a qualitative level, the main observations from Fig. 3 are the following. Both branches in the spectra show density dependence, part of which can be attributed to a Hartree shift. Adding the binding energy *E*_{B} to the quasi-particle branch yields the two-body expectation for the threshold position. This picture is applicable to the whole spectrum in Fig. 3A, which corresponds to a measurement on the BEC side of the crossover. By contrast, for the spectrum displayed in Fig. 3B, corresponding to the crossover regime, we observed Δ*E* ~ *E*_{B} only in the outer regions of the cloud, where the density is low enough that only the two-body bound state plays a role. Toward the center of the cloud, Δ*E* begins to exceed *E*_{B} and shows a strong dependence on the local density (*E*_{F}), indicating that pairing in this regime is a many-body phenomenon. At very low temperatures, the measurement of Δ*E* is difficult because the occupation of the free branch is too low, and for this work, we were unable to prepare a spin-imbalanced sample at temperatures below *T*/*T*_{F} ~ 0.4 (*21*). However, for a balanced gas, we qualitatively observed that the threshold position of the bound branch increases continuously with decreasing temperature, even as we crossed the superfluid transition. This indicates that in the crossover regime, a many-body gap opens in the normal phase rather than at *T*_{c} ≈ 0.17*T*_{F} as expected from BCS theory (fig. S8) (*21*). This observation is the first main result of this work.

To quantitatively study the change in the nature of pairing from the BEC to the BCS side, we measured the spectra at different magnetic fields and extracted Δ*E* in units of the two-body binding energy *E*_{B}. In Fig. 4A, we plot the temperature dependence of Δ*E*/*E*_{B} for different interaction strengths, and the variation of Δ*E*/*E*_{B} as a function of ln(*k*_{F}*a*_{2D}) is shown in Fig. 4B for a fixed ratio *T*/*T*_{F} ≈ 0.5. This constitutes an extremely high-temperature regime even in the context of ultracold fermionic superfluidity, where the largest observed critical temperatures are *T*_{c}/*T*_{F} ≈ 0.17 (*8*, *9*). We performed our measurements with both and mixtures (Fig. 4B, blue and red points) in an overlapping interaction regime. The two mixtures differ in their final state interaction strengths, yet they show similar values of Δ*E* around ln(*k*_{F}*a*_{2D}) ≈ 0.5, demonstrating the robustness of the quantity Δ*E* against final state effects. For larger ln(*k*_{F}*a*_{2D}), the two mixtures allow us to probe complementary regions of the crossover. Details of the experimental parameters used for the two mixtures are tabulated in (table S1) (*21*).

In Fig. 4, we observe that for ln(*k*_{F}*a*_{2D}) ≤ 0.5 the spectra are well described by two-body physics. By contrast, the pronounced density-dependent gap exceeding *E*_{B} for ln(*k*_{F}*a*_{2D}) ≥ 0.5 signals the crossover to a many-body pairing regime. In particular, we observed that Δ*E*/*E*_{B} peaks at ln(*k*_{F}*a*_{2D}) ~ 1, where Δ*E* ~ 2.6*E*_{B} and is a considerable fraction of *E*_{F}(0.6*E*_{F}). The identification of this strongly correlated many-body pairing regime and the observation of many-body–induced pairing at temperatures several times the critical temperature is the second main result of this work. For larger ln(*k*_{F}*a*_{2D}), we saw a downward trend in Δ*E*/*E*_{B}, and for ln(*k*_{F}*a*_{2D}) > 1.5, we observed only a single branch in the spectra near ω_{RF} ~ 0, suggesting the absence of a gap larger than the scale of our experimental resolution (fig. S6) (*21*). Our qualitative observation of a vanishing gap for weaker attraction is consistent with the picture of the normal phase in the BCS limit being a gapless Fermi liquid (*30*). The nonmonotonous behavior of Δ*E* as a function of ln(*k*_{F}*a*_{2D}), as shown in Fig. 4B, is also qualitatively predicted by finite-temperature BCS theory (fig. S4) (*21*) for the superfluid phase.

Here, we discuss our results in the context of current theoretical understanding and previous experimental work. In (*15*), Sommer *et al*. performed trap-averaged RF spectroscopy in the 3D-2D crossover and found good agreement with the mean-field two-body expectation in the regime ln(*k*_{F}*a*_{2D}) ≤ 0.5. In (*16*), Feld *et al*. observed signatures of pairing in the normal phase using momentum-resolved (but trap-averaged) spectroscopy, in a similar interaction regime as (*15*), which were interpreted as a many-body pseudogap. However, subsequent theoretical work based on two-body physics only (*18*, *19*) was consistent with that of many of the observations in (*16*). Beyond this previously explored regime, our measurements reveal that many-body effects enhance the pairing energy far above the critical temperature, with the maximum enhancement occurring at ln(*k*_{F}*a*_{2D}) ≈ 1, where a reliable mean-field description is not available. With regard to the long-standing question concerning the nature of the normal phase of a strongly interacting Fermi gas (*7*, *17*, *31*–*33*), our experiments reveal the existence of a state in the phase diagram whose behavior deviates from both Bose Liquid and Fermi liquid descriptions. Finding a complete description of this strongly correlated phase is an exciting challenge for both theory and experiment.

## Supplementary Materials

www.sciencemag.org/content/359/6374/452/suppl/DC1

Materials and Methods

Supplementary text

Fig. S1 to S8

Table S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We gratefully acknowledge insightful discussions with M. Parish, J. Levinsen, N. Defenu, and W. Zwerger. We thank T. Lompe for discussions and for a critical reading of the manuscript. This work has been supported by the European Research Council consolidator grant 725636 and the Heidelberg Center for Quantum Dynamics and is part of the Deutsche Forschungsgemeinschaft (DFG) Collaborative Research Centre “SFB 1225 (ISOQUANT).” I.B. acknowledges support from DFG and grant BO 4640/1-1. P.M.P. acknowledges funding from European Union’s Horizon 2020 program under the Marie Sklodowska-Curie grant agreement 706487. Supporting data can be found in the supplementary materials. Raw data are available upon request. P.A.M and G.Z. initiated the project. P.A.M., M.N., R.K., and M.H. performed the measurements and analyzed the data. I.B. and T.E. provided theory support and assistance with preparing the manuscript. P.M.P. and S.J. supervised the project. All authors contributed to the interpretation and discussion of the experimental results.