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High-temperature pairing in a strongly interacting two-dimensional Fermi gas

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Science  26 Jan 2018:
Vol. 359, Issue 6374, pp. 452-455
DOI: 10.1126/science.aan5950

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 (Tc) (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 (36).

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 EB exists for arbitrarily small attractions between the atoms. The interactions in the many-body system are captured by the dimensionless parameter ln(kFa2D), where kF is the Fermi momentum and a2D is the 2D scattering length. As we tune the interaction strength from the BEC [large negative ln(kFa2D)] to the BCS side [large positive ln(kFa2D)], 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 a2D is of the same order as the interparticle spacing (~kF–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 (1014) and 2D (15, 16) systems, a consensus is yet to emerge (3, 7, 1720).

Fig. 1 Exploring fermion pairing in a strongly interacting 2D Fermi gas.

(A) Schematic phase diagram of the BEC-BCS crossover. In this work, we investigated the nature of pairing in the normal phase of the crossover regime between the weakly interacting Bose and Fermi liquids. (B) Illustration of RF spectroscopy of a 2D two-component Fermi gas. Pairing and many-body effects shift the atomic transition frequencies between the hyperfine states Embedded Image, which results in observable signatures in the RF response of the system. (C) Absorption images of the cloud [taken at ln(kFa2D) ≈ 1.5 and T/TF ~ 0.3] without RF (reference) and with RF at a particular frequency, and the difference between the two images. The ring feature in δn(r) reveals the density dependence of the RF response. (D) Spatially resolved spectral response function reconstructed from absorption images taken at different RF frequencies. At low temperatures in the spin-balanced sample, the occupation of the free-particle branch is too low to be observable, which makes it difficult to distinguish between mean-field shifts and pairing effects.

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 6Li atoms with ~3 × 104 particles per spin state that were loaded into a single layer of an anisotropic harmonic optical trap. The trap frequencies were ωz ≈ 2π × 6.95 kHz and ωr ≈ 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 (T), and chemical potential (μ) are smaller than the axial confinement energy (21, 22). We tuned the scattering length a2D 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 6Li, which at low magnetic fields are given by Embedded Image, Embedded Image, and Embedded Image. We started with a two-component mixture of atoms in hyperfine states Embedded Image or Embedded Image (fig. S1) (21). A RF pulse transferred atoms from state Embedded Image to the third unoccupied hyperfine state Embedded Image, and we subsequently imaged the remaining density distribution in Embedded Image. 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 Embedded Image or Embedded Image 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 EF(r) = (2πℏ2/m)n(r), where m is the mass of a 6Li atom (24). As a consequence, the thermodynamic quantities T/TF and ln(kFa2D) 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 (τRF = 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 δωRF ≈ 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 Embedded Image 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 functionI(r, ωRF) = [n0(r) – n′(r, ωRF)]/n0(r)where n0(r) and n′(r, ωRF) are the density distribution of atoms in state Embedded Image without and with the RF pulse, respectively. An example of the tomographically reconstructed spectra, taken at ln(kFa2D) ~ 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/TF < 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 ΔE. In our experiments, we created a slight spin-imbalance Embedded Image using a sequence of Landau-Zener sweeps (21), where Embedded Image and Embedded Image are densities in hyperfine states Embedded Image and Embedded Image, respectively. We show typical density profiles of majority and minority components in Fig. 2A.

Fig. 2 Quasi-particle spectroscopy in the BEC and BCS limits.

(A) We created a slightly imbalanced mixture of hyperfine states so as to artificially populate the free-particle branch. The density distributions of the majority and minority spins are shown, as well as the corresponding local imbalance (inset). (B and C) Schematic illustration of single-particle dispersion relations in the BEC and BCS limits at zero temperature. Paired atoms reside in the lowest branch (Bound) and are transferred to the continuum of unoccupied states. The excess majority atoms are unpaired and occupy the upper quasi-particle (Free) branch in the spectrum preferentially at k ~ 0 (BEC) and k ~ kF (BCS). The energy difference between the free-particle dispersion in a noninteracting system (blue dashed line) and the continuum (blue solid line) is the bare hyperfine transition energy and serves as the reference for (D) and (E). (D and E) The transition of paired atoms into the continuum yields an asymmetric response with a sharp threshold in the RF spectral function. The quasi-particle transition contributes another peak, which appears at ωRF = 0 on the BEC side and ωRF = –Δ on the BCS side, where Δ is the BCS gap parameter. Their relative difference yields the pairing energy ΔE, which reveals the distinction between two-body (ΔE ~ EB) and many-body pairing (ΔE > EB) in the two limits.

The pairing energy ΔE allows us to distinguish between two different pairing scenarios. If ΔE coincides with the energy EB of the dimer state, we are in the two-body regime. By contrast, we associate the situation of a density (EF)–dependent ΔE, exceeding EB, 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 ~ kF 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, ΔE ~ EB independent of local density, ΔE ~ Δ + EB on the BCS side, where Embedded Image is the many-body gap parameter from BCS theory. In the latter case, ΔE is necessarily larger than EB 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 ΔE—particularly its deviation from EB—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 Embedded Image mixture, which corresponds to central values of ln(kFa2D) ~ –0.5 and ln(kFa2D) ~ 1, respectively. The response from unpaired quasiparticles appears at frequency ωRF ~ 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.

Fig. 3 From two-body dimers to many-body pairing.

The spatially resolved response function I(r, ωRF) shows qualitatively different behavior for two different scattering lengths. (A and B) I(r, ωRF) for central ln(kFa2D) ~ –0.5 and 1.0, respectively. The gray lines correspond to local T/TF ~ 0.7 in (A) and T/TF ~ 1 in (B). The 3D visualization was obtained by using a linear interpolation between 3000 data points, each of which is an average of 30 realizations. The black solid line is the peak position of the free branch, the red line is the threshold position of the bound branch, and the black dashed line is displaced from the free peak by the two-body binding energy EB. The energy difference between free and bound branches is the pairing energy ΔE, which is seen to agree with EB in (A) (BEC regime) but exceeds EB in (B) (crossover regime). The differential density dependence of the energy of the two branches implies that the pair wave function is strongly modified by the many-body system. (C and D) Local spectra at a fixed radius indicated by gray lines in (A) and (B) corresponding to a homogeneous system with T/TF ~ 0.7 and 1, respectively. The solid blue curves are the fits to the data; the black and red curves are Gaussian and threshold fits to the two branches (21).

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 EB 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 ΔE ~ EB 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, ΔE begins to exceed EB and shows a strong dependence on the local density (EF), indicating that pairing in this regime is a many-body phenomenon. At very low temperatures, the measurement of ΔE 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/TF ~ 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 Tc ≈ 0.17TF 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 ΔE in units of the two-body binding energy EB. In Fig. 4A, we plot the temperature dependence of ΔE/EB for different interaction strengths, and the variation of ΔE/EB as a function of ln(kFa2D) is shown in Fig. 4B for a fixed ratio T/TF ≈ 0.5. This constitutes an extremely high-temperature regime even in the context of ultracold fermionic superfluidity, where the largest observed critical temperatures are Tc/TF ≈ 0.17 (8, 9). We performed our measurements with both Embedded Image and Embedded Image 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 ΔE around ln(kFa2D) ≈ 0.5, demonstrating the robustness of the quantity ΔE against final state effects. For larger ln(kFa2D), 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).

Fig. 4 Normal phase in the 2D BEC-BCS crossover regime.

(A) Pairing energy ΔE in units of EB plotted as a function of T/TF for different interaction strengths [central ln(kFa2D)]. Each point in (A) is the result of fits to local spectra (Fig. 3, C and D), which are averaged over 30 shots. (B) Many-body–induced high-temperature pairing. We plot ΔE/EB as a function of ln(kFa2D) for fixed ratio T/TF ~ 0.5. Red and blue circles correspond to measurements taken with Embedded Image and Embedded Image mixtures, respectively. The dashed black line is a guide to the eye. The errors indicated as shaded bands in (A) and bars in (B) are obtained from the fitting procedure explained in (21). For ln(kFa2D) ≤ 0.5 (strong attraction), we have ΔE/EB ~ 1, with negligible density dependence, indicating two-body pairing. For larger ln(kFa2D) (less attraction), ΔE/EB exceeds 1 and reaches a maximum of 2.6 before showing a downward trend. At ln(kFa2D) ~ 1, we have a critical temperature of Tc ~ 0.17TF (8), which indicates the onset of many-body pairing at temperatures several times Tc.

In Fig. 4, we observe that for ln(kFa2D) ≤ 0.5 the spectra are well described by two-body physics. By contrast, the pronounced density-dependent gap exceeding EB for ln(kFa2D) ≥ 0.5 signals the crossover to a many-body pairing regime. In particular, we observed that ΔE/EB peaks at ln(kFa2D) ~ 1, where ΔE ~ 2.6EB and is a considerable fraction of EF(0.6EF). 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(kFa2D), we saw a downward trend in ΔE/EB, and for ln(kFa2D) > 1.5, we observed only a single branch in the spectra near ωRF ~ 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 ΔE as a function of ln(kFa2D), 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(kFa2D) ≤ 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(kFa2D) ≈ 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, 3133), 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

References (34, 35)

References and Notes

  1. Materials and methods are available as supplementary materials.
  2. EF/h has typical values of 7.5 kHz at the center of the cloud.
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.
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