QUANTUM GASES

High-temperature pairing in astrongly interacting two-dimensionalFermi gasPuneet A. Murthy,1*† Mathias Neidig,1† Ralf Klemt,1† Luca Bayha,1 Igor Boettcher,2

Tilman Enss,3 Marvin Holten,1 Gerhard Zürn,1 Philipp M. Preiss,1 Selim Jochim1

The nature of the normal phase of strongly correlated fermionic systems is an outstandingquestion in quantum many-body physics. We used spatially resolved radio-frequencyspectroscopy to measure pairing energy of fermions across a wide range of temperaturesand interaction strengths in a two-dimensional gas of ultracold fermionic atoms. Weobserved many-body pairing at temperatures far above the critical temperature forsuperfluidity. In the strongly interacting regime, the pairing energy in the normal phaseconsiderably exceeds the intrinsic two-body binding energy of the system and shows aclear dependence on local density. This implies that pairing in this regime is driven bymany-body correlations, rather than two-body physics. Our findings show that pairingcorrelations in strongly interacting two-dimensional fermionic systems are remarkablyrobust against thermal fluctuations.

Fermion pairing is the key ingredient forsuperconductivity and superfluidity in fer-mionic systems (1). In a system with s-waveinteractions, two scenarios can occur: Inthe first one, as realized for weakly attract-

ive fermions that are described by the theory ofBardeen-Cooper-Schrieffer (BCS), formation andcondensation of pairs both take place at thesame critical temperature (Tc) (2). In the secondscenario, fermion pairing accompanied by a sup-pression of the density of states at the Fermisurface occurs at temperatures exceeding thecritical temperature. Finding a description ofthis so-called pseudogap phase, especially fortwo-dimensional (2D) systems, is thought tobe a promising route to understanding the com-plex physics of high-temperature superconduc-tivity (3–6).The Bose-Einstein condensation (BEC)–BCS

crossover of ultracold atoms constitutes a ver-satile framework with which to explore thenormal phase of strongly correlated fermions(Fig. 1A). The crossover smoothly connects twodistinct regimes of pairing: the BEC regime oftightly bound molecules and the BCS regimeof weakly bound Cooper pairs. In 2D (unlike3D) systems with contact interactions, a two-body bound state with binding energy EB ex-ists for arbitrarily small attractions betweenthe atoms. The interactions in the many-bodysystem are captured by the dimensionless pa-rameter ln(kFa2D), where kF is the Fermi mo-mentum and a2D is the 2D scattering length.As we tune the interaction strength from theBEC [large negative ln(kFa2D)] to the BCS side

[large positive ln(kFa2D)], the character of thesystem smoothly changes from bosonic tofermionic (7). A strongly interacting regionlies in between these two weakly interactinglimits, where a2D is of the same order as the inter-particle spacing (~kF

–1). Previously, a matter-wave focusing method was used to measurethe pair momentum distribution of a 2D Fermigas across the crossover, leading to the ob-servation of the Berezinskii-Kosterlitz-Thouless(BKT) transition at low temperatures (8, 9). Anoutstanding question concerns the nature of thenormal phase above the critical temperature—specifically, how does the normal phase cross overfrom a gapless Fermi liquid on the weakly in-teracting BCS side to a Bose liquid of two-bodydimers on the BEC side? Is there an interactionregime in which pairing is driven by many-bodyeffects rather than the two-body bound state?Although previous cold atom experiments haveexplored this problem both in 3D (10–14) and2D (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 Fermigas trapped in a harmonic potential. The un-derlying potential leads to an inhomogeneousdensity distribution, and therefore we can usethe local density approximation to directlymeasure the density dependence of many-bodyproperties. We performed our experiments witha two-component mixture of 6Li atoms with~3 × 104 particles per spin state that were loadedinto a single layer of an anisotropic harmonicoptical trap. The trap frequencies were wz ≈ 2p ×6.95 kHz and wr ≈ 2p × 22 Hz in the axial andradial directions, leading to an aspect ratio ofabout 300:1. We reached the kinematic 2D re-gime by ensuring that the thermodynamic energyscales, temperature (T), and chemical potential(m) are smaller than the axial confinement en-

ergy (21, 22). We tuned the scattering length a2Dby means of a broad magnetic Feshbach reso-nance (23).To investigate fermion pairing in our system,

we used radio-frequency (RF) spectroscopy. Weperformed experiments with the three lowest-lying hyperfine states of 6Li, which at low mag-netic fields are given by j1i = jF = 1

2 ;mF = � 12i,

j2i = j 12 ; 12i, and j3i = j 32 ;� 32i. We started with a

two-component mixture of atoms in hyperfinestates jaijbi ≡ j1ij2ior jaijbi ≡ j1ij3i (fig. S1) (21).A RF pulse transferred atoms from state jbi tothe third unoccupied hyperfine state jci , andwe subsequently imaged the remaining den-sity distribution in jbi . The idea underlyingthis technique is that the atomic transition fre-quencies between hyperfine states are shiftedby interactions or pairing effects in an ensemble.For example, a state of coexisting pairs and freeatoms (Fig. 1B) will lead to two energeticallyseparated branches in the RF spectrum, fromwhich we can gain quantitative information onpairing and correlations in the many-body sys-tem. Creating initial samples in either j1ij2i orj1ij3i allows us to access a wide range of inter-action strengths and minimize final state inter-action effects (21).In our inhomogeneous 2D system, the local

Fermi energy depends on the local densityn(r) in each spin state according to EF(r) =(2pℏ2/m)n(r), where m is the mass of a 6Li atom(24). As a consequence, the thermodynamicquantities T/TF and ln(kFa2D) also vary spatiallyacross the cloud. We applied the thermometrydeveloped in (21, 25) to extract these local ob-servables. We measured the local spectral re-sponse (26) by choosing a RF pulse duration(tRF = 4 ms) that is sufficiently short to preventdiffusion of transferred atoms, but also suf-ficiently long that we obtained an adequateFourier limited frequency resolution dwRF ≈ 2p ×220 Hz (fig. S2) (21). In Fig. 1C, we show atypical absorption image of the 2D cloud that isused as a reference and another with a RF pulseapplied at a particular frequency. The differencebetween the two images features a spatial ringstructure, which qualitatively shows that for agiven frequency, the depletion of atoms in initialstate jbi occurs at a well-defined density/radius.By performing this measurement for a range ofRF frequencies, we can tomographically recon-struct the spatially resolved spectral responsefunction

I(r, wRF) = [n0(r) – n′(r, wRF)]/n0(r)

where n0(r) and n′(r, wRF) are the density dis-tribution of atoms in state jbi without and withthe RF pulse, respectively. An example of the to-mographically reconstructed spectra, taken atln(kFa2D) ~ 1.5, is shown in Fig. 1D. The frequen-cy of maximum response depends smoothly onthe radius and therefore the local density. Suchdensity-dependent shifts may arise from pairingeffects, in which the effective binding energybetween fermions is dependent on the densityof the medium, or Hartree shifts, which are

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1Physics Institute, Heidelberg University, Heidelberg,Germany. 2Department of Physics, Simon Fraser University,Burnaby, BC, Canada. 3Institute for Theoretical Physics,Heidelberg University, Heidelberg, Germany.*Corresponding author. Email: murthy@physi.uni-heidelberg.de†These authors contributed equally to this work.

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offsets in the spectrum caused by the mean-fieldinteraction energy with no influence on the bind-ing energy between fermions. The position ofthe RF absorption peak alone (Fig. 1D) does notserve as a reliable observable by which to dis-tinguish between these two effects because itlacks a suitable reference energy that already in-corporates Hartree shifts (21). One way to obtainthis reference scale is to measure the RF transi-tions from both bound and free branches to thethird unoccupied state (21). However, we foundthat in the temperature regime (T/TF < 1.5) ex-plored in our experiments, the thermal occupa-tion of the free (unpaired) branch is too low tobe observed.In order to achieve a sufficient population

of the unpaired branch, we applied the quasi-particle spectroscopy method pioneered in (27)for the measurement of the superfluid gap ofa 3D Fermi gas. Although our system is in thenormal phase, the same technique can be usedto determine the pairing gap. The key idea ofthis method lies in creating a slightly spin-imbalanced mixture so that the excess majorityatoms necessarily remain unpaired owing to thedensity mismatch. These unpaired atoms (ordressed quasi-particles) contribute a second ab-sorption maximum in the RF response functionin addition to the one from pairs. We refer tothe energy difference between the two branchesin the spectrum as the pairing energy DE. In ourexperiments, we created a slight spin-imbalanceP = ðnjbi�njaiÞ=ðnjbi+ njaiÞ ≲ 0:15 using a se-quence of Landau-Zener sweeps (21), wherenjbi andnjai are densities in hyperfine states jaiand jbi, respectively. We show typical densityprofiles of majority and minority componentsin Fig. 2A.The pairing energy DE allows us to distin-

guish between two different pairing scenarios.If DE coincides with the energy EB of the dimerstate, we are in the two-body regime. By con-trast, we associate the situation of a density(EF)–dependent DE, exceeding EB, with many-body pairing, in which the relative pair wavefunction is strongly altered by the presence ofthe surrounding medium of interacting fermions.In Fig. 2, B and C, we illustrate these two sce-narios using ideal single-particle dispersionrelations in the BEC and BCS limits at zerotemperature; both limits have free and boundbranches. The RF photons drive transitions fromthese branches to the continuum. The transi-tion of bound pairs occurs with a sharp onsetat a threshold RF frequency at which the disso-ciated fragments have no relative momenta.Higher-frequency RF photons provide relativemomenta to the transferred particles, which leadsto a slowly decaying tail in the spectrum (21).This leads to the highly asymmetric feature seenin the spectrum in Fig. 2, D and E. On the otherhand, the transition of unpaired particles leadsto a symmetric peak because it does not involvea dissociation process.The crucial difference between the BEC and

BCS regimes arises from the fact that the energyminimum of the free branch occurs at k ~ 0 on

Murthy et al., Science 359, 452–455 (2018) 26 January 2018 2 of 4

Fig. 1. Exploring fermion pairing in a strongly interacting 2D Fermi gas. (A) Schematic phasediagram of the BEC-BCS crossover. In this work, we investigated the nature of pairing in the normalphase of the crossover regime between the weakly interacting Bose and Fermi liquids. (B) Illustrationof RF spectroscopy of a 2D two-component Fermi gas. Pairing and many-body effects shift theatomic transition frequencies between the hyperfine states jbi � jci, which results in observablesignatures in the RF response of the system. (C) Absorption images of the cloud [taken atln(kFa2D) ≈ 1.5 and T/TF ~ 0.3] without RF (reference) and with RF at a particular frequency, and thedifference between the two images. The ring feature in dn(r) reveals the density dependence ofthe RF response. (D) Spatially resolved spectral response function reconstructed from absorptionimages taken at different RF frequencies. At low temperatures in the spin-balanced sample, theoccupation of the free-particle branch is too low to be observable, which makes it difficult todistinguish between mean-field shifts and pairing effects.

Fig. 2. Quasi-particle spectroscopy in the BEC and BCS limits. (A) We created a slightlyimbalanced mixture of hyperfine states so as to artificially populate the free-particle branch. Thedensity distributions of the majority and minority spins are shown, as well as the corresponding localimbalance (inset). (B and C) Schematic illustration of single-particle dispersion relations in the BECand BCS limits at zero temperature. Paired atoms reside in the lowest branch (Bound) and aretransferred to the continuum of unoccupied states. The excess majority atoms are unpaired andoccupy 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 (bluedashed line) and the continuum (blue solid line) is the bare hyperfine transition energy and serves asthe reference for (D) and (E). (D and E) The transition of paired atoms into the continuum yields anasymmetric response with a sharp threshold in the RF spectral function. The quasi-particle transitioncontributes another peak, which appears at wRF = 0 on the BEC side and wRF = –D on the BCS side,where D is the BCS gap parameter. Their relative difference yields the pairing energy DE, which revealsthe distinction between two-body (DE ~ EB) and many-body pairing (DE > EB) in the two limits.

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the BEC side and k ~ kF on the BCS side.Although the RF spectra in the two limits seemqualitatively similar, the fundamental differencein their dispersion appears as an energy dif-ference between the two branches. Whereason the BEC side, DE ~ EB independent of localdensity, DE ~ D + EB on the BCS side, whereD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

2EBEFp

is the many-body gap parameterfrom BCS theory. In the latter case, DE is nec-essarily larger than EB and density dependent.Although this idealized picture provides someintuition for RF spectroscopy in a 2D Fermi gas,the actual dispersion relations in the stronglyinteracting region and at high temperaturesmay not follow this mean-field description. How-ever, the behavior of DE—particularly its devia-tion from EB—is still a reliable indicator forpairing beyond two-body physics.In Fig. 3, A and B, we show the measured

spectra I(r, w) for magnetic fields 670 and 690 Gusing a j1ij3imixture, which corresponds to cen-tral values of ln(kFa2D) ~ –0.5 and ln(kFa2D) ~ 1,respectively. The response from unpaired quasi-particles appears at frequency wRF ~ 0, whereasthe pairing branch with an asymmetric lineshape appears at larger frequencies. Examplesof spectra at fixed radii are shown in Fig. 3, Cand D. We fit these local spectra with a com-bined fit function that includes a symmetricGaussian (for the quasiparticle peak) and anasymmetric threshold function (for the pairedpeak) that is convolved with a Gaussian to ac-count for spectral broadening arising from finiteRF frequency resolution and final state effects(28). We present a detailed account of our dataanalysis in (21). The choice of fit function hasa systematic effect on the quantitative resultspresented here, which cannot be eliminated atthis point because a reliable theoretical pre-diction of the shape of the spectral functionexists only in the weakly coupled BEC (18) andBCS (29) limits.At a qualitative level, the main observations

from Fig. 3 are the following. Both branchesin the spectra show density dependence, partof which can be attributed to a Hartree shift.Adding the binding energy EB to the quasi-particlebranch yields the two-body expectation for thethreshold position. This picture is applicable tothe whole spectrum in Fig. 3A, which corre-sponds to a measurement on the BEC side ofthe crossover. By contrast, for the spectrum dis-played in Fig. 3B, corresponding to the crossoverregime, we observed DE ~ EB only in the outerregions of the cloud, where the density is lowenough that only the two-body bound stateplays a role. Toward the center of the cloud, DEbegins to exceed EB and shows a strong de-pendence on the local density (EF), indicatingthat pairing in this regime is a many-body phe-nomenon. At very low temperatures, the measure-ment of DE is difficult because the occupationof the free branch is too low, and for this work,we were unable to prepare a spin-imbalancedsample at temperatures below T/TF ~ 0.4 (21).However, for a balanced gas, we qualitatively ob-served that the threshold position of the bound

Murthy et al., Science 359, 452–455 (2018) 26 January 2018 3 of 4

Fig. 3. From two-body dimers to many-body pairing.The spatially resolved response function I(r, wRF)shows qualitatively different behavior for two different scattering lengths. (A and B) I(r, wRF) forcentral 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 datapoints, each of which is an average of 30 realizations. The black solid line is the peak position of thefree branch, the red line is the threshold position of the bound branch, and the black dashed line isdisplaced from the free peak by the two-body binding energy EB. The energy difference between free andbound branches is the pairing energy DE, which is seen to agree with EB in (A) (BEC regime) butexceeds EB in (B) (crossover regime). The differential density dependence of the energy ofthe 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 ahomogeneous system with T/TF ~ 0.7 and 1, respectively.The solid blue curves are the fits to the data; theblack and red curves are Gaussian and threshold fits to the two branches (21).

Fig. 4. Normal phase in the 2D BEC-BCS crossover regime. (A) Pairing energy DE in units of EBplotted as a function of T/TF for different interaction strengths [central ln(kFa2D)]. Each point in (A) isthe 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 DE/EB as a function of ln(kFa2D) for fixed ratio T/TF ~ 0.5.Red and blue circles correspond to measurements taken with j1ij3i and j1ij2i 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 DE/EB ~ 1,with negligible density dependence, indicating two-body pairing. For larger ln(kFa2D)(less attraction), DE/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 onsetof many-body pairing at temperatures several times Tc.

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branch increases continuously with decreasingtemperature, even as we crossed the superfluidtransition. This indicates that in the crossoverregime, a many-body gap opens in the normalphase rather than at Tc ≈ 0.17TF as expectedfrom BCS theory (fig. S8) (21). This observationis 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 magneticfields and extracted DE in units of the two-bodybinding energy EB. In Fig. 4A, we plot the tem-perature dependence of DE/EB for different in-teraction strengths, and the variation of DE/EB

as a function of ln(kFa2D) is shown in Fig. 4B fora fixed ratio T/TF ≈ 0.5. This constitutes anextremely high-temperature regime even in thecontext of ultracold fermionic superfluidity, wherethe largest observed critical temperatures areTc/TF ≈ 0.17 (8, 9). We performed our measure-ments with both j1ij2i and j1ij3imixtures (Fig.4B, blue and red points) in an overlapping inter-action regime. The two mixtures differ in theirfinal state interaction strengths, yet they showsimilar values of DE around ln(kFa2D) ≈ 0.5,demonstrating the robustness of the quantity DEagainst final state effects. For larger ln(kFa2D),the two mixtures allow us to probe complemen-tary regions of the crossover. Details of the ex-perimental parameters used for the two mixturesare tabulated in (table S1) (21).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-dependentgap exceeding EB for ln(kFa2D) ≥ 0.5 signalsthe crossover to a many-body pairing regime.In particular, we observed that DE/EB peaks atln(kFa2D) ~ 1, where DE ~ 2.6EB and is a con-siderable fraction of EF(0.6EF). The identificationof this strongly correlated many-body pairing re-gime and the observation of many-body–inducedpairing at temperatures several times the criticaltemperature is the second main result of thiswork. For larger ln(kFa2D), we saw a downwardtrend in DE/EB, and for ln(kFa2D) > 1.5, we ob-served only a single branch in the spectra nearwRF ~ 0, suggesting the absence of a gap largerthan the scale of our experimental resolution(fig. S6) (21). Our qualitative observation of avanishing gap for weaker attraction is con-sistent with the picture of the normal phase in

the BCS limit being a gapless Fermi liquid (30).The nonmonotonous behavior of DE as a func-tion of ln(kFa2D), as shown in Fig. 4B, is alsoqualitatively predicted by finite-temperature BCStheory (fig. S4) (21) for the superfluid phase.Here, we discuss our results in the context

of current theoretical understanding and previousexperimental work. In (15), Sommer et al. per-formed trap-averaged RF spectroscopy in the3D-2D crossover and found good agreement withthe mean-field two-body expectation in the re-gime ln(kFa2D) ≤ 0.5. In (16), Feld et al. observedsignatures of pairing in the normal phase usingmomentum-resolved (but trap-averaged) spectros-copy, in a similar interaction regime as (15),which were interpreted as a many-body pseudo-gap. However, subsequent theoretical work basedon two-body physics only (18, 19) was consistentwith that of many of the observations in (16).Beyond this previously explored regime, ourmeasurements reveal that many-body effectsenhance the pairing energy far above the criticaltemperature, with the maximum enhancementoccurring at ln(kFa2D) ≈ 1, where a reliable mean-field description is not available. With regardto the long-standing question concerning thenature of the normal phase of a strongly inter-acting Fermi gas (7, 17, 31–33), our experimentsreveal the existence of a state in the phase dia-gram whose behavior deviates from both BoseLiquid and Fermi liquid descriptions. Findinga complete description of this strongly correlatedphase is an exciting challenge for both theoryand experiment.

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ACKNOWLEDGMENTS

We gratefully acknowledge insightful discussions with M. Parish,J. Levinsen, N. Defenu, and W. Zwerger. We thank T. Lompe fordiscussions and for a critical reading of the manuscript. This workhas been supported by the European Research Councilconsolidator grant 725636 and the Heidelberg Center for QuantumDynamics 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 2020program under the Marie Sklodowska-Curie grant agreement706487. Supporting data can be found in the supplementarymaterials. Raw data are available upon request. P.A.M andG.Z. initiated the project. P.A.M., M.N., R.K., and M.H. performedthe measurements and analyzed the data. I.B. and T.E.provided theory support and assistance with preparing themanuscript. P.M.P. and S.J. supervised the project. All authorscontributed to the interpretation and discussion of theexperimental results.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/359/6374/452/suppl/DC1Materials and MethodsSupplementary textFig. S1 to S8Table S1References (34, 35)

5 May 2017; accepted 6 December 201710.1126/science.aan5950

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

M. Preiss and Selim JochimPuneet A. Murthy, Mathias Neidig, Ralf Klemt, Luca Bayha, Igor Boettcher, Tilman Enss, Marvin Holten, Gerhard Zürn, Philipp

originally published online December 21, 2017DOI: 10.1126/science.aan5950 (6374), 452-455.359Science 

, this issue p. 452ScienceThe formation of atomic pairs occurred at much higher temperatures in the unitary regime than previously thought.

studied this crossover in a gas of fermions confined to two dimensions.et al.strongly interacting unitary regime. Murthy example, cause the gas to undergo a crossover from weakly interacting fermions to weakly interacting bosons via a

Cold atomic gases are extremely flexible systems; the ability to tune interactions between fermionic atoms can, forTuning the atomic pairing

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REFERENCES

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