Spectral response and contact of the unitary Fermi gas

TL;DR

This study measures the radiofrequency spectra of a homogeneous unitary Fermi gas across various temperatures, revealing the evolution of spectral features, contact, and non-Fermi-liquid behavior near the superfluid transition.

Contribution

It provides the first comprehensive RF spectral measurements of a homogeneous unitary Fermi gas across the superfluid transition, highlighting the evolution of spectral peaks and contact.

Findings

Spectral peak shifts from atomic resonance to Fermi energy with cooling.

Peak width indicates scattering rate at high T and pair size at low T.

Contact increases rapidly below the superfluid transition.

Abstract

We measure radiofrequency (rf) spectra of the homogeneous unitary Fermi gas at temperatures ranging from the Boltzmann regime through quantum degeneracy and across the superfluid transition. For all temperatures, a single spectral peak is observed. Its position smoothly evolves from the bare atomic resonance in the Boltzmann regime to a frequency corresponding to nearly one Fermi energy at the lowest temperatures. At high temperatures, the peak width reflects the scattering rate of the atoms, while at low temperatures, the width is set by the size of fermion pairs. Above the superfluid transition, and approaching the quantum critical regime, the width increases linearly with temperature, indicating non-Fermi-liquid behavior. From the wings of the rf spectra, we obtain the contact, quantifying the strength of short-range pair correlations. We find that the contact rapidly increases as the gas is cooled below the superfluid transition.

Equations

$$\underset{\omega \to \infty }{\lim }I(\omega )=\left(\frac{C}{N{k}_{\mathrm{F}}}\right)\frac{1}{2\sqrt{2}\pi (1+\hslash \omega /E_b)}{\left(\frac{E_{\mathrm{F}}}{\hslash \omega }\right)}^{3/2},$$

Full text

Understanding fermion pairing and pair correlations is of central relevance to strongly interacting Fermi systems such as nuclei Hen and others (2014); Weiss et al. (2015), ultracold gases Ketterle and Zwierlein (2008); Giorgini et al. (2008); Zwerger (2016); Zwierlein (2014), liquid 3He Leggett (1972), high temperature superconductors Lee et al. (2006), and neutron stars Baym et al. (1969). Strong interactions on the order of the Fermi energy challenge theoretical approaches, especially methods that predict dynamic properties such as transport or the spectral response at finite temperature Enss (2012). Atomic Fermi gases at Feshbach resonances realize a paradigmatic system where the gas becomes as strongly interacting as allowed by unitarity Ketterle and Zwierlein (2008); Giorgini et al. (2008); Zwerger (2016); Randeria and Taylor (2014); Zwierlein (2014). Here, the system becomes universal, requiring only two energy scales: the Fermi energy $E_{F}$ and thermal energy $k_{B}T$, where $k_{B}$ is the Boltzmann constant and $T$ is the temperature. The corresponding length scales are the interparticle spacing $\lambda_{F}\,{=}\,n^{-1/3}$ and the thermal de Broglie wavelength $\lambda_{T}\,{=}\,h/\sqrt{2\pi mk_{B}T}$, where $m$ and $n$ are the mass and number density of the atoms respectively. When the two energy scales are comparable, the system enters a quantum critical regime separating the high temperature Boltzmann gas from the fermionic superfluid Nikolic and Sachdev (2007). Quantum criticality is often associated with the absence of quasiparticles Nikolic and Sachdev (2007); Enss (2012); Frank et al. (2018), spurring a debate on the applicability of Fermi liquid theory to the degenerate normal fluid below the Fermi temperature $T_{F}=E_{F}/k_{B}$ but above the superfluid transition temperature $T_{c}\approx 0.167\,T_{F}$ Ku et al. (2012); Nascimbène et al. (2011); Rothstein and Shrivastava (2019). It has been conjectured that preformed pairs exist above $T_{c}$, up to a pairing temperature $T^{*}$ Ketterle and Zwierlein (2008); Zwerger (2016); Gaebler et al. (2010); Sagi et al. (2015); Perali et al. (2002); Hu et al. (2010); Magierski et al. (2011); Randeria and Taylor (2014).

Radio frequency (rf) spectroscopy measures the momentum integrated, occupied spectral function, providing a powerful tool for studying interactions and correlations in Fermi gases Gupta et al. (2003); Regal and Jin (2003); Regal et al. (2003); Chin et al. (2004); Schunck et al. (2008); Haussmann et al. (2009). Here, a particle is ejected from the interacting many-body state and transferred into a weakly interacting final state. Shifts in rf spectra indicate attractive or repulsive interactions in the gas. At high temperatures, the width of the rf spectrum reflects the scattering rate in the gas, while at low temperatures, the width has been used to infer the pair size of superfluid fermion pairs Schunck et al. (2008).

The high frequency tails of the rf spectra are sensitive to the spectral function at high momenta, and therefore are governed by short range correlations quantified by the contact, which also determines the change of the energy with respect to the interaction strength Tan (2008a, b, c). From the momentum distribution within nuclei Hen and others (2014); Weiss et al. (2015) to the frequency dependence of the shear viscosity in ultracold fermionic superfluids Taylor and Randeria (2010); Enss et al. (2011), the contact is central to Fermi gases dominated by short-range interactions. Since the contact is proposed to be sensitive to the superfluid pairing gap, it could signal a pseudogap regime above $T_{c}$ Pieri et al. (2009); Palestini et al. (2010); Enss et al. (2011); Mueller (2017). Although the temperature dependence of the contact near $T_{c}$ has been the subject of many theoretical predictions, a consensus has not been reached Hu et al. (2011); Enss et al. (2011); Goulko and Wingate (2016); Rossi et al. (2018).

Initial studies of unitary Fermi gases using rf spectroscopy were affected by inhomogeneous densities in harmonic traps, yielding doubly peaked spectra that were interpreted as observations of the pairing gap Chin et al. (2004); Schunck et al. (2007), and from the influence of interactions in the final state, which caused significantly narrower spectra and smaller shifts than expected Gupta et al. (2003); Schunck et al. (2007); Baym et al. (2007); Punk and Zwerger (2007). Measurements of the contact, made using both rf Stewart et al. (2010); Shkedrov et al. (2018) and Bragg Kuhnle et al. (2010, 2011); Hoinka et al. (2013) spectroscopy, were also broadened by inhomogeneous potentials. To avoid trap broadening, tomographic techniques have been used to measure local rf spectra, yielding measurements of the superfluid gap Schirotzek et al. (2008), the spectral function Gaebler et al. (2010); Sagi et al. (2015), and the contact Sagi et al. (2012). A recent advance has been the creation of uniform box potentials Gaunt et al. (2013); Mukherjee et al. (2017); Hueck et al. (2018). These are ideal for rf spectroscopy and precision measurements of the contact: since the entire cloud is at a constant density, global probes such as rf address all atoms, and benefit from a stronger signal.

In this Letter, we report on rf spectroscopy of the homogeneous unitary Fermi gas in a box potential. A single peak is observed for all temperatures from the superfluid regime into the high temperature Boltzmann gas. The tails of the rf spectra reveal the contact, which shows a rapid rise as the temperature is reduced below $T_{c}$.

We prepare ${}^{6}\mathrm{Li}\;$atoms in two of the three lowest hyperfine states $\ket{\downarrow}{=}\ket{1}$ and $\ket{\uparrow}{=}\ket{3}$ at a magnetic field of 690 G, where interspin interactions are resonant. A uniform optical box potential with cylindrical symmetry is loaded with $N\,{\sim}\,10^{6}$ atoms per spin state (with Fermi energies $E_{F}\,{\sim}\,h\,{\times}\,10$ kHz), creating spin-balanced homogeneous gases at temperatures ranging from $T/T_{F}\,{=}\,0.10$ to $3.0$ Mukherjee et al. (2017). A square rf pulse transfers atoms from state $\ket{\downarrow}$ into state $\ket{f}=\ket{2}$. Final state interactions between atoms in state $\ket{f}$ and atoms in states $\ket{\uparrow}$ and $\ket{\downarrow}$ are small ($k_{F}a_{f}\,{\lesssim}\,0.2$, where $a_{f}$ is the scattering length characterizing collisions between atoms in the final and initial states, and $\hbar k_{F}\,{=}\,\sqrt{2mE_{F}}$ is the Fermi momentum) Schunck et al. (2008). After the rf pulse, we measure the atom numbers $N_{\downarrow}$ and $N_{f}$ in the initial and final states. Within linear response, according to Fermi’s golden rule, $N_{f}$ is proportional to the pulse time $T_{\mathrm{Pulse}}$, the square of the single-particle Rabi frequency $\Omega_{R}$ and an energy density of states. Thus, we define a normalized, dimensionless rf spectrum as $I(\omega)\,{=}\,(N_{f}(\omega)/N_{\downarrow})(E_{F}/\hbar\Omega_{R}^{2}T_{\mathrm{Pulse}})$ sup ; Yan et al. (2018). Because of the scale invariance of the balanced unitary Fermi gas, this dimensionless function can only depend on $T/T_{F}$ and $\hbar\omega/E_{F}$.

For thermometry, we release the cloud from the uniform potential into a harmonic trap along one direction Yan et al. (2018). Since the cloud expands isoenergetically, the resulting spatial profile after thermalization provides the energy per particle, which can be related to the reduced temperature, $T/T_{F}$, using a virial relation and the measured equation of state Ku et al. (2012). To clearly identify the superfluid transition, we measure the pair momentum distribution by a rapid ramp of the magnetic field to the molecular side of the Feshbach resonance before releasing the gas into a harmonic trap for a quarter period Mukherjee et al. (2017); sup .

Initially, we focus on changes in the line shape for rf frequencies within ${\sim}E_{F}/\hbar$ of the bare (single-particle) resonance (see Fig 1(a)), and follow the changes in the peak position $E_{p}$ (shown in Fig 1(b)). As the hot spin-balanced Fermi gas is cooled below the Fermi temperature, the peak shift decreases from roughly zero for temperatures $T\,{\gtrsim}\,T_{F}$, to $E_{p}\,{\approx}\,-0.8~{}E_{F}$ for temperatures below the superfluid transition temperature (see Fig. 1(b)). At high temperatures, one might naïvely expect a shift on the order of $E_{p}\sim\hbar n\lambda_{T}/m$ due to unitarity-limited interactions in the gas. However, there exists both an attractive and a repulsive energy branch, which are symmetric about zero at unitarity Ho and Mueller (2004), and when $T\,\gg\,T_{F}$, their contributions to the shift cancel Enss et al. (2011); Sun and Leyronas (2015); Fletcher et al. (2017). As to the interpretation of the peak shift at degenerate temperatures, a solution to the Cooper problem in the presence of a $T>0$ Fermi sea shows that it is energetically favorable to form pairs when $T\lesssim 0.5\,T_{F}$ sup , and the resulting pair energy agrees qualitatively with the observed shifts (grey line in Fig. 1(b)). However, it is known that fluctuations suppress the onset of pair condensation and superfluidity to $0.167(13)\,T_{F}$ Nozières and Schmitt-Rink (1985); Ku et al. (2012); Randeria and Taylor (2014); Zwerger (2016). In a zero-temperature superfluid, BCS theory would predict a peak shift given by the pair binding energy $E_{B}=\Delta^{2}/2E_{F}$, where $\Delta$ is the pairing gap Ketterle and Zwierlein (2008). Including Hartree terms is found to result in an additional shift of the peak Schirotzek et al. (2008); Haussmann et al. (2009).

Now, we turn to the widths, $\Gamma$, defined as the full width at half maximum of the rf spectra (see Fig. 1(c)). As the gas is cooled from the Boltzmann regime, the width gradually increases, and attains a maximum of $\Gamma\,{=}\,1.35(5)\,E_{\mathrm{F}}$ near $T\,{=}\,0.44(4)\,T_{F}$. For temperatures much higher than $T_{F}$, the system is a Boltzmann gas of atoms scattering with a unitarity limited cross section $\sigma\sim\lambda_{T}^{2}$. Transport properties and short-range pair correlations are governed by the scattering rate $\Gamma=n_{\downarrow}\sigma\langle v_{\rm rel}\rangle\sim\hbar n_{\downarrow}\lambda_{T}/m$ and a mean-free path $l=(n_{\downarrow}\sigma)^{-1}\sim(n_{\downarrow}\lambda_{T}^{2})^{-1}$, where $n_{\downarrow}$ is the density of atoms in $\left|\downarrow\right>$, and $\langle v_{\rm rel}\rangle\sim\hbar/m\lambda_{T}$ is the thermally averaged relative velocity. This leads to a width that scales as $\Gamma\propto 1/\sqrt{T}$, shown as the dotted-dashed line in Fig. 1(c) Enss et al. (2011).

As the cloud is cooled below $T\,{\approx}\,0.5\,T_{F}$, the width decreases linearly with temperature to $\Gamma\sim 0.52\,E_{\mathrm{F}}/\hbar$ in the coldest gases measured ($T=0.10(1)\,T_{\mathrm{F}}$). For temperatures below $T_{c}$, we expect the gas to consist of pairs of size $\xi$. The rf spectrum will be broadened by the distribution of momenta $\sim\hbar/\xi$ inside each pair, leading to a spread of possible final kinetic energies $\hbar^{2}k^{2}/m\sim\hbar^{2}/m\xi^{2}$ and a corresponding spectral width $\hbar/m\xi^{2}$. At unitarity and at $T=0$, the pair size is set by the interparticle spacing $\lambda_{F}$ Zwerger (2016); Ketterle and Zwierlein (2008); Schunck et al. (2008). Thus the rf width at low temperatures is $\Gamma\sim\hbar n\lambda_{F}/m$.

For temperatures above $T_{c}$, it has been suggested that the normal fluid can be described as a Fermi liquid Nascimbène et al. (2010, 2011). This would imply a quadratic relation between the peak width and the temperature Nozieres and Pines (1966), as observed in the widths of the rf spectra of Fermi polarons at unitarity Yan et al. (2018). However, the measured width of the spin-balanced Fermi gas changes linearly in temperature, implying non-Fermi liquid behavior in the normal fluid. In addition, $\Gamma>E_{F}/\hbar$ for $0.3\,{\lesssim}\,T/T_{F}\,{\lesssim}\,1.2$, indicating a breakdown of well-defined quasiparticles over a large range of temperatures near the quantum critical regime Nikolic and Sachdev (2007); Enss (2012); Frank et al. (2018).

We now consider the rf spectrum at frequencies much larger than $E_{F}/\hbar$, where the rf-coupled high-momentum tails reveal information about the short-range pair correlations between atoms. In a gas with contact interactions, the pair correlation function at short distances is $\lim_{r\rightarrow 0}\langle n_{\uparrow}({\bf{r_{0}}+{\bf r/2}})n_{\downarrow}({\bf{r_{0}}}-{\bf{r/2}})\rangle=C/(4\pi r)^{2}$. The contact $C$ connects a number of fundamental relations, independent of the details of the short-range interaction potential Tan (2008a). In particular, the contact governs the momentum distribution at large momenta: $\lim_{k\rightarrow\infty}n(k)=C/k^{4}$. For rf spectroscopy, the density of final states scales as $\sqrt{\omega}$, and the energy cost to flip a spin at high momenta is $\lim_{k\rightarrow\infty}\hbar\omega=\hbar^{2}k^{2}/m$. Thus, the number of atoms transferred by the rf pulse at high frequencies in linear response is $\propto C/\omega^{3/2}$ Haussmann et al. (2009); Zwerger (2016). Including final state interactions, the general expression for the rf transfer rate in a gas with unitarity-limited initial state interactions is Braaten et al. (2010):

[TABLE]

where $N=N_{\uparrow}+N_{\downarrow}$ is the total number of atoms, and the final state molecular binding energy is $E_{b}\,{=}\,\hbar^{2}/ma_{f}^{2}\,{\approx}\,h\,{\times}\,433$ kHz $\approx 40\,E_{F}$. Figure 2 shows a typical rf spectrum at $T/T_{F}=0.10$, with a fit of Eq. 1 to data with detunings $\hbar\omega>3\,E_{F}$, using the dimensionless contact $\tilde{C}=C/Nk_{\mathrm{F}}$ as the only free parameter. At detunings larger than about 10 $E_{F}$, the data deviate from a typical $\omega^{-3/2}$ tail, and is better described by the full expression Eq. 1 including final state interactions. Here, the Rabi frequency was varied across the plot to ensure small transfers near the peak and a high signal-to-noise ratio at detunings up to $\hbar\omega/E_{F}=31$. The fit of Eq. 1 to the data gives a low-temperature contact of $\tilde{C}=3.07(6)$, consistent with a quantum Monte Carlo result $\tilde{C}=2.95(10)$ Drut et al. (2011), the Luttinger-Ward (LW) calculation $\tilde{C}=3.02$ Haussmann et al. (2009), as well as previous measurements using losses $\tilde{C}=3.1(3)$ Laurent et al. (2017) and Bragg spectroscopy $\tilde{C}=3.06(8)$ Hoinka et al. (2013).

For a more efficient measurement of the contact across a range of temperatures, we vary the pulse time at a fixed detuning of $60$ kHz ($\hbar\omega/E_{F}\gtrsim 6$) that is large compared to the Fermi energy and temperature. sup . Deviations from linear response are observed for transfers as small as 5% (see inset of Fig. 2). We fit the transfers to an exponentially saturating function $A(1-\exp(-T_{\mathrm{Pulse}}/\tau))$, and find the initial linear slope $A/\tau$ in order to extract the contact for each temperature using Eq. 1. This ensures that every measurement is taken in the linear response regime.

In Fig. 3(a), we show the temperature dependence of the contact. As the gas is cooled, the contact shows a gradual increase down to the superfluid transition $T_{c}$. Entering the superfluid transition, the contact rapidly rises by approximately 15%. The changes in the contact reveal the temperature dependence of short-range pair correlations in the spin-balanced Fermi gas. At temperatures far above $T_{F}$, the contact reflects the inverse mean free path in the gas $1/l\sim 1/T$. At lower temperatures, the behavior of the contact is better described by a third-order virial expansion (see inset of  3(a)) Hu et al. (2011). Near $T_{c}$, predictions of the contact vary considerably. In the quantum critical regime, a leading-order 1/$N$ calculation (equivalent to a Gaussian pair fluctuation or Nozières–Schmitt-Rink method) results in a prediction $\tilde{C}(\mu=0,T\approx 0.68~{}T_{F})=2.34$ Enss (2012), which is consistent with our measurement of $\tilde{C}(T=0.65(4)~{}T_{F})=2.29(13)$. For temperatures above the superfluid transition, our data agree well with both a bold diagrammatic Monte Carlo calculation Rossi et al. (2018), and, especially near $T_{c}$, the LW calculation Enss et al. (2011). The contact rises as the temperature is decreased below $T_{c}$, a feature captured by the LW formalism, in which the contact is directly sensitive to pairing: $\tilde{C}\sim(\Delta/E_{F})^{2}$ Pieri et al. (2009); Haussmann et al. (2009). While short-range pair correlations do not necessarily signify pairing Mueller (2017), the rapid rise of the contact below $T_{c}$ is strongly indicative of an additional contribution from fermion pairs, as predicted by LW. At temperatures $T\ll T_{c}$, below the reach of our experiment, phonons are likely the only remaining excitations in the unitary Fermi gas, and are expected to contribute to the contact by an amount that scales as $T^{4}$ Yu et al. (2009).

In conclusion, rf spectroscopy of the homogeneous unitary Fermi gas reveals strong attractive interactions, the non-Fermi-liquid nature of excitations in the gas across the quantum critical regime, and a rapid increase in short-range pair correlations upon entering the superfluid regime. The strong variation with temperature of the position of the spectral peak may serve as a local thermometer in future studies of heat transport in ultracold Fermi gases. Furthermore, these measurements of the contact provide a benchmark for many-body theories of the unitary Fermi gas. The uniform trap enables direct access to homogeneous measurements of thermodynamic quantities, and increases sensitivity to abrupt changes of those quantities near phase transitions. This could be particularly useful in the limit of high spin imbalance, where the nature of impurities suddenly transitions from Fermi polarons to molecules Punk et al. (2009); Schirotzek et al. (2009).

We note that measurements of the temperature dependence of the contact were simultaneously performed at Swinburne using Bragg spectroscopy Carcy et al. (2019). Their data are in excellent agreement with the present results.

We thank C. J. Vale, F. Werner, and W. Zwerger for helpful discussions. This work was supported by the NSF, AFOSR, ONR, the AFOSR MURI on Exotic Phases, and the David and Lucile Packard Foundation. J.S. was supported by LabEX ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.