Goldstone mode and pair-breaking excitations in atomic Fermi superfluids

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Spontaneous symmetry breaking is a central paradigm of elementary particle physics1, magnetism2, superfluidity3 and superconductivity4. According to Goldstone’s theorem, phase transitions that break continuous symmetries lead to the existence of gapless excitations in the long-wavelength limit5. These Goldstone modes can become the dominant low-energy excitation, showing that symmetry breaking has a profound impact on the physical properties of matter. Here, we present a comprehensive study of the elementary excitations in a homogeneous strongly interacting Fermi gas through the crossover from a Bardeen–Cooper–Schrieffer (BCS) superfluid to a Bose–Einstein condensate (BEC) of molecules using two-photon Bragg spectroscopy. The spectra exhibit a discrete Goldstone mode, associated with the broken-symmetry superfluid phase, as well as pair-breaking single-particle excitations. 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class="c-article-title" data-test="article-title" data-article-title="">Goldstone mode and pair-breaking excitations in atomic Fermi superfluids</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Sascha-Hoinka-Aff1" data-author-popup="auth-Sascha-Hoinka-Aff1" data-author-search="Hoinka, Sascha">Sascha Hoinka</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Paul-Dyke-Aff1" data-author-popup="auth-Paul-Dyke-Aff1" data-author-search="Dyke, Paul">Paul Dyke</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-6760-9641"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-6760-9641</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Marcus_G_-Lingham-Aff1" data-author-popup="auth-Marcus_G_-Lingham-Aff1" data-author-search="Lingham, Marcus G.">Marcus G. Lingham</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Jami_J_-Kinnunen-Aff2" data-author-popup="auth-Jami_J_-Kinnunen-Aff2" data-author-search="Kinnunen, Jami J.">Jami J. Kinnunen</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Georg_M_-Bruun-Aff3" data-author-popup="auth-Georg_M_-Bruun-Aff3" data-author-search="Bruun, Georg M.">Georg M. Bruun</a><sup class="u-js-hide"><a href="#Aff3">3</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 6 authors for this article" title="Show all 6 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Chris_J_-Vale-Aff1" data-author-popup="auth-Chris_J_-Vale-Aff1" data-author-search="Vale, Chris J." data-corresp-id="c1">Chris J. Vale<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-4059-0178"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-4059-0178</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/nphys" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Nature Physics</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 13</b>, <span class="u-visually-hidden">pages </span>943–946 (<span data-test="article-publication-year">2017</span>)<a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">8238 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="citation-count"> <p class="c-article-metrics-bar__count">83 <span class="c-article-metrics-bar__label">Citations</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">15 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/nphys4187/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/quantum-fluids-and-solids" data-track="click" data-track-action="view subject" data-track-label="link">Quantum fluids and solids</a></li><li class="c-article-subject-list__subject"><a href="/subjects/ultracold-gases" data-track="click" data-track-action="view subject" data-track-label="link">Ultracold gases</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs2" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs2">Abstract</h2><div class="c-article-section__content" id="Abs2-content"><p>Spontaneous symmetry breaking is a central paradigm of elementary particle physics<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Nambu, Y. & Jona-Lasino, G. Dynamical model of elementary particles based on an analogy with superconductivity. Phys. Rev. 122, 345–358 (1961)." href="/articles/nphys4187#ref-CR1" id="ref-link-section-d149561705e368">1</a></sup>, magnetism<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Holt, Rinehart & Winston, 1976)." href="/articles/nphys4187#ref-CR2" id="ref-link-section-d149561705e372">2</a></sup>, superfluidity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Griffin, A. Excitations in a Bose Condensed Liquid (Cambridge Univ. Press, 1993)." href="/articles/nphys4187#ref-CR3" id="ref-link-section-d149561705e376">3</a></sup> and superconductivity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Anderson, P. W. Random-phase approximation in the theory of superconductivity. Phys. Rev. 112, 1900–1916 (1958)." href="/articles/nphys4187#ref-CR4" id="ref-link-section-d149561705e380">4</a></sup>. According to Goldstone’s theorem, phase transitions that break continuous symmetries lead to the existence of gapless excitations in the long-wavelength limit<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Goldstone, J. Field theories with superconductor solutions. Nuovo Cimento. 19, 154–164 (1961)." href="/articles/nphys4187#ref-CR5" id="ref-link-section-d149561705e384">5</a></sup>. These Goldstone modes can become the dominant low-energy excitation, showing that symmetry breaking has a profound impact on the physical properties of matter. Here, we present a comprehensive study of the elementary excitations in a homogeneous strongly interacting Fermi gas through the crossover from a Bardeen–Cooper–Schrieffer (BCS) superfluid to a Bose–Einstein condensate (BEC) of molecules using two-photon Bragg spectroscopy. The spectra exhibit a discrete Goldstone mode, associated with the broken-symmetry superfluid phase, as well as pair-breaking single-particle excitations. Our techniques yield a direct determination of the superfluid pairing gap and speed of sound in close agreement with strong-coupling theories.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41586-022-05405-6/MediaObjects/41586_2022_5405_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41586-022-05405-6?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41586-022-05405-6">Unitary <i>p</i>-wave interactions between fermions in an optical lattice </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">11 January 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41598-020-65371-9/MediaObjects/41598_2020_65371_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41598-020-65371-9?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s41598-020-65371-9">Linear response of a superfluid Fermi gas inside its pair-breaking continuum </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">14 August 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41586-021-03443-0/MediaObjects/41586_2021_3443_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41586-021-03443-0?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41586-021-03443-0">Transition from an atomic to a molecular Bose–Einstein condensate </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">28 April 2021</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732324254, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Main"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Main</h2><div class="c-article-section__content" id="Sec1-content"><p>When a Hamiltonian is invariant with respect to a continuous symmetry, but the ground state is not, a massless bosonic mode appears in the spectrum of allowed excitations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Goldstone, J. Field theories with superconductor solutions. Nuovo Cimento. 19, 154–164 (1961)." href="/articles/nphys4187#ref-CR5" id="ref-link-section-d149561705e414">5</a></sup>. At temperatures low enough for quantum effects to become prominent, dynamical behaviours, such as superconductivity and superfluidity, are possible only due to the low-energy excitation spectrum. Superfluid and superconducting states break gauge invariance and the resultant Goldstone mode is an oscillation of the phase of the order parameter giving rise to a collective motion of particles that is distinct from single-particle excitations. In superconductors, the Coulomb interaction lifts the collective mode up to the frequency of the classical plasma oscillation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Anderson, P. W. Random-phase approximation in the theory of superconductivity. Phys. Rev. 112, 1900–1916 (1958)." href="/articles/nphys4187#ref-CR4" id="ref-link-section-d149561705e418">4</a></sup>, present in the normal phase, such that the Goldstone mode is generally imperceptible<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Pekker, D. & Varma, C. M. Amplitude/Higgs modes in condensed matter physics. Annu. Rev. Condens. Matter Phys. 6, 269–297 (2015)." href="/articles/nphys4187#ref-CR6" id="ref-link-section-d149561705e422">6</a></sup>. In neutral superfluids, however, the Goldstone mode takes the form of a gapless phonon<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bogoliubov, N. N. On the theory of superfluidity. J. Phys. USSR 11, 23–32 (1947)." href="/articles/nphys4187#ref-CR7" id="ref-link-section-d149561705e426">7</a></sup> and provides a dramatic signature of macroscopic order.</p><p>Ultracold gases of atomic fermions have enabled the creation and study of high-transition-temperature superfluids in the smooth crossover from the BCS to BEC regimes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Zwerger, W. (ed.) BCS–BEC Crossover and the Unitary Fermi Gas (Lecture Notes in Physics, Springer, 2012)." href="/articles/nphys4187#ref-CR8" id="ref-link-section-d149561705e433">8</a></sup>. Both first-sound<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Joseph, J. et al. Measurement of sound velocity in a Fermi gas near a Feshbach resonance. Phys. Rev. Lett. 98, 170401 (2007)." href="/articles/nphys4187#ref-CR9" id="ref-link-section-d149561705e437">9</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013)." href="/articles/nphys4187#ref-CR10" id="ref-link-section-d149561705e440">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Weimer, W. et al. Critical velocity in the BEC-BCS crossover. Phys. Rev. Lett. 114, 095301 (2015)." href="/articles/nphys4187#ref-CR11" id="ref-link-section-d149561705e443">11</a></sup> and second-sound<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013)." href="/articles/nphys4187#ref-CR10" id="ref-link-section-d149561705e447">10</a></sup> propagation have been observed in inhomogeneous Fermi gases, yet the basic elementary excitation spectrum has not been measured. Here, we present a comprehensive study of the low-temperature excitations in a homogeneous Fermi superfluid throughout the whole BCS–BEC crossover. The spectra exhibit both a dominant Goldstone mode, or Bogoliubov–Anderson (BA) phonon, and a single-particle continuum. Our study reveals how the energy and spectral weight of these excitations evolve as a function of the interaction strength. We develop a theory based on the quasiparticle random-phase approximation (QRPA), which provides a good quantitative description of the data.</p><p>The starting point for our experiments is a harmonically trapped gas of fermionic <sup>6</sup>Li atoms in a balanced mixture of the lowest two hyperfine states with tunable <i>s</i>-wave interactions near a broad Feshbach resonance (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). Atoms are cooled to temperatures below the superfluid transition temperature, <i>T</i><sub>c</sub>, across a wide range of the BCS–BEC crossover<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e466">12</a></sup>. We measure the density–density response of these gases using two-photon Bragg spectroscopy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Stenger, J. et al. Bragg spectroscopy of a Bose–Einstein condensate. Phys. Rev. Lett. 82, 4569–4573 (1999)." href="/articles/nphys4187#ref-CR13" id="ref-link-section-d149561705e471">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Steinhauer, J., Ozeri, R., Katz, N. & Davidson, N. Excitation spectrum of a Bose–Einstein condensate. Phys. Rev. Lett. 88, 120407 (2002)." href="/articles/nphys4187#ref-CR14" id="ref-link-section-d149561705e474">14</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Veeravalli, G., Kuhnle, E., Dyke, P. & Vale, C. J. Bragg spectroscopy of a strongly interacting Fermi gas. Phys. Rev. Lett. 101, 250403 (2008)." href="/articles/nphys4187#ref-CR15" id="ref-link-section-d149561705e477">15</a></sup>. This is the cold-atom analogue of inelastic neutron scattering, which has enabled characterization of the dispersion relation, roton minimum, and condensate fraction in superfluid helium<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Griffin, A. Excitations in a Bose Condensed Liquid (Cambridge Univ. Press, 1993)." href="/articles/nphys4187#ref-CR3" id="ref-link-section-d149561705e481">3</a></sup>; and inelastic X-ray scattering, used to measure electronic excitations in strongly correlated materials<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Rueff, J.-P. & Shukla, A. Inelastic X-ray scattering by electronic excitations under high pressure. Rev. Mod. Phys. 82, 847–896 (2010)." href="/articles/nphys4187#ref-CR16" id="ref-link-section-d149561705e485">16</a></sup>. Using tightly focused Bragg lasers that intersect in the centre of a trapped atom cloud, we scatter atoms from a region of near-homogeneous density (<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Fig. 1a</a>). Bragg scattering involves the absorption of a photon with energy <i><span class="stix">ℏ</span>ω</i><sub>a</sub>, where <i><span class="stix">ℏ</span></i> is Planck’s constant, and wavevector <b>k</b><sub>a</sub> from one laser beam, and stimulated emission of a photon with energy <i><span class="stix">ℏ</span>ω</i><sub>b</sub> and wavevector <b>k</b><sub>b</sub> into a second beam, thereby transferring energy <i><span class="stix">ℏ</span>ω</i> = <i><span class="stix">ℏ</span></i>(<i>ω</i><sub>a</sub> − <i>ω</i><sub>b</sub>) and momentum <i><span class="stix">ℏ</span></i> <b>k</b> = <i><span class="stix">ℏ</span></i>(<b>k</b><sub>a</sub> − <b>k</b><sub>b</sub>). Within linear response the total momentum imparted to the cloud, <i>P</i><sub><i>x</i></sub>, is proportional to the imaginary part of the dynamic susceptibility or density–density response function, <img src="//media.springernature.com/lw62/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq1_HTML.gif" style="width:62px;max-width:none;" alt=""> (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Probing homogeneous excitation spectra with focused beam Bragg scattering."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 1: Probing homogeneous excitation spectra with focused beam Bragg scattering.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/nphys4187/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig1_HTML.jpg?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig1_HTML.jpg" alt="figure 1" loading="lazy" width="685" height="726"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p><b>a</b>, Two far-detuned Bragg lasers, focused to 20 μm 1/<i>e</i><sup>2</sup> radii, intersect at an angle of 12.9° in the centre of a harmonically trapped Fermi gas. <b>b</b>,<b>c</b>, Difference between images of atom clouds with and without Bragg scattering of single particles above the pair-breaking continuum at 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.6 in the BCS regime (<b>b</b>), and the BA phonon at unitarity (<b>c</b>). Purple ellipses indicate the size of the expanded clouds and the dashed lines intersect the cloud centre. Each optical density (OD) image is 400 μm by 180 μm. The Bragg laser intensities used in <b>b</b> were 2.4 times higher than those in <b>c</b> due to the weaker response of single particles compared with the collective BA mode. <b>d</b>,<b>e</b>, Calculated excitation spectra for 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.6 (<b>d</b>) and 0.0 (<b>e</b>). The green circles mark the (<i>ω</i>, <i>k</i>) coordinates of the Bragg excitation used in <b>b</b> and <b>c</b> and the red lines indicate the threshold for single-particle excitations (2<i>Δ</i>).</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/nphys4187/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Bragg spectra are obtained by applying a 1.2 ms Bragg pulse and measuring <i>P</i><sub><i>x</i></sub> as a function of the Bragg frequency <i>ω</i>. A Bragg spectrum provides full information on the energy and spectral weight of particle-conserving excitations at momentum <i><span class="stix">ℏ</span></i> <b>k</b> (ref. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Brunello, A., Dalfovo, F., Pitaevskii, L., Stringari, S. & Zambelli, F. Momentum transferred to a trapped Bose–Einstein condensate by stimulated light scattering. Phys. Rev. A 64, 063614 (2001)." href="/articles/nphys4187#ref-CR17" id="ref-link-section-d149561705e666">17</a>). After applying the Bragg pulse, atoms are released from the trap and we determine the centre-of-mass displacement, <i>ΔX</i> <span class="stix">∝</span> <i>P</i><sub><i>x</i></sub>, of the Bragg scattered volume, with respect to the unperturbed atoms, after 2 ms time of flight<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Veeravalli, G., Kuhnle, E., Dyke, P. & Vale, C. J. Bragg spectroscopy of a strongly interacting Fermi gas. Phys. Rev. Lett. 101, 250403 (2008)." href="/articles/nphys4187#ref-CR15" id="ref-link-section-d149561705e680">15</a></sup>. In the measurements that follow, the mean atomic density in the Bragg volume, <img src="//media.springernature.com/lw7/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq2_HTML.gif" style="width:7px;max-width:none;" alt="">, lies in the range <img src="//media.springernature.com/lw114/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq3_HTML.gif" style="width:114px;max-width:none;" alt="">, where <i>n</i><sub>0</sub> is the peak density in the trap centre (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>).</p><p><a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Figure 1b</a> shows the difference between two time-of-flight images of atom clouds, with and without Bragg scattering, on the BCS side of the Feshbach resonance for an interaction strength of 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.6 (875 G), where <img src="//media.springernature.com/lw85/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq4_HTML.gif" style="width:85px;max-width:none;" alt=""> is the Fermi wavevector and <i>a</i> is the <i>s</i>-wave scattering length. Bragg scattering removes atoms from the centre of the cloud, where the lasers overlap, and displaces them along <i>x</i>. Here, <i>ω</i> lies in the single-particle continuum, indicated by the green circle on the excitation spectrum (<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Fig. 1d</a>), calculated for 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.6 using the QRPA theory (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). The Bragg frequency is normalized by the Fermi energy, which is set by the mean density <i>E</i><sub>F</sub> = <i>k</i><sub>B</sub><i>T</i><sub>F</sub> = <i><span class="stix">ℏ</span></i><sup>2</sup><i>k</i><sub>F</sub><sup>2</sup>/(2<i>m</i>), where <i>k</i><sub>B</sub> is Boltzmann’s constant and <i>m</i> is the atomic mass.</p><p><a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Figure 1c</a> shows a difference image taken with <i>ω</i> tuned to the peak of the BA phonon mode (<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Fig. 1e</a>) at unitarity 1/(<i>k</i><sub>F</sub><i>a</i>) = 0.0. Comparing <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Fig. 1b</a> with <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">1c</a> reveals a qualitative difference between single-particle and phonon (collective) excitations. Phonons are a density modulation (sound wave) with the periodicity of the Bragg lattice, excited when the lattice velocity (<i>ω</i>/<i>k</i>) equals the sound velocity <i>c</i><sub>s</sub>. As the phonon propagates along <i>x</i> and the cloud expands, the density continuously decreases and so too does the local speed of sound (gradient of the dispersion). To conserve energy and momentum, the phonon decays into lower-energy modes, whose wavevectors need not be parallel, in a manner reminiscent of Beliaev damping<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Beliaev, S. T. Energy spectrum of a non-ideal Bose gas. Sov. Phys. JETP 2, 299–307 (1958)." href="/articles/nphys4187#ref-CR18" id="ref-link-section-d149561705e822">18</a></sup>. While the density-dependent decay dynamics may be quite complex, the qualitative result is a long-wavelength density modulation propagating along <i>x</i> (<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig1">Fig. 1c</a>). In contrast to the case of single-particle scattering, the density minimum no longer remains fixed at the location of the Bragg lasers.</p><p>We have measured a series of Bragg spectra at <i>k</i> <span class="stix">∼</span> <i>k</i><sub>F</sub>/2, within the linear response regime, that map the low-temperature excitations for a range of interaction strengths, plotted in <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig2">Fig. 2</a> (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). For 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.9 and −1.6, (rightmost traces in <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig2">Fig. 2</a>) we observe only single-particle excitations. Here, the pair correlation length, <i>ξ</i><sub>pair</sub> (ref. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Schunck, C. H., Shin, Y., Schirotzek, A. & Ketterle, W. Determination of the fermion pair size in a resonantly interacting superfluid. Nature 454, 739–743 (2008)." href="/articles/nphys4187#ref-CR19" id="ref-link-section-d149561705e861">19</a>), exceeds 1/<i>k</i> and the phonon mode merges with the single-particle continuum for <i>k</i> < <i>k</i><sub>F</sub>/2(refs <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Marini, M., Pistolesi, F. & Strinati, G. C. Evolution from BCS superconductivity to Bose condensation: analytic results for the crossover in three dimensions. Eur. Phys. J. B 1, 151–159 (1998)." href="/articles/nphys4187#ref-CR20" id="ref-link-section-d149561705e875">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Combescot, R., Kagan, M. Yu. & Stringari, S. Collective mode of homogeneous superfluid Fermi gases in the BEC–BCS crossover. Phys. Rev. A 74, 042717 (2006)." href="/articles/nphys4187#ref-CR21" id="ref-link-section-d149561705e878">21</a>). Closer to the Feshbach resonance, the Goldstone mode (BA phonon) first appears around 1/(<i>k</i><sub>F</sub><i>a</i>) <span class="stix">≲</span> −0.5, where <i>ξ</i><sub>pair</sub> ≈ 1/<i>k</i> (ref. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Schunck, C. H., Shin, Y., Schirotzek, A. & Ketterle, W. Determination of the fermion pair size in a resonantly interacting superfluid. Nature 454, 739–743 (2008)." href="/articles/nphys4187#ref-CR19" id="ref-link-section-d149561705e895">19</a>), and is identified using a Gaussian fit (shaded blue). At unitarity, the BA mode dominates the spectra and the single-particle branch is increasingly suppressed. In the BEC regime single-particle excitations fall below our measurement sensitivity due to the reduced spin-susceptibility<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Hoinka, S., Lingham, M., Delehaye, M. & Vale, C. J. Dynamic spin response of a strongly interacting Fermi gas. Phys. Rev. Lett. 109, 050403 (2012)." href="/articles/nphys4187#ref-CR22" id="ref-link-section-d149561705e899">22</a></sup> as fermion pairs become more tightly bound.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Bragg spectra throughout the BCS–BEC crossover."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 2: Bragg spectra throughout the BCS–BEC crossover.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/nphys4187/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig2_HTML.jpg?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig2_HTML.jpg" alt="figure 2" loading="lazy" width="685" height="282"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Blue and red points are experimental data, solid blue lines are Gaussian fits to the BA mode peak (shaded blue) and the red shaded region indicates the single-particle excitation branch. The Bragg laser intensities are varied as the interactions are tuned from the BCS to BEC regime to maximize the signal-to-noise while remaining in the linear response regime (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). <img src="//media.springernature.com/lw58/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq7_HTML.gif" style="width:58px;max-width:none;" alt=""> plotted above has been scaled by the ratio of the relative two-photon transition probabilities so that all spectra can be displayed on the same scale. Error bars represent the standard deviation of the measured centre-of-mass displacements.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/nphys4187/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p> The local temperature within the Bragg volume is <i>T</i>/<i>T</i><sub>F</sub> = 0.09(1) at unitarity (just over 0.5<i>T</i><sub>c</sub>), determined by fitting the density profile to the known equation of state<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas. Science 335, 563–567 (2012)." href="/articles/nphys4187#ref-CR23" id="ref-link-section-d149561705e949">23</a></sup>. While direct thermometry away from unitarity remains a challenge, tuning to the BCS (BEC) regime by scanning the magnetic field adiabatically around the Feshbach resonance will decrease (increase) <i>T</i>/<i>T</i><sub>F</sub> such that we expect <i>T</i> to remain below <i>T</i><sub>c</sub> for all spectra. Our calculated temperatures following an adiabatic sweep to the limiting interaction strengths of 1/(<i>k</i><sub>F</sub><i>a</i>) = −1.6 and +1.1 are <i>T</i>/<i>T</i><sub>F</sub> = 0.02(1) and 0.10(2), respectively<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e982">12</a></sup>. In the BEC regime, 1/(<i>k</i><sub>F</sub><i>a</i>) <span class="stix">∼</span> +1, we observe condensate fractions exceeding 80%. A recent study of second-sound propagation in a unitary Fermi gas found that the superfluid fraction approaches unity for <i>T</i> <span class="stix">≲</span> 0.1<i>T</i><sub>F</sub> (ref. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013)." href="/articles/nphys4187#ref-CR10" id="ref-link-section-d149561705e999">10</a>). Hence, our measurements in the strongly interacting regime represent the low-temperature (near ground-state) excitations.</p><p> In the range −0.5 ≤ 1/(<i>k</i><sub>F</sub><i>a</i>) ≤ 0, both phonon and single-particle excitations are visible in the individual spectra. Furthermore, for −0.2 ≤ 1/(<i>k</i><sub>F</sub><i>a</i>) ≤ 0, these two branches separate from each other, enabling direct read-off of the superfluid pairing gap <i>Δ</i>. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig3">Figure 3a</a> shows a zoomed view of the Bragg spectrum at unitarity. Filled points are experimental data and the shaded curve is the QRPA theory including the Fourier width of the Bragg pulse. Since a minimum energy of <i><span class="stix">ℏ</span>ω</i> = 2<i>Δ</i> is required to break a pair and produce two free atoms, we associate the sharp onset of single-particle excitations with 2<i>Δ</i> (dotted vertical line). At unitarity we find <i>Δ</i>/<i>E</i><sub>F</sub> = 0.47 ± 0.03. The pairing gap has previously been measured using momentum<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Stewart, J. T., Gaebler, J. P. & Jin, D. S. Using photoemission spectroscopy to probe a strongly interacting Fermi gas. Nature 454, 744–747 (2008)." href="/articles/nphys4187#ref-CR24" id="ref-link-section-d149561705e1042">24</a></sup> and spatially resolved<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Schirotzek, A., Shin, Y., Schunck, C. H. & Ketterle, W. Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy. Phys. Rev. Lett. 101, 140403 (2008)." href="/articles/nphys4187#ref-CR25" id="ref-link-section-d149561705e1046">25</a></sup> radiofrequency spectroscopy. Our localized Bragg measurements are consistent with previous radiofrequency data<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Schirotzek, A., Shin, Y., Schunck, C. H. & Ketterle, W. Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy. Phys. Rev. Lett. 101, 140403 (2008)." href="/articles/nphys4187#ref-CR25" id="ref-link-section-d149561705e1050">25</a></sup>, yet provide the pairing gap directly, free of final-state effects, Hartree energy shifts or density inhomogeneities. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig3">Figure 3b, c</a> shows spectra for 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.11 and 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.21, respectively. A plot of <i>Δ</i>/<i>E</i><sub>F</sub> for these three spectra is provided in <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig3">Fig. 3d</a> along with different calculations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e1081">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Pieri, P., Pisani, L. & Strinati, G. C. BCS–BEC crossover at finite temperature in the broken-symmetry phase. Phys. Rev. B 70, 094508 (2004)." href="/articles/nphys4187#ref-CR26" id="ref-link-section-d149561705e1084">26</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Carlson, J. & Reddy, S. Superfluid pairing gap in strong coupling. Phys. Rev. Lett. 100, 150403 (2008)." href="/articles/nphys4187#ref-CR27" id="ref-link-section-d149561705e1087">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Diener, R. B., Sensarma, R. & Randeria, M. Quantum fluctuations in the superfluid state of the BCS–BEC crossover. Phys. Rev. A 77, 023626 (2008)." href="/articles/nphys4187#ref-CR28" id="ref-link-section-d149561705e1090">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Gubler, P., Yamamoto, N., Hatsuda, T. & Nishida, Y. Single-particle spectral density of the unitary Fermi gas: novel approach based on the operator product expansion, sum rules and the maximum entropy method. Ann. Phys. 356, 467–497 (2015)." href="/articles/nphys4187#ref-CR29" id="ref-link-section-d149561705e1093">29</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Tajima, H. et al. Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev. A 95, 043625 (2017)." href="/articles/nphys4187#ref-CR30" id="ref-link-section-d149561705e1096">30</a></sup>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Bragg spectra and pairing gap near unitarity."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 3: Bragg spectra and pairing gap near unitarity.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/nphys4187/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig3_HTML.jpg?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig3_HTML.jpg" alt="figure 3" loading="lazy" width="685" height="226"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p><b>a</b>, Experimental spectrum at unitarity (filled data points) and QRPA theory (shaded). Vertical dotted lines indicate the onset of the single-particle branch for <i><span class="stix">ℏ</span>ω</i> ≥ 2<i>Δ</i> (<i>E</i><sub>F</sub> = 12.1 kHz). Inset: comparison of experimental and theoretical BA mode peaks. <b>b</b>,<b>c</b>, Zoomed view of Bragg spectra at 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.11 and 1/(<i>k</i><sub>F</sub><i>a</i>) = −0.21. <b>d</b>, Pairing gap <i>Δ</i> determined from Bragg spectra (blue points) along with previous radiofrequency (RF) measurements (green triangles)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Schirotzek, A., Shin, Y., Schunck, C. H. & Ketterle, W. Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy. Phys. Rev. Lett. 101, 140403 (2008)." href="/articles/nphys4187#ref-CR25" id="ref-link-section-d149561705e1150">25</a></sup> and theoretical predictions: BCS (black dotted line), T-matrix (green dashed)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Pieri, P., Pisani, L. & Strinati, G. C. BCS–BEC crossover at finite temperature in the broken-symmetry phase. Phys. Rev. B 70, 094508 (2004)." href="/articles/nphys4187#ref-CR26" id="ref-link-section-d149561705e1154">26</a></sup>, Luttinger–Ward (black solid)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e1158">12</a></sup>, Gaussian fluctuation theory (red dash-dotted)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Diener, R. B., Sensarma, R. & Randeria, M. Quantum fluctuations in the superfluid state of the BCS–BEC crossover. Phys. Rev. A 77, 023626 (2008)." href="/articles/nphys4187#ref-CR28" id="ref-link-section-d149561705e1162">28</a></sup>, Monte Carlo (brown cross)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Carlson, J. & Reddy, S. Superfluid pairing gap in strong coupling. Phys. Rev. Lett. 100, 150403 (2008)." href="/articles/nphys4187#ref-CR27" id="ref-link-section-d149561705e1166">27</a></sup>, operator product expansion (red asterisk)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Gubler, P., Yamamoto, N., Hatsuda, T. & Nishida, Y. Single-particle spectral density of the unitary Fermi gas: novel approach based on the operator product expansion, sum rules and the maximum entropy method. Ann. Phys. 356, 467–497 (2015)." href="/articles/nphys4187#ref-CR29" id="ref-link-section-d149561705e1171">29</a></sup>, and extended T-matrix (purple dash-dot-dotted)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Tajima, H. et al. Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev. A 95, 043625 (2017)." href="/articles/nphys4187#ref-CR30" id="ref-link-section-d149561705e1175">30</a></sup>. Error bars in <b>a</b>–<b>c</b> denote the standard deviation of the measured centre-of-mass displacements. Error bars in <b>d</b> are calculated by combining the uncertainty in determining <img src="//media.springernature.com/lw7/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq8_HTML.gif" style="width:7px;max-width:none;" alt=""> with the finite spectral resolution of the measurements.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/nphys4187/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The experimentally determined pairing gap can serve as an input parameter for our diagrammatic theory, which uses the chemical potential as a fitting parameter (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). We find good agreement between our QRPA calculation and experiment over the full excitation spectrum (shaded curves, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig3">Fig. 3</a>), particularly in the frequency of the BA mode. When the theory is scaled to match the amplitude of the experimentally measured single-particle branch, the calculated BA mode is consistently more than twice as high and approximately two-thirds the width of the measured peak (inset, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig3">Fig. 3a</a>). Damping mechanisms<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Kurkjian, H., Castin, Y. & Sinatra, A. Concavity of the collective excitation branch of a Fermi gas in the BEC–BCS crossover. Phys. Rev. A 93, 013623 (2016)." href="/articles/nphys4187#ref-CR31" id="ref-link-section-d149561705e1216">31</a></sup>, not included in the theory, may explain this discrepancy.</p><p>As <i>k</i> → 0, the BA mode displays linear dispersion with a gradient set by the sound velocity <i>c</i><sub>s</sub>. The centre frequency of the BA mode <i>ω</i><sub>BA</sub> is found from a Gaussian fit to the BA peak (solid blue lines, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig2">Fig. 2</a>) providing a measure of <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> = <i>ω</i><sub>BA</sub>/(<i>kv</i><sub>F</sub>), where <i>v</i><sub>F</sub> = <i><span class="stix">ℏ</span>k</i><sub>F</sub>/<i>m</i> is the Fermi velocity. This is plotted in <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig4">Fig. 4</a> (blue circles) for interactions where we obtain a reliable fit. In the momentum range used here, 0.4 <span class="stix">≲</span> <i>k</i>/<i>k</i><sub>F</sub> <span class="stix">≲</span> 0.6 (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>), the dispersion remains close to linear near unitarity, but becomes concave (convex) in the BCS (BEC) regime<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Combescot, R., Kagan, M. Yu. & Stringari, S. Collective mode of homogeneous superfluid Fermi gases in the BEC–BCS crossover. Phys. Rev. A 74, 042717 (2006)." href="/articles/nphys4187#ref-CR21" id="ref-link-section-d149561705e1281">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Pieri, P., Pisani, L. & Strinati, G. C. BCS–BEC crossover at finite temperature in the broken-symmetry phase. Phys. Rev. B 70, 094508 (2004)." href="/articles/nphys4187#ref-CR26" id="ref-link-section-d149561705e1284">26</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Kurkjian, H., Castin, Y. & Sinatra, A. Concavity of the collective excitation branch of a Fermi gas in the BEC–BCS crossover. Phys. Rev. A 93, 013623 (2016)." href="/articles/nphys4187#ref-CR31" id="ref-link-section-d149561705e1287">31</a></sup>. We can estimate this curvature for finite <i>k</i>/<i>k</i><sub>F</sub> using the QRPA calculation and correct for it to estimate <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> as <i>k</i> → 0 (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). The corrected data (orange diamonds in <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig4">Fig. 4</a>) agree well with theoretical calculations of <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> throughout the BCS–BEC crossover<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e1325">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Diener, R. B., Sensarma, R. & Randeria, M. Quantum fluctuations in the superfluid state of the BCS–BEC crossover. Phys. Rev. A 77, 023626 (2008)." href="/articles/nphys4187#ref-CR28" id="ref-link-section-d149561705e1328">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Tajima, H. et al. Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev. A 95, 043625 (2017)." href="/articles/nphys4187#ref-CR30" id="ref-link-section-d149561705e1331">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Manini, N. & Salasnich, L. Bulk and collective properties of a dilute Fermi gas in the BCS–BEC crossover. Phys. Rev. A 71, 033625 (2005)." href="/articles/nphys4187#ref-CR32" id="ref-link-section-d149561705e1334">32</a></sup> and also with previous experiments<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Joseph, J. et al. Measurement of sound velocity in a Fermi gas near a Feshbach resonance. Phys. Rev. Lett. 98, 170401 (2007)." href="/articles/nphys4187#ref-CR9" id="ref-link-section-d149561705e1338">9</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013)." href="/articles/nphys4187#ref-CR10" id="ref-link-section-d149561705e1341">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Weimer, W. et al. Critical velocity in the BEC-BCS crossover. Phys. Rev. Lett. 114, 095301 (2015)." href="/articles/nphys4187#ref-CR11" id="ref-link-section-d149561705e1344">11</a></sup>. Our measurements are the first to probe sound propagation in a homogeneous Fermi gas, allowing direct comparison with theory, free of complications arising from density inhomogeneities and partial hydrodynamics across the wavefront.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Speed of sound across the BCS–BEC crossover."><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 4: Speed of sound across the BCS–BEC crossover.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/nphys4187/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig4_HTML.jpg?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_Fig4_HTML.jpg" alt="figure 4" loading="lazy" width="685" height="560"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Solid blue points are found using the raw BA mode frequency and orange diamonds show corrected estimates of <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> as <i>k</i> → 0 based on the calculated curvature of the BA mode dispersion at non-zero <i>k</i>/<i>k</i><sub>F</sub> (see <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a>). At unitarity, <i>v</i><sub>F</sub> = 0.040 m s<sup>−1</sup>. Also shown are theoretical predictions for <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub>: BCS theory (black dotted line), Beliaev (black dashed)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Beliaev, S. T. Energy spectrum of a non-ideal Bose gas. Sov. Phys. JETP 2, 299–307 (1958)." href="/articles/nphys4187#ref-CR18" id="ref-link-section-d149561705e1397">18</a></sup>, density functional (green dash-dash-dotted)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Manini, N. & Salasnich, L. Bulk and collective properties of a dilute Fermi gas in the BCS–BEC crossover. Phys. Rev. A 71, 033625 (2005)." href="/articles/nphys4187#ref-CR32" id="ref-link-section-d149561705e1401">32</a></sup>, Luttinger–Ward (black solid)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haussmann, R., Rantner, W., Cerrito, S. & Zwerger, W. Thermodynamics of the BCS–BEC crossover. Phys. Rev. A 75, 023610 (2007)." href="/articles/nphys4187#ref-CR12" id="ref-link-section-d149561705e1405">12</a></sup>, Gaussian fluctuation theory (red dash-dotted)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Diener, R. B., Sensarma, R. & Randeria, M. Quantum fluctuations in the superfluid state of the BCS–BEC crossover. Phys. Rev. A 77, 023626 (2008)." href="/articles/nphys4187#ref-CR28" id="ref-link-section-d149561705e1410">28</a></sup>, and extended T-matrix (purple dash-dot-dotted)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Tajima, H. et al. Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev. A 95, 043625 (2017)." href="/articles/nphys4187#ref-CR30" id="ref-link-section-d149561705e1414">30</a></sup>. Inset shows measured BA mode frequency for clouds at different <i>k</i>/<i>k</i><sub>F</sub> at unitarity demonstrating linear dispersion. Error bars represent the combined uncertainties in <img src="//media.springernature.com/lw7/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq9_HTML.gif" style="width:7px;max-width:none;" alt="">, the fitted <i>ω</i><sub>BA</sub> and the angle between the Bragg lasers.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/nphys4187/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>By varying the power of our trapping laser, we can tune <img src="//media.springernature.com/lw7/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq5_HTML.gif" style="width:7px;max-width:none;" alt=""> (and hence <i>k</i>/<i>k</i><sub>F</sub>) to map the dispersion of the BA mode. Measured BA mode frequencies for different densities reveal linear dispersion at unitarity in the range 0.45 <span class="stix">≲</span> <i>k</i>/<i>k</i><sub>F</sub> <span class="stix">≲</span> 0.7 (<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/nphys4187#Fig4">Fig. 4</a>, inset) with a slope corresponding to <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> = 0.36 ± 0.01. At unitarity, <img src="//media.springernature.com/lw86/springer-static/image/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_Article_BFnphys4187_IEq6_HTML.gif" style="width:86px;max-width:none;" alt=""> as <i>T</i> → 0, where <i>ξ</i> is the Bertsch parameter (ratio of the internal energy of a resonantly interacting Fermi gas to that of an ideal Fermi gas). Our measured value of <i>c</i><sub>s</sub>/<i>v</i><sub>F</sub> yields <i>ξ</i> = 0.39 ± 0.02 consistent with thermodynamic measurements<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas. Science 335, 563–567 (2012)." href="/articles/nphys4187#ref-CR23" id="ref-link-section-d149561705e1510">23</a></sup>.</p><p>Focused beam Bragg spectroscopy provides a means to characterize the canonical excitations of homogeneous atomic superfluids that are difficult to access in other systems such as superconductors. This paves the way for studies of elementary excitations in a range of settings, including low-dimensional, lattice-confined and spin–orbit-coupled Fermi gases, as well as finite-temperature phenomena such as pseudogap pairing. By tuning the angle between the Bragg lasers, dynamical properties of strongly correlated quantum gases spanning different momenta can be mapped out providing robust benchmarks for theories of many-body systems.</p></div></div></section><section data-title="Data availability."><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Data availability.</h2><div class="c-article-section__content" id="Sec2-content"><p>The data that support the plots in this paper and other findings of this study that are not detailed in the <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/nphys4187#MOESM1">Supplementary Information</a> are available from the corresponding author on request.</p></div></div></section><section data-title="Additional Information"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4">Additional Information</h2><div class="c-article-section__content" id="Sec4-content"><p>Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section> </div> <div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1"><p class="c-article-references__text" id="ref-CR1">Nambu, Y. & Jona-Lasino, G. 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Zwerger, G. Strinati, L. Salasnich and Y. Ohashi for sharing their data and comments on the manuscript and P. Hannaford for fruitful discussions. This work was supported by ARC Discovery Project DP130101807. G.M.B. wishes to acknowledge the support of the Villum Foundation via Grant No. VKR023163 and ARC Discovery Project DP160102739.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Centre for Quantum and Optical Sciences, Swinburne University of Technology, Melbourne, 3122, Australia</p><p class="c-article-author-affiliation__authors-list">Sascha Hoinka, Paul Dyke, Marcus G. Lingham & Chris J. Vale</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Department of Applied Physics, COMP Centre of Excellence, Aalto University School of Science, FI 00076 Aalto, Finland</p><p class="c-article-author-affiliation__authors-list">Jami J. Kinnunen</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">Institut for Fysik og Astronomi, Aarhus Universitet, 8000 Aarhus C, Denmark</p><p class="c-article-author-affiliation__authors-list">Georg M. Bruun</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Sascha-Hoinka-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Sascha Hoinka</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Sascha%20Hoinka" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Sascha%20Hoinka" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Sascha%20Hoinka%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Paul-Dyke-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Paul Dyke</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Paul%20Dyke" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Paul%20Dyke" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Paul%20Dyke%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Marcus_G_-Lingham-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Marcus G. 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Kinnunen</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Jami%20J.%20Kinnunen" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Jami%20J.%20Kinnunen" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Jami%20J.%20Kinnunen%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Georg_M_-Bruun-Aff3"><span class="c-article-authors-search__title u-h3 js-search-name">Georg M. 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Vale</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Chris%20J.%20Vale" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Chris%20J.%20Vale" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Chris%20J.%20Vale%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>S.H., P.D., M.G.L. and C.J.V. conducted the experimental work and data analysis. J.J.K. and G.M.B. performed the theoretical calculations. All authors contributed to the manuscript preparation.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:cvale@swin.edu.au">Chris J. Vale</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading">Competing interests</h3> <p>The authors declare no competing financial interests.</p> </div></div></section><section data-title="Supplementary information"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5">Supplementary information</h2><div class="c-article-section__content" id="Sec5-content"><div data-test="supplementary-info"><div id="figshareContainer" class="c-article-figshare-container" data-test="figshare-container"></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM1"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary information" href="https://static-content.springer.com/esm/art%3A10.1038%2Fnphys4187/MediaObjects/41567_2017_BFnphys4187_MOESM1_ESM.pdf" data-supp-info-image="">Supplementary information</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>Supplementary information (PDF 372 kb)</p></div></div></div></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"><p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=Goldstone%20mode%20and%20pair-breaking%20excitations%20in%20atomic%20Fermi%20superfluids&author=Sascha%20Hoinka%20et%20al&contentID=10.1038%2Fnphys4187&copyright=Springer%20Nature%20Limited&publication=1745-2473&publicationDate=2017-06-26&publisherName=SpringerNature&orderBeanReset=true">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1038/nphys4187" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1038/nphys4187" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Hoinka, S., Dyke, P., Lingham, M. <i>et al.</i> Goldstone mode and pair-breaking excitations in atomic Fermi superfluids. <i>Nature Phys</i> <b>13</b>, 943–946 (2017). https://doi.org/10.1038/nphys4187</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/nphys4187?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span 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Goldstone mode and pair-breaking excitations in atomic Fermi superfluids | Nature Physics