Abstract

Higgs and Goldstone modes are possible collective modes of an order parameter on spontaneously breaking a continuous symmetry. Whereas the low-energy Goldstone (phase) mode is always stable, additional symmetries are required to prevent the Higgs (amplitude) mode from rapidly decaying into low-energy excitations. In high-energy physics, where the Higgs boson1 has been found after a decades-long search, the stability is ensured by Lorentz invariance. In the realm of condensed-matter physics, particle–hole symmetry can play this role2 and a Higgs mode has been observed in weakly interacting superconductors3,4,5. However, whether the Higgs mode is also stable for strongly correlated superconductors in which particle–hole symmetry is not precisely fulfilled or whether this mode becomes overdamped has been the subject of numerous discussions6,7,8,9,10,11. Experimental evidence is still lacking, in particular owing to the difficulty of exciting the Higgs mode directly. Here, we observe the Higgs mode in a strongly interacting superfluid Fermi gas. By inducing a periodic modulation of the amplitude of the superconducting order parameter Δ, we observe an excitation resonance at the frequency 2Δ/h. For strong coupling, the peak width broadens and eventually the mode disappears when the Cooper pairs turn into tightly bound dimers signalling the eventual instability of the Higgs mode.

Main

Spontaneous symmetry breaking occurs when an equilibrium state exhibits a lower symmetry than the corresponding Hamiltonian describing the system. The system then spontaneously picks one of the energetically degenerate choices of the order parameter and due to the specific energy landscape this process is accompanied by new collective modes. The typical picture, which exemplifies spontaneous symmetry breaking, uses a Mexican-hat-shaped energy potential (see Fig. 1a) that suggests the emergence of two distinct collective modes: the gapless ‘Goldstone mode’, which is associated with long-wavelength phase fluctuations of the order parameter, and an orthogonal gapped mode, the ‘Higgs mode’, which describes amplitude modulations of the order parameter. While Goldstone modes, such as phonons, appear necessarily when continuous symmetries are broken, stable Higgs modes are scarce, since decay channels might be present. The best-known example of a Higgs mode appears in the standard model of particle physics where this mode gives elementary particles their mass1.

Fig. 1: Principle of the Higgs mode excitation.
Fig. 1

a, Mexican hat potential of the free energy as a function of the real and imaginary parts of the complex order parameter Δ. The equilibrium state order parameter takes spontaneously one of the values at the energy minima. b, We employ rf dressing of the paired superfluid by off-resonant coupling to an unoccupied state \(\left|3\right\rangle\). c,d, This results in a periodic modulation of both the occupation of the state \(\left|3\right\rangle\) (c) and the superconducting gap (d). Here, N3 is the number of atoms in state \(\left|3\right\rangle\) and N1 (t = 0) is the initial number of atoms in state \(\left|1\right\rangle\). Shown are numerical simulations for a coupling constant 1/(kFa) = −0.6704, ħΩR = 0.0353EF and ħδ = −0.3247EF. e, By adjusting the modulation frequency, we achieve an excitation of the Higgs mode in the Mexican hat.

In the non-relativistic low-energy regime usually encountered in condensed-matter physics, the existence of a stable Higgs mode cannot be taken for granted6. However, under certain conditions, other symmetries, such as particle–hole symmetry, can play the role of Lorentz invariance and induce a stable Higgs mode. A notable example of a low-energy particle–hole symmetric theory hosting a stable Higgs mode is the famous Bardeen–Cooper–Schrieffer (BCS) Hamiltonian describing weakly interacting superconductors2,12. Evidence for the Higgs mode has been found in conventional BCS superconductors3,4,5. However, experimental detections have been solely indirect as the Higgs mode does not couple directly to electromagnetic fields owing to the gauge invariance required for its existence. The far-reaching importance of the Higgs mode is further illustrated by its observation in a variety of specially tuned systems such as antiferromagnets13, liquid 3He (ref. 14), ultracold bosonic atoms near the superfluid/Mott-insulator transition15,16, spinor Bose gases17 and Bose gases strongly coupled to optical fields18. In contrast, weakly interacting Bose–Einstein condensates (BECs) do not exhibit a stable Higgs mode6,10,11.

In recent years, research has focused on advanced materials exhibiting superconductivity far beyond the conventional BCS description, such as cuprates, pnictides and the unitary Fermi gas. Many of these materials are characterized by strong fermionic correlations. Even though in this context the existence of a Higgs mode has been the topic of theoretical debates7,8,9,10,11, experimental evidence for the Higgs mode in systems exhibiting strong correlations between fermions is still absent.

Here, we spectroscopically excite the Higgs mode in a superfluid Fermi gas in the crossover between a weakly interacting BCS superfluid and a BEC of strongly coupled dimers (Fig. 1). We induce a periodic modulation of the amplitude of the superconducting order parameter Δ and find an excitation resonance near twice the superconducting gap value. On the BCS side, the spectroscopic feature agrees with the theoretical expectation of the Higgs mode. On the BEC side of the crossover, we find strong broadening beyond the predictions of BCS theory and, eventually, the disappearance of the mode as predicted for a weakly interacting BEC6,10,11.

Our measurements are conducted in an ultracold quantum gas of ~ 4 × 106 6Li atoms prepared in a balanced mixture of the two lowest hyperfine states \(\left|1\right\rangle\) and \(\left|2\right\rangle\) of the electronic ground state. The gas is trapped in a harmonic potential with frequencies of (ωx, ωy, ωz) = 2π × (91, 151, 235) Hz and is subjected to a homogeneous magnetic field, which is varied in the range of 740–1,000 G to tune the s-wave scattering length a near the Feshbach resonance located at 834 G. This results in an adjustment of the interaction parameter of the gas in the range of −0.8  1/(kFa)  1, (that is, across the whole BCS/BEC crossover). The Fermi energy in the centre of the gas is \({E}_{{\rm{F}}}\simeq h\times (34\pm 3)\) kHz at each of the considered interaction strengths and sets the Fermi wavevector \({k}_{{\rm{F}}}=\sqrt{8{\pi }^{2}m{E}_{{\rm{F}}}{\rm{/}}{h}^{2}}\), where m denotes the mass of the atom and h is Planck’s constant.

Excitation of the Higgs mode requires a scheme that couples to the amplitude of the order parameter rather than creating phase fluctuations or strong single-particle excitations. Previous theoretical proposals8,19,20 for exciting the Higgs mode in ultracold Fermi gases have focused on a modulation of the interaction parameter 1/(kFa); however, experimentally only single-particle excitations have been observed from such a modulation21. We have developed a novel excitation scheme employing a radiofrequency (rf) field, dressing the state \(\left|2\right\rangle\) with the initially unoccupied hyperfine state \(\left|3\right\rangle\) thereby modulating the pairing between the \(\left|1\right\rangle\) and \(\left|2\right\rangle\) states (see Fig. 1b,c). Previous experiments investigating ultracold gases with rf spectroscopy22,23,24,25 have focused on studying single-particle excitations. To this end, there, the duration of the rf pulse τ had been chosen shorter than the inverse of the Rabi frequency ΩR, such that the spectra could be interpreted in the weak-excitation limit using Fermi’s golden rule. In contrast, here, we employ an rf drive far red-detuned from single-particle resonances in the interacting many-body system and in the long-pulse limit \({\it{\Omega} }_{{\rm{R}}}\tau \gg 1\), to couple to the amplitude of the order parameter. To illustrate this, consider first an isolated two-level system of the \(\left|2\right\rangle\) and the \(\left|3\right\rangle\) state coupled by a Rabi frequency ΩR with detuning δ from the resonance. The occupation probability of the atoms in the \(\left|2\right\rangle\) state is p2 = 1 − \({\it{\Omega} }_{{\rm{R}}}^{2}{/{\it{\Omega}} }_{{\rm{R}}}^{{\prime} 2}{{\rm{\sin }}}^{2}\left({\it{\Omega}}_{{\rm{R}}}^{{\prime} }t{\rm{/}}2\right)\); that is, the continuous rf dressing leads to a time-periodic modulation of the occupation of the \(\left|2\right\rangle\)-state with the effective Rabi frequency \({\it{\Omega}}_{{\rm{R}}}^{{\prime} }=\sqrt{{\it{\Omega}}_{{\rm{R}}}^{2}+{\delta }^{2}}\).

In the many-body problem of the BCS/BEC crossover, the situation is complicated by the dispersion of the (quasi) particles and the presence of interactions. In particular, a continuum of excitations typically occurs above the energy of the lowest single-particle excitation to state \(\left|3\right\rangle\) (see Fig. 2a). Deep in the BCS regime, the continuum of excitations is related to the different momentum states and the excitation scheme can be approximated by coupling each occupied momentum state of the BCS quasiparticles in level \(\left|2\right\rangle\) to the corresponding momentum state in state \(\left|3\right\rangle\) since the rf dressing transfers negligible momentum. The effective Rabi frequency \({\it{\Omega}}_{{\rm{R}},k}^{{\prime} }=\sqrt{{\it{\Omega}}_{{\rm{R}}}^{2}+{\delta }_{k}^{2}}\) and therefore the excitation probability becomes momentum dependent by the many-body detuning ħδk = ħδ − Ek − ξk, where ξk is the single-particle dispersion, and \({E}_{k}=\sqrt{{\xi }_{k}^{2}+{\left|{\Delta}\right|}^{2}}\) is the quasiparticle dispersion and Δ is the s-wave superconducting order parameter. A red-detuned rf drive, as employed here, avoids resonant coupling to the single-particle excitations, however, still modulates off-resonantly the occupation of the excited states as shown in Fig. 2b.

Fig. 2: Illustration of the excitation scheme for one modulation frequency.
Fig. 2

a, The rf field is red-detuned from the single-particle excitation of the interacting system. It creates an off-resonant excitation to the state \(\left|3\right\rangle\) with a varying detuning for different momenta. b, Time evolution of the momentum-resolved occupation of the \(\left|3\right\rangle\) state with momentum k for a fixed value of \(\frac{1}{{k}_{{\rm{F}}}a}\) = −0.63, a Rabi frequency ħΩR = 0.038EF and a detuning ħδ = −0.34EF. Blue: \({k}{\rm{/}}{k}_{{\rm{F}}}=0\), red: \({k}{\rm{/}}{k}_{{\rm{F}}}=0.8\), green: \({k}{\rm{/}}{k}_{{\rm{F}}}=1.1\). c, Spectral weight of the momentum-resolved gap \(A_{\rm{k}}(\omega)\) (see Methods). The circles indicate the Higgs mode, the stars mark the response to the modulation frequency and the crosses indicate the quasiparticle excitations at 2Ek. The position of the star at k = 0 approximately represents the effective modulation frequency for the chosen parameters. Inset: Fourier spectra (momentum integrated) of the occupation of the \(\left|3\right\rangle\) state (red) and \(\left|{\Delta}\right|\) (blue). The dashed line is the expected location of the Higgs mode at 2\(\left|{\Delta}\right|\). Panels b, c and the inset of c are for the same driving and detuning parameters.

The off-resonant periodic modulation of the occupation of the state \(\left|2\right\rangle\) with controllable frequency \({\Omega}_{{\rm{R}},k}^{\prime}\) induces a modulation of the amplitude of the order parameter \(\left|{\Delta}\right|\) (Fig. 1d,e, for details see Methods) and hence couples directly to the Higgs mode. To illustrate this mechanism, we numerically solve the minimal set of coupled equations of motion describing the evolution of the order parameter in the presence of an rf coupling to state \(\left|3\right\rangle\) (see Methods and Supplementary Information). We see that the Fourier spectrum of \(\left|{\Delta}\right|\) for one modulation frequency displays—aside from a response corresponding to the modulation frequencies \({\Omega}_{{\rm{R}},k}^{\prime}\)—a sharp peak at the gap value \(2\left|{\Delta}\right|\) (see Fig. 2c inset). By the momentum-resolved representation (Fig. 2c), we identify this peak to be dominated by the Higgs excitation with a momentum-independent dispersion. The amplitude of the Higgs peak is maximal when the averaged effective drive frequency \(\hslash {\bar{\Omega}}_{{\rm{mod}}}\approx 2\left|{\Delta }\right|\) in accordance with its expected frequency. The Higgs mode is a collective mode of the system and even for the harmonically trapped gas exhibits a unique frequency. Numerical studies in the BCS limit have shown that in harmonically trapped systems, the Higgs mode should occur at the frequency of twice the superconducting gap evaluated at the maximum density of the gas8,26,27,28,29 and hence we use this as our reference for the value of the gap to compare with theory and other experiments.

In the experiment, we search for the Higgs mode by measuring the energy absorption spectrum of the fermionic superfluid in the \(\left|1,2\right\rangle\) states for different interactions. Using ΩR and δ as adjustable parameters, we dress the \(\left|2\right\rangle\) state by the \(\left|3\right\rangle\) state with adjustable modulation frequency given by the effective Rabi frequency. We choose a drive frequency in the single-particle excitation gap. For our experiments, we measure the modulation frequency Ωmod and amplitude α of the time-dependent population of the \(\left|3\right\rangle\) state (for calibration, see Methods and Supplementary Information). We then use a constant excitation amplitude \({p}_{\left|3\right\rangle }\simeq 0.5 \%\) and apply the modulation for a fixed period of 30 ms. After the excitation, we conduct a rapid magnetic field sweep onto the molecular side of the Feshbach resonance and convert Cooper pairs into dimers and measure the condensate fraction of the molecular condensate in time-of-flight imaging. The change in the condensate fraction provides us with a sensitive measure of the excitation of modes in the quantum gas. In Fig. 3a, we plot the measured spectra as a function of the modulation frequency for different values of 1/(kFa). On the BCS side of the Feshbach resonance up to unitarity, 1/(kFa) < 0, we observe clear resonances for which the condensate fraction reduces, signalling the excitation of a well-defined mode. For 1/(kFa) > 0, the energy absorption peak is gradually washed out and broadens significantly. Far on the BEC side, for \(1{\rm{/}}({k}_{{\rm{F}}}a)\simeq 1\), we cannot observe a resonance and conclude that the Higgs mode is absent. The resonances generally exhibit an asymmetric line shape, which we fit with a Gaussian to the high-frequency side to extract the peak position and width. A contribution to the asymmetric peak shape stems from the momentum-dependence of the effective Rabi frequencies \({\Omega}_{{\rm{R}},k}^{\prime}\). As indicated in Fig. 2a,b, the effective detuning (and hence the modulation frequency) varies with increasing momentum k. Therefore, a resonant excitation at the Higgs mode frequency can be achieved for high momenta k even though for low momenta the modulation frequency is below the resonant excitation.

Fig. 3: Excitation spectra of the Higgs mode.
Fig. 3

a, Excitation spectra of the Higgs mode for different interaction strengths, 1/(kFa), as labelled in the figure. The different levels of background condensate fraction are due to the different 1/(kFa). The solid lines shows the Gaussian fit to the high-frequency side of the spectra. The error bars show the standard deviation of approximately four measurements. b, Time-of-flight image of the condensate with the thermal background subtracted at 1/(kFa) ≈ −0.43. Rings indicate momentum intervals of 0.02kF. c, Momentum-resolved analysis of the Higgs excitation inside the condensate by averaging the optical density in the colour-coded rings in b for different modulation frequencies. The resonance occurs at the same modulation frequency for all momenta.

To demonstrate the collective-mode nature of our resonance, we perform a number of checks. First, we verify that the excitation resonance frequency (to within 4%) and shape is independent of the modulation strength in the range of 0.001 < α < 0.2 and modulation duration between τ = 0.5 ms and τ = 30 ms (for the definition of α, see Methods and Supplementary Information). Second, we have confirmed that the observed resonance peak is not caused by single-particle excitations by measuring the excitation probability to the \(\left|3\right\rangle\) state versus modulation frequency Ωmod and finding a featureless broad spectrum. This and the following checks have been performed with a modulation amplitude of α = 0.2 and a modulation time of 0.5 ms, which is much shorter than the trap period of ~ 5 ms. Hence, the measurement is insensitive to thermalization effects and/or density redistribution within the cloud. Third, we check the momentum-dependence of the resonance. After the modulation, we perform a time-of-flight expansion for a period of 15 ms, which is approximately a quarter period of the residual harmonic potential during ballistic expansion. This procedure maps the initial momentum states to positions in the absorption image30. We analyse the detected condensate density in momentum intervals of 0.02 kF and find that the excitation resonance is at the same frequency for all momentum intervals (see Fig. 3b,c). Finally, we have searched for possible quasiparticle excitations resulting from our interaction modulation by employing standard rf spectroscopy24,25,31 after the interaction modulation. The spectra show only a very weak and broad background independent of the modulation frequency. This behaviour is not unexpected since the contribution of quasiparticle excitations is smeared out in the presence of a trap as we confirmed numerically using the local-density approximation.

In Fig. 4a, we plot the position of the fitted peak of the energy absorption spectra versus the interaction parameter 1/(kFa) evaluated at the centre of the sample. It has been suggested8 that the Higgs mode frequency is close to twice the superconducting gap in the BCS/BEC crossover and can therefore be used as an approximative measure of the gap. In the crossover regime, the exact value of \(\left|{\Delta}\right|\) is yet unknown and both experiments and numerical calculations are challenging. We compare our data to gap measurements using different methods31,32 and several numerical calculations33,34,35,36,37,38. As compared to the previous experimental results, our extracted value is larger. We note that previous gap measurements rely on fitting the onset of a spectral feature, whereas our method is based on fitting a Gaussian to a slightly skewed spectral feature, and both methodologies could be susceptible to small systematic uncertainties. An upper bound is provided by the theoretical result of mean-field theory (dashed line), which is known to overestimate the superconducting gap.

Fig. 4: Observation of the Higgs mode.
Fig. 4

a, Measured peak positions of the energy absorption spectra (black circles). For comparison, we show numerical simulations of the gap parameter multiplied by 2: BCS mean-field theory (blue dashed line), ref. 33 (red), ref. 34 (green), ref. 35 (orange), ref. 36 (purple), ref. 37 (brown), ref. 38 (pink). The grey symbols show the experimental data of ref. 32. b, Measured full-width at half-maximum (FWHM) of the absorption peaks (black circles). For comparison, the BCS mean-field-theory gap is also shown (blue dashed line). The error bars in a and b represent the standard errors.

In Fig. 4b, we plot the full-width at half-maximum of the Gaussian fits to the energy absorption peaks. Utilizing the BCS model for the momentum-dependent excitation discussed above, we estimate the width of our excitation resonance to be of the order Δ (see Supplementary Information). Hence, we cannot directly interpret the linewidth of our spectra as the decay rate of the Higgs mode but only as a lower limit of the lifetime. On the BCS side of the resonance, we find good agreement with our model and towards the BEC side the measured width far exceeds the prediction, indicating that the Higgs mode becomes strongly broadened, for example, due to the violation of particle–hole symmetry resulting in a decay into Goldstone modes6,10,11,39. Extending, in the future, our novel experimental scheme with a better momentum resolution will provide a route to finally explore the decay mechanisms of the Higgs mode, the understanding of which is a cornerstone in both high-energy particle physics and condensed-matter physics.

Methods

Preparation

Using standard techniques of laser cooling and sympathetic cooling in a mixture with sodium atoms in a magnetic trap, we prepare ~ 5 × 107 cold fermionic lithium atoms in a crossed-beam optical dipole trap of wavelength 1,070 nm in an equal mixture of the two lowest hyperfine states \(\left|1\right\rangle\) and \(\left|2\right\rangle\). Using subsequent evaporative cooling in a homogeneous magnetic field of 795 G, in the immediate vicinity of the Feshbach resonance at 834 G, we produce a condensate in the BCS/BEC crossover regime with a temperature of T/TF = 0.07 ± 0.02. After preparation of the fermionic superfluid, the magnetic offset field is adiabatically adjusted in the range between 740 G and 1,000 G to control the interaction parameter 1/(kFa) in the range of −0.8 < 1/(kFa) < 1 (that is, across the whole BCS/BEC crossover region).


Calibration of spectroscopy and analysis

We experimentally calibrate the modulation frequency and amplitude to take into account energy shifts owing to interaction effects of the initial and final states and the efficiency of the rf antenna set-up. To this end, we drive Rabi oscillations with set values of detuning δ and power and measure the population \({p}_{\left|3\right\rangle }\) as a function of time during the rf drive \({p}_{\left|3\right\rangle }=\alpha {{\rm{\sin }}}^{2}\left({{\Omega }}_{{\rm{mod}}}t{\rm{/}}2\right)\). This provides us with a direct measurement of the modulation frequency and amplitude. To model the data, we assume a Lorentzian line shape \(\alpha =\frac{{\Omega}_{{\rm{R}}}^{2}}{{\Omega}_{{\rm{R}}}^{2}+{\left(\delta -{\delta }_{0}\right)}^{2}}\); however, we allow for a frequency shift δ0(kFa) by which the detuning δ is corrected as compared to the Zeeman-energy resonance of the free atom. The fit parameter δ0 absorbs the effects of interactions in the final state of the spectroscopy, the condensation energy of the initial state and the averaging of different momentum states and densities in the trap. Experimentally, the calibration is performed at a value of \(\alpha \simeq 4 \%\) for which we obtain agreement with the Lorentzian model to a few per cent.

We check for unpaired atoms in the \(\left|2\right\rangle\) state for red- and blue-detuned rf with respect to δ0 as a result of the modulation. This has been achieved by rapidly ramping the field to 450 G with approximately 4 G μs−1, which allows for the detection of free atoms rather than paired atoms. In the case of a red-detuned rf modulation, no enhancement of the signal of unpaired atoms could be observed over the whole range of modulation frequencies. However, blue-detuned rf modulation increases significantly the number of unpaired atoms due to single-particle excitations to the continuum (see Supplementary Fig. 1).

We also vary the driving strength α and observe that the resonance position of the peak with respect to the modulation frequency does not vary (see Supplementary Fig. 2).


Theoretical modelling

The experimental system can be described taking three different fermionic levels into account. Initially the system is prepared in a balanced mixture of states \({\vert}1{\rangle}\) and \(\left|2\right\rangle\). Since we are mainly interested in the excitation mechanism and for this mainly the presence of a difference in the interaction strength between states (\(\left|1\right\rangle\) and \(\left|2\right\rangle\)) and (\(\left|1\right\rangle\) and \(\left|3\right\rangle\)) is needed, we take here only the interaction between these two states into account and decouple this term within the s-wave BCS channel. Using the rotating-wave approximation for the coupling between the states \(\left|2\right\rangle\) and \(\left|3\right\rangle\), we obtain the Hamiltonian

$$H={H}_{{\rm{BCS}}}+\sum _{{\mathbf{{k}}}}\left({\varepsilon }_{{\mathbf{{k}}}}-\hslash \delta \right){n}_{{\mathbf{{k}}},3}+\frac{\hslash {\it{\Omega} }_{{\rm{R}}}}{2}\sum _{{\mathbf{{k}}}}\left({c}_{{\mathbf{{k}}},3}^{\dagger }{c}_{{\mathbf{{k}}},2}+{c}_{{\mathbf{{k}}},2}^{\dagger }{c}_{{\mathbf{{k}}},3}\right)$$

with

$${H}_{{\rm{BCS}}}=\sum _{{\mathbf{{k}}}}{\varepsilon }_{{\mathbf{{k}}}}\left({n}_{{\mathbf{{k}}},1}+{n}_{{\mathbf{{k}}},2}\right)+\sum _{{\mathbf{{k}}}}\left\{{{\it{\Delta }}}^{* }{c}_{-{\mathbf{{k}}},2}{c}_{{\mathbf{{k}}},1}+{\it{\Delta }}{c}_{{\mathbf{{k}}},1}^{\dagger }{c}_{-{\mathbf{{k}}},2}^{\dagger }\right\}$$
(1)

Here \({\it{\Delta }}=\frac{g}{V}{\sum }_{{\mathbf{{k}}}}\left\langle {c}_{-{\mathbf{{k}}},2}{c}_{{\mathbf{{k}}},1}\right\rangle\), ΩR is the Rabi frequency, g is the interaction strength, V is the volume and the momentum-independent detuning is \(\hslash \delta =\hslash {\omega }_{{\rm{rf}}}-\left({\varepsilon }_{3}^{0}-{\varepsilon }_{2}^{0}\right)\), where \({\varepsilon }_{n}^{0}\) is the bare energy for the state n = 2, 3 and εk = ħ2k2/(2m) is the single-particle dispersion. We determine g from the scattering length using the expression provided in ref. 23.

To determine the time evolution of the order parameter, we derive a closed set of equations for the expectation values

$$\begin{array}{lll}\hslash \frac{\partial }{\partial t}\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle & = & i\left\{-2{\epsilon }_{{{\mathbf{k}}}}\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle -\frac{\hslash {\it{\Omega} }_{{\rm{R}}}}{2}\left\langle {c}_{-{{\mathbf{k}}},3}{c}_{{{\mathbf{k}}},1}\right\rangle \right.\\ & & \left.+{\it{\Delta }}\left({n}_{{{\mathbf{k}}},1}+{n}_{-{{\mathbf{k}}},2}-1\right)\right\}\\ \hslash \frac{\partial }{\partial t}\left\langle {c}_{-{{\mathbf{k}}},3}{c}_{{{\mathbf{k}}},1}\right\rangle & = & i\left\{-\frac{\hslash {\it{\Omega} }_{{\rm{R}}}}{2}\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle \right.\\ & & \left.-\left(2{\epsilon }_{{{\mathbf{k}}}}-\hslash \delta \right)\left\langle {c}_{-{{\mathbf{k}}},3}{c}_{{{\mathbf{k}}},1}\right\rangle +{\it{\Delta }}\left\langle {c}_{-{{\mathbf{k}}},2}^{\dagger }{c}_{-{{\mathbf{k}}},3}\right\rangle \right\}\\ \hslash \frac{\partial }{\partial t}\left\langle {c}_{-{{\mathbf{k}}},2}^{\dagger }{c}_{-{{\mathbf{k}}},3}\right\rangle & = & i\left\{{{\it{\Delta }}}^{* }\left\langle {c}_{-{{\mathbf{k}}},3}{c}_{{{\mathbf{k}}},1}\right\rangle +\hslash \delta \left\langle {c}_{-{\boldsymbol{k}},2}^{\dagger }{c}_{-{{\mathbf{k}}},3}\right\rangle \right.\\ & & \left.-\frac{\hslash {\it{\Omega} }_{{\rm{R}}}}{2}\left({n}_{-{{\mathbf{k}}},2}-{n}_{-{{\mathbf{k}}},3}\right)\right\}\\ \hslash \frac{\partial }{\partial t}{n}_{{{\mathbf{k}}},1} & = & -2{\rm{Im}}\left({{\it{\Delta }}}^{* }\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle \right)\\ \hslash \frac{\partial }{\partial t}{n}_{-{{\mathbf{k}}},2} & = & -2{\rm{Im}}\left({{\it{\Delta }}}^{* }\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle \right)\\ & & +\hslash {\it{\Omega} }_{{\rm{R}}}{\rm{Im}}\left(\left\langle {c}_{-{{\mathbf{k}}},2}^{\dagger }{c}_{-{{\mathbf{k}}},3}\right\rangle \right)\\ \hslash \frac{\partial }{\partial t}{n}_{-{{\mathbf{k}}},3} & = & -\hslash {\it{\Omega} }_{{\rm{R}}}{\rm{Im}}\left(\left\langle {c}_{-{{\mathbf{k}}},2}^{\dagger }{c}_{-{{\mathbf{k}}},3}\right\rangle \right)\end{array}$$

where the number densities are defined as \({n}_{{{\mathbf{k}}},m}=\left\langle {c}_{{{\mathbf{k}}},m}^{\dagger }{c}_{{{\mathbf{k}}},m}\right\rangle\) with m = 1, 2, 3. We solve these equations numerically discretizing both time t and momentum k and using the self-consistency condition \({\it{\Delta }}=\frac{g}{V}{\sum }_{{{\mathbf{k}}}}\left\langle {c}_{-{{\mathbf{k}}},2}{c}_{{{\mathbf{k}}},1}\right\rangle\) at each time step ensuring both the convergence for the time step dt and the momentum spacing. Typical values taken are dk/kF = 5 × 10−4, dt = 5 × 10−4ħ/EF and the cutoff for the momentum sum is Ec = 100EF.

The momentum-resolved spectral weight of the gap shown in Fig. 2c is computed as

$${A}_{{\bf{k}}}(\omega )=V{\rm{/}}g\left|{\mathscr{F}}\left\{\left|{{\it{\Delta }}}_{{{\mathbf{k}}}}(t)\right|-\frac{1}{T}{\int }_{0}^{T}{\rm{d}}t\left|{{\it{\Delta }}}_{{{\mathbf{k}}}}(t)\right|\right\}\right|$$
(2)

with the momentum-dependent order parameter \(\Delta_{\rm{k}}=\left(g/V\right)\left\langle C_{-{\rm{k}},2}C_{{\rm{k}},1}\right\rangle\). We use T = 400ħ/EF for the calculation.


Time evolution of the population of state \(\left|3\right\rangle\)

We compare the theoretical evolution of the population of atoms in state \(\left|3\right\rangle\) (see Fig. 1c) during the application of the rf dressing to the experimental results. Supplementary Fig. 3 shows the population of the atoms in state \(\left|3\right\rangle\) normalized to the initial atom number in state \(\left|1\right\rangle\). The simulation and experiment were performed with the same effective modulation frequency, Ωmod, and maximum atom transfer. Both curves show damped oscillations of the population of state \(\left|3\right\rangle\) with time. The initial time behaviour up to approximately three oscillations agrees well between theory and experiment, which means that the dominant damping mechanism is due to a dephasing of the different momentum components. Afterwards, the experimental results show a stronger damping that we attribute to other damping mechanisms, such as, for example, the presence of collisions, which are not considered in the theoretical description.


Time evolution of the condensate fraction

We show the evolution of the condensate fraction during the application of the rf dressing in Supplementary Fig. 4. After different durations of the drive, the rapid mapping to the BEC side was performed and the condensate fraction was measured. The drive amplitude was chosen to be 5%. As a response, an oscillation of the condensate fraction close to the expected Rabi frequency can be observed over several oscillation periods with an amplitude of the order of 5%.


Comparison of the experimental and theoretical spectra

In Supplementary Fig. 5, we show a comparison of the experimentally measured spectra with the theoretical simulation. To gain insight into the structure expected from the excitation scheme, we theoretically extract the weight of the Higgs excitation for different effective modulation frequencies and plot these in the lower panel of Supplementary Fig. 5 for 1/(kFa) = −0.63. To evaluate the area under the Higgs, for each momentum, we integrate around the Higgs excitation peak (shown in Fig. 2c) and then sum over all momenta along the Higgs excitation line. Let us note that this procedure leads to the artefact that at high modulation frequencies still a non-vanishing contribution to the weight is found, which, however, can be attributed to the excitation of quasiparticles in a homogeneous system and would vanish in a trapped system as considered experimentally. More importantly, we see that even though the Higgs mode has a very sharp frequency (as shown in Fig. 2c) and therefore a long lifetime, the resulting spectra show a much broader peak. The width of the peak is due to the excitation procedure. In particular, a resonant excitation of the Higgs mode is already possible if the effective modulation frequency lies below the sharp frequency of the Higgs mode, since then already some of the Rabi frequencies of the higher momentum components (compare stars in Fig. 2c) can resonantly excite the Higgs mode. Thus, the broadening of the spectral feature is mainly due to the particular excitation scheme and not a measurement of the lifetime of the Higgs mode. Let us conclude by pointing out that the full-width at half-maximum in both the theoretical and the experimental spectra is approximately \(\left|{\Delta }\right|\).


Local-density approximation for the quasiparticle excitations

To study the effect of the harmonic trapping on the quasiparticle excitations, we performed a calculation of the system dynamics within the local-density approximation. Within the local-density approximation, we treat points of different density as effectively homogeneous systems with rescaled interaction 1/[kF(r)a], Fermi energy EF(r) and chemical potential consistent with the system’s density profile. We assume the density profile to be the one for non-interacting fermions as typically the profiles change only slightly for the considered interactions. The time evolution of the superconducting order parameter of the homogeneous system is performed locally for each point in the trap and rescaled to give \({\it{\Delta }}({\bf{r}},t){\rm{/}}{E}_{{\rm{F}}}({\bf{r}}=0)\). We then take the density-weighted average of its Fourier transform. It is important to note that the Higgs mode—due to its collective nature—cannot be treated in this formalism, so we remove the Higgs peak in each Fourier transform by hand before we take the trap average. Integrating the resulting spectrum gives the background excitation weight (see Supplementary Fig. 6). In contrast to the peaked quasiparticle structure of a homogeneous system, we find the trap-averaged background excitation weight to be significantly broadened resulting in a featureless, broad background.


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The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank E. Demler, W. Zwerger and M. Zwierlein for fruitful discussion. This work has been supported by BCGS, the Alexander-von-Humboldt Stiftung, ERC (grant nos 616082 and 648166), DFG (SFB/TR 185 project B4), ITN COMIQ and Studienstiftung des Deutschen Volkes.

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Author notes

  1. These authors contributed equally: A. Behrle, T. Harrison.

Affiliations

  1. Physikalisches Institut, University of Bonn, Bonn, Germany

    • A. Behrle
    • , T. Harrison
    • , K. Gao
    • , M. Link
    •  & M. Köhl
  2. HISKP, University of Bonn, Bonn, Germany

    • J. Kombe
    • , J.-S. Bernier
    •  & C. Kollath

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Contributions

The study was conceived by C.K. and M.K.; the experimental set-up was designed and constructed by A.B., T.H., K.G. and M.K.; data collection was performed by A.B., T.H., K.G. and M.L.; data analysis was performed by T.H.; numerical modelling and analysis was performed by J.K., J.-S.B. and C.K.; the manuscript was written by C.K. and M.K. with contributions from all co-authors.

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The authors declare no competing interests.

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Correspondence to K. Gao or M. Köhl.

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https://doi.org/10.1038/s41567-018-0128-6