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arXiv:1811.07796 · Physical Review A

Phononic collective excitations in superfluid Fermi gases at nonzero temperatures

Serghei N. Klimin, Jacques Tempere, Hadrien Kurkjian

Published Dec 23, 2019• 22 citations
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Abstract

We study the phononic collective modes of the pairing field and their corresponding signature in both the order-parameter and density response functions for a superfluid Fermi gas at all temperatures below ${T}_{c}$ in the collisionless regime. The spectra of collective modes are calculated within the Gaussian pair fluctuation approximation. We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a nonperturbative analytic continuation of the pairing field propagator. At low temperature, we recover the known exponential temperature dependence of the damping rate and velocity shift of the Anderson-Bogoliubov branch. In the vicinity of ${T}_{c}$, we find analytically a weakly damped collective mode whose velocity vanishes with a critical exponent of $1/2$, and whose quality factor diverges logarithmically with ${T}_{c}\ensuremath{-}T$, thereby clarifying an existing debate in the literature [Andrianov and Popov, Theor. Math. Phys. 28, 829 (1976); Ohashi and Takada, J. Phys. Soc. Jpn. 66, 2437 (1997)]. A transition between these two phononic branches is visible at intermediary temperatures, particularly in the BCS limit where the phase-phase response function displays two maxima.

DOI
10.1103/physreva.100.063634

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We study the phononic collective modes of the pairing field and their corresponding signature in both the order-parameter and density response functions for a superfluid Fermi gas at all temperatures below Tc{T}_{c}Tc​ in the collisionless regime. The spectra of collective modes are calculated within the Gaussian pair fluctuation approximation. We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a nonperturbative analytic continuation of the pairing field propagator. At low temperature, we recover the known exponential temperature dependence of the damping rate and velocity shift of the Anderson-Bogoliubov branch. In the vicinity of Tc{T}_{c}Tc​, we find analytically a weakly damped collective mode whose velocity vanishes with a critical exponent of 1/21/21/2, and whose quality factor diverges logarithmically with Tc\ensuremath−T{T}_{c}\ensuremath{-}TTc​\ensuremath−T, thereby clarifying an existing debate in the literature [Andrianov and Popov, Theor. Math. Phys. 28, 829 (1976); Ohashi and Takada, J. Phys. Soc. Jpn. 66, 2437 (1997)]. A transition between these two phononic branches is visible at intermediary temperatures, particularly in the BCS limit where the phase-phase response function displays two maxima.
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature Tc{T}_{c}Tc​ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below Tc{T}_{c}Tc​ in the system's response to a driving pairing field. When approaching Tc{T}_{c}Tc​ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a relatively low energy-momentum region. Elsewhere they are replaced by the quadratically dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein condensate regime, this mode is a weakly damped propagating mode associated with a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower edge of the pairing continuum. At intermediate temperatures between 0 and Tc{T}_{c}Tc​ we unify the newly found collective phenomena near Tc{T}_{c}Tc​ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.
We calculate the sound velocity and the damping rate of the collective excitations of a 2D fermionic superfluid in a non-perturbative manner. Specifically, we focus on the Anderson–Bogoliubov excitations in the BEC-BCS crossover regime, as these modes have a sound-like dispersion at low momenta. The calculation is performed within the path-integral formalism and the Gaussian pair fluctuation approximation. From the action functional, we obtain the propagator of the collective excitations and determine their dispersion relation by locating the poles of this propagator. We find that there is only one kind of collective excitation, which is stable at T = 0 and has a sound velocity of v F / 2 for all binding energies, i.e., throughout the BEC-BCS crossover. As the temperature is raised, the sound velocity decreases and the damping rate shows a non-monotonous behavior: after an initial increase, close to the critical temperature T C the damping rate decreases again. In general, higher binding energies provide higher damping rates. Finally, we calculate the response functions and propose that they can be used as another way to determine the sound velocity.
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature TcT_cTc​ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below TcT_cTc​ in the system's response to a driving pairing field. When approaching TcT_cTc​ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a very low energy-momentum region when the pair correlation length reaches its critical divergent behavior ξpair∝∣Tc−T∣−1/2\xi_{\rm pair}\propto|T_c-T|^{-1/2}ξpair​∝∣Tc​−T∣−1/2; elsewhere, they are replaced by the quadratically-dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein Condensate regime, this mode is a weakly-damped propagating mode associated to a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower-edge of the pairing continuum. At intermediate temperatures between 0 and TcT_cTc​, we unify the newly found collective phenomena near TcT_cTc​ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.
Abstract We study the signatures of the collective modes of a superfluid Fermi gas in its linear response functions for the order-parameter and density fluctuations in the Random Phase Approximation (RPA). We show that a resonance associated to the Popov-Andrianov (or sometimes “Higgs”) mode is visible inside the pair-breaking continuum at all values of the wavevector q, not only in the (order-parameter) modulus-modulus response function but also in the modulus-density and density-density responses. At nonzero temperature, the resonance survives in the presence of thermally broken pairs even until the vicinity of the critical temperature Tc, and coexists with both the Anderson-Bogoliubov modes at temperatures comparable to the gap Δ and with the low-velocity phononic mode predicted by RPA near Tc. The existence of a Popov-Andrianov-“Higgs” resonance is thus a robust, generic feature of the high-energy phenomenology of pair-condensed Fermi gases, and should be accessible to state-of-the-art cold atom experiments.
Abstract The Leggett collective excitations for a two-band Fermi gas with s-wave pairing and Josephson interband coupling in the BCS-BEC crossover at finite temperatures are investigated within the Gaussian pair fluctuation approach. Eigenfrequencies and damping factors for Leggett modes are determined in a nonperturbative way, using the analytic continuation of the fluctuation propagator through a branch cut in the complex frequency plane, as in Kurkjian et al (2019 Phys. Rev. Lett. 122 093403). The treatment is performed beyond the low-energy expansion, which is necessary when the collective excitation energy reaches the pair-breaking continuum edge. The results are applied in particular to cold atomic gases at the orbital Feshbach resonance and in a regime far from BEC, which can be relevant for future experiments.
Abstract We consider collective excitations in the superfluid state of Fermi condensed charged gases. The dispersion and damping of collective excitations at nonzero temperatures are examined, and the coexistence and interaction of different branches of collective excitations: plasma oscillations, pair-breaking Higgs modes, and Carlson–Goldman phonon-like excitations are taken into account. The path integral methods for superfluid Fermi gases and for Coulomb gas are combined into a unified formalism that extends the Gaussian fluctuation approximation to account for plasmonic modes. This approximation of Gaussian pair and density fluctuations is able to describe all branches of collective excitations existing in a charged superfluid. The spectra of collective excitations are determined in two ways: from the spectral functions and from the complex poles of the fluctuation propagator. A resonant avoided crossing of different modes is shown. It is accompanied by resonant enhancement of the response provided by the pair-breaking modes due to their interaction with plasma oscillations. This may facilitate the experimental observation of the pair-breaking modes.