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arXiv:1209.2025 · Physical Review A

Rapid ramps across the BEC-BCS crossover: A route to measuring the superfluid gap

R. G. Scott, F. Dalfovo, Лев П. Питаевский, S. Stringari

Published Nov 7, 2012• 47 citations
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Abstract

We investigate the response of superfluid Fermi gases to rapid changes of the three-dimensional $s$-wave scattering length $a$ by solving the time-dependent Bogoliubov--de Gennes equations. In general the magnitude of the order parameter $\left|\ensuremath{\Delta}\right|$ performs oscillations, which are sometimes called the ``Higgs'' mode, with the angular frequency $2{\ensuremath{\Delta}}_{\text{gap}}/\ensuremath{\hbar}$, where ${\ensuremath{\Delta}}_{\text{gap}}$ is the gap in the spectrum of fermionic excitations. First, we excite the oscillations with a linear ramp of $1/a$ and study the evolution of $\left|\ensuremath{\Delta}\right|$. Second, we continously drive the system with a sinusoidal modulation of $1/a$. In the first case, the oscillations in $\left|\ensuremath{\Delta}\right|$ damp according to a power law. In the second case, the continued driving causes revivals in the oscillations. In both cases, the excitation of the oscillations causes a reduction in the time-averaged value of $\left|\ensuremath{\Delta}\right|$. We propose two experimental protocols, based around the two approaches, to measure the frequency and damping of the oscillations, and hence ${\ensuremath{\Delta}}_{\text{gap}}$.

DOI
10.1103/physreva.86.053604

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We investigate the response of superfluid Fermi gases to rapid changes of the three-dimensional sss-wave scattering length aaa by solving the time-dependent Bogoliubov--de Gennes equations. In general the magnitude of the order parameter ∣\ensuremathΔ∣\left|\ensuremath{\Delta}\right|∣\ensuremathΔ∣ performs oscillations, which are sometimes called the ``Higgs'' mode, with the angular frequency 2\ensuremathΔgap/\ensuremathℏ2{\ensuremath{\Delta}}_{\text{gap}}/\ensuremath{\hbar}2\ensuremathΔgap​/\ensuremathℏ, where \ensuremathΔgap{\ensuremath{\Delta}}_{\text{gap}}\ensuremathΔgap​ is the gap in the spectrum of fermionic excitations. First, we excite the oscillations with a linear ramp of 1/a1/a1/a and study the evolution of ∣\ensuremathΔ∣\left|\ensuremath{\Delta}\right|∣\ensuremathΔ∣. Second, we continously drive the system with a sinusoidal modulation of 1/a1/a1/a. In the first case, the oscillations in ∣\ensuremathΔ∣\left|\ensuremath{\Delta}\right|∣\ensuremathΔ∣ damp according to a power law. In the second case, the continued driving causes revivals in the oscillations. In both cases, the excitation of the oscillations causes a reduction in the time-averaged value of ∣\ensuremathΔ∣\left|\ensuremath{\Delta}\right|∣\ensuremathΔ∣. We propose two experimental protocols, based around the two approaches, to measure the frequency and damping of the oscillations, and hence \ensuremathΔgap{\ensuremath{\Delta}}_{\text{gap}}\ensuremathΔgap​.
In quantum many-body systems with spontaneous breaking of continuous symmetries, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs mode has been observed in the ultracold Fermi gas. In the present paper, we use the time-dependent Bogoliubov-de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a Fermi gas induced by a rapid change of the s{\it s}s-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length. We find that the energy of the Higgs mode coincides with the threshold energy of the pair-breaking excitation, and exponent of the power-low decay of the Higgs mode γγγ continuously changes between γ=−1/2γ=-1/2γ=−1/2 and γ=−3/2γ=-3/2γ=−3/2 through the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation (BCS-BEC) crossover. Moreover, we propose the optimal ramp speed of the scattering length for observing the clearest Higgs oscillations.
A new field of research, which gained interest in the past years, is the field of non- equilibrium physics of strongly interacting fermionic systems. In this thesis we present a novel apparatus to study an ultracold strongly interacting superfluid Fermi gas driven out of equilibrium. In more detail, we study the excitation of a collective mode, the Higgs mode, and the dynamics occurring after a rapid quench of the interaction strength. Ultracold gases are ideal candidates to explore the physics of strongly interacting Fermi gases due to their purity and the possibility to continuously change many different system parameters, such as the interaction strength. The novel apparatus enabling the experiments outlined below is built from scratch. The setup is described and characterized in detail. We produce a fermionic superfluid of 4 × 10^6 6Li atoms at a temperature of T/T_F = 0.07 ± 0.02, where T_F is the Fermi temperature. This is achieved within 23 s by a combination of laser cooling, radio frequency evaporation of 23Na and simultaneous sympathetic cooling of 6Li in an optical plugged magnetic trap and subsequent evaporative cooling of 6Li in a dipole trap. In this thesis we provide the first experimental evidence of the Higgs mode in a strongly interacting Fermi gas in the crossover from a BCS superfluid to a molecular Bose-Einstein condensate. We develop a novel excitation method, which directly couples to the amplitude of the order parameter. This is achieved by continuously changing the population of one spin component by driving a radio frequency transition to a previously unoccupied third hyperfine state. This effectively modulates the interaction strength and the amplitude of the order parameter. We spectroscopically observe a resonance behavior at twice the gap frequency. For strong coupling, the peak width broadens and eventually the Higgs mode disappears when the Cooper pairs turn into tightly bound dimers signaling the instability of the Higgs mode. It has been suggested that the Higgs mode frequency is a precise measure of the superconducting gap in the BEC-BCS crossover, where the exact value of the gap is yet unknown and numerical calculations are challenging. Hence, our novel method provides a unique technique to determine the superconducting gap. Moreover, we perform rapid variations of the interaction parameter and vary both the initial interaction strength and the amplitude of the quench. The rapid quenches are performed by rapidly transferring one spin component of a two-component spin mixture into a third state with a different interaction strength by using a radio frequency transition. Using this novel method, we can perform the quenches in half the Fermi time, which is experimentally very close to a sudden change and to date faster than any other research group performing interaction quenches with ultra cold gases. In the experiment, we observe a fast relaxation to a zero order parameter for large quenches, whereas for small quenches we observe, after a sudden drop, a revival of the order parameter and equilibration to a long-term superfluid steady state. Our measurement provides the first evidence of the collapse and subsequent revival of order in a strongly interacting fermionic system.
Experimental evidence of the Higgs mode in strongly interacting superfluid Fermi gases had not been observed until recently [Behrle et al., Nat. Phys. 14, 781 (2018).]. Due to the coupling with other collective modes and quasiparticle excitations, generating stable Higgs-mode oscillations is challenging. We study how to excite long-lived Higgs-mode oscillations in a homogeneous superfluid Fermi gas in the BCS-BEC crossover. We find that the Higgs mode can be excited by time-periodically modulating the scattering length at an appropriate amplitude and frequency. However, even for a modulation frequency below twice the pairing-gap energy, quasiparticles are still excited through the generation of higher harmonics due to nonlinearity in the superfluid. More importantly, we find that persistent Higgs-mode oscillations with almost constant amplitude can be produced by removing particles at an appropriate momentum, and the oscillation amplitude can be controlled by the number of removed particles. Finally, we propose two ways to experimentally realize particle removal.
The Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensate (BEC) crossover problem is solved for stationary gray solitons via the Boguliubov–de Gennes equations at zero temperature. These crossover solitons exhibit a localized notch in the gap and a characteristic phase difference across the notch for all interaction strengths, from BEC to BCS regimes. However, they do not follow the well-known Josephson-like sinusoidal relationship between velocity and phase difference except in the far BEC limit: at unitarity, the velocity has a nearly linear dependence on phase difference over an extended range. For a fixed phase difference, the soliton is of nearly constant depth from the BEC limit to unitarity and then grows progressively shallower into the BCS limit, and on the BCS side, Friedel oscillations are apparent in both gap amplitude and phase. The crossover soliton appears fundamentally in the gap; we show, however, that the density closely follows the gap, and the soliton is therefore observable. We develop an approximate power-law relationship to express this fact: the density of gray crossover solitons varies as the square of the gap amplitude in the BEC limit and as a power of about 1.5 at unitarity.
We study the response of a Fermi superfluid to a rapid change of the interaction strength. For a broad range of quench parameters, the order parameter exhibits damped oscillations, however, we also identify quench regimes in which these oscillations are absent. By comparing our data to a numerical model we find that the damping time constants are explainable by a dephasing in a local density approximation, however, the oscillation frequencies, while being comparable to the superconducting order parameter, display in detail significant differences to the integrable BCS (Bardeen-Cooper-Schrieffer) model.
Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent T-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.