Long-lived Higgs mode in a two-dimensional confined Fermi system

The Higgs mode corresponds to the collective motion of particles due to the vibrations of an invisible field. It plays a fundamental role in our understanding of both low- and high-energy physics, giving elementary particles their mass and leading to collective modes in condensed-matter and nuclear systems. The Higgs mode has been observed in a limited number of table-top systems, where it however is characterized by a short lifetime due to decay into a continuum of modes. A major goal which has remained elusive so far is, therefore, to realize a long-lived Higgs mode in a controllable system. Here, we show how an undamped Higgs mode can be observed unambiguously in a Fermi gas in a two-dimensional trap, close to a quantum phase transition between a normal and a superfluid phase. We develop a first-principles theory of the pairing and the associated collective modes, which is quantitatively reliable when the pairing energy is much smaller than the trap level spacing, yet simple enough to allow the derivation of analytical results. The theory includes the trapping potential exactly, which is demonstrated to stabilize the Higgs mode by making its decay channels discrete. It is shown to be realistic to realize the proposed system using atoms in microtraps. The unrivaled flexibility of atomic gases combined with the quantitatively accurate theory opens the door to a systematic study of the Higgs mode, including the role of confinement and finite-size effects.

Figure 1

The free energy of the system as a function of the pairing field and the associated Higgs and Goldstone modes (arbitrary units). (a) The weak-coupling regime where the system is in the normal phase (${\epsilon }_B<{\epsilon }_B^c$). (b) The strong-coupling superfluid regime (${\epsilon }_B>{\epsilon }_B^c$). The Higgs mode corresponds to amplitude fluctuations around the minimum of the effective action. At the quantum phase transition point between the two phases, ${\epsilon }_B={\epsilon }_B^c$, the Gaussian amplitude fluctuations around $\Delta =0$ cost zero energy. The Goldstone mode corresponds to phase fluctuations in the superfluid phase.

Figure 2

For the closed-shell case, there is only pairing for strong enough attraction, ${\epsilon }_B>{\epsilon }_B^c$, so it is energetically favorable to excite Cooper pairs from the highest occupied to the lowest empty shell. The collective mode in the normal phase is formed by making coherent excitation pairs across the chemical, and the mode frequency goes to zero when the system becomes unstable to pairing.

Figure 3

The Higgs mode for the open-shell case. We plot the Higgs amplitude mode energy as a function of the two-body binding energy per particle, ${\epsilon }_B$, for the open-shell case with $n_F=10$. The solid line is obtained by numerically solving (13), the dashed line is $\omega =2{\Delta }_{n_F}$ with ${\Delta }_{n_F}$ given by (6), and the dotted line is $\omega =2{\epsilon }_B$.

Figure 4

The Higgs mode for the closed-shell case. We plot the Higgs amplitude mode energy as a function of the two-body binding energy per particle, ${\epsilon }_B$, for the closed-shell case with $n_F=10$. The solid line is obtained by numerically solving (13), and the dashed lines are (14) for ${\epsilon }_B<{\epsilon }_B^c$, and $\omega =2{\Delta }_{n_F}$ with ${\Delta }_{n_F}$ given by (8) for ${\epsilon }_B>{\epsilon }_B^c$. The critical two-body binding energy ${\epsilon }_B^c$ is given by (7). The dotted line is the perturbative result $\omega =2{\omega }_{\perp }-4{\epsilon }_B$.