Quantum oscillations in ultracold Fermi gases: Realizations with rotating gases or artificial gauge fields

We consider the angular momentum of a harmonically trapped, noninteracting Fermi gas subject to either rotation or to an artificial gauge field. The angular momentum of the gas is shown to display oscillations as a function of the particle number or chemical potential. This phenomenon is analogous to the de Haas–van Alphen oscillations of the magnetization in the solid-state context. However, key differences exist between the solid-state and ultracold atomic gases that we point out and analyze. We explore the dependence of the visibility of these oscillations on the physical parameters and propose two experimental protocols for their observation. Due to the very strong dependence of the amplitude of the oscillations on temperature, we propose their use as a sensitive thermometer for Fermi gases in the low-temperature regime.

Figure 1

The ratio $R$, for ${\alpha }_z=0.05$, $\alpha =0.01$, and $T/T_F=0.03$ (in blue). The black dashed line is the monotonic contribution, and the red dotted line is the oscillating part. At high particle number, the oscillating contribution is negligible, and $R$ coincides with its monotonic part (17), as they both go to unity.

Figure 2

Dashed curve (red line, left scale): magnetization per particle $M/N{\mu }_B$ of a two-dimensional electron gas at zero temperature, as a function of $N{\Phi }_0/\Phi$, with $N$ the number of electrons and $\Phi /{\Phi }_0$ the magnetic flux in units of the flux quantum ${\Phi }_0=h/e$. A discontinuous jump is observed each time a Landau level is completely filled. Plain curve (blue line, right scale): dependence of the angular momentum $L_z/N^2\hslash$ on $N{\omega }_{-}/{\omega }_{+}$ for a zero-temperature rotating trapped fermionic gas, in the limit of fast rotation. The angular momentum displays cusps at the discrete values $N{\omega }_{-}/{\omega }_{+}=p(p+1)/2=1,3,6,10,...$, corresponding to the complete filling of an integer number of $n_{+}$ levels. At these points, the derivative of the angular momentum versus $N{\omega }_{-}/{\omega }_{+}$ displays discontinuous jumps. This illustrates the similarities, but also the differences between the solid-state and ultracold fermionic gas contexts, regarding the phenomenon of de Haas–van Alphen quantum oscillations.

Figure 3

Sketch explaining the origin of the quantum oscillations for a two-dimensional gas at zero temperature. (The case of low confinement is similar. In this case, each point of the graph will account for the particles that populate the ${\omega }_z$ levels allowed by the energy constraint.) Each level is labeled by its integer coordinate $(n_{+},n_{-})$ and a projection onto the $\vartheta$ axis with slope ${\omega }_{+}/{\omega }_{-}$ gives the single-particle energy $\varepsilon -{\varepsilon }_0$. The occupation of the levels is fixed by the chemical potential (green line). Green circles represent filled levels at zero temperature (satisfying the energy constraint), and the green situation depicted corresponds to the ground state of the system with $N=15$ particles. The angular momentum corresponds to the projection onto the axis labeled by ${\ell }_z/\hslash$ as indicated by the dashed-dotted line. The two different colors for the axis and the corresponding shaded area indicate different sign of ${\ell }_z$. The particles have been attributed a number (inside the green circle) corresponding to their order of appearance as the chemical potential is increased. Inset: The inset shows the evolution of the total angular momentum of the gas (in blue) with the number of particles, and the angular momentum of each particle (orange). The arrows point to the total angular momentum for $N=16$, and the contribution from the supplementary particle added to the system by changing the chemical potential from the green to the red line.

Figure 4

Angular momentum and its classical counterpart as a function of $N$, for ${\alpha }_z=0.05$, $T/T_F=0.01$, and $\alpha =0.05$. The shell filling effect in the angular momentum is magnified due to the difference between $n_{+}$ and $n_{-}$ in its expression, and results in an oscillating behavior. The inset shows the same two quantities, at low particle number. The shell filling effects are visible in the classical angular momentum only at low particle number.

Figure 5

$R$ vs ${\alpha }_{z}N$, at low temperature $T/T_F=0.03$ and $\alpha =0.05$, for ${\alpha }_z=0.01$, ${\alpha }_z=0.02$, ${\alpha }_z=2$, and ${\alpha }_z=10$. Even though at low particle numbers, the amplitude of the oscillation is strongest for the intermediate aspect ratio ${\alpha }_z=2$, the oscillations remain visible for larger particle numbers in the weak confinement case. This indicates that a compromise between these two feature might be a favorable choice of parameter to observe the oscillation of angular momentum.

Figure 6

(a) The ratio $R$ vs particle number, at low rotation frequency $\alpha =0.05$ for $T/T_F=0.03$, $T/T_F=0.05$, $T/T_F=0.07$, and $T/T_F=0.1$. Panel (a) shows a weak aspect ratio of the confinement ${\alpha }_z=0.02$ and panel (b) an intermediate aspect ratio ${\alpha }_z=2$. In both cases, the low-temperature curve displays the most pronounced oscillations, and the oscillating behavior remains visible for larger samples. The stronger confinement leads to larger oscillation amplitudes, but the oscillations disappear for lower particle numbers. The insets shows the Fourier transform of $R$ with respect to $\mu$ for the corresponding parameters. It shows that the amplitude of the oscillation modes decreases with the temperature, but the oscillation period remains unchanged.

Figure 7

Amplitude of the oscillation of the ratio $R$ vs $T/T_F$, for $\alpha =0.05$ and for ${\alpha }_{z}N=11.5$ (blue) and ${\alpha }_{z}N=4.5$ (black) with respect to the monotonic contribution (16). As expected, at fixed confinement, the oscillation becomes less pronounced for bigger particle numbers. The strong damping effect due to the temperature is clearly visible in both cases.

Figure 8

$R$ vs particle number, at zero temperature (upper panel) and $T/T_F=0.03$ (lower panel) and weak trapping ${\alpha }_z=0.075$, for several angular frequencies. Slow rotation favors the appearance of quantum oscillations. At zero temperature, fast oscillations due to the contribution of the ${\omega }_z$ levels appear.

Figure 9

Fourier transform of $R$ vs $x\hslash {\omega }_0$ at zero temperature (upper panel) and $T/T_F=0.03$ (lower panel), for fixed ${\alpha }_z=0.075$. The successive harmonics of the fundamental frequencies are visible. The arrows indicate the position of the two characteristic “frequencies” $\frac{1}{\hslash {\omega }_{\pm }}$. Inset: Fourier transform of $R$ at zero temperature showing the peak corresponding to $\frac{1}{\hslash {\omega }_z}$, indicated by an arrow.

Figure 10

Sketch of the two different protocols for measuring the angular momentum of the fermion gas (see text for description). (a) Measurement of the precession angle. The stirring beam (in green) is also used to create the anisotropic perturbation, which provokes the appearance of quadrupole modes, and are responsible for a precession angle $\phi$. Observing this angle provides a direct measurement of the angular momentum of the gas. (b) Generation of artificial rotation with lasers with spectroscopic measurement of the angular momentum. The two copropagating beams carry different angular momenta, and the effective rotation is proportional to the angular momentum difference. The angular momentum is then measured by spectroscopy as suggested in [30, 31].