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Observation of Fermi surface deformation in a dipolar quantum gas

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Science  19 Sep 2014:
Vol. 345, Issue 6203, pp. 1484-1487
DOI: 10.1126/science.1255259

Aligning a magnetic atomic gas

When a bunch of fermions get together, they obey the Pauli exclusion principle: No two fermions can be in the same quantum state. The fermions populate the available states, starting from those lowest in energy. The boundary between the empty and filled states is called the Fermi surface (FS). For cold gases of fermionic atoms in the lab, the FS is usually spherical. Now, Aikawa et al. observe the FS squishing in a gas of Er atoms, which behave like tiny magnets and align with their magnetic field environment. The squishing reflects the very directional interactions between the Er atoms.

Science, this issue p. 1484

Abstract

In the presence of isotropic interactions, the Fermi surface of an ultracold Fermi gas is spherical. Introducing anisotropic interactions can deform the Fermi surface, but the effect is subtle and challenging to observe experimentally. Here, we report on the observation of a Fermi surface deformation in a degenerate dipolar Fermi gas of erbium atoms. The deformation is caused by the interplay between strong magnetic dipole-dipole interaction and the Pauli exclusion principle. We demonstrate the many-body nature of the effect and its tunability with the Fermi energy. Our observation provides a basis for future studies on anisotropic many-body phenomena in normal and superfluid phases.

The Fermi-liquid theory, formulated by Landau in the late 1950s, is one of the most powerful tools in modern condensed-matter physics (1). It captures the behavior of interacting Fermi systems in the normal phase, such as electrons in metals and liquid 3He (2). Within this theory, the interaction is accounted by dressing the fermions as quasi-particles with an effective mass and an effective interaction. The ground state is the so-called Fermi sea, in which the quasi-particles fill one-by-one all the states up to the Fermi momentum, kF. The Fermi surface (FS), which separates occupied from empty states in k-space, is a sphere of radius kF for isotropically interacting fermions in uniform space. The FS is crucial for understanding system excitations and Cooper pairing in superconductors. When complex interactions act, the FS can get modified. For instance, strongly correlated electron systems violate the Fermi-liquid picture, giving rise to a deformed FS, which spontaneously breaks the rotational invariance of the system (3). Symmetry-breaking FSs have been studied in connection with electronic liquid crystal phases (4) and Pomeranchuk instability (5) in solid-state systems. Particularly relevant is the nematic phase, in which anisotropic behaviors spontaneously emerge and the system acquires an orientational order, while preserving its translational invariance (3).

A completely distinct approach to studying FSs is provided by ultracold quantum gases. These systems are naturally free from impurities and do not have a crystal structure, realizing a situation close to the ideal uniform case; therefore, the shape of the FS can directly reveal the fundamental interactions among particles. Studies of FSs in strongly interacting Fermi gases have been crucial in understanding the Bose-Einstein condensation (BEC)–to–Bardeen-Cooper-Schrieffer (BCS) crossover, in which the isotropic s-wave (contact) interaction causes a broadening of the always-spherical FS (6). Recently, Fermi gases with anisotropic interactions have attracted much attention in the context of p-wave superfluidity (7, 8) and dipolar physics (9). Many theoretical studies have focused on dipolar Fermi gases, predicting the existence of a deformed FS (1015). These studies also include an extension of the Landau Fermi-liquid theory to the case of anisotropic interactions (16). Despite recent experimental advances in polar molecules and magnetic atoms (1720), the observation of anisotropic FSs has so far been elusive.

Here, we present the direct observation of the deformed FS in dipolar Fermi gases of strongly magnetic erbium (Er) atoms. By virtue of the anisotropic dipole-dipole interaction (DDI) among the particles, the FS is predicted to be deformed into an ellipsoid. To minimize the system’s energy, the FS elongates along the direction of the maximum attraction of the DDI, where the atomic dipoles have a “head-to-tail” orientation. To understand the origin of the Fermi surface deformation (FSD), one has to account for both the action of the DDI in k-space and the Pauli exclusion principle, which imposes antisymmetry on the many-body wave function. In the Hartree-Fock formalism, the FSD comes from the exchange interaction among fermions, known as the Fock term (10, 14, 21). Our observations agree very well with parameter-free calculations based on the Hartree-Fock theory (10, 13, 15). We demonstrate that the degree of deformation, related to the nematic susceptibility in the liquid-crystal vocabulary, can be controlled by varying the Fermi energy of the system and vanishes at high temperatures.

Our system is a single-component quantum degenerate dipolar Fermi gas of Er atoms. Like other lanthanoids, a distinct feature of Er is a large permanent magnetic dipole moment μ of 7 Bohr magneton, which causes a strong DDI between the fermions. Similarly to our previous work (20), we take advantage of elastic dipole-dipole collisions to drive efficient evaporative cooling in spin-polarized fermions. The sample is confined into a three-dimensional optical harmonic trap and typically contains 7 × 104 atoms at a temperature of 0.18(1) TF, with TF = 1.12(4) μK (21). We control the alignment of the magnetic dipole moments by setting the orientation of an external polarizing magnetic field. We label β as the angle between the magnetic field and the z axis (Fig. 1, inset).

Fig. 1 AR of an expanding dipolar Fermi gas as a function of the angle β.

In this measurement, the trap frequencies are (fx, fy, fz) = (579, 91, 611) Hz. The data are taken at tTOF = 12 ms. Each individual point is obtained from about 39 independent measurements. The error bars indicate SEM. For comparison, the calculated values are also shown for 0° and 90° (crosses). (Inset) Schematic illustration of the geometry of the system. Gravity is along the z direction. The atomic cloud is imaged with an angle of 28° with respect to the y axis (21). The magnetic field orientation is rotated within the plane that forms an angle of 14° with the xz plane. Schematic illustrations of the deformed FS are also shown above the panel; kF is the Fermi momentum for an ideal Fermi gas.

To explore the impact of the DDI on the momentum distribution, we performed time-of-flight (TOF) experiments. Since its first use as “smoking-gun” evidence for BEC (22, 23), this technique has proved its power in revealing many-body quantum phenomena in momentum space (6, 24). TOF experiments are based on the study of the expansion dynamics of a gas after it has been released from a trap. For a sufficiently long expansion time, the size of the atomic cloud is dominated by the velocity dispersion and, in the case of ballistic (free) expansions, the TOF images purely reflect the momentum distribution in the trap.

In our experiment, we first prepared the ultracold Fermi gas with a given dipole orientation and then let the sample expand by suddenly switching off the optical dipole trap (ODT). From the TOF images, we derived the cloud aspect ratio (AR), which is defined as the ratio of the vertical to horizontal radius of the cloud in the imaging plane (21). The AR for various values of β are shown in Fig. 1. For vertical orientation (β = 0°), we observed a clear deviation of the AR from unity with a cloud anisotropy of ~3%. TOF images show that the cloud has an ellipsoidal shape, with elongation in the direction of the dipole orientation. When changing β, we observed that the cloud follows the rotation of the dipole orientation, keeping the major axis always parallel to the direction of the maximum attraction of the DDI. In a second set of experiments, we recorded the time evolution of the AR during the expansion for β = 0° and β = 90° (Fig. 2). For both orientations, the AR differs from unity at long expansion times. Our results are strikingly different from the ones in conventional Fermi gases with isotropic contact interactions, in which the FS is spherical (AR = 1) and the magnetic field orientation has no influence on the cloud shape (6).

Fig. 2 Time evolution of the AR of the atomic cloud during the expansion.

Measurements are performed for two dipole angles, β = 0° (squares) and β = 90° (circles) under the same conditions as in Fig. 1. The error bars are SEM of about 17 independent measurements. The possible origin of the small fluctuations of the data points at short expansion time is discussed in (21). The theoretical curves show the full numerical calculations (solid lines), which include both the FSD and the NBE effects, and the calculation in the case of ballistic expansions (dashed lines), in the absence of the NBE effect. For comparison, the calculation for a noninteracting Fermi gas is also shown (dot-dashed line).

The one-to-one mapping of the original momentum distribution in the trap and the density distribution of the cloud after long expansion time strictly holds only in the case of pure ballistic expansions. In our experiments, the DDI is acting even during the expansion and could potentially mask the observation of the FSD. We evaluated the effect of the nonballistic expansion (NBE) by performing numerical calculations based on the Hartree-Fock mean-field theory at zero temperature and the Boltzmann-Vlasov equation for expansion dynamics (13, 15, 21). In Fig. 2, the theoretical curves do not have any free parameters and are calculated both in the presence (Fig. 2, solid lines) and absence (Fig. 2, dashed lines) of the NBE effect. The comparison between ballistic and nonballistic expansion reveals that the latter plays a minor role in the final AR, showing that the observed anisotropy dominantly originates from the FSD. The agreement between experiment and theory implies that our model accurately describes the behavior of the system.

Theoretical works have predicted that the degree of deformation depends on the Fermi energy and the dipole moment (10, 1216). In the limit of weak DDI, the magnitude of the FSD in a trapped sample is expected to be linearly proportional to the ratio of the DDI to the Fermi energy, η = nd2/EF (14). Here, n = 4π(2m EF/h2)3/2/3 is the peak number density at zero temperature, h is the Planck constant, m is the mass, d2 = μ0μ2/(4π) is the coupling constant for the DDI, and μ0 is the magnetic constant. For a harmonically trapped ideal Fermi gas, the Fermi energy EF depends on the atom number N and the mean trap frequency Embedded Image, Embedded Image. Given that Embedded Image, the FSD can be tuned by varying EF.

To test the theoretical predictions, we first numerically studied the degree of cloud deformation Δ, defined as Δ = AR – 1, as a function of the trap anisotropy, Embedded Image, and/or Embedded Image. To distinguish the effect of the FSD and of the NBE, we keep the two contributions separated in the calculations (Fig. 3, A and B). Our results clearly convey the following information: (i) The FSD gives the major contribution to Δ; (ii) the FSD is independent of the trap anisotropy but increases with Embedded Image; and (iii) the NBE effect reflects the trap anisotropy and vanishes for a spherical trap (13).

Fig. 3 Δ = AR – 1 for various trap geometries.

(A and B) We consider a cigar-shaped trap with fx = fz in the calculations and show the behaviors of the FSD (dashed lines) and the NBE (dotted lines) separately as (A) a function of the trap anisotropy Embedded Image at Embedded Image = 400 Hz and (B) a function of Embedded Image at Embedded Image. (C) Experimentally observed Δ at tTOF = 12 ms are plotted as a function of η, together with the full calculation (solid line) and the calculation considering only FSD (dashed line). The shaded area shows the uncertainty originating from the uncertainty in determining η in our experiments. The sample contains 6 × 104 atoms at a typical temperature of T/TF = 0.15(1). The error bars represent SEM of about 15 independent measurements. The variation of the trap anisotropy in the experiment is indicated on the top axis. (D and E) Visualization of the FSD at η = 0.009 from (D) the experimental TOF image and (E) the fitted image.

In the experiment, we explored the dependence of Δ on the trap geometry for β = 0° by keeping the axial frequency (fy) constant and varying the radial frequencies (fx = fz within 5%) (Fig. 3C). This leads to a simultaneous variation of both the trap anisotropy and Embedded Image. We observed an increase of Δ with η, which is consistent with the theoretically predicted linear dependence (14).

In analogy with studies in superconducting materials (25), we graphically emphasized the FSD in the measurements at η = 0.009 by subtracting the TOF absorption image taken at β = 90° from the one at β = 0° (Fig. 3D). The resulting image exhibits a cloverleaf-like pattern, showing that the momentum spread along the orientation of the dipoles is larger than in the other direction. For comparison, the same procedure is applied for images obtained by a fit to the observed cloud (Fig. 3E). At η = 0.009, the trap anisotropy is so small that the NBE effect is negligibly small, and the deformation is caused almost only by the FSD.

Last, we investigated the temperature dependence of Δ (Fig. 4). We prepared samples at various temperatures by stopping the evaporative cooling procedure at various points. The final trap geometry is kept constant. When reducing the temperature, we observed the emergence of the FSD, which becomes more and more pronounced at low temperatures and eventually approaches the zero-temperature limit. The qualitative behavior of the observed temperature dependence is consistent with a theoretical result at finite temperatures (14), although further theoretical developments are needed for a more quantitative comparison.

Fig. 4 Δ as a function of the temperature of the cloud.

Measurements are performed for two dipole angles, β = 0° (squares) and β = 90° (circles), under the same conditions as in Fig. 1. The error bars are SEM of about 26 independent measurements. The solid lines show the numerically calculated values at zero temperature for β = 0° and β = 90°.

Our observation clearly shows the quantum many-body nature of the FSD and sets the basis for future investigations on more complex dipolar phenomena, including collective excitations (13, 15, 26, 27) and anisotropic superfluid pairing (28, 29). Taking advantage of the wide tunability of cold-atom experiments, dipolar Fermi gases are ideally clean systems for exploring exotic and topological phases in a highly controlled manner (9).

Supplementary Materials

www.sciencemag.org/content/345/6203/1484/suppl/DC1

Materials and Methods

Fig. S1

References (3041)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We are grateful to A. Pelster, M. Ueda, M. Baranov, R. Grimm, T. Pfau, B. L. Lev, and E. Fradkin for fruitful discussions. This work is supported by the Austrian Ministry of Science and Research (BMWF) and the Austrian Science Fund (FWF) through a START grant under project Y479-N20 and by the European Research Council under project 259435. K.A. is supported within the Lise-Meitner program of the FWF.
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