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arXiv:1011.1327 · Physical Review Letters

Crossover from 2D to 3D in a Weakly Interacting Fermi Gas

Paul Dyke, E. D. Kuhnle, S. Whitlock, Hui Hu, Manfred J. Mark, Sascha Hoinka, Marcus Lingham, Peter Hannaford, Chris J. Vale

Published Mar 11, 2011• 132 citations
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Abstract

We have studied the transition from two to three dimensions in a low temperature weakly interacting 6Li Fermi gas. Below a critical atom number N(2D) only the lowest transverse vibrational state of a highly anisotropic oblate trapping potential is occupied and the gas is two dimensional. Above N(2D) the Fermi gas enters the quasi-2D regime where shell structure associated with the filling of individual transverse oscillator states is apparent. This dimensional crossover is demonstrated through measurements of the cloud size and aspect ratio versus atom number.

DOI
10.1103/physrevlett.106.105304

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Quasi-2D degenerate Fermi gases provide a rich physical environment to explore since many effects that are present do not exist in 3D. This thesis describes the first experiments carried out in a quasi-2D Fermi gas across the broad 6Li Feshbach resonance at 834 G. A description of the apparatus used to form molecular Bose-Einstein condensates and degenerate Fermi gases in the two lowest hyperfine states of 6Li, |F = 1/2,mf = 1/2i and |F = 1/2,mf = −1/2i, is presented. An anisotropic 2D optical potential is formed by an elliptical Gaussian beam produced by tightly focusing a circular Gaussian beam in one direction with a cylindrical lens. The trapping frequencies are ∼3 kHz in the axial direction and ∼50 Hz in the radial direction giving an aspect ratio of ∼60. For a non-interacting Fermi gas this allows ∼1700 atoms to populate the lowest transverse harmonic oscillator state and to be considered as quasi-2D. We present the transition from a three dimensional gas to a two dimensional gas at ∼2000 atoms agreeing within experimental uncertainty with the non-interacting atom number. The observations are made through measuring the radial and axial widths of the gas at three different magnetic fields, 810 G, 834 G, and 991 G. The widths σ are then plotted against the atom number N. The curve, log σ(N) = n logN+a, is then fit to the data in the 3D regime and in the 2D regime. The expansion of a non-interacting Fermi gas in the 2D regime changes from a scaling of N1/4 to N1/6 in the 3D regime and for a Bose gas it changes to N1/5. A value for n is extracted in each of these regimes. For each magnetic field we then obtain in the 2D regime: n = 0.27(0.04) at 810 G, n = 0.23(0.02) at 834 G and n = 0.26(0.01) at 991 G. In the 3D regime we obtain: n = 0.195(0.01) at 810 G, n = 0.154(0.01) at 834 G and n = 0.175(0.01 at 991 G. We also observe a change in the aspect ratio as we approach the 2D regime which is not present in our 3D single beam optical dipole trap and is absent in any 3D system. These first results indicate that we have a quasi-2D Fermi gas. The main experimental results are the first indications of a confinement induced resonance in a 6Li Fermi gas through the measurement of the cloud width in the tightly confined direction after a fixed expansion time. We observe a significant deviation in the transverse width of the 2D interacting Fermi gas from the 3D case due to the increased tight confinement. The confinement induced resonance should appear when the 3D scattering length approaches the transverse harmonic oscillator length, az = p~/mωz, where m is the mass and ωz is the tight transverse trapping frequency. For our experimental parameters this leads to a 3D scattering length of 1.03 μm, where the observation is at 819(1) G. In the weakly confined direction there is no such increase in the width due to the free motion in the xy plane. Instead the cloud width across the Feshbach resonance increases monotonically, similar to that of a 3D gas and as described theoretically for a quasi-2D Fermi gas by Lin et al. [1].

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We have studied the transition from two to three dimensions in a low temperature weakly interacting 6Li Fermi gas. Below a critical atom number N(2D) only the lowest transverse vibrational state of a highly anisotropic oblate trapping potential is occupied and the gas is two dimensional. Above N(2D) the Fermi gas enters the quasi-2D regime where shell structure associated with the filling of individual transverse oscillator states is apparent. This dimensional crossover is demonstrated through measurements of the cloud size and aspect ratio versus atom number.
Quasi-2D degenerate Fermi gases provide a rich physical environment to explore since many effects that are present do not exist in 3D. This thesis describes the first experiments carried out in a quasi-2D Fermi gas across the broad 6Li Feshbach resonance at 834 G. A description of the apparatus used to form molecular Bose-Einstein condensates and degenerate Fermi gases in the two lowest hyperfine states of 6Li, |F = 1/2,mf = 1/2i and |F = 1/2,mf = −1/2i, is presented. An anisotropic 2D optical potential is formed by an elliptical Gaussian beam produced by tightly focusing a circular Gaussian beam in one direction with a cylindrical lens. The trapping frequencies are ∼3 kHz in the axial direction and ∼50 Hz in the radial direction giving an aspect ratio of ∼60. For a non-interacting Fermi gas this allows ∼1700 atoms to populate the lowest transverse harmonic oscillator state and to be considered as quasi-2D. We present the transition from a three dimensional gas to a two dimensional gas at ∼2000 atoms agreeing within experimental uncertainty with the non-interacting atom number. The observations are made through measuring the radial and axial widths of the gas at three different magnetic fields, 810 G, 834 G, and 991 G. The widths σ are then plotted against the atom number N. The curve, log σ(N) = n logN+a, is then fit to the data in the 3D regime and in the 2D regime. The expansion of a non-interacting Fermi gas in the 2D regime changes from a scaling of N1/4 to N1/6 in the 3D regime and for a Bose gas it changes to N1/5. A value for n is extracted in each of these regimes. For each magnetic field we then obtain in the 2D regime: n = 0.27(0.04) at 810 G, n = 0.23(0.02) at 834 G and n = 0.26(0.01) at 991 G. In the 3D regime we obtain: n = 0.195(0.01) at 810 G, n = 0.154(0.01) at 834 G and n = 0.175(0.01 at 991 G. We also observe a change in the aspect ratio as we approach the 2D regime which is not present in our 3D single beam optical dipole trap and is absent in any 3D system. These first results indicate that we have a quasi-2D Fermi gas. The main experimental results are the first indications of a confinement induced resonance in a 6Li Fermi gas through the measurement of the cloud width in the tightly confined direction after a fixed expansion time. We observe a significant deviation in the transverse width of the 2D interacting Fermi gas from the 3D case due to the increased tight confinement. The confinement induced resonance should appear when the 3D scattering length approaches the transverse harmonic oscillator length, az = p~/mωz, where m is the mass and ωz is the tight transverse trapping frequency. For our experimental parameters this leads to a 3D scattering length of 1.03 μm, where the observation is at 819(1) G. In the weakly confined direction there is no such increase in the width due to the free motion in the xy plane. Instead the cloud width across the Feshbach resonance increases monotonically, similar to that of a 3D gas and as described theoretically for a quasi-2D Fermi gas by Lin et al. [1].
We investigate theoretically the transition from two to three dimensions in a weakly interacting ultracold Fermi gas. This transition was recently observed by Dyke et al. [Phys. Rev. Lett. 106, 105304 (2011)] in a cloud of fermionic 6{}^{6}6Li atoms trapped in an oblate harmonic potential. We calculate the density distribution and cloud size of the Fermi cloud by numerically solving three-dimensional Hartree-Fock mean-field equations. Local density approximation is used to ease the calculations in the weakly confined radial direction. Our theoretical results are shown to be in good agreement with the experimental measurements.
We consider a two-component gas of fermionic atoms confined to a quasi-two-dimensional (quasi-2D) geometry by a harmonic trapping potential in the transverse direction. We construct a mean-field theory of the BCS-BEC crossover at zero temperature that allows us to extrapolate to an infinite number of transverse harmonic oscillator levels. In the extreme BEC limit, where the confinement length exceeds the dimer size, we recover 3D dimers confined to two dimensions with weak repulsive interactions. However, even when the interactions are weak and the Fermi energy is less than the confinement frequency, we find that the higher transverse levels can substantially modify fermion pairing. We argue that recent experiments on pairing in quasi-2D Fermi gases [Y. Zhang et al., Phys. Rev. Lett. 108, 235302 (2012)] have already observed the effects of higher transverse levels.
We study equilibrium properties of a cold two-component Fermi gas confined in a quasi-one-dimensional trap of the transverse size ${l}_{\ensuremath{\perp}}$. In the dilute limit ($n{l}_{\ensuremath{\perp}}\ensuremath{\ll}1$, where nnn is the 1D density) the problem is exactly solvable for an arbitrary 3D fermionic scattering length aF{a}_{F}aF​. When ${l}_{\ensuremath{\perp}}/{a}_{F}$ goes from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$, the system successively passes three regimes: weakly interacting Fermi gas, hard core Bose gas, and weakly coupled Bose gas. The regimes are separated by two crossovers at ${a}_{F}\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}n{l}_{\ensuremath{\perp}}^{2}$. In conclusion, we discuss experimental implications of these results.
Ultracold Fermi gases subject to tight transverse confinement offer a highly controllable setting to study the two-dimensional (2D) BCS to Berezinskii-Kosterlitz-Thouless superfluid crossover. Achieving the 2D regime requires confining particles to their transverse ground state which presents challenges in interacting systems. Here, we establish the conditions for an interacting Fermi gas to display 2D kinematics. Transverse excitations are detected by measuring the transverse expansion rate which displays a sudden increase when the atom number exceeds a critical value N2D{N}_{2D}N2D​ signifying a density driven departure from 2D kinematics. For weak interactions N2D{N}_{2D}N2D​ is set by the aspect ratio of the trap. Close to a Feshbach resonance, however, the stronger interactions reduce N2D{N}_{2D}N2D​ and excitations appear at lower density.
We present an experimental investigation of collective oscillations in harmonically trapped Fermi gases through the crossover from two to three dimensions. Specifically, we measure the frequency of the radial monopole oscillation or breathing mode in highly oblate gases with tunable interactions. The breathing mode frequency is set by the adiabatic compressibility and probes the thermodynamic equation of state. In 2D, a dynamical scaling symmetry for atoms interacting via a δ potential predicts the breathing mode to occur at exactly twice the harmonic confinement frequency. However, a renormalized quantum treatment introduces a new length scale which breaks this classical scale invariance resulting in a so-called quantum anomaly. Our measurements deep in the 2D regime lie above the scale-invariant prediction for a range of interaction strengths providing evidence for the quantum anomaly and signifying the breakdown of an elementary δ-potential model of atomic interactions. By varying the atom number we can tune the chemical potential and see the breathing mode frequency evolve smoothly between the 2D to 3D thermodynamic limits.