Detection of Spatial Correlations in an Ultracold Gas of Fermions

Spatial correlations are observed in an ultracold gas of fermionic atoms close to a Feshbach resonance. The correlations are detected by inducing spin-changing rf transitions between pairs of atoms. We observe the process in the strongly interacting regime for attractive as well as for repulsive atom-atom interactions and both in the regime of high and low quantum degeneracy. The observations are compared with a two-particle model that provides theoretical predictions for the measured rf transition rates.

Figure 1

Two-particle model of spin-changing rf transitions between interacting pairs of atoms in the $m_f=-5/2$ and $m_f=-9/2$ spin state. After the transition the spin state of the first atom is changed to $m_f=-7/2$. In addition, the relative momentum of the atoms at large internuclear separation can be changed by $2\delta k$ in this process if the applied radio frequency ${\nu }_{\mathrm{rf}}$ is detuned with respect to the $m_f=-5/2\to m_f=-9/2$ transition of bare atoms. The total kinetic energy of the free atoms is changed by an amount equal to the excess energy of the rf photon.

Figure 2

Atom pair signal for different magnetic-field values $B$ near the Feshbach resonance. The fraction of atoms transferred from the $m_f=-5/2$ into the $m_f=-7/2$ spin state is measured by spin-selective time-of-flight absorption imaging after an rf pulse with a Rabi frequency of $\Omega =2\pi \times 32\mathrm{kHz}$ is applied for $100\mu \mathrm{s}$. For each $B$, ${\nu }_{\mathrm{rf}}$ is chosen so that the rf field is kept at a constant large detuning $\Delta =-2\pi \times 100\mathrm{kHz}$ with respect to the bare atom transition. A calculated $6.3\%$ of off-resonantly transferred bare atoms has been subtracted. The solid line is a Lorentzian fit with an amplitude of $0.41\pm 0.03$ and a width of $1.2\pm 0.2\mathrm{G}$. It is shifted to the repulsive side of the resonance (dotted line) by $0.13\pm 0.04\mathrm{G}$. The temperature of the atoms in this measurement was $T=0.33\pm 0.06T_F$ and the peak density was $n_p=(1.2\pm 0.6)\times {10}^{13}{\mathrm{cm}}^{-3}$ per spin state. The dashed line shows a theoretical plot for a two-body multichannel scattering theory including the rf field. The calculated maximum fraction of atoms is 0.25 and the width is 0.79 G.

Figure 3

Fraction of atoms transferred from the $m_f=-5/2$ into the $m_f=-7/2$ spin state versus the duration of the applied rf pulse. This measurement is performed on the attractive side of the resonance at a magnetic field of $B=224.30\mathrm{G}$, with an rf detuning of $\Delta =-2\pi \times 100\mathrm{kHz}$. The solid line is a fit to an exponentially saturating growth curve. The curve saturates at a fraction of $0.49\pm 0.03$ with a time constant of $61\pm 8\mu \mathrm{s}$.

Figure 4

Increase of the kinetic energy $\delta E_{\mathrm{kin}}$ of the transferred atoms versus the frequency ${\nu }_{\mathrm{rf}}$ of the applied $100\mu \mathrm{s}$ rf pulse, on the attractive side of the resonance. The solid line is a linear fit to the data yielding a slope of $-0.59\pm 0.05$. This result is close to the expected value for an atom pair process. The vertical line indicates the expected resonant rf transition frequency for bare atoms at ${\nu }_{\mathrm{0}}=50.882\pm 0.005\mathrm{MHz}$, neglecting mean-field shifts that are on the order of $-1\mathrm{kHz}$. The measured zero crossing at ${\nu }_{\mathrm{rf}}=50.893\pm 0.008\mathrm{MHz}$ agrees fairly well with this value.

Figure 5

Fraction of atoms transferred by an $100\mu \mathrm{s}$ rf pulse with a detuning of $\Delta =-2\pi \times 100\mathrm{kHz}$ for various temperatures $T$ on the attractive side of the resonance ($B=224.37\mathrm{G}$). No explicit temperature or degeneracy dependence is observed. The data points at low temperature show a density dependence, where $n_p=2.8\times {10}^{12}{\mathrm{cm}}^{-3}$ (triangle), $5.0\times {10}^{12}{\mathrm{cm}}^{-3}$ (square), and $1.2\times {10}^{13}{\mathrm{cm}}^{-3}$ (open circle). The solid points have an average density of $n_p=5\times {10}^{12}{\mathrm{cm}}^{-3}$.