Dimer-projection contact and the clock shift of a unitary Fermi gas

Abstract

Understanding the dynamics of short-range correlations is a central challenge in strongly interacting Fermi gases. In ultracold gases, these correlations are quantified by the contact parameter, yet measurements to date have been limited to equilibrium systems or relatively slow, global dynamics. Here, we introduce a rapid spectroscopic technique based on projection of the interacting state onto an alternate scattering channel with a low-lying dimer state. We demonstrate contact measurements on the microsecond timescale -- faster than the inverse Fermi energy. Using $^{40}$K near a broad $s$-wave Feshbach resonance, we show that the strength of the dimer-projection feature scales proportionally with the contact parameter extracted from the high-frequency tail of radio-frequency spectroscopy, in agreement with coupled-channels calculations. Analysis of the spectra further reveals that the dimer feature provides the dominant contribution to the clock shift of the unitary Fermi gas, allowing the first experimental bound on this quantity. The observed deviations from universal predictions highlight the importance of multichannel effects. Our results open new avenues for studying contact correlators, hydrodynamic attractors, and quantum critical behavior.

Figures (6)

• Figure 1: Spectral regimes. (a) The dimensionless rf transfer rate $\widetilde{\Gamma} = 2 E_F \Gamma /(\pi \hbar \Omega_{23}^2 N)$ is shown versus rf detuning $\omega$ from the single-particle resonance in three regimes: dimer feature (green circles), near-resonant transfer (blue diamonds) and HFT transfer (orange squares). Lines are intended to be guides for the eye. (b) HFT transfer is shown versus frequency with a $\omega^{-3/2}$ fit as the solid orange line. Here, $U_t$ (dashed line) denotes the trap depth. Blue circles use a positive measure of $N_3$, whereas orange squares use loss from $N_2$, as described in the text. The approximate noise floor is denoted by a shaded region.
• Figure 2: Dimer feature. (a) The peak transfer frequency $\omega_d$ (points) at various magnetic fields $B$ is compared to models (lines) for $E_B/\hbar = \hbar \kappa^2/m$. (b) At $202\,$G, the lineshape $\widetilde{\Gamma} (\omega)$ of the dimer feature depends on the pulse time. For long pulses ($t_\mathrm{rf} = 640\,\mu\mathrm{s} \approx 50 \tau_F$, blue circles), the profile reveals the width of the continuum. The blue line has a width comparable to $E_F$, and serves as a guide to the eye. For short pulses ($t_\mathrm{rf} = 10\,\mu\mathrm{s} \approx 1.1 \tau_F$, red squares), the profile is determined by the rf pulse, and insensitive to $T$. The red line shows the $\mathrm{sinc}^2$ function described in the main text. (c) The fractional transfer at $\omega=\omega_d$ divided by $\Omega_{23}^2 t_\mathrm{rf}^2/4$ approaches a constant for short pulse times. In the short-pulse limit, this ratio gives the dimer weight $I_d$ through Eq. \ref{['eq:RabiDI']}. The data from (b) are highlighted as blue and red points. The blue line models the expected $I_d$ from convolving the long-pulse lineshape with $\mathrm{sinc}^2$ functions of various durations.
• Figure 3: Dimer spectral weight versus contact. (a) $\widetilde{C}$ (orange squares) measured at a single frequency in the HFT is shown at various $T/T_F$. A calculation of the harmonic-trap-averaged contact is shown as a dashed orange line, with a shaded band representing the systematic uncertainty of thermometry. Inset: examples of the linear-response calibration for $T/T_F \approx 0.3$ (upper) and $T/T_F \approx 0.6$ (lower). (b) Single-frequency measurements of the dimer spectral weight (green circles) are shown versus measured $\widetilde{C}$ from HFT. Data are compared to three model predictions: the universal z.r. limit (dash-dotted black), the SqW model with a finite effective range (dashed purple), and a molecular CC calculation (solid magenta).
• Figure S1: Effective-range effects on the $s$-wave dimer pole. (a) The dimer pole energy, $-E_B/h$ in the $13$ channel of $^{40}$K, versus field. The measurements at unitarity were taken at the $12$ resonance, indicated by the gray arrow, and $\sim$22 G below the $13$ resonance. (b) Finite-range effect on the dimer pole: $\kappa$ versus $r_{e}/a_S$, for both the SqW theory (red dashed line) and the T-matrix pole (green solid line). (c) $a_{13}/a_0$ versus magnetic field. (d) $r_{e,13}/a_0$ versus magnetic field. (e) $r_{e,13}/a_{13}$ versus magnetic field.
• Figure S2: Linear response calibration. (a) Dimer transfer plotted against rf power. (b) HFT transfer for various rf power detuned $100$ kHz from the 2-to-3 resonance (the detuning used in the HFT in Fig. 3). Solid lines are fits to the saturating transfer equation, and the dashed and dotted lines show extrapolated linear response. The color bar indicates the $T/T_F$ of the initial gas for each data set.