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arXiv:1111.0999 · Physical Review Letters

Clock Shift in a Strongly Interacting Two-Dimensional Fermi Gas

Christian B. Langmack, Marcus Barth, W. Zwerger, Eric Braaten

Published Feb 9, 2012• 49 citations
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Abstract

We derive universal relations for the rf spectroscopy of a two-dimensional Fermi gas consisting of two spin states interacting through an $S$-wave scattering length. The rf transition rate has a high-frequency tail that is proportional to the contact and displays logarithmic scaling violations, decreasing asymptotically like $1/({\ensuremath{\omega}}^{2}{ln}^{2}\ensuremath{\omega})$. Its coefficient is proportional to ${ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$, where ${a}_{2\mathrm{D}}$ and ${a}_{2\mathrm{D}}^{\ensuremath{'}}$ are the two-dimensional scattering lengths associated with initial-state and final-state interactions. The clock shift is proportional to the contact and to $\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$. If $|\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})|\ensuremath{\gg}1$, the clock shift arises as a cancellation between much larger contributions proportional to ${ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})$ from bound-bound and bound-free rf transitions.

DOI
10.1103/physrevlett.108.060402

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We derive universal relations for the rf spectroscopy of a two-dimensional Fermi gas consisting of two spin states interacting through an SSS-wave scattering length. The rf transition rate has a high-frequency tail that is proportional to the contact and displays logarithmic scaling violations, decreasing asymptotically like 1/(\ensuremathω2ln2\ensuremathω)1/({\ensuremath{\omega}}^{2}{ln}^{2}\ensuremath{\omega})1/(\ensuremathω2ln2\ensuremathω). Its coefficient is proportional to ln2(a2D\ensuremath′/a2D){ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})ln2(a2D\ensuremath′​/a2D​), where a2D{a}_{2\mathrm{D}}a2D​ and a2D\ensuremath′{a}_{2\mathrm{D}}^{\ensuremath{'}}a2D\ensuremath′​ are the two-dimensional scattering lengths associated with initial-state and final-state interactions. The clock shift is proportional to the contact and to ln(a2D\ensuremath′/a2D)\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})ln(a2D\ensuremath′​/a2D​). If ∣ln(a2D\ensuremath′/a2D)∣\ensuremath≫1|\mathrm{ln}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})|\ensuremath{\gg}1∣ln(a2D\ensuremath′​/a2D​)∣\ensuremath≫1, the clock shift arises as a cancellation between much larger contributions proportional to ln2(a2D\ensuremath′/a2D){ln}^{2}({a}_{2\mathrm{D}}^{\ensuremath{'}}/{a}_{2\mathrm{D}})ln2(a2D\ensuremath′​/a2D​) from bound-bound and bound-free rf transitions.
Universal relations that hold for any state provide powerful constraints on systems consisting of fermions with two spin states interacting with a large scattering length. In radio-frequency (rf) spectroscopy, the mean shift in the rf frequency and the large-frequency tail of the rf transition rate are proportional to the contact, which measures the density of pairs with small separations. We show that these universal relations can be derived and extended by using the short-time operator product expansion of quantum field theory. This is a general method for identifying aspects of many-body physics that are controlled by few-body physics.
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.
We show that strong pairing correlations in Fermi gases lead to the appearance of a gaplike structure in the rf spectrum, both in the balanced superfluid and in the normal phase above the Clogston-Chandrasekhar limit. The average rf shift of a unitary gas is proportional to the ratio of the Fermi velocity and the scattering length with the final state. In the strongly imbalanced case, the rf spectrum measures the binding energy of a minority atom to the Fermi sea of majority atoms. Our results provide a qualitative understanding of recent experiments by Schunck et al.
Low-dimensional ultracold gases are created in the laboratory by confining three-dimensional (3D) gases inside highly anisotropic trapping potentials. Such trap geometries not only provide access to simulating 1D and 2D physics, but also can be used to study how the character of excitations cross over to three dimensions at higher energy. In this work we study the signature in radio-frequency (rf) spectroscopy for both the 1D-to-3D and the 2D-to-3D crossovers, in spin-polarized Fermi gases. We solve the two-body scattering TTT matrix in the presence of strong harmonic confinement and use it to evaluate both the high-frequency rf transfer rate and the two-body bound-state energies. These general expressions are then compared both to the low-dimensional limit and to the 3D limit. We find that in order to understand the dimensional crossover for spin-polarized Fermi gases with ppp-wave interactions, one needs to take into account an emergent sss-wave interaction.
rf spectroscopy is one of the most powerful probing techniques in the field of ultracold gases. We report on a novel rf spectroscopy scheme with which we can detect very weak signals of only a few atoms. Using this method, we extended the experimentally accessible photon-energies range by an order of magnitude compared to previous studies. We directly verify a universal property of fermions with short-range interactions which is a power-law scaling of the rf spectrum tail all the way up to the interaction scale. We also determine, with high precision, the trap average contact parameter for different interaction strength. Finally, we employ our technique to precisely measure the binding energy of Feshbach molecules in an extended range of magnetic fields. These data are used to extract a new calibration of the Feshbach resonance between the two lowest energy levels of ^{40}K.
The contact is an important concept that characterizes the universal properties of a strongly interacting quantum gas. It appears in both thermodynamic (energy, pressure, etc.) and dynamic quantities (radio-frequency and Bragg spectroscopies, etc.) of the system. Very recently, the concept of contact was extended to higher partial waves; in particular, the ppp-wave contacts have been experimentally probed in recent experiments. So far, discussions on ppp-wave contacts have been limited to three dimensions. In this paper, we generalize the ppp-wave contacts to two dimensions and derive a series of universal relations, including the adiabatic relations, high-momentum distribution, virial theorem, and pressure relation. At the high-temperature and low-density limit, we calculate the ppp-wave contacts explicitly using virial expansion. A formula which directly connects the shift of the breathing-mode frequency and the ppp-wave contacts is given in a harmonically trapped system. Finally, we also derive the relationships between interaction parameters in three- and two-dimensional Fermi gases and discuss possible experimental realization of a two-dimensional Fermi gas with ppp-wave interactions.