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Physical Review A

Sum-rule analysis of radio-frequency spectroscopy of ultracold Fermi gas

Shizhong Zhang, Anthony J. Leggett

Published Mar 14, 2008• 30 citations
Publisher

Abstract

We carried out an analysis based on sum rules and determined the radio-frequency spectroscopy shift observed in Chin et al.'s experiment [C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Denschlag, and R. Grimm, Science 305, 1128 (2004)]. It is shown that such a shift can be interpreted as spin correlations peculiar to a BCS-type state. An analytical form for the shift is obtained which enables us to make quantitative comparisons with the experiment throughout the crossover. We also calculated the width of the resonance. An interesting consequence is noticed, which can possibly be tested by future experiments.

DOI
10.1103/physreva.77.033614

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We carried out an analysis based on sum rules and determined the radio-frequency spectroscopy shift observed in Chin et al.'s experiment [C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Denschlag, and R. Grimm, Science 305, 1128 (2004)]. It is shown that such a shift can be interpreted as spin correlations peculiar to a BCS-type state. An analytical form for the shift is obtained which enables us to make quantitative comparisons with the experiment throughout the crossover. We also calculated the width of the resonance. An interesting consequence is noticed, which can possibly be tested by future experiments.
The last chapter focuses on radio-frequency spectroscopy of multi component fermion gases. Rf experiments of 6Li and 40K are first reviewed. We show that rf experiments essentially measure spin correlation functions, and internal symmetries of the interparticle interactions constrain the rf spectrum through sum rules. We self-consistently generate the spin correlation functions for 6Li within Hartree-Fock-BCS approximation. Our calculation reproduces the mean field shift observed in experiments for normal gases, and confirms that a second peak detected at higher frequencies in the rf spectrum at sufficiently low temperatures arises from paired fermions. We define a mean frequency shift by the sum rules, and show that it is proportional to the square of BCS pairing gap. The discrepancy between the mean shift and the peak frequency shift observed in experiment for the condensed phase lies in the broad width of the rf spectrum function.
We present a theoretical interpretation of radio frequency (RF) pairing gap experiments in trapped atomic Fermi gases, over the entire range of the BCS--Bose-Einstein condensation (BEC) crossover, for temperatures above and below Tc{T}_{c}Tc​. Our calculated RF excitation spectra, as well as the density profiles on which they are based, are in semiquantitative agreement with experiment. We provide a detailed analysis of the physical origin of the two different peak features seen in RF spectra, one associated with nearly free atoms at the edge of the trap, and the other with (quasi-)bound fermion pairs.
Radio-frequency techniques were used to study ultracold fermions. We observed the absence of mean-field "clock" shifts, the dominant source of systematic error in current atomic clocks based on bosonic atoms. This absence is a direct consequence of fermionic antisymmetry. Resonance shifts proportional to interaction strengths were observed in a three-level system. However, in the strongly interacting regime, these shifts became very small, reflecting the quantum unitarity limit and many-body effects. This insight into an interacting Fermi gas is relevant for the quest to observe superfluidity in this system.
In this paper we present an overview of radio frequency (RF) spectroscopy in the atomic Fermi superfluids. An ultimate goal is to suggest new directions in the cold gas research agenda from the condensed matter perspective.Our focus is on the experimental and theoretical literature of cold gases and photoemission spectroscopy of the cuprates particularly as it pertains to areas of overlap. This paper contains a systematic overview of the theory of RF spectroscopy, both momentum integrated and momentum resolved. We discuss the effects of traps, population imbalance, final state interactions over the entire range of temperatures and compare theory and experiment. We show that this broad range of phenomena can be accomodated within the BCS-Leggett description of BCS-BEC crossover and that this scheme also captures some of the central observations in photoemission experiments in the cuprates. In this last context, we note that the key themes which have emerged in cuprate photoemission studies involve characterization of the fermionic self energy, of the pseudogap and of the effects of superconducting coherence (in passing from above to below the superfluid transition temperature, TcT_cTc​).These issues have a counterpart in the cold Fermi gases and it would be most useful in future to use these atomic systems to address these and the more sweeping question of how to describe that anomalous superfluid phase which forms in the presence of a normal state excitation gap.
This thesis studies radio-frequency spectroscopy of superfluid alkali Fermi gases. Radio-frequency and laser fields provide a coherent and well understood tool for manipulating atomic gases. However, new phenomena are expected and have already been seen experimentally when the alkali atoms become strongly interacting. Here we use a simple perturbative treatment of the atom-rf-field coupling for studying the spectroscopy of a strongly interacting gas. At ultracold temperatures fermionic atoms become paired in analogy to electrons in superconductors. The resulting binding energy, or the pairing gap, can be observed as a shift in the radio-frequency spectrum. With strong interactions, the fermions begin pairing already above the critical superfluid transition temperature. The radio-frequency spectroscopy grants a way to study these fluctuation effects quantitatively. The inhomogeneous trapping potential of the atom cloud also adds its own flavour, resulting in distinct mesoscopic effects in the radio-frequency spectra. Our theoretical models are in very good agreement with the experimental results. First order perturbation theory provides qualitatively correct lineshapes, and our generalised perturbative approach gives quantitatively correct magnitudes for the transfer rates. The inhomogeneity plays an important part also in more exotic systems in which the simple BCS-type pairing of atoms is prevented by polarising the gas. Superfluidity and phase separation in such spin-imbalanced gases has been experimentally observed. We have suggested the use of radio-frequency spectroscopy for observing exotic polarised superfluid states, present at the edges of the atom cloud.
The spin response functions measured in multicomponent fermion gases by means of rf transitions between hyperfine states are strongly constrained by the symmetry of the interatomic interactions. Such constraints are reflected in the spin fff-sum rule that the response functions must obey. In particular, only if the effective interactions are not fully invariant in SU(2)SU(2)SU(2) spin space, are the response functions sensitive to mean field and pairing effects. We demonstrate, via a self-consistent calculation of the spin-spin correlation function within the framework of Hartree-Fock-BCS theory, how one can derive a correlation function explicitly obeying the fff-sum rule. By contrast, simple one-loop approximations to the spin response functions do not satisfy the sum rule, except in special cases. As we show, the emergence of a second peak at higher frequency in the rf spectrum, as observed in a recent experiment in trapped 6Li^{6}\mathrm{Li}6Li, can be understood as the contribution from the paired fermions, with a shift of the peak from the normal particle response proportional to the square of the BCS pairing gap.