High-Sensitivity rf Spectroscopy of a Strongly Interacting Fermi Gas

Constantine Shkedrov, Yanay Florshaim, Gal Ness, Andrey Gandman, and Yoav Sagi

Phys. Rev. Lett. 121, 093402 – Published 30 August 2018

Abstract

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 ${}^{40}K$ .

Received 7 March 2018
Revised 25 June 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.093402

Physics Subject Headings (PhySH)

Fermi gases Radio frequency techniques

Feshbach resonance

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Constantine Shkedrov, Yanay Florshaim, Gal Ness, Andrey Gandman, and Yoav Sagi^*

Physics Department, Technion—Israel Institute of Technology, Haifa 32000, Israel

^*yoavsagi@technion.ac.il

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Issue

Vol. 121, Iss. 9 — 31 August 2018

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Images

Figure 1
High sensitivity rf spectroscopy. (a) A gas of ${}^{40}K$ atoms is prepared in a balanced mixture of states $| 1 ⟩ = | F = 9 / 2$ , $m = - 9 / 2 ⟩$ and $| 2 ⟩ = | 9 / 2, - 7 / 2 ⟩$ . (b) We apply a pulse of rf radiation with a frequency $ν$ close to 47 MHz. This pulse transfers a small fraction of the atoms to state $| 3 ⟩ = | 9 / 2, - 5 / 2 ⟩$ . (c) Next, we transfer all the atoms in $| 3 ⟩$ to state $| 4 ⟩ = | 7 / 2, - 3 / 2 ⟩$ using MW sweep whose central frequency is typically around 1.6 GHz. (d) Then the optical trap is turned off and a quadrupole magnetic field is ramped on. Because of the different signs of their magnetic dipole moments, atoms at $| 1 ⟩$ , $| 2 ⟩$ , and $| 3 ⟩$ are antitrapped while the atoms at $| 4 ⟩$ are trapped by this field. (e) After a waiting time, the MOT laser beams are switched on. The fluorescence signal of the atoms captured in the MOT is recorded with a sensitive CMOS camera. The detection sensitivity can, in principle, reach a single atom, although in our apparatus, it is limited to $\sim 5$ atoms due to background scattering from surrounding optical surfaces.
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Figure 2
rf transition rate, $Γ (ν)$ , taken at different magnetic fields in the vicinity of the Feshbach resonance at $B_{0} = 202.2 G$ (dashed line). The frequency axis is plotted relative to $ν_{0}$ —the transition frequency for noninteracting atoms, which is measured with a spin polarized gas. For clarity, only a range of 1 MHz is shown here. The solid line is the theoretical Feshbach molecule binding energy given by Eq. (2) using $a (B)$ with the calibration reported in Ref. [31]. The data shown here were taken at $B = 202.86$ , 202.207, 201.07, 200.446, 199.718, 199.006 G (the $1 σ$ uncertainty in the field is 8 mG).
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Figure 3
rf line shapes for three different interaction strengths in the BCS-BEC crossover [42]: $1 / k_{F} a = 0$ (unitarity), $1 / k_{F} a = 0.49$ (BEC), and $1 / k_{F} a = - 0.53$ (BCS). Linear scaling shows that the data follows a power law at high frequencies. The inset shows the power-law exponent extracted by fitting the tail of each data set with $c_{1} / ν^{n}$ , starting from $ν > 5 E_{F} / h$ for $B > B_{0}$ and $ν > E_{b} + 20 E_{F} / h$ for $B < B_{0}$ , where $E_{b}$ is given by Eq. (2). The error bars correspond to one $σ$ confidence interval of the fit parameter.
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Figure 4
Trap averaged contact, $C$ , versus the interaction strength $1 / k_{F} a$ , extracted by fitting the tail of the rf line shape with Eq. (1). The solid line is the theory of Ref. [24]. The $1 σ$ error bars include the combined uncertainties in the fitting parameter, magnetic field, and normalization of the line shape.
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Figure 5
The binding energy (squares) of the Feshbach molecules is extracted by fitting the rf line shape with the molecular spectral function given by Eq. (3). The inset shows a typical data set taken at $B = 199.371 (8)$ and the fit (solid line), which yields $E_{b} / h = 331 (2) kHz$ with $1 σ$ confidence level. The theory of Eq. (2) (dotted blue line) with the most recent calibration of the Feshbach resonance parameters [ $Δ = 7.04 (10) G$ , $B_{0} = 202.20 (2) G$ and $a_{b g} = 174 a_{0}$ [31, 46] ] shows an increasing systematic deviation from the data farther away from the resonance. We have verified that the mean-field shift is smaller than our experimental resolution. Using the previously measured values $a_{b g} = 169.7 a_{0}$ [52], $C_{6} = 3897$ a.u. [51], and $δ μ / h = 2.35 MHz / G$ [49] in a two-coupled-channels calculation based on the model of Refs. [49, 50], we obtain a new calibration: $B_{0} = 202.14 (1) G$ , $Δ = 6.70 (3) G$ (solid red line). Equation (2), with the new calibration, also fits the data well (dashed black line).
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