Many-Body Treatment of the Collisional Frequency Shift in Fermionic Atoms

Recent experiments have measured collisional frequency shifts in polarized fermionic alkaline-earth atoms using ${}^{1}S_0\mathrm{\text{−}}{}^{3}P_0$ Rabi spectroscopy. Here, we provide a first-principles nonequilibrium theoretical description of the interaction frequency shifts starting from the microscopic many-body Hamiltonian. Our formalism describes the dependence of the frequency shift on excitation inhomogeneity, interactions, temperature, and many-body dynamics, provides a fundamental understanding of the effects of the measurement process, and explains the observed density shift data.

Figure 1

$\delta {\omega }_{eg}^R/(2\pi )$ (i.e., CFS at resonant transfer) for different $a_{eg}^{-}$ (in Bohr radii, $a_0$). The time-averaged ground state fraction was varied by changing $t_f$. The solid lines were calculated by thermally averaging $N_g$ computed from Eq. (3), and the dots by a single realization of $\overrightarrow{\mathit{\nu }}$ randomly chosen out of those satisfying $\Delta \Omega (\overrightarrow{\mathit{\nu }})=⟨\Delta \Omega {⟩}_T$, $\overline{\Omega }(\overrightarrow{\mathit{\nu }})=⟨\overline{\Omega }{⟩}_T$, and $\overline{U}(\overrightarrow{\mathit{\nu }})=⟨\overline{U}{⟩}_T$. The dashed lines show Eq. (5) evaluated at thermally averaged parameters. Here, $N=7$, $T=1\mu \mathrm{K}$, $⟨\Delta \Omega {⟩}_T/⟨\overline{\Omega }{⟩}_T=0.2$ and $\overline{\Omega }=6\pi /(80\mathrm{ms})$. The inset shows $\int {\varphi }_{{\nu }_1}^2(x){\varphi }_{{\nu }_2}^2(x)dx$ in arbitrary units.

Figure 2

CFS for different pulse areas, $\mathcal{A}$. The final ground state fraction, $N_g(t_f)/N$, was varied by changing the detuning. Here four spatial modes at the corners of a square in the ${\nu }_x\mathrm{\text{−}}{\nu }_y$ plane, ${\mathit{\nu }}_1=(30,43)$, ${\mathit{\nu }}_2=(101,71)$, ${\mathit{\nu }}_3=(30,71)$, and ${\mathit{\nu }}_4=(101,43)$, were assumed with $N=2$ atoms occupying ${\mathit{\nu }}_1$ and ${\mathit{\nu }}_2$ at $t=0$. The dashed blue (solid red) lines, for $\overline{U}N/\overline{\Omega }=1$ (3.25) and $t_f=\pi /{\Omega }_0=7\mathrm{ms}$, were obtained using Eq. (2), which allows to populate ${\mathit{\nu }}_3$ and ${\mathit{\nu }}_4$. The empty squares (dots) were obtained using ${\widehat{H}}_S$ with an effective ${\Omega }_0^{\mathrm{eff}}={\Omega }_0/2$ and $t_f^{\mathrm{eff}}=\pi /{\Omega }_0^{\mathrm{eff}}$.

Figure 3

CFS predicted by our spin model for $T=1(3)\mu \mathrm{K}$ and $⟨\Delta \Omega {⟩}_T/⟨\overline{\Omega }{⟩}_T=0.15$ (0.42), determined from the Rabi flopping curves [see inset]. The blue hatched (pink shaded) area shows the uncertainty in CFS for $T=1(3)\mu \mathrm{K}$, assuming a variation in pulse area of $⟨\mathcal{A}{⟩}_T/\pi =1\pm 0.1$. We used ${\Omega }_0^{\mathrm{eff}}=4\pi /80\mathrm{ms}$, $t_f=\pi /{\Omega }_0^{\mathrm{eff}}$, $N=15$ [i.e., $\rho \sim {10}^{11}{\mathrm{cm}}^{-3}$], and $a_{eg}^{-}=200a_0$. The circles (triangles) show the $T=1(3)\mu \mathrm{K}$ experimental data of Ref. 2 at $t_f=80\mathrm{ms}$ and $\rho \sim {10}^{11}{\mathrm{cm}}^{-3}$. Inset: The shaded area shows Rabi flopping curves calculated at $⟨\Delta \Omega {⟩}_T/⟨\overline{\Omega }{⟩}_T=0.15$ for $a_{eg}^{-}=0-600a_0$. The dashed black and solid blue lines correspond to two specific scattering lengths, $a_{eg}^{-}=0$ and $200a_0$, respectively. The dotted purple line is for $⟨\Delta \Omega {⟩}_T/⟨\overline{\Omega }{⟩}_T=0.05$ and $a_{eg}^{-}=0$, while the blue circles are experimental data points from Ref. 2.