Spin noise spectroscopy to probe quantum states of ultracold fermionic atom gases

We theoretically demonstrate that optical measurements of electron spin noise can be a spectroscopic probe of the entangled quantum states of ultracold fermionic atom gases and unambiguously reveal the detailed nature of the underlying interatomic correlations. Different models of the effective interatomic interactions predict entirely new sets of resonances in the spin noise spectrum. Once the correct effective interatomic interaction model is identified, the detailed noise line shapes of the spin noise can be used to constrain this model. We estimate the magnitude of spin noise signals expected in ultracold fermionic atom gases via noise measurements in classical alkali vapors, which demonstrate the feasibility of this approach.

Figure 1

(Color online) (Top) Zeeman structure of ${}^{40}\mathrm{K}$ $(I=4)$. In the two-level theoretical model of the condensate, only the two lowest hyperfine states are occupied. In the three-level model [6], states 1, 2, and 18 are occupied. (Bottom) The one-atom matrix elements $\mid ⟨i\mid {\sigma }_z\mid j⟩\mid$, which determine the spin-response function $S\left(\omega \right)$ [see Eq. (4)], and corresponding frequencies, at the magnetic field of the Feshbach resonance $(B=202\mathrm{G})$. In the two-level model, lines 17–18 and 3–18 are forbidden, and line 1–2 has zero strength. Lines 3–18 and 2–17 are almost degenerate.

Figure 2

(Color online) Calculated line shapes for occupied-to-unoccupied transitions in the two-level model, as a function of the dimensionless parameter $\eta ={\left(k_{F}a\right)}^{-1}$ (where $k_F$ is the Fermi momentum and $a$ is the $s$-wave scattering length). Occupied-to-occupied transitions have zero strength in the two-level model [13]. In an ultracold gas of potassium atoms with a density $\left({\rho }_0\right)$ of ${10}^{13}{\mathrm{cm}}^{-3}$ the Fermi energy is ${\epsilon }_F\approx 5.6\mathrm{kHz}$.

Figure 3

(Color online) Examples of line shapes in the three-level model discussed in Ref. 6 for ${}^{40}\mathrm{K}$: $S^{1\to 2}$ and $S^{1\to 18}$ transitions, respectively. The line shapes depicted in panel (a) have no correspondent in the two-level model, whereas the line shapes depicted in panel (b) correspond to the line shapes shown in Fig. 2. The intensity reduction by an order of magnitude of the lines in the three-level model with respect to the case of the two-level model is a direct consequence of the sum rule obeyed by $S\left(\omega \right)$, $\int d\omega S\left(\omega \right)=2\pi$, i.e., the presence of more lines in the noise spectrum leads to a reduced intensity. Finally, the different energy scale of the line shapes in panels (a) and (b) reflect the long-versus short-range nature of the correlations involving the open and closed channels, respectively (see Ref. 6 for further discussions).

Figure 4

Spectra of spin noise, measured via Faraday rotation, in a ${}^{41}\mathrm{K}$ vapor at $184°\mathrm{C}$ (density $N=7.3\times {10}^{13}{\mathrm{cm}}^{-3}$). Peaks correspond to $\Delta m_F=\pm 1$ transitions. The probe laser $\left(4\mathrm{mW}\right)$ is detuned $100\mathrm{GHz}$ from the D1 transition ($4S_{1∕2}\text{−}4P_{1∕2}$; $770\mathrm{nm}$), and $B_{\perp }=25.6\mathrm{G}$. The vapor is contained in a $10\text{−}\mathrm{mm}$-long cell, and the laser beam diameter (to $1∕e$ peak intensity) is $65\mu \mathrm{m}$. The noise data are shown in units of voltage $\left(\mathrm{nV}{\mathrm{Hz}}^{-1∕2}\right)$, and Faraday rotation fluctuations $\left(\mathrm{nrad}{\mathrm{Hz}}^{-1∕2}\right)$. The white-noise floor is primarily from photon shot noise and amplifier noise. The discrete peaks show noise from electron-spin fluctuations and have integrated values of 1.3, 2.3, 2.6, and $2.1\mu \mathrm{rad}$. Similar signal magnitudes are expected in ultracold atom systems (see the text).