The spin correlation function,, has a Fourier transform S(ω) that is proportional to the measured power spectrum of δθ F (t).A typical noise spectrum from rubidium vapor is shown in Fig. 1b, taken with the laser detuned 25 GHz from the D1 transition. The sharp peaks at frequencies Ω=869and 1303 kHz are due to random spin fluctuations which are precessing in the small 1.85G transverse magnetic field, effectively generating spontaneous spin coherences between ground-state Zeeman sublevels. These coherences precess with effective g- atomic ground state into two hyperfine F-levels with total spin and g-factor, where is the free electron g-factor. Thus, the nuclear spin of 2 ≅ J g 85 Rb (I=5/2) and 87 Rb (I=3/2) may be directly measured from spin fluctuations in thermal equilibrium. Noise spectra acquired near the D2 transition show similar peaks (inset, Fig. 1b), which move as expected with magnetic field. The 13 kHz measured width of these noise peaks indicates an effective transverse spin dephasing time ~100µs, much less than the known Rb spin lifetime (~1s), due largely to the transit time of atoms across the ~100 µm laser diameter. The spectral density of the spin noise is small --the 87 Rb peak in Fig. 1b = ∆That the off-resonant laser non-perturbatively probes spin fluctuations is evidenced in Fig. 2 (Fig. 3c,d Finally, we show that fluctuation correlation spectra can reveal detailed information about complex magnetic ground states arising from, e.g., nuclear magnetism and hyperfine interactions. Figure 4a shows the spin noise spectrum in 39 K.With increasing magnetic field, the noise peak broadens and eventually splits into 4 resolvable spin coherences. This is the well-known quadratic Zeeman effect, which originates in the gradual decoupling of the electron and nuclear spin (the hyperfine interaction) by the applied magnetic field. Within each hyperfine level (see Fig. 4b), the 2F+1 Zeeman levels become unequally spaced, resulting in 2F distinct coherences . The photodiode difference current is converted to voltage (transimpedance gain=40V/mA, or ~20V/mWatt of unbalanced laser power at 790 nm) and measured in a spectrum analyzer. Above a few kilohertz, the measured noise floor arises primarily from photon shot noise, which contributes ~175 nV/ Hz of spectrally flat (white) noise at a typical laser power of 200 µW. The photodiodes and amplifier contribute an additional 65 nV/ Hz of uncorrelated white noise. All noise spectral densities are root-mean-square (rms) values, and spectra were typically signal-averaged for 10-20 minutes. The accuracy of measured hyperfine constants and g-factors was imposed by the gauss resolution of the Hall bar magnetic field sensor. Measurement of inter-hyperfine spin coherence (Fig. 4d) was performed with a higher-bandwidth, lower gain amplifier (~0.70 V/mA), and a typical laser power of 3.5 mW. Unless otherwise stated, measurements of rubidium (potassium) vapor were performed with 250 (125) Torr of nitrogen buffer gas, which broadens the linewidth of the D1 and D2 optical tran...

We measure magnetic quantum oscillations in the underdoped cuprates YBa2Cu3O6+x with x = 0.61, 0.69, using fields of up to 85 T. The quantum-oscillation frequencies and effective masses obtained suggest that the Fermi energy in the cuprates has a maximum at p ≈ 0.11 − 0.12. On either side, the effective mass may diverge, possibly due to phase transitions associated with the T = 0 limit of the metal-insulator crossover (low-p side), and the postulated topological transition from small to large Fermi surface close to optimal doping (high p side).

Per the fluctuation-dissipation theorem, the information obtained from spin fluctuation studies in thermal equilibrium is necessarily constrained by the system's linear response functions. However, by including weak radio frequency magnetic fields, we demonstrate that intrinsic and random spin fluctuations even in strictly unpolarized ensembles can reveal underlying patterns of correlation and coupling beyond linear response, and can be used to study nonequilibrium and even multiphoton coherent spin phenomena. We demonstrate this capability in a classical vapor of (41)K alkali atoms, where spin fluctuations alone directly reveal Rabi splittings, the formation of Mollow triplets and Autler-Townes doublets, ac Zeeman shifts, and even nonlinear multiphoton coherences.