Measurements of spin properties of atomic systems in and out of equilibrium via noise spectroscopy

We explore the applications of spin noise spectroscopy (SNS) for detection of the spin properties of atomic ensembles in and out of equilibrium. In SNS, a linearly polarized far-detuned probe beam on passing through an ensemble of atomic spins acquires the information of the spin correlations of the system which is extracted using its time-resolved Faraday-rotation noise. We measure various atomic, magnetic and sub-atomic properties as well as perform precision magnetometry using SNS in rubidium atomic vapor in thermal equilibrium. Thereafter, we manipulate the relative spin populations between different ground state hyperfine levels of rubidium by controlled optical pumping which drives the system out of equilibrium. We then apply SNS to probe such spin imbalance nonperturbatively. We further use this driven atomic vapor to demonstrate that SNS can have better resolution than typical absorption spectroscopy in detecting spectral lines in the presence of various spectral broadening mechanisms.

The measured Faraday rotation fluctuation θ F (t)θ F (0) (θ F (t) is the Faraday rotation angle at time t) is a direct probe of the magnetization fluctuation M z (t)M z (0) of the system in thermal equilibrium and its Fourier transform to spectral frequency ν is the power spectral density P (ν) of the spin noise. Therefore, where we have used Eq. 1 in the last line. So, the SN power spectrum has a Lorentzian lineshape centred at ν L in frequency domain (refer to the peaks in the SN amplitude spectrum P (ν > 0) in Fig. 1(b) for 87 Rb or 85 Rb) and its full width at half maxima (FWHM) is proportional to 1/T 2 .
The energy E F,m F of hyperfine F -levels for alkali atoms in an arbitrary magnetic field B ⊥ has an exact expression following Breit and Rabi [33], where h∆ hf , g I and m F are the zero-field hyperfine separation between the levels F = I + 1 2 and F = I − 1 2 , the nuclear g-factor and the magnetic quantum number, respectively. Here, x = (g J − g I )µ B B ⊥ /(h∆ hf ) where g J is the Lande g-factor and the nuclear spin I = 3/2 for 87 Rb.
Since, the SNS detects the spin coherences between different Zeeman sub-levels ( m F = ±1), the frequencies of different magnetic resonance peaks have a nonlinear dependence on B ⊥ [34].
The integrated SN power over frequency, χ ≡ dνP (ν > 0), depends on the probe detuning as χ ∝ δ −2 p [17] and is symmetric about the atomic resonance frequency for a far-detuned probe beam (where δ p Γ, Γ being the width of the absorption spectra). However, this integrated SN power χ becomes asymmetric over δ p for δ p ≈ 0 due to the non-vanishing coherences between the ground and excited state hyperfine levels of the atoms [35]. This asymmetry in χ is shown later in this paper in Fig. 8(b,d). The asymmetry in χ also depends on the homogeneous and inhomogeneous broadening present in the medium [24,35].

III. EXPERIMENTAL SET-UP AND METHODS
The schematic of the experimental set-up is shown in Fig. 1  operates in a mode-hop free regime. The frequency ν p of the probe beam is measured using a commercial wavelength meter (HighFinesse, model-WSU2) with a relative accuracy of ± 1 MHz.
To transfer atom from F = 1 to F = 2 The two ground state hyperfine levels (F = 1 and F = 2) are separated by ∼ 6.8 GHz. The probe laser frequency νp is detuned by δp from the F = 2 → F = 3 transition, i.e., where α = (g J − g I )/(g J + 3g I ). In (1, 0) ↔ (1, −1)) in a series of measurements. equilibrium systems where we use optical fields to manipulate spin populations in the different ground state hyperfine levels. Recently, the SNS was employed to detect couplings and correlations between different spin coherences in a non-equilibrium atomic vapor [37]. In the experiment in [37], the Zeeman sub-levels of the ground state hyperfine levels of 41 K are driven by a weak radio frequency magnetic field which brings the vapor out of equilibrium. In contrast, we apply an optical control beam between the ground state hyperfine levels and the excited state hyperfine levels to drive as well as control the spin populations. The control beam is linearly polar-ized and almost co-propagating with the probe beam.
In our experiment, the control beam is derived from an independent tunable external cavity diode laser. The Rb atoms in the vapor cell are optically pumped to the desired ground state hyperfine levels by tuning the frequency ν c and the intensity I c of the pump laser. Substantial modifications of the SNS signals are observed depending upon the relative ground state hyperfine level populations of the vapor.

A. Detection of spin imbalance
Here, we implement the off-resonant SNS to probe the spin imbalance in an optically driven system without applying further perturbation [38]. In the absence of the pump beam, six spin coherences are seen in the SN spectrum in Fig. 7(b) as is expected from an ensemble of atoms with population in both the ground state hyperfine levels (F = 1, 2). On setting the frequency ν c of the control beam on resonance to the F = 1 → F = 2 transition of 87 Rb (see Fig. 2), a fraction of atoms is pumped out of the ground F = 1 level depending on the intensity I c of the pump. This is evident in Fig. 7(a) where only four SN peaks related to F = 2, m F = ±1 are observed at the highest I c . On the other hand, when the atoms are pumped out of the ground F = 2 level using a pump beam resonant with the F = 2 → F = 2 transition, the SN spectrum reduces to two peaks related to F = 1, m F = ±1 spin coherences. This is shown in Fig. 7(c) for different pump beam intensities. This clearly shows that the spin populations in different ground state hyperfine levels is reflected in the SN spectra. Such relatively non-invasive detection of spin states in a non-equilibrium atomic system may find applications in atom interferometry [1], atomic clocks [3] and gravimetry [39].

B. Resolving spectral lines
The enriched 87 Rb vapor cell used in our experiments contained a buffer gas (neon), and thus the conventional absorption spectra that we measure suffers broadening mainly due to homogeneous pressure broadening [40][41][42] and modestly due to inhomogeneous Doppler broadening.
The transmission of the probe through the atomic vapor at 90 • C is studied in the absence and presence of an optical control beam as shown in Fig. 8(a,c). The detuning δ p of the probe beam was varied over a wide range (−10 GHz to 12 GHz) which covers both the F = 2 → F and F = 1 → F transition lines. In the absence of a pump beam in Fig. 8(a), the probe transmission as a function of δ p shows a single dip situated between the transition lines. The integrated SN power χ from the Rb vapor also shows a single dip in Fig. 8(b) in the absence of optical pumping. Thus, both the absorption spectroscopy and SNS fail to detect F = 2 → F and F = 1 → F transition lines separately. Nevertheless, the dip in χ seems to indicate a red-detuned F = 2 → F transition as expected in the presence of neon buffer gas [42].
In the case of an optically pumped atomic ensemble, the probe transmission (Fig. 8(c)) can detect the above two optical transitions independently. In Fig. 8(d), we show the integrated SN power χ with the probe detuning δ p when the atoms are optically pumped to either F = 2 or F = 1 level by a pump intensity I c > 50I sat . We  observe a dip in each integrated SN power near F = 2 → F or F = 1 → F transition lines. Therefore, the SNS can also be used to resolve the spectral lines in a driven atomic system [24]. Moreover, SNS has a better resolution (around three times in Fig. 8(d) than Fig. 8(c)) over the absorption spectroscopy [24].
We can also detect these transitions by tuning the frequency ν c of the pump beam instead of the probe beam.
Here we keep the probe detuning δ p fixed at −10 GHz from F = 2 → F transition (and around −16.8 GHz de- tuned from F = 1 → F transition). Therefore, most of the contribution in the SN signal comes from the F = 2 level. We tune the frequency ν c of the pump beam from −10 GHz to 10 GHz around F = 2 → F = 3 transition. We plot the integrated SN power χ as a function of the pump beam detuning in Fig. 9, and observe a clear dip near F = 2 → F transition and a prominent peak near F = 1 → F transition. Therefore, unlike conventional spectroscopy, in the case of SNS, we have the freedom to scan the pump beam for detecting the spectral lines instead of applying the pump beam at a particular known frequency as in Fig. 8(c,d). This, we believe, will be of particular advantage, when we wish to probe local environment-induced energy level shifts, or in resolving ground state levels in complex molecular and condensed matter systems where one has incomplete knowledge of energy levels.

VI. CONCLUSION AND OUTLOOK
We explore the SNS technique in atomic vapor of Rb in thermal equilibrium as well as in a system driven out of equilibrium by optical pumping. Optical pumping is a commonly used technique in atomic and optical science and technology. To the best of our knowledge, the present study is the first implementation of SNS in an optically pumped atomic system. Our effort to combine the SNS and the optical pumping has the potential for use in magnetometry with alkali atoms [43]. There are important applications of such precision magnetometry, e.g., in cold atom experiments where narrow Feshbach resonances [44] are used extensively with the resonances occurring at magnetic fields ranging from a few Gauss to a few hundred Gauss. In those experiments, measuring the external fields with ultra-high precision will be hugely beneficial in fixing the interaction energy scale.
We also extend the applicability of SNS in precision measurements of various atomic, nuclear and magnetic properties in equilibrium systems.
The relatively non-perturbative nature of SNS makes it a versatile non-invasive detection technique which can be utilized in a wide range of physical systems in atomic, molecular and condensed matter systems. Recently, some measurements of spin polarization in ultracold atoms via Faraday rotation of a far-detuned probe beam were carried out as a non-destructive imaging technique [45][46][47][48]. We are interested to further implement SNS using Faraday rotation noise in ultracold atoms and Bose-Einstein condensates where it may have significant application in quantum non-demolition measurements. However, acquiring sufficient time-resolved Faraday-rotation noise from such ultracold atomic systems poses a challenge.