Rubidium atoms are widely used in, for example, atomic cooling, information storage, spectroscopy, and magnetometry [1,2]. Miniaturization of alkali vapor cells is important for many applications [3–6]. Atoms placed in an external magnetic field undergo shifts of the energy levels and changes in transition probabilities; therefore precise knowledge of the behavior of the atomic transitions is very important [7]. In the case of alkali atomic vapors, a sub-Doppler resolution is needed to study separately each individual atomic transition between hyperfine (hf) Zeeman sublevels of the ground and excited states (in the case of a natural mixture of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ the number of closely spaced atomic transitions can reach several tens). To eliminate the Doppler broadening, in some cases the selective reflection spectroscopy can be useful [8]. Also, the well-known saturation absorption (SA) technique allows one to reach the sub-Doppler resolution; however, in this case the complexity of the Zeeman spectra in a magnetic field arises primarily from the presence of strong crossover resonances, which are also split into many components. That is why the SA technique is applicable only for $B<10–15\mathrm{mT}$ [9]. Recently it was shown that a one-dimensional nanometric-thin cell (NTC) with the thickness of a Rb atomic vapor column $L=\lambda$, with $\lambda =794\mathrm{nm}$ being the wavelength of the laser radiation resonant with the Rb $D_1$ line, is a good tool to reach sub-Doppler spectral resolution. Spectrally narrow velocity-selective optical pumping (VSOP) resonances located exactly at the positions of atomic transitions appear in the transmission spectrum of NTC at the laser intensities $\sim 10\mathrm{mW}/{\mathrm{cm}}^2$ [4,6,10]. When NTC is placed in a weak magnetic field, the VSOPs are split into several components depending on the total angular momentum quantum numbers $F=I+J$, with amplitudes and frequency positions depending on the $B$ field, which makes it convenient to study separately each individual atomic transition. In this Letter we describe a simple and robust system based on NTC and permanent magnets, which allows the achieving of magnetic fields up to 0.7 T, sufficient to observe a hyperfine Paschen–Back (HPB) regime [11]. The magnetic field required to decouple the nuclear and electronic spins is given by $B\gg A_{hfs}/{\mu }_B$ $\sim 0.2\mathrm{T}$ for ${}^{87}\mathrm{Rb}$, and $\sim 0.07\mathrm{T}$ for ${}^{85}\mathrm{Rb}$, where $A_{\mathrm{hfs}}$ is the ground-state hyperfine coupling coefficient for ${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$ and ${\mu }_B$ is the Bohr magneton [12]. For such a large magnetic field the eigenstates of the Hamiltonian are described in the uncoupled basis of $J$ and $I$ projections $(m_J,m_I)$. In Fig. 1 six transitions of ${}^{85}\mathrm{Rb}$ labeled 4–9 and four transitions of ${}^{87}\mathrm{Rb}$ labeled 1–3 and 10 are shown in the case of ${\sigma }^{+}$ polarized laser excitation for the HPB regime.

 

Fig. 1. Diagram of ${}^{85}\mathrm{Rb}$ ($I=5/2$) and ${}^{87}\mathrm{Rb}$ ($I=3/2$) transitions for ${\sigma }^{+}$ laser excitation in the HPB regime. The selection rules: $\mathrm{\Delta }{m}_J=+1$; $\mathrm{\Delta }{m}_I=0$.

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The sketch of the experimental setup is shown in Fig. 2. The circularly (${\sigma }^{+}$) polarized beam of extended cavity diode laser (ECDL, $\lambda =794\mathrm{nm}$, $P_L=30\mathrm{mW}$, ${\gamma }_L<1\mathrm{MHz}$) resonant with the Rb $D_1$ line was directed at normal incidence onto the Rb NTC (2) with the vapor column thickness $L=\lambda =794\mathrm{nm}$. A typical example of a recent version of the NTC is described in [6]. The NTC was placed in a special oven with two openings. The temperature of the NTC side arm determining the density of Rb atoms was kept at $\sim 135°\mathrm{C}$, while for the window the temperature was higher by 20 °C (to prevent atomic vapor condensation on the NTC windows). The transmission signal was detected by a photodiode (4) and was recorded by a four-channel digital storage oscilloscope. To purify the initial linear polarization of the laser radiation, we used a polarizing beam splitter (PBS) followed by a $\lambda /4$ plate (1) to produce a circular polarization. The magnetic field was directed along the laser radiation propagation direction $\mathbf{k}$. About 30% of the pump power was branched to the reference unit with an auxiliary Rb NTC. The fluorescence spectrum of the latter at $L=\lambda /2$ was used as a frequency reference for $B=0$. The magnetic field was measured by a calibrated Hall gauge. An extremely small thickness of NTC is advantageous for the application of very strong magnetic fields with the use of permanent ring magnets (PRM) otherwise unusable because of strong inhomogeneity of the magnetic field: in NTC, the variation of the $B$ field inside the atomic vapor column is by several orders less than the applied $B$ value. The ring magnets are mounted on a $50\mathrm{mm}\times 50\mathrm{mm}$ cross-section Π-shaped holder made from soft stainless steel (see the inset of Fig. 2). Additional form-wounded Cu coils allow the application of extra $B$ fields (up to $\pm 0.1\mathrm{T}$). NTC is placed between PRMs. The linearity of the scanned frequency was tested by simultaneously recorded transmission spectra of a Fabry–Pérot etalon (not shown). The nonlinearity has been evaluated to be about 1% throughout the spectral range. The imprecision in the measurement of the absolute $B$-field value is $\pm 5\mathrm{mT}$. The recorded transmission spectrum of the Rb NTC with $L=\lambda$ for ${\sigma }^{+}$ laser excitation and $B=0.605\mathrm{T}$ is shown in Fig. 3. The VSOP resonances labeled 1–10 demonstrate increased transmission at the positions of the individual Zeeman transitions. In the case of HPB the energy of the ground $5S_{1/2}$ and upper $5P_{1/2}$ levels for the Rb $D_1$ line is given by the following simple formula [11]:

 

Fig. 2. Sketch of the experimental setup. ECDL, diode laser; FI, Faraday isolator; $1-\lambda /4$ plate; 2, NTC in the oven; PBS, polarizing beam splitter; 3, permanent ring magnets; 4, photodetectors; 5, stainless steel Π-shape holder (shown in the inset). The longitudinal size of the NTC including the oven is 8 mm.

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Fig. 3. Transmission spectrum of Rb NTC with $L=\lambda$ for $B=0.605\mathrm{T}$ and ${\sigma }^{+}$ laser excitation. Well-resolved VSOP resonances located at atomic transitions are labeled 1–10 (six transitions, 4–9, belonging to ${}^{85}\mathrm{Rb}$, and four transitions, 1, 2, 3, and 10, belonging to ${}^{87}\mathrm{Rb}$). The change in probe transmission is $\mathrm{\Delta }T=4\%$. The lower gray curve is the fluorescence spectrum of the reference NTC with $L=\lambda /2$, showing the positions of ${}^{87}\mathrm{Rb}$, $F_g=1\to F_e=1$, two transitions for $B=0$, labeled as $1-1^{\prime }$ and $1-2^{\prime }$ (all frequency shifts are measured from $F_g=1\to F_e=2$). Frequency interval between resonances 1 and 2 belonging to ${}^{87}\mathrm{Rb}$, and the interval between resonances 8 and 9 belonging to ${}^{85}\mathrm{Rb}$ are 1.7 GHz and 0.6 GHz, correspondingly, while the interval between resonances 3 (${}^{87}\mathrm{Rb}$) and 7 (${}^{85}\mathrm{Rb}$) is 0.1 GHz.

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The values for nuclear ($g_I$) and fine structure ($g_J$) Landé factors and hyperfine constants $A_{\mathrm{hfs}}$ are given in [13]. The dependence of the frequency shifts on the magnetic field for components 4–9 (${}^{85}\mathrm{Rb}$) is shown in Fig. 4 (solid lines: HPB theory; symbols: experiment, inaccuracy does not exceed 2%). The similar dependence for ${}^{87}\mathrm{Rb}$ (components 1–3 and 10) is shown in the inset of Fig. 4. The HPB regime condition is fulfilled better for $B>0.6\mathrm{T}$. As it is seen from Eq. (1), and also confirmed experimentally, the dependence slope is the same for all the transition components of both ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$: [$g_J(5S_{1/2})m_J+g_J(5P_{1/2})m_J$] ${\mu }_{B}B=18.6\mathrm{MHz}/\mathrm{mT}$ (as $g_I\ll g_J$ we ignore its contribution). The onset of this value is indicative of the Rb $D_1$ line HPB regime. Note that in our previous study for $B\sim 20\mathrm{mT}$ [14] and under the same conditions we observed 32 transitions, as opposed to 10 remaining in the HPB regime.

 

Fig. 4. Magnetic field dependence of the frequency shifts for the transition components labeled 4–9 (${}^{85}\mathrm{Rb}$) and 1–3, 10 (${}^{87}\mathrm{Rb}$, in the inset). Solid lines: theory; symbols: experiment. The inaccuracy is $\le 2\%$ for components 4–9, 1, and 10, and $\le 5\%$ for components 2 and 3. The larger error for 3 and 2 is caused by the closely located strong transition 7 and the location on the transmission spectrum wing, respectively.

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Rb NTC could be implemented for mapping strongly inhomogeneous magnetic fields by local submicron spatial resolution. Particularly, for the $0.1\mathrm{T}/\mathrm{mm}$ gradient, the displacement of NTC by 5 μm results in a 10 MHz frequency shift of VSOP resonance, which is easy to detect. Also the development of a frequency reference based on NTC and PRM, which is $B$-field tunable in the range over 10 GHz, is of high interest. The preceding studies and techniques can be successfully implemented also for HPB studies of $D$ lines of Na, K, Cs, and other atoms.