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Abstract
Optically-dense alkali-metal vapor cells with low optical losses are widely used in quantum precision measurements, yet atomic behaviors in coated cells still need further study. We developed an octadecyltrichlorosilane (OTS) anti-relaxation and dielectric anti-reflection coated multipass optical cell filled with rubidium vapor at high density, and investigated the spin relaxation as well as the spin noise of the atomic ensembles. High contrast spin noise spectrum was observed and we find that the quantum coherence lifetime is equal to the classical transverse spin relaxation time. Number density measurements give the spin-exchange possibility of 14% between rubidium atoms trapped in the coating.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultra-sensitive magnetometry has made essential help to fundamental studies [1–8], biomedical research [9–14], practical applications such as materials characterization [15–17], field stablization [18], and investigations of the earth or space science [19–21]. To realize an atomic magnetometer with both ultra-high sensitivity and small sensor size, one of the effective ways is the employment of multipass cells that can significantly increase the probe beam’s optical path, which has been used in many research and applications such as optical absorption spectroscopy of weak atomic transitions [22], optical data storage [23], molecular Faraday rotation [24] and so on. They have been used to generate very large optical rotations [25–28] and substantially help vector or scalar magnetometers achieve ultra high sensitivity as 0.54 fT/Hz$^{1/2}$ and surpass the quantum limit set by spin-exchange collisions [29]. The sensor’s size can possibly be minimized while high sensitivity can be retained, which promotes applications with sensor arrays of, for example, detecting nuclear quadrupole resonance [30], magnetoencephalography(MEG) operating with warm atoms to obtain high resolution 2-D or 3-D magnetic field images [12,14,31–34] as a beneficial supplement to electroencephalogram (EEG) [13], and so on.
The state-of-the-art magnetometers can be performed as gradiometers working in either spin-exchange relaxation-free (SERF) regime [35–37] or scalar mode [29,38] to eliminate the common-mode magnetic field noise. It is expected that smaller cell volume brings better spatial resolution [39]. But since the atom’s spin phase correlation with other atoms is disarranged while being influenced by the local random magnetic fields upon each collision and surface dwelling [40], the spin lifetime and the magnetometer’s sensitivity will significantly degrade if the cell volume is too small [35]. Moreover, vapor cells with high surface to volume ratio, for example the microfabricated vapor cells, will have much higher spin relaxation rate which would make the performance even worse [41]. The side effect of using bare glass cell wall is that alkali-metal atoms may react with the sodium oxide contained in glass [42].
Anti-relaxation wall coatings [43] and buffer gases [44] are often used to reduce the depolarizing wall collisions or dephasing interactions with magnetic impurities in the cell wall [45,46]. However, it is known that buffer gases will broaden the optical resonance signal [47–49], and the presence of buffer gases can cause temperature-dependent frequency shifts [50], interferences of atomic spin alignment [51], and even gas contamination [52]. The use of anti-relaxation coatings is effective to reduce wall relaxation and minimize the amount of buffer gases, of which another benefit is that faster atomic velocity may result in motional averaging of magnetic field inhomogeneties and then reducing the magnetic resonance linewidth [45]. Paraffin coatings were firstly introduced six decades ago [43] which could reduce the spin wall relaxation rate to as low as 0.5 Hz [46] or two orders lower [53]. The coating characteristics at higher possible temperature was studied [54]. However, since paraffin coatings only work at a low temperature range up to about 95 $^{\circ }$C, high temperature durable coatings are necessary in those ultra-sensitive magnetometry requiring vapor temperature greater than 150 $^{\circ }$C either working in SERF regime or scalar mode. Octadecyltrichlorosilane (OTS) was demonstrated as an effective anti-relaxation cell wall coating in magnetometry [55,56] at higher temperature up to 170 $^{\circ }$C [57,58]. Seltzer et al [55] obtained longitudinal spin relaxation time $T_1$ = 145 ms of potassium atoms, corresponding to 2100 collisions with the cell wall before spin depolarization. The microfabricated cell with OTS coating was also realized [52]. It is feasible that highly dense atoms in a very small cell without buffer gases can be utilized in magnetometry to achieve high sensitivity and spatial resolution.
Although the anti-relaxation coating is very beneficial, however, it had been found that atoms may be absorbed or diffusing in the organic layers in the vapor cell, and practically imperfection of the OTS coating structure may decrease the effect of anti-relaxation [59,60]. As a consequence, the atomic number density measured in coated cells could be lower than that in uncoated cells at the same temperature, and the absorption would not significantly affect the spin-relaxation [61]. The atomic spin-relaxation at the solid surface had been investigated [62,63] and research also showed that the concentration of Rb atoms on the bare glass would contribute to the longitudinal relaxation time [64]. The relaxation probability of Rb atoms on cell glass surfaces with antirelaxation coatings was analyzed theoretically [65]. Spin noises are fundamental noises in the form of random spin fluctuations in thermal equilibrium [66] and can be measured through polarization analysis of the probe light [67]. The random process is similar to the case of telegraph noise [63,68] while the atoms get in and out of the probe beam in the cell. Recently the spin noise spectrum in a hot atomic vapor with different beam parameters was studied and quantitative analysis of the lineshape was firstly performed [69]. Since a few interaction mechanisms between atoms and anti-relaxation coatings are still unknown, it is of great interest to take advantage of high optical depth and high contrast spin noise obtained through multipass cells and to investigate the microscopic changes of the atomic spin-exchange with the anti-relaxation wall coatings while without buffer gases.
We investigated the atomic spin relaxation and quantum spin noise spectrum at different temperatures by an optical magnetometer working in the scalar mode. A newly designed buffer-gas-free and multipass optical cell with both OTS anti-relaxation and dielectric anti-reflection coatings helped the magnetometer operate at a wide range of temperature with much less optical power loss. The lineshape of the spin noise spectrum varies with the total number of atoms in the cell. The spin relaxation time was determined through the independent measurements of spin decay and noise linewidth, and the methods exhibited the same results. The atomic number density was determined through the Faraday rotation measurement and the integral of spin noise area. The certain differences indicate that a small portion of atoms in the OTS coating may have spin exchange with the atoms in the vapor.
2. Experimental setup and procedures
The scalar magnetometer worked in the pump-probe configuration and the general schematic block-diagram of the experimental apparatus is shown in Fig. 1(a). A stabilized pump laser with the D1 resonant wavelength of 795 nm was initially amplified to 600 mW by a tapered amplifier, and was sent to the central area of a rubidium atomic vapor cell along $z$-direction after a $\lambda$/4 waveplate. The pump laser beam was not intentionally expanded, compared to those beams commonly used with large cross section in multipass cells, for example, in Ref. [1,25].
Fig. 1. (a) Schematic experimental setup. M$_1$ and M$_2$: Front and rear mirrors in the multipass cell with a central aperture in the front mirror M$_1$, PBS: polarization beam splitter, PD: photo diode. (b) Cross-sectional view of the vapor cell with flat side windows, showing the OTS anti-relaxation coating as well as anti-reflection coating. (c) Photo of the vapor cell used in the experiment.
A linearly polarized and tunable probe laser with adjustable power intensity was used to detect the atomic polarization and spin precession around the quantized axis. The optically rotational signal after the cell was measured by a polarimeter with low-noise and high-bandwidth photodiodes. The converted multichannel electric signals were subtracted and digitized by a fast and ultra low-noise data acquisition system.
The vapor cell was heated up by a non-magnetic oven with an ac current. A uniform magnetic field was generated by a three-dimensional coil driven by a precision current power source. The multipass cell was put in a three-layer $\mu$-metal magnetic shield with an attenuation factor up to 10$^4$, to minimize ambient magnetic noises.
2.1 Vapor cell fabrication
The designed cylindrical vapor cell was made of high-temperature durable Pyrex glass with an average diameter of 0.62 cm and a length of 5.17 cm, giving the inner cell volume of 1.55 cm$^3$. Two circular side windows had more than 99.4% of power transmission rate with dielectric anti-reflection coatings and silicon dioxide (SiO$_2$) layers, for minimizing the power loss of the weak probe beam during multiple reflections. The uniformity and flatness of the optical windows were much important for the free-space vapor cell, however, in the fabrication process it could be very challenging to maintain the firmness and reliability of the vacuum cell while preserving the anti-reflection coating. We took advantage of the anodic-bonding technique [70], by which a silicon wafer was utilized to bond the highly polished glass window surfaces. The processes were done through a well controlled and sufficiently powerful electrostatic field. The cross-sectional view of the fabricated vapor cell in the experiment is shown in Fig. 1(b) and the cell photo is demonstrated in Fig. 1(c).
The whole inner cell including the side windows were coated with the OTS layer through similar major procedures as those in Ref. [57]. In our case specifically, the cell was firstly carefully cleaned using detergent, and was exposed to a mixture solution of hydrogen peroxide and sulfuric acid for an hour, to get rid of any organic dirt on the wall surface as well as the optical windows. After being rinsed with de-ionized water and methanol, the cell was baked for an hour in order to evaporate any residual solutions. The cell was kept in a humid and clean air flow for an hour to get wet uniformly by a monolayer of moisture on the cell wall. A mixed solution was prepared in a clean beaker with the OTS, chloroform and hexanes by a mix ratio of 0.8:1:4, respectively. Then the cell was filled with the solution and after the exposure to the solution for about 5 minutes, the cell was rinsed again by chloroform to minimize the amount of excessive solutions. Finally, the cell was baked at about 150 $^\circ$C for 24 hours to cure the OTS coating.
The cell was filled with a mixture of rubidium-87 enriched to 99% and potassium of natural abundance. The mixture ratio was about 1:16 for obtaining high polarization by the possible use of hybrid pumping [71]. The OTS coated cell was designed to be without buffer gasses, so the atomic polarization was possible to be well maintained as well as that the rubidium atoms could diffuse quickly enough that they sampled a much larger cell volume before getting completely depolarized. Therefore, the whole cell volume could be used as an active measurement volume. Another advantage was to reduce the change of the lineshape due to atomic diffusion in the collimated probe light region, and there would be no changes in the tightly focused beam [72].
2.2 Multipass cell setup and optical tests
The technique of setting up optical multipass cells was inherited from what we had previously developed [25], as shown in Fig. 1(a). The cylindrical mirrors in the multipass cell had a diameter of 2.54 cm and a curvature of 50 cm. The front mirror M$_1$ and rear mirror M$_2$ were placed apart from each other by a distance of $d_0$ = 16.9 cm, and the relatively rotational angle was about $\theta _R$ = 24$^\circ$. The mirror M$_1$ had a central aperture size of about $\phi _A$ = 1 mm which was designed to reduce beam clipping in the multiple reflections. The probe laser beam had an initial beam waist of 0.4 mm, and it entered the multipass cell by a tilted angle of about 1$^\circ$. A dense beam pattern and 26 passes were obtained in our configuration. Figure 2(a) and Fig. 2(b) show the captured pattern images of the beam spots on mirrors M$_1$ and M$_2$.
Fig. 2. (a) Photo of the beam pattern on mirror M$_1$ in the 26-pass cell, as well as the image of the central aperture. (b) Photo of the beam pattern on mirror M$_2$, with a distance of $d_0$ = 16.9 cm from mirror M$_1$. (c1) Simulated beam pattern with the beam waist (blue circles) on mirror M$_1$. (c2) Cross-sectional power intensity distribution of the probe beam on mirror M$_1$ indicated by the color spectrum from purple (low) to red (high). The orange-colored contour plots are the numerically calculated beam waist, showing the effective cross-sectional beam area A$_{e}$. (d1) Simulated beam pattern on mirror M$_2$. (d2) Beam power intensity distribution on M$_2$. (e1) Simulated beam pattern in the middle of the cell at $d_0$/2. The blue and red circles represent the incoming beam and outgoing beam relatively. (e2) Beam power intensity distribution in the middle of the cell.
We had developed the program with algorithms to calculate and visualize the characteristics of beam spot patterns in the multipass cell using the software of ${Mathematica}$. With the same mirror parameters in the experiment, Fig. 2(c1) shows a simulated probe beam spots on the front mirror M$_1$, where the blue-colored circles represent the individual beam waists. Figure 2(c2) shows the cross-sectional intensity distribution of the probe beam on M$_1$ from the lowest (purple) to the highest (red), indicated by the continuous color spectrum respectively. The actual number of atoms participating in the measurement depended on the transverse intensity profile $I(x,y)$ of the probe beam, determined by summing noise variance from different parts of the beam. The orange-colored contour plots show the effective cross-sectional beam area. The sum of the effective area A$_{e}$ was numerically calculated by the integral of the intensity $I(x,y)$ and the beam area $dxdy$, as in Eq. (1) [73]:
It gives the effective beam areas on the mirrors M$_1$ and M$_2$ of A$_{e(front)}$ = 7.51 mm$^2$ and A$_{e(rear)}$ = 7.53 mm$^2$, respectively. Figure 2(d1) and Fig. 2(d2) show the similar case on the mirror M$_2$. Figure 2(e1) shows the simulated cross-sectional beam pattern in the middle of the multipass cell at $d_{mid}$ = $d_0$/2 = 8.45 cm. The blue and red circles represent the incoming and outgoing beam waists respectively, where beam overlaps can be seen. The corresponding intensity distribution is shown in Fig. 2(e2) and the effective area is A$_{e(mid)}$ = 10.25 mm$^2$. We calculated other effective areas at different $d$, and they were between 44% and 65% of the total beam area.
The experimentally measured reflectivity of the mirrors was from 98.6% to 99.0%. In presence of the OTS coated vapor cell, the transmitted light power remained to be about 25% of the input power after 26 passes. It indicated that each side of the cell window had the optical power transmission rate of around 98.9% to 99.0% which was sacrificed less than 0.5% after coated with OTS layer. The optical power loss in the multipass cell could also partially come from the beam-clipping at the small aperture in the mirror M$_1$ by a few percent, independent of the number of passes.
We carefully performed the optical tests and calibration to the multipass cell. The actual number of beam passes in the vapor cell was verified by measuring the phase shift between the incident probe beam and the exit one. The two sampling points were set to 60 cm and 80 cm away from mirror M$_1$ on the way of probe beam propagation, before beam incidence and after beam exit respectively. The probe laser was tuned to 795 nm and modulated by an electro-optics modulator (EOM) at a frequency of 10 MHz, and the transmitted signal was received and sent to a radio-frequency lock-in amplifier. The experimental result showed that the phase shift between the incident and exit beams was $\Delta _p$ = 73$^\circ$, consistent with the theoretically calculated $\Delta _p'$ = 70.2$^\circ$ by taking account of the 26 passes and the total optical path in and out of the multipass cell.
The degree of polarization of the probe beam is important to the sensitivity and accuracy of polarimeters. However, impure linear polarization could be noticed in the exit probe beam regardless of the highly linear polarization in the incident beam. This was probably due to the birefringence either on the mirror surfaces or in the rubidium atomic vapor. We measured the total effect at different cell temperature by applying magnetic fields in forms of dc amperage on a Faraday rotator, shown in Fig. 1(a) with the rotational parameter of 0.0133 rad/A. The polarization differences between the incident and the exit beams were recorded, and the results indicated that the differences were between 7.5% and 15.8%, which would be used as a calibration in the following measurements.
3. Magnetometer measurements and discussion
3.1 Spin relaxation time
The longitudinal spin relaxation time $T_1$ of alkali atoms in a vapor cell is given by all of the rates that affect the expectation value of the spin component, which is generally in the form of
where $R_{other}$ is the rate contributed by spin-destruction, photon absorption rate, optical pumping rate and so on. If the cell is buffer-gas free and the off-resonant probe beam is weak, this term may be insignificant. $R_{wall}$ is the wall relaxation rate which may dominate all other spin-relaxation mechanisms unless it is suppressed. For example in a vapor cell with bare glass wall, $R_{wall}$ becomes more pronounced in smaller cells as $T_1$ scales linearly with the characteristic length of the cell, the magnetic linewidth will be broadened.We measured the longitudinal relaxation time $T_1$ of rubidium atoms in the vapor cell at 140 $^\circ$C by a magnetic field of 127.5 mG tilted in $x$-$z$ direction. The pump laser was turned on for polarizing the rubidium atoms and turned off for probing the spin precession signal. The probe laser was tuned to 794.42 nm with the power of 370 $\mu$W. As shown in the inset in Fig. 3, the spin decay signal can be fitted with a reversely exponential function, giving a longitudinal relaxation time of $T_1$ = 5.0 ms. It is equivalent to approximately 450 bounces before the atomic spin got completely depolarized. The number of bounces observed is comparable to those in typical OTS coated cells [55,58,74], which has significant difference from that in bare glass cells.
Fig. 3. Experimentally measured atomic spin decay signals at 140 $^\circ$C. Black curve: experimental data of transverse relaxation. Red curve: exponential fitting of the envelope, giving the time constant of $T_2$ = 0.30 ms. Inset: experimental data of longitudinal relaxation (black curve), and exponential fitting (red curve), giving the time constant of $T_1$ = 5.0 ms.
The average transverse spin relaxation time $T_2$ is contributed by the longitudinal $T_1$, spin exchange rate $R_{SE}$ and broadening $R_{Grad}$ due to field gradient:
where $a$ is the broadening factor of spin-exchange, depending on the magnetic field and atomic density. The $R_{Grad}$ may be neglected if a small size cell is used. When the polarized atoms interact with the wall coating, they may be trapped and diffusing. The dwelling time $t_{dw}$ = $t_{min} e^{E_{a}/k_{B}T}$ [75], where $t_{min}$ is the minimum dwell time possible, $E_{a}$ is the absorption energy, $k_{B}$ is the Boltzmann Constant and $T$ is the absolute temperature. The atomic diffusion is proportional to $e^{-E_{act}/k_{B}T}$ [76], where $E_{act}$ is the activation energy of the potential well in the coating.Experimental results showed that the longitudinal $T_1$ at above 120 $^\circ$C was much longer more than the transverse relaxation time $T_2$, indicating that the atomic spin exchange dominated the relaxation process. The $T_2$ was measured when the magnetometer operated in the scalar mode at low initial atomic polarization. A magnetic field of $B_{z}$ = 127.5 mG was applied along the direction of the pump laser beam, and the rubidium atoms were optically pumped for a period of 5 ms, followed by a resonant oscillating field at a frequency of 89.3 kHz in $y$-direction for 33.6 $\mu$s. We varied the body temperatures of the vapor cell from 100 to 160 $^\circ$C, but it should be noted that the cell stem’s temperature is lower than that of the cell body. As an example, Fig. 3 shows the experimental atomic spin decay signal at the temperature of 140 $^\circ$C. The fitting of the envelope gives that $T_2$ = 0.30 ms. The experimental values of $T_2$ at other cell temperatures are shown in Table 1. Note that after hours at high temperature exceeded 160 $^\circ$C, continuously increasing relaxation rate could be observed, which was possibly due to the slow degradation of the OTS coating [57] and that the degradation was not reversible.
Table 1. Calculation of the atomic spin relaxation time from spin decay and noise linewidth, and number density calculation from Faraday rotation and spin noise area integral at different cell temperatures.
In order to measure the spin noise of rubidium atoms in the vapor cell, the pump laser was turned off, and a static magnetic field of $B_{z}$ = 13.7 $\mu$T was applied. The weak and linearly polarized probe beam with the power of 387 $\mu$W at the wavelength of $\lambda$ = 795.01 nm was used to detect the optical angular fluctuation noise. The noise spectrum at the cell temperature of 120 $^\circ$C is shown in Fig. 4. The black points is the experimental power subtracted noise spectrum showing a narrow peak and broad wings. To understand the lineshape of the noise for the vacuum cell with the anti-relaxation coating, we utilize the Langevin’s diffusion model by which the spectrum function $S(f)$ can be described as [69]
where $\delta ^2\phi$ is the total angular noise power contributed by the atoms that lead to Faraday rotation in the probe beam, $f_0$ is the Larmor precessing frequency in the field, and the atom’s probability $C_{d}(t)$ in the probe beam, as analytically derived in detail in Ref. [69], is expressed as a spin diffusion autocorrelation function which is closely related with the radius of the probe beam $r_{p}$ and of the cell $r_{c}$, as well as the mean free path $\lambda _{fmp}$.Fig. 4. Spin noise spectrum data (black points) at 120$^\circ$C consisting of a narrow peak and broad wings. The data points are fitted with Langevin’s diffusion model (red solid curve). The inset shows the Ramsey peak fitted by a Lorentzian function and a remaining background.
The upper inset of Fig. 4 shows the narrow Ramsey peak substructure of the spectrum which is mainly due to the noise of rubidium atoms that bounce back and forth for many times on the cell wall before getting depolarized, and the peak can be well fitted by a single Lorentzian function and a remaining background. The broad wings are contributed by the rubidium atoms that only bounce one or a couple of times before depolarized. The data fitting by the Langevin’s diffusion model gives a rotational angle of $\phi _R$ = 14.8 $\mu$rad for the Ramsey peak and a full-width at half-maximum (FWHM) of 453.5 Hz which is equivalent to the coherent time of 0.70 ms. The free mean path is calculated as $\lambda _{fmp}$ = 2.87 mm, and the lower inset of Fig. 4 shows the $C_{d}(t)$ versus the time $t$. The probe beam’s equivalent radius $r_{p}$ = 1.9 mm according to the fitting result which is the same as that obtained from $A_{e}$. Since $\lambda _{fmp}$ is bigger than $r_{p}$ but smaller than $r_{c}$, the Langevin’s diffusion model is required [69] and the results are self-consistent. The spin noise linewidth at different cell temperatures from 100 $^\circ$C to 160 $^\circ$C was measured in the similar way and is shown in Table 1. and all of the $T_2$ measurement results are the same as those measured from spin decay time, as expected by the fluctuation-dissipation theorem.
3.2 Atomic number density
The number density of atomic vapor varies with different temperatures and under some conditions it can be calculated base on empirical formulas [77]. However, the observed density is possibly much less than expected especially in cells with anti-relaxation coatings [58]. The vapor density can be more accurately determined by direct Faraday rotation measurement, based on that the interaction between birefringent atoms and light in the magnetic field induces the proportional optical rotation with regard to the amplitude of field. The Faraday rotation is a product of $n$, $l$ and $\lambda$. As described in Ref. [78], the benefit of the method is that it does not require the assumption of thermodynamic equilibrium, and if the probe light is far detuned from the central wavelength $\lambda _0$, the rotation angle can be put in the form of
where $\mu _B$ = 9.27$\times$10$^{-24}$ J/T is the Bohr Magneton, $B$ is the magnetic field, $h$ = 6.63$\times$10$^{-34}$ m$^2$kg/s is the Planck’s constant and $\lambda _0$ = 794.98 nm is the central D1 resonance of rubidium.In the experiment a weak and linearly polarized probe beam was generated along the $x$-direction with the incident power of 370 $\mu$W. Two different probe wavelengths were used for comparison. Since the Faraday rotation angle could be as small as $\sim$mrad, we utilized a lock-in amplifier and a low-noise power amplifier for detection as well as for modulating the magnetic field which varied from 30.8 mG to 308 mG. The small sinusoidal modulation was also applied in the $x$-direction at a frequency of 153.5 Hz. The optical signal after the polarimeter was demodulated by the lock-in amplifier and the Faraday rotation angles with regard to the changes of the magnetic field are shown in Fig. 5(a), taking the measurement at the temperature of 140 $^\circ$C for example. The experimental data can be well fitted by simple linear functions with different slopes, giving the same result of the atomic vapor density of $n$ = 2.21$\times$10$^{13}$ cm$^{-3}$ with errors less than 1%. Results shown in Table 1 at other temperatures are calculated by the same method.
Fig. 5. (a) Faraday rotation v.s. magnetic field at two different probe laser wavelengths at the cell temperature of 140 $^\circ$C. Both experimental data fitting gives a consistent density of $n$ = 2.21$\times$10$^{13}$ cm$^{-3}$. (b) Atomic vapor density obtained from the spin noise area integral. Three groups of data were taken using different off-resonant wavelengths and different probe power at the same temperature of 140 $^\circ$C. The experimental results are consistent and the fit gives a density of 1.50$\times$10$^{13}$ cm$^{-3}$. (c) Ratio between the densities obtained from the spin noise area and the Faraday rotation measurement. The fit gives an approximate ratio of 78.7% in the temperature range from 100 $^\circ$C to 160 $^\circ$C.
As an important alternative, the vapor density can also be determined by the total rotational angle of spin noise which is written as [73]
where $n$ is the atomic vapor density, $r_e$ = 2.82$\times$10$^{-13}$ cm is the classical electron radius, $f_D$ = 0.34 is the oscillator strength of the D1 line, $l$ = 134.42 cm is the effective length of the probe beam in the cell, $N = n L A_{e}$ is the probed number of atoms where $L$ is the length of the cell, and $\delta F_x$ is the probe laser detuning from the level F = 2 or F = 1. The total $\phi (n)$ can be determined from the data fitting and the integral of the spin noise spectrum, so that the atomic number density $n$ is derived. As an example, Fig. 5(b) shows the measured density using different probe wavelengths detuned from −40.0 GHz to 17.6 GHz and using different probe power ranging from 0.29 mW to 2.08 mW at the cell temperature of 140 $^\circ$C, and the measurement results give a consistent atomic vapor density of $\sim$1.5$\times$10$^{13}$ cm$^{-3}$. The atomic number density measurement results by the similar way at temperatures from 100 $^\circ$C to 160 $^\circ$C are listed in Table 1.It is interesting to see that the Rb densities shown in Table 1 from the spin noise area and that from the Faraday rotation have non-negligible difference, and the trend is that the former is always smaller than the latter, as shown in Fig. 5 (c) by an average factor of 78.7%. We presume that it is possibly due to the depolarization in the cell stem or the spin exchange with the OTS coating in the vapor cell.
During the fabrication process of the vapor cell we intentionally left a long stem as a rubidium reservoir connected to the middle of the cell, as shown in the photo in Fig. 6 (a) and in Fig. 6 (b) for sizes, to prevent excessive atoms condensing on the inner cell coating, and to reduce the atoms from assembling on the inner windows. Most of the stem is coated with OTS, yet due to the small diameter and the area near the terminal tip where the original coating has been destroyed, the fluctuation of atomic spin polarization is likely to be affected, for example to be suppressed, in case atoms enter the stem as indicated in Fig. 6 (c). In our vacuum vapor cell the atoms diffuse and bounce very quickly and the possibility of entering the stem depends on several factors including the ratio of the cell stem and body volumes, the ratio of the aperture and the cell surface areas, and the temperature difference in the body and the stem. We estimate the possibility by the volume ratio of up to 7.3%. However, the thermal dynamic equilibrium is a complicated process in this case and at present only the above overall statistical analysis is taken into consideration.
Fig. 6. (a) Overview of the vapor cell and the stem. (b) Inner form factors of the cell body and stem. (c) Demonstration of the interaction between atoms and those in the OTS coating as well as in the cell stem.
The possibility of atoms running into the stem may change the lineshape and reduce the equivalent area in the Ramsey peak, and thus affect the calculation of the number density. But this issue only takes about 1/3 of the total impact. We also consider another possibility of the interaction between the atom and the OTS coating, or between free atoms and atoms in the coating. As initiated in the section of introduction, there had been some observations that the experimentally measured atomic vapor density was deviated gradually from the theoretical expectation of saturated density, which was thought to be that the atoms were pumped by the coating [61] or that it was assumed to be a change of absorption energy and diffusion rate of the atoms [53]. In our case, the expected saturated vapor density is 2$\sim$6 times higher than the actual measured one in the vacuum cell, varying with the temperature. The atomic ensembles would reach an equilibrium state after a long time, and experimentally we did not notice significant changes in vapor density at each temperature below 160 $^\circ$ C after hours. It has been proved that at some conditions, such as light-inducing, atoms may be desorbed from the coating [76,79] or still be dwelled in the coating while remaining polarization for a certain period [80]. The free atoms are possible to interact with those atoms in the OTS coating by spin exchange, as shown in Fig. 6 (c), and we can infer that the possibility would be around 14% in our cell, varying slightly with the temperature.
4. Conclusions
In summary, we developed an OTS anti-relaxation and dielectric anti-reflection coated multipass optical cell filled only with rubidium vapor, and investigated the spin relaxation and quantum spin noise spectrums of the atoms. High contrast of spin noise spectrums were observed due to high optical depth obtained by the multipass cell. Low optical power losses in the multiple beam paths were realized by the anti-reflection coating on the cell windows of which the evenness was guaranteed with the help of anodic-bonding technique. The inner wall of the vacuum vapor cell was coated with high-temperature durable OTS coatings so that the atoms may diffuse very quickly in the vacuum cell and could bounce many times before depolarized within the relaxation time, which greatly reduced the needs of large area pump beams. In contrast to cells using buffer gas, we find that the quantum coherence lifetime in the vacuum cell is also equal to the classical transverse spin relaxation time, in accordance with the fluctuation-dissipation theorem. We measured the rubidium atomic vapor density using the approaches of Faraday rotation measurement and the area integral of the spin noise spectrum, and find the results are reasonably close. The approximate 14% of discrepancy indicates the possibility of spin-exchange between rubidium atoms trapped on the surface of the OTS coating.
This work would help further understandings about the interaction between atoms and surface coatings, where the spin relaxation would be pronounced when it comes to a much smaller physical package. It is important to preserve the atomic polarization with effective high atomic density, and to reduce the power loss to minimize the probe power and photon shot noises. The exemption of beam spanning for fiber optics and the easy setup for multipass cells with highly-flat transparent optical windows would practically enhance the fabrication of miniature sensors [81,82], which can also benefit the applications in high-resolution magnetocardiogram (MCG) and MEG measurements with magnetometer arrays [83,84]. The work would probably help further investigation of magnetometers or frequency references using cells filled with other alkali-metals such as cesium [85] or sodium [86].
Funding
National Natural Science Foundation of China (11204273, U21A20437); Zhejiang Provincial Basic Public Welfare Research Program (LGF22F010009).
Acknowledgments
The authors would appreciate Professor M. V. Romalis and Professor D. Sheng for all the help and useful discussions, and thank N. Dural for providing the anodic bonding technique. S. Li thanks DARPA for help discovering connections and cooperations for this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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