We investigated the operation of an all-optical rubidium-87 atomic magnetometer with amplitude-modulated light. To study the suppression of spin-exchange relaxation, three schemes of pumping were implemented with room-temperature and heated paraffin coated vacuum cells. Efficient pumping and accumulation of atoms in the F=2 ground state were obtained. However, the sought-for narrowing of the resonance lines has not been achieved. A theoretical analysis of the polarization degree is presented to illustrate the absence of light narrowing due to radiation trapping at high temperature.

Magnetometry is an important branch of modern metrology. In order to detect magnetic fields, a variety of devices are utilized, including fluxgates, proton-precession magnetometers and superconducting quantum interference devices (SQUIDs).1 With the rapid development of theory and experimental techniques of light interaction with atoms, optically pumped atomic magnetometers2 have recently achieved sensitivities similar to that of SQUIDs and are currently used in fields such as geology,3 fundamental physics4 and medical diagnostics.5 However, the operational requirements of optically pumped atomic magnetometers restricts their further applications. For example, though the Spin Exchange Relaxation Free (SERF) atomic magnetometers reach a sensitivity of sub-fT level,6 they typically require high temperature of the cells (up to 200°C) and magnetic field of no more than ∼100 nT, much less than the Earth field of ∼50 μT. At Earth field, atomic magnetometers usually operate in the regime where the Larmor precession rate is much higher than the spin-exchange collision rate (opposite to the requirement for SERF). Developing optical magnetometers with sensitivity comparable to that of SERF magnetometers, but operating in the Earth-field range remains an important goal due to the great demand for practical applications.7 

The fundamental spin projection noise limited sensitivity is given by8 

δB1γ1nVT2t,
(1)

where γ denotes the alkali-atom gyromagnetic ratio which is often treated as a constant (for 87Rb: γ/2π≈7 Hz/nT, for 85Rb: γ/2π≈4.6 Hz/nT), n is the alkali-vapor density in the measurement volume V, and the transverse spin-relaxation time is T2 with measurement time t>>T2. Therefore, high-sensitivity magnetometers require maximizing the product of atomic density and transverse relaxation time. Under certain relevant conditions, transverse relaxation involves at least five components: spin-exchange relaxation, spin-destruction relaxation, wall-collision relaxation, optical power broadening and magnetic-field gradient broadening. Among these, spin-exchange relaxation and wall-collision relaxation are dominant components in most cases.9 Spin-exchange relaxation results from spin-exchange collisions among alkali atoms that conserve the total spin, while spin-destruction relaxation is caused by spin-destruction collisions that do not conserve the total spin between alkali atoms or with buffer gases. Spin-exchange relaxation is typically much faster than spin-destruction relaxation because spin-exchange cross-sections are 2-4 orders of magnitude larger than those of spin destruction.

Three approaches are adopted to improve the fundamental sensitivity. First, new coating materials are used to reduce wall-collision relaxation and increase T2. The state-of-the-art maximum T2 time of 77 s is achieved in an alkene-coated glass vapor cell.10 Buffer gas can also be used to avoid broadening due to wall collisions, however, it causes the resulting sensor to be more sensitive to magnetic-field gradients, which is an unwanted side effect.11 The effects of collisions due to background gas in paraffin coated cells need future investigation.12 Second, it is possible to obtain relatively high atomic density, by operating at high temperature. Recent work shows that antirelaxation wall coating is still effective up to 95°C.13 Third, one can further increase T2 by eliminating spin-exchange relaxation using light narrowing.14 

In thermal atom samples, spin-exchange relaxation redistributes the population among Zeeman sublevels15 and therefore reduces the sensitivity of atomic magnetometers.16 At magnetic fields that are sufficiently high so the SERF regime cannot be achieved, one approach to suppress spin-exchange relaxation is light narrowing. In the light-narrowing regime, almost all atoms are optically pumped to the stretched state (the state with the maximum or minimum magnetic quantum number). Spin-exchange relaxation is suppressed due to total angular momentum conservation during spin-exchange collisions (SEC), resulting in a narrowed magnetic-resonance linewidth.

The light narrowing phenomenon was first experimentally observed in 1981 with a linewidth-narrowing factor of 2.5 in dense cesium vapor with buffer gas.17 In 1999, gas cells with ∼10 atm pressure at 20°C were utilized and a considerable linewidth narrowing was obtained.14 A potential alternative for atomic clocks was proposed: instead of the traditional 0-0 transition, end resonances could be used to avoid the disadvantages of population dilution, spin-exchange broadening, and poor pumping efficiency at high gas pressure.18 A detailed theoretical analysis was performed for potassium atomic magnetometers in weak radio-frequency magnetic fields.19 With the light narrowing technique, hyperfine and Zeeman resonance linewidths could be decreased from ΔwRSE to ΔwRSERSD where the spin-destruction rate RSD is much smaller than the spin-exchange rate RSE, leading to a linewidth narrowing factor of 10-100. When an atomic magnetometer is limited by photon shot noise, theoretical and experimental results show that its sensitivity improves as the slope of the dispersive part of the magnetic resonance, from σSE1 to σSE3/4σSD1/4 where the spin-exchange cross-section σSE far surpasses the spin-destruction cross-section σSD.20 In a miniaturized cesium cell with 170 mbar nitrogen buffer gas and volume of only 9.9 mm3, a shot noise limited sensitivity of 42 fT/Hz1/2 at μT-level magnetic field was achieved by preparing 95% of the atoms in the stretched state and optimizing pump power, cell temperature, and magnetic fields.21 The same research group operated in the light-narrowing regime at 50 μT magnetic field and eliminated the side effect of light shift by averaging two magnetometers with oppositely circularly polarized channels.22 A sequence of coupled pump pulses repeated at the Larmor frequency suppressed spin-exchange collisions at near-Earth field (0.1 G) in 267 mbar nitrogen buffer gas rubidium-87 cells.23 Both spin orientation up to 92% and narrowing of radio-frequency resonances were observed in paraffin coated cells near room temperature, in a synchronously pumped magnetometer with a small angle (∼10°) between the pump beam and the magnetic field.24–26 

In this paper we investigated the possibility to realize light narrowing in paraffin coated cells without buffer gas in a temperature range from 22°C up to 50.5°C. The apparatus operated in a perpendicular arrangement of the pump beam and the magnetic field to increase the magnetic-resonance amplitude. However, even employing different pumping schemes, we were unable to observe convincing light narrowing effects. A theoretical analysis suggests that the underlying cause is radiation trapping, i.e. reabsorption of spontaneously emitted pump photons in the vapor, which results in depolarization.

The experiments are carried out in a paraffin coated rubidium-87 vapor cell without buffer gas, using amplitude-modulated nonlinear magneto optical rotation (NMOR).

Optical pumping with amplitude-modulated circularly polarized light generates atomic orientation along its propagation direction. In the magnetic field, oriented atoms precess around the magnetic field at the Larmor frequency. Pumping pulses enhance the polarization every cycle provided the modulation frequency matches the Larmor frequency. The magnetization precession causes the light polarization to oscillate. The probe light detects the projection of precessing angular momentum, resulting in optical rotation signal via the atoms’ circular birefringence.27 

Three schemes are implemented to study light narrowing. All schemes probe the F=2 ground state but differ in the way they pump the atomic vapor. Scheme #1 employs a single amplitude-modulated pump laser resonant with the D1 line F=1→F=1,2 transitions (abbreviation as F=1 in the following). Scheme #2 adds an additional amplitude-modulated pump laser on the F=2→F=1,2 transitions (abbreviation as F=2 in the following). Scheme #3 uses an amplitude-modulated pump beam on the F=2 Zeeman manifold and incorporates a non-modulated de-pump laser tuned to F=1 with varied power to depopulate the F=1 ground state. More details on the experimental parameters are presented in Table I and a schematic of the experiment is depicted in Fig. 1.

TABLE I.

Atomic transitions and power modulation for the three schemes.

CategoryScheme #1Scheme #2Scheme #3
Pump #1 Tuning D1 F=1 D1 F=1 D1 F=1 
Pump #2 Tuning N/A D1 F=2 D1 F=2 
Probe Transition Detuned from D1 F=2 500 MHz above D2 F=2 500 MHz above D2 F=2 
Pump #1 Modulation Synchronous Synchronous No modulation 
Pump #2 Modulation N/A Synchronous Synchronous 
CategoryScheme #1Scheme #2Scheme #3
Pump #1 Tuning D1 F=1 D1 F=1 D1 F=1 
Pump #2 Tuning N/A D1 F=2 D1 F=2 
Probe Transition Detuned from D1 F=2 500 MHz above D2 F=2 500 MHz above D2 F=2 
Pump #1 Modulation Synchronous Synchronous No modulation 
Pump #2 Modulation N/A Synchronous Synchronous 
FIG. 1.

Schematic of the apparatus. DAVLL: Polarization spectroscopy optics for frequency stabilization. PID: SIM960 PID controller. POL: Linear polarizer. PD: Thorlabs PDA36A photo detector. AOM: ISOMET 1205C-2 AOM with a 532C driver. BS: Beam splitter. LIA: SRS830 lock-in amplifier. PC: Personal computer. PBS: Polarizing beam splitter. Polarimeter: Balanced differential photodiodes. Source 1: Thorlabs LDC20mA current drivers. Source 2: DS345 function generator and EIN 3200L radio-frequency power amplifier to drive the cell heater. Source 3: DS345 function generator.

FIG. 1.

Schematic of the apparatus. DAVLL: Polarization spectroscopy optics for frequency stabilization. PID: SIM960 PID controller. POL: Linear polarizer. PD: Thorlabs PDA36A photo detector. AOM: ISOMET 1205C-2 AOM with a 532C driver. BS: Beam splitter. LIA: SRS830 lock-in amplifier. PC: Personal computer. PBS: Polarizing beam splitter. Polarimeter: Balanced differential photodiodes. Source 1: Thorlabs LDC20mA current drivers. Source 2: DS345 function generator and EIN 3200L radio-frequency power amplifier to drive the cell heater. Source 3: DS345 function generator.

Close modal

A 795 nm Distributed Bragg Reflector laser (DBR, manufactured by Photodigm Inc.) driven with a Thorlabs ITC4001 controller is used as pump laser #1; the beam diameter is 1.5 mm. A dichroic atomic vapor laser lock (DAVLL) is used to stabilize laser frequency.28 The laser power is also stabilized with an acousto-optical modulator (AOM) in a feedback loop. A second AOM modulates the pump light power before the light is sent through polarization optics (a polarizer and a quarter-wave plate for circular polarization) to the vapor cell.

A second 795 nm Distributed Feedback laser (DFB) driven by Thorlabs LDC201 and TED200 controllers is used as pump laser #2. It produces 10∼30 mW light in a 1.5 mm beam diameter. In scheme #2, both pump #1 and pump #2 beams are modulated and pass through the same AOM. In scheme #3, this AOM is moved into the beam path of pump #2. Pump #1 and pump #2 beams are combined on a beam splitter before the polarization optics.

The probe beam propagates along the y-direction. In scheme #1, a 795 nm DFB laser is used while in scheme #2&3, it is replaced by a 780 nm External Cavity Diode Laser (ECDL, ULDC252S from Vitawave Inc.). Both probe lasers have a 2.5 mm diameter beam output, which is split into two beams: one is used for DAVLL frequency stabilization and the other delivers linearly polarized light into the magnetic shields. In scheme #1, the probe light is kept at ∼3 μW detuned away from D1 line F=2. In scheme #2&3, it is ∼50 μW detuned 500 MHz above the transition frequency to avoid pumping by the probe light.29 After exiting the shields, the optical rotation signal of the probe beam is measured by a balanced polarimeter.30 For small optical rotation, the rotation angle θ is given by:

θ=P1P22(P1+P2),
(2)

where P1 and P2 represent signals of the two photodiodes in the polarimeter. The differential value is acquired with a lock-in amplifier (LIA) referenced to the pump beam modulation provided by a function generator. The demodulated signal from the LIA is acquired by a personal computer for analysis.

The rubidium-87 atomic vapor is contained in a cylindrical glass cell of 20 mm diameter and 20 mm length. The inner walls are coated with paraffin to reduce wall-collision relaxation.

The residual magnetic field inside the four-layer cylinder mu-metal magnetic shields is reduced to <1 nT.31 A set of three orthogonal magnetic-field coils inside the innermost shield is designed to provide bias magnetic field and gradient compensation. Before the experiment, triaxial magnetic field compensation is performed and the static magnetic field is set to 280 nT in the z-direction, perpendicular to the plane spanned by the pump and probe beams.

In order for the spin-exchange relaxation to be visible, it is necessary to increase the atomic vapor density by heating the cell. The heater is driven with an alternative current at 317 kHz, so that the heater doesn’t disturb the magnetic field at the frequencies the magnetometer is sensitive to.32 The heater consists of a resistive printed circuit board (PCB, 50 Ohm), and is placed next to the cell inside the innermost shield, heating the air in the entire innermost-shield volume. The PCB is wired in a retracing pattern with four layers.33 The heater is switched off during the measurements to reduce residual magnetic noise. The cell stem temperature is kept 2∼5°C lower than the wall temperature to avoid the alkali condensation on the inner walls.34 The temperature is measured with OMEGA thermocouples, read with an OMEGA HH223 thermocouple reader and recorded via a 16-bit NI6211 board.

We first consider the configuration with a single pump beam. To explore the dependence on the pumping transition, several experiments are performed with pump #1 light on resonance with the transitions from the F=1 and F=2 ground states. The pump beam is modulated with a 20% duty cycle in both cases. Light power averaged over a duty cycle is measured with a power-meter. To investigate the effect of spin-exchange collisions, the experiments are repeated with different temperatures (room temperature of 22°C and a higher temperature of 44°C) and therefore different vapor densities. We scan the AOM modulation frequency around the Larmor frequency and observe the magnetic resonance in polarization rotation of the probe beam with a lock-in amplifier. To extract the full width at half maximum (FWHM) magnetic-resonance linewidth and amplitude, we fit the frequency response peak with a Lorentzian line-shape. Linewidths as narrow as 2.9 Hz FWHM are observed at room temperature.

As can be seen from Fig. 2, tuning the pump light to F=1 results in a narrower linewidth than tuning it to F=2 at both temperatures of 22°C and 44°C. The pump light on resonance with F=1 transition depopulates F=1 ground state manifold. The power broadening is essentially absent because the pump light is far detuned from transitions of the probed ground state manifold.

FIG. 2.

Measured FWHM in scheme #1, pumping from D1 F=1 and F=2, at 22°C and 44°C.

FIG. 2.

Measured FWHM in scheme #1, pumping from D1 F=1 and F=2, at 22°C and 44°C.

Close modal

During atomic precession, the pump light polarizes the atoms at each pumping pulse. The amplitude and width of the magnetic resonance depend on the duty cycle. This dependence is investigated by varying the duty cycle with the pump beam tuned to F=1. Figure 3 depicts FWHM of the magnetic resonance pumped with 20% and 50% duty cycle at 22°C and 44°C. Amplitude of maximum optical rotation angle is calculated based on Eq. (2) and represented in Fig. 4.

FIG. 3.

Measured FWHM in scheme #1, pumping from D1 F=1 with 20% and 50% duty cycle, at 22°C and 44°C.

FIG. 3.

Measured FWHM in scheme #1, pumping from D1 F=1 with 20% and 50% duty cycle, at 22°C and 44°C.

Close modal
FIG. 4.

Measured amplitude in scheme #1, pumping from D1 F=1 with 20% and 50% duty cycle, at 22°C and 44°C.

FIG. 4.

Measured amplitude in scheme #1, pumping from D1 F=1 with 20% and 50% duty cycle, at 22°C and 44°C.

Close modal

When the pump power is >10 μW and the temperature is 22°C, a 20% duty cycle results in smaller FWHM and larger amplitude of the magnetic resonance. However, when the cell is heated to 44°C, an opposite relationship is observed that a 50% duty cycle has better performance of smaller FWHM and larger amplitude. This is not immediately intuitive, but a further study of this effect is beyond the scope of this paper and will be addressed in future work.

To determine if spin-exchange relaxation is visible in the transverse relaxation at high temperature, we roughly estimate the spin-exchange collision linewidth ΔwSE as35 

ΔwSE=RSEπqSE=nσSEvπqSE,
(3)

where σSE is the spin-exchange collision cross-section, v is the average relative velocity of the atoms, and qSE is the spin-exchange broadening factor. σSE =1.9×10-14 cm2,36 and qSE = 8.31 The calculated spin-exchange linewidth is around 0.27 Hz at 22°C and 2.5 Hz at 44°C.

Due to the existence of longitudinal relaxation, errors of temperature measurement and other factors, the calculated spin-exchange relaxation rate doesn’t account for the transverse relaxation rate observed in the experiment. At 44°C, spin-exchange collisions significantly contribute to the magnetic-resonance linewidth so the light-narrowing effect should be possible to detect. However, no narrowing of the linewidth is observed at 44°C in scheme #1 with an increase in the pump power.

Since no light narrowing is observed with a single pump laser, we presume that the depolarization of the atoms takes place in the probed Zeeman manifold. To investigate this, we replace the main pump beam by a synchronously modulated pump #2 beam on resonance with F=2 and vary the power of the additional synchronously modulated pump #1 beam, which is resonant with F=1. Both beams go through the same AOM and have a duty cycle of 6.8%. Pump #2 beam operates at a constant average power of 21 μW, while the cell is heated to 55°C. As the cell passively cools down, the magnetic-resonance linewidth is recorded as the power of pump #1 beam is varied from 0 to 540 μW. At 36.4°C, spin-exchange collisions make significant contributions to the magnetic-resonance linewidth. Figure 5 illustrates a hint of linewidth narrowing around 25 μW at different temperatures. As such, the narrowing effect is unremarkable and almost negligible at high temperature in dense vapor. Though the narrowed linewidth compared with scheme #1 at 44°C indicates a hint of light narrowing, it would be more convincing to obtain narrowed linewidth with the increase of pump power in the same pumping scheme.

FIG. 5.

Measured FWHM in scheme #2 with fixed pump #2 power tuned to F=2 and varied pump #1 power resonant with F=1 at different temperatures.

FIG. 5.

Measured FWHM in scheme #2 with fixed pump #2 power tuned to F=2 and varied pump #1 power resonant with F=1 at different temperatures.

Close modal

Next, pump #1 has the power of 243 μW, and is tuned to F=1. Pump #2 laser is tuned to F=2, and its power is varied from 0 to 256.8 μW. Both beams are synchronously modulated with a 6.8% duty cycle. The results are represented in Fig. 6. Power broadening increases the linewidth with the increase in pump #2 power. This reveals an incomplete polarization of the atoms because the stretched state (as a dark state for this transition) should not experience power broadening.

FIG. 6.

Measured FWHM in scheme #2 with fixed pump #1 power tuned to F=1 and varied pump #2 power resonant with F=2 at different temperatures.

FIG. 6.

Measured FWHM in scheme #2 with fixed pump #1 power tuned to F=1 and varied pump #2 power resonant with F=2 at different temperatures.

Close modal

Since we do not observe light narrowing in scheme #2, we presume that the depopulation of the F=1 ground state is incomplete. To increase the depopulation of this manifold, we add a non-modulated pump #1 beam on resonance with F=1 and vary its power from 19.45 μW to 1.06 mW at different temperatures. Pump #2 beam is tuned to F=2 and is amplitude modulated with a duty cycle of 1.14% at a fixed average power of 40 μW. The linewidth is measured as a function of pump #1 power analogously to scheme #2, and the results are presented in Fig. 7. For all measured temperatures, the linewidth increases with pump power. A selection of the expected spin-exchange collision linewidths based on Eq. (3) and measured FWHM are listed in Table II. Even though spin-exchange collisions contribute to the linewidth greatly at high temperature (for example, at 41.2°C), no light narrowing is observed. The possible reason lies in the fraction of population in the stretched state, as will be analyzed below.

FIG. 7.

Measured FWHM in scheme #3 with varied pump #1 power tuned to F=1 and a modulated constant-power pump #2 beam resonant with F=2 at different temperatures.

FIG. 7.

Measured FWHM in scheme #3 with varied pump #1 power tuned to F=1 and a modulated constant-power pump #2 beam resonant with F=2 at different temperatures.

Close modal
TABLE II.

Comparison of calculated spin-exchange linewidth and measured FWHM at different temperatures.

Temperature /°C 22 36.4 41.2 44 45.3 50.5 
Calculated spin-exchange linewidth /Hz 0.27 1.2 2.0 2.5 2.9 4.5 
Measured FWHM /Hz 3.3 4.0 6.0 11 8.1 13 
Ratio of calculated spin-exchange linewidth to measured FWHM 8% 30% 33% 23% 36% 35% 
Temperature /°C 22 36.4 41.2 44 45.3 50.5 
Calculated spin-exchange linewidth /Hz 0.27 1.2 2.0 2.5 2.9 4.5 
Measured FWHM /Hz 3.3 4.0 6.0 11 8.1 13 
Ratio of calculated spin-exchange linewidth to measured FWHM 8% 30% 33% 23% 36% 35% 

The experiment is conducted with relatively dense rubidium-87 vapor (n>1011 cm-3) in a paraffin coated cell. The cell does not contain any buffer gas. Figure 8 illustrates the principle of the light-narrowing regime and the corresponding distribution of atoms. The steady state population for ground states is roughly estimated based on rate equations.37 Strong optical pumping depopulates sublevels of the F=1 ground state. Through relaxation, atoms get redistributed towards thermal equilibrium on F=1 and F=2 sublevels. Spin-exchange collisions provide a redistribution mechanism in the ground states, and the pumping rate needs to exceed the spin-exchange rate to fully deplete F=1 sublevels. Modestly repumping from F=2 assists in driving atoms to the stretched state where spin-exchange relaxation is forbidden by selection rules. When most of the atoms accumulate in the stretched state, they maintain orientation. Due to angular momentum conservation, elastic collisions fail to redistribute atoms to other sublevels. Therefore, spin-exchange relaxation is suppressed.

FIG. 8.

(a) The conventional configuration of pumping from F=2; (b) Light-narrowing configuration of mainly pumping from F=1. Thickness of arrows shows pump power and grey blocks represent the roughly estimated ground-state population.

FIG. 8.

(a) The conventional configuration of pumping from F=2; (b) Light-narrowing configuration of mainly pumping from F=1. Thickness of arrows shows pump power and grey blocks represent the roughly estimated ground-state population.

Close modal

To calculate the limit of light-narrowed linewidth, we start from the transverse relaxation time T2 and corresponding linewidth Δw given by14 

1T2=πΔw=ROP2I+1+RSERSDROPGw0,RSE,
(4)

where ROP is the optical pumping rate. For rubidium-87, the nuclear spin is I = 3/2 and the ground state hyperfine splitting whf = 6835 MHz. The Larmor frequency in the experiment is w0 ∼ 2 kHz and RSE < 100 s-1. The approximate value for parameter G (w0, RSE) when the polarization degree is close to 1 is:

Gw0,RSE=ReRSE+8iw02/whf5RSE+16iw02/whf15.
(5)

As a result, the minimum linewidth in Eq. (4) is obtained in the light-narrowing regime. The light-narrowed linewidth ΔwLN is given by:

1T2,LN=πΔwLN=RSERSD5,
(6)

where σSD = 1.6×10-17 cm2.38 

Considering Eq. (3) and (6), the theoretical light narrowing factor is ΔwSEwLN ≈ 10 regardless of temperature, because atomic density and velocity cancel out. This describes the extent by how much spin-exchange relaxation is suppressed when polarization degree approximates 1. Considering a variety of factors such as temperature non-reproducibility, coating dependence on temperature and changing field gradients, a small discrepancy of linewidths in the above three schemes is observed, but the remarkable linewidth narrowing is absent.

Taking all factors into consideration, the reason for the absence of light narrowing in dense vapor is likely low polarization degree without quenching gas. The atoms are polarized by absorbing pump photons, and decaying atoms emit partially polarized light through spontaneous emission, which propagates in random direction. When these photons are reabsorbed by other atoms, they become depolarized. Because of the continuous processes of photon emission and reabsorption before emitted photons ultimately escape the optical path, radiation trapping is likely to be established, and the polarization degree is reduced.39 In particular, high temperature and high density contribute to enhancing the effect of radiation trapping. In the experiment, it means that the population in the F=2 sublevels fails to concentrate in the stretched state and the linewidth is not narrowed.

If this is the case, two approaches need to be considered to prepare and keep the atomic ensemble in a polarized state. One is the use of quenching gas to provide a non-radiative decay route back into the ground state and avoid reabsorption of spontaneous emission.40 However, this method reduces the benefits of anti-relaxation coated cells. Another approach is using a different pumping method, for example, introducing another alkali species to generate polarization via inter-species spin-exchange collisions.41 A calculation of the polarization degree is presented below to evaluate the first approach.

Three processes are considered in the calculation: pumping polarization, spontaneous emission, and emitted photons reabsorption. The probability to reabsorb a photon is proportional to the ratio of the cell size and the characteristic absorption length.42 The polarization degree p is given by:

dpdt=1pROPpRrelpROPLchar2Labsp,
(7)

where Rrel is the polarization relaxation rate, Lchar is the characteristic cell length (wall to wall distance) and Labsp is the absorption length.43 

The stationary solution for this rate equation is represented in Fig. 9. With quenching gas, polarization remains effective as expected. Without quenching gas, the degree of polarization decreases dramatically as the cell temperature increases while at room temperature the polarization degree is still high (∼1). In vapor cells without buffer gas, when spin-exchange relaxation contributes significantly to the transverse relaxation, almost half the atoms get depolarized due to radiation trapping. The calculation explains why light narrowing phenomenon is absent at high temperature in the appropriate regime. Applying high-resolution radio-frequency spectroscopy to measure the polarization degree based on quadratic Zeeman effects is a possible method for experimentally quantifying the relationship between the degree of atomic polarization and the linewidth.

FIG. 9.

Calculated polarization degree dependence on the cell temperature with and without quenching gas.

FIG. 9.

Calculated polarization degree dependence on the cell temperature with and without quenching gas.

Close modal

In summary, we present an amplitude-modulated NMOR atomic magnetometer with paraffin coated rubidium-87 vapor cells. To demonstrate the principles of light narrowing in dense vapor, three schemes are attempted: a single pump beam, synchronously modulated dual pump beams, as well as dual pump beams of which only one beam is modulated. However, light narrowing is not observed in our experiments. The analysis of polarization degree in vacuum cells reveals that the low orientation caused by radiation trapping accounts for the absence of light narrowing. Therefore, for light narrowing to be possible, it is necessary to find a way to eliminate radiation trapping. While the standard way to accomplish this is adding quenching gas such as nitrogen, this is not an acceptable solution for situations where the absence of buffer gas is explicitly desired, for instance, to reduce the sensitivity of the atomic magnetometer to magnetic-field gradients. Our findings provide useful input for the design of the future high-sensitivity atomic magnetometers in the Earth magnetic-field range.

This work was supported by Independent Research Program of Tsinghua University (52300300917) and China Scholarship Council (501100004543). DCH was supported by DARPA Contract D16 PC00191. The authors thank Andrejs Jarmola for his help with the experiment.

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