Combined effect of non-linear optical and collisional processes on absorption saturation in a dense rubidium vapour

We study non-linear absorption of intense monochromatic light through a dense natural rubidium (Rb) vapour. We measure transmission through a 10 cm long heated vapour cell for atom densities up to 3 × 1019 m−3 and saturation parameters up to 104, for linear and circular polarisation, close to resonance on the 87Rb D2 F = 1 → F′ = 0, 1, 2 transition. The strong absorption at low intensity is frustrated by an interplay of optical non-linearities (saturation and optical pumping) and non-linear effects due to the high atom density (collisional broadening and collisional depumping). To understand the results of the transmission measurements, we developed a model that incorporates these non-linear effects into the optical absorption. The model takes into account the absolute line strengths of all transitions from both hyperfine levels of the ground state of both isotopes of naturally abundant Rb. Doppler and collisional broadening are included in the Voigt profiles for the resonances. We show the effect of each of the non-linear processes on the calculation results of the model, and from comparison with experiment we conclude that all non-linear effects are necessary for a quantitative agreement.


Introduction
The study of optical properties of dense atomic vapours has resulted in the observation of fascinating non-linear effects on the propagation of light through such media [1]. An originally opaque dense vapour becomes transparent when excited with an intense control beam of laser light [2,3]. This electromagnetically induced transparency (EIT) is performed in various schemes with resonant [4,5] or off-resonant light [6,7] and is applied in all-optical switching techniques [8][9][10]. An extreme way of controlling the propagation is in slow 1 Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. light experiments, in which the group velocity of the light is reduced by several orders of magnitudes compared to the speed of light in vacuum [11,12]. In dense atomic vapours the velocity reduction has a broad bandwidth, making them suitable for slowing down picosecond pulses [13] and delaying nanosecond pulses by tens of pulse widths in a tunable alloptical delay line [14][15][16]. The dramatic reduction of speed to almost zero sparked research into storing light pulses in dense alkali vapours [17][18][19][20][21], which shows great promise in quantum network communication and synchronisation [22,23].
In recent years focus has been on propagation through extremely dense vapours. In heated rubidium millicells, the transmission at low intensity was demonstrated to undergo collisional broadening [24,25]. At frequencies below resonance and at high atomic densities, the propagation of the optical eld was found to mimic the time evolution of a 2D-Bose gas with repulsive interactions [26,27].
In the understanding of the effects of the experiments above, a proper description of the absorptive and refractive properties of the medium is pivotal. For speci c ranges in intensity, atom density and detuning, these properties have been reported in earlier work: absorption has been studied at low laser intensities for high vapour densities [24,28,29], absorption with optical pumping at medium intensities [30][31][32][33], and absorption and refraction at high intensity and high atom density for large detunings [26,34]. We capture these properties in a comprehensive model applicable over the whole range of detunings, saturation parameters and atomic densities. This makes the model suitable for all of the above mentioned areas of research. The principles of the model are applicable to any transition in an atomic vapour. We validate our model by comparing it to transmission measurements in a dense Rb vapour at high intensities close to resonance.
First we describe our experimental setup and present the measured on-resonance transmission curves as a function of saturation parameters up to 2.5 × 10 4 for temperatures from 60 to 130 • C. These results inspire us to construct a model to calculate the transmission, which includes the following nonlinear effects: collisional broadening, optical pumping to the dark ground state, collisional depumping and saturation. We show the in uence of each of the effects on the resulting transmission curve and show all effects are necessary to achieve quantitative agreement with the experimental results.

Experimental setup and transmission curves
The experimental setup to measure transmission through a heated cell containing rubidium with a natural isotope ratio is shown in gure 1. The 10 cm long vapour cell is wrapped with heating wires, isolated and placed in an aluminium heat shield. The temperature of the cell is controlled by the current Figure 2. Transmission for circular (circles) and linear (triangle) polarisation for different cell temperatures. The bare peak saturation parameter is given by I peak /I s , where I peak = 2P in /(πw 2 0 ) and I s is 1.6 (2.5) mW cm −2 for circular (linear) polarisation. Transmission is measured at ω = 384 234 576MHz (4170 MHz detuning in table 1), the transition strength weighted average of the 87 Rb F = 1 → F ′ = 0, 1, 2 manifold. Temperatures are measured by a thermocouple.
through the wires and monitored by a thermocouple in direct contact with the cell wall.
Our light source is a continuous wave external cavity diode laser. An anamorphic prism pair ensures a circular beam and a 60 dB Faraday isolator prevents unwanted re ections from destabilising the laser. Part of the light is split off by a beam splitter for characterisation of the laser frequency using a wavemeter and through a spectroscopy cell at room temperature for reference. After the beam splitter the light can be sent directly through the heated cell for low intensity calibration measurements, in which the light is collected by a photodiode and recorded on the oscilloscope together with the signal from the reference cell. For high intensity measurements, the beam is ampli ed in a tapered ampli er (TA) to an output power of 1.3 W. Afterwards a small portion of the light is detected by a photodiode and analysed using an electronic spectrum analyser to ensure the diode laser has no multimode sidebands. Spatial ltering converts the elongated TA output mode into a Gaussian beam with a waist w 0 of 850 µm. At the entrance window of the heated cell the beam power is 500 mW, which can be attenuated to lower powers by absorptive neutral density lters just before cell entry. For circular polarisation measurements a quarter wave plate is placed in front of these attenuators. The transmission is collected and recorded by a power metre directly after the vapour cell.
The measured transmission for circular and linear polarisation at several cell temperatures is shown in gure 2. To avoid the possibility of self-focusing effects which occur above the resonance, transmission traces were taken slightly below resonance. The bare peak s 0 is determined by I peak /I s where I peak = 2P in /(πw 2 0 ) and the saturation intensity I s is 1.6 (2.5) mW cm −2 for circular (linear) polarisation. The sudden rise from zero to non-zero of the transmission suggests a saturation effect. At the highest experimental s 0 the transmission levels off to a value depending on the atom density. This suggests a non-linear effect in atom density.

Transmission model
The intensity of light propagating through the dense rubidium vapour is subject to the Lambert-Beer equation where I is the intensity, z the propagation direction, ρ at the atomic density and α the absorption coef cient. Refraction has a negligible effect in on-resonance experiments and is therefore not included in this treatment of the transmission.
To determine the absorption coef cient α we take into account all allowed transitions that make up the Rb D 2 line [29]. For the correct transition frequencies of these transitions, we calculate the hyper ne frequency shifts for the ground state and the excited state of both isotopes [35,36]. Table 1 shows the resulting frequencies for all transitions from F g = i to F e = j in the D 2 manifold. The frequencies ω i j 0 are listed with respect to the excitation frequency of 85 Rb, as determined without hyper ne splitting. There we also list the transition strength coef cients R ij with respect to the cycling transition. The absorption coef cient α is calculated by summing over all the allowed transitions as where 3λ 2 2π is the light-atom scattering cross-section for a twolevel system, λ is the transition wavelength, ρ i is the density of atoms in ground state i, and s i j 0 = R i j I/I s is the transition speci c saturation parameter with I s the saturation intensity for the cycling transition. V(δ, s 0 ) is the Voigt pro le, which is a convolution of the Doppler pro le and a (collisionally and power) broadened Lorentz pro le.
Use of the Voigt pro le is required, because at high intensities power broadening makes the broadened Lorentz width comparable to the Doppler width and the Voigt pro le thus ensures the models validity over a large range of detunings: from far above to far below the Doppler width ( gure 3).
In the convolution only the axial velocity distribution needs to be taken into account, because the velocity distributions in different directions are decoupled. The Doppler integral for the Voigt pro le is thus where v th = k B T/m, γ is the linewidth of the transition and the detuning δ is dependent on the atom velocity v as δ(v) = δ 0 − v/λ, with δ 0 the original detuning of the excitation and λ the transition wavelength.
The Voigt pro les are obtained using the real part of the Faddeevva function w(z), for which we use the python implementation scipy.special.wofz, as follows: where The homogeneous linewidth γ is broadened at high densities due to collisional broadening and is given by γ = γ 0 + βρ at . For the rubidium D 2 line the natural linewidth γ 0 = 6.06MHz and the collisional broadening coef cient β = (1.10 ± 0.17) × 10 −19 m 3 MHz is determined in previous experimental work [25,37]. The saturation parameter is de ned as s 0 = 2Ω 2 /γ 2 with Ω the Rabi frequency, and since the linewidth broadening is a homogeneous effect, the bare peak saturation parameter needs to be adjusted by a factor γ 2 /γ 2 0 in the dense Rb vapour. The response of the medium to a beam of light with intensity I and frequency ω, now only depends on the temperature and the density. The density can be expressed as a function of temperature by [38] ρ at (T) = p k B T , with log 10 (p) = 5.0057 + 4.312 − 4040/T, where p is the vapour pressure and k B is the Boltzmann constant. We measure the temperature on a particular part of the cell wall, but the atoms in the vapour cell thermalise to the coldest point of the cell. Therefore we perform a calibration to relate the equilibrium temperature of the atoms to the measured temperature of the cell wall.

Atom temperature calibration
To calibrate the atom temperature in the vapour cell, we measure the transmission spectra of a low intensity beam and t the spectra using the model above ( gure 4). The transmission signals from the heated cell and the reference cell are recorded on an oscilloscope. In this experiment the laser frequency is varied by modulating the current through the laser diode. Intensity changes due to this modulation are compensated for by tting the background slope of the reference signal to second order in detuning.
As the transmission in a dense rubidium vapour is zero for large parts of the spectrum around the resonances, the frequency axis is calibrated by tting the reference signal with our calculation model as a function of temperature. The result for room temperature measurements in gure 4 shows a good t with a temperature of 18.9 • C. The t procedure was conrmed to produce undiscernible results for a t function with pure Doppler pro les, which is equally appropriate for low intensity and a tting range within a few Doppler widths of the resonances. We also veri ed the t function by reproducing the spectral ts of transmission through high density vapours in millicells [39].
At higher temperatures the transmission is zero over a large range around the resonances. The parts of the spectrum useful for tting are therefore located in the wings, where only the Voigt-pro les are valid. The ts in gure 4 show good agreement with the data. The table shows the relation between the temperature obtained from the spectral t (T sp ) and the temperature measured by the thermocouple (T tc ). We use a linear t to assign a spectral temperature to all the temperatures at which transmision is measured in gure 2 and nd the relation T sp = (5.9 ± 4.6) • C + (0.636 ± 0.028) T tc .
This equation provides a good working value for the spectral temperature to calculate the transmission for high density and high temperature conditions of our measurements, despite the uncertainty in the relation.

Saturation and optical pumping
Even though the above model includes saturation and power broadening, it is not yet suitable to use at high intensities, where optical pumping comes into play. In optical pumping a fraction of the atoms is pumped into an optical dark state, leaving fewer atoms available for excitation in the bright state. The effect is most prominent, when pumping is on resonance with one of the transitions from a particular ground hyper ne states. The strong laser beam pumps atoms from the resonant, bright hyper ne ground state to the excited state, from which they have a possibility of falling into the other, dark hyper ne ground state. The process is depicted in the Jablonski diagram in gure 5(a) and can be described by Einstein rate equations. The pumping rate from ground state i to the combined excited level e is given by where δ ij = ω − ω ij . The chance to fall back into the lower (upper) hyper ne ground state is given by the branching ratio br ei , 1/2 (1/2) for 85 Rb and 4/9 (5/9) for 87 Rb [36].
Since we work at high atomic densities, collisions occur frequently. The collisional depumping rates Γ 12 and Γ 21 , which redistribute the population over the ground states, therefore need to be included. The rates are not equal because of a difference in multiplicity m i and are given by   Here we have introduced the collision rate Γ col = ρ at v col σ col , where v col = 2v th is the most probable interatomic speed and σ col is the collisional cross section.
Collision rates and collisional cross sections speci cally are notoriously dif cult to determine and inherently speci c for the process under investigation. Collision rates have been determined for ultracold Rb atoms in magnetic and magnetooptical traps [40] and spin relaxation and excitation-transfer collisions between Rb and buffer gases [41,42], but not for the hyper ne ground state changing collision of concern here. To estimate this speci c cross section we determine the relevant impact parameter by calculating the minimum distance between atoms for which the energy exchange exceeds the hyper ne splitting of the 87 Rb ground state. This is the largest energy difference to be overcome for a hyper ne state change, which can therefore occur during a collision between any two atoms. The energy transfer for a given impact parameter depends on the type of interaction involved in the collision. The collision between identical atoms is described by a van der Waals potential C 6 /R 6 , for which the C 6 coef cient of Rb is 4.47 × 10 3 in atomic units [43,44]. The estimate for the impact parameter b in this case is 1.43 nm and the collisional cross section is σ 6 = πb 2 = 6.42 nm 2 , which is of the same order of magnitude as the total elastic scattering cross section Q = 23.6 nm 2 [40]. Dipole-dipole-type collisions, with van der Waals potenial C 3 /R 3 , between excited and ground state can be neglected, because the occupation of the excited state is severely reduced by optical pumping up to saturation parameters in the order of 10 5 , when the power broadening becomes of the order of the ground state hyper ne splitting.
We solve the Einstein rate equations for the above values and nd the populations of the ground states of both isotopes shown in gure 5(b). The population of the bright ground state reduces strongly when the pumping laser is on resonance with this bright state. We also notice this effect in the transmission spectra. The drop in bright state population strongly enhances the transmission.
The populations we nd are the ingredient for a nal adjustment of our treatment of saturation. The transitionspeci c saturation parameter depends on the occupation of the excited state in question. At high intensities, the contribution to the excitation from the other ground state cannot be neglected anymore, as due to power broadening their absorption overlaps signi cantly. The transition-speci c s i j 0 will therefore also include a contribution from the other ground state, which we write formally as a sum over all other ground states as where γ k j = γ 1 + R k j s 0 is the transition speci c power broadened linewidth. With the saturation parameters s i j 0 dependent on the populations ρ i , and the populations dependent on the saturation parameter, the model needs to calculate these quantities in an iterative procedure, for which convergence is reached after 3 iterations.

Transmission at resonance
The interplay of all the non-linear processes described above has a particularly intriguing effect on the transmission at resonance. We apply our model to the transmission at the 87 Rb F = 1 → F ′ = 0, 1, 2 transition over a wide range of saturation parameters s 0 for one typical temperature (111 • C). To accurately describe the experimental conditions, we simulate the propagation of a Gaussian beam by discretizing the Gaussian in 20 regions and propagating these regions in parallel in 10 steps. For comparison with experimentally obtained transmission, we compensate for losses from passing the windows of the cell by a factor of √ 0.85 at each window. This factor is chosen to match the measured transmission of 85% far offresonance at δ = 20 GHz. To illustrate the in uence of the separate non-linear effects, we turn the effects on one-by-one, as shown in gure 6.
We start with the curve taking into account only saturation (solid line). The threshold at which transmission becomes non-zero is at s 0 ≃ 10 5 , far from the experimentally observed value of ≃ 10 3 , and the curve has no intermediate plateau.
With the introduction of optical pumping (dashed-dotted line) a plateau appears for s 0 < 10 5 , caused by the near depletion of the bright state. As there are no collisions to repopulate the bright state, the bright state is only repopulated by the reverse optical pumping from the dark state, which is much weaker because it is much further detuned. In the steady state solution the population of the bright state is therefore only a few percent (see gure 5(b)), which is nevertheless responsible for signi cant absorption. As this fraction of the atom density is stable for all s 0 (up to s 0 ∼ 10 5 ), the amount of absorption and thus the level of transmission is determined by the atom density. The height of this plateau is therefore dependent on the atom density, in accordance with experimental observations.
With the introduction of collisions into our model (dotted line) the curve resembles the shape of the measurement data, as the effect of optical pumping are suppressed by collisional depumping for Γ 21 > Γ 1e br e2 . This criterion determines the threshold, where the transmission becomes non-zero. For the collisional rate estimated above, this occurs at s 0 ∼ 20 and is not in agreement with the experiments. We therefore adjust the collision rate through the estimated collisional cross section such that the model matches all of our measurements simultaneously.
The collisional cross section is an intrinsic property of the Rb vapour and independent on the atom density and temperature, for the range of temperatures that we have explored. The optimum value for the cross section is therefore best recovered by simultaneously matching our model to all of the measurements in gure 2, using a single value of the cross section as the only free parameter, for the temperature relation in equation (7). The resulting curves are judged on retrieval of the correct curve slopes and the correct thresholds for the value of s 0 at which transmission rst becomes non-zero. The results of this procedure for are shown in gures 6 and 7 for an optimum value of σ col = 67 nm 2 . In particular the correct slope for different temperatures and the correct threshold values for s 0 are captured. Note that this should not be interpreted as a measurement of σ col , as the uncertainty in the coef cients of the temperature calibration leads to a large uncertainty in σ col . The lower bound of the temperature yields an optimum for σ col = 322 nm 2 , whereas the upper bound gives σ col = 19 nm 2 . Since we are here concerned with the resulting optical properties, more accurate knowledge of T sp and thus σ col is not required.
The model only deviates from the measurement at T tc = 90 • C, at which the spectral temperature is more uncertain, as it is far away from the calibration points (19,166,183,188 and 202 • C) for the linear temperature t of equation (7). The performed calibration was mainly aimed at high temperatures, where non-linear effects are most prominent. But more importantly, at these intermediate temperatures the optical pumping scheme of gure 5, which neglects magnetic sublevels and hyper ne states of the excited state, might be an oversimplication. This notion is supported by the difference in behaviour for different polarisations, which is expected to happen when more subtle pumping effects including magnetic sublevels of the ground and excited states become important. For higher temperatures the model accurately describes the non-linear transmission through a dense rubidium vapour for s 0 < 10 4 . Figure 6. Transmission measurements at T tc = 166 • C and calculation at T at = 111 • C (from equation (7)). In the plots the non-linear effects are turned on one-by-one. With only saturation turned on (solid line) the threshold s 0 , where transmission becomes non-zero, is too high. With the introduction of optical pumping (dash-dotted line) an the atom density dependent plateau appears and transmission stays non-zero even for low s 0 . Addition of collisions to the model introduces a new theshold value for non-zero transmission, which is too low for the estimated collisional cross section of 6.4 nm 2 . With a collision rate of 67 nm 2 , adjusted to t our measurements, the model (dashed line) shows the correct threshold value of s 0 ≃ 10 3 . This collisional cross section is chosen to match the calculations to all experiments in gure 7 simultaneously. Temperatures measured by the thermocouple T tc and the corresponding temperatures T sp from absorption spectrum analysis (equation (7)) are used as input temperature for the model calculations (solid lines). The collisional cross section is set to σ col = 67 nm 2 .
For higher s 0 the effects of non-linear refraction and nonnegligible population of the excited state need to be taken into account.

Conclusion
We developed a model to describe the absorption of a continuous light source in a dense rubidium vapour at high intensity. The model includes non-linear effects in intensity (optical pumping and saturation) and atomic density (collisional broadening and collisional depumping). The principles of the model are applicable to any transition in an atomic vapour. The model is particularly straight forward to transfer to the D 2 line of another alkali metal, as only values of the atomic properties need to be adjusted. We compare the results of our model to measurements of transmission through a 10 cm cell containing a dense vapour of a natural isotope mixture of rubidium. The transmission for light resonant with the highest line in the Rb D 2 manifold shows good agreement for saturation parameters s 0 < 10 4 and temperatures 90 • C < T sp < 130 • C, when we assume a collisional cross section of σ col = 67 nm 2 . We show that all the non-linear effects included in the model are essential to explain the characteristics of the transmission curves.