Theoretical and experimental study of the 6S-8S two-photon absorption cross-section in cesium atoms

Michael Caracas Núñez; Miguel A. Gonzalez; Mayerlin Núñez Portela

doi:10.1364/OE.496654

1. Introduction

Two-photon absorption (TPA) is a multiphotonic process that was first analyzed theoretically by Göppert-Mayer in 1930 where the absorption probability of two simultaneous photons in atoms was calculated [1]. This probability is quantified by means of the TPA cross-section, $\sigma$. Measurements of this cross-section when exciting with laser light, i.e., classical TPA, have many applications in different fields [2]. Recently, the probability of producing TPA in atoms and molecules using light sources with different statistical properties has acquired an increasing interest due to the possibility to enhance this process [3–6]. As a result, TPA has also been theoretically and experimentally studied in different samples using thermal light [7], squeezed light [8–11] and entangled photon pairs [12–16]. In particular, results of Entangled TPA (ETPA) experiments have many questions regarding the difficulty to isolate the ETPA signal in organic molecules [17,18]. An alternative to detect an ETPA process is to study it in systems with a narrow bandwidth, such as atomic samples. In these, ETPA can be measured from the spectrum of the two-photon transition. The first step towards this experiment is to determine the values of the classical TPA cross-section $\sigma$ in atomic samples [15].

Theoretical values of $\sigma$ have been calculated for different molecules and atoms using second-order perturbation theory and model potentials [19–25]. Experimentally, the TPA cross-section can be measured in molecules by direct methods e.g., z-scan [26], or by indirect methods that quantify secondary effects produced from the TPA process, e.g., thermal lensing [27], multiphoton ionization [28] and two-photon induced fluorescence (TPIF) [29–31]. In atoms, the TPA cross section has been measured mainly by TPIF in Rubidium [32], Oxygen [33] and Germanium [34].

For Cesium (Cs) the theoretical and experimental values of the TPA cross-section have not been reported. In this work the classical TPA cross-section for the $6S_{1/2}\rightarrow 8S_{1/2}$ transition in Cs atoms is calculated using second order perturbation theory. An experiment is set up to measure this cross-section based on the quadratic dependence between the TPA transition rate and the intensity of the laser light interacting with a thermal atomic vapor. These results are the starting point for ETPA experiments in Cs. The information presented here complements the spectroscopic information of the $6S_{1/2}\rightarrow 8S_{1/2}$ transition and has potential applications in experiments regarding Four-Wave Mixing [35,36], atomic clocks [37], the generation of light sources from alkali vapors [38], among others.

2. Theoretical framework

In order to calculate the classical TPA cross-section, the Hamiltonian, $\hat {H}$, of a hydrogen-like atom interacting with an electromagnetic wave is considered. In general,

(1)$$\hat{H}=\hat{H}_0+\hat{H}_{int},$$

with $\hat {H}_0$ the Hamiltonian of the unperturbed atomic system. The eigenstates and eigenvalues of $\hat {H}_0$ are $\left \{ | {\psi _n}\rangle \right \}$ and $\left \{ E_n \right \}$, respectively. The interaction with the electromagnetic field is a perturbation described by $\hat {H}_{int}$. Considering an initial atomic state $| {\psi _i}\rangle$, the state of the system for a given time $t$ can be written as $| {\psi (t)}\rangle = \hat {U}(t,t_0)| {\psi _i}\rangle$, where $\hat {U}(t,t_0)$ is the temporal evolution operator. In the interaction picture, $\hat {H}_{int}^{(I)}(t)$ is defined as

(2)$$\hat{H}_{int}^{(I)}(t) = e^{i\hat{H}_0t/\hbar}\hat{H}_{int}(t)e^{{-}i\hat{H}_0t/\hbar},$$

and $\hat {U}^{(I)}(t,t_0)$ can be expanded in a Dyson series in terms of $\hat {H}_{int}^{(I)}$ as [39]

(3)$$\hat{U}^{(I)}(t,t_0)=1-\frac{i}{\hslash}\int_{t_0}^{t} \text{d}t' \hat{H}_{int}^{(I)}\left(t'\right)+\left(-\frac{i}{\hslash}\right)^2\int_{t_0}^{t}\text{d}t'\int_{t_0}^{t'}\text{d}t^{\prime\prime}\hat{H}_{int}^{(I)}(t')\hat{H}_{int}^{(I)}(t^{\prime\prime})+\dots.$$

The second-order term in the expansion is related to a two-photon excitation of the atom. The transition probability amplitude between an initial state $| {\psi _i}\rangle$ and a final state $| {\psi _f}\rangle$ for this process is therefore,

A_{i\rightarrow f}^{(2)}(t)=\left(-\frac{i}{\hbar}\right)^2\sum_m\int_{t_0}^{t}\text{d}t'\int_{t_0}^{t'}\text{d}t^{\prime\prime}\langle{\psi_f}|\hat{H}_{int}(t')|{\psi_m}\rangle{\langle}{\psi_m}|\hat{H}_{int}(t^{\prime\prime})|{\psi_i}{\rangle}e^{i\omega_{fm}t'}e^{i\omega_{mi}t^{\prime\prime}},

where the sum runs over all possible eigenstates $| {\psi _m}\rangle$ with energy $E_m$. The frequencies $\omega _{mi}=\omega _m-\omega _i=(E_m-E_i)/\hbar$ and $\omega _{fm}=\omega _f-\omega _m=(E_f-E_m)/\hbar$ correspond to the transition frequencies between the $| {\psi _i}\rangle\rightarrow | {\psi _m}\rangle$ states and $| {\psi _m}\rangle\rightarrow | {\psi _f}\rangle$ states, respectively.

The interaction of the atom with an electromagnetic (EM) monochromatic plane wave, $\mathbf {E}(\mathbf {r},t)$, can be described in terms of the electric dipole operator $e\hat {\mathbf {r}}$, such that

(4)$$\hat{H}_{int}(t)={-}e\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r},t)={-}\frac{eE_0 \boldsymbol{\hat{\mathbf{r}}}\cdot \boldsymbol{\epsilon}}{2}\left[e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}+e^{{-}i(\mathbf{k}\cdot\mathbf{r}-\omega t)}\right] .$$

In this expression, $\mathbf {k}$ is the wave vector, $\omega$ the frequency of the EM radiation, $E_0$ the amplitude of the electric field and $\boldsymbol {\epsilon }$ the direction of the polarization of the electromagnetic wave [39]. The first term in Eq. (4) corresponds to stimulated absorption of photons and the second to stimulated emission. Considering only the absorption term and the rotating wave approximation, the transition probability amplitude can be written as

(5)$$A_{i\rightarrow f}^{(2)}(t)=\left(\frac{eE_0}{2\hslash}\right)^{2}\sum_{m}\frac{1}{3}\frac{\mu_{fm}\mu_{mi}}{\omega_{mi}-\omega}\left[\frac{e^{i(\omega_{mi}+\omega_{fm}-2\omega)t}-1}{\omega_{mi}+\omega_{fm}-2\omega}\right].$$

Here $\frac {1}{3}|\mu _{mg}|^2=|\langle \psi _m|\boldsymbol {\epsilon }\cdot \hat {\mathbf {r}}|\psi _g\rangle |^2$ indicates the electric dipole moment matrix elements and the factor $1/3$ accounts for all possible orientations of the atom with respect to the polarization of the field [40]. The probability for the two-photon transition is therefore,

(6)$$\big{|}A_{i\rightarrow f}^{(2)}(t)\big{|}^2= 4\left(\frac{eE_0}{2\hslash}\right)^{4}\left | \sum_{m}\frac{1}{3}\frac{\mu_{fm}\mu_{mi}}{\omega_{mi}-\omega} \right |^{2}\left[\frac{\sin\left( t(\omega_{fi}-2\omega)/2 \right)}{\omega_{fi}-2\omega}\right]^2.$$

When the frequency of the EM radiation is close to half the resonance frequency of the transition, $2\omega \rightarrow \omega _{if}$, it is possible to approximate

(7)$$\lim_{t\to\infty}\left[\frac{\sin\left( t(\omega_{fi}-2\omega)/2\right)}{\omega_{fi}-2\omega}\right]^2=\frac{\pi t}{2}\delta(\omega_{fi}-2\omega).$$

The transition rate of the TPA process, $\Gamma _{i\rightarrow f}^{(2)}$, is obtained from the probability in Eq. (6) as

(8)$$\Gamma_{i\rightarrow f} ^{(2)}= \frac{\mathrm{d}\big{|}A_{i\rightarrow f}^{(2)}(t)\big{|}^2 }{\mathrm{d}t}=\left( \frac{2\pi}{\hslash^{4}} \right)\left( \frac{eE_0}{2} \right)^{4}\left | \sum_{m}\frac{1}{3}\frac{\mu_{fm}\mu_{mi}}{\omega_{mi}-\omega} \right |^{2}\delta(\omega_{fi}-2\omega).$$

The $\delta (\omega _{fi}-2\omega )$ function can be replaced by a Lorentzian function that describes the line-shape of the transition. In this case [39],

(9)$$\delta(\omega_{fi}-2\omega)=\lim_{\gamma_{f}\to 0}\frac{1}{2\pi}\frac{\gamma_{f}}{(\omega_{fi}-2\omega)^{2}+\gamma_{f}^{2}/4},$$

with $\gamma _f=1/\tau$ the natural line-width of the transition and $\tau$ the lifetime of the excited state.

For a system with $\mathcal {N}$ atoms per unit of volume, the total transition rate of the TPA process is $\Gamma _{T}^{(2)}=\mathcal {N} \Gamma _{i\rightarrow f}^{(2)}$. According to the propagation equation for an EM wave through a medium, the total transition rate can be written in terms of the TPA cross-section $\sigma$ as [41]

(10)$$\Gamma_{T}^{(2)}= \sigma\frac{\mathcal{N}I^{2}}{2\hbar^{2}\omega^{2}},$$

where $I=\frac {c\epsilon _0}{2}E_0^{2}$ is the intensity of the light beam, $\epsilon _0$ is the vacuum permittivity, $c$ is the speed of light in vacuum and $\omega =2\pi \nu$ with $\nu$ the true frequency.

From Eqs. (8) and (10), the TPA cross-section takes the explicit form,

(11)$$\sigma (\omega) = \frac{e^{4}\mu_0\omega}{2\epsilon_0 \hbar^{3}}\left[ \sum_{m,j}\frac{1}{9} \frac{\mu_{ij}\mu_{jf}\mu_{fm}\mu_{mi}}{(\omega_{mi} - \omega)(\omega_{ji} - \omega_{int})} \right]\frac{\gamma_{f}}{(\omega_{fi}-2\omega)^{2}+\gamma_{f}^{2}/4}.$$

This expression considers all the possible paths in the system between the ground state and the excited state. It also gives the dependence of $\sigma$ with respect to $\omega$, the frequency of the EM radiation. The frequency $\omega _{int}$ in Eq. (11) is a fixed value corresponding to half of the energy of the two-photon decay process [20].

For an atomic system with hyperfine structure (HFS), the TPA cross-section must be calculated considering the hyperfine energy levels. In this case, $\sigma (\omega )$ is proportional to the line-strength, $S_{m,n}$, between the hyperfine transitions, such that [42],

(12)$$\mu_{fm}\mu_{mi} \longrightarrow \mu_{fm}\mu_{mi}\times \sum_n S_{m,n}.$$

The values of $S_{m,n}$ are quantified by the $6-j$ Wigner symbols as

(13)$$\begin{aligned}S_{m,n} &=({-}1)^{J_f+J_m+2I+F_i+F_{m,n}+2p}(2F_{m,n}+1)\sqrt{(2F_i+1)(2F_f+1)}\\ &\times \begin{Bmatrix} J_m & F_{m,n} & I \\ F_i & J_i & p \end{Bmatrix}\begin{Bmatrix} J_f & F_f & I \\ F_{m,n} & J_m & p \end{Bmatrix}, \end{aligned}$$

where $n$ denotes the possible hyperfine energy levels for the intermediate state $| {\psi _m}\rangle$, $J$ corresponds to the total angular momentum of the electron, $I$ to the nuclear spin, and $F=J+I$, to the total angular momentum of the atom. From the selection rules for $\Delta J$ and $\Delta F$, $p$ can take the following values:

(14)$$\begin{aligned}\Delta J=0\rightarrow p &= 0 ,\\ \Delta J\neq 0\rightarrow p &= 1 . \end{aligned}$$

Therefore, the total TPA cross-section for the transition $| {\psi _i,F_i}\rangle\rightarrow | {\psi _f,F_f}\rangle$ is given by

(15)$$\sigma_{F_i,F_f}(\omega) = \frac{\mu_0 e^{4}\omega}{2\epsilon_0 \hbar^{3}} \frac{\gamma_{f}}{(\omega_{fi}-2\omega)^{2}+\gamma_{f}^{2}/4}\sum_{m,j,n}\frac{1}{9} \frac{\mu_{ij}\mu_{jf}\mu_{fm}\mu_{mi}}{(\omega_{mi} - \omega)(\omega_{ji} - \omega_{int})}\times S_{m,n}\times S_{j,n}.$$

In the following section, the values of the TPA cross-section for the $6S_{1/2}\rightarrow 8S_{1/2}$ transition in Cs atoms, including the hyperfine levels, are obtained.

3. Theoretical values of the TPA cross-section in Cs atoms

The energy levels of Cs considered in this work are presented in Fig. 1. The one-photon transitions are represented by solid lines with the respective wavelengths. Blue lines correspond to transitions relevant for the experimental detection of the TPA process. Dashed lines show the wavelengths of both the two-photon transition and a single photon from the EM radiation. The branching ratios for the different decay paths from the $8S_{1/2}\rightarrow 6S_{1/2}$ transition are also displayed. The HFS of the $6S_{1/2}$ and $8S_{1/2}$ energy levels is presented in the right part of Fig. 1. The $8P, 9P, 10P,\ldots,$ states are not considered as a first approximation since the terms $1/(\omega -\omega _{ij})$ and $1/(\omega _{ij}-\omega _{int})$ in Eqs. (11) and (15) for these energy levels are small in comparison with those for the $6P$ and $7P$ states.

Fig. 1. Energy levels of atomic Cs relevant for this work. Solid lines represent one-photon transitions. Dashed lines indicate two-photon transitions. The values for the branching ratios and the wavelengths are presented. Right part of the image shows the hyperfine splitting of the $6S_{1/2}$ and $8S_{1/2}$ energy levels.

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Table 1 summarizes the spectroscopic information of Cs relevant to calculate the values of the TPA cross-section for the $6S_{1/2}\rightarrow 8S_{1/2}$ transition according to Eqs. (11) and (15). The line-width of this transition is $\gamma _f=9.62\times 10^{6}$ s$^{-1}$ obtained from the lifetime of the $8S_{1/2}$ state, $\tau =104~$ns [43]. Frequencies presented are derived from the wave numbers reported in the references. Dipole moment matrix elements are converted from atomic units to SI units. Values of the branching ratios are taken from [44].

Table 1. Summary of the transition frequencies, $\nu$, dipole moments per unit charge, $\mu _{ij}$, and branching ratios relevant for the theoretical study of $\sigma$ with the respective references.^a

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The cross-sections, $\sigma (\omega )$ and $\sigma _{F_i,F_f}(\omega )$, are calculated near the resonance frequency of the TPA transition. Figure 2(a) shows the cross-section as a function of the detuning of two times the laser frequency with respect to the resonance frequency, $\Delta =2\nu -\nu _{6S_{1/2}\rightarrow 8S_{1/2}}$. Figure 2(b) presents TPA cross-section as a function of $\Delta$ when considering the HFS of the $6S_{1/2}$ and $8S_{1/2}$ energy levels. The maximum in Fig. 2(a) corresponds to the value of the TPA cross-section at the resonance frequency such that $\sigma (\Delta =0)=\sigma _0$. The maximum of the curves in Fig. 2(b) corresponds to the values for the hyperfine transitions.

Fig. 2. TPA cross-section $\sigma (\Delta )$ as a function of the frequency detuning. (a) TPA cross-section for the $6S_{1/2}\longrightarrow 8S_{1/2}$ transition in Cs atoms according to Eq. (11). (b) TPA cross-section for the allowed hyperfine transitions $F_i=3\rightarrow F_f=3$ and $F_i=4\rightarrow F_f=4$ according to Eq. (15).

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Equations (11) and (15) are modified when considering a thermal atomic vapor. In this case, the line-shape of the TPA transition is dominated by a Gaussian distribution due to Doppler broadening. Since this effect does not change the area in the absorption curve, the maximum value for the TPA cross-section decreases [42]. Therefore, the Doppler broadened TPA cross-section, $\widetilde \sigma (\omega )$, is described by [42],

(16)$$\widetilde\sigma(\omega) = \widetilde\sigma_{0} \exp\left[-\left(\frac{2\omega-\omega_{fi}}{\Gamma_D}\right)^{2}\right],$$

where, $\widetilde \sigma _{0}=\frac {\gamma _f\sqrt {\pi }}{2\Gamma _D}\sigma _0$ is the Doppler broadened cross-section in resonance. In this expression $\Gamma _D$ is the Doppler width, and $\delta \omega _D$ is the FWHM of the lineshape given by

(17)$$\delta\omega_D=2\sqrt{\ln 2}\Gamma_D = \frac{\omega_{fi}}{c}\sqrt{\frac{8k_BT \ln 2}{M}}$$

with $M$ the molar mass of the atoms, $T$ the temperature of the gas and $k_B$ the Boltzmann constant. Figure 3 shows the values of the TPA cross-section as a function of $\Delta$ when considering Doppler broadening for a vapor cell at room temperature ($T=287$ K). Figure 3(a) plots $\widetilde \sigma$ for the $6S_{1/2} \longrightarrow 8S_{1/2}$ transition and Fig. 3(b) shows the results when considering the HFS. The latter are compared with the experimental results presented in the following section.

Fig. 3. TPA cross-section $\widetilde \sigma$ as a function of the detuning $\Delta$ when considering Doppler broadening for a thermal atomic vapor at $287$ K. (a) TPA cross-section for the Doppler broadened $6S_{1/2}\longrightarrow 8S_{1/2}$ transition in Cs atoms according to Eq. (16). (b) TPA cross-section for the Doppler broadened hyperfine transitions $F_i=3\rightarrow F_f=3$ and $F_i=4\rightarrow F_f=4$ according to Eq. (16).

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4. Experimental setup

Experimentally, the TPA cross-section can be obtained by measuring the dependence of the total transition rate, $\Gamma _{T}^{(2)}$, with the intensity of the laser light, according to Eq. (10). In a thermal atomic vapor, the total rate of the TPA process, $R_{TPA}$, is proportional to the integral of $\Gamma _{T}^{(2)}$ over the excitation volume. This volume is defined by the waist of the laser light in the sample and two times the Rayleigh range. The intensity, $I$, is quantified by the power of the laser light, $P$, and the waist of the laser beam [2,31]. Therefore,

(18)$$R_{TPA}={-}\mathcal{N}~ \widetilde\sigma_{0}\frac{2P^{2}}{hc}.$$

This rate is proportional to the maximum value of the spectrum of the two-photon transition. In this work, the spectrum is measured via TPIF and the maximum of the fluorescence signal, $F$, is

(19)$$F = \left | R_{TPA}\cdot\eta\right |,$$

where $\eta$ accounts for all the efficiencies involved in the detection process.

The experimental setup employed to perform such measurements is presented in Fig. 4. An extended cavity diode laser (LP820P100) in Littrow configuration is employed to drive the two-photon transition. An optical isolator reduces the intensity of the back reflections to the laser diode. A small fraction of the laser light is reflected by a beam-splitter (BS) and coupled to a single mode fiber connected to a wavelength-meter (WLM) (High Finesse WS/6-200). Lenses $f_1=25$ mm and $f_2=100$ mm generate a magnification of the incident laser beam. The laser light is then focused, using $f_3=50$ mm, obtaining a beam waist of $9.0\pm 0.3~\mu$m. A Cs vapor cell from Thorlabs (GC25075-CS) is placed in the waist of the laser beam. The power of the laser light is changed using 6 different neutral density filters (NDF) with optical densities $ND=0.1-0.6$. The TPIF signal is measured using a photomultiplier tube (PMT Hamamatsu R5929). The PMT pulses generated from the single-photon detection are counted with a time to digital converter (TDC quTAU H). A spectral filter (SF) from Semrock (FF01-715/SP-25) is placed in front of the PMT to detect the fluorescence from the transitions $7P_{1/2}\longrightarrow 6S_{1/2}$ and $7P_{3/2}\longrightarrow 6S_{1/2}$ in Fig. 1. A long-pass filter (LPF) and a black box are employed to reduce the background noise in the PMT signal associated with the external light of the laboratory.

Fig. 4. Experimental setup implemented to detect TPIF in Cs atoms. A diode laser at 822 nm is sent to the vapor cell. Lenses $f_1$, $f_2$ and $f_3$ are employed to set the waist of the light in the sample. The power of the laser beam is changed by means of neutral density filters. The fluorescence signal is detected with a PMT. The frequency of the laser light is constantly monitored with a wavelength-meter (WLM).

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The frequency of the laser light can be centered at either of the hyperfine transitions by changing the current and the temperature of the laser diode. Once the frequency of the transition is reached, a 1.5 GHz scan of the laser frequency around $\nu _{fi}/2$ is implemented by modulating the laser diode current. This modulation varies the current over a range of $0.4$ mA changing the total power of the laser light by $0.2$ mW. This corresponds to a fluctuation in the laser light intensity below 1%.

5. Results and discussion

The fluorescence signals induced from the $6S_{1/2}\longrightarrow 8S_{1/2}$ hyperfine transitions are measured as a function of the laser light frequency for different values of the power of the laser light. Figures 5(a) and (b) show the spectra for the $F_i=4\longrightarrow F_f=4$ transition, and Figs. 5(c) and (d) show the spectra for the $F_i=3\longrightarrow F_f=3$ transition. The power of the laser light is indicated in the inset of the figures. Since these measurements are performed in a gas cell at room temperature, Gaussian profiles are fitted to each set of data. Solid lines in the figures correspond to these curves. From the fitting parameters, the average FWHM obtained from all the measurements is $\delta \omega _D=2\pi (766873 \pm 1$ kHz), consistent with the temperature of the laboratory, $\approx 287$ K.

Fig. 5. Two-photon induced fluorescence signal measured as a function of the detuning for different laser intensities. (a) and (b) show the spectrum for the $F_i=4\longrightarrow F_f=4$ transition. (c) and (d) show the spectrum for the $F_i=3\longrightarrow F_f=3$ transition. Solid lines correspond to a Gaussian function fit to each data set.

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The maximum photon count rate from each fluorescence signal in Fig. 5 is plotted as a function of the power of the laser beam in Fig. 6. In this figure, data represented by stars correspond to the $F_i=3\longrightarrow F_f=3$ transition and the dots correspond to the $F_i=4\longrightarrow F_f=4$ transition. Solid lines are the fits of quadratic functions of the form $aP^2+c$ to each set of data. The values of $\widetilde \sigma _{0}$ are obtained from Eqs. (18) and (19) using the fitting parameter $a$ and the values of $\eta$ and $\mathcal {N}$.

Fig. 6. Maximum of TPIF signal as a function of the power of the laser light. Dots and stars correspond to the experimental data for the $F_i=4\longrightarrow F_f=4$ and $F_i=3\longrightarrow F_f=3$ transitions respectively. Solid lines correspond to quadratic fits to the data.

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Table 2 shows the relevant information to obtain the cross-section from the experimental data in Fig. 6. The values $a_{3\rightarrow 3}$ and $a_{4\rightarrow 4}$ correspond to the fitting parameters of the quadratic functions. The value of $\mathcal {N}$ is estimated from the vapor pressure of Cs at $287$ K and the excitation volume. The vapor pressure is calculated by means of the Antoine equation with the coefficients reported in [51]. The value of $\eta$ depends on two main factors, the probability of producing a fluorescence signal from a TPA process, and the efficiency in the detection scheme. The first one, $\eta _{BR}$, is quantified by the branching ratios of the detected transitions showed as blue lines in Fig. 1. The second factor includes the solid angle covered by the PMT, $\eta _G$, the sensitivity and cathode efficiency of the PMT, $\eta _{PMT}$, and the transmission efficiency of the spectral filter, $\eta _F$. The value of $\eta$ is therefore $\eta =\eta _{BR}\cdot \eta _{PMT}\cdot \eta _{G}\cdot \eta _{F}$.

Table 2. Quantities required to obtain the TPA cross-section from the experiment according to Eqs. (18) and (19).^a

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With the information in Table 2 the experimental values obtained for the two hyperfine transitions are $\widetilde \sigma _{3,3} = (3.6\pm 1.1)\times 10^{-24}$(cm$^{4}$/W) and $\widetilde \sigma _{4,4} = (4.6\pm 1.1)\times 10^{-24}$(cm$^{4}$/W). Table 3 summarizes the results of the TPA cross-section obtained in this work. Theoretical results include hyperfine structure and Doppler broadening. The theoretical and experimental values for the Doppler broadened cross-section for both hyperfine transitions agree within the uncertainty of the experiment. Notice that the theoretical values are below the experimental measurements. A similar difference between theory and experiment has been reported for the 5S-5D two-photon transition in rubidium [20,32]. This difference can be reduced by taking into account additional states ($8P,9P,10P$) in the theoretical calculations.

Table 3. Theoretical and experimental results obtained for the TPA cross-section of the $6S_{1/2}\rightarrow 8S_{1/2}$ transition in Cs atoms.

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In addition, the relative strength of the two hyperfine transitions is the same for theory and experiment when taking the ratio between $\widetilde \sigma _{3,3}$ and $\widetilde \sigma _{4,4}$. Moreover, the difference between the two resonance frequencies of the two hyperfine transitions measured in the laboratory is $2\times (4156\pm 1)$ MHz. This measurement is in good agreement with the experimental value reported by [52]. This is the first time that these TPA cross-sections are reported for the $6S_{1/2}\longrightarrow 8S_{1/2}$ transition in Cs.

6. Conclusions

In this work, we calculated the theoretical values of the TPA cross-section for the $6S_{1/2}\longrightarrow 8S_{1/2}$ transition in Cs atoms using second order perturbation theory. We reported values of the cross-section that considered both the hyperfine structure of the energy levels and the effects of Doppler broadening. An experiment was set up to measure the TPA cross-section for the $6S_{1/2}\longrightarrow 8S_{1/2}$ hyperfine transitions. The experiment exploited the quadratic dependence between the TPA transition rate and the intensity of the laser light. The values obtained from the measurements are in agreement with the theory for the two hyperfine transitions, $F_i=3\longrightarrow F_f=3$ and $F_i=4\longrightarrow F_f=4$. This is the first time that these cross-sections are calculated and measured. The results reported here are the starting point to design experiments of TPA in Cs atoms using entangled photon pairs.

Funding

Facultad de Ciencias, Universidad de los Andes (INV-2020-105-2081).

Acknowledgments

The authors thank the members of the Experimental Quantum Optics Group, Department of Physics, Universidad de los Andes, Colombia, for useful discussions regarding the results presented in this paper.

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 corresponding author upon reasonable request.

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Transition	Frequency $[$ THz $]$	$μ_{i j}$ [ $10^{- 10}$ m]	Branching Ratio	Ref.
$6 S_{1 / 2} \to 6 P_{1 / 2}$	335.116048807(41)	1.6840	1	[45,46]
$6 S_{1 / 2} \to 6 P_{3 / 2}$	351.72571850(11)	2.3700	1	[46,47]
$6 S_{1 / 2} \to 7 P_{1 / 2}$	652.50461	0.14859	0.29	[48,49]
$6 S_{1 / 2} \to 7 P_{3 / 2}$	657.93241	0.30586	0.36	[48,49]
$6 P_{1 / 2} \to 8 S_{1 / 2}$	393.89197	0.5429	0.21	[48,50]
$6 P_{3 / 2} \to 8 S_{1 / 2}$	377.28410	0.7731	0.37	[48,50]
$7 P_{1 / 2} \to 8 S_{1 / 2}$	76.501147	4.928	0.14	[48,50]
$7 P_{3 / 2} \to 8 S_{1 / 2}$	71.0774 $^{a}$	7.443	0.27	[50]
$6 S_{1 / 2} \to 8 S_{1 / 2}$	729.009799	TPA	TPA	[48]
$F_{i} = 3 \to F_{f} = 3$	729.014477	TPA	TPA	[37]
$F_{i} = 4 \to F_{f} = 4$	729.006161	TPA	TPA	[37]

Parameters	Symbol	Value
Fitting parameter	$a_{3 \to 3}$	$1.1414 \pm 0.0005$ (mW $^{2}$ s) $^{- 1}$
Fitting parameter	$a_{4 \to 4}$	$1.4854 \pm 0.0006$ (mW $^{2}$ s) $^{- 1}$
Atomic density	$N$	$1.51 \times 10^{16}$ atoms $/$ m $^{3}$
Branching Ratios	$η_{B R}$	0,1378
PMT efficiency	$η_{P M T}$	$0.11 \pm 0.02$
Geometric efficiency	$η_{G}$	$0.017 \pm 0.007$
Infrared filter	$η_{F}$	$0.80 \pm 0.02$

TPA Cross-Section	$6 S_{1 / 2} \to 8 S_{1 / 2}$	$F_{i} = 3 \to F_{f} = 3$	$F_{i} = 4 \to F_{f} = 4$
Doppler free [cm $^{4}$ /W]	$4.58 \times 10^{- 21}$	$1.05 \times 10^{- 21}$	$1.23 \times 10^{- 21}$
Doppler broadened [cm $^{4}$ /W]	$1.36 \times 10^{- 23}$	$3.10 \times 10^{- 24}$	$3.65 \times 10^{- 24}$
Experimental [cm $^{4}$ /W]		$(3.6 \pm 1.1) \times 10^{- 24}$	$(4.6 \pm 1.1) \times 10^{- 24}$

Transition	Frequency $[$ THz $]$	$μ_{i j}$ [ $10^{- 10}$ m]	Branching Ratio	Ref.
$6 S_{1 / 2} \to 6 P_{1 / 2}$	335.116048807(41)	1.6840	1	[45,46]
$6 S_{1 / 2} \to 6 P_{3 / 2}$	351.72571850(11)	2.3700	1	[46,47]
$6 S_{1 / 2} \to 7 P_{1 / 2}$	652.50461	0.14859	0.29	[48,49]
$6 S_{1 / 2} \to 7 P_{3 / 2}$	657.93241	0.30586	0.36	[48,49]
$6 P_{1 / 2} \to 8 S_{1 / 2}$	393.89197	0.5429	0.21	[48,50]
$6 P_{3 / 2} \to 8 S_{1 / 2}$	377.28410	0.7731	0.37	[48,50]
$7 P_{1 / 2} \to 8 S_{1 / 2}$	76.501147	4.928	0.14	[48,50]
$7 P_{3 / 2} \to 8 S_{1 / 2}$	71.0774 $^{a}$	7.443	0.27	[50]
$6 S_{1 / 2} \to 8 S_{1 / 2}$	729.009799	TPA	TPA	[48]
$F_{i} = 3 \to F_{f} = 3$	729.014477	TPA	TPA	[37]
$F_{i} = 4 \to F_{f} = 4$	729.006161	TPA	TPA	[37]

Parameters	Symbol	Value
Fitting parameter	$a_{3 \to 3}$	$1.1414 \pm 0.0005$ (mW $^{2}$ s) $^{- 1}$
Fitting parameter	$a_{4 \to 4}$	$1.4854 \pm 0.0006$ (mW $^{2}$ s) $^{- 1}$
Atomic density	$N$	$1.51 \times 10^{16}$ atoms $/$ m $^{3}$
Branching Ratios	$η_{B R}$	0,1378
PMT efficiency	$η_{P M T}$	$0.11 \pm 0.02$
Geometric efficiency	$η_{G}$	$0.017 \pm 0.007$
Infrared filter	$η_{F}$	$0.80 \pm 0.02$

Theoretical and experimental study of the 6S-8S two-photon absorption cross-section in cesium atoms

Abstract

1. Introduction

2. Theoretical framework

3. Theoretical values of the TPA cross-section in Cs atoms

4. Experimental setup

5. Results and discussion

6. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (3)

Equations (20)

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