Measurement of the atom-surface van der Waals interaction by transmission spectroscopy in a wedged nano-cell

TL;DR

This paper introduces a transmission spectroscopy method in a wedged nano-cell to precisely measure atom-surface van der Waals interactions, enabling better characterization of atom-based nano-devices.

Contribution

The study presents a novel experimental approach to measure atom-surface interactions with high precision using a wedged nano-cell and theoretical modeling.

Findings

Measured van der Waals coefficient C3 for cesium in sapphire cell.

Demonstrated high-precision determination of atom-surface interaction coefficients.

Validated the method's effectiveness for characterizing atom-surface interactions.

Abstract

We demonstrate a method for measuring atom-surface interactions using transmission spectroscopy of thermal vapors confined in a wedged nano-cell. The wedged shape of the cell allows complementary measurements of both the bulk atomic vapor and atoms close to surfaces experiencing strong van der Waals atom-surface interaction. These are used to tightly constrain the dipole-dipole collisional parameters of a theoretical model for transmission spectra that accounts for atom-surface interactions, cavity effects, collisions with the surface of the cell and atomic motion. We illustrate this method on a cesium vapor in a sapphire cell, find $C_3=1.3\pm0.1$\,kHz.$\mu$m$^3$ and demonstrate that even the weakest of the van der Waals atom-surface interaction coefficients - for ground-state alkali atom transitions - can be determined with a very good precision. This result paves the way towards a precise quantitative characterization of atom-surface interactions in a wide range of atom-based nano-devices.

Equations

$${\rho }_{\pm }(z,v>0)$$

$${\rho }_{\pm }(z,v<0)$$

$${\Lambda }_{\pm }^{\mathrm{bulk}}(z)=\left[({\Gamma }_{\mathrm{P}}+{\Gamma }_0)/2-i({\Delta }_{F{F}^{\prime }}\mp kv)\right]z,$$

$$E_{\mathrm{A}\pm }(z)=\frac{t_2}{1-r_2^2{e}^{2ikL}}\frac{ik}{2{\epsilon }_0}{\int }_0^L\mathrm{d}{z}^{\prime }P(z^{\prime },\omega )\exp [\pm ik(z-z^{\prime })],$$

$${\Lambda }_{\pm }(z)$$

Full text

Atoms close to surfaces offer new possibilities for engineered atom-atom and atom-light interactions through light confinement and surface mode excitations Chang et al. (2018); Lodahl et al. (2017); Stehle et al. (2014); Mitsch et al. (2014), ultimately down to the single-photon level Chang et al. (2007); Aoki et al. (2010); Chang et al. (2014). This has stimulated a recent growth in the number of platforms where atoms are kept close to surfaces, ranging from nano-fibers Kien et al. (2004); Vetsch et al. (2010); Deasy et al. (2014); Mitsch et al. (2014); Patterson et al. (2018) and nano-cells Sarkisyan et al. (2001); Peyrot et al. (2019a) to waveguides Ritter et al. (2018) and microtoroidal optical resonators Aoki et al. (2010). Simultaneously, shrinking the dimensions of atom-based sensors Knappe et al. (2005); Wakai et al. (2012); Wade et al. (2017) increases the number of atoms close to a surface relative to atoms in the bulk. Atom-surface interactions are therefore becoming increasingly important: they may limit the ultimate achievable precision of atom vapor sensors and they are crucial in understanding the dynamics in each new platform Aoki et al. (2010); Deasy et al. (2014); Patterson et al. (2018). However, despite their significance, direct measurements of atom-surface interactions are scarce.

Measuring the van der Waals (vdW) atom-surface interaction — that scales with the distance $z$ to the surface as $1/z^{3}$ Hinds (1991) — is challenging as it requires placing the atoms in a given internal state at a distance $z<\lambda/(2\pi)$ from the surface 111At larger distances interaction potential retardation effects have to be taken into account, resulting in Casimir-Polder potential $\propto z^{-4}$. See e.g. the review Hinds (1991). Here $\lambda$ is the wavelength of the strongest atomic transition from the considered state. Previous experiments on vdW atom-surface interactions used sophisticated techniques like reflections of cold atoms on a surface Landragin et al. (1996); Mohapatra and Unnikrishnan (2006), high-lying atomic states Fichet et al. (2007) or both Sandoghdar et al. (1992). High-lying states allow easier access to the vdW regime as: (i) transitions among higher-lying states correspond to longer wavelengths $\lambda$, relaxing the constrain on the atom-surface distance; and (ii) these transitions have larger dipole matrix elements, resulting in a stronger vdW coefficient $C_{3}$ in the atom-surface potential $V(z)=-C_{3}/z^{3}$. However, for many applications Lodahl et al. (2017); Stehle et al. (2014); Mitsch et al. (2014); Chang et al. (2007); Aoki et al. (2010); Deasy et al. (2014); Mitsch et al. (2014); Patterson et al. (2018); Sarkisyan et al. (2001); Peyrot et al. (2019a); Ritter et al. (2018), it is the properties of the ground-state atom-surface potential that are of most interest.

Spectroscopy of thermal vapors contained in cells Oria et al. (1991); Failache et al. (1999); Fichet et al. (2007); Sargsyan2017 is an attractive method for the measurement of atom-surface interactions, since it can be used for a large range of vapors, atomic or molecular, and surfaces. However, measurements have low precision for weak vdW interaction strengths of ground state atoms, mainly limited by the uncertainty in estimating collisional processes in dense vapors Oria et al. (1991); Fichet et al. (2007). A recent method that measured the ground state vdW interaction based on fluorescence spectroscopy in low-density thermal vapors Whittaker et al. (2014) raised debate Bloch (2015) about the absolute achievable precision of the measurements, since the theoretical model used Whittaker et al. (2014) neglected atomic motion in the spatially varying atom-surface potential. Finding simple and precise methods that would allow reliable extraction of the ground-state atom-surface potential parameters, and characterization of the atom dynamics in the proximity of surfaces in new platforms, remains an open goal.

Here, we demonstrate a new method for measuring the atom-surface interaction. Using a wedged nano-cell we obtain transmission spectra for a vapor thicknesses range of $L=50-275$ nm. This allows access both to the region where vdW interactions have strong effects on the total transmission signal, and the region where atoms in the bulk, not affected by the vdW induced shifts, dominantly contribute. Spectra from the thick region yield the collisional parameters for the bulk atomic vapor, thus allowing reliable fitting of thin-region spectra using our model. The model includes atomic motion in a spatially varying atom-surface potential, in addition to other surface-induced transient effects Peyrot et al. (2019b). This procedure allows measurement of the atom-surface potential even for the ground state transitions with a good precision. We illustrate it by a measurement of the vdW induced shift of the Cesium $6S_{1/2}\rightarrow 6P_{1/2}$ transition in the presence of a Sapphire surface. Our model significantly improves on a phenomenological one based on fitted line shift and broadening parameters, and shows excellent agreement with the measured spectra.

The experimental setup Peyrot et al. (2019b) is presented in Fig. 1(a). We use an external cavity diode laser (ECDL) to probe the $D_{1}$ ($\lambda=894$ nm) transition of a cesium vapor contained in an ultra-thin wedged sapphire cell, with thickness varying from $L=30$ nm to 2 $\mu$m. We measure the in-situ vapor thickness $L$ using interference resulting from the far off-resonant reflections from the wedge. With a beam waist of 50 $\mu$m ($1/\mathrm{e}^{2}$ intensity), the uncertainty in the determination of $L$ is 5 nm. The frequency scan of the ECDL is linearized using a Fabry-Perot cavity, and referenced to a standard 7 cm Cesium spectroscopic cell. The laser intensity is stabilized during the scan by an acousto-optic modulator-based noise eater. The wedged nano-cell is placed in a double oven: the bottom part containing the cesium reservoir is kept at 235 ∘C, and sets the atom number density $\mathcal{N}$; the top part with the wedged sapphire windows is kept at a temperature 30 ∘C higher to avoid vapor condensation. The oven is temperature stabilised to $\sim$5 ∘C. A low-noise photodiode is used to obtain the transmission signal $T$ subtracted from the background noise.

The theoretical analysis of transmission spectra in nano-cells is significantly more complicated than for transmission through bulk vapor Zentile et al. (2015). With the reduction of the vapor thickness $L$, a number of effects starts to play a role [Fig. 1(b)]: (i) for micrometer thick layers, the cell walls act as a low-finesse cavity, resulting in level shifts Peyrot et al. (2018); (ii) the cell windows also cause dephasing of atoms upon direct collisions Schuurmans (1976). Atoms flying off the walls experience transient dynamics during a time $1/\Gamma$, with $\Gamma$ the collisionally-broadened linewidth. For a cell thickness below $v/\Gamma$ ($v$ is the average atom velocity), a significant number of probed atoms experience the transient regime, which significantly modifies the measured transmission Peyrot et al. (2019b). For cesium atoms at a temperature of $\sim 200~{}^{\circ}$C this corresponds to a distance of $\sim 1~{}\mu$m. Finally, (iii) at atom-surface distances $z<\lambda/(2\pi)$ atomic energy levels experience a vdW shift $V=-C_{3}/z^{3}$ [Fig. 1(c)]. This atom-surface interaction comes from the coupling between an atomic dipole and its image dipole in the sapphire surface Fichet et al. (1995). To extract this vdW interaction at small atom-surface distances (iii), it is necessary to first account for the effects (i) and (ii) that modify the atom dynamics in thin cells even outside the range of the atom-surface potential.

We have recently developed a model for the atom response in the bulk Peyrot et al. (2019b), away from the influence of the vdW potential, following Refs. Bloch and Ducloy (2005); Dutier et al. (2003). To account for the transient atoms dynamics  (ii) described above, we solve the optical Bloch equations Authors for the atom coherence field $\rho_{21}(z,v,\omega)=\rho_{+}(z,v,\omega)e^{ikz}+\rho_{-}(z,v,\omega)e^{-ikz}$, for each atomic velocity $v$ class, where $k=2\pi/\lambda=\omega/c$ is the laser field wave-vector in vacuum. Assuming a loss of coherence in atom-wall collisions  Schuurmans (1976), one obtains

[TABLE]

where $E_{\pm}$ are the co- and counter-propagating driving fields along the $z$ direction, and $d_{FF^{\prime}}$ is the dipole moment for the hyperfine transition $F\rightarrow F^{\prime}$. In a bulk cell, away from the range of the atom-surface potential, a level shift $\Delta_{\rm P}$ and a broadening $\Gamma_{\rm P}$ of the transition arise due to the atom-atom collisional interaction. Thus, for $z>\lambda/(2\pi)$, $\Lambda_{\pm}$ in Eqs. (1-2) is

[TABLE]

for a laser detuning $\Delta_{FF^{\prime}}=\omega-\omega_{FF^{\prime}}-\Delta_{\rm P}$ from the transition at frequency $\omega_{FF^{\prime}}$ with radiative linewidth $\Gamma_{0}/(2\pi)\approx 4.6$ MHz for the Cs D1 line Authors . Equations (1-2) can then be analytically solved, and the polarisation of the medium $P(z,\omega)=\sum_{F,F^{\prime}}\int_{-\infty}^{\infty}{\rm{d}}v~{}2\mathcal{N}M_{\rm b}(v)d_{FF^{\prime}}\rho(z,v,\omega)$ is obtained by summing the contributions from all velocity classes in the Maxwell-Boltzman distribution $M_{\rm b}(v)$.

In order to include the cavity effects (i), we take into account the transmission $t_{1}=2n/(1+n)$ and reflection $r_{2}=(1-n)/(1+n)$ coefficients of the driving field $E_{0}$ at the two sapphire cell walls (refractive index $n$). Assuming an optically dilute atomic medium, we neglect the contribution of atoms to the driving field $E_{\pm}$ inside the cavity, so that $E_{+}e^{ikz}+E_{-}e^{-ikz}\simeq t_{1}/[1-r_{2}^{2}\exp(2ikL)]E_{0}\{\exp[ikz]+r_{2}\exp[ik(2L-z)]\}$. Similarly, the radiated atomic fields $E_{\rm A+}$ and $E_{\rm A-}$ initially co- or counter-propagating along the $z$-axis respectively Authors , can be reflected multiple times inside the cavity Peyrot et al. (2019b). They give rise to an atom induced field outside the cavity

[TABLE]

in the direction of initial emission, and fields $r_{2}\exp[2ik(L-z)]E_{\rm A+}$ and $r_{2}\exp(2ikz)E_{\rm A-}$ respectively in the opposite direction Authors , where $t_{2}=2/(1+n)$. Finally, we obtain the transmission factor through the thin cell system $T=\left|E_{\rm T}/E_{0+}\right|^{2}$, where $E_{\rm T}=E_{0+}+E_{\rm A+}+r_{2}\exp(2ikz)E_{\rm A-}$ is a superposition of the transmitted atom-induced field and transmitted driving field $E_{0+}=t_{1}t_{2}/[1-r_{2}^{2}\exp(2ikL)]E_{0}\exp(ikz)$.

The model described above (from now on Model A) fits well the measured transmission spectra [Fig. 2(a)] for cell thicknesses $L\geq 150$ nm [see for example Fig. 2(b)], where the signal is dominated by atoms far from the surface. In this region, fitting the temperature, level shift $\Delta_{\rm P}$ and broadening $\Gamma_{\rm P}$ allows us to obtain the collisional self-broadening $\Gamma_{\rm P0}/(2\pi)=830\pm 10$ MHz [Fig. 2(c)] and line shift $\Delta_{\rm P0}/(2\pi)=-10\pm 20$ MHz [Fig. 2(d)] in the bulk vapor, arising solely from atom-atom collisions. The value and error bar are the mean and standard error of the fitted values for cell thicknesses $L\geq 175$ nm. The measured broadening is in good agreement with theoretical predictions Weller et al. (2011) at $235~{}^{\circ}$C.

For cell thicknesses $L\lesssim 150$ nm, the contribution from atoms close to the surface becomes significant. The vdW interaction [Fig. 1(c)] offsets the $6S_{1/2}$ and $6P_{1/2}$ levels by different amounts due to the different vdW $C_{3}$ coefficients for the two states. Model A captures this atom-surface interaction phenomenologically as a cell thickness dependent shift and broadening, as shown in Fig. 2(c-d). However, this phenomenological fit does not provide direct access to the $C_{3}$ vdW coefficient. In addition, Fig. 3 indicates that fine features of thin-cell spectra are not captured by Model A. This motivates extending the model to explicitly include the spatially varying vdW potential.

To do so, we now make the transition frequency spatially dependent: $\omega_{FF^{\prime}}\rightarrow\omega_{FF^{\prime}}-V(z)$. The atom-surface position dependent shift of the D1 transition frequency is $V(z)=-C_{3}[1/z^{3}+1/(L-z)^{3}]$, where $C_{3}=C_{3}[6P_{1/2}]-C_{3}[6S_{1/2}]$. In summing independently the potentials of each surface, we account only for the interaction of the atomic dipole with its image from the two surfaces. Image dipoles of image dipoles have contributions smaller than $\sim 3$% Authors . Solving the Bloch equations leads to the solution for the atomic coherence field of the same form as in Eqs. (1-2), except that now

[TABLE]

Here we set the collisional broadening $\Gamma_{\rm P0}$ to be the same as in the bulk, which is justified as we are far from polaritonic resonances of the crystal surface Failache et al. (2002). The first part of Eq. (5) describes the bulk [Eq. (3)], while the second line is due to the atom-surface vdW potential. By using Eq (5) in Eq. (1-2), we include the internal atomic dynamics as the atoms move relative to the surface, thus experiencing time-varying level shift due to the vdW interaction $V(z)$, to obtain the coherence field for each velocity class at location $z$. In previous works based on fluorescence measurement Whittaker et al. (2014), the motion was not accounted for Bloch (2015). For details of the numerical integration see Supplemental Authors . We call this Model B.

The result of the fitting of the data for a thin cell with $L=60$ nm is shown in Fig. 3. Model B has only two free parameters, the temperature and $C_{3}$, as $\Gamma_{\rm P0}$ and $\Delta_{\rm P0}$ are constrained to their bulk extracted values. Yet, it shows outstanding agreement compared to Model A that only phenomenologically accounts for atom-surface vdW interactions. In particular, the red-side asymmetry cannot be retrieved without taking properly the vdW shift into account. The value for $C_{3}$ is extracted from fitting the spectra for each cell thickness $L$ (Fig. 4 main panel). Thin cells yield a larger fraction of atoms close to the surface, where the vdW potential induces large level shifts, and allow tighter constraints on the fitted parameters. This is visible on the map of the sum of squared residuals (Fig. 4 top insets), where the region that minimizes $\sigma^{2}$ is more tightly localized along the $C_{3}$-axis for thin cells. The error bars on $C_{3}$ (main panel of Fig. 4) are systematic and result from propagating errors on the cell thickness and the fitted bulk vapor parameters $\Gamma_{\textrm{P0}}$ and $\Delta_{\textrm{P0}}$ Authors . Assuming no dependence on $L$, the final value for $C_{3}$ is obtained as an average of the fitted values for different cell thicknesses weighted by their individual error bars Hughes and Hase (2010). We obtain $C_{3}=1.3\pm 0.1$ kHz.$\mu$m3, in good agreement (Fig. 4 side scale) with the theoretical value $C_{3}^{\rm th}=1.10\pm 0.03$ kHz.$\mu$m3 predicted from the Lifshitz theory of vdW interactions Lifshitz (1956); Authors , based on the refractive index of sapphire Weber (2003) and the latest calculations of dipole matrix elements for low-lying cesium transitions Safronova et al. (2016); Šibalić et al. (2016).

In conclusion, we have demonstrated a method to measure the interaction between an atom and a surface. It relies on comparing the transmission of a laser beam through a wedged nano-cell filled with an atomic vapor to a theoretical model including the atom-surface interaction, cavity effects, collisions with surfaces of the cell and atomic motion. The wedged cell provides simultaneous access to the bulk vapor properties, tightly constraining theoretical parameters, and to thin cell regions with strong vdW interaction. We have illustrated the method with thermal cesium atoms confined within a sapphire cell and measured the D1 transition shift due to the cesium-sapphire interaction. Our model yields transmission predictions in excellent agreement with the experimental data. The measured $C_{3}$ coefficient is consistent with the non-resonant Lifshitz theory. Further improvements in precision are expected from adapting this work to fluorescence measurements, as better signal to noise ratio lead to a better determination of collisional properties as well as a reduction of statistical errors in extremely thin regions. Finally, this work opens the way to quantitative analysis of atom dynamics close to surfaces in new platforms Aoki et al. (2010); Deasy et al. (2014); Patterson et al. (2018), the search for predicted atom-surface repulsive potential Failache et al. (1999) and bound-states Lima et al. (2000), and the examination of short- Vargas and Mochán (1998) and long-range Carvalho et al. (2018) limits of atom-surface potential, using thermal vapor spectroscopy. The data and the code for theoretical model are both available PaperData ; PaperCode .