A Guide for Future Experiments

Abstract

This chapter aims to provide a guide for experimentalists seeking to observe the effects we predicted in earlier chapters. Section 8.1 presents an overview of the different geometries explored in the previous chapters. For each of these geometries, Sect. 8.2 calculates and compares the relevant friction forces acting on the particle. Section 8.3 discusses a number of experimentally-accessible configurations and calculates cooling times and equilibrium temperatures that can be expected in each situation. The first appendix to this chapter is a technical note discussing electric fields inside dielectrics and the origin of the Clausius–Mossotti equation that describes the response of a bulk dielectric to an applied electric field. Finally, two appendices then follow that discuss, respectively, some problems encountered when calculating electric fields inside microscopic hemispherical mirrors, and general expressions for the force acting on an atom inside an arbitrary monochromatic field, ignoring delay effects.

[...] [I]t is more important to have beauty in one’s equations than to have them fit experiment. P. A. M. Dirac, Scientific American 208, 5 (1963)

Appendices

Appendix: Electric Fields Inside Dielectrics

Within a simple model of the dielectric of volume V as a collection of closely spaced dipoles at random positions, its response to an electric field can be embodied entirely in the relative permittivity $\epsilon$, defined by the relation

$$\begin{aligned} \varvec{\mathcal P }=V(\epsilon -1)\epsilon _0\varvec{\mathcal E }. \end{aligned}$$

(8.12)

The Clausius–Mossotti relation, as quoted in Eq. (8.1), connects the susceptibility $\chi$ of the dielectric to the bulk refractive index of the dielectric. Many derivations of this relation (see, for example, Ref. [20], Sect. 4.5) divide the dielectric into two regions: a spherical section of the dielectric, large enough to contain several dipoles but small enough that the polarisation is practically constant within; and the rest of the dielectric. One objection to this argument is that it depends critically on the first region chosen as being spherical, an assertion that has no real physical justification and no immediate connection to the main assumption, mentioned below, present in the Clausius–Mossotti relation. Hannay [21] presented an alternative derivation of this same expression that does not make use of this model. Let us briefly recapitulate Hannay’s argument. The electric field $\varvec{\mathcal E }$ produced by an ideal point-like dipole $\varvec{p}$ is, as a function of the displacement $\varvec{r}$ from the dipole,

$$\begin{aligned} \varvec{\mathcal E }=\frac{1}{4\pi \epsilon _0}\Biggl [\frac{3(\varvec{p}\cdot \varvec{r})\varvec{r}}{|\varvec{r}|^5}-\frac{\varvec{p}}{|\varvec{r}|^3}\Biggr ]-\frac{\varvec{p}}{3\epsilon _0}\delta (\varvec{r})\,. \end{aligned}$$

(8.13)

The central assumption that leads to the Clausius–Mossotti equation is that a “test” dipole inserted at a random position in the dielectric experiences an electric field that is free from the influence of the $\delta$-spikes that occur at each of the dipoles making up the dielectric. These $\delta$-spikes, of which there are N, contribute a spatially-averaged field

$$\begin{aligned} -\frac{N\varvec{p}}{3V\epsilon _0}=-\frac{\varvec{\mathcal P }}{3V\epsilon _0}\,, \end{aligned}$$

(8.14)

defining the macroscopic polarisation of the medium by $\varvec{\mathcal P }\equiv N\varvec{p}$. Thus, the field experienced by the test dipole—and, therefore, the field inside the medium—is equal to the “normalised” field

$$\begin{aligned} \varvec{\mathcal E }_\text{ norm}=\varvec{\mathcal E }+\frac{\varvec{\mathcal P }}{3V\epsilon _0}, \end{aligned}$$

(8.15)

where $\varvec{\mathcal E }$ is the local microscopic spatial average of the electric field that the dielectric is immersed in. Finally, $\varvec{\mathcal E }_\text{ norm}$ is related to $\varvec{\mathcal P }$ through the definition of $\chi$ ([20], Sect. 4.5)

$$\begin{aligned} \varvec{\mathcal P }=\epsilon _0\chi \varvec{\mathcal E }_\text{ norm}=\epsilon _0\chi \Biggl (\varvec{\mathcal E }+\frac{\varvec{\mathcal P }}{3V\epsilon _0}\Biggr )\,. \end{aligned}$$

(8.16)

Thus,

$$\begin{aligned} \varvec{\mathcal P }=V(\epsilon -1)\epsilon _0\varvec{\mathcal E }=\frac{\epsilon _0\chi \varvec{\mathcal E }}{1-\chi /\bigl (3V\bigr )}\,, \end{aligned}$$

(8.17)

which can be rearranged to give the Clausius–Mossotti equation:

$$\begin{aligned} \chi =3V\frac{\epsilon -1}{\epsilon +2}. \end{aligned}$$

(8.18)

For a particle of volume V made from a typical dielectric with a refractive index $n=1.5$, $\chi =0.9V\approx V$. An intuitive understanding of $\chi$ is therefore possible as the volume of dielectric that is polarised by an incoming field. It is perhaps interesting to note that this means that each single molecular dipole in the dielectric has an effective susceptibility

$$\begin{aligned} \chi _\text{ eff}\approx \frac{4}{3}\pi a^3\,, \end{aligned}$$

(8.19)

where 2a is the mean distance between the molecules making up the dielectric; $a\sim 10^{-10}$ m is typically several orders of magnitude larger than the molecular radius itself. However, $\chi _\text{ eff}$ is of the same order as the susceptibility of a free molecule [22]. In other words, the polarisation of an individual dipole due to an off-resonant electric field is of about the same order whether that dipole is isolated or in a bulk solid.

This $\chi$ can be used to calculate the power dissipated by a small dielectric sphere, modelled as a single point dipole, due to blackbody radiation. At a temperature $T$ there are $n_k=1/\bigl \{\exp \bigr [\hbar ck/\bigl (k_\text{ B}T\bigr )\bigr ]-1\bigr \}$ photons with a wavevector $\varvec{k}$ and wavenumber $k$. These photons produce an equivalent electric field of optical power $P_k=\hbar kc^2n_k\sigma _\text{ L}/V_\text{ q}$, $V_\text{ q}$ being the quantisation volume, and therefore lead to an absorbed power due to that mode

$$\begin{aligned} P_{\text{ abs},k}\approx \frac{\hbar k^2c^2n_k}{V_\text{ q}}\operatorname{Im}\{\chi \}\,, \end{aligned}$$

(8.20)

for small $|\chi |$. We must now sum over every mode to obtain the total absorbed power $P_\text{ abs}=\sum _{\varvec{k}}P_{\text{ abs},k}$, which quickly becomes cumbersome since the number of modes becomes infinite as the quantisation volume grows indefinitely. By assuming that the modes are evenly distributed in $\varvec{k}$-space, with a density $(2\pi )^3/V_\text{ q}$, we can transform this sum into a three-dimensional integral

$$\begin{aligned} P_\text{ abs}&=\frac{2V_\text{ q}}{(2\pi )^3}\int \int \int \frac{\hbar k^2c^2n_k}{V_\text{ q}}\operatorname{Im}\{\chi \}\,\mathrm d ^3\varvec{k}\nonumber \\&=\frac{\hbar c^2}{4\pi ^3}\operatorname{Im}\{\chi \}\int \limits _0^{2\pi }\int \limits _0^\pi \int \limits _0^\infty \frac{k^4}{\exp \bigr [\hbar ck/\bigl (k_\text{ B}T\bigr )\bigr ]-1}\sin (\theta )\,\mathrm d k\,\mathrm d \theta \,\mathrm d \phi \nonumber \\&=\frac{24\zeta (5)}{\pi ^2 c^3\hbar ^4}\operatorname{Im}\{\chi \}\bigl (k_\text{ B}T\bigr )^5\,, \end{aligned}$$

(8.21)

where the extra factor of $2$ accounts for the two polarisations, where $\chi$ was assumed to be independent of $k$, and where $\zeta (5)$ is the Riemann zeta function. The $k$-integral is performed by appealing to the definition of the $\zeta (z)\equiv \zeta (z,1)$ ([23], Sect. 9.51). Our final step is to note that in thermal equilibrium, $P_\text{ abs}$ is equal to the power dissipated by the sphere, $P_\text{ diss}$.

Appendix: Calculating the Electric Field Inside Hemispherical Mirrors

Hemispherical voids templated on gold surfaces are a good system to work with experimentally: once made, they require no further alignment; and regular, close-packed arrays can be made with several tens (for large diameters) up to several hundreds (for diameters of the order of 1 $\upmu$m [24]) of dimples. However, the analysis of the electromagnetic fields inside the dimples presents a challenge. The most direct route to exploring these fields is through the numerical solution of Maxwell’s equations. A large number of software packages are available, with several operating either on finite element method (FEM) or finite-difference time-domain (FDTD) principles or employing Mie theory (see Ref. [25] for a recent review of such techniques). Analyses of scattering of electromagnetic radiation off spherical particles are usually performed using Mie theory [20], which exploits the fact that the vector spherical harmonics form a complete orthogonal set of modes on the sphere. In the case of a hemisphere, however, no such set of modes exists—and the situation is even worse for truncated hemispheres—and FEM or FDTD techniques are more desirable. It is interesting to note that this problem can be formally circumvented in certain situations. For example, the authors of Ref. [3] implicitly assume knowledge of the field outside the spherical region defined by a perfect truncated hemispherical mirror to explore the behaviour of the vacuum field inside this same spherical region. In the absence of a compact analytical solution, we use an open-source software package called MEEP [26] for our analysis.

We show an example of such a simulated field in Fig. 8.5. Close to the geometrical focus of the dimple, a strong focus is found; the size and peak intensity of this focus can be used to explore the possibility of producing single- or few-atom dipole traps inside such cavities. 2D arrays of dipole traps were demonstrated [14] a number of years ago, and recently used to perform site-selective manipulation of atoms [15]—both these experiments relied on a purpose-built refractive micro-lens array. Using templated surfaces offers a number of advantages over the micro-lens array, not least ease of manufacture and the possibility of integration into so-called atom chips [16]; in this regard, the use of reflective rather than refractive optical elements is of paramount importance.

The dimples in the first surface produced for the experimental investigations in Southampton were not grown to full hemispheres. Rather, latex spheres with a diameter of 100 $\upmu$m were used and gold was only templated up to a depth of 15 $\upmu$m. Part of the resulting surface is shown in Fig. 8.1, with the field inside one such dimple simulated in Fig. 8.6. Note that aliasing artifacts are more apparent in this figure than in Fig. 8.5, the reason being that the 0.05 $\upmu$m resolution possible in the case of the latter was not possible in simulating the larger sample, due to computer memory constraints. In the case of Fig. 8.6, a resolution of 0.17 $\upmu$m was used. In both cases, the incoming field was a plane wave with a wavelength of $\lambda =780$ nm.

Fig. 8.5

Finite-difference time-domain analysis of the electric field intensity on a 2D slice of a 10 $\upmu$m diameter hemispherical void in an ideal metal substrate. The incident field is a linearly polarised plane wave, with a wavelength of 780 nm, propagating downwards. Two sections, both intersecting the focus, are also shown: the vertical (horizontal) section runs along the vertical (horizontal) dashed line

Fig. 8.6

Electric field intensity inside a $100$ $\upmu$m diameter hemispherical void in an ideal metal substrate. The size of the hemispherical template and the depth of the void ($15$ $\upmu$m vertically from the bottom to the lip) match the surface in Fig. 8.1

Appendix: Force Acting on an Atom Inside an Arbitrary Monochromatic Field

In a remarkable piece of work dating to 1980, Gordon and Ashkin [27] give several useful expressions for the force and diffusion experienced by an atom inside what they called a ‘radiation trap’—essentially an arbitrary (monochromatic) electric field. Implicit in their work is the assumption that the system has no ‘memory’; we cannot directly apply their equations to mirror-mediated or external cavity cooling systems, for example. Nonetheless, such a model is perhaps the easiest way of exploring the behaviour of atoms inside fields as complex as those in hemispherical voids on a metal surface. Unfortunately, the authors of Ref. [27] do not give explicit general formulae for the velocity-dependent force acting on an atom; we will now generalise their expressions [specifically, Eqs. (8.14) and (8.15)] to the case when the atom is not motionless. Let us first briefly introduce the notation we will use in this section⁴: $\varvec{\mathsf F }$ is the force acting on the atom; $D$ is the difference in the populations of the upper and lower states; $\sigma$ is the atomic lowering operator; $\Delta =\omega -\omega _\text{ a}$, with $\omega$ the frequency of the driving field and $\omega _\text{ a}$ the atomic resonance frequency; $\gamma =\Gamma -\mathrm i \Delta$; $\varvec{\mu }_{21}$ the atomic dipole operator; $\varvec{v}$ the atomic velocity; and $\mathcal E e^{-\mathrm i \omega t}$ the classical incident electric field. We also define

$$\begin{aligned} g=\tfrac{\mathrm i }{\hbar }\varvec{\mu }_{21}\cdot \mathcal E \,, \end{aligned}$$

(8.22)

and the saturation parameter $s=2|g|^2/|\gamma |^2$. It is also convenient to define the vectors $\varvec{\alpha }$ and $\varvec{\beta }$ by $\operatorname{grad}g=(\varvec{\alpha }+\mathrm i \varvec{\beta })g$. In practice, numerical simulation gives us knowledge of $\mathcal E$, assuming that the perturbation of the atom on the field is quite small. By specifying the magnetic field, we can determine $\omega _\text{ a}$, and therefore $\Delta$. Given the atomic species, we also know $\Gamma$ and $\varvec{\mu }_{21}$; knowledge of $\gamma$, $g$, $s$, $\varvec{\alpha }$, $\varvec{\beta }$ and $\langle \sigma \rangle$ then follows, as we will see. Finally, this determines $\langle \varvec{\mathsf F }\rangle$, the classical force acting on the atom.

To first order in $\varvec{v}$, we have

$$\begin{aligned} \langle \varvec{\mathsf F }\rangle&=-\mathrm i \hbar \bigl [\langle \sigma \rangle ^*\operatorname{grad}g-\langle \sigma \rangle \operatorname{grad}g^*\bigr ]\,,\end{aligned}$$

(8.23)

$$\begin{aligned} \langle \dot{\sigma }\rangle +\gamma \langle \sigma \rangle&=\langle D\rangle g\,,\end{aligned}$$

(8.24)

$$\begin{aligned} \langle \dot{D}\rangle +2\Gamma \langle D\rangle&=2\Gamma -2(g^*\langle \sigma \rangle +g\langle \sigma \rangle ^*)\,,\end{aligned}$$

(8.25)

$$\begin{aligned} \langle \dot{D}\rangle&=-\tfrac{2s}{1+s}(\varvec{v}\cdot \varvec{\alpha })\langle D\rangle \,,\end{aligned}$$

(8.26)

$$\begin{aligned} \langle \dot{\sigma }\rangle&=\bigl [(\varvec{v}\cdot \varvec{\alpha })\tfrac{1-s}{1+s}+\mathrm i (\varvec{v}\cdot \varvec{\beta })\bigr ]\langle \sigma \rangle \,\text{,} \text{ and}\end{aligned}$$

(8.27)

$$\begin{aligned} \dot{g}&=\varvec{v}\cdot (\varvec{\alpha }+\mathrm i \varvec{\beta })g\,, \end{aligned}$$

(8.28)

directly from Ref. [27]. Thus, to the same order,

$$\begin{aligned} \langle D\rangle =\bigl [1-\tfrac{1}{\Gamma }(g^*\langle \sigma \rangle +g\langle \sigma \rangle ^*)\bigr ]+\tfrac{1}{\Gamma }\tfrac{s}{1+s}(\varvec{v}\cdot \varvec{\alpha })\bigl [1-\tfrac{1}{\Gamma }(g^*\langle \sigma \rangle +g\langle \sigma \rangle ^*)\bigr ]\,, \end{aligned}$$

(8.29)

whereby

$$\begin{aligned} \langle \sigma \rangle \bigl [\gamma +(\varvec{v}\cdot \varvec{\alpha }) \tfrac{1-s}{1+s}+\mathrm i (\varvec{v}\cdot \varvec{\beta })\bigr ] =&\bigl [1-\tfrac{1}{\Gamma }(g^*\langle \sigma \rangle +g\langle \sigma \rangle ^*)\bigr ]g\nonumber \\&+\tfrac{1}{\Gamma }\tfrac{s}{1+s}(\varvec{v}\cdot \varvec{\alpha }) \bigl [1-\tfrac{1}{\Gamma }(g^*\langle \sigma \rangle +g\langle \sigma \rangle ^*)\bigr ]. \end{aligned}$$

(8.30)

We can solve this for $\langle \sigma \rangle$ to obtain

$$\begin{aligned} \langle \sigma \rangle =&\frac{g}{\gamma (1+s)}+\frac{g}{|\gamma |^2(1+s)}\bigl (\tfrac{1}{2\Gamma }\gamma ^*+\tfrac{1-s}{1+s}\bigr )(\varvec{v}\cdot \varvec{\alpha })\nonumber \\&-2\bigl [\Gamma (1-s)+\tfrac{1}{\Gamma }|g|^2\bigr ]\frac{g}{\gamma |\gamma |^2(1+s)^3}(\varvec{v}\cdot \varvec{\alpha })\nonumber \\&+\frac{\bigl [2\Delta -\mathrm i \gamma (1+s)\bigr ]g}{\gamma |\gamma |^2(1+s)^2}(\varvec{v}\cdot \varvec{\beta })\,, \end{aligned}$$

(8.31)

which we can plug into Eq. (8.23), together with $g$, to obtain the (velocity-dependent) force acting on the atom.