Improved Effective Range Expansion for Casimir-Polder potential

TL;DR

This paper introduces an improved effective range expansion method for analyzing scattering in Casimir-Polder potentials, enhancing accuracy at low energies by leveraging Liouville transformations and potential decomposition.

Contribution

The authors develop a novel effective range expansion technique that incorporates Liouville transformations and potential decomposition for better low-energy scattering analysis.

Findings

More accurate low-energy scattering amplitudes

Effective decomposition into elementary problems

Enhanced theoretical understanding of Casimir-Polder interactions

Abstract

We study the effective range expansion of scattering on a real Casimir-Polder potential. We use Liouville transformations which transform the potential landscape while preserving the reflection and transmission amplitudes. We decompose the scattering calculation in two more elementary problems, one for the homogeneous 1/z^4 potential and the other one for the correction to this idealization. We use the symmetries of the transformed problem and the properties of the scattering matrices to derive an improved effective range expansion leading to a more accurate expansion of scattering amplitudes at low energy.

Tables

Table 1. Table 1: Typical length scales ℓ 4 subscript ℓ 4 \ell_{4} , ℓ 3 subscript ℓ 3 \ell_{3} and scattering length ℓ ℓ \ell calculated for the real Casimir-Polder potential between antihydrogen and helium or silica surfaces (atomic units).

	${\ell }_4$	$\ell$	${\ell }_3$
He	$75.51$	$44.78-34.90i$	$16.54$
SiO2	$194.7$	$272.7-77.04i$	$321.3$

Table 2. Table 2: Coefficients in the expansion of ρ 𝜌 \rho deduced by fitting the numerical results known for He and SiO 2 surfaces.

	He	SiO2
${\rho }_0$	$0.158+0.336i$	$0.064+0.138i$
${\rho }_1$	$-0.009+0.011i$	$-0.001-0.004i$
${\rho }_2$	$0.098-0.117i$	$0.026+0.034i$
${\rho }_3$	$-0.513+0.611i$	$-0.359+0.469i$
${\rho }_4$	$-0.083-0.487i$	$0.204-0.740i$

Table 3. Table 3: Standard deviations σ ^ ^ 𝜎 \hat{\sigma} of the residuals in the fits, calculated for real and imaginary parts of ρ 𝜌 \rho , for different surfaces, as a function of the fit method – based on the even terms and the full expansions.

${10}^6\hat{\sigma }$	He		SiO2
	Re($\rho$)	Im ($\rho$)	Re($\rho$)	Im ($\rho$)
Even terms expansion	$31$	$36$	$41$	$80$
Full expansion	$2$	$2.4$	$1.0$	$5.5$

Table 4. Table 4: Modified values for the scattering length ℓ ℓ \ell and the coefficient α ~ 2 subscript ~ 𝛼 2 \tilde{\alpha}_{2} computed for helium and silica surfaces.

	He	SiO2
$\ell$	$44.78-34.90i$	$272.7-77.04i$
${\tilde{\alpha }}_2$	$2.54-2.51i$	$1.73-3.67i$

Table 5. Table 5: Maximum relative error for the old effective range theory and the improved effective range expansion for helium and silica surfaces.

	He	SiO2
Modified ERT	$9.1\times {10}^{-4}$	$2.5\times {10}^{-3}$
Improved ERT	$4.8\times {10}^{-5}$	$4.3\times {10}^{-5}$

Equations

$$\begin{array}{rl}& V(z)\simeq -C_3/z^3\equiv V_3(z),z\to 0^{+}\text{leavevmode},\\ & V(z)\simeq -C_4/z^4\equiv V_4(z),z\to +\infty \text{leavevmode}.\end{array}$$

$$\begin{array}{r}{\psi }^{\prime \prime }(z)+F(z)\psi (z)=0\\ F(z)\equiv \frac{2m(E-V(z))}{{\hslash }^2}.\end{array}$$

$$\psi (z)\underset{z\to +\infty }{\sim }{e}^{-ikz}+r(k)\text{leavevmode}{e}^{ikz}\text{leavevmode}.$$

$$\tilde{\mathcal{A}}(k)=-i\frac{1+r(k)}{1-r(k)},\underset{k\to 0}{\lim }\frac{\tilde{\mathcal{A}}(k)}{k}=-i\ell .$$

$${\varphi }_{dB}(z)={\int }_{z_{\varphi }}^z\sqrt{F(z^{\prime })}d{z}^{\prime }\text{leavevmode},$$

$$\underset{z\to +\infty }{\lim }({\varphi }_{dB}(z)-kz)=\varphi .$$

$$\begin{array}{rl}\mathbf{z}& =\frac{{\varphi }_{dB}(z)}{\sqrt{k{\ell }_4}}\\ \mathbf{\Psi }(\mathbf{z})& =\sqrt{{\mathbf{z}}^{\prime }(z)}\text{leavevmode}\psi (z)\text{leavevmode}.\end{array}$$

$${\mathbf{z}}_{\varphi }\equiv {\int }_{z_{\varphi }}^{z_0}\sqrt{\frac{F(z^{\prime })}{k{\ell }_4}}\text{leavevmode}d{z}^{\prime }\text{leavevmode}.$$

$$\begin{array}{rl}{\mathbf{\Psi }}^{\prime \prime }& (\mathbf{z})+\mathbf{F}(\mathbf{z})\mathbf{\Psi }(\mathbf{z})=0\text{leavevmode},\\ & \mathbf{F}(\mathbf{z})=\mathbf{E}-\mathbf{V}(\mathbf{z})\text{leavevmode},\end{array}$$

$$\begin{array}{rl}\mathbf{F}(\mathbf{z})& =\frac{F(z)-\frac{1}{2}\{\mathbf{z},z\}}{{\mathbf{z}}^{\prime }(z{)}^2}\\ \{\mathbf{z},z\}& =\frac{{\mathbf{z}}^{\prime \prime \prime }(z)}{{\mathbf{z}}^{\prime }(z)}-\frac{3}{2}\frac{{\mathbf{z}}^{\prime \prime }(z{)}^2}{{\mathbf{z}}^{\prime }(z{)}^2}\equiv 2F(z)Q(z)\text{leavevmode}.\end{array}$$

$$\begin{array}{rl}\mathbf{E}& =k{\ell }_4\text{leavevmode},\\ \mathbf{V}(\mathbf{z})& =k{\ell }_{4}Q(z)\text{leavevmode},\\ \mathbf{k}& =\sqrt{k{\ell }_4}\text{leavevmode}.\end{array}$$

$${\mathbf{z}}_M\equiv \mathbf{z}(z=\zeta )={\mathbf{z}}_{\ast }+{\mathbf{z}}_{\varphi }\text{leavevmode},\zeta \equiv \sqrt{\frac{{\ell }_4}{k}}$$

$${\mathbf{z}}_{\ast }\equiv 2\left[{}_2{F}_1\left(\frac{1}{2},-\frac{1}{4};\frac{3}{4};-1\right)-\frac{1}{\sqrt{2}}\right]=\frac{1}{\sqrt{\pi }}\Gamma {\left(\frac{3}{4}\right)}^2\text{leavevmode},$$

$$\begin{array}{rl}{\mathbf{V}}_4(\mathbf{z})& =\frac{5}{8{\cosh }^3(2u)},u\equiv \ln \frac{z}{\zeta }\text{leavevmode},\\ \mathbf{z}& ={\int }_0^u\sqrt{2\cosh (2u^{\prime })}\text{d}{u}^{\prime }\text{leavevmode},\end{array}$$

$${\mathbf{V}}_4(\mathbf{z})\underset{\mathbf{z}\to \pm \infty }{\simeq }\frac{5}{{\mathbf{z}}^6},\mathbf{z}\underset{u\to \infty }{\simeq }{e}^u\text{leavevmode}.$$

$${\mathbf{V}}_3(x)=3x\frac{1+16x^3}{16(1+x^3{)}^3}\text{leavevmode},$$

$$\mathbf{z}=3x\left[{}_2{F}_1\left(\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{1}{x^3}\right)-\frac{2}{3}\sqrt{1+\frac{1}{x^3}}\right]\text{leavevmode}.$$

$$\mathbf{V}(\mathbf{z})\underset{\mathbf{z}\to -\infty }{\simeq }{\mathbf{V}}_3(\mathbf{z})\underset{\mathbf{z}\to -\infty }{\simeq }\frac{3}{4{\mathbf{z}}^2}\text{leavevmode},$$

$$\mathbf{z}\underset{x\to 0}{\simeq }-2/\sqrt{x}\text{leavevmode}.$$

$$\left(\begin{array}{c}{a}_{+}^{\mathrm{out}}\\ a_{-}^{\mathrm{out}}\end{array}\right)=\mathcal{S}\left(\begin{array}{c}{a}_{+}^{\mathrm{in}}\\ a_{-}^{\mathrm{in}}\end{array}\right),\mathcal{S}=\left(\begin{array}{cc}\overline{t}& r\\ \overline{r}& t\end{array}\right)\text{leavevmode}.$$

$$\begin{array}[]{lrcl}\Pi:&\left(\begin{matrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{matrix}\right)&\mapsto&\frac{1}{m_{11}}\left(\begin{matrix}1&-m_{12}\\ m_{21}&\text{det}(M)\end{matrix}\right)\\ \end{array}$$

$$\mathcal{T}=\Pi (\mathcal{S}),\mathcal{S}=\Pi (\mathcal{T}),\Pi \circ \Pi =\mathcal{I}\text{leavevmode}.$$

$$({\mathcal{S}}_a,{\mathcal{S}}_b)\mapsto {\mathcal{S}}_a\star {\mathcal{S}}_b\equiv \Pi [\Pi ({\mathcal{S}}_a)\times \Pi ({\mathcal{S}}_b)]\text{leavevmode}.$$

$$\mathcal{S}={\mathcal{S}}_{\rho }\star {\mathcal{S}}_4\equiv \left(\begin{array}{cc}\overline{t}& r\\ \overline{r}& t\end{array}\right).$$

$${\mathcal{S}}_4^{-1}=\Pi ([\Pi ({\mathcal{S}}_4){]}^{-1})$$

$${\mathcal{S}}^{†}\mathcal{S}=\mathcal{I}\text{leavevmode},$$

$$|r{|}^2+|t{|}^2=|\overline{r}{|}^2+|\overline{t}{|}^2=1,{\overline{t}}^{\ast }r+{\overline{r}}^{\ast }t=0\text{leavevmode}.$$

$${\mathcal{S}}_4^{\ast }=M{\mathcal{S}}_4^{-1}M,M=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\text{leavevmode},$$

$$|\text{det}\mathcal{S}{|}^2=|t\overline{t}-r\overline{r}{|}^2=1\text{leavevmode}.$$

$$\overline{r_4}=r_4,\overline{t_4}=t_4\text{leavevmode}.$$

$$S_4=\left(\begin{array}{cc}{t}_4& r_4\\ r_4& t_4\end{array}\right)\text{leavevmode}.$$

$$\begin{array}{rl}{r}_4& =-i\frac{\sinh (\sigma )}{\sinh (\sigma +i\pi \tau )}\\ t_4& =\frac{\sin (\pi \tau )}{\sinh (\sigma +i\pi \tau )}\end{array}$$

$${\tilde{\Psi }}^{ϵ}(\tilde{z})=\sum _{n=-\infty }^{\infty }(-1{)}^n{A}_n^{\tau }{J}_{ϵ(n+\tau )}(\mathbf{k}{e}^{\tilde{z}})J_{ϵn}(\mathbf{k}{e}^{-\tilde{z}})$$

Full text

I Introduction

The effective range theory was developed long ago Hamermesh1947 ; Blatt1949 ; Bethe1949 ; Barker1949 ; Teichmann1951 ; Brueckner1951 ; Salpeter1951 for studying nucleon scattering with scattering amplitudes described at low energies by a small number of parameters, namely a scattering length and an effective range in the simplest cases. In the present paper, we focus our attention on the study of atom scattering on the Casimir-Polder potential above a surface Lennard-Jones1936III ; Lennard-Jones1936IV ; Casimir1946 ; Casimir1948 . The Casimir-Polder potential varies rapidly in the vicinity of the surface, which leads to quantum reflection even though the potential well is attractive Yu1993 ; Berkhout1993 ; Carraro1998 ; Shimizu2001 ; Druzhinina2003 ; Pasquini2004 ; Oberst2005 ; Zhao2008 .

Quantum reflection should play a key role in the GBAR experiment that will test the weak equivalence principle on antihydrogen atoms Debu2012 ; Indelicato2014 ; Perez2015 ; Mansoulie2019 as it prevents the detection of antihydrogen atoms through annihilation on the detector Voronin2005jpb ; Voronin2011 ; Voronin2012pra . It can also be turned into a tool for reaching better experimental accuracy Dufour2014shaper ; Dufour2015ahep . In a new quantum measurement methods recently proposed to improve the accuracy of GBAR experiment Crepin2019 , a precise knowledge of the Casimir-Polder shifts of quantum gravitational states Nesvizhevsky2002 ; Nesvizhevsky2005 is required, and this requirement can be met by mastering the effective range expansion for scattering amplitudes at low energy Crepin2017 .

We explain in the next section the motivation for building an improved effective range expansion in the case of real Casimir-Polder potentials. We then propose a new derivation based on Liouville transformations of the Schrödinger equation which map the physical potential landscape into another potential more convenient to study quantum reflection. We use symmetries and composition properties of scattering amplitudes to derive an improved effective range expansion, leading to more accurate predictions at low energy and opening new perspectives for precise spectroscopic measurement of quantum gravitational states.

II Motivations for an improved effective range expansion

An (anti)hydrogen atom of mass $m$ at height $z$ above a horizontal surface is submitted to the gravity potential and the Casimir-Polder potential which dominates at the distances considered, of the order of a few micrometers. The Casimir-Polder potential $V(z)$ is attractive at all distances, with characteristic asymptotic power laws, the so called Van der Waals limit near the surface for $z\to 0$ and retarded limit far from it $z\to+\infty$ :

[TABLE]

The long-range length scale corresponding to this potential is $\ell_{4}=\sqrt{2mC_{4}}/\hbar$ while the short-range length scale is $\ell_{3}=2mC_{3}/\hbar^{2}$. Typical values for helium and silica surfaces are given in table 1.

The scattering amplitudes for an atom falling down onto the surface can be calculated by solving the one-dimensional stationary Schrödinger equation obeyed by the wave function Berry1972 ; Friedrich2004  :

[TABLE]

The equation (2) can be solved numerically by imposing absorbing boundary conditions $\psi(0)=0$ on the antihydrogen atom, annihilated at contact on the surface. The potential vanishes at large distances and the wave function is written ($E=\hbar k^{2}/(2m)$ is the energy and $k$ the asymptotic wavevector) :

[TABLE]

The reflection amplitude $r(k)$ which describes the quantum reflection of the atom on the Casimir-Polder potential has been computed for antihydrogen falling down on different surfaces Dufour2013qrefl ; Dufour2013porous ; CrepinEPL2017 . It goes to $-1$ at the limit of low energies, with an asymptotic approach to this limit described by a scattering length :

[TABLE]

For the ideal homogeneous potential $V_{4}$, the length $\ell$ coincides with the length scale $\ell_{4}$ OMalley1961 ; Arnecke2006 , that is real. This is why we chose to call $\ell$ the scattering length while the traditional scattering length – often denoted by $a\equiv-i\ell$ – is purely imaginary in that case. For a real Casimir-Polder potential in contrast, it is not the case though it might have been expected that the low-energy behavior of $r(k)$ is determined by the long-range part of the potential. As shown in table 1, the scattering length computed for real potentials with antihydrogen on helium or silica surfaces significantly differs from this expectation.

Not only the scattering length but also the Taylor expansion of $\mathcal{A}(k)$ at low energies is required to compute the precise positions of gravitational quantum states Crepin2017 . This effective range expansion is known for the ideal $V_{4}$ potential OMalley1961 ; Arnecke2006 . The parameters of the expansion can be modified when assuming that the difference between the real potential $V$ and the long-range limit $V_{4}$ is a short-range potential. However this assumption is not valid for the real Casimir-Polder potential, where the difference $V-V_{4}$ is known to behave essentially as another long-range potential $V_{3}$.

In the following, we will introduce Liouville transformations of the Schrödinger equation which change the potential landscape while exactly preserving reflection amplitudes Dufour2015jpb ; Dufour2015epl . We will see that this allows one to bypass the mathematical intricacies in the modified effective range theory for Casimir-Polder potentials, and then to get an improved asymptotic expansion of reflection amplitudes at low energies.

III Liouville transformations

We consider now Liouville transformations which transform the potential and remove its divergence in the vicinity of the surface Dufour2015jpb ; Dufour2015epl .

We first introduce the WKB phase :

[TABLE]

where $z_{\phi}$ is a reference point to be fixed by choosing a phase origin for the wave function $\phi$ at $z\rightarrow+\infty$ where the WKB phase is linear :

[TABLE]

We define the Liouville transformation as a related change of coordinate and wavefunction scaling  :

[TABLE]

The phase $\phi$ now corresponds to a translation length $\mathbf{z}_{\phi}$ in the new coordinates to be adjusted by symmetry considerations discussed later on :

[TABLE]

We can now rewrite the Schrödinger equation in the transformed coordinates :

[TABLE]

with $\mathbf{F}(\mathbf{z})$ given by a Schwarzian derivative $\{\mathbf{z},z\}$ :

[TABLE]

A significant badlands function $Q(z)$ indicates the badlands where the WKB approximation fails and quantum reflection occurs. The transformed energy, potential and wave-vector can also be written :

[TABLE]

For the ideal homogeneous potential $V_{4}(z)$ the transformed $\mathbf{V}_{4}(\mathbf{z})$ reaches a maximum at $\mathbf{z}_{M}$ Dufour2015jpb  :

[TABLE]

where ${}_{2}F_{1}$ is the hypergeometric function that relates $\mathbf{z}$ and $z$ for homogeneous potentials Dufour2015 . In order to make the potential $\mathbf{V}_{4}$ an even function of $\mathbf{z}$, we chose $\mathbf{z}_{M}=0$, that is also $\mathbf{z}_{\phi}=-\mathbf{z}_{*}$.

The potential $\mathbf{V}_{4}$ shows universal properties which do not depend on the amplitude $C_{4}$ of the $V_{4}$ potential after the change of coordinates is done as in (7). It can be rewritten as :

[TABLE]

and its asymptotic behavior deduced :

[TABLE]

The new potential in Liouville coordinates is plotted as the black full line in figure 1 with its asymptotic behavior shown as dashed red lines.

The real Casimir-Polder potential $V$ departs from the ideal form $V_{4}$ at not too large distances, which breaks the symmetry and modifies the transformed $\mathbf{V}$ after Liouville transformation, as can be seen in figure 2. The change is particularly significant in the left part $\mathbf{z}<0$ which corresponds to $z<\zeta$.

For $\mathbf{z}\rightarrow-\infty$, $\mathbf{V}$ behaves asymptotically, as $\mathbf{V}_{3}$ and the expression of the latter potential is known (as for all homogeneous potentials $\mathbf{V}_{n}$; see Dufour2015 ) :

[TABLE]

From (15), we deduce the asymptotic behaviour of $\mathbf{V}$ shown as the dashed red line in figure 2 :

[TABLE]

IV Two-step scattering process

After the Liouville transformation, the potential landscape is much smoother than the original one so that we can decompose the scattering process into two steps. The first step is the reflection on the universal long-range part $\mathbf{V}_{4}$ felt by the atom when he falls down from large distances, while the second one is the reflection on the inner $\mathbf{V}$ differing from $\mathbf{V}_{4}$ when the atom has been transmitted trough the first barrier.

The scattering matrix $\mathcal{S}$ connects the amplitudes of waves propagating out and in and it has a general form for one-channel scattering :

[TABLE]

It is also useful to define the transfer matrix $\mathcal{T}$ that relates left and right waves. $\mathcal{S}$ and $\mathcal{T}$ matrices are related by an operation Genet2003 defined for all matrices with a non-zero coefficient $m_{1,1}$ :

[TABLE]

This operation $\Pi$ is an involution transforming $\mathcal{S}$ into $\mathcal{T}$ as well as $\mathcal{T}$ into $\mathcal{S}$ :

[TABLE]

For a two-step scattering process such as the one discussed here, the full process is described by a mere product for $\mathcal{T}$ matrices that is also by a $\star$ operation for elementary $S$ matrices defined as follows Genet2003  :

[TABLE]

Here, the first step corresponds to reflection on the $\mathbf{V}_{4}$ potential, described by a known matrix $\mathcal{S}_{4}$, while the second step is the reflection above the tail $\mathbf{z}<0$ of the real potential $\mathbf{V}$, that will be discussed below as a matrix $\mathcal{S}_{\rho}$ matrix.

The whole process, schematized in figure 3, is then described by the $\mathcal{S}$ matrix :

[TABLE]

This decomposition might appear as artificial, as the two processes take place in the same space region. This argument can however be bypassed as $\mathcal{S}$ and $\mathcal{S}_{4}$ can both be computed numerically and the $\mathcal{S}_{\rho}$ matrix then defined without ambiguity as $\mathcal{S}_{\rho}=S\star\mathcal{S}_{4}^{-1}$ where $\mathcal{S}_{4}^{-1}$ is the inverse matrix of $\mathcal{S}_{4}$ for the $\star$ law :

[TABLE]

with $[\Pi(\mathcal{S}_{4})]^{-1}$ the inverse of $\Pi(\mathcal{S}_{4})$ for the usual product law of matrices. This gives a proper formal definition of $\mathcal{S}_{\rho}$ which matches the physical intuition of a two-step process. We will see below that this is the key to the improved effective range expansion obtained in this paper.

We first focus attention on the symmetries of the scattering problem, namely unitarity and reciprocity which are general symmetries and then space parity which holds for the specific potential $\mathbf{V}_{4}$. Unitarity of the $\mathcal{S}-$matrices associated with current conservation is valid for all processes considered here :

[TABLE]

Reciprocity is the result of time-reversal symmetry :

[TABLE]

Space parity is a property of the transformed $\mathbf{V}_{4}$ potential. It implies that the reflection and transmission amplitudes are the same if we consider a scattered wave coming from the left or from the right :

[TABLE]

As a consequence of these symmetries, the $S_{4}$ matrix has a simple form :

[TABLE]

Its eigenvalues $s_{\pm}=t_{4}\pm r_{4}$ have unit modulus (unitarity) and they can be written as pure dephasings $s_{\pm}=e^{i\delta_{\pm}}$ ($\delta_{\pm}\in\mathbb{R}$). The known solution of the scattering problem for the $V_{4}$ potential gives Dufour2015jpb  :

[TABLE]

where $\tau$ is a Mathieu characteristic exponent and $\sigma$ the ratio $\ln\frac{\tilde{\psi}^{-}(0)}{\tilde{\psi}^{+}(0)}$, corresponding to the two solutions ($\epsilon=\pm 1$) of the Mathieu equation Dufour2015jpb  :

[TABLE]

We may now write the reflection amplitude for the full scattering process, using these properties and the reflection amplitude $\rho$ (still to be calculated) :

[TABLE]

where we have used (23), (24) and (25) to rewrite the determinant of the $S_{4}-$matrix :

[TABLE]

We can also reverse the problem and express $\rho$ from expressions of $r$ and $r_{4}$ :

[TABLE]

V Derivation of the effective range expansion

We now derive an expansion of $\tilde{\mathcal{A}}$ which is invariant under the Liouville transformation :

[TABLE]

The expansion of $\tilde{\mathcal{A}}_{4}$ is known OMalley1961  :

[TABLE]

and that of $\tilde{\mathcal{A}}_{4}(\mathbf{k})$ deduced from (36) ($\mathbf{k}=\sqrt{k\ell_{4}}$) :

[TABLE]

Using (29) and (32), we deduce :

[TABLE]

It is interesting to note that if $\tilde{\mathcal{A}}_{4}(k)\in\mathbb{R}$, then $\tilde{\mathcal{A}}=\tilde{\mathcal{A}}_{4}$ independently of the value of $\rho$. We see from (34) that we need an expansion of $\tilde{\mathcal{A}}(\mathbf{k})$ up to order 6 in $\mathbf{k}$. if we want an expansion of $\tilde{\mathcal{A}}(k)$ up to order 3 in $k$. Since $\tilde{\mathcal{A}}_{4}^{*}-\tilde{\mathcal{A}}_{4}\sim\mathbf{k}^{2}$, this aim only requires an expansion of $\rho(\mathbf{k})$ up to the order 4 in $\mathbf{k}$.

As the difference $\mathbf{V}-\mathbf{V}_{4}$ is regular and decreases fast enough to apply Lippmann-Schwinger equations in scattering theory, it is natural to postulate that $\rho$ has a regular Taylor expansion in $\mathbf{k}$ :

[TABLE]

This postulate can then be checked out and the coefficients $\rho_{i}$ deduced through a fit using the exact expression known for $\mathbf{r}_{4}(\mathbf{k})$ and the numerical results of $\mathbf{r}(\mathbf{k})$ calculated for different surfaces.

We have performed these fits in the range $k\ell\in[2\times 10^{-3},10^{-1}]$. The numerical noise is indeed too large for $k\ell<2\times 10^{-3}$, while the truncated Taylor expansion ceases to be valid for $k\ell>10^{-1}$. We have taken uniformly 1000 points in the interval to build the discreet set of points to be fitted. The results of this fit are presented in table 2.

This polynomial expansion in $\mathbf{k}$ introduces square root terms in the original expansion in $k$. Such terms would have been absent in a naive approach keeping only integer powers of $k$, that is equivalently even powers of $\mathbf{k}$ in the expansion (36). We may compare the quality of the fits corresponds to the full Taylor expansion– truncated after the fourth order – and to even terms expansion restricted to even powers of $\mathbf{k}$ by looking at the standard deviations of the residuals. These standard deviations $\hat{\sigma}$ calculated for real and imaginary parts of $\rho$, and different surfaces (He or SiO2), are presented in table 3.

The ratio $\hat{\rho}/\rho$ of the estimated $\hat{\rho}$ for both even terms and full expansions to the numerically known $\rho$ is another illustration of the quality of the two fits plotted in figure 4. Both table 3 and figure 4 show a large improvement of the quality of the fit with the full Taylor expansion. The residual linear behavior in figure 4 reveals missing terms in the expansion of $\rho$ – the odd terms in the case of the even terms expansion and terms from the order 5 in the case of the full expansion. Liouville transformations thus reveal the existence of square root terms which could not have been discovered otherwise and neatly improve the precision of the estimators.

We can finally derive the expansion of $\tilde{\mathcal{A}}$ knowing the ones for $\tilde{\mathcal{A}}_{4}$ and $\rho$ :

[TABLE]

The last equation is a renormalization of the scattering length $\ell$ differing from the long-range length scale $\ell_{4}$ due to the non vanishing value of $\rho_{0}$. The other constants in the expansion are deduced from $\alpha_{i}$ in (33) and $\rho_{i}$ :

[TABLE]

VI Discussion of the new expansion

We conclude our discussion of the improved effective range expansion by showing its benefit with respect to the modified effective range theory OMalley1961 .

In the latter, two parameters had to be modified in the expansion of $\tilde{\mathcal{A}}$: the scattering length $\ell$ and the effective range $\mathcal{R}_{0}$, or equivalently $\tilde{\alpha}_{2}$ :

[TABLE]

with the numerical values of the scattering length $\ell$ and $\tilde{\alpha}_{2}$ for helium and silica in table 4.

In the improved effective range expansion proposed in this paper, five parameters have to be determined, $\rho_{i}$ for $i\in\llbracket 0,4\rrbracket$, some of them leading to new terms in the expansion of $\tilde{\mathcal{A}}$. We calculate the error due to the two expansions by comparing the values of the expansions to the precise numerical one.

We plotted in figure 5 the ratio $\tilde{\mathcal{A}}_{th}/\tilde{\mathcal{A}}_{num}$ of $\tilde{\mathcal{A}}_{th}$ in the complex plan obtained theoretically for the preceding effective range theory and the improved one, over $\tilde{\mathcal{A}}_{num}$ known numerically. It is clear that the improved effective range theory reproduces the energy dependance of $\tilde{\mathcal{A}}$ with a much better accuracy than the original theory. To be more quantitative, we define a norm as the maximum deviation of the predicted and numerically known values :

[TABLE]

and compare $\left|\left|\frac{\tilde{\mathcal{A}}_{th}}{\tilde{\mathcal{A}}_{num}}-1\right|\right|$ for the two theories in table 5. These results confirm what was already visible in the figure 5. The new improved expansion is more than 10 times more accurate than the expansion derived from the original effective range theory.

The improvement is partly due to the fact that the improved expansion allows more degrees of freedom than the old one (5 instead of 2). But the key improvement is the addition of new terms in the expansion (37) which represent a much better physical understanding of the two-step scattering process. These terms come from the better defined scattering problem after Liouville transformation. The coefficients $\rho_{i}$ obtained with the new fit are sufficient to compute the expansion of $\tilde{\mathcal{A}}$ with a very high accuracy, needed to determine precisely the Casimir-Polder shifts of quantum gravitation states Crepin2017 and, then, to take full benefit of the recently proposed quantum interference method Crepin2019 .

*Aknowledgments * Thanks are due to G. Dufour, A. Lambrecht, V. Nesvizhevsky and A. Voronin for insightful discussions during this work and also to colleagues in the GBAR and GRANIT collaborations.