An excited atom interacting with a Chern insulator: toward a far-field resonant Casimir–Polder repulsion

Abstract

We investigate the resonant Casimir–Polder interaction of an excited atom which has a single (electric dipole) transition with a Chern insulator, using the approach of quantum linear response theory. The Chern insulator has a nonzero, time-reversal symmetry breaking Hall conductance, leading to an additional contribution to the resonant Casimir–Polder interaction which depends on the coupling between the Hall conductance and the circular polarization state of the atomic transition. We find that the resonant Casimir–Polder shift can be significantly enhanced if the atomic de-excitation frequency is near a value associated with a van Hove singularity of the Chern insulator. Furthermore, we find that the resonant Casimir–Polder force can become monotonically decaying and repulsive for a relatively large atom–surface distance. This happens if the atomic dipole transition is right (left) circularly polarized and the Chern number of the Chern insulator is $-1$ ($+1$), and the atomic de-excitation energy is comparable to the bandgap energy of the Chern insulator. This has potential implications for the design of atom–surface interaction which can be tuned repulsive over a relatively large range of separations.

Graphic abstract

Appendices

Appendix A: Fresnel coefficients

In this appendix, we present the formulas (with dimensions restored) for the Fresnel coefficients obtained for the case of a single Chern insulator [37]. We denote by the symbols $r_{ss}$, $r_{sp}$, $r_{ps}$ and $r_{pp}$, respectively, the reflection coefficients for an incident s-polarized wave which is reflected as an s-polarized wave, an incident s-polarized wave which is reflected as an p-polarized wave, an incident p-polarized wave which is reflected as an s-polarized wave, and an incident p-polarized wave which is reflected as an p-polarized wave. For the transmission coefficients, we replace the symbol r by the symbol t.

$$\begin{aligned} r_{ss}= & {} -\frac{1}{\Delta } \bigg ( \frac{4\pi ^2}{c^2} (\sigma _{xx}^2 + \sigma _{xy}^2) + \frac{2\pi }{c} \big ( 1 - (ck_\parallel /\omega )^2 \big )^{-1/2} \sigma _{xx} \bigg ),\nonumber \\ \end{aligned}$$

(A1a)

$$\begin{aligned} r_{ps}= & {} r_{sp} = - \frac{2\pi }{c} \frac{\sigma _{xy}}{\Delta }, \end{aligned}$$

(A1b)

$$\begin{aligned} r_{pp}= & {} \frac{1}{\Delta } \bigg ( \frac{4\pi ^2}{c^2} (\sigma _{xx}^2 + \sigma _{xy}^2) + \frac{2\pi }{c} \big ( 1 - (ck_\parallel /\omega )^2 \big )^{1/2} \sigma _{xx} \bigg ),\nonumber \\ \end{aligned}$$

(A1c)

$$\begin{aligned} t_{ss}= & {} \frac{1}{\Delta } \bigg ( 1 + \frac{2\pi }{c} \big ( 1 - (ck_\parallel /\omega )^2 \big )^{1/2} \sigma _{xx} \bigg ), \end{aligned}$$

(A1d)

$$\begin{aligned} t_{ps}= & {} t_{sp} = - \frac{2\pi }{c} \frac{\sigma _{xy}}{\Delta }, \end{aligned}$$

(A1e)

$$\begin{aligned} t_{pp}= & {} \frac{1}{\Delta } \bigg ( 1 + \frac{2\pi }{c} \big ( 1 - (ck_\parallel /\omega )^2 \big )^{-1/2} \sigma _{xx} \bigg ), \end{aligned}$$

(A1f)

where $\Delta \equiv 1 + \frac{2\pi }{c} \Big ( \big ( 1 - (ck_\parallel /\omega )^2 \big )^{1/2} + \big ( 1 - (ck_\parallel /\omega )^2 \big )^{-1/2} \Big ) \sigma _{xx} + \frac{4\pi ^2}{c^2} \big ( \sigma _{xx}^2 + \sigma _{xy}^2 \big )$. By defining a dimensionless conductivity $\widetilde{\sigma }\equiv 2\pi \sigma /c$, we can put the reflection coefficients above in the form Eqs. (2.9).

Appendix B: derivation of energy shifts in the presence of a nonreciprocal medium

In Ref. [30], it was shown using perturbation theory methods that for a quantum emitter interacting with the radiation field via the dipole interaction $-\varvec{\mu }\cdot \textbf{D}$, the shift $\delta E_m'$ in the energy $E_m$ of atomic state $|m\rangle$ is given by

$$\begin{aligned}&\delta E_m' = - \mathcal {P} \left\{ \sum _{n} \int _{-\infty }^\infty \textrm{d}\omega \, \frac{\mu _a^{mn} \mu _b^{nm} }{\omega + \omega _{nm}}\right. \nonumber \\&\left. \left[ \frac{1}{\hbar } \sum _{B,N} p(B) D_a^{BN}({\textbf{r}}_0) D_b^{NB}({\textbf{r}}_0) \, \delta (\omega - \omega _{NB}) \right] \right\} . \nonumber \\\end{aligned}$$

(B1)

Here, $\mathcal {P}$ denotes the principal value, $a, b = 1,2,3$ label Cartesian axes, m, n label atomic states, B, N label field states, p(B) denotes the prior probability to prepare the field in state $|B\rangle$, $\omega _{nm} \equiv (E_n - E_m)/\hbar$, $\mu _a^{mn} \equiv \langle m | \mu _a | n \rangle$ denotes the dipole transition matrix element from atomic state $|n\rangle$ to $|m\rangle$, and $D_a^{BN} \equiv \langle B | D_a | N \rangle$ denotes the dipole transition matrix element from field state $|N\rangle$ to $|B\rangle$. As we are interested in the surface-induced correction to the energy shift, we have ignored a contribution which arises purely from the electromagnetic fluctuations in free space.

In the presence of a nonreciprocal medium, it turns out that the term enclosed by the square brackets in Eq. (B1) is proportional to the anti-Hermitian part of the dyadic Green function. In what follows, we are going to derive this result. We first recall that the dyadic Green tensor $\mathbb {G}({\textbf{r}},{\textbf{r}}',t)$ connects the dipole $\textbf{p}(t')$ at time $t'$ and position ${\textbf{r}}'$ to the expectation value of its displacement field response, $\langle \textbf{D}({\textbf{r}},t) \rangle$, via

$$\begin{aligned} \langle D_a({\textbf{r}},t) \rangle = \int _{-\infty }^t \textrm{d}t' \, G_{ab}({\textbf{r}},{\textbf{r}}',t-t') \, p_b(t'), \end{aligned}$$

(B2)

with the dyadic Green tensor being given by [47]

$$\begin{aligned}&G_{ab}({\textbf{r}},{\textbf{r}}',t) = \frac{i}{\hbar } \sum _{B,N} p(B) \big [ \langle B | D_a({\textbf{r}},t) | N \rangle \langle N | D_b({\textbf{r}}',0) |B\rangle \nonumber \\&\quad - \langle B | D_b({\textbf{r}}' , 0) | N \rangle \langle N | D_a({\textbf{r}},t) |B\rangle \big ] \Theta (t), \end{aligned}$$

(B3)

where $\Theta (t)$ is the Heaviside step function, being equal to 1 if $t > 0$ and 0 if $t < 0$, and the operator $D_a({\textbf{r}},t)$ is in the interaction picture. The operator in the interaction picture is related to its counterpart $D_a({\textbf{r}})$ in the Schrödinger picture by $D_a({\textbf{r}},t) = e^{iH_0 t/\hbar } D_a({\textbf{r}}) \, e^{-iH_0 t/\hbar }$, where $H_0$ is the free Hamiltonian for the field in the absence of the dipole interaction. In terms of the Schrödinger picture, the dyadic Green tensor is given by

$$\begin{aligned}{} & {} G_{ab}({\textbf{r}},{\textbf{r}}',t) = \frac{i}{\hbar } \sum _{B,N} p(B)\nonumber \\{} & {} \quad \big [ e^{-i\omega _{NB}t} D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}') \nonumber \\ {}{} & {} \quad - e^{i\omega _{NB}t} D_b^{BN}({\textbf{r}}') D_a^{NB}({\textbf{r}}) \big ] \Theta (t).\nonumber \\ \end{aligned}$$

(B4)

By relabeling $B \leftrightarrow N$ in the second term, we obtain

$$\begin{aligned}{} & {} G_{ab}({\textbf{r}},{\textbf{r}}',t)\nonumber \\{} & {} \quad = \frac{i}{\hbar } \sum _{B,N} e^{-i\omega _{NB}t} \big [ p(B) - p(N) \big ] D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}') \Theta (t).\nonumber \\ \end{aligned}$$

(B5)

The Fourier transform is given by

$$\begin{aligned}{} & {} G_{ab}({\textbf{r}},{\textbf{r}}',\omega ) = \int _{-\infty }^{\infty } \textrm{d}t \, e^{i\omega t} G_{ab}({\textbf{r}},{\textbf{r}}',t) \nonumber \\{} & {} \quad =-\frac{1}{\hbar } \sum _{B,N} \big [ p(B) - p(N) \big ] \frac{D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}')}{\omega - \omega _{NB} + i0} \nonumber \\{} & {} \quad = -\frac{1}{\hbar } \sum _{B,N}\big [ p(B) - p(N) \big ] D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}')\nonumber \\{} & {} \qquad \times \left[ \mathcal {P} \left( \frac{1}{\omega - \omega _{NB}} \right) - i\pi \delta (\omega - \omega _{NB}) \right] . \end{aligned}$$

(B6)

As the displacement field is a physical observable, $\widehat{\textbf{D}} = \widehat{\textbf{D}}^\dagger$, which implies $(D_{a}^{BN})^*= D_{a}^{NB}$. This implies

$$\begin{aligned} G_{ba}^*({\textbf{r}}',{\textbf{r}},\omega )= & {} -\frac{1}{\hbar } \sum _{B,N}\big [ p(B) - p(N) \big ] D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}')\nonumber \\{} & {} \times \left[ \mathcal {P} \left( \frac{1}{\omega - \omega _{NB}} \right) + i\pi \delta (\omega - \omega _{NB}) \right] .\nonumber \\ \end{aligned}$$

(B7)

Noting that $p(B) - p(N) = p(B) (1 - e^{-\beta \hbar \omega _{NB}})$ and making use of the Dirac delta to change $\omega _{NB}$ to $\omega$, we obtain

$$\begin{aligned}{} & {} G_{ab}({\textbf{r}},{\textbf{r}}',\omega ) - G_{ba}^*({\textbf{r}}',{\textbf{r}},\omega ) = \frac{2\pi i}{\hbar } \sum _{B,N} p(B) D_a^{BN}({\textbf{r}}) \nonumber \\{} & {} \quad \times D_b^{NB}({\textbf{r}}') \big ( 1 - e^{-\beta \hbar \omega } \big ) \delta (\omega - \omega _{NB}), \end{aligned}$$

(B8)

which implies

$$\begin{aligned}&\frac{1}{\hbar } \sum _{B,N} p(B) D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}') \delta (\omega - \omega _{NB}) \nonumber \\&= \frac{1}{2\pi i} \frac{G_{ab}({\textbf{r}},{\textbf{r}}',\omega ) - G_{ba}^*({\textbf{r}}',{\textbf{r}},\omega )}{1 - e^{-\beta \hbar \omega }}. \end{aligned}$$

(B9)

We can rewrite the denominator of the rightmost term in terms of the Bose-Einstein distribution function, $n(T,\omega )=(e^{\beta \hbar \omega }-1)^{-1}$ as follows

$$\begin{aligned}&\frac{1}{\hbar } \sum _{B,N} p(B) D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}') \delta (\omega - \omega _{NB}) \nonumber \\&=-\frac{n(T,-\omega )}{2\pi i} \bigg (G_{ab}({\textbf{r}},{\textbf{r}}',\omega ) - G_{ba}^*({\textbf{r}}',{\textbf{r}},\omega )\bigg ). \end{aligned}$$

(B10)

When the temperature of the system is at or close to $T=0K$, the Bose-Einstein distribution function can be approximated in terms of Heaviside function as $n(T,\omega )\rightarrow -\Theta (-\omega )$, and hence we have that $n(T,-\omega )\rightarrow -\Theta (\omega )$:

$$\begin{aligned}&\frac{1}{\hbar } \sum _{B,N} p(B) D_a^{BN}({\textbf{r}}) D_b^{NB}({\textbf{r}}') \delta (\omega - \omega _{NB}) \nonumber \\&= \frac{\Theta (\omega )}{2\pi i} \bigg (G_{ab}({\textbf{r}},{\textbf{r}}',\omega ) - G_{ba}^*({\textbf{r}}',{\textbf{r}},\omega )\bigg ). \end{aligned}$$

(B11)

Using this expression, we can rewrite the energy shift $\delta E'_m$ in Eq. (B1) as

$$\begin{aligned}{} & {} \delta E_m'=-\frac{1}{2\pi i} \mathcal {P} \left( \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _{-\infty }^{\infty } \textrm{d}\omega \,\right. \nonumber \\{} & {} \quad \left. \frac{\Theta (\omega )\big (G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) - G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega )\big )}{\omega + \omega _{nm}} \right) .\nonumber \\ \end{aligned}$$

(B12)

Since the Heaviside function $\Theta (\omega )$ has values of 1 and 0 if its argument $\omega$ takes on positive and negative values respectively, only the positive frequency integral will survive. Hence we get

$$\begin{aligned} \delta E_m'= & {} -\frac{1}{2\pi i} \mathcal {P} \left( \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _0^\infty \textrm{d}\omega \right. \nonumber \\{} & {} \left. \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) - G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega )}{\omega + \omega _{nm}} \right) \nonumber \\= & {} -\frac{1}{2\pi i} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _0^\infty \textrm{d}\omega \nonumber \\{} & {} \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) - G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega )}{\omega + \omega _{nm} + i\eta } \nonumber \\{} & {} -\frac{1}{2} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _0^\infty \textrm{d}\omega \big ( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega )\nonumber \\{} & {} - G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \big ) \delta (\omega + \omega _{nm}). \end{aligned}$$

(B13)

In the above, on going to the second step, we have made use of the Plemelj relation, $\mathcal {P}(1/x) = 1/(x+i\eta ) + i\pi \delta (x)$. Next we make use of the Schwarz reflection property $G_{ba}^*(\omega ) = G_{ba}(-\omega ^*)$ (which is simply a statement about the reality of the Green function in real time domain) to obtain

$$\begin{aligned} \delta E_m'= & {} -\frac{1}{2\pi i} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _0^\infty \textrm{d}\omega \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega )}{\omega - \omega _{mn} + i\eta } \nonumber \\{} & {} -\frac{1}{2\pi i} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _{-\infty }^0 \textrm{d}\omega \frac{G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; \omega )}{\omega + \omega _{mn} - i\eta } \nonumber \\{} & {} -\frac{1}{2} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _0^\infty \textrm{d}\omega \, \delta (\omega - \omega _{mn}) G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega )\nonumber \\{} & {} +\frac{1}{2} \sum _n \mu _{a}^{mn} \mu _{b}^{nm} \int _{-\infty }^0 \textrm{d}\omega \, \delta (\omega + \omega _{mn}) G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; \omega )\nonumber \\ \end{aligned}$$

(B14)

Fig. 11

Integration contours on the complex frequency plane. There are two poles, which are located at $\omega = \pm (\omega _{mn} - i\eta )$

To evaluate the first term, we consider integrating around the contour $C_+$ in Fig. 11, which does not enclose any pole, and the segment $R_+$ tends radially to infinity. By the residue theorem we have

$$\begin{aligned}{} & {} \oint _{C_+} \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \textrm{d}\omega }{\omega - \omega _{mn} + i\eta }\nonumber \\{} & {} \quad = \Big ( \int _0^\infty + \int _{R_+} - \int _0^{i\infty } \Big ) \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \textrm{d}\omega }{\omega - \omega _{mn} + i\eta } = 0.\nonumber \\ \end{aligned}$$

(B15)

By Jordan’s Lemma, the integral over $R_+$ vanishes as $|\omega | \rightarrow \infty$ for $G_{ab}$ becomes exponentially suppressed. Thus,

$$\begin{aligned} \int _0^\infty \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \, \textrm{d}\omega }{\omega - \omega _{mn} + i\eta } = i \int _0^\infty \frac{G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \, \textrm{d}\xi }{- \omega _{mn} + i \xi + i\eta }. \end{aligned}$$

(B16)

Next, to evaluate the second term in Eq. (B14), we consider integrating around contour $C_-$ in Fig. 11, which encloses a pole at $\omega = -\omega _{mn} + i\eta$, and the segment $R_-$ tends radially to infinity. By the residue theorem we have

$$\begin{aligned}{} & {} \oint _{C_-} \frac{G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \, \textrm{d}\omega }{\omega + \omega _{mn} - i\eta } \\{} & {} \quad = 2\pi i \, G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; -\omega _{mn} + i\eta ) \, \Theta (\omega _{mn}) \\{} & {} \quad = \left( \int _{-\infty }^0 + \int _0^{i\infty } + \int _{R_-} \right) \frac{G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \textrm{d}\omega }{\omega + \omega _{mn} - i\eta }. \end{aligned}$$

Again, Jordan’s Lemma implies we can neglect the integral over $R_-$, and we thus have

$$\begin{aligned}{} & {} \int _{-\infty }^0 \frac{G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; \omega ) \, \textrm{d}\omega }{\omega + \omega _{mn} - i\eta } \nonumber \\{} & {} \quad = - i \int _0^{\infty } \frac{G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \, \textrm{d}\xi }{\omega _{mn} + i\xi - i\eta }\nonumber \\{} & {} \qquad + 2\pi i \, G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; -\omega _{mn} + i\eta ) \, \Theta (\omega _{mn}) .\nonumber \\ \end{aligned}$$

(B17)

As $\int _0^\infty \textrm{d}\omega \, \delta (\omega - \omega _{mn}) G_{ab}(\omega ) = G_{ab}(\omega _{mn})\Theta (\omega _{mn})$ and $\int _{-\infty }^0 \textrm{d}\omega \, \delta (\omega + \omega _{mn}) G_{ba}(\omega ) = G_{ba}(- \omega _{mn}) \Theta (\omega _{mn})$, and using Eqs. (B16) and (B17), we can express Eq. (B14) as

$$\begin{aligned} \delta E_m'= & {} \frac{1}{\pi } \sum _n \mu _a^{mn} \mu _b^{nm} \int _0^\infty \textrm{d}\xi \nonumber \\{} & {} \quad \left( \frac{\omega _{mn} \left( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) + G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \right) }{2(\omega _{mn}^2 + \xi ^2)} \right. \nonumber \\{} & {} \left. - \frac{\xi \left( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) - G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \right) }{2i(\omega _{mn}^2 + \xi ^2)} \right) \nonumber \\{} & {} - \frac{1}{2} \sum _n \mu _a^{mn} \mu _b^{nm} \Theta (\omega _{mn}) ( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{mn})\nonumber \\{} & {} + G_{ba}({\textbf{r}}_0,{\textbf{r}}_0; -\omega _{mn})) \nonumber \\= & {} \frac{1}{\pi } \sum _n \mu _a^{mn} \mu _b^{nm} \int _0^\infty \textrm{d}\xi \nonumber \\{} & {} \quad \left( \frac{\omega _{mn} \left( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) + G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \right) }{2(\omega _{mn}^2 + \xi ^2)} \right. \nonumber \\{} & {} \left. - \frac{\xi \left( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) - G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; i\xi ) \right) }{2i(\omega _{mn}^2 + \xi ^2)} \right) \nonumber \\{} & {} - \frac{1}{2} \sum _n \mu _a^{mn} \mu _b^{nm} \Theta (\omega _{mn})\nonumber \\{} & {} \quad \left( G_{ab}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{mn}) + G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega _{mn}) \right) .\nonumber \\ \end{aligned}$$

(B18)

On going to the second equality, we have again made use of the Schwarz reflection property: $G_{ba}^*(\omega ) = G_{ba}(-\omega ^*)$. For (real) imaginary frequencies, this implies $G_{ba}(-\omega ) = G_{ba}^*(\omega )$ ($G_{ba}(i\xi ) = G_{ba}^*(i\xi )$). Equation (B18) generalizes the formula for the atomic energy level shift to nonreciprocal media.

The first two terms of Eq. (B18) represent the nonresonant contribution to the Casimir–Polder shift, while the third term represents the resonant contribution. For a two-level atom with a single (dipole) transition, the third term leads to $\delta E_1^{\textrm{cl}}$ and $\delta E_0^{\textrm{cl}}$ in Eq. (2.2).

We can check that for reciprocal normal insulators (for which $G_{ba}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega ) = G_{ab}^*({\textbf{r}}_0,{\textbf{r}}_0; \omega )$), we recover Eq. (2.22) of Ref. [30]. Our result (B18) also agrees with the result obtained from an atomic dynamics-based approach.²

Appendix C: Green tensor contractions for various dipole configurations

In this appendix, we collect the results for the contractions of the reflection Green tensor with the dipole orientation vector for three different dipole configurations.

1.1 1 dipole polarization perpendicular to the surface

For the case $\textbf{n}^{\textrm{T}} = (0,0,1)$, we find

$$\begin{aligned} n_a \big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b^*= & {} 2 {\textrm{Re}}\, \big ( G_{zz}^R \big ), \end{aligned}$$

(C1a)

$$\begin{aligned} n_a^*\big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b= & {} 2 {\textrm{Re}}\, \big ( G_{zz}^R \big ), \end{aligned}$$

(C1b)

$$\begin{aligned} n_a \big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b^*= & {} 2i {\textrm{Im}}\, \big ( G_{zz}^R \big ), \end{aligned}$$

(C1c)

$$\begin{aligned} n_a^*\big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b= & {} 2i {\textrm{Im}}\, \big ( G_{zz}^R \big ). \end{aligned}$$

(C1d)

1.2 2 dipole polarization parallel to the surface

For the case $\textbf{n}^{\textrm{T}} = (1,0,0)$, we find

$$\begin{aligned} n_a \big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b^*= & {} 2 {\textrm{Re}}\, \big ( G_{xx}^R \big ), \end{aligned}$$

(C2a)

$$\begin{aligned} n_a^*\big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b= & {} 2 {\textrm{Re}}\, \big ( G_{xx}^R \big ), \end{aligned}$$

(C2b)

$$\begin{aligned} n_a \big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b^*= & {} 2i {\textrm{Im}}\, \big ( G_{xx}^R \big ), \end{aligned}$$

(C2c)

$$\begin{aligned} n_a^*\big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b= & {} 2i {\textrm{Im}}\, \big ( G_{xx}^R \big ). \end{aligned}$$

(C2d)

1.3 3 right circular dipole polarization

Finally, for the case $\textbf{n}^{\textrm{T}} = (1,i,0)/\sqrt{2}$, we find

$$\begin{aligned} n_a \big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b^*= & {} 2 \big ( {\textrm{Re}}\, G_{xx}^R + {\textrm{Im}}\, G_{xy}^R \big ), \end{aligned}$$

(C3a)

$$\begin{aligned} n_a^*\big ( G_{ab}^R + G_{ba}^{R*} \big ) n_b= & {} 2 \big ( {\textrm{Re}}\, G_{xx}^R - {\textrm{Im}}\, G_{xy}^R \big ), \end{aligned}$$

(C3b)

$$\begin{aligned} n_a \big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b^*= & {} 2i \big ( {\textrm{Im}}\, G_{xx}^R - {\textrm{Re}}\, G_{xy}^R \big ), \end{aligned}$$

(C3c)

$$\begin{aligned} n_a^*\big ( G_{ab}^R - G_{ba}^{R*} \big ) n_b= & {} 2i \big ( {\textrm{Im}}\, G_{xx}^R + {\textrm{Re}}\, G_{xy}^R \big ). \end{aligned}$$

(C3d)

Appendix D: Enhancement of the resonant Casimir–Polder shift at $\widetilde{\omega }_{10}=1.9$

Fig. 12

a Frequency dependences of the imaginary part of $\widetilde{\sigma }_{xx}$ (red circles) and the real part of $\widetilde{\sigma }_{xy}$ (blue squares); b the $\widetilde{k}_\parallel$-dependence of $|r_{ss}|$ (blue, dashed), $|r_{pp}|$ (red, dot-dashed) and $|r_{ps}|=|r_{sp}|$ (green, dotted) for the dispersive case with $\widetilde{\omega }_{10} = 1.9$. The behaviors of the corresponding reflection coefficients in the nondispersive limit are plotted in black: $|r_{ss}|$ (dashed), $|r_{pp}|$ (dot-dashed), and $|r_{ps}|= |r_{sp}|$ (dotted)

In this appendix, we consider a circularly polarized dipole near a $C = -1$ Chern insulator, and look more deeply into how the resonant Casimir–Polder shift $\delta \widetilde{\omega }_{10}^{\textrm{res}}$ becomes greatly enhanced at $\widetilde{\omega }_{10} = 1.9$, compared to the nondispersive limit. From Eq. (2.6), the magnitude of $\delta \widetilde{\omega }_{10}^{\textrm{res}}$ is determined by the magnitude of the reflection Green tensor, and from Eq. (2.7), the latter’s magnitude is in turn determined by the magnitude of the reflection coefficients.

The magnitudes of the reflection coefficients depend on the frequency in the conductances $\widetilde{\sigma }_{xx}$ and $\widetilde{\sigma }_{xy}$. The frequency dependence of the conductances is studied and plotted in Sec. II C of Ref. [37], where we found that for $\widetilde{\omega }_{10} < 2$, the real part of $\widetilde{\sigma }_{xx}$ and the imaginary part of $\widetilde{\sigma }_{xy}$ both vanish. For the reader’s convenience, we quote here the formulaic results we found using the Kubo formula for the conductivity tensor [37]:

$$\begin{aligned} \sigma _{xx}(\omega )= & {} -\frac{1}{\hbar } \int _{BZ} \frac{d^2\textbf{k}}{(2\pi )^2} {\textrm{Re}} \, [\langle + | j_x | - \rangle \langle - | j_x | + \rangle ] \frac{f(d) - f(-d)}{2d} \nonumber \\{} & {} \times \bigg ( \pi \big ( \delta (\hbar \omega + 2d) + \delta (\hbar \omega - 2d) \big ) + \frac{2 i \hbar \omega }{\hbar ^2\omega ^2 - 4d^2} \bigg ),\nonumber \\ \end{aligned}$$

(D1a)

$$\begin{aligned} \sigma _{xy}(\omega )= & {} \frac{1}{\hbar } \int _{BZ} \frac{d^2\textbf{k}}{(2\pi )^2} \, {\textrm{Im}} \, [\langle + | j_x | - \rangle \langle - | j_y | + \rangle ] \frac{f(d) - f(-d)}{2d} \nonumber \\{} & {} \times \bigg ( \frac{4d}{4d^2-\hbar ^2\omega ^2} + i \pi \big ( \delta (\hbar \omega - 2d) - \delta (\hbar \omega + 2d) \big ) \bigg ),\nonumber \\ \end{aligned}$$

(D1b)

where $d = \sqrt{d_x^2 + d_y^2 + d_z^2}$, $d_x = t \sin k_x a$, $d_y = t \sin k_y a$, $d_z = t (\cos k_x a + \cos k_y a) + u$, $f(d) = (\exp (\beta d) + 1)^{-1}$, and

$$\begin{aligned}{} & {} {\textrm{Im}}\, [\langle + | j_x | - \rangle \langle - | j_y | + \rangle ] \nonumber \\{} & {} \quad = - {\textrm{Im}}\, [\langle - | j_x | + \rangle \langle + | j_y | - \rangle ] \nonumber \\{} & {} \quad =\frac{t^3 (ae)^2}{4\,d (d_x^2+d_y^2)} \left( \left( \cos k_x a \sin ^2 k_y a + \cos k_y a \sin ^2 k_x a \right) \right. \nonumber \\{} & {} \quad \qquad \left( 2(d_x^2+d_y^2 + t^2) - t^2 (\cos 2k_x a + \cos 2k_y a) \right) \nonumber \\{} & {} \qquad \left. + 4td_z \cos k_x a \cos k_y a \left( \sin ^2 k_x a + \sin ^2 k_y a \right) \right) ,\nonumber \\ \end{aligned}$$

(D2a)

$$\begin{aligned}{} & {} \qquad \quad {\textrm{Re}}\, [\langle + | j_x | - \rangle \langle - | j_x | + \rangle ] \nonumber \\{} & {} \quad =\frac{(tae)^2}{16\,d^2 (d_x^2+d_y^2)} \left( \sin ^2 k_x a \left( t^2 (\cos 2k_x a + \cos 2k_y a) \right. \right. \nonumber \\{} & {} \qquad \left. - 2((d_x^2+d_y^2) + t^2) \right) \nonumber \\{} & {} \qquad \times \left( t^2 (\cos 2k_x a + \cos 2k_y a) - 2(d_x^2+d_y^2 + t^2) \right. \nonumber \\{} & {} \qquad \left. - 8 t d_z \cos k_x a) + 4 t^2 d_z^2 \sin ^2 2k_x a \right. \nonumber \\{} & {} \qquad \left. + 16 t^2 d^2 \cos ^2 k_x a \sin ^2 k_y a \right) . \end{aligned}$$

(D2b)

As in Ref. [37], we show in Fig. 12(a) the plots of the frequency dependence of the conductivity components which are non-vanishing, i.e., the imaginary part of $\widetilde{\sigma }_{xx}$ (red circles) and the real part of $\widetilde{\sigma }_{xy}$ (blue squares). As we explained in Ref. [37], for $\widetilde{\omega }_{10} < 2$, ${\textrm{Im}} \, \widetilde{\sigma }_{xx}$ is zero and ${\textrm{Re}} \, \widetilde{\sigma }_{xy} = -\alpha \approx -1/137$, a feature that we also see in the plot.

Figure 12(b) shows plots for the $\widetilde{k}_\parallel$-dependence of the magnitudes of the reflection coefficients for the dispersive case where $\widetilde{\omega }=1.9$, and also plots for the magnitudes of the reflection coefficients corresponding to the nondispersive limit. In the nondispersive limit, the reflection coefficients have constant values, with $r_{ss} = - r_{pp} \approx -\alpha ^2 \sim \mathcal {O}(10^{-4})$ and $r_{ps} = r_{sp} \approx \alpha \sim \mathcal {O}(10^{-2})$. On the other hand, we see that the reflection coefficients for the dispersive case with $\widetilde{\omega }_{10} = 1.9$ depend on $\widetilde{k}_\parallel$, and are generally much larger than those in the nondispersive limit.

Finally, let us express $\widetilde{G}_{xx}^R$ and $\widetilde{G}_{xy}^R$ in the form

$$\begin{aligned} \widetilde{G}_{xx}^R = \int _0^\infty \textrm{d}\widetilde{k}_\parallel \, g_1(\widetilde{k}_\parallel ), \quad \widetilde{G}_{xy}^R = \int _0^\infty \textrm{d}\widetilde{k}_\parallel \, g_2(\widetilde{k}_\parallel ), \end{aligned}$$

(D3)

which allows us to express Eq. (3.1) in the form

$$\begin{aligned} \delta \widetilde{\omega }_{10}^{\textrm{res}} = - \frac{3}{4} \int _0^\infty \textrm{d}\widetilde{k}_\parallel \Big ( {\textrm{Re}}\, g_1(\widetilde{k}_\parallel ) + {\textrm{Im}}\, g_2(\widetilde{k}_\parallel ) \Big ). \end{aligned}$$

(D4)

In Fig. 13, we plot the behavior of ${\textrm{Re}}\, g_1(\widetilde{k}_\parallel ) + {\textrm{Im}}\, g_2(\widetilde{k}_\parallel )$ for positive values of $\widetilde{k}_\parallel$, for both the dispersive case with $\widetilde{\omega }_{10} = 1.9$ (blue) and the nondispersive limit (black). We see that over the range plotted, ${\textrm{Re}}\, g_1(\widetilde{k}_\parallel ) + {\textrm{Im}}\, g_2(\widetilde{k}_\parallel )$ for the dispersive case generally has a much larger magnitude than the nondispersive limit. For values of $\widetilde{k}_\parallel$ much larger than 1, ${\textrm{Re}}\, g_1(\widetilde{k}_\parallel ) + {\textrm{Im}}\, g_2(\widetilde{k}_\parallel )$ becomes rapidly exponentially suppressed.

To summarize: in the nondispersive limit, $\widetilde{\sigma }_{xx} = 0$ and $\widetilde{\sigma }_{xy} \approx -1/137$, whereas the magnitudes of both $\widetilde{\sigma }_{xx}$ and $\widetilde{\sigma }_{xy}$ are of the order of unity in the dispersive case with $\widetilde{\omega }_{10} = 1.9$. This dispersion contributes to an enhancement in the reflection coefficients, which directly results in the enhancement of $\delta \widetilde{\omega }_{10}^{\textrm{res}}$.

Fig. 13

Behavior of ${\textrm{Re}}\, g_1 + {\textrm{Im}}\, g_2$ as a function of $\widetilde{k}_\parallel$ for the dispersive case with $\widetilde{\omega }_{10} = 1.9$ (blue) and the nondispersive limit (black). The functions $g_1$ and $g_2$ are defined in Eq. (D3)

Appendix E: Asymptotic regimes for the resonant Casimir–Polder shift

In this appendix, we obtain the asymptotic behavior of the resonant Casimir–Polder shift $\delta \widetilde{\omega }_{10}^{\textrm{res}}$ in the near-field ($\eta \rightarrow 0$) and far-field limit $\eta \gg 1$. For the dipole transitions which are polarized perpendicular to the surface, parallel to the surface, and right circularly polarized in the plane of the surface, this amounts to looking at the asymptotic behavior of ${\textrm{Re}}\,G_{xx}^R$, ${\textrm{Re}}\,G_{zz}^R$ and ${\textrm{Im}}\,G_{xy}^R$. For our asymptotic analysis, we follow the same procedure as the one described in Ref. [37]. Firstly, we scale out the dependence on $\eta$ in the exponential factors in Eqs. (2.8) by defining a new dimensionless variable $t \equiv {\widetilde{k}}_z \eta$ for the range $0 \le {\widetilde{k}}_\parallel < 1$ (recalling that ${\widetilde{k}}_z = (1 - {\widetilde{k}}_\parallel ^2)^{1/2}$), whence ${\widetilde{k}}_\parallel \textrm{d}{\widetilde{k}}_\parallel = - {\widetilde{k}}_z \textrm{d}{\widetilde{k}}_z = - t \, \textrm{d}t/\eta ^2$. For this range, $0 \le {\widetilde{k}}_z < 1$ and thus $0 \le t < \eta$. Then for the range $1 \le {\widetilde{k}}_\parallel < \infty$, we have $0 \le {\widetilde{k}}_z < i\infty$. If we define ${\widetilde{k}}_z \equiv i \ell$, then $0 \le \ell < \infty$, and we can define $t \equiv \ell \eta$, whereupon ${\widetilde{k}}_\parallel \textrm{d}{\widetilde{k}}_\parallel = \ell \, \textrm{d}\ell = t \, \textrm{d}t/\eta ^2$. After rescaling and using Eqs. (2.9), Eqs. (2.8) become

$$\begin{aligned}{} & {} {\widetilde{G}}_{xx}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) = g_{1}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) + h_{1}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}), \nonumber \\{} & {} g_{1}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) = - \frac{i}{2\eta } \int _0^\eta \textrm{d}t\nonumber \\{} & {} \quad \frac{ [(1 + (\frac{t}{\eta })^2)(\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2) + (\frac{\eta }{t} + (\frac{t}{\eta })^3)\widetilde{\sigma }_{xx}]e^{it} }{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + (\frac{\eta }{t} + \frac{t}{\eta })\widetilde{\sigma }_{xx}}, \nonumber \\{} & {} h_{1}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) =- \frac{1}{2\eta } \int _0^\infty \textrm{d}t \nonumber \\{} & {} \quad \frac{[(1 - (\frac{t}{\eta })^2)(\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2) - i ((\frac{t}{\eta })^3 + \frac{\eta }{t})\widetilde{\sigma }_{xx}]e^{-t}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + i (\frac{t}{\eta } - \frac{\eta }{t})\widetilde{\sigma }_{xx}}; \end{aligned}$$

(E1a)

$$\begin{aligned}{} & {} {\widetilde{G}}_{xy}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) = {\widetilde{G}}_{yx}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \nonumber \\{} & {} \quad = g_{2}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) + h_{2}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}), \nonumber \\{} & {} g_{2}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) = - \frac{i}{\eta ^2} \int _0^\eta \textrm{d}t \, t\nonumber \\{} & {} \quad \frac{\widetilde{\sigma }_{xy}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + (\frac{\eta }{t} + \frac{t}{\eta })\widetilde{\sigma }_{xx}} e^{it}, \nonumber \\{} & {} h_{2}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) = - \frac{i}{\eta ^2} \int _0^\infty \textrm{d}t \, t \nonumber \\{} & {} \quad \frac{\widetilde{\sigma }_{xy}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + i (\frac{t}{\eta } - \frac{\eta }{t})\widetilde{\sigma }_{xx}} e^{-t}; \end{aligned}$$

(E1b)

$$\begin{aligned}{} & {} {\widetilde{G}}_{zz}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) = g_{3}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) + h_{3}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}), \nonumber \\{} & {} g_{3}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) = \frac{i}{\eta } \int _0^\eta \textrm{d}t \, \bigg ( 1 - \frac{t^2}{\eta ^2} \bigg ) \nonumber \\{} & {} \quad \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + \frac{t}{\eta } \widetilde{\sigma }_{xx}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + (\frac{\eta }{t} + \frac{t}{\eta })\widetilde{\sigma }_{xx}} e^{it}, \nonumber \\{} & {} h_{3}({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10}) = \frac{1}{\eta } \int _0^\infty \textrm{d}t \, \bigg ( 1 + \frac{t^2}{\eta ^2} \bigg )\nonumber \\{} & {} \quad \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + i \frac{t}{\eta } \widetilde{\sigma }_{xx}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + i (\frac{t}{\eta } - \frac{\eta }{t})\widetilde{\sigma }_{xx}} e^{-t}. \end{aligned}$$

(E1c)

In the above, we have expressed each Green tensor component as the sum of a contribution with an oscillatory integrand, $g({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10})$, and a contribution with an exponentially decaying integrand, $h({\textbf{r}}_0,{\textbf{r}}_0; \omega _{10})$.

1.1 1 Near-field asymptotics

Let us consider the asymptotic behavior of the resonant Casimir–Polder shift in the near-field limit in the low and high frequency regimes. This amounts to considering the behavior of the contributions ${\textrm{Re}}\,{\widetilde{G}}_{xx}^R = {\textrm{Re}}\, g_1 + {\textrm{Re}}\, h_1$, ${\textrm{Im}}\,{\widetilde{G}}_{xy}^R = {\textrm{Im}}\, g_2 + {\textrm{Im}}\, h_2$ and ${\textrm{Re}}\,{\widetilde{G}}_{zz}^R = {\textrm{Re}}\, g_3 + {\textrm{Re}}\, h_3$ to the terms in Eq. (E1) in the limit that $\eta \rightarrow 0$.

As the resonant Casimir–Polder shift for our considered dipole polarizations involves ${\widetilde{G}}_{xx}^R$, ${\widetilde{G}}_{xy}^R$ and ${\widetilde{G}}_{zz}^R$, let us derive the near-field limits for the functions $g_1$, $g_2$, $g_3$, $h_1$, $h_2$ and $h_3$, for the case that the de-excitation frequency is not close to frequencies associated with van Hove singularities. Since $\eta$ is the upper limit in the integration over t in the functions $g_1$, $g_2$ and $g_3$ in Eqs. (E1), we have that $t/\eta \rightarrow 1$, $\eta /t \rightarrow 1$ and $e^{it} \rightarrow 1$ as $\eta \rightarrow 0$. We obtain

$$\begin{aligned} g_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} -\frac{i}{2\eta } \int _0^\eta \textrm{d}t\, \frac{2(\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2) + 2\widetilde{\sigma }_{xx}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}} \nonumber \\{} & {} \quad =- \frac{i(\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + \widetilde{\sigma }_{xx})}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}}, \end{aligned}$$

(E2a)

$$\begin{aligned} g_2({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} - \int _0^\eta \textrm{d}t \frac{i}{\eta ^2} \frac{t \widetilde{\sigma }_{xy}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}} \nonumber \\{} & {} \quad =-\frac{i}{2} \frac{\widetilde{\sigma }_{xy}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}}, \end{aligned}$$

(E2b)

$$\begin{aligned} g_3({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} \frac{i}{\eta } \int _0^\eta \textrm{d}t \, \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + \widetilde{\sigma }_{xx}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}} \Big (1 - \frac{t^2}{\eta ^2}\Big ) \nonumber \\{} & {} \quad =\frac{2i}{3} \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + \widetilde{\sigma }_{xx}}{1 + \widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2 + 2\widetilde{\sigma }_{xx}}. \end{aligned}$$

(E2c)

Next, let us consider the near-field limits for the functions $h_1$, $h_2$ and $h_3$. We can perform a perturbation expansion in powers of $\eta /t$ in the integrand and retain the leading order term. This leads to

$$\begin{aligned} h_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} \frac{1}{2\eta ^3} \int _0^\infty \textrm{d}t \, t^2 e^{-t} = \frac{1}{\eta ^3}, \end{aligned}$$

(E3a)

$$\begin{aligned} h_2({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} - \frac{1}{\eta } \int _0^\infty \textrm{d}t \, \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} e^{-t} = - \left( \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} \right) \frac{1}{\eta },\nonumber \\ \end{aligned}$$

(E3b)

$$\begin{aligned} h_3({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} \frac{1}{\eta ^3} \int _0^\infty \textrm{d}t\, t^2 e^{-t} = \frac{2}{\eta ^3}, \end{aligned}$$

(E3c)

From the above, we see again that the near-field limiting values of ${\textrm{Re}}\,h_1$, ${\textrm{Im}}\,h_2$ and ${\textrm{Re}}\,h_3$ dominate the limiting values of ${\textrm{Re}}\,g_1$, ${\textrm{Im}}\,g_2$ and ${\textrm{Re}}\,g_3$, which implies that ${\textrm{Re}}\,{\widetilde{G}}_{xx}^R \approx {\textrm{Re}}\, h_1$, ${\textrm{Im}}\,{\widetilde{G}}_{xy}^R \approx {\textrm{Im}}\, h_2$ and ${\textrm{Re}}\,{\widetilde{G}}_{zz}^R \approx {\textrm{Re}}\, h_3$.

Using Eq. (3.7), we obtain the near-field limit of the resonant Casimir–Polder shift for a dipole transition polarized perpendicular to the surface:

$$\begin{aligned} \delta \widetilde{\omega }_{10}^{\textrm{res}} \approx -\frac{3}{4} {\textrm{Re}} \, h_3({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx -\frac{3}{2\eta ^3} \,, \end{aligned}$$

(E4)

Similarly, using Eq. (3.13) we obtain the near-field limit of the resonant Casimir–Polder shift for a dipole transition polarized parallel to the surface:

$$\begin{aligned} \delta \widetilde{\omega }_{10}^{\textrm{res}} \approx -\frac{3}{4} {\textrm{Re}} \, h_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx -\frac{3}{4\eta ^3} \,. \end{aligned}$$

(E5)

Finally, the near-field limit of the resonant Casimir–Polder shift for a right circular dipole polarization, Eq. (3.1), is

$$\begin{aligned} \delta \widetilde{\omega }_{10}^{\textrm{res}}\approx & {} -\frac{3}{4} \left( {\textrm{Re}} \, h_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) + {\textrm{Im}} \, h_2({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \right) \nonumber \\\approx & {} -\frac{3}{4\eta ^3}. \end{aligned}$$

(E6)

The above near-field limits are the same for all the frequency regimes.

1.2 2 Far-field asymptotics

In the far-field limit, $t/\eta \ll 1$. We can perform a perturbation expansion in powers of $t/\eta$ in the integrands of the functions $g_1$, $g_2$, $g_3$, $h_1$, $h_2$ and $h_3$ from Eqs. (E1), and retain the leading order term. We obtain

$$\begin{aligned}{} & {} g_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx -\frac{i}{2\eta } \int _0^\eta \textrm{d}t \, e^{it} = \frac{1}{2\eta } \left( 1 - \cos \eta - i \sin \eta \right) ,\nonumber \\ \end{aligned}$$

(E7a)

$$\begin{aligned}{} & {} g_2({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx -\frac{i}{\eta ^3} \int _0^\eta \textrm{d}t \, t^2 e^{it} \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}}\nonumber \\{} & {} \quad = - \frac{1}{\eta ^3} \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} \left( 2 + \left( \eta ^2 -2 + 2 i \eta \right) e^{i \eta } \right) , \end{aligned}$$

(E7b)

$$\begin{aligned}{} & {} g_3({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx \frac{i}{\eta ^2} \int _0^\eta \textrm{d}t \, t \, e^{it} \left( \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2}{\widetilde{\sigma }_{xx}} \right) \nonumber \\{} & {} \quad =- \frac{i}{\eta ^2} \left( \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2}{\widetilde{\sigma }_{xx}} \right) \left( 1 - \left( 1 - i \eta \right) e^{i \eta } \right) , \end{aligned}$$

(E7c)

$$\begin{aligned}{} & {} h_1({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx -\frac{1}{2\eta } \int _0^\infty \textrm{d}t \, e^{-t} = - \frac{1}{2\eta }, \end{aligned}$$

(E7d)

$$\begin{aligned}{} & {} h_2({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx \frac{1}{\eta ^3} \int _0^\infty \textrm{d}t \, t^2 e^{-t} \left( \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} \right) = \frac{2}{\eta ^3} \left( \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} \right) ,\nonumber \\ \end{aligned}$$

(E7e)

$$\begin{aligned}{} & {} h_3({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx \frac{i}{\eta ^2} \int _0^\infty \textrm{d}t \, t \, e^{-t} \left( \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2}{\widetilde{\sigma }_{xx}} \right) \nonumber \\{} & {} \quad = \frac{i}{\eta ^2} \left( \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2}{\widetilde{\sigma }_{xx}} \right) . \end{aligned}$$

(E7f)

As ${\widetilde{G}}^R = g + h$, we have

$$\begin{aligned} {\widetilde{G}}_{xx}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} - \frac{\cos \eta + i \sin \eta }{2\eta } , \end{aligned}$$

(E8a)

$$\begin{aligned} {\widetilde{G}}_{xy}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} - \frac{1}{\eta ^3} \frac{\widetilde{\sigma }_{xy}}{\widetilde{\sigma }_{xx}} \left( \eta ^2 -2 + 2 i \eta \right) e^{i \eta } , \end{aligned}$$

(E8b)

$$\begin{aligned} {\widetilde{G}}_{zz}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10})\approx & {} \frac{i}{\eta ^2} \left( \frac{\widetilde{\sigma }_{xx}^2 + \widetilde{\sigma }_{xy}^2}{\widetilde{\sigma }_{xx}} \right) \left( 1 - i \eta \right) e^{i \eta }. \nonumber \\\end{aligned}$$

(E8c)

In the low frequency regime ($\omega _{10} < 2(2t-|u|)/\hbar$) and high frequency regime ($\omega _{10} > 2(2t+|u|)/\hbar$), $\widetilde{\sigma }_{xx} = i\widetilde{\sigma }_{xx}''$ and $\widetilde{\sigma }_{xy} = \widetilde{\sigma }_{xy}'$ [37]. The resonant Casimir–Polder shifts for our considered dipole polarizations involve ${\textrm{Re}}\, {\widetilde{G}}_{xx}^R$, ${\textrm{Im}}\, {\widetilde{G}}_{xy}^R$ and ${\textrm{Re}}\, {\widetilde{G}}_{zz}^R$. Using (E8), we obtain their far-field limits:

$$\begin{aligned}{} & {} {\textrm{Re}}\, {\widetilde{G}}_{xx}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \approx - \frac{\cos \eta }{2\eta } , \end{aligned}$$

(E9a)

$$\begin{aligned}{} & {} {\textrm{Im}}\, {\widetilde{G}}_{xy}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \nonumber \\{} & {} \quad \approx - \frac{1}{\eta ^3} \frac{\widetilde{\sigma }_{xy}'}{\widetilde{\sigma }_{xx}''} \left( 2\cos \eta + 2\eta \sin \eta - \eta ^2\cos \eta \right) , \end{aligned}$$

(E9b)

$$\begin{aligned}{} & {} {\textrm{Re}}\, {\widetilde{G}}_{zz}^R({\textbf{r}}_0,{\textbf{r}}_0;\omega _{10}) \nonumber \\{} & {} \quad \approx - \frac{1}{\eta ^2} \left( \frac{(\widetilde{\sigma }_{xx}'')^2 - (\widetilde{\sigma }_{xy}')^2}{\widetilde{\sigma }_{xx}''} \right) \left( \cos \eta + \eta \sin \eta \right) .\nonumber \\ \end{aligned}$$

(E9c)