Quantum friction in the presence of a perfectly conducting plate

It is well known that, when a neutral but polarizable particle sits near a perfectly conducting (PC) plate, it feels a force normal to the surface, pulling it towards the plate. This attractive force is often named after Casimir and Polder, who predicted it back in 1948 [1]. And the Casimir-Polder force was first experimentally confirmed by measuring the deflection of a sodium atom beam passing through a gold cavity [2]. There have been many experiments confirming the existence of the Casimir-Polder force ever since. Another ingenious method, which is suitable for detecting the thermal effects at larger atom-surface separation, is through the measurement of the center-of-mass oscillation frequencies of a rubidium atom Bose-Einstein condensate [3, 4]. But, will a force parallel to the surface of the PC plate arise when the particle moves parallel to the plate?

Even though the subject of quantum friction (QF) with a dielectric surface has been much discussed in the literature, this more idealized case involving a PC plate seems to have been largely ignored. The lack of discussion of this case may be due to an incorrect intuition arising from the image particle picture. One might think that the interaction between the particle and the PC plate can be entirely mimicked by the particle’s interaction with its image. As the particle moves above the plate, the image moves below the plate. Because the plate is perfectly conducting, the image keeps up with the particle and is always located at the mirror position of the particle. Consequently, any interaction between the two would only lie in the direction normal to the surface of the plate and no force in the transverse directions could possibly arise. This reasoning sounds convincing except that it ignores one important aspect: the particle interacts with the blackbody radiation surrounding it even when the plate is taken away. The nonrelativistic discussion of frictional force on particles moving in free space filled with only blackbody radiation can be traced back to the works of Mkrtchian et al. [5] or even Einstein and Hopf [6]. Ever since, there has been considerable interest in the subject of blackbody friction/quantum vacuum friction (QVF) [7, 8, 9, 10]. Recently, we have also investigated such quantum vacuum frictional effects on a particle moving with relativistic velocities, be it a nondissipative atom [11] or an intrinsically dissipative nanoparticle [12]. Now, when a PC plate is added into the configuration, the vacuum field in the vicinity of the plate will be different from that of the free space considered in Refs. [11, 12]. We therefore expect the QVF to be modified and become spatially varying in the normal direction. For convenience of presentation, we will refer to this quantum frictional force on a neutral particle passing above a PC plate as QFPC.

In this paper, we focus on calculating the QFPC for a nondissipative atom. This is somewhat simpler than the calculation of QFPC for a dissipative nanoparticle, where the temperatures of the particle and of the environment enter the problem independently. The discussion of QFPC for the dissipative nanoparticle is postponed to a subsequent paper.

Throughout the paper, we set $k_{B}=c=\hbar=1$ in the analytic expressions. SI units are reinstated in the numerical evaluations.

The physical situation we consider is illustrated in Fig. 1. A PC plate lies in the $x$-$y$ plane. An atom is at a distance $a$ from the plate and moves in the $x$ direction with constant velocity $v$. The polarizability of the atom is $\bm{\alpha}(\omega)$, which could be dispersive in frequency and have different components corresponding to different polarization states of the atom. Since the atom we consider is intrinsically nondissipative, $\bm{\alpha}(\omega)$ is a real quantity. The radiation background is at finite temperature $T$. We assume the PC plate is in thermal equilibrium with the radiation background. Because of its motion, the atom is not in equilibrium with the radiation. However, it is guaranteed, by the optical theorem [13], to be in the nonequilibrium steady state (NESS) [14], and it does not have an independent temperature [11].

The friction on an atom moving above a general planar surface has actually been calculated in Ref. [11] and tabulated for different polarization states in Appendix D therein. We find the formula for quantum friction can be recast into the following more convenient form:

where $\gamma=1/\sqrt{1-v^{2}}$ is the relativistic dilation factor. The atom’s polarizability, $\bm{\alpha}$, is defined in its own rest frame, $\mathcal{P}$, while the inverse temperature, $\beta$, is defined in the rest frame of the radiation, $\mathcal{R}$. The $\bm{g}^{\prime}$ in Eq. (1) is not the usual reduced Green’s dyadic, $\bm{g}$, defined in frame $\mathcal{R}$, but, rather, its Lorentz transform to frame $\mathcal{P}$. Each component of $\bm{g^{\prime}}$ can be expressed in terms of a combination of different components of $\bm{g}$, which we detail in Appendix A. The explicit form of $\bm{g}$ for a general planar background is recorded in Appendix B. Finally, $\Im\bm{g}$ refers to the anti-Hermitian part of the Green’s dyadic, of which the components are

The quantum friction in Eq. (1) is in fact the $x$ component of the Lorentz force on a moving dipole quantized using the fluctuation-dissipation theorem (FDT). Because the atom is intrinsically nondissipative, the frictional force is second order in $\bm{\alpha}$ like that discussed in Ref. [11]. There are two contributions to the force: the $\bar{k}_{x}$ term comes from the field fluctuations directly while the $k_{x}$ term comes from the induced dipole fluctuations. Although entering the friction formula with different signs, the two contributions do not cancel each other due to the Doppler shifting of the frequency in the $\coth$ factor. For an ordinary point particle, the polarizability of which is reciprocal, a relative velocity between the particle and the surrounding blackbody background is necessary for any transverse force to arise.¹¹1For a nonreciprocal point particle which is not in thermal equilibrium with the environment, self propulsion can be induced even if the particle is initially at rest. We will explore the fluctuation-induced effects for such a nonreciprocal particle in Ref. [15]. In addition, self propulsion is also possible for extended objects made up of reciprocal but nonuniform materials. See, for example, Ref. [16] by Reid et al. Even at zero temperature, the quantum friction does not vanish in general. But we have learned from our previous investigations that if the background is just free space, the resultant QVF does vanish at zero temperature [11, 12].

The matrix structure under the trace in the integrand is in general complicated. Even for an isotropic atom, there will be contributions to the quantum friction that mix the different diagonal polarization states of the atom and pick up the off-diagonal components of $\bm{g}^{\prime}$.²²2The effective polarizability, $\hat{\bm{\alpha}}=\bm{\alpha}\cdot\bm{\Gamma}\cdot{\bm{\alpha}}$, as defined in Ref. [11], can acquire off-diagonal components through the off-diagonal components of the Green’s dyadic, even when the intrinsic polarizability of the atom, $\bm{\alpha}$, is diagonal. Moreover, each component of $\bm{g}^{\prime}$ is still to be re-expressed as a combination of the different components of $\bm{g}$ in the actual calculation. However, the special background indicated in Fig. 1 has several features which greatly help to simplify the calculation. In the presence of the PC plate, $\bm{g}$ is found using the general expressions in Appendix B to be

with $\kappa=\sqrt{k^{2}-\omega^{2}}$. Similar to the vacuum background, the permittivity of a PC plate, $\varepsilon\to\infty$, is invariant under a Lorentz transformation in the $x$ direction. As a result, $\bm{g}^{\prime}$ is found to be identical to $\bm{g}$ when applying the transformations listed in Appendix A:

That is, for this special case, we can replace $\bm{g}^{\prime}$ in Eq. (1) with $\bm{g}$ in Eq. (3). In addition, terms containing the product of $g_{xy}$ and $g_{yx}$ or $g_{yz}$ and $g_{zy}$ do not contribute to the friction due to their oddness in $k_{y}$. If the polarizability tensor of the atom is diagonal, the only contribution to the friction involving the off-diagonal components of the Green’s dyadic, which mixes different components of the polarizability tensor is

Here and in the rest of the paper, we use a superscript on $F$ to specify contributions from different polarization states of the atom. In fact, $F^{\rm{XZ}}$ turns out to be the most interesting contribution to the frictional force, because it actually corresponds to a push instead of a drag.

Before presenting the results we obtain for QFPC, let us stress that no QFPC arises at zero temperature. In Appendix C, we prove that the zero temperature QF is absent not only for the vacuum case and the PC case, but also for the broader class of diaphanous materials when a nontrivial permeability ($\mu\neq 1$) is taken into account.

We see from the starting formula Eq. (1) that each contribution to the frictional force is proportional to the product of two nonvanishing components of the polarizability tensor. For simplicity, let us assume the polarizability tensor to be diagonal. We have ruled out most of the contributions that mix the components of the polarizability tensor based on the symmetry of the integrand. Then there are only four nonvanishing contributions left. They are proportional to $\alpha_{xx}^{2}$, $\alpha_{yy}^{2}$, $\alpha_{zz}^{2}$, $\alpha_{xx}\alpha_{zz}$ and will be denoted as $F^{\rm{XX}}$, $F^{\rm{YY}}$, $F^{\rm{ZZ}}$ and $F^{\rm{XZ}}$, respectively.

Crucial to the calculation is finding the anti-Hermitian part of the relevant components of $\bm{g}^{\rm{PC}}$. It can be seen from Eq. (3) that $\Im\,\bm{g}^{\rm{PC}}=\bm{0}$ unless the propagation wave number $\kappa$ develops an imaginary part. Since the integrand in Eq. (1) involves the product of two Green’s dyadics evaluated at $(\omega,\bm{k}_{\perp})$ and $(\omega,\bar{\bm{k}}_{\perp})$, respectively, the integration is restricted to regions where the propagation wave numbers associated with both Green’s dyadics become imaginary,

The branches need to be chosen so that the Green’s dyadic is retarded. The anti-Hermitian parts of the relevant components of the Green’s dyadic therefore read

The off-diagonal components of $\Im\bm{g}$ are different from the diagonal components in several respects. First of all, they are purely imaginary. Second, they are odd in $k_{x}$. As a result, in Eq. (1), $F^{\rm{XZ}}$ contributes to the total friction through the $-k_{x}$ term, while $F^{\rm{XX}}$, $F^{\rm{YY}}$ and $F^{\rm{ZZ}}$ all contribute through the $\bar{k}_{x}$ term. It is precisely the apparent minus sign in the $-k_{x}$ term that renders $F^{\rm{XZ}}$ positive, corresponding to a push instead of a drag.³³3Physically, the $-k_{x}$ term comes from the induced dipole fluctuations, while the $\bar{k}_{x}$ term comes from the direct field fluctuations. In the QVF case, these two different contributions also occur and they are called $F_{\rm{II}}^{\prime}$ and $F_{\rm{I}}^{\prime}$, respectively, in Ref. [11]. There, $F_{\rm{II}}^{\prime}$ vanishes, which reflects the fact that the dipole radiation emitted by the atom is isotropic. Here, the fact that $F^{\rm{XZ}}$ exists through the $-k_{x}$ term indicates that the dipole radiation in the $x$ direction is no longer isotropic when the PC plate is present. Furthermore, the positive sign of this contribution reflects that the corresponding dipole radiation emitted backward must exceed that emitted forward. Third, they do not contain terms independent of the atom-plate separation, $a$, as those in the diagonal components. These terms reflect the vacuum contributions. So, the off-diagonal components do not contribute to the QVF discussed in Ref. [11].

Without any further assumptions, we insert Eq. (7) into Eq. (1) and integrate $k_{x}$, $k_{y}$ and $\bar{k}_{y}$ analytically. With a further change of variable, $\bar{k}_{x}=\omega u$, we find the contribution to the QFPC from the $\rm{PQ}$ polarization states can be written as

and for each contribution, $\mathcal{F}^{\rm{PQ}}$ reads

Here, we have introduced a dimensionless frequency scaled by the distance $a$,

as well as a dimensionless inverse temperature also scaled by $a$,

So far, the expressions we have for QFPC in Eq. (9) are exact and involve the dynamical polarizability of the atom. For frequencies smaller than the lowest excitation energy of the atom, the dynamical polarizability, $\bm{\alpha}(\omega)$, can be replaced by the static polarizability [17], $\bm{\alpha}(0)$. Due to the common exponential factors in Eq. (9a)–(9d), the high frequency modes with $\beta\omega=xz\gg 1$ will be cut off and do not significantly contribute to the $\omega$ integral. Therefore, so long as the temperature is not high enough to excite the atom to its higher energy states, we can work in the static limit, where we substitute the polarizability with its static value. This allows us to take the product of the polarizabilities out of the $\omega$ integral in Eq. (8):

where the dimensionless functions $f^{\rm{PQ}}$ now characterize contributions to QFPC from different polarization states.

Note the magnitude of $z$ determines the dominating modes of the $x$ integral in Eq. (12). For $z\ll 1$, it is dominated by the large $x$ modes, where the complicated $x$ dependences in the integrands become subdominant and drop out, except for the common factor of $\frac{x^{7}}{e^{x\gamma(1+uv)z}-1}$. As a result, the diagonal contributions $F^{\rm{XX}}$, $F^{\rm{YY}}$ and $F^{\rm{ZZ}}$ become distance independent and proportional to $T^{8}$. Indeed, for $z\ll 1$, the diagonal contributions of QFPC precisely reduce to the corresponding contributions of QVF in Ref. [11]. On the other hand, $F^{\rm{XZ}}$, which is proportional to $T^{4}/a^{4}$, becomes completely negligible in comparison to the diagonal contributions. To sum up, the contributions to QFPC in the small $z$ limit read

As is shown in Eq. (13), unlike the diagonal contributions which monotonically increase with velocity, we find $F^{\rm{XZ}}$ vanishes when the velocity approaches the speed of light. Since the small $z$ limit of $F^{\rm{XZ}}$ has not been worked out in Ref. [11], we show how to obtain it analytically in Appendix E.

It is not so surprising that the small $z$ limit of QFPC coincides with QVF. Small $z$ values correspond to large distances or high temperatures. When the atom is far away from the PC plate, it is obvious that QFPC should reduce to QVF. In the case of high temperatures (but not so high to ionize the atom), the atom interacts with photons of very high frequency. It therefore mainly probes the very short distances around it and, effectively, does not feel the PC plate. That is, in the high temperature limit, the distribution of energy eigenvalues of photons interacting with the atom is insensitive to the presence of the plate.

Since quantum vacuum friction has been explored for a nondissipative atom in Ref. [11], the new physics really lies in the large $z$ limit, the short-distance or low-temperature behavior of QFPC. For $z\gg 1$, the small $x$ modes dominate the integrals. We can therefore expand the integrands in powers of $x$ before carrying out the integrals. Quite interestingly, the integrands for various polarization states exhibit different leading power behavior in $x$, which determines the distance and temperature dependences of their contributions to QFPC. After expansion in $x$, both the $x$ and $u$ integrals can be done exactly if we keep only the leading in $z$ terms. (Appendix E contains an approach to derive the asymptotic expression for $F^{\rm{XZ}}$ in the large $z$ limit as well.) For $z\gg 1$, the resultant QFPC is found to be

Since the results shown in Eq. (14) are for the large $z$ limit, it is apparent that $F^{\rm{ZZ}}$ dominates over the contributions from the other polarizations. In this limit, $F^{\rm{ZZ}}$ is independent of distance $a$ and proportional to $T^{8}$, just as is the case for QVF. In fact, we find $F^{\rm{ZZ}}$ is precisely four times the corresponding QVF contribution shown in Eq. (13). The next leading contribution, $F^{\rm{XZ}}$, is proportional to $a^{2}T^{10}$ with an overall positive sign, suggesting that this particular contribution corresponds to a push instead of a drag. The smallest contributions, $F^{\rm{XX}}$ and $F^{\rm{YY}}$, are both proportional to $a^{4}T^{12}$. On closer examination of Eq. (14), we also observe that $f^{\rm{YY}}$ is always greater than $f^{\rm{XX}}$, for arbitrary velocities.

Interestingly, these behaviors of QFPC may be easily understood from the image particle picture criticized in the Introduction. In fact, there is nothing wrong with replacing the PC plate by an image particle moving synchronously with the actual particle. We only need to keep in mind that both particles would interact with the surrounding photon bath, so that a frictional force does indeed arise. Following this line of reasoning, the image particle would double the normal component of the fluctuation-induced field, $E_{z}$, but eliminate the tangential components, $E_{x}$ and $E_{y}$, at the surface of the PC plate. Since these fluctuation-induced frictional forces are proportional to the product of the relevant fields, $F^{\rm{ZZ}}$ is therefore quadrupled while the other contributions are all suppressed when the distance between the particle and the PC plate approaches zero.

We have been advised by Matthias Krüger that the physics here is analogous to a classical situation in hydrodynamics. For example, the authors of Ref. [18] studied colloidal particles driven through a suspension of mutually noninteracting Brownian particles and the corresponding frictional force induced by the nonequilibrium fluid structure. (The flow field comoving with the colloidal particles is not in equilibrium with the Brownian particles.) They found that the frictional force on a single colloidal particle traveling along a wall (analogous to the PC plate in our case) is precisely the same as that on two colloidal particles driven side by side. The authors also found an enhancement of the friction due to the wall/image colloidal particle in comparison to the friction on an isolated colloidal particle. From the density plot of the solute Brownian particles, they interpret this increase in friction as the result of more Brownian particles aggregating in front of the colloidal particles when the wall/image particle is present. An analogous interpretation applies to what we see here in this paper. That is, the electromagnetic energy density is stronger near the PC plate.

So far, both the small $z$ results in Eq. (13) and the large $z$ results in Eq. (14) are exact in velocity. Another question is whether we can obtain the nonrelativistic (NR) limit analytically without assuming anything about $z$. This is possible as long as the dynamical polarizability is still replaced by its static value. We illustrate the procedure of obtaining the NR limit for $F^{\rm{XZ}}$, valid for all $z$ values in Appendix D. It turns out that all contributions to QFPC start with a term linear in $v$ in the NR limit.

As one of the contributions, $F^{\rm{XZ}}$, is positive (a push), while the others are all negative (a drag), a natural question arises: could the overall “frictional” force on an atom ever flip sign and therefore become a push? Of course, from Eq. (13) and Eq. (14), we can already conclude that the overall QFPC is negative definite in both the small $z$ (vacuum/high-temperature) limit and large $z$ (short-distance/low-temperature) limit. But, there is no convincing argument just from the analytic results suggesting that QFPC cannot switch sign in the intermediate $z$ regime. Therefore, we resort to numerical methods to ascertain the sign of QFPC.

We will here mainly consider atoms in their ground states, the polarizability of which is normally quite isotropic⁴⁴4Closed-shell atoms are almost exactly isotropic [19]. Even for open-shell atoms, the anisotropy is typically small compared to the isotropic part of the polarizability. Among the elements in a certain period, the anisotropy is largest when the first $p$ electron is added [17]. and can be well approximated by its static value, $\bm{\alpha}(\omega)=\alpha(0)\bm{1}$. For such isotropic atoms, the sign of the QFPC is determined by the sum of the dimensionless functions introduced in Eq. (12):

We show the absolute value of these dimensionless functions across their transition region in Fig. 2. Starting from small $z$ values, the total frictional force on the isotropic particle is dominated almost evenly between the $\rm{ZZ}$ and $\rm{YY}$ contributions. But as $z$ grows larger, the weight of the $\rm{YY}$ contribution decays so that the $\rm{ZZ}$ contribution solely dominates the entire frictional force. As for the unique positive contribution from the $\rm{XZ}$ polarization, it is completely negligible when $z$ is small but it eventually surpasses the contributions from the $\rm{XX}$ and $\rm{YY}$ polarizations for large $z$. Nonetheless, it never dominates the $\rm{ZZ}$ polarization. The asymptotic (in $z$) expressions in Eq. (13) and Eq. (14) are consistent with these behaviors and the agreement with the numerical data in their supposedly valid regimes are also clearly illustrated in the figure. So, we can conclude that the total QFPC on an isotropic atom is always a drag, since it cannot change sign even in the intermediate $z$ regime.

Another interesting aspect of the force is, of course, its magnitude. Fluctuation induced forces are typically small. But, is the QFPC possibly accessible to experiment? Here, we estimate QFPC on a cesium (Cs) atom, which has the largest static polarizability,⁵⁵5Within a period, the alkali metal atoms generally have the biggest polarizabilities. They are also supposed to have very tiny anisotropy because their valence electrons are in $s$ states [17]. Cs has the largest polarizability among the alkali metal atoms. according to Ref. [19], $\alpha_{\rm{Cs}}(0)=59.3\,\mathrm{\mathring{A}}^{3}$. Because the expression (12) we use for numerical calculation is obtained in the static limit, the corresponding numerical results are only expected to be appropriate when the atom is in its ground state, that is, up to the temperature that corresponds to the first excitation energy of the Cs atom, $T_{1}=16\,100\,\rm{K}$,⁶⁶6This temperature and the ionization temperature used later are obtained from the first excitation energy of Cs listed in Ref. [20]. beyond which a model for its dynamical polarizability is needed. In Fig. 3, we show the magnitude of the total frictional force on a Cs atom up to $T_{1}$, fixing velocity and distance. The friction clearly exhibits a power-law dependence on temperature. This is no surprise because we already know that the frictional force should behave as $T^{8}$ in both the large $z$ (low $T$) and small $z$ (high $T$) regimes.

Of course, QFPC also depends on the distance between the atom and the plate, distinguishing it from QVF. Considering the size of the Cs atom⁷⁷7Cesium also has the largest covalent radius ($244\,\rm{pm}$) among the nonradioactive atoms according to Ref. [21]., we should keep the distance greater than $1\,\rm{nm}$ to avoid additional surface effects. We therefore show the magnitude of QFPC for a Cs atom as a function of distance in Fig. 4, from $1\,\rm{nm}$ to $1\,\rm{\mu m}$, fixing the velocity at $v=0.5$ and temperature at $T=T_{1}$. It is seen that the total friction is only doubled when the distance is reduced from $1\,\rm{\mu m}$ to $1\,\rm{nm}$. This can be well understood from the asymptotic behavior of the dominant contributions: the $\rm{ZZ}$ contribution quadruples, yet the $\rm{YY}$ contribution vanishes at small distances, which is also clearly illustrated in the figure.

Finally, QFPC depends on the velocity of the atom. As is shown in Fig. 5a, the magnitude of the frictional force is linear in $v$ for very small velocities; however, the velocity dependence becomes more prominent for larger velocities. In Fig. 5b, we not only plot the total frictional force at the first excitation temperature, $T_{1}=16\,100\,\rm{K}$, but also extrapolate our numerical results to the ionization temperature of the cesium atom, $T_{i}=45\,100\,\rm{K}$ [20]. Above $T_{i}$, the outermost electron will be stripped off the atom so that the cesium atom cannot stay neutral. It is therefore not feasible experimentally to detect the quantum friction on an atom above its ionization temperature. In between $T_{1}$ and $T_{i}$, the atom can be excited, though not ionized. Now, the frequencies corresponding to the transition of the atom’s internal energy levels become important in evaluating QFPC. At these frequencies, the polarizability of the atom develops an imaginary part [22], which results in a QFPC that is first order in the polarizability. This effect is not included in the results we show for $T=T_{i}$. In addition, by employing the static value for the polarizability, we underestimate the magnitude of the second order QFPC, because atoms in excited states, e.g., Rydberg atoms⁸⁸8Even though Rydberg atoms possess much larger polarizabilities, which presumably will enhance the resulting frictional effect, we are unsure whether they could be appropriate candidates for experimental consideration, because blackbody radiation induces transitions to lower $n$ states and reduces the lifetime of the Rydberg states. Even at room temperature, transitions induced by blackbody radiation can contribute more to the decay rate than the spontaneous transitions [23]. At higher temperatures, the transition rate induced by blackbody radiation is even larger., tend to have much larger polarizabilities.

In this paper, we calculate the frictional force induced by fluctuations of the electromagnetic field on a neutral, nondissipative atom moving parallel to a perfectly conducting (PC) plate, which we term QFPC for brevity. This friction exists in second order in the polarizability of the atom and reduces to the quantum vacuum friction previously explored [11] in the limit of large distance from the plate or high temperature. At short distances or low temperatures, however, the PC plate modifies the behavior of the frictional force. For an isotropic atom, the frictional force is found to be negative definite (a drag) and twice the magnitude of the quantum vacuum friction felt by the same atom moving through blackbody radiation without a PC plate. Interestingly, the contribution to QFPC from a particular polarization state of the atom is positive (a push). However, this contribution turns out to be always subdominant to the negative contributions from the other polarization states. As a result, the total transverse force on the moving atom remains a drag. The magnitude of the friction seems to be too tiny to be observed unless the atom is made to move in a very hot background, at very high velocities. It is then fitting to make some comments about the experimental conditions on temperature and velocity.

First, the temperature is absolutely bounded by the ionization temperature of the atom, $T_{i}$, because the atom will no longer stay neutral above $T_{i}$. In this paper, we have used a static polarizability for the atom in the numerical calculations, which can only be justified if the atom remains in its ground state. The temperature is therefore further bounded by the temperature, $T_{1}$, corresponding to the first excitation energy of the atom. In principle, however, one can calculate QFPC on the atom up to its ionization temperature if its dynamical polarizability is known for a sufficiently wide frequency spectrum.

Of course, for experiments, the material which approximates a PC plate is very likely to give a more restrictive bound on the temperature. For example, the standard candidate, gold, will melt at $1337\,\rm{K}$. Even if one could imagine using liquid metals to mimic the perfectly conducting plate, the temperature is still bounded by the boiling point of the metal. Tungsten has the highest boiling point among metals, $6203\,\rm{K}$, which is still much lower than the typical ionization temperature, $T_{i}$, of an atom. (For the cesium atom discussed in the paper, $T_{i}=45\,100\,\rm{K}$.) This reality might motivate us to study QFPC for situations when the plate is not in thermal equilibrium with the radiation background. For the Casimir-Polder force (the force normal to the plate), such a scenario has been studied both theoretically [24] and experimentally [4].

Another apparent challenge to any feasible experiment is accelerating neutral atoms to relativistic velocities. But, in fact, it is possible nowadays to manipulate the conventional ion accelerators so that fast ions can be converted to neutral atoms with little change in momentum. For example, in Ref. [25], the maximum kinetic energy obtained for a copper atom is $1\,\rm{MeV}$, which is equivalent to a velocity of $0.0058$, after conversion using the relativistic formula for kinetic energy, $K=(\gamma-1)m$.

This paper only considers a very idealized background with the PC plate. For a surface with a real finite index of refraction, $n$, there will be induced Cherenkov friction [26] on the moving particle if it moves at a velocity above the Cherenkov threshold, $v>1/n$. If one further allows the surface to have dissipation, which is unavoidable in reality and perhaps induces an even greater frictional effect, the problem becomes complicated by the presence of several different mechanisms that give rise to friction. Ref. [27] provides a recent overview of this complicated subject with many useful references; however, it mainly focuses on only zero-temperature effects. Other works, like Ref. [28], do include finite-temperature effects but the discussion is restricted to only the nonrelativistic regime. In the future, we intend to calculate the quantum friction associated with a dispersive and dissipative surface fully for arbitrary temperatures and relativistic velocities. Alternatively, one could still assume a perfectly conducting boundary but allow the moving particle itself to be intrinsically dissipative. The resulting QFPC will then be a modification of the quantum vacuum friction studied in Ref. [12], where an independent temperature of the particle comes into play. We will discuss such QFPC in a subsequent paper.