## 1. Introduction

The interaction of atoms with material surfaces is a long-explored area. At atom-surface separations exceeding one or two nanometers it is determined to a large extent by the zero-point and thermal fluctuations of the electromagnetic field. This is a particular case of the van der Waals forces which act between two atoms, an atom and a surface (or between two surfaces), and originate from fluctuating electric dipole moments [

1]. The first theory of atom-surface interaction was created by London on the basis of nonrelativistic quantum mechanics [

2]. It was generalized by taking into account an existence of the zero-point fluctuations and retardation of the electromagnetic interaction by Casimir and Polder, who studied a polarizable atom in the vicinity of an ideal metal wall [

3]. Presently, the atom-wall force is often referred to by their names.

A more general theory taking into account the material properties of the wall at nonzero temperature in thermal equilibrium with the environment was developed in [

4] on the basis of the Lifshitz theory [

5,

6]. In the framework of this theory, the free energy of the Casimir-Polder interaction and respective force were expressed via the frequency-dependent atomic polarizability and dielectric permittivity of the wall material. In succeeding years, the fluctuation-induced forces between atoms, molecules, and material surfaces found numerous applications in atomic physics, condensed matter physics, as well as in biology and chemistry [

1,

7]. The advent of magnetic traps and ultracold cooling has opened up new opportunities for detailed study of the Casimir-Polder force in high-precision experiments.

Thus, the phenomenon of quantum reflection, that is, a reflection of ultra-cold atoms under an action of the attractive atom-surface force was demonstrated for both liquid and solid surfaces [

8,

9,

10,

11,

12,

13,

14,

15,

16,

17]. Another phenomenon is the Bose-Einstein condensation in dilute gases cooled to very low temperatures and confined in the magnetic trap near a material surface [

18,

19,

20,

21,

22]. Although the Casimir-Polder force between a unit atom and a surface is extremely small and not directly measurable, it should be accounted for in precise experiments with atomic clouds involving an abundance of atoms. This stimulated applications of the Lifshitz theory to calculation of the Casimir-Polder force for different atoms and cavity walls, taking into account the atomic dynamic polarizabilities and dielectric properties of wall materials [

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

37,

38,

39,

40,

41,

42,

43,

44,

45].

During the last few years, special attention was paid to the nonequilibrium Casimir-Pol-der forces which arise in situations when the surface (cavity wall) and the environment are kept at different temperatures. It is usually assumed that the temperature of the environment remains unchanged, whereas the cavity wall is heated to some higher temperature. The Casimir-Polder force under these conditions was discussed in [

46], whereas the generalization of the Lifshitz theory to nonequilibrium situations, including the case of atom-wall interaction, was performed in [

47,

48]. This theory was used for interpretation of the measurement data of the experiment [

49], where the thermal Casimir-Polder force between the ground state

${}^{87}$Rb atoms, belonging to the Bose-Einstein condensate, and a silica glass wall heated up to 605 K was measured for the first time. The measure of agreement between the experimental data of [

49] and theoretical predictions was used for constraining the Yukawa-type corrections to Newtonian gravity [

50], and the coupling constants of axions to nucleons [

51].

In this article, using the formalism of [

47,

48], we consider another physical situation where the nonequilibrium effects are gaining in importance, that is, the Casimir-Polder force between different atoms and subjected to a heating cavity wall which is made of the phase-change material. In fact, the equilibrium Casimir forces acting between a Au-coated sphere and either a VO

${}_{2}$ film deposited on a sapphire substrate [

52] or a AgInSbTe film on a Si substrate [

53] have already been considered in the literature. It has been known that VO

${}_{2}$ undergoes the phase transition from the dielectric to metallic phase with increasing temperature above

${T}_{c}=341\phantom{\rule{3.33333pt}{0ex}}$K. As to an amorphous AgInSbTe film, it undergoes the phase transition to the crystalline phase as a result of annealing. It was shown that the equilibrium Casimir force between a Au sphere and a VO

${}_{2}$ film experiences an abrupt jump resulting from the phase transition [

52]. The same holds for the Casimir force between a sphere and an AgInSbTe film in different phase states [

53]. Thus, it would be interesting to consider the combined effect of the nonequilibrium and phase-change conditions on the Casimir-Polder force for the wall material which undergoes phase transition with increasing temperature.

Here, we consider the atoms of metastable helium He

${}^{*}$, Na, Rb, and Cs interacting with a VO

${}_{2}$ film deposited on a sapphire wall for the atom–film separations varying from 5 to 10

$\mathsf{\mu}$m, as much as in the experiment [

49] for

${}^{87}$Rb atoms and a SiO

${}_{2}$ wall. We calculate the nonequilibrium Casimir-Polder forces for all these atoms at different wall temperatures, both below and above the critical temperature

${T}_{c}$ at which the VO

${}_{2}$ film transforms from the dielectric to metallic phase, leaving the environmental temperature constant. It is shown that the magnitude of the Casimir-Polder force decreases monotonously with increasing atom-wall separation, but takes a much larger value at each separation if the wall temperature, although slightly, exceeds the critical temperature. When the atom-wall separation is fixed, the force magnitude increases monotonously with increasing wall temperature in the region

$T<{T}_{c}$, experiences a jump at

$T={T}_{c}$, and increases further at

$T>{T}_{c}$.

By way of example, for a Rb atom 5

$\mathsf{\mu}$m apart from the wall made of the phase-change material, the magnitudes of equilibrium Casimir-Polder force at 300 K, 340 K (i.e., before the phase transition), and 342 K (i.e., after the phase transition) are equal to 1.92, 2.17, and 2.68 (

${10}^{-13}$ fN), respectively. If, however, the environmental temperature remains equal to 300 K and only the wall is heated, one obtains the force magnitudes of 1.92, 5.05, and 8.29 (

${10}^{-13}\phantom{\rule{3.33333pt}{0ex}}$fN) at the same respective wall temperatures. We have also compared the nonequilibrium Casimir-Polder forces and their gradients between a Rb atom and either a VO

${}_{2}$ film on a sapphire wall, which undergoes the phase transition, or a dielectric SiO

${}_{2}$ wall employed in the experiment [

49]. Using the experimental parameters, it is shown, for instance, that the nonequilibrium Casimir-Polder force between a Rb atom 7

$\mathsf{\mu}$m apart from the wall made of the phase-change material reaches at

$T=415\phantom{\rule{3.33333pt}{0ex}}$K wall temperature the same magnitude as is reached at

$T=605\phantom{\rule{3.33333pt}{0ex}}$K in the case of a dielectric wall.

The structure of this article is the following. In

Section 2, we briefly outline the used formalism.

Section 3 contains the calculation results for the nonequilibrium Casimir-Polder force for different atoms and phase-change wall material as functions of atom-wall separation and wall temperature. In

Section 4, the nonequilibrium Casimir-Polder force and its gradient for the phase-change wall material are compared with those for a dielectric wall used in the experimental configuration [

49].

Section 5 contains a discussion of the obtained results. In

Section 6, the reader will find our conclusions.

## 2. Nonequilibrium Casimir-Polder Force in the Micrometer
Separation Range

Here, we summarize the main expressions, allowing calculation of the nonequilibrium force between a ground-state atom and a cavity wall spaced at separations of a few micrometers, obtained in [

47,

48]. In doing so, the atom is described by its static polarizability

$\alpha \left(0\right)$, and the wall material by its dielectric properties at vanishing frequency. In the case of phase-change wall material, these properties are very specific and endow the nonequilibrium Casimir-Polder force with new and useful applications.

Let the temperature of the environment be

${T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K, and the wall temperature be

${T}_{W}\u2a7e{T}_{E}$. The atom-wall separation is denoted by

a. Then the nonequilibrium Casimir-Polder force can be presented in the form [

47,

48]

Here,

${F}_{\mathrm{eq}}$ is the well-known equilibrium Casimir-Polder force given by the Lifshitz formula, and

${F}_{\mathrm{neq}}$ is the proper nonequilibrium contribution. As is seen from (

1), in the state of thermal equilibrium

${T}_{W}={T}_{E}$, the nonequilibrium terms cancel each other out, and the Casimir-Polder force reduces to the equilibrium contribution alone.

The explicit expressions for both terms on the right-hand side of (

1) at all separations can be found in [

47,

48] (see also the monograph [

54]). Keeping in mind the values of the experimental parameters [

49], here we deal with atom-wall separations in the region from 5 to 10

$\mathsf{\mu}$m. In this region, one can obtain simple asymptotic expressions for both

${F}_{\mathrm{eq}}$ and

${F}_{\mathrm{neq}}$. Thus, for an ideal dielectric wall possessing finite dielectric permittivity at zero frequency

$\epsilon \left(0\right)<\infty $, one has [

54,

55]

where

${k}_{B}$ is the Boltzmann constant.

It is common knowledge, however, that at any nonzero temperature, real dielectric materials possess some small conductivity (so-called dc conductivity) which vanishes with

T exponentially fast [

56,

57]. As a result, for real dielectric materials, one obtains

$\epsilon \left(\omega \right)\to \infty $ when

$\omega $ goes to zero, and instead of (

2), one has

The same result was derived for the equilibrium Casimir-Polder force between an atom and a metallic wall at separations exceeding a few micrometers. Note that (

3) is given by the zero-frequency term of the Lifshitz formula. In so doing, the omitted sum over all nonzero Matsubara frequencies, which describes contribution of the zero-point oscillations to the force and some extra thermal contribution, is equal to 6.8% of (

3) at

$a=5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m,

$T=300\phantom{\rule{3.33333pt}{0ex}}$K, and quickly decreases with increasing separation and/or temperature.

Equations (

2) and (

3) invite some further information. The point is that measurements of the equilibrium Casimir force between a Au-coated sphere and a dielectric plate at separations of a few hundred nanometers were found in disagreement with theoretical predictions of the Lifshitz theory if the dc conductivity of plate material is taken into account in computations [

58,

59,

60,

61]. An agreement between the theory and the measurement data is reached only if the dc conductivity is disregarded [

58,

59,

60,

61]. The measured equilibrium Casimir-Polder force is also in good agreement with theoretical predictions if the dc conductivity of the SiO

${}_{2}$ wall is omitted in computations [

49], but excludes these predictions found with taken into account dc conductivity [

62]. What is even more surprising is that the Casimir-Polder entropy corresponding to the equilibrium Casimir-Polder force found in the framework of the Lifshitz theory violates the Nernst heat theorem if the dc conductivity of the dielectric wall is included in calculation [

63,

64,

65,

66,

67]. This conundrum is not yet resolved. Because of this, in the computations below, we consider both cases of taken into account and disregarded dc conductivity of the cavity wall at temperatures below

${T}_{c}$ when VO

${}_{2}$ is in the dielectric phase.

A similar problem was revealed for the equilibrium Casimir force measured between the metallic sphere and metallic plate. It was found that theoretical predictions of the Lifshitz theory are excluded by the measurement data if the relaxation properties of conduction electrons are taken into account in calculations (see, e.g., [

54,

55,

68,

69] for a review and [

70] for an attempts to solve this puzzle). For the interaction of nonmagnetic atoms with the metallic wall, however, the equilibrium Casimir-Polder force given by the Lifshitz formula does not depend on the relaxation properties of conduction electrons and, thus, the problem becomes immaterial.

Now, we consider the nonequilibrium contributions on the right-hand side of (

1). At separations of a few micrometers between an atom and a dielectric wall, the asymptotic expression for

${F}_{\mathrm{neq}}$ was derived under a condition that the thermal frequency

${\omega}_{T}={k}_{B}T/\hslash $ is much less than the characteristic frequency of the wall material [

47,

71] (see also review [

72])

This result was obtained with disregarded dc conductivity of the dielectric wall

${\sigma}_{0}\left(T\right)$ connected with the dielectric permittivity according to

We emphasize, however, that for the nonequilibrium contribution to the Casimir-Polder force, the inclusion in the calculation of the dc conductivity

${\sigma}_{0}\left(T\right)\ll {\omega}_{T}$ leads to practically the same values of

${F}_{\mathrm{neq}}$, as are given in (

4) [

62]. This is distinct from the equilibrium contribution to the Casimir-Polder force between an atom and a dielectric wall.

For an atom interacting with a metallic wall possessing the conductivity

${\sigma}_{m}\left(T\right)$ under a condition

${\omega}_{T}\ll {\sigma}_{0}\left(T\right)$, it holds that [

47]

where

$\zeta \left(z\right)$ is the Riemann zeta function. It should be stressed that this equation is derived by taking into account the relaxation properties of conduction electrons by means of the Drude model. If the relaxation properties of conduction electrons in metals are disregarded, the theoretically meaningless result for

${F}_{\mathrm{neq}}$ follows, which is also in contradiction with the measurement data. Because of this, one can conclude that the puzzling problems discussed above refer to only the Casimir and Casimir-Polder forces in a state of thermal equilibrium. Note also that following [

47,

71], we consider atoms in their ground states and assume that they cannot absorb the thermal radiation. Possible effects caused by the atomic absorption spectrum and radiation pressure are discussed in [

72].

In the next sections, the above results are applied to different atoms interacting with a wall made of the phase-change material.

## 3. The Casimir-Polder Force between Different Atoms and
VO${}_{\mathbf{2}}$ Film on a Sapphire Wall

Here, we consider the atoms of metastable He${}^{*}$ and ground state Na, Rb, and Cs interacting with VO${}_{2}$ film of thickness 100 nm deposited on a bulk sapphire wall. The atoms under consideration are characterized by their static atomic polarizabilities with the following values:

${\alpha}^{{\mathrm{He}}^{*}}\left(0\right)=4.678\times {10}^{-29}\phantom{\rule{3.33333pt}{0ex}}{m}^{3}$ [

73,

74],

$\phantom{\rule{1.em}{0ex}}{\alpha}^{\mathrm{Na}}\left(0\right)=2.411\times {10}^{-29}\phantom{\rule{3.33333pt}{0ex}}{m}^{3}$ [

75],

${\alpha}^{\mathrm{Rb}}\left(0\right)=4.73\times {10}^{-29}\phantom{\rule{3.33333pt}{0ex}}{m}^{3}$ [

76], and

$\phantom{\rule{1.em}{0ex}}{\alpha}^{\mathrm{Cs}}\left(0\right)=5.981\times {10}^{-29}\phantom{\rule{3.33333pt}{0ex}}{m}^{3}$ [

75,

77].

It is known that both vanadium dioxide (VO

${}_{2}$) crystals and thin films undergo the phase transition from dielectric, monoclinic, to a metallic tetragonal phase when the temperature increases up to

${T}_{c}=341\phantom{\rule{3.33333pt}{0ex}}$K [

78]. The dielectric permittivity of VO

${}_{2}$ film that is 100 nm thick deposited on a sapphire wall was measured in the frequency region from

$3.8\times {10}^{14}$ to

$7.6\times {10}^{15}\phantom{\rule{3.33333pt}{0ex}}$rad/s and fitted to the isotropic oscillator representation in [

79,

80] both before and after the phase transition.

According to the obtained results, the static dielectric permittivity of VO

${}_{2}$ film in the dielectric phase on a sapphire wall is equal to

$\epsilon \left(0\right)=9.909$. The characteristic absorption frequency of a VO

${}_{2}$ film on a sapphire wall before the phase transition is of the order of

$1.5\times {10}^{15}\phantom{\rule{3.33333pt}{0ex}}$rad/s—that is, it is much larger than the thermal frequency

${\omega}_{T}=3.9\times {10}^{13}\phantom{\rule{3.33333pt}{0ex}}$rad/s at room temperature

$T=300\phantom{\rule{3.33333pt}{0ex}}$K and also at higher temperatures up to 600 K considered below. Because of this, Equation (

4) is applicable in this case. The dc conductivity in the dielectric phase is of the order of

${\sigma}_{0}\sim {10}^{11}\phantom{\rule{3.33333pt}{0ex}}{s}^{-1}$. In the region around room temperature, it is almost temperature-independent but goes to zero exponentially fast with vanishing

T. As was noted in

Section 2, an account of this conductivity does not make an impact on the result (

4).

In the metallic phase at

$T=355\phantom{\rule{3.33333pt}{0ex}}$K, the conductivity value of the film was measured to be

${\sigma}_{m}\left(T\right)=2.03\times {10}^{15}\phantom{\rule{3.33333pt}{0ex}}{s}^{-1}$ [

79], that is, by the four orders of magnitude higher than the dc conductivity in the dielectric phase. The conductivity

${\sigma}_{m}\left(T\right)$ is well-described in the framework of the Drude model as

where

${\omega}_{p}=5.06\times {10}^{15}\phantom{\rule{3.33333pt}{0ex}}$rad/s is the plasma frequency and

$\gamma (T=355\phantom{\rule{0.166667em}{0ex}}K)=1.0\times {10}^{15}\phantom{\rule{3.33333pt}{0ex}}$rad/s is the relaxation parameter. In the temperature region of our interest, the linear dependence of the relaxation parameter on

T is determined by the electron-phonon interaction [

81]

It is seen that for VO

${}_{2}$ film in a metallic phase,

${\omega}_{T}\ll {\sigma}_{m}\left(T\right)$ when

T varies between 300 K and 600 K. Thus, the application condition of Equation (

6) is satisfied.

Now, we compute the Casimir-Polder force between a Rb atom and a VO${}_{2}$ film on a sapphire wall as a function of separation between them, assuming that the temperature of the environment is ${T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K, whereas the wall temperature ${T}_{W}$ is either equal to ${T}_{E}$ (the situation of thermal equilibrium) or ${T}_{W}>{T}_{E}$ (the out-of-thermal-equilibrium situation)—specifically, ${T}_{W}=340\phantom{\rule{3.33333pt}{0ex}}$K, 345 K, and 385 K. In doing so, the first of these temperatures corresponds to a VO${}_{2}$ film in the dielectric phase, whereas the next two, in the metallic phase.

The computational results for the force magnitude are shown in

Figure 1 in the logarithmic scale as functions of separation by the lower and upper pairs of solid and dashed lines found at

${T}_{W}={T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K and

${T}_{W}=340\phantom{\rule{3.33333pt}{0ex}}$K, respectively, and by the two solid lines next to them computed at

${T}_{W}=345\phantom{\rule{3.33333pt}{0ex}}$K and 385 K. At thermal equilibrium (the lower pair of solid and dashed lines), the computations are made by Equations (

2) and (

3), respectively, with disregarded and taken into account dc conductivity. The upper pair of solid and dashed lines (out-of-thermal equilibrium situation, but VO

${}_{2}$ film is in the dielectric phase) are computed by Equations (

1), (

2), (

4), and (

1), (

3), (

4), respectively, with disregarded and taken into account dc conductivity in the equilibrium contribution. The top two solid lines in

Figure 1 (out-of-thermal-equilibrium situation, but VO

${}_{2}$ film is in the metallic phase) were computed by Equations (

1), (

3), and (

6)–(

8) at respective temperatures. Note that (

6) and (

8) take into account the dependence of the conductivity of metal on temperature as a parameter, as it should be done also for nonequilibrium Casimir force between two parallel plates [

82].

As is seen in

Figure 1, the magnitude of negative (attractive) Casimir-Polder force decreases with increasing atom-wall separation and increases with increasing temperature. If the dc conductivity is taken into account in computations (the dashed lines in the lower and upper pairs of lines related to a VO

${}_{2}$ film in the dielectric phase), the larger magnitudes of the Casimir-Polder force are obtained. There is a jump in the force magnitude in the temperature region from

${T}_{W}=340\phantom{\rule{3.33333pt}{0ex}}$K to 345 K, which includes the critical temperature

${T}_{c}=341\phantom{\rule{3.33333pt}{0ex}}$K of the phase transition.

To gain a better insight, in

Figure 2 we present the same computational results for the magnitude of the Casimir-Polder force, as in

Figure 1, times the factor

${a}^{3}/{\alpha}^{\mathrm{Rb}}\left(0\right)$. This allows for plotting of the figure in a homogeneous scale, preserving the meaning and disposition of all lines. What is more,

Figure 2 does not depend on the type of atom and gives the possibility to obtain the magnitudes of the nonequilibrium Casimir-Polder force for any atom interacting with a VO

${}_{2}$ film deposited on a sapphire wall, and specifically, for the atoms of He

${}^{*}$, Na, and Cs. For this purpose, one should multiply the data of

Figure 2 by the factor

$\alpha \left(0\right)/{a}^{3}$ using the value of

$\alpha \left(0\right)$ for the desirable atom.

Now, using the same equations as explained above, we compute the nonequilibrium Casimir-Polder force between a Rb atom and a VO

${}_{2}$ film on a sapphire wall spaced at

$a=5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m separation as a function of wall temperature. The computational results are shown in

Figure 3 by the lower pair of solid and dashed lines (wall temperature varies from 300 K to 340 K, VO

${}_{2}$ is in the dielectric phase, the dc conductivity is disregarded and taken into account, respectively) and by the bottom solid line (wall temperature varies from 342 K to 380 K, VO

${}_{2}$ is in the metallic phase). This computation is made at the environmental temperature

${T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K. For comparison purposes, the upper pair of solid and dashed lines and the top solid line in the region of higher temperatures show similar equilibrium results computed under a condition

${T}_{E}={T}_{W}$, that is, when the environment is heated up to the same temperature as the wall (measurement of the difference Casimir force in such a situation was proposed in [

83]).

As is seen in

Figure 3, the jump in the Casimir-Polder force due to the phase transition is much more pronounced in the nonequilibrium case than in the equilibrium one. Thus, the equilibrium Casimir-Polder force at room temperature (

${T}_{E}={T}_{W}=300\phantom{\rule{3.33333pt}{0ex}}$K), just before the phase transition (

${T}_{E}={T}_{W}=340\phantom{\rule{3.33333pt}{0ex}}$K) and just after the phase transition (

${T}_{E}={T}_{W}=342\phantom{\rule{3.33333pt}{0ex}}$K) are equal to –1.92, –2.17, and –2.68 (

${10}^{-13}$ fN), respectively (the dc conductivity in the dielectric phase is disregarded). If, however, the situation can be nonequilibrium, that is,

${T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K, but

${T}_{W}=300\phantom{\rule{3.33333pt}{0ex}}$K, 340 K, and 342 K, one finds larger (in magnitude) values of the Casimir-Polder force –1.92, –5.05, and –8.29 (

${10}^{-13}$ fN), respectively (the dc conductivity is again disregarded).

One more conclusion following from

Figure 3 does not depend on whether the plate material undergoes the phase transition or is an ordinary dielectric or metal. In all these cases, the magnitude of the equilibrium Casimir-Polder force obtained when both the plate and the environment are heated up to some temperature

T > 300 K is substantially smaller than the magnitude of the nonequilibrium force in the case when only the plate is heated up to the temperature

T, whereas the environment preserves its temperature of 300 K.

The nonequilibrium Casimir-Polder force as a function of wall temperature was also computed for atoms of Na, He

${}^{*}$ and Cs spaced 5

$\mathsf{\mu}$m apart of VO

${}_{2}$ film on a sapphire wall, whereas the environment was kept at

${T}_{E}=300\phantom{\rule{3.33333pt}{0ex}}$K. The results are presented in

Figure 4 in the same way as for Rb atoms in the nonequilibrium case in

Figure 3. The three pairs of solid and dashed lines from top to bottom are for Na, He

${}^{*}$ and Cs atoms interacting with a VO

${}_{2}$ film in the dielectric phase (as above, the solid lines disregard dc conductivity and the dashed lines take it into account). The three solid lines from top to bottom are for the same respective atoms interacting with a VO

${}_{2}$ film in the metallic phase. In all these cases, the magnitude of the Casimir-Polder force increases significantly in out-of-thermal equilibrium conditions.

## 4. Comparison between the Nonequilibrium Casimir-Polder Forces for the Phase-Change and Dielectric Materials

From

Figure 1,

Figure 3, and

Figure 4, it is seen that both the equilibrium and nonequilibrium Casimir-Polder forces are extremely small. In fact, the forces below 1 fN are not accessible to a direct experimental observation. However, as noted in

Section 1, the atom-wall forces lead to measurable effects in precise experiments, with atomic clouds incorporating a great number of atoms. Thus, the Bose–Einstein condensate of

${}^{87}$Rb atoms was produced in a magnetic trap near a silica glass (SiO

${}_{2}$) surface and resonantly driven into a dipole oscillation [

49]. It was shown [

47,

48] that both in the equilibrium (

${T}_{W}={T}_{E}$) and nonequilibrium (

${T}_{W}>{T}_{E}$) situations, the Casimir-Polder forces lead to quite measurable shifts in the oscillation resonant frequency. These shifts were measured and recalculated into the gradients of the Casimir-Polder force [

49].

Below, we compare the values of the nonequilibrium Casimir-Polder forces and their gradients, which were obtained in the experimental configuration [

49] using the dielectric SiO

${}_{2}$ wall, with those found for the sapphire wall covered with a VO

${}_{2}$ film, which undergoes dielectric-to-metal phase transition with increasing temperature. The computations of the Casimir-Polder force in both cases were made using Equations (

1)–(

7), where the values of all necessary parameters are indicated above. For the static dielectric permittivity of SiO

${}_{2}$, one has

${\epsilon}^{{\mathrm{SiO}}_{2}}\left(0\right)=3.8$ [

84]. The temperature of the environment

${T}_{E}=310\phantom{\rule{3.33333pt}{0ex}}$K is used as in [

49].

In

Figure 5, the nonequilibrium Casimir-Polder forces between a Rb atom and a wall at 7

$\mathsf{\mu}$m separation are shown as the functions of wall temperature by the upper pair of solid and dashed lines for a SiO

${}_{2}$ wall and by the lower pair of solid and dashed lines continued to higher temperatures by the bottom solid line for a VO

${}_{2}$ film deposited on a sapphire wall. In both cases, the solid and dashed lines incorporated in pairs are computed with disregarded and taken into account dc conductivity of a dielectric material, respectively. As is seen in

Figure 5, in the case of phase-change wall material, the magnitudes of the Casimir-Polder force reach much larger values than for a dielectric SiO

${}_{2}$ wall. For example, at

${T}_{W}=415\phantom{\rule{3.33333pt}{0ex}}$K, the Casimir-Polder force between a Rb atom and a VO

${}_{2}$ film on a sapphire wall reaches the same magnitude as it reaches at

${T}_{W}=605\phantom{\rule{3.33333pt}{0ex}}$K in the case of a SiO

${}_{2}$ wall.

In

Figure 6, the comparison between the Casimir-Polder forces in the cases of phase-change and dielectric walls is made on an enlarged scale in the vicinity of the transition temperature

${T}_{c}=341\phantom{\rule{3.33333pt}{0ex}}$K. It is seen that although the phase transition contributes essentially to the magnitude of the Casimir-Polder force at

$T>{T}_{c}$, the slope of the force lines for a phase-change wall is, in any case, larger than for a dielectric wall. This makes the phase-change wall material preferable for measurements of the nonequilibrium Casimir-Polder force in precise experiments.

To confirm this conclusion, we also compare the gradients of the Casimir-Polder forces between Rb atoms and either the VO

${}_{2}$ on sapphire or SiO

${}_{2}$ walls, taking into account that the force gradients are directly connected with the measured frequency shift [

49]. The gradient of the nonequilibrium Casimir-Polder force is obtained by differentiating (

1) with respect to separation

Here, the equilibrium contribution to the force gradient for a dielectric wall with disregarded dc conductivity is obtained from (

2)

and with taken into account dc conductivity from (

3)

The last result is also valid for a wall material in the metallic phase.

The nonequilibrium contribution to (

9) is obtained from (

4) for a dielectric wall

and from (

6) for a metallic wall

Computations of the gradient of the nonequilibrium Casimir-Polder force between a Rb atom and either a SiO

${}_{2}$ wall or a VO

${}_{2}$ film on a sapphire wall at 7

$\mathsf{\mu}$m separation were made by Equations (

9)–(

13). The computational results are shown in

Figure 7 as the functions of wall temperature by the lower pair of solid and dashed lines for a SiO

${}_{2}$ wall and by the upper pair of solid and dashed lines continued to higher temperatures by the top solid line for a VO

${}_{2}$ film deposited on a sapphire wall. In both cases, the solid and dashed lines incorporated in pairs are computed with disregarded and taken into account dc conductivity of the dielectric material, respectively. The environmental temperature

${T}_{E}=310\phantom{\rule{3.33333pt}{0ex}}$K is used in computations. In the inset, the range of temperatures in the vicinity of the critical temperature

${T}_{c}=341\phantom{\rule{3.33333pt}{0ex}}$K, at which the phase transition occurs, is shown on an enlarged scale.

As is seen in

Figure 7, in the case of a VO

${}_{2}$ film on a sapphire wall, one obtains much larger gradients of the nonequilibrium Casimir-Polder force than for a SiO

${}_{2}$ wall under the same conditions. Thus, at

${T}_{W}=350\phantom{\rule{3.33333pt}{0ex}}$K and 400 K, the gradients of the Casimir-Polder force for a VO

${}_{2}$ film on a sapphire wall exceed those for a SiO

${}_{2}$ wall by factors of 4.46 and 6.02, respectively. The value of the force gradient reached in the experiment [

49] at

${T}_{W}=605\phantom{\rule{3.33333pt}{0ex}}$K would be reached for a VO

${}_{2}$ film on a sapphire wall at a much lower temperature,

${T}_{W}=342\phantom{\rule{3.33333pt}{0ex}}$K, just after the phase transition (compared with 415 K found for the respective values of the magnitude of the Casimir-Polder force in

Figure 5). This confirms that the use of phase-change wall materials is advantageous in experiments involving measuring the nonequilibrium Casimir-Polder forces.