Anomalous Doppler Effect Singularities in the Radiative Heat Generation,   Interaction Force and Frictional Torque for two Rotating Nanoparticles

A.I.Volokitin

arXiv:1702.02491·quant-ph·August 9, 2017

Anomalous Doppler Effect Singularities in the Radiative Heat Generation, Interaction Force and Frictional Torque for two Rotating Nanoparticles

A.I.Volokitin

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TL;DR

This paper investigates the divergence in heat generation, force, and torque between two rotating nanoparticles caused by anomalous Doppler effect singularities, revealing critical behavior at specific angular velocities.

Contribution

It introduces the concept of anomalous Doppler effect singularities in rotating nanoparticles, showing divergence in radiative heat transfer, force, and torque at specific velocities, a novel insight into nanoparticle interactions.

Findings

01

Heat generation rate diverges at specific angular velocities.

02

Singularities occur due to the anomalous Doppler effect and mutual polarization.

03

Results are relevant for biomedical applications.

Abstract

We calculate the quantum heat generation, the interaction force and the frictional torque for two rotating spherical nanoparticles with a radius $R$ . In contrast to the static case, when there is an upper limit in the radiative heat transfer between the particles, for two rotating nanoparticles the quantum heat generation rate diverges when the angular velocity becomes equal to the poles in the photon emission rate. These poles arise for the separation $d < d_{0} = R (3/ ε^{''} (ω_{0}))^{1/3}$ (where $ε^{''} (ω_{0})$ is the imaginary part of the dielectric function for the particle material at the surface phonon or plasmon polariton frequency $ω_{0}$ ) due to the anomalous Doppler effect and the mutual polarization of the particles and they exist even for the particles with losses. Similar singularities exist also for the interaction force and the frictional torque. The…

Equations144

p_{i} (t) = \int_{- \infty}^{\infty} \frac{d ω}{2 π} p_{i} (ω) e^{- iω t} .

p_{i} (t) = \int_{- \infty}^{\infty} \frac{d ω}{2 π} p_{i} (ω) e^{- iω t} .

p_{1} (ω) = α_{1} (ω) E_{12} (ω) + p_{1}^{f} (ω),

p_{1} (ω) = α_{1} (ω) E_{12} (ω) + p_{1}^{f} (ω),

E_{12} (ω) = \frac{3 p _{2 z} ( ω ) z ^}{d ^{3}} - \frac{p _{2} ( ω )}{d ^{3}},

E_{12} (ω) = \frac{3 p _{2 z} ( ω ) z ^}{d ^{3}} - \frac{p _{2} ( ω )}{d ^{3}},

p_{2}^{'} (ω) = α_{2} (ω) E_{21}^{'} (ω) + p_{2}^{f'} (ω),

p_{2}^{'} (ω) = α_{2} (ω) E_{21}^{'} (ω) + p_{2}^{f'} (ω),

\mathbf{p}^{\prime}_{2\perp}(t)=\left(\begin{array}[]{cc}\cos\Omega t&\sin\Omega t\\ -\sin\Omega t&\cos\Omega t\end{array}\right)\mathbf{p}_{i\perp}(t),

\mathbf{p}^{\prime}_{2\perp}(t)=\left(\begin{array}[]{cc}\cos\Omega t&\sin\Omega t\\ -\sin\Omega t&\cos\Omega t\end{array}\right)\mathbf{p}_{i\perp}(t),

p_{2 ⊥}^{'} (ω) = \overset{e}{^}^{' +} p_{2}^{-} (ω^{+}) + \overset{e}{^}^{' -} p_{2}^{+} (ω^{-})

p_{2 ⊥}^{'} (ω) = \overset{e}{^}^{' +} p_{2}^{-} (ω^{+}) + \overset{e}{^}^{' -} p_{2}^{+} (ω^{-})

E_{21 ⊥}^{'} (ω) = \overset{e}{^}^{' +} E_{21}^{-} (ω^{+}) + \overset{e}{^}^{' -} E_{21}^{+} (ω^{-}),

E_{21 ⊥}^{'} (ω) = \overset{e}{^}^{' +} E_{21}^{-} (ω^{+}) + \overset{e}{^}^{' -} E_{21}^{+} (ω^{-}),

E_{21} (ω) = \frac{3 p _{1 z} ( ω ) z ^}{d ^{3}} - \frac{p _{1} ( ω )}{d ^{3}},

E_{21} (ω) = \frac{3 p _{1 z} ( ω ) z ^}{d ^{3}} - \frac{p _{1} ( ω )}{d ^{3}},

p_{2 z} (ω) = \frac{2 α _{2} ( ω ) p _{1 z} ( ω )}{d ^{3}} + p_{2 z}^{f} (ω),

p_{2 z} (ω) = \frac{2 α _{2} ( ω ) p _{1 z} ( ω )}{d ^{3}} + p_{2 z}^{f} (ω),

p_{2 x} (ω) + i p_{2 y} (ω) = - \frac{α _{2} ( ω ^{+} ) [ p _{1 x} ( ω ) + i p _{1 y} ( ω )]}{d ^{3}} + p_{2}^{f' +} (ω^{+}),

p_{2 x} (ω) + i p_{2 y} (ω) = - \frac{α _{2} ( ω ^{+} ) [ p _{1 x} ( ω ) + i p _{1 y} ( ω )]}{d ^{3}} + p_{2}^{f' +} (ω^{+}),

p_{2 x} (ω) - i p_{2 y} (ω) = - \frac{α _{2} ( ω ^{-} ) [ p _{1 x} ( ω ) - i p _{1 y} ( ω )]}{d ^{3}} + p_{2}^{f' -} (ω^{-}),

p_{2 x} (ω) - i p_{2 y} (ω) = - \frac{α _{2} ( ω ^{-} ) [ p _{1 x} ( ω ) - i p _{1 y} ( ω )]}{d ^{3}} + p_{2}^{f' -} (ω^{-}),

p_{1 z} (ω) = \frac{p _{1 z}^{f} ( ω ) + 2 α _{1} ( ω ) p _{2 z}^{f} ( ω ) / d ^{3}}{1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6}},

p_{1 z} (ω) = \frac{p _{1 z}^{f} ( ω ) + 2 α _{1} ( ω ) p _{2 z}^{f} ( ω ) / d ^{3}}{1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6}},

p_{2 z} (ω) = \frac{p _{2 z}^{f} ( ω ) + 2 α _{2} ( ω ) p _{1 z}^{f} / d ^{3}}{1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6}},

p_{2 z} (ω) = \frac{p _{2 z}^{f} ( ω ) + 2 α _{2} ( ω ) p _{1 z}^{f} / d ^{3}}{1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6}},

p_{1 x} (ω) = \frac{1}{2} [\frac{p _{1}^{f +} ( ω ) - α _{1} ( ω ) p _{2}^{f' +} ( ω ^{+} ) / d ^{3}}{D ^{+}} + \frac{p _{1}^{f -} ( ω ) - α _{1} ( ω ) p _{2}^{f' -} ( ω ^{-} ) / d ^{3}}{D ^{-}}],

p_{1 x} (ω) = \frac{1}{2} [\frac{p _{1}^{f +} ( ω ) - α _{1} ( ω ) p _{2}^{f' +} ( ω ^{+} ) / d ^{3}}{D ^{+}} + \frac{p _{1}^{f -} ( ω ) - α _{1} ( ω ) p _{2}^{f' -} ( ω ^{-} ) / d ^{3}}{D ^{-}}],

p_{1 y} (ω) = \frac{1}{2 i} [\frac{p _{1}^{f +} ( ω ) - α _{1} ( ω ) p _{2}^{f' +} ( ω ^{+} ) / d ^{3}}{D ^{+}} - \frac{p _{1}^{f -} ( ω ) - α _{1} ( ω ) p _{2}^{f' -} ( ω ^{-} ) / d ^{3}}{D ^{-}}],

p_{1 y} (ω) = \frac{1}{2 i} [\frac{p _{1}^{f +} ( ω ) - α _{1} ( ω ) p _{2}^{f' +} ( ω ^{+} ) / d ^{3}}{D ^{+}} - \frac{p _{1}^{f -} ( ω ) - α _{1} ( ω ) p _{2}^{f' -} ( ω ^{-} ) / d ^{3}}{D ^{-}}],

p_{2 x} (ω) = \frac{1}{2} [\frac{p _{2}^{f' +} ( ω ^{+} ) - α _{2} ( ω ^{+} ) p _{1}^{f +} ( ω ) / d ^{3}}{D ^{+}} + \frac{p _{2}^{f' -} ( ω ^{-} ) - α _{2} ( ω ^{-} ) p _{1}^{f -} ( ω ) / d ^{3}}{D ^{-}}],

p_{2 x} (ω) = \frac{1}{2} [\frac{p _{2}^{f' +} ( ω ^{+} ) - α _{2} ( ω ^{+} ) p _{1}^{f +} ( ω ) / d ^{3}}{D ^{+}} + \frac{p _{2}^{f' -} ( ω ^{-} ) - α _{2} ( ω ^{-} ) p _{1}^{f -} ( ω ) / d ^{3}}{D ^{-}}],

p_{2 y} (ω) = \frac{1}{2 i} [\frac{p _{2}^{f' +} ( ω ^{+} ) - α _{2} ( ω ^{+} ) p _{1}^{f +} ( ω ) / d ^{3}}{D ^{+}} - \frac{p _{2}^{f' -} ( ω ^{-} ) - α _{2} ( ω ^{-} ) p _{1}^{f -} ( ω ) / d ^{3}}{D ^{-}}],

p_{2 y} (ω) = \frac{1}{2 i} [\frac{p _{2}^{f' +} ( ω ^{+} ) - α _{2} ( ω ^{+} ) p _{1}^{f +} ( ω ) / d ^{3}}{D ^{+}} - \frac{p _{2}^{f' -} ( ω ^{-} ) - α _{2} ( ω ^{-} ) p _{1}^{f -} ( ω ) / d ^{3}}{D ^{-}}],

< p_{ij}^{f} (ω) p_{ik}^{f *} (ω^{'}) >= 2 π δ (ω - ω^{'}) < p_{ij}^{f} p_{ik}^{f}) >_{ω}

< p_{ij}^{f} (ω) p_{ik}^{f *} (ω^{'}) >= 2 π δ (ω - ω^{'}) < p_{ij}^{f} p_{ik}^{f}) >_{ω}

< p_{ij}^{f} p_{ik}^{f} >_{ω} = ℏ Im α_{i} (ω) coth (\frac{ℏ ω}{2 k _{B} T _{i}}) δ_{j k} .

< p_{ij}^{f} p_{ik}^{f} >_{ω} = ℏ Im α_{i} (ω) coth (\frac{ℏ ω}{2 k _{B} T _{i}}) δ_{j k} .

M_{z} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < [p_{1 x} E_{12 y} - p_{1 y} E_{12 x}] >_{ω}

M_{z} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < [p_{1 x} E_{12 y} - p_{1 y} E_{12 x}] >_{ω}

M_{z} = \frac{ℏ}{π d ^{6}} \int_{- \infty}^{\infty} d ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})

M_{z} = \frac{ℏ}{π d ^{6}} \int_{- \infty}^{\infty} d ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})

M_{z Q} = - \frac{2ℏ}{π d ^{6}} \int_{0}^{Ω} d ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}}

M_{z Q} = - \frac{2ℏ}{π d ^{6}} \int_{0}^{Ω} d ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}}

P_{1} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < j_{1} \cdot E_{12} >_{ω} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < - iω p_{1} \cdot E_{12} >_{ω}

P_{1} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < j_{1} \cdot E_{12} >_{ω} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < - iω p_{1} \cdot E_{12} >_{ω}

= \frac{ℏ}{π d ^{6}} \int_{- \infty}^{\infty} d ω ω [2 \frac{Im α _{1} ( ω ) Im α _{2} ( ω )}{∣1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})

= \frac{ℏ}{π d ^{6}} \int_{- \infty}^{\infty} d ω ω [2 \frac{Im α _{1} ( ω ) Im α _{2} ( ω )}{∣1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})

+ \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})]

+ \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} - coth \frac{ℏ ω}{2 k _{B} T _{1}})]

P_{1 Q} = - \frac{2ℏ}{π d ^{6}} \int_{0}^{Ω} d ω ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} .

P_{1 Q} = - \frac{2ℏ}{π d ^{6}} \int_{0}^{Ω} d ω ω \frac{Im α _{1} ( ω ) Im α _{2} ( ω ^{-} )}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} .

F_{1 z} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < p_{1} \cdot \frac{d}{d z} E_{12} (z \to 0) >_{ω}

F_{1 z} = \int_{- \infty}^{\infty} \frac{d ω}{2 π} < p_{1} \cdot \frac{d}{d z} E_{12} (z \to 0) >_{ω}

= \frac{ℏ}{π d ^{7}} \int_{- \infty}^{\infty} d ω [\frac{6}{∣1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6} ∣ ^{2}} (Im α_{1} (ω) Re α_{2} (ω) coth \frac{ℏ ω}{2 k _{B} T _{1}} + Re α_{1} (ω) Im α_{2} (ω) coth \frac{ℏ ω}{2 k _{B} T _{2}})

= \frac{ℏ}{π d ^{7}} \int_{- \infty}^{\infty} d ω [\frac{6}{∣1 - 4 α _{1} ( ω ) α _{2} ( ω ) / d ^{6} ∣ ^{2}} (Im α_{1} (ω) Re α_{2} (ω) coth \frac{ℏ ω}{2 k _{B} T _{1}} + Re α_{1} (ω) Im α_{2} (ω) coth \frac{ℏ ω}{2 k _{B} T _{2}})

+ \frac{3}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (Re α_{1} (ω) Im α_{2} (ω^{-}) coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} + Im α_{1} (ω) Re α_{2} (ω^{-}) coth \frac{ℏ ω}{2 k _{B} T _{1}})]

+ \frac{3}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} (Re α_{1} (ω) Im α_{2} (ω^{-}) coth \frac{ℏ ω ^{-}}{2 k _{B} T _{2}} + Im α_{1} (ω) Re α_{2} (ω^{-}) coth \frac{ℏ ω}{2 k _{B} T _{1}})]

F_{1 z}^{A D} = \frac{ℏ}{π d ^{7}} \int_{0}^{Ω} d ω \frac{3}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} [Im α_{1} (ω) Re α_{2} (ω^{-}) - Re α_{1} (ω) Im α_{2} (ω^{-})] .

F_{1 z}^{A D} = \frac{ℏ}{π d ^{7}} \int_{0}^{Ω} d ω \frac{3}{∣1 - α _{1} ( ω ) α _{2} ( ω ^{-} ) / d ^{6} ∣ ^{2}} [Im α_{1} (ω) Re α_{2} (ω^{-}) - Re α_{1} (ω) Im α_{2} (ω^{-})] .

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Full text

Anomalous Doppler Effect Singularities in Radiative

Heat Generation, Interaction Forces and Frictional Torque for two Rotating Nanoparticles

A.I.Volokitin1,2∗

1Samara State Technical University, Physical Department, 443100 Samara, Russia

2Peter Grünberg Institut, Forschungszentrum Jülich, D-52425, Germany

Abstract

We calculate the quantum heat generation, the interaction force and the frictional torque for two rotating spherical nanoparticles with a radius $R$ . In contrast to the static case, when there is an upper limit in the radiative heat transfer between the particles, for two rotating nanoparticles the quantum heat generation rate diverges when the angular velocity becomes equal to the poles in the photon emission rate. These poles arise for the separation $d<d_{0}=R(3/\varepsilon^{\prime\prime}(\omega_{0}))^{1/3}$ (where $\varepsilon^{\prime\prime}(\omega_{0})$ is the imaginary part of the dielectric function for the particle material at the surface phonon or plasmon polariton frequency $\omega_{0}$ ) due to the anomalous Doppler effect and the mutual polarization of the particles and they exist even for the particles with losses. Similar singularities exist also for the interaction force and the frictional torque. The obtained results can be important for biomedical applications.

PACS: 42.50.Lc, 12.20.Ds, 78.67.-n

I Introduction

At present a great deal of attention is devoted to the study of rotating nanoparticles in the context of wide variety of physical, chemical and biomedical applications. The most important are related to using of rotating nanoparticles for targeting of cancer cells Nanomed2014 ; TrendsBioTech2011 ; ACSNano2014 . The frictional forces due to quantum fluctuations acting on a small sphere rotating near a surface were studied in Ref. PendryPRL2012 ; DedkovEPL2012 . Different experimental setups for trapping and rotating nanoparticles were discussed recently in Refs. nacommun2011 ; nanano2013 ; nanolett2014 .

Two arbitrary media in relative motion or at rest and separated by a vacuum gap continually exchange energy and momentum via a fluctuating electromagnetic field which is always present in the vacuum gap due to thermal and quantum fluctuations inside media VolokitinRMP2007 . This energy and momentum transfer is responsible for the radiative heat transfer and non-contact friction. At the nanoscale, these phenomena are enhanced by many orders of magnitude due to the contribution from evanescent electromagnetic waves. Further enhancement occurs if the media can support surface phonon- or plasmon- polariton modes. The possibility of using localized photon tunnelling between adsorbate vibrational modes for heating of the molecules was discussed in Ref.PerssonPRB2007 . All these phenomena raised a fundamental question. Are there limits which restrict the efficiently of energy and momentum transfer betweens bodies? For the static case in the far-field the radiative heat transfer is maximal for blackbodies when it is described by the Stefan-Boltzmann law. In the near-field the upper limit for the radiative heat transfer is determined by the transmission coefficient for photon tunnelling which can not exceed unity in the static case PendryJPCM1997 ; VolokitinJETPLett2003 ; JoulainPRB2010 . However for two sliding plates the photon emission rate can diverge at the resonant conditions due to the anomalous Doppler effect JacobJOpt2014 ; JacobOptExp2014 ; VolokitinPRB2016 .

In this article we calculate the frictional torque, the interaction force and the heat generation for two rotating nanoparticles using fluctuation electrodynamics. We determine the resonance conditions under which these quantities have singularities due to the mutual polarization of the particles and the anomalous Doppler effect.

II Theory

We consider two spherical particles 1 and 2 located along the $\hat{z}$ -axis at $\textbf{r}_{1}=(0,0,0)$ and $\textbf{r}_{2}=(0,0,d)$ (see Fig. 1). They are characterized by the different temperatures $T_{1}$ , $T_{2}$ and have the frequency-dependent polarizabilities $\alpha_{1,2}(\omega)$ . We introduce two reference frames $K$ and $K^{\prime}$ . In the $K$ frame the particle 1 is at rest while the particle 2 rotates around the axis passing through it with an angular velocity $\Omega$ . The $K^{\prime}$ frame is the rest reference frame for the particle 2. The orientation of the rotation axis for the particle 2 can be arbitrary but in the present study we consider the most symmetric cases when the rotation axis is along $\hat{z}$ or $\hat{x}^{\prime}$ axes as on Figures 1a and 1b, respectively. In the comparison with the general case for these limiting cases the calculations are much simpler and the obtained results are qualitatively the same.

II.1 Rotation axis along the $\hat{z}$ axis (see Fig. 1a)

According to fluctuation electrodynamicsVolokitinRMP2007 , the dipole moment for a polarizable particle $\mathbf{p}=\mathbf{p}^{f}+\mathbf{p}^{ind}$ where $\mathbf{p}_{i}^{f}$ is the fluctuating dipole moment due to quantum and thermal fluctuations inside the particle, $\mathbf{p}_{i}^{ind}$ is the induced dipole moment. In the $K$ frame the Fourier transformation is determined by

[TABLE]

In the $K$ frame the dipole moment for particle 1 satisfies equation

[TABLE]

where the electric field created by particle 2 at the position of particle 1 $\mathbf{E}_{12}$ is given by

[TABLE]

where the first and second terms in the right side of Eq. (2) determine the induced and fluctuating dipole moments of particle 1, respectively, $\alpha_{1}(\omega)$ is the polarizability for the particle 1. In the $K^{\prime}$ frame the components of the dipole moments $\mathbf{p}_{i}^{\prime}$ satisfy to the equation similar to the Eq. (2)

[TABLE]

where $\mathbf{E}^{\prime}_{21}$ is the electric field created by the particle 1 at the position of the particle 2. The relations between the dipole moment of the particle 2 in the $K$ and $K^{\prime}$ frames are determined by the equations: $p_{2z}^{\prime}(t)=p_{2z}(t)$ and

[TABLE]

where $\mathbf{p}^{\prime}_{2\perp}=(p_{x},p_{y})$ , and for the Fourier components: $p_{2z}^{\prime}(\omega)=p_{iz}(\omega)$ ,

[TABLE]

where $\omega^{\pm}=\omega\pm\Omega$ , $\hat{e}^{\prime\pm}=(\hat{x}^{\prime}\pm i\hat{y}^{\prime})/\sqrt{2}$ , $p_{2}^{\pm}=(p_{2x}\pm ip_{2y})/\sqrt{2}$ . The same relations are valid for the $\mathbf{E}^{\prime}_{21}$ : $E_{21z}^{\prime}(\omega)=E_{21z}(\omega)$ ,

[TABLE]

where $E_{21}^{\pm}=(E_{21x}\pm iE_{21y})/\sqrt{2}$ ,

[TABLE]

Using these relations in Eq. (4) and taking into account that $(\hat{e}^{\pm}\cdot\hat{e}^{\mp})=1$ , $(\hat{e}^{\pm}\cdot\hat{e}^{\pm})=0$ and $(\hat{e}_{z}\cdot\hat{e}^{\pm})=(\hat{e}^{\pm}\cdot\hat{e}_{z})=0$ we get

[TABLE]

where $p_{2}^{f\pm}(\omega^{\pm})=p_{2x}^{f\prime}(\omega^{\pm})\pm ip_{2y}^{f\prime}(\omega^{\pm})$ . From Eqs. (2) and (9)-(11) we get:

[TABLE]

where $D^{\pm}=1-\alpha_{1}(\omega)\alpha_{2}(\omega^{\pm})/d^{6}$ , $p_{1}^{f\pm}(\omega)=p_{1x}^{f}(\omega)\pm ip_{1y}^{f}(\omega)$ . The spectral density of the fluctuations of the dipole moment of the $i$ -th particle in the rest reference frame of the particle is determined by the fluctuation dissipation theorem

[TABLE]

where

[TABLE]

The torque acting on the particle 1 along the $\hat{z}$ -axis can be written in the form

[TABLE]

where $\mathbf{E}_{12}$ is the electric field created by the particle 2 at the position of the particle 1. Using Eqs. (12)-(18) we get

[TABLE]

The contribution to the torque from the quantum fluctuations (quantum friction) which exists even for $T_{1}=T_{2}=0$ K

[TABLE]

The heat generated in the particle 1 by a fluctuating electromagnetic field is determined by

[TABLE]

and the heat generated by the quantum fluctuations is given by

[TABLE]

The force acting on the particle 1 along the $\hat{z}$ -axis is given by

[TABLE]

The contribution to $F_{1z}$ from the frequency region corresponding to the anomalous Doppler effect in Eq.(25) is determined by the integration in the interval $0<\omega<\Omega$ and for $T_{1}=T_{2}=0$ K is given by

[TABLE]

II.2 The rotation axis is along the $\hat{x}^{\prime}$ axis (see Fig. 1b)

The details of the calculations for the case when the rotation axis is along the $\hat{x}^{\prime}$ axis are given in Appendix A. These calculations are more involved in comparison with the case when the rotation axis is along the $\hat{z}$ axis. Using Eqs. (61)-(62) we get the resulting formulas for $M_{x},\,P_{1},\,F_{1z}$ and $F_{1y}$ :

[TABLE]

where $D_{1}^{\pm}=1-4\alpha_{1}(\omega)\alpha_{2}(\omega^{\pm})/d^{6}$ , $D_{2}^{\pm}=1-\alpha_{1}(\omega)\alpha_{2}(\omega^{\pm})/d^{6}$ and $\Delta=D^{+}_{1}D^{-}_{2}+D^{-}_{1}D^{+}_{2}$ .

III Resonant heat transfer and heat generation due to quantum friction

For $\Omega=0$ the transmission coefficient for the photon tunnelling for two identical particles is restricted by the conditionVolokitinRMP2007 ; VolokitinJETPLett2003

[TABLE]

Thus $P\leq P_{max}$ where

[TABLE]

The radiative heat transfer between two particles is strongly enhanced in the case of the resonant photon tunnelling VolokitinRMP2007 ; VolokitinJETPLett2003 . For a spherical particle of radius $R$ the particle polarizability is given by

[TABLE]

where $\varepsilon_{i}$ is the dielectric function for a material of sphere. A particle has the resonance at $\varepsilon^{\prime}(\omega_{i})=-2$ where $\varepsilon^{\prime}$ is the real part of $\varepsilon$ . For a polar dielectric $\omega_{i}$ determines the frequency of the surface phonon-polariton. Close to the resonance for $\omega\approx\omega_{i}$ the particle polarizability can be written in the form

[TABLE]

where

[TABLE]

Close to the resonance for two identical particles ( $\omega_{1}=\omega_{2}=\omega_{0}$ , $a_{1}=a_{2}=a$ ) the transmission coefficient can be written in the form

[TABLE]

where $\omega_{\pm}=\omega_{0}\pm a(R/d)^{3}$ . For $a(R/d)^{3}>\Gamma$ the resonant heat transfer is given by

[TABLE]

where $n_{i}(\omega)=[\exp(\hbar\omega/k_{B}T_{i})-1]^{-1}$ . For $\hbar\omega_{0}<k_{B}T_{i}$ $P_{res}\approx 6\Gamma k_{B}(T_{2}-T_{1})$ and for $T_{2}\gg T_{1}$

[TABLE]

For $a(R/d)^{3}<\Gamma$

[TABLE]

Another resonance is possible in the condition of the anomalous Doppler effect when $\omega_{1}-\Omega=-\omega_{2}$ VolokitinRMP2007 ; VolokitinPRL2011 ; VolokitinEPL2013 ; JacobJOpt2014 ; JacobOptExp2014 . At this resonant condition, taking into account that

[TABLE]

the denominators in the integrands in Eqs. (22), (24) and (3) contain the factor

[TABLE]

At the resonance $|\alpha_{1}(\omega_{1})\alpha_{2}(\omega_{2})|/d^{6}$ can be larger than unity thus the denominator is equal to zero at

[TABLE]

what means that for $d<d_{0}$ the friction torque, heat generation and force interaction can diverge. The origin of this divergence is related to the creation below critical separation $d_{0}$ of the resonance at frequency determined by the pool of the photon emission rate for two rotating particle. This resonance can be lossless even in the case when the surface phonon- polariton modes for the isolated particles have losses. At such critical conditions the amplitude of electric field increases infinitely with time which gives rise to the divergence of the heat generation and interaction forces SilveirinhaNJP2014 .

Substituting Eq.(34) in Eq.(42) for the critical separation we get

[TABLE]

(for example, for silicon carbide (SiC) $d_{0}=2.57R$ (see below)) and the polarizabilities for the particles 1 and 2 for $\omega\approx\omega_{1}$ and $\omega-\Omega\approx\omega_{2}$ are given by Eq. (34) and by equation

[TABLE]

respectively. In this resonant case the photon emission rate for $0<\omega<\Omega$ is given by equation

[TABLE]

where $\Omega_{0}=\omega_{1}+\omega_{2}$ ,

[TABLE]

For two identical particles the transmission coefficient diverges at $\omega=\omega_{c}=\omega_{0}$ and $\Omega=\Omega^{\pm}$ where

[TABLE]

Close to the resonance when

[TABLE]

using Eq. (45) in Eq. (24) we get

[TABLE]

At $\Omega=\Omega_{0}$ the photon emission rate diverges at $\omega=\omega_{1}$ and $d=d_{0}$ . Close to this resonance quantum heat generation behaves as

[TABLE]

As an example, consider two nanoparticles of silicon carbide (SiC). The optical properties of this material can be described using an oscillator model Palik

[TABLE]

with $\varepsilon_{\infty}=6.7$ , $\omega_{L}=1.8\cdot 10^{14}$ s*-1*, $\omega_{T}=1.49\cdot 10^{14}$ s*-1*, and $\Gamma=8.9\cdot 10^{11}$ s*-1*. The frequency of surface phonon polaritons is determined by the condition $\varepsilon^{\prime}(\omega_{0})=-2$ and from (51) we get $\omega_{0}=1.73\cdot 10^{14}$ s*-1*. From Eq.(42) we get the critical distance $d_{0}=2.57R$ .

For a particle rotating around the $x^{\prime}$ axis the denominators in the integrands in Eqs. (27)-( $\ref{Fyx})$ contain the factor $\Delta=D^{+}_{1}D^{-}_{2}+D^{-}_{1}D^{+}_{2}$ (see Sec.II.2). Under the resonance conditions when $\omega\approx\omega_{0}$ and $\omega-\Omega\approx-\omega_{0}$ we can put $D^{+}_{1}\approx D^{+}_{2}\approx 1$ . Thus, a resonance occurs when

[TABLE]

From this equation we get that for the SiC particles the divergence in the photon emission rate occurs for $d<d_{0}=3R$ . For an arbitrary orientation of the rotation axis, the critical separation for SiC particles is in the range: $2.57R<d_{0}<3R$ .

Fig.2 shows the dependence of a) the quantum heat generation rate for a particle 1 and b) the interaction force between the particles on the angular velocity of the particle 2 for $d\geq 2.6R>d_{0}$ . In accordance with the above theoretical analysis these dependences have sharp resonance for $d\rightarrow d_{0}$ . For static particles at $T_{2}=300$ K and $T_{1}=0$ K from Eq.(37) follows that the resonant photon tunneling contribution to the radiative heat transfer $P_{res}\approx 10^{-9}$ W. In sharp contrast to the static case, for rotating particles the heat generation rate diverges at the resonance at $d=d_{0}$ and $\Omega=\Omega_{0}=2\omega_{0}$ . At the resonance the stationary rotation of a particle is impossible, since in this case the friction force increases unrestrictedly with time. However, near the resonance the stationary rotation with an arbitrarily high heat generation rate due to conversion of the mechanical energy into heat is possible. Near the resonance frequency, the interaction force changes sign (see Fig.2b). In the static case, the van der Waals force between two particles is given by formula

[TABLE]

where according to Ref.SvetovoyPRB2014 the Hamaker constant for the SiC-SiC system $A_{H}=16.5\cdot 10^{-20}$ J. For $d=2.6R=1.3$ nm $F_{vdW}=5.7\cdot 10^{-12}$ N. For rotating particles near resonance the interaction force can be arbitrarily large. Thus tuning of the interaction force is possible by changing the angular velocity of a particle.

Fig.3 shows the dependences of a) the heat generation rate and b) the interaction forces between the particles on the separation between the particles for $d\geq 2.6R>d_{0}$ for $\Omega=\Omega_{0}$ (red curve) and $\Omega=\Omega_{0}(1+0.003)$ (green curve). In accordance with the above theoretical analysis these dependences have divergences at the critical angular velocity $\Omega_{0}$ .

The condition for the validity of the dipole approximation for two particles is determined by $2R/d\ll 1$ . For SiC particles, the multipole expansion parameter for $d\approx 2.6R\approx d_{0}$ is equal to 0.8 and 0.7 for the rotation axis directed along and perpendicular to the z axis, respectively. Therefore the numerical calculations given above play the role of a qualitative estimation of the effect. Its quantitative description for SiC particles requires consideration of multipole effects.

IV Summary

Fluctuation electrodynamics was used to calculate the heat generation, the interaction force and the frictional torque for two rotating nanoparticles, taking into account the mutual polarization of the particles. In a sharp contrast to the static case, all these quantities diverge at the resonant conditions even for the case when there are losses in the particles. The origin of these features is related to the divergence of the photon emission rate under the conditions of the anomalous Doppler effect. The obtained results can found broad application in nanotechnology. In particular, they can be used for tuning of the interaction forces and the heat generation by changing the angular velocity. These processes can be used for targeting cancer cells. For practical application of the predicted effects, it is necessary to search for or create materials with a low frequency of the plasmon or phonon polaritons and a small imaginary part of the dielectric function at this frequency. InSb semiconductor has a frequency of the surface plasmon-phonon polaritons in the THz regionPalik . However, the dielectric function for this material has a large imaginary part at this frequency, which leads to a small value for the critical distance. On the other hand metamaterials can have a frequency of the plasmon polaritons in the GHz regionPendryJPCM1998 .

V Acknowledgement

The study was supported by the Russian Foundation for Basic Research (Grant No. 16-02-00059-a).

Appendix A The rotation axis is along the $\hat{x}^{\prime}$ axis (see Fig. 1b)

In the case of the rotation axis directed along the $\hat{x}^{\prime}$ axis instead of Eqs. (9)-(11) we get

[TABLE]

Using Eq. (2) in Eqs. (55) and (56) we get the set of equations:

[TABLE]

where $D_{1}^{\pm}=1-4\alpha_{1}(\omega)\alpha_{2}(\omega^{\pm})/d^{6}$ , $D_{2}^{\pm}=1-\alpha_{1}(\omega)\alpha_{2}(\omega^{\pm})/d^{6}$ , $P_{2}^{f\pm}=\alpha_{2}(\omega^{\pm})[2p_{1z}^{f}(\omega)\mp ip_{1y}^{f}(\omega)]/d^{3}+p_{2}^{f\prime\pm}(\omega^{\pm})$ . From Eqs. (2), (54), (57) and (58) we get

[TABLE]

where $P_{1z}^{f\pm}=p_{1z}^{f}(\omega)\mp 2ip_{1y}^{f}(\omega)+2\alpha_{1}(\omega)p_{2}^{f\prime\pm}(\omega^{\pm})/d^{3}$ , $P_{1y}^{f\pm}=p^{f}_{1y}(\omega)\pm ip^{f}_{1z}(\omega)/2\pm i\alpha_{1}(\omega)p_{2}^{f\prime\pm}(\omega^{\pm})/d^{3}$ , $\Delta=D^{+}_{1}D^{-}_{2}+D^{-}_{1}D^{+}_{2}$ .

∗[email protected]

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Anomalous Doppler Effect Singularities in Radiative

Abstract

I Introduction

II Theory

II.1 Rotation axis along the z^\hat{z}z^ axis (see Fig. 1a)

II.2 The rotation axis is along the x^′\hat{x}^{\prime}x^′ axis (see Fig. 1b)

III Resonant heat transfer and heat generation due to quantum friction

IV Summary

V Acknowledgement

Appendix A The rotation axis is along the x^′\hat{x}^{\prime}x^′ axis (see Fig. 1b)

II.1 Rotation axis along the $\hat{z}$ axis (see Fig. 1a)

II.2 The rotation axis is along the $\hat{x}^{\prime}$ axis (see Fig. 1b)

Appendix A The rotation axis is along the $\hat{x}^{\prime}$ axis (see Fig. 1b)