Introduction

It is well established that light carries angular momentum1. The simplest evidence of this phenomenon is the possibility of spinning nanostructures by illuminating them with circularly polarized light2,3. The angular momentum of light is also manifested in Casimir interactions4 that originate from the vacuum and thermal fluctuations of the electromagnetic field5,6. For instance, two parallel plates made of birefringent materials with in-plane optical anisotropy have been shown to experience a Casimir torque that rotates them to a configuration in which the two optical axes are aligned7,8,9,10,11. A similar phenomenon is predicted to occur for two nanostructured plates12, or for a nanorod placed above a birefringent plate13.

Vacuum and thermal fluctuations also produce friction on rotating nanostructures; for instance, it has been predicted that a nanoparticle rotating in vacuum experiences a Casimir torque that slows its angular velocity and eventually stops it14,15. A similar effect is also expected for a rotating pair of atoms16. The Casimir torque can be enhanced by placing the particle near a substrate17,18, which also leads to a lateral Casimir force19,20, or by using materials with magneto-optical response21. The origin of this torque can be found in the imbalance of the absorption and emission of left-handed and right-handed photons caused by the rotation of the particle. Although Casimir interactions produce friction and stiction that affect the dynamics of nanomechanical devices22, they also offer new opportunities for the noncontact transfer of momentum; for example, if a rotating nanostructure is placed near another structure, the Casimir torque that slows the first one down will necessarily accelerate the other one23,24,25,26,27,28,29,30,31.

In this paper, we investigate the noncontact transfer of angular momentum at the nanoscale by analyzing the Casimir torque acting on a chain of rotating nanoparticles, which contains an arbitrary number of elements. This system can be thought of as a Casimir analog to a chain of particles rotating in a viscous medium, whose motion is coupled through a drag force. We obtain analytical expressions describing the Casimir torque and exploit them to study the free rotational dynamics of chains with different geometries. We show that the synchronization of the angular velocities of the particles can happen in timescales as short as seconds for realistic structures. Furthermore, we predict exotic behaviors in chains with inhomogeneous particle sizes and separations, including “rattleback”-like dynamics32,33, in which the sense of rotation of a particle changes several times before synchronization, as well as configurations for which angular momentum is never transferred to a selected particle in the chain. We analyze, as well, the steady-state distribution of angular velocities in chains in which one or multiple particles are externally driven at a constant angular velocity. The results of this work show that the Casimir torque provides a mechanism for the efficient transfer of angular momentum between nanoscale objects without requiring them to be in contact.

Results

Theoretical model

The system under study is depicted in Fig. 1. It consists of a chain of N spherical nanoparticles with diameters Di separated by center-to-center distances dij. Each nanoparticle can rotate with an angular velocity Ωi around the axis of the chain, which we choose as the z-axis. We assume that the size of the particles is much smaller than the relevant wavelengths of the problem, which are determined by the temperature, the material properties, and the angular velocities of the particles, and consider geometries obeying \(d_{ij} \ge \frac{3}{2}{\mathrm{max}}(D_i,D_j)\). This allows us to model each particle as a point electric dipole with a frequency-dependent polarizability αi(ω). Within this approximation, the Casimir torque acting on particle i can be written as \(M_i = \langle {\mathbf{p}}_i \times {\mathbf{E}}_i\rangle \cdot \widehat {\mathbf{z}}\)34, where the brackets stand for the average over vacuum and thermal fluctuations, while pi and Ei are, respectively, the self-consistent dipole and field at particle i, which originate from: (i) the fluctuations of the dipole moment of each particle \({\mathbf{p}}_j^{{\mathrm{fl}}}\) and (ii) the fluctuations of the field \({\mathbf{E}}_j^{{\mathrm{fl}}}\). Then, solving pi and Ei in terms of \({\mathbf{p}}_j^{{\mathrm{fl}}}\) and \({\mathbf{E}}_j^{{\mathrm{fl}}}\) and taking the average over fluctuations using the fluctuation–dissipation theorem35,36,37, one can write the Casimir torque as \(M_i = M_i^ + - M_i^ -\), where \(M_i^ \pm\) is given by (see the “Methods” section for the detailed derivation)

$$M_i^ \pm = - \frac{{2\hbar }}{\pi }{\int_0}^\infty {\mathrm{{d}}} \omega \mathop {\sum}\limits_{j = 1}^N {\left[ {{\mathrm{\Gamma }}_j^{0 \pm }\delta _{ij} + {\mathrm{\Gamma }}_{ij}^ \pm } \right]} \left[ {n_j(\omega _j^ \mp ) - n_0(\omega )} \right].$$
Fig. 1
figure 1

System schematics. The system under study consists of a chain of N nanoparticles with diameters Di separated by a center-to-center distance dij. The nanoparticles are assumed to rotate around the axis of the chain

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Here, \(\omega _i^ \pm = \omega \pm {\mathrm{\Omega }}_i\) and ni(ω) = [exp(ℏω/kBTi)−1]−1 is the Bose–Einstein distribution at temperature Ti, with T0 being the temperature of the surrounding environment. Furthermore, \({\mathrm{\Gamma }}_i^{0 \pm } = \frac{{2\omega ^3}}{{3c^3}}{\mathrm{Re}}\left\{ {2A_{ii}^ \pm - 1} \right\}\chi _i^ \pm (\omega )\) corresponds to the contribution to the Casimir torque produced by the environment, while

$${\mathrm{\Gamma }}_{ij}^ \pm = \delta _{ij}{\mathrm{Im}}\left\{ {S_{ii}^ \pm } \right\}\chi _i^ \pm (\omega ) - \left| {S_{ij}^ \pm } \right|^2\chi _i^ \pm (\omega )\chi _j^ \pm (\omega )$$

is the contribution arising from the particle–particle interaction. Physically, the first of these contributions corresponds to the exchange of angular momentum between each particle and the environment, while the second one is associated with the exchange of angular momentum between the particles. In these expressions, \(A_{ij}^ \pm\) are the components of the N × N matrix \({\mathbf{A}}^ \pm = \left[ {{\cal{I}} - \alpha ^ \pm {\mathbf{G}}} \right]^{ - 1}\), where α± is a diagonal matrix whose components are the effective polarizabilities of the particles \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega )\) as seen from the frame at rest, G is the dipole–dipole interaction matrix with components \(G_{ij} = (1 - \delta _{ij}){\mathrm{exp}}(ikd_{ij})[(kd_{ij})^2 + ikd_{ij} - 1]/d_{ij}^3\), and k = ω/c is the wave number. Furthermore, \(\chi _i^ \pm (\omega ) = {\mathrm{Im}}\left\{ {\alpha _{{\mathrm{eff}},i}^ \pm (\omega )} \right\} - \frac{2}{3}k^3\left| {\alpha _{{\mathrm{eff}},i}^ \pm (\omega )} \right|^2\) and \(S_{ij}^ \pm = {\sum}_{m = 1}^N {A_{mi}^ \pm } G_{mj}\) (see the “Methods” section). We want to remark that our results include radiative corrections, which are crucial to correctly describe the contributions to the torque produced by the environment38. It is also important to notice that the Casimir torque described here is a dissipative effect, and therefore it depends strongly on the temperature of the particles and the environment.

The calculation of the effective polarizability \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega )\) is subtle. In the past, this quantity has been computed assuming that the intrinsic response of the system was independent of the rotation, which resulted in an effective polarizability \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega ) = \alpha _i(\omega _i^ \mp )\) that only accounted for the Doppler-shift produced by the rotation14,15,17,19,28,39,40,41. However, recently21,42, it has been pointed out that the inclusion of the Coriolis and centrifugal effects gives rise to corrections that, for the case of spherical particles, cancel the effect of the Doppler-shift and introduce a dependence on Ω2. As a consequence of this, the effective polarizability of a rotating sphere becomes \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega ) = \alpha _i(\omega ) + {\cal{O}}({\mathrm{\Omega }}^2)\), as shown in the Supplementary Note 1. Notice that the component of the polarizability along the rotation axis, which does not contribute to the Casimir torque, is not affected by the rotation.

Rotational dynamics

Under realistic conditions, Ωi is smaller than both the thermal frequency kBT/ℏ (≈6 THz at room temperature) and the frequencies of the optical modes of the nanoparticles (≈28 THz for SiC). This allows us to expand \(M_i^ \pm\) in powers of Ωi and retain the lowest nonvanishing order, which is the linear one. This linear approximation is accurate for angular velocities up to Ωi/2π~100 GHz, as shown in Supplementary Fig. 1. Importantly, within this limit, the corrections to the effective polarizability discussed above become irrelevant. Assuming all particles and the environment have the same finite temperature, we can write the following equation for their angular velocities:

$${\dot{\mathrm{\Omega }}}_i = \mathop {\sum}\limits_{j = 1}^N {H_{ij}} {\mathrm{\Omega }}_j,$$

where the matrix H has components \(H_{ij} = h_i^0\delta _{ij} + h_{ij}\) with

$$\left( {\begin{array}{*{20}{l}} {h_i^0} \hfill \\ {h_{ij}} \hfill \end{array}} \right) = \frac{{4\hbar }}{{\pi I_i}}{\int_0}^\infty {{\mathrm{{d}}}\omega } \left( {\begin{array}{*{20}{l}} {{\mathrm{\Gamma }}_i^0} \hfill \\ {{\mathrm{\Gamma }}_{ij}} \hfill \end{array}} \right)\frac{{\partial n(\omega )}}{{\partial \omega }}.$$

Here, \(I_i = \pi \rho D_i^5/60\) is the moment of inertia of the nanoparticles, and ρ their mass density. The expressions for \({\mathrm{\Gamma }}_i^0\) and Γij are obtained from the corresponding definitions above by setting all angular velocities to zero.

We can hence study the rotation dynamics of a chain by analyzing its natural decay rates and modes given, respectively, by the eigenvalues and eigenvectors of H. As an initial example, we analyze a chain with N = 5 SiC spheres, all of them with identical diameter D = 10 nm, that are uniformly distributed with a center-to-center distance d = 1.5D. Throughout this work, unless stated otherwise, we assume that all of the particles remain at the same temperature as the environment, and set that temperature to 300 K. This assumption is a good approximation for the systems under consideration, as discussed in the Supplementary Note 2 and Supplementary Fig. 2. The polarizability of the particles is obtained from the dipolar Mie coefficient43 with the dielectric function of SiC modeled as \(\varepsilon (\omega ) = \varepsilon _\infty \left[ {1 + (\omega _{\mathrm{L}}^2 - \omega _{\mathrm{T}}^2)/(\omega _{\mathrm{T}}^2 - \omega ^2 - i\omega \gamma )} \right]\), with ε∞ = 6.7, ħωT = 98.3 meV, ħωL = 120 meV, and ħγ = 0.59 meV 44. With this choice of size and material, the optical response of the nanoparticles is dominated by a phonon polariton at ≈28 THz. Figure 2a shows the different components of H for the chain under analysis. As a consequence of the prevalent role of the near-field coupling, the matrix is almost tridiagonal. Furthermore, the diagonal elements are all negative, with the contribution of the environment, \(h_i^0\), being five orders of magnitude smaller than that of the particle–particle interaction hij, as shown in Fig. 2b. Interestingly, H is a negative definite matrix (i.e., all of its eigenvalues are strictly negative), which ensures that, in absence of external driving, the angular velocities decay to zero at large times.

Fig. 2
figure 2

Rotational dynamics. a Hij for a chain of N = 5 identical particles with D = 10 nm made of SiC, which are uniformly distributed with a center-to-center distance d = 1.5D. b Diagonal components of hij (blue) and the corresponding \(h_i^0\) (red). Notice the different scale. c Natural modes of the chain obtained by diagonalizing Hij and the corresponding decay rates. d–f Temporal evolution of the angular velocity of each particle in the chain for three different initial conditions

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The natural modes of the chain, together with the corresponding decay rates, are displayed in Fig. 2c. The first natural mode corresponds to all particles rotating with the same angular velocity, which makes the contribution arising from the particle–particle interaction vanish, leaving only the Casimir torque produced by the environment. This results in a decay rate λ1 = −0.95 × 10−6 s−1 (in perfect agreement with the stopping time calculated in ref. 14 for single particles), much smaller than those of the remaining modes, which are all on the order of s−1. In these other modes, the particles rotate with angular velocities of different magnitude and sign, but, as expected from the symmetry of the system, the modes are either even or odd with respect to the central particle, which, consequently, is at rest in the odd modes.

Figure 2d–f show the temporal evolution of the angular velocities of the particles for three different initial conditions. Specifically, in Fig. 2d, only particle 1 (black) is initially rotating with Ω1/2π = 10 GHz, an angular velocity that is within experimental reach, as recently demonstrated45,46. As time evolves, the rotation is transferred through the chain to the other particles, resulting in a synchronized rotation dynamics after a few seconds, in which the initial angular momentum is uniformly distributed among all particles. The situation is different when particle 5 (yellow) is also set to rotate initially at Ω5 = −Ω1. In this case, since the initial angular momentum of the system is zero, the rotation of the particles ends after a few seconds, as shown in Fig. 2e. It is important to notice that any initial state with finite angular momentum must have a nonzero overlap with the first natural mode. In these cases, after synchronization, the angular velocity of the particles gradually decreases and eventually stops on a much larger timescale ≈|λ1|−1, as a consequence of the Casimir torque produced by the environment (i.e., \(h_i^0\)). This can be seen in Fig. 2f, where we plot the temporal evolution of the chain when initialized in the first natural mode.

The rotation dynamics of chains with arbitrarily large N can be understood in a similar way by analyzing the corresponding decay rates and natural modes. In Fig. 3, we plot the decay rates of chains with different N as a function of an effective momentum keff, defined as keffd/π = (n−1)/(N−1), where n is the index labeling the natural modes. The effective momentum is directly proportional to the number of nodes of the natural mode, i.e., the number of times that its components change sign. Examining Fig. 3, we observe that, as N increases, the decay rates converge to a curve that resembles the dispersion relation of the transversal mode of the infinite chain47. As in the N = 5 case analyzed before, the first mode always corresponds to all particles rotating with the same angular velocity (see Supplementary Fig. 3), which means that only the environment contributes to the Casimir torque, thus resulting in a decay rate λ1 with values ≈10−6 s−1 for any N, as shown in the inset. As keff increases, the corresponding modes show a more complicated pattern in which neighboring particles rotate with increasingly different angular velocities (see Supplementary Fig. 3). This makes the contribution arising from the particle–particle interaction increase, leading to much faster decay rates.

Fig. 3
figure 3

Natural decay rates. Evolution of the natural decay rates for chains with different N (as indicated in the legend) plotted as a function of the effective momentum keff. The inset shows the evolution of the slowest decay rate λ1 as a function of N. In all of the cases, D = 10 nm and d = 1.5D

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Exotic dynamics

We can use the insight obtained from the analysis of the natural modes and decay rates to study the behavior of chains with more exotic rotation dynamics. In particular, by breaking the uniformity in the particle separation, it is possible to obtain a “rattleback”-like rotation dynamics32,33, in which a particle reverses its sense of rotation multiple times during the synchronization process. This is shown in Fig. 4a for a chain of N = 3 SiC particles with D = 10 nm, in which the central particle is, respectively, at distance 2d and d (with d = 1.5D) from the left and right particles, as depicted in the inset. Analyzing the dynamics of the system, we observe that particle 2 (red) changes its sense of rotation two times before all particles synchronize. This happens because this particle synchronizes first with particle 3, due to the smaller distance that separates them, which results in a larger coupling, and, therefore, Casimir torque, among them. After that, both particles 2 and 3 have to synchronize with particle 1 and, since the total initial angular momentum is positive, the synchronized angular velocities have to be positive. Interestingly, it is possible to obtain more reversals in the sense of rotation using chains with larger N, as shown in Supplementary Fig. 4 for a chain with five particles. In all cases, after synchronization, the rotation velocities of the particles decay as a consequence of the Casimir torque produced by the environment.

Fig. 4
figure 4

Exotic rotational dynamics. a Temporal evolution of the angular velocity of the particles of a chain with N = 3, arranged as shown in the inset, where D = 10 nm and d = 1.5D. b Crossing of the second and third decay rates (i.e., λ2 and λ3) of the N = 3 chain shown in the inset of panel c when varying the ratio D′/D. In this case, D = 10 nm and d = 3D. c Temporal evolution of the angular momentum, L(t) = IΩ(t), for each of the three particles of the chain depicted in the inset, assuming D′/D = 1.914. We analyze two different cases with initial angular velocities given by Ω1/2π = 10 GHz + Ωc/2π, Ω2/2π = −0.39 GHz + Ωc/2π, and Ω3 = Ωc. In one case, Ωc = 0 while, in the other, Ωc/2π = 5 GHz. However, since there is no transfer of angular momentum to particle 3, both cases produce exactly the same results

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Another interesting situation arises when the particles in the chain have different sizes. In particular, it is possible to make the decay rates of two different modes equal, as shown in Fig. 4b for the N = 3 chain depicted in the inset of Fig. 4c, with D = 10 nm and d = 3D. Clearly, as the diameter of the central particle D′ increases, the decay rates of the second (green curve) and third (yellow curve) natural modes cross each other. At the crossing point D′ = 1.914D, the degeneracy of the natural modes allows us to prepare the system in an initial state such that angular momentum is only transferred between the central and one of the side particles, without altering the dynamics of the remaining one. We analyze two different examples of this behavior in Fig. 4c, where we plot the temporal evolution of the angular momentum, L(t) = IΩ(t), for each of the three particles. The corresponding initial angular velocities are: Ω1/2π = 10 GHz + Ωc/2π, Ω2/2π = −0.39 GHz + Ωc/2π, and Ω3 = Ωc (see the Supplementary Note 3). In the first example, we choose Ωc = 0, whereas in the second one, Ωc/2π = 5 GHz. In both cases, as expected, the angular momentum of particles 1 (black curve) and 2 (red curve) changes identically, but with opposite signs, while the angular momentum of particle 3 remains completely unchanged (blue curve).

Driven dynamics

So far, we have analyzed systems in free rotation, in which the initial angular momentum is transferred among the particles in the chain and is eventually dissipated into the surrounding environment. The situation is different when one or more particles in the chain are externally driven so their angular velocities remain constant. These particles act as a continuous source of angular momentum, which is transferred to the rest of the particles of the chain, and, as a consequence, after some transient evolution, the whole system reaches a steady-state rotation dynamics. As an initial example, we consider the chain of N = 30 identical SiC nanoparticles shown in the inset of Fig. 5a, in which particle 1 is externally driven at a constant angular velocity Ω1/2π = 10 GHz. We consider two different combinations of D and d for which we calculate the corresponding steady-state angular velocities, which are displayed in Fig. 5a. These velocities are determined by the interplay between the contributions to the Casimir torque arising from the environment and particle–particle interactions, with the former being the mechanism that dissipates angular momentum out of the chain, and the latter being the one that mediates its transfer between the particles. For D = 10 nm and d = 1.5D (black curve), the environment contribution is much smaller than that of the particle–particle interaction and, consequently, the angular velocities are almost identical. However, for D = 50 nm and d = 3D (red curve), the difference between the two contributions is reduced (see Supplementary Fig. 5), and, therefore, the angular velocities significantly decrease as we move away from the driven particle. This demonstrates that an efficient transfer of angular momentum is possible in systems for which \(|h_{ii}| \gg |h_i^0|\).

Fig. 5
figure 5

Driven rotational dynamics. a Steady-state angular velocity for the particles of a N = 30 chain, in which particle 1 is externally driven at an angular velocity 10 GHz. All of the particles have the same diameter D and are uniformly distributed with a center-to-center distance d, as shown in the inset. b Same as a, but, in this case, two particles, which are indicated with small arrows (see legend), are externally driven at angular velocities 10 and −10 GHz, respectively

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Another interesting situation arises when two particles in the chain are driven at opposite angular velocities. In this case, as shown in Fig. 5b, the particles located between them have steady-state angular velocities with values uniformly distributed between those of the driven particles. On the other hand, the particles outside display almost constant velocities, which are determined by the separation between the driven particles.

Discussion

In summary, we have shown that the Casimir torque enables an efficient transfer of angular momentum at the nanoscale. To that end, we have analyzed the dynamics of chains of rotating nanoparticles with an arbitrary number of elements. We have found that this noncontact interaction leads to rotational dynamics happening on timescales as fast as seconds for nanoscale particles, which corresponds to torques on the order of 10−27 N m for angular velocities of 10 GHz, both of which are within experimental reach13,45,46,48,49. We have derived an analytical formalism describing the rotational dynamics of these systems, which is based on the analysis of the natural modes of the chain, and exploited it to reveal unexpected behaviors. These include a “rattleback”-like dynamics, in which the sense of rotation of a particle changes multiple times before synchronization, as well as configurations for which a selected particle is left out of the angular momentum transfer process. Furthermore, by analyzing the steady-state angular velocities of systems in which one or more particles are externally driven, we have established the conditions under which an efficient transfer of angular momentum can be achieved. A possible experimental verification of our results would involve the use of nanoparticles trapped with optical tweezers45,46,48, or molecules with rotational degrees of freedom, such as as chains of fullerenes50, whose angular velocity can be detected through measurements of rotational frequency shifts51,52. The presented results describe a mechanism for the noncontact transfer of angular momentum at the nanoscale, which brings new opportunities for the control of nanomechanical devices.

Methods

The system under consideration is depicted in Fig. 1. It consists of a linear chain with N spherical nanoparticles. Each of the particles has a diameter Di and is separated from its neighbors by a center-to-center distance dij. All of the particles are allowed to rotate with arbitrary angular velocity Ωi around the axis of the chain, which we choose to be the z-axis. As discussed in the paper, we assume that the size of the particles is much smaller than the relevant wavelengths of the problem, which are determined by the temperature, the angular velocities, and the material properties of the particles, and use geometries obeying \(d_{ij} \ge \frac{3}{2}{\mathrm{max}}(D_i,D_j)\). Within these limits, we can model each particle as a point electric dipole with a frequency-dependent polarizability αi(ω). This allows us to write the torque acting on particle i as \(M_i = \langle{\mathbf{p}}_i(t) \times {\mathbf{E}}_i(t)\rangle \cdot \widehat {\mathbf{z}}\), where 〈〉 represents the average over fluctuations. Working in the frequency domain ω, defined via the Fourier transform \({\mathbf{p}}_i(t) = \mathop {\int}\limits_{ - \infty }^\infty {\frac{{{\mathrm{{d}}}\omega }}{{2\pi }}} {\mathbf{p}}_i(\omega ){\mathrm{{e}}}^{ - i\omega t}\), for the dipole moment, and similarly for other quantities, and using the circular basis defined as: \(\widehat {\mathbf{e}}_ \pm = (1/\sqrt 2 )(\widehat {\mathbf{x}} \pm i\widehat {\mathbf{y}})\), and \(\widehat {\mathbf{e}}_0 = \widehat {\mathbf{z}}\), we can rewrite the torque as \(M_i = M_i^ + - M_i^ -\) with

$$M_i^ \pm = i {\int_{ - \infty }^{\infty}} {\frac{{{\mathrm{{d}}}\omega {\mathrm{{d}}}\omega \prime }}{{4\pi ^2}}} {\mathrm{{e}}}^{ - i(\omega - \omega \prime )t}\left\langle {p_i^{ \pm \ast }(\omega \prime )E_i^ \pm (\omega )} \right\rangle .$$
(1)

I-n this expression, * denotes the complex conjugate, while \(p_i^ \pm (\omega )\) and \(E_i^ \pm (\omega )\) are the self-consistent dipole moment and electric field in particle i, which originate from the vacuum and thermal fluctuations of: (i) the dipole moments of the particles in the chain \(p_j^{{\mathrm{fl}}, \pm }(\omega )\) and (ii) the electric field \(E_j^{{\mathrm{fl}}, \pm }(\omega )\). We can write \(p_i^ \pm (\omega )\) and \(E_i^ \pm (\omega )\) in terms of \(p_j^{{\mathrm{fl}}, \pm }(\omega )\) and \(E_j^{{\mathrm{fl}}, \pm }(\omega )\) as

$$\begin{array}{*{20}{l}} {p_i^ \pm (\omega )} \hfill & = \hfill & {p_i^{{\mathrm{fl}}, \pm }(\omega ) + \alpha _{{\mathrm{eff}},i}^ \pm (\omega )E_i^{{\mathrm{fl}}, \pm }(\omega ) + \alpha _{{\mathrm{eff}},i}^ \pm (\omega )\mathop {\sum}\limits_{j \ne i}^N {G_{ij}} (\omega )p_j^ \pm (\omega ),} \hfill \\ {E_i^ \pm (\omega )} \hfill & = \hfill & {E_i^{{\mathrm{fl}}, \pm }(\omega ) + \mathop {\sum}\limits_{j \ne i}^N {G_{ij}} (\omega )p_j^ \pm (\omega ) + G_0(\omega )p_i^ \pm (\omega ),} \hfill \end{array}$$

where \(G_{ij}(\omega ) = (1 - \delta _{ij}){\mathrm{exp}}(ikd_{ij})[(kd_{ij})^2 + ikd_{ij} - 1]/d_{ij}^3\) is the dipole–dipole interaction between particles i and j, k = ω/c, and \(G_0(\omega ) = \frac{{2i}}{3}k^3\). Notice that the inclusion of G0(ω) is necessary to account for radiative corrections38. Furthermore, \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega )\) is the effective polarizability of particle i as seen from the frame at rest. Following the notation of ref. 53, these equations can be solved simultaneously as

$$\begin{array}{*{20}{l}} {p_i^ \pm (\omega )} \hfill & = \hfill & {\mathop {\sum}\limits_{j = 1}^N {A_{ij}^ \pm } (\omega )p_j^{{\mathrm{fl}}, \pm }(\omega ) + \mathop {\sum}\limits_{j = 1}^N {B_{ij}^ \pm } E_j^{{\mathrm{fl}}, \pm }(\omega ),} \hfill \\ {E_i^ \pm (\omega )} \hfill & = \hfill & {\mathop {\sum}\limits_{j = 1}^N {C_{ij}^ \pm } (\omega )p_j^{{\mathrm{fl}}, \pm }(\omega ) + \mathop {\sum}\limits_{j = 1}^N {D_{ij}^ \pm } (\omega )E_j^{{\mathrm{fl}}, \pm }(\omega ),} \hfill \end{array}$$

where \(A_{ij}^ \pm (\omega )\) are the matrix elements of the N × N matrix defined as \({\mathbf{A}}^ \pm (\omega ) = \left[ {{\cal{I}} - \alpha ^ \pm (\omega ){\mathbf{G}}(\omega )} \right]^{ - 1}\), with α± being a diagonal matrix whose components are \(\alpha _{{\mathrm{eff}},i}^ \pm (\omega )\), and G the dipole–dipole interaction matrix with components Gij(ω). Similarly, the other matrices are defined as \(B_{ij}^ \pm (\omega ) = A_{ij}^ \pm (\omega )\alpha _{{\mathrm{eff}},j}^ \pm (\omega )\), \(C_{ij}^ \pm (\omega ) = \mathop {\sum}\limits_{k = 1}^N {\left[ {\delta _{ik}G_0(\omega ) + G_{ik}(\omega )} \right]} A_{kj}^ \pm (\omega )\), and \(D_{ij}^ \pm (\omega ) = \delta _{ij} + C_{ij}^ \pm (\omega )\alpha _{{\mathrm{eff}},j}^ \pm (\omega )\). Using these expressions, we can write Eq. (1) in terms of averages over the dipole and field fluctuations

$$M_i^ \pm = \hskip 2pt i{\int_{ - \infty }^\infty} \frac{{{\mathrm{{d}}}\omega {\mathrm{{d}}}\omega {\prime}}}{{4\pi ^2}}{\mathrm{{e}}}^{ - i(\omega - \omega {\prime})t}\mathop {\sum}\limits_{j,k = 1}^N \left[ A_{ij}^{ \pm \ast }(\omega {\prime})C_{ik}^ \pm (\omega )\langle p_j^{{\mathrm{fl}}, \pm \ast }(\omega {\prime})p_k^{{\mathrm{fl}}, \pm }(\omega )\rangle \right. \\ \left. + \, B_{ij}^{ \pm \ast }(\omega {\prime})D_{ik}^ \pm (\omega )\langle E_j^{{\mathrm{fl}}, \pm \ast }(\omega {\prime})E_k^{{\mathrm{fl}}, \pm }(\omega )\rangle \right] .$$
(2)

Notice that there are no cross terms involving dipole and field fluctuations since these are uncorrelated14. In order to evaluate the averages over fluctuations, we use the fluctuation–dissipation theorem14,35,36, which, for dipole fluctuations, taking into account the rotation of the particles, reads

$$\left\langle {p_j^{{\mathrm{fl}}, \pm \ast }(\omega {\prime})p_k^{{\mathrm{fl}}, \pm }(\omega )} \right\rangle = 4\pi \hbar \delta (\omega - \omega {\prime})\chi _j^ \pm (\omega )\delta _{jk}\left[ {n_j(\omega _j^ \mp ) + \frac{1}{2}} \right],$$
(3)

where \(\omega _i^ \pm = \omega \pm {\mathrm{\Omega }}_i\) and we use \(\chi _i^ \pm (\omega ) = {\mathrm{Im}}\left\{ {\alpha _{{\mathrm{eff}},i}^ \pm (\omega )} \right\} - \frac{{2\omega ^3}}{{3c^3}}\left| {\alpha _{{\mathrm{eff}},i}^ \pm (\omega )} \right|^2\) instead of \({\mathrm{Im}}\{ {\alpha _{{\mathrm{eff}},i}^ \pm (\omega )} \}\) to account for the radiative corrections in the response of the nanoparticle54,55. Similarly, for the field fluctuations

$$\langle E_j^{{\mathrm{fl}}, \pm \ast }(\omega {\prime})E_k^{{\mathrm{fl}}, \pm }(\omega )\rangle = 4\pi \hbar \delta (\omega - \omega {\prime}){\mathrm{Im}}\left\{ {\delta _{jk}G_0(\omega ) + G_{jk}(\omega )} \right\}\left[ {n_0(\omega ) + \frac{1}{2}} \right].$$
(4)

In these expressions, ni(ω) = [exp(ℏω/kBTi)−1]−1 is the Bose–Einstein distribution at temperature Ti, while T0 is the temperature of the surrounding vacuum.

With these tools, we can evaluate the averaged dipole and field fluctuations occurring in Eq. (2). We begin with the first term in that equation, which involves the dipole fluctuations. After using the fluctuation–dissipation theorem given in Eq. (3), we get

$$M_{i,p}^ \pm = \frac{{i\hbar }}{\pi } {\int_{ - \infty }^\infty} \mathrm{d} \omega \mathop {\sum}\limits_{j = 1}^N {A_{ij}^{ \pm \ast }} (\omega )C_{ij}^ \pm (\omega )\chi _j^ \pm (\omega )\left[ {n_j(\omega _j^ \mp ) + \frac{1}{2}} \right].$$

We can simplify the integral over frequency by noting that \(n_i( - \omega \pm {\mathrm{\Omega }}_i) + \frac{1}{2} = - n_i(\omega \mp {\mathrm{\Omega }}_i) - \frac{1}{2}\) and that, due to causality, \(G_{ij}( - \omega ) = G_{ij}^ \ast (\omega )\) and \(\alpha _{{\mathrm{eff}},i}^ \pm ( - \omega ) = \alpha _{{\mathrm{eff}},i}^{ \mp \ast }(\omega )\), which also imply that \(T_{ij}^ \pm ( - \omega ) = T_{ij}^{ \mp \ast }(\omega )\), where T = A, B, C, or D. By doing so, we obtain

$${M_{i,p}^ \pm = - \frac{{2\hbar }}{\pi }{\int_0^\infty} \mathrm{d}\omega \mathop {\sum}\limits_{j = 1}^N {\left[ {\frac{{2k^3}}{3}\left| {A_{ij}^ \pm (\omega )} \right|^2 + \mathop {\sum}\limits_{k = 1}^N {{\mathrm{Im}}} \left\{ {A_{ij}^{ \pm \ast }(\omega )A_{ki}^ \pm (\omega )G_{jk}(\omega )} \right\}} \right]} \chi _j^ \pm (\omega )\left[ {n_j(\omega _j^ \mp ) + \frac{1}{2}} \right].}$$

Using \(\mathop {\sum}\limits_{m = 1}^N {A_{im}^ \pm } (\omega )\alpha _{{\mathrm{eff}},m}^ \pm (\omega )G_{mj}(\omega ) = A_{ij}^ \pm (\omega ) - \delta _{ij}\), the expression above reduces to

$$\begin{array}{*{20}{l}} {M_{i,p}^ \pm } \hfill & = \hfill & { - \frac{{2\hbar }}{\pi }{\int_0^\infty} {\mathrm{{d}}} \omega \left[ {\frac{{2k^3}}{3}{\mathrm{Re}}\left\{ {2A_{ii}^ \pm (\omega ) - 1} \right\} + {\mathrm{Im}}\left\{ {S_{ii}^ \pm (\omega )} \right\}} \right]\chi _i^ \pm (\omega )\left[ {n_i(\omega _i^ \mp ) + \frac{1}{2}} \right]} \hfill \\ {} \hfill & {} \hfill & { + \frac{{2\hbar }}{\pi }{\int_0^\infty} {\mathrm{{d}}} \omega \mathop {\sum}\limits_{j = 1}^N {\left| {S_{ij}^ \pm (\omega )} \right|^2} \chi _i^ \pm (\omega )\chi _j^ \pm (\omega )\left[ {n_j(\omega _j^ \mp ) + \frac{1}{2}} \right],} \hfill \end{array}$$
(5)

where \(S_{ij}^ \pm (\omega ) = {\sum_{m = 1}^N} {A_{mi}^ \pm } (\omega )G_{mj}(\omega )\).

The second term in Eq. (2) can be computed in a similar way using, in this case, Eq. (4). This leads to

$$M_{i,E}^ \pm = \frac{{i\hbar }}{\pi } {\int_{ - \infty }^\infty} {\mathrm{{d}}} \omega \mathop {\sum}\limits_{j,k = 1}^N {B_{ij}^{ \pm \ast }} (\omega )D_{ik}^ \pm (\omega ){\mathrm{Im}}\left\{ {G_0(\omega )\delta _{jk} + G_{jk}(\omega )} \right\}\left[ {n_0(\omega ) + \frac{1}{2}} \right].$$

Following the same steps as above and using the definitions of \(B_{ij}^ \pm (\omega )\) and \(D_{ij}^ \pm (\omega )\), this expression reduces to

$$M_{i,E}^{\pm} = {\frac{{2\hbar }}{\pi } {\int_0^{\infty}} {\mathrm{d}} \omega \left[ {\frac{{2k^3}}{3}{\mathrm{{Re}}}\left\{ {2A_{ii}^ \pm (\omega ) - 1} \right\} + {\mathrm{Im}}\left\{ {S_{ii}^ \pm (\omega )} \right\}} \right]\chi _i^ \pm (\omega )\left[ {n_0(\omega ) + \frac{1}{2}} \right]} \\ - \frac{{2\hbar }}{\pi } {\int_0^\infty} {\mathrm{{d}}} \omega \mathop {\sum}\limits_{j = 1}^N {\left| {S_{ij}^ \pm (\omega )} \right|^2} \chi _i^ \pm (\omega )\chi _j^ \pm (\omega )\left[ {n_0(\omega ) + \frac{1}{2}} \right].$$
(6)

Finally, combining Eqs. (5) and (6), we obtain the expression of the Casimir torque given in the paper.