Thermal equilibrium spin torque: Near-field radiative angular momentum transfer in magneto-optical media

We consider a single- or two-cube system made of nonreciprocal materials shown in Figure 1. We focus on the AM-resolved thermal radiation on the vacuum side of the systems. For this purpose, momentum and AM will be studied in the context of the thermal radiation. These quantities are important because they are conserved and thus follow the conservation laws. First, the radiative momentum at the observation point can be quantified by Poynting flux $\mathbf{P}$ and the momentum flux density $\mathbf{\Sigma}$ [49]:

where $\otimes$ denotes the outer product of two vectors, and $\left\langle...\right\rangle$ denotes the thermodynamic ensemble average. All the quantities in Eq. 1 and Eq. 2 are dependent on the position $\mathbf{r}$ and time $t$. The radiation momentum transfer leads to a force $\mathbf{F}$ on the objects that can be obtained by the conservation law of momentum:

A nonreciprocal medium can lead to a nonzero radiation AM flux and thus results in a torque $\tau$ on objects. This torque can be obtained by the conservation law of AM, which is the integral form of the continuity equation in Eq. 6:

All the quantities throughout the paper are described in SI units. Next, we express the physical quantities in terms of their Fourier transforms, such as

for electromagnetic field, with similar notation for other quantities. Then above quantities $\mathbf{Q}=\left\{\mathbf{\Sigma},\mathbf{M},\mathbf{F},\tau\right\}$ are to be integrated over frequency to obtain the total flux/force/torque as $\mathbf{Q}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\mathbf{Q}(\omega)e^{-i\omega t}$. The electromagnetic field correlations required for calculating densities and flux rates above are obtained from the FDT. We can separate the contributions into two parts: the first one accounts for the fluctuations of particle dipole moments that will further induce electromagnetic field; the second part involves environmental field fluctuations. The FDT in our case gives [50]

for the electric dipole fluctuations and

for the environmental electric-field fluctuations, where $k_{0}=\omega/c$ is the magnitude of the vacuum wave vector. Here $\hat{\alpha}$ is the polarizability of the objects and $\mathbf{G}_{EE}(\mathbf{r},\mathbf{r}^{\prime})$ is the free space electric-electric dyadic Green’s tensor [50]. The temperature of the particle $T_{p}$ and the vacuum $T_{0}$ enter these expressions through the Bose-Einstein distribution $n_{l}(\omega)=1/(e^{\hbar\omega/k_{B}T_{l}}-1).$

To numerically compute the physical quantities introduced in Eq. 5 and Eq. 7, we extend the TDDA approach for MO objects based on Ref. [40]. Here we summarize the important equations we have developed. More details can be found in Appendix A.

Considering a two-body system interacting with a thermal bath, we use a collection $N_{p}$ (for object p) electric point dipoles to describe the system. Each dipole is characterized by a volume $V_{i,p}$ and a polarizability tensor $\hat{\alpha}_{i,p}$, where p=1,2 denotes the body that the dipole belongs to and i=1,2,…,$N_{p}$ indicates the i-th subvolume in that object. We group the electric dipoles and electric fields inside bodies in a compact form:

The notation $\bar{\mathbf{C}}$ ($\mathbf{C}=\{\mathbf{p},\mathbf{E},...\}$) indicates that the dipoles or the field are inside bodies and in the following discussion, we denote $\mathbf{C}$ without the overhead bar as the quantities in vacuum.

To obtain the total radiative AM flux in the near field or far field (obsevation point), we need to compute the statistical average of the AM flux density introduced in Eq. (5). Making use of the Fourier transforms, the average of the AM flux density can be expressed as

Using FDT from Eq. 10 and Eq. 11, the above expression can be reduced to the integration of the terms containing $\left\langle\mathbf{E}(\mathbf{r},\omega)\otimes\mathbf{E}^{*}(\mathbf{r},\omega)\right\rangle$ and $\left\langle\mathbf{H}(\mathbf{r},\omega)\otimes\mathbf{H}^{*}(\mathbf{r},\omega)\right\rangle$. As we discussed in the last section, Eq. 10 and Eq. 11 introduce two different sources that contribute to the total field correlations. The first one is the fluctuating particle dipoles determined by the temperatures of bodies. In the TDDA approach, it can be written as:

where $T_{1}$ and $T_{2}$ are the temperatures of the two objects, and $\hat{n}_{\mathrm{B}}\left(\omega,T_{1},T_{2}\right)$ is a diagonal tensor with $3N$ elements given by the Bose-Einstein distribution. $\hat{\chi}$ is a tensor combining the imaginary part of the polarization tensor and the radiative correction [51, 50]. Using the general TDDA equations and some algebraic manipulations, we obtain the following expressions for the correlations of electric and magnetic fields at the observation point outside bodies:

where $\mathbf{T}_{ij}=\delta_{ij}\mathbb{I}-\left(1-\delta_{ij}\right)k_{0}^{2}\bar{\alpha_{i}}\bar{\mathbb{G}}_{EE,ij}$ and $\mathbf{G}_{HE}$ is the magnetic-electric field Green’s tensor (Eq. 37). More details about the derivation are shown in Appendix A.1.

The second source is the environmental field fluctuation. The fluctuational electric fields in the vacuum can induce electric dipoles on the objects and then generate electromagnetic fields at the observation point. In this case, the field-field correlations at the observation point outside bodies are given by (Appendix A.2):

where

are the vacuum electric-electric field correlation inside bodies, the vacuum electric-electric field correlation between the object position and observation point, and the vacuum electric-magnetic field correlation between the object position and observation point, respectively. Generally, these two sources have opposite contributions to the total far field radiations and at global thermal equilibrium, they should cancel each other to satisfy the detailed balance of the radiative AM flux.

As shown in figure 2(a) and 3(a), a single cube made of InSb is in the presence of an uniform magnetic field of 1 T along the x direction. The first issue we want to address now is the description of the thermal AM radiation from the single cube using the formalism detailed in Section II. Instead of using the SAM density $\mathbf{S}(\mathbf{r})=\frac{\epsilon_{0}}{2\omega}\operatorname{Im}\left\langle\mathbf{E}^{*}(\mathbf{r})\times\mathbf{E}(\mathbf{r})\right\rangle+\frac{\mu_{0}}{2\omega}\operatorname{Im}\left\langle\mathbf{H}^{*}(\mathbf{r})\times\mathbf{H}(\mathbf{r})\right\rangle$ that was utilized to quantify the spin component of the thermal radiation in our former work [31], here we define a tensor of angular-momentum flux density $\mathbf{M}$ (Eq. 5) in analogy to the well-known Maxwell stress tensor. It allows us to compute the total flux $\Phi_{tot}$ of the radiative AM in the near-field and far-field, making a better connection between the thermal radiation and the induced torques. This definition includes the spin and orbital parts of the electromagnetic field. It reveals the whole story of the AM in the radiation and will not contradict the debate about the angular-momentum separation for electromagnetic fields [56, 57, 58].

Thermal equilibrium: We first consider a single cube in equilibrium with vacuum ($T_{1}=T_{0}=300$ K) and show the balance of the total AM flux $\Phi_{tot}$ in the far field (Fig. 2). To clarify the origin of the balance, we separate the total flux $\Phi_{tot}$ into two parts: $\Phi_{tot}=\Phi_{p}+\Phi_{E}$. $\Phi_{p}$ is induced by the fluctuational particle dipoles of the cube that is determined by the body temperature $T_{1}$ (Appendix A.1), while $\Phi_{E}$ comes from the environmental field fluctuations which are dependent on the environment temperature $T_{0}$ (Appendix A.2). Then we compute each AM flux $\Phi_{i}$ (for i= x,y,z) across the plane we defined. For this purpose, the AM flux is written as

which describes the integrated flux across a differential section d$A$ perpendicular to the radial vector $\mathbf{R}$. Here we choose a surface that encloses the cube to compute the total AM flux radiated to the far field. As shown in Figure 2(b), the spectra of the thermal AM flux $\Phi_{x,p}$ and $\Phi_{x,E}$ have the same magnitude but opposite signs at each frequency. It is noted that the background thermal radiation in vacuum has no net flux. The non-zero $\Phi_{x,E}$ originates from the scattering of the incident background thermal radiation by the particle. The total flux $\Phi_{x,tot}$ is zero as $\Phi_{x,p}$ and $\Phi_{x,E}$ perfectly cancel each other. The vanishing of $\Phi_{x,tot}$ satisfies the detailed balance of the AM flux and it indicates that there is no radiative torque applied on the cube at thermal equilibrium.

For $\Phi_{y}$ and $\Phi_{z}$ perpendicular to the gyrotropy axis, both of them are zero regardless of the temperatures and hence result in a zero total flux $\Phi_{i,tot}$ (for $i=y,z$). Since $\Phi_{p}$ and $\Phi_{E}$ have no contribution to the perpendicular component of the AM flux, the torques along y and z directions in this case will always remain zero no matter it is at thermal equilibrium or not. Thus, in the following discussions on the non-equilibrium case of a single cube, we focus on only the AM flux along the magnetic field (gyrotropy axis).

Thermal non-equilibrium: Here we show that thermal non-equilibrium can lead to a net radiative AM flux along the gyrotropy axis. As shown in Fig. 3 (a), We assume that the environment is at 0 K while the cube is kept at the room temperature $T_{1}$ = 300 K. Since the temperature of the cube is higher than the environment, $\Phi_{p}$ has a larger magnitude than $\Phi_{E}$ at each frequency and thus their summation gives a net AM flux along the magnetic field (Fig. 3(b)). In addition, due to the zero-point fluctuation, the background radiation still exists at 0 K and thus the AM flux $\Phi_{E}$ (originating from the environment) in this case has a considerable value. Figure 3(d) and (e) depict the spatial distributions of the AM flux across yz-planes on the two sides of the cube. AM flux at 24.75 meV is shown here as a typical example. The AM loss through the two planes (in front of and behind the cube) has an equal contribution to the total AM flux, giving a non-zero value of $\Phi_{x}$. The other components that are perpendicular to the gyrotropy axis ($\Phi_{y}$ and $\Phi_{z}$) still remain zero at thermal non-equilibrium. In total, a torque along the external magnetic fields is induced by the AM loss due to the thermal non-equilibrium between the cube and environment.

In Figure 3(c), we compute the thermal AM torque as a function of the cube size, and compare our numerical calculations to the dipole approximation. In TDDA-125, we divide each cube into 125 subvolumes, while for TDDA-1, we use a single point dipole to represent the object. At small sizes, the results from TDDA-1 and TDDA-125 show a good agreement. When the size of the cube increases, TDDA-125 has a different magnitude as it takes the shape effect into consideration, which should be more precise than TDDA-1 for larger-size objects. Therefore, for following examples, we focus on only the numerical results obtained by TDDA-125.

In the previous section, we showed that, for a single cube, there is no net AM flux going to the far field at thermal equilibrium. The non-vanishing AM flux along the gyroelectric axis exists only when the cube and the environment have different temperatures. However, things can be different for a two-body MO system. In a two-cube system, we find that a net AM flux can also exist in the near field between two cubes when the magnetic fields on each cube are misaligned. Such AM transfer induces an equilibrium torque between the cubes, which can be tuned by changing the angle between the magnetic fields.

Thermal equilibrium: In Fig. 4, the two-cube system is at global thermal equilibrium where $T_{1}=T_{2}=T_{0}=300$ K. $T_{1}$, $T_{2}$ and $T_{0}$ denote the temperatures of the top cube, bottom cube and vacuum, respectively. To compute the total AM flux radiating from the top cube, we choose a box enclosing the top cube and calculate the flux across the surface, $\mathbf{M}\cdot\hat{n}_{\mathbf{r}}$. If the magnetic fields on two cubes are parallel (Fig. 4 (a)-(b)), there is no AM flux going out from either cube, which is similar to the single cube case. However, once the magnetic fields are misaligned, the gyrotropy axes of the cubes are along different directions (Fig 4 (c)-(e)). And then surprisingly, there is a net AM flux along the z direction (perpendicular to the plane formed by the two gyrotropy axes) which is being exchanged between two cubes. Meanwhile, $\Phi_{x}$ and $\Phi_{y}$ still remain zero regardless of the angle $\beta$.

Such AM transfer of $\Phi_{z}$ is interesting but non-intuitive since it can occur despite the global thermal equilibrium. It is also important to point out that the presence of the non-zero radiative AM transfer at the global thermal equilibrium does not lead to any thermodynamic contradictions. To demonstrate this argument, we separately compute the AM flux $\Phi_{z}$ across different xy-planes to reveal the origins of the AM transfer (Fig. 5). Here, we typically choose three planes to show the flux $\Phi_{z}$: (b) The mid plane between two cubes; (c) the top plane above the top cube; (d) the bottom plane below the bottom cube. The mid xy-plane shows the near field AM transfer between the cubes, while the top and bottom planes show the AM flux going to the far field from each cube, respectively. Here we do not show the AM flux leaking through the other surfaces since their contributions to $\Phi_{z}$ are negligible compared to xy-planes we showed above. From Fig. 5, it is easy to find that the AM flux $\Phi_{z}$ mainly transfers through the mid xy-plane, with almost two-order larger magnitude than the top and bottom planes. Moreover, the fluxes $\Phi_{z}$ across the top and bottom planes have the same magnitude but different signs (directions) at each frequency. Therefore,considering the total flux radiated to the environment from the combined two-cube system, (c) and (d) cancel each other and give a zero net flux of $\Phi_{z}$. This means that such AM transfer is a localized phenomenon and won’t cause any net flux that transfers between the system and the surrounding environment, conserving AM globally.

With a net AM flux exchanged between the cubes, an equilibrium torque is induced on each cube. Such equilibrium torque can be tuned by changing the angle between the magnetic fields and vanishes when the magnetic fields are parallel. Fig. 6 plots magnetic field dependence of the equilibrium torque. Here we assume that the magnetic field on the bottom cube is fixed while we change the angle $\beta$ of the B field on the top cube. Similar to what we did in the last subsection, we separate the torques into two parts originating from the particle dipole fluctuations ($M_{p}$) and the environment field fluctuations ($M_{E}$), respectively.

First we only consider the torques within the xy-plane where the magnetic fields are applied. In this case, $M_{p}$ and $M_{E}$ have the same magnitude but different signs and hence result in a zero net torque within the xy-plane on each cube. The vanishing of the torque in xy-planes is independent of $\beta$ and is always true at global thermal equilibrium. Unlike the single-body system where the $M_{p}$ and $M_{E}$ should be exactly along its gyrotropy axis (magnetic field), here they can be slightly misaligned with their own gyrotropy axis. Such misalignment originates from the interaction between the cubes since $M_{p}$ ($M_{E}$) of the bottom cube in a fixed magnetic field varies when we tune the direction of the B field on the top cube (the left and middle plots in Fig. 6 (b)). The torque that is perpendicular to the external magnetic field has a much smaller magnitude than the parallel component, indicating that the interaction is weak at global thermal equilibrium.

Along the z direction perpendicular to the plane formed by the magnetic fields, a torque is induced by the AM transfer between two cubes despite global thermal equilibrium (the right plots in Fig. 6). When the magnetic fields on two cubes are parallel or anti-parallel, there is no torque applied because the AM transfer is prohibited. Otherwise, each cube feels a torque along the z direction trying to align their gyrotropy axes parallel to each other. Such torques have the same magnitude but opposite signs and, therefore, there is no net torque on the combined system, conserving the global AM at thermal equilibrium. Here we want to note that the ”thermal equilibrium” discussed in this work refers to only the global equality of temperature. Due to the nonzero torque induced by the near-field AM transfer, the condition of mechanical equilibrium is broken. The full mechanical dynamics of the particles accounting for all forces including the attractive Casimir force between them is beyond the scope of this work, and will be considered in a future work.

At thermal equilibrium, there are two distinct zero-torque configurations: parallel gyrotropy axes ($\beta=0$) and anti-parallel gyrotropy axes ($\beta=\pi$). If the system starts from the anti-parallel configuration ($\beta=\pi$), two cubes have a tendency to relax back to the parallel configuration. This means that $\beta=\pi$ is an unstable equilibrium point. On the other hand, if two cubes are left in the parallel configuration ($\beta=0$), they will remain in it and tend to go back after a small disturbance.

Finally, we plot the temperature dependence and distance dependence of the torque $\tau_{z}$ in Figure 7. Temperature affects the AM flux through the mean thermal energy $\Theta(\omega,T)=\hbar\omega/2+\hbar\omega/\left[\exp\left(\hbar\omega/k_{B}T\right)-1\right]$. Since the mean thermal energy is approximately constant over the frequency range of interest, the magnitude of $\tau_{z}$ increases proportionately with the temperature. The left panel in Fig. 7(b) demonstrates the spectrum for $\Phi_{z}$ as a function of distance between two cubes. At each frequency, the sign (the direction of the radiative AM flux) stays the same while the magnitude decays as a function of distance. The right panel in Fig. 7(b) shows the torque along the z direction after doing integral of $\Phi_{z}$ over frequency. Similarly, the magnitude of the torque $\tau_{z}$ decreases as a function of distance while its direction remains unchanged. This confirms that the radiative AM transfer for $\Phi_{z}$ at global thermal equilibrium originates from the near-field interaction between the cubes, as the interaction strength decays with increasing separation between them.

Thermal non-equilibrium: Above we showed that, for a two-cube system at global thermal equilibrium ($T_{1}=T_{2}=T_{0}$), AM transfer can happen only along z direction. And we have also pointed out that there is no torque within xy-plane at global thermal equilibrium due to the balance between $M_{p}$ and $M_{E}$, which are slightly misaligned with their gyrotropy axes. Here we are going to show that non-equilibrium ($T_{1}\neq T2$) helps amplify the interaction between the cubes, making the AM transfer between two cubes much stronger than the equilibrium case.

Figure 8 depicts the magnetic field dependence of the radiative AM torques at thermal non-equilibrium. As shown in Figure 8 (a), we assume that the top cube is heated up to $T_{1}=500$ K, while the bottom cube remains at equilibrium with the environment ($T_{2}=T_{0}=300$ K). Similarly, we tune the direction of the magnetic field on the top cube and compute the non-equilibrium torque as a function of $\beta$. In this case, the torques on each cube are strongly modified due to the near-field AM transfer (Figure 8 (b)-(d)). First, the torques within the xy-plane no longer follow the direction of the external magnetic fields. On the contrary, the torques in the xy-pane have a big departure from the gyrotropy axes. Second, the torque in the xy-plane on the bottom cube (in a fixed magnetic field) also shows a strong dependence of the magnetic field $B_{1}$ applied on the top cube. At each angle $\beta$, the torques on two cubes have comparable magnitudes and opposite signs. Their summation gives a smaller total torque on the combined system, which is one order smaller than the torque on each cube. Third, the torques along the z direction are amplified by two orders compared with the equilibrium case, directly indicating that the radiative AM transfer is strongly amplified because of the unequal temperatures of two cubes.