Moving media as photonic heat engine and pump

A system consisting of two slabs with different temperatures can exhibit a nonequilibrium lateral Casimir force on either one of the slabs when Lorentz reciprocity is broken in at least one of the slabs. This system constitutes a photonic heat engine that converts radiative heat into work done by the nonequilibrium lateral Casimir force. Reversely, by sliding two slabs at a sufficiently high relative velocity, heat is pumped from the slab at a lower temperature to the other one at a higher temperature. Hence the system operates as a photonic heat pump. In this paper, we study the thermodynamic performance of such a photonic heat engine and pump via the fluctuational electrodynamics formalism. The propulsion force due to the nonreciprocity and the drag force due to the Doppler effect were revealed as the physical mechanisms behind the heat engine. We also show that in the case of the heat pump, the use of nonreciprocal materials can help reduce the required velocity. We present an ideal material dispersion to reach the Carnot efficiency limit. Furthermore, we derive a relativistic version of the thermodynamic efficiency for our heat engine and prove that it is bounded by the Carnot efficiency that is independent of the frame of reference. Our paper serves as a conceptual guide for the realization of photonic heat engines based on fluctuating electromagnetic fields and relativistic thermodynamics and shows the important role of electromagnetic nonreciprocity in operating them.

Figure 1

Two semi-infinite parallel slabs separated by a vacuum gap $d$ that are moving relative to each other at constant velocity $V$ along the $x$ direction. Two slabs are at the proper temperatures $T_1$ and $T_2^{\prime }$ in the rest and comoving frames, respectively. The physical quantities with the prime mark are defined in the comoving frame. (a) The system operating as a heat engine where the net heat flux ${\phi }_{1\to 2}$ from slab 1 to slab 2 is converted into work driven by nonequilibrium lateral Casimir force $f_{x,2}$ on slab 2 under proper directions of external magnetic fields. (b) The system operating as a heat pump where the external work done by the force $f_{\mathrm{ext}}$ pumps heat from slab 2 at a low proper temperature to slab 1 at a higher proper temperature. External magnetic fields are not necessary.

Figure 2

(a) Radiative heat flux from slab 1 to slab 2, (b) shear stress on slab 2, and (c) thermodynamic efficiency of the heat engine as a function of the velocity of relative motion. Two slabs made of $n$-InSb are at $T_1=305$ K and $T_2^{\prime }=300$ K and separated by $d=10$ nm. The results for different magnitudes of external magnetic fields are shown. The dash-dotted line in panel (b) describes $f_{x,2}=0$. The results of the linear approximation Eqs. (12) and (14) are plotted in the dashed lines.

Figure 3

(a) Spectral shear stress on slab 2 for different velocities of relative motion. (b) Plot of the transmission coefficient $\tau$ as a function of the frequency $\omega$ and $x$ component of the wave vector $q_x$ when two slabs are at rest. The transmission coefficient is plotted for the modes with in-plane wave vectors along the $x$ direction, i.e., $\mathit{q}=\left(q_x,0\right)$. The magnetic fields are $B_1=-B_2=3\mathrm{T}.$ (c) The ratio of the net transferred thermal photon number with relative motion $\mathrm{\Delta }{n}_B=n_B\left(\omega ,T_1\right)-n_B\left({\omega }^{\prime },T_2^{\prime }\right)$ to the one without relative motion $\mathrm{\Delta }{n}_B=n_B\left(\omega ,T_1\right)-n_B\left(\omega ,T_2^{\prime }\right)$ for fixed $q_x=\pm {10}^8{\mathrm{m}}^{-1}$ at $V={10}^2$ and $1.4\times {10}^3$ m/s. (d) Plot of mode-resolved radiative heat flux ${\tilde{\phi }}_{1\to 2}\left(\omega ,q_x,q_y\right)$ defined as ${\phi }_{1\to 2}={\int }_0^{\infty }d\omega {\int }_{-\infty }^{\infty }d\mathit{q}{\tilde{\phi }}_{1\to 2}\left(\omega ,q_x,q_y\right)$ for the same condition as panel (b). The unit is $\left[{10}^{-15}\frac{\mathrm{W}\mathrm{s}}{\mathrm{rad}}\right]$. The other physical parameters are the same as those in Fig. 2.

Figure 4

(a) Radiative heat flux from slab 1 to slab 2, (b) shear stress on slab 2, and (c) coefficient of performance of the heat pump as a function of the velocity of relative motion. The physical parameters are the same as those in Fig. 2. The dash-dotted line in panel (a) describes ${\phi }_{1\to 2}=0$. The COP is only defined for the region of ${\phi }_{1\to 2}<0$ where the system operates as the heat pump. The black dashed line in panel (a) represents the radiative heat flux from the modes with ${\omega }^{\prime }<0$ only.

Figure 5

Spectral radiative heat flux between two slabs under antiparallel static magnetic fields of 3 T for different velocities of relative motion. The other physical parameters are the same as those in Fig. 2.

Figure 6

(a, b) Plot of the transmission coefficient $\tau$ as a function of the frequency $\omega$ and $x$ component of the wave vector $q_x$ at $V={10}^4$ and $1.5\times {10}^5$ m/s, respectively. (c, d) Plot of mode-resolved radiative heat flux ${\tilde{\phi }}_{1\to 2}\left(\omega ,q_x,q_y=0\right)$ defined in the caption of Fig. 3. The unit is $\left[{10}^{-15}\frac{\mathrm{W}·\mathrm{s}}{\mathrm{rad}}\right]$. The physical parameters are the same as those in Fig. 2. The black lines in panels (c) and (d) are the constraint for cooling in Eq. (22). The dashed line in panel (d) is the line below which the angular frequency in the comoving frame is negative, i.e., ${\omega }^{\prime }<0$. The white dashed lines are the solutions of the dispersion relation Eq. (23) in the lossless limit.

Figure 7

(a) Application of the first law of relativistic thermodynamics on the control volume enclosed by the dashed line. slab 2 is hypothetically separated into the engine and heat sink, both of which are moving at the velocity $V$. (b) The evolution of the energy of slab 2 and the work done to slab 2 for acceleration $W_a$ and the work extracted by decelerating slab 2 $W_d$. For both panels, we consider the observer is in the rest frame with slab 1.