Adiabatic Thermal Radiation Pumps for Thermal Photonics

TL;DR

This paper introduces a method to control and enhance thermal radiation directionality and magnitude between bodies at equilibrium using adiabatic pumping and wave interference, with practical circuit validation.

Contribution

It presents a novel approach employing adiabatic pumping and scattering matrix analysis to manipulate thermal radiation directionality at equilibrium, highlighting the role of wave interference.

Findings

Adiabatic pumping can significantly enhance directional heat transfer.

Wave interference is critical in controlling thermal radiation.

Experimental circuit setup confirms theoretical predictions.

Abstract

We control the direction and magnitude of thermal radiation, between two bodies at equal temperature (in thermal equilibrium), by invoking the concept of adiabatic pumping. Specifically, within a resonant near-field electromagnetic heat transfer framework, we utilize an {\it instantaneous} scattering matrix approach to unveil the critical role of wave interference in radiative heat transfer. We find that appropriately designed adiabatic pumping cycling near diabolic singularities can dramatically enhance the efficiency of the directional energy transfer. We confirm our results using a realistic electronic circuit set-up.

Equations

$$ı\frac{d}{dt}\Psi$$

$${\theta }^{(-)}$$

$$⟨{\theta }_n^{(+)}(\omega )({\theta }_m^{(+)}({\omega }^{\prime }){)}^{\ast }⟩=\frac{\hslash \omega }{2\pi }{\Theta }_T(\omega )\delta (\omega -{\omega }^{\prime }){\delta }_{nm}$$

$$S=-I_2-ıW^T{G}_{\mathrm{eff}}W,G_{\mathrm{eff}}=\frac{1}{H_{\mathrm{eff}}-\omega I_{N_s}}$$

$$\bar{\mathcal{I}}\equiv \frac{\Omega }{2\pi }\int \frac{d\omega }{2\pi }\hslash \omega {\Theta }_T(\omega )Q(\omega ),Q(\omega )\equiv \underset{\mathcal{A}\to 0}{\lim }\frac{{\int }_0^{\frac{2\pi }{\Omega }}dt{\mathcal{I}}_{x_0}\left(t,\omega \right)}{\mathcal{A}}$$

$$S^t$$

$$Q\left(\omega \right)=\underset{\mathcal{A}\to 0}{\lim }\frac{1}{\mathcal{A}}\frac{\partial }{\partial \omega }{\int }_0^{2\pi /\Omega }dt{R}^t\frac{d{\alpha }^t}{dt}=\frac{\partial }{\partial \omega }\left|\frac{\partial \left(R^t,{\alpha }^t\right)}{\partial \left(u^t,v^t\right)}\right|$$

$$R^t={\left|S_{11}^t\right|}^2,$$

$$H_0(t;u^t,v^t)$$

$$Q\left(\omega \right)\equiv \frac{\partial }{\partial \omega }P\left(\omega ,d,\{w_n\}\right)$$

$$\bar{\mathcal{I}}=-\frac{\Omega }{2\pi }\int \frac{d\omega }{2\pi }\frac{\partial \left[\hslash \omega {\Theta }_T(\omega )\right]}{\partial \omega }P(\omega ,d,\{w_n\})$$

$$\bar{\mathcal{I}}\propto \sum _n\mathrm{Res}\left[\frac{\partial f(\omega )}{\partial \omega }P(\omega );{\omega }_n\right];\mathcal{I}m\left({\omega }_n\right)>0$$

$$\bar{\mathcal{I}}\propto {\sum _n}^{\prime }\mathrm{Res}\left[P(\omega );{\omega }_n\right]$$

$$\tilde{\mathcal{I}}$$

Full text

Introduction – Understanding the rules that dictate thermal radiation and the development of novel schemes that allow us to tame its flow, has offered over the years an exciting arena of research VP07 ; BLID11 ; HSM11 ; F17 . From one side, there are fundamental challenges associated with basic constraints that need to be understood in order to manage thermal radiation F17 ; MZF17 ; ZF14 ; SWL98 ; HSA16 . At their core are questions associated with the importance of thermal electromagnetic fluctuations and their implications in directional thermal radiation. On the other hand, there is a wide range of applications that can benefit from advances in thermal radiation management. In fact, in close connection with the rapid developments that we are witnessing in the field of nanophotonics, the subfield of thermal photonics has emerged RIBCJSJ11 ; AB14 ; SNC09 ; RSJVCCG09 ; OQWALMRTW11 ; KHZMFH12 ; GGZFL14 ; SLLNS15 ; Kim15 ; SGSTFFFVCRM15 ; GZFL16 ; STFGRM16 and promises to revolutionize modern energy technologies. Examples include thermophotovoltaics LMF03 ; NC03 ; BZF09 ; IJJCS12 , thermal imaging WFCGLJMCG06 ; KHPBRH05 , thermal circuits OLF10 ; BF11 ; ZOF12 , and radiative cooling GOPFL12 .

In this paper we propose to manipulate thermal radiation by introducing the concept of an adiabatic thermal radiation pump. The pump operates between two reservoirs that are maintained at the same temperature - as opposed to common approach where heat flow requires a temperature gradient. A possible setup of a thermal radiation pump is depicted in Fig. 1a. By slow periodic modulation of the eigenfrequencies of two resonators, coupled to two separate reservoirs at equal temperature $T$, one can transfer thermal radiation from one reservoir to the other (see green arrows). This process of creating a directional radiation flow may be termed adiabatic thermal radiation pumping. The amount of heat pumped in one cycle depends on the details of the modulation process. In particular, we show that the existence of a diabolic point (i.e. an exact degeneracy) in the spectrum of the system’s Hamiltonian leads to a dramatic enhancement of the effect, for an appropriately chosen modulation cycle. Our theoretical results are based on a coupled-mode-theory (CMT) approach to resonant thermal radiation and are backed up by detailed numerical simulations using realistic circuit set-ups, see Fig. 1b. Our approach unveils the importance of wave interference in radiative heat transfer by connecting the pumped thermal radiative current with the instantaneous reflection phase. This connection opens up new possibilities in the field of thermal photonics. Our concept of the adiabatic thermal radiation pumping is inspired by adiabatic charge pumping in condensed matter, where a DC current in response to a slowly varying time-periodic potential has been proposed T83 ; AG99 ; Levinson2002 ; Brouwer1998 ; AEGS01 ; C03 and experimentally demonstrated SMCG99 .

CMT Modeling of Thermal Radiation– We consider a photonic circuit supporting a finite number of resonant modes $N_{s}$ described by a Hermitian Hamiltonian $H_{0}$. The system is in contact with two separate heat baths at constant temerature (see Fig. 1a where $N_{s}=3$) via leads with the system-lead coupling described by an operator ${\hat{W}}$.

At thermal equilibrium, the two baths generate photons at frequency $\omega$ with mean number $\Theta_{T}(\omega)=\left(e^{\hbar\omega\over k_{B}T}-1\right)^{-1}$ given by the Bose-Einstein statistics. The radiative thermal energy exchange between the two heat baths can be studied using a time-dependent CMT ZSOFSL13 ; KJ15

[TABLE]

where $\Psi=\left(\psi_{1},\psi_{2},\cdots,\psi_{N_{s}}\right)^{T}$ describes the modal amplitude of the field and it is normalized in a way that $|\psi_{s}|^{2}$ represents the energy of the $s^{th}$ mode. The variables $\theta_{n}^{(\pm)}(\omega)$ (frequency domain of $\theta^{\pm}$ in Eq. (Adiabatic Thermal Radiation Pumps for Thermal Photonics)) indicate the flux-amplitudes of the incoming (+) and outgoing (-) waves from and towards the reservoir $n=1,2$ via leads. At thermal equilibrium the incoming flux (from the reservoirs) satisfies the correlation relation ZSOFSL13

[TABLE]

and, therefore, the outgoing power from the $n^{th}$ heat bath is given by the double integral over frequency $\omega$ of this correlation function.

The associated scattering matrix $S$, connecting the outgoing $\theta^{-}(\omega)$ to the incoming $\theta^{+}(\omega)$ waves, can be evaluated using Eq. (Adiabatic Thermal Radiation Pumps for Thermal Photonics). We have Li2017

[TABLE]

where $I_{n}$ is the $n\times n$ identity matrix. The matrix $W$, describes the coupling to the leads (in frequency domain), and has dimensions $N_{s}\times 2$. Its elements are $W_{s,n}=\sqrt{v_{g}}w_{n}\delta_{sn}$ where $w_{n}$ are dimensionless coupling strengths, and $v_{g}={\partial\omega(k)\over\partial k}$. Finally, the matrix $\Lambda$ appearing in $H_{\rm eff}$ (see Eq. (Adiabatic Thermal Radiation Pumps for Thermal Photonics)) is a renormalization term due to the coupling of the system with the leads and it is specific to the properties of the leads.

Thermal Radiation Pumps– Next, we consider a system whose Hamiltonian $H_{0}(u^{t},v^{t})$ depends on two time-varying independent parameters $(u^{t},v^{t})$. We further assume that these parameters are periodically modulated in time with frequency $\Omega$, such that $H_{0}\left(t\right)=H_{0}\left(t+2\pi/\Omega\right)$. During one period of the modulation these parameters form a closed cycle in the $(u^{t},v^{t})$ parameter space. The associated enclosed “pumping area” is ${\mathcal{A}}\equiv\int_{0}^{2\pi/\Omega}dtu^{t}\frac{dv^{t}}{dt}$. We want to analyze the net radiative energy flux from one bath to another during one pumping circle.

We consider circumstances where the variation of $(u^{t},v^{t})$ is small such that ${\mathcal{A}}\rightarrow 0$. We want to evaluate the total radiative (time-averaged) thermal energy flux per pumping area $\bar{\mathcal{I}}$. The latter is:

[TABLE]

where $Q(\omega)$ is the radiative energy density (i.e. per area in the parameter space) and $\mathcal{I}_{x_{0}}(t,\omega)$ is the dimensionless (normalized) time-dependent directional net energy current, at some observation cross section at $x=x_{0}$ within the leads. The latter is evaluated under the condition of two uncorrelated counter propagating incoming waves of frequency $\omega$ and unit flux. In the case where $H_{0}$ is static, the net thermal radiative current $\mathcal{I}$ at each lead is zero. From Eq. (4) it is clear that an understanding of $Q\left(\omega\right)$ is essential for the analysis and control of $\bar{\mathcal{I}}$ note2 .

Adiabatic Pumping– In the adiabatic limit, $\Omega\rightarrow 0$, the study of radiative thermal energy $Q\left(\omega\right)$ boils down to the analysis of the instantaneous scattering matrix $S^{t}$ Brouwer1998 . The latter is given in terms of Eq. (3) with the superscript $t$ indicating the parametric dependence of the matrix elements of $S$ at a specific instant $t$ during the pumping cycle. It can be generally parametrized in terms of three independent parameters: the instantaneous reflectance $R^{t}$, and the instantaneous reflection and transmission phases $\alpha^{t},\varphi^{t}\in\mathcal{R}$ respectively. Specifically we have:

[TABLE]

Using this parametrization, we write $Q\left(\omega\right)$ as

[TABLE]

which applies whenever the period of the driving is larger than the delay time that the “photons" dwell inside the scatterer. It is important to stress that Eq. (6) allows us to connect wave interference phenomena (imprinted via the reflection phase $\alpha^{t}$) with the thermal radiation problem. Furthermore, it opens up new directions in the field of thermal photonics. Using Eqs. (5,3) we have that

[TABLE]

where the subscripts indicate the matrix elements of $S^{t}$.

Direct inspection of Eqs. (5,7) indicates that the pumped thermal radiation energy will be affected by the proximity of resonant modes where $R$ and $\alpha$ experience an abrupt change in $\omega$. One would expect that higher order spectral singularities, like diabolic points (DP), could lead to a dramatic enhancement of $Q\left(\omega\right)$. We will show that their effect in $\bar{\mathcal{I}}$ is controlled by the position of the adiabatic cycle in the parameter space, with respect to such spectral singularities.

A prototype CMT model with DPs– We consider a prototype system of three coupled resonant modes, that can support a DP degeneracy. The system is described by the CMT Hamiltonian

[TABLE]

where $u^{t}=u^{(0)}+\delta u^{t}$, $v^{t}=v^{(0)}+\delta v^{t}$ with $\left(u^{(0)},v^{(0)}\right)=\left(d\cos\theta,d\sin\theta\right)$ and $\delta u^{t}=\delta u^{t+2\pi/\Omega},\,\delta v^{t}=\delta v^{t+2\pi/\Omega}$ . The pair $\left(u^{(0)},v^{(0)}\right)$ determines the center of the adiabatic cycle, reparameterized with $d$ and $\theta$ in polar form.

When $u^{t}=v^{t}=0$, the eigen-frequencies of Hamiltonian Eq. (8) are $a_{1}^{(0)}=a_{2}^{(0)}=\omega_{\rm DP}=\omega_{0}-1$ (DP degeneracy) and $a_{3}^{(0)}=\omega_{0}+2$. This DP degeneracy can be lifted in two ways: (a) by introducing $\left(u^{(0)},v^{(0)}\right)\neq(0,0)$; (b) by coupling the system to the leads. In the latter case, the eigen-modes turn into resonant modes. The resonant frequencies are the real parts of the poles of the $S$-matrix which are identified with the complex eigenvalues $\left\{a_{n}\right\}$ of the effective Hamiltonian $H_{\rm eff}^{(0)}$, see Eqs. (Adiabatic Thermal Radiation Pumps for Thermal Photonics,3) in the absence of modulation. Their corresponding imaginary part is the resonance line-width which describes the decay rate of these modes to the reservoirs. In other words the degenerate eigenmodes $a_{1,2}^{(0)}$ move apart from one-another and turn to $a_{1}\neq a_{2}$ whenever the above two mechanisms are in effect. The degree of repulsion between $(a_{1},a_{2})$ is controlled by the interplay of the proximity of the adiabatic cycle to the DP and the coupling with the leads.

Effects of DP in Pumped Thermal Radiation– In the presence of small time-periodic variations $(\delta v^{t},\delta u^{t})$, the Green’s function $G_{\rm eff}=(H_{\rm eff}-\omega I_{N_{s}})^{-1}$ which appears in the evaluation of $S^{t}$ (and therefore in $Q(\omega)$), involves the instantaneous effective Hamiltonian $H_{\rm eff}=H_{\rm eff}^{(0)}+\Delta^{t}$ where $\Delta^{t}={\rm diag}\left(\delta v^{t},0,\delta u^{t}\right)$. We expand $G_{\rm eff}$ in a series keeping terms up to first order in $\Delta^{t}$ i.e. $G_{\rm eff}=G_{\rm eff}^{(0)}-G_{\rm eff}^{(0)}\Delta^{t}G_{\rm eff}^{(0)}+\cdots$ where $G_{\rm eff}^{(0)}=\left(H_{\rm eff}^{(0)}-\omega I_{N_{s}}\right)^{-1}$. We can make further progress by representing $G_{\rm eff}$ in the bi-orthogonal basis $\left\{\left(a_{i}\right|,\left|a_{i}\right)\right\}(i=1,2,3)$ of $H_{\rm eff}^{(0)}$ where $G_{\rm eff}^{(0)}$ is diagonal. The reflection coefficient $R^{t}$ and phase $\alpha^{t}$ are evaluated (up to first order in $\delta u^{t},\delta v^{t}$) by direct substitution of $G_{\rm eff}$ into Eq. (3). Finally, the adiabatic thermal radiation energy $Q$ is calculated using Eq. (6). Specifically, using Eqs. (5, 6, 3, 7), we find that

[TABLE]

where $A^{ij}=\imath\frac{W^{T}\left|a_{i}\right)\left(a_{j}\right|W}{\sqrt{a_{i}-\omega}\sqrt{a_{j}-\omega}}$ and $\tilde{\triangle}_{ij}^{t}=\frac{\left(a_{i}\right|\triangle^{t}\left|a_{j}\right)}{\sqrt{a_{i}-\omega}\sqrt{a_{j}-\omega}}$. The implicit subscript (not shown) of matrix $A^{ij}$ denotes the entry while the trace operation is with respect to the dummy variable, e.g., $\mathrm{Tr}\left(\tilde{\triangle}^{t}A_{11}\right)=\sum_{i,j}\tilde{\triangle}_{ij}^{t}A_{11}^{ji}$.

Direct substitution of Eq. (9) in Eq. (4) allows us to perform an integration by parts and express the thermal energy flux per pumping area ${\bar{\mathcal{I}}}$ as

[TABLE]

where for near-resonant thermal radiation the boundary contributions (associated with the integration by part) are neglected. Using the residue theorem we get

[TABLE]

where $f(\omega)\equiv\hbar\omega\Theta(\omega,T)$ and $\omega_{n}$ are the poles of $P(\omega)$ (note that $f(\omega)$ is an analytic function). In the case of near-field resonant thermal transport, these poles can be associated with the poles of the scattering matrix Eq. (3). They originate from the DP degeneracies of the isolated system once they move to the complex frequency plane due to coupling with the leads.

In fact, the singular behavior of $R^{t}$ and $d\alpha^{t}/dt$ will be significant in the presence of a diabolic point degeneracy. In this case a further progress can be made for the evaluation of $\bar{\mathcal{I}}$. Specifically,

[TABLE]

where the summation is restricted to poles near the DP.

Examples of Adiabatic Pumps – The above theoretical considerations can be directly tested using the CMT system of Eq. (8). We chose $\theta=45^{0}$, $\delta u^{t}=r\cos\Omega t$ and $\delta v^{t}=r\sin\Omega t$. Furthermore we assume (left and right) tight-binding leads with dispersion $\omega=\omega_{0}-2\cos k$ that are coupled to the scattering target with the same coupling constant $w_{L}=w_{R}=\epsilon$. We get the following expression for the total radiative (time-averaged) thermal energy flux $\bar{\mathcal{I}}$ per pumping area (see Appendix supl )

[TABLE]

In Fig. 2a we report Eq. (13), together with the numerical calculations using Eq. (4,6). We have performed similar calculations for different angles $\theta$ in order to validate the general features of Eq. (13). Specifically, in Fig. 2b, we report the behavior of $\bar{\mathcal{I}}$ versus the control parameter $d$ for $\theta=135^{0}$. In both cases we have found that $\bar{\mathcal{I}}$ diminishes for large values of the control parameter $d$ (see also Eq. (13)). In this limit, the Hamiltonian $H_{0}$ of the isolated system does not support a DP. As $|d|\rightarrow 0$, the DP is re-established, leading to an enhanced radiative thermal energy flux. This enhancement can be further boosted by decreasing the coupling $\epsilon$ between the system and the leads. As discussed previously, the coupling shifts the degenerate levels to the complex plane, thus lifting the DP degeneracy and therefore deteriorating the performance of the pump. We point out that typically, the extrema of $\bar{\mathcal{I}}$ is in the vicinity of the DP corresponding to $H_{0}(d=0;r=0)$ (see other cases at the Appendix supl ). This is due to the reminiscent effects of the coupling, which results in differences between $H_{0}$ and $H_{\rm eff}$(see Eq. (Adiabatic Thermal Radiation Pumps for Thermal Photonics)). Nevertheless, there can be appropriate choices of the control parameters $d,\epsilon$ for which these reminiscent effects disappears completely and the extrema of $\bar{\mathcal{I}}$ occurs at $d=0$, see Fig. 2b.

Adiabatic pumping using Circuits– An experimental demonstration of the effects of DPs on the adiabatically pumped thermal radiation, can be achieved using the electrical circuit shown in Fig. 1b. The system consists of a chain of three LC resonators with resonant frequency $\omega_{0}=1/\sqrt{LC}$. The resonators are capacitively coupled along the chain via capacitances $\lambda C$. Moreover, the first and last resonators are coupled through a mutual inductance $M=\mu L$, leading to system coupling similar to Eq. 8. The circuit is connected in series to two terminal capacitors $C_{e}=\epsilon C$ which we initially consider as grounded, i.e. $V_{L(R)}=0$. For concreteness, we will set $\omega_{0}/2\pi=1GHz$, $\lambda=0.2$, $\epsilon=0.1$, and $z_{0}=\omega_{0}L=1/\omega_{0}C=70$ Ohm. When $\mu=\mu_{\rm DP}\approx 0.13$, the eigen-frequencies of the grounded system demonstrate a DP degeneracy (see Supplement supl ).

We now turn the system into a scattering set-up by coupling each terminal capacitor $C_{e}$ to independent reservoirs at the same temperature $T$. These reservoirs are represented by a model for bandwidth limited Thevenin equivalent TEM transmission lines with characteristic impedance $Z_{0}=50$ Ohms. The noise sources $V_{n}$ are synthesized such that $\langle V_{n}(\omega)V^{*}_{m}(\omega^{\prime})\rangle=\frac{2Z_{0}}{\pi}\Phi(\omega)\delta(\omega-\omega^{\prime})\delta_{nm}$ where $\Phi(\omega)=k_{B}T\Theta(\omega)$. For demonstration purposes, we set $\Theta(\omega)=\sqrt{1-\left(\omega\over 2\omega_{c}\right)^{2}}$, with $\omega_{c}\approx 0.47$ GHz.

The pumping scheme is chosen to always enclose the DP when $\mu=\mu_{DP}$. Specifically, we consider a periodic modulation of the capacitances at left and right resonators such that $C_{1}(t)=C\left(1+r\sin(\Omega t)\right),\quad C_{2}(t)=C\left(1+r\cos(\Omega t)\right)$.

Next we inject into the circuit uncorrelated incoming waves of the same frequency $\omega$ and power $P_{s}=V_{s}^{2}/(8Z_{0})$ from the left (L) and right (R) reservoir. The average (over a cycle) net power flowing through the node $L(R$) is obtained from the voltage $v_{L(R)}(t)$ and current $i_{L(R)}(t)$ sampled at the respective node $L(R)$ (see Supplement supl ). Specifically, $Q(\omega)$ is evaluated using Eq. (4) where the time-dependent energy current is ${\cal I}_{L(R)}(t,\omega)=v_{L(R)}(t,\omega)i_{L(R)}(t,\omega)/P_{s}$. In our simulations, we made sure the system reached a stationary state before the evaluation. The results from the time domain simulations are shown in Fig. 3a together with the outcome from the instantaneous $S^{t}$-matrix approach, see Eq. (6).NGspice In the latter case we have extracted the instantaneous reflectance $R^{t}$ and reflection phase $\alpha^{t}$ using a standard scattering approach (see Supplement supl ) LSEK12 .

Having at our disposal the total radiative energy density $Q(\omega)$ for the circuit set-up, we are now able to incorporate $\Phi(\omega)$ for the total radiative thermal energy flux (per pumping area) passing through the system, $\bar{\cal I}$ versus $\mu$. In Fig. 3b we report our findings using the instantaneous $S^{t}$ matrix and the direct time-domain approaches. The data nicely demonstrates the enhancement in ${\cal I}$ due to the presence of the diabolic point as expected from the predictions of CMT.

*Conclusions – * We have introduced the concept of adiabatic thermal radiation pumps as a means to manage the direction of net radiative energy current for bodies in equilibrium. We addressed this problem by appropriately adopting, and establishing in the framework of resonant near-field thermal radiation, an instantaneous scattering matrix formalism borrowed from mesoscopic condensed matter. Using this tool, we highlighted the importance of wave interference effects in the field of thermal photonics for thermal radiation management. As an example, we demonstrated the impact of diabolic point spectral singularities in such framework. Our results have been tested against realistic simulations using electronic circuits, with the techniques directly applicable to systems where temporal control of resonant coupling is possible. An exciting application of our proposal might involve tunable superconducting resonators AKD16 ; BHG16 which will enable new forms of superconducting Q-bit manipulation. A future interesting direction is the study of the full counting statistics for thermal radiation. It will also be interesting to extend this study to other types of spectral singularities, like exceptional points CLEK17 . These questions will be addressed in a subsequent publication.

Acknowledgements –(HL, LJFA, TK) acknowledge partial support by an ONR Grant No. N00014-16-1-2803, by a DARPA NLM program via Grant No. HR00111820042, by an AFOSR Grant No. FA 9550-14-1-0037, and by a NSF Grant No. EFMA-1641109. (BS) acknowledges the hospitality of Wesleyan Univ. where this work has been performed.