Programmable Thermal Metamaterials Based on Optomechanical Systems

Zhengyang Zhou,^1,2,* Xiangying Shen,^1,3,* Chenchao Fang³ and Jiping Huang^1,*

¹ Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200433, China
² Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³ Department of Physics, Chinese University of Hong Kong, Hong Kong 999077, China

* Email: zhengyang.zhou@riken.jp (Z. Hou); xiangyingshen@cuhk.edu.hk (X. Shen); jphuang@fudan.edu.cn (J. Huang) 

Abstract

Controlling heat flow is crucial to improving human's production efficiency and daily life. Although thermal metamaterials have played important roles in heat manipulations, as the growing demands of intelligent and multi-functional apparatuses, the requirements of more advanced metamaterials are still far from being satisfactory. Moreover, the popular thermal metamaterials such as cloak, concentrator, rotator, etc., are essentially based on particularly spatial distributions of thermal conductivity. Hence, the designs of thermal meta devices with certain functions are greatly restrained by the positive and definite conductivities of natural materials. In this article, we propose an approach to building thermal metamaterial by utilizing the optomechanical system as an elemental unit. After deriving the relationship between thermal conductivities and pump light in one such unit, we can program the value of conductivity at an arbitrary point of an array system consisting of optomechanical cells by directly adjusting the pump power. This characteristic ensures great freedoms and flexibilities in designing metamaterials and permits people achieving any effects of existing thermal devices by using only one array system.

Table of Content

A thermal metadevice is proposed by utilizing the optpmechanical system, which can assemble multiple function to one device.

 

 

 

 

Keywords: Optomechanical systems; Thermal metamaterial; Adjustable thermal conductivities.

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1. Introduction

Manipulating heat flow is significant in energy saving, chips cooling, thermoelectric effects, and industrial manufacturing. Thermal metamaterials have been proved to be powerful tools in controlling heat flow at will.^1–21 This sort of artificial materials are able to show some unusual behaviors that can not be found in natural materials. Nowadays,advanced applications in prosthetics, soft deformable material, and wearable technology have become more and more popular. Demands of programmable materials with intelligence²² and multifunctions are urgently needed in thermal energy storage²³ and serving as building blocks of that fantastic implementations.²⁴ However, due to the architecture and thermal conductivity are crucial to metamaterials’properties and functionalities, different types of thermal devices always have different geometrical structures and conductivity tensors. As a result, there exist large gaps between different implements. Essentially, most of the popular thermal meta devices are constructed by tailoring the spatial distribution of the devices’ thermal conductivities, which is a positive and definite value for certain materials. Although we can combine different materials to fit desired conductivity under the instruction of effective medium approximation (EMA) theory, the upper and lower bound of the fitted conductivity are constrained to the highest and lowest conductivities of chosen materials respectively, which also restrict the freedom of designing devices. Moreover, for those thermal meta devices that emphasize noninvasively, i.e., the existence of the device makes no distortions to the temperature gradient, the effective conductivities of such metamate-rials should match with the one of background, which also narrows the application potential due to the different environments require different constituents to build the device. Therefore, an improved adjustable and pro-grammable multi-functional thermal metamaterial with a wide range of tunable thermal conductivity is of great value to be investigated.

Fortunately, several experiments have shown that the microstructures can change the thermal conductivity by manipulating the phonon modes.^{25, 26} Inspired by these results, we investigate the possibility of construct-ing adjustable and multi-functional metamaterial with optomechanical systems driven by a light pump. Besides the structures of the materials, the electromagnetic field can also influence the phonon modes. For example, optomechanical systems are such kind of devices that can change the phonon modes of oscillators with light²⁷ and are usually formed by oscillating mirrors(the oscillators) coupled to light field. The amplitude (average phonon number) of the mechanical oscillators can either be increased²⁸ or decreased²⁹ by the light, and the rigidity of the mechanical oscillator³⁰ is also possible to be adjusted. Thus, by changing the phonon modes, the thermal conductivity of the optomechanical systems can vary accordingly. In this article, we will first show that changing the phonon modes in optomechanical systems can be an effective way to adjust the thermal conductivity of thermal metamaterial. Then, based on optomechanical systems, we will design a new practical method to build programmable thermal metamaterials, whose functions (as mentioned before, functions are always defined by the thermal conductivities) can be controlled by the intensity of the pump light.

 

2. Thermal conductivity of single device

The thermal conducting unit we propose is an optome-chanical system with two mechanical modes coupled to a common optical mode. This system can be realized with an optical cavity formed by two movable mirrors as shown in Fig. 1. The displacement of the mirror will change the frequency of the cavity, which results in the coupling be-tween optical modes and mechanical modes.27 In this case, the heat flux through the device is produced by the phonon exchanges between the two mechanical oscilla-tors. The Hamiltonian of the cavity can be expressed as

                (1)

where ω is the frequency of the cavity mode which de-pends on the displacement of the two mechanical oscil-lators; x_i , i = 1, 2 is the i displacement of the mechanical oscillator i ; a† (a) is the bosonic creation (annihilation) operator of the cavity optical mode. The photon number a†a is a conserved quantity of the Hamiltonian H in Eq. (1).  So we can substitute the photon number operator a†a with the average photon number n_ph . The photon number, which is determined by the pump intensity and the decay of the cavity, can be calculated with

               (2)

where F is the finesse of the cavity, P is the pump power, and Δω_FSR is the free spectral range of the cavity.²⁷ Then we expand the Hamiltonian in Eq. (1) to the second order of the oscillator displacement.

  (3)

The first five terms, which change the equilibrium positions or the frequencies of the oscillators, can be absorbed into the parameters of the oscillators. Only the last term in Eq. (3) contributes to the coupling between the two mechanical oscillators. As a result, we focus on the last term, and obtain the following Hamiltonian 

where we define the second derivative of cavity frequency ${\omega }^{\left(2\right)}\equiv \frac{{\vartheta }^2}{\vartheta x_1\vartheta x_2}\omega$. Now we quantize the mechanical oscillator. To make the case easy, we consider the single mode situation.

(4)

The Ω_m and H_m here are the frequencies and Hamiltonian of the m m mechanical oscillators, respectively. ${b_i}^{+}\left(b_i\right)$ is the bosonic creation (annihilation) operator of the phonon mode (i =1,2). The x_ZPF is the ZPF zero-point fluc-tuation amplitude of the mechanical oscillator.

Fig. 1 The illustration of the thermal conducting device. The cavity in our scheme has two movable mirrors. These two mirrors are connected to two identical mechanical oscillators. Both mechanical modes are coupled to a thermal bath indicated by the red and blue cuboid representing the heat and cold source respectively. The intensity of the light in the cavity is controlled by the pump intensity.

 

with m_eff the effective mass of the mechanical oscillator.²⁷ Under eff rotating wave approximation, the Hamiltonian of the coupling between two mechanical os-cillators in Eq. (4) becomes

(5)

where we have defined the coupling strength between the two mechanical modes as $g\equiv n_{ph}{\omega }^{\left(2\right)}{x_{ZPF}}^2$. Now, we come to the dynamics of the  mechanical modes. From Heisenberg equation we get

(6)

The energy of the mechanical mode,$E=h{\Omega }_m({b_i}^{+},b_i)$ with i=1,2, described by the product of the eigenenergy of mode and the phonon number. Thus we can get the heat flow from the time derivative of the number operator $\frac{d}{dt}{b_i}^{+}{b}_i$ in Eq. (6). Note that Eq. (6) ensures which $\frac{d}{dt}{b_1}^{+}{b}_1=-\frac{d}{dt}{b_2}^{+}{b}_2$ is the conservation of the energy. As a result, we can focus on the equation of $\frac{d}{dt}{b_1}^{+}{b}_1$ in the following derivation. The annihilation operators can be formally solved as

(7)

With the formal solution Eq. (7), the time derivative of the number operators can be expressed as

(8)

Then, we assume the coupling between the mechanical modes and the heat bathes to be strong, so that both mechanical oscillators can achieve thermal equilibrium with the bathes in a very short time. Thus we have  Here, is the average phonon number at temperature T, the quanty T describes the temperature of the bath i coupled to mechanical mode i, and (b_i⁺(t)b_j(s)) is the expectation value of operator b_i⁺(t)b_j(s). Then, the expectation value of Eq. (8) can be estimated,

Equation (9) provides the average number of the phonon that leaves the mechanical mode 1 per unit time. These phonons go to the mode 2 since we have shown $\frac{d}{dt}{b_1}^{+}{b}_1=-\frac{d}{dt}{b_2}^{+}{b}_2$. Note that the right side of Eq. (9) has an additional dimension of time, so that the both sides have the same dimension. This is the result of the relation ${\int }_0^{t}ds{e}^{-i{\Omega }_m(t-s)}\delta (t-s)-n\left(T_i\right)=\overline{n}\left(T_i\right)$. The energy flux carried by phonon is

Where ΔT ≡ (T₁ − T₂ ) is the temperature difference between the two bathes. Eq. (10) describes the rela-tion between the heat flux through the single device and the temperature difference in the two sides of the device. This relation has the same form as an ordinary object, but the coupling strength g can be changed by the inten-sity of the pump light. If we turn off the light, the heat flux through the device can be shut down. In the high temperature limit k_BT$\gg$hΩ_m, Eq. (10) becomes a simpler form

(11)

Regularly, thermal metamaterials work at room temperature. Thus, it always satisfies k_BT$\gg$hΩ_m. There-fore, in the rest of this article, we will always use the high-temperature limit. In a word, this simple optome-chanical system is the elemental unit of our adjustable thermal metamaterial just like the split-ring resonators (SRRs) adopted in the electro-magnetic cloak case.

 

3. Thermal conductivity tuned by array of ptomechanical systems

We can arrange the optomechanical systems to form an array as shown n Fig. 2. Considering a two-dimensional array, the squares in the figure epresent the heat bathes, which provide the local temperatures at ifferent points. The thermal energy can conduct among those squares hrough the optomechanical systems connecting them. ow, we are in a position to show that the new array system can ork as controllable thermal metamaterial, and to derive the Fourier hermal conduction equation of it. Let’s assume the distance between he nearby squares is L in both X and Y direction. We first consider the eat flux density in X direction. Suppose there is a line with length 1 in  direction at coordinate (x, y), the number of the optomechanical ystems that carry heat flux through it is $\frac{1}{L}$. If the heat flux carried by ach optomechanical Lsystem is Φ_x then the flux density in X direction t that point should be

 

Here, Δ T_x(x,y) is the temperature difference between two closest bathes n X direction. The coupling strength g_x(x,y) can be adjusted by the ump light on the op-tomechanical system at （x,y） in X direction. We an approximate the quantity $\frac{△T_x(x,y)}{L}$with the derivative $\frac{\partial }{\partial x}T(x,y)$,so that the heat flux density in X direction can be expressed as $J_X(x,y)=-k_{B}g{x}^2(x,y)\frac{\partial }{\partial x}T(x,y)$ In the same way we can get the expression for the heat flux density in Y direction, $J_X(x,y)=-k_{B}g{x}^2(x,y)\frac{\partial }{\partial y}T(x,y)$, With the heat flux density in both direction, the Fourier thermal conduction equation for the array can be obtained, 

(12)

where Pi(x,y) is the pump power on the optomechanical system at (x,y) in I direction i=x,y. We can find from Eq. (12) that the array of optomechanical systems has the thermal conductivity related to the square of the pump power. If we choose an uniform pump power i n some area, this array has the