A Versatile Electronic Dimer Exhibiting PT and Anti-PT Symmetry

Abstract

We propose a versatile electronic dimer cooperatively coupled by means of mutual induction, capacitance, and resistance. In a lot of related works, the electronic dimer is inductively coupled, with one resonator characterized by positive resistance (dissipation) and the other by negative resistance (amplification). We go beyond this picture by considering capacitive and resistive coupling, and by exploring cases where both resistances are positive, as well as a case where the resonant frequencies of the individual resonators are different. Based on analytical derivation and numerical calculations, we obtain and observe the properties of parity-time (PT), quasi-PT (QPT) and quasi-anti-PT (QAPT) symmetry by adjusting the constitutive parameters of the system. This study provides a versatile and feasible platform for observing PT/anti-PT (APT) symmetry-based phenomena and provides a foundation for further studies on finding PT/APT symmetry in more sophisticated circuits.

1. Introduction

The recently emerging parity-time (PT) symmetry, transplanted from quantum systems [1,2] into classical wave systems, has prompted comprehensive investigations to seek novel phenomena and applications. In quantum mechanics, a PT-symmetric system consists of two subsystems, among which one has an outward current density of probability while the other has an inward current density of probability of an equal value. The Hamiltonian of the system H and the joint operator PT commute: [PT, H] = 0, where P is the parity operator while T is the time reversal operator. In classical wave systems, the outward and inward current density of probability was mapped to the loss and gain. Taking optics for example, the loss and gain of the system is represented by a complex conjugate pair of permittivity, i.e., $\epsilon (-r)={\epsilon }^{\ast }(r)$ [3,4]. The most commonly investigated phenomenon is the phase transition from an unbroken PT phase to a broken PT symmetry phase, which occurs at the so-called exceptional points (EPs). Across the EPs, the energy eigenvalue changes from two real values into a complex conjugate pair, resulting in a pair of exponentially decaying and amplifying eigenmodes [4]. Due to the phase transition, a lot of novel phenomena have been observed—for example, exotic lasing [5,6], unidirectional invisibility [7], optical isolation and nonreciprocity [8], robust wireless power transfer [9], and so on. Hot topics also include PT-symmetric waveguides and asymmetric power oscillations [10,11] and PT symmetry breaking in scattering systems, which generates the so-called coherent perfect absorption and lasing (CPAL) point [12,13]. In addition, sensing at exceptional points has become another important application of PT-symmetric structures [14]. Differing from frequency splitting close to a diabolic point (DP), frequency splitting close to an EP is more sensitive. PT optomechanics has also promoted the study of the mechanisms of interaction between optical fields and mechanical motion. New topics in this area include phonon lasing and optomechanically induced transparency [15]. In addition to EPs, exceptional surfaces have also recently been observed [16].

A similar but quite different concept is the anti-PT (APT) symmetry. In an APT-symmetric system, the Hamiltonian anti-commutes with the PT operator: {PT, H} = 0 [14]. An APT-symmetric dimer commonly comprises two equally amplifying oscillators at energy levels of equal value but opposite signs. Related novel phenomena include the spontaneous phase transition of the scattering matrix, a total transmission band, and a continuous lasing spectrum [17]. Specifically, Peng et al. conducted the first experimental demonstration of optical APT symmetry in a warm atomic-vapor cell [18]. Ge et al. proposed photonic structures with balanced positive- and negative-index materials [19], Yang et al. realized APT symmetry in dissipatively coupled systems comprising three optical waveguides [20].

An electronic circuit is a good platform for studying non-Hermitian physics, especially PT and APT symmetry. The electronic dimer is a basic candidate among various prototypes. Schindler et al. investigated a PT-symmetric electronic dimer with mutual inductance coupling and capacitive coupling [21,22] and conducted an experimental study. Choi et al. and Yang et al. independently studied resistively coupled LRC circuits [17,23]. Choi focused on the observation of exceptional point and energy-difference conserving dynamics, while Yang investigated the coherent-perfect-absorber and laser (CPAL) point at which the zero and pole of the scattering matrix S coexist at the same real frequency, both in the APT-symmetric scenario. Tabeu et al. investigated the PT-symmetric electronic dimer with an imaginary resistor, coupled using an ZC combination (Z stands for an imaginary resistor) [24]. Zhou et al. observed perturbed eigenvalues of a PT-symmetric resonator system where the dimer is coupled via mutual induction [25]. Wang et al. observed the two PT transitions in a similar configuration [26]. Tagouegni et al. proposed a PT-symmetric dimer without a gain material, which is coupled using an imaginary resistor and a capacitor [27]. Importantly, Cao et al. realized a fully integrated PT-symmetric dimer with capacitive coupling, which promoted the development of integrated circuit technology [28]. Above, we present a few representative works on electronic dimers, but this is not a complete collection. These proposed dimers are coupled either by mutual induction only, by a capacitor only, or by a pair comprising a resistor and capacitor. It is still necessary to analyze a versatile model which incorporates all kinds of coupling channels.

2. Model and Description

We propose a generic and versatile model which incorporates all the coupling channels into an LRC dimer, including mutual induction M, a resistor R_c, and a capacitor C_c. The electronic parameters of the LRC resonators are denoted as inductors L_i, capacitors C_i, and resistors R_i (i = 1,2). The sketch of the prototype is shown in Figure 1.

The node voltages V₁ and V₂ and currents I₁ and I₂ satisfy the following equations:

$$I_1+{\displaystyle \frac{V_1}{R_1}}+C_1{\displaystyle \frac{\mathrm{d}{V}_1}{\mathrm{d}t}}+C_{\mathrm{c}}{\displaystyle \frac{\mathrm{d}(V_1-V_2)}{\mathrm{d}t}}+{\displaystyle \frac{V_1-V_2}{R_{\mathrm{c}}}}=0$$

(1)

$$I_2+{\displaystyle \frac{V_2}{R_2}}+C_2{\displaystyle \frac{\mathrm{d}{V}_2}{\mathrm{d}t}}+C_{\mathrm{c}}{\displaystyle \frac{\mathrm{d}(V_2-V_1)}{\mathrm{d}t}}+{\displaystyle \frac{V_2-V_1}{R_{\mathrm{c}}}}=0$$

(2)

Here, $V_1=L_1{\displaystyle \frac{\mathrm{d}{I}_1}{\mathrm{d}t}}+M{\displaystyle \frac{\mathrm{d}{I}_2}{\mathrm{d}t}}$ and $V_2=L_2{\displaystyle \frac{\mathrm{d}{I}_2}{\mathrm{d}t}}+M{\displaystyle \frac{\mathrm{d}{I}_1}{\mathrm{d}t}}$. Note that we add resistive coupling according to [21]. After some algebraic manipulations, we reach the following matrix equation (assuming the time harmonic factor is ${\mathrm{e}}^{\mathrm{i}\omega t}$):

$$\left(\begin{array}{cc}{\displaystyle \frac{L_2}{\mathrm{i}\omega (L_1{L}_2-M^2)}}+{\displaystyle \frac{1}{R_1}}+\mathrm{i}\omega (C_1+C_{\mathrm{c}})+{\displaystyle \frac{1}{R_{\mathrm{c}}}}& -{\displaystyle \frac{M}{\mathrm{i}\omega (L_1{L}_2-M^2)}}-\mathrm{i}\omega C_{\mathrm{c}}-{\displaystyle \frac{1}{R_{\mathrm{c}}}}\\ -{\displaystyle \frac{M}{\mathrm{i}\omega (L_1{L}_2-M^2)}}-\mathrm{i}\omega C_{\mathrm{c}}-{\displaystyle \frac{1}{R_{\mathrm{c}}}}& {\displaystyle \frac{L_1}{\mathrm{i}\omega (L_1{L}_2-M^2)}}+{\displaystyle \frac{1}{R_2}}+\mathrm{i}\omega (C_2+C_{\mathrm{c}})+{\displaystyle \frac{1}{R_{\mathrm{c}}}}\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=0$$

(3)

By introducing the factors $c_i=C_{\mathrm{c}}/C_i$, $m_i=M/L_i$, ${\gamma }_i=R_i^{-1}{C}_i^{-1}$, ${\gamma }_{\mathrm{c}i}=R_{\mathrm{c}}^{-1}{C}_i^{-1}$ and ${\omega }_i^2=L_i^{-1}{C_i}^{-1}$ (i = 1, 2), Equation (3) can be recast as

$$\left(\begin{array}{cc}{\displaystyle \frac{{\omega }_1^2}{\mathrm{i}\omega (1-m_1{m}_2)}}+{\gamma }_1+\mathrm{i}\omega (1+c_1)+{\gamma }_{\mathrm{c}1}& -{\displaystyle \frac{{\omega }_1^2{m}_2}{\mathrm{i}\omega (1-m_1{m}_2)}}-\mathrm{i}\omega c_1-{\gamma }_{\mathrm{c}1}\\ -{\displaystyle \frac{{\omega }_2^2{m}_1}{\mathrm{i}\omega (1-m_1{m}_2)}}-\mathrm{i}\omega c_2-{\gamma }_{\mathrm{c}2}& {\displaystyle \frac{{\omega }_2^2}{\mathrm{i}\omega (1-m_1{m}_2)}}+{\gamma }_2+\mathrm{i}\omega (1+c_2)+{\gamma }_{\mathrm{c}2}\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=0.$$

(4)

The matrix in Equation (4), denoted as N, describes the full dynamics of the generic dimer, but is generally neither PT-symmetric nor APT-symmetric. In the following, we attempt to adjust the parameters to reach PT or APT symmetry based on Equation (4).

2.1. PT Symmetry

For a PT-symmetric dimer, the matrix N should satisfy PT N (PT)⁻¹ = N. Here, the P operator is represented by the Pauli matrix ${\sigma }_x$, which reverses the system’s orientation, while the T operator stands for complex conjugation. Imposing the requirement for PT symmetry on the matrix N, we found it necessary to add additional conditions for the matrix N to reach PT symmetry. These conditions are as follows: ${\omega }_1^2={\omega }_2^2={\omega }_0^2$, ${\gamma }_2=-{\gamma }_1={\gamma }_0>0$, $c_1=c_2=c_0$, $m_1=m_2=m_0$ and ${\gamma }_{\mathrm{c}1}={\gamma }_{\mathrm{c}2}=0$. With the above conditions imposed, the matrix N reaches its PT-symmetric form:

$$N_{\mathrm{PT}}=\left(\begin{array}{cc}{\displaystyle \frac{1}{\tilde{\omega }(1-m_0^2)}}-\tilde{\omega }(1+c_0)-\mathrm{i}{\tilde{\gamma }}_0& -{\displaystyle \frac{m_0}{\tilde{\omega }(1-m_0^2)}}+\tilde{\omega }{c}_0\\ -{\displaystyle \frac{m_0}{\tilde{\omega }(1-m_0^2)}}+\tilde{\omega }{c}_0& {\displaystyle \frac{1}{\tilde{\omega }(1-m_0^2)}}-\tilde{\omega }(1+c_0)+\mathrm{i}{\tilde{\gamma }}_0\end{array}\right).$$

(5)

Here, ${\tilde{\gamma }}_0={\gamma }_0/{\omega }_0$, $\tilde{\omega }=\omega /{\omega }_0$. Then, Equation (4) becomes $N_{\mathrm{PT}}V=0$, where $V={(V_1,V_2)}^{\mathrm{T}}$. The equation for calculating eigenfrequencies is as follows:

$${\left[{\displaystyle \frac{1}{\tilde{\omega }(1-m_0^2)}}-\tilde{\omega }(1+c_0)\right]}^2+{{\tilde{\gamma }}_0}^2-{\left[-{\displaystyle \frac{m_0}{\tilde{\omega }(1-m_0^2)}}+\tilde{\omega }{c}_0\right]}^2=0.$$

(6)

Next, we numerically investigate Equation (6). Firstly, we set m₀ = 0.1 and $c_0=0.5$ and plot the dependence of eigenfrequency on the dissipation factor ${\tilde{\gamma }}_0$ in Figure 2. Apparently, Equation (6) has four solutions. We choose the two solutions with a positive real part without loss of generality. It is clearly observed that the figure can be divided into three regions. The first region ranges from ${\tilde{\gamma }}_0=0$ to ${\tilde{\gamma }}_0\approx 0.3$, where the system is in the unbroken PT phase. The eigenfrequencies are a pair of real values with vanishing imaginary parts. It can be seen that $\mathrm{Re}(\tilde{\omega })<1$, indicating that the eigenfrequency is always below the resonant frequency of an independent LRC resonator due to the dimer coupling. The second region ranges from ${\tilde{\gamma }}_0\approx 0.3$ to ${\tilde{\gamma }}_0\approx 2.4$. In this region, the system is in the PT broken phase. The two branches of the real part of $\tilde{\omega }$ merge into one line at the exceptional point at ${\tilde{\gamma }}_0\approx 0.3$. The imaginary part bifurcates into complex conjugate pairs, which is an obvious phenomenon of PT symmetry breaking. The third region starts at ${\tilde{\gamma }}_0\approx 2.4$. In this region, the real part of $\tilde{\omega }$ vanishes. The imaginary part of the upper branch in the second region bifurcates again into two branches. Then, the eigenfrequencies turn into pure imaginary positive values, which indicates a pure decaying process of both eigenmodes. From the above analysis, we can distinguish these dimers from common dimers with two coupling oscillating resonators, where only two different regions usually exist. It should be noted that the condition ${\gamma }_2=-{\gamma }_1={\gamma }_0>0$ requires negative resistance in one of the LRC resonators, which can be reached using an operational amplifier [21].

2.2. Quasi-PT Symmetry

Rigorous PT symmetry requires one resonator to dissipate while the other amplifies. An amplifying resonator is not easily achievable, usually requiring operational amplifiers, as mentioned above. However, by adjusting the system parameters, we do not need an amplifying resonator, instead only needing a pair of damping oscillators to reach PT symmetry. In this scenario, we realize the so-called quasi-PT symmetry (QPT) by imposing a transformation on the state vector ${(V_1,V_2)}^{\mathrm{T}}$. Firstly, let us impose the following assumptions on the model in Equation (4): ${\omega }_1^2={\omega }_2^2={\omega }_0^2$, ${\gamma }_{1\mathrm{c}}={\gamma }_{2\mathrm{c}}=0$, ${\gamma }_2\ne {\gamma }_1$, ${\gamma }_1>0$, ${\gamma }_2>0$, and $c_1=c_2=c_0$. The conditions ${\gamma }_1>0$ and ${\gamma }_2>0$ imply that only positive resistance is involved. Introducing the variable ${\Omega }_0^2={\omega }_0^2/(1+c_0)$ and assuming ${\Omega }_0^2/\omega -\omega \approx 2({\Omega }_0-\omega )$, Equation (4) is reduced to

$$\left(\begin{array}{cc}({\Omega }_0-\omega )+\mathrm{i}{\Gamma }_1& \kappa \\ \kappa & ({\Omega }_0-\omega )+\mathrm{i}{\Gamma }_2\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=0.$$

(7)

Here, $\kappa ={\omega }_0{\displaystyle \frac{c_0-m_0}{2(1+c_0)}}$, ${\Gamma }_i={\displaystyle \frac{{\gamma }_i}{2(1+c_0)}}$ i = 1, 2. Therefore, the Hamiltonian H can be defined as

$$H=\left(\begin{array}{cc}{\Omega }_0+\mathrm{i}{\Gamma }_1& \kappa \\ \kappa & {\Omega }_0+\mathrm{i}{\Gamma }_2\end{array}\right).$$

(8)

The eigenvalue equation now reads as $HV=\omega V$. Introducing ${({V^{\prime }}_1,{V^{\prime }}_2)}^{\mathrm{T}}={\mathrm{e}}^{({\Gamma }_1+{\Gamma }_2)t/2}{(V_1,V_2)}^{\mathrm{T}}$, the eigenvalue equation can be written as $H_{\mathrm{QPT}}{V}^{\prime }={\omega }^{\prime }{V}^{\prime }$, where the transformed Hamiltonian H_QPT is PT-symmetric:

$$H_{\mathrm{QPT}}=\left(\begin{array}{cc}{\Omega }_0+\mathrm{i}({\Gamma }_1-{\Gamma }_2)/2& \kappa \\ \kappa & {\Omega }_0-\mathrm{i}({\Gamma }_1-{\Gamma }_2)/2\end{array}\right).$$

(9)

The eigenvalue ${\omega }^{\prime }$ can be derived from the equation det $\left|{\omega }^{\prime }I-H_{\mathrm{QPT}}\right|$ = 0 and the final results can be written as ${\omega }^{\prime }={\Omega }_0\pm \sqrt{{\kappa }^2-{\left[({\Gamma }_1-{\Gamma }_2)/2\right]}^2}$. Rewriting the eigenfrequency in a dimensionless form,

$${\tilde{\omega }}^{\prime }={\displaystyle \frac{{\omega }^{\prime }}{{\omega }_0}}={\displaystyle \frac{1}{\sqrt{1+c_0}}}\pm {\displaystyle \frac{1}{2(1+c_0)}}\sqrt{c_0-m_0-{\displaystyle \frac{1}{4}}{({\tilde{\gamma }}_1-{\tilde{\gamma }}_2)}^2}.$$

(10)

Here, ${\tilde{\gamma }}_i={\gamma }_i/{\omega }_0$, i = 1, 2. We set $c_0=0.5$ and m₀ = 0.1 and plot ${\tilde{\omega }}^{\prime }$ vs. $\left|{\tilde{\gamma }}_1-{\tilde{\gamma }}_2\right|$ in Figure 3.

2.3. Quasi-APT (QAPT) Symmetry

Rigorous APT symmetry requires two equally amplifying oscillators with energy levels of the same value but opposite signs. It is not feasible to directly implement negative energy in the Hamiltonian. Here, we achieve QAPT with the proposed electronic dimer. By setting $c_i=0$, Equation (4) becomes

$$\left(\begin{array}{cc}{\displaystyle \frac{{\omega }_1^2}{\mathrm{i}\omega (1-m_1{m}_2)}}+{\gamma }_1+\mathrm{i}\omega +{\gamma }_{\mathrm{c}1}& -{\displaystyle \frac{{\omega }_1^2{m}_2}{\mathrm{i}\omega (1-m_1{m}_2)}}-{\gamma }_{\mathrm{c}1}\\ -{\displaystyle \frac{{\omega }_2^2{m}_1}{\mathrm{i}\omega (1-m_1{m}_2)}}-{\gamma }_{\mathrm{c}2}& {\displaystyle \frac{{\omega }_2^2}{\mathrm{i}\omega (1-m_1{m}_2)}}+{\gamma }_2+\mathrm{i}\omega +{\gamma }_{\mathrm{c}2}\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=0.$$

(11)

Assuming that m₁ and m₂ are very small, such that $1-m_1{m}_2\approx 1$, and using the approximation ${\omega }_i^2/\omega -\omega \approx 2({\omega }_i-\omega )$, i = 1, 2. Equation (11) can be approximated as

$$\left(\begin{array}{cc}({\omega }_1-\omega )+\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}& -\mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}\\ -\mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}& ({\omega }_2-\omega )+\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=0$$

(12)

Here, we have additionally supposed that ${\gamma }_{\mathrm{c}1}={\gamma }_{\mathrm{c}2}={\gamma }_{\mathrm{c}0}$, ${\gamma }_2={\gamma }_1={\gamma }_0$. Rewriting Equation (12) in a form like Schrodinger’s Equation [14], Equation (12) can be denoted as

$$\left(\begin{array}{cc}-{\omega }_1-\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}& \mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}\\ \mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}& -{\omega }_2-\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}\end{array}\right)\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)=\mathrm{i}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\begin{array}{c}{V}_1\\ V_2\end{array}\right)$$

(13)

Note that the coupling coefficients in this Hamiltonian are purely imaginary. Comparing with a standard APT Hamiltonian, the natural frequencies ${\omega }_1$ and ${\omega }_2$ still do not satisfy ${\omega }_1=-{\omega }_2$. We conduct a transformation to the state vector ${(V_1,V_2)}^{\mathrm{T}}$, namely ${({V^{\prime }}_1,{V^{\prime }}_2)}^{\mathrm{T}}={\mathrm{e}}^{-\mathrm{i}{\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}t}{(V_1,V_2)}^{\mathrm{T}}$, which leads to the following dynamic equation with respect to the state vector ${({V^{\prime }}_1,{V^{\prime }}_2)}^{\mathrm{T}}$:

$$\left(\begin{array}{cc}{\displaystyle \frac{{\omega }_2-{\omega }_1}{2}}-\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}& \mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}\\ \mathrm{i}{\displaystyle \frac{{\gamma }_{\mathrm{c}0}}{2}}& -{\displaystyle \frac{{\omega }_2-{\omega }_1}{2}}-\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}\end{array}\right)\left(\begin{array}{c}{V^{\prime }}_1\\ {V^{\prime }}_2\end{array}\right)=\mathrm{i}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\begin{array}{c}{V^{\prime }}_1\\ {V^{\prime }}_2\end{array}\right)$$

(14)

Now, the Hamiltonian in Equation (14) has standard APT symmetry. The eigenvalue can be written as $\omega =\mathrm{i}{\displaystyle \frac{({\gamma }_0+{\gamma }_{\mathrm{c}0})}{2}}\pm {\displaystyle \frac{1}{2}}\sqrt{{({\omega }_2-{\omega }_1)}^2-{{\gamma }_{\mathrm{c}0}}^2}$. Using dimensionless variables, $\tilde{\omega }=\omega /\left|{\omega }_2-{\omega }_1\right|$, ${\tilde{\gamma }}_0={\gamma }_0/\left|{\omega }_2-{\omega }_1\right|$, ${\tilde{\gamma }}_{\mathrm{c}0}={\gamma }_{\mathrm{c}0}/\left|{\omega }_2-{\omega }_1\right|$, the eigenfrequency solution can be written as $\tilde{\omega }=\mathrm{i}{\displaystyle \frac{({\tilde{\gamma }}_0+{\tilde{\gamma }}_{\mathrm{c}0})}{2}}\pm {\displaystyle \frac{1}{2}}\sqrt{1-{{\tilde{\gamma }}_{\mathrm{c}0}}^2}$. Note that in the scenario of APT symmetry, which is different from PT symmetry, eigenfrequencies with the same imaginary part and two branches of real parts denote the broken symmetry phase [29]. In Figure 4, eigenfrequency vs. coupling coefficient ${\tilde{\gamma }}_{\mathrm{c}0}$ is plotted with ${\tilde{\gamma }}_0$ = 0.1. It can be observed that in the broken APT symmetry phase (from ${\tilde{\gamma }}_{\mathrm{c}0}$ = 0 to ${\tilde{\gamma }}_{\mathrm{c}0}$ = 1), the two eigenfrequencies share the same single imaginary part and possess real parts of opposite signs. In the APT symmetry phase, the real part of the two eigenvalues vanishes, resulting in a pair of purely imaginary values.

3. Conclusions

In summary, we have proposed a versatile prototype of an electronic dimer comprising a pair of LRC resonators coupled using all the three kinds of channels: mutual induction, capacitance, and resistance. Based on the proposed dimer, we achieved PT symmetry, QPT symmetry, and QAPT symmetry. This shows that the proposed electronic dimer is a versatile and feasible platform for observing PT/APT symmetry-related phenomena. Possible further study should be focused on PT/APT-symmetric electronic resonator chains which could exhibit more and richer novel physical phenomena.