Non-reciprocal transmission of coupled LC resonators through parity-time symmetry breaking

Non-reciprocal devices that allow a signal to be transmitted only in one direction are important for full-duplex communications. Due to the requirements of miniaturized systems, there has been an increase interest in non-magnetic non-reciprocal devices in recent years. Based on parity-time (PT) symmetric inductors-capacitors (LC) resonators, this paper has proposed non-reciprocal transmission configurations by PT-symmetry breaking. In the configuration, the coupled capacitance between the two coupled LC resonators can be adjusted so that the transmission frequency is tunable. At the same time, the resonant frequency and transmission frequency have been discriminated to optimize the non-reciprocal transmission. The configuration has been implemented by utilizing discrete components on a printed circuit board (PCB). It demonstrates that the center operation frequency of 14.05 MHz with the bandwidth 4 MHz, the insertion loss 0.32 dB, and the isolation 11 dB is adjusted to the center operation frequency of 14.95 MHz with the bandwidth 4.6 MHz, the insertion loss 0.716 dB, and the isolation 14.5 dB.


Introduction
Non-reciprocal components capable of one-way transmission, such as isolators and circulators, are a key for the readout of qubits in superconducting quantum computing applications [1,2] and modern communication technology [3]. However, the current magnetic non-reciprocal components are bulky and expensive. Due to the requirements of miniaturized systems, attempts have been made to achieve non-magnetic non-reciprocity in recent years. They are usually classified as active transistors or nonlinearity, temporal modulation of permittivity, temporal modulation of conductivity, and hybrid acoustic-electronic components [4]. For the non-reciprocal transmission based on the nonlinearity of devices and circuits, the spatial asymmetry causes signals incident from one port to enter the nonlinear region with greater amplitude, leading to more attenuation, while it causes signals from the other port to enter the nonlinear region with lesser amplitude, leading to less attenuation. Obviously, it is important to construct nonlinear circuits that exhibit spatial asymmetry. Two capacitively coupled LC resonators, one with gain and the other with equivalent loss, can form PT-symmetric circuits [5]. Depending upon the coupling relative to the loss, the PT-symmetric circuits have two distinguished regimes, i.e., an exact PT-symmetric regime with real frequencies and a broken PT-symmetric regime with complex-conjugate frequencies. Exceptional points (EPs) at which eigenfrequencies coalesce separate the exact regime from the broken regime. For the PT-symmetric circuits operating at the exact symmetric regime, unidirectional transparency for some characteristic frequencies has been observed [6]. For those operating at the EPs, the enhanced sensitivity of sensors constructed by LC resonators has been implemented [7][8][9]. Moreover, PT-symmetric direct electrical transmission lines show a phase transition from real to complex eigenvalues, and the emerging Anderson-like localized modes in the broken phase can be utilized in applications for controlling the flow of energy such as switching [10,11]. In particular, for those operating at the broken PT-symmetric regime, the complex-conjugate frequencies behave with the same real eigenfrequencies but the opposite signs of the imaginary parts, leading to oscillations magnitudes with an exponentially growing mode and an Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. exponentially decaying mode. Due to the nonlinear gain of the PT-symmetric circuits, the exponentially growing mode eventually enters the stable oscillations. By using this phenomenon, non-reciprocal transmission has been attempted for acoustic wave [12], electronic signals [13], and optic wave [14][15][16][17]. However, such a nonreciprocal transmission is operated at a fixed frequency. Signal transmission in different frequency bands is usually desired in communications. Therefore, tunable nonreciprocal systems needs to be explored. Based on the variable coupled capacitor of the broken PT-symmetric LC resonators, this paper presents a nonreciprocal transmission system with tunable frequency. In section 2, the operation principle for PT-symmetric LC resonators is given. The forward transmission coefficient and the backward transmission coefficient are derived. In section 3, using the ADS (Advanced Design System) software, the negative resistance providing gain, implemented by an operational amplifier and resistors, is simulated. The non-reciprocal ratio as a function of input power is simulated and optimized for different coupling coefficients. In section 4, experiments are performed by utilizing discrete components on a printed circuit board (PCB), measurement results and discussions are then given. Finally, some conclusions are made. Figure 1 shows the PT-symmetric LC system [5]. It consists of a pair of capacitively coupled LC resonators, one with gain and the other with loss, attached to transmission line (TL) leads with characteristic impedance Z 0 . R L , L L , and C L are the resistance, inductance and capacitance of the loss side, respectively. R G , L G , and C G are the negative resistance, inductance and capacitance of the gain side, respectively. The negative resistance providing gain is implemented by an operational amplifier and resistors. C C is the coupled capacitance between the loss and gain resonators, which is designed here to be adjustable. The PT symmetry is satisfied by setting R G = R L = R, L G = L L = L and C G = C L = C.

Non-reciprocal transmission
Application of Kirchhoff's law to the coupled resonators of figure 1 yields the set of equations is the voltage of the loss (gain) side, and I L (I G ) is the current of the loss (gain) side, respectively. ω is the angular frequency. Eliminating the currents from the above equations, scaling frequency by Figure 1. The schematic diagram of the PT-symmetric LC system, where adjustable capacitive coupling (C C ) is used to connect two LC resonators, one with gain −R G and the other one with equal loss R L . The negative resistance is implemented by an operational amplifier and resistors. In the broken PT-symmetric regime, when the signal transits from the loss side to the gain side, the signal has less amplitude attenuation, while when the signal transits from the gain side to the loss side, the signal amplitude is greatly attenuated.
Higher transmission is achieved in the forward direction than the backward direction.
It is apparent from equation (5) that swapping the indices and changing the sign of j leave the equations unchanged, indicating that the system is PT-symmetric. By solving the secular equation resulting from equation (5), the eigenfrequencies of the system can be written as [5] c c c 1 2 1 1 2 1 The eigenfrequencies depend upon the coupling relative to the loss. They have two distinguished regimes, i.e., an exact PT-symmetric regime with real frequencies and a broken PT-symmetric regime with complexconjugate frequencies. Exceptional points (EPs) at which eigenfrequencies coalesce separate the exact regime from the broken regime. The coupling coefficient at EP (the PT symmetry breaking point) is identified as.
For systems in exact phase (c c PT > ), the two eigenfrequencies are real. When the coupling coefficient is equal to the exceptional point (c c PT = ), the two eigenfrequencies are merged into one real. When in broken phase (c c PT < ), the two eigenfrequencies are two complex with the same real and opposite imaginary parts. For the non-reciprocal transmission studied here, the system is operated in the broken PT-symmetric regime.
As shown in figure 1, the forward direction of propagation is thought to be from the loss resonator to the gain resonator while the backward direction is from the gain resonator to the loss resonator. For the broken PTsymmetric regime, the eigenfrequencies are coming in complex conjugate pairs with the same real parts. It is ultimately dominated by an exponentially growing mode, and ends up at a saturation level due to nonlinear gains. The negative resistance circuit shown in figure 1 features nonlinear gain. For a high input signal, the operational amplifier operates in the saturation region at which its output voltage is clamped by the power supply voltages. Hence, the non-reciprocal transmission is expected to be dependent of input signal power.
Based on the TL theory, the voltage determining the amplitudes of the traveling-wave components at the loss and gain side is given by, respectively are the voltage amplitudes of incident and reflected waves at the loss (gain) side. ω b is the resonant frequency of the coupled LC resonators (the real part of the eigenfrequency for the broken PTsymmetric regime).
is the loss factor of the saturation mode during backward transmission, which is a function of the input saturation power P, where -R(P) is the saturated negative resistance. h is the coefficient related to the characteristic impedance Z 0 of the transmission line.
According to equations (8) and (9), the forward transmission coefficient The non-reciprocal ratio r NR , which is defined as the ratio of the forward transmission coefficient to the backward transmission coefficient, is given by the operation frequency ω T can be non-analytically derived for the maximum nonreciprocal ratio. Obviously, for the non-reciprocal transmission configurations, the resonant frequency (ω b ) of the coupled LC resonators and the operation frequency for the maximum non-reciprocal ratio (ω T ) are different. The operation frequency for the maximum non-reciprocal ratio can be tuned by adjusting the coupling coefficient c and the loss factor of the saturation mode P . g ( )

Simulation
As shown in the insertion of figure 2, the equivalent negative resistance circuit providing gain is implemented by an operational amplifier and resistors. The negative resistance is simply given by R G = −R 1 R 2 /R 3 in the linear operation region. The ADS software was used here to simulate the resistance as a function of input power levels [see appendix A for more details]. The equivalent resistance can be obtained from S 11 parameters. Figure 2 shows the equivalent resistance as a function of input power levels. Depending upon the input power levels, it is divided into three regions: The linear region (white area), negative resistance saturation region (light blue area), and positive resistance saturation region (dark blue area). It is clear that under the saturation power point (−32 dBm in figure 2) the equivalent negative resistance operates in the linear region where the resistance is constant, above the saturation power point it operates in the negative resistance saturation region where the resistance is dependent of input power, and as input power further increases it eventually goes into positive resistance saturation region where it is clamped by the power supply voltage.
In the broken phase (c < c PT ), as shown in equations (6) and (7), eigenfrequencies are two complex with the same real and opposite imaginary parts. For the non-reciprocal transmission studied here, the system is operated in the broken PT-symmetric regime. Therefore, the eigenfrequencies as a function of the coupling coefficient are simulated and measured in figure 3(a). It shows that the experimental data consists with simulation results. In our following experiments, c PT is approximately equal to 4. In order to intuitively understand the critical point, both c PT and PT w as a function of the loss are plotted in figure 3(b). It shows that the non-reciprocal transmission with tunable frequencies can be achieved within some frequency bands as long as c < c .

PT
The schematic diagram of PT-symmetric LC resonators for ADS software simulation is shown in figure 4. All setting parameters are also given in the diagram. The negative resistance in figure 2 is connected in parallel to the LC resonator to form a gain resonator, and then the gain resonator is connected to the loss resonator through a coupling capacitor to form a PT-symmetrical system. C8, C9, C11 and C14 are used to isolate DC signals in order to prevent DC signals from interfering with the system. R14, R15 and R16 are used to clamp the DC supply voltage (VDC/2) to the in-phase input of the operational amplifier. In order to optimize the non-reciprocal ratio at tunable operation frequencies, the coupling coefficient c that ensures the system to be in the broken regime is varied in the simulation. Figure 5 shows the non-reciprocal ratio as a function of input power level for different coupling coefficients. The simulations show that the nonreciprocal ratio is related to the input power level. The    non-reciprocal ratios maximize at moderate inputs where the saturable gains exhibit the largest difference between forward and backward inputs, which can be explained by figure 2.
The forward and backward transmission as a function of operation frequency at the optimal input power are simulated in figure 6. It shows that the non-reciprocal transmission is possible with the tunable operation frequencies by adjusting the coupling coefficients. Overall, the non-reciprocal ratios at the optimal input power depend upon the coupling coefficient and the small signal gain. A weaker coupling coefficient leads to a higher non-reciprocal ratio, however, increases the insertion loss which is related to the forward transmission coefficient. And, a larger gain in the gain resonator improves the non-reciprocal ratios, however, decreases the isolation which is related to the backward transmission coefficient. Hence, there is a trade-off among the various parameters.

Experiments and results
To demonstrate the non-reciprocal transmission with tunable frequencies, according to the optimization by simulation, PT-symmetric LC resonators were fabricated on a PCB (5 cm×5 cm) through printed inductors and discrete capacitors and resistors, as displayed in figure 7. All the parameters used for components are listed in table 1. The coupling capacitors between gain and loss resonators were regulated by light touch switches so that the capacitance with 100 pF, 75 pF and 51 pF were obtained, respectively. The negative resistance in the gain resonator side in the linear region corresponds to that at the loss resonator side (51 Ω).
Test setup is given in appendix B. A DC power supply was used to provide the bias voltage for the negative resistance circuit. The amplitude and frequency of the input signal were provided by an AC signal source, and the input signals were coupled to the port of the PCB circuit by a TL with the characteristic impedance of 50 Ω. The output was connected to an oscilloscope through a 50 Ω TL. The network analyzer was used to measure S parameters. The low-pass filter (LPF) was connected between the output port of the system and the input port of the oscilloscope to remove the high-frequency self-excited oscillation signal of the circuit. To ensure the negative resistance circuits to operate in the saturation region, a sine wave with the peak-to-peak value of 1200 mV was applied to the input. Figure 8 shows the measured non-reciprocal ratio as a function of input signal frequency at different coupling coefficients. The simulated results are also plotted for comparison. The theoretical predictions are in a qualitative agreement with the measurements. The deviations are due to component tolerances and parasitic effects of the PCB circuit. Since the low-pass filter is not an ideal filter, the passband is not perfectly flat, thus resulting in a leftward shift of the non-reciprocal frequency. It can be seen from figure 8 that as the coupling coefficient decreases the non-reciprocal ratio increases and the frequency at which the non-reciprocal ratio achieves maximum increasingly shifts. Hence, the non-reciprocal transmission with tunable frequencies is achieved.
The frequency at which the non-reciprocal ratio achieves maximum is chosen here as the operation frequency. The bandwidth is given by the difference between the upper and lower frequency at which the maximum non-reciprocal ratio drops 3 dB. Figure 9(a) shows the measured upper and lower frequencies as a    The insertion loss and isolation are defined as the forward and backward transmission coefficient [18], they are IL Log T 20

= -
(| |) respectively. Figure 9(b) shows the measured insertion loss and isolation as a function of coupling coefficient at the operation frequency. As the coupling coefficient increases, both the insertion loss and isolation decreases. For the strong coupling (c = 2), they are 0.32 dB and 11 dB, respectively. For the weak coupling (c = 1), they are 0.716 dB and 14.5 dB, respectively. The practical applications require the low insertion loss and high isolation. Hence, there is a trade-off between the parameters for the non-reciprocal transmission with tunable frequencies.

Conclusion
In conclusions, this paper has designed, simulated, and experimentally verified the non-reciprocal transmission with tunable frequencies by coupled LC resonators through PT-symmetry breaking. The input signal frequency and power have been optimized for non-reciprocal ratios. The tunable frequencies are achieved by adjusting the coupling capacitance between the coupled resonators. In our experiments, we used the coupling capacitors regulated by light touch switches. They could be implemented by a micromachined capacitor providing continuous capacitance regulations. Furthermore, the PT-symmetric LC resonators could be fabricated using application-specific integrated circuits instead of discrete components. The frequency could be extended to a few gigahertz by scaling down the components. Our configuration is potential for wireless communication.

Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 62274030).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Appendix A. Simulations of negative resistance circuits Figure A1 shows the schematic diagram of the negative resistance constructed by ADS software simulation using a THS4304 operational amplifier. When the op-amp operates in the linear region, the resistance is determined by R1, R2 and R3. The expression for the negative resistance in the linear region is given by For a system with γ = 2, the negative resistance value in the linear region is set to −51 Ω by setting the DC supply voltage VDC = 2.7 V, R1 = 17 Ω , R2 = 300 Ω and R3 = 100 Ω. The negative resistance is only applied with AC signals, hence, C8, C9, C11 (2.2 μF) and C14 (100 nF) are used to isolate DC signals in order to prevent DC signals from interfering with the system. R14, R15 and R16 (10 kΩ) are used to clamp half of the DC supply voltage (V DC /2) to the in-phase input of the operational amplifier. A power source (PORT1) with an internal resistance of 50 Ω is used to provide a 10 MHz AC signal. We set the ADS software simulation to Large-Signal S-Parameters simulation (LSSP) and a voltage sweep interval of 0.3 mV to 800 mV. The equivalent resistance can be obtained from S11 parameters.
The S 11 parameters are scanned so that the negative resistance can be solved from S 11 . The power is then converted to voltage amplitude to obtain the curve of the negative resistance with the change of AC signal amplitude. Figure B1 shows photo of test instruments. They are Waveform generator (Agilent 33220 A), signal oscilloscope (Agilent MSO-X 3032 A), and network analyzer (Agilent N5224A). A DC power supply (2.7 V) was used to provide the bias voltage for the negative resistance circuit. The low-pass filter (LPF) was connected between the Figure A1. Schematic diagram of negative resistance circuits constructed by ADS. Figure B1. Photo of test instruments. output port of the system and the input port of the oscilloscope to remove the high-frequency self-excited oscillation signal of the circuit. The input/output nodes of LC resonators were coupled to the port of the PCB circuit by a transmission line with a characteristic impedance of 50 Ω.