Microwave Tunneling and Robust Information Transfer Based on Parity-Time-Symmetric Absorber-Emitter Pairs

Robust signal transfer in the form of electromagnetic waves is of fundamental importance in modern technology, yet its operation is often challenged by unwanted modifications of the channel connecting transmitter and receiver. Parity-time- (PT-) symmetric systems, combining active and passive elements in a balanced form, provide an interesting route in this context. Here, we demonstrate a PT-symmetric microwave system operating in the extreme case in which the channel is shorted through a small reactance, which acts as a nearly impenetrable obstacle, and it is therefore expected to induce large reflections and poor transmission. After placing a gain element behind the obstacle, and a balanced lossy element in front of it, we observe full restoration of information and overall transparency to an external observer, despite the presence of the obstacle. Our theory, simulations, and experiments unambiguously demonstrate stable and robust wave tunneling and information transfer supported by PT symmetry, opening opportunities for efficient communication through channels with dynamic changes, active filtering, and active metamaterial technology.

1. Scattering properties of parity-time (PT) symmetric wave tunneling and information transfer system

Ideal PT-symmetric wave tunneling and information transfer circuit
The transfer matrix of the two-port network can be expressed as: where 0 Z is the characteristic impedance of the transmission line,  is the time delay in the transmission line,  is the signal frequency, 0 L is the inductance of the shunt inductor. By simplifying Eq. (1), the following four matrix elements are obtained: According to the relation between scattering matrix and transfer matrix: (  )   11  12  11  12  0  21 0  22  11 22  12 21   21  22  11  12  0  21 0  22  11  12  0  21 0  22   /2  1  ,  / 2 /

SS T T Z T Z T T T T T S SS T T Z T Z T T T Z T Z T
S-parameters are given by where 0  is the designed frequency of the circuit,  7) and (8) confirms this assessment. It is worth mentioning that the PT tunneling resonance is irrelevant to whether if the system is in exact PT phase or in broken PT phase regimes. When the loss and gain is balanced, the wave tunneling in PTsymmetric system is very similar to electron or photon tunneling phenomena.
Meanwhile, the wave tunneling can also happen on the second port where 22 0 S = . The solutions are As we see from the above equation, the solutions always exist. But the tunneling frequencies are inherently different from Eq. (6). The smaller  is, the larger  becomes. In this case, the 12 21 1 SS == and 11 0 S  . There are four backward wave tunneling points, they are: 0 7. 56, 7.82, 9.44, 9.88   = . These tunneling points are consistent with Eqs. (6) and (9).

PT-symmetric wave tunneling and information transfer circuit with one-pole NIC
For practical implementation of the PT-symmetric wave tunneling and information transfer, we make two major modifications over the ideal model. First, the transmission line is replaced with a π model LC tank with finite transmission window to shrink the form factor of the board. Second, the negative impedance is implemented with an amplifier feedback circuit.
We apply the transfer matrix formalism: The corresponding scattering matrix can be obtained by substitute the above equation into Eq.
(3). Implementation of the PCB board involves consideration of parasitic effects, wave leakage in the circuit channel and many more. In this part, we demonstrate our simulation results of our PCB board with ADS and Modelithics package. Figure

Stability analysis of an ideal PT-symmetric wave tunneling and information transfer circuit
In this section, we provide a detailed derivation of the transfer function and stability analysis of an ideal PTsymmetric wave tunneling circuit. To analyze the influence of parameter detuning, we assume two small perturbations on the circuit: relative time delay perturbation    on the second segment of transmission and small perturbation 0 Z Z  on the ideal negative impedance.
At port 2, the effective load impedance is a negative impedance in parallel with the characteristic impedance where 1  is the time delay on the first transmission line. As a result, the transfer function which is defined as the voltage ratio on the negative impedance versus the generator can be written as The above equation is the transfer function of an ideal PT-symmetric wave tunneling and information transfer circuit with parameter detuning. We apply simple numerical calculations and figure out the poles of the transfer function. Figure S8 demonstrates the pole locations with parameter detuning. Our study indicates that the circuit remains stable for any value 0   based on the assumption that there is no parameter detuning. If imperfection exists, the coupling coefficient  should be larger than 0.1 to ensure stable operation.
It is also important to verify the stability issue in time domain. Figure S9 demonstrates the impulse responses of the scattering systems with various detuning schemes. The finite amplitude of impulse responses is consistent with the numerical results in Fig. S8.
In summary, the ideal PT-symmetric tunneling circuit is stable for any obstacle strength. To ensure a more robust operation [5% tolerance on impedance and delay detuning], the coupling coefficient should be larger than 0.1. It is important to note that the EP is irrelevant to the stability issue.

Robustness analysis with dynamic obstacle
In this section, we will graphically prove the robustness of system with dynamic obstacle. From Figs S10a-c, we can see that the system will reach steady state in 0.5 μs when the obstacle is static. In Figs S10d-f, we apply a square wave modulation on the obstacle at 0.5 μs. The source and load sides will reach to steady in a longer time, which is approximately 1 μs. These graphs prove that our system is capable of retuning to steady state when the obstacle dynamically changes in time.     . Note that the number of poles is infinite, and we only show the poles that will potentially cause stability problems.