PT-symmetric tunneling and robust information transfer are schematically sketched in Figure 1(a), where the signal is transported wirelessly via an air channel from a transmitting to a receiving antenna. When a reflective obstacle is inserted within the transmission channel, information transfer efficiency is expected to drop dramatically. To restore full-wave transmission, in this work, we add a gain element -Z_0 behind the reflecting obstacle and a conjugate lossy element Z_0 in front of it to form a PT-symmetric conjugate pair, where Z_0 is the characteristic impedance of the transmission channel (see Figure 1(a)). As we show in the following, this combination can largely restore transmission through the channel, despite the presence of the largely reflective obstacle. In order to prove this principle, we model the wireless information transmission channel with a basic transmission line, as shown in Figure 1(b), and the highly reflective obstacle with a small inductor L_0, close to a short and therefore highly reflective. Since we utilize the resonant tunneling feature of the PT-symmetric system, the transmission line is set at half-wavelength of the impinging wave. As a result, the loss element Z_0, the first quarter-wavelength transmission channel, and the highly reflective obstacle behave like a Salisbury screen absorber. The obstacle and the second quarter-wavelength transmission channel, together with the gain element -Z_0, work as a synchronized information transmitter, a time-reversed replica of a Salisbury absorber, which emits in sync with the absorbing portion. We show that in a steady state the whole PT-symmetric system works as a tunneling channel for impinging waves, mimicking a lossless, ideally transmitting channel.

To begin with, we analyze the resonant transmission mode and frequency of the PT-symmetric wave tunneling and information transfer system. We define the steady-state voltage on the transmitting node, obstacle, and receiving node as V_{\text{source}}, V_{\text{obstacle}}, and V_{\text{load}}, which form the state vector \left|\psi \right.〉={\left[V_{\text{source}}{V}_{\text{obstacle}}{V}_{\text{load}}\right]}^{\text{T}}. The characteristic equation reads \text{PT}\left|\psi \right.〉=\lambda \left|\psi \right.〉, which supports even and odd eigenstates \left|{\psi }_{+}\right.〉={\left[\begin{array}{ccc}1& -1& 1\end{array}\right]}^{\text{T}} and \left|{\psi }_{-}\right.〉={\left[\begin{array}{ccc}1& 0& -1\end{array}\right]}^{\text{T}} and eigenvalues {\lambda }_{\pm }=\pm 1, where

is the parity operator, T is the time reversal operator (equivalent to complex conjugation for a steady state), and \text{T} is the transpose operator. As we can infer from the above analysis, the obstacle is a short and contributes no phase delay at odd resonant tunneling states. Therefore, the total phase delay of the transmitted wave is \omega \left(T_0/2\right)=\left(2k+1\right)\pi with \omega =\left(2k+1\right){\omega }_0,k=0,1,2,3\cdots, and {\omega }_0=2\pi /T_0 being the resonant frequency of the fundamental standing mode. For the even scattering state, we sum up the phase delay on the transmission line and the inductor:

which can be simplified to the characteristic equation

where \alpha ={\omega }_0{L}_0/Z_0 is the coupling strength between the Salisbury screen absorber and its conjugate emitter and Z_0 is the characteristic impedance of the transmission line. The eigenequation of even standing modes is a transcendental equation, which requires a numerical evaluation. As \alpha approaches zero, the circuit is shorted by the obstacle and the wave cannot tunnel through the system; as \alpha approaches infinity, the load can be neglected and the even mode has the close-form solution \omega =2k{\omega }_0,k=0,1,2,3\cdots. An alternative approach to find the resonant frequencies of the system is to employ the transfer matrix method and the generalized conservation law, with detailed calculations presented in the Supplementary Materials. The standing even/odd mode solutions \left|{\psi }_{\pm }\right.〉 explicitly show that the absorber and emitter exhibit identical amplitudes and 0/\pi phase difference, which is direct evidence of synchronized wave absorption at the input port and emission at the output port. We demonstrate the standing wave profile of this PT-symmetric absorber-emitter pair in Figure 1(c). The unusual standing mode profile, considering the presence of the reflective obstacle, is a direct manifestation of PT symmetry and Neumann boundary conditions at the ports. We also show the power transmission spectrum in Figure 1(d), which demonstrates full-wave tunneling at the resonant frequencies. The tunneling mechanism satisfies the generalized flux conservation law applicable to any two-port PT-symmetric network [33], which reads T-1=\left|r_{\text{L}}{r}_{\text{R}}\right|e^{i\left({\varphi }_{\text{L}}-{\varphi }_{\text{R}}+\pi \right)}, where T is the power transmission coefficient, r_{\text{L}} and r_{\text{R}} are the amplitude reflection coefficients on the left and right ports of the system, and {\varphi }_{\text{L}} and {\varphi }_{\text{R}} are the corresponding phases of the amplitude reflection coefficients. As a result, the system demonstrates superunitary power transmission inside the region bounded by any pair of even and odd tunneling eigenfrequencies. Note that, due to the presence of gain in the system, the transmission may go over unity at frequencies close to the tunneling frequency; however, the system remains stable.

To better understand the PT-symmetric wave tunneling, it is of interest to investigate the scattering properties at resonance and analyze the PT phase transition of the scattering state. Rigorous transfer matrix analysis enables us to express the scattering properties of even and odd resonant states of this two-port information transfer network as

where S_{+} and S_{-} are scattering matrices of even and odd resonant modes, respectively, obeying the symmetry relation {PS}_{\pm }^{\ast }\left(\omega \right){PS}_{\pm }\left(\omega \right)=\mathbf{I} [29]. Wave tunneling happens at the resonant frequencies, and it is totally independent of the obstacle strength \alpha or its type (inductive, capacitive, even resistive, or any combination of these) between the Salisbury absorber and the synchronized emitter, since the obstacle is effectively shorted in a steady state. This offers unique flexibility to maintain the transmissivity of the proposed system for arbitrary variations of the obstacle, ensuring stable operation. Even for dynamically varying obstacles, the system is capable of self-tuning to a stable full-transmission region (see Supplementary Materials).

The scattering matrix in the equation has two eigenvalues {\lambda }_{1,2}=\left(-j\pm \sqrt{{\left(\alpha \left(\omega /{\omega }_0\right)\right)}^2-1}\right)/\left(\alpha \left(\omega /{\omega }_0\right)\right). At \alpha ={\left(\omega /{\omega }_0\right)}^{-1}, these two eigenvalues coalesce, supporting a non-Hermitian degeneracy. This exceptional point (EP) separates the scattering system from its PT-symmetric phase when \alpha >{\left(\omega /{\omega }_0\right)}^{-1} and its broken PT-symmetric phase when \alpha >{\left(\omega /{\omega }_0\right)}^{-1}. We substitute the eigenfrequencies of even and odd scattering states into the expression \alpha ={\left(\omega /{\omega }_0\right)}^{-1} to find the EPs:

Depending on the reflectivity of the obstacle, i.e., how close to zero is its reactance, the resonant tunneling state can be in the exact PT symmetry phase with unitary eigenvalues \left|{\lambda }_1\right|=\left|{\lambda }_2\right|=1 or in the broken PT symmetry phase with \left|{\lambda }_1\right|=1/\left|{\lambda }_2\right|>1 [29]. The PT tunneling functionality is independent of the PT symmetry phase transition point. The stability analysis is presented in the Supplementary Materials, which shows inherent stability for any finite value of the obstacle reflectivity [].

Quite interestingly, the proposed system is robust even when the time delay on the transmission line is asymmetric (see the detailed discussion in the Supplementary Materials). As it was mentioned above, regardless of how small or large the inductive load is (i.e., how reflective the inductive load is), the structure will still fully retrieve the input signal at the output once it reaches a steady state. In fact, the obstacle can be inductive, capacitive, or even resistive, and yet the PT-symmetric setup will still recover the input signal at the output. In the general case of a finite-size obstacle, the 1-D transmission line model implemented here does not apply, because of scattering into the continuum of radiation. However, if the beamwidth of the input wave is smaller than the size of the obstacle and the obstacle does not scatter energy from its edges, scattering loss can be modeled as a small resistive load in our transmission line model and we still expect to efficiently recover the input signal at its output.

We experimentally demonstrate a robust proof-of-concept PT-symmetric microwave tunneling prototype (see Figures 2(a) and 2(b)). The gain unit is realized through a dispersive negative impedance converter (NIC) based on a noninverting feedback amplifier configuration (see Figure 2(a)). A compensating inductor L_{\text{c}} in series with the noninverting port realizes purely negative impedance -Z_0 at the operational frequency. The dispersion relation of the NIC follows

where A_0 is the open-loop gain of the amplifier, {\omega }_{\text{p}} is the pole frequency of the amplifier, \beta =R_{\text{f}}/\left(R_{\text{f}}+R_{\text{g}}\right) is the feedback factor, Z_{\text{F}}=2Z_0 is the feedback resistor, and s is the complex frequency. We choose a compensating inductor L_{\text{c}}=3Z_0/{\omega }_{3\text{dB}} and the operational frequency {\omega }_0={\omega }_{3\text{dB}}/\sqrt{3}, where {\omega }_{3\text{dB}}=A_0{\omega }_{\text{p}}/2 is the 3 dB frequency of the closed-loop gain coefficient. The NIC dispersion is shown in Figure 2(c), showing a negative impedance -Z_0 at the operational frequency {\omega }_0.

Our prototype is designed to work at the fundamental odd scattering state. We realize the transmission line through a \pi-type LC resonator to reduce the form factor (see Figure 2(a)), which offers unitary transmission as well as \pi/2 phase delay at resonant frequency to mimic a quarter-wavelength transmission line segment. The impedance is automatically matched at both transmitting and receiving nodes, as we choose L=Z_0/{\omega }_0 and C=1/Z_0/{\omega }_0. This compact design significantly reduces the form factor of our device from \lambda /2 to a deeply subwavelength scale \lambda /50. The fabricated device is shown in Figure 2(b), with a size of approximately 2 cm by 2 cm.

We confirm our theoretical analysis with a cosimulation between ADS and the Modelithics package, which shows excellent PT-symmetric tunneling at 48.7 MHz. The scattering properties are demonstrated in Figure 2(d) with a tunneling point of -0.01 dB transmission and -28.8 dB reflection. Finally, we confirm the PT symmetry of our designed circuit by graphing the spectral properties of the eigenvalues of the scattering matrix in Figure 2(e). With the coupling coefficient \alpha =0.13, the eigenvalues demonstrate unitary property \left|{\lambda }_1\right|=1/\left|{\lambda }_2\right| at {\omega }_0, which is a typical hallmark of PT symmetry in the broken symmetry phase. No other PT symmetry point is found in the full spectra.

To better understand how this PT-symmetric wave tunneling circuit works, we plot the temporal response at the source, obstacle, and load nodes in Figures 2(f)–2(h). We place a microwave generator V_{\text{g}}=\sin {\omega }_{0}t with internal impedance Z_0 in the source node. The voltage at this port initially experiences a small reflection and then rapidly reaches a steady state V_{\text{source}}=\left(1/2\right)\sin {\omega }_{0}t, indicating that the impinging wave is fully absorbed by the shunt resistor Z_0. The obstacle node demonstrates a purely decaying response, confirming our previous steady-state analysis in the ideal PT-symmetric configuration V_{\text{obstacle}}=0. The load port is essentially an emitter, and it demonstrates an exponentially growing trend towards a steady state V_{\text{load}}=-\left(1/2\right)\sin {\omega }_{0}t. Since it operates at the fundamental odd resonant tunneling state, the emitted wave has a \pi phase shift with respect to the impinging wave.

Apart from the PT-symmetric full-wave tunneling functionality, our design is stable and robust to parameter detuning and reasonable fabrication errors [34–36]. More interestingly, our system can self-tune to the stable region even if the obstacle dynamically changes in time (see Supplementary Materials). To study the stability issue, it is important to investigate the voltages at the source, obstacle, and load in the complex frequency domain. Application of Kirchhoff’s current and voltage laws leads to a set of linear equations:

where {\tilde{V}}_{\text{g}}\left(s\right) is the voltage of the microwave generator and {\tilde{V}}_{\text{source}}\left(s\right), {\tilde{V}}_{\text{obstacle}}\left(s\right), and {\tilde{V}}_{\text{load}}\left(s\right) are the complex voltages on the source, obstacle, and load nodes, respectively. Here, we define the transfer function as the voltage ratio between the load and generator: \tilde{H}\left(s\right)={\tilde{V}}_{\text{load}}\left(s\right)/{\tilde{V}}_{\text{g}}\left(s\right) (see the detailed expression in Materials and Methods). To maintain a stable operation, the impulse response should have a finite energy, which is equivalent to requiring that all the poles of the transfer function lie on the left hemi-Riemann sphere [36]. Equation (7) indicates that the relative pole locations only depend on the strength of the obstacle. As \alpha approaches to zero, the transfer function turns to zero and it is meaningless to study the stability issue. As \alpha goes to infinity, the emitter is directly connected with the absorber, creating a marginally stable point on the north pole (see Figure 3(a)). In this scenario, any small perturbation in the system will move the marginally stable point and transform it into eight poles on the right hemi-Riemann sphere. Numerical computation indicates that our circuit operates robustly as the coupling strength varies between 0.1 and 0.3. As our realistic design involves both dispersive negative feedback and a \pi-type transmission line, they place a more stringent condition on stability compared with the ideal model where the stable condition is . In our fabricated prototype, we carefully choose \alpha =0.13 to ensure a stable operation within expected fabrication imperfections. Figure 3 demonstrates the pole distributions as well as the impulse response of our design. There are eight poles on the left hemi-Riemann sphere when \alpha =0.13, indicating that our fabricated prototype is stable. To further confirm the robustness of our implemented prototype, we study the influence of feedback factor \beta and feedback resistor Z_{\text{F}} perturbation on the pole distribution and impulse responses in Materials and Methods.

Figures 4(a) and 4(b) show the experimentally measured and theoretically calculated forward transmission spectrum and phase. A PT-symmetric unitary transmission was observed at 44.8 MHz with a 180-degree phase, which agrees with our simulation. Meanwhile, the reflection is fully suppressed at this point, as shown in Figures 4(c) and 4(d). The experimental reflection phase at resonance is essentially undefined due to the zero amplitude, causing fast oscillations around the resonance, as shown in Figure 4(d). It is worth mentioning that at resonance, the experimental backward reflection amplitude \left|S_{22}\right| is 15 dB and the phase \text{Arg}\left[S_{22}\right] is -90 degrees, confirming that our PT-symmetric system has asymmetric reflections and satisfies the generalized conservation law. Since the system is linear and time-invariant, reciprocity is satisfied and the transmission coefficients in forward and backward directions are identical. The observed tunneling frequency is smaller than the simulation results, which may stem from fabrication errors where the inductance or capacitance values in the transmission line are smaller than the designed ones. The scattering parameters in a narrower or wider spectrum are demonstrated in the extended data figure section in the Supplementary Materials (Figs. S3 and S4). In the extended Fig. S5, we show the linearity of our tunneling circuit, which demonstrates an excellent linear relation between input and output power at tunneling frequency. This graph clearly shows that although at the steady state, the output is completely disconnected from the input (since the inductive load is fully shorted), the output can still linearly follow the input. In other words, if after reaching the steady state, any change happens to the input signal, the system goes into the transient regime for a very short period of time, during which the information of the input signal is communicated towards the output port and then the output reaches the steady state again and fully replicates the input signal.

The design of the circuit and fabricated prototype is shown in Figures 2(a) and 2(b). In the NIC part, we use the amplifier OPA 355-Q1 from Texas Instruments. This amplifier has a 200 MHz gain bandwidth product, with an open-loop gain A_0={10}^5 and pole frequency {\omega }_{\text{p}}=2\pi \times 2000\text{Hz}. The effective impedance of the NIC in the Fourier domain can be easily inferred from Equation (6) by replacing complex frequency s with physical frequency j\omega:

To have purely negative impedance -Z_0, the real part of the above equation should be -Z_0 and the imaginary part of the effective impedance should be 0. Then, we have the following operational conditions:

where Z_0=50\Omega, Z_{\text{F}}=100\Omega, \beta =R_{\text{f}}/\left(R_{\text{f}}+R_{\text{g}}\right)=0.5, and R_{\text{f}}=R_{\text{g}}=560\Omega. Here, the feedback resistor Z_{\text{F}} is chosen as 2Z_0 due to the dispersion of the one-pole model amplifier. We substitute these values into Equation (9) and get the operational frequency f=\omega /2\pi =57.74\text{MHz} of the negative impedance converter and the compensating inductor L_{\text{c}}=238.7\text{nH}. In circuit simulation, we use two inductors with inductance value 120 nH. Then, we cosimulate our circuit with ADS and Modelithics to check the effective negative impedance. Our simulations demonstrate a −50.9  Ω negative impedance at 50.2 MHz. The shift of the design frequency can be contributed to the parasitic effect and possible high-order poles in the amplifier. The inductor L=150\text{nH} and capacitor C=68\text{pF} in the transmission line are chosen to match both the operational frequency of NIC and the characteristic impedance of the port. The reflective obstacle is modeled as an inductor L_0=20\text{nH}, leading to a coupling factor \alpha =0.13 and ensuring robust operation of the system.

The circuit is fabricated on a 0.062-inch-thick FR-4 substrate with relative permittivity 4.4 and dissipation factor 0.017. The following components are surface mounted on the PCB board: (1) characteristic impedance Z_0 from KOA Speer with part number RK73B1ETTP510J; (2) feedback resistor Z_{\text{F}} from KOA Speer with part number RK73B1ETTP910J; (3) feedback resistor R_{\text{f}} from KOS Speer with part number RK73B1ETTP561J; (4) two variable resistors from Bourns Inc. with part numbers 3223W-1-101ETR-ND and 3223W-1-200ETR-ND; (5) capacitor in the resonator from Murata Inc. with part number GRM1552C2A680GA01#; (6) inductors from Coilcraft with part numbers 0402HPHR15X, 0402CS20N, and 0402CS12X; (7) amplifier from Texas Instruments with part number OPA355QDBVRQ1; and (8) port connectors from Amphenol with part number 132322.

A linear, time-invariant, causal circuit is stable if and only if the impulse response is absolutely integrable, which means

where H\left(t\right)={\sum }_{n=0}^{\infty }{c}_n{e}^{p_{n}t} is the impulse response, p_n is the nth pole of the transfer function H\left(s\right)={\sum }_{n=0}^{\infty }{c}_n/\left(s-p_n\right), c_n is the amplitude coefficient of each Laplace component [36]. Therefore, to operate in the stable region, all the real parts of the poles must be negative:

which is the stability condition of a linear, causal, and time-invariant circuit.

In our circuit, we assume that the system is excited with a power source V_{\text{g}} which has an internal impedance Z_0. The transfer function \tilde{H}\left(s\right) can be defined as the ratio between {\tilde{V}}_{\text{load}}\left(s\right) and {\tilde{V}}_{\text{g}}\left(s\right). According to Kirchhoff’s current and voltage laws, we express currents and voltages in the Laplace domain:

where Z_{\text{NIC}}=2Z_0\left[\left(s+\sqrt{3}{\omega }_0\right)/\left(s-\sqrt{3}{\omega }_0\right)+\sqrt{3}s/2{\omega }_0\right]. We solve the above linear equation set and get the transfer function:

Simple numerical calculation is employed to solve the pole locations of the above transfer function. Our calculation indicates that for stable operation of the PT-symmetric circuit, the coupling coefficient must meet the following condition:

To verify the robustness of our PT-symmetric wave tunneling circuit, we investigate the influence of parameter detuning on pole locations. Extended data Fig. S1 demonstrates pole locations with feedback factor detuning and feedback resistance detuning, which shows excellent stability under reasonable perturbation.

It is also equally important to investigate the impulse response and confirm the stability of our PT-symmetric system in the time domain. We substitute the solutions of the poles into the following equation of impulse response:

where “Res” means the residue of a complex function. Extended data Fig. S2 in the supplementary document demonstrates the impulse responses with parameter detuning, which shows that the impulse response has a finite energy and evolves in a stable fashion in the time domain. Both frequency and time domain analyses prove that our system is immune from reasonable unwanted perturbation and fabrication error and maintains stable operation. It is important to note that the stability issue inherently relies on the measurement circuit (where the generator, source, and load impedance are included), rather than the isolated PT-symmetric tunneling system.

The scattering properties of our device are measured with the Agilent Technologies network analyzer with part number E5071C, which can analyze S parameters from 9 kHz to 8.5 GHz with great precision. One of the DC bias ports of the amplifier is grounded, and the other port is supplied with a 1.3 V DC voltage. The source is from Agilent Technologies with part number E3631A. When the measurement range is from 20 MHz to 80 MHz, the step size of the excitation signal is 0.0375 MHz.