Active mode-locking (ML) is an important technique in laser science, which greatly shortens the laser pulse. Here, we construct an anti-parity-time (anti-PT) symmetric Su–Schrieffer–Heeger frequency lattice by two ring resonators with antisymmetric amplitude (AM) modulations. We find that the temporal width of the generated pulse can be greatly shortened by the phase-mismatching of the AM modulations. In addition, the pulse shortening shows extremely high sensitivity to the phase transition point, at which the anti-PT symmetry of the system is completely broken. This work exploits the concept of anti-PT symmetry in a laser field to realize ML, and will have broad application prospects in ultrafast spectroscopy and ultra-high sensitive sensors.

In real life, physical systems always exchange energy with surrounding environments, which can be described by a non-Hermitian Hamiltonian.^{1–7} Recently, the exotic phenomena in the non-Hermitian physics with parity-time (PT) symmetry,^{4,8–10} anti-parity-time (anti-PT) symmetry,^{11–15} and non-Hermitian skin effect (NHSE)^{5,16,17} have attracted intense interest. The most special feature for the PT and anti-PT symmetric systems is that they can possess real eigenvalues, in the cases of the PT symmetry maintained and the anti-PT symmetry broken, despite they are non-Hermitian systems. The eigenvalues and the eigenmodes of the systems will be coalesce at their exceptional points (EPs).^{10,11,18} After the phase transition with an EP, the band structures of the PT and anti-PT symmetric systems will show time-windows with gain, which indicates their abilities to generate a short pulse.^{19–22} In addition, thanks for the sensitivity of the EPs, the systems with PT and anti-PT symmetry have application prospects for sensors with ultrahigh sensitivity.^{23–25}

In the last few decades, the concept of the synthetic dimensions has been a versatile platform for the studies of the physical system with a complex structure such as non-Hermitian topological photonics.^{26–28} The ring resonator with dynamic modulation will make the resonant frequency modes coupled and form a virtual frequency dimension.^{26,29,30} The previous works have applied the dynamic modulated ring resonator to the actively mode-locked system, which have shown that the PT phase transition can be used to shorten the pulse width of a laser,^{31} and a turbulent behavior of the laser pulse emission will take place when the PT symmetry is broken.^{32} For an anti-PT symmetric system, the existence of a time-window with gain also suggests its potential applications in mode-locking (ML), which will generate ultrashort pulses and will have major applications in ultrafast science such as ultrafast spectroscopy and high-speed optical communications.

Here, we construct an anti-PT symmetric Su–Schrieffer–Heeger (SSH) lattice in the frequency dimension based on two ring resonators with antisymmetric amplitude (AM) dynamic modulations. The energy band of the anti-PT symmetric system can be flexibly controlled by the forms of the applied AM modulations. That is to say, the time-window with gain of the anti-PT symmetric double-ring system can be adjusted conveniently to generate ultra-short pulses and realize active ML. Moreover, the shortening of the pulse shows extremely high sensitivity in the vicinity of the phase transition point at which the anti-PT symmetry of the system is completely broken.

*v*and

_{g}*L*are the group velocity of light and the length of the ring, respectively. In an experiment, the AM modulation can be achieved by an AM modulator with period-modulated loss along with an amplifier with constant gain over time.

^{29}Here, the dynamic modulations applied to each ring have opposite phase-mismatching and are in the following forms:

*J*

_{1}and

*J*

_{2}are the modulated amplitudes.

*κ*and

*δ*are the coupling strength between the two rings and the phase-mismatching of the modulations, respectively. As the two rings are mutually coupled, the resonant frequency

*ω*of the two rings hybridizes into symmetric ( $\omega n+=\omega n+\kappa \u2261\omega am$) and antisymmetric ( $\omega n\u2212=\omega n\u2212\kappa \u2261\omega bm$) supermodes, respectively. Then the applied antisymmetric AM dynamic modulations will make the frequency sites to be coupled, and a frequency lattice will be formed. Here, we ignore the group velocity dispersion and apply the representation transformation and rotation wave approximation to the coupled mode equations of the double-ring system. The corresponding dynamics of the system in the frequency domain is described as

_{n}^{30,33}

*a*and

_{m}*b*correspond to the amplitudes of the

_{m}*m*-th symmetric and antisymmetric supermodes, respectively. $TR=2\pi /\Omega $ is the round trip time.

*T*is the slow time variable, which labels the number of round trips that the pulse goes through. Equations (2) and (3) indicate that the symmetric (

*a*) and antisymmetric (

_{m}*b*) supermodes in the frequency dimension form a strip with nonuniform alternating spacing $2\kappa $ and $\Omega \u22122\kappa $. In addition, the phase-mismatching of the AM modulations makes the two sites in a unit cell to have the opposite on-site potentials

_{m}*δ*, as described in Fig. 1(b). The detailed derivations for the anti-PT symmetric SSH frequency lattice and the validity of the tight-binding model with the rotating wave approximation (RWA) are shown in the supplementary material. Therefore, a non-Hermitian SSH lattice with alternating imaginary coupling strengths is obtained in the frequency dimension, and the couplings and the on-site potentials can be controlled by the modulated amplitudes flexibly.

*t*is the fast time variable that varies over the cavity round trip interval ( $\u2212TR/2\u2264t<TR/2$). Then, the dynamics of the double-ring system in the time domain can be transformed from Eqs. (2) and (3) as

*H*(

*t*) is anti-PT symmetric, i.e., $(PT)H=\u2212H(PT)$, where the parity operator

*P*is defined as $PH(t)P\u22121=\sigma xH(\u2212t)\sigma x$ (

*σ*is the Pauli matrix), and the time-reversal operator

_{x}*T*is defined as $TH(t)T\u22121=H*(\u2212t)$.

^{14,34}The Bloch momentum of the anti-PT symmetric system is represented by

*t*.

^{35}Thus, we are able to define the energy band of the double-ring system by the eigenvalues $\lambda \xb1$. By analogy with the crystal lattice of a solid, the band energy of the above anti-PT symmetric double-ring system is expressed as

*δ*, and the point of $\delta =J2+J1$ is the phase transition point for the completely broken of the anti-PT symmetric system. In addition, the $\delta =J2+J1$ is also a transition point for vanishing of the time-window of gain. Therefore, we pay close attention to the evolution of the band structure for various values of

*δ*around $J2+J1$. For intuition, the real and imaginary parts of $\lambda \xb1$ for various

*δ*values are plotted in Figs. 2(a) and 2(b), respectively. We can see that when $\delta \u2264J2\u2212J1$ (marked by red dot), the eigenvalues of the system are pure imaginary, meaning that the anti-PT symmetry is kept for all

*t*. In this case, the anti-PT symmetric double-ring system exhibits a time-window of gain for the whole of the band structure. As

*δ*increases and is above $J2\u2212J1$, the anti-PT symmetry will be partly broken at $\Omega t=\xb1\pi $. As shown in Fig. 2(c), when $\delta =0.9(J2+J1)>J2\u2212J1$, the band structure of the anti-PT symmetric system is complex and displays a time-window with gain with the center of $\Omega t=0$. When

*δ*is in close proximity to $J2+J1$ [Fig. 2(d)], the time-window with gain gets very narrow, indicating the arising of pulse shortening. The narrowing of the time-window with gain will continue until $\delta =J2+J1$ [marked by red dot in Figs. 2(a) and 2(b)]. Once the value of

*δ*is above $J2+J1$, the anti-PT-symmetry is completely broken for all

*t*, and the eigenvalues become real. In this situation, the time-windows for gain and loss are disappeared, and the further increasing of

*δ*will widen the bandgap of the anti-PT symmetric system, as shown in Figs. 2(e) and 2(f). In conclusion, by adjusting the phase-mismatching

*δ*, the band structure of the anti-PT symmetric system can be controlled flexibly, which indicates the regulation of the ML in the double-ring system.

^{19,32,36}

*g*and

*l*are the saturated gain and loss per transit in the rings, respectively.

*H*(

*t*) is the anti-PT symmetric Hamiltonian defined in Eq. (5). Ω

_{g}is the gain bandwidth of the rings. For a slow-gain medium, saturated gain

*g*obeys the rate equation $dgdT=\u2212\gamma \u2225(g\u2212g0+gP)$, where

*g*

_{0}is the small-signal gain from pumping, $\gamma \u2225=TR/\tau $ is the ratio between cavity transit time

*T*and upper laser level lifetime

_{R}*τ*$(\tau \u226aTR)$, and $P=1TR\u222b\u2212TRTRdt|\Phi (T,t)2|$ is the average laser power normalized to the saturation power.

*δ*for the regime of $\delta <J2+J1$. The corresponding pulse width is

*δ*is close to $J1+J2$, which agrees with the narrowed time-window for gain with the center of $\Omega t=0$ in Fig. 2. To demonstrate that, we checked the theoretical analysis by numerical simulations with the initial condition of small random amplitude of the intracavity field $\Psi (0,t)$ at the initial round trip. As shown in Fig. 3(a), for $\delta =0.9(J2+J1)$, the numerically computed evolution of the normalized pulse intensity in Eq. (7) is centered at $\Omega t=0$, just meeting with our expectation. As the value of

*δ*increases and is close to $(J1+J2)$, the pulse width $\Delta tA\u2212PT$ becomes quite narrow [Fig. 3(b)], corresponding to the narrowed time-window for gain in Fig. 2(d). Therefore, the output pulses generated by the double-ring system are greatly shortened with increasing phase-mismatching

*δ*in the case of $\delta <J1+J2$. To make sure that the numerically computed results under the RWA are valid, we show the corresponding numerically computed evolutions for the full time-dependent dynamical equations in the supplementary material, which are consistent with the evolutions in Fig. 3.

*t*in the anti-PT symmetric system. In this regime, active ML also occurs due to the effective phase modulation introduced by the modulators.

^{37}In this case, the ML will take place at $\Omega t=0$ and $\xb1\pi $. Here, we focus on the analysis in the vicinity of $\Omega t=0$ and

*π*, as the solution near $\Omega t=\u2212\pi $ is similar to that for $\Omega t=\pi $. Thus, the steady-state solutions are still a Gaussian and in the forms of $\Psi (T,t)=e\u2212(1+i)t2/2\Delta tAPT2$ and $\Psi (T,t)=e\u2212(1+i)(t\u2212TR/2)2/2\Delta tAPT2$ for $\Omega t=0$ and

*π*, respectively. By performing a Taylor expansion near $\Omega t=0$ and

*π*, we obtain the saturated gains for the two cases as

*l*. The corresponding pulse widths for $\Omega t=0$ and

*π*are

*δ*will increase the pulse widths for both two cases, and the pulse width for $\Omega t=0$ is narrower than that for $\Omega t=\pi $. In addition, Eq. (14) suggests that the pulse can be greatly shortened at the phase transition point of $\delta =(J2\u2212J1)$ for $\Omega t=\pi $. However, from the band energy in Fig. 2, we can know that the broad time-window with gain will make the energy of the light for $\Omega t=\pi $ to be obscured by the larger gain near $\Omega t=0$. Figure 3(c) shows the numerically computed evolution of the normalized pulse intensity for $\delta =1.005(J1+J2)$, and the pulse centered at $\Omega t=0$ is much narrower than that for $\Omega t=\pi $. When $\delta =1.1(J1+J2)$, the corresponding pulse width near $\Omega t=0$ gets wider, as shown in Fig. 3(d), in agreement with our predictions. The corresponding numerically computed evolutions for the full time-dependent dynamical equations are also shown in the supplementary material.

Moreover, we plot the pulse width parameter $\Delta tAPT$ and the saturated gain *g _{s}* for the anti-PT symmetric coupled-ring system, as the value of

*δ*changes from 0 to $2(J1+J2)$. In Fig. 4,

*g*and $\Delta tAPT$ are normalized to the corresponding values for

_{s}*δ*= 0. In the regime of $\delta \u2264J1+J2$, the analytical solution for Eq. (10) [blue solid curve in Fig. 4(a)] exhibits a narrowing of the temporal width of the pulse with the increasing of

*δ*, which agrees well with the numerical simulation results for $\Delta tA\u2212PT$ (marked by red circle). We can notice that the analytical solutions of the $\Delta tAPT$ for $\Omega t=\pi $ are disappeared as $\delta \u2264J1+J2$, which is because that the time-window with gain makes the energy of the light for $\Omega t=\pi $ to be obscured by the larger gain. Meanwhile, the numerical simulation results show that the saturated gain increases gradually with

*δ*and are consistent with the calculated

*g*in Eq. (9), as shown for the regime of $\delta \u2264J1+J2$ in Fig. 4(b). The narrowing of the pulse and the increasing of the saturated gain simultaneously end when the value of

_{s}*δ*is larger than $J1+J2$. The calculated pulse width parameters $\Delta tAPT$ for $\delta \u2265J1+J2$ in Eqs. (13) and (14) are plotted in Fig. 4(a) and show that the further increasing of

*δ*broadens the width of the pulse. In addition, the pulse width in the vicinity of $\Omega t=0$ [green solid curve in Fig. 4(a)] displays a great shortening as

*δ*is approaching to $J1+J2$, and is much narrower than that for $\Omega t=\pi $ [black solid curve in Fig. 4(a)]. Therefore, the temporal width of the pulse generated by the double-ring system is sensitive to the phase transition point of $\delta =J1+J2$, at which the anti-PT symmetry of the system is completely broken, while the saturated gain for the regime of $\delta \u2265J1+J2$ is almost constant, in accordance with the analytic theories for Eqs. (11) and (12). Moreover, the numerical simulation results for $\delta >J1+J2$ (red circle) are still in agreement with our theoretical analysis. Thus, we can conclude that our analytic theory can well describe the active ML of the anti-PT symmetric double-ring system. In addition, the output pulses experience a great shortening in the vicinity of the phase transition point for the completely anti-PT symmetry broken, which shows high sensitivity and suggests the application in the sensitive sensor.

In summary, we constructed an anti-PT symmetric SSH lattice in the frequency dimension based on a double-ring system with the antisymmetric AM dynamic modulations. In the time domain, the time-window with gain of the anti-PT symmetric system can be controlled by the phase-mismatching of the AM modulations, which is flexibly and conveniently able to be used to shorten the temporal width of the pulse. In addition, the shortening of the pulse is highly sensitive to the phase transition point, at which the anti-PT symmetry is completely broken. This work exploits the concept of anti-PT symmetry in lasing systems to realize the active ML and shows broad application prospects in ultrafast spectroscopy and ultra-high sensitive sensors. Additionally, this work provides a convenient way to construct various non-Hermitian topological insolators in the synthetic frequency dimension, which is difficult to be realized in realistic systems. Then, we can expect to explore the interesting physical phenomenon of the synthetic non-Hermitian topological lattices in further studies.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the detailed derivations of the anti-PT symmetric SSH frequency lattice, the validity of the tight-binding model with RWA, and the numerically computed evolutions for the full time-dependent dynamical equations.

This work was supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ23A040001), the general project of Zhejiang Provincial Education Department (No. Y202146469), and the Open Foundation Project of Hubei Provincial Key Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of Technology (No. 202203).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yiling Song:** Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Software (equal); Writing – original draft (lead); Writing – review & editing (lead). **Shaolin Ke:** Funding acquisition (supporting); Investigation (supporting). **Yuelan Chen:** Investigation (supporting). **Mingfeng Wang:** Investigation (supporting); Project administration (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.