We consider an actively mode-locked laser system with parity-time symmetry. The system consists of two ring cavities, each incorporating an amplitude modulator operating at the same modulation frequency, but with the modulation phases differing by *π*. We show that spontaneous parity-time symmetry breaking can be used to shorten the temporal width of the pulse generated through active mode locking in this system. Our work highlights the importance of applying the concept of parity-time symmetry in pulsed laser systems.

## I. INTRODUCTION

Starting from the pioneering work in Ref. 1 suggesting that a non-Hermitian Hamiltonian with parity-time ($PT$) symmetry can have a purely real eigenvalue spectrum, there are now a large body of studies exploring physical phenomena related to $PT$ symmetry in a variety of physical systems.^{2–23} As one significant recent development, lasing action in $PT$-symmetric systems has been explored, where enhanced control of continuous-wave lasing is achieved due to spontaneous breaking of the $PT$ symmetry.^{24–32}

In this article, as a step further to exploit the concept of $PT$-symmetry in lasing systems, we consider an actively mode-locked laser system exhibiting $PT$ symmetry. Active mode locking is an important technique for generating short pulses from a laser.^{33–40} A standard actively mode-locked system consists of a ring resonator incorporating an amplitude modulator. Short pulses can be generated when the modulation frequency matches the free spectral range of the ring. Although the shortest pulses in mode-locked systems are generated with passive mode locking techniques, in many situations, active mode locking is still extensively applied. For example, it has been noted that active mode locking is the method of choice for generating short pulses in quantum cascade lasers due to the unique gain characteristics of quantum cascade lasers.^{40–44}

In our work, we consider a system consisting of two rings coupled together. Within each ring, there is an amplitude modulator. Such a system can be described by a Hamiltonian that has a $PT$ symmetry when the phases of the modulators differ by *π*. We show that, with the same modulation frequency and gain bandwidth, a much shorter pulse can be generated in the two-ring system as compared to the one-ring system, due to a spontaneous breaking of the $PT$ symmetry. Related to but distinct from our work, Ref. 45 studied active mode locking in a single cavity with both amplitude modulation and frequency modulation and noted the signatures of $PT$ phase transition in the pulse dynamics. However the system does not possess $PT$ symmetry. Moreover, our main observation here is that $PT$ phase transition can be used to shorten the pulse in an actively mode-locked system. This observation has not been previously noted.

## II. THEORETICAL ANALYSIS FOR TWO-RING SYSTEM WITH PARITY-TIME SYMMETRY

To illustrate the physics of active mode locking in a parity-time symmetric system, we first briefly review a single ring resonator with free-spectral-range Ω_{R} incorporating an amplitude modulator [see Fig. 1(a)]. In this article, we assume that light travels along only one direction in the rings and that the waveguide forming the ring has zero group velocity dispersion. The variation of the temporal shape of a pulse after a round trip is described by^{46}

where Ψ(*T*, *t*) is related to the electric field amplitude. Here *T*_{R} = 2*π*/Ω_{R} is the round-trip time. *T* is the slow time variable that labels the number of round trips that the pulse goes through. *t* is the fast time variable (−*T*_{R}/2 < *t* < *T*_{R}/2). *κ*, Ω, and *ϕ* are the modulation depth, the modulation frequency, and the modulation phase, respectively. In Eq. (1), we place *i* in both sides so that the equation has the same form as the Schrödinger equation. We set Ω = Ω_{R} throughout the paper. Since cos(Ω*t* + *ϕ*) oscillates between positive and negative values, practically, Eq. (1) can be implemented as a system with a time-independent background gain and a modulator with a time-dependent loss. The dynamics in Eq. (1) can alternatively be written in the frequency domain.^{47–51} For this purpose, we expand Ψ(*T*, *t*) by

and rewrite Eq. (1) as

Equation (3) shows that such a modulated ring is described by a one-dimensional tight-binding lattice model with a non-Hermitian Hamiltonian having a complex coupling constant, as shown schematically in Fig. 1(b).

Since the system of Eq. (3) is periodic along the frequency axis, analogous to a regular periodic structure in real space, one can define a band structure relating the eigenvalue of the Hamiltonian to the wavevector along the frequency axis. The eigenvalues of the Hamiltonian correspond to the Floquet quasi energy of the modulated system of Eq. (1), whereas the wavevector along the frequency axis in fact is the fast time variable *t* in Eq. (1). For the system, the band structure is in fact *λ*(*t*) as defined in Eq. (1). The band structure of this system with *ϕ* = 0 from Eq. (1) is shown in Fig. 2(a). The quasi energy is purely imaginary. And its imaginary part oscillates sinusoidally between the positive and negative values, indicating the time windows with either gain or loss. The existence of a time window where the system exhibits net gain is essential to active mode locking in such a system.^{36,37,52}

Now, we consider two identical ring resonators. Each ring is modulated in the same fashion as above [see Fig. 1(c)]. The two rings couple to each other evanescently with a coupling strength *η*. We assume that the modulation phases [*ϕ* in Eq. (1)] are chosen to be 0 and *π* for the two rings, respectively. The system can be described by

Here *a*_{n} and *b*_{n} correspond to the amplitudes of the *n*Ω frequency components in the two rings, respectively. The different signs in the coupling constant along the frequency axis arise from the *π* phase difference in the modulation phase. Equations (4) and (5) show that such a two-ring system can be described by a synthetic lattice structure along the frequency axis, as shown in Fig. 1(d).

where the Hamiltonian *H*(*t*) = *ησ*_{x} + *i*2*κ* cos(Ω*t*)*σ*_{z}, with *σ*_{x,z} being the Pauli matrices. The Hamiltonian *H*(*t*) is $PT$-symmetric with the definition of an anti-unitary time-reversal operator $T$ defined by $TH T \u2009\u2009 \u2212 1 = H * $, and a unitary parity operator $P$ defined as $PH P \u2212 1 = \sigma x H \sigma x \u2212 1 $.

The eigenvalues of *H*(*t*) are given by

Recall the equivalence of the fast time variable *t* and the wavevector along the frequency axis. Equation (7) thus defines the band structure of this system. For the eigenstates of *H*, the $PT$ symmetry is preserved if |2*κ* cos(Ω*t*)| ≤ *η* for which case the eigenvalues are real, while the $PT$ symmetry is spontaneously broken if |2*κ* cos(Ω*t*)| > *η* and the eigenvalues become imaginary.^{4} We plot both the real and imaginary parts of *λ*_{±}(*t*) in Figs. 3(a) and 3(b) for various *η* values. When *η* = 0, the $PT$-symmetry is spontaneously broken for all *t*. And the system exhibits a time window with gain with a width of *T*_{R}/2. As *η* increases, the $PT$-symmetry remains broken near *t* = 0 and *t* = 0.5*T*_{R}. Within a time window centered at *t* = 0.25*T*_{R}, on the other hand, the $PT$-symmetry is preserved where imaginary parts of the eigenvalues become zero. Thus, as *η* increases, the time window with gain narrows. This process continues until *η* > 2*κ*, where the $PT$-symmetry is preserved for all *t*. The eigenvalues then become completely real, and there is no longer any time window with either gain or loss.

This effect of the narrowing of the time-window with gain is directly related to the $PT$ symmetry of the system. In Figs. 3(c) and 3(d), we plot the cases where we vary the differences in the modulation phases of the rings. As the modulation phase difference deviates from *π*, the system no longer has $PT$ symmetry. While various bifurcation phenomena remain, imaginary parts of the eigenvalues remain non-zero for the entire modulation period.

## III. ACTIVE MODE LOCKING IN PARITY-TIME SYMMETRIC SYSTEM

The ability of a mode-locked system to generate a short pulse is directly related to the existence of a time-window with gain. Therefore, intuitively, we expect that the narrowing of the time window with gain in the $PT$ symmetric system should translate into the capability of the $PT$ symmetric system to generate a shorter pulse as compared with a regular actively mode-locked system. To confirm such an intuition, we now provide a direct calculation of the pulse width in the $PT$ symmetric system, using the standard equations for the actively mode-locked system which incorporates the effects of gain saturation and finite spectral bandwidth for the gain.

We first briefly review the formalism of solving a single mode-locking resonator for pedagogical purposes. We consider a single ring resonator under the amplitude modulation in Fig. 1(a). Inside the resonator, a gain medium is introduced. This system can be described by^{36–38,45,52}

Here we assume that the gain has a Lorentzian lineshape. Ω_{g} is the bandwidth of the gain medium. *g* is the saturated gain at the center of the gain spectral line and will be determined self-consistently. *l* is the loss of the resonator, which includes the loss due to output coupling. Near *t* = 0, which is the center of the time window with gain, as shown in Fig. 1(a), one can approximate Eq. (8) as^{36,37,52}

The steady-state solution is

with the saturated gain

and with the pulse width parameter

Now we consider the $PT$-symmetric mode-locking system described by

For two rings under the $PT$-symmetric modulation with 0 < *η* < 2*κ*, the time windows with gain are centered at *t* = 0 and *t* = 0.5*T*_{R} [see Fig. 3(b)]. Here we focus on the analysis in the vicinity of *t* = 0. The analysis for the solution near *t* = 0.5*T*_{R} is similar. Equations (13) and (14) can be rewritten as

where $\xd4 ( t ) \u2261i g \u2212 l \u2212 2 \kappa + g \Omega g 2 \u2202 2 \u2202 t 2 $, *H*(*t*) is defined in Eq. (6), and $|\Phi ( T , t ) \u2261 \Psi A , \Psi B T $. We define the left eigenstates ⟨±| of *H*(*t* = 0) with ⟨±|*H*(*t* = 0) ≡ ⟨±|*λ*_{±}. From Eq. (15),

where we have expanded *H*(*t*) in the vicinity of *t* = 0. We expect the steady state to be dominated by its component in |+⟩ which experiences gain, since its component in |−⟩ experiences loss and hence that component decays as the system evolves. Therefore, we ignore the second term in the brace in Eq. (16) to obtain

where Ψ(*T*, *t*) = ⟨+|Φ(*T*, *t*)⟩. The first term in the brace in Eq. (16) gives the last term in Eq. (17). The steady solution of Eq. (17) again is a Gaussian of the form of Eq. (10), corresponding to a saturated gain,

with a pulse width parameter

In obtaining the last result in Eq. (19), we note that in both the single-ring and two-ring cases, *g*_{s} is dominated by the cavity loss rate *l*. And hence as an approximation, we can assume *g*_{s} to be the same in both cases. Equation (19) thus predicts significant shortening of the pulse due to the $PT$ symmetry breaking. In order to obtain Eq. (19), in Eq. (17), we made an expansion of *H*(*t*). This expansion is valid only when the resulting pulse fits in the gain time window, i.e.,

or

Hence, Eq. (19) should be valid except in the regime where *η* → 2*κ*. The pulse width in time does not decrease to zero.

When *η* > 2*κ*, the $PT$-symmetry is preserved for all *t* in the system. In this regime, active mode locking also occurs due to the effective phase modulation introduced by the modulators. The largest gain occurs at *t* = ±*T*_{R}/4. From Eqs. (13) and (14), performing a Taylor expansion near *t* = ±*T*_{R}/4 results in

From which, we obtain the steady-state solution,^{52}

with the pulse width parameter

The saturated gain is

Therefore, further increasing *η* beyond 2*κ* increases the pulse width in time.

As an illustration of the theory, we compare it to the results from numerical simulations of Eqs. (13) and (14). In the simulations, *g*, *l*, and *κ* denote changes to the field amplitudes per round trip. Their numerical values are taken from the experiments on the actively mode-locked quantum cascade laser system.^{42} The gain linewidth is Ω_{g} = 6 THz. The cavity round-trip length is *L* = 1 m, and the modulation frequency is Ω = 300 MHz which is equal to the free-spectral range of such a cavity. The round-trip loss is *l* = 0.5, and the modulation depth is 2*κ* = 0.1. These choices of parameters result in the saturated round-trip gain *g*_{s} ≈ 0.5 and Δ*t* = 7.9 ps with a single ring, which gives a full width at half maximum of 13.2 ps consistent with the theory and experiment.^{40–44}

In Fig. 4, we consider the coupled-ring system and plot the pulse width parameter $\Delta t P T $ as well as the saturated gain *g*_{s} as we vary *η* from 0 to 3*κ*. $\Delta t P T $ and *g*_{s} are normalized to Δ*t* in Eq. (12) and $ g s 0 $ in Eq. (11) at *η* = 0, respectively. In the $PT$ symmetry breaking regime (*η* < 2*κ*), as *η* increases, we observe the narrowing of the temporal width of pulse. In this regime, the saturated gain *g*_{s} increases as the pulse width in time narrows. The narrowing ends when *η* is immediately below 2*κ* [the inset in Fig. 4(a)]. The analytic theory agrees very well with the numerical calculations and also provides a fairly accurate prediction of the value of *η* where the pulse narrowing stops. The use of the coupled ring scheme results in a maximum reduction of the pulse length by a factor of 4.4, as compared to the single ring system. When applied to the quantum cascade laser system, this scheme should result in a pulse with a full width at half maximum of 3 ps. When *η* ≥ 2*κ*, $PT$-symmetry is preserved. Increasing *η* broadens the pulse width in time, whereas the saturated gain remains a constant, in consistency with the analytic theory.

## IV. DISCUSSION AND CONCLUSION

In the theory above, for simplicity, we do not consider the effects of the group-velocity dispersion. For typical values of the group-velocity dispersion in actively mode-locked quantum cascade laser systems,^{53} the estimated pulse broadening due to the group-velocity dispersion is ∼3 orders of magnitude smaller than the standard pulse duration in actively mode-locked quantum cascade laser systems.^{40–44} Moreover, it has been shown that incorporating the effects of the group-velocity dispersion and the self-phase modulation properly can lead to further shortening of the pulse in a single ring system.^{54} For further work, it will be worthwhile to seek to combine the $PT$-symmetry concept, with the engineering of self-phase modulation and group-velocity dispersion, to further shorten the pulse. In addition, in the derivation, we assume that the two rings have the same length. In practice, the results would apply as long as the resonant frequencies of the two rings match with each other over a substantial range of frequencies. Thus one can use a number of tuning mechanisms to try to align these frequencies even in the case when the length of the rings is not the same.

In summary, we have shown that the concept of $PT$ symmetry can be applied in the actively mode-locked system to achieve a significant reduction of the pulse width in time. Our work expands the potential application of the concept of $PT$ symmetry in the lasing system by showing that these concepts can be fruitfully applied in the pulsed laser system.

## ACKNOWLEDGMENTS

This work is supported by a Vannevar Bush Faculty Fellowship from the U.S. Department of Defense (Grant No. N00014-17-1-3030) and the U.S. Air Force Office of Scientific Research Grant No. FA9550-16-1-0010).