Non-reciprocal frequency conversion in a non-Hermitian multimode nonlinear system

Abstract

Nonlinear optics has become the workhorse for countless applications in classical and quantum optics, from optical bistability to single photon pair generation. However, the intrinsic weakness of optical nonlinearity and reciprocity of nonlinear interactions generally places stringent limits on the efficiency of nonlinear optical processes and their ability to be tailored for advanced applications in multimode systems. Here, motivated by recent advances in using non-Hermitian photonics and gain/loss engineering to enable non-reciprocal light transport, we explore how the interplay between non-Hermiticity and optical nonlinearity leads to a fundamentally new regime of nonlinear frequency conversion. We show how non-Hermitian coupling between discrete frequency modes can result in non-reciprocal flow of energy in a frequency dimension, closely resembling the non-Hermitian skin effect (NHSE). Applying our theory to a multimode nonlinear cavity supporting cascaded nonlinear processes, we demonstrate chiral energy flow in a frequency dimension, leading to long-range frequency shifts of quasi-continuous wave sources, shaped frequency combs robust to defects and disorder, terahertz (THz) generation far exceeding the Manley-Rowe limit, and nonlinear multimodal limit cycles for multi-frequency pump-probe spectroscopy.

Introduction

Nonlinear optical systems have long been a cornerstone of advanced photonics, enabling a wide range of applications from frequency conversion^1,2 and high-speed communication³ to quantum information processing^4,5. More recently, the focus has shifted to multimode nonlinear systems, which offer the potential for more complex interactions through nonlinear processes. For instance, frequency combs, which consist of many discrete frequency modes, have found widespread application in precision metrology and high-capacity data transmission^6,7,8. However, traditional nonlinear optical systems are constrained by the intrinsic optical properties of nonlinear materials, which are generally weak and obey laws of reciprocity. This restricts conversion efficiencies and the ability to manipulate nonlinear energy flow for more sophisticated functionalities. Achieving control over nonlinear interactions in multimode systems could enable breakthroughs such as overcoming the standard efficiency limits in frequency conversion^9,10,11 and realizing new topological phenomena in real and synthetic frequency dimensions^12,13,14. Despite these promising applications, control over multimode nonlinear systems has been difficult to achieve, owing to high dimensionality and the intrinsic complexity of many nonlinear interactions.

The introduction of non-Hermitian elements into optical systems has opened new avenues for controlling light-matter interactions and energy flow in real and synthetic dimensions^{15,16,17,18,19,20,21}. These systems can exhibit unique phenomena not seen in traditional Hermitian systems, such as exceptional points^22,23,24, non-reciprocity²², parity-time (${{{\mathcal{P}}}}{{{\mathcal{T}}}}$) symmetry breaking²⁵, and the non-Hermitian skin effect (NHSE)^26,27,28. Recent studies have shown that breaking reciprocity in a discrete synthetic frequency dimension can realize non-Hermitian Hamiltonians and topological windings in the photonic band structure^29,30. However, the intersection of non-reciprocity in a frequency dimension and nonlinear frequency conversion has not been explored, although exciting related work in the optomechanical domain has recently been reported³¹, in addition to reports of nonlinear skin modes and shaping of quantum noise in non-reciprocal Hatano-Nelson systems with on-site Kerr nonlinearity^32,33. Overall, considerable attention has been given to breaking spatial reciprocity in optical and optomechanical systems^{34,35,36,37,38,39,40,41}, with a notable gap existing in the interplay between optical nonlinearity and non-reciprocal coupling in multimode nonlinear systems.

In this work, we investigate a multimode nonlinear cavity system that leverages both Hermitian and non-Hermitian interactions to achieve enhanced control over nonlinear energy flow. Our system features a cavity supporting multiple frequency modes that are nonlinearly coupled to a common idler mode. This configuration supports cascaded parametric upconversion and downconversion processes, forming a frequency comb. We demonstrate that the interplay between nonlinear Hermitian coupling, dissipation, and anti-Hermitian amplitude modulation can be utilized to shape nonlinear frequency conversion processes. By fine-tuning the balance between nonlinearity, dissipation, and amplitude modulation, we achieve non-reciprocal frequency conversion and enhanced energy localization in a chiral mode. This boosts the energy conversion efficiency for both the chiral mode as well as the idler mode orders of magnitude above efficiencies imposed by reciprocity. Our results further reveal how nonlinear frequency conversion can be controlled using system parameters to generate frequency combs of arbitrary asymmetry, with the asymmetry remaining robust to defects and disorder in the frequency comb. We also identify different behaviors (phases) in the nonlinear, non-Hermitian system’s mean-field dynamics, demonstrating stable multimodal limit cycles that could provide a new resource for time-domain multiplexing and multi-frequency pump-probe spectroscopy⁴².

Results

Theory and system description

The system under consideration is depicted in Fig. 1a, representing a multimodal nonlinear cavity that supports cascaded three-wave mixing processes mediated by a common idler bath mode at frequency ω_T. A pump mode at ω₀ initiates cascaded parametric upconversions and downconversions mediated by the common idler mode to create blueshifted and redshifted modes relative to the pump mode, forming a comb with spacing ω_T. Energy conservation for each three-wave mixing process reads ω_n = ω_n−1 − ω_T, where ω_n>0 corresponds to the comb mode associated with the generation of n idler photons, so that ω_n = ω₀ − nω_T.

Fig. 1: Shaping nonlinear energy flow in a non-Hermitian nonlinear multimode cavity.

a Tight-binding schematic of nearest-neighbor couplings in a frequency multimode system with second-order nonlinearity. The interplay between nonlinear Hermitian coupling (NL) mediated by a common idler mode and the anti-Hermitian coupling generated by amplitude modulation (AM) at the idler frequency breaks reciprocity in the frequency conversion (κ⁺ ≠ κ⁻), which can create unidirectional energy flow in the frequency dimension. b The presence of anti-Hermitian coupling through AM or Hermitian coupling through NL alone is insufficient to break symmetry in the frequency conversion process. However, the presence of both couplings simultaneously generates an effective non-Hermitian system that can bias frequency downconversions and create an asymmetric frequency comb. c In the nonlinear, non-Hermitian system, the interplay between NL and AM is reflected in the temporal dynamics of the modal amplitudes since the nonlinear coupling is time-dependent, βa_T(t). The initial frequency conversion is symmetric since κ⁺= − κ⁻ (purely AM coupling). As the idler mode is populated, βa_T → κ. A threshold is reached where reciprocity in upconversion/downconversion is broken and an NHSE-type phenomenon in the frequency dimension occurs, with all energy flowing to the chiral mode a_N. In (b, c), 2N + 1 = 19, β₀= 2 × 10⁻⁴ J^−1/2, κ = 7.8 × 10⁹ s⁻¹, Q₀= 9 × 10⁵, μ = 0.1γ, Q_T= 10³, and ℏω₀∣s₀∣²= 5 MW. In this system, the pump frequency ω₀= 2π ⋅ 282 THz, and the idler frequency ω_T= 2π ⋅ 1.06 THz.

The system additionally undergoes amplitude modulation at frequency ${\omega }_{{{{\rm{mod}}}}}={\omega }_{T}$, generating anti-Hermitian coupling between neighboring modes (we consider the case ${\omega }_{{{{\rm{mod}}}}}\ll {\omega }_{T}$ in the Supplementary Information). We assume here that the modulation is provided by an external drive that gives rise to a modulation index J ≪ 1. However, amplitude modulation can also be self-started by the cascaded optical nonlinearity without an external drive at ${\omega }_{{{{\rm{mod}}}}}$, as shown in the Supplementary Information. When externally driven, ${\omega }_{{{{\rm{mod}}}}}$ sets the idler frequency ${\omega }_{T}={\omega }_{{{{\rm{mod}}}}}$ as long as the three-wave mixing process {ω₀, ω₁, ω_T} lies within the nonlinear crystal’s phase matching bandwidth, since the system is not seeded⁴³. When self-driven, the difference frequency ω_T = ω₀ − ω₁ naturally sets the modulation frequency ${\omega }_{{{{\rm{mod}}}}}$. The composite Hamiltonian reads (see Methods)

$$\frac{{{{\mathcal{H}}}}}{\hslash }= ({\omega }_{T}-i({\gamma }_{T}+{\mu }_{T})){\hat{a}}_{T}^{{{\dagger}} }{\hat{a}}_{T}+{\sum }_{n}({\omega }_{n}-i(\gamma+\mu )){\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n}\\ +i{\sum }_{n}({\beta }_{T,n,n-1}{\hat{a}}_{T}^{{{\dagger}} }+\kappa ){\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n-1}\\ -i{\sum }_{n}({{\beta }}_{T,n,n-1}^{*}{\hat{a}}_{T}-{\kappa }^{*}){\hat{a}}_{n}{\hat{a}}_{n-1}^{{{\dagger}} }\\ +i\sqrt{2\gamma }{s}_{0}({\hat{a}}_{0}^{{{\dagger}} }-{\hat{a}}_{0}).$$

(1)

From this Hamiltonian, equations of motion for the mean field amplitudes of the comb modes a_n (with energy E_n = ℏω_n∣a_n∣²) and idler bath mode a_T (with energy E_T = ℏω_T∣a_T∣²) in the interaction picture read

$${\dot{a}}_{T}= {\sum }_{n}{\beta }_{T,n,n-1}{a}_{n}^{*}{a}_{n-1}-({\gamma }_{T}+{\mu }_{T}){a}_{T}\\ {\dot{a}}_{n}= -({\beta }_{T,n,n+1}^{*}{a}_{T}-{\kappa }^{*}){a}_{n+1}+({\beta }_{T,n,n-1}{a}_{T}^{*}+\kappa ){a}_{n-1}\\ -(\gamma+\mu ){a}_{n}+\sqrt{2\gamma }{s}_{n}{\delta }_{n,0},$$

(2)

where ${\beta }_{T,n,n-1}={\beta }_{0}\sqrt{\hslash {\omega }_{T}{\omega }_{n}{\omega }_{n-1}}$ is the nonlinear coupling strength with β₀ a constant dependent on the second-order nonlinearity (see Supplementary Information), κ the strength of amplitude modulation, s_n the strength of external pumping, and γ, μ the outcoupling and intrinsic losses. We provide numerical estimates for these physical parameters in the Supplementary Information. Although we simulate systems in the text with the full frequency dependence of β_T,n,n−1, because the comb span is generally much smaller than the frequency of an individual comb mode, we will approximate β_T,n,n−1 → β as constant for simplicity in further expressions. As we will show, the interplay between on-site non-Hermiticity (loss) and non-Hermitian intermodal coupling in this model will enable us to have a new degree of freedom in controlling nonlinear energy flow in a frequency dimension and shaping nonlinear frequency conversion. We note that, in the Hamiltonian description of this system, a unitary transformation can be used to move the off-diagonal non-Hermiticity in the frequency mode basis onto the diagonal (see Methods). Specifically, this shows how nonlinearity can play the role of gain/loss in the frequency dimension, which will be crucial for non-reciprocal frequency conversion.

The nonlinear system of differential equations specified by Eq. (2) can be numerically solved as a function of time starting from an initial condition where the fields are zero. If the system reaches steady state, the power conversion efficiency in a given mode can be computed as (see Supplementary Information for derivation using temporal coupled mode theory)

$${\eta }_{k\ne 0}=\frac{2{\omega }_{k}{\gamma }_{k}| {a}_{k}{| }^{2}}{{\sum }_{{k}^{\prime} }{\omega }_{{k}^{\prime} }{\left(-{s}_{{k}^{\prime} }+\sqrt{2{\gamma }_{{k}^{\prime} }}{a}_{{k}^{\prime} }\right)}^{2}},$$

(3)

where ${k}^{\prime}$ indexes over all modes in the system, k denotes any mode that is not coherently pumped, and μ ≪ γ. The denominator of this expression includes both the input pump power and the power drawn from the amplitude modulator. It is useful to compare the power generation efficiency to the Manley-Rowe (MR) limit⁴⁴. The MR limit describes the optimal efficiency in a difference-frequency generation process where every high-energy pump photon produces a lower-energy signal and idler photon. In the case of the idler mode in our system, the MR limit is then given by ω_T/ω₀ (see Supplementary Information for derivation).

When the system does not reach a steady state, interesting behaviors such as stable limit cycles may exist, which we will also explore.

Chiral energy flow in nonlinear multimode cavities

A key prediction of non-Hermitian point gap topology is the non-Hermitian skin effect (NHSE), which occurs when the hopping reciprocity in a lattice is broken, enabling bulk eigenstates of the Hamiltonian to be localized on the boundary of the lattice^45,46,47. In one dimension, the Hatano-Nelson model is a simple example exhibiting the NHSE, where reciprocity is broken via unequal nearest-neighbor rightward and leftward hoppings^48,49. In our system, we realize a nonlinear analog of the Hatano-Nelson model, where nonlinear Hermitian coupling and non-Hermitian amplitude modulation combine to give non-reciprocal hopping amplitudes in the frequency dimension. In Fig. 1b, c, we show how this can result in unidirectional (or chiral) energy flow towards the lowest-frequency mode in our frequency lattice. This chiral energy flow is not accessible with conventional optical nonlinearity or electro-optic modulation on its own.

The chiral energy flow is associated with the formation of an asymmetric frequency comb centered around ω₀. As described in Methods, we can quantify the comb asymmetry as

$$\xi=\frac{\log | {r}_{1}|+\log | {r}_{2}| }{2\log | {r}_{2}| },$$

(4)

where we analytically solve the steady state of Eq. (2) using a non-Hermitian Bloch wave ansatz a_−n+1 = r₂a_−n, a_n+1 = r₁a_n for the blueshifted (n < 0) and redshifted (n > 0) branches of the comb, respectively.

We now show how the asymmetry in the frequency comb can be controlled through the interplay between nonlinear and non-Hermitian coupling. In Fig. 2a, we plot the asymmetry parameter ξ over a sweep of the quality factor of the frequency comb modes Q₀. When the comb modes are very leaky (low Q₀), the nonlinear coupling remains weak and a symmetric comb is created through the amplitude modulation (Fig. 2b, (I)). As Q₀ is increased, nonlinear energy can more effectively flow in the frequency space lattice. The asymmetry initially increases slowly, then undergoes a sharp increase as the redshifted part of the comb (n > 0) is pulled upwards. This occurs because βa_T → κ, amplifying coupling to lower-frequency modes while simultaneously suppressing the nearest-neighbor hopping to higher-frequency modes.

Fig. 2: Asymmetric and highly efficient frequency conversion through non-reciprocal frequency conversion.

a For low Q₀ (strong dissipation in comb modes), the nonlinear coupling is very weak and AM creates a symmetric comb, ξ ≈ 0 (because γ ≫ κ, modes other than the pump mode are negligibly occupied). As Q₀ is increased, the interplay between AM and NL begins to dominate the system dynamics, entering the regime of a high-finesse cavity in frequency space and suppressing frequency upconversion through nonreciprocity. This biases the frequency comb towards redshifted modes. Finally, for the highest Q₀, nearly complete asymmetry is obtained, ξ ≈ 1. The efficiency parameter η_N quantifies how much power is converted into the chiral mode. A sharp increase near the transition to high asymmetry (ξ ≈ 0.5) is observed. b Varying levels of asymmetry in the modal energy distribution for three different values of the Q₀ sweep shown in (a). The interference pattern in (III) emerges due to non-open boundary conditions at the frequency boundaries ω_±N, which results in Bloch mode interference in the frequency space lattice. c As the asymmetry is increased through improving the finesse of the frequency space cavity, so too is the THz conversion efficiency. The non-reciprocity strongly biases THz generation (downconversion) over THz annihilation (upconversion) processes. In these simulations, ℏω₀∣s₀∣²= 1 MW, β₀= 10⁻⁴ J^−1/2, κ = 2π ⋅ 1060 MHz, Q_T= 10⁴, and N = 10. An intrinsic loss μ = 0.01γ was assumed. In the absence of AM (NL only), a seed ℏω_T∣s_T∣²= 760 W was added to form the frequency comb, matching the drive power necessary to ensure J ~ 0.1 in the presence of AM. Here, ω₀= 2π ⋅ 282 THz, ω_T= 2π ⋅ 1.06 THz. In (c), in the limit μ → 0, η_THz for the NL only case ranges from 8 × 10⁻⁸ to 6 × 10⁻⁶ over the range of Q₀, effectively zero on the plot.

When the asymmetry reaches ξ = 0.5, hopping to higher frequency modes is completely suppressed, κ = βa_T, and a flat comb for the redshifted cascading orders is created through a balance between gain and loss for each mode. The steady state energy of the blueshifted modes continue to decay from the pump mode ω₀ as ∣r₂∣^−2∣n∣ (Fig. 2b, (II)). For the highest Q₀ considered, nearly perfect asymmetry in the frequency comb is generated (Fig. 2b, (III)). The interference pattern in the steady state modal energy distribution for this last case is due to Bloch interference in the frequency lattice that emerges because the boundaries of this system in the frequency dimension are not impedance matched (i.e., are partially reflecting)⁵⁰. The interference exists because we only change Q₀ in these simulations, but can be removed by additionally tuning the boundary condition Q_N for example (as shown in the Supplementary Information).

Also shown in Fig. 2a is a plot of the power conversion efficiency for the chiral mode η_N. We observe a sharp increase in η_N near the transition to high asymmetry. η_N asymptotes near 86% for high Q₀, which is orders of magnitude larger than the efficiency for a system with only nonlinearity and no amplitude modulation. Note that the latter system requires a seed to generate comb modes and the THz idler mode, so we set ℏω_T∣s_T∣² equal to the input power of the drive for amplitude modulation in the former system at a modulation index J ~ 0.1 (see Supplementary Information for details).

High efficiency terahertz generation

The unidirectional energy flow enabled by non-reciprocal frequency conversion in the frequency dimension biases frequency downconversions that produce idler photons at ω_T (and suppresses frequency upconversions that annihilate idler photons) in the cascaded nonlinear system. When ω_T lies at THz frequency, the system can realize high efficiency THz generation.

This concept is shown in Fig. 2c, where the efficiency of conversion into the propagating THz idler mode outcoupled from the cavity increases with Q₀. Notice that the curve of THz efficiency plateaus near ξ ≈ 0.5. At this value of the asymmetry parameter, upconversions are perfectly suppressed, a flat IR comb is generated, and THz generation is enhanced by the number of comb modes. Here, the THz photon generation rate is equally contributed by three-wave mixing between each pair of nearest-neighbor comb modes. As Q₀, ξ increases, the THz generation becomes dominated by three-wave mixing at the low frequency end of the frequency comb, which is populated strongly due to the non-reciprocal frequency conversion.

We observe an order of magnitude enhancement in THz conversion efficiency for the non-reciprocal frequency comb compared to the reciprocal frequency comb without amplitude modulation. We attribute this enhancement to the unique ability to suppress upconversions in the non-Hermitian system and therefore break symmetry between THz-generating and THz-annihilating processes. The critical transition to high asymmetry and chiral mode efficiency (black dotted line in Fig. 2a) is accompanied by a transition to high THz conversion efficiencies.

In Fig. 3, we explore how THz conversion efficiencies strongly surpassing the Manley-Rowe limit can be achieved through (1) longer combs and (2) appropriate dissipation engineering, reaching enhancement factors exceeding 32 times above the Manley-Rowe limit (η_THz > 12%). The former enables more cascading steps to generate THz photons from a single pump photon, while the latter shapes the steady state energy distribution in the frequency lattice to maximize THz-generating nonlinear interactions. In Fig. 3, the dissipation engineering corresponds to partially open boundary conditions and low occupation of the blueshifted comb modes (which correspond to THz annihilation processes). As noted in the Supplementary Information, dissipation engineering is another resource to control nonlinear energy flow, and new capabilities can arise when dissipation engineering is combined with non-reciprocal frequency conversion.

Fig. 3: Breaking the Manley-Rowe limit through non-reciprocal frequency conversion.

Efficiency of terahertz idler mode generation η_THz as a function of comb length with (solid lines) and without (dashed line) dissipation engineering, showing strong enhancement with respect to the Manley-Rowe (MR) limit. In these simulations, ℏω₀∣s₀∣²= 10 kW, β₀= 3 ×  10⁻⁴ J^−1/2, Δ = 2π ⋅ 6.34 MHz, Q_T= 10⁴, and μ = 10⁻²γ. With dissipation engineering, the pump and boundary modes have quality factor Q_N= 4 × 10⁵ and the blueshifted modes have quality factor Q_b= 10². The case with only nonlinearity (NL only) included a seed ℏω_T∣s_T∣²= 0.484 mW, equal to the THz drive power necessary for modulation index J ~ 0.03 (free spectral range FSR = 1 GHz) in the presence of amplitude modulation. The following efficiencies are not shown because they are effectively zero on the plot: η_THz≈ 0 (NL only with Q₀= 10⁷, 4 × 10⁷ and no dissipation engineering), η_THz= 1.38 × 10⁻⁷ (NL only with Q₀= 10⁷ and dissipation engineering). The case for the non-reciprocal system (AM + NL) with ${Q}_{\max }=4\times 1{0}^{7}$ without dissipation engineering is not plotted because it did not reach a steady state within 1 μs.

Note that in Fig. 3 we simulate a pump power of 10 kW, which is achievable with microsecond pulses. This is too low to generate appreciable THz using cascaded nonlinear conversion alone, but nevertheless achieves high η_THz when combined with amplitude modulation to break reciprocity in nonlinear frequency conversion. This highlights the ability of our mechanism to overcome the conventional limitations of high pump power that usually plagues high-efficiency frequency generation owing to the weakness of optical nonlinearity. The high-power THz generation in the nonlinear, non-Hermitian system is enabled through the interplay between optical nonlinearity and non-Hermitian coupling between frequency modes. Although chiral energy flow can be achieved through combined amplitude and phase modulation without optical nonlinearity (as described in the Supplementary Information), the latter is necessary to simultaneously generate idler photons with high efficiency. It is this nonlinear coupling that actually makes the comb robust to chiral energy flow towards lower frequency instead of higher frequency. Reciprocity is only broken when ∣βa_T∣ ~ ∣κ∣, which necessitates preference towards downconversion processes that generate idler photons. Thus, chiral energy flow towards higher energy is not generally allowed by this mechanism, but can be accessed using combined amplitude and phase modulation (see Supplementary Information).

Robustness to defect and disorder

In this section, we show how the non-reciprocal frequency conversion we observe in the frequency dimension preserves the robustness to disorder observed for NHSE⁵¹. In Fig. 4a, we introduce a random leaky mode in the frequency mode, which has a Q factor lower than the other modes in the comb, Q_defect < Q₀. The original skin effect is preserved for large defect strengths, even when the defect mode has a Q factor over 90% below Q₀. When the defect mode becomes too leaky (Q_defect/Q₀ ≲ 1%), the comb terminates at the defect mode, and the new chiral mode becomes the comb mode just before the defect mode.

Fig. 4: Robustness of non-reciprocal frequency conversion to defect and disorder.

a A random low Q-factor defect is introduced in the frequency comb. The non-reciprocal frequency conversion is robust to strong defects, though the strongest defects result in the skin effect terminating just before the defect mode. Here, the modal energy E_n = ℏω_n∣a_n∣² is plotted for various defect strengths Q_defect/Q₀= x, where Q₀ denotes the Q factor of the remaining modes in the frequency comb. b Robustness of non-reciprocal frequency conversion to disorder. Here, the Q factor of all frequency modes in the comb is sampled uniformly from [xQ₀, Q₀]. N_samp= 100 samples are drawn from this uniform distribution and individually simulated, with the shaded regions denoting  ± 1σ standard deviation from the mean modal energies. Shown in the inset is the robustness of the energy of the chiral mode a_N, plotted as the standard deviation normalized by the mean (coefficient of variation), f_N ≡ σ_N/〈E_N〉, where 〈E_N〉 denotes the ensemble average over N_samp samples. In the absence of amplitude modulation κ = 0, the disorder affects f_N much more strongly than the case κ ≠ 0, where reciprocity breaking helps stabilize the modal energy in the chiral mode. System parameters include 2N + 1 = 19, β₀= 5 × 10⁻⁴ J^−1/2, κ = 2π ⋅ 350 MHz, Q₀= 8.86 × 10⁹, Q_T= 500, μ = 0.9γ, ℏω₀∣s₀∣²= 5 MW, ω₀= 2π ⋅ 282 THz, and ω_T= 2π ⋅ 1.06 THz.

In Fig. 4b, we consider disorder in the Q factor distribution for all of the frequency modes in the comb. To model this disorder, we sample each mode’s Q factor individually and uniformly from the distribution [xQ₀, Q₀] with 0 ≤ x ≤ 100%. We see how the skin effect is resistant to disorder, preserving the unidirectional flow of energy in frequency space even for large amounts of disorder 1 − x > 50%. To show that this emerges from the non-Hermitian coupling in the system, we plot in the inset of Fig. 4b the standard deviation in the modal energy of the chiral mode normalized to its mean in the presence (κ ≠ 0) and absence (κ = 0) of amplitude modulation. (The mean and standard deviation are computed over 100 simulated systems with Q factors drawn from the aforementioned uniform Q factor distribution.) The variation in the chiral mode’s energy is larger and scales more sharply with the dimensionless disorder parameter x in the case κ = 0, showing the robustness of the non-reciprocal frequency conversion (κ ≠ 0) to the disorder considered here. This can have a very strong impact for practical applications conventionally limited by dispersion or impurities. In the Supplementary Information, we explore how the system’s non-reciprocal frequency conversion remains robust to comb-cavity mode detuning and dispersion. However, the shaping of the comb profile by dispersion suggests dispersion engineering as an exciting tool, together with the dissipation and non-reciprocity engineering we have explored here, to enable precise control over energy flow in frequency dimensions.

Temporal dynamics and stable limit cycles

We now take a closer look at the temporal dynamics of states supported by the nonlinear, non-Hermitian multimode systems considered here. In Fig. 5, we show how these states can generally be classified as steady state solutions or limit cycles (LCs). Thus far, we have focused on combs generated in steady state. LCs are a particularly interesting class of solutions unique to nonlinear systems that are characterized by stable oscillations around a fixed point⁵². As shown in Fig. 5a, the LC regime can be entered by gradually increasing the quality factor of the idler mode, Q_T. The steady state solutions are distinguished by an approach to equilibrium characterized by damped relaxation oscillations (ROs) and Bloch oscillations, whereas the limit cycle regime is characterized by periodic oscillations in the modal amplitudes that never relax to a steady state. We numerically verified the existence of a fixed point in between the extrema of the LCs. Zooming in on the LC regime, we observe that the repetition rate of the LCs appears to be governed by the amplitude modulation strength κ, while the pulse width is governed by the interplay between on-site loss and gain/loss due to intermodal coupling. For each individual mode, the LCs resemble picosecond pulses with GHz repetition rates. The high powers in this system, particularly near the LC regime, are limited by nonlinear saturation effects, which we model in our simulations by inclusion of a two-photon absorption term in the rate equations with β_TPA ~ 0.1 cm/GW.

Fig. 5: Limit cycles and temporal dynamics in nonlinear, non-Hermitian multimode systems.

a For low quality factors of the idler bath mode (Q_T), the system initially features damped relaxation oscillations (ROs) and the amplitude modulation drives the system towards a stable steady state that features non-reciprocal frequency conversion. As Q_T increases, the system eventually transitions into a regime of stable limit cycles that feature periodic oscillations in modal energy. Finally, larger Q_T cause a return back to damped ROs and Bloch oscillations, with the nonlinearity and amplitude modulation enabling a gain-loss equilibrium. Oscillations in modal energy in the limit cycle regime have a GHz repetition rate, roughly corresponding to the amplitude modulation strength κ. b Phase diagram showing steady state (SS) and limit cycle (LC) regimes as a function of the nonlinear and amplitude modulation strengths. For small κ, a symmetric comb in steady state is produced (ξ, η ≈ 0, with reciprocal frequency conversion (RFC SS)). For larger κ, an asymmetric comb in steady state is produced by the non-reciprocal frequency conversion (NRFC SS, ξ, η > 0). For the largest κ, LCs are present. Along the transition boundary, the temporal dynamics feature long-lasting Bloch solitons where energy bounces back and forth in the frequency space cavity. c Representative time traces of the power in the outcoupled field ${s}_{{{{\rm{out}}}}}(t)={\sum }_{n}{s}_{n,{{{\rm{out}}}}}(t){e}^{-i{\omega }_{n}t}$ corresponding to the phases in (b). Simulation parameters used in (a) are Q₀= 1.1 × 10⁷, κ = 2π ⋅ 100 MHz, β₀= 10⁻⁴ J^−1/2. The pump power is 1 kW and the comb length is 2N + 1 = 21.

We can sweep over the nonlinear and amplitude modulation strengths β, κ to create a phase diagram portraying the distinct temporal dynamics, as shown in Fig. 5b. For small κ, symmetric combs are produced in a steady state irrespective of β (reciprocal frequency conversion, RFC SS). As κ is increased, the non-reciprocal frequency conversion drives the system towards an asymmetric comb in steady state (NRFC SS). The nonlinearity acts as a loss mechanism to balance the gain from amplitude modulation, so that for small nonlinearity β, gain induced by κ drives the system to very high powers that would be limited by nonlinear saturation effects (e.g., TPA). Finally, the largest values of κ in nonlinearity-stabilized systems yield LCs. Like the sharp transition to non-reciprocal frequency conversion, the transition to LCs appears to be quite sensitive to κ, switching sharply from steady state solutions to LCs for almost all values of β. Therefore, beyond a certain threshold, the amplitude modulation stabilizes the LCs, and the LC phenomenon is relatively insensitive to the nonlinear strength. The phase transition boundary is characterized by sustained Bloch oscillations in the frequency space cavity defined by the frequency comb, as shown in the inset⁵³. These oscillations resemble solitons in a frequency dimension, where nonlinear energy flow bounces back and forth inside the frequency space cavity (outside of the LC regime, these oscillations are damped and relax to a steady state).

Finally, in Fig. 5c, we explore the power in the outcoupled field ${s}_{{{{\rm{out}}}}}(t)={\sum }_{n}{s}_{n,{{{\rm{out}}}}}(t){e}^{-i{\omega }_{n}t}$. Both the RFC and NRFC SS phases can show approximate continuous wave operation when one mode has most of the system’s energy (the pump mode (blue) in the former case, the frequency-shifted chiral mode (red) in the latter case). NRFC SS can additionally create flat combs with sinc-type pulse profiles (orange). The LC regime features pulses with GHz repetition rate and THz subpulse dynamics, providing the opportunity to encode information at two different timescales in the optical field.

Discussion

The interplay between nonlinearity and non-Hermitian coupling in the frequency comb systems studied here offers numerous exciting avenues of research for both fundamental science and photonic applications. First, the non-reciprocal frequency conversion we explore enables long-range frequency shifting, realizing coherent sources whose frequency can be tuned simply by shifting the boundary condition along the frequency dimension. In the quantum optical regime, the interaction between optical nonlinearity and non-Hermiticity remains to be explored and may enable the creation of topologically-protected multimode quantum and non-Gaussian states of light⁵⁴. In addition, tailoring non-reciprocal hopping in a frequency dimension offers an exciting resource for quantum walk studies. Furthermore, the multimode limit cycles explored in this work, in addition to being used for multi-frequency pump-probe spectroscopy and time multiplexing, suggest the possibility of harnessing multistability for switching, entangled state generation, and more⁵⁵. Finally, we envision that combining non-reciprocal conversion with techniques such as dissipation engineering^10,56 will lead to even more precise control over nonlinear energy flow in multimode systems.

We briefly comment on experimental platforms capable of realizing the effects described here. Most of the systems we simulate in the main text are based on centimeter-scale free space/monolithic cavities pumped by microsecond pulsed or nanosecond Q-switched lasers, which realize quasi-continuous wave operation at kilowatt and megawatt-level powers, respectively. A further discussion of typical parameters expected in this platform is provided in the Supplementary Information. The THz amplitude modulation (AM) we consider can come in two forms (1) externally-driven AM and (2) self-driven AM. In the first case, a weak THz drive (to ensure a modulation index J ≪ 1) is injected into the AM. The steady state THz energies are several orders of magnitude larger, showcasing the ability of our system to generate high-energy THz radiation from a weak THz seed with high efficiency. In the second case, no external THz radiation is injected into the cavity. Rather, the system is pumped and seeded at two infrared frequencies to initiate the cascaded three-wave mixing, and the resulting THz photons are recycled to drive the AM. We provide a more detailed discussion with specific results for this case in the Supplementary Information. To our knowledge, this is the first proposal of intracavity terahertz amplification by recycling idler photons for electro-optic modulation in an optically driven cavity. Exciting steps in this direction have been taken in demonstrations of THz time-domain electro-optic sampling⁵⁷.

Ring resonators have recently received significant attention in studies of topological effects in synthetic frequency dimensions^29,58,59. For example, one such system used Mach-Zehnder-based amplitude modulation of a fiber ring resonator to realize non-Hermitian band topology. We view this platform as an excellent candidate to realize the effects described here when combined with lithium niobate photonics (or another nonlinear material) to generate nonlinear coupling. On-chip nonlinear microring resonators have also been extensively developed and suggest exciting potential for developing integrated non-reciprocal frequency conversion setups^60,61,62. Advances in THz electro-optic modulation further suggest exciting possibilities for using THz amplitude modulators to generate the THz-spaced non-reciprocal combs considered in this work^63,64,65.

We emphasize that our theory is applicable to different frequency regimes. In particular, THz modulation frequencies were considered here to highlight the well-known difficulty in high-efficiency THz generation that our mechanism addresses. However, optomechanical and circuit quantum electrodynamics platforms are ideal platforms to realize the physics described here at microwave frequencies⁶⁶. Furthermore, MHz/GHz amplitude modulation with THz-based cascaded nonlinearity can lead to excitation of long-range Bloch modes, band structures with very large winding numbers, and formation of a series of flat combs at IR and mid-IR/THz frequencies, as described in the Supplementary Information. Exploiting these two disparate frequency ranges offers the potential for exploring higher-dimensional non-Hermitian topology.

In summary, we have uncovered the phenomenon of non-reciprocal frequency conversion in multimode systems supporting simultaneous non-Hermitian and nonlinear intermodal coupling. We envision widespread adoption of the physical concepts described here in both fundamental studies of non-Hermitian topology in nonlinear systems as well as applications to new classes of high-efficiency nonlinear photonic devices.

Methods

Hamiltonian description of multimode nonlinear system

The non-interacting multimode driven-dissipative Hamiltonian describing our system is the sum of the bare Hamiltonians for each mode and reads⁶⁷:

$${{{{\mathcal{H}}}}}_{0}/\hslash= i\sqrt{2\gamma }{s}_{0}({\hat{a}}_{0}^{{{\dagger}} }-{\hat{a}}_{0})+{\sum }_{n}\left[{\omega }_{n}-i(\gamma+\mu )\right]{\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n}\\ +\left[{\omega }_{T}-i({\gamma }_{T}+{\mu }_{T})\right]{\hat{a}}_{T}^{{{\dagger}} }{\hat{a}}_{T},$$

(5)

where ${\hat{a}}_{n}$ is the annihilation operator for the comb mode at angular frequency ω_n and ${\hat{a}}_{T}$ is the annihilation operator for the idler bath mode at frequency ω_T. This Hamiltonian is non-Hermitian due to on-site loss (γ for outcoupling loss and μ for intrinsic loss) that emerges from the coupling of each frequency mode to the environment. Here, we only consider the mode at ω₀ to be externally pumped (s₀ ≠ 0). The nonlinear interaction between the frequency modes is described by a Hermitian Hamiltonian⁶⁸

$${{{{\mathcal{H}}}}}_{{{{\rm{NL}}}}}=i\beta {\sum }_{n}{\hat{a}}_{T}^{{{\dagger}} }{\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n-1}+{{{\rm{h.c.}}}},$$

(6)

where h.c. denotes the Hermitian conjugate, and, as noted in the main text, we drop the frequency dependence of β for simplicity in the mathematical expressions. We now add amplitude modulation at the frequency ${\omega }_{{{{\rm{mod}}}}}={\omega }_{T}$ to this system, generating an anti-Hermitian Hamiltonian⁶⁹

$${{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}=i\kappa {\sum }_{n}{\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n-1}-{{{\rm{h.c.}}}}.$$

(7)

We consider the case ${\omega }_{{{{\rm{mod}}}}}\ll {\omega }_{T}$ in the Supplementary Information (SI), demonstrating a completely distinct regime of operation. Here, the composite Hamiltonian ${{{\mathcal{H}}}}={{{{\mathcal{H}}}}}_{0}+{{{{\mathcal{H}}}}}_{{{{\rm{NL}}}}}+{{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}$ is non-Hermitian and, from it, we can derive Heisenberg equations of motion in the mean field for the annihilation operators ${\hat{a}}_{n},{\hat{a}}_{T}$, as quoted in the main text:

$${\dot{a}}_{T}= \beta {\sum }_{k}{a}_{k}^{*}{a}_{k-1}-({\gamma }_{T}+{\mu }_{T}){a}_{T}\\ {\dot{a}}_{n}= -({\beta }^{*}{a}_{T}+{C}^{*}-{\kappa }^{*}){a}_{n+1}+(\beta {a}_{T}^{*}+C+\kappa ){a}_{n-1}-({\gamma }_{n}+{\mu }_{n}){a}_{n}+\sqrt{2{\gamma }_{n}}{s}_{n}.$$

(8)

where s_n≠0 = 0. Here, we include for completeness the Hermitian coupling C that can be achieved through phase modulation, though, for examples in the main text, C = 0. An example of asymmetric comb generation using amplitude and phase modulation is provided in the Supplementary Information.

Two-level model with onsite gain and loss

In this section, we show how the non-Hermitian coupling in the multimode nonlinear system manifests as gain and loss. Considering a two-mode basis for neighboring modes {a_n, a_n+1}, we can write the unit cell Hamiltonian as

$${{{\mathcal{H}}}}= \left(\begin{array}{cc}{\omega }_{0}-i\gamma &\kappa+\beta {a}_{T}\\ \kappa -\beta {a}_{T}&{\omega }_{0}-i\gamma \end{array}\right)\\= ({\omega }_{0}-i\gamma ){{{\bf{I}}}}+\left(\begin{array}{cc}0\hfill&\kappa+\beta {a}_{T}\\ \kappa -\beta {a}_{T}&0\hfill\end{array}\right).$$

(9)

One can think of this Hamiltonian as the one-photon transition matrix in the synthetic frequency dimension. Denote the basis in which this Hamiltonian is written $\{\left\vert 0\right\rangle,\left\vert 1\right\rangle \}$ such that $\left\vert 0\right\rangle \equiv {a}_{n}^{{{\dagger}} }\left\vert {{{\rm{vac}}}}\right\rangle,\left\vert 1\right\rangle \equiv {a}_{n+1}^{{{\dagger}} }\left\vert {{{\rm{vac}}}}\right\rangle$, where $\left\vert {{{\rm{vac}}}}\right\rangle$ denotes the (multimode) vacuum state. The matrix elements of H can be derived as $\left\langle m\right\vert {{{\mathcal{H}}}}\left\vert n\right\rangle$, with ${{{\mathcal{H}}}}$ the composite Hamiltonian written above.

Performing a change of basis $\left\vert 0\right\rangle,\left\vert 1\right\rangle \to \frac{\left\vert 0\right\rangle \pm i\left\vert 1\right\rangle }{\sqrt{2}}$, we find the interaction Hamiltonian now reads

$$\begin{array}{r}{{{\mathcal{H}}}}^{{\prime} }_{{{{\rm{int}}}}}=\left(\begin{array}{cc}-i\beta {a}_{T}&i\kappa \\ -i\kappa &i\beta {a}_{T}\end{array}\right),\end{array}$$

(10)

so that the nonlinearity now manifests as on-site gain and loss, and the amplitude modulation acts as a Hermitian coupling.

Asymmetry parameter for non-reciprocal frequency conversion

To quantify the asymmetry in the redshifted and blueshifted branches of the comb around the pump mode ω₀, we will assume a long frequency comb with negligible boundary effects (see Supplementary Information for boundary-related effects and further details on the following method). In this case, we can solve the rate equations in steady state using an ansatz a_n+1 ∝ r₁a_n, a_−n+1 ∝ r₂a_−n, respectively, for the redshifted and blueshifted branches (n > 0 here). This ansatz is physically motivated by the non-Hermitian analog of Bloch bands (where in the Hermitian case r₁ = r₂ = e^ik with k the wavenumber). Both r₁, r₂ must satisfy the steady state condition

$$\left(\kappa -\beta {a}_{T}\right)r+\left(\kappa+\beta {a}_{T}\right){r}^{-1}-\gamma=0,$$

(11)

where a_T denotes the steady state THz mode amplitude. From the solutions of this equation (see Supplementary Information), we can calculate the asymmetry parameter ξ used in the main text. Graphically, in a logarithmic plot of modal energy, ξ represents a normalized ratio of the sum of the slopes of the redshifted (2∣r₁∣) and blueshifted (2∣r₂∣) branches. When amplitude modulation is absent and dissipation exceeds the nonlinear rate (κ = 0, γ ≫ ∣βa_T∣), a symmetric comb with ξ = 0 is generated. When the strength of amplitude modulation approaches the nonlinear rate, κ ~ βa_T, reciprocity is broken in the upconversion and downconversion processes, biasing energy flow towards redshifted modes and creating an asymmetric comb with ξ ≠ 0. We confirmed the validity of ξ as a figure of merit for asymmetry by numerically performing exponential fits to the modal energy distributions (see Supplementary Information for details).