Persistent dynamics in coupled non-degenerate parametric oscillators: pump saturation prevents mode competition

Shai Ben-Ami; Igal Aharonovich; Avi Pe’er

doi:10.1364/OE.482828

1. Introduction

In a (doubly-resonant) parametric oscillator, the cavity fields (signal and idler) experience parametric amplification due to the nonlinear interaction with the strong pump of frequency $\omega _p$, inducing an oscillation in two conjugate signal ($\omega _s$) and idler ($\omega _i$) frequency modes of the cavity, whose frequency sums to the pump ($\omega _p = \omega _s + \omega _i$). Due to the coherent interaction, the phases of the fields fulfill: $\left ( \varphi _i + \varphi _s \right ) / 2 = \varphi _p/2 + \pi n$, $(n\in \{0,1\})$, which indicates inherent bi-stability of the phase-sum between two opposite values $0,\pi$.

Although recent works have suggested exploiting this bi-stability to realize a coherent Ising machine (CIM) [1–9], we find the unique dynamics of coupled parametric oscillators far richer, and we point out that only under certain, rather narrow conditions can a network of coupled parametric oscillators help solving the Ising problem [10–14]. These works show that a steady state solution to a network of coupled parametric oscillators can be coherent, everlasting beats of power, as also manifested in this work.

This result of permanent beats also contradicted the standard intuition of coupled active oscillators, which expects pump saturation to quickly eliminate coherent multi-mode dynamics due to the competition for pump resources between the participating modes. Indeed, even in [10,11] the persistent beating regime appeared only near the oscillation threshold, where pump saturation is low, and raising the pump-power further resulted in collapsing of both POs to a synchronized oscillation at a single frequency. Here we show that when both POs share a single pump, the pump saturation does not affect the oscillators’ dynamics. Specifically, the competition for pump resources in this case does not eliminate beating, but rather enforces it, causing the beats to prevail for any pump power.

We demonstrate this competition effect in an RF experiment, and elaborate on its origin through a comprehensive numerical simulation along with an approximate analytic discussion. In our experiment, a digital FPGA grants us full control over the coupling properties between the different modes, which allows us to explore, for instance, both energy preserving and energy dissipating couplings, and test the conditions for beats to appear and their robustness.

2. Results

We employed the different frequency modes of a single radio-frequency (RF) cavity to realize two separate non-degenerate parametric oscillators (see scheme in Fig. 1). The parametric gain was provided by an RF frequency mixer, pumped by a strong, coherent pump, giving rise to a degenerate mode and several pairs of signal-idler modes within the cavity (see Fig. 2). Our experiment involves the degenerate mode at ${26.89}\,\textrm{MHz}$ and a signal-idler mode-pair at ${32.25}\,\textrm{MHz}, {21.53}\,\textrm{MHz}$, which are all modes of the cavity, separated by multiples of its free spectral range (FSR). To couple the different modes in a flexible and arbitrary fashion, both in amplitude and in phase, we used a high-bandwidth digital field programmable gate array (FPGA), which sampled the circulating field in real time, performed homodyne detection on the frequencies of interest, coupled their complex amplitudes arbitrarily, and converted the signals back to an analog field (see Fig. 1 inset for a box-diagram of the FPGA algorithm). The coupling matrix realized by the FPGA is of the form:

(1)$$\left(\begin{array}{@{}c|cc@{}} \begin{array}{@{}ccc@{}} t & \alpha + r & 0 \\ \alpha - r & t & \alpha - r \\ 0 & \alpha + r & t \end{array} & & \mbox{0} \\ \hline 0 & & \begin{array}{@{}ccc@{}} t & \alpha + r & 0 \\ \alpha - r & t & \alpha - r \\ 0 & \alpha + r & t \end{array} \end{array}\right),$$

where $t$ is the transmission coefficient, and $\alpha, r$ are the symmetric and anti-symmetric coupling coefficients, respectively. This matrix multiplies the vector $\begin {pmatrix} \begin {matrix} x_i & x_d & x_s & y_i & y_d & y_s \end {matrix} \end {pmatrix}^T$, which contains the real ($x$) and imaginary ($y$) parts of the idler, degenerate and signal ($i,d,s$) fields.

Fig. 1. Schematic block diagrams of the experimental system. A frequency mixer is used as a nonlinear parametric amplifier, which enables parametric oscillation. The coupling is realized digitally using an FPGA. Inset: A schematic block diagram of the FPGA used to couple the different modes. The input field is sampled by the A/D converter. It is then split into its three frequency components (idler, degenerate and signal) by multiplying by local oscillators ($\omega _i, \omega _d, \omega _s$) and filtering the outcome with a moving average filter (MAF). Then, the different modes are coupled to each other in a completely general and controllable manner through the coupler module (that implements a general complex $3 \, \times \, 3$ matrix). Next, each mode is modulated back to its carrier frequency and all modes are added together. Finally, the D/A converter generates the feedback analog signal.

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Fig. 2. Illustration of POs modes in the frequency domain. A degenerate mode is located at exactly half the pump frequency ($\omega _d\!=\!\omega _p/2$), and signal-idler pairs are spaced around it at integer multiples of $\omega _0$, the repetition frequency of the cavity. Our work here employs only the degenerate mode and a single mode-pair

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An example of the measured temporal amplitudes is shown in Fig. 3, where the two parametric oscillators (degenerate and signal-idler) beat coherently when coupled with energy preserving coupling, similar to the case of coupled individually-pumped degenerate POs [10]. We focus our discussion on the effects of the pump power (non-linear gain and saturation) and of the coupling properties (energy-preserving or dissipative) on the dynamics of the coupled oscillators. We will also address the feasibility of this configuration to act as an Ising simulator.

Fig. 3. Experimental beats in the time domain with purely anti-symmetric (energy preserving) coupling between the modes. When the degenerate mode (green, top graph) is depleted, the signal-idler mode (red, bottom graph) is in full oscillation and vice-versa, in a complete orthogonal manner. Left: Long time window, where the beating dynamics is visible. Right: short time window (highlighted by dashed blue lines on the left figure), where the carriers are visible. In the top (degenerate) case, the carrier is exactly at the degenerate frequency ($\omega _d$), while in the bottom (signal-idler) case, the carrier is comprised of two frequencies ($\omega _i = \omega _d - \Delta \omega$, $\omega _s = \omega _d + \Delta \omega$), and therefore it beats rapidly. In this measurement $P_{\text {Pump}} = -8.3$ dBm (pump power), $\text {DC} = {66}\,\textrm{mV}$ (pump DC offset, used as linear gain), $t = 0.45$ (transmission coefficient of the coupling), $a = 1.35$, $s = 0$ (anti-symmetric and symmetric coupling coefficients between the POs, respectively).

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The equations governing the coupled POs dynamics can be approximated in the following form (see the derivation in the Appendix):

(2)$$\tau \frac{d}{dt} \begin{pmatrix} A \\ B \end{pmatrix} = \begin{pmatrix} G & R_{AB} \\ R_{BA} & G \end{pmatrix} \begin{pmatrix} A \\ B \end{pmatrix},$$

where $A,B$ are the envelope amplitudes of the POs, $\tau$ is the repetition time in the cavity, $t$ is time, $G$ is the overall gain in a single round-trip, which settles to zero in steady state, and $R_{AB}$ ($R_{BA}$) is the coupling coefficient from PO B (A) to A (B). The coupling coefficients may be anti-symmetric ($R_{AB} = -R_{BA} \equiv a$), in which case energy is preserved; they could be symmetric ($R_{AB} = R_{BA} \equiv s$), causing energy dissipation, or generally, any mixture of symmetric and anti-symmetric ($R_{AB} = s-a$, $R_{BA} = s+a$). The solution of the equations above is:

(3a)$$\begin{bmatrix}A(t)\\B(t)\end{bmatrix} = \begin{bmatrix} A(0)\cosh{(\sqrt{s^2-a^2}t/\tau)}+\sqrt{\frac{s+a}{s-a}}B(0)\sinh{(\sqrt{s^2-a^2}t/\tau)} \\ B(0)\cosh{(\sqrt{s^2-a^2}t/\tau)}+\sqrt{\frac{s-a}{s+a}}A(0)\sinh{(\sqrt{s^2-a^2}t/\tau)} \end{bmatrix} \,\,\, \text{for} \, |s|>|a|$$

(3b)$$\begin{bmatrix}A(t)\\B(t)\end{bmatrix} = \begin{bmatrix} A(0)\cos{(\sqrt{a^2-s^2}t/\tau)}-\sqrt{\frac{a+s}{a-s}}B(0)\sin{(\sqrt{a^2-s^2}t/\tau)} \\ B(0)\cos{(\sqrt{a^2-s^2}t/\tau)}-\sqrt{\frac{a-s}{a+s}}A(0)\sin{(\sqrt{a^2-s^2}t/\tau)} \end{bmatrix} \,\,\,\,\,\,\,\,\,\, \text{for} \, |s|<|a|$$

Clearly, when the symmetric coupling dominates ($|s|>|a|$), the fields experience exponential gain (until depletion kicks in to stabilize the gain), but if $|a|>|s|$, the steady state shows persistent, full-scale beats of the fields’ amplitudes at a frequency $f_{\text {Beats}} = \sqrt {a^2-s^2}/2\pi \tau$. Furthermore, the beats in one PO are exactly out of phase with respect to the other.

Using the FPGA, we were able to couple the parametric oscillators in an arbitrary manner. We used this ability to introduce both symmetric and anti-symmetric coupling components ($s,a$ respectively). Our main findings are summarized in Fig. 4 that illustrates the oscillation steady state (beating or phase locked) as we vary the saturation level (pump power) and the coupling nature. In the experiment we fixed the value of the symmetric (dissipative) coupling, and measured the beat-frequency (where beats are observed) as a function of both the pump power and the strength of the anti-symmetric coupling. The results are presented as 2D heatmaps in Fig. 4(a) along side with two simulation scenarios - one where the two POs share the same pump (as in our experiment, described by Eq. (4) and depicted in Fig. 4(b)), and the other for two independent pumps (as in [10], described by Eq. (5) and depicted in Fig. 4(c)). Thus, the simulation in Fig. 4(b) assumes that the coupled POs compete for a single pump (since they exist in a single cavity), whereas Fig. 4(c) simulates coupling of POs in different cavities that are pumped independently. The equations describing the evolution of coupled POs with a common pump or with separate pumps, are, respectively:

(4a)$$\tau \frac{dA}{dt} = \left[ g_0^A - \alpha \left( A^2 + B^2 \right) \right]A^* - L_A A + R_{AB} B $$

(4b)$$\tau \frac{dB}{dt} = \left[ g_0^B - \alpha \left( A^2 + B^2 \right) \right]B^* - L_B B + R_{BA} A $$

(5a)$$\tau \frac{dA}{dt} = \left( g_0^A - \alpha A^2 \right)A^* - L_A A + R_{AB} B $$

(5b)$$\tau \frac{dB}{dt} = \left( g_0^B - \alpha B^2 \right)B^* - L_B B + R_{BA} A, $$

where $A, B, \tau, t, R_{AB}, R_{BA}$ are defined above (Eq. (2)), $g_0^A$ ($g_0^B$) is the linear gain in cavity $A$ ($B$), $\alpha$ is the saturation coefficient, describing the pump depletion rate, and $L_A$ ($L_B$) is the linear loss in cavity $A$ ($B$). The essence of the difference between the common pump and the separate pumps cases is in the form of the saturation term (pump depletion). Specifically, when the pump is shared, its saturation corresponds to the total power of both POs (as opposed to separate pumps, that saturate according to the power of each PO separately).

Fig. 4. Beat frequency as a function of pump intensity and anti-symmetric coupling. Dark blue indicates sub-threshold operation, light blue indicates “synchronization” (constant amplitudes, no beats) and the rest indicates beats with various frequencies. White dashed lines show the constant symmetric coupling term ($s = 0.167$) in comparison with the anti-symmetric one. In (a),(b) the two POs co-exist in a single cavity (“common pump”), while in (c) each PO exists in a separate cavity (“separate pumps”). In (a),(b), One can see that in the region where $s > a$ ($a$ being the anti-symmetric coupling), there are no beats, and when $a > s$, there are beats that become faster as $a$ increases. The pump intensity has almost no effect on the beats behavior. In (c), analyzed in [10], the saturation causes the beats to vanish for strong enough pump. Experimentally, $\text {DC} = {66}\,\textrm{mV}$, $t = 0.49$, repetition rate $={2.75}\,\textrm{MHz}$. In the simulation: non-linear gain $g=3.9$, $t=0.5$, repetition rate $={31.25}\,\textrm{MHz}$.

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Looking at the three maps of Fig. 4, it is clear that near the threshold both simulations agree with the experiment: the oscillators synchronize when the symmetric coupling dominates and start beating when the anti-symmetric coupling dominates (the dashed white line marks the equality borderline in Fig. 4). However, as the pump power is increased above threshold the maps differ: For independent pumps the simulation predicts that the beating frequency should decay until finally forcing the oscillators to synchronize at a specific saturation level (solid white line in Fig. 4(c)). This suppression of the beats can be expected due to the common tendency of saturation to give preference for a single winning coupled mode. The experiment shows a drastically different behaviour: when the competition between modes is further increased by letting the oscillators compete for the same pump, the beats are not further suppressed, but rather prevail for any pump power, well above threshold, as reconstructed by the simulation in Fig. 4(b).

To understand this counter-intuitive tendency to preserve the beats when the oscillators compete for the pump, let us consider the field oscillation in time and its inter-relations with the pump. Since the nonlinear interaction between the pump and the down-converted field is instantaneous, pump depletion can occur only when the field intensity in the cavity is high. Thus, in the beating regime, each oscillator harvests the pump resources only during 50% of the time (near the beat peaks). For independent pumps, this indicates a waste of 50% of the pump resources, which is wasteful of pump power, especially high above threshold, where the pump depletion is increased (saturation). Consequently, the beats deform as the pump is increased until finally synchronization is forced. However, for a shared pump this 50% penalty of pump power no longer exists for the beats, since the two POs beat in quadrature, alternating their amplitudes, and the pump depletion is alternated accordingly. Thus, the beating regime is no different in terms of pump-depletion from a CW synchronized oscillation, which eliminates the drive to synchronization. Note that since the wavelength in our RF cavity is much longer than the physical dimensions of the nonlinear mixer (as opposed to optical nonlinear media, where $\lambda \ll$ [nonlinear medium length]), the nonlinear interaction can be considered instantaneous in both space and time. Consequently, the dispersion within the mixer can be completely neglected and no phase matching limitations are imposed on the interaction between the 3-waves in the RF regime.

The measured beat frequency near the oscillation threshold (whenever beating was observed) is presented in Fig. 5 as a 2D heatmap of both coupling components, for the experiment (Fig. 5(a)); for a complete numerical simulation (Fig. 5(b)) of the nonlinear evolution in the cavity (see Eq. (9), derived in the Appendix); and for an analytic linearized approximation (Fig. 5(c) that represents Eq. (3(b)), derived in the theoretical discussion lateron. All the maps of Fig. 5 agree very well with each other, showing beats whenever the anti-symmetric coupling dominates over the symmetric coupling, similar to [10]. When synchronization is observed, it is interesting to consider whether the oscillators converge to the Ising solution dictated by the coupling, or not. This question was fully addressed in [12], which found that, even in cases where the oscillators synchronize, they do not always settle to the Ising solution. Particularly, if the saturation is common to all oscillators (as in Figs. 4(a), 4(b)) the coupled oscillation always converges to the eigen vector with the maximal real part eigenvalue of the coupling matrix, $J_{ij}$ (which generally does not coincide with the Ising ground state). On the other hand, if the oscillators saturate independently (as in Fig. 4(c)), the probability for the coupled PO-network to obtain the Ising ground state is finite $P<1$, and power-dependent.

Fig. 5. Beats vs. coupling maps. 2D heatmaps, showing the beats frequency as a function of both symmetric and anti-symmetric coupling (energy dissipative and energy preserving, respectively) for (a) experiment, (b) simulation and (c) theory (Eq. (3(b)). We see good qualitative agreement - the beats appear only when the anti-symmetric coupling is dominant, and their frequency increases linearly with its strength. We note that while positive and negative symmetric coupling seem to be identical, they represent “ferromagnetic” and “anti-ferromagnetic” coupling, respectively, which manifests in the phase of the oscillator, as can be seen in Fig. 6. In (a) $P_{\text {Pump}} = -8.3$ dBm, $\text {DC} = {66}\,\textrm{mV}$, $t = 0.45$, repetition rate $={2.75}\,\textrm{MHz}$. In (b) non-linear gain $g=3.9$, $t=0.5$, repetition rate $={31.25}\,\textrm{MHz}$.

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Figure 6 shows the relative phase between the two POs $\left ( \left | \varphi _d - \frac {\varphi _i + \varphi _s}{2} \right | \right )$ for both simulation (6(a)) and experiment (6(b)) as a function of the coupling strength for a pure symmetric (Ising) case. The Ising phases are $\varphi _d$ and $\frac {\varphi _i+\varphi _s}{2}$, and they can be either $0$ ($\uparrow$) or $\pi$ ($\downarrow$). Experimentally the phases were calculated by measuring the field circulating in the cavity directly, and then performing homodyne detection digitally in real time for each frequency independently (using the FPGA). The simulation indeed shows perfect correlation between the relative Ising phase and the sign of the coupling. The experiment unfortunately suffers from an inherent ambiguity in the measurement of the phases of the individual frequencies, which are defined only up to a $\pi$ offset in the expression $\frac {\varphi _i+\varphi _s}{2}$. This prevents identification of the absolute sign of the amplitude in the experiment. Thus the experiment can identify the phase-locking due to the coupling and the transition near null coupling, but cannot confirm the phase-alignment.

Fig. 6. The relative Ising phase $\left ( \left | \frac {\varphi _i+\varphi _s}{2} - \varphi _d \right | \right )$ as a function of the symmetric coupling, in the absence of anti-symmetric coupling term, both in (a) simulation, and in (b) experiment. The phases comprising this expression are the carriers’ phases of the two POs (degenerate, $\varphi _d$ and signal-idler, $\frac {\varphi _i+\varphi _s}{2}$), which represents their Ising state (“$0$” or “$\pi$”). One can see direct disagreement between experiment and simulation, as the simulation predicts Ising behavior (correlation between the coupling’s sign and the relative Ising phase), but experimentally no such correlation is observed.

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3. Conclusions

We examined the effect of pump depletion and competition on the dynamics of coupled parametric oscillators. We find that contrary to “common wisdom”, competition for the pump does not suppress the coherent multi-mode beating dynamics, but rather enforces it. Our experiment employed a new, frequency-domain approach to coupling parametric oscillators that used the different frequency modes of a single cavity as independent POs, which were coupled arbitrarily with an FPGA. Compared to the standard space [10,11,13,14] or time [1–4,7–9] coupling methods, this frequency approach expands the research of coupled POs in two aspects: First, the implementation of several POs within a single cavity that share a single pump allows to examine the role of pump competition and depletion, as we discussed. In addition, our frequency approach reveals the role of non-degenerate (two-mode) POs, that can now be considered on equal footing with degenerate POs, exploring the implications of the additional phase freedom for non-degenerate POs (the envelope phase) on the coupled dynamics.

A recent work [15] proposes the realization of D dimensional hyper-spins by D independent POs that share the same pump, but are not coupled. One may see our experiment as the realization of a single two dimensional hyper-spin, where we explore the effect of internal coupling on its performance. Initially, one may think that the ideal hyper-spin requires no coupling whatsoever, as such coupling will cause the spin to collapse to a specific vector value. Indeed, symmetric coupling within the hyper-spin is deleterious, as it will set the relative amplitudes of the POs and force the hyper-spin to a specific vector direction. However, the situation of no coupling at all is unrealistic in practical systems, since any slight deviation from absolute homogeneity between the POs in the hyper-spin would also define a preferred direction to align to, which will interfere with any effort to find the ground state out of a set of close, low energy states. Here the anti-symmetric coupling may prove very powerful, as the persistent beating between the POs within a hyper-spin represents a rotation of the hyper-spin at a constant angular velocity. This rotation automatically averages over all the possible spin configurations, which effectively homogenizes the dynamics. Assuming that all the hyper-spins rotate at the same frequency (the beat frequency, dictated by the coupling strength only), they can still be coupled to each other to form a network of rotating spins, whose network state can be easily read off by moving to a rotating frame (at the beat frequency).

Appendix. Dynamical equations of coupled non-degenerate parametric oscillators

The equation governing the evolution of the fields in a parametric oscillator is the nonlinear wave equation, derived from Maxwell’s equations under the assumption that the electric polarization is a nonlinear series of the electric field [16,17]. Specifically, we assume that the polarization is quadratic in the electric field, which produces second order nonlinearities, among them is parametric amplification.

Skipping some common derivations, we begin with the coupled differential equations for the slow-varying amplitudes of the fields involved in the parametric amplification process (pump, and multiple pairs of signal and idler). We assume that the input to the nonlinear gain medium is a pump field at frequency $\omega _p$, an idler-signal pair at frequencies $\omega _i, \omega _s$ respectively, maintaining $\omega _i+\omega _s=\omega _p$, and a degenerate field at frequency $\omega _d$, maintaining $\omega _p=2\omega _d$.

(6a)$$ \frac{dA_i}{dz} = \left( i \frac{2d_{\text{eff}}\omega_i^2}{k_i c^2} A_p \right) A_s^* \exp{\left( i \Delta k_{is} z \right)} $$

(6b)$$ \frac{dA_s}{dz} = \left( i \frac{2d_{\text{eff}}\omega_s^2}{k_s c^2} A_p \right) A_i^* \exp{\left( i \Delta k_{is} z \right)} $$

(6c)$$ \frac{dA_d}{dz} = \left( i \frac{2d_{\text{eff}}\omega_d^2}{k_d c^2} A_p \right) A_d^* \exp{\left( i \Delta k_{d} z \right)} $$

(6d)$$ \frac{dA_p}{dz} = i \frac{2d_{\text{eff}}\omega_p^2}{k_p c^2} \bigl[ A_i A_s \exp{\left({-}i \Delta k_{is} z \right)} + A_d^2 \exp{\left({-}i \Delta k_{d} z \right)} \bigr], $$

where $z$ is the propagation dimension (assuming no interesting dynamics in the transverse components $x,y$); $A_{\alpha }$ $\left ( \alpha \in \{i,s,d,p\} \right )$ is the complex amplitude of the electric field of frequency $\omega _{\alpha }$ and wave number $k_{\alpha }$ $\left ( E_{\alpha }(z,t) = A_{\alpha }(z) \exp {\left ( ik_{\alpha }z-i\omega _{\alpha } t \right )} \right )$; $t$ is the time coordinate; $d_{\text {eff}}$ is the second-order nonlinear susceptibility; $c \approx {3\times10^{8}}\,\textrm{ms}^{-1}$ is the speed of light in vacuum; $\Delta k_{is} \equiv k_p-k_i-k_s$ is the idler-signal phase mismatch; and $\Delta k_{d} \equiv k_p-2k_d$ is the degenerate phase mismatch.

The assumptions that were taken to arrive to these equations from the nonlinear wave equation are (1) that all the fields are polarized in the same direction so that the dynamics are all scalar, and (2) that the amplitudes are varying slowly (“slowly varying amplitudes approximation”). To simplify the calculations, we make a few extra approximations, namely that there is no phase mismatch $\left ( \Delta k_{is} = \Delta k_{d} = 0 \right )$, and that we can treat the variables $A_i,A_s,A_d$ in the RHS of Eq. (6d) as constants to obtain a set of simple and linear equations (justified by the assumption that in a parametric oscillator the single-pass gain is rather small). All of the assumptions and approximations taken here are self consistent and exist in our experimental setup.

Adopting the transformation $\tilde {A}_{\alpha } \equiv \frac {2d_{\text {eff}}}{nc} \cdot A_{\alpha }$ for convenience, we obtain:

(7)$$\tilde{A}_{\alpha}(l)-\tilde{A}_{\alpha}(0) = \left[ i \omega_{\alpha} \tilde{A}_p(0) l - \frac{1}{2}\omega_{\alpha}\omega_p \left( \tilde{A}_i \tilde{A}_s + \frac{1}{2} \tilde{A}_d^2 \right) l^2 \right] \tilde{A}_{p-\alpha}^*,$$

where $l$ is the length of the nonlinear gain medium and $\alpha \in \{i,s,d\}$ stands for the dynamical fields in the cavity. The subscript $p-\alpha$ stands for the “parametric partner” of the mode $\alpha$ (for the idler, the “parametric partner” is the signal, for the degenerate the partner is the degenerate itself and for the signal it is the idler).

The final expression can be converted to a time-dependent equation describing the evolution of the field in the cavity, and a loss term can be added to resemble a realistic setup:

(8)$$\tau \frac{d\tilde{A}_{\alpha}}{dt} = \biggl[ i \omega_{\alpha} \tilde{A}_p l - \frac{1}{2}\omega_{\alpha}\omega_p \left( \tilde{A}_i \tilde{A}_s + \frac{1}{2} \tilde{A}_d^2 \right) l^2 \biggr] \tilde{A}_{p-\alpha}^* - L \tilde{A}_{\alpha},$$

where $\tau$ is the round-trip time in the cavity. In this simple dynamical equation one can identify some important properties of a parametric oscillator. The first term in the square brackets accounts for the nonlinear gain. focusing on the phase of this term, we see that in order for it to represent amplification, it has to acquire the same phase as the LHS. Therefore $\varphi _{\alpha } = \frac {\pi }{2} + \varphi _p - \varphi _{p-\alpha }$, or $\varphi _{\alpha } + \varphi _{p-\alpha } = \varphi _p + \frac {\pi }{2}$. By convention we define the phase $\varphi _p + \frac {\pi }{2}$ to be zero, so that $\varphi _{\alpha } + \varphi _{p-\alpha } = 2 \pi n$, or $\frac {\varphi _{\alpha } + \varphi _{p-\alpha }}{2} = \pi n$. This relation is well known as the bi-stability of the parametric oscillator; While the phase of each frequency component by itself is random, the average of the pair’s phases can acquire only one of the two phases $0$ or $\pi$. The second term in the square brackets accounts for saturation. Under the specified phase relations, one can see that indeed the higher the amplitudes of the fields, the higher the saturation term, which acts to stabilize the oscillation. Moreover, the saturation term in this analysis contains both parametric pairs, indicating the “common pump” mode of operation (see Fig. 4).

In our simulation we generated an initial random vector of $32$ bins, representing modes of the cavity, which we then progressed repeatedly by applying Eq. (8) as the nonlinear gain and saturation mechanism, multiplied by a coupling matrix of choice, and inserted some linear loss and noise to each such round-trip. We concatenated the output vector of each round-trip to one another for a large number of repetitions, until steady state was achieved, which resulted in a representative temporal waveform of the circulating field in the cavity.

Let us now analyze the beating regime and the beats frequency in a system of coupled POs. For simplicity we will focus on the case of degenerate POs. Each one of the two oscillators (labeled $A$ and $B$ here), is governed by Eq. (8), and a coupling term is added as follows:

(9a)$$\tau \frac{dA}{dt} = \left( g_0^A - \alpha A^2 \right)A^* - L_A A + R_{AB} B $$

(9b)$$\tau \frac{dB}{dt} = \left( g_0^B - \alpha B^2 \right)B^* - L_B B + R_{BA} A, $$

where $A, B$ replace $A_{\alpha }$ in Eq. (8) for the two oscillators; $g_0^A = i \omega _d \tilde {A}_p^A l$ is the parametric gain of A, and similarly for B; $\alpha = \frac {1}{4} \omega _d \omega _p l^2$ is the saturation coefficient of A, and similarly for B; $L_A$ ($L_B$) is the loss in cavity $A$ ($B$); and $R_{AB}$ ($R_{BA}$) is the coupling coefficient from cavity $B$ ($A$) to cavity $A$ ($B$).

The phase relations that were introduced above imply that, defining the phases correctly, we can treat $A$, $B$ as real. Additionally, assuming that the two cavities are identical, and therefore $g_0^A = g_0^B \equiv g_0$ and $L_A = L_B \equiv L$, and also assuming that the saturation term is negligible (working close to threshold), we obtain:

(10a)$$\tau \frac{dA}{dt} = G A + R_{AB} B $$

(10b)$$\tau \frac{dB}{dt} = G B + R_{BA} A, $$

where $G = g_0 - L$. As of the coupling coefficients, there are several possible cases. In the anti-symmetric case ($R_{AB} = -R_{BA} \equiv a$) the coupling would be totally energy preserving. In the symmetric case ($R_{AB} = R_{BA} \equiv s$) the coupling would be totally energy dissipative. Most generally, $R_{AB}$, $R_{BA}$ could be any real numbers, and therefore the coupling would be partially energy preserving. In this case we would use the definitions $R_{AB} = s-a$, $R_{BA} = s+a$. The general solution of the set of ODEs introduced above is:

(11a)$$\begin{aligned}\begin{bmatrix}A(t)\\B(t)\end{bmatrix} &= e^{Gt/\tau} \begin{bmatrix} A(0)\cosh{(\sqrt{s^2-a^2}t/\tau)}+\sqrt{\frac{s+a}{s-a}}B(0)\sinh{(\sqrt{s^2-a^2}t/\tau)} \\ B(0)\cosh{(\sqrt{s^2-a^2}t/\tau)}+\sqrt{\frac{s-a}{s+a}}A(0)\sinh{(\sqrt{s^2-a^2}t/\tau)} \end{bmatrix} \,\,\, \text{for} \, |s|>|a| \end{aligned}$$

(11b)$$\begin{aligned}\begin{bmatrix}A(t)\\B(t)\end{bmatrix} &= e^{Gt/\tau} \begin{bmatrix} A(0)\cos{(\sqrt{a^2-s^2}t/\tau)}-\sqrt{\frac{a+s}{a-s}}B(0)\sin{(\sqrt{a^2-s^2}t/\tau)} \\ B(0)\cos{(\sqrt{a^2-s^2}t/\tau)}-\sqrt{\frac{a-s}{a+s}}A(0)\sin{(\sqrt{a^2-s^2}t/\tau)} \end{bmatrix} \,\,\,\,\,\,\,\,\,\, \text{for} \, |s|<|a| \end{aligned}$$

One can see that if $|s|>|a|$, the solution grows exponentially (until saturation takes over and our approximation no longer holds), and if $|s|<|a|$, the solution becomes oscillatory, which describes coherent beats between the two oscillators. In steady state the gain and the loss equate and we obtain $G = 0$, so that the exponential growth disappears.

Funding

United States-Israel Binational Science Foundation (2017743, 2020790).

Acknowledgments

The authors would like to thank Dr. Leon Bello and Mallachi Meller for many fruitful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Persistent dynamics in coupled non-degenerate parametric oscillators: pump saturation prevents mode competition

Abstract

1. Introduction

2. Results

3. Conclusions

Appendix. Dynamical equations of coupled non-degenerate parametric oscillators

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (20)

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