Transcript of the presentation Networks of Optical Parametric Oscillators: From Ising Machines to Quantum Photonics, given at the NTT Upgrade 2020 Research Summit, September 29, 2020

Good morning, Good afternoon, good evening everyone. I should thank NTT research and the OCI for putting together this program and also the opportunity to speak here. My name is Alireza Marandi and I’m from Caltech and today I’m going to tell you about the work that we have been doing on Networks of Optical Parametric Oscillators and how we have been using the for Ising Machines and how we’re pushing them toward Quantum Photonics. Today I acknowledge my team at Caltech which is now eight graduate students and five researcher and postdocs as well as collaborators from all over the world including NTT Research and also the funding from different places including NTT.

So this talk is primarily about networks of resonators and these networks are everywhere from nature, for instance the brain which has a network of oscillators, all the way to optics and photonics. And some of the biggest examples are metamaterials which is an array of small resonators and more recently the field of topological photonics, which is trying to implement a lot of the topological behaviors of models in the condensed matter physics, in photonics. And if you want to extend it even further, some of the implementations of quantum computing are technically networks of quantum oscillators. So, we started thinking about these things in the context of Ising machines, which is based on the Ising problem, which was based on the Ising model, which is this simple summation over the spins, and spins can be either up or down, and the couplings is given by the Jij. And the Ising problem is if you know Jij, what is the spin configuration that gives you the ground state. And this a problem is shown to be an NP-Hard problem. So it’s computationally important, because it’s a representative of the NP problems. And NP problems are important because first they’re hard in standard computers if you use a brute force algorithm, and they’re everywhere on the application side. That’s why there is this demand for making a machine that can target these problems. And, hopefully, it can provide some meaningful computational benefit, compared to the standard digital computers.

So I have been building these Ising machines based on this building block, which is a Degenerate Optical Parametric Oscillator. And what it is, is a resonator with nonlinearity in it. And we pumped these resonators, and we generate the signal at half the frequency of the pump. One photon on pump splits into two identical photons of signal. And they have some very interesting phase out frequency locking behaviors. And if you look at the phase locking behavior you realize that it can actually have two possible phase states as the escalation result of these OPO, which are off by pi. And that’s one of the important characteristics of them. So I want to emphasize a little more on that. And I have this mechanical analogy, which are basically two simple pendula, but they are parametric oscillators because I’m going to modulate a parameter of them in this video, which is the length of the string. And by that modulation which is an omega pump I’m going to make them oscillate at omega signal which is half the frequency of the pump. And I have two of them to show you that they can acquire these phase states. So the two phase on frequency lock through the pump but it can oscillate in either the zero or pi phase state. And the idea is to use this binary phase to represent the binary Ising spin. So each OPO is going to represent spin, which can be either zero pi, or up or down. And to implement a network of these resonators, we use the time-multiplexed scheme. And the idea is that we put n pulses into the cavity. These pulses are separated by the repetition period that you put in or TR and you can think about these pulses in one resonator is as n temporarily separated synthetic resonators.

If you want to couple these resonators through each other, now we can introduce these delays each of which is a multiple of TR. If you look at the shortest delay, it couples the resonator one to two, two to three, and so on. If you look at the second delay, which is two times the rotation period, the couples one, two, three, and so on. And if you have n minus one delay lines, then you can have any potential couplings among these synthetic resonators. And if I can introduce these modulators in those delay lines so that I can have strength I can control the strength and the phase of these couplings at the right time, then I can have a programmable all-to-all connected network in this time-multiplexed scheme. And the whole physical size of this system scales linearly with a number of pulses.

So the idea of OPO-based Ising machine is then having these OPOs, each of them can be either zero pi and they can arbitrarily connect them to each other, and then I start with programming this machine to a given Ising problem by just setting the couplings and setting the controllers in each of those delay lines. And now I have a network, which represents an Ising problem. Then the Ising problem maps to finding the phase state that satisfy maximum number of coupling constraints. And the way it happens is that the Ising Hamiltonian maps through the linear loss of the network. And if I start adding gain by just putting pumping into the network, then the OPOs are expected to oscillate into lowest loss state.

And we have been doing these in the past six or seven years. And I’m just going to quickly show you the transition especially what happened in the first implementation, which was using a free space optical system. And then the guided wave implementation in 2016 and a measurement feedback idea which led to increasing the size and doing actual computation with these machines. So I just want to make this distinction here that the first implementation was an all-optical interaction. We also had an n = 16 implementation, and then we transitioned to this measurement feedback idea, which I’ll tell you quickly what it is. And there’s still a lot of ongoing work, especially on the NTT side, to make larger machines using the measurement feedback. But I’m going to mostly focus on the all-optical networks on how we’re using all optical networks to go beyond simulation of Ising Hamiltonians both in the linear and the nonlinear side and also how we were working on miniaturization of these OPO networks.

So the first experiment, which was a four-OPO machine, It was a free space implementation and this is the actual picture of the machine. And we implemented a small n = 4 Max-Cut problem on the machine. So one problem for one experiment, and we ran the machine a thousand times and we looked at the state and we always saw its oscillate in one of these ground states of the Ising Hamiltonian. So then the Measurement-Feedback idea was to replace those couplings and the controller with a simulator. So we basically simulated all those coherent attractions on an FPGA ,and we’ve replicated a coherent pulse with respect to all those measurements and then we injected it back into the cavity. And nonlinearity would still remain so it still is a nonlinear dynamical system, but the linear side is all simulated.

So there lots of questions about, if this system is preserving important information or not, or if it’s going to behave better computationally or worse, and that’s still a lot of ongoing studies; but nevertheless the reason that this implementation was very interesting is that you don’t need the N minus one delay lines so you can just use one and you can implement a larger machine and then you can run several thousands of problems in the machine and then you can compare to performance from the computational perspective.

So I’m going to split this idea of OPO-Based Ising Machine into two parts. One is the Linear part which is if you take out the nonlinearity out of the resonator and just think about the connections you can think about this as a simple Matrix Multiplication scheme. And that’s basically gives you the Ising Hamiltonian modeling. So the optical loss of this network corresponds to the Ising Hamiltonian. And if I just want to show you the example of the N = 4 experiment on all those phase states and the histogram that we saw, you can actually calculate the loss of each of those states, because all those interferences and beam splitters and delay lines are going to give you different losses. And then you will see that the ground states corresponds to the lowest loss of the actual optical network. If you add the nonlinearity, the simple way of thinking about what the non-linearity does is that it provides the gain, and then you start bringing up to your gain so that it hits the loss then you go through the gain saturation or the threshold which is going to give you this phase bifurcation. So you’d go either to zero or pi phase states. And the expectation is that the network oscillates into lowest possible loss state.

So there are some challenges associated with this intensity-driven phase transition, which I’m going to briefly talk about. And I’m also going to tell you about other types of non-linear dynamics that we were looking at on the nonlinear side of these networks. So if you just think about the linear network we’re actually interested in looking at some topological behaviors in these networks. And the difference between looking at the topological behaviors on the Ising machine is that now, first of all, we were looking at the type of Hamiltonians that are a little different than the Ising Hamiltonian. And one of the biggest differences is that most of these topological Hamiltonians that require breaking the time reversal symmetry, meaning that you go from one side to another side, and then you have one phase, and if you go backward you get a different phase.

And the other thing is that we’re not just interested in finding the ground state. We’re actually now interesting and looking at all sorts of states and looking at the dynamics and the behaviors of all these states in the network. So we started with the simplest implementation, of course, which is a one D chain of these resonators, which corresponds to us so-called SSH model in topological work. And we get the similar energy to the loss mapping, and now we can actually look at the band structure, and this is an actual measurement that we get with this SSH model and see how it reasonably how well it actually follows the prediction and the theory. One of the interesting things about the time multiplexing implementation is that now you have the flexibility of changing the network as you’re running the machine and that’s something unique about this time multiplex implementation.

So we can actually look at the dynamics. And one example that we have looked at is we can actually go through the transition of going from topological to the standard and to the trivial behavior of the network. We can then look at the edge states and we can also see the trivial edge states and topological edge states actually showing up in this network. We have just recently implemented a 2D network with a Harper Hofstadter model, and I don’t have the results here, but we’re one of the other important characteristics of time multiplexing is that you can go to higher and higher dimensions and keeping that flexibility on dynamics. And we can also think about adding non-linearity both in the classical and quantum regimes, which is going to give us a lot of exotic classical and quantum nonlinear behaviors in these networks.

So I told you about the linear side mostly, let me just switch a few and talk about the non-linear side of the network. And the biggest thing that I talked about so far in the Ising machine is this phase transition at threshold. So below threshold we have squeezed state in these OPOs and if we increase the pump, we’re go through this intensity-driven phase transition. And then we get a phase states above the threshold. And this is basically the mechanism of the computation and these OPOs, which is through this phase transition, below to above threshold. So one of the characteristics of this phase transition is that below threshold you expect to see quantum states; above threshold you expect to see more classical states or coherent states. And that’s basically corresponding to the intensity of the driving pump.

So it’s really hard to imagine that it can go above threshold or you can have this phase transition happening all in the quantum regime. And there are also some challenges associated with the intensity homogeneity of the network, which for example is if one OPO starts oscillating and then its intensity goes really high then it’s going to ruin this collective decision-making of the network because of the intensity-driven phase transition nature. So the question is can we look at other phase transitions? Can we utilize them for both computing, and also can we bring them to quantum regime? And I’m going to specifically talk about the phase transition in the spectral domain, which is the transition from the so-called degenerate regime, which is what I mostly talked about, to the non-degenerate regime, which happens by just tuning the phase of the cavity.

And what is interesting is that these phase transition corresponds to a distinct phase noise behavior. So in the degenerate regime, which we call it the order state, you’re going to have the phase being locked at a phase of the pump as I talked about. In the non-degenerate, however the phase is mostly dominated by the quantum diffusion of the phase, which is limited by the so-called shallow [unintelligible]. And I can see that transition from degenerate to non-degenerate which also has distinct symmetry differences. And this transition corresponds to a symmetry breaking. In the non-degenerate case the signal can acquire any of those phases on the circle so it has a U-1 symmetry. And if you go to the degenerate case, then that symmetry is broken and you only have zero pi phase states that were looked at.

So now the question is, can we utilize this phase transition, which is a phase-driven phase transition, and can we use it for similar computational schemes? So that’s one of the questions that we’re also thinking about, and it’s not just this phase transition is not just important for computing. It’s also interesting from sensing potentials. And this phase transition you can easily bring it below threshold and just operated into quantum regime, either Gaussian or non-Gaussian. If you make a network of OPOs, now we can see all sorts of more complicated and more interesting phase transitions in the spectral domain. One of them is the first sort of phase transition, which you get by just coupling two OPOs. And that’s a very abrupt phase transition, compared to the to the single OPO phase transition.

And if you do the couplings right, you can actually get a lot of non-Hermition dynamics and exceptional points, which are actually very interesting to explore both in the classical and the quantum regime. And I should also mention that you can think about the couplings to be also non-linear couplings. And that’s another behavior that it can see specially in the nonlinear in a degenerate. So with that I basically told you about these OPO networks, how we can think about the linear scheme and the linear behaviors, and how we can think about the rich nonlinear dynamics and nonlinear behaviors both in the classical and quantum regime. And I want to switch gear and tell you a little bit about the miniaturization of these OPO networks.

And, of course, the motivation is if you look at the electronics and what we had 60 or 70 years ago we’d vacuum tubes and how we transitioned from relatively small scale computers in the order of thousands of nonlinear elements to billions of nonlinear elements, where we are now we’d optics is probably very similar to seven years ago which is a tabletop implementation. And the question is how can we utilize nanophononics? I’m going to just briefly show you the two directions on that which we’re working on. One is based on thin-film lithium niobate, and the others based on even smaller resonators. So the work on nanophotonic lithium niobate was started in collaboration with Harvard, Marko Loncar, and also M.M. Fejer at Stanford.

And we could show that you can do the periodic polling in lithium niobate and you get all sorts of very highly non-linear processes happening in this nanophotonic periodic loophole lithium niobate. And now we’re working on dueling OPOs based on that nanophotonic lithium niobate. And these are some examples of the devices that we have been building in the past few months, which I’m not going to tell you more about but, you know OPOs and the OPO networks are in the works. And that’s not the only way of making large networks, but also I want to point out that the reason that these nanophotonic PPLNs are actually exciting is not just because you can make large networks and it can make a compact in a in a small footprint, they also provide some opportunities in terms of the operation regime.

And one of them is about making cat states in OPOs, which is, can we have the quantum superposition of easier on pi states that I talked about. And the nanophotonic with lithium niobate provides some opportunities to actually get closer to that regime because of this spatial temporal confinement that it can get these wave guides. So we’re doing some theory on that. We are confident that the type of nonlinearity to losses that it can get with these platforms are actually much higher than what they can get with other platform, other existing platforms. And to go even smaller we have been asking the question of what is the smallest possible OPO that it can be? Then you can think about really wavelengths-scale type resonators and adding the chi-2 non-linearity and see how and when you can get the OPO to operate.

And recently in collaboration with USC, we have in actually USC and Creole, we have demonstrated that you can use nano-lasers and get some spin Hamiltonian implementations on those networks. So if we can build the OPOs, we know that there’s a path for implementing OPO networks on such a nanoscale. So we have looked at these calculations and we try to estimate the threshold of OPOs, let’s say for Mie Resonator. And it turns out that it can actually be even lower than the type of bulk PPLN OPOs that we have been building in the past 50 years or so. So we’re working on the experiments and we’re hoping that we can actually make even larger and larger scales OPO networks.

So let me summarize the talk I told you about the OPO networks and our work that has been going on Ising machines and the measuring feedback. And I told you about the ongoing work on all-optical implementations those on the linear side and also on the nonlinear behaviors. And I also told you a little bit about the efforts on miniaturization and going to the to the nano scale. So with that I would like to stop here and thank you for your attention.