Optics in 2022 and Beyond

I was recently asked to contribute to Optics & Photonics News a short paragraph on predictions on what will be the biggest advances in topological photonics in 2022. Here are my initial thoughts (to be refined):

Prominent trends in 2021 have been:

• Demonstrations of topological defect modes including Jackiw-Rossi modes and free-form disclination waveguides.
• Increased interest in topological designs among non-specialists in topology, studying important questions including how to quantify scattering losses of topological waveguides (e.g. due to sidewall roughness), how to efficiently couple between topological and non-topological modes, and the design of functional components such as power splitters, directional couplers, and absorbers using topological modes.
  
• Many theoretical and experimental studies of nonlinear and quantum effects in topological systems, including topological lasers and frequency combs.
• Optical fiber loops and 3D printed waveguide arrays and photonic crystals have emerged as flexible new platforms for implementing topological models. Most prominently, the former has allowed the first demonstration of various non-Hermitian topological effects. At the same time, the surging interest in quantum computing (in particular photonic approaches) mean that equipment for probing topological wave systems with quantum states of light should become cheaper and more accessible in the near future.
• Links between topological photonics and other hot topics (bound states in the continuum, structured light, singular optics, transformation optics, leaky mode theory) are becoming better-appreciated.

Already a few years ago there were discussions at conferences on how the field is starting to mature, meaning that some useful practical applications of topological photonics need to be demonstrated to sustain interest among researchers, high impact journal editors, and funding agencies. Therefore I think in the coming year there will be a growing focus on demonstrating topological waveguides and cavities with superior performance compared to conventional designs using well-established figures of merit.

I would like to focus on a specific application (e.g. lasers or integrated waveguides) to give my final paragraph more of a punch.

Did I overlook any important lines of research? I welcome comments and criticism.

Unfinished projects

Inspired by my recent post on knotting of energy bands, I have been digging through my backup drive storing old files from my PhD and earlier postdocs. I was quite surprised to see how many unfinished projects I had with semi-polished write-ups. Some of the topics covered included vortex knots in modal superpositions, accelerating solutions of the Dirac equation, coupled mode theory for plasmonic optical fibres, peculiar (topological?) edge states in certain tight binding models, non-Hermitian degeneracies in photonic lattices, and more. In some cases similar ideas ended up being published later in high impact journals by others.

It would be nice to wrap up some of these unfinished projects when I have the time. Especially, theoretical and experimental capabilities have advanced considerably over the past several years, making some of the research lines more feasible.

There are many reasons why projects which looked promising at first may end up being abandoned:

• Other commitments taking over, such as needing to write up the PhD thesis
• Preliminary work being trashed by reviewers, killing motivation to continue the line of research
• Neat theoretical ideas that unfortunately seem impractical to realize in any experiment
• Inability to come up with a more general theory to describe novel behaviour seen in specific examples
•  Shifting research or funding agency interests

These are all temporary problems. Dead projects may be revived later. The bottom line (PhD students take note!): Keep good records. Ideally include readable figures and a discussion of your motivation. Be diligent with commenting your code in case future you needs to figure out how it works.

More arxiv papers

I didn't have a chance to blog this week since I wanted time to wrap up some long-delayed projects. A few highlights from arXiv today:

Persistent homology of quantum entanglement studies phase transitions in the Ising and XXZ spin models. Their approach considers each spin as a point in an abstract high-dimensional space, with "distances" between pairs of spin given by their quantum mutual information, which measures the degree to which the two spins are entangled. Changes in the barcode diagram constructed from the models' ground states can be used to detect quantum phase transitions. The authors speculate that the information provided by the resulting persistence diagrams can be used to guide the construction of efficient numerical approximations such as matrix product states.

Cavity optomechanics with Anderson-localized optical modes reports the observation of optomechanical amplification and phonon lasing in an air-hole photonic crystal waveguide. This provides a way to exploit the unavoidable disorder (surface roughness) present in nanofabricated devices to achieve strong photon-phonon interactions.

An exponentially more efficient optimization algorithm for noisy quantum computers presents a qubit-efficient alternative to the popular quantum approximate optimization (QAOA) algorithm for solving MaxCut problems. In QAOA, each binary classical variable is mapped to a single qubit, making it hard to map problem instances of practical importance onto near-term devices. The author of this article proposes a scheme by which the N binary variables are mapped onto log(N) continuous variables and a variational scheme to optimize these variables to find the maximum number of cuts of the problem graph.

Diversity measures for discrete optimization by the Google quantum team proposes measures to quantify the how distinct different approximate solutions to hard optimization problems are. This seems an important step towards quantifying potential advantages offered by NISQ-compatible quantum optimization algorithms.

arxiv papers

A few arXiv preprints that caught my eye this week:

Topology in shallow-water waves: A spectral flow perspective: This intriguing paper demonstrates a novel form of the topological bulk-boundary correspondence for continuous wave systems such as water waves, for which the more conventional bulk-boundary fails. In this case, the edge invariant describes a peculiar form of topological pumping driven by changes in boundary conditions. The authors write: "We are not aware of any physical evidence of such a quantity, but because of its stability there might be some way to observe it in some clever device." Light propagation in a suitable homogeneous medium described by Maxwell's equations seems like a prime candidate. Who will be first?

 
A Qubit-Efficient Encoding Scheme for Quantum Simulations of Electronic Structure: The authors propose a method to map quantum chemistry problems onto a number of qubits that grows only logarithmically with the number of orbitals simulated. Such qubit-efficient mappings are particularly important for solving practical problems on near-term quantum processors supporting a limited number of qubits. The trade-off, however, is that the design of ansatz circuits (for variational quantum algorithms) becomes a lot harder, so far limited to so-called hardware-efficient circuits that may be challenging to optimise to find the ground state of the system.

Topological Wannier cycles. This work was mentioned by Prof. Jiang at the METANANO conference last month. Threading a flux through a single unit cell of higher-order topological insulators induces a novel form of topological pumping that can be used to distinguish certain higher-order topological phases that cannot be distinguished via the bulk-edge correspondence.

Non-randomness of Google's quantum supremacy benchmark. The authors apply various statistical tests to the bitstrings sampled in Google's famous quantum supremacy paper. Due to noise and imperfections in the gates, the circuit does not precisely sample the family of random unitary matrices, and therefore the sampled bitstrings exhibit subtle biases. For example, qubits are 2.8% more likely to be sampled in their ground state. This suggests it may be possible to classically spoof the sampling performed by the quantum processor, challenging the claimed quantum supremacy.

 

The Complexity of Bipartite Gaussian Boson Sampling. Xanadu and collaborators analyze a variant of their programmable quantum photonic chip demonstrated earlier this year, in which different unitary transformations U and V are applied to each group of two-mode squeezed states. In practice, this would likely involve the use of wavelength or polarization-dependent couplers to direct signal and idler photons to different tunable beamsplitter meshes. Then, thanks to the singular value decomposition, the output photon number distribution can sample from distributions given by permanents of arbitrary matrices, enabling the classically-intractable regime to be reached using fewer photons (or in this case less squeezing).

The future of qubits

Will future fault-tolerant quantum computers be scaled-up versions of existing quantum processors, or will they be based on completely different materials with intrinsic fault tolerance? Dave Bacon (formerly of IonQ and Google Quantum) envisions a middle ground in which current noisy qubits may be cleverly arranged to create intrinsically-tolerant topological qubits without requiring the massive overheads of quantum error-correcting codes: Quantum Computing's Middle Way

On a lighter note: The Quantum Hype Scorecard

Gradient catastrophe of nonlinear topological edge states

Highlighting our research published in Physical Review Research a few days ago: Gradient catastrophe of nonlinear photonic valley-Hall edge pulses

In this paper we studied the propagation dynamics of topological edge wavepackets in the presence of Kerr nonlinearity, describing light-induced shifts to the refractive index of the medium. Prior works studying the nonlinear dynamics of topological edge states largely focused on the analysis of plane wave-like edge states using the nonlinear Schrodinger equation, which can describe their self-focusing dynamics and modulational instability. However, the nonlinear Schrodinger equation is a purely scalar wave equation, in which the origin of the edge states (e.g. whether they are topologically protected) does not qualitatively affect the dynamics.

We carried out a theoretical analysis based on the more complete nonlinear Dirac equation, which takes into account the spin-like degrees of freedom required to create topological band gaps and edge states. Assuming the ambient medium exhibits a simple Kerr nonlinearity, we derive effective nonlinear wave equations governing the dynamics of edge wavepackets. Interestingly, a pure Kerr nonlinearity gives rise to a novel higher-order self-steepening term in the effective nonlinear wave equation, which means that an initially-symmetric wavepacket will become increasingly asymmetric as it propagates, eventually acquiring a shock-like discontinuity (gradient catastrophe).

We confirmed the validity of our effective nonlinear wave equations using direct numerical simulations of the paraxial wave equation describing 2D topological waveguide lattices. Similar phenomena may also be observed in related nonlinear wave systems, including exciton-polariton condensates in microcavities.

Enhancing free-electron radiation using photonic flatbands

Highlighting the preprint "Observation of enhanced free-electron radiation from photonic flatband resonances" that appeared on arXiv the other day.

 

The efficiency at which free electrons radiate electromagnetic energy is dictated by phase matching between the electron and photons. In other words, the two should remain in phase. Phase matching requires an intersection in momentum (k) space between the electronic and photonic dispersion relations. 

In a uniform medium, phase matching can occur when the electron travels faster than the speed of light in the medium, corresponding to Cherenkov radiation. For periodic media, phase matching is no longer limited to a single wavevector, but can occur at a discrete set of points thanks to the periodicity of the Brillouin zone, known as Smith-Purcell resonances. 

Fine-tuning the band structure of a photonic crystal, one can achieve phase matching along a continuous line of momenta in the Brllouin zone. The authors predict orders-of-magnitude enhancement of the radiation intensity due to a flatband resonance, observing a 100-fold enhancement in their experiments. Discrepancies between simulations and observations are attributed to electron beam-induced doping of their photonic crystal.

I found this study particularly noteworthy for a few reasons:

• One of the main selling points of photonic flatbands is their ability to enhance nonlinear optical effects, which are most commonly considered in the framework of nonlinear Schrodinger-type equations describing phenomena such as solitons and lasing. This work establishes a new application of flatbands: mediating efficient interactions between free-electrons and light by bridging their distinct spatial scales.
• Flatbands are usually studied under the tight binding approximation, where they occur exactly. In photonic crystals the tight binding approximation doesn't hold, making it hard to achieve a sufficiently flat dispersion throughout the entire Brillouin zone (we tried this unsuccessfully in the past). However, enhancement of free electron-light interaction does not require a perfectly flat 2D dispersion relation, but merely flat dispersion along 1D line in the Brillouin zone. Such line degeneracies occur more generically at critical points between elliptic and hyperbolic isoenergy contours. This makes it easier to design and harness flatbands in practical systems.

Tying knots in energy bands

The generation of knots in optical systems has fascinated me since my PhD days. Influential early works focused on optical vortex knots. Optical vortices trace out curves in 3D space; in certain circumstances the curves trace out knotted trajectories. Influential early works include the generation of knotted vortices in light beams using a spatial light modulator, large-scale statistical analysis of the probability of knots forming in random optical beams, and the spontaneous formation of vortex knots in the tails of self-trapped nonlinear optical beams. 

More recently, with the surge in interest in topological photonics, interest has turned to the study of knots in other objects, including the energy bands of periodic systems.

An article just published in Nature reports the observation of braided energy bands using a pair of coupled modulated ring resonators. Braids are closely related to knots (knots can be obtained by connecting the ends of non-trivial braids). Braids are more natural to study in the context of band structures, owing to the periodicity of the Brillouin zone.

In order to generate knots or braids, one needs to consider curves in a 3D space. In this work, one of the dimensions is provided by the momentum (k) space, while the other two dimensions are played by the real and imaginary parts of the complex energy eigenvalues E. Using phase and amplitude modulators it is now possible to implement near-arbitrary non-Hermitian Hamiltonians using coupled ring resonators, allowing exquisite of the trajectories of the energy eigenvalues in the complex plane.

To observe the energy band braiding, the authors probed their ring resonator system using a continuous wave laser beam, measuring the time-dependent transmission, with time playing the role of k. Scanning the frequency of the probe beam, a peak in transition occurs at frequencies corresponding to resonances (real parts of the eigenvalues) of the system, with the linewidth of the resonance corresponding to the imaginary part of the eigenvalue. This allows the reconstruction of various braids, including those corresponding the simplest Hopf link (two threaded rings) and more complex objects such as trefoil knots.

Quantum error correction with an ion trap quantum processor

Published in Nature a few days ago: Fault-tolerant control of an error-corrected qubit

This study by Christopher Monroe's team at the University of Maryland demonstrates error-corrected single qubit operations using an ion trap quantum processor. The publication is quite timely, given that the group's spin-off IonQ just became a publicly-traded company.

This is a milestone achievement. For a long time, quantum computing skeptics pointed to the lack of any demonstration of even a single logical qubit protected against native errors as evidence that large scale quantum computing is infeasible. Here, the authors demonstrate logical operations with error probability less than that of its constituent qubits. In theory, by increasing the number of physical qubits used per logical qubit one can exponentially suppress errors and build a fault-tolerant quantum computer.

In practice, it's not that simple and there are still significant hurdles to overcome. The present study has demonstrated fault-tolerant single qubit operations. Next steps will be to demonstrate error-corrected two-qubit gates and multiple rounds of error correction, requiring improvements to the native gate fidelities and the ability to perform mid-circuit measurements on ancilla qubits. And while ion trap quantum processors have very low gate errors (making this demonstration possible), their main drawback is that the gates are very slow (e.g. 0.2 ms for a native two-qubit gate). So even when large scale error-correction is achieved, you might be waiting a while for the calculation to run!

More on the physics: the error correction scheme used is based on products of 3-qubit entangled GHZ states. Since the GHZ states are decoupled from one another, a local error in one GHZ state can be detected (leading to a cubic suppression of the logical error-rate via post-selection) and/or corrected (giving a quadratic error suppression). The measured error rates (compared against a non-fault-tolerant circuit in which there is entanglement between the different GHZ steps) agree with this predicted scaling.

IPS Meeting Day 2

Some highlights from Day 2 of the IPS Meeting:

• Prof. Javier G. Fernandez (SUTD) gave a plenary talk about bio-inspired manufacturing, including the 3D printing of low density yet strong structures inspired by cellulose and chitin. His main take-home message was that nature finds a way to make the most of available materials using intricate multi-scale design principles; details from the nanoscale up to hundreds of micrometers all contribute to bulk structural properties. We need to follow a similar paradigm to develop next-generation sustainable materials, and for colonies on other planets to be viable.
• Prof. José Ignacio Latorre (NUS) covered the fundamentals and future of quantum technologies in his plenary talk. He emphasised how quantum algorithms will be very different from the classical algorithms we are more familiar with. For example, computing 2 x 3 = 6 is easy for a classical computer, but such a computation cannot be performed using unitary quantum evolution (making it hard for quantum computers) because it is a non-invertible operation. For example, since 1 x 6 = 6 as well, the inverse transformation is ill-defined. Another important principle is the no-cloning theorem, meaning that it is impossible to save or copy quantum data. Finally, measurements are intrinsically random. These differences make it hard to find useful quantum algorithms. The known quantum algorithms with exponential speed-ups compared to classical algorithms are specific to problems with special structure (such as periodicity), which allows quantum interference to be tailored to find the solution faster.
• Prof. Cesare Soci (NTU) talked about the fabrication of superconducting single photon detectors and efforts to develop detectors for infrared frequencies using microbridge designs. Conventional nanowire detectors are based on an incident photon breaking a Cooper pair, resulting in a cascade of Joule heating which destroys the superconducting state. On the other hand, in microbridge detectors the incident photon creates vortex-antivortex pairs in the superconducting order parameter; this differing mechanism allows for designs more compatible with lower frequency detectors. 
• Ximeng Zheng (NTU) discussed hollow core fibre-based atomic vapor cells. The idea is that by placing atoms inside the hollow fibre core, the strong light confinement can be used to enhance the light-matter interactions. One challenge is that since the fibre core is so small, it takes a long time (weeks or months!) for the atoms to be loaded into the fibre. On the other hand, once they are in there they can remain for a few years...
• Prof. Murray Barret (NUS) gave an overview of his team's efforts to integrate ion clocks into silicon chips, an essential step towards making them more practical and affordable. For example, commercially-available ion traps cost hundreds of thousands of dollars, whereas mass-produced integrated ion chips could be as cheap as $350 each. But getting comparable performance (i.e. timing precision of 10^-18!) in integrated devices is challenging, due to the shallower confining potentials and stronger thermal effects.
• The QEP 2.0 panel discussion emphasised the need for outreach and involvement of stakeholders and potential industry end-users in order to discover more potential use-cases of quantum technologies. Development of open-source frameworks such as QIBO will be essential to get full value out of future quantum devices.
  

Overall, despite all the social distancing requirements it was an enjoyable conference. Definitely being in the room watching the talks live makes it easier to stay focused on the presentations and avoid distractions. Watching some of the presentations (particularly the poster pitch session) was a good reminder that, when presenting your work to a general physics audience, less is more. It is hard to distill your months or years of work down to a few minutes, but essential to get your message across. In hindsight, even after cutting my talk heavily I still tried to include too much. Something to remember for the next conference...