A busy January

There's been a lot going on here...

Machine Learning & Physics

Unsupervised learning of quantum many-body scars using intrinsic dimension - Now available on arXiv! We applied manifold learning techniques to identify scar states in the PXP model The take-home message: manifold learning techniques are a powerful alternative to more popular deep learning methods, especially in physics problems where you might not have access to enough training data for deep learning to work well.

Identifying topology of leaky photonic lattices with machine learning - Just published in Nanophotonics! We apply various machine learning methods to distinguish different topological phases in a photonic lattice, assuming one only has access to intensity measurements. This can serve as an alternative to full state tomography or phase retrieval methods, but one needs to be careful when training the models on ideal / pristine systems and then applying them to disordered systems. The journal also published a press release on WeChat!

Quantum Computing

Computing electronic correlation energies using linear depth quantum circuits - Finally published in Quantum Science & Technology, after more than a year and a half working through the peer review system. We use perturbation theory to determine electronic correlation energies in small molecular systems (hydrogen, lithium hydride, etc.) using a large set of shallow circuits, giving an alternative to existing methods which require deeper circuits infeasible for current quantum processors. We also tested the algorithm on cloud quantum processors, observing the detrimental impacts of noise. It would be interesting to run this again now to see how much (or how little) the performance from the different cloud providers has improved!

Landscape approximation of low-energy solutions to binary optimization problems - Published in Physical Review A. We present a method to obtain approximate solutions to binary optimization problems using the localization landscape, a function which is able to place bounds on the regions of Anderson localized eigenstates in disordered media without solving the underlying eigenvalue problem. We lay out the conditions required for these bounds to hold, outline how a quadratic unconstrained binary optimization problem can be transformed to fit these conditions, and provide details on how the quantum state representing the landscape function can be produced and sampled using techniques developed for near-term quantum devices.

 

On a related note, I was interested to see this month a new arXiv preprint in which the localization landscape was used to engineer multifractal resonances in SiN membranes!

Photonic Flatband Resonances

Photonic Flatband Resonances in Multiple Light Scattering - Published in Physical Review Letters. We reveal that flatbands can emerge as collective resonances in fine-tuned arrays of Mie-resonant nanoparticles, leading to giant values of the Purcell factor for dipolar emitters. The article was also highlighted with a Synopsis in Physics Magazine!

IPS Meeting 2023

A few things I learned attending the first two days of this year's IPS Meeting, held right here at NUS:

Prof. Giovanni Vignale gave a plenary talk on bulk currents and edge accumulation in anomalous Hall systems. In the conventional quantum Hall effects, accumulation of charge at the edges of the sample are driven by the bulk quantized Hall conductivity. Anomalous quantum Hall systems, on the other hand, do not show an accumulation of spin or valley densities at their edges, despite their corresponding bulk spin or valley Hall conductivities being nonzero. In the case of spin Hall systems it's because bulk electrons will flip their spin when reflecting off the edge of the sample. Thus, the edges accumulate a nonzero charge density, but their spin density remains zero. Interestingly, a similar argument does not hold for the case of valley Hall systems because the applied electric field that drives the current also induces coupling between the valleys in the bulk. Further details can be found here.

The second plenary talk by Prof. Silvija Gradecak focused on the use of imperfect or novel materials to develop new components. A striking example given was the use of 2D materials as diffusion barriers in nanoscale metal contacts in integrated circuits, which promises the ability to further miniaturize electronic components.

Dr. Sen Mu talked about Kardar-Parisi-Zhang (KPZ) physics in the Anderson localization of two-dimensional wavepackets. The KPZ equation describes fluctuations that arise in the density fluctuations of expanding waves in the presence of disorder. These fluctuations are universal and arise in a variety of wave systems, including the spreading of coffee poured out onto a napkin, which he demonstrated for us live! arXiv preprint

Weitao Chen discussed critical dynamics in one-dimensional disordered systems with long range coupling. In critical systems the eigenstates exhibit multifractality, meaning that the different moments of the eigenstates scale with different non-integer exponents with the system size. This is a bit abstract and hard to measure directly in an experiment, but remarkably this multifractality can also be observed by exciting a single site of the lattice and measuring the time-dependent return probability! arXiv preprint

Prof. Di Zhu in another plenary surveyed integrated photonics for the generation, manipulation, and detection of quantum states of light. A recurring theme was that many of the improvements required to scale up integrated quantum photonic systems can be found by looking back to scientific literature from the 1960s! One neat example he gave was scaling up superconducting nanowire single photon detectors: Putting many of them one one chip is hard, because each coaxial microwave read-out line also conducts heat in - if you have too many you will no longer be able to keep the chip cool enough for the detectors to work. The solution? Move from detection based on a lumped circuit model to a transmission line detector, which can (with a bit of signal processing) perform spatially-resolved detection of multiple single photons. A demonstration of this idea was published this year in Physical Review Applied after spending quite some time under peer review by the looks of it.

There were many other interesting talks and posters that I didn't take enough notes on to write about, but it was nevertheless great to see the breadth of physics being done at the different universities and research institutes in Singapore.

The localization landscape

Localization of waves due to destructive interference in disordered media - Anderson localization - has been a subject of intense investigation for more than 50 years. The original paper has now been cited more than 15,000 times according to Google Scholar. Being a property of linear, non-interacting Hamiltonians subjected to a random potential, one might think that everything there is to know about this problem would have already been studied to death a long time ago, with current work focusing on understanding peculiarities arising under special circumstances.

Delightfully, this is not the case! Back in February, this preprint caught my attention due to its keywords of many-body localization and persistent homology. Specifically, the authors have used persistent homology to characterize the shape of the "localization landscape" of a many-body Hamiltonian. 

What is the localization landscape?

The localization landscape was first proposed by Filoche and Mayboroda in an article in PNAS which, despite its broad applicability to understanding wave transport in a variety of settings (condensed matter, photonics, acoustics), did not attract as much interest as other arguably more specialized areas such as non-Hermitian systems or topological edge states. In short, the localization landscape u of a Hamiltonian $\mathcal{H}$ satisfying certain properties is the solution of the linear equation

$$\mathcal{H} u = \mathcal{1},$$

where $\mathcal{1}$ is a vector with all elements equal to 1. u corresponds to the steady-state response of the medium to a uniformly-distributed source.

Left: The localization landscape of a disordered two-dimensional wave medium. Right: Five lowest energy eigenstates of the Hamiltonian, which are localized to distinct peaks of the landscape bounded by lines of small $u$ (red). For more details, read the paper!

Remarkably, it can be shown that $u$ bounds the spatial extend of the low energy eigenmodes of $\mathcal{H}$. Thus, instead of solving the eigenvalue problem and plotting its low energy modes individually to find out where they are localized, plotting $u$ alone is enough to find all the effective low-energy valleys in a medium. Moreover, as a solution to a linear system of equations, $u$ is easier to obtain than the eigenstates themselves. A more rigorous discussion (including the properties that must be satisfied by $\mathcal{H}$ can be found in the PNAS article, which is a pleasant and accessible read.

Interest is now growing in localization landscapes, thanks to recent generalizations that can bound the spatial extent of modes residing in the middle of the energy spectrum, eigenvectors of real symmetric matrices and modes of many-body interacting quantum systems. The latter remarkably shows that useful landscape functions are not limited to low-dimensional wave media, but can also bound the spreading of wavefunctions in the high-dimensional Fock space of many-body quantum systems.

A natural question that arises from these recent works is whether the localization landscape might have some potential applications in the context of quantum computing and noisy intermediate-scale quantum (NISQ) processors. For example, finding the ground state of a generic many-body Hamiltonian is a computationally challenging problem, and methods for NISQ devices such as the variational quantum eigensolver may fail to converge due to the presence of vanishing gradients or sub-optimal local minima. 

Can the localization landscape offer an easier, more hardware-efficient way to sample from low-energy solutions of a hard-to-solve Hamiltonian? Watch this space to find out!

arXiv highlights

Some arXiv preprints that caught my attention over the past week:

Fock lasers based on deep-strong coupling of light and matter

Researchers at MIT predict that the deep strong coupling regime achievable in state-of-the-art superconducting quantum circuits can be used to design novel light sources of N photon Fock states with N ~100. The mechanism is a suppression of stimulated emission via a many-body photon blockade. Large N Fock states are a key ingredient in certain quantum sensing and superresolution schemes, but are extremely hard to make using conventional quantum light sources. This proposal focuses on the generation of microwave photons. A related preprint considers the extension to optical frequencies.

Eigenvalue topology of non-Hermitian band structures in two and three dimensions.

This is a theory follow-up to the authors recent publication demonstrating knotting and braiding of non-Hermitian energy bands. In 2D systems the braid group describes non-contractible loops in the Brillouin zone which can either encircle non-Hermitian degeneracy points or wrap around the whole Brillouin zone. In 3D it is the non-Hermitian degeneracies themselves that form curves that can be knotted or braided. The group theory used to classify general N-band non-Hermitian systems seems rather heavy at first glance.

Anderson localization of a Rydberg electron.

Rydberg atoms host highly excited electrons residing in orbitals with large principal quantum numbers n. Thanks to spherical symmetry, each n corresponds to n^2 degenerate orbital angular momentum modes, forming a large set of available modes. Using now-sophisticated optical tweezer technology it the authors propose placing strong scatterers in the vicinity of the Rydberg orbitals, inducing coupling between them. By controlling the effective inter-orbital coupling and energies by the scatterer positions' the authors propose the realization of the Anderson tight binding model describing localization of electrons in disordered systems. This is a neat idea as it means a single Rydberg atom may be used as a toolbox for exploring various fundamental models from condensed matter physics.

Superconducting optomechanics in topological lattices.

Topological lattice designs are now attracting growing experimental interest in optomechanics, following seminal theoretical proposals several years ago. Here 1D Su-Schrieffer-Heeger lattices and 2D honeycomb lattices are implemented. Optomechanics is a promising setting for exploring both nonlinear and quantum topological phenomena in lattices.

Finally, some self-promotion:

Nonlinear signatures of Floquet band topology.

We finally finished the follow-up to our PRL published earlier this year on measuring Chern numbers using nonlinear modulational instability. The approach also works for periodically-driven Floquet systems, at least for identifying Chern insulator phases. The detection of anomalous Floquet phases is a little more tricky.

 

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.

Dynamical versus spectral localization in dissipative systems

A short summary of this paper which was published in Nature Photonics last week.

Waves in random or disordered media can exhibit Anderson localization. In Anderson localization, interference between different wave scattering paths (and in particular, constructive interference of backscattering) results in a complete suppression of wave propagation; waves remain localized around their sources indefinitely, with an amplitude (or intensity) decaying exponentially with the distance from the source.

Anderson localization is a universal phenomenon. It was originally predicted in a 1958 publication analyzing electrons in disordered crystalline materials, and has since been observed for a wide variety of waves including matter waves (Bose-Einstein condenstates), optics, and acoustics.

Historically, a thorny issue complicating the observation of Anderson localization in classical wave systems such as optics and acoustics has been the question of how to distinguish Anderson localization from absorption-induced localization; both lead to a similar exponential decay of the wave amplitude.

The present work concerns the generalization of Anderson localization to disordered dissipative optical wave systems with random distributions of gain and/or loss, and the subtle distinction between spectral localization and dynamical localization.

Spectral localization refers to localization of the modes of the medium. Each mode has a specific energy (frequency). The mode may be excited by placing a source (e.g. a speaker in the case of acoustic waves) in the medium tuned to that frequency, in which case the amplitude profile of the generated wave will match the profile of the correspond mode.

Dynamical localization refers to the time evolution behaviour of wavepackets comprising a range of frequencies, excited by switching the source on for a short time.

In wave systems that are Hermitian (conservative, i.e. no gain or loss of energy), spectral and dynamical localization coincide because the time evolution of any wavepacket can be obtained by expanding it as a sum of the medium's modes. The paper shows that this is not the case for wave systems with dissipative disorder; dynamical delocalization can occur despite spectral localization. In other words, waves generated by a monochromatic source will have an exponentially localized amplitude profile, whereas waves emitted by a broadband source will spread to distant parts of the system.

To demonstrate this dynamical delocalization, the authors had to carefully distinguish between energy transport and energy loss due to absorption. In particular, for lossy media the wave amplitudes are always decaying exponentially in time. At any moment in time we can consider the shape of the wave's amplitude distribution (e.g. by increasing the sensitivity of the camera or microphone used to detect the waves) and how it decays with separation from the source. 

In dissipative wave media the modal expansion is still valid, but different modes will have different loss rates. Therefore, the relative amplitudes of the different modes in the expansion will change in time, leading to large changes and in particular spreading in the normalized wave amplitude profile - dynamical delocalization. This is the main result of the study, which observes the phenomenon using a cleverly-designed system of coupled optical fibres.