Artificial Lattices and Designer Matter

An artificial lattice is a deliberately assembled array in which experimentally identifiable modes play the roles of sites and controllable couplings play the roles of bonds. The sites may be electronic quantum dots, minima of an optical standing wave, optical waveguides, exciton–polariton micropillars, microwave resonators, superconducting qubits, or surface-state regions confined by positioned molecules. These systems are called designer matter because geometry, hopping, interactions, flux, and boundaries can be adjusted more directly than in an ordinary crystal.

The same hopping graph can describe very different experiments. Electrons are fermions; atoms may be bosons or fermions; a classical optical beam probes wave interference without preparing an interacting many-photon state. This page explains what carries over between platforms and what changes with particle statistics, preparation and loss.

Required background. Tight-Binding Models supplies the localized-orbital Hamiltonian and its hopping conventions. Quantum Dots supplies discrete levels and charging scales; Optical Lattices supplies the mapping from a light-generated potential to matter-wave lattice parameters. Lindblad–GKSL Equation supplies the qualified Markovian description of drive, loss and decoherence.

From sites and bonds to a quantum model

A lattice diagram specifies connections. A physical model also needs a hopping amplitude on each connection, the particle statistics, interactions and the occupied state. Tight-Binding Models supplies the orbital construction; here we compare how different devices realize it.

A common target Hamiltonian

A broad family of lattice targets can be written

$$\begin{aligned} H_{\mathrm{tar}} ={}& H_{\mathrm{site}} + H_{\mathrm{hop}} + H_{\mathrm{int}} + H_{\mathrm{spin}} + H_{\mathrm{drive}}(t), \\ H_{\mathrm{site}} ={}& \sum_{i,\alpha} \epsilon_{i\alpha}n_{i\alpha}, \\ H_{\mathrm{hop}} ={}& - \sum_{i<j,\alpha} \left( \tau_{ij}^{(\alpha)} c_{i\alpha}^{\dagger}c_{j\alpha} + \mathrm{h.c.} \right), \\ \tau_{ij}^{(\alpha)} ={}& J_{ij}^{(\alpha)} e^{i\phi_{ij}^{(\alpha)}}, \\ H_{\mathrm{int}} ={}& \frac{1}{2} \sum_{i,j} V_{ij}n_i(n_j-\delta_{ij}) . \end{aligned}$$

For symmetric density couplings $V_{ij}=V_{ji}$, the subtraction at $i=j$ excludes a particle interacting with itself: the onsite term is $V_{ii}n_i(n_i-1)/2$. A convention using $n_i^2$ instead must absorb the extra one-body term into the onsite energy. Each hopping sum with its Hermitian conjugate uses $i<j$, so each bond is counted once; the reverse amplitude is fixed by complex conjugation. The index $\alpha$ can label spin, polarization, orbital, valley, or an internal atomic state. The operators $c_{i\alpha}$ may be fermionic or bosonic. The same graph and hopping matrix can therefore support different many-body Hilbert spaces.

When a time-local Markovian description is adequate, an open, imperfect implementation can be modeled by the Lindblad–GKSL equation with nonnegative rates $\Gamma_\mu$,

$$\begin{aligned} \frac{d\rho}{dt} ={}& -\frac{i}{\hbar} \left[ H_{\mathrm{impl}}, \rho \right] + \sum_{\mu} \Gamma_\mu \mathcal D[L_\mu]\rho, \\ H_{\mathrm{impl}} ={}& H_{\mathrm{tar}} + \delta H_{\mathrm{cal}} + \delta H_{\mathrm{leak}}, \\ \mathcal D[L]\rho ={}& L\rho L^\dagger - \frac{1}{2} \left\{ L^\dagger L,\rho \right\}. \end{aligned}$$

Here $\delta H_{\mathrm{cal}}$ contains static detuning, unwanted links, amplitude errors, and phase errors. The term $\delta H_{\mathrm{leak}}$ couples the intended active space to higher orbitals, bands, modes, or circuit levels. The rates $\Gamma_\mu$ describe loss, dephasing, heating, or imperfect pumping. For an effectively closed simulator, these rates must be negligible on the observation time; for a driven-dissipative simulator, they are part of the target and must be measured rather than hidden.

Four devices can implement a similar hopping graph while assigning different meanings to a site, particle, interaction, clock, and detector. The hopping graph alone does not specify particle statistics, the prepared state or the measured observable.

What a site means in each platform

Platform	Site and particle	Principal coupling	Interaction	Natural readout
quantum-dot array	confined electronic orbital; fermionic electron or hole	gate-controlled tunnelling	charging, exchange, and longer-range Coulomb terms	charge sensing and electrical transport
optical lattice	Wannier orbital; bosonic or fermionic atom	wavefunction tunnelling	contact, dipolar, or mediated interaction	time of flight, spectroscopy, or site-resolved imaging
photonic lattice	waveguide, ring, or cavity mode; optical field or photon	evanescent or resonant coupling	often negligible; Kerr, emitter-mediated, or excitonic when engineered	output intensity, phase, spectrum, and correlations
polariton lattice	microcavity mode; mixed photon–exciton boson	photon hopping plus excitonic hybridization	exciton-mediated nonlinearity and saturation	angle-, energy-, time-, and position-resolved emission
circuit lattice	resonator or nonlinear circuit mode; microwave photon or encoded excitation	capacitive, inductive, or parametrically activated exchange	Josephson anharmonicity and engineered couplers	site-resolved microwave readout and state tomography

The table is a translation, not an equivalence proof. Matching the intended $J_{ij}$ does not establish matching statistics, interactions, reservoirs, preparation, or measurement operators.

Quantum-Dot Arrays

From artificial atoms to an extended Hubbard array

For one active orbital per dot, a common fermionic model is

$$\begin{aligned} H_{\mathrm{QD}} ={}& H_{\epsilon} + H_t + H_C, \\ H_{\epsilon} ={}& \sum_{i,\sigma} \widetilde{\epsilon}_i n_{i\sigma}, \\ H_t ={}& - \sum_{\langle i,j\rangle,\sigma} \left( t_{ij} d_{i\sigma}^{\dagger}d_{j\sigma} + \mathrm{h.c.} \right), \\ H_C ={}& \sum_i U_i n_{i\uparrow}n_{i\downarrow} + \sum_{i<j} V_{ij}n_i n_j . \end{aligned}$$

The onsite energy $\widetilde{\epsilon}_i$ includes gates, static disorder, and fixed external or background charges. Occupation-dependent shifts from the active electrons elsewhere in the array belong to the explicit $V_{ij}n_i n_j$ interaction and are not added again to $\widetilde{\epsilon}_i$. A Hartree approximation or a conditional addition-energy description is a separate construction and must retain its consistent interaction subtraction. The offsite terms $V_{ij}$ are often appreciable because Coulomb interactions are long ranged. A minimal onsite-only Hubbard model is therefore a hypothesis to test, not a default consequence of drawing an array of dots.

Gate voltages are not local coordinates. To first order,

$$\delta\boldsymbol{\epsilon} = \mathbf A\, \delta\mathbf V_{\mathrm g},$$

where the lever-arm matrix $\mathbf A$ is generally dense. Experimental virtual gates apply an approximate inverse of this matrix so that one control coordinate changes a chosen onsite energy or barrier while compensating crosstalk. The inverse is only local in parameter space: charge rearrangements and gate-dependent capacitances require recalibration.

In the singly occupied, large-interaction regime, virtual hopping produces antiferromagnetic exchange. For a symmetric two-site sector with nearest-neighbor repulsion,

$$J_{\mathrm{ex}} \simeq \frac{4|t|^2}{U-V}, \qquad |t|\ll U-V.$$

This relation is a useful scale estimate, but it does not certify a Heisenberg array. Valley states, spin–orbit coupling, direct exchange, charge admixture, and unequal dot energies can change the effective spin Hamiltonian.

Finite arrays and correlation scales

Hensgens and collaborators tuned a finite dot array through a collective Coulomb-blockade crossover, a finite-size analog of a Mott transition. This distinction matters: contact resistance can suppress current even when the array itself is compressible, and a few sites do not establish a thermodynamic transition.

Local charge sensing tests filling, while excitation spectroscopy and spin correlations test the inferred interaction scales. The electron temperature must be compared with the exchange scale, which can be much smaller than the charging energy. Exercise 2 illustrates this separation.

Optical Lattice Analogs

Clean potentials do not guarantee cold many-body states

Interfering laser fields create periodic potentials whose lowest-band projection can realize Bose–Hubbard, Fermi–Hubbard, spin, and synthetic-gauge models. The detailed construction and Wannier integrals are developed in Optical Lattices. In the simplest single-band case,

$$\begin{aligned} H_{\mathrm{atom}} ={}& H_J + H_U + H_{\mathrm{trap}}, \\ H_J ={}& - \sum_{\langle i,j\rangle,\sigma} \left( J_{ij} e^{i\phi_{ij}} a_{i\sigma}^{\dagger}a_{j\sigma} + \mathrm{h.c.} \right), \\ H_U ={}& \sum_i \frac{U_i}{2} n_i(n_i-1), \\ H_{\mathrm{trap}} ={}& \sum_i V_i^{\mathrm{trap}}n_i . \end{aligned}$$

The displayed interaction assumes state-independent onsite density coupling and no spin-changing collisions; it includes the single-component bosonic model. Fermionic mixtures require the appropriate intercomponent scattering parameters.

The platform’s strengths are low microscopic disorder, controllable statistics and interactions, flexible geometry, and isolation from a solid-state bath. Quantum gas microscopy can measure occupation and correlations site by site. Periodic driving and laser-assisted tunnelling can add Peierls phases and synthetic flux.

Its limiting resources are preparation entropy, slow equilibration, particle loss, residual harmonic confinement, calibration of $J/U$, and heating from lattice light or drive. A visually uniform optical potential need not produce a homogeneous state because the smooth trap changes the local chemical potential. Likewise, a slow lattice ramp relative to a band gap can still be nonadiabatic relative to a small many-body gap.

Which state was prepared?

Greiner and collaborators observed the bosonic superfluid–Mott transition, and Mazurenko and collaborators measured antiferromagnetic correlations in a finite two-dimensional Fermi–Hubbard system. Their observables concern different many-body states, not merely different pictures of a lattice.

Site-resolved measurements can reveal filling, doublons and spin correlations, but their parity and loss responses must be included when inferring those quantities. Even a precisely known Hamiltonian need not be in its ground state. The analogy also does not automatically include a solid’s phonons, Coulomb tails or electrical conductivity.

Photonic and Polaritonic Lattices

Propagation can emulate time

In a weakly guiding waveguide array, the paraxial equation maps longitudinal propagation to Schrödinger evolution. After a modal projection,

$$i\frac{d\boldsymbol{\psi}}{dz} = \mathbf K(z)\boldsymbol{\psi},$$

Here $z$ has length units and the entries of $\mathbf K$ have inverse-length units. Evolution with $z$ emulates time evolution; it is not evolution in laboratory time. This mapping makes waveguide arrays exceptionally direct laboratories for interference, edge propagation, Floquet band engineering, and compact localized states. It does not turn an optical beam into an equilibrium electron gas: the observed output is the propagated field from a chosen input condition.

Ring resonators, photonic crystals, and cavity arrays instead evolve in laboratory time. A semiclassical driven array often obeys

$$i\frac{d\alpha_i}{dt} ={} \left( \omega_i - i\frac{\kappa_i}{2} \right) \alpha_i - \sum_j J_{ij}\alpha_j + U_i|\alpha_i|^2\alpha_i + F_i(t).$$

Here $\alpha_i$ is normalized so that $|\alpha_i|^2$ is a mode occupation; $\omega_i$, $J_{ij}$, $U_i$ and $\kappa_i$ have angular-frequency units, unlike the energy-valued hopping above. $F_i$ has amplitude-per-time units. Loss, pumping and the occupation-dependent shift affect both the spectrum and the spatial pattern. For few-photon claims, the operator master equation and normally ordered correlation functions are required; a classical coupled-mode fit is insufficient.

Exciton–polaritons add a matter fraction that strengthens interactions but also introduces exciton disorder, saturation, reservoir dynamics, and branch-dependent loss. Their emitted light provides unusually rich momentum- and position-resolved access to the mode, while simultaneously making the detector part of an open-system inference problem. Driven-Dissipative Matter develops the pump–loss field theory and the distinction among condensation, lasing, and superfluid response.

What topological photonics demonstrates

Coupled silicon rings and helical waveguide arrays have demonstrated boundary-localized propagation robust against selected defects. Photonic Lieb lattices have displayed compact flat-band localization. Polariton honeycomb lattices have resolved Dirac cones and flat bands, and magnetically biased polariton lattices have shown chiral edge-mode propagation and reversal with field.

These are substantive topological-wave and band-engineering results. They do not imply quantized electronic Hall conductance, a filled Fermi sea, or immunity to arbitrary disorder. The bulk dispersion, boundary spectrum and response to controlled defects test different parts of the band model. Ozawa and collaborators explain the distinction between topological wave propagation and electronic Hall transport. Light following an edge alone does not measure a Chern number.

Circuit Lattices

Josephson nonlinearity makes photons interact

Arrays of superconducting resonators, transmons, or hybrid qubit–resonator cells can implement bosonic hopping models with strong, adjustable nonlinearities. A useful rotating-frame model is

$$\frac{H_{\mathrm{circ}}}{\hbar} ={} \sum_i \left[ \Delta_i n_i + \frac{K_i}{2} n_i(n_i-1) \right] - \sum_{i<j} \left( J_{ij}e^{i\phi_{ij}} b_i^\dagger b_j + \mathrm{h.c.} \right) + \frac{H_{\mathrm{pump}}}{\hbar}.$$

The Josephson element supplies the Kerr scale $K_i$ or an effective spin nonlinearity. For a transmon, the leading Kerr coefficient is negative; the quartic model is then a low-occupation approximation, not a Hamiltonian to extrapolate to arbitrarily many photons. Higher circuit levels and the full Josephson potential limit that approximation. Tunable couplers and parametric modulation control $J_{ij}$ and can imprint synthetic Peierls phases. Microwave control enables local preparation and time-resolved measurement inaccessible in most natural materials.

The same components create limitations: fabrication spread in $\Delta_i$, leakage beyond a qubit subspace, drive-induced Stark shifts, residual couplings, correlated control errors, relaxation, and dephasing. For coherent dynamics of an individual excitation, useful scale comparisons are

$$\begin{aligned} \Gamma_{\mathrm{inc}} &\equiv \max_i \left\{ \kappa_i,\gamma_{\phi i} \right\}, \\ \hbar\Gamma_{\mathrm{inc}} &\ll E_{\mathrm{dyn}}, \\ \tau_{\mathrm{sequence}} \Gamma_{\mathrm{inc}} &\ll 1, \end{aligned}$$

unless dissipation is intentionally included in the target. These per-site comparisons do not bound a whole array’s error: with $N$ excitations and identical independent loss rate $\kappa$, the probability of no loss during $t$ is $e^{-N\kappa t}$ for number-conserving dynamics. Exercise 6 compares local and global survival.

Small superconducting circuits have measured chiral currents of interacting microwave photons under synthetic flux. Larger circuit-QED chains have revealed driven dissipative switching and spectroscopic signatures of localization. Ma and collaborators used reservoir engineering to stabilize an incompressible one-photon-per-site state and follow defect removal in a Bose–Hubbard array. The phrase “Mott insulator of photons” in that experiment refers to a deliberately stabilized open-system state; it should not be silently reinterpreted as the equilibrium ground state of a closed photon-number-conserving solid.

Designer Flat Bands and Topology

Geometry is a control knob, not a conclusion

Lieb, kagome, stub, dice, and line-graph geometries can suppress dispersion through destructive interference. Honeycomb and dimerized graphs can host Dirac crossings or topological band inversions. Artificial platforms are valuable because bonds and boundaries can be changed without synthesizing an entirely new bulk crystal.

The gauge-invariant flux through an oriented loop $\mathcal C$ is

$$\Phi_{\mathcal C} = \left[ \sum_{(ij)\in\mathcal C} \phi_{ij} \right]_{\!2\pi}.$$

Individual bond phases depend on the site basis; loop fluxes do not. A topological design therefore requires phase calibration around independent loops, not merely a programmed waveform that was intended to create complex hopping.

The term flat band also needs an operational definition. Measure the full relevant dispersion and quote its bandwidth $W$, isolation gap $\Delta_{\mathrm{iso}}$, linewidth or disorder scale $\Gamma$, interaction scale $U_{\mathrm{eff}}$, and temperature. A density-of-states peak or a localized input pattern is not enough to determine all five. The canonical distinctions among spectral flatness, compact states, quantum geometry, and correlated Chern bands are developed in Flat Bands.

Atom-by-atom and electrostatic electronic lattices

Scanning-probe manipulation of repulsive CO molecules on Cu(111) has confined surface-state electrons into artificial honeycomb, Lieb, and fractal geometries. Spectroscopy and wavefunction maps established designer Dirac behavior, a Lieb-lattice flat band, and geometry-dependent fractal electronic states. These experiments offer atomic control and direct local spectroscopy, but the substrate continuum, finite structures, tip response, and weak interaction scale delimit the analogy to an isolated many-body lattice.

Electrostatically patterned two-dimensional electron gases offer electrical transport and Coulomb interactions. In either implementation, measuring the one-particle spectrum and measuring interacting correlations answer different questions. Programmability is most informative when a changed coupling predicts a new observable that was not used to fit the model.

Exercises

1. Gauge transformations and loop flux

Let the site operators transform as $c_i\mapsto e^{i\chi_i}c_i$. Show how a hopping phase $\phi_{ij}$ transforms and prove that $\Phi_{\mathcal C}$ is unchanged around a closed oriented loop.

Solution

The hopping operator transforms as

$$c_i^\dagger c_j \longmapsto e^{-i\chi_i} e^{i\chi_j} c_i^\dagger c_j.$$

The same Hamiltonian form is retained if

$$\phi_{ij} \longmapsto \phi_{ij} + \chi_i - \chi_j.$$

Summing around the loop gives

$$\sum_{(ij)\in\mathcal C} \left( \phi_{ij} + \chi_i - \chi_j \right) = \sum_{(ij)\in\mathcal C} \phi_{ij},$$

because every site phase appears once with each sign. The loop flux modulo $2\pi$ is therefore gauge invariant, even though no individual bond phase is.

2. Exchange scale in a double dot

A symmetric double dot has $U=1.20\,\mathrm{meV}$, nearest-neighbor repulsion $V=0.20\,\mathrm{meV}$, and $|t|=30\,\mu\mathrm{eV}$. Estimate $J_{\mathrm{ex}}$ and check the small-hopping condition.

Solution

The virtual charge-excitation cost is

$$U-V = 1.00\,\mathrm{meV}.$$

Therefore

$$\begin{aligned} J_{\mathrm{ex}} &\simeq \frac{4(0.030\,\mathrm{meV})^2} {1.00\,\mathrm{meV}} \\ &= 0.0036\,\mathrm{meV} = 3.6\,\mu\mathrm{eV}. \end{aligned}$$

The ratio $|t|/(U-V)=0.03$ is small, so the leading second-order estimate is plausible. It would still need corrections for detuning, valley states, spin–orbit coupling, and higher orbitals in a real device.

3. Diagnose a finite-size insulator claim

A six-site dot chain shows a conductance minimum at one electron per site. The minimum deepens as $U/t$ increases, but no local charge sensor or temperature estimate is available. Which claim level is supported, and what three measurements would most improve it?

Solution

The data support an interaction-dependent transport suppression in a finite array. They are compatible with collective Coulomb blockade or a finite-size Mott analog, but they do not establish uniform filling, bulk incompressibility, or a thermodynamic phase.

The most useful additions are:

1. site-resolved or sufficiently local charge sensing to establish the occupation pattern;
2. compressibility or addition spectroscopy showing an interaction gap rather than a contact bottleneck;
3. calibrated electron temperature and a parameter sweep comparing the gap with $t$, $U$, and disorder.

Repeating the measurement for several chain lengths and contact configurations would then address scaling and lead effects.

4. Coherent cycles in a circuit lattice

A circuit lattice has $J/(2\pi)=20\,\mathrm{MHz}$, Kerr nonlinearity $|K|/(2\pi)=100\,\mathrm{MHz}$, and photon-loss rate $\kappa/(2\pi)=1.0\,\mathrm{MHz}$. Compute $|K|/J$, $\kappa/J$, and the number of reference coupling periods $2\pi/J$ within one lifetime $1/\kappa$.

Solution

Ratios are unchanged when all angular frequencies are divided by $2\pi$:

$$\frac{|K|}{J} = 5, \qquad \frac{\kappa}{J} = 0.05.$$

The stipulated reference coupling period is $2\pi/J=1/(20\,\mathrm{MHz})=50\,\mathrm{ns}$, while the lifetime is $1/\kappa=1/(2\pi\times1\,\mathrm{MHz})\simeq159\,\mathrm{ns}$. Thus

$$\frac{1/\kappa}{2\pi/J} = \frac{J}{2\pi\kappa} \simeq 3.18.$$

Only about three reference periods fit within one lifetime. This is a stated comparison clock, not a universal population-oscillation period: an isolated resonant dimer with off-diagonal angular coupling $J$ has population period $\pi/J$. The energy ratio $\kappa/J=0.05$ can look small while the available real-time propagation window remains short; the convention for rates and periods matters.

5. A compact state on a Lieb plaquette

Consider one square face of an ideal nearest-neighbor Lieb lattice. Four occupied edge sites sit at the midpoints of the face’s sides, and four connector sites sit at its corners. Each edge couples with equal real hopping $J$ to its two adjacent corner connectors. Set all edge onsite energies to zero and all other amplitudes initially to zero. Construct a normalized compact eigenstate and verify cancellation at every connector, including the absence of leakage into the surrounding lattice.

Solution

Label the four edges cyclically and choose

$$\boldsymbol{\psi}_{\mathrm{edge}} =\frac12(1,-1,1,-1).$$

The norm is one. At corner $\ell$, the two adjacent occupied edges contribute opposite amplitudes, so, with cyclic indices,

$$J(\psi_\ell+\psi_{\ell+1})=0, \qquad \ell=1,2,3,4.$$

All four connector amplitudes therefore remain zero. Each occupied edge is coupled only to those zero-amplitude connectors and has zero onsite energy, so $H\psi=0$ on the edges as well. Outside the square, unoccupied edges receive no amplitude because their connector amplitudes vanish. This verifies a compact zero-energy eigenstate of the full ideal graph, not merely cancellation at one central site. A common nonzero edge onsite energy would shift its eigenvalue without changing this cancellation.

The four-edge pattern around an empty face is the geometry used in Mukherjee and collaborators’ Lieb-lattice experiment. Four arms around a single central connector would cancel there but generally leak into outer connectors. Unequal couplings, onsite disorder, next-neighbor hopping, or loss imbalance generally spoil exact compactness. Observing localization from this input tests interference, but a full flat-band claim still requires the mode spectrum or dispersion.

6. Local lifetime and whole-array survival

Take eight photons in a number-conserving circuit array, with independent single-photon loss at the same rate $\kappa$ on every site and no pump. Using $\kappa/J=0.05$, find the probability that no photon is lost during $t=2\pi/J$. Compare with one photon. Does hopping change this result?

Solution

In a fixed-$N$ sector, the total jump rate is $\kappa\sum_i\langle n_i\rangle=N\kappa$. Therefore

$$P_0(t)=e^{-N\kappa t},\qquad P_0(2\pi/J)=e^{-2\pi N(0.05)}.$$

For one photon this is $0.7304$; for eight it is $0.0810$. Number-conserving hopping redistributes the photons but preserves the total loss rate when all sites have the same $\kappa$. Unequal rates, pumping or interactions that change particle number require a different calculation. A small local loss-to-hopping ratio alone does not ensure high whole-state survival.

Further connections

Flat Bands explains compact states and band geometry. Open-System Circuit QED develops microwave measurement and dissipation, and Floquet Quantum Matter develops the role of micromotion and heating in periodically driven lattices.

References

• I. Bloch, J. Dalibard, and W. Zwerger, “Many-Body Physics with Ultracold Gases,” Reviews of Modern Physics 80, 885–964 (2008), doi:10.1103/RevModPhys.80.885.
• M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, “Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice,” Physical Review X 7, 011016 (2017), doi:10.1103/PhysRevX.7.011016.
• K. K. Gomes et al., “Designer Dirac Fermions and Topological Phases in Molecular Graphene,” Nature 483, 306–310 (2012), doi:10.1038/nature10941.
• M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum Phase Transition from a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms,” Nature 415, 39–44 (2002), doi:10.1038/415039a.
• M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging Topological Edge States in Silicon Photonics,” Nature Photonics 7, 1001–1005 (2013), doi:10.1038/nphoton.2013.274.
• T. Hensgens et al., “Quantum Simulation of a Fermi–Hubbard Model Using a Semiconductor Quantum Dot Array,” Nature 548, 70–73 (2017), doi:10.1038/nature23022.
• T. Jacqmin et al., “Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons,” Physical Review Letters 112, 116402 (2014), doi:10.1103/PhysRevLett.112.116402.
• S. N. Kempkes et al., “Design and Characterization of Electrons in a Fractal Geometry,” Nature Physics 15, 127–131 (2019), doi:10.1038/s41567-018-0328-0.
• S. Klembt et al., “Exciton–Polariton Topological Insulator,” Nature 562, 552–556 (2018), doi:10.1038/s41586-018-0601-5.
• R. Ma et al., “A Dissipatively Stabilized Mott Insulator of Photons,” Nature 566, 51–57 (2019), doi:10.1038/s41586-019-0897-9.
• A. Mazurenko et al., “A Cold-Atom Fermi–Hubbard Antiferromagnet,” Nature 545, 462–466 (2017), doi:10.1038/nature22362.
• S. Mukherjee et al., “Observation of a Localized Flat-Band State in a Photonic Lieb Lattice,” Physical Review Letters 114, 245504 (2015), doi:10.1103/PhysRevLett.114.245504.
• T. Ozawa et al., “Topological Photonics,” Reviews of Modern Physics 91, 015006 (2019), doi:10.1103/RevModPhys.91.015006.
• M. C. Rechtsman et al., “Photonic Floquet Topological Insulators,” Nature 496, 196–200 (2013), doi:10.1038/nature12066.
• P. Roushan et al., “Chiral Ground-State Currents of Interacting Photons in a Synthetic Magnetic Field,” Nature Physics 13, 146–151 (2017), doi:10.1038/nphys3930.
• P. Roushan et al., “Spectroscopic Signatures of Localization with Interacting Photons in Superconducting Qubits,” Science 358, 1175–1179 (2017), doi:10.1126/science.aao1401.
• M. R. Slot et al., “Experimental Realization and Characterization of an Electronic Lieb Lattice,” Nature Physics 13, 672–676 (2017), doi:10.1038/nphys4105.