Hamiltonian tomography of photonic lattices

In this paper we introduce an approach to Hamiltonian tomography of noninteracting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites $\alpha$ and $\beta$ may be obtained directly from $S_{\alpha \beta }\left(\omega \right)$, the (suitably normalized) two-port measurement between sites $\alpha$ and $\beta$ at frequency $\omega$. This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is a topological property of the two-port spectrum. We further provide extensions of this technique for measurement of band projectors in finite, disordered systems with good band flatness ratios, and apply the tool to direct real-space measurement of the Chern number. Our approach demonstrates the extraordinary potential of microwave quantum circuits for exploration of exotic synthetic materials, providing a clear path to characterization and control of single-particle properties of Jaynes-Cummings-Hubbard lattices. More broadly, we provide a robust, unified method of spectroscopic characterization of linear networks from photonic crystals to microwave lattices and everything in between.

Figure 1

Truncation of integration region. To avoid the divergence in the integration of the tails of the transmission and reflection spectra, the symmetric tails of the spectra should be identified by the locations in the spectra where the lattice response has reduced to a simple Lorentzian, decaying as $1/\mathrm{\Delta }$ (for $S_{\alpha \alpha }$, the decay is $1/{\mathrm{\Delta }}^2$; for $S_{\alpha \beta }$, $\alpha \ne \beta$), and their values are identical. Here $\mathrm{\Delta }$ is the detuning of the probe frequency to the manifold of resonances being considered. The integration then should only be performed between these two points (area in gray), and the divergence of the tails (area in gray stripes) will then cancel. In practice, choosing the cutoff location is a trade-off between ensuring that one is far enough from the resonant features to be in regions with $1/\mathrm{\Delta }$ (or $1/{\mathrm{\Delta }}^2$) decay, but not so far out that other resonator modes (corresponding to parasitic resonances outside of the effective model as shown at the left, or other bands within the effective model) become important. The impact of these other modes on the integration can be further reduced using the technique outlined in Appendix pp1.

Figure 2

Three site spectroscopy. (a)–(c) Reflection spectra of sites 1, 2, and 3, respectively, of a three-site tight-binding chain, whose outer sites are at equal energies, while the central site is detuned [(a), inset]. (d)–(f) Transmission spectra for 1-2, 2-3, and 1-3, respectively. All spectra are plotted as the absolute value of the amplitude, and share a common (arbitrary) normalization. The spectra exhibit a low-energy doublet resulting from second-order coupling between sites 1 and 3, and an isolated high-energy peak arising from the central site. (g)–(i) Application of the Hamiltonian estimation technique of Eq. (4) to the three-site chain, as a function of the numerical integration range. We fix the lower limit of integration at $-150$ MHz and vary the upper limit between $-150$ MHz and 150 MHz. (g) On-site energy of site 1, converging to within $J^2/\delta \omega$ of $-\delta \omega /2$ once the integration range includes the doublet, and the rest of the way once the high-energy feature is included. (h) The tunneling matrix element between sites 1 and 2, converging only once all spectral features are included. (i) The tunneling matrix element between sites 1 and 3, which is zero in our model. For (g)–(i) we plot only the real part of the estimated matrix elements, as the imaginary part converges less rapidly.

Figure 3

Measuring Chern numbers in real space. To measure the Chern number of a disordered band in the bulk of a Chern insulator, a bulk region large compared to the unit cell size (magnetic length in the Hofstadter model, whose band projector onto an arbitrary bulk site is shown as gray-scale squares for $\alpha =\frac{1}{4}$) is partitioned into three similarly sized regions (red circles, green triangles, and blue squares). The difference of triple band-projector products $\text{red}\to \text{green}\to \text{blue}$ and $\text{blue}\to \text{green}\to \text{red}$, summed over all sites in each region, is equal to the Chern number $C/\left(12\pi i\right)$. There are a number of ways to spectroscopically measure this projector, discussed in the text.

Figure 4

Technical limitations of Chern number measurement. (a) Numerically computed dependence of the spectroscopically extracted Chern number of the lowest band upon region radius for an array of lossless resonators coupled in an $\alpha =\frac{1}{4}$ Hofstadter configuration. Once the radius exceeds a magnetic length ($\frac{1}{\alpha }=4$ sites), the measured value approaches the translationally invariant TKNN value $C=1$; it drops at the system edge (gray bar). (b) Numerically computed Chern number vs the rms variation in on-site energy. The error band shows the variation over disorder realizations; the Chern number is robust to disorder up to $\sim 0.1\times J$, comparable to the band splitting of the model. (c) Numerically computed Chern number vs resonator linewidth (in units of tunneling energy), using a frequency integration of the two port measurement $S_{\alpha \beta }\left(\omega \right)$, as described in the text, for a region with five site radius. A loss rate $\le 0.03\times \mathrm{tunneling}$ provides a fidelity $\ge 0.95$. (d)–(f) Identical calculations for the two middle bands of the $\alpha =\frac{1}{4}$ Hofstadter lattice, which touch at Dirac points and thus must be analyzed together. The extracted Chern number $C=-2$ is consistent with the TKNN formula.