Imaging Photon Lattice States by Scanning Defect Microscopy

Microwave photons inside lattices of coupled resonators and superconducting qubits can exhibit surprising matterlike behavior. Realizing such open-system quantum simulators presents an experimental challenge and requires new tools and measurement techniques. Here, we introduce scanning defect microscopy as one such tool and illustrate its use in mapping the normal-mode structure of microwave photons inside a 49-site kagome lattice of coplanar waveguide resonators. Scanning is accomplished by moving a probe equipped with a sapphire tip across the lattice. This locally perturbs resonator frequencies and induces shifts of the lattice resonance frequencies, which we determine by measuring the transmission spectrum. From the magnitude of mode shifts, we can reconstruct photon field amplitudes at each lattice site and thus create spatial images of the photon-lattice normal modes.

Popular Summary

While measurements of global system properties are often readily obtained, measurements of more detailed local properties can provide a deeper understanding of the system at hand on a microscopic level. However, measuring local properties is often difficult. One solution is to use scanning probe measurements in which a probe is dragged across the surface of the system to create a local disturbance or defect. The effects of this defect can be tracked as a systematic change in a global measurement, allowing one to indirectly reconstruct an image of the local properties. Here, we introduce a new scanning probe tool that allows us to image the location of photons inside a lattice of microwave cavities.

We study a two-dimensional, 49-site lattice of superconducting microwave cavities that can serve as a backbone for realizing quantum phase transitions of light. The resonators in this cavity are connected via three-way capacitors. Our scanning probe consists of a sapphire dielectric roughly 2 mm on a side that we bring into close proximity with a lattice cavity, thereby shifting its frequency (by up to 663 MHz) and effectively “poking a hole” in the lattice; the transmission of photons through the lattice depends on the defect strength. By monitoring the changes in transmission of a coherent microwave tone while probing each cavity, we can image the amplitudes of electromagnetic normal modes on each lattice site. Once qubits are integrated into such lattices, mediated photon-photon interactions can give rise to emergent many-body states of light. We note that understanding such states through global measurements alone will be a tremendous challenge.

We expect that the local information accessed by our scanning probe will be useful for studying quantum phase transitions.

Figure 1

Scanning defect microscopy of a photon lattice. A dielectric probe is scanned across a photon lattice of coplanar waveguide resonators. The dielectric locally perturbs the electric field and shifts the frequency of a target resonator, creating a lattice defect. The transmission of photons through the lattice systematically depends on the defect strength and can be used to image the photon occupancy on individual lattice sites.

Figure 2

Verification of probe performance and calibration for a single resonator. (a) Measurement data of the resonator frequency shift as a function of probe position along the resonator agree well with predictions from finite-element simulation (using a $2\mu \mathrm{m}$ probe height). In the experiment, the probe is centered over and placed into mechanical contact with the resonator for each data point. The frequency shifts follow the field distribution of the $\lambda /2$ mode, with shift maxima for probe positions close to the resonator ends where the field is strongest and a shift minimum at the resonator midpoint. (b) The calibration of the resonator frequency shift (defect size) with respect to vertical distance of the probe from the resonator is performed at the lateral position of maximum shift. After adjusting for an offset in the $z$ direction, finite-element simulation and measurement data agree over most of the vertical distance range. Deviations observed at the smallest probe heights may be due to a small tilt of the dielectric probe with a misalignment angle as small as 0.02°, as indicated by simulations shown in the inset.

Figure 3

Device picture of photon lattice, consisting of 49 coupled coplanar-waveguide resonators. (a) The resonators are arranged to form a kagome lattice for photons (the dual of the honeycomb geometry visible when viewing the physical resonators). The device is equipped with a port used to drive the lattice with a coherent microwave tone (yellow triangle), and three possible ports to detect the transmitted signal (red triangles). (b) Resonators have a meandering section (to save space on the chip) and are coupled by three-way capacitors. Small capacitors (c) connect four of the edge resonators to ports and (d) terminate the remaining edge resonators by coupling to a high-frequency resonator stub. (e) Isolated ground planes within the lattice are connected with aluminum bridges evaporated on top of insulating pads.

Figure 4

Transmission and mode-shift data from scanning defect microscopy. (a) The transmission spectrum of the kagome lattice exhibits a multitude of resonances corresponding to normal modes that can be excited and detected through the selected input and output ports. (b) Positioning the scanning probe above one resonator and changing the vertical probe distance $z$ produces shifts in mode frequencies, here seen as “bending” of maxima in the measured transmission S21 (color-coded), especially for small probe distances. (c) Calibration allows conversion of the probe height $z$ into the corresponding defect size $\mathrm{\Delta }{\omega }_n$ of site $n$. For small defect size, the normal-mode shifts $\mathrm{\Delta }{\mathrm{\Omega }}_{\mu |n}$ are linear in the $\mathrm{\Delta }{\omega }_n$. Normal-mode weights are directly determined from the slope of each curve.

Figure 5

Normal-mode weights for three select modes, $\mu =35$, 36, and 49. For each lattice site $n$, the mode weight $W_{\mu n}=|(n|\mu ){|}^2$ is depicted as an overlay of two color-coded disks. Inner disks represent the theory prediction for the weights, the outer disks the weights determined experimentally by scanning defect microscopy. The bottom panels show deviations in normal-mode weights as lines connecting experimental data (black dots) and theory prediction (blue dots). Comparison shows good agreement with noticeable deviations in only a few sites in each case.

Figure 6

Transmission spectra for different lateral probe positions. (a) Starting on a ground plane, the probe is moved across one lattice resonator, traverses the ground plane, and stops when it is centered on a second lattice resonator. Significant shifts of the lowest-frequency mode result when the probe is centered on one resonator; no shifts occur when it is in contact with a ground plane. (b) Starting at the edge of a three-way coupling capacitor, the probe is moved across the capacitor, then off the capacitor and onto a lattice resonator. When the probe covers a three-way capacitor, three lattice sites are perturbed simultaneously, resulting in multiple modes shifting down in frequency.