In-situ Characterization of Light-Matter Coupling in Multimode Circuit-QED Systems

Abstract

Multimode cavity-QED systems can be leveraged to explore a wide range of physical phenomena; however, a complex multimode environment makes systematic characterization of light-matter interactions challenging. Here we present a general measurement protocol, applicable to both atomic and synthetic cavity-QED systems, that enables the determination of coupling to individual photonic modes. The method leverages measurements of the AC-Stark and Kerr effects, along with known detuning dependencies, to eliminate the need for single-photon resolution, independent photon-number calibration, or insertion-loss calibration. We demonstrate the method using a superconducting transmon qubit coupled to a one-dimensional microwave resonator lattice. We validate the consistency of the extracted light-matter couplings $g$ determined at multiple qubit detunings, and from the self-Kerr and cross-Kerr shifts for three photon modes, which provide separate measurements of $g$ for each of the three modes.

Figures (10)

• Figure 1: CPW lattice. (a) CAD of three of the nine unit cells of the CPW lattice device used in this work. The resonators forming the lattice are shown in black, and the qubit used to perform the protocol is highlighted with a yellow square. (Full device CAD is shown in Fig. \ref{['fig:mode_spectroscopy']} in the Supplemental Material.) (b) Schematic of the tight-binding lattice realized by this device overlaid on the physical layout of the resonator chain. Gold dots indicate lattice sites with dark blue lines indicating nearest-neighbor coupling. The light blue lines in the background represent the physical resonators. (c) Expected density of states for an infinite version of the quasi-1D tight-binding lattice displayed in (b). The dispersive bands are shown in light blue, and the flat bands, which extend above the scale of the plot, are shown in dark blue. The protocol is demonstrated using lattice modes between $4.96$ and $5.00$ GHz and qubit frequencies between $\sim 4.2$ and $\sim 4.6$ GHz.
• Figure 2: AC-Stark and Kerr shifts. Measurements of AC-Stark and Kerr shifts carried out using a drive mode with frequency $\omega_D/2\pi = 4.997$ GHz, a monitor mode with frequency $\omega_M/ 2\pi = 4.969$ GHz, and a qubit tuned to $\omega_q/2 \pi = 4.593$ GHz. (a) Schematic of the AC-Stark-shift measurement. Mode $D$ is driven while the AC-Stark shift of the qubit frequency, $\omega_q'$, is measured. (b) Two-tone spectroscopy measurements Blais:revmodphys of the qubit with varying drive powers to mode $D$. (See Section \ref{['app:TwoTone']} of the Supplemental Material for measurement details.) As the drive power increases, the effective qubit frequency (labeled with a black dot) is AC-Stark shifted down accordingly. (c) The measured qubit frequency shift versus the drive power. The slope of a linear fit to this data gives $\partial_{P} \omega_q'/2\pi= -32.2$ MHz/nW. (d) Schematic of the Kerr-shift measurement. Mode $D$ is driven while the self-Kerr and cross-Kerr shifts on $\omega_D'$ and $\omega_M'$, mediated by the qubit, are simultaneously measured. (e) Two-tone spectroscopy measurements of the drive mode $D$, taken by measuring transmission of a probe tone through mode $M$, with varying drive tone powers to mode $D$. As the drive power increases, the effective frequency of the drive mode (labeled with a black dot) is self-Kerr shifted down accordingly. The cross-Kerr shift on the monitor mode is inferred from the strength of the response at this point. (f) The measured frequency shift of the drive and monitor modes versus the drive power. A linear fit to the self-Kerr data yields a slope $\partial_{P} \omega_D'/2\pi = -32.0$ kHz/nW, and similarly for the cross-Kerr data, $\partial_{P} \omega_M'/2\pi = -42.1$ kHz/nW.
• Figure 3: Consistency of qubit-mode coupling versus detuning. Results from repeated iterations of the $g$-verification protocol using a drive mode with frequency $\omega_D/2\pi = 4.997$ GHz, a monitor mode with frequency $\omega_M/2\pi = 4.960$ GHz, and with the qubit frequency swept from $\sim 4.2$ GHz -- $\sim 4.6$ GHz. The slopes (a) $\partial_{P} \omega_q'$ and (b) $\partial_{P} \omega_D'$ and $\partial_{P} \omega_M'$ at each detuning, determined from AC-Stark-shift and Kerr-shift measurements, respectively. Fit error bars are smaller than markers and not shown. (c) Calculated qubit-mode couplings for the drive mode ($g_D$) and the monitor mode ($g_M$). Error bars indicate expected systematic error determined from repeated measurement at two fixed detunings (see Section \ref{['app:stability']} of the Supplemental Material). The calculated couplings at different detunings are consistent with a drive-mode coupling $g_D/2\pi$ of approximately $16$ MHz and a monitor-mode coupling $g_M/2\pi$ of approximately $12$ MHz.
• Figure S1: Device layout and photonic modes. (a) CAD of the CPW lattice device used in this work with colors added to highlight details. The lattice resonators are shown in black, with the exception of the resonators in the central unit cell which are shown in red. Pockets created in the side of each CPW where qubits can be added are shown in pink. The three embedded flux-tunable transmon qubits are highlighted with squares. The flux-tunable transmon used to collect the data in this work is highlighted with a yellow square and the other two transmons are highlighted with gray squares. Their corresponding flux-bias lines are shown in orange. Input/output ports in the bottom-left and top-right corners of the device allow microwave transmission measurements. (b) High-power two-tone spectroscopy measurement of the CPW lattice spectrum (a detailed description of this measurement technique can be found in Ref. OBrien2025). The three modes used to test the coupling verification protocol are indicated with gray arrows ($\omega_A/2 \pi = 4.960$ GHz, $\omega_B/2 \pi = 4.969$ GHz, $\omega_C/2 \pi = 4.997$ GHz). (c)-(d) Transmission spectroscopy of mode $B$ showing the magnitude and phase of the transmitted microwave tone respectively as a function of signal frequency. A correction is applied to the measured phase response of the mode to account for linear phase delay from the transmission line.
• Figure S2: Generalized two-tone spectroscopy. (a) Schematic depicting general two-tone spectroscopy measurements. The phase of a probe tone transmitted through the sensor mode is measured while the frequency of a pump tone is swept over a target feature, either a qubit or a second normal mode. The strongest phase response $\delta \phi$ occurs when the pump frequency is resonant with the effective frequency of the target $\omega_t'$, indicating that the target frequency is shifted by $\delta \omega_t$ from its original frequency $\omega_t$. (b) Schematic depicting the phase response function of the sensor mode to a probe tone for two cases: (i) where no pump tone is applied to the target feature (solid line), and (ii) where a pump tone resonant with the target feature is applied (dashed line), shifting the center frequency of the probed mode by $\delta \omega_s$ from its original value $\omega_s$. As the expected response of the sensor is known prior to the measurement, the phase difference $\delta \phi$ observed in two-tone spectroscopy is used to determine $\delta \omega_s$. (c) Schematic of the AC-Stark-shift measurement highlighting the roles of the microwave tones involved. Two variants of the two-tone spectroscopy measurement used to track the effective qubit frequency are shown: (i) the method used to collect the data presented in this work, and (ii) a modification for systems where the transmitted signal through the drive mode is negligible. In the first case, mode $D$ functions as the sensor mode, and the probe tone through the mode doubles as the drive producing the AC-Stark shift on the qubit frequency. In the second, mode $M$ functions as the sensor mode; a probe tone is sent through mode $M$ and a dedicated drive tone is applied to mode $D$. In both cases, the qubit is the target feature and is excited by a pump tone. (d) Schematic of the Kerr-shift measurement highlighting the roles of the microwave tones involved. A probe tone is sent on resonance with mode $M$, which functions as the sensor mode. Mode $D$ functions as the target feature and the frequency of a pump tone is swept over the mode. This pump tone doubles as the drive producing the self-Kerr shift on mode $D$ and the cross-Kerr shift on mode $M$.