Qubit measurement and backaction in a multimode nonreciprocal system

Abstract

High fidelity qubit readout is a cornerstone for quantum information protocols. In traditional superconducting qubit readout, a chain of microwave amplifiers and nonreciprocal components aid in detecting the qubit's state with tolerable added noise and backaction. However, the loss, size, and magnetic field of standard nonreciprocal components have sparked a decades-long search for more efficient and scalable alternatives. One prominent approach employs networks of parametrically coupled modes to achieve nonreciprocity. While this class of devices can be directly integrated with the qubit's readout cavity, current understanding of the resulting single quantum system is substantially lacking. Here we provide a first-principles theoretical tool to understand and design networks of linear modes integrated with embedded qubits. We utilize this theory to inform and analyze the experimental implementation of a qubit readout with an integrated three-mode nonreciprocal system. In doing so, we achieve excellent agreement between the experimental and theoretical qubit measurement and dephasing rates. We then theoretically analyze the same system operated as an integrated nonreciprocal amplifier, predicting high efficiency for reasonable experimental parameters.

Figures (22)

• Figure 1: Integrating the readout network. (a) Conceptual diagram of a dispersive superconducting qubit measurement with a traditional readout chain consisting of a qubit (Q) and cavity (C) subsystem $\hat{\rho}_{\scaleobj{0.8}{\mathrm{Q}},\scaleobj{0.8}{\mathrm{C}}}$, as well as an arbitrary resonant parametric amplifier $\hat{\rho}_{\mathrm{amp}}$, separated by an isolator. The amplifier is comprised of a network of coupled modes labeled $\mathrm{M}_{n}$, $n \in \mathbb{N}$. (b) When the intermediate isolator is removed, the cavity mode must be included in the network of modes comprising the amplifier. The qubit and readout network therefore combine into a single quantum system described by the inseparable density matrix $\hat{\rho}_{\scaleobj{0.8}{\mathrm{Q}},\scaleobj{0.8}{\mathrm{C}},\mathrm{amp}}$.
• Figure 2: A three-mode readout network for efficient qubit measurement. The amplifier (A), buffer (B), and cavity (C) modes form an interferometer to directionally route and amplify readout signals containing information about the qubit (Q). The arrows with color gradients represent beam-splitter interactions between modes J and K with strength $g_{\scaleobj{0.8}{\mathrm{J}}\scaleobj{0.8}{\mathrm{K}}}$. The AC beam splitter is arbitrarily chosen to be the only beam splitter with nonzero phase, entirely constraining the interferometer phase $\phi$. The dashed arrow represents dispersive coupling of strength $\chi_{\scaleobj{0.8}{\mathrm{C}}}$, and all other dispersive shifts are neglected in this diagram. The solid gray single-sided arrow represents single mode squeezing at the A mode with strength $\lambda$ and phase $\theta$. The oscillating, double-sided arrow represents the input/output coupling to mode B, facilitating a drive of amplitude $\varepsilon$ and phase $\varphi$.
• Figure 3: Device overview.(a) Sliced COMSOL model of the device. A transmon qubit is housed on a chip to the left, a $\lambda/4$ coaxial cavity in the center constitutes the cavity (C) mode, and a secondary chip on the right contains the amplifier (A) and buffer (B) modes. This chip also includes a dc SQUID and its inductively-coupled bias line, as well as a port to drive both the qubit and the B mode. (b) Measured and simulated frequencies of each mode and the qubit as a function of the DC flux bias through the SQUID. The black dashed line marks the operating flux bias of 0.246 $\Phi_0$. (c) Table of the mode frequencies, linewidths, and dispersive couplings with the qubit for each mode at the operating flux bias ($\chi_{\scaleobj{0.8}{\mathrm{Q}}}$ denotes the self-Kerr or anharmonicity of the transmon).
• Figure 4: Experimental sequences to quantify qubit measurement.(a)$\Gamma_{\mathrm{d}}$ is extracted with interleaved $T_1$ (blue dashed box) and $T_2^\mathrm{echo}$ (purple dashed box) measurements in the presence of arbitrary linear operations on the modes, denoted $U$. In reality, the operations in $U$ begin before the qubit state preparation to ensure the multimode system is in the steady state. The B mode rail is drawn thicker to represent the mode's comparatively large linewidth. Example $T_1$ and $T_2^\mathrm{echo}$ traces in the absence of any readout network drives or parametric interactions ($U=\bm{I}_3$) are shown below the sequence diagram. (b) To extract $\Gamma_{\mathrm{meas}}$, the qubit is prepared in either the ground or excited state, a set of operations $U$ that contains a weak B mode drive is initiated, then the steady state signal is averaged over time $\tau$. Single shot histograms (plotted for the case of the interferometer with $\phi=0.55\pi$ and $\tau$ = 700 ns) are fit to extract $\mathrm{SNR}^2$ as a function of $\tau$, producing a linear relationship with slope $\Gamma_{\mathrm{meas}}$.
• Figure 5: Mean thermal occupancy extraction. Experimental data and theoretical fits (with one free parameter each) of the qubit's parasitic dephasing rate $\Gamma_{\mathrm{d,p}}$ as a function of (a) AC beam-splitter strength to extract $n^{\mathrm{th}}_{\scaleobj{0.8}{\mathrm{A}}} = 0.33$ and (b) BC beam-splitter strength to extract $n^{\mathrm{th}}_{\scaleobj{0.8}{\mathrm{B}}} = 0.0030$.