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Strong parametric dispersive shifts in a statically decoupled two-qubit cavity QED system


Qubits in cavity quantum electrodynamic (QED) architectures are often operated in the dispersive regime, in which the operating frequency of the cavity depends on the energy state of the qubit, and vice versa. The ability to tune these dispersive shifts provides additional options for performing either quantum measurements or logical manipulations. Here we couple two transmon qubits to a lumped-element cavity through a shared superconducting quantum interference device (SQUID). Our design balances the mutual capacitive and inductive circuit components so that both qubits are statically decoupled from the cavity with low flux sensitivity, offering protection from decoherence processes. Parametric driving of the SQUID flux enables independent, dynamical tuning of each qubit’s interaction with the cavity. As a practical demonstration, we perform pulsed parametric dispersive readout of both qubits. The dispersive frequency shifts of the cavity mode follow the theoretically expected magnitude and sign. This parametric approach creates an extensible, tunable cavity QED framework with various future applications, such as entanglement and error correction via multi-qubit parity readout, state and entanglement stabilization, and parametric logical gates.

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Fig. 1: A flux- and parametrically tunable two-qubit cavity QED system.
Fig. 2: Cavity spectrum and parametric dispersive shifts of the R transmon.
Fig. 3: Static dispersive shifts of the R transmon.
Fig. 4: Cavity photon-induced dephasing.
Fig. 5: Demonstration of joint state readout of both L and R transmons.

Data availability

The data that support the findings of this study are available from the NIST Public Data Repository at Source data are provided with this paper.


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This work was partially performed with financial assistance from award no. 70NANB18H006 from the US Department of Commerce, National Institute of Standards and Technology. T.N., Z.X., E.D., L.R., L.G. and A.K. received support from the Department of Energy under grant no. DE-SC0019461. We thank D. Slichter and A. Sirois for commenting on the paper.

Author information

Authors and Affiliations



T.N. conducted the experiment. T.N. and R.W.S. analysed the data. R.W.S. designed and K.C. fabricated the device. X.Y.J. and J.A. contributed to the measurement set-up and software. Z.X., E.D., L.C.G.G., L.R. and A.K. provided theoretical support and suggestions for the experiment. T.N. and R.W.S. wrote the paper and supplementary information. R.W.S. conceived the experiment and supervised the project. All authors contributed to the preparation of the paper.

Corresponding authors

Correspondence to T. Noh or R. W. Simmonds.

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Nature Physics thanks Christopher Axline and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Cavity spectrum and parametric dispersive shifts of the L transmon.

As with Fig. 2 in the main text for the R transmon, we performed the same measurements with the L transmon. (a) A typical cavity spectrum while sweeping the pump frequency ωp. The upper and lower panel shows the result with the L transmon in the ground state \(\left\vert {g}_{L}\right\rangle\) and the first excited state \(\left\vert {e}_{L}\right\rangle\), respectively. (b) The dispersive shifts as a function of the pump detuning frequency, ΔpL = ωp − (ωC − ωL), at various calibrated pump amplitudes, δϕp = 0.028 (red), 0.056 (orange), 0.084 (yellow), 0.112 (green), 0.14 (blue), and 0.168 (purple). The solid lines are the fits to the data from the theory. See the main text for more details.

Source data

Extended Data Fig. 2 Static dispersive shifts of the L transmon.

As with Fig. 3 in the main text for the R transmon, we performed the same measurements with the L transmon. (a) The static dispersive shift 2χsL/2π as a function of the L transmon frequency. For empty squares, the dispersive shift was measured by comparing the cavity response with the L transmon in either state \({\left\vert g\right\rangle }_{L}\) or \({\left\vert e\right\rangle }_{L}\). For filled circles, the dispersive shift was extracted by measuring additional dephasing ΓnL while driving the cavity with a weak coherent state. The solid line represents a fit to the data based on the circuit model. (b) Additional dephasing as a function of average photon number \(\bar{n}\) of a weak coherent drive at three different flux biases where the R tranmon frequencies are 5.784 (red square), 5.826 (yellow circle), and 5.858 GHz (green triangle), respectively. Solid lines are linear fits to the data.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–9, Discussion and Table I.

Source data

Source Data Fig. 1

Cavity, R transmon, and L transmon spectroscopy data for Fig. 1d.

Source Data Fig. 2

Cavity spectrum at various pump frequencies for top 2D color maps in Fig. 2a. Cavity spectrum at two pump frequencies for bottom plots in Fig.2a. Dispersive shift as a function of pump frequency at various pump amplitudes for Fig. 2b. Parametric coupling strength as a function of pump amplitude for Fig. 2c.

Source Data Fig. 3

Static dispersive shift as a function of R transmon frequency for Fig. 3a. Additional dephasing of R transmon as a function of the average photon number at three different flux biases for Fig. 3b.

Source Data Fig. 4

T2* as a function of R transmon frequency at various average photon numbers.

Source Data Fig. 5

Cavity spectrum with four different two-transmon states for Fig. 5a, b, and c.

Source Data Extended Data Fig. 1

Cavity spectrum at various pump frequencies for top 2D color maps in Extended Data Fig. 1a. Dispersive shift as a function of pump frequency at various pump amplitudes for Extended Data Fig. 1b.

Source Data Extended Data Fig. 2

Static dispersive shift as a function of L transmon frequency for Extended Data Fig. 2a. Additional dephasing of L transmon as a function of the average photon number at three different flux biases for Extended Data Fig. 2b.

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Noh, T., Xiao, Z., Jin, X.Y. et al. Strong parametric dispersive shifts in a statically decoupled two-qubit cavity QED system. Nat. Phys. 19, 1445–1451 (2023).

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