Quantum squeezing amplification with a weak Kerr nonlinear oscillator

Nat. Commun. 17, 970 (2025) · DOI: 10.1038/s41467-025-67699-0

License: CC BY 4.0.

Abstract

Quantum squeezed states, with biased quantum noise, have been widely utilized in quantum sensing and quantum error correction applications. However, generating and manipulating these nonclassical states with a large squeezing degree typically requires strong nonlinearity, which inevitably induces additional decoherence that diminishes the overall performance. Here, we demonstrate the generation and amplification of squeezed states in a superconducting microwave cavity with weak Kerr nonlinearity. By subtly engineering an off-resonant microwave drive, we observe cyclic dynamics of the quantum squeezing evolution in a displaced frame of the cavity. Furthermore, we deterministically realize quantum squeezing amplification by alternately displacing the Kerr oscillator using the Trotterization technique, achieving a maximum squeezing degree of 14.6 dB and a squeezing rate of 0.28 MHz. Our demonstrated displacement-enhanced squeezing operation offers a hardware-efficient approach for generating large squeezed states, promising potential applications in quantum-enhanced sensing and quantum information processing.

Figures

33 panels with data across 3 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF. Page 2 of the PDF.

Fig. 2

• panel (b-1): Measured Wigner function $W(\alpha)$ at $t = 0$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-2): Measured Wigner function $W(\alpha)$ at $t = 600$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-3): Measured Wigner function $W(\alpha)$ at $t = 800$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-4): Measured Wigner function $W(\alpha)$ at $t = 1000$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-5): Measured Wigner function $W(\alpha)$ at $t = 1300$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-6): Measured Wigner function $W(\alpha)$ at $t = 1600$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-7): Measured Wigner function $W(\alpha)$ at $t = 2250$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Top row (Exp.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-8): Numerically simulated Wigner function $W(\alpha)$ at $t = 0$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-9): Numerically simulated Wigner function $W(\alpha)$ at $t = 600$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-10): Numerically simulated Wigner function $W(\alpha)$ at $t = 800$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-11): Numerically simulated Wigner function $W(\alpha)$ at $t = 1000$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-12): Numerically simulated Wigner function $W(\alpha)$ at $t = 1300$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-13): Numerically simulated Wigner function $W(\alpha)$ at $t = 1600$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (b-14): Numerically simulated Wigner function $W(\alpha)$ at $t = 2250$ ns of the first cycle ($C = 1$), against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$; the dashed grey line is the state trajectory. Bottom row (Sim.) of printed panel (b); the map covers only the region around the state, as deposited. data.csv
• panel (c-1): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 1$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. data.csv
• panel (c-2): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 2$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. data.csv
• panel (c-3): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 3$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. data.csv
• panel (c-4): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 4$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. data.csv
• panel (c-5): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 5$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The deposited map reaches $\mathrm{Im}(\alpha) = \pm 5$; the print shows $\pm 4$. data.csv
• panel (c-6): Measured Wigner function $W(\alpha)$ of the squeezed vacuum state at the end of cycle $C = 6$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The deposited map reaches $\mathrm{Im}(\alpha) = \pm 5$; the print shows $\pm 4$. data.csv

Fig. 3

• panel (b-1): Measured one-dimensional Wigner cuts $W(\alpha)$ after $M = 0$ Trotter steps: along the real axis, $\mathrm{Im}(\alpha) = 0$ (blue), and along the imaginary axis, $\mathrm{Re}(\alpha) = 0$ (orange), with the solid lines of the global fit to both. The pale band marks a width of $1/2$ about zero. Subplot 1 of 4 in printed panel (b). data.csv
• panel (b-2): Measured one-dimensional Wigner cuts $W(\alpha)$ after $M = 3$ Trotter steps: along the real axis, $\mathrm{Im}(\alpha) = 0$ (blue), and along the imaginary axis, $\mathrm{Re}(\alpha) = 0$ (orange), with the solid lines of the global fit to both. The pale band marks a width of $1/2$ about zero. Subplot 2 of 4 in printed panel (b). data.csv
• panel (b-3): Measured one-dimensional Wigner cuts $W(\alpha)$ after $M = 6$ Trotter steps: along the real axis, $\mathrm{Im}(\alpha) = 0$ (blue), and along the imaginary axis, $\mathrm{Re}(\alpha) = 0$ (orange), with the solid lines of the global fit to both. The pale band marks a width of $1/2$ about zero. Subplot 3 of 4 in printed panel (b). data.csv
• panel (b-4): Measured one-dimensional Wigner cuts $W(\alpha)$ after $M = 9$ Trotter steps: along the real axis, $\mathrm{Im}(\alpha) = 0$ (blue), and along the imaginary axis, $\mathrm{Re}(\alpha) = 0$ (orange), with the solid lines of the global fit to both. The pale band marks a width of $1/2$ about zero. Subplot 4 of 4 in printed panel (b). data.csv
• panel (c): Squeezing parameter $|\xi|$ against the number of Trotter steps $M$ for displacement amplitudes $|\beta|^2 = 1$, 4, 16, 36 and 64: measured (circles, error bars are 95% confidence intervals of the fits) and simulated (dashed lines). The print's top axis (time, $\mu$s) and right axis (squeezing level, dB) are not shown. data.csv
• panel (c-inset): Inset of printed panel (c): squeezing rate (MHz) against the displacement amplitude $|\beta|^2$, measured (circles, one colour per $|\beta|^2$ as in the main panel), simulated (dashed grey line) and the ideal rate $K\beta^2$ (solid black line). data.csv
• panel (d-1): Measured Wigner function $W(\alpha)$ of the squeezed state generated with $M = 1$ Trotter steps at $|\beta|^2 = 100$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The fidelity and squeezing level the print writes on the tile are not shown. data.csv
• panel (d-2): Measured Wigner function $W(\alpha)$ of the squeezed state generated with $M = 4$ Trotter steps at $|\beta|^2 = 100$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The fidelity and squeezing level the print writes on the tile are not shown. data.csv
• panel (d-3): Measured Wigner function $W(\alpha)$ of the squeezed state generated with $M = 7$ Trotter steps at $|\beta|^2 = 100$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The fidelity and squeezing level the print writes on the tile are not shown. data.csv
• panel (d-4): Measured Wigner function $W(\alpha)$ of the squeezed state generated with $M = 10$ Trotter steps at $|\beta|^2 = 100$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The fidelity and squeezing level the print writes on the tile are not shown. data.csv
• panel (d-5): Measured Wigner function $W(\alpha)$ of the squeezed state generated with $M = 13$ Trotter steps at $|\beta|^2 = 100$, against $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$. The fidelity and squeezing level the print writes on the tile are not shown. data.csv

Fig. 4

• panel (b): Allan deviation $\sigma_\alpha$ of the displacement estimate against the number of independent measurements $N_{\mathrm{meas}}$, for squeezed states made with $M = 1$ (red) and $M = 13$ (green) Trotter steps: measured (circles, standard deviations of repeated experiments as error bars), with the Cramer-Rao bounds from the classical (dashed) and quantum (solid) Fisher information. data.csv
• panel (c): Allan deviation $\sigma_\alpha$ of the displacement estimate against the number of Trotter steps $M$ at $N_{\mathrm{meas}} = 3.2\times 10^4$: measured (circles, standard deviations as error bars), the Cramer-Rao bounds from the classical (dashed) and quantum (solid) Fisher information, and the standard quantum limit (grey). In absolute units; the print scales the axis by $10^{-3}$. data.csv

Cite

Yanyan Cai et al. (11 authors). Quantum squeezing amplification with a weak Kerr nonlinear oscillator. Nat. Commun. 17, 970 (2025). https://doi.org/10.1038/s41467-025-67699-0

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