Dissipative Stabilization of Squeezing Beyond 3 dB in a Microwave Mode

While a propagating state of light can be generated with arbitrary squeezing by pumping a parametric resonator, the intraresonator state is limited to 3 dB of squeezing. Here, we implement a reservoir-engineering method to surpass this limit using superconducting circuits. Two-tone pumping of a three-wave-mixing element implements an effective coupling to a squeezed bath, which stabilizes a squeezed state inside the resonator. Using an ancillary superconducting qubit as a probe allows us to perform a direct Wigner tomography of the intraresonator state. The raw measurement provides a lower bound on the squeezing at about $6.7\pm 0.2$ dB below the zero-point level. Further, we show how to correct for resonator evolution during the Wigner tomography and obtain a squeezing as high as $8.2\pm 0.8$ dB. Moreover, this level of squeezing is achieved with a purity of $0.91\pm 0.09$.

Popular Summary

Vacuum squeezing, which lowers the variance on one field quadrature below vacuum fluctuations, is an essential resource for quantum information processing and metrology. Squeezing is routinely used with propagating waves and now reaches factors above 10 dB. For a stationary mode however, similar techniques are fundamentally limited to 3 dB of squeezing in the steady state.

While dissipation has long been seen as the nemesis of quantum properties, it can be harnessed and it provides a valuable tool for quantum information processing. Here, we use dissipation engineering to generate stationary microwave squeezing in excess of 8 dB. Borrowing from a technique developed for mechanical resonators, we use a driven nonlinear superconducting device, and manage to turn the environment of a resonator into an effective squeezed bath. Hence, at equilibrium with this nonclassical environment, the resonator gets stabilized into a vacuum squeezed state. Using one of the many tools offered by superconducting circuits, we perform a direct quantum-state tomography of the resonator. It reveals a state purity of about 0.9 for 8 dB of squeezing and allows us to observe the squeezing dynamics when turning on and off the device.

Beyond the stabilization of record steady-state squeezing for microwave resonators, the technique presented here could be an essential component of bosonic codes, a system of choice for fault-tolerant quantum computing. This technique could also be used to generate highly entangled many-body states inside of an array of resonators.

Figure 1

(a) Principle of the experiment. A cavity mode (green) at frequency ${\omega }_c$ is coupled to a dump mode at frequency ${\omega }_d$ (orange) via a Josephson ring modulator (JRM, in purple). The dump mode is strongly coupled to a cold transmission line through which the JRM is pumped at both frequencies ${\omega }_{+}={\omega }_c+{\omega }_d$ (two-mode squeezing) and ${\omega }_{-}={\omega }_d-{\omega }_c$ (photon conversion). A squeezed vacuum state is stabilized into the cavity as a result. An ancillary qubit with an ancillary readout resonator is used as a Wigner tomograph. The contours of the Wigner functions of each mode are shown as colored regions in the quadrature phase space, while a dashed circle represents the vacuum state. (b) Frequencies of the involved modes and drives. (c) Pulse sequence. The sum pump at ${\omega }_{+}$ with amplitude $g_{+}$ and the difference pump at ${\omega }_{-}$ with amplitude $g_{-}$ are applied for a time $t_s$. After a waiting time $t_w$, the Wigner function of the cavity $W(\alpha )$ is measured using a cavity displacement by $-\alpha$ followed by a parity measurement [32, 33, 34].

Figure 2

Characterization of the stabilized squeezed state. (a) Top panels: measured steady-state squeezing $S_{-}=⟨X_{-}^2⟩/X_0^2$ (left) and antisqueezing $S_{+}=⟨X_{+}^2⟩/X_0^2$ (right) factors. Bottom panels: theoretical prediction for $S_{\pm }$ using Eq. (4). Green dashed lines correspond to the value $g_{-}^{\mathrm{opt}}$ as a function of $g_{+}$ that minimizes the squeezing $S_{-}$ according to Eq. (4) (i.e., calculated by neglecting Kerr nonlinearities). (b) Purity $\mathcal{P}$ (top, green), squeezing $S_{-}$ (bottom, blue) and antisqueezing $S_{+}$ (bottom, orange) factors as a function of $g_{+}/g_{-}$ for a fixed value $g_{-}/2\pi =1.85$ MHz [cut along the arrow in (a)]. Circles are the normalized eigenvalues of the covariance matrix of the measured Wigner functions at each pump amplitude as shown in (c) and reach a squeezing factor as low as $S_{-}=-6.7\pm 0.2$ dB. Points with error bars are the values obtained when correcting for cavity evolution during Wigner tomography (see Appendix 11), which reveals a stabilized squeezing reaching as low as $S_{-}=-8.2\pm 0.8$ dB. Solid lines come from the model Eq. (4). (c) Selected measured Wigner functions along the same axis $g_{-}/2\pi =1.85$ MHz, for various $g_{+}/g_{-}$ ratios as indicated in the labels. The star indicates the Wigner function at optimum squeezing.

Figure 3

(a) Number photon-distribution measurement using qubit spectroscopy. Solid lines: measured probability $P_{|e⟩}$ that the qubit gets excited by a 200-ns-wide hyperbolic secant $\pi$ pulse of frequency $\omega$ after a cavity state is stabilized. Blue: near vacuum state when no pumps are applied. Orange: thermal state when only a pump $g_{+}$ is applied. Green: squeezed vacuum state when both pumps $g_{+}$ and $g_{-}$ are applied. Vertical lines indicate the qubit resonance frequency conditioned on the cavity having $n$ photons. Filled circles: numerical simulations. (b) Dots with error bars: Klyshko number $K_n$ (see main text) calculated from the qubit spectroscopy. Orange bars: expected Klyshko number for a thermal state at any temperature. Green bars: Klyshko numbers predicted with the model described in the text.

Figure 4

Dynamics of the squeezing factors. (a) Measured squeezing (dots) and antisqueezing (diamonds) factors normalized by their steady-state values $S_{\pm }^{\mathrm{ss}}$ as a function of the stabilization time $t_s$ for $g_{+}/2\pi =1.16$ MHz and various $g_{-}$. The data are shifted by $1$ for each value of $g_{-}$ ranging in $g_{-}/2\pi =[1.48,1.85,2.22,2.59,2.96,3.33]$ MHz (from light green to dark blue). Solid lines: results of the numerical simulation. Rectangles indicate the predicted characteristic stabilization times ${\kappa }_d/{\mathcal{G}}^2$ assuming 2% relative uncertainty on $g_{+}$ and $g_{-}$. (b) Selected measured Wigner tomograms for $g_{-}/2\pi =1.85\mathrm{MHz}$ and $g_{+}/2\pi =1.16$ MHz after various stabilization times $t_s$.

Figure 5

Decay of the squeezed state towards thermal equilibrium. (a) Dots: measured squeezing factor $S_{-}$ (blue) and antisqueezing factor $S_{+}$ (orange) as a function of the waiting time $t_w$ during which the pumps are turned off after they are at $g_{+}/2\pi =1.42$ MHz and $g_{-}/2\pi =2.6$ MHz. Solid lines: numerical simulations using $K/{\kappa }_c=0.5$. Dashed lines: same simulations but without self-Kerr ($K=0$). (b) Measured Wigner functions after various pump off times $t_w$ from 20 ns to $15.8\mu \mathrm{s}$ as indicated on each label.

Figure 6

Schematic of the measurement setup. The color of the rf sources refers to the frequency of the matching element in the device up to a modulation frequency. Multiple instances of a microwave source with the same color represent a single instrument with split outputs. The sum (blue) and difference (red) pumps are obtained by mixing the cavity and dump rf sources to ensure phase stability. The TWPA [79] is provided by Lincoln Labs.

Figure 7

Calibration of the rate $g_{-}$. Dot: measured mean photon number in the cavity as a function of pump-pulse width $\sigma$ with a hyperbolic secant shape for three amplitudes ${\mathcal{A}}_{-}$. Solid lines: prediction of the photon number using a master equation using $g_{-}/{\mathcal{A}}_{-}=74$ MHz/V.

Figure 8

Calibration of the rate $g_{+}$. Dots: measured mean photon number $n_c^{\mathrm{th}}$ as a function of the amplitude ${\mathcal{A}}_{+}$. Numerical simulations allow us to extract the rate $g_{+}$ (top axis) that leads to a given $n_c^{\mathrm{th}}$ (see text). Solid line: third-order polynomial fit of $g_{+}$ as a function of ${\mathcal{A}}_{+}$ that is used as an empirical calibration.

Figure 9

(a),(b) Representation of the map $f_W^{-1}$. Premeasurement squeezing $S_{-}^i$ and antisqueezing $S_{+}^i$ as a function of the measured squeezing and antisqueezing factors $S_{\pm }^W$. Circles: simulated values. Colors: linear interpolation of $f_W^{-1}$. (c),(d) Version of Fig. 2 corrected for the measurement error during Wigner tomography. (e),(f) Color: retropredicted uncertainty on the squeezing factors $\mathrm{\Delta }{S}_{-}^i$ and $\mathrm{\Delta }{S}_{+}^i$ owing to a measurement uncertainty of $\pm 0.2$ dB on $S_{\pm }^W$.

Figure 10

(a)–(c) Crosses, retropredicted squeezing factors for $g_{-}/2\pi =1.85$ MHz as in Fig. 2. Shaded areas correspond to different models with 2% uncertainty on $g_{+}$ and $g_{-}$; in (a), Kerr-free analytical model, in (b), steady-state simulations with $K/2\pi =20\pm 2\mathrm{kHz}$, in (c), steady-state simulations, including, in addition to the Kerr effect, some JRM four-wave mixing terms, $K_{\mathrm{cd}}/2\pi =250$ kHz, $K_{\mathrm{pc}}|p_{-}{|}^2/2\pi =172\pm 4$ kHz, and $K_{\mathrm{pd}}|p_{-}{|}^2/2\pi =172\pm 4$ kHz.

Figure 11

Color: simulated squeezing factor $S_{-}$ as a function of waiting time $t_w$ and cavity self-Kerr rate $K$ for an initial state with squeezing factor $S_{-}=-5.7$ dB and antisqueezing of $S_{+}=6.2\mathrm{dB}$. The dashed line indicates our device parameter $K/2\pi =20\mathrm{kHz}$.