Extended Data Fig. 10: Tunable single-mode and effective two-mode squeezing in the squeezing dimer.

a, Intra-resonator squeezing as a function of the beamsplitter coupling J. Two values Φ = 0, π of the flux are shown for equal single-mode squeezing strengths η₁ = η₂ = 0.5 kHz. The level of single-mode squeezing is expressed by the ratio of the smallest (\(\Delta {R}_{{\rm{sq}}}^{2}\)) and largest (\(\Delta {R}_{{\rm{a}}}^{2}\)) eigenvalues of the covariance matrix of the quadrature amplitudes recorded for each resonator. These eigenvalues indicate the amplitude variance along the squeezed and anti-squeezed principal quadrature components, respectively. For Φ = π, where the squeezed (anti-squeezed) quadratures X_i (Y_i) of both resonators are coupled (see Fig. 2d), the slight initial imbalance in variance ratio is reduced as J increases while the value of the variance ratio remains low. By contrast, for Φ = 0—when the squeezed quadrature X_i in one resonator is coupled to the anti-squeezed quadrature Y_j in the other—we observe cancellation of single-mode squeezing as the variance ratio tends to 1 with increasing J. This agrees well with theory (dashed line), where for simplicity we have assumed equal dissipation rates \(\bar{\gamma }=2.2\,{\rm{kHz}}\) equal to the average of the experimental losses γ_i = {2.6, 1.9} kHz, as well as equal bath occupations. Owing to dynamical (optothermal) backaction, for this particular experiment the effective bath occupations \({\bar{n}}_{1}\approx {\bar{n}}_{2}\) only differed by a few per cent. b, Two-mode squeezing observed in the cross-resonator amplitude distribution of quadratures X₁ and Y₂ for Φ = 0, J = 3.5 kHz and η₁ = η₂ = 0.5 kHz. The dashed ellipse depicts the standard deviation of the principal components of the quadrature covariance matrix and shows positive correlations between X₁ and Y₂ (covariance σ(X₁, Y₂) = 0.08). c, Covariance of the coupled quadrature pairs X₁Y₂ and Y₁X₂ as a function of J, with η₁ = η₂ = 0.5 kHz. No correlations are found for flux Φ = π, when single-mode squeezing is strongest and independent of J (compare with a). However, for Φ = 0, positive correlations σ(X₁, Y₂), σ(Y₁, X₂) > 0 are found when J is increased, as predicted by theory (dashed line). A trade-off between the squeezing axes rotation towards the standard two-mode squeezing limit and the decrease in the overall squeezing level as J is increased leads to a maximum covariance (although not optimal squeezing level for the rotated quadratures) at a coupling J_opt. For the simple theory model with equal dissipation and bath occupation that we use it is given by \({J}_{{\rm{opt}}}^{2}=({\gamma }^{2}-4{\eta }^{2})/4\). Error bars in a and c reflect statistical uncertainty and control parameter stability (Methods section ‘Error estimation’).