Squeezing Enhances Quantum Synchronization

It is desirable to observe synchronization of quantum systems in the quantum regime, defined by the low number of excitations and a highly nonclassical steady state of the self-sustained oscillator. Several existing proposals of observing synchronization in the quantum regime suffer from the fact that the noise statistics overwhelm synchronization in this regime. Here, we resolve this issue by driving a self-sustained oscillator with a squeezing Hamiltonian instead of a harmonic drive and analyze this system in the classical and quantum regime. We demonstrate that strong entrainment is possible for small values of squeezing, and in this regime, the states are nonclassical. Furthermore, we show that the quality of synchronization measured by the FWHM of the power spectrum is enhanced with squeezing.

Figure 1

Classical phase plane diagram. (a) $\eta /{\gamma }_1=0$, (b) $\eta /{\gamma }_1=1$, and (c) $\eta /{\gamma }_1=1.5$; the blue and orange curves show $R$- and $\varphi$-nullclines, respectively. For a small squeezing parameter, only a single fixed point exists (solid black circle), while for large enough $\eta /{\gamma }_1$, two new fixed points are created, one unstable (empty white circle) and one stable as displayed in (c) and discussed in the main text. The other parameters are $F/{\gamma }_1=1$, $\mathrm{\Delta }/{\gamma }_1=1$, $\theta =\pi /4$, and ${\gamma }_2/{\gamma }_1=3$.

Figure 2

Bifurcation of the Wigner function. For column (a)–(g) $\eta /{\gamma }_1=0$, for column (b)–(h) $\eta /{\gamma }_1=1$, and for column (c)–(i) $\eta /{\gamma }_1=3$. Row (a)–(c): Undriven vdP when $F/{\gamma }_1=0$, $\mathrm{\Delta }/{\gamma }_1=0$. Squeezing acts to split the Wigner function into two localized lobes symmetric around $\mathrm{Im}(\alpha )=0$. Row (d)–(f): $F/{\gamma }_1=1$, $\mathrm{\Delta }/{\gamma }_1=0$. Similar to the undriven case, the oscillator displays symmetric bifurcation with increasing $\eta$. The difference is that the oscillator develops a definite preferred phase when $\eta /{\gamma }_1=0$. Row (g)–(i): $F/{\gamma }_1=1$, $\mathrm{\Delta }/{\gamma }_1=1$. Detuning breaks the symmetry of the above two cases as one of the lobes of the Wigner function nearly completely vanishes. All plots are in the regime of a few excitations ${\gamma }_2/{\gamma }_1=3$, and squeezing is along the position quadrature $\theta =0$.

Figure 3

Entrainment of squeezing- and harmonically driven vdP. Ratio of dissipative processes for all subplots is ${\gamma }_2/{\gamma }_1=3$. (a) Four slices of the Arnold tongue at various squeezing parameters $\eta$ show that squeezing produces stronger entrainment when compared with a harmonic drive, shown in (b). (c),(d) Power spectrum $S(\omega )$ when $\mathrm{\Delta }/{\gamma }_1=0.3$. Stronger squeezing produces narrower frequency distribution, while the harmonic drive has the opposite effect and causes broadening. This is highlighted by the solid black lines representing FWHM $\sigma$ of $S(\omega )$ in (e) and (g). The shaded regions of (e) and (f) mark where $Q_M$ (dashed orange line) is negative. (f) Harmonic drive, on the other hand, produces a steady state of vdP for which $Q_M$ is negative for all considered values of $F$.