Phase synchronization in coupled bistable oscillators

We introduce a simple model system to study synchronization theoretically in quantum oscillators that are not simply in limit-cycle states but rather display a more complex bistable dynamics. Our oscillator model is purely dissipative, with a two-photon gain balanced by single- and three-photon loss processes. When the gain rate is low, loss processes dominate and the oscillator has a very low photon occupation number. In contrast, for large gain rates, the oscillator is driven into a limit-cycle state where photon numbers can become large. The bistability emerges between these limiting cases with a region of coexistence of limit-cycle and low-occupation states. Although an individual oscillator has no preferred phase, when two of them are coupled together a relative phase preference is generated which can indicate synchronization of the dynamics. We find that the form and strength of the relative phase preference varies widely depending on the dynamical states of the oscillators. In the limit-cycle regime, the phase distribution is $\pi$ periodic with peaks at 0 and $\pi$, while in the low-occupation regime $\pi$-periodic phase distributions can be produced with peaks at $\pi /2$ and $3\pi /2$. Tuning the coupled system between these two regimes reveals a region where the relative phase distribution has $\pi /2$ periodicity.

Figure 1

(a) The three dissipative processes of the oscillator: two-photon gain, single-photon loss, and three-photon loss, at rates ${\kappa }_2,{\kappa }_1$, and ${\kappa }_3$ respectively. (b) Steady-state properties as a function of ${\kappa }_1/{\kappa }_2$ for fixed ${\kappa }_3={\kappa }_2\times {10}^{-2}$. A color scale shows the photon-number distribution $P_n$ (for $n>0$), with the average photon number $\langle n\rangle$ calculated numerically (white line), location of the peak in $P_n$ away from $n=0$ where it exists (solid black line), and the mean-field prediction $n_{+}$ (gray diamonds), superposed. Large photon number states are occupied when the gain is sufficiently large (black cross), the fixed point state is predominantly occupied if the loss dominates (black circle), and a bimodal distribution appears in an intermediate region (black star). Also shown is the second moment ${\mu }_{\left(2\right)}$ (solid magenta line). The corresponding Wigner functions, $W\left({\alpha }_r,{\alpha }_i\right)$, are for (c) limit-cycle (${\kappa }_1/{\kappa }_2={10}^{0.5}$), (d) bistability (${\kappa }_1/{\kappa }_2={10}^{1.25}$) and (e) fixed point (${\kappa }_1/{\kappa }_2={10}^2$).

Figure 2

Steady-state behavior of (a) the second moment ${\mu }_{\left(2\right)}$ and (b) average occupation number $\langle n\rangle$ overlaid with boundaries between the fixed point (FP), limit-cycle (LC), and bistable (B) regimes obtained by analyzing the peaks in the radial Wigner distribution $W(r=|\alpha \left|\right)$ (see Appendix pp2 for details). The FP-B boundary (upper dashed curve) agrees well with the appearance of the stable nonzero mean field solution, $n_{+}$ [Eq. (4)] (full line). The LC-B boundary is shown using two different approaches (see main text for details): The dotted line indicates where the peak at the origin of the Wigner function disappears entirely, while the (lower) dashed line indicates the edge of a bistable region in which the Wigner peak at the origin remains non-negligible in size. The second moment is maximal within the bistable region where the corresponding LC contains a large occupation number (i.e ${\kappa }_2\gg {\kappa }_3$); elsewhere, it is rather smooth. The average photon number distribution $\langle n\rangle$ is largest deep within the LC regime (red) and lowest for the fixed point regions (white), but does not provide any direct indication of bistability.

Figure 3

(a) The slowest timescales ${\tau }_k$ of the oscillator for $k=0,1,2$ with ${\kappa }_3={\kappa }_2\times {10}^{-2}$ and (b) the metastability $M$ [Eq. (5)] plotted on a logarithmic scale. The phase boundaries obtained using Wigner functions (dashed lines) and the mean field calculation (full line) are shown.

Figure 4

Sample quantum trajectories for each of the oscillator states illustrating the frequency of the different jump processes for (top) limit-cycle (${\kappa }_1={\kappa }_2\times {10}^{1/2}$), (middle) bistable (${\kappa }_1={\kappa }_2\times {10}^{5/4})$, and (bottom) fixed-point (${\kappa }_1={\kappa }_2\times {10}^{7/4}$) states (with ${\kappa }_3={\kappa }_2\times {10}^{-7/4}$ throughout). The individual jump processes involving one-photon loss (rate ${\kappa }_1$), two-photon gain (rate ${\kappa }_2$), and three-photon loss (rate ${\kappa }_3$) are indicated. The bistable oscillator can be seen to flip intermittently between LC-like and FP-like behavior. In the FP state, the one-photon loss jumps occur in pairs soon after each two-photon gain jump (see magnified portion of the lower panel).

Figure 5

(a) $[P\left(\varphi \right)-1/2\pi ]({\kappa }_2^2/J^2)$ as a function of ${\kappa }_1/{\kappa }_2$ (calculated with perturbation theory), with $J/{\kappa }_2={10}^{-2}$ and ${\kappa }_3={\kappa }_2\times {10}^{-1}$. The range of ${\kappa }_1/{\kappa }_2$ shown spans the three states (FP, B, and LC). Very weak single-photon loss (${\kappa }_1\ll {\kappa }_2$) leads to coupled limit-cycles with peaks at 0 and $\pi$. As the single-photon loss rate is increased, this $\pi$-periodic pattern vanishes before then reappearing with peaks at $\pi /2$ and $3\pi /2$. Eventually, very strong single-photon loss (${\kappa }_1\gg {\kappa }_2$) again suppresses the pattern. (b) The dominant Fourier coefficients $F_2$ (red) and $F_4$ (blue), defined in the main text (here units are chosen such that ${\kappa }_2=1$). $F_2$ accounts for the $\pi$-periodic component of the relative phase distribution and its sign determines the position of the peaks; $F_4$ is the next largest, though it is much less important than $F_2$, except for the region in which $F_2$ passes through zero. The corresponding $\frac{\pi }{2}$-periodic $P\left(\varphi \right)$ distribution for the case where $F_2$ is zero is shown in the inset; the peak-to-peak height is ${10}^{-4}$ in units of ${(J/{\kappa }_2)}^4$.

Figure 6

Behavior of (a) $F_2/J^2$ and (b) $F_4/J^4$, the two most dominant Fourier coefficients of the relative phase distribution scaled with coupling strength as a function of the relative loss-gain rates (in units such that ${\kappa }_2=1)$.

Figure 7

(a) Mapping of the contrast in the Wigner function across the region of bistability, plotted on a log scale with a series of contours from ${10}^{-18}$ to ${10}^{-2}$. The contrast is defined as $\text{min}(W_0,W_{+})-W_{-}$, where $W_0$ and $W_{+}$ are the values of $W\left(r\right)$ at the two maxima of the bistability at $r=0$ and $r>0$ respectively, and $W_{-}$ is the value of $W\left(r\right)$ at the local minimum between them. The white regions indicate Wigner functions with only a single maximum, the dashed black line indicates the locus where $W^{″}{|}_{r=0}=0$, and the solid black line shows the mean-field boundary [Eq. (4)]. (b) Second derivative of the Wigner distribution at $r=0$ along the line ${\kappa }_3={\kappa }_2\times {10}^{-1.5}$. The point where $W^{″}{|}_{r=0}=0$ indicates the disappearance of the local maximum at the origin. (c) Illustrations of the Wigner function corresponding to the points marked in panel (a): by a blue-cross [(c)i], FP state with a single peak in $W\left(r\right)$ at the origin; red-cross [(c)ii], bistable state with two peaks in the Wigner function (the contrast $\text{min}(W_0,W_{+})-W_{-}$ is also shown; it is $\approx {10}^{-2}$ in this case); and green-cross [(c)iii], limit-cycle state which has a single peak at $r\ge 0$.