Transient dynamics of the quantum Stuart-Landau oscillator

TL;DR

This paper addresses the transient dynamics of the quantum Stuart-Landau oscillator and shows how a Lindblad model with one- and two-quantum processes leads to a quantum limit cycle. It develops a Wigner-function description with an equation of motion resembling a truncated Kramers-Moyal expansion, identifies a classical-regime eligibility criterion based on energy balance, and demonstrates that the steady-state Wigner function is sharply localized around $r_{lc}=\sqrt{(\kappa_1-\gamma_1)/\gamma_2}$ in the classical regime. The work reveals how decoherence drives $\langle a\rangle$ to zero while preserving phase information statistically along the limit cycle, and it shows that Wigner negativity can transiently increase due to two-quantum dissipation. It also analyzes the Lindbladian spectrum and steady-state times for different initial states, finding fast-convergence regions for diagonal states that do not generally track the Liouvillian gap, with coherent states exhibiting notably slower convergence.

Abstract

We investigate the transient dynamics of the quantum Stuart-Landau oscillator, a paradigmatic quantum system exhibiting a quantum limit cycle and synchronization. From the energy dynamics, we determine a condition for the classical regime of transient dynamics and the limit cycle. Additionally, we formulate a guess function that fits the classical-regime steady-state Wigner function. The equation of motion for the Wigner function is derived and compared to the Kramers-Moyal equation for stochastic processes. We then characterize the classical-like behavior as the system evolves from a coherent state, noting the slow decay of neighboring-level coherence. We also study the evolution of the Wigner negativity as an indicator of nonclassicality, showing its temporary increase for some specific cases. To quantify the evolution speed, we examined the system's Lindbladian spectra, particularly the Liouvillian gap. Finally, we record the time it takes to reach the steady state for some Fock, thermal, and coherent states. The parameter dependence of the steady-state time may differ from the Liouvillian gap, and the limit-cycle attraction is significantly slower for coherent states compared to Fock or thermal states. For the diagonal states, there are \revision{fast convergence regimes} for which the steady-state time is locally minimized. This study provides a deeper insight into the transient behavior of self-sustained quantum systems.

Figures (8)

• Figure 1: Log-scale plots of the ratio between the steady-state energy $\left\langle{\hat{a}^\dagger\hat{a}}\right\rangle^\mathrm{(ss)}$ and the corresponding SL oscillator's energy $\left|\alpha_\mathrm{c,lc}\right|^2=(\kappa_1-\gamma_1)/2\gamma_2=\mathcal{B}/2$ in different working regimes. The left and right plots are different views of the same result. The blacked-out regions correspond to the damped regime below the Hopf bifurcation $\mathcal{A}<\mathcal{B}$ and to $\mathcal{A}>100$ for the left and right plots, respectively. The green region marks where the ratio is less than or equal to $1.05$. The ratio never gets smaller than $1$. The basis parameter is $\kappa_1=1$. Scaling $\kappa_1$ by some constant leaves the plots unchanged.
• Figure 2: Steady-state Wigner function $p=0$ cross-section $W^\mathrm{(ss)}(x\geq 0,p=0)$ and the corresponding occupation probability distribution $P_n^\mathrm{(ss)}$ of the first few occupied energy levels, for different parameters $(\kappa_1,\gamma_1,\gamma_2)$: (a) $(1, 0.9, 0.2)$; (b) $(1, 0.9, 0.005)$; (c) $(1, 0.1, 0.045)$. The value of $\left\langle{\hat{a}^\dagger\hat{a}}\right\rangle^\mathrm{(ss)}$ is numerically calculated from $\rho^\mathrm{(ss)}$ to be approximately (a) 1.46, (b) 13.08, (c) 10.50. For case (c), the guess function for $W^\mathrm{(ss)}$ given by Eq. \ref{['eq:fit_wss']} is additionally plotted as the dashed red line.
• Figure 3: Wigner function $W(x,p)$ representation of the evolution of a coherent state $\left|{\beta}\right\rangle$ under different system parameters $(\kappa_1,\gamma_1,\gamma_2,\beta)$: (a) $(1, 0.1, 0.05, 5)$; (b) $(1, 0.1, 0.05, 1)$; (c) $(1, 0.9, 0.005, 5)$; (d) $(1, 0.9, 0.1, 5)$; taken at different time points specified on top of each plot over the simple harmonic period of $2\pi$, alongside the trace distance $d_\mathrm{tr}$ toward the steady state given by Eq. \ref{['eq:tracedist_to_ss']}. For each case, the time points are arbitrarily chosen to show: (1) the initial state, (2) some short time into the evolution before the SL phase point reaches its limit cycle, (3) the SL phase point reaching its limit cycle, (4) significant angular spreading of the Wigner function, and (5) when steady state is reached, as indicated by Eq. \ref{['eq:tracedist_to_ss']}. For each plot, the color mapping is normalized against the Wigner function maximum at the given time point. The solid green lines show the Wigner function marginals, where the value 1 is set to be half the plot dimension. The dotted fuchsia circle is the classical SL limit cycle for the same parameters, given by $x^2+p^2=(\kappa_1-\gamma_1)/\gamma_2$. The red dot gives the value of $\left\langle{a}\right\rangle$ obtained from $\rho(t)$ evolved under Eq. \ref{['eq:lme']}, while the dotted red line is its trail some time into the past. The orange square is the classical SL phase point $\alpha_\mathrm{c}(t)$ obtained by solving Eq. \ref{['eq:expval_aop_evo_semiclassical']}, while the dashed orange line is its trail some time into the past. Along the rightmost column are plots of the expected energy $\left\langle{\hat{a}^\dagger\hat{a}}\right\rangle$, and half the radial distance of the classical SL phase point from the origin, which gives the energy of the classical SL oscillator, $E_\mathrm{c}=|\alpha_\mathrm{c}|^2$. The horizontal lines show the stationary values, while the vertical dotted black lines mark the time points of the Wigner function plots along the same row.
• Figure 4: Tile plots showing the difference $\left(\Delta\rho(t)\right)_{mn}$ between the absolute value of the density matrix elements corresponding to case (a) in Fig. \ref{['fig3']} and their respective steady-state values, i.e. $P_n^\mathrm{(ss)}$ as given by Eq. \ref{['eq:pnss']} for the diagonal elements and zero for the off-diagonal elements. The green-gradient color map is normalized against the highest value in the given plot, except when the highest value is lower than $10^{-2}$, in which case it is normalized against $10^{-2}$ instead. The diagonal tiles turn red once the values become smaller than $10^{-3}$.
• Figure 5: Early evolution of the negative volume $\mathcal{V}$ of the Wigner function given by Eq. \ref{['eq:negative_volume']} of some selected states $\left|{\psi}\right\rangle$, with $(\kappa_1,\gamma_1,\gamma_2)=(0.01,0.009,1)$ and $\mathcal{N}$ some normalization constant. The inset shows the evolution of the cat states' $\mathcal{V}$ for the first hundredth of the simple harmonic period. For the other three states, the evolution of $\mathcal{V}$ over the same time window is simply a monotonous decrease and is not shown. The horizontal axis of the main plot is log-scaled, while that of the inset is linear. A time step of $2\pi/200$ and $2\pi/10000$ is used for the main plot and the inset, respectively. More delicate shapes may be obtained with smaller time steps.