General Linearized Theory of Quantum Fluctuations around Arbitrary Limit Cycles

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, being the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. Here, we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a test bed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (many-body) systems.

Figure 1

Limit cycle emerging for $\mathrm{\Delta }=\sqrt{0.4}$ and $F=\sqrt{0.1}$. (a) We show in gray the closed trajectory described in phase space. The arrows refer to the direction of the Floquet eigenvectors in selected points of the cycle. (b) Time evolution of the cycle’s absolute value and phase. (c) Evolution of the variance of $\theta$, see Eqs. (3) and (8). Note that $\gamma$, which sets how relevant quantum fluctuations are, appears just as an absolute scale for the variance, whose dependence on time is set by the limit cycle’s shape.

Figure 2

Steady-state Wigner functions of the driven VdP oscillator, Eq. (1), for $\mathrm{\Delta }=\sqrt{0.4}$, $F=\sqrt{0.1}$, and two values of $\gamma$, 0.1 and 0.01. In (a) and (b) we show the exact solutions, to be compared with the mixture of Gaussians (c),(d) obtained through linearization, see Eq. (12). In (e) and (f) we show a few of these Gaussian states at selected points of the trajectory, in order to see how these change along the limit cycle. Note how, as shown explicitly in (e), all the Gaussians carry vacuum fluctuations along the direction defined by the ${\mathbf{q}}_0(\tau )$ Floquet eigenvector (black arrows), with varying fluctuations along the ${\mathbf{p}}_1(\tau )$ direction (gray arrows).