Dissipative Complex Quantum Networks

Abstract

Most of the classical literature about synchronization phenomena in networks deals with self-sustained phase oscillators in Kuramoto-type models, or with identical nonlinear oscillators studied through the master stability formalism [1]. We instead continue focusing on synchronization during the relaxation dynamics of different linear oscillators driven out of equilibrium and exploring the key role of dissipation. A first step to characterize quantum spontaneous synchronization, considering quantum fluctuations and correlations beyond the classical limit, has been considered in Chap. 4 where synchronization between one pair of damped quantum harmonic oscillators has been reported. We have already seen that, depending on the damping, a pair of oscillators with different frequencies can exhibit synchronous evolution emerging after a transient, as well as robust (slowly decaying) non-classical correlations [2]. This connection has been extended in Chap. 5, where we showed that synchronization may occur between three oscillators or in a single pair depending on the symmetries of the system [3], discussing both transient and relaxation effects.

Appendices

Appendix

A.1   Master Equation for Nodes

From the master equation in the basis of normal modes after the (post-trace) rotating wave approximation, given in Sect. 6.1 [Eq. (6.7)] (see also Appendix A.2 in Chap. 5), we can derive an equivalent expression turning to the basis of the original oscillators by simply applying the change of basis matrix $\mathcal {F}$, defined by diagonalization of ${\mathcal {H}}$. Rearranging terms one obtains

$$\begin{aligned} \frac{d \rho (t)}{dt}= & {} -i[\hat{H}_S,{\rho }(t)]- \nonumber \\- & {} \frac{1}{4 \hslash ^2} \sum _{j k} i \tilde{\varGamma }_{j k} \left( [\hat{q}_j,\{\hat{p}_k,\rho (t)\}] -[\hat{p}_k,\{\hat{q}_j,\rho (t)\}] \right) + \nonumber \\+ & {} \tilde{D}_{j k}^a [\hat{q}_j,[\hat{q}_k,\rho (t)]] - \tilde{D}_{j k}^b [\hat{p}_j,[\hat{p}_k,\rho (t)]]. \end{aligned}$$

(A.1)

Here we have introduced new master equation coefficients denoted by a tilde and defined from the previous ones as

$$\begin{aligned}&\tilde{\varGamma }_{j k}&= \sum _{n} {\mathcal {F}}_{j n} {\mathcal {F}}_{k n} \varGamma _n, \end{aligned}$$

(A.2)

$$\begin{aligned}&\tilde{D}_{j k}^a&= \sum _{n} {\mathcal {F}}_{j n} {\mathcal {F}}_{k n} D_n, \end{aligned}$$

(A.3)

$$\begin{aligned}&\tilde{D}_{j k}^b&= \sum _{n} {\mathcal {F}}_{j n} {\mathcal {F}}_{k n} \frac{D_n}{\varOmega _n^2}. \end{aligned}$$

(A.4)

Those are valid for all the cases considered in the paper, namely, common bath, local bath and separate baths, with the proper definitions of the untilded coefficients for each case (see Sect. 6.1). Note however that for the case of separate baths (assuming an Ohmic frequency spectral distribution with sharp cutoff in the bath) the damping coefficients in the master equation reduce simply to $\tilde{\varGamma }_{i j} = \gamma \delta _{i j}$, i.e. all the nodes in the network dissipate through their own bath at the same rate, determined by the equivalence of the separate baths. This further simplification in the case of separate baths marks its difference from the common or local bath cases, producing a different structure for the friction terms in the equations of motion, as we will see in the next sections of this Appendix.

A.2   Equations for the First- and Second-Order Moments

For Gaussian states, the full dynamics of the oscillators is embedded in the first- and second-order moments [25] and the former give the classical limit of this quantum system, obtained neglecting quantum fluctuations. From the master equation we obtain the evolution of the first-order moments

$$\begin{aligned} \frac{d}{dt} \langle \hat{Q}_n \rangle&= \langle \hat{P}_n \rangle - \frac{\varGamma _{n}}{2 \hslash } \langle \hat{Q}_n \rangle , \end{aligned}$$

(A.5)

$$\begin{aligned} \frac{d}{dt} \langle \hat{P}_n \rangle&= - \varOmega _n^2 \langle \hat{Q}_n \rangle - \frac{\varGamma _{n}}{2 \hslash } \langle \hat{P}_n \rangle , \end{aligned}$$

(A.6)

where the first term corresponds to the free evolution of uncoupled oscillators and the second one is a damping term stemming from the influence of the bath. For the second order moments we obtain the more complicated expressions:

$$\begin{aligned} \frac{d}{dt} \langle \hat{Q}_n \hat{Q}_m \rangle&= \frac{1}{2} \langle \{\hat{Q}_n , \hat{P}_m\} + \{\hat{P}_n, \hat{Q}_m\} \rangle \nonumber \\ {}&- \left( \frac{\varGamma _{n} + \varGamma _{m}}{2 \hslash } \right) \langle \hat{Q}_n \hat{Q}_m \rangle + D_{n} \frac{\delta _{n m}}{2 \varOmega _n^2}, \end{aligned}$$

(A.7)

$$\begin{aligned} \frac{d}{dt} \langle \hat{P}_n \hat{P}_m \rangle&= - \frac{\varOmega _n^2}{2} \langle \{\hat{Q}_n , \hat{P}_m \} \rangle - \frac{\varOmega _m^2}{2} \langle \{\hat{Q}_m , \hat{P}_n \} \rangle \nonumber \\ {}&- \left( \frac{\varGamma _{n} + \varGamma _{m}}{2 \hslash } \right) \langle \hat{P}_n \hat{P}_m \rangle + D_{n} \frac{\delta _{n m}}{2}, \end{aligned}$$

(A.8)

$$\begin{aligned} \frac{d}{dt} \langle \{ \hat{Q}_n ,\hat{P}_m \} \rangle&= 2 \langle \hat{P}_n \hat{P}_m \rangle - 2 \varOmega _m^2 \langle \hat{Q}_n \hat{Q}_m \rangle \nonumber \\ {}&- \left( \frac{\varGamma _{n} + \varGamma _{m}}{2 \hslash } \right) \langle \{\hat{Q}_n , \hat{P}_m\} \rangle , \end{aligned}$$

(A.9)

where the first two terms arise from the reduced motion of the free normal modes, and the last ones are induced by the environmental action, which combines damping and diffusion effects.

We also notice that a common environment gives rise to a rather symmetric damping, also known as diffusive coupling (apart from an irrelevant change of sign) [2]. This kind of diffusive coupling is a typical phenomenological assumption when synchronization is modeled in classical systems [46]. This can be seen by looking at the first order moments, for which we obtain different expressions in the case of common, local and separate baths. In the first two cases we have

$$\begin{aligned} \frac{d}{dt} \langle \hat{q}_n \rangle&= \langle \hat{p}_n \rangle -\frac{1}{2 \hslash } \sum _k \tilde{\varGamma }_{n k} \langle \hat{q}_k \rangle , \end{aligned}$$

(A.10)

$$\begin{aligned} \frac{d}{dt} \langle \hat{p}_n \rangle&= - \omega _n^2 \langle \hat{q}_n \rangle -\sum _k \lambda _{n k} \langle \hat{q}_k \rangle - \frac{1}{2 \hslash } \sum _k \tilde{\varGamma }_{n k} \langle \hat{p}_k \rangle . \end{aligned}$$

(A.11)

while for the separate baths case the expression transforms into:

$$\begin{aligned} \frac{d}{dt} \langle q_n \rangle&= \langle \hat{p}_n \rangle - \frac{1}{2 \hslash } \tilde{\varGamma } \langle \hat{q}_n \rangle , \end{aligned}$$

(A.12)

$$\begin{aligned} \frac{d}{dt} \langle \hat{p}_n \rangle&= - \omega _n^2 \langle \hat{q}_n \rangle - \sum _k \lambda _{n k} \langle \hat{q}_k \rangle - \frac{1}{2 \hslash } \tilde{\varGamma } \langle \hat{p}_n \rangle . \end{aligned}$$

(A.13)

It is immediately seen that the presence of a common bath, a local bath or N separate (even if identical) baths, leads to different friction terms in the dynamical equations. While the damping of oscillators in the common and local bath cases depends on all the network oscillators weighed by the effective couplings ($\kappa _n^2$) through the tilded damping coefficients of Eq. (A.2), in the separate bath case each oscillator decays independently from the rest of the network, being coupled only through the Hamiltonian part of the dynamical evolution.

A.3   Three-Oscillator Motif Details

Here we give the analytical expressions for the synchronization of the three-oscillators linear motif, i.e. an open chain of three oscillators embedded in a bigger network. We are able to give the specific parameter relations that have to be fulfilled in order to obtain a non-dissipative mode, that is, to make the effective coupling for a motif mode $\kappa _\sigma = 0$.

By solving Eq. (6.16) for this particular case, we obtain:

$$\begin{aligned} \mathcal {F}_{a \sigma }&= C \left( \frac{\lambda _{a c}}{\varOmega _\sigma ^2 - \omega _a^2} \right) , \end{aligned}$$

(A.1)

$$\begin{aligned} \mathcal {F}_{b \sigma }&= C \left( \frac{\lambda _{b c}}{\varOmega _\sigma ^2 - \omega _b^2} \right) , \end{aligned}$$

(A.2)

$$\begin{aligned} \mathcal {F}_{c \sigma }&= C, \end{aligned}$$

(A.3)

where $C^{2} =1 /\left( 1 + \left( \frac{\lambda _{a c}}{\varOmega _\sigma ^2 - \omega _a^2} \right) ^2 + \left( \frac{\lambda _{b c}}{\varOmega _\sigma ^2 - \omega _b^2} \right) ^2\right)$.

Now we can obtain a explicit expression for the effective coupling of the normal mode $Q_\sigma$ to the heat bath:

$$\begin{aligned} \kappa _\sigma = C \left( 1 + \frac{\lambda _{a c}}{\varOmega _\sigma ^2 - \omega _a^2} + \frac{\lambda _{b c}}{\varOmega _\sigma ^2 - \omega _b^2} \right) , \end{aligned}$$

(A.4)

that enables a dissipation-free channel, i.e. no coupling with the bath $(\kappa _\sigma = 0)$ when

$$\begin{aligned} \frac{\lambda _{a c}}{\varOmega _\sigma ^2 - \omega _a^2} + \frac{\lambda _{b c}}{\varOmega _\sigma ^2 - \omega _b^2} = -1. \end{aligned}$$

(A.5)

This last condition gives another different expression for the synchronization frequency in this regime:

$$\begin{aligned} \varOmega _\sigma ^2&= \frac{\omega _a^2 + \omega _b^2}{2} - \frac{\lambda _{a c} + \lambda _{b c}}{2} \\&\pm \sqrt{ \left( \frac{\omega _a^2 - \omega _b^2}{2} \right) ^2 + \left( \frac{\lambda _{a c} + \lambda _{b c}}{2}\right) ^2 - \frac{(\omega _a^2 - \omega _b^2) (\lambda _{a c} - \lambda _{b c})}{2} }, \nonumber \end{aligned}$$

(A.6)

where we have to check that $\varOmega _\sigma ^2$ is real and positive, i.e. that $(\omega _a^2 - \omega _b^2)^2 + (\lambda _{a c} + \lambda _{b c})^2 > 2 (\omega _a^2 - \omega _b^2)(\lambda _{a c} - \lambda _{b c})$.

From the explicit expression of $\varOmega _\sigma$ and the previous equations, a consistency relation for the selected natural frequencies and coupling of the a, b and c oscillators follows by substituting the expression of $\varOmega _\sigma ^2$ into the equation

$$\begin{aligned} \varOmega _\sigma ^2 - \omega _c^2 = \frac{\lambda _{a c}^2}{\varOmega _\sigma ^2 - \omega _a^2} + \frac{\lambda _{b c}^2}{\varOmega _\sigma ^2 - \omega _b^2}, \end{aligned}$$

(A.7)

whose solution for $\lambda _{a c}$, is

$$\begin{aligned} \lambda _{a c}&= \frac{\lambda _{b c}^2 - \lambda _{b c}(\omega _a^2 - \omega _b^2)}{2 \lambda _{b c} - \omega _a^2 + \omega _c^2} \\&\pm \frac{(\lambda _{b c} - \omega _a^2 + \omega _c^2) \sqrt{\lambda _{b c}^2 - (\omega _a^2 - \omega _b^2)(\omega _b^2-\omega _c^2)}}{2 \lambda _{b c} - \omega _a^2 + \omega _c^2}, \nonumber \end{aligned}$$

(A.8)

corresponding to two different branches of solutions. These two bran-ches intersect when we have that $\lambda _{b c} = \omega _a^2 - \omega _c^2$ or equivalently $\lambda _{a c} = \omega _b^2 - \omega _c^2$, in this case we have the simpler relation for the couplings $\lambda _{a c} - \lambda _{b c} = \omega _b^2 - \omega _a^2$ and here the mode $\hat{Q}_\sigma$ is degenerated, i.e. there are two non-dissipative normal modes with different frequencies. It is worth noticing that when we have different branches it is necessary to impose the condition $\lambda _{b c}^2 > (\omega _a^2 - \omega _b^2) (\omega _b^2 - \omega _c^2)$ in order to obtain $\lambda _{a c}$ real.