Chimera States in Quantum Mechanics

Abstract

Classical chimera states are paradigmatic examples of partial synchronization patterns emerging in nonlinear dynamics. These states are characterized by the spatial coexistence of two dramatically different dynamical behaviors, i.e., synchronized and desynchronized dynamics. Our aim in this contribution is to discuss signatures of chimera states in quantum mechanics. We study a network with a ring topology consisting of N coupled quantum Van der Pol oscillators. We describe the emergence of chimera-like quantum correlations in the covariance matrix. Further, we establish the connection of chimera states to quantum information theory by describing the quantum mutual information for a bipartite state of the network.

Appendix

In this appendix we discuss the Gaussian quantum fluctuations of the quartic harmonic oscillator

$$\begin{aligned} H=\frac{p^2}{2m}+\frac{m\omega q^2}{2}+\frac{\lambda }{4}q^4=\omega \left( a^{\dagger }a+\frac{1}{2}\right) +\lambda \left( \frac{1}{4m\omega }\right) ^2(a^{\dagger }+a)^4. \end{aligned}$$

(16.33)

For simplicity, we consider units in such a way that $\hbar =1$. As in the main text, we consider the decomposition $a=\tilde{a}+\alpha (t)$, where $\tilde{a}$ describes quantum fluctuations and $\alpha (t)$ is the mean field. We now consider the time-dependent displacement operator [81]

$$\begin{aligned} D\left[ \alpha (t)\right]&=\exp \left[ \alpha (t)a^{\dagger }-\alpha ^{*}(t) a\right] =\exp \left[ \alpha (t)\tilde{a}^{\dagger }-\alpha ^{*}(t) \tilde{a}\right] \nonumber \\&=\exp \left[ -\frac{|\alpha (t)|^{2}}{2} \right] \exp \left[ \alpha (t)\tilde{a}^{\dagger }\right] \exp \left[ -\alpha ^{*}(t)\tilde{a}\right] . \end{aligned}$$

(16.34)

Under this Gauge transformation, the Schrödinger equation $\mathrm {i}\partial _{t}|\varPsi (t)\rangle =H|\varPsi (t)\rangle$ is transformed to $\mathrm {i}\partial _{t}|\varPsi _{\alpha }(t)\rangle =H^{(\alpha )}(t)|\varPsi _{\alpha }(t)\rangle$, where $|\varPsi _{\alpha }(t)\rangle =D^{\dagger }\left[ \alpha (t)\right] |\varPsi (t)\rangle$ and

$$\begin{aligned} \hat{H}^{(\alpha )}(t)=D^{\dagger }\left[ \alpha (t)\right] (H-\mathrm {i}\partial _t)D\left[ \alpha (t)\right] . \end{aligned}$$

(16.35)

For later purposes, we need to use the identity

$$\begin{aligned} \mathrm {i}D^{\dagger }\left[ \alpha (t)\right] \partial _t D\left[ \alpha (t)\right] =\frac{\mathrm {i}}{2}[\dot{\alpha }(t)\alpha ^{*}(t)-\alpha (t)\dot{\alpha }^{*}(t)]+\mathrm {i}[\dot{\alpha }(t)\tilde{a}^{\dagger }-\dot{\alpha }^{*}(t)\tilde{a}]. \end{aligned}$$

(16.36)

After some algebraic manipulations we can write

$$\begin{aligned} \hat{H}^{(\alpha )}(t)&=\frac{\lambda }{16}\left( \frac{1}{m\omega }\right) ^2(\tilde{a}^{\dagger }+\tilde{a})^4+\frac{1}{2}\left( \frac{1}{m\omega }\right) ^2(\tilde{a}^{\dagger }+\tilde{a})^3\text {Re}[\alpha (t)] \nonumber \\&+\omega \tilde{a}^{\dagger }\tilde{a}+\frac{3\lambda }{2}\left( \frac{1}{m\omega }\right) ^2(\tilde{a}^{\dagger }+\tilde{a})^2(\text {Re}[\alpha (t)])^{2} \nonumber \\&-\mathrm {i}[\dot{\alpha }(t)\tilde{a}^{\dagger }-\dot{\alpha }^{*}(t)\tilde{a}]+\omega (\alpha ^{*}\tilde{a}+\alpha \tilde{a}^{\dagger })+2\lambda \left( \frac{1}{m\omega }\right) ^2(\tilde{a}^{\dagger }+\tilde{a})(\text {Re}[\alpha (t)])^{3} \nonumber \\&-\frac{\mathrm {i}}{2}[\dot{\alpha }(t)\alpha ^{*}(t)-\alpha (t)\dot{\alpha }^{*}(t)]+\omega |\alpha (t)|^2+\lambda \left( \frac{1}{m\omega }\right) ^2(\text {Re}[\alpha (t)])^{4}. \end{aligned}$$

(16.37)

To study the quantum fluctuations about a semiclassical trajectory, we assume that $|\alpha (t)|\gg 1$. To obtain the quadratic fluctuations we must neglect the non-Gaussian terms in Eq. (16.37). In addition, the center of the co-moving frame $\alpha (t)$ must satisfy the condition

$$\begin{aligned} \dot{\alpha }(t)=-\mathrm {i}\left( \omega \alpha (t) +2\lambda \left( \frac{1}{m\omega }\right) ^2(\text {Re}[\alpha (t)])^{3}\right) , \end{aligned}$$

(16.38)

which corresponds to the classical equations of motion. This condition is satisfied if the linear terms in the quantum fluctuations $\tilde{a}$ of Eq. (16.37) vanish [78, 79]. We can go a step further and define the classical Hamiltonian function

$$\begin{aligned} H(\alpha ,\alpha ^*)&=\omega |\alpha (t)|^2+\lambda \left( \frac{1}{m\omega }\right) ^2(\text {Re}[\alpha (t)])^{4} \nonumber \\&=\omega \alpha ^{*}(t)\alpha (t)+\lambda \left( \frac{1}{m\omega }\right) ^2\left[ \frac{\alpha ^{*}(t)+\alpha (t)}{2}\right] ^{4}, \end{aligned}$$

(16.39)

from which we obtain the equations of motion Eq. (16.38). In so doing we define the Poisson bracket $\{F(\alpha ,\alpha ^*),G(\alpha ,\alpha ^*)\}=-\mathrm {i}(\partial _{\alpha }F\partial _{\alpha ^{*}}G-\partial _{\alpha }G\partial _{\alpha ^{*}}F)$. By using this definition we obtain the equations of motion Eq. (16.38) as $\dot{\alpha }(t)=-\mathrm {i}\partial _{\alpha ^{*}}H(\alpha ,\alpha ^*)$.

Now the role of each one of the terms in Eq. (16.37) is clear. In contrast, the quadratic terms give us the first quantum corrections about the semiclassical trajectory. To study these quantum fluctuations we need to study the quadratic Hamiltonian

$$\begin{aligned} \hat{H}_{\text {Q}}^{(\alpha )}(t)=\omega \tilde{a}^{\dagger }\tilde{a}+\frac{3\lambda }{2}\left( \frac{1}{m\omega }\right) ^2(\tilde{a}^{\dagger }+\tilde{a})^2(\text {Re}[\alpha (t)])^{2}+L(\alpha ,\alpha ^*), \end{aligned}$$

(16.40)

where $L(\alpha ,\alpha ^*)=-\frac{\mathrm {i}}{2}[\dot{\alpha }(t)\alpha ^{*}(t)-\alpha (t)\dot{\alpha }^{*}(t)]+H(\alpha ,\alpha ^*)$ is the Lagrangian.