Towards a Theory of Metastability in Open Quantum Dynamics

By generalizing concepts from classical stochastic dynamics, we establish the basis for a theory of metastability in Markovian open quantum systems. Partial relaxation into long-lived metastable states—distinct from the asymptotic stationary state—is a manifestation of a separation of time scales due to a splitting in the spectrum of the generator of the dynamics. We show here how to exploit this spectral structure to obtain a low dimensional approximation to the dynamics in terms of motion in a manifold of metastable states constructed from the low-lying eigenmatrices of the generator. We argue that the metastable manifold is in general composed of disjoint states, noiseless subsystems, and decoherence-free subspaces.

Figure 1

Example of metastability in a three-level system. (a) Level scheme and transitions. (b) Spectrum of $\mathcal{L}$ showing the separation of time scales between $({\lambda }_1,{\lambda }_2)$ (full and dashed) and $\{{\lambda }_{k>2}\}$ (shaded), for the case $\kappa =4{\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2={\mathrm{\Omega }}_1/10$. (c) Illustration of the distance of the state $\rho (t)$ to the MM. We consider $\rho (t)$ starting from pure states corresponding to the eigenvectors of $L_2$ with maximal [top (red)] and minimal eigenvalues [bottom (blue)] $c_2^{\max }$ and $c_2^{\min }$. The full curves indicate the nearest state on the MM, ${\rho }_{\mathrm{MS}}(t)$, to the full state $\rho (t)$. The shaded region indicates the scale of the “error” $\parallel \delta \rho (t)\parallel$ with $\delta \rho (t)≔\rho (t)-{\rho }_{\mathrm{MS}}(t)$. On times of order ${\tau }^{\prime \prime }$ (open circle) the state $\rho (t)$ relaxes to the MM (in this case to either of the EMSs ${\tilde{\rho }}_{1,2}$), as seen by the shaded region decreasing to zero. On times of order $\tau$ (filled circle) there is an eventual relaxation to the stationary state ${\rho }_{\mathrm{SS}}$ (central, black line). Since $m=2$, in this case ${\tau }^{\prime }=\tau$. (d) The MM is a one-dimensional simplex. (e) Normalized autocorrelation $C(t)$ of the observable $|1⟩⟨1|-|2⟩⟨2|$, in the stationary state. For decreasing ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1$ (i.e., decreasing gap), metastability in the regime ${\tau }^{\prime \prime }$ (open symbols) to $\tau$ (filled symbols) is increasingly pronounced.

Figure 2

Example of a coherent metastable manifold. (a) Spectrum of $\mathcal{L}$ for example II (at ${\gamma }_1=4{\gamma }_2$, ${\mathrm{\Omega }}_1=2{\mathrm{\Omega }}_2={\gamma }_2/50$). The first four eigenvalues ($m=4$) split from the rest (shaded) and define the MM. Note the further splitting between $({\lambda }_1,{\lambda }_2)$ and $({\lambda }_3,{\lambda }_4)$ (which are almost degenerate). (b) The MM is a qubit. The dots represent the metastable states reached from random initial pure states. They map out under Eq. (4) an affinely transformed Bloch ball (shaded). The large dot (green) is ${\rho }_{\mathrm{SS}}$; the curves indicate paths in the MM taken by the states evolving from the extreme eigenvectors of $L_2$ (red and blue), and $L_3$ and $L_4$ (purple) towards ${\rho }_{\mathrm{SS}}$. (c) Time evolution in the MM (affinely transformed to a Bloch ball—the planes are projections in the direction of the eigenbasis of ${\rho }_{\mathrm{SS}}$ and another orthogonal direction): the MM contracts towards a one-dimensional simplex before relaxing eventually to ${\rho }_{\mathrm{SS}}$, due to the splitting between the first two eigenvalues and the next two, see panel (a). (d) Normalized autocorrelation $C(t)$ for the observable ${\sigma }_1^z-{\sigma }_2^z$ (green, solid). Same for the case where there is an extra perturbing Hamiltonian $\mathrm{\Delta }H=\mathrm{\Omega }{\sigma }_1^x\otimes {\sigma }_2^x$, which induces a rotation in the MM, manifesting in oscillations in $C(t)$ in the metastable regime (black, dashed). This realizes in a metastable system the proposal of Refs. [46, 47] for implementing operations in a DFS.