Time-dependent variational principle for dissipative dynamics

We extend the time-dependent variational principle to the setting of dissipative dynamics. This provides a locally optimal (in time) approximation to the dynamics of any Lindblad equation within a given variational manifold of mixed states. In contrast to the pure-state setting, there is no canonical information geometry for mixed states, and this leads to a family of possible trajectories—one for each information metric. We focus on the case of the operationally motivated family of monotone Riemannian metrics and show further that, in the particular case where the variational manifold is given by the set of fermionic Gaussian states, all of these possible trajectories coincide. We illustrate our results in the case of the Hubbard model subject to spin decoherence.

Figure 1

Time-dependent variational principle with respect to a variational manifold of mixed states $\mathcal{V}=\left\{\rho \right(\mathbf{x}\left)\right|\mathbf{x}\in \mathbb{R}\}$. A mixed quantum state $\rho \left(\mathbf{x}\right)$ within a variational manifold $\mathcal{V}$ evolves in time according to an arbitrary physical process described by a CPT map ${\rho }_t={\mathcal{E}}_t\left({\rho }_0\right)$. In general, such a process leads out of the tangent space of the variational manifold at the point $\rho \left(\mathbf{x}\right)$. Hence, we have to “project back” into $\mathcal{V}$ using an appropriate measure of distance.

Figure 2

(a) Antiferromagnetic (AF) order in the ground state ${\rho }_0$ of the Hubbard model for $u=4,\mu =-2$. The steady state ${\rho }_s$ shows a ferromagnetic (FM) order. (b) Comparison of the real-time evolution (time $t$ in units of $1/\kappa$) of the dissipative process with the Gaussified version. We present the difference in the CM and the two states $d\Gamma \left(t\right)=\parallel {\Gamma }_{{\rho }_G\left(t\right)}-{\Gamma }_{\rho \left(t\right)}{\parallel }_2,d\rho \left(t\right)={\parallel {\rho }_G\left(t\right)-\rho \left(t\right)\parallel }_2$. (c) Purity, $p_{\rho }=\text{tr}\left[{\rho }^2\right]$, for solid: the exact and dashed: the Gaussified evolution ${\rho }_G$. (d) Evolution of the particle number $n$ for the two spin states ${\sigma }_s=\uparrow ,\downarrow$ for the real ($\rho$), and for the Gaussified (${\rho }_G$) process.