Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems

We present a new variational method based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus, allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that, indeed, the stationary state is often well described by a MPO of very moderate dimensions.

Figure 1

Left: Correlation $⟨S_y^2⟩$ for the low-dimensional Dicke model, as a function of $g/\gamma$ for system sizes $N=10–100$ (in increasingly darker shades). The solid line is the result of finite size extrapolation, linear in $1/N$, as explicitly shown in the inset for several values of $g/\gamma$. Right: Purity of the converged steady state for the same system sizes. For large dissipation (shaded region), we show only results for the smallest systems, for which the simple variational search converges reliably. The inset shows the explicit dependence of the purity on the system size for several coupling values.

Figure 2

Left: Antiferromagnetic order parameter $\sqrt{⟨M_z^2⟩}$ in the steady state of the Ising model with local dissipation, for several system sizes, and fixed $\gamma =1$, $V=5$, $\mathrm{\Omega }=1.5$, as a function of $\mathrm{\Delta }$. The inset shows the exponential decay of correlations $C_{zz}^{N/2}(n)=⟨{\sigma }_{N/2}^z{\sigma }_{N/2+n}^z⟩-⟨{\sigma }_{N/2}^z⟩⟨{\sigma }_{N/2+n}^z⟩$, for $N=50$ in the various regions of $\mathrm{\Delta }$. Right: Purity of the converged steady state for the same system sizes. The inset shows the local polarization ${\sigma }_n^z$ for the $N=10$ case. At $\mathrm{\Delta }=0$, the antiferromagnetic ordering can be appreciated, while for larger values of $|\mathrm{\Delta }|$, the steady state approaches total polarization.