Relaxation Control of Open Quantum Systems

Abstract

A fundamental problem in experiments with open quantum systems is to ensure steady-state convergence within a given operational time window. Here, we devise a general state preparation recipe to control relaxation timescales and achieve steady-state convergence within experimental run times. We do so by constructing a unitary operation that cancels the desired relaxation modes. We provide an example in a few-body interacting system (long-range qubit chain), taking into account limitations of experimentally accessible unitary operations in quantum simulators.

Figures (7)

• Figure 1: Schematics representing the selective suppression of eigenmodes of the Liouvillian operator (cf. Eq. \ref{['eq:eigendecomp']}) and exponential speedup of the relaxation. (a) A unitary rotation allows to cancel the projection of an arbitrary initial state $\rho_0$ along undesired decaying modes, which is here illustrated as the rotation of a three-dimensional vector $\rho_0$ into the vector $\rho_\perp$, perpendicular to the undesired directions $r_1,r_2$. (c) Spectrum of the Liouvillian for the long-range Ising chain sketched in panel (b) and defined in Eqs. \ref{['eq:model-H']} and \ref{['eq:model-L']} ($N{=}5$, $h_x{=}1$, $J{=}1.25$, $\alpha{=}0.5$ and $\gamma{=}1$). While the relaxation timescale $\tau{=}\abs{\Re\lambda_2^{-1}}$ is regulated by the second dominant eigenvalue, eigenvalues $\lambda_{3},\dots,\lambda_{12}$ have comparable real parts (red shaded area). In this scenario, suppressing the second eigenmode would not provide any practical advantage. (d) Speedups (in terms of $\Re(\lambda_i)/\Re(\lambda_2)$) achievable through multi-mode suppression. Notice the considerable gap dividing the $12^{\textrm{th}}$ and $13^{\textrm{th}}$ eigenvalues in the example. (e) Comparison of the different relaxations to the steady state $\rho_{\infty}$ obtained by applying our recipe to the fully down-polarized initial state $\ket{\downarrow,\dots,\downarrow}$ (cf. Eq. \ref{['eq:distance_measure']}). We selectively suppress the projection of the initial density matrix $\rho_0$ along the first $n=2,3,\dots,12$ slowest-decaying modes. While cancellation up to the $11^{\textrm{th}}$ mode does not sensibly change the relaxation timescale, simultaneous suppression of the first 12 modes provides a faster convergence, $\Re(\lambda_{13})/\Re(\lambda_{2}) \approx 2$.
• Figure 2: Speedup of the relaxation dynamics for different parameters $\alpha,h_x,J$ of the Liouvillian in Eqs. \ref{['eq:model-H']},\ref{['eq:model-L']}. The three figures display the relative time gain achieved by the transformed state $\rho_\perp$, as compared to the original state $\rho_0$ (cf. Eq. \ref{['eq:rel-time-gain']}). The time gain is generally a significant fraction of the original time (i.e., 50%), except for few isolated cases where the projection of the original state $\rho_0$ on the slowest Liouvillian eigenmodes already happens to be small.
• Figure 3: Speedup of the relaxation dynamics toward the steady state, obtained through the experimentally accessible unitary transformations in Eq. \ref{['eq:U-phys']}, in the open quantum Ising chain in Eq. \ref{['eq:model-H']},\ref{['eq:model-L']} ($N{=}5$, $h_x{=}1$, $J{=}1.25$, $\alpha{=}0.5$ and $\gamma{=}1$). The plot shows the distance to the steady state $\rho_\infty$ of the initial state $\rho_0$ and the transformed states, $\check\rho_\perp^{(n)}$, obtained with the experimentally accessible unitary (black and dashed curves, respectively; cf. Eq. \ref{['eq:distance_measure']}). For reference, we also plot the evolution of $\rho_\perp^{(12)}$ from Fig. \ref{['fig:1']}d, obtained with the geodesic unitary (solid red curve) supplemental. Despite experimental limitations, the recipe produces a speedup in the steady-state convergence that progressively increases with the number $n$ of suppressed modes.
• Figure 4: Slowdown of the relaxation dynamics toward the steady state in the open quantum Ising chain in Eqs. \ref{['eq:model-H']},\ref{['eq:model-L']} ($N{=}5$, $h_x{=}1$, $J{=}1.25$, $\alpha{=}0.5$ and $\gamma{=}1$). We use the recipe introduced in the main text to selectively suppress the projection of the initial density matrix $\rho_0$ along all except the slowest-decaying mode. The plot shows the distance to the steady state $\rho_\infty$ of the initial and transformed states $\rho_0(t),\rho_\parallel(t),\check\rho_\parallel(t)$ (black, solid-red and dashed-red curves, respectively). The dashed line refers to the state preparation obtained with the single-qubit transformation in Eq. \ref{['eq:U-phys']}. This result demonstrates the efficacy of the recipe for the slowdown of the relaxation dynamics.
• Figure 5: Schematic representation of the projection and rescaling operations for the control of the relaxation timescale (see main text). The real vectors $\vb n,\tilde{\vb n},\dots$ represent different operators $\rho_0,\tilde{\rho}_\perp,\dots$ on the basis of traceless hermitian matrices $\vb S$. Here, we represent vectors on the two-dimensional plane $\mathcal{X}$ spanned by $\tilde{\vb v}_s,\tilde{\vb v}$; we use dashed arrows for vectors $\vb n, \vb v_s, \vb v$, as they do not lie on the plane $\mathcal{X}$. Panel (a) represents the projection operation $\rho_0 \mapsto \tilde{\rho}_\perp$ (step one of the recipe). Panels (b) and (c) represent the two alternative situations occurring during the rescaling operation $\tilde{\rho}_\perp \mapsto \bar{\rho}_\perp$ (step two of the recipe). In case (b), $\abs{\bar{\vb v}_s} < \abs{\vb n}$ and the coefficients $\alpha_\pm$ are guaranteed to be real. Geometrically, the reality of coefficients $\alpha_\pm$ is tied with the existence of intersections between the sphere of radius $\abs{\vb n}$ and the dashed straight line (passing through the endpoint of $\bar{\vb v}_s$ with direction $\tilde{\vb v}'$). In case (c), $\abs{\bar{\vb v}_s} > \abs{\vb n}$ and the coefficients $\alpha_\pm$ are real if and only if the condition \ref{['eq:reality-condition']} is satisfied. The inequality \ref{['eq:reality-condition']} has a simple geometrical interpretation, and it can be rewritten as $\theta < \theta_\mathrm{m}$. Here, $\theta$ is the angle between $\bar{\vb v}_s$ and $\tilde{\vb v}'$; $\theta_\mathrm{m}$ is the angle between $\bar{\vb v}_s$ and its tangent to the sphere of radius $\abs{\vb n}$. If $\theta > \theta_\mathrm{m}$, then there is no intersection between the sphere of radius $\abs{\vb n}$ and the dashed straight line; hence, $\alpha_\pm\not\in\mathbb R$.