Dissipative diffusion in quantum state preparation: from passive cooling to system-bath engineering

TL;DR

The paper compares two particle-number-conserving dissipative schemes—thermal coupling to a cold bath and engineered dissipation—for preparing the ground state of a dimerized topological model. Using a mix of exact diagonalization, tensor-network methods, quantum trajectories, and a truncated Hilbert-space approach, the authors extract Liouvillian gaps and cooling times across system sizes, revealing diffusion-dominated relaxation with a universal $\tau \propto N^2$ scaling. A mean-field and a classical random-walk model capture the essential physics: diffusion controls the late-time approach to the ground state, and the engineered protocol yields a unique, robust dark state that accelerates cooling relative to the thermal scheme. The work clarifies the role of diffusion and dark-state uniqueness in dissipative state preparation, with implications for scalable cooling in quantum simulators and quantum information platforms.

Abstract

We investigate and compare two particle number conserving protocols for the preparation of a topologically nontrivial state. The first is derived from thermally coupling the system to a cold bath, while the second is based on engineered dissipation. We numerically study the time required to reach the target state as well as its robustness against physically important perturbations. Crucially, in both protocols the cooling capability is limited by dissipatively induced diffusion processes. The resulting quadratic scaling of the cooling time with system size is corroborated also analytically using mean-field approximations and a purely classical random walk model. Furthermore, we find that the engineered protocol admits a unique and stable dark state, which contributes to an ongoing discussion regarding the applicability of dissipative state preparation to many-body systems.

Figures (16)

• Figure 1: (a) Parent Hamiltonian $H$ from \ref{['eq:common_hamiltonian']} with yellow $a$ (green $b$) sub-lattice sites, and inter-cell hoping $t$. (b) Generic cooling protocol based on local system-bath couplings with a system governed by the parent Hamiltonian and a cold thermal environment inducing intra-cell (inter-cell) cooling $L^{\uparrow\mathrel{\mspace{-3.5mu}}\downarrow}$ ($L^{\rightleftarrows}_{a,b}$). (c) Engineered dissipative state preparation protocol targeting the ground state of a specific parent Hamiltonian in a non-equilibrium fashion, resulting in Lindblad jump operators $L^\pm$ that pump toward the steady state without relying on coherent Hamiltonian dynamics.
• Figure 2: Plot of the spectral density $J(\omega; \beta)$ in \ref{['eq:spectral_density']} for different values of $\beta$ in units of the typical energy scales of the model $\Lambda_0$. For small temperatures $T \lesssim \Lambda_0/5$ ($\beta\Lambda_0\gtrsim 5$), for which the ground state constitutes a macroscopic portion of the steady state, the contributions increasing energy at $\omega = -\Lambda_0$ are heavily suppressed.
• Figure 3: The cost function $C(\tilde{\lambda})$ for the engineered dissipation Liouvillian with $N=8$. The eigenvalues are marked by stars, which coincide perfectly with the minima of the cost function. Crucially, this landscape is absent of anomalous minima which motivates the spectral gradient search strategy. There, we start with the trust region $(\alpha, \beta)$ and successively shrink the interval using binary search.
• Figure 4: The excited population fraction $\nu = \mathop{\mathrm{tr}}\nolimits (\rho_\infty \mathcal{N}_+) / \langle \mathcal{N} \rangle$ in the steady state $\rho_\infty$ for both thermal protocols at different inverse temperatures $\beta$ for system size $N=12$. Around $\beta \approx 4$, the excited population is effectively described by the detailed balance with respect to the level spacing $\Delta$ as enforced by the condition in \ref{['eq:kubomartinschwinger']}.
• Figure 5: Comparison of the evolution of the excited population $\nu = \langle \mathcal{N}_+ \rangle / \langle \mathcal{N} \rangle$ for $m=10000.0$ quantum trajectories (of which $50$ are shown) for the three different protocols in the thermal (a) Davies and (b) formulation as well as for (c) the engineered protocol. For each sample, the mean and $1\sigma$ ($2\sigma$) region corresponding to $68.3%$ ($95.4%$) of measurements is shown. Clearly, the aggregate trajectory converges towards the targeted state with $\nu = 0$. Both the Davies and the formulation yield virtually indistinguishable results. In all dissipation protocols, impact ionization events can be identified as illustrated for the orange trajectory in (c) which become increasingly insignificant at late times.