Fast and direct preparation of a genuine lattice BEC via the quantum Mpemba effect

Abstract

We demonstrate that dissipative state preparation protocols in many-body systems can be substantially accelerated via the quantum Mpemba effect. Our approach exploits weak symmetries to analytically identify a class of simple, experimentally-realizable states that converge exponentially faster to the steady state than typical random initializations. In particular, we study the preparation of a lattice Bose-Einstein condensate (BEC), where the depletion can be controlled via the dissipation strength. We also show how to tune the momentum of the created high-fidelity BEC by combining superfluid immersion with lattice shaking. Our theoretical predictions are confirmed by numerical simulations of the dissipative dynamics. This protocol paves the way to unlock the enormous potential of a dissipative preparation of highly entangled states in analog quantum simulators.

Figures (3)

• Figure 1: Mpemba- effect- assisted preparation of a BEC in an optical lattice. Panel a): A condensate of interacting bosonic particles can be prepared by combining coherent hopping with dissipation- mediated tunneling via a superfluid environment in which the optical lattice is immersed. Panel b) left: we consider a Lindbladian $\hat{\mathcal{L}}$ featuring a weak $\mathds Z_2$ symmetry corresponding to reflections about the center of the lattice. This endows the Lindbladian with a block-diagonal structure, with one block corresponding to evenly transforming states and one to oddly transforming states. Spectral analysis shows that the slowest-decaying mode belongs to the oddly transforming block. Panel b) right: Based on this symmetry argument, we identify a class of (evenly-transforming) product states that converge exponentially faster to the target .
• Figure 2: Panel a): Simulating the dissipative preparation of a finite- momentum in a lattice. We show the condensate depletion as a function of the inverse total particle number $N$ for $|k_0|<\pi/2$. The extrapolations to $1/N \to 0$ (gray lines) were performed with a third- order polynomial in $1/N$. Inset: Momentum space representation of the eigenvector $\psi$ corresponding to the eigenvalue $N_0$ of $\bm\gamma$ for characteristic Lindbladian momenta $k_0=0$ and $k_0=\pi/2$ (see \ref{['eq:jump:op']}). Panel b): parameter regions in which symmetric initial states \ref{['eq:symmetric:states']} yield a Mpemba speedup (red) and in which they don't (blue). These two regions correspond to the slowest- decaying mode of the Lindbladian transforming oddly and evenly under the inversion symmetry \ref{['eq:inversion:symmetry']}, respectively. We consider $L=6$ and $N=3$ at $k_0<\pi/2$ and employ . Inset: Criterion to numerically validate the occurrence of the Mpemba effect. In the limit $\kappa\to0$, the imaginary part of $\lambda_2$ (green dots) converges to the eigenvalue $e_0$ of the first excited eigenspace of $\hat{\mathcal{H}}$ in perturbation theory (see \ref{['eq:vectorized:unitary:part:of:L']}). For the extrapolation we employ a quadratic fit (orange line) and consider $U=0$.
• Figure 3: Mpemba- speedups in the preparation of . Panel a): The SL state (red lines), where all particles are initially located on the central site(s), converges exponentially faster to the steady state $\hat{\rho}_\mathrm{ss}$ than random initial product states (blue lines). Random states are generated by distributing $N$ particles on the lattice, sampling the positions from a uniform distribution over the sites. Line styles indicate different bosonic interaction strengths and characteristic momenta $k_0$. All calculations were performed with system parameters $L=10$, $N=10$, local dimension $d=N+1$, and $\kappa=2J$, and we averaged over $5$ product state realizations. Panel b): The corresponding speedups $S(\epsilon) = t_\mathrm{random}/t_\mathrm{symmetric}$ as a function of the total particle number. Note that the speedup is dictated by the spectrum of the Lidbladian, namely $S(\epsilon) \to \mathrm{Re}\, \lambda_3/\mathrm{Re}\, \lambda_2$ as $\epsilon\to 0$. $t_\mathrm{random}$ and $t_\mathrm{symmetric}$ are the times at which the $L_2$-distance from the steady state has dropped below a precision threshold $\epsilon=10^{-4}$ for the random and the states, respectively.