Preparing Spin Squeezed States via Adaptive Genetic Algorithm

TL;DR

This work addresses robust preparation of spin-squeezed states for quantum metrology in open quantum systems by developing an adaptive genetic algorithm (GA) that optimizes sequences of square control pulses in a Hamiltonian $\\hat{H}=\\hat{H}_0+\\sum_k f_k(t)\\hat{H}_k$, specifically targeting $\\hat{H}/\\hbar=\\kappa\\hat{J}_z^{2}+\\Omega_x(t)\\hat{J}_x$ with $\\kappa=1$. The GA iteratively evolves pulse sequences to maximize spin squeezing, quantified by $\\xi_Z^2=\\frac{4\\Delta \\hat{J}_z^{2}}{N}$, under Lindblad dissipation, achieving final-state fidelity $|\\eta|^2>0.99$ for $N$ in $[20,100]$ and exhibiting robust performance against dissipation and thermal noise. When optimized for metrological enhancement, the GA yields scalable squeezing with exponents approaching the Heisenberg limit in ideal conditions (e.g., $\\xi^2_\perp \sim N^{-0.71}$) and maintains strong scaling under dissipation and thermal noise (e.g., $N^{-0.69}$ to $N^{-0.65}$). The framework outperforms constant-control strategies and rivals reinforcement learning, is experimentally feasible in atomic BEC systems, and is adaptable to other quantum platforms, with the GA module replaceable by alternative optimizers such as PSO, ACO, or Firefly algorithms.

Abstract

We introduce a novel strategy employing an adaptive genetic algorithm (GA) for iterative optimization of control sequences to generate quantum nonclassical states. Its efficacy is demonstrated by preparing spin-squeezed states in an open collective spin model governed by a linear control field. Inspired by Darwinian evolution, the algorithm iteratively refines control sequences using crossover, mutation, and elimination strategies, starting from a coherent spin state within a dissipative and dephasing environment. We rigorously benchmark our method against constant control protocols and reinforcement learning, demonstrating competitive and robust performance. Furthermore, we showcase the GA's versatility by directly optimizing for metrologically relevant squeezing, achieving scalable performance, even in the presence of dissipation and thermal noise. The proposed strategy demonstrates a high state-preparation fidelity, exceeding 0.99, and provides a long time window for maintaining the spin squeezed state, even under dissipative conditions. We discuss feasible experimental implementations and potential extensions to alternative quantum systems, and the adaptability of the GA module. This research establishes the foundation for utilizing GA-like strategies in controlling quantum systems and achieving desired nonclassical states.

Figures (8)

• Figure 1: The GA scheme is used to optimize control pulses for squeezing a spin system in an open quantum system. It starts with a randomly generated population of individuals, each representing a unique control pulse sequence. Individuals undergo crossover and mutation operations to create new offspring. Fitness is evaluated by simulating the open quantum system dynamics under each control pulse, with higher squeezing leading to greater fitness. The GA iteratively selects individuals based on their fitness, promoting the propagation of optimal pulse sequences across generations until the set number of iteration limit is reached.
• Figure 2: (a) Evolution of the spin squeezing parameter while 100 rectangular pulses are taken across each evolution time interval $[0,2]$ and a total of 20 consecutive training rounds are shown with 10 iteration curves (one curve is displayed every two generations). (b) Schematic representation of the three-level ($\Omega(t) =1,0,-1$) square control pulses used in (a). (c) The evolution of average spin squeezing parameter with different pulse numbers and corresponding variance of squeezed state at $t=2$ during iterations. (d) Evolutionary trend of the statistical distribution, average, and median of final-state squeezing parameter values for individuals within uniform-sampled 10 generations, and each 'violin' pattern corresponds to one generation as well as to one population. Solid black balls represent the mean, and white bars represent the median. (e)The real part of the density matrix of the initial CSS and($\rm e_1$) is the Wigner representation of it. (f)The real part of the matrix of the spin squeezed state at $t=2$ and ($\rm f_1$) is its corresponding Wigner representation. $\gamma/\kappa = 0.001$ and $\gamma_z/\kappa = 0.001$ in these results.
• Figure 3: (a) Mean evolution of the last iteration sample of $\xi_{Z}^2$ in 5 repetitions every 30 generations. The shaded regions indicate the variance for different frequencies of applying rectangular pulses. Control time interval $[0,2]$ is divided into the number of specific segments at which the square pulses with three levels($\Omega(t)=1,0,-1$) are applied. Samples were all obtained from the final generation results of each training session. (b) Evolution of the spin squeezing parameter for different numbers of pulse gears when 100 total pulses are applied, and the correspondence between the number of pulses and their intensity is as follows: $\{\Omega(t)\}$= $\{(1, 0, -1)|~actions=3\}$, $\{(1, 0.5, 0, -0.5, -1)|~actions=5\}$, $\{(1, 0.67, 0.33, 0, -0.33, -0.67, -1)|~actions=7\}$, $\{(1, 0.75, 0.5, 0.25, 0, -0.25, -0.5, -0.75, -1)|~actions=9\}$.
• Figure 4: Evolutions for the last iteration of $\xi_{Z}^2$ every 30 generations for different sizes of the collective spin system $N = 2J$ under three-type ($\Omega(t)=1,0,-1$) control. The error zone corresponding to variance is calculated by five repetitions. The samples are picked in the same manner as those in FIG. \ref{['DoubleCFeacc']} (a) and number of segments is 100.
• Figure 5: The average evolutions of the last iteration of $\xi_{Z}^2$ in 5 repetitions every 30 generations with variance for different thermal excitations: the average number of photons for a mode with frequency $\omega$ in the reservoir. The samples are picked in the same manner as those shown in FIG. \ref{['RLlearning3size']}. The subgraph shows the squeezing parameter versus average thermal excitation at $t=2$.