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The act of a quantum measurement seems to evade the accessibility of the Schrödinger equation and its unitary time evolution [1]. In this work, we explore whether a quantum measurement on a subsystem by an apparatus can be simulated within an experimentally realizable quantum mechanical many-body system by the unitary time evolution of the many-body Schrödinger equation.
For the numerical simulation, we consider a model inspired by the Fermi-Hubbard model, which is realizable experimentally with ultracold atoms [2] containing a fixed number of indistinguishable fermionic environmental particles and a single distinguishable impurity. The impurity is assumed to be initially in a spatial superposition but not entangled with the state of the environment. A key advantage of this system lies in the high tunability of its parameters, both experimentally and in simulations.
To simulate the process of a measurement within the many-body Schrödinger equation, the environmental particles are assumed to act as both a measuring apparatus on the impurity and as the environment in a more conventional sense. We determine the exact quantum mechanical time evolution by exact diagonalization of mesoscopically sized systems. We calculate decoherence times and correlation functions for different system sizes, and we study the reduced-density matrix of the impurity and the surrounding environmental particles. We investigate how the spectral properties of the system determine whether the interaction between the impurity and the environment performs a measurement on the impurity or whether the interaction leads to thermalization [3]. To analyze this, the spectral distribution of the eigenvalues of the entire system is calculated as a measure of quantum chaos [4]. The key aim is to identify which degrees of freedom of the surrounding particles act as the measuring apparatus—thereby encoding information about the state of the impurity—and which act as the environment, inducing decoherence.
References:
[1] Emanuel Schwarzhans, Felix C Binder, Marcus Huber, and Maximilian PE Lock. Quantum measurements and equilibration: the emergence of objective reality via entropy maximisation. arXiv preprint arXiv:2302.11253, 2023.
[2] Immanuel Bloch. Ultracold quantum gases in optical lattices. Nature physics, 1(1):23–30, 2005.
[3] Armando Relano. Decoherence framework for wigner’s-friend experiments. Physical Review A, 101(3):032107, 2020.
[4] Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer, and Iva Brezinova. Canonical density matrices from eigenstates of mixed systems. Entropy, 24(12):1740, 2022.
