Exploration of the decay processes of a quantum state loosely coupled to a tank of finite size

Sketch of a quantum state (white dot) loosely coupled to the discrete levels of a chaotic quantum dot (black dots connected by lines). Credit: Micklitz et al

In quantum physics, Fermi’s golden rule, also known as the golden rule of time-dependent perturbation theory, is a formula that can be used to calculate the rate at which an initial quantum state changes to a final state, which is composed of a continuum of states (a so-called “bathroom”). This valuable equation has been applied to numerous physics problems, particularly those for which it is important to consider how systems respond to imposed perturbations and stabilize in stationary states over time.

Fermi’s golden rule applies specifically to cases where an initial quantum state is loosely coupled to a continuum of other final states, which overlap its energy. Researchers from Centro Brasileiro de Pesquisas Físicas, Princeton University and Universität zu Köln recently decided to investigate what happens when a quantum state is instead coupled to a set of discrete end states with a mean-level spacing other than zero, as observed in many body physics studies.

“The decay of a quantum state into a continuum of final states (i.e. a ‘bath’) is commonly associated with inconsistent decay processes, as described by Fermi’s golden rule,” Tobias Micklitz, one of the researchers who conducted the study, he told Phys.org. “A standard example for this is an excited atom emitting a photon into an infinite vacuum. Current experiments, on the other hand, systematically implement composite systems involving quantum states coupled to actually finite-sized reservoirs that are composed of discrete sets of final states rather than a continuum “.

While several past studies have identified systems in which quantum states are coupled to finite-sized reservoirs, understanding the conditions under which this occurs, allowing finite-sized reservoirs to effectively act as “baths” is a challenging task. The key goal of the recent work by Micklitz and his colleagues was to better understand the process by which a quantum state decays when coupled to a finite-sized reservoir.

“Our starting point was to consider generic tanks of finite size with no specific symmetries,” explained Micklitz. “Such systems usually exhibit quantum chaotic behavior and can be modeled from random matrices for which powerful analytical tools are available.”

To carry out their analyzes, Micklitz and his colleagues used a combination of effective matrix integral techniques, which are commonly used in studies applying the theory of random matrices, a theory that summarizes the different properties of matrices with items randomly extracted from different probability distributions. To compare the results of their analyzes, they then used exact diagonalization, a powerful numerical technique often used by physicists to study single quantum many-body systems.

“Initially we didn’t expect the decay in a finite-sized tank to be described by such a complex time dependence,” Micklitz said. “We found that the probability of residing in the loosely coupled level exhibits a non-motonic time dependence with an initial decay, followed by an increase, before saturating to a constant value. The time profile follows (over a wide range of parameters) the “Spectral form factor”, a well-studied object in the quantum chaos community, which encodes information about the correlations of the energy level in the reservoir. This makes a lot of sense in retrospect. “

Now published in Physical Review Letters, the recent study by this group of researchers offers a fully analytical description of a crucial and fundamental physical problem. More specifically, it offers a connection between the problem of how a quantum state decays into a set of discrete end states to the statistics associated with energy levels and wave functions in chaotic quantum systems.

“We link the residence probability time profile to the spectral form factor and the relationship between the minimum and saturation values ​​of the probability with the eigenfunctions of the tank,” added Micklitz. “Our work focuses on a fundamental but also quite basic example of relaxation in a finite-dimensional reservoir. We are now trying to tackle more complex systems, such as spin sets coupled to a quantum dot. Hopefully, progress can be made using similar methods such as those employed in our recent article. ”

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More information:
Tobias Micklitz et al, The Emergence of Fermi’s Golden Rule, Physical Review Letters (2022). DOI: 10.1103 / PhysRevLett.129.140402

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Citation: Exploration of the decay processes of a quantum state loosely coupled to a finite-sized reservoir (2022, 19 October) recovered on 19 October 2022 from https://phys.org/news/2022-10-exploring-quantum-state- weakly- coupled.html

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