# Studies in a Random Noise Model of Decoherence

## TL;DR

This paper investigates how random noise affects decoherence in a double-potential well quantum system, combining numerical simulations and analytical models to understand the influence of noise characteristics on qubit stability.

## Contribution

It introduces a comprehensive formalism for modeling noise-induced decoherence, extending existing formulas and analyzing the impact of low-frequency noise on quantum systems.

## Key findings

- Decoherence rate depends critically on low-frequency noise power.
- Analytical two-state model agrees with full numerical simulations.
- Low-frequency noise absence leads to negligible decoherence.

## Abstract

We study the effects of noise and decoherence for a double-potential well system, suitable for the fabrication of qubits and quantum logic elements. A random noise term is added to the hamiltonian, the resulting wavefunction found numerically and the density matrix obtained by averaging over noise signals. Analytic solutions using the two-state model are obtained and found to be generally in agreement. In particular, a simple formula for the decoherence rate in terms of the noise parameters in the two-state model is reviewed and verified for the full simulation with the multi-level system. The formalism is extended to describe multiple sources of noise or different "dephasing" axes at the same time. Furthermore, the old formula for the "Turing-Watched Pot" effect is generalized to the case where the environmental interactions do not conserve the "quality" in question. Various forms for the noise signal are investigated. An interesting result is the importance of the noise power at low frequency. If it vanishes there is, in leading order, no decoherence. This is verified in a numerical simulation where two apparently similar noise signals, but differing in the power at zero frequency, give strikingly different decoherence effects. A short discussion of situations dominated by low frequency noise is given.