On dissipative stochastic equations in a Hilbert space

Alberto Frigerio

Hans Christian Öttinger

We propose a piecewise deterministic Markovian jump process in Hilbert space such that the covariance matrix of this stochastic process solves the thermodynamic quantum master equation. The proposed stochastic process is particularly simple because the normalization of the vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearity of the thermodynamic master equation, the construction of stochastic trajectories involves the density matrix as a running ensemble average. We identify a principle of detailed balance and a fluctuation-dissipation relation for our Markovian jump process.

O. Nicrosini, A. Rimini

Sayak Adhikari, Roi Baer

The Lindblad master equation is a fundamental tool for describing the evolution of open quantum systems, but its computational complexity poses a significant challenge, especially for large systems. This article introduces a stochastic representation of the Lindblad dissipator that addresses this challenge by bundling the Lindblad operators. The stochastic dissipator maintains the Lindblad form, ensuring completely positive and trace-preserving dynamics. We demonstrate the effectiveness of this method by considering a Morse oscillator coupled to a spin bath. Our numerical experiments show that a small number of stochastically bundled operators can accurately capture the system's dynamics, even when the Hilbert space dimension is large. This method offers a new perspective on open quantum systems and provides a computationally efficient way to simulate their dynamics.

Forrest G. Hall

Nicolas Gisin, M. B. Cibils

The authors develop the tool of quantum diffusion (i.e. Hilbert-space-valued stochastic differential equations) for dissipative quantum systems. The aims are to find possible limitations to this approach and to investigate new pictures of open quantum systems. They are guided by the relaxation process for arbitrary spin and the associated natural rotational symmetry. They also impose the condition that the spin-coherent states remain coherent during the dissipative evolution. They present a new quantum diffusion equation that satisfies the above conditions and that is the unique quantum diffusion satisfying Percival's condition (d psi ) 2 =0.