[Paper Review] Stochastic Schrödinger equations and memory

Alberto Barchielli, Paolo Di Tella+2 more|arXiv (Cornell University)|Jun 18, 2010

Quantum Information and Cryptography22 references3 citations

TL;DR

This paper introduces a non-Markovian extension of the stochastic Schrödinger equation by incorporating memory through stochastic adapted coefficients and colored noise, such as Ornstein-Uhlenbeck processes. The approach preserves complete positivity and provides an unravelling of master equations with memory kernels, enabling numerical simulations while maintaining a measurement-theoretic interpretation for open quantum systems with memory effects.

ABSTRACT

By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.

Motivation & Objective

• To develop a non-Markovian extension of quantum trajectory theory that preserves complete positivity of the dynamics.
• To incorporate memory effects in open quantum systems by introducing stochastic coefficients dependent on the system's past history.
• To provide a measurable, simulation-friendly framework for non-Markovian dynamics using diffusive stochastic Schrödinger equations with memory.
• To establish a connection between the proposed stochastic equations and exact master equations with memory kernels via projection techniques.
• To ensure the resulting dynamics remains physically valid by preserving complete positivity, even in non-Markovian regimes.

Proposed method

• The authors generalize the linear stochastic Schrödinger equation by introducing stochastic, adapted coefficients that depend on the system's past, thereby embedding memory into the dynamics.
• They employ a diffusive stochastic Schrödinger equation with a d-dimensional Wiener process and bounded Lindblad-type operators to model continuous measurements with memory.
• A specific model uses an Ornstein-Uhlenbeck process as the driving noise, leading to a random Hamiltonian evolution and a non-Markovian unraveling of the dynamics.
• The mean evolution of the system is derived using the Nakajima-Zwanzig projection technique, yielding a closed master equation with a memory kernel.
• The resulting master equation includes an inhomogeneous term and a time-nonlocal integral kernel, both arising from the stochastic coefficients and noise correlations.
• An approximation is introduced by replacing the stochastic evolution propagator with a deterministic one based on the mean Liouville generator, simplifying the memory kernel and stochastic term.

Experimental results

Research questions

• RQ1 Can memory effects be consistently introduced at the Hilbert space level in stochastic Schrödinger equations while preserving complete positivity?
• RQ2 How can non-Markovian quantum dynamics be unraveled via stochastic Schrödinger equations with memory?
• RQ3 What is the form of the master equation corresponding to the proposed non-Markovian stochastic Schrödinger equation?
• RQ4 Can a measurement-theoretic interpretation be maintained in the non-Markovian regime using this framework?
• RQ5 What are the implications of using colored noise, such as Ornstein-Uhlenbeck, in the stochastic evolution of open quantum systems?

Key findings

• The proposed stochastic Schrödinger equation with adapted coefficients and colored noise preserves complete positivity of the dynamical maps, ensuring physical validity.
• The mean evolution of the system satisfies a closed master equation with a memory kernel, derived via the Nakajima-Zwanzig projection technique.
• The memory kernel is explicitly given by the expectation of a time-ordered product of stochastic operators, capturing non-Markovian correlations.
• The stochastic equation provides a valid unravelling of the non-Markovian master equation, enabling numerical simulations through quantum trajectory methods.
• An approximation using the deterministic mean evolution propagator yields a simplified master equation with a memory kernel involving the exponential of the modified Liouvillian.
• For the specific model with constant operators, the memory kernel reduces to a time-convolution form with a decaying exponential, reflecting the correlation time of the Ornstein-Uhlenbeck noise.