[Paper Review] A Wave Function approach to dissipative processes

Yvan Castin, Jean Dalibard+1 more|ArXiv.org|May 26, 2008

Quantum Mechanics and Applications2 references4 citations

TL;DR

This paper introduces a Monte Carlo Wave Function (MCWF) approach to model dissipative quantum systems coupled to a Markovian reservoir, using stochastic quantum jumps and non-Hermitian Hamiltonians to simulate open quantum dynamics. The method is formally equivalent to the master equation but offers computational advantages for large systems and provides direct physical insight into individual quantum trajectories, linking simulations to real experimental measurements of single systems.

ABSTRACT

Due to growing interest in quantum measurement, control and feedback, we reproduce a manuscript from 1992, presenting a simple physical and mathematical derivation of stochastic differential equations for wave functions of probed quantum systems. V. P. Belavkins seminal quantum filtering theory with similar equations, developed in the 1980es, was not known to the authors at the time of writing of the present manuscript.

Motivation & Objective

• To develop a stochastic wave function approach for open quantum systems in the Markovian regime, avoiding the computational cost of full density matrix evolution.
• To establish equivalence between the MCWF method and the standard master equation approach for relaxation processes.
• To provide a physically intuitive framework where quantum jumps model dissipation and fluctuations, linking simulation to actual measurement sequences.
• To enable efficient simulation of complex quantum optics and atomic physics systems with large Hilbert spaces (N ≫ 1).
• To connect simulated quantum trajectories to experimental observables, such as quantum noise in single-system measurements.

Proposed method

• The system evolves under a non-Hermitian Hamiltonian H = H_S - iħ/2 ∑ₘ Cₘ†Cₘ, which induces a non-unitary, norm-decreasing evolution over small time steps δt.
• At each time step, a quantum jump occurs with probability δpₘ = δt ⟨ϕ(t)|Cₘ†Cₘ|ϕ(t)⟩, corresponding to the operator Cₘ acting on the wave function.
• If a jump occurs, the wave function is updated as |ϕ(t+δt)⟩ = Cₘ|ϕ(t)⟩ / ||Cₘ|ϕ(t)⟩||, followed by normalization.
• For the case of continuous-time limit, the method is reformulated as a stochastic differential equation using Gaussian-distributed noise terms Δζₘ with zero mean and variance √Δt.
• The continuous limit is derived by considering a large number of infinitesimal jump components Dₘ,ε, leading to a stochastic equation equivalent to the Itô form.
• The method is validated by showing that ensemble averages of wave function trajectories reproduce the master equation dynamics exactly.

Experimental results

Research questions

• RQ1 Can a stochastic wave function approach reproduce the dynamics of a dissipative quantum system governed by a master equation?
• RQ2 How can quantum jumps be used to model both dissipation and fluctuations in open quantum systems?
• RQ3 What is the connection between the MCWF method and actual experimental measurement sequences on individual quantum systems?
• RQ4 How does the MCWF method compare computationally to the standard master equation approach for large Hilbert spaces?
• RQ5 Can the discrete jump process be mapped to a continuous stochastic differential equation in the limit of small time steps and many jump channels?

Key findings

• The MCWF method is formally equivalent to the master equation approach, with ensemble averages of wave function trajectories yielding the same reduced density matrix evolution.
• The method reduces computational cost from O(N²) for density matrices to O(N) for wave functions, making it efficient for systems with large Hilbert space dimension N ≫ 1.
• Quantum jumps are directly linked to physical measurement processes: each trajectory corresponds to a possible history of a single quantum system under continuous observation.
• The quantum noise observed in MCWF simulations of an observable A corresponds exactly to the quantum noise expected in a real experiment on a single system.
• The discrete jump process converges to a continuous Itô-type stochastic differential equation in the limit of infinitesimal time steps and a large number of jump channels.
• The continuous limit is achieved by introducing complex jump operators Dₘ,ε with ε = ±1, ±i, and the resulting equation matches known stochastic wave function equations in the literature.