Quantum Brownian Motion Mapped to Classical Phase Space Dynamics at Any Temperature

The behaviour of quantum particles subject to environmental noise remains a fundamental challenge in physics, and understanding how quantum Brownian motion emerges is crucial for developing future quantum technologies. Dmitriy Kondaurov from the Russian Quantum Centre and the Moscow Institute of Physics and Technology, along with Evgeny Polyakov from the Russian Quantum Centre, now demonstrate a surprising connection between the quantum and classical worlds. Their work establishes that the seemingly complex dynamics of a quantum Brownian particle can be accurately described by a classical, albeit non-Markovian, stochastic process in phase space, even at any temperature. This achievement simplifies the modelling of quantum systems interacting with their environment, offering a powerful new framework for analysing complex driven-dissipative protocols and potentially paving the way for more efficient simulations of open quantum systems.

This mapping allows scientists to formulate a closed set of equations describing the particle’s coordinate and momentum, bypassing the need to solve complex quantum master equations. The resulting classical equations of motion accurately reproduce all quantum characteristics of the Brownian particle, including its position and momentum distributions, and incorporate the effects of both dissipation and fluctuations. This approach reveals a direct connection between quantum and classical descriptions of Brownian motion, demonstrating that the quantum dynamics can be fully captured by a suitably defined classical stochastic process, even with strong environmental coupling.

The research focuses on a Markovian stochastic process in phase space, which describes the dynamics of a particle interacting with a thermal bath. Starting with a correlated thermal equilibrium state between the particle and the bath, the team demonstrates that this correspondence is exact for quadratic potentials, regardless of the initial quantum state of the particle. For more general, smooth potentials, the investigation identifies a natural small parameter, revealing that the density matrix becomes strongly quasidiagonal in the coordinate representation. This quasidiagonality manifests as a shrinking off-diagonal width as the bath’s spectral cutoff increases, providing a controlled parameter for accurate approximation of the system’s behaviour. The team successfully shows that, for a particle experiencing a quadratic potential, the quantum dynamics can be accurately represented by a classical stochastic process, regardless of temperature. This simplification allows for the simulation of complex quantum systems by averaging classical trajectories, significantly reducing the computational demands associated with solving the full quantum problem. The approach accurately reproduces known analytical results and captures the rapid decoherence and thermalization effects caused by environmental fluctuations, phenomena often missed by simpler approximations.

The research highlights the importance of considering non-local effects when modeling quantum dissipation, demonstrating that the environment’s influence extends beyond simple friction. The team identified the ratio of the equilibrium coherence length to the classical scale as a key parameter governing the accuracy of the approximation for smooth, non-quadratic potentials. This provides a clear path for future improvements, potentially through perturbation theory or hybrid numerical methods that combine accurate quantum treatment within the coherence length with stochastic sampling of larger-scale classical motion. The team demonstrates that for a particle experiencing quadratic potentials, this mapping is exact, regardless of temperature, when starting from a physically realistic thermal equilibrium state. This allows for the simulation of complex quantum systems by generating and averaging classical trajectories, significantly simplifying the computational challenge posed by exponentially large quantum Hilbert spaces.

The method accurately reproduces known analytical results and, importantly, captures the rapid decoherence and thermalization effects caused by environmental fluctuations, phenomena often missed by simpler, high-temperature approximations. Furthermore, the researchers identified the ratio of the equilibrium coherence length to the classical scale as a key parameter governing the accuracy of the approximation for smooth, non-quadratic potentials. This provides a clear path for future improvements, potentially through perturbation theory or hybrid numerical methods that combine accurate quantum treatment within the coherence length with stochastic sampling of larger-scale classical motion. In conclusion, this work delivers a versatile computational framework for studying quantum Brownian motion across a wide range of temperatures, offering a conceptually simpler and numerically efficient alternative to existing methods.