Elsevier

Physics Letters A

Volume 306, Issues 2–3, 30 December 2002, Pages 97-103
Physics Letters A

Quantum hydrodynamic analysis of decoherence: quantum trajectories and stress tensor

https://doi.org/10.1016/S0375-9601(02)00602-3Get rights and content

Abstract

Quantum trajectories, obtained by integrating equations of motion for elements of the probability fluid, are used to analyze decoherence in a model two-mode system. Analysis of trajectories, flux maps, and the stress tensor for two composite systems, in one of which the system is uncoupled from the environment, leads to a hydrodynamic interpretation of the decoherence process.

Introduction

Decoherence is the physical process by which the density matrix for a superposition of quantum states evolves into a result which is very similar to that obtained for a statistical mixture where interference effects are prohibited [1], [2], [3]. The relationship between quantum and classical descriptions of physical processes, quantum measurement theory and quantum computation are some areas where decoherence plays a fundamental role. It is generally accepted that decoherence occurs either when a quantum system interacts with a many degree of freedom heat bath or when it is entangled with an environment with a few degrees of freedom 2. Conventionally, influence functionals 4, quantum master equations 5, or decoherent histories 6 have been used to predict and analyze the destruction of interference effects attributed to decoherence.

The de Broglie–Bohm version of quantum hydrodynamics [7], [8], [9], [10] provides a unique window for viewing the time-dependent dynamics of the decoherence process. In this study, we investigate the decoherence of a superposition of quantum states by comparing the hydrodynamic fields for two composite systems, in one of which the system–environment coupling is turned off. The decoherence mechanism emerges through analysis of these fields, especially quantum trajectories for the fluid elements and the inter-related flux and stress maps for the quantum fluid. Hydrodynamic analysis of decoherence was studied using quantum trajectories based upon the Caldeira–Legget master equation for the reduced density matrix [11], [12]. However, the decoherence studies reported here are the first quantitative results on a model system obtained using the hydrodynamical formulation without the coarse graining or averaging that was inherent in all previous works.

Section snippets

Model Hamiltonian and initial wavefunction

In this study, we emphasize new insights gleaned from the hydrodynamic analysis of an interference experiment bearing analogies to the double-slit experiment. Initially, a coherent superposition of two well separated Gaussian wave packets was prepared in a composite system involving a system mode, x, coupled to a harmonic bath mode, y. The Hamiltonian for the composite system is decomposed into system, (harmonic) bath and coupling contributions: H=Hs+Hb+Hc=px22m0+12py2m+ky2+cxy. The coupling

Hydrodynamic equations of motion

The hydrodynamic formulation [7], [8], [9], [10] is initiated by substituting the polar form of the wavefunction, Ψ(r,t)=R(r,t)expiS(r,t)/ℏ, into the time-dependent wave equation. We then obtain the continuity equation, ρ̇=−ρ∇·ν, which connects the probability density, ρ=R2, and the velocity by ν=j/ρ=(∇S)/m, where j is the probability flux. The second equation is ν̇=−∇(V+Q)/m, in which the flow acceleration is produced by the sum of the classical force, fc=−∇V, and the quantum force, fq=−∇Q.

Quantum trajectories and probability density

We will first compare an ensemble of 200 quantum trajectories and the resultant probability density for the uncoupled and coupled cases in Fig. 2 (at t=225 time steps). The quantum potential causes streamlines followed by the fluid elements to expand outward from the two localized components of the initial distribution. For the uncoupled case (a), trajectories in the central region merge near the attractor, the mid-plane between the two separated initial wave packets. The probability density

Quantum Navier–Stokes equation and the stress tensor

We now investigate a pair of inter-related hydrodynamic fields, the flux vector distribution and the stress tensor for the quantum fluid. In classical hydrodynamics, the Navier–Stokes (NS) equation 24 governing the change in the momentum density, ρ·, is given by ∂(ρmνi)∂t=−jjΠj,i−ρ∇iV, where the last term is the ‘external’ force density arising from the potential V and where Π is the stress tensor (units of pressure, force/area, or momentum flux, momentum/(area×time)). The classical stress

Symptoms of decoherence

Tracing out the bath degrees of freedom is not a necessary condition for the appearance of decoherence. However, many previous studies are based upon features of the reduced density matrix and the corresponding Wigner function. System–bath interplay registered by the reduced density matrix for the system and the Wigner function is described separately 28, so only a brief summary of results will be mentioned here. For the decoherent system, even when the system is coupled to only one bath

Conclusions

In this study, comparisons were made of the time-dependent hydrodynamic fields for a system initiated in a coherent superposition, without and with coupling to one bath mode. From the viewpoint of the quantum Navier–Stokes equation, the uncoupled and coupled systems differ through only one term, the coupling force density −ρVc on the right side of this equation. This term plays a crucial role in the decoherence mechanism. For the uncoupled system, internal stress is relieved as the initially

Acknowledgements

This research was supported in part by the Robert Welch Foundation and the National Science Foundation. We thank Justin Briggle for assistance with the graphics and the Texas Advanced Computing Center for providing access to the Cray SV1.

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