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arXiv:1712.05163 · Physical Review Letters

Rotational Friction and Diffusion of Quantum Rotors

Benjamin A. Stickler, Björn Schrinski, Klaus Hornberger

Published Jul 23, 2018• 49 citations
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Abstract

We present the Markovian quantum master equation describing rotational decoherence, friction, diffusion, and thermalization of planar, linear, and asymmetric rotors in contact with a thermal environment. It describes how an arbitrary initial rotation state decoheres and evolves toward a Gibbs-like thermal ensemble, as we illustrate numerically for the linear and the planar top, and it yields the expected rotational Fokker-Planck equation of Brownian motion in the semiclassical limit.

DOI
10.1103/physrevlett.121.040401

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We present the Markovian quantum master equation describing rotational decoherence, friction, diffusion, and thermalization of planar, linear, and asymmetric rotors in contact with a thermal environment. It describes how an arbitrary initial rotation state decoheres and evolves toward a Gibbs-like thermal ensemble, as we illustrate numerically for the linear and the planar top, and it yields the expected rotational Fokker-Planck equation of Brownian motion in the semiclassical limit.
We study decoherence, diffusion, friction, and how they thermalize a planar rotor in the presence of an external potential. Representing the quantum master equation in terms of auxiliary Wigner functions in periodic phase space not only illustrates the thermalization process in a concise way, but also allows for an efficient numerical evaluation of the open quantum dynamics and its approximate analytical description. In particular, we analytically and numerically verify the existence of a steady state that, in the high-temperature regime, closely approximates a Gibbs state. We also derive the proper classical limit of the planar rotor time evolution and present exemplary numerical studies to verify our results.
In the present work, I identify the pointer states of quantum Brownian motion as rotated Gaussian states in phase space. By means of a particular piecewise deterministic unraveling, the pointer state unraveling, I determine the trajectories of the position and momentum expectation of the pointer states. They turn out to be described by a diffusion process in phase space and, in the semiclassical limit, they turn into the Langevin equation of classical Brownian motion. For collisional decoherence, the non-dissipative limit of the quantum linear Boltzmann equation, which describes the dynamics of a marker particle in an ideal gaseous environment, I examine the width of the pointer states and thereby extend previous results to more than one dimension. Crucially, it follows that pointer states do not exist for all parameter values in that case. By expanding the Lindblad operators of the quantum linear Boltzmann equation, I derive a new Lindblad master equation, dissipative collisional decoherence, which shows a decoherence behavior similar to that of collisional decoherence and also incorporates frictional as well as diffusion effects. Moreover this equation exhibits soliton-like solutions as candidates for the pointer states.
We characterize the pointer states generated by the master equation of quantum Brownian motion and derive stochastic equations for the dynamics of their trajectories in phase space. Our method is based on a Poissonian unraveling of the master equation whose deterministic part exhibits soliton-like solutions that can be identified with the pointer states. In the semiclassical limit, their equations of motion in phase space turn into those of classical diffusion, yielding a clear picture of the quantum-classical transition induced by environmental decoherence.
A nonlinear stochastic Schr\"odinger equation for pure states describing non-Markovian diffusion of quantum trajectories and compatible with non-Markovian master equations is presented. This provides an unraveling of the evolution of any quantum system coupled to a finite or infinite number of harmonic oscillators without any approximation. Its power is illustrated by several examples, including measurementlike situations, dissipation, and quantum Brownian motion. Some examples treat this environment phenomenologically as an infinite reservoir with fluctuations of arbitrary correlation. In other examples the environment consists of a finite number of oscillators. In such a quasiperiodic case we see the reversible decay of a macroscopic quantum-superposition (``Schr\"odinger cat''). Finally, our description of open systems is compatible with different positions of the ``Heisenberg cut'' between system and environment.