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arXiv:1712.05596 · New Journal of Physics

Quantum angular momentum diffusion of rigid bodies

Birthe Papendell, Benjamin A. Stickler, Klaus Hornberger

Published Nov 10, 2017• 30 citations
arXivHTMLPDF

Abstract

We show how to describe the diffusion of the quantized angular momentum vector of an arbitrarily shaped rigid rotor as induced by its collisional interaction with an environment.We present the general form of the Lindblad-type master equation and relate it to the orientational decoherence of an asymmetric nanoparticle in the limit of small anisotropies.The corresponding diffusion coefficients are derived for gas particles scattering off large molecules and for ambient photons scattering off dielectric particles, using the elastic scattering amplitudes.

DOI
10.1088/1367-2630/aa99d1

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We show how to describe the diffusion of the quantized angular momentum vector of an arbitrarily shaped rigid rotor as induced by its collisional interaction with an environment.We present the general form of the Lindblad-type master equation and relate it to the orientational decoherence of an asymmetric nanoparticle in the limit of small anisotropies.The corresponding diffusion coefficients are derived for gas particles scattering off large molecules and for ambient photons scattering off dielectric particles, using the elastic scattering amplitudes.
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We show how the ro-translational motion of anisotropic particles is affected by the model of continuous spontaneous localization (CSL), the most prominent hypothetical modification of the Schrödinger equation restoring realism on the macroscale. We derive the master equation describing collapse-induced spatio-orientational decoherence and demonstrate how it leads to linear- and angular-momentum diffusion. Since the associated heating rates scale differently with the CSL parameters, the latter can be determined individually by measuring the random motion of a single levitated nanorotor.
The mechanism of decoherence for a mesoscopic quantum system with rotational degrees of freedom is studied. From a simple model of elastic scattering, we show that the nondiagonal density-matrix elements of the system exponentially decay. The decay rate depends on the difference of scattering amplitudes for different rotational configurations, leading to the gradual loss of quantum coherence between the pointer states in the orientational space. For a dielectric ellipsoid immersed in a photon-gas environment (assuming no absorption), the decay rate is found to be proportional to the seventh power of the temperature. For an ellipsoidal object interacting with massive particles, the decay rate is proportional to the 5/2 power of the temperature. Both are different from the case of translational decoherence induced by the same environment scattering. For photon scattering, the coherence time in the rotational degrees of freedom is shown to be much shorter than that in the translational degrees of freedom.
Levitated nanoparticles are a promising platform for sensing applications and for macroscopic quantum experiments. While the nanoparticles' motional temperatures can be reduced to near absolute zero, their uncontrolled internal degrees of freedom remain much hotter, inevitably leading to the emission of heat radiation. The decoherence and motional heating caused by this thermal emission process is still poorly understood beyond the case of the center-of-mass motion of point particles. Here, we present the master equation describing the impact of heat radiation on the motional quantum state of arbitrarily sized and shaped dielectric rigid rotors. It predicts the localization of spatio-orientational superpositions only based on the bulk material properties and the particle geometry. A counterintuitive and experimentally relevant implication of the presented theory is that orientational superpositions of optically isotropic bodies are not protected by their symmetry, even in the small-particle limit.
The free translational dynamics of an n-body quantum system has many applications, e.g. in molecular and nuclei systems, where it is common to classify the degrees of freedom of the system in 3 rotational and 3n - 6 shape coordinates. It is known that there exists an interaction between the shape and orientation degrees of freedom. In particular, changes of the shape could induce an orientation change. In this work, it is shown a rotating quantum system which does not deform its shape probability density and with vanishing angular momentum. The orientation change is monitored with a localized orientation wavefunction. We characterize the localized wavefunctions and study their evolution under a rigid rotor-like Hamiltonian, concluding that this kind of wavefunctions may rotate by their own.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.