Deformation of a Quantum Many-Particle System by a Rotating Impurity

During the past 70 years, the quantum theory of angular momentum has been successfully applied to describing the properties of nuclei, atoms, and molecules, and their interactions with each other as well as with external fields. Because of the properties of quantum rotations, the angular-momentum algebra can be of tremendous complexity even for a few interacting particles, such as valence electrons of an atom, not to mention larger many-particle systems. In this work, we study an example of the latter: a rotating quantum impurity coupled to a many-body bosonic bath. In the regime of strong impurity-bath couplings, the problem involves the addition of an infinite number of angular momenta, which renders it intractable using currently available techniques. Here, we introduce a novel canonical transformation that allows us to eliminate the complex angular-momentum algebra from such a class of many-body problems. In addition, the transformation exposes the problem’s constants of motion, and renders it solvable exactly in the limit of a slowly rotating impurity. We exemplify the technique by showing that there exists a critical rotational speed at which the impurity suddenly acquires one quantum of angular momentum from the many-particle bath. Such an instability is accompanied by the deformation of the phonon density in the frame rotating along with the impurity.

Popular Summary

The concepts of rotation and angular momentum play a crucial role in areas of physics as diverse as nuclear spectroscopy, molecular collisions, and precision measurements. In quantum mechanics, rotations possess peculiar properties that make it challenging to understand the behavior of even a few interacting particles with nonzero angular momentum. In real-life experiments, quantum systems are almost never isolated; they are, instead, perturbed by their surrounding environment (e.g., a solution, a gas, or lattice vibrations in a crystal). In such settings, angular momentum can be redistributed between infinitely many quantum particles, which renders such problems intractable using currently available theoretical approaches. Here, we introduce a technique that paves the way for understanding the properties of angular momentum in the context of many-particle systems.

Our approach is based on a novel canonical transformation that makes it possible to drastically simplify the many-body problem and bypass the otherwise-intractable angular-momentum algebra. We consider a rotating impurity immersed in a many-body bosonic medium. Such systems occur naturally in state-of-the-art experiments of molecules in superfluid helium nanodroplets, and they can also serve as building blocks for more complex condensed-matter structures. We find that this rotating impurity, which can be prepared experimentally, is able to change the state of the system as a whole.

We expect that our findings can be applied to a broad range of systems from atomic, condensed-matter, and chemical physics, such as molecules trapped inside solids or excited-state electrons in crystals and Bose-Einstein condensates.

Figure 1

Action of the canonical transformation, Eq. (4), on the many-body system. Left: In the laboratory frame, $(x,y,z)$, the molecular angular momentum $\mathbf{J}$ combines with the bath angular momentum $\mathbf{\Lambda }$ to form the total angular momentum of the system $\mathbf{L}$. Right: After the transformation, the bath degrees of freedom are transferred to the rotating frame of the molecule, $(x^{\prime },y^{\prime },z^{\prime })$. As a result, the molecular angular momentum in the transformed space coincides with the total angular momentum of the system in the laboratory frame.

Figure 2

Change of the angulon spectral function $A_L(\omega )$, where $\omega =E-BL(L+1)$, with the rotational constant $B$, for three lowest total angular-momentum states. The $L>0$ states show an instability in the spectrum. The red dashed line shows the deformation energy, Eq. (9), which is independent of $L$. The circles indicate the points for which the phonon density modulation is shown in Fig. 3.

Figure 3

Phonon density in the impurity frame for selected values of $\log [B/u_0]$, which are specified in the right-hand top corner of the panels and (a)–(d) as labeled in Fig. 2; (e) far to the right from the instability, at $\log [B/u_0]=3.5$ for $L=1$ and at $\log [B/u_0]=3.0$ for $L=2$. The coordinates $(x,z)$ are in units of $(m{u}_0{)}^{-1/2}$.

Figure 4

Detection of the angulon self-energy ${\mathrm{\Sigma }}_L$ using (a) photoassociation spectroscopy [38] and (b) shift of $p$- and $d$-wave Feshbach resonances [39].