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

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Deformation of a Quantum Many-Particle System by a Rotating Impurity PHYSICAL REVIEW X 6, 011012 (2016) Deformation of a Quantum Many-Particle System by a Rotating Impurity † Richard Schmidt1,2,* and Mikhail Lemeshko3, 1ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA 2Physics Department, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA 3IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria (Received 14 July 2015; revised manuscript received 8 December 2015; published 12 February 2016) 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. DOI: 10.1103/PhysRevX.6.011012 Subject Areas: Atomic and Molecular Physics, Chemical Physics, Mesoscopics I. INTRODUCTION pointlike particles. The latter is justified by the separation of the energy scales inherent to the impurity and the An important part of modern condensed matter physics surrounding bath. A well-known example is that of Bose deals with so-called “impurity problems,” aiming to under- and Fermi polarons realized in cold atomic gases by a stand the behavior of individual quantum particles coupled – to a complex many-body environment. The interest in number of groups [7 16]. There, the spherically symmetric quantum impurities goes back to the classic works of ground state of an alkali atom lies hundreds of THz lower Landau, Pekar, Fröhlich, and Feynman, who showed that than any of its electronically excited states. Given ultracold propagation of electrons in crystals is largely affected by collision energies, such an energy gap renders all the the quantum field of lattice excitations and can be ration- processes happening inside of an atom irrelevant. alized by introducing the quasiparticle concept of the More complex systems, such as molecules, are extended polaron [1–4]. In turn, the properties of a quantum many- objects and therefore possess a number of fundamentally body system can be drastically modified by the presence of different types of internal motion. The latter stem from the impurities. The most well-known examples are the Kondo relative motion of the nuclei, such as rotation and vibration, effect [5]—suppression of electron transport due to mag- which couple to each other as well as to the electronic spin – netic impurities in metals—and the Anderson orthogonality and orbital degrees of freedom [17 20]. This results in a catastrophe, which leads to the edge singularities in the x-ray rich low-energy dynamics which is highly susceptible to absorption spectra of metals [6]. external perturbations. Moreover, in many experimental In many instances, the impurities—even those possess- realizations molecular rotation is coupled to a phononic ing an internal structure—can be accurately described as bath pertaining to the surrounding medium, such as super- fluid helium [21], a rare-gas matrix [22], or a Coulomb crystal formed in an ion trap [23], which needs to be *[email protected] properly accounted for by a microscopic theory. † [email protected] The concept of orbital angular momentum, however, goes far beyond physically rotating systems and is being Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- used to describe, e.g., the excited-state electrons in solids, bution of this work must maintain attribution to the author(s) and whose motion is perturbed by lattice vibrations [24],or the published article’s title, journal citation, and DOI. Rydberg atoms immersed into a Bose-Einstein condensate 2160-3308=16=6(1)=011012(13) 011012-1 Published by the American Physical Society RICHARD SCHMIDT and MIKHAIL LEMESHKO PHYS. REV. X 6, 011012 (2016) X [25,26]. Despite the ubiquitous use of the angular- ˆ Jˆ2 ω ˆ † ˆ H ¼ B þ kbkλμbkλμ momentum concept in various branches of physics, a λμ X k versatile theory describing the redistribution of orbital à ˆ ˆ ˆ † ˆ ˆ ˆ Uλ k Y θ; ϕ b Yλμ θ; ϕ b λμ ; angular momentum in quantum many-body systems has þ ð Þ½ λμð Þ kλμ þ ð Þ k ð1Þ λμ not yet been developed. k Recently, we have undertaken the first step towards such θˆ ϕˆ where Yλμð ; Þ are the spherical harmonics [28]P depend-R a theory by deriving a generic Hamiltonian that describes θˆ ϕˆ ≡ the coupling of an SOð3Þ-symmetric impurity—a quantum ing on the molecular angle operators and , k dk, ℏ ≡ 1 rotor—with a bath of harmonic oscillators [27].Wehave and . shown that the problem can be approached most naturally The first term of Eq. (1) corresponds to the kinetic energy — of the translationally localized linear-rotor impurity, with B by introducing the quasiparticle concept of the angulon a ˆ quantum rotor dressed by a quantum field. The angulon is the rotational constant and J the angular-momentum an eigenstate of the total angular momentum of the system, operator. In the absence of an external bath, the impurity which remains a conserved quantity in the presence of eigenstates jj; mi are labeled by the angular momentum j the impurity-bath interactions. It was found that even and its projection m onto the laboratory-frame z axis. single-phonon excitations of the bath alone are capable Unperturbed rotational states form ð2j þ 1Þ-fold degener- 1 of drastically modifying the rotational spectrum of the ate multiplets with energies Ej ¼ Bjðj þ Þ [17,19,20]. impurity, which manifests itself in the emerging many- The second term of Eq. (1) represents the kinetic energy body-induced fine structure [27]. of the bosonic bath, where the corresponding creation and ˆ † ˆ Here, we demonstrate that rotation of an anisotropic annihilation operators, bk and bk, are expressed in the ˆ † ˆ k λ μ impurity can, in turn, substantially alter the collective state spherical basis, bkλμ and bkλμ. Here, k ¼j j, while and of a many-particle system. The effects are most significant define, respectively, the boson angular momentum and its in the regime of strong correlations, which, however, projection onto the laboratory z axis; see Appendix A for requires adding an infinite number of angular-momentum details. vectors pertaining to possible many-body states. The result- The last term of Eq. (1) describes the interaction between ing angular-momentum algebra involves Wigner 3nj sym- the impurity and the bath. The angular-momentum- bols [28] of an arbitrarily high order and is therefore dependent coupling strength UλðkÞ depends on the micro- intractable using standard techniques. In order to overcome scopic details of the two-body interaction between the this problem, here we introduce a canonical transformation, impurity and the bosons. For example, in Ref. [27] we which, to our knowledge, has never appeared in the literature showed that, for a linear rotor immersed into a Bose gas, before. The transformation renders the Hamiltonian inde- the couplings are given by pendent of the impurity coordinates, thereby eliminating Z 2 1=2 the complex angular-momentum algebra from the many- 8k ϵkρ 2 UλðkÞ¼uλ drr fλðrÞjλðkrÞ: ð2Þ body problem. Furthermore, the transformation singles out ω ð2λ þ 1Þ the conserved quantities of the many-body problem and k renders it solvable exactly in the limit of a slowly rotating This assumes that in the impurity frame the interaction impurity. between the rotor and a bosonic atom is expanded as The transformation makes it apparent that there exists a X critical rotational speed that leads to an instability, accom- r0 0 Θ0 Φ0 Vimp-bosð Þ¼ uλfλðr ÞYλ0ð ; Þ; ð3Þ panied by a discontinuity in the many-particle spectrum. λ Unlike in the vortex instability, originating from the 0 rotation of a condensate around a given axis [29], the with uλ and fλðr Þ giving the strength and shape of the instability we uncover here corresponds to the finite potential in the corresponding angular-momentum channel. transfer of three-dimensional angular momentum between The prefactor of Eq. (2) depends on the bath density ρ, the the impurity and the bath. It exists solely due to the discrete kinetic energy of the bare atoms ϵk, and the dispersion energy spectrum inherent to quantum rotation. We dem- relation of the bosonic quasiparticles ωk. Since the angulon onstrate that the emerging instability is ushered by a Hamiltonian Eq. (1) describes the interactions between a macroscopic deformation of the surrounding bath, i.e., quantum rotor and a bosonic bath of, in principle, any kind, the phonon density modulation in the frame corotating with we approach it from an entirely general perspective, the impurity.
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