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

PHYSICAL REVIEW X 6, 011012 (2016)

† 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 superfluid 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 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. exemplifying the couplings by UλðkÞ of Eq. (2). Many-body problems such as given by the Hamiltonian Eq. (1) are typically hard to solve. The conventional II. CANONICAL TRANSFORMATION approaches to tackle them include, when applicable, We start from the general Hamiltonian of the angulon perturbation theory, renormalization group, or in principle problem, as defined in Ref. [27]: uncontrolled methods such as those based on the selective

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z ^ diagram resummations, as well as purely numerical tech- S niques. An alternative, actively used since the development z of classical mechanics, involves canonical transformations z’ z’ of the underlying Hamiltonian [30,31]. Here, the idea is to x’ x x’ partially diagonalize the Hamiltonian and/or to expose the y constants of motion, which allows us to reveal some of the ’ eigenstates properties exactly. In the context of impurity x y’ y’ problems, typical approaches employ the collective bath y variables as a generator of the symmetry transformations, J +=L J 2 = L2 as it has been used, e.g., in the polaron theory [32–34]. In the angulon problem discussed in this paper, the total angular momentum is a good quantum number. However, due to the coupling of bath degrees of freedom with the FIG. 1. Action of the canonical transformation, Eq. (4), on the impurity coordinates, as given by the third term of Eq. (1), many-body system. Left: In the laboratory frame, ðx; y; zÞ, the this conservation law is not apparent. Here, we introduce a molecular angular momentum J combines with the bath angular canonical transformation that makes this constant of motion momentum Λ to form the total angular momentum of the system explicit and allows us to achieve several other goals listed L. Right: After the transformation, the bath degrees of freedom 0 0 0 below. The corresponding operator Sˆ uses the composite are transferred to the rotating frame of the molecule, ðx ;y;zÞ. angular momentum of the bath as a generator of rotation, As a result, the molecular angular momentum in the transformed space coincides with the total angular momentum of the system in which transfers the environment degrees of freedom into the laboratory frame. the frame corotating along with the quantum rotor. The transformation is given by natural to introduce two coordinate frames, as schemati- ˆ − ϕˆ ⊗Λˆ − θˆ⊗Λˆ − γˆ⊗Λˆ S ¼ e i z e i y e i z : ð4Þ cally shown in Fig. 1. The laboratory frame ðx; y; zÞ is singled out by the collective state of the bosons, while the The angle operators ðϕˆ ; θˆ; γˆÞ act in the Hilbert space of the rotating impurity frame ðx0;y0;z0Þ is defined by the instan- rotor, and taneous orientation of the molecular axes. The relative X orientation of the two frames is given by the eigenvalues of ˆ ˆ † λ ˆ ϕˆ θˆ γˆ Λ ¼ b σμνb λν ð5Þ the Euler angle operators ð ; ; Þ acting in the impurity kλμ k ˆ kλμν Hilbert space. The S operator transforms the many-body state of the bosons into the rotating molecular frame, using is the collective angular-momentum operator of the many- Λˆ as a generator of quantum rotations. In turn, as we show body bath, acting in the Hilbert space of the bosons. below, the molecular state in the transformed frame σλ Here, denotes the vector of matrices fulfilling the becomes an eigenstate of the total angular momentum of angular-momentum algebra in the representation of angular the system, which is a constant of motion. λ momentum . Introducing the body-fixed coordinate frame bound to The transformation brings the Hamiltonian Eq. (1) into the impurity makes explicit an additional quantum number, the following form: n, which gives the projection of the angular momentum X 0 −1 † onto the rotor axis z . The angular-momentum basis states Hˆ ≡ ˆ ˆ ˆ Jˆ0 − Λˆ 2 ω ˆ ˆ ˆ2 S H S ¼ Bð Þ þ kbkλμbkλμ jj; m; ni are therefore the eigenstates of the J , Jˆ , and Jˆ 0 λμ z z X k operators, as given by Eqs. (C1)–(C3) of Appendix C. ˆ † ˆ þ VλðkÞ½b þ b λ0: ð6Þ For a linear-rotor molecule in the absence of a bath, the kλ0 k ˆ ˆ ˆ ˆ kλ total angular momentum L ¼ J þ Λ coincides with J. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Therefore, Lˆ is perpendicular to the molecular axis z0, 2λ 1 4π Jˆ0 Here, VλðkÞ¼UλðkÞ ð þ Þ=ð Þ and is the n 0 “ ” resulting in ¼ . With the bosons present, the total anomalous angular-momentum operator acting in the angular momentum is no longer perpendicular to z0, ˆ0 rotating frame of the impurity. Since the components of J providing the molecular state with nonzero n in the trans- act in the body-fixed frame, they obey anomalous com- formed frame. In other words, the transformation Eq. (4) “ ” mutation relations [20,35] as opposed to the ordinary converts a linear-rotor molecule into an effective “sym- ˆ angular-momentum operator, J of Eq. (1), which acts metric top” [20] by dressing it with a boson field. in the laboratory frame. The details of the derivation, as Compared to the original Hamiltonian, Eq. (1), the well as the properties of the Jˆ0 operator, are presented in transformed Hamiltonian, Eq. (6), possesses the following Appendix B. properties. We now discuss the physical meaning of the trans- (1) Hˆ is explicitly expressed through the total angular formation Sˆ. In order to describe the composite system, it is momentum, which is a constant of motion. Because

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of the isotropy of space, the eigenstates of the with Wkλ ¼ ωk þ Bλðλ þ 1Þ. This transformation removes original Hamiltonian Hˆ are simultaneous eigenstates the terms linear in the bosonic operators, replacing them by of the total angular-momentum operators Lˆ 2 and Lˆ , the deformation energy of the bath: z X and thus can be labeled as jL; Mi. The transformed 2 −1 E ¼ − VλðkÞ =W λ: ð9Þ states Sˆ jL; Mi are hence the eigenstates of the def k kλ transformed Hamiltonian Hˆ . As detailed in Appen- dix C, these transformed states are also eigenstates As a consequence, in the limit of B ¼ 0, the vacuum of 2 0 of the Jˆ0 operator with the eigenvalues LðL þ 1Þ, phonon excitations j i becomes the exact ground state of corresponding to the total angular momentum. Eq. (7). On the other hand, such a coherent shift trans- 2 formation corresponds to a macroscopic deformation of the Consequently, the Jˆ0 operator in Eq. (6) can be bath, and could not be easily performed on the original replaced by the classical number LðL þ 1Þ. Hamiltonian Eq. (1) where the impurity coordinates are (2) Hˆ does not contain the impurity coordinates ðθˆ; ϕˆ Þ, strongly coupled with the bath degrees of freedom. which allows us to bypass the intractable angular- Here, we are interested in the effect of a slowly rotating momentum algebra, arising from the impurity- impurity on the many-body state of the environment. bath coupling. The angle operators of the original Therefore, we introduce a variational ansatz based on Hamiltonian, Eq. (1), couple the impurity states with single-phonon excitations on top of the bosonic state every single boson excitation, which results in the macroscopically deformed by the operator Uˆ : problem of adding an infinite number of angular X momenta in three dimensions. The latter involves ˆ † jψi¼g j0ijLM0iþ α λ b j0ijLMni: ð10Þ working with Wigner 3nj symbols of an arbitrarily LM k n kλn kλn large order. In the transformed Hamiltonian, on the other hand, the problem is reduced to adding the The states of an isolated symmetric-top molecule are angular-momentum projections of the impurity and characterized by three quantum numbers: the angular the bath. There, the impurity-bath coupling, Jˆ0 · Λˆ , momentum L, its projection M onto the laboratory-frame has the form of spin-orbit interaction and does not z axis, and its projection n onto the molecular symmetry lead to an involved angular-momentum algebra. axis z0. For a linear-rotor molecule, the angular momentum (3) Hˆ can be solved exactly in the limit of a slowly vector is always perpendicular to the molecular axis and rotating impurity, B → 0; see Sec. III. therefore n is identically zero. The transformation Eq. (4), (4) Hˆ allows us to find the eigenstates containing an however, transfers the bosons to the molecular frame, infinite number of phonon excitations, which is thereby creating an effective “many-body symmetric-top” crucial, e.g., to account for the macroscopic defor- state. The latter consists of a linear-rotor impurity dressed mation of the condensate. This follows directly from by the field of bosons carrying finite angular momentum. point (2) above, and is detailed in Sec. III. As a result, the total angular momentum of such a (5) Hˆ contains information about the deformation of the symmetric top is no longer perpendicular to the linear- condensate in the rotating impurity frame. Com- rotor axis and provides the finite values of the projection n. pared to the laboratory frame, where the deformation See Appendix C for more details. of the bath is averaged over the angles, this provides It is worth emphasizing that the nontransformed many- an additional insight into the nature of the many- body wave function corresponding to Eq. (10) is given by body state and, consequently, into the origin of the jϕi¼Sˆ · Uˆ jψi. Therefore, it is a highly involved object angulon instability, discussed in Sec. III. with an infinite number of degrees of freedom entangled with each other. The simple ansatz of Eq. (10) is made possible by the consecutive canonical transformations, III. MACROSCOPIC DEFORMATION OF THE Eqs. (4) and (8). Furthermore, it is straightforward to BATH AND THE EMERGING INSTABILITY extend Eq. (10) to bath excitations of higher order, since In the limit of a slowly rotating impurity, B → 0, the this does not generate any complexities related to the Hamiltonian Eq. (6) can be solved exactly by means of an angular-momentum algebra. additional canonical transformation: Performing the variational solution for the energy, E ¼hψjℋˆ jψi=hψjψi, we obtain the condition ℋˆ ¼ Uˆ −1Hˆ U;ˆ ð7Þ −E þ BLðL þ 1Þ − ΣLðEÞ¼0; ð11Þ where which has the form of a Dyson equation with self-energy X ΣLðEÞ [36], as given by Eq. (D13); see Appendix D for a ˆ VλðkÞ ˆ − ˆ † U ¼ exp ðbkλ0 bkλ0Þ ; ð8Þ detailed derivation. Equation (11) can be rewritten in terms λ Wkλ −1 k of the angulon Green’s function as ½GLðEÞ ¼ 0, where

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−1 0 −1 − Σ ½GLðEÞ ¼½GLðEÞ LðEÞ; ð12Þ threshold at zero energy. Here, this is no longer the case, since the transformation Uˆ of Eq. (8) introduces an infinite 0 −1 − 1 with ½GLðEÞ ¼ E þ BLðL þ Þ. number of phonon excitations into the variational ansatz. The ground- and excited-state properties of the This leads to an energetic renormalization of the phonon system are contained in the spectral function, ALðEÞ¼ emission threshold providing all the excited angulon states þ Im½GLðE þi0 Þ. Without restricting the generality of what with decay channels for phonon emission. This, in turn, follows, we assume potentials whose angular-momentum leads to a finite lifetime for any magnitude of the impurity- expansion, Eq. (3), is given by the Gaussian form factors, bath coupling. −3=2 −r2=ð2r2Þ → 0 fλðrÞ¼ð2πÞ e λ , and nonzero magnitudes, u0 In the limit of B the molecule is not rotating and is and u1, in two lowest angular-momentum channels. inducing an anisotropic deformation of the bath, corre- We assume an anisotropy ratio of u1=u0 ¼ 5, a range sponding to the mean-field-like deformation energy, Eq. (9). −1=2 B r0 ¼ r1 ¼ 15ðmu0Þ , and set the interactions with λ > 1 The magnitude of the deformation energy decreases with monotonically and determines the general shape of the to zero. Furthermore,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we use a Bogoliubov-type dispersion 2 spectrum. Apart from the deformation energy which is relation, ω ¼ ϵ ðϵ þ 2g nÞ, where ϵ ¼ k =2m with m k k k bb k identical for all L’s, the energy of the angulon acquires the mass of a boson. We choose the boson-boson interaction an additional contribution due to phonon excitations in the 418 3 −1=2 0 014 3=2 gbb ¼ ðm u0Þ and density n ¼ . ðmu0Þ . This surrounding medium. The latter corresponds to the rota- choice of parameters reproduces the speed of sound in 4 tional Lamb shift discussed in Ref. [27], which has been superfluid He for u0 ¼ 2π × 100 GHz [37]. Figure 2 observed as the renormalization of the rotational spectrum shows the dependence of the spectral function on the for molecules in superfluid helium nanodroplets [21]. Most rotational constant B for the three lowest rotational states. importantly, we find that for the excited states with L>0 The width of the lines reflects the lifetimes of the corre- there exists a critical rotational constant, where a disconti- sponding levels. In Ref. [27], we studied the nontransformed nuity in the rotational spectrum occurs. This effect corre- Hamiltonian Eq. (1) using a variational ansatz based on sponds to a transfer of one quantum of angular momentum single-phonon excitations. Using this ansatz we found that from the bath to the impurity. One can see that the faster the angulon states become stable after crossing the phonon the rotation (i.e., the larger L), the earlier this instability occurs. Such an instability has been briefly discussed in Ref. [27], where it was referred to as many-body-induced L = 2 (c) (d) 0 fine structure of the second kind. While the instability can be detected using spectroscopy 0

/u in the laboratory frame, an insight into its origin can be ω (a) gained by making use of the canonical transformation, –0.2 (b) Eq. (4). Namely, in the frame corotating with the impurity, the instability manifests itself as a change of the phonon ˆ † ˆ L = 1 (c) (d) density hbrbri; for analytic expressions, see Appendix E. 0 Figure 3 shows the phonon density for L ¼ 1 and 2 at five

0 different values of the impurity rotational constant. Darker /u (a) ω shade corresponds to higher density. Far to the left of the (b) –0.2 instability, Fig. 3(a), the impurity is rotating slowly and the bosons are able to adiabatically follow its motion. As a result, the surrounding bath becomes polarized, which L = 0 0 manifests itself in a highly asymmetric phonon density. The shape of the density modulation is given by the first 0

/u spherical harmonic, which arises due to the λ ¼ 1 term in ω the impurity-boson potential included in our model. Closer –0.2 to the instability, Fig. 3(b), the phonon density increases, signaling the onset of the resonant phonon excitations. –8 –6 –4 –2 0 2 At the right edge of the instability, Fig. 3(c), the phonon log[B/u 0 ] density drops drastically. Farther away from the instability, the density distribution becomes more symmetric the faster FIG. 2. Change of the angulon spectral function ALðωÞ, where ω ¼ E − BLðL þ 1Þ, with the rotational constant B, for three the impurity rotates, as illustrated in Figs. 3(d) and 3(e). lowest total angular-momentum states. The L>0 states show In other words, when the rotational constant exceeds the an instability in the spectrum. The red dashed line shows the critical value given by the instability, it becomes energeti- deformation energy, Eq. (9), which is independent of L. cally unfavorable for the bosons to follow the motion of The circles indicate the points for which the phonon density the impurity. As a consequence, the bosonic bath does not modulation is shown in Fig. 3. possess finite angular momentum, which results in the

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L = 1 L = 2 TABLE I. Comparison of the angulon instability with the vortex instability. (a) –4.0 –4.5 20 Angulon Vortex 10 ˆ 2 ˆ Corresponding rotation Spherical, L Planar, Lz ℏℏ 0 Angular momentum transfer per particle Circulation Zero integer × 2πℏ=m –10

–20 spherically symmetric density distribution. Thus, the phonon density in the transformed frame can serve as a fingerprint of the angular-momentum transfer from the bath (b) –2.0 –2.5 20 to the impurity which takes place at the instability point. It is important to note that the “angulon instability” we 10 discuss here is fundamentally different from the vortex 0 instability [29], also associated with rotation. The comparison between the two is summarized in Table I. First, the –10 rotation of the impurity is inherently three dimensional and –20 does not involve any specific rotation axis. This is different for a vortex line, which singles out a particular direction in space. Second, the formation of a vortex requires a transfer (c) –1.0 –1.5 of one unit of angular momentum per particle in the bath. In 20 the angulon instability, on the other hand, a finite (small) 10 number of rotational quanta is shared between the impurity and the collective state of the many-particle environment. 0 Finally, the vortex instability leads to a finite circulation –10 around the vortex line, which is absent for the angulon instability. –20 IV. EXPERIMENTAL IMPLEMENTATION (d) 1.5 1.0 The described effects can be observed experimentally 20 both with molecules trapped in strongly interacting super- 10 fluids, such as helium droplets [21], and with molecular

0 impurities immersed in weakly interacting Bose-Einstein condensates [29]. The dependence of the angulon self- –10 energy, ΣL of Eq. (12), on the many-body parameters can –20 be revealed by measuring the relative shift between the rotational states of a diatomic molecule. Since the effects will be most pronounced for the molecular states possess- (e) 3.5 3.0 ing a small rotational constant B, experiments involving 20 molecules in highly excited vibrational states provide the 10 most natural setup. In the context of ultracold gases, the latter include photoassociation spectroscopy [38] and

zzzzz 0

–10 (a) (b)

–20 closed channel L=0 –20 –10 0 10 20 –20 –10 0 10 20 ΣL xx open channel L=0 L=2 Σ FIG. 3. Phonon density in the impurity frame for selected values L of log½B=u0, 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 FIG. 4. Detection of the angulon self-energy Σ using (a) photo- 3 5 1 3 0 L instability, at log½B=u0¼ . for L ¼ and at log½B=u0¼ . association spectroscopy [38] and (b) shift of p- and d-wave −1=2 for L ¼ 2. The coordinates ðx; zÞ are in units of ðmu0Þ . Feshbach resonances [39].

011012-6 DEFORMATION OF A QUANTUM MANY-PARTICLE SYSTEM … PHYS. REV. X 6, 011012 (2016) measuring nonzero angular-momentum Feshbach resonan- Eq. (1). This resonates with Wegner’s idea of the continu- ces [39]. In both cases, the shifts of the spectroscopic lines ous unitary transformations [50], which underlies one of will be proportional to the angulon self-energy, as sche- the Hamiltonian formulations of the renormalization group matically illustrated in Fig. 4. An alternative possibility is approach [51]. measuring ΣL as a shift of the microwave lines in the Finally, the impurity problem we consider here can be spectra of weakly bound molecules [40], prepared using used as a building block of a general theory describing the one of these techniques. In the frequency domain, at redistribution of orbital angular momentum in quantum sufficiently low temperatures the width of the lines will many-particle systems. This opens up a perspective of correspond to the angulon lifetime. The instability shown in applying the techniques of this article to the several Fig. 2 corresponds to the vanishing quasiparticle weight problems in condensed matter [24] and chemical [52] with a related emergence of a broad incoherent background physics. and therefore can be detected as a line broadening with increasing impurity-bath interactions. In the time domain, ACKNOWLEDGMENTS on the other hand, the angulon Green’s function can be detected using Ramsey and spin-echo techniques [41,42]. We are grateful to Eugene Demler, Jan Kaczmarczyk, In such a measurement, the angulon instability leads to Laleh Safari, and Hendrik Weimer for insightful discus- dephasing dynamics with a related pronounced decay of the sions. The work was supported by the NSF through a grant Ramsey and spin-echo contrast [41,42]. for the Institute for Theoretical Atomic, Molecular, and While in superfluid helium the interactions cannot be Optical Physics at Harvard University and Smithsonian tuned as easily as in ultracold gases, the range of chemical Astrophysical Observatory. species amenable to trapping is essentially unlimited [21]. The latter, combined with advances in the theory of APPENDIX A: ANGULAR-MOMENTUM molecule-helium interactions [43], paves the way to study- REPRESENTATION ing angulon physics in a broad range of parameters. The creation and annihilation operators of Eq. (1) are expressed in the angular momentum representation, which V. CONCLUSIONS is related to the Cartesian representation as In this paper, we study the redistribution of orbital Z k angular momentum between a quantum impurity and a ˆ † Φ Θ Θ ˆ † λ Θ Φ bkλμ ¼ 3 2 d kd k sin kbki Yλμð k; kÞ; ðA1Þ many-particle environment. We introduce a technique that ð2πÞ = allows us to drastically simplify the problem of adding an 2π 3=2 X infinite number of angular momenta which occur in the ˆ † ð Þ ˆ † −λ Θ Φ — bk ¼ bkλμi Yλμð k; kÞ: ðA2Þ regime of strong interactions. The essence of the method k λμ a novel canonical transformation—paves the way to eliminating the complex angular-momentum algebra from the The quantum numbers λ and μ define, respectively, the ’ problem, as well as to exposing the problem s constants of angular momentum of the bosonic excitation and its projec- ’ motion. We exemplifiy the technique s capacity by study- tion onto the laboratory-frame z axis. Equations (A1) ing an instability that occurs in the spectrum of the many- and (A2) correspond to the following commutation relations: particle system due to the interaction between the bath and the rotating impurity. Such an instability should be detect- ˆ ˆ † 2π 3δð3Þ k − k0 ½bk; bk0 ¼ð Þ ð Þ; ðA3Þ able with molecules in superfluid helium droplets [21] and might be responsible for the long time scales emerging in ˆ ˆ † 0 ½b λμ; b 0 0 0 ¼δðk − k Þδλλ0 δμμ0 : ðA4Þ molecular rotation dynamics in the presence of an environ- k k λ μ ment [44], which presently lacks even a qualitative explan- In the coordinate space, the transformation between the ation. Moreover, the rotating impurities can be prepared representations is defined as experimentally in perfectly controllable settings, based on Z ultracold molecules immersed into a Bose or Fermi gas ˆ † Φ Θ Θ ˆ † λ Θ Φ [17,18,45] and cold molecular ions inside Coulomb crystals brλμ ¼ r d rd r sin rbri Yλμð r; rÞ; ðA5Þ [23]. It is important to note that the transformation, as defined by Eq. (4), is quite general, and can be applied to 1 X ˆ † ˆ † −λ br ¼ b i YλμðΘ ; Φ Þ; ðA6Þ extended Fröhlich Hamiltonians [36], to impurities with r rλμ r r complex rotational structure [20], Rydberg molecules λμ [25,46–48], as well as to the case of a fermionic bath [49]. The ultimate goal of our approach is to find a series of with the corresponding commutation relations, canonical transformations that would lead to exact solu- ˆ ˆ † δð3Þ r − r0 tions to the many-body Hamiltonians of the same class as ½br; br0 ¼ ð Þ; ðA7Þ

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ˆ ˆ † 0 In the laboratory frame, the angular-momentum vector is ½b λμ; b 0 0 0 ¼δðr − r Þδλλ0 δμμ0 : ðA8Þ r r λ μ Jˆ ˆ ˆ ˆ defined by its spherical components, ¼fJ−1; J0; Jþ1g, The operators in the coordinate and momentum space are where related through the Fourier transform, ˆ ˆ J0 ¼ J ; ðB4Þ Z z d3k ˆ † ˆ † ik·r 1 br ¼ 3 bke ; ðA9Þ ˆ ˆ ˆ ð2πÞ J 1 ¼ − pffiffiffi ðJ þ iJ Þ; ðB5Þ þ 2 x y from which one can obtain the corresponding relation for ˆ 1 ˆ ˆ the angular-momentum components, J−1 ¼ pffiffiffi ðJ − iJ Þ; ðB6Þ 2 x y rffiffiffi Z 2 ˆ † λ ˆ † b i r kdkjλ kr b ; see Refs. [28,35]. We use the analogous notation for rλμ ¼ π ð Þ kλμ ðA10Þ the components of the total angular momentum of the Λˆ Λˆ Λˆ Λˆ with jλðkrÞ the spherical Bessel function [53]. bosons ¼f −1; 0; þ1g, Eq. (5). The operators, Eqs. (B4)–(B6), obey the following commutation relations with each other, APPENDIX B: CANONICAL TRANSFORMATION pffiffiffi ˆ ˆ − 2 1;iþk ˆ Here, we provide details on the derivation of the trans- ½Ji; Jk¼ C1;i;1;kJiþk; ðB7Þ formed Hamiltonian, Eq. (6). In the angular-momentum representation, the boson where i, k ¼f−1; 0; þ1g, and with the rotation operators, ˆ † ˆ creation and annihilation operators, b λμ and bkλμ, k ˆ λ ϕˆ θˆ γˆ are defined as irreducible tensors of rank λ [28]. ½Jk;Dμνð ; ; Þ ˆ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Consequently, they are transformed by the S operator of kþ1 λ;μ−k λ ˆ ˆ ¼ð−1Þ λðλ þ 1ÞCλ μ;1 − Dμ− νðϕ; θ; γˆÞ; ðB8Þ Eq. (4) in the following way: ; ; k k; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X λ λ;μþk λ † † ˆ ϕˆ θˆ γˆ λ λ 1 ϕˆ θˆ γˆ ˆ −1 ˆ ˆ λ ϕˆ θˆ γˆ ˆ ½Jk;Dμνð ; ; Þ ¼ ð þ ÞCλ;μ;1;kDμþk;νð ; ; Þ: S bkλμS ¼ Dμνð ; ; Þbkλν; ðB1Þ ν ðB9Þ X ˆ −1 ˆ ˆ λ ˆ ˆ ˆ S b λμS Dμν ϕ; θ; γˆ b λν: B2 l3;m3 k ¼ ð Þ k ð Þ Here, C ; are the Clebsch-Gordan coefficients [28]. ν l1;m1 l2;m2 By using the latter property, one can show that the λ ˆ ˆ operators Eqs. (B4)–(B6) transform under Eq. (4) in the Here, Dμνðϕ; θ; γˆÞ are Wigner D matrices [28] whose following way: arguments are the angle operators defining the relative orientation of the impurity frame with respect to the X Jˆ ≡ ˆ −1 ˆ ˆ ˆ − 1 ϕˆ θˆ γˆ Λˆ laboratory frame. These expressions can also be derived i S JiS ¼ Ji Dik ð ; ; Þ k: ðB10Þ −1 0 1 using the explicit expression for the angular momentum of k¼ ; ; the bosons, Eq. (5). After some angular-momentum algebra, we obtain the The Wigner rotation matrix appearing in Eq. (B1) is following expression for the square of the angular momen- complex conjugate with respect to the one of Eq. (B2) and tum in the transformed frame: therefore corresponds to an inverse rotation. As a result, ˆ −1 ˆ2 ˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ0 ˆ 2 X X S J S ≡ J 0 − J 1J −1 − J −1J 1 ¼ðJ − ΛÞ : ˆ −1 ˆ † ˆ ˆ ˆ † ˆ þ þ S bkλμbkλμ S ¼ bkλμbkλμ; ðB3Þ ðB11Þ μ μ Here, Jˆ0 is the angular-momentum operator in the and the second term of Eq. (1) does not change under the rotating molecular (i.e., body-fixed) coordinate frame transformation. [20,35], which can be expressed via the laboratory-frame Similarly, in the last term of Eq. (1) we use that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi components as θˆ ϕˆ 2λ 1 4π λ ϕˆ θˆ 0 Yλμð ; Þ¼ ð þ Þ=ð ÞDμ0ð ; ; Þ, which leads to X cancellation of the Wigner D matrices. In such a way, the Jˆ 0 ¼ D1 ðϕˆ ; θˆ; γˆÞJˆ : ðB12Þ ˆ i k;i k transformation S eliminates the molecular angle variables k from the Hamiltonian. The transformation of the molecular rotational The spherical components of Jˆ0 are expressed through the Hamiltonian BJˆ2 turns out to be slightly more cumbersome. Cartesian components using the relations analogous to

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ˆ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eqs. (B4)–(B6). Note that this makes the J operators j;n−k Jˆ 0 jj; m; ni¼ð−1Þk jðj þ 1ÞC jj; m; n − ki; ðC6Þ different from the so-called contravariant angular- k j;n;1;−k momentum components used by Varshalovich et al. [28]. where k ¼f−1; 0; þ1g. Thus, in the molecular frame the The molecular-frame angular-momentum operators raising operators lower the projection quantum number n obey the anomalous commutation relations with one and the lowering operators raise it. another [19,35], Unlike for nonlinear polyatomic molecules [20], the pffiffiffi angular momentum of a linear rotor is always perpendicular ˆ 0 ˆ 0 2 1;iþk ˆ 0 ½Ji; Jk¼ C1;i;1;kJiþk; ðB13Þ to the internuclear axis (defining z0), and therefore n is identically zero. However, this is the case only before the and the following commutation relations with the rotation transformation Sˆ is applied. Let us consider the most matrices, general many-body state in the nontransformed frame: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ 0 λ ϕˆ θˆ γˆ − λ λ 1 λ;νþk λ ϕˆ θˆ γˆ X ½Jk;Dμνð ; ; Þ ¼ ð þ ÞCλ;ν;1;kDμ;νþkð ; ; Þ; i L;M 0 ⊗ λμ jL; Mi¼ akλjCj;m;λ;μjjm i jk ii: ðC7Þ ðB14Þ kλμ jm;i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ 0 λ ϕˆ θˆ γˆ −1 k λ λ 1 λ;ν−k λ ϕˆ θˆ γˆ ½Jk;Dμνð ; ; Þ ¼ ð Þ ð þ ÞCλ;ν;1;−kDμ;ν−kð ; ; Þ: The molecular states jjm0i are the eigenstates of the ðB15Þ molecular angular-momentum operator, as given by Eqs. (C1) and (C2). The same relations are fulfilled for ˆ 2 It is worth noting that, in the case of a linear-rotor the collective bosonic states: Λ jkλμi¼λðλ þ 1Þjkλμi and ˆ molecule, the molecule-boson interaction does not depend Λˆ jkλμi¼μjkλμi, where Λ is defined by Eq. (5), and k is γˆ z on the third Euler angle, . However, this angle must be the linear momentum. The index i labels all the possible preserved in Eq. (4), as well as in all the derivations boson configurations resulting in a collective state jkλμi, described above, in order to keep the transformation spanning the complete many-body Hilbert space of the unitary. bosonic bath. It is straightforward to show that the state, Eq. (C7), is an eigenstate of the total angular-momentum operator, APPENDIX C: MOLECULAR STATES IN THE ˆ Lˆ ¼ Jˆ þ Λ: TRANSFORMED SPACE ˆ 2 In the main text and Fig. 1 we introduce two coordinate L jL; Mi¼LðL þ 1ÞjL; Mi; ðC8Þ frames: the laboratory one, ðx; y; zÞ, and the molecular one, 0 0 0 ˆ ðx ;y;zÞ. A general molecular state, therefore, can be LzjL; Mi¼MjL; Mi: ðC9Þ characterized by three quantum numbers: the magnitude of ˆ −1 angular momentum j, its projection m onto the laboratory- By acting on jL; Mi with S , after some angular- frame z axis, and its projection n onto the molecular-frame momentum algebra, we obtain the state in the transformed z0 axis: frame: X −1 ˆ2 ˆ i ⊗ λ J jj; m; ni¼jðj þ 1Þjj; m; ni; ðC1Þ S jL; Mi¼ fkλnjLMni jk nii; ðC10Þ kλni ˆ Jzjj; m; ni¼mjj; m; ni; ðC2Þ where the coefficients are given by X i λþn i j;0 ˆ 0 f λ ¼ð−1Þ a λ C − ;λ : ðC11Þ Jzjj; m; ni¼njj; m; ni: ðC3Þ k n k j L; n ;n j In the angular representation, the corresponding wave We see that the transformation effectively transfers the functions are given by [20] angular momentum of the bosons to the molecular frame. rffiffiffiffiffiffiffiffiffiffiffiffiffi This is reflected by the fact that the transformed state, 2 1 ˆ −1 ϕ θ γ j þ j ϕ θ γ S jL; Mi, becomes an eigenstate of the body-fixed angu- h ; ; jj; m; ni¼ 2 Dmnð ; ; Þ: ðC4Þ 8π lar momentum operator Jˆ02, with the eigenvalues of the total angular-momentum operator Lˆ 2; i.e., The action of the space-fixed and molecule-fixed components of angular momentum is given by the general Jˆ02 ˆ −1 1 ˆ −1 formula [19,35], ðS jL; MiÞ ¼ LðL þ ÞðS jL; MiÞ: ðC12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Each state LMn in the superposition of Eq. (C10) is an ˆ 1 j;mþk j i Jkjj; m; ni¼ jðj þ ÞCj;m;1;kjj; m þ k; ni; ðC5Þ effective symmetric-top state [20], with the projection of

011012-9 RICHARD SCHMIDT and MIKHAIL LEMESHKO PHYS. REV. X 6, 011012 (2016) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi total angular momentum on the molecular axis entirely − 1 ω α − 2 1 α ½ E þ BLðL þ Þþ k k1;1 B LðL þ Þ k10 determined by the boson field. X ξ ξ α þ B k1 k01 k01;1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik APPENDIX D: DERIVATION OF THE DYSON − 1 ξ EQUATION FROM THE VARIATIONAL ¼ B LðL þ Þ k1gLM: ðD6Þ PRINCIPLE By symmetry we expect jαk11j¼jαk1−1j; however, if We minimize the energy obtained from the expectation αk11 ¼ −αk1−1 were true, Eq. (D5) would imply value of Eq. (7) with respect to the variational state: αk10 ¼ 0. This in turn would lead to a contradiction in X Eq. (D6), which shows that αk11 ¼ αk1−1. ψ 0 0 α ˆ † 0 j i¼gLMj ijLM iþ kλnbkλnj ijLMni: ðD1Þ Thus, from Eq. (D5) we obtain kλn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2B 2LðL þ 1Þ Minimization with respect to α and g yields the α α kλn LM k10 ¼ − ω 1 2 k11: ðD7Þ following equations: E þ k þ BLðL þ Þþ B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX Let us now define the inverse propagator, ½−E þ BLðL þ 1ÞgLM þ B LðL þ 1Þ ξkλαkλn ¼ 0 kλ P ðkÞ¼BLðL þ 1Þ − E þ ω ðD2Þ E k 4B2LðL þ 1Þ − ; ðD8Þ and −E þ ωk þ BLðL þ 1Þþ2B X λ L ½−E þ BLðL þ 1ÞþWkλαkλn − 2B σnνηnναkλν and rewrite Eq. (D6) as ν pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X δ ξ ξ α Bξ 1 B LðL þ 1Þξ 1 þ B n;1 kλ k0λ0 k0λ0n k k αk11 ¼ − ξk01αk011 − gLM: ðD9Þ k0λ0 P ðkÞ P ðkÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E k0 E ¼ −B LðL þ 1Þξ λg δ 1; ðD3Þ k LM n; χ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In addition it is convenient to introduce the variable as δ δ δ ξ λ λ 1 X where we define n;1 ¼ n;1 þ n;−1, kλ ¼ ð þ ÞVλðkÞ= L ˆ0 gLMχ ¼ ξk1αk11: ðD10Þ Wkλ, Wkλ ¼ ωk þ Bλðλ þ 1Þ,andηnν ¼hLMnjJ jLMνi.In what follows, we show that Eqs. (D2) and (D3) can be solved k in closed form. After multiplying Eq. (D9) with ξk1 and integration over k, First, the angular-momentum coupling term of Eq. (D3) we find is given by R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ dkξ2 =P k 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χ − 1 0 R k1 Eð Þ λ L 2 ¼ B LðL þ Þ ∞ 2 : ðD11Þ σ νη ν ¼ n δ ν þ λðλ þ 1Þ − νðν þ 1Þ 1 ξ n n n 2 þ B 0 dk k1=PEðkÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − ν ν 1 δ × LðL þ Þ ð þ Þ n;νþ1 Finally, this yields the Dyson equation: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ λðλ þ 1Þ − νðν − 1Þ − 1 − Σ 0 2 E þ BLðL þ Þ LðEÞ¼ ; ðD12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − ν ν − 1 δ × LðL þ Þ ð Þ n;ν−1: ðD4Þ where the self-energy is given by ≠ 0 λ 0 R Assuming that VλðkÞ for ¼ , 1 only, we obtain that ∞ dkξ2 =P ðkÞ α 0 λ 0 Σ E B2L L 1 0 R k1 E Eqs. (D2) and (D3) are solved by kλn ¼ for ¼ . Lð Þ¼ ð þ Þ 1 ∞ ξ2 ðD13Þ α α þ B 0 dk k1=PEðkÞ Consequently, gLM, k11, and k10 are the only variational parameters. and For αk10, we obtain pffiffiffi − 1 ω 2 α V1ðkÞ ½ E þ BLðL þ Þþ k þ B k10 ξk1 ¼ 2 : ðD14Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωk þ 2B − B 2LðL þ 1Þðαk11 þ αk1−1Þ¼0: ðD5Þ Furthermore, we absorb the deformation energy Edef, α For the k11 components, we find two identical Eq. (9), which is identical for all the L levels, into equations: the definition of E. Note that if B=uλ ≫ 1, the self-energy

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Σ → 1 0 ˆ † ˆ L BLðLþ Þ and the Dyson equation is solved by E¼ . nðrÞ ≡ hbrbri This means that for weak interactions the impurity levels 1 X −λþλ0 Θ Φ Θ Φ ˆ † ˆ are shifted by the mean-field deformation energy only. ¼ 2 i Yλμð r; rÞYλ0μ0 ð r; rÞhbrλμbrλ0μ0 i: The self-energy of Eq. (D13) can be partially evaluated r λμ analytically. It is convenient to define λ0μ0 ðE1Þ ω ¼ E − BLðL þ 1ÞðD15Þ Σret ω ≡ Using Eq. (A10), we evaluate the partial-wave and to rewrite the retarded self-energy, L ð Þ þ contributions: ΣLðωþi0 Þ,as Z 2 2 χLðωÞ ˆ † ˆ λ−λ0 2 Σret ω 2 1 hb λμb λ0μ0 i¼i r kdk L ð Þ¼ B LðL þ Þ 1 2χ ω ; ðD16Þ r r π þ Lð Þ Z 0 0 0 ˆ † ˆ where × k dk jλðkrÞjλ0 ðk rÞhbkλμbk0λ0μ0 i: ðE2Þ Z ∞ 2 1 χ ω V1ðkÞ Lð Þ¼ dk 2 : ðD17Þ We calculate the expectation values h i with respect 0 ½ω þ 2B Pω 0þ ðkÞ k þi to the states in the transformed frame, jϕi¼Uˆ jψi, ˆ ψ The integrand of χ ðωÞ possesses poles at the momenta k0 where U and j i are given by Eqs. (8) and (10) of the L ˆ † ˆ ω ω 0 b b 0λ0μ0 satisfying k0 ¼ for L ¼ and at the momenta k1;2 main text. Finally, the expectation values of h kλμ k i are ω ω 2 ω ω −2 1 satisfying k1 ¼ þ BL and k2 ¼ BLðL þ Þ for given by states with L>0. Using the relation 1=ðx þ i0þÞ¼ 0 P 1 − πδ † Vλ k Vλ0 k Vλ k ð =xÞ i ðxÞ, this reveals the onset of the scattering ˆ ˆ δ δ 3 ð Þ ð Þ − α ð Þ hbkλμbk0λ0μ0 i¼ μ0 μ00 gLM k0λ00 continua in the spectral function. W λ W 0λ0 W λ k k k For L 0, one finds 0 ¼ Vλ0 k Vλ k − α ð Þ ð Þ α 2 gLM kλ0 þj kλμj : ðE3Þ χ ω πθ ω ζ Wk0λ0 Wkλ Im L¼0ð Þ¼ ð Þ 0; ðD18Þ where 2 ∂ω −1 V1ðk0Þ k ζ0 ; ¼ ω 2 2 ∂ ðD19Þ ½ þ B k k¼k0 [1] L. D. Landau, Über die Bewegung der Elektronen in Kristalgitter, Phys. Z. Sowjetunion 3, 664 (1933). while for L>0, one has [2] L. D. Landau and S. I. Pekar, Effective Mass of a Polaron, Zh. Eksp. Teor. Fiz. 18, 419 (1948) [Ukr. J. Phys. 53,71 π 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2008)]. ImχL>0ðωÞ¼ θðωk1Þ 1 − ζ1 2 1 þ 4LðL þ 1Þ [3] H. Fröhlich, Electrons in Lattice Fields, Adv. Phys. 3, 325 (1954). π 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [4] R. P. Feynman, Slow Electrons in a Polar Crystal, Phys. þ θðωk2Þ 1 þ ζ2; 2 1 þ 4LðL þ 1Þ Rev. 97, 660 (1955). [5] J. Kondo, Resistance Minimum in Dilute Magnetic Alloys, ðD20Þ Prog. Theor. Phys. 32, 37 (1964). [6] P. W. Anderson, Infrared Catastrophe in Fermi Gases with where Local Scattering Potentials, Phys. Rev. Lett. 18, 1049 (1967). 2 ∂ω −1 [7] A. P. Chikkatur, A. Görlitz, D. M. Stamper-Kurn, S. Inouye, V1ðk0Þ k ζ1 2 ¼ : ðD21Þ S. Gupta, and W. Ketterle, Suppression and Enhancement of ; ½ω þ 2B2 ∂k k¼k1;2 Impurity Scattering in a Bose-Einstein Condensate, Phys. Rev. Lett. 85, 483 (2000). χ ω Finally, the real part of Lð Þ follows from the principal [8] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein, value integration. Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms, Phys. Rev. Lett. 102, 230402 (2009). APPENDIX E: DEFORMATION OF THE [9] S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Quantum PHONON DENSITY Transport through a Tonks-Girardeau Gas, Phys. Rev. Lett. 103, 150601 (2009). From Eq. (A6) we obtain the expression for the phonon [10] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, density in the rotating impurity frame: P. Massignan, G. M. Bruun, F. Schreck, and R. Grimm,

011012-11 RICHARD SCHMIDT and MIKHAIL LEMESHKO PHYS. REV. X 6, 011012 (2016)

Metastability and Coherence of Repulsive Polarons in a [28] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonski, Strongly Interacting Fermi Mixture, Nature (London) 485, Quantum Theory of Angular Momentum (World Scientific, 615 (2012). Singapore, 1988). [11] M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. [29] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensa- Feld, and M. Köhl, Attractive and Repulsive Fermi tion (Clarendon, Oxford, 2003). Polarons in Two Dimensions, Nature (London) 485, 619 [30] M. Wagner, Unitary Transformations in Solid State Physics (2012). (Elsevier, New York, 1986). [12] N. Spethmann, F. Kindermann, S. John, C. Weber, D. [31] L. D. Landau and E. M. Lifshitz, Course of Theoretical Meschede, and A. Widera, Dynamics of Single Neutral Physics: Mechanics, 3rd ed. (Butterworth-Heinemann, Impurity Atoms Immersed in an Ultracold Gas, Phys. Rev. London, 1976), Vol. 1. Lett. 109, 235301 (2012). [32] T. D. Lee, F. E. Low, and D. Pines, The Motion of Slow [13] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Electrons in a Polar Crystal, Phys. Rev. 90, 297 (1953). Schauß, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi, [33] M. Girardeau, Motion of an Impurity Particle in a Boson C. Gross, I. Bloch, and S. Kuhr, Quantum Dynamics of a Superfluid, Phys. Fluids 4, 279 (1961). Mobile Spin Impurity, Nat. Phys. 9, 235 (2013). [34] J. T. Devreese, Fröhlich Polarons. Lecture Course Includ- [14] R. Scelle, T. Rentrop, A. Trautmann, T. Schuster, and M. K. ing Detailed Theoretical Derivations, arXiv:1012.4576. Oberthaler, Motional Coherence of Fermions Immersed in a [35] L. C. Biedenharn and J. D. Louck, Angular Momentum Bose Gas, Phys. Rev. Lett. 111, 070401 (2013). in Quantum Physics (Addison-Wesley, Reading, MA, [15] M. Cetina, M. Jag, R. S. Lous, J. T. M. Walraven, R. Grimm, 1981). R. S. Christensen, and G. M. Bruun, Decoherence of Impu- [36] S. P. Rath and R. Schmidt, Field-Theoretical Study of the rities in a Fermi Sea of Ultracold Atoms, Phys. Rev. Lett. Bose Polaron, Phys. Rev. A 88, 053632 (2013). 115, 135302 (2015). [37] R. J. Donnelly and C. F. Barenghi, The Observed Properties [16] P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons, of Liquid Helium at the Saturated Vapor Pressure, J. Phys. Dressed Molecules, and Itinerant Ferromagnetism in Chem. Ref. Data 27, 1217 (1998). Ultracold Fermi Gases, Rep. Prog. Phys. 77, 034401 [38] J. Ulmanis, J. Deiglmayr, M. Repp, R. Wester, and M. (2014). Weidemüller, Ultracold Molecules Formed by Photoasso- [17] M Lemeshko, R. Krems, J. Doyle, and S. Kais, Manipu- ciation: Heteronuclear Dimers, Inelastic Collisions, and lation of Molecules with Electromagnetic Fields, Mol. Phys. Interactions with Ultrashort Laser Pulses, Chem. Rev. 112, 111, 1648 (2013). 4890 (2012). [18] Cold Molecules: Theory, Experiment, Applications, edited [39] T. Köhler, K. Góral, and P. S. Julienne, Production of Cold by R. V. Krems, W. C. Stwalley, and B. Friedrich Molecules via Magnetically Tunable Feshbach Resonances, (CRC Press, Boca Raton, FL, 2009). Rev. Mod. Phys. 78, 1311 (2006). [19] P. F. Bernath, Spectra of Atoms and Molecules, 2nd ed. [40] M. Mark, F. Ferlaino, S. Knoop, J. G. Danzl, T. Kraemer, C. (Oxford, New York, 2005). Chin, H.-C. Nägerl, and R. Grimm, Spectroscopy of [20] H. Lefebvre-Brion and R. W. Field, The Spectra and Ultracold Trapped Cesium Feshbach Molecules, Phys. Dynamics of Diatomic Molecules (Elsevier, New York, Rev. A 76, 042514 (2007). 2004). [41] M. Knap, A. Shashi, Y. Nishida, A. Imambekov, D. A. [21] J. P. Toennies and A. F. Vilesov, Superfluid Helium Drop- Abanin, and E. Demler, Time-Dependent Impurity in Ultra- lets: A Uniquely Cold Nanomatrix for Molecules and cold Fermions: Orthogonality Catastrophe and Beyond, Molecular Complexes, Angew. Chem., Int. Ed. Engl. 43, Phys. Rev. X 2, 041020 (2012). 2622 (2004). [42] R. Schmidt, H. R. Sadeghpour, and E. Demler, A Meso- [22] S. Cradock and A. J. Hinchcliffe, Matrix Isolation: A scopic Rydberg Impurity in an Atomic Quantum Gas, Technique for the Study of Reactive Inorganic Species arXiv:1510.09183. (Cambridge University Press, Cambridge, England, [43] K. Szalewicz, Interplay between Theory and Experiment in 1975). Investigations of Molecules Embedded in Superfluid Helium [23] S. Willitsch, Coulomb-Crystallised Molecular Ions in Nanodroplets, Int. Rev. Phys. Chem. 27, 273 (2008). Traps: Methods, Applications, Prospects, Int. Rev. Phys. [44] D. Pentlehner, J. H. Nielsen, A. Slenczka, K. Mølmer, and Chem. 31, 175 (2012). H. Stapelfeldt, Impulsive Laser Induced Alignment of [24] G. D. Mahan, Many-Particle Physics, Physics of Solids and Molecules Dissolved in Helium Nanodroplets, Phys. Rev. Liquids (Plenum, New York, 1990). Lett. 110, 093002 (2013). [25] C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, [45] D. S. Jin and J. Ye, Introduction to Ultracold Molecules: Creation of Polar and Nonpolar Ultra-Long-Range Ryd- New Frontiers in Quantum and Chemical Physics, Chem. berg Molecules, Phys. Rev. Lett. 85, 2458 (2000). Rev. 112, 4801 (2012). [26] J. B. Balewski, A. T. Krupp, A. Gaj, D. Peter, H. P. Buchler, [46] V. Bendkowsky, B. Butscher, J. Nipper, J. P. Shaffer, R. Low, S. Hofferberth, and T. Pfau, Coupling a Single R. Low, and T. Pfau, Observation of Ultralong-Range Electron to a Bose-Einstein Condensate, Nature (London) Rydberg Molecules, Nature (London) 458, 1005 (2009). 502, 664 (2013). [47] M. A. Bellos, R. Carollo, J. Banerjee, E. E. Eyler, P. L. [27] R. Schmidt and M. Lemeshko, Rotation of Quantum Gould, and W. C. Stwalley, Excitation of Weakly Bound Impurities in the Presence of a Many-Body Environment, Molecules to Trilobitelike Rydberg States, Phys. Rev. Lett. Phys. Rev. Lett. 114, 203001 (2015). 111, 053001 (2013).

011012-12 DEFORMATION OF A QUANTUM MANY-PARTICLE SYSTEM … PHYS. REV. X 6, 011012 (2016)

[48] A. T. Krupp, A. Gaj, J. B. Balewski, P.Ilzhöfer, S. Hofferberth, [51] S. Kehrein, The Flow Equation Approach to Many-Particle R. Löw, T. Pfau, M. Kurz, and P.Schmelcher, Alignment of D- Systems (Springer, New York, 2006). StateRydbergMolecules,Phys.Rev.Lett.112, 143008 (2014). [52] Encyclopedia of Chemical Physics and Physical Chemistry, [49] F. Chevy, Universal Phase Diagram of a Strongly Interact- edited by J. H. Moore and N. D. Spencer (CRC Press, Boca ing Fermi Gas with Unbalanced Spin Populations, Phys. Raton, FL, 2001). Rev. A 74, 063628 (2006). [53] Handbook of Mathematical Functions, edited by [50] F. Wegner, Flow-Equations for Hamiltonians, Ann. Phys. M. Abramowitz and I. A. Stegun (Dover, New York, (Berlin) 506, 77 (1994). 1972).

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