Dynamics of Finite Momentum Bose Polarons MASSACHUSETTS INSTITI ITF OF TECHNOLOGY Kushal Seetharam JUN 1 3 B.S.E., Duke University (2014) LIBRARIES Submitted to the ARCHIVES Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 @2019. Kushal Seetharam. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature of Author: ..... Signature redacted ...................... Department of Electrical Engineering and Computer Science Signature redacted May 23, 2019 C ertified by :............ ............................. Eugene Demler Professor of Physics Supervisor Signature redacted Thesis A ccepted by:............. ........................... Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Dynamics of Finite Momentum Bose Polarons by Kushal Seetharam Submitted to the Department of Electrical Engineering and Computer Science on May 23, 2019 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Electrical Engineering and Computer Science Abstract We study the behavior of a finite-momentum impurity immersed in a weakly interacting Bose-Einstein con- densate (BEC) of ultra-cold atoms near an interspecies Feshbach resonance. Using the time-dependent variational approach, we study both ground state properties and quench dynamics of the system after a sudden immersion of the impurity into the BEC. We find evidence of a ground state phase transition when the impurity has a velocity greater than that of the sound velocity (Landau critical velocity) associated with Bogoliubov quasiparticle excitations of the BEC. As we cross from the subsonic regime to the supersonic regime, we get a breakdown of the polaron quasiparticle description of the system, emission of Cherenkov phonons, and a sound-like dispersion of the system. This phase transition manifests in several ways during real-time dynamics of the system and showcases a rich interplay between polaronic physics and Cherenkov physics. One key signature, dissipation in the the supersonic regime, can be seen in experimental protocols where the impurity and BEC are made to move relative to each other through an external force. We sug- gest one such experimental protocol to measure the polaron's effective mass as long as the impurities are subsonic. While the measurement scheme becomes error-prone due to dissipation when the impurities are allowed to become supersonic, this sensitivity suggests a way to experimentally probe the Cherenkov physics of supersonic impurities immersed in a BEC. Thesis Supervisor: Eugene Demler Title: Professor of Physics 3 4 Contents 1 Introduction 7 2 The System 9 2.1 Microscopic Model ......... .......................................... 9 2.2 Equations of M otion ................ ................ ......... 14 2.3 Observables and Distribution Functions ............... ............... 15 3 Ground State 17 4 Real-time Dynamics 20 5 Experimental Connection 25 6 Summary 26 A Local Density Approximation (LDA) 27 A.1 External Potentials and Forces ............. ................ ...... 27 A.2 Inhomogeneous BEC .............. ........................... 27 A.3 Impurity Potentials/Forces Included in Simulation ....................... 28 A.3.1 Impurity's confining potential ................... 28 A.3.2 Polaron energy potential . ........... .......... .......... ... 28 5 List of Figures 3.1 Ground state energy. The energy is initially quadratic (subsonic region) and then becomes linear at a critical momentum (Cherenkov region). (a) Ground state energy. (b) First deriva- tive of ground state energy. (c) Second derivative of ground state energy. (d) Average impurity velocity and polaron velocity. Both are equal to each other and the speed of sound regardless of interaction strength. (e) Mass enhancement of polaron. The effective mass increases with interaction strength. .......... .............................. 17 3.2 Phonon Number. (a) After the transition, there is a large increase in number of phonon excitations. (b) If we increase the infrared cutoff of the momentum grid in numerics, the number of phonons decreases as the system puts weight in the lowest energy modes accessible, which now occur at larger momentum. ............................... 18 3.3 Impurity speed distribution. (a) Example distribution showing coherent delta peak (red line) and incoherent part (black). The blue line shows the full-width half-max (FWHM) of the incoherent part. (b) Quasiparticle residue. Drops to zero at the transition. (c) FWHM of incoherent part. The sharp dip at the transition indicates that the incoherent part of the impurity speed distribution gets sharply peaked at the critical momentum ........... 18 3.4 Ground state phase diagram. The critical momentum increases with interaction strength and is related to polaronic mass enhancement ........................... 19 3.5 Quasiparticle residue in subsonic region. The residue drops towards zero as we increase interaction strength. .......... .............................. 20 4.1 Observables. (a) Loschmidt echo. Curves from the subsonic region saturate to a finite value at long times. Curves in the Cherenkov region decay to zero as a power-law at long times. (b) Average impurity momentum. Curves from the subsonic region saturate to different subsonic values at long times. Curves in the Cherenkov region all approach the speed of sound at long tim es. ....... ............... ................ ........... 21 4.2 Exponents characterizing long-time behavior observables. The power-law decay only starts after a critical momentum which depends on interaction strength. ............ 22 4.3 Final impurity speed. For weak interactions, the impurity ends up at the speed of sound if we start it supersonic. For strong interactions, the impurity will always end up subsonic regardless of its initial velocity. ................ ................... 22 4.4 Final Loschmidt echo. For all interaction strengths, there is a range of initial impurity velocities where the the long-time dynamical overlap is nonzero followed by a range of ini- tial impurity velocities where the long-time dynamical overlap is zero. The location of the discontinuous transition between these two regions depends on interaction strength and mass ratio............... ... ........................................... 23 5.1 Effective mass protocol fully in subsonic region. (a) Average impurity speed. The impurity velocity increases linearly while the constant external force is applied. (b) Mass enhancement. We 'measure' a larger effective mass as we increase interaction strength. .... 25 5.2 Effective mass protocol partially in supersonic region. (a) Average impurity speed. We see curvature corresponding to dissipation as the external force pushes the impurity past the speed of sound. (b) Mass enhancement. There is error in the 'measured' effective mass even at weak interactions. .. ............... ................ ..... 26 6 1 Introduction The non-equilibrium dynamics of quantum many-body systems is amongst the most actively researched areas of modern physics. One of the most prototypical problems that underlies the physics behind several such systems is that of the quantum impurity, where an impurity degree of freedom interacts with a surrounding many-body system. In certain cases, we can describe this scenario using the notion of a polaron quasiparticle which is adiabatically connected to the free impurity. The polaron describes an impurity that has been dressed by a cloud of excitations of the surrounding bath which, in equilibrium, results in a renormalization of its mass, energy, and other properties. A toy analogy can be made to a ball rolling around a rubber sheet; the ball creates a deformation in the sheet that moves along with the ball and modifies its motion. The ball represents the impurity and the deformation in the sheet represents the cloud of particles in the many-body bath that dresses the impurity. Depending on whether the excitations dressing the impurity are bosonic or fermionic, we call the resulting quasiparticle a Bose polaron or a Fermi polaron. Landau introduced the original manifestation of the polaron in 1933 when he examined an electron getting trapped by a crystal lattice [18]. The codification of an electron trapped by a crystal lattice as a phonon-dressed quasiparticle called a polaron was done by Pekar in 1946 and then further developed in a joint paper between Pekar and Landau in 1948 [26, 17]. While the initial treatment of the polaron was in the strong-coupling regime, it was soon extended to the weak coupling regime by Frohlich [5]. Since these original investigations, it has been discovered that polaronic physics is related to electron mobility in semiconductors, charge transport in organic solar cells, mechanisms behind high- Tc superconductivity, and many other salient systems [2, 6, 13, 27]. Polarons also serve as an archetype upon which we can build an understanding of other types of quantum impurity problems [29]. In modern times, ultracold atomic gases have provided an experimental platform that allow a much more detailed probing of polaronic physics. Control of host atom species allows specification of fermionic or bosonic statistics