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Condensed Matter Physics

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Full-Text Articles in Quantum Physics

Stereographic Visualization Of Bose-Einstein Condensate Clouds To Measure The Gravitational Constant, Ed Wesley Wells Jan 2016

Stereographic Visualization Of Bose-Einstein Condensate Clouds To Measure The Gravitational Constant, Ed Wesley Wells

College of Graduate Studies: Theses & Dissertations

This thesis describes a set of tools that can be used for the rapid design of atom interferometer schemes suitable for measuring Newton's Universal Gravitation constant also known as "Big G". This tool set is especially applicable to Bose--Einstein--condensed systems present in NASA's Cold Atom Laboratory experiment to be deployed to the International Space Station in 2017. These tools include a method of approximating the solutions of the nonlinear Schrödinger or Gross--Pitaevskii equation (GPE) using the Lagrangian Variational Method. They also include a set of software tools for translating the approximate solutions of the GPE into images of the optical …

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First Results Of Magnetic Field Penetration Measurements Of Multilayer Sis Structures, O. B. Malyshev, L. Gurran, R. Valizadeh, S. Pattalwar, N. Pattalwar, K. D. Dumbell, A. Gurevich Jan 2016

First Results Of Magnetic Field Penetration Measurements Of Multilayer Sis Structures, O. B. Malyshev, L. Gurran, R. Valizadeh, S. Pattalwar, N. Pattalwar, K. D. Dumbell, A. Gurevich

Physics Faculty Publications

The performance of superconducting RF cavities made of bulk Nb is limited by a breakdown field of Bp ≈200 mT, close to the superheating field for Nb. A potentially promising solution to enhance the breakdown field of the SRF cavities beyond the intrinsic limits of Nb is a multilayer coating suggested in [1]. In the simplest case, such a multilayer may be a superconductor-insulator-superconductor (S-I-S) coating, for example, bulk niobium (S) coated with a thin film of insulator (I) followed by a thin layer of another superconductor (S) which could be e.g. dirty niobium [2]. Here we report the …

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Hohenberg-Kohn Theorems In Electrostatic And Uniform Magnetostatic Fields, Xiao-Yin Pan, Viraht Sahni Nov 2015

Hohenberg-Kohn Theorems In Electrostatic And Uniform Magnetostatic Fields, Xiao-Yin Pan, Viraht Sahni

Publications and Research

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby, a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle …

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Hohenberg-Kohn Theorems In Electrostatic And Uniform Magnetostatic Fields, Xiao-Yin Pan, Viraht Sahni Oct 2015

Hohenberg-Kohn Theorems In Electrostatic And Uniform Magnetostatic Fields, Xiao-Yin Pan, Viraht Sahni

Publications and Research

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle …

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Experimental Design And Construction For Critical Velocity Measurement In Spin-Orbit Coupled Bose-Einstein Condensates, Ting-Wei Hsu, Yong P. Chen Aug 2015

Experimental Design And Construction For Critical Velocity Measurement In Spin-Orbit Coupled Bose-Einstein Condensates, Ting-Wei Hsu, Yong P. Chen

The Summer Undergraduate Research Fellowship (SURF) Symposium

Quantum simulation using ultra-cold atoms, such as Bose-Einstein Condensates (BECs), offers a very flexible and well controlled environment to simulate physics in different systems. For example, to simulate the effects of spin orbit coupling (SOC) on electrons in solid state systems, we can make a SOC BEC which mimics the behavior of SOC electrons. The goal of this project is to see how the superfluid property of BECs change in the presence of SOC. In particular, we plan to measure the critical velocity of an 87Rb BEC with and without SOC by stirring it with a laser. This laser needs …

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Static And Dynamical Properties Of Ferroelectrics And Related Material In Bulk And Nanostructure Forms, Zhigang Gui Jul 2015

Static And Dynamical Properties Of Ferroelectrics And Related Material In Bulk And Nanostructure Forms, Zhigang Gui

Graduate Theses and Dissertations

Ferroelectrics (FE) and multiferroics (MFE) have attracted a lot of attentions due to their rich and novel properties. Studies towards FE and MFE are of both fundamental and technological importance. We use a first-principles-based effective Hamiltonian method, conventional ab-initio packages and linear-scale three-dimension fragment method to investigate several important issues about FE and MFE. Tuning the properties of FE and MFE films are essential for miniaturized device applications, which can be realized through epitaxial strain and growth direction. In this dissertation, we use the effective Hamiltonian method to study (i) BaTiO 3 films grown along the (110) pseudocubic direction on …

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Mirror Buckling Transitions In Freestanding Graphene Membranes Induced Through Scanning Tunneling Microscopy, James Kevin Schoelz May 2015

Mirror Buckling Transitions In Freestanding Graphene Membranes Induced Through Scanning Tunneling Microscopy, James Kevin Schoelz

Graduate Theses and Dissertations

Graphene has the ability to provide for a technological revolution. First isolated and characterized in 2004, this material shows promise in the field of flexible electronics. The electronic properties of graphene can be tuned by controlling the shape of the membrane. Of particular interest in this endeavor are the thermal ripples in graphene membranes. Years of theoretical work by such luminaries as Lev Landau, Rudolf Peierls, David Mermin and Herbert Wagner have established that 2D crystals should not be thermodynamically stable. Experimental research on thin films has supported this finding. Yet graphene exists, and freestanding graphene films have been grown …

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Observation Of Antiferromagnetic Correlations In The Hubbard Model With Ultracold Atoms, Russell Hart, Pedro Duarte, Tsung-Lin Yang, Xinxing Liu, Thereza Paiva, Ehsan Khatami, Richard Scalettar, Nandini Trivedi, David Huse, Randall Hulet Mar 2015

Observation Of Antiferromagnetic Correlations In The Hubbard Model With Ultracold Atoms, Russell Hart, Pedro Duarte, Tsung-Lin Yang, Xinxing Liu, Thereza Paiva, Ehsan Khatami, Richard Scalettar, Nandini Trivedi, David Huse, Randall Hulet

Faculty Publications

Ultracold atoms in optical lattices have great potential to contribute to a better understanding of some of the most important issues in many-body physics, such as high-temperature superconductivity. The Hubbard model—a simplified representation of fermions moving on a periodic lattice—is thought to describe the essential details of copper oxide superconductivity. This model describes many of the features shared by the copper oxides, including an interaction-driven Mott insulating state and an antiferromagnetic (AFM) state. Optical lattices filled with a two-spin-component Fermi gas of ultracold atoms can faithfully realize the Hubbard model with readily tunable parameters, and thus provide a platform for …

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Multiscale Examination And Modeling Of Electron Transport In Nanoscale Materials And Devices, Douglas R. Banyai Jan 2015

Multiscale Examination And Modeling Of Electron Transport In Nanoscale Materials And Devices, Douglas R. Banyai

Dissertations, Master's Theses and Master's Reports - Open

For half a century the integrated circuits (ICs) that make up the heart of electronic devices have been steadily improving by shrinking at an exponential rate. However, as the current crop of ICs get smaller and the insulating layers involved become thinner, electrons leak through due to quantum mechanical tunneling. This is one of several issues which will bring an end to this incredible streak of exponential improvement of this type of transistor device, after which future improvements will have to come from employing fundamentally different transistor architecture rather than fine tuning and miniaturizing the metal-oxide-semiconductor field effect transistors (MOSFETs) …

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Computing And The Electrical Transport Properties Of Coupled Quantum Networks, Casey Andrew Cain Jan 2015

Computing And The Electrical Transport Properties Of Coupled Quantum Networks, Casey Andrew Cain

Doctoral Dissertations

In this dissertation a number of investigations were conducted on ballistic quantum networks in the mesoscopic range. In this regime, the wave nature of electron transport under the influence of transverse magnetic fields leads to interesting applications for digital logic and computing circuits. The work specifically looks at characterizing a few main areas that would be of interest to experimentalists who are working in nanostructure devices, and is organized as a series of papers. The first paper analyzes scaling relations and normal mode charge distributions for such circuits in both isolated and open (terminals attached) form. The second paper compares …

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Wigner High-Electron-Correlation Regime Of Nonuniform Density Systems: A Quantal-Density-Functional-Theory Study, Douglas Achan, Lou Massa, Viraht Sahni Aug 2014

Wigner High-Electron-Correlation Regime Of Nonuniform Density Systems: A Quantal-Density-Functional-Theory Study, Douglas Achan, Lou Massa, Viraht Sahni

Publications and Research

The Wigner regime of a system of electrons in an external field is characterized by a low electron density and a high electron-interaction energy relative to the kinetic energy. The low-correlation regime is in turn described by a high electron density and an electron-interaction energy smaller than the kinetic energy. The Wigner regime of a nonuniform-electron-density system is investigated via quantal density functional theory (QDFT). Within QDFT, the contributions of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects are separately delineated and explicitly defined. The nonuniform-electron-density system studied is that of the Hooke's atom in …

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Environmental Testing Of Lasers For Jpl's Cold Atom Laboratory, Carey L. Baxter Aug 2014

Environmental Testing Of Lasers For Jpl's Cold Atom Laboratory, Carey L. Baxter

STAR Program Research Presentations

NASA’s Cold Atom Lab (CAL) is a multi-user facility designed to study ultra-cold quantum gases in the microgravity environment of the International Space Station (ISS). One of the main goals of CAL is to explore the unknown territory of extremely low temperatures—possibly as low as the picokelvin range!—where new and fascinating quantum phenomena can be observed. At such temperatures matter stops behaving as particles and instead becomes macroscopic matter waves. CAL will be remotely controlled to perform a multitude of experiments and is scheduled to launch in 2016. In order to anticipate problems that might occur during and post-launch, including …

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Wave Function For Harmonically Confined Electrons In Time-Dependent Electric And Magnetostatic Fields, Hong-Ming Zhu, Jin-Wang Chen, Xiao-Yin Pan, Viraht Sahni Jan 2014

Wave Function For Harmonically Confined Electrons In Time-Dependent Electric And Magnetostatic Fields, Hong-Ming Zhu, Jin-Wang Chen, Xiao-Yin Pan, Viraht Sahni

Publications and Research

We derive via the interaction “representation” the many-body wave function for harmonically confined electrons in the presence of a magnetostatic field and perturbed by a spatially homogeneous time-dependent electric field—the Generalized Kohn Theorem (GKT) wave function. In the absence of the harmonic confinement – the uniform electron gas – the GKT wave function reduces to the Kohn Theorem wave function. Without the magnetostatic field, the GKTwave function is the Harmonic Potential Theorem wave function. We further prove the validity of the connection between the GKT wave function derived and the system in an accelerated frame of reference. Finally, we provide …

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Demonstrating Entanglement By Testing Bell's Theorem In Majorana Wires, David E. Drummond, Alexey Kovalev, Chang-Yu Hou, Kirill Shtengel, Leonid P. Pryadko Jan 2014

Demonstrating Entanglement By Testing Bell's Theorem In Majorana Wires, David E. Drummond, Alexey Kovalev, Chang-Yu Hou, Kirill Shtengel, Leonid P. Pryadko

Department of Physics and Astronomy: Faculty Publications

We propose an experiment that would establish the entanglement of Majorana zero modes in semiconductor nanowires by testing the Bell and Clauser-Horne-Shimony-Holt inequalities. Our proposal is viable with realistic system parameters, simple “keyboard” gating, and projective measurement. Theoretical models and simulation results indicate entanglement can be demonstrated with moderately accurate gate operations. In addition to providing further evidence for the existence of the Majorana bound states, our proposal could be used as an experimental stepping stone to more complicated braiding experiments.

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Wave Function For Time-Dependent Harmonically Confined Electrons In A Time-Dependent Electric Field, Yu-Qi Li, Xiao-Yin Pan, Viraht Sahni Sep 2013

Wave Function For Time-Dependent Harmonically Confined Electrons In A Time-Dependent Electric Field, Yu-Qi Li, Xiao-Yin Pan, Viraht Sahni

Publications and Research

The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the Harmonic Potential Theorem wave function for the case of the time-independent harmonic confining potential.

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Multiband S -Wave Topological Superconductors: Role Of Dimensionality And Magnetic Field Response, Shusa Deng, Gerardo Ortiz, Lorenza Viola May 2013

Multiband S -Wave Topological Superconductors: Role Of Dimensionality And Magnetic Field Response, Shusa Deng, Gerardo Ortiz, Lorenza Viola

Dartmouth Scholarship

We further investigate a class of time-reversal-invariant two-band s-wave topological superconductors introduced earlier [Deng, Viola, and Ortiz, Phys. Rev. Lett. 108, 036803 (2012)]. Provided that a sign reversal between the two superconducting pairing gaps is realized, the topological phase diagram can be determined exactly (within mean field) in one and two dimensions as well as in three dimensions upon restricting to the excitation spectrum of time-reversal-invariant momentum modes. We show how, in the presence of time-reversal symmetry, Z2 invariants that distinguish between trivial and nontrivial quantum phases can be constructed by considering only one of the Kramers’ sectors …

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Density Functional Theory And The Calculation Of Tcmg2O4 Spinel Lattice Parameters, Jon Karlo Macias Mar 2013

Density Functional Theory And The Calculation Of Tcmg2O4 Spinel Lattice Parameters, Jon Karlo Macias

Physics

The cohesive energy, lattice constant and bulk modulus of two simple HCP crystal structures of magnesium and technetium were calculated using the vienna ab initio simulation package (VASP) which is based on density functional theory (DFT). The same properties were determined for TcMg2O4 spinel. The theoretical results of the lattice constant of the pure crystals were compared to experimental results and found to be in excellent agreement with a difference of less than 2%. The results for the lattice constant of the TcMg2O4 spinel were found to be in excellent agreement as well with …

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Comment On "Density And Physical Current Density Functional Theory", Xiao-Yin Pan, Viraht Sahni Jan 2013

Comment On "Density And Physical Current Density Functional Theory", Xiao-Yin Pan, Viraht Sahni

Publications and Research

In this letter to the editor, the authors comment on an earlier article they had published, "Density and Physical Current Density Functional Theory" (Pan, X.-Y. and Sahni, V. (2010), Density and physical current density functional theory. Int. J. Quantum Chem., 110: 2833–2843. doi: 10.1002/qua.22862).

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Electron–Phonon Coupling And Structural Phase Transitions On Au/Mo(112), Keisuke Fukutani Nov 2012

Electron–Phonon Coupling And Structural Phase Transitions On Au/Mo(112), Keisuke Fukutani

Department of Physics and Astronomy: Dissertations, Theses, and Student Research

The electronic structures, many-body interactions and Fermi surface topologies of Au/Mo(112) were investigated in detail and were found to play important roles in the newly discovered order-disorder structural phase transition of the system. First, the high-resolution angle-resolved photoemission spectroscopy was utilized to characterize the electronic band structure of Mo(112) in far greater details than before. This elucidated the existence of several surface-derived states and their dispersion relations in high precisions near the Fermi level, as well as the symmetries of the bulk and surface electronic states, which are in good quantitative agreement with the ab-initio calculations. Such thorough understanding of …

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Hohenberg-Kohn Theorem Including Electron Spin, Xiao-Yin Pan, Viraht Sahni Oct 2012

Hohenberg-Kohn Theorem Including Electron Spin, Xiao-Yin Pan, Viraht Sahni

Publications and Research

The Hohenberg-Kohn theorem is generalized to the case of a finite system of N electrons in external electrostatic epsilon(r) = -del nu(r) and magnetostatic B(r) = del x A(r) fields in which the interaction of the latter with both the orbital and spin angular momentum is considered. For a nondegenerate ground state a bijective relationship is proved between the gauge invariant density rho(r) and physical current density j(r) and the potentials {nu(r), A(r)}. The possible many-to-one relationship between the potentials {v(r), A(r)} and the wave function is explicitly accounted for in the proof. With the knowledge that the basic variables …

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Nanomechanical Resonator Coupled Linearly Via Its Momentum To A Quantum Point Contact, Latchezar L. Benatov, Miles P. Blencowe Aug 2012

Nanomechanical Resonator Coupled Linearly Via Its Momentum To A Quantum Point Contact, Latchezar L. Benatov, Miles P. Blencowe

Dartmouth Scholarship

We use a Born-Markov approximated master equation approach to study the symmetrized-in-frequency current noise spectrum and the oscillator steady state of a nanoelectromechanical system where a nanoscale resonator is coupled linearly via its momentum to a quantum point contact (QPC). Our current noise spectra exhibit clear signatures of the quantum correlations between the QPC current and the back-action force on the oscillator at a value of the relative tunneling phase (η=−π/2) where such correlations are expected to be maximized. We also show that the steady state of the oscillator obeys a classical Fokker-Planck equation, but can experience thermomechanical noise squeezing …

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The Effect Of Polarization And Ingan Quantum Well Shape In Multiple Quantum Well Light Emitting Diode Heterostructures, Patrick M. Mcbride Jun 2012

The Effect Of Polarization And Ingan Quantum Well Shape In Multiple Quantum Well Light Emitting Diode Heterostructures, Patrick M. Mcbride

Master's Theses

Previous research in InGaN/GaN light emitting diodes (LEDs) employing semi-classical drift-diffusion models has used reduced polarization constants without much physical explanantion. This paper investigates possible physical explanations for this effective polarization reduction in InGaN LEDs through the use of the simulation software SiLENSe. One major problem of current LED simulations is the assumption of perfectly discrete transitions between the quantum well (QW) and blocking layers when experiments have shown this to not be the case. The In concentration profile within InGaN multiple quantum well (MQW) devices shows much smoother and delayed transitions indicative of indium diffusion and drift during …

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Hohenberg-Kohn And Percus-Levy-Lieb Proofs Of Density-Functional Theory, Viraht Sahni, Xiao-Yin Pan May 2012

Hohenberg-Kohn And Percus-Levy-Lieb Proofs Of Density-Functional Theory, Viraht Sahni, Xiao-Yin Pan

Publications and Research

The premise of density-functional theory is that knowledge of the ground-state density uniquely determines the Hamiltonian, and thereby, via solution of the corresponding time-independent Schrodinger equation, all the properties of the system. The density therefore constitutes a basic variable of quantum mechanics. There are at present two paths from the density to the Hamiltonian: the Hohenberg and Kohn proof of the bijectivity between the external potential and the basic variable, and the Percus, Levy, and Lieb constrained-search proof. We argue the Hohenberg- and Kohn-type proof to be the more fundamental, and that this is the case in general when both …

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Light Vs. Quantum Gravity, Irving Martinez^* Apr 2012

Light Vs. Quantum Gravity, Irving Martinez^*

COURI Symposium Abstracts, Spring 2012

No abstract provided.

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Demonstration Of The Gunnarsson-Lundqvist Theorem And The Multiplicity Of Potentials For Excited States, Yu-Qi Li, Xiao-Yin Pan, Biao Li, Viraht Sahni Mar 2012

Demonstration Of The Gunnarsson-Lundqvist Theorem And The Multiplicity Of Potentials For Excited States, Yu-Qi Li, Xiao-Yin Pan, Biao Li, Viraht Sahni

Publications and Research

The Gunnarsson-Lundqvist (GL) theorem of density functional theory states that there is a one-to-one relationship between the density of the lowest nondegenerate excited state of a given symmetry and the external potential. As a consequence, knowledge of this excited state density determines the external potential uniquely. [The GL theorem is the equivalent for such excited states of theHohenberg-Kohn (HK) theorem for nondegenerate ground states.] For other excited states, there is no equivalent of the GL or HK theorem. For these states, there thus exist multiple potentials that generate the excited-state density. We show, by example, the satisfaction that the GL …

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Majorana Modes In Time-Reversal Invariant S -Wave Topological Superconductors, Shusa Deng, Lorenza Viola, Gerardo Ortiz Jan 2012

Majorana Modes In Time-Reversal Invariant S -Wave Topological Superconductors, Shusa Deng, Lorenza Viola, Gerardo Ortiz

Dartmouth Scholarship

We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes. The multiband character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.

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Wave-Function Functionals For The Density, Marlina Slamet, Xiao-Yin Pan, Viraht Sahni Nov 2011

Wave-Function Functionals For The Density, Marlina Slamet, Xiao-Yin Pan, Viraht Sahni

Publications and Research

We extend the idea of the constrained-search variational method for the construction of wave-function functionals psi[chi] of functions chi. The search is constrained to those functions chi such that psi[chi] reproduces the density rho(r) while simultaneously leading to an upper bound to the energy. The functionals are thereby normalized and automatically satisfy the electron-nucleus coalescence condition. The functionals psi[chi] are also constructed to satisfy the electron-electron coalescence condition. The method is applied to the ground state of the helium atom to construct functionals psi[chi] that reproduce the density as given by the Kinoshita correlated wave function. The expectation of single-particle …

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Unitary-Quantum-Lattice Algorithm For Two-Dimensional Quantum Turbulence, Bo Zhang, George Vahala, Linda L. Vahala, Min Soe Jan 2011

Unitary-Quantum-Lattice Algorithm For Two-Dimensional Quantum Turbulence, Bo Zhang, George Vahala, Linda L. Vahala, Min Soe

Electrical & Computer Engineering Faculty Publications

Quantum vortex structures and energy cascades are examined for two-dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice algorithm, is employed to simulate the Bose-Einstein condensate governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincare recurrence time. It is demonstrated that such short recurrence times are destroyed by stronger nonlinear interaction. The similar loss of Poincare recurrence is also seen in the 3D GP equation. Various initial conditions are considered in an attempt to discern if 2D QT exhibits inverse …

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Band Structure Of Fe Based Superconductors., Teh Yee Lin Jan 2010

Band Structure Of Fe Based Superconductors., Teh Yee Lin

Student Works (2010-2019)

Several iron containing superconductors are discovered in 2008. The discovery introduced a new class of high-temperature superconductors, which are iron-based. The iron-based superconductors do not resemble conventional superconductors and have properties on their own which differ from the cuprates. We have studied the iron selenium, FeSe compound by removing some Se atoms from the lattice forming Fe:Se = 2:1 and 8:6 and also by replacing the Se vacancy with tellurium, samarium and oxygen atoms forming Fe:Se:Te = 2:1:1 and 8:6:2, Fe:Se:O = 8:6:2 and 8:4:2, FeSeSmO, FeSeTeO, Fe2SeSm2O2, FeSeTeSmO and FeTeSmO. The band structures for each lattice are calculated by …

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Damping And Decoherence Of A Nanomechanical Resonator Due To A Few Two-Level Systems, Laura G. Remus, Miles P. Blencowe, Yukihiro Tanaka Nov 2009

Damping And Decoherence Of A Nanomechanical Resonator Due To A Few Two-Level Systems, Laura G. Remus, Miles P. Blencowe, Yukihiro Tanaka

Dartmouth Scholarship

We consider a quantum model of a nanomechanical flexing beam resonator interacting with a bath comprising a few damped tunneling two-level systems. In contrast with a resonator interacting bilinearly with an ohmic free oscillator bath (modeling clamping loss, for example), the mechanical resonator damping is amplitude dependent, while the decoherence of quantum superpositions of mechanical position states depends only weakly on their spatial separation.

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