Philip C Stamp

Professor

Relevant Degree Programs

 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Excitonic modes and phonons in biological molecules (2018)

There are two kinds of environmental modes in open quantum systems: the delocalized modeswhich can typically be modeled by "oscillator bath" models and the localized modes which can be mapped to "spin bath" modes. To understand the quantum phenomena in photosynthetic energy transfer, we at first conduct thorough studies of the proper modeling of light harvesting molecules as well as their interactions with the central system. These modes can couple to the system by either modulating the on-site energy (Holstein coupling) or modulating the hopping amplitude (Peierls coupling). Only the Holstein couplings of delocalized modes have been extensively studied. The importance of other types of couplings is rarely discussed in the literature. For the spin bath, we study a particle hopping around a general lattice, coupled to a spin bath. Analytical results are found for the dynamics of the influence functional and for the reduced density matrix of the particle in various parameter regimes. Spin baths behave qualitatively differently from oscillator baths and dissipation and decoherence happen independently in different parameter regimes. For the Peierls couplings, we start with a dimer model for light harvesting molecules, which contains a reaction center and both types of phonon couplings. We find that the effect of Peierls type coupling on the transfer rate can be significant even when it is not noticeable in the spectrum. Our study suggests that Peierls couplings cannot be easily neglected in light harvesting molecules in which the energy difference between the sites is usually much larger than the hopping amplitude. We apply our method to a real light harvesting model. Although we do not have much detailed information of the Peierls couplings in vivo, we find that vibrational phonons can affect the path-selecting of the central particles as well as increasing the transfer rate.

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Fluctuations and phase transitions in quantum Ising systems (2017)

The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.

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Equation of motion of a quantum vortex (2011)

Quantum vortices are an important excitation in a wide variety of systems. They are a basic ingredient in our understanding of superfluids and superconductors --- indeed, the very definition of these phases relies heavily on the existence of quantum vortices. Despite this,the equation of motion of a quantum vortex remains controversial. In this thesis, we derive the two dimensional equation of motion of a vortex in superfluid helium, and also discuss adapting our derivation for a vortex in a ferromagnet dot.In addition to the undisputed superfluid Magnus force and vortex inertia, we derive the controversial Iordanskii force, a pair of memory forces, and the associated fluctuating force. The memory forces include a generalization of the usual longitudinal damping force, a frequency dependent inertial force, and a higher order, frequency dependent correction to the Iordanskii force. We quantify the slow limit in which these forces become local or frequency independent. In a superfluid, the motion is frequency dependent, manifest primarily through a suppression of the damping rate of the vortex motion. Magnetic vortex motion is typically at much lower frequencies and the memory effects can so far be ignored. Our analysis involves a careful separation of vortex and quasiparticle degrees of freedom. We prove definitively that there are no interactions that are first order in quasiparticle variables: therefore, all resulting forces on the vortex resulting from interactions with the quasiparticles are temperature-dependent. We calculate the vortex influence functional resulting from a velocity-dependent quadratic coupling with perturbed quasiparticles that have already been perturbed by the presence of the static vortex. From the vortex influence functional and the bare vortex action, we derive the full quantum equation of motion of a vortex.We relate our arguments and results to the wealth of ideas presented in the superfluid and magnetic literature. We discuss extensions of this work: on including normal fluid viscosity, dynamics of a multiple vortex configuration, to a finite system, and to a three-dimensional system.

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Polaron physics beyond the Holstein model (2011)

Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.

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Master's Student Supervision (2010 - 2018)
Eikonal analysis of linearized quantum gravity : a functional approach (2017)

The low energy effective theory of quantized gravity is currently our most successful attempt at unifying general relativity and quantum mechanics. It is expected to serve as the universal low energy limit of any future microscopic theory of quantum gravitation, so it is crucial to properly understand its low frequency, long wavelength, "infrared" limit. However, this effective theory suffers from the same kind of infrared divergences as theories like quantum electrodynamics. It is the aim of this work to characterize these divergences and isolate the infrared behavior of quantum gravity using functional methods. We begin with a review of infrared divergences, and how they are treated in QED. This includes a brief overview of the known applications of functional methods to the problem. We then discuss the construction of the effective field theory of quantum gravity in the linearized limit, coupled to scalar matter. Proceeding to the main body of the work, we employ functional techniques to derive the form of the scalar propagator after soft graviton radiation is integrated out. An eikonal form for the generating functional of the theory is then presented. In the final chapter, we use this generating functional to derive the soft graviton theorem and the eikonal form of the two-body scalar scattering amplitude. The result is a concise derivation of multiple known results, as well as a demonstration of the factorization of soft graviton radiation against the eikonal amplitude. We conclude with some comments on how these results can be extended, and we argue that the functional framework is the best candidate for a unified understanding of all relevant infrared features of quantum gravity.

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Manifestly gauge invariant transition amplitudes and thermal influence functionals in QED and linearized gravity (2017)

Einstein’s theory of General Relativity tells us that gravity is not a forcebut rather it is the curvature of spacetime itself. Spacetime is a dynamicalobject evolving and interacting similar to any other system in nature. Theequivalence principle requires everything to couple to gravity in the sameway. Consequently, as a matter of principle it is impossible to truly isolate asystem|it will always be interacting with the dynamical spacetime in whichit resides. This may be detrimental for large mass quantum systems sinceinteraction with an environment can decohere a quantum system, renderingit effectively classical. To understand the effect of a ‘spacetime environment’, we compute the Feynman-Vernon influence functional (IF), a usefultool for studying decoherence. We compute the IF for both the electromagnetic and linearized gravitational fields at finite temperature in a manifestlygauge invariant way. Gauge invariance is maintained by using a modification of the Faddeev-Popov technique which results in the integration overall gauge equivalent configurations of the system. As an intermediate stepwe evaluate the gauge invariant transition amplitude for the gauge fieldsin the presence of sources. When used as an evolution kernel the transitionamplitude projects initial data onto a physical (gauge-invariant) subspace ofthe Hilbert space and time-evolves the states within that physical subspace.The states in this physical subspace satisfy precisely the same constraintequations which one implements in the constrained quantization method ofDirac. Thus we find that our approach is the path-integral equivalent ofDirac’s. In the gauge invariant computation it is clear that for gauge theories the appropriate separation between system and environment is not a)matter and gauge field, but rather b) matter (dressed with a coherent field)and radiation field. This implies that only the state of the radiation fieldcan be traced out to obtain a reduced description of the matter. We stressthe importance of gauge invariance and the implementation of constraintsbecause it resolves the disagreement between results in reported recent literature in which influence functionals were computed in different gaugeswithout consideration of constraints.

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The effect of deuteration on receptor-ligand binding (2017)

In this thesis, we use a particle coupled to a phonon bath to accurately model biologicaland chemical reactions. The path decomposition expansion (PDX) formalism is used todetermine the tunneling dynamics of the particle. By decomposing the potential energylandscape into the classically allowed and classically forbidden regions, we can calculate thepath integrals associated with each region and connect them to evaluate the full Green'sfunction. We will also discuss how deuteration of ligand molecules may affect enzyme-substratebinding in GPCR systems. It has been theorized that binding may be dependent on amolecular vibrational component. We investigate this in the β-adrenergic receptor systemusing the deuterated and non-deuterated forms of the ligand epinephrine. The measurementfor successful binding is determined by the amounts of second messenger cyclic-AMPproduced. However, our results proved inconclusive and a discussion of possible problemsas well as recommendations is included.

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