# Physics

35 Sloane Physics Laboratory, 203.432.3607

http://physics.yale.edu

M.S., M.Phil., Ph.D.

**Chair**

Paul Tipton

**Director of Graduate Studies**

Sean Barrett (SPL 24, 203.432.6928, graduatephysics@yale.edu)

**Professors** Robert Adair (*Emeritus*), Charles Ahn (*Applied Physics*), Yoram Alhassid, Thomas Appelquist, Charles Bailyn (*Astronomy*), O. Keith Baker, Charles Baltay, Sean Barrett, Hui Cao (*Applied Physics*), Richard Casten (*Emeritus*), Paolo Coppi (*Astronomy*), David DeMille, Michel Devoret (*Applied Physics*), Frank Firk (*Emeritus*), Bonnie Fleming, Marla Geha (*Astronomy*), Steven Girvin, Leonid Glazman, Jack Harris, John Harris, Karsten Heeger, Victor Henrich (*Applied Physics*), Jay Hirshfield (*Adjunct*), Jonathon Howard (*Molecular Biophysics & Biochemistry*), Francesco Iachello, Sohrab Ismaill-Beigi (*Applied Physics*), Steve Lamoreaux, Samuel MacDowell (*Emeritus*), Simon Mochrie, Vincent Moncrief, Priyamvada Natarajan (*Astronomy*), Peter Parker (*Emeritus*), Daniel Prober (*Applied Physics*), Nicholas Read, Jack Sandweiss (*Emeritus*), Robert Schoelkopf (*Applied Physics*), Ramamurti Shankar, Witold Skiba, Charles Sommerfield (*Emeritus*), A. Douglas Stone (*Applied Physics*), Paul Tipton, Thomas Ullrich (*Adjunct*), C. Megan Urry, Pieter van Dokkum (*Astronomy*), John Wettlaufer (*Geology & Geophysics*), Robert Wheeler (*Emeritus*), Werner Wolf (*Emeritus*), Michael Zeller (*Emeritus*)

**Associate Professors** Helen Caines, Sarah Demers, Thierry Emonet (*Molecular, Cellular & Developmental Biology*), Walter Goldberger, Reina Maruyama, Daisuke Nagai, Corey O’Hern (*Mechanical Engineering & Materials Science*), Nikhil Padmanabhan, David Poland

**Assistant Professors** Murat Acar (*Molecular, Cellular & Developmental Biology*), Meng Cheng, Damon Clark (*Molecular, Cellular & Developmental Biology*), Liang Jiang (*Applied Physics*), David Moore, John Murray (*Psychiatry*), Nir Navon, Laura Newburgh, Peter Rakich (*Applied Physics*)

## Fields of Study

Fields include atomic physics and quantum optics; nuclear physics; particle physics; astrophysics and cosmology; condensed matter; biological physics; quantum information physics; applied physics; and other areas in collaboration with the School of Engineering & Applied Science, and the departments of Applied Physics; Mathematics; Chemistry; Molecular Biophysics and Biochemistry; Molecular, Cellular, and Developmental Biology; Geology and Geophysics; and Astronomy.

## Special Admissions Requirements

The prerequisites for work toward a Ph.D. degree in physics include a sound undergraduate training in physics and a good mathematical background. The GRE General Test and the Subject Test in Physics are required.

## Integrated Graduate Program in Physical and Engineering Biology (PEB)

Students applying to the Ph.D. program in Physics may also apply to be part of the PEB program. See the description under Non-Degree-Granting Programs, Councils, and Research Institutes for course requirements, and http://peb.yale.edu for more information about the benefits of this program and application instructions.

## Special Requirements for the Ph.D. Degree

To complete the course requirements students are expected to take a set of nine term courses: six core courses, an advanced course, and two electives. The six core courses (PHYS 500, Advanced Classical Mechanics; PHYS 502, Electromagnetic Theory I; PHYS 506, Mathematical Methods of Physics; PHYS 508 and PHYS 608, Quantum Mechanics I and II; and PHYS 512, Statistical Physics I) serve to complete the student’s undergraduate training in classical and quantum physics. Students select an advanced course from a set of five (PHYS 538, Introduction to Relativistic Astrophysics and General Relativity; PHYS 609, Relativistic Field Theory I; PHYS 610, Quantum Many-Body Theory; PHYS 628, Statistical Physics II; and PHYS 630, Relativistic Field Theory II) that provide an introduction to modern physics and research. Certain equivalent course work and successful completion of a pass-out examination may reduce the course requirement or allow substitution of elective courses for individual students. In addition, all students are required to be proficient and familiar with advanced laboratory techniques. This requirement can be met either by taking PHYS 504, Modern Physics Measurements, or PHYS 990, Special Investigations. In addition to all other requirements, students must successfully complete PHYS 590, Responsible Conduct in Research for Physical Scientists, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

Students who have completed their course requirements with satisfactory grades (a grade of Honors in PHYS 990, Special Investigations, may be counted toward the Graduate School requirement of two grades of Honors), passed the qualifying examination, and submitted an acceptable thesis prospectus are recommended for admission to candidacy. The qualifying examination, normally taken at the beginning of the third term (and no later than the beginning of the fifth term), is a six-hour written examination covering the six core courses as described above. Students normally submit the dissertation prospectus before the end of the third year of study.

There is no foreign language requirement. Teaching experience is regarded as an integral part of the graduate training program. During their study students are expected to serve four terms as teaching fellows at the TF-10 level, usually in the first two years. Formal association with a dissertation adviser normally begins in the fourth term after the qualifying examination has been passed and required course work has been completed. An adviser from a department other than Physics can be chosen in consultation with the director of graduate studies (DGS), provided the dissertation topic is deemed suitable for a physics Ph.D.

## Master’s Degrees

**M.Phil.** Students who have successfully advanced to candidacy qualify for the M.Phil. degree.

**M.S. (en route to the Ph.D.)** Students who complete the first-year graduate courses with a satisfactory record (including two Honors or four High Passes) qualify for the M.S. degree.

Program materials are available upon request to the Director of Graduate Studies, Department of Physics, Yale University, PO Box 208120, New Haven CT 06520-8120; e-mail, graduatephysics@yale.edu; website, http://physics.yale.edu.

## Courses

**PHYS 500a, Advanced Classical Mechanics** Yoram Alhassid

Newtonian dynamics, Lagrangian dynamics, and Hamiltonian dynamics. Rigid bodies and Euler equations. Oscillations and eigenvalue equations. Classical chaos. Introduction to dynamics of continuous systems.

TTh 11:35am-12:50pm

**PHYS 502b, Electromagnetic Theory I** A. Douglas Stone

Classical electromagnetic theory including boundary-value problems and applications of Maxwell equations. Macroscopic description of electric and magnetic materials. Wave propagation.

TTh 9am-10:15am

**PHYS 504b, Modern Physics Measurements** Reina Maruyama

A laboratory course with experiments and data analysis in soft and hard condensed matter, nuclear and elementary particle physics.

HTBA

**PHYS 506a, Mathematical Methods of Physics** Nicholas Read

Survey of mathematical techniques useful in physics. Includes vector and tensor analysis, group theory, complex analysis (residue calculus, method of steepest descent), differential equations and Green's functions, and selected advanced topics.

MW 9am-10:15am

**PHYS 508a, Quantum Mechanics I** Thomas Appelquist

The principles of quantum mechanics with application to simple systems. Canonical formalism, solutions of Schrödinger's equation, angular momentum, and spin.

MW 11:35am-12:50pm

**PHYS 512b, Statistical Physics I** Meng Cheng

Review of thermodynamics, the fundamental principles of classical and quantum statistical mechanics, canonical and grand canonical ensembles, identical particles, Bose and Fermi statistics, phase transitions and critical phenomena, enormalization group, irreversible processes, fluctuations.

TTh 2:30pm-3:45pm

**PHYS 517b / ENAS 517b / MB&B 517b / MCDB 517b, Methods and Logic in Interdisciplinary Research** Staff

This half-term PEB class is intended to introduce students to integrated approaches to research. Each week, the first of two sessions is student-led, while the second session is led by faculty with complementary expertise and discusses papers that use different approaches to the same topic (for example, physical and biological or experiment and theory). Counts as 0.5 credit toward graduate course requirements. ½ Course cr

HTBA

**PHYS 522a, Introduction to Atomic Physics** David DeMille

The course is intended to develop basic theoretical tools needed to understand current research trends in the field of atomic physics. Emphasis is given to laser-spectroscopic methods including laser cooling and trapping. Experimental techniques discussed when appropriate.

TTh 1pm-2:15pm

**PHYS 523b / CB&B 523b / ENAS 541b / MB&B 523b, Biological Physics** Simon Mochrie

The course has two aims: (1) to introduce students to the physics of biological systems and (2) to introduce students to the basics of scientific computing. The course focuses on studies of a broad range of biophysical phenomena including diffusion, polymer statistics, protein folding, macromolecular crowding, cell motion, and tissue development using computational tools and methods. Intensive tutorials are provided for MATLAB including basic syntax, arrays, for-loops, conditional statements, functions, plotting, and importing and exporting data.

TTh 1pm-2:15pm

**PHYS 528b / ENAS 848b, Soft Condensed Matter Physics** Eric Brown

An introduction to the physics and phenomenology of soft condensed matter: classical systems with mesoscale structure where thermal fluctuations and interfacial forces play essential roles. Discussion of applications to materials science/engineering, nanotechnology, and molecular/cellular biology. Essential concepts from statistical thermodynamics, classical mechanics, and electricity and magnetism are reviewed/developed as needed.

MW 11:35am-12:50pm

**PHYS 530a, Scientific Teaching for Physical Sciences** Rona Ramos and Kaury Kucera

The course covers fundamentals of learning theory and practical strategies for teaching in the physical sciences. Students will practice teaching scientific concepts, manage classroom dynamics, and implement strategies for effective and inclusive teaching. In the second half of the course, will students (1) apply these principles as they develop and evaluate instructional materials for a college level science course and (2) develop a peer reviewed and polished teaching statement. Pre-reqs: Completed one semester of required teaching at Yale (n/a for postdocs).

T 9am-11am

**PHYS 538b, Introduction to Relativistic Astrophysics and General Relativity** Vincent Moncrief

Basic concepts of differential geometry (manifolds, metrics, connections, geodesics, curvature); Einstein's equations and their application to such areas as cosmology, gravitational waves, black holes.

MW 2:30pm-3:45pm

**PHYS 548a / APHY 548a / ENAS 850a, Solid State Physics I** Victor Henrich

A two-term sequence (with PHYS 549) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.

TTh 1pm-2:15pm

**PHYS 549b / APHY 549b / ENAS 851b, Solid State Physics II** Vidvuds Ozolins

A two-term sequence (with PHYS 548) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.

TTh 1pm-2:15pm

**PHYS 561a / CB&B 561a / MB&B 561a / MCDB 561a, Introduction to Dynamical Systems in Biology** Thierry Emonet and Kathryn Miller-Jensen

Study of the analytic and computational skills needed to model genetic networks and protein signaling pathways. Review of basic biochemical concepts including chemical reactions, ligand binding to receptors, cooperativity, and Michaelis-Menten enzyme kinetics. Deep exploration of biological systems including: kinetics of RNA and protein synthesis and degradation; transcription activators and repressors; lyosogeny/lysis switch of lambda phage and the roles of cooperativity and feedback; network motifs such as feed-forward networks and how they shape response dynamics; cell signaling, MAP kinase networks and cell fate decisions; bacterial chemotaxis; and noise in gene expression and phenotypic variability. Students learn to model using MATLAB in a series of in-class hackathons that illustrate biological examples discussed in lectures.

TTh 2:30pm-3:45pm

**PHYS 562b / AMTH 765b / CB&B 562b / ENAS 561b / INP 562b / MB&B 562b / MCDB 562b, Dynamical Systems in Biology** Damon Clark, Thierry Emonet, and Jonathon Howard

This course covers advanced topics in computational biology. How do cells compute, how do they count and tell time, how do they oscillate and generate spatial patterns? Topics include time-dependent dynamics in regulatory, signal-transduction, and neuronal networks; fluctuations, growth, and form; mechanics of cell shape and motion; spatially heterogeneous processes; diffusion. This year, the course spends roughly half its time on mechanical systems at the cellular and tissue level, and half on models of neurons and neural systems in computational neuroscience. Prerequisite: MCDB 561 or equivalent, or a 200-level biology course, or permission of the instructor.

TTh 2:30pm-3:45pm

**PHYS 570a / ASTR 570a, High-Energy Astrophysics** Priyamvada Natarajan

A survey of current topics in high-energy astrophysics, including accreting black hole and neutron star systems in our galaxy, pulsars, active galactic nuclei and relativistic jets, gamma-ray bursts, and ultra-high-energy cosmic rays. The basic physical processes underlying the observed high-energy phenomena are also covered.

TTh 4pm-5:15pm

**PHYS 590b / APHY 590b, Responsible Conduct in Research for Physical Scientists** Staff

Required seminar for all first-year students.

HTBA

**PHYS 608b, Quantum Mechanics II** Jack Harris

Approximation methods, scattering theory, and the role of symmetries. Relativistic wave equations. Second quantized treatment of identical particles. Elementary introduction to quantized fields.

MW 11:35am-12:50pm

**PHYS 609a, Relativistic Field Theory I** Walter Goldberger

The fundamental principles of quantum field theory. Interacting theories and the Feynman graph expansion. Quantum electrodynamics including lowest order processes, one-loop corrections, and the elements of renormalization theory.

MW 11:35am-12:50pm

**PHYS 610b / APHY 610b, Quantum Many-Body Theory** Yoram Alhassid

Identical particles and second quantization. Electron tunneling and spectral function. General linear response theory. Approximate methods of quantum many-body theory. Dielectric response, screening of long-range interactions, electric conductance, collective modes, and photon absorption spectra. Fermi liquid; Cooper and Stoner instabilities; notions of superconductivity and magnetism. BCS theory, Josephson effect, and Majorana fermions in condensed matter; superconducting qubits. Bose-Einstein condensation; Bogoliubov quasiparticles and solitons.

TTh 11:35am-12:50pm

**PHYS 624b, Group Theory** Francesco Iachello

Lie algebras, Lie groups, and some of their applications. Representation theory. Explicit construction of finite-dimensional irreducible representations. Invariant operators and their eigenvalues. Tensor operators and enveloping algebras. Boson and fermion realizations. Differential realizations. Quantum dynamical applications.

MW 9am-10:15am

**PHYS 628a / APHY 628a, Statistical Physics II** Meng Cheng

An advanced course in statistical mechanics. Topics may include mean field theory of and fluctuations at continuous phase transitions; critical phenomena, scaling, and introduction to the renormalization group ideas; topological phase transitions; dynamic correlation functions and linear response theory; quantum phase transitions; superfluid and superconducting phase transitions; cooperative phenomena in low-dimensional systems.

TTh 2:30pm-3:45pm

**PHYS 630b, Relativistic Field Theory II** Walter Goldberger

An introduction to non-Abelian gauge field theories, spontaneous symmetry breakdown, and unified theories of weak and electromagnetic interactions. Renormalization group methods, quantum chromodynamics, and nonperturbative approaches to quantum field theory.

TTh 1pm-2:15pm

**PHYS 633b / APHY 633b, Introduction to Superconductivity** Daniel Prober

The fundamentals of superconductivity, including both theoretical understandings of basic mechanism and description of major applications. Topics include historical overview, Ginzburg-Landau (mean field) theory, critical currents and fields of type II superconductors, BCS theory, Josephson junctions and microelectronic and quantum-bit devices, and high-Tc oxide superconductors.

MW 11:35am-12:50pm

**PHYS 634a / APHY 634a, Mesoscopic Physics I** Michel Devoret

Introduction to the physics of nanoscale solid state systems, which are large and disordered enough to be described in terms of simple macroscopic parameters like resistance, capacitance, and inductance, but small and cold enough that effects usually associated with microscopic particles, like quantum-mechanical coherence and/or charge quantization, dominate. Emphasis is placed on transport and noise phenomena in the normal and superconducting regimes.

MW 9am-10:15am

**PHYS 675a / APHY 675a, Principles of Optics with Applications** Hui Cao

Introduction to the principles of optics and electromagnetic wave phenomena with applications to microscopy, optical fibers, laser spectroscopy, nanophotonics, plasmonics, and metamaterials. Topics include propagation of light, reflection and refraction, guiding light, polarization, interference, diffraction, scattering, Fourier optics, and optical coherence.

TTh 11:35am-12:50pm

**PHYS 762a or b / CHEM 562La or b, Laboratory in Instrument Design and the Mechanical Arts** Kurt Zilm and David Johnson

Familiarization with modern machine shop practices and techniques. Use of basic metalworking machinery and instruction in techniques of precision measurement and properties of commonly used metals, alloys, and plastics.

HTBA

**PHYS 816a / APHY 816a, Techniques Microwave Measurement** Robert Schoelkopf

An advanced course covering the concepts and techniques of radio-frequency design and their application in making microwave measurements. The course begins with a review of lumped element and transmission line circuits, network analysis, and design of passive elements, including filters and impedance transformers. We continue with a treatment of passive and active components such as couplers, circulators, amplifiers, and modulators. Finally, we employ this understanding for the design of microwave measurement systems and techniques for modulation and signal recovery, to analyze the performance of heterodyne/homodyne receivers and radiometers.

TTh 2:30pm-3:45pm

**PHYS 990a, Special Investigations** Staff

Directed research by arrangement with individual faculty members and approved by the DGS.

HTBA

**PHYS 991a / ENAS 991a / MB&B 591a / MCDB 591a, Integrated Workshop** Corey O'Hern, Lynne Regan, Simon Mochrie, Christine Jacobs-Wagner, Scott Holley, and Megan King

This required course for students in PEB involves hands-on laboratory modules with students working in pairs. A biology student is paired with a physics or engineering student; a computation/theory student is paired with an experimental student. The modules are devised so that a range of skills is acquired, and students learn from each other. Modules are hosted in faculty laboratories.

MW 9am-10:15am