Stephen Clark

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Stephen Clark

Visiting researcher, Lecturer at University of Bath

Research topics

Major themes for my research revolve around non-equilibrium phenomena in many-body systems ranging from ultra-cold atoms to strongly correlated electron materials. Specifically, I am interested in:

  • Understanding the nature of entanglement, correlations and quantum mutual information in ground states and thermal states of commonly encountered many-body systems with striking and deep connections to their simulability.
  • Exploiting and further developing sophisticated tensor network theory techniques for efficiently simulating many-body quantum systems. Currently this most prominently includes the density matrix renormalization group (DMRG) method and its generalization to time-dependent phenomena via the time-evolving block decimation (TEBD) algorithm applicable to 1D systems. A major long term effort to extend the success of these methods to 2D quantum systems is underway.
  • Developing a comprehensive and highly optimized freely available open-source software library for tensor network theory algorithms which can be found at
  • Connecting tensor network theory to other extremely successful techniques in condensed matter physics such as density functional theory and dynamical mean-field theory.
  • Applying these toolbox of methods to realistically model systems of ultra-cold atoms and strongly correlated electron systems which are driven far from equilibrium. In particular I am interested in how strongly driving systems can stabilize new phases not possible in equilibrium and permit the controlled manipulation of material properties giving quantum enhanced functionality.
  • Exploring foundational issues regarding quantum theory including non-locality and quantifying quantumness, as well as connections to thermodynamics of small systems and fluctuation relations.

Academic Biography

  • Senior Researcher, 03/2014 - present
    Atomic and Laser Physics Dept., Clarendon Laboratory, University of Oxford, UK.
  • Career Development Fellow, 10/2013 - present
    Keble College, University of Oxford, UK.
  • Visiting Scientist, 03/2014 - present
    Max Planck Institute for the Structure and Dynamics of Matter, University of Hamburg, Germany.
  • Senior Research Fellow, 01/2013 - 03/2014
    Centre for Quantum Technologies, National University of Singapore, Singapore.
  • Research Fellow and Tutor, 10/2009 - 10/2013
    Keble College, University of Oxford, UK.
  • Visiting Researcher, 04/2009 - 03/2014
    Atomic and Laser Physics Dept., Clarendon Laboratory, University of Oxford, UK.
  • Research Fellow, 04/2009 - 01/2013
    Centre for Quantum Technologies, National University of Singapore, Singapore.
  • College Lecturer, 10/2008 - 10/2009
    Keble College, University of Oxford, UK.
  • Post Doctoral Researcher, 10/2007 - 04/2009
    Atomic and Laser Physics Dept., Clarendon Laboratory, University of Oxford, UK.


  • Post-graduate Diploma in Teaching and Learning in Higher Education, 10/2010 - 09/2011
    Oxford Learning Institute, University of Oxford, UK.
  • DPhil in Physics, 10/2003 - 09/2007
    Atomic and Laser Physics Dept., Clarendon Laboratory and Trinity College, University of Oxford, UK.
  • Part III Maths Tripos, 10/2002 - 06/2003
    DAMTP and Selwyn College, University of Cambridge, UK.
  • MSci in Physics, 10/1998 - 06/2002
    H.H. Wills Physics Laboratory, University of Bristol, UK.


You can find my latest publication list on the arXiv, or alternatively using the department's Symplectic database list available on the righthand side menu bar. At the risk of being overly narcissistic here is my Google Scholar page, my Web of Knowledge ResearcherID page, my ResearchGate page, and my ORCID profile page, giving a full array of statistics and metrics we are supposed to care about these days.

Media coverage

You can find a nice article about one of my recent papers, Phys. Rev. Lett. 110, 230601 (2013), on a proposal to experimentally test work statistics and fluctuation relations at the site. Further to this our theoretical proposal has now been implemented in an NMR experiment.

Editorial and conference work

I am an editorial board member for Nature's new open access multidisciplinary journal Scientific Reports. Also I was on the organising committee for a recent workshop on the Dynamics of Complex Quantum Systems held at Windsor. The next workshop in this series is scheduled for the 3rd-6th of August 2015.

Other home pages

You can also find pages about me at the Centre for Quantum Technologies, at Keble College, Oxford as well as our old group website.

Second year tutorials

Mathematical Methods and A3: Quantum Mechanics

Michaelmas term

This term consists of 6 tutorials covering both Mathematical Methods and A3: Quantum Mechanics.

Teaching material

The main books for this course are:

  • Binney and Skinner: "The Physics of Quantum Mechanics"
  • Gasiorowicz: "Quantum Physics" Stephen Gasiorowicz (3rd ed) (Supplementary Material)
  • Shankar: "Principles of Quantum Mechanics" R. Shankar (2nd ed)

Several copies of each book are available in the Keble library. The first book will be the basis for your lectures by Prof. Binney, but you are advised to read around from all three. It is reasonable to expect that you should have read a book like Gasiorowicz entirely during the second year (it is a short book after all). The tutorial plan lists those chapters of the book which should be read in preparation for the corresponding tutorial. Reading the book by Shankar is optional for those of you wishing to gain a better understanding of some of the more mathematical aspects of quantum mechanics. With this said I also encourage you to look around for other books which you might prefer, for example the one by Griffiths is very popular. Please discuss with me if you want advice about using another book.

Lecture Notes
You might find some of the older quantum lecture notes useful, as well as Prof. Binney's book:

Problem sets
These are the full problem sheets from which I have extracted the questions you attempt each week:


Please hand in all written work promptly by 5pm two days before the tutorials (i.e. the Tuesday) by placing it in the "C" pigeon hole in the Clarendon Laboratory. Any work handed in late will be recorded as such and will potentially be left unmarked. A tutorial session starts at 2pm with a class of approximately 1.5 - 2.0 hours in the room specified below. This is followed by paired tutorials of 30 minutes each in the office DB101 which I share with Dr. Brian Smith. The written work is returned at the beginning of each class and is required for the paired tutorials. There are no paired tutorials in revision class sessions or when only returning vacation work. Below is a guide to the work I expect for each tutorial - please read it carefully since it may differ from previous tutors you have had so far:

Problem sheets
I have constructed an individual problem sheet for each class. These are composed of a subset of questions extracted from the major problem sheets for the maths methods and quantum courses available in the links above. For the quantum mechanics questions you will notice that I use problems from several sources to give you a wide flavour of problem styles. Note that the problems not included in the tutorial work are also important so when revising for collections you should go back and attempt them as well.

Self-assessed and Main questions?
When examining the problem sheet you will notice that they are separated into two parts - self-assessed and main questions. You should read all the questions before starting. Write the answers to each section separately in your script. The self-assessed components is made up of questions for you to practise essential mathematical skills and reproduce important bookwork derivations. Work on this section must be handed in, as evidence that you have attempted them, but will not be marked. Instead I will give you an answer sheet in the tutorial for you to mark your own work afterwards. The main questions are those which should be attempted and handed in. These questions are usually a little harder and will be marked by me.

What are "Essay" questions?
The purpose of essay questions is for you explore your notes and textbooks for material discussing a basic question or issue. You should then formulate a written explanation including relevant mathematical content from what you find. As a guide you should not write more than one handwritten page per question. Moreover to deter you from copying and pasting from Google/Wikipedia I will only accept handwritten answers. To get an idea of what an answer should look like I have made a basic model answer to an essay like question here (see also comments in the pdf).

What is the class "Question"?
The points referred to under Questions are points that will be discussed in class. I will expect you to have researched something about the issues involved beforehand. In particular you should hand in some brief bullet point in your script outlining your ideas and potential contribution to the class discussion. I'm not necessarily looking for correctness here or fully formed answers, just evidence of some thought. Indeed if something puzzles you about the question write it down as well. As a guide here is a basic model answer to a class question (see also comments in the pdf).   

Class project
In the week 5 tutorial I will split the class into two groups of four and assign one seminal research paper in the field of quantum information to each group. Both of these papers are short and herald the discover of a simple but profound concept, namely the “no-cloning theorem” and “quantum teleportation”. By the last tutorial in week 7 we will have covered enough of the basics to have a meaningful discussion about these powerful results and each group will spend about 10 minutes accurately explaining their content to the rest of the class. To help you out I have heavily annotated the pdf files of the papers to give you clues as to what you need to focus on. Also I have written some additional notes here to give you a brief guide to qubits, tensor products and entanglement which will be very useful for understanding these papers.

Tips for your answer script
It is a good idea for you to make a cover sheet for your tutorial work in which you write down unresolved problems or issues you had while tackling the any part of the work we have done so far. I will attempt to help answer these in our tutorial. 

Mathematical methods and A3: Quantum Mechanics

MT Week 1, Thursday (Seminar room 1)

  • Material: Linear Algebra; Vector spaces; Introduction to QM and Dirac notation
  • Reading: Gasiorowicz Ch 1; Shankar Ch 2, 3
  • Question:
    Why can the double slit experiment with electrons not be described by classical wave theory?
  • Problem sheet: MT-week-1
  • Essay:
    Discuss the way probabilities enter into quantum mechanics. How does this differ from classical physics?

MT Week 2, Thursday (Seminar room 1)

  • Material: Linear Algebra; Hermitian matrices and diagonalization; The Schroedinger equation
  • Reading: Gasiorowicz Ch 2, 3.1, 3.2; Shankar Ch 4.3
  • Question:
    What are stationary states and how are they related to diagonalization?
  • Problem sheet: MT-week-2
  • Essay:
    How does the Schroedinger equation differ from the classical wave and diffusion equations? What are the consequences?

MT Week 3, Thursday (Seminar room 1)

  • Material: Hermiticity, orthogonality, Sturm-Liouville problem; Particle in a box
  • Reading: Gasiorowicz Ch 3.3 to 3.6; Shankar Ch 5.2, 9
  • Question:
    Is the discrete "quantized" spectrum found in some quantum systems surprising? Can it occur in classical systems?
  • Problem sheet: MT-week-3
  • Class project work: study a classic research paper:
    - W.K. Wootters and W.H. Zurek, "A single quantum cannot be cloned", Nature 299, 802 (1982) Download;
    - C.H. Bennett et al, "Teleporting an unknown quantum state via a dual classical and EPR channels", Phys. Rev. Lett. 70, 1895 (1993) Download

MT Week 5, Friday (Stafford Crane room)

  • Material: PDE's; Operators
  • Reading: Gasiorowicz Ch 3.2, 3.3, 3.6, 9; Shankar Ch 1, 5.2
  • Question:
    Why are Hermitian and unitary operators important in quantum mechanics?
  • Problem sheet: MT-week-5

MT Week 6, Thursday (Seminar room 1)

  • Material: Fourier analysis; Operators and measurements
  • Reading: Gasiorowicz Ch 5, 6.1; Shankar Ch 4, 9
  • Questions:
    How do measurements change the state of a system if the outcome is not revealed? Is there a connection between the Heisenberg uncertainty principle and the Fourier transformation?
  • Problem sheet: MT-week-6
  • Essay:
    Discuss the energy-time uncertainty relation (open the QMMT problem sheet and do Q5.8).

MT Week 7, Friday (Stafford Crane room)

  • Material: The quantum harmonic oscillator
  • Reading: Gasiorowicz Ch 4.7, 6.4; Shankar Ch 7
  • Question:
    Why is the quantum harmonic oscillator important?
  • Problem sheet: MT-week-7
  • Essay:
    Describe the connection between classical motion in a quantum harmonic oscillator and coherent states.
  • Presentation of group project work.

Christmas Vacation - to be handed in Monday of Week 1 Hilary Term

  • Material: One-dimensional potentials, entanglement
  • Reading: Xmas, Gasiorowicz Ch 4, 20; Shankar Ch 5
  • Problem sheet: MT-vacation
  • Essay:
    Describe the phenomena of quantum entanglement and how it is illustrated with Schroedinger cats.

Revision for Hilary term collection!
The collection paper you will have at the start of Hilary term will include all the Maths methods and Quantum Mechanics done in the tutorials this term. It will not include the topics covered in the vacation work. Use the themes for the material covered in the tutorials as a guide for what to revise.

Second year tutorials

Mathematical Methods and A3: Quantum Mechanics

Hilary term

This term consists of 5 tutorials covering A3: Quantum Mechanics.

Teaching material

Lecture Notes
In addition to the materials listed on the MT Teaching page this term we also have the following lecture notes:

Problem sets
This term the full problem sheets from which I have extracted the questions you attempt each week are:

A3: Quantum Mechanics

HT Week 1, Friday (Seminar room 1)

  • Material: 3D potential and Angular momentum
  • Reading: Gasiorowicz Ch 7; Shankar Ch 12
  • Question:
    What defines a set of operators to be angular momentum operators?
  • Problem sheet: HT-week-1
  • Essay:
    Describe the most common 3D coordinate systems and discuss with examples when it is most appropriate to use each of them.

HT Week 2, Friday (Seminar room 6)

  • Material: The Hydrogen atom, spin
  • Reading: Gasiorowicz Ch 8, 10; Shankar Ch 13, 14
  • Question:
    Is spin different from classical rotation of a particle? If so, how?
  • Problem sheet: HT-week-3
  • Essay:
    Discuss the interpretation of spin as intrinsic angular momentum.

HT Week 4, Friday (Stafford Crane)

  • Material: Addition of angular momenta, two particles
  • Reading: Gasiorowicz Ch 10-5, 10-A; Shankar Ch 15
  • Question:
    What are triplet and singlet states?
  • Problem sheet: HT-week-4
  • Essay:
    Discuss Clebsch-Gordan coefficients giving their physical interpretation.

HT Week 6, Friday (Seminar room 1)

  • Material: The "real" Hydrogen atom, perturbation theory
  • Reading: Gasiorowicz Ch 8, 11, 12
  • Question:
    When does perturbation theory work? Is it ever guaranteed to work?
  • Problem sheet: HT-week-6
  • Essay:
    Discuss the main physical effects ignored with our earlier treatment of the H-atom and explain qualitatively how they alter the spectrum.

MT Week 7, Friday (Seminar room 1)

  • Material: Helium atom and Zeeman effect
  • Reading: Gasiorowicz Ch 14
  • Questions:
    Which physical effects cause the states of the first excited level in Helium to split?
  • Problem sheet: HT-week-8
  • Essay:
    Explain how atomic states and emission lines are split in the presence of a static B field.

Easter Vacation - to be handed in Monday of Week 1 Trinity Term

  • Material: Exchange symmetry
  • Reading: Gasiorowicz Ch 13; Shankar 10
  • Problem sheet: HT-vacation
  • Essay:
    Look up and discuss the spin-statistics theorem. Relate what you find to the statistical physics of identical particles?

Revision for Trinity term mock exam
Will we be having a one-day revision session on the Saturday of revision week. This should kick-start your own full revision over easter in preparation for the mock exam, and of course the finals.

Second year tutorials

Mathematical Methods and A3: Quantum Mechanics

Trinity term

This term consists of 1 tutorial covering A3: Quantum Mechanics, a mock exam and an associated revision class.

Teaching material

Lecture Notes
In addition to the materials listed on the MT and HT Teaching pages this term we also have the following lecture notes:

Problem sets
This is the full problem sheet from which I have extracted the questions you attempt this term:

  • Quantum mechanics: FQMTT

Mathematical methods and A3: Quantum Mechanics

TT Week 2, Friday (Audrey Wood seminar room - Clarendon Laboratory) - 3pm to 5pm (!)

  • Material: Selection rules, magnetic resonances, radiation - see atomic dipole transitions applet for nice visuals (c.f. with picture at top left of this page)
  • Reading: Gasiorowicz Ch 15, 16, 17, 18
  • Question:
    When can an atom emit a photon? What determines the properties of the emitted photon?
  • Problem sheet: TT-week-1 (see file attachment on side bar)
  • Essay:
    Discuss dipole radiation (see FQMTT 7.4, 7.5)

TT Week 3, Friday (Arco roof terrace room)

  • Mock exam paper from 2:00pm until 5:00pm. Please arrive at least 5 minutes before.

TT Week 4, Friday (Arco roof terrace room)

  • Revision class going over the mock exam paper from 2:00pm until 4:00pm.

Good luck for your finals!