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Find below a list and summary of the research work and projects I have worked on, you can open the links to the papers and reports for more details. Here is my Research statement that summarizes my work and my research objectives as of December 2012.

## Quantum Hall Effect in a Spinning disk geometry

Work I did as part of my Senior Year thesis, it was a year long study and was supervised by Professor Pervez Hoodbhoy. A preprint version of the paper is being worked on, you can find a link to the Thesis here, the mid year presentation here, final presentation here, and a poster that was presented around mid year here.

What I have discovered, while working on my senior year thesis, is that by spinning a Quantum Hall system you get another really nice quantised result. Another energy quantisation in addition to the Landau quantisation is obtained and the degeneracy of the states in the Landau levels is lifted. The spacing between these levels is dependent on the frequency of spinning the disk and we have shown by calculations that you don’t need to spin it at ridiculously high speeds to observe it. In fact, it can easily be done in the lab. This splitting causes an overall broadening of Landau levels and we can use it to mimic the broadening we would get, had there been impurities in the sample. The impurities are essential in quantisation of conductance and there role is still not properly understood, but now we can control the broadening and overlap of peaks through the spinning frequency and can better understand this dependence. Apart from helping in understanding the conduction plateaus in terms of impurity broadening, this work could also be used to make extremely sensitive and robust accelerometers and gyroscopes.

After we got this interesting energy expression we wanted to see if spinning the Quantum Hall disk sample also affects the quantised conductance as it changes the quantised Energy spectrum. We first understood how to get the Kubo’s formula of conductance from perturbation theory and how we can relate it to Topology and Chern numbers. Then, we looked at the TKNN invariant form of topologically protected conductance and saw how it represents a Torus in momentum space. But having the second perturbation of spinning the disk in addition to the magnetic field really complicated the matter and we had to tackle it with a different approach. So we used Many Body Theory and the Diagrammatic Perturbation Theory and finally got a nice expression for the conductance in a spinning Quantum Hall disk. We can simulate these expressions and see how the jumps in the conductance plateaus become less sharp with impurity broadening of Landau levels.

What I have discovered, while working on my senior year thesis, is that by spinning a Quantum Hall system you get another really nice quantised result. Another energy quantisation in addition to the Landau quantisation is obtained and the degeneracy of the states in the Landau levels is lifted. The spacing between these levels is dependent on the frequency of spinning the disk and we have shown by calculations that you don’t need to spin it at ridiculously high speeds to observe it. In fact, it can easily be done in the lab. This splitting causes an overall broadening of Landau levels and we can use it to mimic the broadening we would get, had there been impurities in the sample. The impurities are essential in quantisation of conductance and there role is still not properly understood, but now we can control the broadening and overlap of peaks through the spinning frequency and can better understand this dependence. Apart from helping in understanding the conduction plateaus in terms of impurity broadening, this work could also be used to make extremely sensitive and robust accelerometers and gyroscopes.

After we got this interesting energy expression we wanted to see if spinning the Quantum Hall disk sample also affects the quantised conductance as it changes the quantised Energy spectrum. We first understood how to get the Kubo’s formula of conductance from perturbation theory and how we can relate it to Topology and Chern numbers. Then, we looked at the TKNN invariant form of topologically protected conductance and saw how it represents a Torus in momentum space. But having the second perturbation of spinning the disk in addition to the magnetic field really complicated the matter and we had to tackle it with a different approach. So we used Many Body Theory and the Diagrammatic Perturbation Theory and finally got a nice expression for the conductance in a spinning Quantum Hall disk. We can simulate these expressions and see how the jumps in the conductance plateaus become less sharp with impurity broadening of Landau levels.

thesis-1.pdf | |

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final_year_presentation.pptx | |

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mid_year_presentation.pptx | |

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poster.jpg | |

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## Topological Insulators

This was the initial topic of my senior year thesis, but when we were working on it we got side tracked into other interesting stuff and were able to do new work in Quantum Hall effects in spinning disk geometries. So we never got to finishing this work.

My thesis adviser is Dr. Pervez Hoodbhoy and the project is expected to be done by August 2012. A brief introduction on Topological Insulators. Topological Insulators are a newly discovered phase of matter that have become one of the hottest topics in condensed-matter physics. They are phases of matter that are insulators on the inside but can conduct on the surface. They are very robust and the properties don't change due to any imperfections on the surface or impurities. Moreover, the conducting electrons arrange themselves into spin-up electrons traveling in one direction, and spin down electrons traveling in the other. A interesting article on Topological Insulators in Physics Today by Kane and Moore is here.

My project is to understand these Topological Insulators from the point of view of Berry Phases and topology. You can find links to my project synopsis report and a power point presentation of the synopsis below.

My thesis adviser is Dr. Pervez Hoodbhoy and the project is expected to be done by August 2012. A brief introduction on Topological Insulators. Topological Insulators are a newly discovered phase of matter that have become one of the hottest topics in condensed-matter physics. They are phases of matter that are insulators on the inside but can conduct on the surface. They are very robust and the properties don't change due to any imperfections on the surface or impurities. Moreover, the conducting electrons arrange themselves into spin-up electrons traveling in one direction, and spin down electrons traveling in the other. A interesting article on Topological Insulators in Physics Today by Kane and Moore is here.

My project is to understand these Topological Insulators from the point of view of Berry Phases and topology. You can find links to my project synopsis report and a power point presentation of the synopsis below.

**A proposed outline of my thesis is:**

Chapter 1: The Adiabatic approximation, Berry phases, Relation to the Aharanov Bohm effect, relation to the magnetic monopoles.

Chapter 2: Solve single spin 1/2 particles in a magnetic field and calculate the Berry phase, do it for spin 1 particles (3x3 matrices).

Chapter 3: Understanding Fractional Quantum Hall Effect from the point of view of Berry Phases, this is the Hamiltonian approach (paper by Shankar).

Chapter 4: Topology and it's applications in Condensed Matter Physics.

Chapter 5: Understanding Topological Insulators from the point of view of Berry phases and forms.

Chapter 1: The Adiabatic approximation, Berry phases, Relation to the Aharanov Bohm effect, relation to the magnetic monopoles.

Chapter 2: Solve single spin 1/2 particles in a magnetic field and calculate the Berry phase, do it for spin 1 particles (3x3 matrices).

Chapter 3: Understanding Fractional Quantum Hall Effect from the point of view of Berry Phases, this is the Hamiltonian approach (paper by Shankar).

Chapter 4: Topology and it's applications in Condensed Matter Physics.

Chapter 5: Understanding Topological Insulators from the point of view of Berry phases and forms.

synopsis_report_on_topological_insulators.pdf | |

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topological_insulators.pptx | |

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## Water Management in Ayubia National Park

This was a project funded by the Coca Cola foundation and WWF. Coca Cola wants to be water neutral and the tons of water it uses in it's various manufacturing processes, it wants to return that water back to nature. Over the years the natural water channels have been drying out due to deforestation and other factors. Our goal was to come up with mathematical models and experimental techniques to quantitatively measure how reforestation and various other projects by Coca Cola foundation in the area had affected and what else can be done. We were supported in the project by WWF and various other departments at Lums SSE. After a comprehensive literature survey we wrote a proposal which got accepted by WWF, we designed our own measuring apparatus and spent the summers of 2010 doing field work in Ayubia national park. We collected alot of data and soil samples, analysed them and applied different statistical analysis. We gave our results and report to WWF and it was presented at a Coca Cola conference. Later this work was published and we also presented it at "International water Sustainability conference held at UET, Lahore". A link to the project proposal and a power point presentation is below.

proposal1.pdf | |

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water_management_in_ayubia_national_park.ppt | |

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## International Hydrology Conference

Infilterometer being used for measuring soil infiltration rates.

The work we did in Ayubia was published and we also presented it at the International Conference on Water Resources Engineering and management, UET Lahore which was held 7-8th March, 2011. Here is the abstract of our paper. And you can download the published paper here and by the link below.

**Abstract:**This research aimed to study the affect deforestation on soil infiltration rates and effective porosity in an effort to gauge the sensitivity of the soils of Ayubia National Park to deforestation Furthermore we formulated a mathematical model that related soil infiltration rates and water retention capacities of the soil to the vegetation (trees, shrub and grass cover) while accounting for variables such as slope, soil texture, canopy cover etc through multi-variable linear regression. All work is based to the maximum on self made empirical observations. The cheapest and simplest methods of observing and calculating variables were chosen given the financial and time restraints. A total of 18 sites, ten from forested areas and 8 from deforested areas were documented and these two data sets were compared. The sites were in located in the larger Namli Mera District. Lastly, the Coca Cola Company as part of its comprehensive social responsibility program, wants to become water neutral. In addition to other endeavors, it has funded WWF in its reforestation program in the areas surrounding Ayubia National Park. The company wished to know how much water it will save by reforesting several target locations, one of which is also located in Namli Mera. We have attempted to provide a possible framework for calculating this value. water_management_in_ayubia_national_park.pdf | |

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title_pages.pdf | |

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## Mathematical modeling of the influence of Heat shock Proteins on Cancer Growth

Poster on Math model of cancer growth

As part of the Mathematics department undergraduate summer research program we did a project in Mathematical Biology with Dr. Adnan Khan and Dr. Usman Qazi. The project involved after programming and solving Partial differential equations based on a biological model and then tweaking the paramters to math data and coming up with results. At the end of the summer program there was a poster display and the judges awarded us the first prize. You can download the poster from the link below. Following is the abstract of the project.

**Abstract:**Every year millions of people die of various forms of cancer around the world, so curing cancer becomes our social responsibility. Hypothesis: One of the major groups involved in the process of cancer metastasis is taht of Heat Shock Proteins (HSPs). HSP 90 particularly proposed to be responsible for triggering cancerous cell invasion. OBjective: Modelling the effect of the concentration of HSP 90 on the concentration of cancer cells and matrix degrading enzymes. Method: Assume a basic model of a set of partial differential equations describing the dynamics of cancer cells and numerically anlysing them using automated software. math-bio-poster.jpg | |

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## An Experiment to Determine the Affect of varying Temperature on the Resistance of Strain Gage for use as a calorimeter in a ratiometer

My experimental setup for high temperatures

This was a experimental physics project I did with Dr. Sabieh Anwar. It involved designing a experiment for measuring the change of properties of strain gauges at high and low temperatures and use the results to calibrate the strain gauge as a sensitive negative feedback heating device so it could be used in a ratio meter which measures the ratios of the quantum states of hydrogen, ortho and para hydrogen. The paper is expected to get published in a South Asian physics journal, Resonance. You can see the link to the project report below and on the Physics Lab website.

**Abstract:**We had to perform this experiment as a sub module for another experiment; in which we were trying to design a ratiometer. A ratiometer is a device that determines the percentage of ortho and para hydrogen, the two different quantum states of hydrogen. Based on the difference of thermal conductivity of the two states we can determine the ratio of the two gases. We needed some device for calculating the energy Q which travels between the hot plate and the cold plate, to determine the conductivity of the gas in our ratio meter. When the temperature of the metal plate (hot plate) is changed, there is a resistance change in the strain gauge, by a negative feedback system we pass current through the strain gauge to compensate the change and maintain a constant resistance, the strain gauge is used as a sensitive heating element. However in the experiment under discussion we will be testing strain gauge as a temperature sensor. straingaugereport.pdf | |

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## Making a Low Field NMR Spectrometer

My NMR experimental setup

This was a project I did with Dr. Sabieh Anwar, The ultimate goal of this long project is building an NMR spectrometer with the magnet, electronic detection circuitry and software. We like to see the FID (Fourier induction decay) signal from a test tube containing water and should be able to measure the relaxation times of the spins. We have designed a low field magnet casing using a Hallbach array after simulations through softwares like vizimag, the probe is also complete which can be used to send and receive RF pulses, but the project is still in progress and we are programming the wave generators. Find below the report of the work done till now.

Here is a link to Dr. Sabieh's interesting and very informative lectures on NMR spectroscopy.

Here is a link to Dr. Sabieh's interesting and very informative lectures on NMR spectroscopy.

nmr_spectrometer.pdf | |

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## Principles and Applications of Superconducting Quantum Interference Devices (SQUIDs)

This was a part of our experimental physics course with Dr. Sabieh Anwar. This was a very interested experiment to explore the different properties of superconductors and play around with their applications. In this experiment we will use a DC SQUID magnetometer to demonstrate the quantization of flux, the DC Josephson effect, detecting the superconducting phase transition and in the end use the SQUID with a FLL (Flux Lock Loop) circuit to make a sensitive DC/AC voltmeter. Attached is the report for this experiment.

squids.pdf | |

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