1. Theoretical Investigations of Magnetic Field Sensitivity of an Area Chirp Array of Atom Interferometers via the Aharonov Bohm Effect Abstract Introduction Methods Results Discussion References A B  r   Interferometry is a fundamental technique used in physics that has been around for over a century. An interferometer splits a wave source into two paths and recombines these paths at a detector, where a phase shift occurs if there is a difference in the lengths of the paths. This project focused on simulating Aharonov Bohm phase shifts of matter waves passing through a multi-ring interferometer to determine the sensitivity of these structures to changes in magnetic fields. The matter waves that pass through the interferometer are split in position space in order to properly analyze the geometric phase shift that occurs at the detector. Using the solutions to the appropriate Schrodinger equation, transfer matrices were acquired for the system from which transmission as a function of phase can be obtained. The slopes of the rapidly oscillating transmission graphs were analyzed to determine their sensitivity to phase. The parameters that were varied include kl (wave number * circumference), N (Number of Rings), and γ (Chirp Factor). By introducing various chirp factors into the system, the optimal geometry can be found. The minimum phase shift found from the analysis of the transmission function can be used to determine the structures’ limits for detecting changes in magnetic fields. Ring Interferometry Obtaining The Transmission Coefficient The wave functions were found from solving the appropriate Schrodinger Equation (taking into consideration the kinetic energy and the energy associated with the magnetic vector potential in the Hamiltonian) Boundary conditions were set (wave continuity and current conservation) and a system of linear equations was solved to find the refection amplitude, r, and the transmission amplitude, t The reflection and transmission amplitudes were entered into the 2x2 transfer matrix (see A) The transfer matrix of each ring in the system was multiplied to find the transfer matrix for the whole system (see B) The Transmission coefficient was obtained by using the transmission amplitude of the final transfer matrix (see C) The figure above illustrates wave propagation through a single ring A) B) C) Figures A and B illustrate Transmission vs. Phase graphs for 9 rings and.25π kl These graphs were made varying the parameters kl (wave number * circumference), N (number of rings), and chirp factor, γ From each graph, the steepest peak can be found and the max slope can be determined Figure A illustrates chirp=0 Figure B illustrates chirp=.25 A) Transmission vs. Phase for an Unchirped Ring The figures above plot kl vs. chirp vs. Max Slope for 3 rings, 5 rings, and 7 rings, respectively. The parameter k is multiplied by π and represents the wave number times the circumference as a unitless variable. As the number of rings increases, so does the maximum slope Maximum slopes occur at different parameters Maximum slope for every N when the chirp factor is equal to zero is shown N1357911 kl.45π.40π.27π.40π.27π.15π Chirp2.00E-011.50E-01 5.00E-021.00E-011.50E-01 T4.15E-017.22E-017.47E-017.49E-017.52E-018.11E-01 3.24E+004.80E+025.45E+041.56E+054.64E+058.48E+06 1.52E-049.34E-077.98E-092.78E-099.30E-104.62E-11 1.88E-051.16E-079.90E-103.45E-101.15E-105.73E-12 1.88E-101.16E-129.90E-153.45E-151.15E-155.73E-17 Using the maximum slopes for each ring, the sensitivity to small changes in magnetic fields and magnetic fluxes can be determined by setting the following two equations equal to each other: The above table shows the maximum slope for each ring and the parameters at which it occurs as well as the smallest changes in magnetic field and flux that the sensor can read. While these are just preliminary results, the data shows that these multi-ring sensors (where the center ring diameter is 10 microns) would be able to read changes in magnetic field in the picotesla range. These sensors which have applications to mesoscopic physics show a very promising future in investigating various quantum properties. Minimum Changes the Sensor can Measure at Varying Parameters 1. Toland, J. R., Arouh, S. J., Diggins, C. J., Sorrentino, C., & Search, C. P.,, Phys. Rev. A, 85(4) (2012). 2. C. P. Search, J.R.E.Toland, M. Zivkovic,, Phys. Rev. A 79, 053607 (2009). 3. Alexander D. Cronin, Jorg Schmiedmayer, David E. Pritchard (2009). "Optic Interferometry with Atoms and Molecules".RevModPhys.81.1051. 4. W.Y. Cui, S.Z.Wu, G. Jina, X. Zhaob, and Y.Q. Ma.(2007). Eur. Phys. J. B 59, 47–54. 5. Jian-Bai Xia(1992). "Quantum Waveguide theory for Mesoscopic Structures". Chinese Center of Advanced Science and Technology". Phys. Rev. B. Sponsors National Aeronautics and Space Administration (NASA) NASA Goddard Space Flight Center (GSFC) NASA Goddard Institute for Space Studies (GISS) NASA New York City Research Initiative (NYCRI) Department of Education – The Alliance for Continuous Learning Environments in STEM (CILES) grant# P031C110158 LaGuardia Community College (LAGCC) Acknowledgements Dr. Jorge Gonzalez, ME Professor, PI of CILES grant, CCNY Dr. Yasser Hassebo, EE Professor, LAGCC Dr. Reginald Eze, ME Professor, LAGCC Vector Potential, A The Aharonov Bohm Effect Chirp Factor The magnetic field B, is defined as the curl of the potential, A. This figure illustrates the magnetic vector potential outside a toroidal inductor. Wikipedia. Wikimedia Foundation, n.d. Web. 03 July 2014.. Magnetic vector potential couples with the phase of the matter wave The wave enters the ring at point A and splits into two beams The waves recombine at point B A phase shift occurs at the detector or screen Each ring is approximately 10 microns in diameter Multi-ring System with a Negative Chirp about the Center Ring. B) Transmission vs. Phase for a Chirped Ring Matter Waves pass through two slits and pass a cylindrical solenoid The Chirp Factor, γ, is the ratio of magnetic fluxes from the center ring outwards If γ>0, then the rings will increase in flux If γ<0, then the rings will decrease in flux Multi-ring Interferometer solenoid Mentor: Dr. John R.E. Toland Tabitha Rivera (LaGuardia Researcher) Angelo Angeles (Educator, The School for Legal Studies)