*Parity-Time Symmetry in Nanophotonics and Nanoscale Charge Transport*

*Dr. Yogesh Joglekar (Co-PI), Physics, NSF Award # 1054020.*

In the past decade, memristors [1] and parity-time (PT) symmetric, on-chip optics [2,3] have revolutionized investigation of electrical and optical charge transport, respectively, in materials at the nanoscale. In the following, two available REU projects are described, one in each area:

Project 1: Memristors, realized by Hewlett Packard in metal-oxide (TiO2) thin-films [1], hold the promise for low-power, high-density nonvolatile memory, with implications to artificial neurons [4]. The dynamics of a single memristor is well understood in terms of non-linear dopant drift [5]; however, their networks [6] are poorly understood due to a non-linear, hysteretic response. In this REU project, a student will investigate the I-V characteristics of such a network in response to transient or periodic stimuli, and its power profile. The two goals of this project are to a) model a small circuit of memristors and other passive elements that exhibits a limit-cycle and thus mimics a neuron above threshold, b) explore the network of such units, with focus on power considerations for memory storage in such a network. Using Kirchoff laws and internal-state dynamics of a memristor, the problem can be reduced to a set of first-order, coupled, ordinary differential equations that may be tackled by MATLAB. Therefore, this project is accessible to incoming sophomore or junior students; for example, the work [5] by a starting sophomore student in the Co-PI's group has received more than 400 citations in the last 7 years.

Project 2: PT symmetric optics is a novel, rapidly developing research area where properties of light in nanoscale structures are tailored by using both absorbing and gain media [2,3]. The dynamics of light in such structures, in the paraxial approximation, is described by an effective, non-Hermitian, PT symmetric Hamiltonian. When the eigenvalues of such a Hamiltonian change from purely real to complex conjugates, the light-propagation properties are dramatically altered, and can lead to unidirectional invisibility [7], coherent absorption [8], and loss-induced lasing [9]. The primary goal of this project is to construct and investigate model PT Hamiltonians that are realizable in coupled waveguides or microrings. This problem, at its heart, reduces to eigen-properties of non-Hermitian, possibly defective, finite-dimensional matrices, and time-evolution determined by them. Therefore, it is easily accessible to young students; the high-school and undergraduate students in the co-PI's group have co-authored half-a-dozen peer-reviewed papers on PT symmetric lattice models [10], with a total of more than 100 citations.

These two projects will provide students an opportunity to engage in cutting-edge research with implications to nanotechnology, and, based on the Co-PI's track record, have a very high likelihood of success. Students will learn modeling, analytical and numerical techniques, and critical thinking in interpreting their results and connecting them to the real work of optical and electronic nanoscale devices.

Project 1: Memristors, realized by Hewlett Packard in metal-oxide (TiO2) thin-films [1], hold the promise for low-power, high-density nonvolatile memory, with implications to artificial neurons [4]. The dynamics of a single memristor is well understood in terms of non-linear dopant drift [5]; however, their networks [6] are poorly understood due to a non-linear, hysteretic response. In this REU project, a student will investigate the I-V characteristics of such a network in response to transient or periodic stimuli, and its power profile. The two goals of this project are to a) model a small circuit of memristors and other passive elements that exhibits a limit-cycle and thus mimics a neuron above threshold, b) explore the network of such units, with focus on power considerations for memory storage in such a network. Using Kirchoff laws and internal-state dynamics of a memristor, the problem can be reduced to a set of first-order, coupled, ordinary differential equations that may be tackled by MATLAB. Therefore, this project is accessible to incoming sophomore or junior students; for example, the work [5] by a starting sophomore student in the Co-PI's group has received more than 400 citations in the last 7 years.

Project 2: PT symmetric optics is a novel, rapidly developing research area where properties of light in nanoscale structures are tailored by using both absorbing and gain media [2,3]. The dynamics of light in such structures, in the paraxial approximation, is described by an effective, non-Hermitian, PT symmetric Hamiltonian. When the eigenvalues of such a Hamiltonian change from purely real to complex conjugates, the light-propagation properties are dramatically altered, and can lead to unidirectional invisibility [7], coherent absorption [8], and loss-induced lasing [9]. The primary goal of this project is to construct and investigate model PT Hamiltonians that are realizable in coupled waveguides or microrings. This problem, at its heart, reduces to eigen-properties of non-Hermitian, possibly defective, finite-dimensional matrices, and time-evolution determined by them. Therefore, it is easily accessible to young students; the high-school and undergraduate students in the co-PI's group have co-authored half-a-dozen peer-reviewed papers on PT symmetric lattice models [10], with a total of more than 100 citations.

These two projects will provide students an opportunity to engage in cutting-edge research with implications to nanotechnology, and, based on the Co-PI's track record, have a very high likelihood of success. Students will learn modeling, analytical and numerical techniques, and critical thinking in interpreting their results and connecting them to the real work of optical and electronic nanoscale devices.

**References:**

###### D.B. Strukov, G.S. Snider, D.R. Stewart, and R. Stanley Williams, "The Missing Memristor Found," Nature, 453, 80, 2008.

###### L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, "Single-mode Laser by Parity-Time Symmetry Breaking," Science, 346(6212), 972-975, 2014.

###### H. Hodaei, M.-A. Miri, M. Heinrich, D.N. Christodoulides, and M. Khajavikhan, "Parity Time-Symmetric Microring Lasers," Science 346(6212), 975-978, 2014.

###### L. Chua, V. Sbitnev, and H. Kim, "Hodgkin-Huxley Axon is Made of Memristors," International Journal of Bifurcation and Chaos, 22, 123011, 2012.

###### Y.N. Joglekar and S.J. Wolf*, "The Elusive Memristor: Properties of Basic Electrical Circuits," European Journal of Physics, 30, 661, 2009.

###### M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adams, K. K. Likharev, and D. B. Strukov, "Training and Operation of an Integrated Neuromorphic Network Based on Metal-Oxide Memristors," Nature, 521, 61-64, 2015.

###### Z. Lin et al., "Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures," Physics Review Letters, 106, 213901, 2011; A. Regensburger et al., "Parity-time Synthetic Photonic Lattices," Nature 488, 167, 2012.

###### W. Wan, et al., "Time-reversed Lasing and Interferometric Control of Absorption," Science 331(6019), 889-892, 2011.

###### B. Peng, et al., "Loss-induced Suppression and Revival of Lasing," Science, 346(6207), 328-332, 2014.

###### V. Gewin, "Turning Point: Yogesh Joglekar," Nature, 494, 139, 2013.