Bharath H M

Graduate Student

Education

- MSc (Integrated) in Physics, Indian Institute of Technology, Kanpur, 2013
- Ph.D in Physics, Georgia Institute of Technology (exp. 2018)
- M.S in Mathematics, Georgia Institute of Technology (exp. 2018)

Research Interests

My broad, long term research objective is to use the platform of trapping and controlling ultracold atoms to explore the fundamental properties of matter and develop any resulting technical applications. Often, fundamental properties of a physical system come from abstract properties of the mathematical structure describing the system. For instance, Berry’s geometric phase comes from the geometrical properties of the phase space; topological quantum numbers come from the topological properties of the phase space. Quantum entanglement comes from the algebra of how we multiply Hilbert spaces. Therefore, my research recipe is to pick up an abstract mathematical idea, understand how it transcends into the theoretical model of a physical system and explore in the laboratory, any experimentally observable consequence of it.

**Research Projects**

**Geometric phases in spin-1 systems:**

In this project, we develop a new non-Abelian geometric phase for spin-1 systems and observe it in a sample of ultracold 87Rb atoms. The quantum state of a spin-1/2 system is, up to an overall phase, uniquely specified by its spin vector which lies on the Bloch sphere. Berry's geometric phase is defined as the phas efactor picked up the quantum state when the spin vector is transported along a loop on the Bloch sphere. In contrast, the spin vector of a spin-1 system can lie anywhere on or *inside* the Bloch sphere. In addition, a spin-1 quantum state can not be uniquely represented by its spin vector --- typically, infinitely many spin-1 quantum states share any given spin vector. This ambiguity is broken by considering the second order spin moments i.e., the spin fluctuation tensor, which can be represented by an ellipsoid surrounding the spin vector. We show that this ellipsoid picks up a geometric phase when the spin vector is transported along a loop inside the Bloch sphere. We formulate this geometric phase as an SO(3) operator (see https://arxiv.org/abs/1702.08564). In the experiment, we use coherent control of ultracold 87Rb atoms in an optical trap to observe this geometric phase (see https://arxiv.org/abs/1801.00586).

**Many body entanglement and real algebraic geometry: **

In this project, we are working towards experimentally characterizing the many-body entanglement generated in an interacting BEC, using ideas imported from *real algebraic geometry*. The experimentally accessible observables of a many-body bosonic system are the spin expectation values of the spin operators Si and their anti-commutators {Si, Sj}, which, upon truncating at rank two, are 9 independent quantities. The basic problem is to determine the entanglement generated in an interacting spin-1 BEC using the above 9 numbers obtained experimentally. An entanglement criterion that utilizes two out of these 9 parameters has been developed recently. In the present project, we rephrase this problem in the language of real algebraic geometry in order to develop an entanglement criterion that utilizes all of the 9 parameters.

If the parent quantum state is un-entangled, these 9 numbers are moments of a probability distribution function; if it is entangled, we may have to admit negative values for the probability distribution function. If we consider these 9 numbers as representing a point in a 9 dimensional space, those with an un-entangled parent state lie within a convex domain, also known as the *moment cone*. Therefore, the task at hand is to locate the point in this space corresponding to the measured values, and determine whether it lies outside this domain; this falls under a class of problems known as *truncated K-moment problem* in real algebraic geometry.

- "Staircase in magnetization and entanglement entropy of spin squeezed

condensates",**H. M. Bharath**, M. S. Chapman, and C. A. R. Sa de Melo, 2018, arXiv link: https://arxiv.org/abs/1804.03745 - "Singular loops and their non-abelian geometric phases in spin-1 ultracold atoms",
**H. M. Bharath**, M. Boguslawski, M. Barrios, Lin Xin and M. S. Chapman, 2018, arXiv link: https://arxiv.org/abs/1801.00586 - "Non-Abelian geometric phases carried by the spin fluctuation tensor",
**H. M. Bharath**, J. Math. Phys,**59**, 062105 (2018) link: https://aip.scitation.org/doi/abs/10.1063/1.5018188 - "Adiabatic quenches and characterization of amplitude excitations in a continuous quantum phase transition", T. M. Hoang,
**H. M. Bharath**, M. Boguslawski, M. Anquez, B. A. Robbins and M. S. Chapman, PNAS, vol. 113, No. 34, pp. 9475, 2016. Link: http://www.pnas.org/content/113/34/9475.short - "Quantum Kibble-Zurek mechanism in a spin-1 Bose-Einstein condensate", M. Anquez, B. A. Robbins,
**H. M. Bharath**, M. Boguslawski, T. M. Hoang and M. S. Chapman, Phys. Rev. Lett.,**116**, 155301, 2016. Link: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.155301 - "Quantum enhanced precision in a collective measurement", H. M. Bharath and Saikat Ghosh, 2014, arXiv link: https://arxiv.org/abs/1411.5090
- "Classical simulation of entangled states",
**H. M. Bharath**and V. Ravishankar, Phys. Rev. A.,**89**, 062110, 2014, Link: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.062110 - "Benford's law: a theoretical explanation for base 2",
**H. M. Bharath**, 2012, arXiv link: https://arxiv.org/abs/1211.7008 - "Non-holonomic constrain force postulates",
**H. M. Bharath**, 2010, arXiv link: https://arxiv.org/abs/1010.3902