Quantum Fisher Information And Geometric Phase In Quantum Optics

Since Feynman et al introduced the concept of quantum computation and the EPR paradox raised by Einstein et al showed the possibility to use quantum entanglement to transmit informa-tion, the field of quantum computer and quantum communication has been developing rapidly and unified into the subject of Quantum Information. As Moore’s law reaches its limit and the need for secure communication, quantum computer and quantum communication are now becoming reality. This thesis is devoted to study a sub-field of quantum information, which is quantum metrology.The content of thesis is as follows:(1) In Chapter 2, we review the definition of quantum fisher information based on symmetric logarithmic derivative operators, and showed that the inverse of quantum fisher information gives the upper bound of the accuracy of estimation with rigorous proof. As for multiparameter estimation problem, we derived the quantum fisher information matrix and pointed out its relationship with metric in Hilbert space.(2) In Chapter 3, we reviewed the notions in classical differential geometry, especially the parallel transport law and the holonomy, and we introduced the geometric phase together with Berry curvature. For the major forms of generalization of geometric phase, we give the derivation based on physical fact and from the math based on Fiber bundle. Meanwhile, we introduced the quantum geometric tensor and its relation with fidelity and Berry curvature.(3) In Chapter 4, we review the classical decoherence model based on dipole interaction be-tween atom and light field, i.e., Jaynes-Cummings model and Rabi model, with emphasis on the rotating-wave approximation. Under the Lorentzian spectrum at zero temperature, we calculated the geometric phase of a qubit under decoherence with rotating-wave approximation analytically, and we found that under non-Markovian dynamics and strong coupling, the geometric phase has nodal structure. Utilizing Hierarchy equation of motion, we investigated the evolution of qubit be-yond rotating-wave approximation and we found the nodal structure in geometric phase is gone. Thus we concluded that the nodal structure of geometric phase is a result of rotating-wave approx- imation.(4) In Chapter 5, we review the derivation of parameter generators and use them to express the quantum fisher information and Berry curvature. Within the framework of estimating param-eters under unitary evolution of pure state, we derived an inequality with the product of different parameters’ quantum Fisher information on onside and the four times of square of Berry curva-ture on the other side, and upon this we introduced the concept of quantum Fisher information squeezing. Based on Robertson-Schrodinger inequality we derived another inequality that involves off-diagonal elements of quantum Fisher information matrix. We test the inequality to a specific parameter estimation problem with spin-coherent state and the inequality works satisfactorily.(5) Within the Appendix are the derivations that is too long to plug into the main body, some particularly related subject involves the matrix inequality, adiabatic theorem and the introduction to spin-coherent state.The last Chapter contains the summary and possible directions to push forward.