On Non-Hermitian QM and Dynamical Model forWavefunction Collapse

Singh, Gurpahul (2023) On Non-Hermitian QM and Dynamical Model forWavefunction Collapse. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Gurpahul Singh (18MS106)) Thesis_18MS106.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

Hermiticity has always been one of the important postulates of Quantum Mechanics (QM). But in late 1990s, this was challenged by a new class of non-Hermitian Hamiltonians which were PT-symmetric and admitted a real spectra. Since then, non-Hermitian Quantum Mechanics has been an exciting topic of theoretical research. With experiments showing up in classical non-Hermitian physics, the quantum domain on the practical side was explored with regard to non-Hermitian Hamiltonians fairly recently. In the first part of this thesis, after a detailed description about the inner products used in non-Hermitian QM, we discuss the case of Rabi Oscillations for a non-Hermitian Hamiltonian driving a state. We find that for real eigenvalues of the Hamiltonian, the transition probabilities are oscillatory (much like the Hermitian case), but for complex eigenvalues, the probabilities diverge to ∞ and − ∞. The von Neumann equation is altered in case of a non-Hermitian Hamiltonian such that it is nonlinear in ρ. Its analysis shows that a non-Hermitian Hamiltonian with complex eigenvalues will have an attractor eigenstate— one which has the largest imaginary part of the eigenvalue. We use this fact and apply it to a very fundamental problem in QM which constitutes the second part of the thesis. Ever since the formulation of quantum mechanics, there is very little understanding of the process of the collapse of a wavefunction. We have proposed a dynamical model to emulate the measurement postulates of quantum mechanics. We postulate that a non-Hermitian Hamiltonian operates during the process of measurement, which evolves any state to an attracting equilibrium state (which is its eigenstate), thus, mimicking a “collapse”. We demonstrate this using a 2-level system and then extend it to an N-level system. For a 2-level system, we also demonstrate that the dynamics generated by the Lindblad master equation can be replicated as an incoherent sum of the evolution by two separate non-Hermitian Hamiltonians. In order to obtain an interaction Hamiltonian that operates in the system-ancilla space and brings about entanglement between system and ancilla during the measurement interval, we use the Naimark Dilation protocol. Using this formalism, we obtain non-Hermitian evolution in the system subspace while the full system-ancilla space evolves in a unitary fashion. We obtain a final state which has the probability amplitudes (that come from Born’s rule) scaled down, although the ratio are the same. We also have two extra “dump” states in the ancilla that come with the protocol. Lastly, we have shown that a particular non-Hermitian Hamiltonian can generate the same steady state as one given by the Lindblad master equation, given that we choose appropriate jump operator.

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