Study of Temporal Correlations in Non-Unitary Quantum Dynamics

Varma, Anant Vijay (2022) Study of Temporal Correlations in Non-Unitary Quantum Dynamics. PhD thesis, Indian Institute of Science Education & Research Kolkata.

Text (PhD thesis of Anant Vijay Varma (16RS020)) 16RS020.pdf - Submitted Version Restricted to Repository staff only Download (15MB)

Abstract

The dynamics of a quantum system coupled to a bath or an environment can conform to a variety of correlations among measurement outcomes of observables measured at two different times. Bounds on the ensemble averages of these correlations which are set by foundational aspects of quantum theory can lead to the identification of the dynamics of the system as classical or quantum. All possible dynamics (including unitary) of a quantum system that are completely positive and trace preserving (CPTP) are considered physical within the standard framework of quantum mechanics. However, a system undergoing dynamics which is outside the realm of CPTP dynamical maps i.e. strictly positive, is generally believed to be unphysical and hence has not been explored much. It turns out that these dynamics can also be motivated within the framework of quantum mechanics provided the dynamics are effectively generated from the post-selected ensemble of either unitary dynamics or Lindbladian dynamics. Non- Hermitian dynamics is one such class of dynamics which is only positive. One of the profound manifestations of these dynamics is the violation of the bounds set by CPTP dynamics on the ensemble averages of correlations measured at two different times and is the central focus of this thesis. In this thesis, we begin with the study of a Two-Level System (TLS) and demonstrate when can such a system behave classically i.e. follows both Macro-realism & non-invasive measurability. We probe the classical nature of this system by the celebrated test of Macrorealism known as “Leggett-Garg Inequality” (LGI) or “temporal Bell’s Inequality”. It is known that violation of this inequality can be considered as testament to the non-classical behaviour of the system, given that all the measurements are performed in a non-invasive manner [1]. We also illustrate this fact for a TLS undergoing unitary dynamics. However, systems where the dynamics of the state of a TLS is governed by non-Hermitian Hamiltonian with real eigenvalues (“PT-symmetric”) lead to a violation beyond the bound set by quantum dynamics of an isolated TLS on LGI. This violation is a clear indication of extreme temporal correlations which are unexpected both in standard classical and quantum theories of few degrees of freedom. We present a clear understanding of this elusive phenomenon in terms of trajectories of state evolution and their corresponding speed of state evolution. We further move on to show that the class of quantum systems represented by Hamiltonians with imaginary pair of eigenvalues (“PT-broken”), exhibits unexpected temporal correlations beyond quantum bound and violates LGI maximally (algebraic bound). We identify underlying responsible factor for this violation as the combination of maximum and minimum speed of evolution of the state, where minimum speed of evolution can serve as an order parameter which is finite on the PT-symmetric (real eigenvalues) side and identically zero on the PT-broken side. Utilizing Naimark’s dilation method [2], we then demonstrate that how these non-Hermitian dynamics can be fitted well within the framework of quantum mechanics by “embedding” a TLS into a higher dimensional Hilbert space. In particular, we discuss three different recipes to simulate the non-Hermitian dynamics. We also demonstrate the possibility of experimentally realizing both PT-symmetric and PT-broken quantum TLS dynamics using three atomic levels coupled to the cavity modes representing an open quantum system undergoing Lindbladian dynamics. Since we found that the non-Hermitian dynamics of a TLS is experimentally feasible, it is natural to extend this idea of “embedding” to simulate many-body non-Hermitian dynamics within the framework of quantum mechanics. To this end we apply this “embedding” technique to realize the non-Hermitian dynamics of “N” spin-1/2 system, leading to “N+1” spin-1/2, “all body” interacting system and probe its many-body states for “quantumness”. We show that such a system exhibits quantum effects at macroscopic scale and manifests as Orthogonality Catastrophe in the thermodynamic limit. However, this “embedding” technique is limited to non-Hermitian systems with real eigenspectrums (“PT-symmetric”). We finally contrast the nH dynamics mentioned so far (which do not conform to a CPTP map) with the dynamics which conform to CPTP maps. We start with a two level quantum system coupled to additional quantum degrees of freedom (DOF). We derive an algebraic expression for the LGI for the case where the nature of the interaction between TLS and the additional quantum DOFs is such that the corresponding dynamical map is unital (purity of the state of TLS is strictly a non-increasing function of time). This analysis divides the unital maps into two sets where, in one of the sets temporal correlations can be simulated using classical set-ups, and in another contrasting set the dynamics is quantum mechanical. With the aid of “No-signaling in time” (NSIT) criterion, the necessary and sufficient conditions are also derived for the violation of Macro-realism (classicality) for a TLS, whose dynamics conforms to unital maps and is Markovian in nature as well.

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