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arXiv:1909.10418 · Annals of Physics

A new approach for open quantum systems based on a phonon number representation of a harmonic oscillator bath

M. Tokieda, K. Hagino

Published Nov 2, 2019• 12 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2019.168005

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This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. The obtained propagator is then used to derive the time evolution equation for the density matrix describing the system. The method used to derive the propagator of the open quantum system considered in this paper shows promise for analyzing open quantum systems, due to its ease of use and mathematical rigor.
Recent progress on qubit manipulation allows application of periodic driving signals on qubits. In this study, a harmonic driving field is added to a Rabi dimer to engineer photon and qubit dynamics in a circuit quantum electrodynamics device. To model environmental effects, qubits in the Rabi dimer are coupled to a phonon bath with a sub-Ohmic spectral density. A nonperturbative treatment, the Dirac-Frenkel time-dependent variational principle together with the multiple Davydov D2 ansatz, is employed to explore the dynamical behavior of the tunable Rabi dimer. In the absence of the phonon bath, the amplitude damping of the photon number oscillation is greatly suppressed by the driving field, and photons can be created, thanks to the resonance between the periodic driving field and the photon frequency. In the presence of the phonon bath, one can still change the photon numbers in two resonators and indirectly alter the photon imbalance in the Rabi dimer by directly varying the driving signal in one qubit. It is shown that qubit states can be manipulated directly by the harmonic driving. The environment is found to strengthen the interqubit asymmetry induced by the external driving, opening up a new venue to engineer the qubit states.
This thesis is concerned with the study of quantum systems coupled linearly to a continuous bath of oscillators examples of which are the spin-boson model and the Nelson model. The main theme throughout is to develop a better understanding of the properties of the bath of oscillators such that more efficient representations of it can be made to facilitate the understanding of the system dynamics. The main difficulty in simulating the system dynamics is that the bath of oscillators composes an infinite number of degrees of freedom. In this thesis, we investigate the mathematical properties of an approach in which the bath modes are written as a semi-infinite chain of nearest neighbour interacting harmonic oscillators such that the efficient time dependent density-matrix renormalisation group (t-DMRG) methods can be applied for simulation. In the first section, we show how there are many different ways to represent the bath as semiinfinite chains and prove that seemingly unrelated methods can all be achieved using the same mathematical formalism. We show that in an iterative process the bath can be transformed into a chain of oscillators with nearest neighbour interactions. This is achieved using the formalism of orthogonal polynomials. This allows one to define a sequence of residual spectral densities at each site along the chain. We show that this sequence of residual spectral densities is provided by the so-called ”sequence of secondary measures”. We derive a systematic procedure to obtain the spectral density of the residual bath in each step. We find that these residual spectral densities are related to an old abstract problem in mathematics known as the ”secondary measures”. We solve this problem from the field of orthogonal polynomials to give an explicit expression for the residual spectral densities and go on to prove that these functions converge under very general conditions. That is, the asymptotic part of the chain is universal, translation invariant with universal spectral density. These results suggest efficient methods for handling the numerical treatment of the residual bath. In the second section, we take a different approach. Rather than studying the properties of residual baths, we look at how system observables are affected by truncating the semi-infinite chain of harmonic oscillators to a finite length chain. By developing locality bounds for the dynamics, we derive an upper bound to the error introduced by such a truncation and show that for all finite times it can be made arbitrarily small by including a sufficient number of harmonic oscillators 3 before truncating. Furthermore, it is shown that the speed at which the system communicates with different harmonic oscillators in the chain is proportional to the maximum frequency of the environment but that this speed also depends on the particular version of the chain. These bounds are given for when the dynamics are calculated in the Interaction picture and in the Schrodinger picture.