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arXiv:2012.01856

Finite-frequency noise in non-Markovian systems: A Markovian embedding approach

Krzysztof Ptaszyński

Published Dec 3, 2020• 0 citations
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Abstract

I present an approach to calculate the finite-frequency quantum noise in systems strongly coupled to structured reservoirs based on the Markovian embedding. This technique consists of mapping of the non-Markovian dynamics of the original system onto the Markovian dynamics of the extended supersystem. The applicability of this method is demonstrated on a single electronic level coupled to a reservoir with an energy-dependent density of states. This model can be mapped onto an equivalent double-level system whose dynamics is governed by an exact Markovian master equation. It is demonstrated that the presented approach provides a way to describe genuinely quantum effects, such as the asymmetry of the absorption and the emission noise, as well as to deal with non-Markovian effects and intra-system interactions on equal footing.

DOI
10.48550/arxiv.2012.01856

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I present an approach to calculate the finite-frequency quantum noise in systems strongly coupled to structured reservoirs based on the Markovian embedding. This technique consists of mapping of the non-Markovian dynamics of the original system onto the Markovian dynamics of the extended supersystem. The applicability of this method is demonstrated on a single electronic level coupled to a reservoir with an energy-dependent density of states. This model can be mapped onto an equivalent double-level system whose dynamics is governed by an exact Markovian master equation. It is demonstrated that the presented approach provides a way to describe genuinely quantum effects, such as the asymmetry of the absorption and the emission noise, as well as to deal with non-Markovian effects and intra-system interactions on equal footing.
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.
Abstract By virtue of the superoperator technique, we solve analytically the non-Markovian master equation that describes single-mode continuous variable system interacting with a structure reservoir. Our analytical solution allows us to investigate the non-Markovian effects on the dynamics of quantum coherence and non-classicality of Schrödinger cat states embedded in various types of structured reservoirs. We show that the dynamical behaviors of the fidelity, entanglement potentials, and higher-order antibunching of such states can be divided into two phases: they decay rapidly within a short time, but would either vanish completely or tend to a nonzero steady-state value in the long-time limit depending on the initial Schrödinger cat states and system parameters. We also show that by examining the Wigner function and higher-order sub-Poissonian statistics, the super-Ohmic environment can better help to protect the non-classicality, whereas the sub-Ohmic environment causes the disappearance of the non-classicality. Our approach is concise and can generalize to other open quantum system coupled to non-Markovian environment at finite-temperature or to linear non-Markovian Gaussian system.
Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.
The exact non-Markovian dynamics of a microcavity strongly coupled to a general reservoir at arbitrary temperature are studied. With the exact master equation for the reduced density operator of the cavity system, we analytically solve the time evolution of the cavity state and the associated physical observables. We show that the non-Markovian dynamics are completely determined by the propagating (retarded) and correlation Green functions. Comparing the non-Markovian behavior at finite temperature with those at the zero-temperature limit or Born-Markov limit, we find that the non-Markovian memory effect can dramatically change the coherent and thermal dynamics of the cavity. We also numerically study the dissipation dynamics of the cavity through the mean mode amplitude decay and the average photon number decay in the microwave regime. It is shown that the strong coupling between the cavity and the reservoir results in a long-time dissipationless evolution to the cavity field amplitude, and its noise dynamics undergo a critical transition from the weak to the strong coupling due to the non-Markovian memory effect.
Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.
We develop an exact framework for describing the non-Markovian dynamics of an open quantum system interacting with an environment modeled by a generalized spectral density function.The approach relies on mapping the initial system onto an auxiliary configuration, comprising the original open system coupled to a small number of discrete modes, which in turn are each coupled to an independent Markovian reservoir.Based on the connection between the discrete modes and the poles of the spectral density function, we show how expanding the system using the discrete modes allows for the full inclusion non-Markovian effects within an enlarged open system whose dynamics is governed by an exact Lindblad master equation.Initially we apply this result to obtain a generalization of the pseudomode method [B.M. Garraway, Phys. Rev. A. 55, 2290 (1997)] in cases where the spectral density function has a Lorentzian structure.For many other types of spectral density function, we extend our proof to show that an open system dynamics may be modeled physically using discrete modes which admit a non-Hermitian coupling to the system and for such cases determine the equivalent master equation to no longer be of Lindblad form.For applications involving two discrete modes, we demonstrate how to convert between pathological and Lindblad forms of the master equation using the techniques of the pseudomode method.