A new quantitative approach to general bosonic environments in open quantum systems : rigorous error bounds and a novel numerical method for finite systems
A Dissertation on open quantum systems, containing a general review of the current standing of the field, in particular regarding systems interacting with Gaussian environments, as well as original analytical and numerical results. The problem of the impact of a variation in the environmental properties on the reduced dynamics of an open system is investigated analytically, and a new computational technique for the simulation of non-perturbative, non-Markovian open quantum system dynamics based on devising error-certified effective environments is presented.
A Dissertation on open quantum systems, containing a general review of the current standing of the field, in particular regarding systems interacting with Gaussian environments, as well as original analytical and numerical results. The problem of the impact of a variation in the environmental properties on the reduced dynamics of an open system is investigated analytically, and a new computational technique for the simulation of non-perturbative, non-Markovian open quantum system dynamics based on devising error-certified effective environments is presented.
Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. A common approach to the analysis of such systems is to approximate the non-Markovian environment by discrete bosonic modes thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been proposed in previous works [D. Tamascelli et al, PRL (2012), A. W. Chin et al, J. of Math. Phys (2010)], the resulting approximation lacks rigorous convergence guarantees. In this paper, we initiate a rigorous study of the convergence properties of these methods. We show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that, for most physically interesting models of non-Markovian environments, the approximation error falls off polynomially with the number of modes. Our results lend rigor to numerical methods used for approximating non-Markovian quantum dynamics and allow for a quantitative assessment of classical as well as quantum algorithms in simulating non-Markovian quantum systems.
In this thesis, we discuss aspects of the simulation, characterisation, and control of the dynamics of open quantum systems. The latter are systems governed by quantum mechanical laws and interacting with an environment, which is typically much larger than the system itself. According to the standard approach, one is usually able to perform a suitable average over the environmental degrees of freedom, resulting in an effective description of the main system dynamics, which accounts for the effects of the interaction with the surroundings. In standard scenarios, usually studied within the so-called Born-Markov approximation, the system-environment coupling is weak and such that we can perform a neat separation of timescales: the environmental dynamics is assumed to be fast compared to the typical evolution timescale of the system of interest. However, nowadays we are able to inspect physical scenarios where the usual Born-Markov approximation breaks down: the interaction between system and environment can be strong, likely leading to non-negligible memory (non-Markovian) effects. In this work, we discuss some instances in which we cannot work in the standard Born-Markov regime. We introduce and analyse some numerical and analytical techniques to characterise and simulate non-standard scenarios. We show that, even in the Markovian regime, the traditional formulation of thermodynamic irreversibility shows flaws and inconsistencies. We thus use phase-space methods to assess the role of initial correlations shared by the two parties of a bipartite harmonic system in the entropy production rate, including non-Markovian effects. Furthermore, we show that certain interactions enable to redraw the boundaries between the system and the environment in an effective manner, resulting in a new picture where the system degrees of freedom are augmented, while the residual environment is rearranged in such a way that the Born-Markov approximation is recovered. This analytical technique, known as reaction coordinate mapping approach, is employed in this work to show that, upon an accurate choice of the parameter regime, a spin-boson model can serve as a quantum analogue simulator of non-Markovian multiphoton Jaynes-Cummings models. These systematic studies shed light on the thermodynamic characterisation of open quantum systems, as well as on their numerical simulation.
Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. The non-Markovian environment is often approximated by discrete bosonic modes, thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been previously proposed, the resulting approximation lacks rigorous and general convergence guarantees. In this Letter, we show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that this approximation error typically falls off polynomially with the number of modes. Our results lend rigor to classical and quantum algorithms for approximating non-Markovian dynamics.
We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.
The subject of this thesis is the modelling of non-Markovian open quantum systems, in particular those that display memory effects due to time-delayed interactions. I study these systems in the context of a model of an open quantum system that interacts with a bosonic environment by way of particle exchanges. While the model is flexible enough to describe quite general memory effects, this work focuses to a large extent on a single prototypical system. This system emits into a coherent feedback loop with a time delay, and it captures the essential phenomena we aim to model. Within the context of this general model, I develop a general approach to spontaneous emission from singly-excited systems and show how to calculate the corresponding emission spectra. I also evaluate two commonly-used perturbative techniques in the context of spontaneous emission from singly-excited systems, coming to the conclusion that perturbation theory is of limited use in modelling open quantum systems with delayed feedback. I then turn attention to exact equations of motion for more general non-Markovian open quantum systems. I derive Heisenberg–Langevin equations that describe a large class of such systems, and solve these equations analytically for a series of simple bosonic systems. I also use the theory of cascaded open quantum systems to simulate systems with delayed coherent feedback in the limit of long time delay, and compare the results of these simulations with the analytic solutions obtained in the Heisenberg picture. Finally, I show that once we have a solution to a given Heisenberg–Langevin equation we can derive an alternative, time-local equation of motion that describes equivalent dynamics, and that these time-local Heisenberg–Langevin equations can in turn be used to derive corresponding time-local master equations. These master equations provide an exact description of the non-Markovian dynamics, but this treatment is limited to linear, bosonic systems—those systems for which the Heisenberg–Langevin equations can be solved in closed form. I discuss the prospect of generalising these master equations to take into account nonlinear and fermionic systems.