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arXiv:2006.03324 · Physical review. B./Physical review. B

Nonequilibrium non-Markovian steady states in open quantum many-body systems: Persistent oscillations in Heisenberg quantum spin chains

Regina Finsterhölzl, Manuel Katzer, Alexander Carmele

Published Nov 16, 2020• 12 citations
arXivHTMLPDF

Abstract

We investigate the effect of a non-Markovian, structured reservoir on an open Heisenberg spin chain. We establish a coherent self-feedback mechanism as the reservoir couples frequency dependently to the spin chain. Thus, loss and driving take place due to the interaction of the spin chain with its own past. This new paradigm of non-Markovian imposed boundary driving allows discussion of a new kind of nonequilibrium steady state. We show that for certain parameters, even in the long-time limit, persistent oscillations occur within the chain. Moreover, we demonstrate that the conditions for these oscillations and excitation trapping depend on the characteristics of the chain, thus making it possible to characterize a chain by detection of its emitted signal under influence of self-feedback.

DOI
10.1103/physrevb.102.174309

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We investigate the effect of a non-Markovian, structured reservoir on an open Heisenberg spin chain. We establish a coherent self-feedback mechanism as the reservoir couples frequency dependently to the spin chain. Thus, loss and driving take place due to the interaction of the spin chain with its own past. This new paradigm of non-Markovian imposed boundary driving allows discussion of a new kind of nonequilibrium steady state. We show that for certain parameters, even in the long-time limit, persistent oscillations occur within the chain. Moreover, we demonstrate that the conditions for these oscillations and excitation trapping depend on the characteristics of the chain, thus making it possible to characterize a chain by detection of its emitted signal under influence of self-feedback.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
Many-body localization, which prevents a many-body quantum systems from reaching thermal equilibrium has opened the possibility to study out-of-equilibrium dynamics of isolated quantum systems, leading to new phases of matter such as the discrete time crystal. However, little is known about these effects when the quantum many-body system is coupled to an external environment. An intuitive assumption is that out-of-equilibrium dynamics and quantum coherences would not survive due to thermalization with external degrees of freedom. The present work focuses on open quantum many-body systems and aims at the question how an external reservoir can preserve such out-of-equilibrium effects. In the first part, common generic reservoirs are assumed to focus on many-body effects of the systems of interest. As a first many-body system, an optically driven many-emitter phonon laser is investigated. The full quantum mechanical treatment reveals additional resonances caused by collective effects of the many-emitter setup. Optically addressing these resonances results in an enhancement of phonon intensities within the acoustic cavity. The Heisenberg spin-chain has achieved the role of a standard model to study many-body localization. As a second many-body system, the model is generalized to an open quantum system via boundary reservoirs which are driving the system out-of equilibrium and a spin-current is induced. When driven far from equilibrium, the transport is very much dependent on the external reservoirs. In contrast, it is demonstrated that a setup with long-range coupling between the spins shows transport behavior independent of the external environment. In the second part of this work, the coupling to the reservoir is structured by assuming a boundary condition. Experimentally this is achieved by placing a distant mirror close to the system. A common Born-Markov approximation leading to factorizing system and reservoir states is insufficient to describe these feedback dynamics. In this work, an approach is used to treat the reservoir states as part of the many-body system which is based on matrix product states using the quantum stochastic Schrödinger equation. Focusing on the reservoir statistics of a single two-level system, it is demonstrated that by combining time-dependent optical excitation with time-delayed feedback via structuring the reservoir, individual photon probabilities are controlled by changing external degrees of freedom. Finally, this method is generalized to a many-body system including a coupling to an external reservoir beyond Born-Markov approximation. It is shown that feedback dynamics stabilize a discrete time crystal against dissipation to the environment. Out-of-equilibrium dynamics and many-body localization of the discrete time crystal survive for long times even in the presence of an external environment and become independent of the coupling to the reservoir. The developed method benefits from small entanglement entropy of the many-body localized system as well as from a suppression of entanglement between system and reservoir due to the feedback dynamics. This allows to consider up to forty spins with individual structured reservoirs in a numerically exact manner.
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We derive a non-Markovian master equation for the evolution of a class of open quantum systems consisting of quadratic fermionic systems coupled to wideband reservoirs. This is done by providing an explicit correspondence between master equations and nonequilibrium Green's function approaches. Our findings permit us to address non-Markovian regimes characterized by negative decoherence rates and to characterize the dynamics with respect to a recently proposed measurement of ``non-Markovianity.'' We study the real-time dynamics and the steady-state solution of two illustrative models: a tight-binding electronic model and XY\mathit{XY}XY spin chains. The rich set of nonequilibrium steady-state phases encountered extends previous results to the non-Markovian regime.
Quantum simulation of open quantum systems offers a pathway towards better understanding various non-equilibrium physics that would otherwise be challenging to study. While most open quantum systems studied are modeled as being memory-less (obeying the Markov approximation), real baths generally are influenced by the system-bath interaction, and some systems existing in structured non-Markovian environments can display novel behavior as a result. Here we utilize a trapped ion quantum simulator to simulate a single spin-1/21/21/2 driven-dissipative system with a non-Markovian dissipation channel, and experimentally compare steady-states to those from an analogous Markovian bath. We observe that a non-Markovian dissipative channel can shift the steady-state even for a single qubit, to a regime inaccessible for Markovian dissipation. The techniques used here are compatible with many-body extensions of the model, which can not be simulated efficiently on a classical computer in general. Our work also opens up new possibilities in quantum reservoir engineering beyond the Markovian regime.
We conjecture that nonequilibrium boundary conditions generically trigger long-range order in nonequilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin-1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long-range spin-spin correlations in an integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long-range order also typically emerges after breaking the integrability of the model.