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arXiv:2104.11202 · SciPost Physics

Fermionic duality: General symmetry of open systems with strong dissipation and memory

Valentin Bruch, Konstantin Nestmann, Jens Schulenborg, M. R. Wegewijs

Published Sep 10, 2021• 2 citations
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Abstract

We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schrödinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamiołkowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.

DOI
10.21468/scipostphys.11.3.053

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We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schrödinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamiołkowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.
We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schr\"odinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamio{\l}kowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.
Open fermion systems with energy-independent bilinear coupling to a fermionic environment have been shown to obey a general duality relation [J. Schulenborg et al., Phys. Rev. B 93, 081411 (2016)] which allows for a drastic simplification of time-evolution calculations. In the weak-coupling limit, such a system can be associated with a unique dual physical system in which all energies are inverted, in particular the internal interaction. This paper generalizes this fermionic duality in two ways: we allow for weak coupling with arbitrary energy dependence and describe both occupations and coherences coupled by a quantum master equation for the density operator. We also show that whenever generalized detailed balance holds (Kolmogorov criterion), the stationary probabilities for the dual system can be expressed explicitly in terms of the stationary recurrence times of the original system, even at large bias. We illustrate the generalized duality by a detailed analysis of the rate equation for a quantum dot with strong onsite Coulomb repulsion, going beyond the commonly assumed wide-band limit. We present predictions for (i) the decay rates for transient charge and heat currents after a gate-voltage quench and (ii) the thermoelectric linear-response coefficients in the stationary limit. We show that even for pronouncedly energy-dependent coupling, all nontrivial parameter dependence in these problems is entirely captured by just two well-understood stationary variables, the average charge of the system and of the dual system. Remarkably, it is the latter that often dictates the most striking features of the measurable quantities (e.g., positions of resonances), underscoring the importance of the dual system for understanding the actual one.
The study of electronic transport through strongly confined, interacting open quantum systems has regained considerable interest over the past years. One main motivation behind this concerns the possibility of time-dependently controlled operations on individual electrons, promising applications in, e.g., metrology and electron-based quantum computing. In particular, fundamental questions of quantum thermodynamics and the practical necessity to recover waste heat from nanocircuits have attracted attention towards electronic energy currents.The research articles covered by this thesis contribute to this topic by deriving and exploring a fundamental symmetry relation -- the fermionic duality. This duality applies to the quantum master equation of any locally interacting, fermionic open quantum system tunnel-coupled to non-interacting reservoirs. It yields a crosslink between modes and amplitudes corresponding to the evolution rates in the time-dependent decay of the open-system state. This crosslink involves a mapping between the system of interest and a dual system with inverted environment potentials, local energies, and thus especially inverted interactions. The duality thereby explains many, at first sight unintuitive, transport features and significantly improves their analytic accessability. In particular, we can understand why charge- and energy currents through quantum dots with strong local Coulomb repulsion in fact exhibit features of electron-electron attraction, both in the time-dependent decay after a sudden switch and in the stationary limit.More fundamental insights are obtained by identifying the duality to be rooted in Pauli's exclusion principle and the parity superselection principle. Namely, this implies that the duality is independent of, and hence combinable with many other general symmetries, including particle-hole symmetry, time-reversal symmetry, detailed balance and Onsager reciprocity. Especially the combination with the latter offers a novel perspective on the thermoelectric response of open, locally interacting electronic nanosystems.
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the NNN-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.
We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom H0{\mathcal{H}}_{0}H0​ of the system, and targeting a unique dark state in H0{\mathcal{H}}_{0}H0​. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace H0{\mathcal{H}}_{0}H0​, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.
This paper presents an exact Grassmann stochastic Schr\"odinger equation for the dynamics of an open fermionic quantum system coupled to a reservoir consisting of a finite or infinite number of fermions. We use this stochastic Schr\"odinger equation as a generic open system tool to derive the exact master equation for an electronic system strongly coupled to fermionic reservoirs. The generality and applicability of this Grassmann stochastic approach are justified and exemplified by several quantum fermionic system problems concerning quantum coherence coupled to vacuum or finite-temperature fermionic reservoirs. Our studies show that the quantum coherence property of a quantum dot system can be profoundly modified by the environmental memory.