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arXiv:2007.03234 · Physical Review A

Discrete memory kernel for multitime correlations in non-Markovian quantum processes

Mathias R. Jørgensen, Felix A. Pollock

Published Nov 4, 2020• 24 citations
arXivHTMLPDF

Abstract

Efficient simulations of the dynamics of open systems is of wide importance for quantum science and technology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multitime measurement scenarios. The transfer-tensor method sets out to compute the state of an open few-body quantum system at long times, given that only short-time system trajectories are available. Here, we show that the transfer-tensor method can be extended to processes which include multiple interrogations (e.g., measurements) of the open system dynamics as it evolves, allowing us to propagate high-order short-time correlation functions to later times, without further recourse to the underlying system-environment evolution. Our approach exploits the process-tensor description of open quantum processes to represent and propagate the dynamics in terms of an object from which any multitime correlation can be extracted. As an illustration of the utility of the method, we study the buildup of system-environment correlations in the paradigmatic spin-boson model and compute steady-state emission spectra, taking fully into account system-environment correlations present in the steady state.

DOI
10.1103/physreva.102.052206

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Efficient simulations of the dynamics of open systems is of wide importance for quantum science and technology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multitime measurement scenarios. The transfer-tensor method sets out to compute the state of an open few-body quantum system at long times, given that only short-time system trajectories are available. Here, we show that the transfer-tensor method can be extended to processes which include multiple interrogations (e.g., measurements) of the open system dynamics as it evolves, allowing us to propagate high-order short-time correlation functions to later times, without further recourse to the underlying system-environment evolution. Our approach exploits the process-tensor description of open quantum processes to represent and propagate the dynamics in terms of an object from which any multitime correlation can be extracted. As an illustration of the utility of the method, we study the buildup of system-environment correlations in the paradigmatic spin-boson model and compute steady-state emission spectra, taking fully into account system-environment correlations present in the steady state.
Abstract The non-Markovianity of an arbitrary open quantum system is analyzed in reference to the multi-time statistics given by its monitoring at discrete times. On the one hand, we exploit the hierarchy of inhomogeneous transfer tensors (TTs), which provides us with relevant information about the role of correlations between the system and the environment in the dynamics. The connection between the TT hierarchy and the CP-divisibility property is then investigated, by showing to what extent quantum Markovianity can be linked to a description of the open-system dynamics by means of the composition of one-step TTs only. On the other hand, we introduce the set of stochastic TT transformations associated with local measurements on the open system at different times and conditioned on the measurement outcomes. The use of the TT formalism accounts for different kinds of memory effects in the multi-time statistics and allows us to compare them on a similar footing with the memory effects present in non-monitored non-Markovian dynamics, as we illustrate on a spin-boson case study.
We introduce a general numerical method to compute the dynamics and multitime correlations of chains of quantum systems, where each system may couple strongly to a structured environment. The method combines the process tensor formalism for general (possibly non-Markovian) open quantum systems with time-evolving block decimation for one-dimensional chains. It systematically reduces the numerical complexity originating from system-environment correlations before integrating them into the full many-body problem, making a wide range of applications numerically feasible. We illustrate the power of this method by studying two examples. First, we study the thermalization of individual spins of a short XYZ Heisenberg chain with strongly coupled thermal leads. Our results confirm the complete thermalization of the chain when coupled to a single bath, and they reveal distinct effective temperatures in low-, mid-, and high-frequency regimes when the chain is placed between a hot and a cold bath. Second, we study the dynamics of diffusion in a longer XY chain, when each site couples to its own bath.
The transfer tensor method is a versatile tool for analyzing and propagating general open quantum systems. It captures in a compact manner all memory effects in a non-Markovian system through a straightforward transformation of a set of dynamical maps. Transfer tensors provide the exact convolutional propagator associated with a given time discretization over the past evolution of an open quantum system. Here we show that, for any finite time discretization, the memory kernel of the Nakajima Zwanzig equation deviates from the exact transfer tensors, although both converge in the continuous-time limit, as expected. We examine this behaviour in the context of an analytically solvable model: a two level atom resonant with a lossy cavity in the Jaynes Cummings limit. The atomic dynamics separate into two decoupled degrees of freedom -- the coherence and the population inversion. We derive exact expressions for the dynamical map, the transfer tensors and the memory kernel governing the coherence, and we relate them to their counterparts for the population inversion. As a function of the ratio between the cavity loss rate and the atom-cavity coupling strength, we identify regions of enhanced non-Markovianity in which the system can be described as fully Markovian for certain time-step choices.
Numerical methods for obtaining exact dynamics of non-Markovian open quantum systems are mostly limited to either small systems or to short-time evolution only. Here, we propose a new algorithm for computing process tensors--matrix product operator (MPO) representations that capture the environment influence--which achieves greatly enhanced computational scalings with system size, while maintaining linear scaling with simulation length. We build on recent developments in the field which allow for long-time evolutions through process tensors which have a time-translational invariance. These can be built for general Gaussian environments and generic coupling operators with the system using infinite time-evolving block decimation (iTEBD). We introduce a modified iTEBD algorithm using intermediate compression steps which bring down the computation time scaling with system size ddd from O(d8)\mathcal{O}(d^8)O(d8) to O(d4)\mathcal{O}(d^4)O(d4), as well as significantly lowering the required memory. To illustrate the power of this method, we apply it to the problem of dispersive qubit readout in circuit QED, which was previously out-of-reach numerically. The full treatment of the measurement resonator, which requires a large system space, combined with the long simulation times precipitated by the separation of timescales between the measurement drive and the environment dissipation, is now possible. The algorithm we introduce not only allows for capturing non-Markovian dynamics in large open quantum systems, but also further extends all the existing capabilities of process tensors, for example in quantum optimal control, or in computation of multi-time correlations or of steady states, to more complex systems with tens of levels.
The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long time scales. The non-Markovian transfer tensor method (TTM) demonstrates the coherent-to-incoherent transition as a function of the strength of quantum dissipation and predicts the noncanonical equilibrium distribution due to the system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig equation and, therefore, can be used to reconstruct the dynamical operators (the system Hamiltonian and memory kernel) from quantum trajectories obtained in simulations or experiments. The concept underlying the approach can be generalized to physical observables with the goal of learning and manipulating the trajectories of an open quantum system.
The process tensor framework to open quantum systems provides the most general description of multi-time correlations in non-Markovian quantum dynamics. A compressed representation of a process tensor in terms of matrix product operators (MPO) can be used for numerically exact calculations of multi-time correlation functions in systems strongly coupled to a non-Markovian reservoir. We show here that the numerical scaling for computing multi-dimensional spectra can be significantly improved using a time-translation invariant MPO representation of the process tensor obtained from the uniform time-evolving matrix product operator (uniTEMPO) method. In particular, this approach provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution. We calculate linear and 2D electronic spectra for an example system and discuss the performance and numerical scaling of our simulations.