We introduce a general numerical method to compute dynamics and multi-time 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 (TEBD) for 1D 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 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 an longer XY chain, when each site couples to its own bath.
We propose simple protocols for performing quantum noise spectroscopy based on the method of transfer tensor maps (TTM), [Phys. Rev. Lett. 112, 110401 (2014)]. The TTM approach is a systematic way to deduce the memory kernel of a time-nonlocal quantum master equation via quantum process tomography. With access to the memory kernel it is possible to (1) assess the non-Markovianity of a quantum process, (2) reconstruct the noise spectral density beyond pure dephasing models, and (3) investigate collective decoherence in multiqubit devices. We illustrate the usefulness of TTM spectroscopy on the …
We calculate the exact many-body time dynamics of polaritonic states supported by an optical cavity filled with organic molecules. Optical, vibrational and radiative processes are treated on an equal footing employing the Time-Dependent Variational Matrix Product States algorithm. We demonstrate signatures of non-Markovian vibronic dynamics and its fingerprints in the far-field photon emission spectrum at arbitrary light-matter interaction scales, ranging from the weak to the strong coupling regimes. We analyse both the single and many-molecule cases, showing the crucial role played by the col…
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-dependent 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 to discuss a new kind of non-equilibrium 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 osci…
The decay of a local spin excitation in an inhomogeneous spin chain is evaluated exactly: I) It starts quadratically up to a spreading time t_{S}. II) It follows an exponential behavior governed by a self-consistent Fermi Golden Rule. III) At longer times, the exponential is overrun by an inverse power law describing return processes governed by quantum diffusion. At this last transition time t_{R} a survival collapse becomes possible, bringing the polarization down by several orders of magnitude. We identify this strongly destructive interference as an antiresonance in the time domain. These …
We propose a simple structure for stationary non-Markovian quantum chains in the framework of collisional dynamics of open quantum systems. To this end, we modify the microscopic Markovian system--reservoir model, consider multiple collisions with each of the molecules with an overlap between the collisional time intervals. We show how the equivalent Markovian quantum chain can be constructed with the addition of satellite quantum memory to the system. We distinguish quantum from classical non-Markovianity. Moreover, we define the counts of non-Markovianity by the required number of satellite …
With continuing improvements on the quality of fabricated quantum devices, it becomes increasingly crucial to analyze noisy quantum process in greater details such as characterizing the non-Markovianity in a quantitative manner. In this work, we propose an experimental protocol, termed Spectral Transfer Tensor Maps (SpecTTM), to accurately predict the RHP non-Markovian measure of any Pauli channels without state-preparation and measurement (SPAM) errors. In fact, for Pauli channels, SpecTTM even allows the reconstruction of highly-precised noise power spectrum for qubits. At last, we also disc…
We study the dynamics of entanglement in the extended cluster spin-1/2 XX chain, equivalent to a 1D spin-1/2 XX model with three-spin interaction (TSI). Selecting the nearest neighbor pair spins as an open quantum system, the rest of the chain plays the role of environment. The two-spin Heisenberg and the TSI interaction are responsible for coupling between system and environment. We show the existence of a critical value in the TSI, where the dynamics of concurrence changes from Markovian to the non-Markovian. In the region with the non-Markovian dynamics, entanglement sudden death in the sys…
We analyze the dynamics of a spin 1/2 subsystem coupled to a spin chain. We simulate numerically the full quantum many-body system for various sets of parameters and initial states of the chain, and characterize the divisibility of the subsystem dynamics, i.e. whether it is Markovian and can be described by a (time dependent) master equation. We identify regimes in which the subsystem admits such Markovian description, despite the many-body setting, and provide insight about why the same is not possible in other regimes. Interestingly, coupling the subsystem at the edge, instead of the center,…
Process Tensor Approaches to Non-Markovian Quantum Dynamics
SUPA, School of Physics and Astronomy, University of St Andrews（圣安德鲁斯大学物理与天文学学院）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）；Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Department of Chemistry, Columbia University（哥伦比亚大学化学系）
In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, non-interacting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor -- a more general representation of a quantum stochastic process -- describing the evolution. We then use this connection to motivate a tensor network algorithm for the simulation of multi-time correlations in open systems, building on recent work where the influence functional is represented in terms of time …
Understanding the physical significance and probing the global invariants characterizing quantum topological phases in extended systems is a main challenge in modern physics with major impact in different areas of science. Here, a quantum-information-inspired probing method is proposed where topological phase transitions are revealed by a non-Markovianity quantifier. The idea is illustrated by considering the decoherence dynamics of an external read-out qubit that probes a Su-Schrieffer-Heeger (SSH) chain with either pure dephasing or dissipative coupling. Qubit decoherence features and non-Ma…
We develop in the weak coupling approximation a quasi-non-Markovian master equation and study the phenomenon of decoherence during the operation of a controlled-not (CNOT) quantum gate in a quantum computer model formed by a linear chain of three nuclear spins system with second neighbor Ising interaction between them. We compare with the behavior of the Markovian counterpart for temperature different from zero (thermalization) and at zero temperature for low and high dissipation rates. At low dissipation there is a very small difference between Markovian and quasi no-Markovian at any temperat…
We show how to learn structures of generic, non-Markovian, quantum stochastic processes using a tensor network based machine learning algorithm. We do this by representing the process as a matrix product operator (MPO) and train it with a database of local input states at different times and the corresponding time-nonlocal output state. In particular, we analyze a qubit coupled to an environment and predict output state of the system at different time, as well as reconstruct the full system process. We show how the bond dimension of the MPO, a measure of non-Markovianity, depends on the proper…
Uniform process tensor approach for the calculation of multi-time correlation functions of non-Markovian open systems
TUD Dresden University of Technology（德累斯顿工业大学）；Institute of Theoretical Physics, TUD Dresden University of Technology（德累斯顿工业大学理论物理研究所）；Institut für Physik und Astronomie, Technische Universität Berlin（柏林工业大学物理与天文学系）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
In this work we provide a recursive method of calculating the wavefunction of a XX spin chain coupled at both ends to non-Markovian reservoirs with arbitrary spectral density. The method is based on the appropriate handling of the time-dependent Schrodinger's equations of motion in Laplace space and leads to closed form solutions of the transformed amplitudes, for arbitrary chain lengths as well as arbitrary initial conditions, within the single-excitation subspace. Results on the dynamical as well as state transfer properties of the system for various combinations of parameters are also prese…
Efficient construction of time-invariant process tensors for simulating high-dimensional non-Markovian open quantum systems
Département de Physique de l’ENS Lyon, ENS de Lyon（里昂高等师范学校物理系）；SUPA, School of Physics and Astronomy, University of St Andrews（圣安德鲁斯大学物理与天文学学院）；Sorbonne Université, Institut des NanoSciences de Paris（索邦大学巴黎纳米科学研究所）；CNRS, Institut des NanoSciences de Paris（法国国家科学研究中心巴黎纳米科学研究所）
Correlations of an environment are crucial for the dynamics of non-Markovian quantum systems, which may not be known in advance. In this paper, we propose a gradient algorithm for identifying the correlations in terms of time-varying damping rate functions in a time-convolution-less master equation for spin chains. By measuring time trace observables of the system, the identification procedure can be formulated as an optimization problem. The gradient algorithm is designed based on a calculation of the derivative of an objective function with respect to the damping rate functions, whose effect…
The classical capacity of a quantum channel with arbitrary Markovian correlated noise is evaluated. For the general case of a channel with long-term memory, which corresponds to a Markov chain which does not converge to equilibrium, the capacity is expressed in terms of the communicating classes of the Markov chain. For an irreducible and aperiodic Markov chain, the channel is forgetful, and one retrieves the known expression for the capacity.
Nanoscale devices - either biological or artificial - operate in a regime where the usual assumptions of a structureless, Markovian, bath do not hold. Being able to predict and study the dynamics of such systems is crucial and is usually done by tracing out the bath degrees of freedom, which implies losing information about the environment. To go beyond these approaches we use a numerically exact method relying on a Matrix Product State representation of the quantum state of a system and its environment to keep track of the bath explicitly. This method is applied to a specific example of inter…
Efficient simulations of the dynamics of open systems is of wide importance for quantum science and tech-nology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multi-time 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 showthat the transfer-tensor method can be extended to processes which include multiple interrogations (e.g. measurements) of the open system dynamics as it evolves, allo…
We study the dissipative dynamics of $N$ quantum spins with Lindblad generator consisting of operators scaling as fluctuations, namely with the inverse square-root of $N$. In the large $N$ limit, the microscopic dissipative time-evolution converges to a non-Markovian unitary dynamics on strictly local operators, while at the mesoscopic level of fluctuations it gives rise to a dissipative non-Markovian dynamics. The mesoscopic time-evolution is Gaussian and exhibits either a stable or an unstable asymptotic character; furthermore, the mesoscopic dynamics builds correlations among fluctuations t…
Quantum transport along a tight-binding chain connected to Markovian reservoirs
Kirensky Institute of Physics, Federal Research Centre KSC SB RAS（基连斯基物理研究所，俄罗斯科学院西伯利亚分院克拉斯诺亚尔斯克科学中心联邦研究中心）；IRC SQC, Siberian Federal University（西伯利亚联邦大学量子系统与量子控制国际研究中心）；School of Engineering Physics and Radio Electronics, Siberian Federal University（西伯利亚联邦大学工程物理与无线电电子学院）
We study non-Markovian enhancement effects in the spontaneous emission of a collective excitation in a linear chain of up to 100 qubits coupled to a 1D waveguide. We find that for a critical separation of qubits, the system exhibits super-superradiant (SSR) behavior leading to collective decay stronger than the usual Dicke superradiance. Here, time-delayed coherent quantum feedback effects are at play on top of the usual Dicke superradiance effects. We find a linear scaling for the SSR decay rate with increasing qubit number $N$ such that $Γ_{\rm SSR} \sim 2.277 N γ_0$, where $γ_0$ is the sing…
We put forward a framework to study the dynamics of a chain of interacting quantum particles affected by individual or collective multi-mode environment, focussing on the role played by the environmental quantum correlations over the evolution of the chain. The presence of entanglement in the state of the environmental system magnifies the non-Markovian nature of the chain's dynamics, giving rise to structures in figures of merit such as entanglement and purity that are not observed under a separable multi-mode environment. Our analysis can be relevant to problems tackling the open-system dyna…
We propose a new approach called Ket-Bra Entangled State (KBES) Method for converting master equation into Schrödinger-like equation. With this method, we investigate decoherence process and entanglement dynamics induced by a $2$-qubit spin chain that each qubit coupled with reservoir. The spin chain is an anisotropy $XYZ$ Heisenberg model in the external magnetic field $B$, the corresponding master equation is solved concisely by KBES method; Furthermore, the effects of anisotropy, temperature, external field and initial state on concurrence dynamics is analyzed in detail for the case that in…
We investigate the behavior in time of the energy current between a quantum spin chain and its surrounding non-Markovian, finite temperature baths, together with its relationship to the coherence dynamics of the system. To be specific, both the system and the baths are assumed to be initially in thermal equilibrium at temperature $T_s$ and $T_b$, respectively. This model plays a fundamental role for the study of quantum system evolution towards thermal equilibrium in an open system. The non-Markovian quantum state diffusion (NMQSD) equation approach is used to calculate the dynamics of the spi…
We consider the problem of energy transport in a chain of coupled dissipative quantum systems in the presence of non-Markovian dephasing. We use a model of non-Markovianity which is experimentally realizable in the context of controlled quantum systems. We show that non-Markovian dephasing can significantly enhance quantum transport, and we characterize this phenomenon in terms of internal coupling strengths of the chain for some chain lengths. Finally, we show that the phenomenon of dephasing-assisted quantum transport is also enhanced in the non-Markovian scenario when compared to the Markov…
We study a harmonic system coupled to chain of first neighbor interacting oscillators. After deriving the exact dynamics of the system, we prove that one can effectively describe the exact dynamics by considering a suitable shorter chain. We provide the explicit expression for such an effective dynamics and we provide an upper bound on the error one makes considering it instead of the dynamics of the full chain. We eventually prove how error, timescale and number of modes in the truncated chain are related.
We investigate the Rabi oscillation of an atom placed inside a quantum cavity where each mirror is formed by a chain of atoms trapped near a one-dimensional waveguide. This proposal was studied previously with the use of Markov approximation, where the delay due to the finite travel time of light between the two cavity mirrors is neglected. We show that Rabi oscillation analogous to that obtained with high-finesse classical cavities is achieved only when this travel time is much larger than the time scale that characterizes the collective response of the atomic chain. Therefore, the delay must…
We consider the quantum correlations (entanglement and quantum discord) dynamics of two coupled spin qubits with Dzyaloshinsky-Moriya interaction influenced by a locally external magnetic field along $z$-direction and coupled to bath spin-1/2 particles as independent non-Markovian environment. We find that with increasing $D_{z}$ and decreasing $J_{z}$, the value of entanglement and quantum discord increase for both antiferromagnetic and ferromagnetic materials. Not that, this growth is more for the ferromagnetic materials. In addition, we perceive that entanglement and quantum discord decreas…
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of $n$ copies of a tripartite state $ρ^{ABC}$, and is transformed to a quantum Markov chain conditioned by $B^n$ with a small error, using a random unitary operation on $A^n$. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy requi…
We derive a sequence of measures whose corresponding Jacobi matrices have special properties and a general mapping of an open quantum system onto 1D semi infinite chains with only nearest neighbour interactions. Then we proceed to use the sequence of measures and the properties of the Jacobi matrices to derive an expression for the spectral density describing the open quantum system when an increasing number of degrees of freedom in the environment have been embedded into the system. Finally, we derive convergence theorems for these residual spectral densities.
A Markov assumption considers a physical system memoryless to simplify its dynamics. Whereas memory effect or the non-Markovian phenomenon is more general in nature. In the quantum regime, it is challenging to define or quantify the non-Markovianity because the measurement of a quantum system often interferes with it. We simulate the open quantum dynamics in a superconducting processor, then characterize and quantify the non-Markovian process. With the complete set of intervening projections and the final measurement of the qubit, a restricted process tensor can be determined to account for th…
Based on tensor network realizations of path integrals reducing exponential memory scaling to polynomial efficiency and a Liouville space implementation of a time-discrete quantum memory, we investigate a quantum system simultaneously exposed to two structured reservoirs. For this purpose, we employ a numerically exact quasi-2D tensor network combining both diagonal and off-diagonal system-reservoir interactions with a twofold memory for continuous and discrete retardation effects. As a possible example, we study the non-Markovian dynamical interplay between discrete photonic feedback and stru…
One-spin and two-spin in a thermodynamic limit spin-1/2 chain as open quantum systems are considered. The dynamics of system is generated by XX Heisenberg interaction and three-spin interaction (TSI) among all the nearest three spins in the chain. Two variety Hamiltonians of the three-spin interaction are considered. Using the fermionization technique and calculating the trace distance, non-Markovianity as a function of the TSI is evaluated, and the results for the two different open quantum systems are compared. In addition, the time behavior of the probability amplitudes are studied. The res…
We investigate the back-flow of information in a system with a second-order structural phase transition, namely a quasi one-dimensional Coulomb crystal. Using standard Ramsey interferometry which couples a target ion (the system) to the rest of the chain (a phononic environment), we study the non-Markovian character of the resulting open system dynamics. We study two different time-scales and show that the back-flow of information pinpoints both the phase transition and different dynamical features of the chain as it approaches criticality. We also establish an exact link between the back-flow…
The theoretical description of quantum dynamics in an intriguing way does not necessarily imply the underlying dynamics is indeed intriguing. Here we show how a known very interesting master equation with an always negative decay rate [eternal non-Markovianity (ENM)] arises from simple stochastic Schrödinger dynamics (random unitary dynamics). Equivalently, it may be seen as arising from a mixture of Markov (semi-group) open system dynamics. Both these approaches lead to a more general family of CPT maps, characterized by a point within a parameter triangle. Our results show how ENM quantum dy…
Using recently proposed measures for non-Markovianity [H. P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. {\bf 103}, 210401 (2009)], we study the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism. We show the existence of a point, in the parameter space of the system, where the qubit dynamics is effectively Markovian and that such a point separates two regions with completely different dynamical behaviors. Indeed, our study demonstrates that the qubit evolution can in principle be tuned from a perfectly forgetful one to a deep non-Markovian regime where …
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 timescales. The non-Markovian Transfer Tensor Method (TTM) demonstrates the coherent…
We present a generic, compact formula for the current flowing in interacting and non-interacting systems which are driven out-of-equilibrium by biased reservoirs described by Lindblad jump operators. We show that, in the limit of high temperature and chemical potential, our formula is equivalent to the well-known Meir-Wingreen formula, which describes the current flowing through a system connected to fermionic baths, therefore bridging the gap between the two formalisms. Our formulation gives a systematic way to address the transport properties of correlated systems strongly driven out of equi…
In the classical domain, it is well-known that divisibility does not imply that a stochastic process is Markovian. However, for quantum processes, divisibility is often considered to be synonymous with Markovianity. We show that completely positive (CP) divisible quantum processes can still involve non-Markovian temporal correlations, that we then fully classify using the recently developed process tensor formalism, which generalizes the theory of stochastic processes to the quantum domain.
Non-Markovian processes can often be turned Markovian by enlarging the set of variables. Here we show, by an explicit construction, how this can be done for the dynamics of a Brownian particle obeying the generalized Langevin equation. Given an arbitrary bath spectral density $J_{0}$, we introduce an orthogonal transformation of the bath variables into effective modes, leading stepwise to a semi-infinite chain with nearest-neighbor interactions. The transformation is uniquely determined by $J_{0}$ and defines a sequence $\{J_{n}\}_{n\in\mathbb{N}}$ of residual spectral densities describing the…
