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Bose A. Quantum correlation functions through tensor network path integral. J Chem Phys 2023; 159:214110. [PMID: 38051096 DOI: 10.1063/5.0174338] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2023] [Accepted: 11/07/2023] [Indexed: 12/07/2023] Open
Abstract
Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman-Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna-Matthews-Olson complex to light.
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Affiliation(s)
- Amartya Bose
- Department of Chemical Sciences, Tata Institute of Fundamental Research, Mumbai 400005, India
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Anto-Sztrikacs N, Ivander F, Segal D. Quantum thermal transport beyond second order with the reaction coordinate mapping. J Chem Phys 2022; 156:214107. [DOI: 10.1063/5.0091133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Standard quantum master equation techniques, such as the Redfield or Lindblad equations, are perturbative to second order in the microscopic system–reservoir coupling parameter λ. As a result, the characteristics of dissipative systems, which are beyond second order in λ, are not captured by such tools. Moreover, if the leading order in the studied effect is higher-than-quadratic in λ, a second-order description fundamentally fails even at weak coupling. Here, using the reaction coordinate (RC) quantum master equation framework, we are able to investigate and classify higher-than-second-order transport mechanisms. This technique, which relies on the redefinition of the system–environment boundary, allows for the effects of system–bath coupling to be included to high orders. We study steady-state heat current beyond second-order in two models: The generalized spin-boson model with non-commuting system–bath operators and a three-level ladder system. In the latter model, heat enters in one transition and is extracted from a different one. Crucially, we identify two transport pathways: (i) System’s current, where heat conduction is mediated by transitions in the system, with the heat current scaling as j q ∝ λ2 to the lowest order in λ. (ii) Inter-bath current, with the thermal baths directly exchanging energy between them, facilitated by the bridging quantum system. To the lowest order in λ, this current scales as j q ∝ λ4. These mechanisms are uncovered and examined using numerical and analytical tools. We contend that the RC mapping brings, already at the level of the mapped Hamiltonian, much insight into transport characteristics.
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Affiliation(s)
- Nicholas Anto-Sztrikacs
- Department of Physics, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - Felix Ivander
- Chemical Physics Theory Group, Department of Chemistry and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George St., Toronto, Ontario M5S 3H6, Canada
| | - Dvira Segal
- Department of Physics, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
- Chemical Physics Theory Group, Department of Chemistry and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George St., Toronto, Ontario M5S 3H6, Canada
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Abstract
The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.
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Affiliation(s)
- Nancy Makri
- Departments of Chemistry and Physics, University of Illinois, 505 S. Mathews Avenue, Urbana, Illinois 61801, United States
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Abstract
The dynamical behaviors of a two-level system (TLS) coupled to a harmonic dissipative bath has been studied extensively using a variety of analytical and numerical methods. The focus of the vast majority of these studies has been on the properties of the TLS, averaged with respect to the bath degrees of freedom. In this work, we use real-time path integral methods to probe the behavior of select bath degrees of freedom during the dynamics of a symmetric two-level system (TLS) coupled to a dissipative bath by calculating system-bath densities (SBD) and coordinate expectation values. Overall, the SBD motion on each diabatic state is simpler than the motion of the total density. In the weak coupling regime, which characterizes the parameters of oscillators that comprise such a bath, the SBD on each TLS state remains primarily compact and Gaussian-like, such that its peak is well characterized by the mode expectation value. In the absence of a dissipative environment, nonadiabatic density depletion leads to spikes in coordinate expectation values. The evolution of the SBD peak trajectory for two discrete modes exhibits Lissajous patterns with frequency-dependent shapes that strongly resemble classical trajectory motion on a torus. These patterns become more complex when the coupling of the mode to the TLS is increased outside of this regime, leading to persistent small amplitude oscillations in the TLS populations characterized by a very slow decay and SBD trajectories that exhibit behaviors reminiscent of chaotic classical systems. Indirect coupling to a dissipative bath has a stabilizing effect on the dynamics, eliminating spikes, synchronizing the SBD motion on the two diabatic states and regularizing the SBD trajectory to simple rectangular Lissajous-like shapes with a slowly shrinking boundary, regardless of the mode frequencies.
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Affiliation(s)
- Sohang Kundu
- Department of Chemistry, University of Illinois, Urbana, Illinois 61801, United States
| | - Nancy Makri
- Department of Chemistry, University of Illinois, Urbana, Illinois 61801, United States.,Department of Physics, University of Illinois, Urbana, Illinois 61801, United States
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Affiliation(s)
- Nancy Makri
- Departments of Chemistry and Physics, University of Illinois, 505 South Mathews Avenue, Urbana, Illinois 61801, United States
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Acharyya N, Ovcharenko R, Fingerhut BP. On the role of non-diagonal system-environment interactions in bridge-mediated electron transfer. J Chem Phys 2020; 153:185101. [PMID: 33187441 DOI: 10.1063/5.0027976] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Bridge-mediated electron transfer (ET) between a donor and an acceptor is prototypical for the description of numerous most important ET scenarios. While multi-step ET and the interplay of sequential and direct superexchange transfer pathways in the donor-bridge-acceptor (D-B-A) model are increasingly understood, the influence of off-diagonal system-bath interactions on the transfer dynamics is less explored. Off-diagonal interactions account for the dependence of the ET coupling elements on nuclear coordinates (non-Condon effects) and are typically neglected. Here, we numerically investigate with quasi-adiabatic propagator path integral simulations the impact of off-diagonal system-environment interactions on the transfer dynamics for a wide range of scenarios in the D-B-A model. We demonstrate that off-diagonal system-environment interactions can have profound impact on the bridge-mediated ET dynamics. In the considered scenarios, the dynamics itself does not allow for a rigorous assignment of the underlying transfer mechanism. Furthermore, we demonstrate how off-diagonal system-environment interaction mediates anomalous localization by preventing long-time depopulation of the bridge B and how coherent transfer dynamics between donor D and acceptor A can be facilitated. The arising non-exponential short-time dynamics and coherent oscillations are interpreted within an equivalent Hamiltonian representation of a primary reaction coordinate model that reveals how the complex vibronic interplay of vibrational and electronic degrees of freedom underlying the non-Condon effects can impose donor-to-acceptor coherence transfer on short timescales.
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Affiliation(s)
- Nirmalendu Acharyya
- Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany
| | - Roman Ovcharenko
- Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany
| | - Benjamin P Fingerhut
- Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany
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Affiliation(s)
- Nancy Makri
- Departments of Chemistry and Physics, University of Illinois, 505 South Mathews Avenue, Urbana, Illinois 61801, United States
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Chatterjee S, Makri N. Real-Time Path Integral Methods, Quantum Master Equations, and Classical vs Quantum Memory. J Phys Chem B 2019; 123:10470-10482. [DOI: 10.1021/acs.jpcb.9b08429] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Sambarta Chatterjee
- Department of Chemistry, University of Illinois, 505 South Mathews Avenue, Urbana, Illinois 61801, United States
| | - Nancy Makri
- Department of Chemistry, University of Illinois, 505 South Mathews Avenue, Urbana, Illinois 61801, United States
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Palm T, Nalbach P. Dephasing and relaxational polarized sub-Ohmic baths acting on a two-level system. J Chem Phys 2019; 150:234108. [PMID: 31228889 DOI: 10.1063/1.5098467] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study a quantum two-level system under the influence of two independent baths, i.e., a sub-Ohmic pure dephasing bath and an Ohmic or sub-Ohmic relaxational bath. We show that cooling such a system invariably polarizes one of the two baths. A polarized relaxational bath creates an effective asymmetry. This asymmetry can be suppressed by additional dephasing noise. This being less effective, the more dominant low frequencies are in the dephasing noise. A polarized dephasing bath generates a large shift in the coherent oscillation frequency of the two-level system. This frequency shift is little affected by additional relaxational noise nor by the frequency distribution of the dephasing noise itself. As our model reflects a typical situation for superconducting phase qubits, our findings can help optimize cooling protocols for future quantum electronic devices.
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Affiliation(s)
- T Palm
- Fachbereich Wirtschaft & Informationstechnik, Westfälische Hochschule, Münsterstrasse 265, 46997 Bocholt, Germany
| | - P Nalbach
- Fachbereich Wirtschaft & Informationstechnik, Westfälische Hochschule, Münsterstrasse 265, 46997 Bocholt, Germany
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