1
|
Liu L, Ren J, Fang W. Improved memory truncation scheme for quasi-adiabatic propagator path integral via influence functional renormalization. J Chem Phys 2024; 161:084101. [PMID: 39171703 DOI: 10.1063/5.0221916] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2024] [Accepted: 08/05/2024] [Indexed: 08/23/2024] Open
Abstract
Accurately simulating non-Markovian quantum dynamics in system-bath coupled problems remains challenging. In this work, we present a novel memory truncation scheme for the iterative quasi-adiabatic propagator path integral (iQuAPI) method to improve accuracy. Conventional memory truncation in iQuAPI discards all influence functional beyond a certain time interval, which is not effective for problems with a long memory time. Our proposed scheme selectively retains the most significant parts of the influence functional using the density matrix renormalization group algorithm. We validate the effectiveness of our scheme through simulations of the spin-boson model across various parameter sets, demonstrating faster convergence and improved accuracy compared to the conventional scheme. Our findings suggest that the new memory truncation scheme significantly advances the capabilities of iQuAPI for problems with a long memory time.
Collapse
Affiliation(s)
- Limin Liu
- Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, 100875 Beijing, People's Republic of China
| | - Jiajun Ren
- Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, 100875 Beijing, People's Republic of China
| | - Weihai Fang
- Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, 100875 Beijing, People's Republic of China
| |
Collapse
|
2
|
Zhou L, Gao X, Shuai Z. A stochastic Schrödinger equation and matrix product state approach to carrier transport in organic semiconductors with nonlocal electron-phonon interaction. J Chem Phys 2024; 161:084118. [PMID: 39212211 DOI: 10.1063/5.0221143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2024] [Accepted: 08/12/2024] [Indexed: 09/04/2024] Open
Abstract
Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward-backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward-backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron-phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size.
Collapse
Affiliation(s)
- Liqi Zhou
- MOE Key Laboratory of Organic OptoElectronics and Molecular Engineering, Department of Chemistry, Tsinghua University, Beijing 100084, China
| | - Xing Gao
- School of Materials, Sun Yat-sen University, Shenzhen, Guangdong 518107, China
| | - Zhigang Shuai
- MOE Key Laboratory of Organic OptoElectronics and Molecular Engineering, Department of Chemistry, Tsinghua University, Beijing 100084, China
- School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
| |
Collapse
|
3
|
Cygorek M, Gauger EM. ACE: A general-purpose non-Markovian open quantum systems simulation toolkit based on process tensors. J Chem Phys 2024; 161:074111. [PMID: 39158046 DOI: 10.1063/5.0221182] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2024] [Accepted: 08/01/2024] [Indexed: 08/20/2024] Open
Abstract
We describe a general-purpose computational toolkit for simulating open quantum systems, which provides numerically exact solutions for composites of zero-dimensional quantum systems that may be strongly coupled to multiple, quite general non-Markovian environments. It is based on process tensor matrix product operators (PT-MPOs), which efficiently encapsulate environment influences. The code features implementations of several PT-MPO algorithms, in particular Automated Compression of Environments for general environments comprised of independent modes as well as schemes for generalized spin boson models. The latter includes a divide-and-conquer scheme for periodic PT-MPOs, which enable million time step simulations for realistic models. PT-MPOs can be precalculated and reused for efficiently probing different time-dependent system Hamiltonians. They can also be stacked together and combined to provide numerically complete solutions of small networks of open quantum systems. The code is written in C++ and is fully controllable by configuration files, for which we have developed a versatile and compact human-readable format.
Collapse
Affiliation(s)
- Moritz Cygorek
- Condensed Matter Theory, Department of Physics, TU Dortmund, 44221 Dortmund, Germany
- SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Erik M Gauger
- SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| |
Collapse
|
4
|
Butler EP, Fux GE, Ortega-Taberner C, Lovett BW, Keeling J, Eastham PR. Optimizing Performance of Quantum Operations with Non-Markovian Decoherence: The Tortoise or the Hare? PHYSICAL REVIEW LETTERS 2024; 132:060401. [PMID: 38394576 DOI: 10.1103/physrevlett.132.060401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2023] [Accepted: 12/19/2023] [Indexed: 02/25/2024]
Abstract
The interaction between a quantum system and its environment limits our ability to control it and perform quantum operations on it. We present an efficient method to find optimal controls for quantum systems coupled to non-Markovian environments, by using the process tensor to compute the gradient of an objective function. We consider state transfer for a driven two-level system coupled to a bosonic environment, and characterize performance in terms of speed and fidelity. This allows us to determine the best achievable fidelity as a function of process duration. We show there can be a trade-off between speed and fidelity, and that slower processes can have higher fidelity by exploiting non-Markovian effects.
Collapse
Affiliation(s)
- Eoin P Butler
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
| | - Gerald E Fux
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
- Abdus Salam International Center for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
| | - Carlos Ortega-Taberner
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
| | - Brendon W Lovett
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Jonathan Keeling
- SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
| | - Paul R Eastham
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, Ireland
| |
Collapse
|
5
|
Wang G, Cai Z. Real-Time Simulation of Open Quantum Spin Chains with the Inchworm Method. J Chem Theory Comput 2023. [PMID: 38039552 DOI: 10.1021/acs.jctc.3c00751] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/03/2023]
Abstract
We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.
Collapse
Affiliation(s)
- Geshuo Wang
- Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
| | - Zhenning Cai
- Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
| |
Collapse
|
6
|
Zhang D, Zuo L, Ye L, Chen ZH, Wang Y, Xu RX, Zheng X, Yan Y. Hierarchical equations of motion approach for accurate characterization of spin excitations in quantum impurity systems. J Chem Phys 2023; 158:014106. [PMID: 36610957 DOI: 10.1063/5.0131739] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.
Collapse
Affiliation(s)
- Daochi Zhang
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lijun Zuo
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lyuzhou Ye
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zi-Hao Chen
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yao Wang
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Rui-Xue Xu
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao Zheng
- Hefei National Research Center for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Research Center for Physical Sciences at the Microscale and iChEM, University of Science and Technology of China, Hefei, Anhui 230026, China
| |
Collapse
|
7
|
Ke Y, Borrelli R, Thoss M. Hierarchical equations of motion approach to hybrid fermionic and bosonic environments: Matrix product state formulation in twin space. J Chem Phys 2022; 156:194102. [DOI: 10.1063/5.0088947] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation (introduced in J. Chem. Phys. 150, 234102, [2019]) to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schrödinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-rank matrices. The new approach facilitates accurate simulations of non-equilibrium quantum dynamics in larger and more complex open quantum systems. The performance of the method is demonstrated for a model of a molecular junction exhibiting current-induced mode-selective vibrational excitation.
Collapse
Affiliation(s)
- Yaling Ke
- Institute of Physics, Albert-Ludwigs-Universität Freiburg, Germany
| | - Raffaele Borrelli
- Department of Agricoltural Science, Università degli Studi di Torino, Italy
| | - Michael Thoss
- University of Freiburg Institute of Physics, Germany
| |
Collapse
|
8
|
Bose A, Walters PL. A multisite decomposition of the tensor network path integrals. J Chem Phys 2022; 156:024101. [PMID: 35032978 DOI: 10.1063/5.0073234] [Citation(s) in RCA: 12] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Tensor network decompositions of path integrals for simulating open quantum systems have recently been proven to be useful. However, these methods scale exponentially with the system size. This makes it challenging to simulate the non-equilibrium dynamics of extended quantum systems coupled with local dissipative environments. In this work, we extend the tensor network path integral (TNPI) framework to efficiently simulate such extended systems. The Feynman-Vernon influence functional is a popular approach used to account for the effect of environments on the dynamics of the system. In order to facilitate the incorporation of the influence functional into a multisite framework (MS-TNPI), we combine a matrix product state (MPS) decomposition of the reduced density tensor of the system along the sites with a corresponding tensor network representation of the time axis to construct an efficient 2D tensor network. The 2D MS-TNPI network, when contracted, yields the time-dependent reduced density tensor of the extended system as an MPS. The algorithm presented is independent of the system Hamiltonian. We outline an iteration scheme to take the simulation beyond the non-Markovian memory introduced by solvents. Applications to spin chains coupled to local harmonic baths are presented; we consider the Ising, XXZ, and Heisenberg models, demonstrating that the presence of local environments can often dissipate the entanglement between the sites. We discuss three factors causing the system to transition from a coherent oscillatory dynamics to a fully incoherent dynamics. The MS-TNPI method is useful for studying a variety of extended quantum systems coupled with solvents.
Collapse
Affiliation(s)
- Amartya Bose
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Peter L Walters
- Department of Chemistry, University of California, Berkeley, California 94720, USA
| |
Collapse
|