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Brenes M, Min B, Anto-Sztrikacs N, Bar-Gill N, Segal D. Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach. J Chem Phys 2024; 160:244106. [PMID: 38916270 DOI: 10.1063/5.0207028] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2024] [Accepted: 06/05/2024] [Indexed: 06/26/2024] Open
Abstract
Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.
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Affiliation(s)
- Marlon Brenes
- Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - Brett Min
- Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - Nicholas Anto-Sztrikacs
- Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - Nir Bar-Gill
- Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem, Israel
- The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel
| | - Dvira Segal
- Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
- Department of Chemistry, University of Toronto, 80 Saint George St., Toronto, Ontario M5S 3H6, Canada
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2
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Link V, Tu HH, Strunz WT. Open Quantum System Dynamics from Infinite Tensor Network Contraction. PHYSICAL REVIEW LETTERS 2024; 132:200403. [PMID: 38829084 DOI: 10.1103/physrevlett.132.200403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2023] [Accepted: 04/09/2024] [Indexed: 06/05/2024]
Abstract
Approaching the long-time dynamics of non-Markovian open quantum systems presents a challenging task if the bath is strongly coupled. Recent proposals address this problem through a representation of the so-called process tensor in terms of a tensor network. We show that for Gaussian environments highly efficient contraction to a matrix product operator (MPO) form can be achieved with infinite MPO evolution methods, leading to significant computational speed-up over existing proposals. The result structurally resembles open system evolution with carefully designed auxiliary degrees of freedom, as in hierarchical or pseudomode methods. Here, however, these degrees of freedom are generated automatically by the MPO evolution algorithm. Moreover, the semigroup form of the resulting propagator enables us to explore steady-state physics, such as phase transitions.
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Affiliation(s)
- Valentin Link
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062, Dresden, Germany
| | - Hong-Hao Tu
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062, Dresden, Germany
| | - Walter T Strunz
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062, Dresden, Germany
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3
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Funo K, Ishizaki A. Dynamics of a Quantum System Interacting with White Non-Gaussian Baths: Poisson Noise Master Equation. PHYSICAL REVIEW LETTERS 2024; 132:170402. [PMID: 38728715 DOI: 10.1103/physrevlett.132.170402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2023] [Revised: 03/25/2024] [Accepted: 04/01/2024] [Indexed: 05/12/2024]
Abstract
Quantum systems are unavoidably open to their surrounding degrees of freedom. The theory of open quantum systems is thus crucial to understanding the fluctuations, dissipation, and decoherence of a quantum system of interest. Typically, the bath is modeled as an ensemble of harmonic oscillators, which yields Gaussian statistics of the bath influence on the quantum systems. However, there are also phenomena in which the bath consists of two-state systems, spins, or anharmonic oscillators; therefore, the non-Gaussian properties of the bath become important. Nevertheless, a theoretical framework to describe quantum systems under the influence of such non-Gaussian baths is not well established. Here, we develop a theory to describe quantum dissipative systems affected by Poisson noise properties of the bath, because the Lévi-Itô decomposition theorem asserts that Poisson noise is fundamental in describing arbitrary white noise beyond Gaussian properties. We introduce a quantum bath model that allows for the consistent description of dissipative quantum systems. The obtained master equation reveals non-Gaussian bath effects in the white noise regime, and provides an essential step toward describing open quantum dynamics under the influence of generic baths.
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Affiliation(s)
- Ken Funo
- Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan
| | - Akihito Ishizaki
- Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan
- Graduate Institute for Advanced Studies, SOKENDAI, Okazaki 444-8585, Japan
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4
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Lednev M, García-Vidal FJ, Feist J. Lindblad Master Equation Capable of Describing Hybrid Quantum Systems in the Ultrastrong Coupling Regime. PHYSICAL REVIEW LETTERS 2024; 132:106902. [PMID: 38518335 DOI: 10.1103/physrevlett.132.106902] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2023] [Accepted: 02/02/2024] [Indexed: 03/24/2024]
Abstract
Despite significant theoretical efforts devoted to studying the interaction between quantized light modes and matter, the so-called ultrastrong coupling regime still presents significant challenges for theoretical treatments and prevents the use of many common approximations. Here we demonstrate an approach that can describe the dynamics of hybrid quantum systems in any regime of interaction for an arbitrary electromagnetic (EM) environment. We extend a previous method developed for few-mode quantization of arbitrary systems to the case of ultrastrong light-matter coupling, and show that even such systems can be treated using a Lindblad master equation where decay operators act only on the photonic modes by ensuring that the effective spectral density of the EM environment is sufficiently suppressed at negative frequencies. We demonstrate the validity of our framework and show that it outperforms current state-of-the-art master equations for a simple model system, and then study a realistic nanoplasmonic setup where existing approaches cannot be applied.
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Affiliation(s)
- Maksim Lednev
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Francisco J García-Vidal
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore 138632, Republic of Singapore
| | - Johannes Feist
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
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5
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Becker T, Schnell A, Thingna J. Canonically Consistent Quantum Master Equation. PHYSICAL REVIEW LETTERS 2022; 129:200403. [PMID: 36461992 DOI: 10.1103/physrevlett.129.200403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2022] [Revised: 08/01/2022] [Accepted: 10/11/2022] [Indexed: 06/17/2023]
Abstract
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system toward a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, we use the exact mean-force Gibbs state to correct the Redfield quantum master equation. By benchmarking it with the exact solution of the damped harmonic oscillator, we show that our method also helps correct the long-standing issue of positivity violation, albeit without complete positivity. Our method of a canonically consistent quantum master equation opens a new perspective in the theory of open quantum systems leading to a reduced density matrix accurate beyond the commonly used Redfield and Lindblad equations, while retaining the same conceptual and numerical complexity.
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Affiliation(s)
- Tobias Becker
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
| | - Alexander Schnell
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
| | - Juzar Thingna
- Department of Physics and Applied Physics, University of Massachusetts, Lowell, Massachusetts 01854, USA
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
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6
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Sánchez-Barquilla M, García-Vidal FJ, Fernández-Domínguez AI, Feist J. Few-mode field quantization for multiple emitters. NANOPHOTONICS 2022; 11:4363-4374. [PMID: 36147197 PMCID: PMC9455278 DOI: 10.1515/nanoph-2021-0795] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/20/2021] [Accepted: 08/08/2022] [Indexed: 06/16/2023]
Abstract
The control of the interaction between quantum emitters using nanophotonic structures holds great promise for quantum technology applications, while its theoretical description for complex nanostructures is a highly demanding task as the electromagnetic (EM) modes form a high-dimensional continuum. We here introduce an approach that permits a quantized description of the full EM field through a small number of discrete modes. This extends the previous work in ref. (I. Medina, F. J. García-Vidal, A. I. Fernández-Domínguez, and J. Feist, "Few-mode field quantization of arbitrary electromagnetic spectral densities," Phys. Rev. Lett., vol. 126, p. 093601, 2021) to the case of an arbitrary number of emitters, without any restrictions on the emitter level structure or dipole operators. The low computational demand of this method makes it suitable for studying dynamics for a wide range of parameters. We illustrate the power of our approach for a system of three emitters placed within a hybrid metallodielectric photonic structure and show that excitation transfer is highly sensitive to the properties of the hybrid photonic-plasmonic modes.
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Affiliation(s)
- Mónica Sánchez-Barquilla
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain
| | - Francisco J. García-Vidal
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain
- Institute of High Performance Computing, Agency for Science, Technology, and Research (A*STAR), Connexis, 138632Singapore, Singapore
| | - Antonio I. Fernández-Domínguez
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain
| | - Johannes Feist
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain
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7
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Wang S, Chuang YT, Hsu LY. Simple but accurate estimation of light-matter coupling strength and optical loss for a molecular emitter coupled with photonic modes. J Chem Phys 2021; 155:134117. [PMID: 34624977 DOI: 10.1063/5.0060171] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Light-matter coupling strength and optical loss are two key physical quantities in cavity quantum electrodynamics (CQED), and their interplay determines whether light-matter hybrid states can be formed or not in chemical systems. In this study, by using macroscopic quantum electrodynamics (MQED) combined with a pseudomode approach, we present a simple but accurate method, which allows us to quickly estimate the light-matter coupling strength and optical loss without free parameters. Moreover, for a molecular emitter coupled with photonic modes (including cavity modes and plasmon polariton modes), we analytically and numerically prove that the dynamics derived from the MQED-based wavefunction approach is mathematically equivalent to the dynamics governed by the CQED-based Lindblad master equation when the Purcell factor behaves like Lorentzian functions.
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Affiliation(s)
- Siwei Wang
- Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
| | - Yi-Ting Chuang
- Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
| | - Liang-Yan Hsu
- Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
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8
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Medina I, García-Vidal FJ, Fernández-Domínguez AI, Feist J. Few-Mode Field Quantization of Arbitrary Electromagnetic Spectral Densities. PHYSICAL REVIEW LETTERS 2021; 126:093601. [PMID: 33750181 DOI: 10.1103/physrevlett.126.093601] [Citation(s) in RCA: 24] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2020] [Accepted: 02/11/2021] [Indexed: 05/23/2023]
Abstract
We develop a framework that provides a few-mode master equation description of the interaction between a single quantum emitter and an arbitrary electromagnetic environment. The field quantization requires only the fitting of the spectral density, obtained through classical electromagnetic simulations, to a model system involving a small number of lossy and interacting modes. We illustrate the power and validity of our approach by describing the population and electric field spatial dynamics in the spontaneous decay of an emitter placed in a complex hybrid plasmonic-photonic structure.
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Affiliation(s)
- Ivan Medina
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, 09210-170 Santo André, Sao Pãulo, Brazil
| | - Francisco J García-Vidal
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
- Donostia International Physics Center (DIPC), E-20018 Donostia/San Sebastián, Spain
| | - Antonio I Fernández-Domínguez
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Johannes Feist
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
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9
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Torres-Sánchez J, Feist J. Molecular photodissociation enabled by ultrafast plasmon decay. J Chem Phys 2021; 154:014303. [DOI: 10.1063/5.0037856] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- José Torres-Sánchez
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Johannes Feist
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
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10
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Zwolak M. Analytic expressions for the steady-state current with finite extended reservoirs. J Chem Phys 2020; 153:224107. [PMID: 33317280 PMCID: PMC8356363 DOI: 10.1063/5.0029223] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Open-system simulations of quantum transport provide a platform for the study of true steady states, Floquet states, and the role of temperature, time dynamics, and fluctuations, among other physical processes. They are rapidly gaining traction, especially techniques that revolve around "extended reservoirs," a collection of a finite number of degrees of freedom with relaxation that maintains a bias or temperature gradient, and have appeared under various guises (e.g., the extended or mesoscopic reservoir, auxiliary master equation, and driven Liouville-von Neumann approaches). Yet, there are still a number of open questions regarding the behavior and convergence of these techniques. Here, we derive general analytical solutions, and associated asymptotic analyses, for the steady-state current driven by finite reservoirs with proportional coupling to the system/junction. In doing so, we present a simplified and unified derivation of the non-interacting and many-body steady-state currents through arbitrary junctions, including outside of proportional coupling. We conjecture that the analytic solution for proportional coupling is the most general of its form for isomodal relaxation (i.e., relaxing proportional coupling will remove the ability to find compact, general analytical expressions for finite reservoirs). These results should be of broad utility in diagnosing the behavior and implementation of extended reservoir and related approaches, including the convergence to the Landauer limit (for non-interacting systems) and the Meir-Wingreen formula (for many-body systems).
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Affiliation(s)
- Michael Zwolak
- Biophysical and Biomedical Measurement Group, Microsystems and Nanotechnology Division, Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
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11
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Luchnikov IA, Vintskevich SV, Grigoriev DA, Filippov SN. Machine Learning Non-Markovian Quantum Dynamics. PHYSICAL REVIEW LETTERS 2020; 124:140502. [PMID: 32338970 DOI: 10.1103/physrevlett.124.140502] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2019] [Revised: 01/28/2020] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
Machine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however, they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of projective single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems.
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Affiliation(s)
- I A Luchnikov
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
| | - S V Vintskevich
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
| | - D A Grigoriev
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
| | - S N Filippov
- Moscow Institute of Physics and Technology, Institutskii Pereulok 9, Dolgoprudny, Moscow Region 141700, Russia
- Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Prospekt 34, Moscow 117218, Russia
- Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia
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12
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Somoza AD, Marty O, Lim J, Huelga SF, Plenio MB. Dissipation-Assisted Matrix Product Factorization. PHYSICAL REVIEW LETTERS 2019; 123:100502. [PMID: 31573298 DOI: 10.1103/physrevlett.123.100502] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2019] [Revised: 06/24/2019] [Indexed: 06/10/2023]
Abstract
Charge and energy transfer in biological and synthetic organic materials are strongly influenced by the coupling of electronic states to a highly structured dissipative environment. Nonperturbative simulations of these systems require a substantial computational effort, and current methods can only be applied to large systems if environmental structures are severely coarse grained. Time evolution methods based on tensor networks are fundamentally limited by the times that can be reached due to the buildup of entanglement in time, which quickly increases the size of the tensor representation, i.e., the bond dimension. In this Letter, we introduce a dissipation-assisted matrix product factorization (DAMPF) method that combines a tensor network representation of the vibronic state within a pseudomode description of the environment where a continuous bosonic environment is mapped into a few harmonic oscillators under Lindblad damping. This framework is particularly suitable for a tensor network representation, since damping suppresses the entanglement growth among oscillators and significantly reduces the bond dimension required to achieve a desired accuracy. We show that dissipation removes the "time-wall" limitation of existing methods, enabling the long-time simulation of large vibronic systems consisting of 10-50 sites coupled to 100-1000 underdamped modes in total and for a wide range of parameter regimes. For these reasons, we believe that our formalism will facilitate the investigation of spatially extended systems with applications to quantum biology, organic photovoltaics, and quantum thermodynamics.
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Affiliation(s)
- Alejandro D Somoza
- Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
| | - Oliver Marty
- Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
| | - James Lim
- Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
| | - Susana F Huelga
- Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
| | - Martin B Plenio
- Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
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13
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Lambert N, Ahmed S, Cirio M, Nori F. Modelling the ultra-strongly coupled spin-boson model with unphysical modes. Nat Commun 2019; 10:3721. [PMID: 31427583 PMCID: PMC6700178 DOI: 10.1038/s41467-019-11656-1] [Citation(s) in RCA: 35] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2019] [Accepted: 07/29/2019] [Indexed: 11/23/2022] Open
Abstract
A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state. However, when the coupling to the environment is ultra-strong the ground-state is expected to become dressed with virtual excitations. This regime is difficult to capture with some traditional methods because of the explosion in the number of Matsubara frequencies, i.e., exponential terms in the free-bath correlation function. To access this regime we generalize both the hierarchical equations of motion and pseudomode methods, taking into account this explosion using only a biexponential fitting function. We compare these methods to the reaction coordinate mapping, which helps show how these sometimes neglected Matsubara terms are important to regulate detailed balance and prevent the unphysical emission of virtual excitations. For the pseudomode method, we present a general proof of validity for the use of superficially unphysical Matsubara-modes, which mirror the mathematical essence of the Matsubara frequencies.
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Affiliation(s)
- Neill Lambert
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan.
| | - Shahnawaz Ahmed
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan
- Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96, Gothenburg, Sweden
| | - Mauro Cirio
- Graduate School of China Academy of Engineering Physics, Haidian District, Beijing, 100193, China.
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan
- Department of Physics, University of Michigan, Ann Arbor, MI, 48109-1040, USA
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14
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Hartmann R, Werther M, Grossmann F, Strunz WT. Exact open quantum system dynamics: Optimal frequency vs time representation of bath correlations. J Chem Phys 2019; 150:234105. [PMID: 31228905 DOI: 10.1063/1.5097158] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Richard Hartmann
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Michael Werther
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Frank Grossmann
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
| | - Walter T. Strunz
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
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15
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Luchnikov IA, Vintskevich SV, Ouerdane H, Filippov SN. Simulation Complexity of Open Quantum Dynamics: Connection with Tensor Networks. PHYSICAL REVIEW LETTERS 2019; 122:160401. [PMID: 31075029 DOI: 10.1103/physrevlett.122.160401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2018] [Indexed: 06/09/2023]
Abstract
The difficulty to simulate the dynamics of open quantum systems resides in their coupling to many-body reservoirs with exponentially large Hilbert space. Applying a tensor network approach in the time domain, we demonstrate that effective small reservoirs can be defined and used for modeling open quantum dynamics. The key element of our technique is the timeline reservoir network (TRN), which contains all the information on the reservoir's characteristics, in particular, the memory effects timescale. The TRN has a one-dimensional tensor network structure, which can be effectively approximated in full analogy with the matrix product approximation of spin-chain states. We derive the sufficient bond dimension in the approximated TRN with a reduced set of physical parameters: coupling strength, reservoir correlation time, minimal timescale, and the system's number of degrees of freedom interacting with the environment. The bond dimension can be viewed as a measure of the open dynamics complexity. Simulation is based on the semigroup dynamics of the system and effective reservoir of finite dimension. We provide an illustrative example showing the scope for new numerical and machine learning-based methods for open quantum systems.
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Affiliation(s)
- I A Luchnikov
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
| | - S V Vintskevich
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
- A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov St. 38, Moscow 119991, Russia
| | - H Ouerdane
- Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, Moscow Region 121205, Russia
| | - S N Filippov
- Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
- Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, Russia
- Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia
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16
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Hughes S, Richter M, Knorr A. Quantized pseudomodes for plasmonic cavity QED. OPTICS LETTERS 2018; 43:1834-1837. [PMID: 29652376 DOI: 10.1364/ol.43.001834] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2017] [Accepted: 03/13/2018] [Indexed: 06/08/2023]
Abstract
We present a quantized pseudomode theory for solving system-level cavity quantum electrodynamics with quantum emitters coupled to plasmonic resonators. Exploiting the Green function response of a silver nanoparticle and an exact solution for the input-output coupling, we study the non-Markovian coupling regimes of multi-mode vacuum Rabi oscillations and coherently pumped multi-photon resonances from the higher lying ladder states of collective plasmon modes. The nonlinear pumping regime also facilitates pronounced population inversion of a single two-level atom through multiphoton resonances dominated by dark plasmons.
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17
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Tamascelli D, Smirne A, Huelga SF, Plenio MB. Nonperturbative Treatment of non-Markovian Dynamics of Open Quantum Systems. PHYSICAL REVIEW LETTERS 2018; 120:030402. [PMID: 29400486 DOI: 10.1103/physrevlett.120.030402] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2017] [Indexed: 06/07/2023]
Abstract
We identify the conditions that guarantee equivalence of the reduced dynamics of an open quantum system (OQS) for two different types of environments-one a continuous bosonic environment leading to a unitary system-environment evolution and the other a discrete-mode bosonic environment resulting in a system-mode (nonunitary) Lindbladian evolution. Assuming initial Gaussian states for the environments, we prove that the two OQS dynamics are equivalent if both the expectation values and two-time correlation functions of the environmental interaction operators are the same at all times for the two configurations. Since the numerical and analytical description of a discrete-mode environment undergoing a Lindbladian evolution is significantly more efficient than that of a continuous bosonic environment in a unitary evolution, our result represents a powerful, nonperturbative tool to describe complex and possibly highly non-Markovian dynamics. As a special application, we recover and generalize the well-known pseudomodes approach to open-system dynamics.
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Affiliation(s)
- D Tamascelli
- Università degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, I-20133 Milano, Italy
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11D-89069 Ulm, Germany
| | - A Smirne
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11D-89069 Ulm, Germany
| | - S F Huelga
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11D-89069 Ulm, Germany
| | - M B Plenio
- Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11D-89069 Ulm, Germany
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18
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Caycedo-Soler F, Schroeder CA, Autenrieth C, Pick A, Ghosh R, Huelga SF, Plenio MB. Quantum Redirection of Antenna Absorption to Photosynthetic Reaction Centers. J Phys Chem Lett 2017; 8:6015-6021. [PMID: 29185757 DOI: 10.1021/acs.jpclett.7b02714] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
The early steps of photosynthesis involve the photoexcitation of reaction centers (RCs) and light-harvesting (LH) units. Here, we show that the historically overlooked excitonic delocalization across RC and LH pigments results in a redistribution of absorption amplitudes that benefits the absorption cross section of the optical bands associated with the RC of several species. While we prove that this redistribution is robust to the microscopic details of the dephasing between these units in the purple bacterium Rhodospirillum rubrum, we are able to show that the redistribution witnesses a more fragile, but persistent, coherent population dynamics which directs excitations from the LH toward the RC units under incoherent illumination and physiological conditions. Even though the redirection does not seem to affect importantly the overall efficiency in photosynthesis, stochastic optimization allows us to delineate clear guidelines and develop simple analytic expressions in order to amplify the coherent redirection in artificial nanostructures.
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Affiliation(s)
- Felipe Caycedo-Soler
- Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm , Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Christopher A Schroeder
- Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm , Albert-Einstein-Allee 11, D-89069 Ulm, Germany
- Joint Quantum Institute, Department of Physics, University of Maryland and National Institute of Standards and Technology , College Park, Maryland 20742, United States
| | - Caroline Autenrieth
- Department of Bioenergetics, Institute of Biomaterials and Biomolecular Systems, University of Stuttgart , Pfaffenwaldring 57, D-70569 Stuttgart, Germany
| | - Arne Pick
- Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm , Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Robin Ghosh
- Department of Bioenergetics, Institute of Biomaterials and Biomolecular Systems, University of Stuttgart , Pfaffenwaldring 57, D-70569 Stuttgart, Germany
| | - Susana F Huelga
- Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm , Albert-Einstein-Allee 11, D-89069 Ulm, Germany
| | - Martin B Plenio
- Institute of Theoretical Physics and Integrated Quantum Science and Technology IQST, University of Ulm , Albert-Einstein-Allee 11, D-89069 Ulm, Germany
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19
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Iles-Smith J, Dijkstra AG, Lambert N, Nazir A. Energy transfer in structured and unstructured environments: Master equations beyond the Born-Markov approximations. J Chem Phys 2016; 144:044110. [DOI: 10.1063/1.4940218] [Citation(s) in RCA: 82] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Affiliation(s)
- Jake Iles-Smith
- Controlled Quantum Dynamics Theory, Imperial College London, London SW7 2PG, United Kingdom
- Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
- Department of Photonics Engineering, DTU Fotonik, Ørsteds Plads, 2800 Kongens Lyngby, Denmark
| | - Arend G. Dijkstra
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | | | - Ahsan Nazir
- Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
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20
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Correa LA, Palao JP, Alonso D. Internal dissipation and heat leaks in quantum thermodynamic cycles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032136. [PMID: 26465455 DOI: 10.1103/physreve.92.032136] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2015] [Indexed: 06/05/2023]
Abstract
The direction of the steady-state heat currents across a generic quantum system connected to multiple baths may be engineered to realize virtually any thermodynamic cycle. In spite of their versatility, such continuous energy-conversion systems are generally unable to operate at maximum efficiency due to non-negligible sources of irreversible entropy production. In this paper we introduce a minimal model of irreversible absorption chiller. We identify and characterize the different mechanisms responsible for its irreversibility, namely heat leaks and internal dissipation, and gauge their relative impact in the overall cooling performance. We also propose reservoir engineering techniques to minimize these detrimental effects. Finally, by looking into a known three-qubit embodiment of the absorption cooling cycle, we illustrate how our simple model may help to pinpoint the different sources of irreversibility naturally arising in more complex practical heat devices.
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Affiliation(s)
- Luis A Correa
- Departament de Física, Universitat Autònoma de Barcelona, E08193 Bellaterra, Spain
| | - José P Palao
- IUdEA Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203, Spain and Departamento de Física, Universidad de La Laguna, 38204, Spain
| | - Daniel Alonso
- IUdEA Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203, Spain and Departamento de Física, Universidad de La Laguna, 38204, Spain
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21
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Shao ZQ, Li YQ, Pan XY. Dissipation-induced transition of a simple harmonic oscillator. J Chem Phys 2014; 141:224110. [PMID: 25494735 DOI: 10.1063/1.4903178] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the dissipation-induced transition probabilities between any two eigenstates of a simple harmonic oscillator. Using the method developed by Yu and Sun [Phys. Rev. A 49, 592 (1994)], the general analytical expressions for the transition probabilities are obtained. The special cases: transition probabilities from the ground state to the first few excited states are then discussed in detail. Different from the previous studies in the literature where only the effect of damping was considered, it is found that the Brownian motion makes the transitions between states of different parity possible. The limitations of the applicability of our results are also discussed.
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Affiliation(s)
- Zong-Qian Shao
- Department of Physics, Ningbo University, Ningbo 315211, China
| | - Yu-Qi Li
- Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
| | - Xiao-Yin Pan
- Department of Physics, Ningbo University, Ningbo 315211, China
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22
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Roden J, Strunz WT, Whaley KB, Eisfeld A. Accounting for intra-molecular vibrational modes in open quantum system description of molecular systems. J Chem Phys 2012. [DOI: 10.1063/1.4765329] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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23
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Huelga SF, Rivas Á, Plenio MB. Non-Markovianity-assisted steady state entanglement. PHYSICAL REVIEW LETTERS 2012; 108:160402. [PMID: 22680702 DOI: 10.1103/physrevlett.108.160402] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2011] [Indexed: 06/01/2023]
Abstract
We analyze the steady state entanglement generated in a coherently coupled dimer system subject to dephasing noise as a function of the degree of Markovianity of the evolution. By keeping fixed the effective noise strength while varying the memory time of the environment, we demonstrate that non-Markovianity is an essential, quantifiable resource that may support the formation of steady state entanglement whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states. This result illustrates possible mechanisms leading to long-lived entanglement in purely decohering, possibly local, environments. We present a feasible experimental demonstration of this noise assisted phenomenon using a system of trapped ions.
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Affiliation(s)
- Susana F Huelga
- Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany
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24
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Kliesch M, Barthel T, Gogolin C, Kastoryano M, Eisert J. Dissipative quantum Church-Turing theorem. PHYSICAL REVIEW LETTERS 2011; 107:120501. [PMID: 22026760 DOI: 10.1103/physrevlett.107.120501] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2011] [Indexed: 05/31/2023]
Abstract
We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.
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Affiliation(s)
- M Kliesch
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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25
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Roden J, Strunz WT, Eisfeld A. Non-Markovian quantum state diffusion for absorption spectra of molecular aggregates. J Chem Phys 2011; 134:034902. [DOI: 10.1063/1.3512979] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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26
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Wiseman HM, Gambetta JM. Pure-state quantum trajectories for general non-Markovian systems do not exist. PHYSICAL REVIEW LETTERS 2008; 101:140401. [PMID: 18851507 DOI: 10.1103/physrevlett.101.140401] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2008] [Indexed: 05/26/2023]
Abstract
Since the first derivation of non-Markovian stochastic Schrödinger equations, their interpretation has been contentious. In a recent Letter [Phys. Rev. Lett. 100, 080401 (2008)10.1103/Phys. Rev. Lett.100.080401], Diósi claimed to prove that they generate "true single system trajectories [conditioned on] continuous measurement." In this Letter, we show that his proof is fundamentally flawed: the solution to his non-Markovian stochastic Schrödinger equation at any particular time can be interpreted as a conditioned state, but joining up these solutions as a trajectory creates a fiction.
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Affiliation(s)
- Howard M Wiseman
- Centre for Quantum Dynamics, School of Science, Griffith University, Nathan 4111, Australia
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27
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Piilo J, Maniscalco S, Härkönen K, Suominen KA. Non-Markovian quantum jumps. PHYSICAL REVIEW LETTERS 2008; 100:180402. [PMID: 18518352 DOI: 10.1103/physrevlett.100.180402] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2008] [Indexed: 05/26/2023]
Abstract
Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics.
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Affiliation(s)
- Jyrki Piilo
- Department of Physics, University of Turku, FI-20014 Turun yliopisto, Finland
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28
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Breuer HP, Gemmer J, Michel M. Non-Markovian quantum dynamics: correlated projection superoperators and Hilbert space averaging. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016139. [PMID: 16486248 DOI: 10.1103/physreve.73.016139] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2005] [Indexed: 05/06/2023]
Abstract
The time-convolutionless (TCL) projection operator technique allows a systematic analysis of the non-Markovian quantum dynamics of open systems. We present a class of projection superoperators that project the states of the total system onto certain correlated system-environment states. It is shown that the application of the TCL technique to this class of correlated superoperators enables the nonperturbative treatment of the dynamics of system-environment models for which the standard approach fails in any finite order of the coupling strength. We demonstrate further that the correlated superoperators correspond to the idea of a best guess of conditional quantum expectations, which is determined by a suitable Hilbert-space average. The general approach is illustrated by means of the model of a spin that interacts through randomly distributed couplings with a finite reservoir consisting of two energy bands. Extensive numerical simulations of the full Schrödinger equation of the model reveal the power and efficiency of the method.
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Affiliation(s)
- Heinz-Peter Breuer
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany.
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29
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Banerjee D, Bag BC, Banik SK, Ray DS. Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects. J Chem Phys 2004; 120:8960-72. [PMID: 15267831 DOI: 10.1063/1.1711593] [Citation(s) in RCA: 53] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.
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Affiliation(s)
- Dhruba Banerjee
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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30
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Nakano M, Kishi R, Nitta T, Yamaguchi K. Monte Carlo wave-function approach to the quantum-phase dynamics of a dissipative molecular system interacting with a single-mode amplitude-squeezed field. J Chem Phys 2003. [DOI: 10.1063/1.1626113] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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32
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Mančal T, May V. Interplay of non-Markovian relaxation and ultrafast optical state preparation in molecular systems: The Laguerre polynomial method. J Chem Phys 2001. [DOI: 10.1063/1.1334619] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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33
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Jack MW, Collett MJ, Walls DF. A non-Markovian quantum trajectory approach to radiation into structured continuum. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/1464-4266/1/4/316] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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34
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35
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