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Journal articles on the topic 'Quantum Dissipative Systems'

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Published: 4 June 2021
Last updated: 14 March 2023

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1

Grigolini, Paolo, and Bruce J. West. "Quantum dissipative systems." Journal of Statistical Physics 77, no. 3-4 (November 1994): 951–52. http://dx.doi.org/10.1007/bf02179474.

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2

Mollai, Maedeh, and Seyed Majid Saberi Fathi. "An Application of the Madelung Formalism for Dissipating and Decaying Systems." Symmetry 13, no. 5 (May 6, 2021): 812. http://dx.doi.org/10.3390/sym13050812.

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This paper is concerned with the modeling and analysis of quantum dissipation and diffusion phenomena in the Schrödinger picture. We derive and investigate in detail the Schrödinger-type equations accounting for dissipation and diffusion effects. From a mathematical viewpoint, this equation allows one to achieve and analyze all aspects of the quantum dissipative systems, regarding the wave equation, Hamilton–Jacobi and continuity equations. This simplification requires the performance of “the Madelung decomposition” of “the wave function”, which is rigorously attained under the general Lagrangian justification for this modification of quantum mechanics. It is proved that most of the important equations of dissipative quantum physics, such as convection-diffusion, Fokker–Planck and quantum Boltzmann, have a common origin and can be unified in one equation.
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3

STEFANESCU, E., A. SǍNDULESCU, and W. GREINER. "QUANTUM TUNNELING IN OPEN SYSTEMS." International Journal of Modern Physics E 02, no. 01 (March 1993): 233–58. http://dx.doi.org/10.1142/s0218301393000078.

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We study the barrier penetrability in the frame of the Lindblad theory of open quantum systems. In addition to the diagonal elements of the density matrix, leading to the Gamow’s formula, new terms, describing energy dissipation and spectral line broadening effects are obtained. It is shown that the presence of a dissipative environment increase the barrier penetrability, in accordance with a very simple physical interpretation: for a system initially found in its ground state the dissipation can lead only to transitions to the reaction channels where lower-energy levels exist.
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4

Harris, Edward G. "Quantum tunneling in dissipative systems." Physical Review A 48, no. 2 (August 1, 1993): 995–1008. http://dx.doi.org/10.1103/physreva.48.995.

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5

Hovhannisyan, Karen V., and Alberto Imparato. "Quantum current in dissipative systems." New Journal of Physics 21, no. 5 (May 7, 2019): 052001. http://dx.doi.org/10.1088/1367-2630/ab1731.

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6

Yan, YiJing, and RuiXue Xu. "QUANTUM MECHANICS OF DISSIPATIVE SYSTEMS." Annual Review of Physical Chemistry 56, no. 1 (May 5, 2005): 187–219. http://dx.doi.org/10.1146/annurev.physchem.55.091602.094425.

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7

Marianer, S., and J. M. Deutsch. "Quantum diffusion in dissipative systems." Physical Review B 31, no. 11 (June 1, 1985): 7478–81. http://dx.doi.org/10.1103/physrevb.31.7478.

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8

Chen, Yu. "Dissipative linear response theory and its appications in open quantum systems." Acta Physica Sinica 70, no. 23 (2021): 230306. http://dx.doi.org/10.7498/aps.70.20211687.

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With the recent development of experimental technology, the ability to control the dissipation in quantum many-body system is greatly enhanced. Meanwhile, many new breakthroughs are achieved in detecting the quantum states and others. All these advances make it necessary to establish a new theory for calculating the dissipative dynamics in strongly correlated sstems. Very recently, we found that by taking the interactions between the system and the bath as a perturbation, a systematic dissipative response theory can be established. In this new approach, the calculation of dissipative dynamics for any physical observables and the entropies can be converted into the calculation of certain correlation functions in initial states. Then we discuss how Markovian approximation at low temperature limit and at high temperature limit can be reached Also, we review the progress of the dissipative dynamics in open Bose-Hubbard model. In the fourth section, we review recent progress of entropy dynamics of quench dynamics of an open quantum system. Finally, we draw a conclusion and discuss possible development in the future.
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9

Cruz-Prado, Hans, Alessandro Bravetti, and Angel Garcia-Chung. "From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics." Quantum Reports 3, no. 4 (October 12, 2021): 664–83. http://dx.doi.org/10.3390/quantum3040042.

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Starting from the geometric description of quantum systems, we propose a novel approach to time-independent dissipative quantum processes according to which energy is dissipated but the coherence of the states is preserved. Our proposal consists of extending the standard symplectic picture of quantum mechanics to a contact manifold and then obtaining dissipation by using appropriate contact Hamiltonian dynamics. We work out the case of finite-level systems for which it is shown, by means of the corresponding contact master equation, that the resulting dynamics constitute a viable alternative candidate for the description of this subclass of dissipative quantum systems. As a concrete application, motivated by recent experimental observations, we describe quantum decays in a 2-level system as coherent and continuous processes.
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10

Cruz-Prado, Hans, Alessandro Bravetti, and Angel Garcia-Chung. "From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics." Quantum Reports 3, no. 4 (October 12, 2021): 664–83. http://dx.doi.org/10.3390/quantum3040042.

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Starting from the geometric description of quantum systems, we propose a novel approach to time-independent dissipative quantum processes according to which energy is dissipated but the coherence of the states is preserved. Our proposal consists of extending the standard symplectic picture of quantum mechanics to a contact manifold and then obtaining dissipation by using appropriate contact Hamiltonian dynamics. We work out the case of finite-level systems for which it is shown, by means of the corresponding contact master equation, that the resulting dynamics constitute a viable alternative candidate for the description of this subclass of dissipative quantum systems. As a concrete application, motivated by recent experimental observations, we describe quantum decays in a 2-level system as coherent and continuous processes.
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11

Öztürk, Fahri Emre, Tim Lappe, Göran Hellmann, Julian Schmitt, Jan Klaers, Frank Vewinger, Johann Kroha, and Martin Weitz. "Observation of a non-Hermitian phase transition in an optical quantum gas." Science 372, no. 6537 (April 1, 2021): 88–91. http://dx.doi.org/10.1126/science.abe9869.

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Quantum gases of light, such as photon or polariton condensates in optical microcavities, are collective quantum systems enabling a tailoring of dissipation from, for example, cavity loss. This characteristic makes them a tool to study dissipative phases, an emerging subject in quantum many-body physics. We experimentally demonstrate a non-Hermitian phase transition of a photon Bose-Einstein condensate to a dissipative phase characterized by a biexponential decay of the condensate’s second-order coherence. The phase transition occurs because of the emergence of an exceptional point in the quantum gas. Although Bose-Einstein condensation is usually connected to lasing by a smooth crossover, the observed phase transition separates the biexponential phase from both lasing and an intermediate, oscillatory condensate regime. Our approach can be used to study a wide class of dissipative quantum phases in topological or lattice systems.
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12

Zloshchastiev, Konstantin G. "Model Hamiltonians of open quantum optical systems: Evolvement from hermiticity to adjoint commutativity." Journal of Physics: Conference Series 2407, no. 1 (December 1, 2022): 012011. http://dx.doi.org/10.1088/1742-6596/2407/1/012011.

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Abstract In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.
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13

Ikeda, Tatsuhiko N., and Masahiro Sato. "General description for nonequilibrium steady states in periodically driven dissipative quantum systems." Science Advances 6, no. 27 (July 2020): eabb4019. http://dx.doi.org/10.1126/sciadv.abb4019.

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Laser technology has developed and accelerated photo-induced nonequilibrium physics, from both the scientific and engineering viewpoints. Floquet engineering, i.e., controlling material properties and functionalities by time-periodic drives, is at the forefront of quantum physics of light-matter interaction. However, it is limited to ideal dissipationless systems. Extending Floquet engineering to various materials requires understanding of the quantum states emerging in a balance of the periodic drive and energy dissipation. Here, we derive a general description for nonequilibrium steady states (NESSs) in periodically driven dissipative systems by focusing on systems under high-frequency drive and time-independent Lindblad-type dissipation. Our formula correctly describes the time average, fluctuation, and symmetry properties of the NESS, and can be computed efficiently in numerical calculations. This approach will play fundamental roles in Floquet engineering in a broad class of dissipative quantum systems from atoms and molecules to mesoscopic systems, and condensed matter.
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14

Harris, Edward G. "Erratum: Quantum tunneling in dissipative systems." Physical Review A 49, no. 1 (January 1, 1994): 629–30. http://dx.doi.org/10.1103/physreva.49.629.2.

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15

Bhattacharya, Samyadeb, and Sisir Roy. "Hartman effect and dissipative quantum systems." Journal of Mathematical Physics 54, no. 5 (May 2013): 052101. http://dx.doi.org/10.1063/1.4803132.

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16

Morikawa, Masahiro. "Classical fluctuations in dissipative quantum systems." Physical Review D 33, no. 12 (June 15, 1986): 3607–12. http://dx.doi.org/10.1103/physrevd.33.3607.

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17

Cerdeira, Hilda A., K. Furuya, and B. A. Huberman. "Lyapunov Exponent for Quantum Dissipative Systems." Physical Review Letters 61, no. 22 (November 28, 1988): 2511–13. http://dx.doi.org/10.1103/physrevlett.61.2511.

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18

Graham, R. "Period doubling in dissipative quantum systems." Physical Review Letters 62, no. 15 (April 10, 1989): 1806. http://dx.doi.org/10.1103/physrevlett.62.1806.

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19

Tarasov, Vasily E. "Stationary states of dissipative quantum systems." Physics Letters A 299, no. 2-3 (July 2002): 173–78. http://dx.doi.org/10.1016/s0375-9601(02)00678-3.

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20

Faupin, Jérémy, and François Nicoleau. "Scattering matrices for dissipative quantum systems." Journal of Functional Analysis 277, no. 9 (November 2019): 3062–97. http://dx.doi.org/10.1016/j.jfa.2019.06.010.

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21

Cubitt, Toby S., Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia. "Stability of Local Quantum Dissipative Systems." Communications in Mathematical Physics 337, no. 3 (April 7, 2015): 1275–315. http://dx.doi.org/10.1007/s00220-015-2355-3.

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22

Graham, R. "Period Doubling in Dissipative Quantum Systems." Europhysics Letters (EPL) 3, no. 3 (February 1, 1987): 259–63. http://dx.doi.org/10.1209/0295-5075/3/3/001.

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23

Hol/yst, J. A., and L/ A. Turski. "Dissipative dynamics of quantum spin systems." Physical Review A 45, no. 9 (May 1, 1992): 6180–84. http://dx.doi.org/10.1103/physreva.45.6180.

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24

Raine, D. J., and D. W. Sciama. "Hawking radiation and dissipative quantum systems." Classical and Quantum Gravity 14, no. 1A (January 1, 1997): A325—A329. http://dx.doi.org/10.1088/0264-9381/14/1a/024.

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25

Bimonte, Giuseppe, Giampiero Esposito, Giuseppe Marmo, and Cosimo Stornaiolo. "Classical brackets for dissipative systems." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2311–18. http://dx.doi.org/10.1142/s0217732303012520.

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We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.
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26

Cai, M. L., Z. D. Liu, Y. Jiang, Y. K. Wu, Q. X. Mei, W. D. Zhao, L. He, X. Zhang, Z. C. Zhou, and L. M. Duan. "Probing a Dissipative Phase Transition with a Trapped Ion through Reservoir Engineering." Chinese Physics Letters 39, no. 2 (February 1, 2022): 020502. http://dx.doi.org/10.1088/0256-307x/39/2/020502.

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Dissipation is often considered as a detrimental effect in quantum systems for unitary quantum operations. However, it has been shown that suitable dissipation can be useful resources in both quantum information and quantum simulation. Here, we propose and experimentally simulate a dissipative phase transition (DPT) model using a single trapped ion with an engineered reservoir. We show that the ion’s spatial oscillation mode reaches a steady state after the alternating application of unitary evolution under a quantum Rabi model Hamiltonian and sideband cooling of the oscillator. The average phonon number of the oscillation mode is used as the order parameter to provide evidence for the DPT. Our work highlights the suitability of trapped ions for simulating open quantum systems and shall facilitate further investigations of DPT with various dissipation terms.
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27

Baskoutas, S. "Wigner Distribution Function for Open Quantum Systems." Modern Physics Letters B 11, no. 09n10 (April 30, 1997): 391–97. http://dx.doi.org/10.1142/s0217984997000487.

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Using the modified biorthonormal Heisenberg equations of motion for non-Hermitian (NH) Hamilton operators, in order to imply a consistent Lie-algebraic structure and also the equivalence between the Heisenberg and Schrödinger pictures, we have obtained the analytical form of the Wigner distribution function which is unavoidable complex. Its imaginary part accounts for the influence of additional degrees of freedom, which are always present in the phenomenological representation of dissipative systems through (NH) Hamiltonians. Applications of the above formalism can be found, for instance, in dissipative macroscopic quantum tunneling (MQT) effect for Josephson junctions, and in the dissipative tunneling of trapped atoms in optical crystals.
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28

Olivera-Atencio, M. L., L. Lamata, S. Kohler, and J. Casado-Pascual. "Universal patterns in multifrequency-driven dissipative systems." Europhysics Letters 137, no. 1 (January 1, 2022): 12001. http://dx.doi.org/10.1209/0295-5075/ac55f3.

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Abstract The response of dissipative systems to multi-chromatic fields exhibits generic properties which follow from the discrete time-translation symmetry of each driving component. We derive these properties and illustrate them with paradigmatic examples of classical and quantum dissipative systems. In addition, some computational aspects, in particular a matrix continued-fraction method, are discussed. Moreover, we propose possible implementations with quantum optical settings.
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29

Cuccoli, Alessandro, Andrea Fubini, Valerio Tognetti, and Ruggero Vaia. "Thermodynamics of quantum dissipative many-body systems." Physical Review E 60, no. 1 (July 1, 1999): 231–41. http://dx.doi.org/10.1103/physreve.60.231.

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30

Liao, Jie-Lou, and Eli Pollak. "Quantum transition state theory for dissipative systems." Chemical Physics 268, no. 1-3 (June 2001): 295–313. http://dx.doi.org/10.1016/s0301-0104(01)00289-0.

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31

Lozano, G. S., H. F. Lozza, and D. Pérez Daroca. "Entanglement in quantum dissipative Ising spin systems." Physica B: Condensed Matter 398, no. 2 (September 2007): 455–59. http://dx.doi.org/10.1016/j.physb.2007.04.059.

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32

Popescu, Bogdan, Hasan Rahman, and Ulrich Kleinekathöfer. "Chebyshev Expansion Applied to Dissipative Quantum Systems." Journal of Physical Chemistry A 120, no. 19 (February 12, 2016): 3270–77. http://dx.doi.org/10.1021/acs.jpca.5b12237.

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33

Naraschewski, M., and A. Schenzle. "Monte Carlo integration of dissipative quantum systems." Zeitschrift f�r Physik D Atoms, Molecules and Clusters 33, no. 2 (June 1995): 79–88. http://dx.doi.org/10.1007/bf01437425.

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34

Schmid, Albert. "Repeated measurements on dissipative linear quantum systems." Annals of Physics 173, no. 1 (January 1987): 103–48. http://dx.doi.org/10.1016/0003-4916(87)90095-9.

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35

Sato, S. A., U. De Giovannini, S. Aeschlimann, I. Gierz, H. Hübener, and A. Rubio. "Floquet states in dissipative open quantum systems." Journal of Physics B: Atomic, Molecular and Optical Physics 53, no. 22 (October 23, 2020): 225601. http://dx.doi.org/10.1088/1361-6455/abb127.

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36

Jaiswal, Ambuja Bhushan, Akhilesh Pandey, and Ravi Prakash. "Universality classes of quantum chaotic dissipative systems." EPL (Europhysics Letters) 127, no. 3 (September 12, 2019): 30004. http://dx.doi.org/10.1209/0295-5075/127/30004.

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37

Cao, Jianshu, Michael Messina, and Kent R. Wilson. "Quantum control of dissipative systems: Exact solutions." Journal of Chemical Physics 106, no. 12 (March 22, 1997): 5239–48. http://dx.doi.org/10.1063/1.473522.

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38

CUGLIANDOLO, LETICIA F. "DISSIPATIVE QUANTUM DISORDERED MODELS." International Journal of Modern Physics B 20, no. 19 (July 30, 2006): 2795–804. http://dx.doi.org/10.1142/s0217979206035308.

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This article reviews recent studies of mean-field and one dimensional quantum disordered spin systems coupled to different types of dissipative environments. The main issues discussed are: (i) The real-time dynamics in the glassy phase and how they compare to the behaviour of the same models in their classical limit. (ii) The phase transition separating the ordered – glassy – phase from the disordered phase that, for some long-range interactions, is of second order at high temperatures and of first order close to the quantum critical point (similarly to what has been observed in random dipolar magnets). (iii) The static properties of the Griffiths phase in random king chains. (iv) The dependence of all these properties on the environment. The analytic and numeric techniques used to derive these results are briefly mentioned.
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39

Bussandri, Diego G., Tristán M. Osán, Pedro W. Lamberti, and Ana P. Majtey. "Correlations in Two-Qubit Systems under Non-Dissipative Decoherence." Axioms 9, no. 1 (February 12, 2020): 20. http://dx.doi.org/10.3390/axioms9010020.

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We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states with maximally mixed marginals independently interacting with non-dissipative decohering environments in different dynamical scenarios of physical relevance. In order to get further insight about the physical meaning of the behavior of these correlation measures we compared our results with those obtained by means of well-known correlation measures such as quantum mutual information and quantum discord. On one hand, we found that the behaviors of total and classical correlations, as assessed by means of the measures introduced in this work, are qualitatively in agreement with the behavior displayed by quantum mutual information and the measure of classical correlations typically used to calculate quantum discord. We also found that the optimization of all the measures of classical correlations depends upon a single parameter and the optimal value of this parameter turns out to be the same in all cases. On the other hand, regarding the measures of quantum correlations used in our studies, we found that in general their behavior does not follow the standard quantum discord D . As the quantification by means of standard quantum discord and the measures of quantum correlations introduced in this work depends upon the assumption that total correlations are additive, our results indicate that this property needs a deeper and systematic study in order to gain a further understanding regarding the possibility to obtain reliable quantifiers of quantum correlations within this additive scheme.
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40

Chakrabarty, Indranil, Subhashish Banerjee, and Nana Siddharth. "A study of quantum correlations in open quantum systems." Quantum Information and Computation 11, no. 7&8 (July 2011): 541–62. http://dx.doi.org/10.26421/qic11.7-8-1.

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In this work, we study quantum correlations in mixed states. The states studied are modeled by a two-qubit system interacting with its environment via a quantum non demolition (purely dephasing) as well as dissipative type of interaction. The entanglement dynamics of this two qubit system is analyzed. We make a comparative study of various measures of quantum correlations, like Concurrence, Bell's inequality, Discord and Teleportation fidelity, on these states, generated by the above evolutions. We classify these evoluted states on basis of various dynamical parameters like bath squeezing parameter $r$, inter-qubit spacing $r_{12}$, temperature $T$ and time of system-bath evolution $t$. In this study, in addition we report the existence of entangled states which do not violate Bell's inequality, but can still be useful as a potential resource for teleportation. Moreover we study the dynamics of quantum as well as classical correlation in presence of dissipative coherence.
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41

Tarasov, V. E. "Quantum dissipative systems. I. Canonical quantization and quantum Liouville equation." Theoretical and Mathematical Physics 100, no. 3 (September 1994): 1100–1112. http://dx.doi.org/10.1007/bf01018575.

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42

Moura, A. R., and P. D. Mesquita. "Thermodynamics of dissipative coherent states." Journal of Physics A: Mathematical and Theoretical 54, no. 50 (November 23, 2021): 505301. http://dx.doi.org/10.1088/1751-8121/ac367b.

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Abstract Almost all traditional physical formalisms are developed by using conservative forces, and the microscopic implementation of dissipation involves a sort of unusual process, mainly in quantum systems. In this work, we study the quantum harmonic model endowed with a non-Hermitian term responsible for dissipation. In addition, we also include an oscillating field that drives the model to a coherent state, which is dominated by fluctuation in a specific frequency, while regular thermal states are lowly occupied. The usual coherent state formalism at zero temperature is extended to treat dissipative models at finite temperature. We define a generating function that is used in the evaluation of the most relevant statistical averages, such as the particle distribution. Then, we successfully employ the developed formalism to discuss two well-known applications; the damped quantum harmonic oscillator, and the precession magnetization in a ferromagnetic sample.
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43

WANG, CHUN-YANG, AN-QI ZHAO, and XIANG-MU KONG. "ENTROPY AND ITS QUANTUM THERMODYNAMICAL IMPLICATION FOR ANOMALOUS SPECTRAL SYSTEMS." Modern Physics Letters B 26, no. 07 (March 20, 2012): 1150043. http://dx.doi.org/10.1142/s0217984911500436.

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The state function entropy and its quantum thermodynamical implication for two typical dissipative systems with anomalous spectral densities are studied by investigating on their low-temperature quantum behavior. In all cases, it is found that the entropy decays quickly and vanishes as the temperature approaches zero. This reveals a good conformity with the third law of thermodynamics and provides another evidence for the validity of fundamental thermodynamical laws in the quantum dissipative region.
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44

Yamaga, Kazuki. "Dissipative Dynamics of Non-Interacting Fermion Systems and Conductivity." Axioms 9, no. 4 (November 3, 2020): 128. http://dx.doi.org/10.3390/axioms9040128.

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In this paper, Non-Equilibrium Steady State that is induced by electric field and the conductivity of non-interacting fermion systems under the dissipative dynamics is discussed. The dissipation is taken into account within a framework of the quantum dynamical semigroup introduced by Davies (1977). We obtain a formula of the conductivity for the stationary state, which is applicable to arbitrary potentials. Our formula gives a justification of an adiabatic factor that is often introduced in practical calculation while using the Kubo formula. In addition, the conductivity of crystals (i.e., periodic potentials) is also discussed.
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45

Grigoryan, Hayk, Hrachya Astsatryan, Tigran Gevorgyan, and Vahe Manukyan. "Cloud Service for Numerical Calculations and Visualizations of Photonic Dissipative Systems." Cybernetics and Information Technologies 17, no. 5 (December 20, 2017): 89–100. http://dx.doi.org/10.1515/cait-2017-0058.

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Abstract Nowadays quantum physics is crucial for several scientific applications, where it is no longer possible to neglect the environmental interaction, like dissipation and decoherence. In these cases, the quantum systems are usually treated as open systems and their time-evolution is described by a density matrix in frames of the master equation, instead of the Hilbert-space vector and the Schrodinger equation. The visualization of such quantum systems allows users to calculate and study the sensitivity of the parameters, like excitation photon numbers or photonnumber distribution functions or Wigner functions. In this paper, a cloud service for numerical calculations and visualization of photonic dissipative systems is presented, which enables numerical simulations and visualizations of a wide variety of Hamiltonians, including those with arbitrary time-dependences widely used in many physics applications. The service allows creating graphics and charts for interacting complex systems and simulating their time evolution with many available timeevolution drivers.
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46

Sewell, Geoffrey L. "W★ Dynamics of Infinite Dissipative Quantum Systems." Reports on Mathematical Physics 87, no. 1 (February 2021): 107–10. http://dx.doi.org/10.1016/s0034-4877(21)00014-8.

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47

Sergi, Alessandro. "Deterministic constant-temperature dynamics for dissipative quantum systems." Journal of Physics A: Mathematical and Theoretical 40, no. 17 (April 11, 2007): F347—F354. http://dx.doi.org/10.1088/1751-8113/40/17/f05.

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48

Werner, Philipp, and Matthias Troyer. "Cluster Monte Carlo Algorithms for Dissipative Quantum Systems." Progress of Theoretical Physics Supplement 160 (2005): 395–417. http://dx.doi.org/10.1143/ptps.160.395.

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49

Liao, Jie-Lou, and Eli Pollak. "Mixed quantum classical rate theory for dissipative systems." Journal of Chemical Physics 116, no. 7 (February 15, 2002): 2718–27. http://dx.doi.org/10.1063/1.1436108.

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50

Shirai, Tatsuhiko, Juzar Thingna, Takashi Mori, Sergey Denisov, Peter Hänggi, and Seiji Miyashita. "Effective Floquet–Gibbs states for dissipative quantum systems." New Journal of Physics 18, no. 5 (May 10, 2016): 053008. http://dx.doi.org/10.1088/1367-2630/18/5/053008.

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