We introduce the Quantum Dissipative Villain (QDV) model as a prototype model to study tunneling in dissipative quantum mechanics. Dissipation is provided by a coupled linear environment. In the QDV model, the discrete character of a tunneling degree of freedom coupled to an environment is explicit, leading to a rich dual structure. We derive general exact mappings of the QDV model on several dual discrete representations, including pairs of self-dual models, for general linear environments and arbitrary temperatures. Self-duality allows to write exact equations for each correlation function of each representation. Analogies with the theory of classical network transformations are also presented. Finally we discuss the fundamental character of the QDV model. For instance, the standard Caldeira-Leggett model, which describes mesoscopic Josephson junctions in a circuit and many other physical systems, is a special QDV model. The self-dual structure of the QDV model allows then the exact generalization of the Schmid approximate self-duality to general linear environments and arbitrary temperatures.We introduce the Quantum Dissipative Villain (QDV) model as a prototype model to study tunneling in dissipative quantum mechanics. Dissipation is provided by a coupled linear environment. In the QDV model, the discrete character of a tunneling degree of freedom coupled to an environment is explicit, leading to a rich dual structure. We derive general exact mappings of the QDV model on several dual discrete representations, including pairs of self-dual models, for general linear environments and arbitrary temperatures. Self-duality allows to write exact equations for each correlation function of each representation. Analogies with the theory of classical network transformations are also presented. Finally we discuss the fundamental character of the QDV model. For instance, the standard Caldeira-Leggett model, which describes mesoscopic Josephson junctions in a circuit and many other physical systems, is a special QDV model. The self-dual structure of the QDV model allows then the exact generalization of the Schmid approximate self-duality to general linear environments and arbitrary temperatures.

arXiv:cond-mat/9812105Dec 7, 19980 citations

A soluble model for quantum mechanical dissipation

N. G. van Kampen

Journal of Statistical PhysicsJan 1, 199566 citations

Memory-induced weak dissipation in fractional-time-derivative quantum Lindblad-based model

Chun Yun Kee, L. K. Ang

Modeling the interaction between a quantum system and its environment is crucial for practical quantum technologies. The Lindblad master equation is the simplest equation to understand this interaction. In this paper, we have extended the traditional Lindblad equation by fractionalizing its time derivative to account for the memory-induced dissipation. Using this approach, we show weak dissipation of selected quantum systems can be reproduced by this proposed time-fractional Lindblad equation without introducing any specific dissipation terms in the model. By varying the order of the time-fractional Lindblad equation without dissipation terms, we can reproduce the results with good agreements to three tested cases: (a) dissipative Rabi oscillation, (b) dissipative Ising model, and (c) collapse and revival in the Jaynes–Cumming model. We believe this proposed time-fractional Lindblad equation may be a useful modeling tool to characterize weakly dissipative quantum systems in practical quantum technologies, especially if the complex dissipation mechanism is not completely known from the traditional approaches.Modeling the interaction between a quantum system and its environment is crucial for practical quantum technologies. The Lindblad master equation is the simplest equation to understand this interaction. In this paper, we have extended the traditional Lindblad equation by fractionalizing its time derivative to account for the memory-induced dissipation. Using this approach, we show weak dissipation of selected quantum systems can be reproduced by this proposed time-fractional Lindblad equation without introducing any specific dissipation terms in the model. By varying the order of the time-fractional Lindblad equation without dissipation terms, we can reproduce the results with good agreements to three tested cases: (a) dissipative Rabi oscillation, (b) dissipative Ising model, and (c) collapse and revival in the Jaynes–Cumming model. We believe this proposed time-fractional Lindblad equation may be a useful modeling tool to characterize weakly dissipative quantum systems in practical quantum technologies, especially if the complex dissipation mechanism is not completely known from the traditional approaches.

APL QuantumMar 1, 20247 citations

Models for dissipation in quantum mechanics

N. G. van Kampen

Lecture notes in physicsJan 1, 19951 citation

Monte Carlo simulations of dissipative quantum Ising models

Iver Bakken Sperstad, Einar B. Stiansen, Asle Sudbø

The dynamical critical exponent $z$ is a fundamental quantity in characterizing quantum criticality, and it is well-known that the presence of dissipation in a quantum model has significant impact on the value of $z$ . Studying quantum Ising spin models using Monte Carlo methods, we estimate the dynamical critical exponent $z$ and the correlation length exponent $\ensuremath{\nu}$ for different forms of dissipation. For a two-dimensional quantum Ising model with Ohmic site dissipation, we find $z\ensuremath{\approx}2$ as for the corresponding one-dimensional case, whereas for a one-dimensional quantum Ising model with Ohmic bond dissipation, we obtain the estimate $z\ensuremath{\approx}1$.The dynamical critical exponent $z$ is a fundamental quantity in characterizing quantum criticality, and it is well-known that the presence of dissipation in a quantum model has significant impact on the value of $z$. Studying quantum Ising spin models using Monte Carlo methods, we estimate the dynamical critical exponent $z$ and the correlation length exponent $\ensuremath{\nu}$ for different forms of dissipation. For a two-dimensional quantum Ising model with Ohmic site dissipation, we find $z\ensuremath{\approx}2$ as for the corresponding one-dimensional case, whereas for a one-dimensional quantum Ising model with Ohmic bond dissipation, we obtain the estimate $z\ensuremath{\approx}1$.

arXiv:1002.3369Physical Review BMar 16, 201014 citations

The dynamical critical exponent

z

is a fundamental quantity in characterizing quantum criticality, and it is well-known that the presence of dissipation in a quantum model has significant impact on the value of

z

. Studying quantum Ising spin models using Monte Carlo methods, we estimate the dynamical critical exponent

z

and the correlation length exponent $\ensuremath{\nu}$ for different forms of dissipation. For a two-dimensional quantum Ising model with Ohmic site dissipation, we find $z\ensuremath{\approx}2$ as for the corresponding one-dimensional case, whereas for a one-dimensional quantum Ising model with Ohmic bond dissipation, we obtain the estimate $z\ensuremath{\approx}1$.