Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Dynamics of simple dissipative systems

F. Guinea

Published Oct 1, 1985• 43 citations
Publisher

Abstract

The dynamics of a two-level system (TLS) coupled to an external heat bath has been studied. Correlation functions are obtained by combining analytical and numerical renormalization-group methods. It is shown that perturbation theory can be applied to obtain the low-energy, long-time behavior of the system. Analytical expressions are given for 〈${\ensuremath{\sigma}}_{z}$(t)${\ensuremath{\sigma}}_{z}$(0)${〉}_{T=0}$ valid in the long- and short-time limits (as compared with the renormalized frequency of the TLS). It is shown that other correlation functions, like 〈${\ensuremath{\sigma}}_{x}$(t)${\ensuremath{\sigma}}_{x}$(0)〉, depend sensitively on the nature of the high-energy cutoff of the environment. Thermodynamic functions, like the susceptibility to a bias potential and the correction to the specific heat of the environment, are also calculated. Finally, it is discussed how, in these systems, the experimental setup may influence the quantity being measured.

DOI
10.1103/physrevb.32.4486

Related papers

Non-Markovian dynamics of a dissipative two-level system: Nonzero bias and sub-Ohmic bath

Congjun Gan, Hang Zheng

The quantum dynamics of the dissipative two-level system with nonzero bias and sub-Ohmic bath is studied by means of the perturbation approach based on a unitary transformation. It has been shown that for the sub-Ohmic bath it is necessary to use the non-Markovian approach, especially for the short time behavior of the coupled system and environment. The nonequilibrium correlation $P(t)$ has been calculated to show that a finite bias may favor the short time coherence. The spectrum of the susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ of the sub-Ohmic case may have a double peak structure in the range of $\ensuremath{\omega}>0$ when the coupling $\ensuremath{\alpha}$ is relatively strong. Besides, the coherence-decoherence transition point ${\ensuremath{\alpha}}_{c}$ is determined for different $0<s\ensuremath{\le}1$ by the condition of ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega}=0)=\ensuremath{\infty}$ when $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$. Finally, we show that Shiba's relation is exactly satisfied in our results.

Physical Review E•Oct 6, 2009•16 citations

Dynamics of open quantum systems

Herbert Schoeller

A tutorial introduction is presented for the calculation of the time dynamics for models of dissipative quantum mechanics where a small quantum system is coupled to noninteracting bosonic or fermionic reservoirs. We discuss the basics and applications of the real-time renormalization group method (RTRG), where the Fourier variable $E$ conjugate to the time variable $t$ is used as flow parameter. It is shown how a well-controlled perturbation theory can be set up in the coupling to the bath where secular and logarithmic terms are resummed to all order of perturbation theory. This is achieved by a set of perturbative RG equations for the effective Liouvillian and the effective vertices. Truncating the RG equations we show how the time dynamics can be obtained analytically for exponentially small, intermediate and exponentially large times. Basic applications are discussed for particle and potential energy exchange within the interacting resonant level model, spin exchange for the nonequilibrium Kondo model, and heat exchange within the ohmic spin boson model at zero bias.

arXiv:1802.10014•Feb 27, 2018•10 citations

Renormalization-Group Theory for Quantum Dissipative Systems in Nonequilibrium

Keil, Markus

We apply the recently developed real-time renormalization-group (RTRG) method to different quantum dissipative systems. In contrast to other renormalization-group techniques the RTRG approach is formulated on the Keldysh contour and therefore allows for the description of nonequilibrium situations. The method is based on an exact kinetic equation which determines the time-dependent reduced density matrix. The kinetic equation involves an integral kernel which describes the phenomenon of dissipation. We calculate this quantity using the RTRG method: After applying Wick's theorem the arising diagrams are accounted for successively with respect to their time scale. During that process the theory is left invariant by renormalizing the operators describing the theory. As a starting point we discuss the polaron problem. We calculate the renormalized ground-state energy and the effective mass. We compare our results of the RTRG with those obtained from the flow equation formalism, which is based on infinitesimal transformations. In the central part of this thesis we consider the spin-boson model which is applicable to various physical and chemical systems. It models a particle in a double-well potential which is coupled to a bosonic environment. We determine static properties such as the static susceptibility. We also present results for the time evolution of the reduced density matrix starting from an arbitrary nonequilibrium state. The asymptotic behaviour of the reduced density matrix can be characterized by an oscillation frequency and two decay constants, all of which are determined. Finally, we calculate equilibrium correlation functions. We find consistency of our results with quantum Monte Carlo simulations and with the Shiba-relation. Additionally, our results show the correct scaling behaviour. Comparisons are also drawn to results of the noninteracting blip approximation and to results obtained for the anisotropic Kondo model. Our approach is very accurate for not too large couplings to the environment. Furthermore, the stationary tunnel current through a double quantum dot coupled to a phonon bath is calculated and compared to experimental data. Eventually, the running couplings of the two-lead Kondo model are discussed.

•Jan 1, 2002•0 citations

Quantum dynamics of the dissipative two-state system coupled with a sub-Ohmic bath

Zhiguo Lü, Hang Zheng

The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function $P(t)$, the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function $C(t)$ are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.

arXiv:quant-ph/0612042•Physical Review B•Feb 2, 2007•77 citations

Characterization of a qubit in presence of dissipation and external driving

Francesco Nesi

In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.

•Jan 1, 2007•1 citation

Thermodynamics of the dissipative two-state system: A Bethe-ansatz study

Theo A. Costi, Gergely Zaránd

The thermodynamics of the dissipative two-state system is calculated exactly for all temperatures and level asymmetries for the case of Ohmic dissipation. We exploit the equivalence of the two-state system to the anisotropic Kondo model and extract the thermodynamics of the former by solving the thermodynamic Bethe-ansatz equations of the latter. The universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$ and static dielectric susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$ are extracted for all dissipation strengths $0<\ensuremath{\alpha}<1.$ The logarithmic corrections to these quantities at high temperatures are found in the Kondo limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}},$ whereas for $\ensuremath{\alpha}<1$ we find the expected power law temperature dependences with the powers being functions of the dissipative coupling \ensuremath{\alpha}. The low-temperature behavior is always that of a Fermi liquid.

arXiv:cond-mat/9903104•Physical review. B, Condensed matter•May 15, 1999•55 citations
The dynamics of a two-level system (TLS) coupled to an external heat bath has been studied. Correlation functions are obtained by combining analytical and numerical renormalization-group methods. It is shown that perturbation theory can be applied to obtain the low-energy, long-time behavior of the system. Analytical expressions are given for 〈\ensuremathσz{\ensuremath{\sigma}}_{z}\ensuremathσz​(t)\ensuremathσz{\ensuremath{\sigma}}_{z}\ensuremathσz​(0)〉T=0{〉}_{T=0}〉T=0​ valid in the long- and short-time limits (as compared with the renormalized frequency of the TLS). It is shown that other correlation functions, like 〈\ensuremathσx{\ensuremath{\sigma}}_{x}\ensuremathσx​(t)\ensuremathσx{\ensuremath{\sigma}}_{x}\ensuremathσx​(0)〉, depend sensitively on the nature of the high-energy cutoff of the environment. Thermodynamic functions, like the susceptibility to a bias potential and the correction to the specific heat of the environment, are also calculated. Finally, it is discussed how, in these systems, the experimental setup may influence the quantity being measured.
The quantum dynamics of the dissipative two-level system with nonzero bias and sub-Ohmic bath is studied by means of the perturbation approach based on a unitary transformation. It has been shown that for the sub-Ohmic bath it is necessary to use the non-Markovian approach, especially for the short time behavior of the coupled system and environment. The nonequilibrium correlation P(t)P(t)P(t) has been calculated to show that a finite bias may favor the short time coherence. The spectrum of the susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ of the sub-Ohmic case may have a double peak structure in the range of $\ensuremath{\omega}>0$ when the coupling $\ensuremath{\alpha}$ is relatively strong. Besides, the coherence-decoherence transition point ${\ensuremath{\alpha}}_{c}$ is determined for different $0<s\ensuremath{\le}1$ by the condition of ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega}=0)=\ensuremath{\infty}$ when $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$. Finally, we show that Shiba's relation is exactly satisfied in our results.
A tutorial introduction is presented for the calculation of the time dynamics for models of dissipative quantum mechanics where a small quantum system is coupled to noninteracting bosonic or fermionic reservoirs. We discuss the basics and applications of the real-time renormalization group method (RTRG), where the Fourier variable EEE conjugate to the time variable ttt is used as flow parameter. It is shown how a well-controlled perturbation theory can be set up in the coupling to the bath where secular and logarithmic terms are resummed to all order of perturbation theory. This is achieved by a set of perturbative RG equations for the effective Liouvillian and the effective vertices. Truncating the RG equations we show how the time dynamics can be obtained analytically for exponentially small, intermediate and exponentially large times. Basic applications are discussed for particle and potential energy exchange within the interacting resonant level model, spin exchange for the nonequilibrium Kondo model, and heat exchange within the ohmic spin boson model at zero bias.
We apply the recently developed real-time renormalization-group (RTRG) method to different quantum dissipative systems. In contrast to other renormalization-group techniques the RTRG approach is formulated on the Keldysh contour and therefore allows for the description of nonequilibrium situations. The method is based on an exact kinetic equation which determines the time-dependent reduced density matrix. The kinetic equation involves an integral kernel which describes the phenomenon of dissipation. We calculate this quantity using the RTRG method: After applying Wick's theorem the arising diagrams are accounted for successively with respect to their time scale. During that process the theory is left invariant by renormalizing the operators describing the theory. As a starting point we discuss the polaron problem. We calculate the renormalized ground-state energy and the effective mass. We compare our results of the RTRG with those obtained from the flow equation formalism, which is based on infinitesimal transformations. In the central part of this thesis we consider the spin-boson model which is applicable to various physical and chemical systems. It models a particle in a double-well potential which is coupled to a bosonic environment. We determine static properties such as the static susceptibility. We also present results for the time evolution of the reduced density matrix starting from an arbitrary nonequilibrium state. The asymptotic behaviour of the reduced density matrix can be characterized by an oscillation frequency and two decay constants, all of which are determined. Finally, we calculate equilibrium correlation functions. We find consistency of our results with quantum Monte Carlo simulations and with the Shiba-relation. Additionally, our results show the correct scaling behaviour. Comparisons are also drawn to results of the noninteracting blip approximation and to results obtained for the anisotropic Kondo model. Our approach is very accurate for not too large couplings to the environment. Furthermore, the stationary tunnel current through a double quantum dot coupled to a phonon bath is calculated and compared to experimental data. Eventually, the running couplings of the two-lead Kondo model are discussed.
The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function P(t)P(t)P(t), the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function C(t)C(t)C(t) are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.
In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.
The thermodynamics of the dissipative two-state system is calculated exactly for all temperatures and level asymmetries for the case of Ohmic dissipation. We exploit the equivalence of the two-state system to the anisotropic Kondo model and extract the thermodynamics of the former by solving the thermodynamic Bethe-ansatz equations of the latter. The universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$ and static dielectric susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$ are extracted for all dissipation strengths $0<\ensuremath{\alpha}<1.$ The logarithmic corrections to these quantities at high temperatures are found in the Kondo limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}},$ whereas for $\ensuremath{\alpha}<1$ we find the expected power law temperature dependences with the powers being functions of the dissipative coupling \ensuremath{\alpha}. The low-temperature behavior is always that of a Fermi liquid.