Dynamics of the Density Matrix in Contact with a Thermal Bath and the Quantum Master Equation

Mori, Toshiyuki; Miyashita, Seiji

doi:10.1143/jpsj.77.124005

Cited by 64 publications

(95 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Up to the second order in the interaction Hamiltonian H SE we have 53) are the first-and second-order correction, respectively. Up to second order in the interaction Hamiltonian H SE , the reduced density matrix therefore reads showing that in the thermal equilibrium state, due to the interaction, we should expect a deviation from the canonical distribution of the system.…”

Section: Resultsmentioning

confidence: 99%

Approach to Equilibrium in Nano-scale Systems at Finite Temperature

et al. 2010

Self Cite

View full text Add to dashboard Cite

We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles. The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment. The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble. The state of the composite system evolves according to the time-dependent Schrödinger equation, the interaction creating entanglement between the system and the environment. It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value. We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases. As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles (<36), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.

show abstract

Section: Resultsmentioning

confidence: 99%

Approach to Equilibrium in Nano-scale Systems at Finite Temperature

et al. 2010

Self Cite

View full text Add to dashboard Cite

show abstract

“…Such equations can give steady state solutions for the density matrix accurate to the lowest, zeroth order only [118,119]. Some progress is made recently [120] in extending the accuracy of the density matrix to second order by a novel analytic continuation without actually solving more complicated fourth order master equation [121,122].…”

Section: Quantum Master Equationsmentioning

confidence: 99%

Nonequilibrium Green’s function method for quantum thermal transport

et al. 2013

View full text Add to dashboard Cite

This review deals with the nonequilibrium Green's function (NEGF) method applied to the problems of energy transport due to atomic vibrations (phonons), primarily for small junction systems. We present a pedagogical introduction to the subject, deriving some of the well-known results such as the Laudauer-like formula for heat current in ballistic systems. The main aim of the review is to build the machinery of the method so that it can be applied to other situations, which are not directly treated here. In addition to the above, we consider a number of applications of NEGF, not in routine model system calculations, but in a few new aspects showing the power and usefulness of the formalism. In particular, we discuss the problems of multiple leads, coupled left-right-lead system, and system without a center. We also apply the method to the problem of full counting statistics. In the case of nonlinear systems, we make general comments on the thermal expansion effect, phonon relaxation time, and a certain class of mean-field approximations. Lastly, we examine the relationship between NEGF, reduced density matrix, and master equation approaches to thermal transport.

show abstract

“…Higherorder terms might become important for very long times. For instance, the terms of O(λ 4 ) may change the stationary state in O(λ 2 ) [17,19]. However, they do not play important roles for the transient relaxation dynamics, which is the major interest of this work.…”

Section: B(t)/2a(t) and Minimizing With Respect To T > 0 We Havementioning

confidence: 99%

Natural correlation between a system and a thermal reservoir

Mori

2014

Phys. Rev. A

Self Cite

View full text Add to dashboard Cite

Non-Markovian corrections to the Markovian quantum master equation of an open quantum system are investigated up to the second order of the interaction between the system of interest and a thermal reservoir. The concept of "natural correlation" is discussed. When the system is naturally correlated with a thermal reservoir, the time evolution of the reduced density matrix looks Markovian even in a short-time regime. If the total system was initially in an "unnatural" state, the natural correlation is established during the time evolution, and after that the time evolution becomes Markovian in a long-time regime. It is also shown that for a certain set of reduced density matrices, the naturally correlated state does not exist. If the initial reduced density matrix has no naturally correlated state, the time evolution is inevitably non-Markovian in a short-time regime.The time evolution of the reduced density matrix ρ S (t) of the system of interest in contact with a large thermal reservoir is described by the quantum master equation [1][2][3]. When the coupling constant λ between the system and the reservoir is very small, the time scale of the relaxation is well separated from the time scale of the microscopic motion of the system and that of the reservoir, and hence the Markov approximation is justified. Indeed, the Markovian quantum master equation is rigorously derived in the van Hove limit, λ → 0 with λ 2 t fixed [4]. However, in some real situations, such as atoms in photonic band gap media [5][6][7][8] and an optical cavity coupled to a structured reservoir [9], the deviation from the van Hove limit is important and the non-Markov effect is not negligible.In this paper, the non-Markovian corrections are investigated up to the order of λ 2 . The key concept discussed in this work is the natural correlation between the system of interest and the thermal reservoir. The following properties are to be clarified. When the total system starts from a naturally correlated state, the time evolution of the system of interest obeys the Markovian quantum master equation even if there is deviation from the van Hove limit. On the other hand, if the initial state is not a naturally correlated state, the time evolution must be non-Markovian in short times t τ R , where τ R is a characteristic time of the motion of the thermal reservoir. Even in this case, after a sufficiently long time t ≫ τ R , the time evolution approximately becomes Markovian since the natural correlation is established during the time evolution.The further observation of this work is that there is a set U of reduced density matrices such that for any ρ S ∈ U, the naturally correlated state does not exist. Therefore, if some ρ S ∈ U is chosen as an initial state of the system of interest, the time evolution is inevitably non-Markovian in the short time regime t τ R . After some time t, ρ S (t) will get out of U and the natural correlation will develop there. * Electronic address: mori@spin.phys.s.u-tokyo.ac.jp Now the setup is explained. The Hamiltonian ...

show abstract

Dynamics of the Density Matrix in Contact with a Thermal Bath and the Quantum Master Equation

Cited by 64 publications

References 37 publications

Approach to Equilibrium in Nano-scale Systems at Finite Temperature

Approach to Equilibrium in Nano-scale Systems at Finite Temperature

Nonequilibrium Green’s function method for quantum thermal transport

Natural correlation between a system and a thermal reservoir

Contact Info

Product

Resources

About