Symmetry of bilinear master equations for a quantum oscillator

Tay, B.A.

doi:10.1016/j.physa.2016.10.067

Cited by 6 publications

(31 citation statements)

References 36 publications

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“…The coherent and squeezed states are minimum uncertainty states that minimize the generalized uncertainty relation. Gaussian states satisfy the generalized uncertainty relation if and only if they have positive time evolution [19,25].…”

Section: Moments Of Transformed Statesmentioning

confidence: 99%

“…S T denotes the transposition operation on S [16,19] that can be obtained by using the invariance of trace under cyclic permutations of operators. We restrict our consideration to S that can be written as a sum of superoperators a i × b i , that is, S = i µ i (a i × b i ), where µ i are complex numbers.…”

Section: Moments Of Transformed Statesmentioning

confidence: 99%

“…[19], there are seven generators that give rise to transformation that preserve the hermiticity and trace of density matrices. Three of the generators among them [19], generate the familiar unitary operators. The first one iL 0 ρ = i[a † a, ρ] generates the time evolution of free oscillator.…”

Section: Unitary Operatorsmentioning

confidence: 99%

“…From the results of Ref. [19], there are four generators that give rise to damping. The four dissipative generators are…”

Section: Dissipative Operatorsmentioning

confidence: 99%

“…There was effort to set up a basis of operators in terms of commutator brackets [18]. Recently, we construct a different basis of seven generators that are bilinear in the creation and annihilation operators of harmonic oscillator [19]. They generate transformations that preserve the hermiticity and the trace of density matrices.…”

Section: Introductionmentioning

confidence: 99%

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Damping modes of harmonic oscillator in open quantum systems

Tay

2019

Physica A: Statistical Mechanics and its Applications

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Through a set of generators that preserves the hermiticity and trace of density matrices, we analyze the damping of harmonic oscillator in open quantum systems into four modes, distinguished by their specific effects on the covariance matrix of position and momentum of the oscillator. The damping modes could either cause exponential decay to the initial covariance matrix or shift its components. They have to act together properly in actual dynamics to ensure that the generalized uncertainty relation is satisfied. We use a few quantum master equations to illustrate the results.

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Section: Moments Of Transformed Statesmentioning

confidence: 99%

Section: Moments Of Transformed Statesmentioning

confidence: 99%

Section: Unitary Operatorsmentioning

confidence: 99%

“…From the results of Ref. [19], there are four generators that give rise to damping. The four dissipative generators are…”

Section: Dissipative Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Damping modes of harmonic oscillator in open quantum systems

Tay

2019

Physica A: Statistical Mechanics and its Applications

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Solutions of generic bilinear master equations for a quantum oscillator—Positive and factorized conditions on stationary states

Tay

2017

Physica A: Statistical Mechanics and its Applications

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We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states for a given initial state isuggests conditions on the coefficients of the master equations that generate positive evolution.When we obtain the master equations of a system in contact with an environment, there are often assumptions made to facilitate the derivation of the effective influence of the environment on the system [1,2]. For instance, we assume that the initial density matrix of the system and environment is factorized, the coupling between the system and its environment is weak, the rotating wave approximation is valid, the memory effects are negligible, or the generators of the master equation should be completely positive, and etc. Appropriate derivations should produce master equations with well-behaved solutions.However, anomaly could arise if we carelessly apply the master equations to situations that are not consistent with the assumptions made in their derivations, such as studying low temperature behavior of a system with the Caldeira-Leggett equation [3,4,5], obtaining an Markovian equation by ignoring the memory effects or using the rotating wave approximation [6,7,8,9], or starting with initial conditions inconsistent with the factorized assumptions [10,11].An issue that has received wide attention is the completely positive [12,13] nature of the generators of the master equations. Initial states that are positive could evolve outside their positive domain during some interval of the time evolution when the master equations do not have completely positive generators. However, there is no a priori reason why master equations must be completely positive [10]. Not completely positive generators with appropriately chosen coefficients could maintain the positivity of the states for a given initial state for all time [14]. The systems are then properly behaved as far as positivity is concerned.It is in this spirit that we start with a generic master equation that satisfies the essential hermitian and trace preserving requirements of a reduced dynamics [15,16,17]. We obtain its explicit solutions in Gaussian form and analyzed the stationary states. We further identify a factorized condition on the stationary states that is strong enough to produce a generic class of master equations. This class includes the well-known master equations as special cases, i.e., the Kossakowski-Lindblad (KL) equation for quantum optical ...

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Eigenvalues of the Liouvillians of quantum master equation for a harmonic oscillator

Tay

2020

Physica A: Statistical Mechanics and its Applications

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Symmetry of bilinear master equations for a quantum oscillator

Cited by 6 publications

References 36 publications

Damping modes of harmonic oscillator in open quantum systems

Damping modes of harmonic oscillator in open quantum systems

Solutions of generic bilinear master equations for a quantum oscillator—Positive and factorized conditions on stationary states

Eigenvalues of the Liouvillians of quantum master equation for a harmonic oscillator

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