On the Structure of Generators for Non-Markovian Master Equations

Kossakowski, Andrzej; Rebolledo, Rolando

doi:10.1142/s1230161209000190

Cited by 37 publications

(41 citation statements)

References 38 publications

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“…For this case a general consistent theoretical framework has not yet been found, and partial results have been obtained as a result of intense efforts (see, e.g., Refs. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). An important step in the development of a general theory consists in the construction of a suitable measure that quantifies the degree of non-Markovianity for a given dynamical evolution [22,23].…”

Section: Introductionmentioning

confidence: 99%

Exact master equations for the non-Markovian decay of a qubit

2010

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Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. Employing the exact solution for the model we determine analytical expressions for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows a detailed investigation and comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, we find that the structure of widely used phenomenological master equations with memory kernel may be incompatible with a non-perturbative treatment of the underlying microscopic model. We discuss several physical implications of our results on the microscopic analysis and the phenomenological modelling of non-Markovian quantum dynamics of open systems.

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Section: Introductionmentioning

confidence: 99%

Exact master equations for the non-Markovian decay of a qubit

2010

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“…It turns out that the popular Markovian approximation which does not take into account memory effects is not sufficient for modern applications and todays technology calls for truly nonMarkovian approach. Non-Markovian dynamics was recently studied in [3][4][5][6][7][8][9][10][11][12][13][14][15]. Interestingly, several measures of non-Markovianity were proposed during last year [16][17][18][19].…”

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confidence: 99%

Non-Markovian Quantum Dynamics: Local versus Nonlocal

2010

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We analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing evolution of such a system may be described either by non-local master equation with memory kernel or equivalently by equation which is local in time. These two descriptions are complementary: if one is simple the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point 't0'. This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular, nevertheless, the corresponding dynamics is perfectly regular. Remarkably, singularities of generator may lead to interesting physical phenomena like revival of coherence or sudden death and revival of entanglement. [2]. It turns out that the popular Markovian approximation which does not take into account memory effects is not sufficient for modern applications and todays technology calls for truly nonMarkovian approach. Non-Markovian dynamics was recently studied in [3][4][5][6][7][8][9][10][11][12][13][14][15]. Interestingly, several measures of non-Markovianity were proposed during last year [16][17][18][19].The standard approach to the dynamics of open system uses the Nakajima-Zwanzig projection operator technique [20] which shows that under fairly general conditions, the master equation for the reduced density matrix ρ(t) takes the form of the following non-local equationin which quantum memory effects are taken into account through the introduction of the memory kernel K(t): this simply means that the rate of change of the state ρ(t) at time t depends on its history (starting at t = t 0 ). Usually, one takes t 0 = 0, however, in this letter we shall keep 't 0 ' arbitrary. An alternative and technically much simpler scheme is the time-convolutionless (TCL) projection operator technique [1,21,22] in which one obtains a first-order differential equation for the reduced density matrix. The advantage of the TCL approach consists in the fact that it yields an equation of motion for the relevant degrees of freedom which is local in time and which is therefore often much easier to deal with than the Nakajima-Zwanzig non-local master equation (1).An essential step to derive TCL from (1) relies on the existence of certain operator inverse [22]. However, this inverse needs not exist and then the method does not work [1,22]. Moreover, even if it exists the corresponding local in time TCL generator is usually defined by the perturbation series (see e.g. detailed discussion in [1]) in powers of the coupling strength characterizing the system. However in general the perturbative approach leads to significant problems. For example the dynamical map needs not be completely positive if one takes only finite number of terms from the perturbative expansion.In the present paper we take a different path. We show that any solution of the non-local equation (1) always satisfies equation which does not involve the integral memory k...

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“…Let us observe that this problem may be reformulated as follows [19]: any completely positive solution Λ t of Eq. (3) may be represented by…”

Section: Preliminary Ideas a Non-markovian Dynamicsmentioning

confidence: 99%

“…where B is a quantum channel (i.e., a trace preserving CP map) [12,18,19]. L t generates a completely positive trace preserving dynamics Λ t if the Laplace transform κ(s) satisfies…”

Section: B Quantum Channelmentioning

confidence: 99%

Long-time memory in non-Markovian evolutions

2010

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If the dynamics of an open quantum systems is non-Markovian, its asymptotic state strongly depends on the initial conditions, even if the dynamics possesses an invariant state. This is the very essence of memory effects. In particular, the asymptotic state can remember and partially preserve its initial entanglement. Interestingly, even if the non-Markovian evolution relaxes to an equilibrium state, this state needs not be invariant. Therefore, the non-invariance of equilibrium becomes a clear sign of non-Markovianity.

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On the Structure of Generators for Non-Markovian Master Equations

Abstract: Complete characterization of complete positivity preserving non-Markovian master equations is presented.

Cited by 37 publications

References 38 publications

Exact master equations for the non-Markovian decay of a qubit

Exact master equations for the non-Markovian decay of a qubit

Non-Markovian Quantum Dynamics: Local versus Nonlocal

Long-time memory in non-Markovian evolutions

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