Non-Markovian Quantum Dynamics: Local versus Nonlocal

Abstract

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...

Decoherence control in different environments

“…The reservoir memory effects are encoded in the time-dependent coefficients. We note in passing that time-local master equations are equivalent to master equations containing a memory kernel, in the sense that the latter ones can always be recast in time-local form [30].…”

Phenomenological memory-kernel master equations and time-dependent Markovian processes

“…Many analytical methods and numerical techniques have been developed to treat non-Markovian processes [2][3][4][5][6][7][8][9][10][11]. In addition to their importance in addressing fundamental questions [12], this is mainly due to the applications non-Markovian systems find in many branches of physics.…”

Optimal state pairs for non-Markovian quantum dynamics

“…During recent years, there has been significant conceptual, theoretical, and experimental progress dealing with non-Markovian processes [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In particular, the very definition of non-Markovianity and quantification of quantum memory effects in the dynamics of open systems has received a lot of interest.…”