Spectral origin of non-Markovianity in an exact finite harmonic model

Ruggero Vasile, Fernando Galve, Roberta Zambrini

IFISC (UIB-CSIC), Instituto de Física Interdisciplinar y Sistemas Complejos, Palma de Mallorca, Spain

Open systems in the quantum formalism are described through an elegant theoretical approach leading to master or to Langevin equations. A fundamental assumption in phenomenological approaches is the form of the spectral density J(ω) embedding all information about the real couplings and frequencies in the complex infinite environment, and the structure of the coupling to the system. Typical approximations to simplify the treatment, such as negligible memory effects (Markovian approximation), system time coarse graining, weak system-bath coupling, large frequency cut-off, Ohmic spectral density, drastically constrain the possible frequency dependence of J(ω). Important deviations from these simple instances, however, are common in several systems and can lead to memory effects and deviations from Markovian dynamics. These effects can be quantified with several measures and recently different approaches have been proposed focusing on deviations from the Lindblad form of the generator of the master equation [1], on flow-back of information from the environment [2], or on entanglement decay with an ancilla [3], to mention some of them. In this talk we identify non-Markovian effects originating in the structure of the system and bath couplings as well as in the distribution of energies considering a microscopic model given by an inhomogeneous harmonic chain, avoiding the limitations of approximate approaches. The case of an oscillator attached to a homogeneous chain was already studied by Rubin to determine the statistical properties of crystals with defects: this configuration leads to a Ohmic dissipation (thus Markovian, at least for large temperature) [4]. Here we will discuss the potential offered by non-homogeneous tunable chains. As a matter of fact, experimental implementation of a tunable chain of oscillators can be obtained through recent progress in segmented Paul traps [5, 6] also allowing tunability of ions couplings and on-site potentials. Other possible setups are based on photonic crystal nanocavities, microtoroid resonators or optomechanic resonators [7]. Furthermore, correlations spectra of the system can be measured to gain insight on the spectral density induced by the rest of the chain [8, 9]. The non-homogeneous harmonic chain we consider in this work represents then a structured and controllable reservoir amenable to experimental realization. Moving to non-homogeneous chain configurations [10] allow us to inquire the origin of non-Markovianity and to distinguish among several independent features quantifying separately different sources of non-Markovian dynamics. When focusing on periodic systems (e.g. dimers), we can engineer spectral densities with finite band-gaps, like in semiconductors or photonic crystals. For suitable couplings we show that the system is actually influenced by the resonant portion of the environment. Memory effects are then evaluated by sweeping the spectral density for a structured bath allowing us to show the effects of the local form of the spectrum. Non-Markovianity is quantified with the mentioned measures of information flow-back and non-divisibility of the system dynamical map [2, 3]. We show strongest memory effect at band-gap edges and provide an interpretation based on energy flow between system and environment. A system weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of the environment as well as the local density of modes are not substantial factors. We show an opposite effect when increasing the temperature inside or outside the spectral band-gap. Further, non-Markovianity arises for larger (negative and positive) powers of algebraic spectral densities, being the Ohmic case not always the most Markovian one. Ongoing developments of this research project will be also presented.

[1] M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Phys. Rev. Lett. 101, 150402 (2008).

[2] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).

[3] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010).

[4] R. J. Rubin , Phys. Rev. 131, 964 (1963), and references therein.

[5] A. Walther et al., Phys. Rev. Lett. 109, 080501 (2012).

[6] R. Bowler et al., Phys. Rev. Lett. 109, 080502 (2012).

[7] M. Aspelmeyer, S. Groeblacher, K. Hammerer, and N. Kiesel, J. Opt. Soc. Am. B 27, 189 (2010).

[8] S. Groeblacher, A. Trubarov, N. Prigge, M. Aspelmeyer and J. Eisert, arXiv:1305.6942 (2013).

[9] W.M. Zhang, P. Y. Lo, H. N. Xiong, M. W. Y. Tu; F.Nori Phys. Rev. Lett. 109, 170402 (2012).

[10] R. Vasile, F. Galve, R. Zambrini, Phys. Rev. A 89, 022109 (2014).