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arXiv:1008.4022 · Physical Review E

Environment-assisted quantum transport and trapping in dimers

Oliver Mülken, Tobias Schmid

Published Oct 21, 2010• 19 citations
arXivHTMLPDF

Abstract

We study the dynamics and trapping of excitations for a dimer with an energy off-set $\ensuremath{\Delta}$ coupled to an external environment. Using a Lindblad quantum master equation approach, we calculate the survival probability $\ensuremath{\Pi}(t)$ of the excitation and define different lifetimes ${\ensuremath{\tau}}_{s}$ of the excitation, corresponding to the duration of the decay of $\ensuremath{\Pi}(t)$ in between two predefined values. We show that it is not possible to always enhance the overall decay to the trap. However, it is possible, even for not too small environmental couplings and for values of $\ensuremath{\Delta}$ of the order $\mathcal{O}(1)$, to decrease certain lifetimes ${\ensuremath{\tau}}_{s}$, leading to faster decay of $\ensuremath{\Pi}(t)$ in these time intervals: there is an optimal environmental coupling, leading to a maximal decay for fixed $\ensuremath{\Delta}$.

DOI
10.1103/physreve.82.042104

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We study the dynamics and trapping of excitations for a dimer with an energy off-set \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ coupled to an external environment. Using a Lindblad quantum master equation approach, we calculate the survival probability \ensuremathΠ(t)\ensuremath{\Pi}(t)\ensuremathΠ(t) of the excitation and define different lifetimes \ensuremathτs{\ensuremath{\tau}}_{s}\ensuremathτs​ of the excitation, corresponding to the duration of the decay of \ensuremathΠ(t)\ensuremath{\Pi}(t)\ensuremathΠ(t) in between two predefined values. We show that it is not possible to always enhance the overall decay to the trap. However, it is possible, even for not too small environmental couplings and for values of \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ of the order O(1)\mathcal{O}(1)O(1), to decrease certain lifetimes \ensuremathτs{\ensuremath{\tau}}_{s}\ensuremathτs​, leading to faster decay of \ensuremathΠ(t)\ensuremath{\Pi}(t)\ensuremathΠ(t) in these time intervals: there is an optimal environmental coupling, leading to a maximal decay for fixed \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ.
The dynamics of a dimer coupled to two different environments, each in a spin star configuration under the influence of decoherence, is studied. The exact analytical expression for the transition probability in the dimer system is obtained for different situations, i.e., independent and correlated environments. In all cases considered, it is shown that there exist well-defined ranges of parameters for which decoherent interaction with the environment assists energy transfer in the dimer system. In particular, we find that correlated environments can assist energy transfer more efficiently than separate baths.
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In this paper, we report on a detailed study of how decoherence assists quantum transport in a dimer system. By carrying out analytical and numerical computations, we find that the decoherence induced by a Markovian (memoryless) environment only assists the maximum probability of quantum transport in the dimer system to 50% not close to the unity, and the decoherence induced by a non-Markovian environment with perfect memory (infinite memory time) can improve the maximum probability of quantum transport to unity. The combination of non-Markovian decoherence and Markovian decoherence assist the quantum transport more effectively. Comparing the classical environment with the quantum environment, we obtain that the classical non-Markovian environment cannot assist the maximum probability of quantum transport close to unity. In addition, enlarging the quantum environment can improve the maximum probability of quantum transport.
Coherent transport of excitations along chains of coupled quantum systems represents an interesting problem with a number of applications ranging from quantum optics to solar cell technology. A convenient tool for studying such processes are quantum walks. They allow us to determine all the process features in a quantitative way. We study the survival probability and the transport efficiency on a simple, highly symmetric graph represented by a ring. The propagation of excitation is modeled by a discrete-time (coined) quantum walk. For a two-state quantum walk, where the excitation (walker) has to leave its actual position to the neighboring sites, the survival probability decays exponentially and the transport efficiency is unity. The decay rate of the survival probability can be estimated using the leading eigenvalue of the evolution operator. However, if the excitation is allowed to stay at its present position, i.e. the propagation is modeled by a lazy quantum walk, then part of the wave-packet can be trapped in the vicinity of the origin and never reaches the sink. In such a case, the survival probability does not vanish and the excitation transport is not efficient. The dependency of the transport efficiency on the initial state is determined. Nevertheless, we show that for some lazy quantum walks dynamical, percolations of the ring eliminate the trapping effect and efficient excitation transport can be achieved.
We study the transport property of diffusion in a finite translationally invari-ant quantum subsystem described by a tight-binding Hamiltonian with a sin-gle energy band. The subsystem interacts with its environment by a coupling expressed in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superop-erator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely pro-portional to the size of the subsystem, the spectrum at given wave number con-tains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. An analytical expres-sion is obtained for the dispersion relation of diffusion. The diffusion coef-ficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffu-sion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling. KEY WORDS: Quantum transport; environment-induced diffusion; quantum
The Lindblad master equation is widely used to describe the reduced dynamics of open quantum systems under Markovian assumptions. Here, we investigate its ability to reproduce the reduced evolution emerging from a microscopic many-body model in which two interacting two-level subsystems are embedded in a larger environment and evolve under fully unitary dynamics. The exact evolution exhibits a clear separation of timescales. At short times, decoherence arises from environmentally induced dephasing, leading to a Gaussian suppression of coherences and a quadratic decay of purity. At intermediate times, collective decoherence channels saturate and a slower, still Gaussian, decay driven by relative environmental fluctuations dominates. At later times the system settles in a complete decohered state. The first two behaviors cannot be reproduced by a Lindblad dynamics with constant coefficients, which always results in an exponential decay: Our work provides a simple example of the breakdown of the effective description relevant in many realistic settings.