Survival Probabilities in Coherent Exciton Transfer with Trapping

Abstract

In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power-law, in contrast to the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.PACS numbers: 05.60. Gg, 32.80.Rm, 34.20.Cf Recent years have seen an upsurge of interest in coherent energy transfer, given the experimental advances in manipulating and controlling quantum mechanical systems. From the theoretical side, such investigations are of long standing; see, e.g., [1]. Here, tight-binding models, which model coherent exciton transfer, are closely related to the quantum walks (QW An appropriate means to monitor transport is to follow the decay of the excitation due to trapping. The long time decay of chains with traps is a well studied problem for classical systems [8,9]: for an ensemble of chains of different length with traps at both ends the averaged exciton survival probability has a stretched exponential form exp(−bt λ ), with λ = 1/3 (see, e.g., [9]). In contrast, quantum mechanical tight-binding models lead to λ = 1/4 [10, 11]. However, up to now only little is known about the decay of the quantum mechanical survival probability at experimentally relevant intermediate times.Here we evaluate and compare the intermediate-time decays due to trapping for both RW and QW situations by employing the similarity of the CTRW and the CTQW formalisms. Without traps, the coherent dynamics of excitons on a graph of connected nodes is modeled by the CTQW, which is obtained by identifying the Hamiltonian H 0 of the system with the CTRW transfer matrix T 0 , i.e., H 0 = −T 0 ; see e.g. [3, 12] (we will set ≡ 1 in the following). For undirected graphs, T 0 is related to the connectivity matrix A 0 of the graph by T 0 = −A 0 , where (for simplicity) all transmission rates are taken to be equal. Thus, in the following we take H 0 = A 0 . The matrix A 0 has as non-diagonal elements A the Rydberg gases considered in the following, the coupling strength is roughly H (0) k,j / 1 MHz, i.e., the time unit for transfer between two nodes is of the order of a few hundred nanoseconds.The states |j associated with excitons localized at the nodes j (j = 1, . . . , N ) form a complete, orthonormal basis set (COBS) of the whole accessible Hilbert space, i.e., k|j = δ kj and k |k k| = 1. In general, the time evolution of a state |j starting at time t 0 = 0 is given by |j; t = exp(−iH 0 t)|j ; hence the transition amplitudes and the probabilities read α kj (t) ≡ k| exp(−iH 0 t)|j and π kj (t) ≡ |α kj (t)| 2 , respectively. In the corresponding classical CTRW case the transition probabilities follow from a master equation as ...

Rabi Oscillations and Excitation Trapping in the Coherent Excitation of a Mesoscopic Frozen Rydberg Gas

“…This effect can be exploited as an exciton trap for continuous time quantum random walk experiments [9]. It also constitutes another constraint to states that are suited for dipole blockade experiments besides the already identified zeroes in Rydberg-Rydberg interactions for certain internal states [24] and atom pair alignments [25].…”

“…We show that an interaction-induced coupling to a larger number of internal states can be used to trap the excitation. Employed in a controlled way, this effect offers future applications in the experimental realization of quantum random walks with exciton trapping [9].…”

“…We show that an interaction-induced coupling to a larger number of internal states can be used to trap the excitation. Employed in a controlled way, this effect offers future applications in the experimental realization of quantum random walks with exciton trapping [9].Our experiments are performed with a magneto-optical trap (MOT) of about 10 7 87 Rb atoms at densities of 10 10 cm −3 and temperatures below 100 µK. Rydberg excitation is achieved with two counterpropagating laser beams at 780 nm and 480 nm (see Fig.…”

“…In the last decade Rydberg physics were extended to laser-cooled atomic gases which allowed the study of a frozen system with controllable, strong interactions and negligible thermal contributions. This has opened a wide field in both experiment and theory covering such diverse areas as resonant energy transfer [2,3], plasma formation [4,5], exotic molecules [6,7], and quantum random walks [8,9]. In addition, these frozen Rydberg systems have been proposed as a possible candidate for quantum information processing [10,11].…”

Generalized Kac lemma for recurrence time in iterated open quantum systems

“…Monitoring the hitting time in this way is a discrete-time analog for the hitting time defined for continuous-time quantum walks, which is defined through the survival time of an excitation in the system where the Hamiltonian includes a trapping site [29][30][31]. …”