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arXiv:1902.01683 · Physical review. B./Physical review. B

From strong to weak temperature dependence of the two-photon entanglement resulting from the biexciton cascade inside a cavity

Tim Seidelmann, F. Ungar, Moritz Cygorek, Alexei Vagov, A. M. Barth, T. Kühn, V. M. Axt

Published Jun 3, 2019• 24 citations
arXivHTMLPDF

Abstract

We investigate the degree of entanglement quantified by the concurrence of photon pairs that are simultaneously emitted in the biexciton-exciton cascade from a quantum dot in a cavity. Four dot-cavity configurations are compared that differ with respect to the detuning between the cavity modes and the quantum dot transitions, corresponding to different relative weights of direct two-photon and sequential single-photon processes. The dependence of the entanglement on the exciton fine-structure splitting $\ensuremath{\delta}$ is found to be significantly different for each of the four configurations. For a finite splitting and low temperatures, the highest entanglement is found when the cavity modes are in resonance with the two-photon transition between the biexciton and the ground state and, in addition, the biexciton has a finite binding energy of a few meV. However, this widely used configuration is rather strongly affected by phonons such that other dot-cavity configurations, that are commonly regarded as less suited for obtaining high degrees of entanglement, become more favorable already at temperatures on the order of $10\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and above. If the cavity modes are kept in resonance with one of the exciton-to-ground-state transitions and the biexciton binding energy is finite, the entanglement drastically drops for positive $\ensuremath{\delta}$ with rising temperatures when $T$ is below $\ensuremath{\simeq}4\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, but is virtually independent of the temperature for higher $T$.

DOI
10.1103/physrevb.99.245301

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We investigate the degree of entanglement quantified by the concurrence of photon pairs that are simultaneously emitted in the biexciton-exciton cascade from a quantum dot in a cavity. Four dot-cavity configurations are compared that differ with respect to the detuning between the cavity modes and the quantum dot transitions, corresponding to different relative weights of direct two-photon and sequential single-photon processes. The dependence of the entanglement on the exciton fine-structure splitting \ensuremathδ\ensuremath{\delta}\ensuremathδ is found to be significantly different for each of the four configurations. For a finite splitting and low temperatures, the highest entanglement is found when the cavity modes are in resonance with the two-photon transition between the biexciton and the ground state and, in addition, the biexciton has a finite binding energy of a few meV. However, this widely used configuration is rather strongly affected by phonons such that other dot-cavity configurations, that are commonly regarded as less suited for obtaining high degrees of entanglement, become more favorable already at temperatures on the order of 10K10\phantom{\rule{0.28em}{0ex}}\mathrm{K}10K and above. If the cavity modes are kept in resonance with one of the exciton-to-ground-state transitions and the biexciton binding energy is finite, the entanglement drastically drops for positive \ensuremathδ\ensuremath{\delta}\ensuremathδ with rising temperatures when TTT is below \ensuremath≃4K\ensuremath{\simeq}4\phantom{\rule{0.16em}{0ex}}\mathrm{K}\ensuremath≃4K, but is virtually independent of the temperature for higher TTT.
We study the quantum properties and statistics of photons emitted by a quantum-dot biexciton inside a cavity. In the biexciton-exciton cascade, fine-structure splitting between exciton levels degrades polarization-entanglement for the emitted pair of photons. However, here we show that the polarization-entanglement can be preserved in such a system through simultaneous emission of two degenerate photons into cavity modes tuned to half the biexciton energy. Based on detailed theoretical calculations for realistic quantum-dot and cavity parameters, we quantify the degree of achievable entanglement.
While, theoretically, the degree of entanglement of a two-qubit state can be quantified by the concurrence, the experimental determination of a two-qubit state, such as via quantum state tomography of the two-photon state generated by the biexciton cascade in a quantum dot, typically involves the summation of many data points corresponding to a time integration. In the prevalent literature on the biexciton cascade in quantum dots, it is therefore common to define the concurrence in terms of single- or double-time-integrated two-time correlation functions. Here, we compare the behavior of the concurrence according to the different definitions found in the literature and study the relation between the time-integrated concurrences and the actual degree of entanglement of the two-photon states generated by the biexciton cascade in a quantum dot embedded in a microcavity. We focus mainly on the often-discussed situation of a dot with finite biexciton binding energy in a cavity tuned to the two-photon resonance. Analytic and numerical calculations reveal that in this case the single-time-integrated concurrence indeed agrees well with the typical value of the concurrence according to its original definition, even when the interaction between the quantum dot and longitudinal acoustic phonons is taken into account. However, the double-time-integrated concurrence shows completely different trends with respect to changes in the exciton fine structure splitting or the cavity loss rate and should not be interpreted as a measure for the degree of entanglement.
We compare three different notions of concurrence to measure the polarization entanglement of two-photon states generated by the biexciton cascade in a quantum dot embedded in a microcavity. The focus of the paper lies on the often-discussed situation of a dot with finite biexciton binding energy in a cavity tuned to the two-photon resonance. Apart from the time-dependent concurrence, which can be assigned to the two-photon density matrix at any point in time, we study single- and double-time integrated concurrences commonly used in the literature that are based on different quantum state reconstruction schemes. In terms of the photons detected in coincidence measurements, we argue that the single-time integrated concurrence can be thought of as the concurrence of photons simultaneously emitted from the cavity without resolving the common emission time, while the more widely studied double-time integrated concurrence refers to photons that are neither filtered with respect to the emission time of the first photon nor with respect to the delay time between the two emitted photons. Analytic and numerical calculations reveal that the single-time integrated concurrence indeed agrees well with the typical value of the time-dependent concurrence at long times, even when the interaction between the quantum dot and longitudinal acoustic phonons is accounted for. Thus the more easily measurable single-time integrated concurrence gives access to the physical information represented by the time-dependent concurrence. However, the double-time integrated concurrence shows a different behavior with respect to changes in the exciton fine structure splitting and even displays a completely different trend when the ratio between the cavity loss rate and the fine structure splitting is varied while keeping their product constant. This implies the nonequivalence of the physical information contained in the time-dependent and single-time integrated concurrence on the one hand and the double-time integrated concurrence on the other hand.
We report on simulations of the degree of polarization entanglement of photon pairs simultaneously emitted from a quantum dot-cavity system that demand revisiting the role of phonons. Since coherence is a fundamental precondition for entanglement and phonons are known to be a major source of decoherence, it seems unavoidable that phonons can only degrade entanglement. In contrast, we demonstrate that phonons can cause a degree of entanglement that even surpasses the corresponding value for the phonon-free case. In particular, we consider the situation of comparatively small biexciton binding energies and either finite exciton or cavity mode splitting. In both cases, combinations of the splitting and the dot-cavity coupling strength are found where the entanglement exhibits a nonmonotonic temperature dependence which enables entanglement above the phonon-free level in a finite parameter range. This unusual behavior can be explained by phonon-induced renormalizations of the dot-cavity coupling g in combination with a nonmonotonic dependence of the entanglement on g that is present already without phonons.
Semiconductor quantum dots are promising sources for polarization-entangled photons. As an alternative to the usual cascaded biexciton-exciton emission, direct two-photon emission from the biexciton can be used. With a high-quality optical resonator tuned to half the biexciton energy, a large proportion of the photons can be steered into the two-photon emission channel. In this case the degree of polarization entanglement is inherently insensitive to the exciton fine-structure splitting. In the present work we analyze the biexciton emission with particular emphasis on the influence of coupling of the quantum-dot cavity system to its environment. Especially for a high-quality cavity, the coupling to the surrounding semiconductor material can open up additional phonon-assisted decay channels. Our analysis demonstrates that with the cavity tuned to half the biexciton energy, the potentially detrimental influence of the phonons on the polarization entanglement is strongly suppressed---high degrees of entanglement can still be achieved. We further discuss spectral properties and statistics of the emitted twin photons.
A quantum dot strongly coupled to a photonic crystal has been recently proposed as a source of entangled photon pairs [R. Johne et al., Phys. Rev. Lett. 100, 240404 (2008)]. The biexction decay via intermediate polariton states can be used to overcome the natural splitting between the exciton states coupled to the horizontally and vertically polarized light modes so that high degrees of entanglement can be expected. We investigate theoretically the features of realistic dot-cavity systems, including the effect of the different oscillator strength of excitons resonances coupled to the different polarizations of light. We show that in this case, an independent adjustment of the cavity resonances is needed in order to keep a high entanglement degree. We also consider the case when the biexciton-exciton transition is also strongly coupled to a cavity mode. We show that a very fast emission rate can be achieved allowing the repetition rates close to the terahertz range. Such fast emission should however be paid for by a very complex tuning of the many strongly coupled resonances involved and by a loss of quantum efficiency. Altogether a strongly coupled dot-cavity system seems to be very promising as a source of entangled photon pairs.