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The meaning of the concurrence obtained from time-integrated two-photon density matrices in the biexciton cascade in quantum dots coupled to microcavities

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

Published Dec 7, 2017• 0 citations
OpenAlex

Abstract

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.

Related papers

Comparison of different concurrences characterizing photon pairs generated in the biexciton cascade in quantum dots coupled to microcavities

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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 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 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 $\ensuremath{\delta}$ with rising temperatures when TTT is below $\ensuremath{\simeq}4\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, but is virtually independent of the temperature for higher TTT.
A full quantum microscopic theory is developed to analyze a biexciton radiative cascade coupled to bulk acoustic phonons in a quantum dot. By considering the phonon sub-system in coherent state representation a new approach is proposed for investigating the phonon effects. Via this approach it is possible to obtain an exact analytical result for the phonon kernel in this system. This approach is introduced in the context of an example: the process of generating polarization-entangled photon pairs from the biexciton cascade in a quantum dot. We calculate the exact density matrix (using quantum state tomography) of photons and their concurrence. We show that the exchange interaction and temperature have remarkable effects on the degree of entanglement of the emitted photons. The approach introduced provides an exact analytical result for finite discrete electron states interacting with phonons.
We theoretically study the polarization entanglement of photons generated by the biexciton cascade in a GaAs/InAs semiconductor quantum dot (QD) located in a nanocavity. A detailed analysis of the complex interplay between photon and carrier coherences and phonons which occurs during the cascade allows us to clearly identify the conditions under which entanglement is generated and destroyed. A quantum state tomography is evaluated for varying exciton fine-structure splittings. Also, by constructing an effective multiphonon Hamiltonian which couples the continuum of the QD-embedding wetting layer states to the quantum confined states, we investigate the relaxation of the biexciton and exciton states. This consistently introduces a temperature dependence to the cascade. Considering typical Stranski-Krastanov grown QDs for temperatures around 80 K the degree of entanglement starts to be affected by the dephasing of the exciton states and is ultimately lost above 100 K.
We discuss generation of two types of entangled states of two photons: noon state, which is entangled in number and polarization, and polarization-entangled state, which is entangled in polarization and frequency. We consider a single quantum dot coupled with a bimodal cavity in a strong coupling regime. We analyze the effect of exciton–phonon coupling on concurrence of the generated entangled states. We find that for both states, concurrence is maximum in the absence of an anisotropic energy gap between exciton states and remains unchanged in the presence of exciton–phonon coupling. However, for finite anisotropic, energy gap concurrence decreases with increasing temperature of the phonon bath.
Photons which are generated in a two-photon cascade process have an underlying time correlation since the spontaneous emission of the upper level populates the intermediate state. This correlation leads to a reduction of the purity of the photon emitted from the intermediate state. Here we characterize this time correlation for the biexciton-exciton cascade of an InAs/GaAs quantum dot. We show that the correlation can be reduced by tuning the biexciton transition in resonance to a planar distributed Bragg reflector cavity. The enhanced and inhibited emission into the cavity accelerates the biexciton emission and slows down the exciton emission thus reduces the correlation and increases the purity of the exciton photon. This is essential for schemes like creating time-bin entangled photon pairs from quantum dot systems.