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Optics Express

Temporal dynamics of zero-delay second order correlation function and spectral entanglement of two photons emitted from ladder-type atomic three-level systems

Kwang Jun Ahn

Published Jan 4, 2020• 4 citations
Publisher

Abstract

We theoretically investigate temporal dynamics of the second order cross correlation function at zero delay time ( G 12 ( 2 ) ( t ) ) and spectral entanglement of two photons emitted from an atomic three-level cascade. In Heisenberg’s picture, a closed set of quantum kinetic equations of motion for G 12 ( 2 ) ( t ) is derived within density matrix formalism with cluster expansion rule. G 12 ( 2 ) ( t ) shows qualitatively distinctive features depending on the spectral entanglement of two photons. Although incoherent photon pairs generated from spontaneous radiation of the excited electron are not entangled, their correlation and anti-correlation properties can be found in G 12 ( 2 ) ( t ) depending on the radiative decay rates. In the coherent excitation regime where the light emitter is located in a high Q-cavity, and its atomic polarizations are predominantly initialized, spectral entanglement between two coherent photons is established. We show that G 12 ( 2 ) ( t ) is well fitted by the entanglement criterion by Duan-Giedke-Cirac-Zoller and explain the close relationship between them by means of the optically forbidden transition in the three-level cascade.

DOI
10.1364/oe.382498

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We theoretically investigate temporal dynamics of the second order cross correlation function at zero delay time ( G 12 ( 2 ) ( t ) ) and spectral entanglement of two photons emitted from an atomic three-level cascade. In Heisenberg’s picture, a closed set of quantum kinetic equations of motion for G 12 ( 2 ) ( t ) is derived within density matrix formalism with cluster expansion rule. G 12 ( 2 ) ( t ) shows qualitatively distinctive features depending on the spectral entanglement of two photons. Although incoherent photon pairs generated from spontaneous radiation of the excited electron are not entangled, their correlation and anti-correlation properties can be found in G 12 ( 2 ) ( t ) depending on the radiative decay rates. In the coherent excitation regime where the light emitter is located in a high Q-cavity, and its atomic polarizations are predominantly initialized, spectral entanglement between two coherent photons is established. We show that G 12 ( 2 ) ( t ) is well fitted by the entanglement criterion by Duan-Giedke-Cirac-Zoller and explain the close relationship between them by means of the optically forbidden transition in the three-level cascade.
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We theoretically investigate resonance fluorescence of a ladder- and a triangular-type three-level atomic system in the full quantization regime. By solving Heisenberg’s quantum kinetic equations of motion of an electron interacting with photons and two coherent laser fields, we obtain the firstorder electric field correlation function, which corresponds to photoluminescence spectrum. When the single electrons in both systems are resonantly excited by two strong cw-laser fields, in contrast to the ladder-type atomic systems emitting mainly incoherent photons, the triangular-type system shows that continuous coherent photons can be selectively generated from three optical transitions by trapping electronic population inversions and manipulated by changing the transition rates.
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We investigate the excitation of a three-level ladder-type atom by a unidirectional field with a pair of indistinguishable photons. Starting from an analytical expression for the two-photon absorption probability, we determine the two-photon state that maximizes the population of the upper atomic state at a chosen time and show that in the limit of an infinitely long pulse, perfect excitation is possible. The optimal state is identified as the time-reversed counterpart of the two-photon state emitted in spontaneous cascade decay. We then compare this ideal excitation strategy with experimentally accessible families of states, including symmetrized Gaussian product states, temporally correlated Gaussian states, and coherent pulses. We analyze how the optimal excitation conditions depend on the ratio of atomic decay rates and on the separation of the atomic transition frequencies. For indistinguishable photons, quantum interference may shift the maxima of the marginal spectral distribution away from the atomic resonances and qualitatively modify the optimal excitation strategy. Our results clarify the role of indistinguishability and correlations in two-photon absorption and provide guidance for designing realistic excitation schemes in quantum-optical light-matter interfaces.
The ladder configuration of atomic levels provides a source for telecom photons (signal) from the upper atomic transition. For rubidium and caesium atoms, the signal field has the range around 1.3–1.5 μm that can be coupled to an optical fibre and transmitted to a remote location. Cascade emission may result in pairs of photons, the signal entangled with the subsequently emitted infrared photon (idler) from the lower atomic transition. This correlated two-photon source is potentially useful in the DLCZ (Duan–Lukin–Cirac–Zoller) protocol for the quantum repeater. We implement the cascade emission to construct a modified DLCZ quantum repeater and investigate the role of the time-frequency entanglement in the protocol. The dependence of the protocol on the resolving and non-resolving photon-number detectors is also studied. We find that the frequency entanglement deteriorates the performance but the harmful effect can be diminished by using shorter pump pulses to generate the cascade emission. An optimal cascade-emission-based DLCZ scheme is realized by applying a pure two-photon source in addition to using detectors of perfect quantum efficiency.
We develop a formalism based on a time-dependent wave-function ansatz to study correlations of photons emitted from a collection of two-level quantum emitters. We show how to simulate the system dynamics and evaluate the intensity of the scattered photons and the second-order correlation function g(2)g^{(2)}g(2) in terms of the amplitudes of the different components of the wave function. Our approach is efficient for considering systems that contain up to two excitations. To demonstrate this we first consider the example of spectral filtering of photons emitted from a single quantum emitter. We show how our formalism can be used to study spectral filtering of the two-photon component of the emitted light from a single quantum emitter for various kinds of filters. Furthermore, as a general application of our formalism, we show how it can be used to study photon-photon correlations in an optically dense ensemble of two-level quantum emitters. In particular we lay out the details of simulating correlated photon transport in such ensembles reported recently by S. Mahmoodian {\it et.al.} [Phys. Rev. Lett. {\bf 121}, 143601 (2018)]. Compared to other existing techniques, the advantage of our formalism is that it is applicable to any generic spectral filter and quantum many-body systems involving a large number of quantum emitters while requiring only a modest computational resource.