Complete and fidelity-robust state analysis for polarization-spatial-time-bin hyperentanglement with double-sided quantum-dot-cavity systems

Yuan-Hao Sun; Yu-Qing Guo; Cong Cao; Cong Cao; Cong Cao

doi:10.1364/JOSAB.495668

Journal of the Optical Society of America B
Vol. 40,
Issue 8,
pp. 2073-2084
(2023)
•https://doi.org/10.1364/JOSAB.495668

Complete and fidelity-robust state analysis for polarization-spatial-time-bin hyperentanglement with double-sided quantum-dot-cavity systems

Yuan-Hao Sun, Yu-Qing Guo, and Cong Cao

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Yuan-Hao Sun, Yu-Qing Guo, and Cong Cao, "Complete and fidelity-robust state analysis for polarization-spatial-time-bin hyperentanglement with double-sided quantum-dot-cavity systems," J. Opt. Soc. Am. B 40, 2073-2084 (2023)

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Abstract

Hyperentanglement represents a key resource in many quantum information processing schemes. Here, we present a complete and fidelity-robust hyperentangled-Bell-state analysis (HBSA) scheme for two-photon systems entangled in polarization, spatial mode, and time-bin degrees of freedom (DOFs). The scheme uses a fidelity-robust quantum nondemolition (QND) detector called BLOCK1, built with a singly charged semiconductor quantum dot (QD) in a double-sided optical microcavity (double-sided QD-cavity system) under the balance condition, and some linear-optical elements. Compared with the previously proposed complete HBSA schemes based on ideal optical giant circular birefringence, our scheme guarantees the robust fidelity and relaxes the requirement on the QD-cavity parameters with the assistance of the balance condition. Different from the schemes using error-detected blocks and exchanging Bell states in different DOFs to get unity fidelity, the BLOCK1 under the balance condition can directly measure the Bell states of polarization and spatial mode so that fewer kinds of linear optical components are needed in our scheme. In addition, our scheme simplifies the discrimination process and reduces the required light–matter interaction by using a self-assisted mechanism. These features indicate that our scheme may be directly generalized to multiphoton hyperentangled Greenberger–Horne–Zeilinger state analysis and more feasible in practical quantum applications based on the photonic hyperentanglement.

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Initial Hyperentangled States	Spin States	${B L O C K 2}_{k}$ Where Photons Are Detected
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$

Initial Hyperentangled States	Detectors Where Photons Are Detected
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$

Initial Hyperentangled States	Spin States	${B L O C K 2}_{k}$ Where Photons Are Detected
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{+} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{0}^{-} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| + ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{+} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{3} o r {B L O C K 2}_{2}, {B L O C K 2}_{4}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| + ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| + ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| - ⟩_{1} \| + ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{y}^{φ_{T}} ⟩_{T} \otimes \| Ω_{1}^{-} ⟩_{S}$	$\| - ⟩_{1} \| - ⟩_{2} \| - ⟩_{3}$	${B L O C K 2}_{1}, {B L O C K 2}_{4} o r {B L O C K 2}_{2}, {B L O C K 2}_{3}$

Initial Hyperentangled States	Detectors Where Photons Are Detected
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{0}^{+} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{0}^{-} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{1}^{+} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{0}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} L_{2}, c_{1} L_{1} d_{2} R_{2}, d_{1} R_{1} c_{2} L_{2}, d_{1} L_{1} c_{2} R_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{0}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} d_{2} R_{2}, c_{1} L_{1} d_{2} L_{2}, d_{1} R_{1} c_{2} R_{2}, d_{1} L_{1} c_{2} L_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{1}^{+} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} L_{2}, c_{1} L_{1} c_{2} R_{2}, d_{1} R_{1} d_{2} L_{2}, d_{1} L_{1} d_{2} R_{2}$
$\| Ω_{1}^{-} ⟩_{P} \otimes \| Ω_{1}^{-} ⟩_{T} \otimes \| Ω_{z}^{φ_{S}} ⟩_{S}$	$c_{1} R_{1} c_{2} R_{2}, c_{1} L_{1} c_{2} L_{2}, d_{1} R_{1} d_{2} R_{2}, d_{1} L_{1} d_{2} L_{2}$

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Journal of the Optical Society of America B