Quantum networking with photons and trapped atoms (Invited)

D. L. Moehring; M. J. Madsen; K. C. Younge; R. N. Kohn, Jr.; P. Maunz; L.-M. Duan; C. Monroe; B. B. Blinov

doi:10.1364/JOSAB.24.000300

Journal of the Optical Society of America B
Vol. 24,
Issue 2,
pp. 300-315
(2007)
•https://doi.org/10.1364/JOSAB.24.000300

Quantum networking with photons and trapped atoms (Invited)

D. L. Moehring, M. J. Madsen, K. C. Younge, R. N. Kohn, P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov

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D. L. Moehring, M. J. Madsen, K. C. Younge, R. N. Kohn, Jr., P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov, "Quantum networking with photons and trapped atoms (Invited)," J. Opt. Soc. Am. B 24, 300-315 (2007)

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Abstract

Distributed quantum information processing requires a reliable quantum memory and a faithful carrier of quantum information. Atomic qubits have very long coherence times and are thus excellent candidates for quantum information storage, whereas photons are ideal for the transport of quantum information as they can travel long distances with a minimum of decoherence. We discuss the theoretical and experimental combination of these two systems and their use for not only quantum information transfer but also scalable quantum computation architectures.

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Quantum State	Probability	Description
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$1 - p p_{e} - p p_{b g}$	No photons
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$p p_{e}$	Good photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p p_{b g}$	Background photon
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{2} p p_{b g}$	Background photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{b g}^{2}$	Background photons
${∣ M ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{b g}^{2}$	Background photons
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{b g}^{2}$	Background photons
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{e} p_{b g}$	$Background + good photons$
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{e} p_{b g}$	$Background + good photons$
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$p p_{2 e}$	Double-excitation photon
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$p p_{2 e}$	Double-excitation photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{2 e}$	Double-excitation photons
${∣ M ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{2 e}$	Double-excitation photons
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{2 e}$	Double-excitation photons

Photonic State	Atomic State	Probability
$({∣ 0 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β}) ({∣ 1 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ Ψ^{β α} ⟩}_{a b}$	$p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} \cos^{4} \frac{χ}{2} + \sin^{2} θ_{a} \cos^{2} θ_{b} \sin^{4} \frac{χ}{2})$

	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} (1 + \frac{\sin^{2} χ}{2}) + \frac{1}{2} p_{b g} p_{e} [1 + (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ p_{2 e} p_{b g} (1 - \frac{\sin^{2} χ}{2}) + p_{2 e} p_{e} [1 - \frac{\sin^{2} χ}{2} + (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β}) ({∣ 0 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{a b}$	$p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} \sin^{4} \frac{χ}{2} + \sin^{2} θ_{a} \cos^{2} θ_{b} \cos^{4} \frac{χ}{2})$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} (1 + \frac{\sin^{2} χ}{2}) + \frac{1}{2} p_{b g} p_{e} [1 - (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ p_{2 e} p_{b g} (1 - \frac{\sin^{2} χ}{2}) + p_{2 e} p_{e} [1 - \frac{\sin^{2} χ}{2} - (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β}) ({∣ 0 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{a a}$	$\frac{1}{4} p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} + \sin^{2} θ_{a} \cos^{2} θ_{b}) \sin^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$\frac{1}{2} p^{2} [\frac{1}{2} p_{b g}^{2} (2 - \frac{\sin^{2} χ}{2}) + p_{b g} p_{e} + p_{2 e} (p_{b g} + p_{e}) \sin^{2} χ + p_{2 e} (1 - \frac{\sin^{2} χ}{2})]$

$({∣ 0 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β}) ({∣ 1 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{b b}$	$\frac{1}{4} p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} + \sin^{2} θ_{a} \cos^{2} θ_{b}) \sin^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$\frac{1}{2} p^{2} [\frac{1}{2} p_{b g}^{2} (2 - \frac{\sin^{2} χ}{2}) + p_{b g} p_{e} + p_{2 e} (p_{b g} + p_{e}) \sin^{2} χ + p_{2 e} (1 - \frac{\sin^{2} χ}{2})]$

$({∣ 0 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β})$ $({∣ 0 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ ↓ ⟩}_{a} {∣ ↓ ⟩}_{b}$	$p^{2} p_{e}^{2} \cos^{2} θ_{a} \cos^{2} θ_{b} \cos^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} + \frac{1}{4} p_{b g} p_{e} (\cos^{2} θ_{a} + \cos^{2} θ_{b}) (1 + \cos^{2} χ)$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$+ p_{2 e} [p_{b g} + p_{e} (\cos^{2} θ_{a} + \cos^{2} θ_{b})] \cos^{2} χ + \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β})$ $({∣ 1 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ ↑ ⟩}_{a} {∣ ↑ ⟩}_{b}$	$p^{2} p_{e}^{2} \sin^{2} θ_{a} \sin^{2} θ_{b} \cos^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} + \frac{1}{4} p_{b g} p_{e} (\sin^{2} θ_{a} + \sin^{2} θ_{b}) (1 + \cos^{2} χ)$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$+ p_{2 e} [p_{b g} + p_{e} (\sin^{2} θ_{a} + \sin^{2} θ_{b})] \cos^{2} χ + \frac{1}{4} p_{2 e} \sin^{2} χ}$

Quantum State	Probability	Description
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$1 - p p_{e} - p p_{b g}$	No photons
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$p p_{e}$	Good photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p p_{b g}$	Background photon
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{2} p p_{b g}$	Background photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{b g}^{2}$	Background photons
${∣ M ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{b g}^{2}$	Background photons
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{b g}^{2}$	Background photons
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{e} p_{b g}$	$Background + good photons$
$\cos θ_{s} {∣ ↓ ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β} + \sin θ_{s} {∣ ↓ ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{e} p_{b g}$	$Background + good photons$
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$p p_{2 e}$	Double-excitation photon
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$p p_{2 e}$	Double-excitation photon
${∣ M ⟩}_{s} {∣ 0 ⟩}_{s}^{α} {∣ 2 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{2 e}$	Double-excitation photons
${∣ M ⟩}_{s} {∣ 2 ⟩}_{s}^{α} {∣ 0 ⟩}_{s}^{β}$	$\frac{1}{4} p^{2} p_{2 e}$	Double-excitation photons
${∣ M ⟩}_{s} {∣ 1 ⟩}_{s}^{α} {∣ 1 ⟩}_{s}^{β}$	$\frac{1}{2} p^{2} p_{2 e}$	Double-excitation photons

Photonic State	Atomic State	Probability
$({∣ 0 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β}) ({∣ 1 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ Ψ^{β α} ⟩}_{a b}$	$p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} \cos^{4} \frac{χ}{2} + \sin^{2} θ_{a} \cos^{2} θ_{b} \sin^{4} \frac{χ}{2})$

	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} (1 + \frac{\sin^{2} χ}{2}) + \frac{1}{2} p_{b g} p_{e} [1 + (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ p_{2 e} p_{b g} (1 - \frac{\sin^{2} χ}{2}) + p_{2 e} p_{e} [1 - \frac{\sin^{2} χ}{2} + (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β}) ({∣ 0 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{a b}$	$p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} \sin^{4} \frac{χ}{2} + \sin^{2} θ_{a} \cos^{2} θ_{b} \cos^{4} \frac{χ}{2})$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} (1 + \frac{\sin^{2} χ}{2}) + \frac{1}{2} p_{b g} p_{e} [1 - (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ p_{2 e} p_{b g} (1 - \frac{\sin^{2} χ}{2}) + p_{2 e} p_{e} [1 - \frac{\sin^{2} χ}{2} - (\cos 2 θ_{a} - \cos 2 θ_{b}) \frac{\cos χ}{2}]$
		$+ \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β}) ({∣ 0 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{a a}$	$\frac{1}{4} p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} + \sin^{2} θ_{a} \cos^{2} θ_{b}) \sin^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$\frac{1}{2} p^{2} [\frac{1}{2} p_{b g}^{2} (2 - \frac{\sin^{2} χ}{2}) + p_{b g} p_{e} + p_{2 e} (p_{b g} + p_{e}) \sin^{2} χ + p_{2 e} (1 - \frac{\sin^{2} χ}{2})]$

$({∣ 0 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β}) ({∣ 1 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ Ψ^{α β} ⟩}_{b b}$	$\frac{1}{4} p^{2} p_{e}^{2} (\cos^{2} θ_{a} \sin^{2} θ_{b} + \sin^{2} θ_{a} \cos^{2} θ_{b}) \sin^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$\frac{1}{2} p^{2} [\frac{1}{2} p_{b g}^{2} (2 - \frac{\sin^{2} χ}{2}) + p_{b g} p_{e} + p_{2 e} (p_{b g} + p_{e}) \sin^{2} χ + p_{2 e} (1 - \frac{\sin^{2} χ}{2})]$

$({∣ 0 ⟩}_{a}^{α} {∣ 1 ⟩}_{a}^{β})$ $({∣ 0 ⟩}_{b}^{α} {∣ 1 ⟩}_{b}^{β})$	${∣ ↓ ⟩}_{a} {∣ ↓ ⟩}_{b}$	$p^{2} p_{e}^{2} \cos^{2} θ_{a} \cos^{2} θ_{b} \cos^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} + \frac{1}{4} p_{b g} p_{e} (\cos^{2} θ_{a} + \cos^{2} θ_{b}) (1 + \cos^{2} χ)$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$+ p_{2 e} [p_{b g} + p_{e} (\cos^{2} θ_{a} + \cos^{2} θ_{b})] \cos^{2} χ + \frac{1}{4} p_{2 e} \sin^{2} χ}$

$({∣ 1 ⟩}_{a}^{α} {∣ 0 ⟩}_{a}^{β})$ $({∣ 1 ⟩}_{b}^{α} {∣ 0 ⟩}_{b}^{β})$	${∣ ↑ ⟩}_{a} {∣ ↑ ⟩}_{b}$	$p^{2} p_{e}^{2} \sin^{2} θ_{a} \sin^{2} θ_{b} \cos^{2} χ$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$p^{2} {\frac{1}{4} p_{b g}^{2} + \frac{1}{4} p_{b g} p_{e} (\sin^{2} θ_{a} + \sin^{2} θ_{b}) (1 + \cos^{2} χ)$
	${∣ M ⟩}_{a} {∣ M ⟩}_{b}$	$+ p_{2 e} [p_{b g} + p_{e} (\sin^{2} θ_{a} + \sin^{2} θ_{b})] \cos^{2} χ + \frac{1}{4} p_{2 e} \sin^{2} χ}$

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Journal of the Optical Society of America B