TEÓRIA DE LA VARIABLE 3

We have now established all of the theory that we need to quantify the performance limits of the atom imaging experiments used in this thesis, as well as the limits of even ideal free-space atom-network links. The biggest challenge to efficiently and coherently couple a light field to an atom in free-space remains the difficulty of capturing a large proportion of the total solid angle, and the simplest way to achieve this is a parabolic reflector [118, 131]. A parabolic mirror of infinite extent encloses the complete solid angle and captures 100% of the atomic fluorescence. But as we saw above, only 92% of the field collected from a ⇡ transition overlaps with a useful fibre mode. The parabolic mirror is even worse at coupling transitions, as optical-spin orbit coupling produces a pupil field with spatially inhomogeneous polarization that is difficult to correct. The fibre coupling efficiency of a transition, infinite-parabola image is only 0.49. Prof. Gerd Leuchs and his team

§4.7 Summary 59

Optic a/⇡ NA ⌦/4⇡ ⌘I dˆ Matched mode ⌘c ⌘

Lens 0.13 0.4 0.04 0.06 /⇡ LG 0 0 0.81 0.1 Tophat 1.00 0.12 0.25 0.7 0.30 0.38 /⇡ LG00 0.81 0.15 Tophat 0.99 0.19 Parabola 0.71 - 0.81 0.76 LG00 0.65 0.49 0.90 ⇡ Doughnut 0.97 0.88

Table 4.1: Key figures of merit for imaging systems referred to in this thesis, including NA 0.4 and NA 0.7 confocal lens systems and a parabolic mirror coupler of the same dimensions as Ref. [111]. For each system we calculate the total solid angle included⌦, the total collected intensity⌘I, the matched mode for each transition image, the mode-matching⌘c, and the total collection efficiency ⌘.

have pioneered the use of parabolic mirror atom-light couplers and have the most efficient free-space atom-light collection yet demonstrated with a parabola of aperture a= 0.71⇡

[111, 133]. The corresponding collection efficiencies for an ideal parabolic mirror of those dimensions are shown in Tab. 4.7.

In this thesis we will consider two variations of a confocal lens imaging system. In Chap. 7 we will image trapped ions with a two-lens system of numerical aperture 0.4. This system collects 12% of the total ion fluorescence, and couples at most 10% into the single-mode fibre network, see Tab. 4.7. The improved imaging system we describe in Chap. 11 combines an asphere with numerical aperture 0.7 and a hemispherical mirror to collect 38% of the total fluorescence and will couple at most 30% into a step-profile single mode fibre, although this could be improved to 38% with a top hat fibre mode. These are critical figures for the efficiency of atom-imaging with these systems and limit our capacity to network these atomic qubits with optical links, but increasing the numerical aperture of a lens system beyond 0.8 is a challenging proposition with diminishing returns. In Chap. 10 we will consider an alternative means of efficiently coupling atom light sources to optical networks with near-hemispheric mirrors that shape the vacuum mode density in the vicinity of an atom to favour efficiently collected spatial modes. In this case it is not necessary to collect a large proportion of the solid angle, but rather to suppress emission outside of the collected aperture.

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Part II

Few atoms

Single photon sources: a review

According to the assumption to be contemplated here, when a ray of light is spread- ing from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized points in space, move without dividing, and can be absorbed or generated only as a whole. – Albert Einstein,Concerning the production of light (1905) Evidence for the quantization of light has existed for over a hundred years [134, 135] and photons have been detected individually for over 60 years [136], but devices capable of producing photons on demand are a relatively recent innovation. Such definite, single- excitation light fields are an elementary tool for quantum information and a key feature in schemes for encoding, manipulating and communicating quantum information. There are two broad applications for single-photon sources in quantum information: networking stationary qubits and as a resource for photonic quantum computing (see Sec. 1.4.4)1_{. We} can draw a further distinction between two networking applications: interfacing modular registers of qubits within a quantum computer, and as flying qubits for long-range quantum key distribution (QKD).

In the following chapters we will implement a trapped-atom single-photon source and consider its properties and limitations. Because trapped atoms are themselves excellent stationary qubits, trapped-atom photon sources are a path to implementing quantum memories, and to networking registers of trapped-atom qubits in a quantum computer as described in Sec. 1.4.1. To put this work in its proper context, we will review in this chapter the various demonstrated single-photon technologies, their advantages and disadvantages as sources for quantum information networks and for optical quantum computing, recent noteworthy results, and the future prospects of each.

5.1

Photon source performance criteria

An ideal single photon source emits on demand a single photon (and never more than one photon) into a well defined spatio-temporal mode, with 100% probability and de- sirably with high brightness (meaning the attainable single-photon rate) [136, 137]. We can measure the photon-number purity with the second-order correlation function g(2)(0) measured in a HBT experiment as introduced in Sec. 2.5.4. Multi-photon components of the optical state contribute to the HBT coincidence rate, and in this sense the HBT anti-correlation parameter_A= 4Pc/Ps2 from Eqn. 2.62 is a measure of the photon-number

purity of states close to the single photon Fock state.

1_{Single-photon interferometry is another noteworthy application of single-photon sources, but we’ll}

restrict this review to the broad applications of photon sources. 64